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# Difference between three quantities
Robert M. Yamaleev
Joint Institute for Nuclear Research, LIT, Dubna, Russia.
Universidad Nacional Autonoma de Mexico, Mexico.
Email: yamaleev@jinr.ru
###### Abstract
The notion of difference between three and more quantities is introduced. The
method is based on one of the remarkable properties of the Vandermonde’s
determinant.
Keywords: Distance, Vandermonde determinant, geometry, matrix, polynomial.
Introduction.
The notion difference between two quantities $a$ and $b$ given by $(a-b)$
plays a basic role in mathematics, consequently in all branches of human
activity where the mathematics is applied. However the long stand question is:
what is the difference between three (or more) quantities? This question
frequently arises, for example, in the physics of systems consisting of many
particles, in economics etc.
The binary operation $[a,b]=(a-b)$ possesses the following principal feature:
with respect to the third quantity $c$ this operation is decomposed into a sum
of the same operations between $a$ and $c$, and $c$ and $b$, i.e.,
$[a,b]=[a,c]+[c,b].$
There were several problems of mathematics and physics where investigators
needed in the notion of the difference between three quantities. Y.Nambu [1]
quantizing the generalized Poisson structure on three dimensional phase space
met the problem of extension of the notion of commutator. In fact,the notion
of commutator can be defined only for the pair of operators $A$ and $B$ as a
difference $AB$ and $BA$. Thus, if one wants to extend this notion for triple
operators he will need on the notion of the difference between three
quantities. In Refs.[2],[3], the following approach has been developed. By
noting that in formula $(a-b)=(a+\theta b$ the value $\theta=-1$ is a
primitive root of quadratic polynomial $x^{2}-1$, the author suggested the
following definition of the difference between three quantities
$[a,b,c]=a+b\theta+c\theta^{2},$
where $\theta$ is a primitive root of polynomial $x^{3}-1$. In Refs.[5],[4],
this formula of difference have been used in order to formulate a notion of
ternary commutator. Apparently, this definition belong to the field of complex
numbers and possesses with the following feature
$[a,b,c]=[a,d,f]+[d,a,f]+[d,f,c].$
In the present paper we suggest a definition of the notion of the difference
between three and more quantities making use of a feature of the Vandermonde
determinant. Denote by $[a,b,c]$ difference between three quantities $a,b,c$.
With respect to additional quantity $d$ this definition of the difference is
decomposed as follows
$[a,b,c]=[d,b,c]+[a,d,c]+[a,b,d].$
All quantities belong to the field of real numbers.
## 1 Difference between three quantities
Let us start with the fraction of type
$\frac{1}{x^{3}-3p_{1}x^{2}+2p_{2}x-p^{2}}=\frac{1}{(x-x_{3})(x-x_{2})(x-x_{1})},$
$None$
where $x_{i},i=1,2,3$ are roots of the cubic polynomial
$P_{3}(x):=x^{3}-3p_{1}x^{2}+2p_{2}x-p^{2}.$ $None$
The following expansion for that fraction holds true
$\frac{1}{x^{3}-3p_{1}x^{2}+2p_{2}x-p^{2}}=\frac{(x_{3}-x_{2})}{V}\frac{1}{x-x_{1}}+\frac{(x_{1}-x_{3})}{V}\frac{1}{x-x_{2}}+\frac{(x_{2}-x_{1})}{V}\frac{1}{x-x_{3}},$
$None$
where by $V$ we denoted the Vandermonde’s determinant of matrix of order
$(3\times 3)$
$V=(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})=Det~{}\left(\begin{array}[]{ccc}1&1&1\\\
x_{1}&x_{2}&x_{3}\\\ x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right).$ $None$
Collecting the terms in the right-hand side of (1.3) we obtain
$\frac{(x_{3}-x_{2})}{V}\frac{1}{x-x_{1}}+\frac{(x_{1}-x_{3})}{V}\frac{1}{x-x_{2}}+\frac{(x_{2}-x_{1})}{V}\frac{1}{x-x_{3}}=$
$\frac{1}{V}\frac{1}{(x-x_{3})(x-x_{2})(x-x_{1})}(~{}$
$(x_{3}-x_{2})(x-x_{2})(x-x_{3})+(x-x_{2})(x-x_{1})(x_{1}-x_{2})+(x-x_{3})(x-x_{1})(x_{3}-x_{1})~{}).$
$None$
From this equality we come to the conclusion, that
$(x_{3}-x_{2})(x-x_{2})(x-x_{3})+(x-x_{2})(x-x_{1})(x_{1}-x_{2})+(x-x_{3})(x-x_{1})(x_{3}-x_{1})=V.$
$None$
This equation displays an interesting feature of Vandermonde’s determinant,
which to our knowledge still has not been revealed [6].
We suggest to use formula (1.6) as a formula of difference between three
amounts. The interval of this formal distance is bounded by three points
$x_{1},x_{2},x_{3}$ and it is divided into three parts by using only one
point, $x$, among them. Thus, the following formula
$V=(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})~{}\mbox{ is an analogue of the
interval}~{}(x_{1}-x_{2}),$
and the property given by formula (1.6) is an analogue of the following
property of the interval between two points
$x_{1}-x_{2}=(x_{1}-x)+(x-x_{2}).$ $None$
The Proof of formula (1.6).
Formula (1.6) is a consequence of one of the properties of Vandermonde’s
determinant. We have to prove that
$Det~{}\left(\begin{array}[]{ccc}1&1&1\\\ x_{1}&x_{2}&x_{3}\\\
x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right)=$
$Det\left(\begin{array}[]{ccc}1&1&1\\\ x&x_{2}&x_{3}\\\
x^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right)+Det\left(\begin{array}[]{ccc}1&1&1\\\
x_{1}&x&x_{3}\\\
x_{1}^{2}&x^{2}&x_{3}^{2}\end{array}\right)+Det\left(\begin{array}[]{ccc}1&1&1\\\
x_{1}&x_{2}&x\\\ x_{1}^{2}&x_{2}^{2}&x^{2}\end{array}\right).$ $None$
Consider the following matrix
$AV:=\left(\begin{array}[]{ccc}1+1&1+1&1+1\\\ x_{1}+x&x_{2}+x&x_{3}+x\\\
x_{1}^{2}+x^{2}&x_{2}^{2}+x^{2}&x_{3}^{2}+x^{2}\end{array}\right),$ $None$
and calculate determinant of this matrix in two ways.
Firstly, let us calculate the determinant on making use of the method of
expansion with respect to lines of the matrix. In this way we find that
$Det(AV)=Det~{}\left(\begin{array}[]{ccc}1+1&1+1&1+1\\\
x_{1}+x&x_{2}+x&x_{3}+x\\\
x_{1}^{2}+x^{2}&x_{2}^{2}+x^{2}&x_{3}^{2}+x^{2}\end{array}\right)$
$=Det~{}\left(\begin{array}[]{ccc}1+1&1+1&1+1\\\ x_{1}&x_{2}&x_{3}\\\
x_{1}^{2}+x^{2}&x_{2}^{2}+x^{2}&x_{3}^{2}+x^{2}\end{array}\right)+Det~{}\left(\begin{array}[]{ccc}1+1&1+1&1+1\\\
x&x&x\\\ x_{1}^{2}+x^{2}&x_{2}^{2}+x^{2}&x_{3}^{2}+x^{2}\end{array}\right)$
$=Det~{}\left(\begin{array}[]{ccc}2&2&2\\\ x_{1}&x_{2}&x_{3}\\\
x_{1}^{2}+x^{2}&x_{2}^{2}+x^{2}&x_{3}^{2}+x^{2}\end{array}\right)=Det~{}\left(\begin{array}[]{ccc}2&2&2\\\
x_{1}&x_{2}&x_{3}\\\
x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right)+Det~{}\left(\begin{array}[]{ccc}2&2&2\\\
x_{1}&x_{2}&x_{3}\\\ x^{2}&x^{2}&x^{2}\end{array}\right).$
The last determinant is equal to zero. In this way we get
$Det(AV)=2V.$ $None$
Secondly, let us expand the determinant with respect to columns.
For that purpose it is convenient to use the following notation of the
Vandermonde determinant:
$V=Det~{}\left(\begin{array}[]{ccc}1&1&1\\\ x_{1}&x_{2}&x_{3}\\\
x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{array}\right)=[x_{1},x_{2},x_{3}].$
In this notation $Det(AV)=[x_{1}+x,x_{2}+x,x_{3}+x]$. Then, the following
expansion holds true
$[x_{1}+x,x_{2}+x,x_{3}+x]=[x_{1},x_{2}+x,x_{3}+x]+[x,x_{2}+x,x_{3}+x]=$
$[x_{1},x_{2},x_{3}+x]+[x_{1},x,x_{3}+x]+[x,x_{2},x_{3}]=[x_{1},x_{2},x_{3}]+[x_{1},x_{2},x]+[x_{1},x,x_{3}]+[x,x_{2},x_{3}].$
$None$
Notice, the first term is the Vandermond’s determinant. Therefore,
$Det(AV)=V+[x_{1},x_{2},x]+[x_{1},x,x_{3}]+[x,x_{2},x_{3}]=2V.$
Hence,
$V=[x_{1},x_{2},x]+[x_{1},x,x_{3}]+[x,x_{2},x_{3}].$ $None$
End of proof.
## 2 Difference between $n\geq 2$ quantities
Since we have found the concept of difference between three quantities this
generalization to the case of $n\geq 3$ quantities is straightforward.
Consider $n$-th order Vandermonde’s matrix
$V_{ik}:=\left(\begin{array}[]{ccccc}1&1&1&...&1\\\
x_{1}&x_{2}&x_{3}&...&x_{n}\\\ x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&...&x_{n}^{2}\\\
x_{1}^{3}&x_{2}^{3}&x_{3}^{3}&...&x_{n}^{3}\\\ ...&...&...&...&...\\\
x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&...&x_{n}^{n-1}\end{array}\right).$ $None$
The determinant of this matrix is given by well-known Vandermonde’s formula:
$V=Det(V_{ij})=\prod_{i>k}(x_{i}-x_{k}).$ $None$
Consider the following auxiliary matrix
$AV(x):=\left(\begin{array}[]{ccccc}1+1&...&1+1&...&1+1\\\
x_{1}+x&...&x_{k}+x&...&x_{n}+x\\\
x_{1}^{2}+x^{2}&...&x_{k}^{2}+x^{2}&...&x_{n}^{2}+x^{2}\\\
...&...&...&...&...\\\
x_{1}^{l}+x^{l}&...&x_{k}^{l}+x^{l}&...&x_{n}^{l}+x^{l}\\\
...&...&...&...&...\\\
x_{n-1}+x^{n-1}&...&x_{k}^{n-1}+x^{n-1}&...&x_{n}^{n-1}+x^{n-1}\end{array}\right).$
$None$
Now, let us prove that the determinant of this matrix equal to $2V$. Firstly,
expand this determinant with respect to $l$-th line. The result is given by
sum of two determinants
$Det(AV(x))=Det(AV_{1}(x,x_{k}^{l}))+Det(AV_{2}(x,x^{l})),$ $None$
where we denoted
$AV_{1}(x,x^{l})=\left(\begin{array}[]{ccccc}1+1&...&1+1&...&1+1\\\
x_{1}+x&...&x_{k}+x&...&x_{n}+x\\\
x_{1}^{2}+x^{2}&...&x_{k}^{2}+x^{2}&...&x_{n}^{2}+x^{2}\\\
...&...&...&...&...\\\ x_{1}^{l}&...&x_{k}^{l}&...&x_{n}^{l}\\\
...&...&...&...&...\\\
x_{1}^{n-1}+x^{n-1}&...&x_{k}^{n-1}+x^{n-1}&...&x_{n}^{n-1}+x^{n-1}\end{array}\right),$
and,
$AV_{2}(x,x^{l})=\left(\begin{array}[]{ccccc}1+1&...&1+1&...&1+1\\\
x_{1}+x&...&x_{k}+x&...&x_{n}+x\\\
x_{1}^{2}+x^{2}&...&x_{k}^{2}+x^{2}&...&x_{n}^{2}+x^{2}\\\
...&...&...&...&...\\\ x^{l}&...&x^{l}&...&x^{l}\\\ ...&...&...&...&...\\\
x_{1}^{n-1}+x^{n-1}&...&x_{k}^{n-1}+x^{n-1}&...&x_{n}^{n-1}+x^{n-1}\end{array}\right).$
The determinant of the second matrix is trivial because there $l$-th line is
proportional to the first one. Continue to expand the first determinant
$AV_{1}(x,x^{l})$ with respect to other lines. At the final step of this
process the determinant of the auxiliary matrix is reduced to the following
form
$Det(AV(x))=Det\left(\begin{array}[]{ccccc}1+1&...&1+1&...&1+1\\\
x_{1}&...&x_{k}&...&x_{n}\\\ x_{1}^{2}&...&x_{k}^{2}&...&x_{n}^{2}\\\
...&...&...&...&...\\\ x_{1}^{l}&...&x_{k}^{l}&...&x_{n}^{l}\\\
...&...&...&...&...\\\
x_{1}^{n-1}&...&x_{k}^{n-1}&...&x_{n}^{n-1}\end{array}\right),$ $None$
which obviously equal to $2V$,
$Det(AV(x))=2V.$ $None$
Now let us calculate the determinant $Det(AV(x))$ by expanding with respect to
columns. For the sake of convenience denote the Vandermonde’s determinant
(2.1) as follows
$V=Det(V[ij])=[x_{1}...x_{k}...x_{n}].$ $None$
Correspondingly, the determinant of the auxiliary matrix will be written in
the form
$Det(AV(x))=[x_{1}+x...x_{k}+x...x_{n}+x].$ $None$
The expansion process with resect to columns of this determinant is worked out
as follows.
$[x_{1}+x...x_{k}+x...x_{n}+x]=[x_{1},x_{2}+x...x_{k}+x...x_{n}+x]+[x,x_{2}+x...x_{k}+x...x_{n}+x].$
$None$
The second term in right-hand side is equal to
$[x,x_{2}+x...x_{k}+x...x_{n}+x]=[x,x_{2}...x_{k}...x_{n}].$
Continue to expand the first of the sum,
$[x_{1},x_{2}+x...x_{k}+x...x_{n}+x]=[x_{1},x_{2},x_{3}+x...x_{k}+x...x_{n}+x]+[x,x_{2},x...x_{k}+x...x_{n}+x].$
The last term is equal to zero. The first one is represented as follows
$[x_{1},x_{2},x,x_{4}+x...x_{k}+x...x_{n}+x]=[x,x_{2},x_{3},x_{4}+x...x_{k}...x_{n}].$
At the final step of this process we come to the following equation
$Det(AV(x))=2V=\sum^{n}_{k=1}[x_{1},x_{2},...x_{k-1},x,x_{k+1}...x_{n}]+V,$
$None$
On the other hand, according to (2.6) $Det(AV(x))=2V$. Hence,
$V=\sum^{n}_{k=1}[x_{1},x_{2},...x_{k-1},x,x_{k+1}...x_{n}].$ $None$
This formula implies one of the important features of the Vandermond’s
determinant. That is the formula which we suggest to use as a definition of
the difference between $n$ quantities.
Concluding remarks.
Ternary algebraic operations and cubic relations have been considered,
although quite sporadically, by several authors already in the XIX-th century,
e.g. by A. Cayley ([7]) and J.J. Sylvester ( [8]. The development of Cayley’s
ideas, which contained a cubic generalization of matrices and their
determinants, can be found in a recent book by M. Kapranov, I.M. Gelfand and
A. Zelevinskii ([9]). A discussion of the next step in generality, the so
called $n-ary$ algebras, can be found in ([10]).
The difference between two quantities has direct geometrical interpretation as
a distance between two points on a straight line. Let $O,A,B$ be a set of
points on the line and let point $O$ be a point on the left-hand side of the
points $A$ and $B$. Let the values $d(OA),d(OB)$ mean distances between points
$A$ and $B$ of the point $O$, correspondingly. Then the difference
$[d(OA),d(OB)]$ does not depend of the motion of the point $O$ and means the
distance between points $A$ and $B$.
In the similar way, let $O$ be a point on the straight line on the left-hand
side of three points $A,B,C$ installed on the same line. Let
$d(OA),d(OB),d(OC)$ be distances from $O$ till points $A,B,C$,
correspondingly. Then the difference $[d(OA),d(OB),d(OC)]$ does not depend of
the motion of the point $O$ along the straight line. This definition of the
difference we suggest use in geometry in order to found a concept of ternary
distance between three points. A generalization to the case of $n\geq 3$
points is straightforward.
## References
* [1] Nambu Y., Generalized Hamilton dynamics. Physical Review D 7, p.2405 (1973)
* [2] Yamaleev R.M. On geometrical models in 3D space with cubic metrics. Communications of Joint Institute for Nuclear Research, P5-89-269, Dubna, 1989.
* [3] Yamaleev R.M. Fractional power of momenta and paragrassmann extension of Pauli equation. Adv.Appl.Clifford Alg.7 (S) (1997)279.
* [4] A. Himbert, Comptes Rendus de l’Acad.Sci. Paris, (1935).
* [5] R. Kerner, “The Cubic Chessboard”, Class. and Quantum Gravity, 14 1A, p. A203 (1997).
* [6] Vein R., Dale P., ”Determinants and their applications in mathematical physics”, Springer-Verlag, New York, Inc., 1999. ISBN 0-387-98558-1.
* [7] A. Cayley , Cambridge Math. Journ. 4, p. 1 (1845)
* [8] J.J. Sylvester, Johns Hopkins Circ. Journ., 3, p.7 (1883).
* [9] M. Kapranov, I.M. Gelfand, A. Zelevinskii, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser ed., (1994)
* [10] L. Vainerman, R. Kerner, Journal of Math. Physics, 37 (5), p. 2553 (1996)
99
|
arxiv-papers
| 2012-09-22T18:55:22 |
2024-09-04T02:49:35.434191
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert M. Yamaleev",
"submitter": "Robert Yamaleev Masgutovich",
"url": "https://arxiv.org/abs/1209.5012"
}
|
1209.5036
|
Proof of Church’s Thesis
by Ramón Casares
Abstract
This is a proof of Church’s Thesis. I am just proposing a more basic thesis
from which Church’s Thesis is implied.
Church’s Thesis
Church’s Thesis, also known as Church-Turing Thesis, says, see [Gandy 1980]:
What is effectively calculable is computable. $None$
What is computable is anything that a Turing Machine, see [Turing 1936], can
compute. There are other definitions, using for example Church’s
$\lambda$-calculus, but all of them are equivalent (see [Turing 1936] for the
case of $\lambda$-calculus). In any case, the term computable is defined with
mathematical rigor.
But, because what can be calculated was considered a vague notion, it was
assumed that Church’s Thesis could not be formally proven. In what follows, I
will made some definitions and assumptions in order to overcome the ambiguity.
Syntax Engine
Syntax, as opposed to Semantics, is concerned with transformations of strings
of symbols, irrespective of the symbols meanings, but according to a precise
and finite set of well-defined rules. We call algorithm this set of rules.
The previous definition of Syntax generalizes the linguistic one, see [Chomsky
1957]: Syntax is the study of the principles and processes by which sentences
are constructed in particular languages. A sentence is a string of words, and
a word is a particular case of symbol.
We will assume here that persons, that is, the members of our own species homo
sapiens, have a syntactical capability. If a person speaks a particular
language, then, according to the linguistic definition, she has a syntactical
capability. Also, most of mathematics is pure Syntax, according to the general
definition, see [Hilbert 1921], and mathematics are produced and consumed by
persons.
We will call syntax engine whatever that implements a syntactical capability.
So, if persons have a syntactical capability, then each person has a syntax
engine. In the case of a general-purpose computer, the central processing unit
(CPU) is its syntax engine.
Finite Turing Machine
A Turing Machine can then be seen as a mathematical model for syntax engines,
see [Chomsky 1959]. The only part of a Turing Machine that cannot be
physically built is its infinite tape. So we will define a Finite Turing
Machine as a Turing Machine with a finite tape instead of the infinite tape.
The tape is just read and write memory.
For each Finite Turing Machine there is one corresponding Turing Machine,
which is identical to the former, except for the tape. Defining processor as
the whole Turing Machine except its tape, then any Finite Turing Machine and
its corresponding Turing Machine have the same processor.
We can also define Finite Universal Turing Machines. A Finite Universal Turing
Machine is a Universal Turing Machine, but with a finite tape, instead of the
infinite tape. As any Universal Turing Machine is a Turing Machine, see
[Turing 1936], any Finite Universal Turing Machine is a Finite Turing Machine.
The corresponding Turing Machine of a Finite Universal Turing Machine is a
Universal Turing Machine; both have the same processor.
With these definitions and considerations, we can say that the general-purpose
computer CPU is, by design, a Finite Universal Turing Machine.
Effective Computation
We will call effective computation any computation done by a Finite Turing
Machine. Effective computations are physically achievable, because Finite
Turing Machines can be physically built, as the general-purpose computer CPU
proves.
Finite Turing Machines will fail on those computations that require more tape
than they have available. Also, in practical terms, and if the tape is long,
time available could be exhausted before reaching a tape end or a HALT
instruction, and then the computation would have to be aborted, failing. These
computations that fail in the Finite Turing Machine because of a lack of
memory or a lack of time would continue in its corresponding Turing Machine,
and they will eventually succeed, or not. Summarizing: computing is more
successful than effectively computing.
But, how much successful? A computation is successful when the Turing Machine
has reached a HALT instruction in a finite time. And, whenever a computation
has reached a HALT instruction in a finite time, the Turing Machine has only
had time to inspect a finite number of tape cells. This means that any
successful computation could be done in some Finite Turing Machine in a finite
time. So the answer is: not too much.
In addition, each Finite Turing Machine computation that HALTs will be run
identically by its corresponding Turing Machine, because not limitations of
memory nor time were found, and there are not any other differences between
the two machines. So any computation done by a Finite Turing Machine will be
identically computed by its corresponding Turing Machine. Summarizing: an
effective computation is a computation.
Finite Universal Turing Machines are not universal because, taking any Finite
Universal Turing Machine, there will always be some computations that fail in
it, but that do not fail in other Finite Turing Machines that have more memory
or more time. On the other hand, any Finite Universal Turing Machine
computation that HALTs, will be run identically by its corresponding Turing
Machine, which is a Universal Turing Machine. And this means that, except for
memory or time limitations, a Finite Universal Turing Machine behaves exactly
as its corresponding Universal Turing Machine.
Calculability
We will assume the following thesis:
What is effectively calculable is what a person can calculate. $None$
This could be denied from a Platonist view of mathematics, because a super-
person, provided with a syntax super-engine, could calculate what a plain
person cannot. But, firstly, we should say that this would be “super-
calculable”, not just “calculable”, and even less “effectively calculable”.
And, secondly, if those super-calculations are outside our plain syntactical
capabilities, then we could never identify those super-calculations as
calculations, nor we could follow nor understand them.
Another objection could be that different persons can have different
calculating capabilities. This is true, but to investigate this issue, let me
present my own thesis:
Persons’ syntax engine is a Finite Universal Turing Machine. $None$
If this is the case, then, for normal persons, that is, persons without mental
disabilities, the difference can only be the amount of memory or time.
Proof
If thesis 3 is true, then, it is easy to show that Church’s Thesis (thesis 1)
is also true.
First, using thesis 2, we substitute ‘what is effectively calculable’ with
‘what a person can calculate’. Second, using the concept of syntax engine,
what a person can calculate is what her syntax engine can compute. Now, if
thesis 3 is true, that is, if persons’ syntax engine is a Finite Universal
Turing Machine, then what a person’s syntax engine can compute is what a
Finite Universal Turing Machine can compute. A Finite Universal Turing Machine
is a Finite Turing Machine, and anything that a Finite Turing Machine can
compute can also be computed by its corresponding Turing Machine, as shown in
the section on effective computation. To close this proof just remember the
first definition: anything that a Turing Machine can compute is computable.
Discussion
From the proof it is easy to see that it is also possible to imply Church’s
Thesis from a weaker assumption:
Persons’ syntax engine is a Finite Turing Machine. $None$
But this thesis 4 does not fit some facts. Because, if thesis 4 were true, but
thesis 3 were false, then some computations would be outside the syntactical
capability of persons. This would mean that persons could not follow some
Turing Machine computations. In fact, we could only calculate one algorithm.
On the other hand, the stronger thesis 3 implies thesis 4, and it is very near
to the Church’s Thesis reversed. Church’s Thesis reversed would be:
What is computable is effectively calculable. $None$
For this thesis 5 to be true, person’s syntax engine should have to be a
Universal Turing Machine, but then persons could calculate any computation,
that is, any algorithm with any data. Sadly, we persons are finite, and
therefore our syntax engine cannot has an infinite memory, nor ourselves can
have an infinite time to do calculations.
Luckily, if thesis 3 is true, and persons’ syntax engines are Finite Universal
Turing Machines, then, except for limitations of memory or time, persons can
calculate any computation. So, taking a small enough part of any computation,
persons can calculate if it is computed rightly or wrongly.
Conclusion
Turing could not had designed his Universal Machine if his own syntax engine
were not, at least, a Finite Universal Turing Machine.
For additional evolutionary arguments based on the theory of the problem
supporting thesis 3, see [Casares 2010], and [Casares 2012].
References
[Gandy 1980] Robin Gandy, “Church’s thesis and principles for mechanisms”; in
The Kleene Symposium (ed. J. Barwise et alii), North-Holland, 1980, pg 123.
[Turing 1936] Alan Turing, “On Computable Numbers, with an Application to the
Entscheidungsproblem”; in Proceedings of the London Mathematical Society,
1937, Volume s2-42, Issue 1, pp 230-265. Received 28 May, 1936. Read 12
November, 1936.
[Chomsky 1957] Noam Chomsky, Syntactic Structures, The Hague, Mouton & Co.,
1957, pg 11.
[Hilbert 1921] David Hilbert, ,,Neubegründung der Mathematik: Erste
Mitteilung“; in Abhandlungen aus dem Seminar der Hamburgischen Universität,
December 1922, Volume 1, Issue 1, pp 157-177. Series of talks given at the
University of Hamburg, July 25-27, 1921.
[Chomsky 1959] Noam Chomsky, “On certain formal properties of grammars”; in
Information and Control, June 1959, Volume 2, Issue 2, pp 137-167.
[Casares 2010] Ramón Casares, El doble compresor, isbn: 978-1-4536-0915-6,
2010.
[Casares 2012] Ramón Casares, Sobre la libertad, isbn: 978-1-4750-4270-2 / On
Freedom, isbn: 978-1-4752-8739-4, 2012.
|
arxiv-papers
| 2012-09-23T07:26:53 |
2024-09-04T02:49:35.440145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ram\\'on Casares",
"submitter": "Ram\\'on Casares",
"url": "https://arxiv.org/abs/1209.5036"
}
|
1209.5072
|
incollectioninproceedings
THE REAL AND COMPLEX TECHNIQUES
IN HARMONIC ANALYSIS
FROM THE POINT OF VIEW OF COVARIANT TRANSFORM
Vladimir V. Kisil
_Dedicated to Prof. Sergei V. Rogosin on the occasion of his 60th birthday_
Key words: wavelet, coherent state, covariant transform, reconstruction
formula, the affine group, $ax+b$-group, square integrable representations,
admissible vectors, Hardy space, fiducial operator, approximation of the
identity, atom, nucleus, atomic decomposition, Cauchy integral, Poisson
integral, Hardy–Littlewood maximal function, grand maximal function, vertical
maximal function, non-tangential maximal function, intertwining operator,
Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO,
Hilbert transform, boundary behaviour, Carleson measure, Littlewood–Paley
theory
AMS Mathematics Subject Classification: Primary 42-02; Secondary 42A20, 42B20,
42B25, 42B35, 42C40, 43A50, 43A80.
Abstract. This paper reviews complex and real techniques in harmonic analysis.
We describe the common source of both approaches rooted in the covariant
transform generated by the affine group.
###### Contents
1. 1 Introduction
2. 2 Two approaches to harmonic analysis
3. 3 Affine group and its representations
4. 4 Covariant transform
5. 5 The contravariant transform
6. 6 Intertwining properties of covariant transforms
7. 7 Composing the covariant and the contravariant transforms
8. 8 Transported norms
9. 9 Conclusion
## 1 Introduction
There are two main approaches in harmonic analysis on the real line. The real
variables technique uses various maximal functions, dyadic cubes and,
occasionally, the Poisson integral [Stein93]. The complex variable technique
is based on the Cauchy integral and fine properties of analytic functions
[Nikolski02a, Nikolski02b].
Both methods seem to have clear advantages. The real variable technique:
1. i.
does not require an introduction of the imaginary unit for a study of real-
valued harmonic functions of a real variable (Occam’s Razor: among competing
hypotheses, the one with the fewest assumptions should be selected);
2. ii.
allows a straightforward generalization to several real variables.
By contrast, access to the beauty and power of analytic functions (e.g.,
Möbius transformations, factorisation of zeroes, etc. [Koosis98a]) is the main
reason to use the complex variable technique. A posteriori, a multidimensional
analytic version was also discovered [McIntosh95a], it is based on the
monogenic Clifford-valued functions [BraDelSom82].
Therefore, propensity for either techniques becomes a personal choice of a
researcher. Some of them prefer the real variable method, explicitly cleaning
out any reference to analytic or harmonic functions [Stein93]*Ch. III, p. 88.
Others, e.g. [Krantz09a, CoifmanJonesSemmes89], happily combine the both
techniques. However, the reasons for switching between two methds at
particular places may look mysterious.
The purpose of the present paper is to revise the origins of the real and
complex variable techniques. Thereafter, we describe the common group-
theoretical root of both. Such a unification deepens our understanding of both
methods and illuminates their interaction.
###### Remark 1.1.
In this paper, we consider only examples which are supported by the affine
group $\mathrm{Aff}$ of the real line. In the essence, $\mathrm{Aff}$ is the
semidirect product of the group of dilations acting on the group of
translations. Thus, our consideration can be generalized to the semidirect
product of dilations and homogeneous (nilpotent) Lie groups, cf. [FollStein82,
Kisil12b]. Other important extensions are the group
$\mathrm{SL}_{2}(\mathbb{R}{})$ and associated hypercomplex algebras, see
Rems. 3.5, 4.10 and [Kisil05a, Kisil12a, Kisil11c]. However, we do not aim
here to a high level of generality, it can be developed in subsequent works
once the fundamental issues are sufficiently clarified.
## 2 Two approaches to harmonic analysis
As a starting point of our discussion, we provide a schematic outline of
complex and real variables techniques in the one-dimensional harmonic
analysis. The application of complex analysis may be summarised in the
following sequence of principal steps:
Integral transforms.
For a function $f\in L_{p}{}(\mathbb{R}{})$, we apply the Cauchy or Poisson
integral transforms:
$\displaystyle[\mathcal{C}f](x+\mathrm{i}y)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-(x+\mathrm{i}y)}\,dt\,,$
(2.1) $\displaystyle[\mathcal{P}f](x,y)$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}{}}\frac{y}{(t-x)^{2}+y^{2}}\,f(t)\,dt\,.$
(2.2)
An equivalent transformation on the unit circle replaces the Fourier series
$\sum_{k}c_{k}e^{\mathrm{i}kt}$ by the Taylor series
$\sum_{k=0}^{\infty}c_{k}z^{k}$ in the complex variable $z=re^{\mathrm{i}t}$,
$0\leq r<1$. It is used for the Abel summation of trigonometric series
[Zygmund02]*§ III.6.
Domains.
Above integrals (2.1)–(2.2) map the domain of functions from the real line to
the upper half-plane, which can be conveniently identified with the set of
complex numbers having a positive imaginary part. The larger domain allows us
to inspect functions in greater details.
Differential operators.
The image of integrals (2.1) and (2.2) consists of functions,belonging to the
kernel of the Cauchy–Riemann operator $\partial_{\bar{z}}$ and Laplace
operator $\Delta$ respectively, i.e.:
$\partial_{\bar{z}}=\frac{\partial}{\partial
x}+\mathrm{i}\frac{\partial}{\partial
y}\,,\qquad\Delta=\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\,.$ (2.3)
Such functions have numerous nice properties in the upper half-plane, e.g.
they are infinitely differentiable, which make their study interesting and
fruitful.
Boundary values and SIO.
To describe properties of the initial function $f$ on the real line we
consider the boundary values of $[\mathcal{C}f](x+\mathrm{i}y)$ or
$[\mathcal{P}f](x,y)$, i.e. their limits as $y\rightarrow 0$ in some sense.
The Sokhotsky–Plemelj formula provides the boundary value of the Cauchy
integral [MityushevRogosin00a]*(2.6.6):
$[\mathcal{C}f](x,0)=\frac{1}{2}f(x)+\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-x}\,dt.$
(2.4)
The last term is a singular integral operator defined through the principal
value in the Cauchy sense:
$\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(t)}{t-x}\,dt=\lim_{\varepsilon\rightarrow
0}\frac{1}{2\pi\mathrm{i}}\int\limits_{-\infty}^{x-\varepsilon}+\int\limits_{x+\varepsilon}^{\infty}\frac{f(t)}{t-x}\,dt\,.$
(2.5)
For the Abel summation the boundary values are replaced by the limit as
$r\rightarrow 1^{-}$ in the series
$\sum_{k=0}^{\infty}c_{k}(re^{\mathrm{i}t})^{k}$.
Hardy space.
Sokhotsky–Plemelj formula (2.4) shows, that the boundary value
$[\mathcal{C}f](x,0)$ may be different from $f(x)$. The vector space of
functions $f(x)$ such that $[\mathcal{C}f](x,0)=f(x)$ is called the Hardy
space on the real line [Nikolski02a]*A.6.3.
Summing up this scheme: we replace a function (distribution) on the real line
by a nicer (analytic or harmonic) function on a larger domain—the upper half-
plane. Then, we trace down properties of the extensions to its boundary values
and, eventually, to the initial function.
The real variable approach does not have a clearly designated path in the
above sense. Rather, it looks like a collection of interrelated tools, which
are efficient for various purposes. To highlight similarity and differences
between real and complex analysis, we line up the elements of the real
variable technique in the following way:
Hardy–Littlewood maximal function
is, probably, the most important component [Koosis98a]*§ VIII.B.1
[Stein93]*Ch. 2 [Garnett07a]*§ I.4 [Burenkov12a] of this technique. The
maximal function $f^{M}$ is defined on the real line by the identity:
$f^{M}(t)=\sup_{a>0}\left\\{\frac{1}{2a}\int\limits^{t+a}_{t-a}\left|f\left(x\right)\right|\,dx\right\\}.$
(2.6)
Domain
is not apparently changed, the maximal function $f^{M}$ is again defined on
the real line. However, an efficient treatment of the maximal functions
requires consideration of tents [Stein93]*§ II.2, which are parametrised by
their vertices, i.e. points $(a,b)$, $a>0$, of the upper half-plane. In other
words, we repeatedly need values of all integrals
$\frac{1}{2a}\int\limits^{t+a}_{t-a}\left|f\left(x\right)\right|\,dx$, rather
than the single value of the supremum over $a$.
Littlewood–Paley theory
[CoifmanJonesSemmes89]*§ 3 and associated dyadic squares technique
[Garnett07a]*Ch. VII, Thm. 1.1 [Stein93]*§ IV.3 as well as stopping time
argument [Garnett07a]*Ch. VI, Lem. 2.2 are based on bisection of a function’s
domain into two equal parts.
SIO
is a natural class of bounded linear operators in $L_{p}{}(\mathbb{R}{})$.
Moreover, maximal operator $M:f\rightarrow f^{M}$ (2.6) and singular integrals
are intimately related [Stein93]*Ch. I.
Hardy space
can be defined in several equivalent ways from previous notions. For example,
it is the class of such functions that their image under maximal operator
(2.6) or singular integral (2.5) belongs to $L_{p}{}(\mathbb{R}{})$
[Stein93]*Ch. III.
The following discussion will line up real variable objects along the same
axis as complex variables. We will summarize this in Table 1.
## 3 Affine group and its representations
It is hard to present harmonic analysis and wavelets without touching the
affine group one way or another. Unfortunately, many sources only mention the
group and do not use it explicitly. On the other hand, it is equally difficult
to speak about the affine group without a reference to results in harmonic
analysis: two theories are intimately intertwined. In this section we collect
fundamentals of the affine group and its representations, which are not yet a
standard background of an analyst.
Let $G=\mathrm{Aff}$ be the $ax+b$ (or the _affine_) group [AliAntGaz00]*§
8.2, which is represented (as a topological set) by the upper half-plane
$\\{(a,b)\,\mid\,a\in\mathbb{R}_{+}{},\ b\in\mathbb{R}{}\\}$. The group law
is:
$(a,b)\cdot(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b).$ (3.1)
As any other group, $\mathrm{Aff}$ has the _left regular representation_ by
shifts on functions $\mathrm{Aff}\rightarrow\mathbb{C}{}$:
$\Lambda(a,b):f(a^{\prime},b^{\prime})\mapsto
f_{(a,b)}(a^{\prime},b^{\prime})=f\left(\frac{a^{\prime}}{a},\frac{b^{\prime}-b}{a}\right).$
(3.2)
A left invariant measure on $\mathrm{Aff}$ is $dg=a^{-2}\,da\,db$, $g=(a,b)$.
By the definition, the left regular representation (3.2) acts by unitary
operators on $L_{2}{}(\mathrm{Aff},dg)$. The group is not unimodular and a
right invariant measure is $a^{-1}\,da\,db$.
There are two important subgroups of the $ax+b$ group:
$A=\\{(a,0)\in\mathrm{Aff}\,\mid\,a\in\mathbb{R}_{+}{}\\}\quad\text{ and
}\quad N=\\{(1,b)\in\mathrm{Aff}\,\mid\,b\in\mathbb{R}{}\\}.$ (3.3)
An isometric representation of $\mathrm{Aff}$ on $L_{p}{}(\mathbb{R}{})$ is
given by the formula:
$[{\rho_{p}}(a,b)\,f](x)=a^{-\frac{1}{p}}\,f\left(\frac{x-b}{a}\right).$ (3.4)
Here, we identify the real line with the subgroup $N$ or, even more
accurately, with the homogeneous space $\mathrm{Aff}/N$ [ElmabrokHutnik12a]*§
2. This representation is known as _quasi-regular_ for its similarity with
(3.2). The action of the subgroup $N$ in (3.4) reduces to shifts, the subgroup
$A$ acts by dilations.
###### Remark 3.1.
The $ax+b$ group definitely escapes Occam’s Razor in harmonic analysis, cf.
the arguments against the imaginary unit in the Introduction. Indeed, shifts
are required to define convolutions on $\mathbb{R}^{n}{}$, and an
_approximation of the identity_ [Stein93]*§ I.6.1 is a convolution with the
dilated kernel. The same scaled convolutions define the fundamental _maximal
functions_ , see [Stein93]*§ III.1.2 cf. Example 7.6 below. Thus, we can avoid
usage of the upper half-plane $\mathbb{C}_{+}{}$, but the same set will anyway
re-invent itself in the form of the $ax+b$ group.
The representation (3.4) in $L_{2}{}(\mathbb{R}{})$ is reducible and the space
can be split into irreducible subspaces. Following the philosophy presented in
the Introduction to the paper [Kisil09e]*§ 1 we give the following
###### Definition 3.2.
For a representation ${\rho}$ of a group $G$ in a space $V$, a _generalized
Hardy space_ $H{}$ is an ${\rho}$-irreducible (or ${\rho}$-primary, as
discussed in Section 7) subspace of $V$.
###### Example 3.3.
Let $G=\mathrm{Aff}$ and the representation ${\rho_{p}}$ be defined in
$V=L_{p}{}(\mathbb{R}{})$ by (3.4). Then the classical Hardy spaces
$H_{p}{}(\mathbb{R}{})$ are ${\rho_{p}}$-irreducible, thus are covered by the
above definition.
Some ambiguity in picking the Hardy space out of all (well, two, as we will
see below) irreducible components is resolved by the traditional preference.
###### Remark 3.4.
We have defined the Hardy space completely in terms of representation theory
of $ax+b$ group. The traditional descriptions, via the Fourier transform or
analytic extensions, will be corollaries in our approach, see Prop. 3.6 and
Example 6.9.
###### Remark 3.5.
It is an interesting and important observation, that the Hardy space in
$L_{p}{}(\mathbb{R}{})$ is invariant under the action of a larger group
$\mathrm{SL}_{2}(\mathbb{R}{})$, the group of $2\times 2$ matrices with real
entries and determinant equal to $1$, the group operation coincides with the
multiplication of matrices. The $ax+b$ group is isomorphic to the subgroup of
the upper-triangular matrices in $\mathrm{SL}_{2}(\mathbb{R}{})$. The group
$\mathrm{SL}_{2}(\mathbb{R}{})$ has an isometric representation in
$L_{p}{}(\mathbb{R}{})$:
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}:\ f(x)\ \mapsto\
\frac{1}{\left|a-cx\right|^{\frac{2}{p}}}\,f\left(\frac{dx-b}{a-cx}\right),$
(3.5)
which produces quasi-regular representation (3.4) by the restriction to upper-
triangular matrices. The Hardy space $H_{p}{}(\mathbb{R}{})$ is invariant
under the above action as well. Thus, $\mathrm{SL}_{2}(\mathbb{R}{})$ produces
a refined version in comparison with the harmonic analysis of the $ax+b$ group
considered in this paper. Moreover, as representations of the $ax+b$ group are
connected with complex numbers, the structure of
$\mathrm{SL}_{2}(\mathbb{R}{})$ links all three types of hypercomplex numbers
[Kisil05a] [Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3, see also Rem. 4.10.
To clarify a decomposition of $L_{p}{}(\mathbb{R}{})$ into irreducible
subspaces of representation (3.4) we need another realization of this
representation. It is called _co-adjoint_ and is related to the _orbit method_
of Kirillov [Kirillov04a]*§ 4.1.4 [Folland95]*§ 6.7.1. Again, this isometric
representation can be defined on $L_{p}{}(\mathbb{R}{})$ by the formula:
$[{\hat{\rho}_{p}}(a,b)\,f](\lambda)=a^{\frac{1}{p}}\,\mathrm{e}^{-2\pi\mathrm{i}b\lambda}f(a\lambda).$
(3.6)
Since $a>0$, there is an obvious decomposition into invariant subspaces of
${\hat{\rho}_{p}}$:
$L_{p}{}(\mathbb{R}{})=L_{p}{}(-\infty,0)\oplus L_{p}{}(0,\infty).$ (3.7)
It is possible to demonstrate, that these components are irreducible. This
decomposition has a spatial nature, i.e., the subspaces have disjoint
supports. Each half-line can be identified with the subgroup $A$ or with the
homogeneous space $\mathrm{Aff}/N$.
The restrictions ${\hat{\rho}^{+}_{p}}$ and ${\hat{\rho}^{-}_{p}}$ of the co-
adjoint representation ${\hat{\rho}_{p}}$ to invariant subspaces (3.7) for
$p=2$ are not unitary equivalent. Any irreducible unitary representation of
$\mathrm{Aff}$ is unitary equivalent either to ${\hat{\rho}^{+}_{2}}$ or
${\hat{\rho}^{-}_{2}}$. Although there is no intertwining operator between
${\hat{\rho}^{+}_{p}}$ and ${\hat{\rho}^{-}_{p}}$, the map:
${J}:\ L_{p}{}(\mathbb{R}{})\rightarrow L_{p}{}(\mathbb{R}{}):\
{f}(\lambda)\mapsto{f}(-\lambda),$ (3.8)
has the property
${\hat{\rho}^{-}_{p}}(a,-b)\circ{J}={J}\circ{\hat{\rho}^{+}_{p}}(a,b)$ (3.9)
which corresponds to the outer automorphism $(a,b)\mapsto(a,-b)$ of
$\mathrm{Aff}$.
As was already mentioned, for the Hilbert space $L_{2}{}(\mathbb{R}{})$,
representations (3.4) and (3.6) are unitary equivalent, i.e., there is a
unitary intertwining operator between them. We may guess its nature as
follows. The eigenfunctions of the operators ${\rho_{2}}(1,b)$ are
$\mathrm{e}^{2\pi\mathrm{i}\omega x}$ and the eigenfunctions of
${\hat{\rho}_{2}}(1,b)$ are $\delta(\lambda-\omega)$. Both sets form
“continuous bases” of $L_{2}{}(\mathbb{R}{})$ and the unitary operator which
maps one to another is the Fourier transform:
$\mathcal{F}:f(x)\mapsto\hat{f}(\lambda)=\int_{\mathbb{R}{}}e^{-2\pi\mathrm{i}\lambda
x}\,f(x)\,dx.$ (3.10)
Although, the above arguments were informal, the intertwining property
$\mathcal{F}{\rho_{2}}(a,b)={\hat{\rho}_{2}}(1,b)\mathcal{F}$ can be directly
verified by the appropriate change of variables in the Fourier transform.
Thus, cf. [Nikolski02a]*Lem. A.6.2.2:
###### Proposition 3.6.
The Fourier transform maps irreducible invariant subspaces $H_{2}{}$ and
$H_{2}^{\perp}{}$ of (3.4) to irreducible invariant subspaces
$L_{2}{}(0,\infty)=\mathcal{F}(H_{2}{})$ and
$L_{2}{}(-\infty,0)=\mathcal{F}(H_{2}^{\perp}{})$ of co-adjoint representation
(3.6). In particular, $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$.
Reflection $J$ (3.8) anticommutes with the Fourier transform:
$\mathcal{F}J=-J\mathcal{F}$. Thus, $J$ also interchange the irreducible
components ${\rho^{+}_{p}}$ and ${\rho^{-}_{p}}$ of quasi-regular
representation (3.4) according to (3.9).
Summing up, the unique rôle of the Fourier transform in harmonic analysis is
based on the following facts from the representation theory. The Fourier
transform
* •
intertwines shifts in quasi-regular representation (3.4) to operators of
multiplication in co-adjoint representation (3.6);
* •
intertwines dilations in (3.4) to dilations in (3.6);
* •
maps the decomposition $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$
into spatially separated spaces with disjoint supports;
* •
anticommutes with $J$, which interchanges ${\rho^{+}_{2}}$ and
${\rho^{-}_{2}}$.
Armed with this knowledge we are ready to proceed to harmonic analysis.
## 4 Covariant transform
We make an extension of the wavelet construction defined in terms of group
representations. See [Kirillov76] for a background in the representation
theory, however, the only treated case in this paper is the $ax+b$ group.
###### Definition 4.1.
[Kisil09d, Kisil11c] Let ${\rho}$ be a representation of a group $G$ in a
space $V$ and $F$ be an operator acting from $V$ to a space $U$. We define a
_covariant transform_ $\mathcal{W}_{F}^{{\rho}}$ acting from $V$ to the space
$L{}(G,U)$ of $U$-valued functions on $G$ by the formula:
$\mathcal{W}_{F}^{{\rho}}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in
V,\ g\in G.$ (4.1)
The operator $F$ will be called a _fiducial operator_ in this context (cf. the
fiducial vector in [KlaSkag85]).
We may drop the sup/subscripts from $\mathcal{W}_{F}^{{\rho}}$ if the
functional $F$ and/or the representation ${\rho}$ are clear from the context.
###### Remark 4.2.
We do not require that the fiducial operator $F$ be linear. Sometimes the
positive homogeneity, i.e. $F(tv)=tF(v)$ for $t>0$, alone can be already
sufficient, see Example 4.7.
###### Remark 4.3.
It looks like the usefulness of the covariant transform is in the reverse
proportion to the dimension of the space $U$. The covariant transform encodes
properties of $v$ in a function $\mathcal{W}_{F}^{{\rho}}v$ on $G$, which is a
scalar-valued function if $\dim U=1$. However, such a simplicity is not always
possible. Moreover, the paper [Kisil12b] gives an important example of a
covariant transform which provides a simplification even in the case $\dim
U=\dim V$.
We start the list of examples with the classical case of the group-theoretical
wavelet transform.
###### Example 4.4.
[Perelomov86, FeichGroech89a, Kisil98a, AliAntGaz00, KlaSkag85,
FeichGroech89a] Let $V$ be a Hilbert space with an inner product
$\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary
representation of a group $G$ in the space $V$. Let
$F:V\rightarrow\mathbb{C}{}$ be the functional $v\mapsto\left\langle
v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$. The vector $v_{0}$ is
often called the _mother wavelet_ in areas related to signal processing, the
_vacuum state_ in the quantum framework, etc.
In this set-up, transformation (4.1) is the well-known expression for a
_wavelet transform_ [AliAntGaz00]*(7.48) (or _representation coefficients_):
$\mathcal{W}:v\mapsto\tilde{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle
v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$ (4.2)
The family of the vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or
_coherent states_. The image of (4.2) consists of scalar valued functions on
$G$.
This scheme is typically carried out for a square integrable representation
${\rho}$ with $v_{0}$ being an admissible vector [Perelomov86, FeichGroech89a,
AliAntGaz00, Fuhr05a, ChristensenOlafsson09a, DufloMoore], i.e. satisfying the
condition:
$0<\left\|\tilde{v}_{0}\right\|^{2}=\int_{G}\left|\left\langle
v_{0},{\rho_{2}}(g)v_{0}\right\rangle\right|^{2}\,dg<\infty.$ (4.3)
In this case the wavelet (covariant) transform is a map into the square
integrable functions [DufloMoore] with respect to the left Haar measure on
$G$. The map becomes an isometry if $v_{0}$ is properly scaled. Moreover, we
are able to recover the input $v$ from its wavelet transform through the
reconstruction formula, which requires an admissible vector as well, see
Example 5.3 below. The most popularized case of the above scheme is provided
by the affine group.
###### Example 4.5.
For the $ax+b$ group, representation (3.4) is square integrable for $p=2$. Any
function $v_{0}$, such that its Fourier transform $\hat{v}_{0}(\lambda)$
satisfies
$\int\limits_{0}^{\infty}\frac{\left|\hat{v}_{0}(\lambda)\right|^{2}}{\lambda}\,d\lambda<\infty,$
(4.4)
is admissible in the sense of (4.3) [AliAntGaz00]*§ 12.2. The _continuous
wavelet transform_ is generated by representation (3.4) acting on an
admissible vector $v_{0}$ in expression (4.2). The image of a function from
$L_{2}{}(\mathbb{R}{})$ is a function on the upper half-plane square
integrable with respect to the measure $a^{-2}\,da\,db$. There are many
examples [AliAntGaz00]*§ 12.2 of useful admissible vectors, say, the _Mexican
hat_ wavelet: $(1-x^{2})e^{-x^{2}/2}$. For sufficiently regular $\hat{v}_{0}$
admissibility (4.4) of $v_{0}$ follows by a weaker condition
$\int_{\mathbb{R}{}}v_{0}(x)\,dx=0.$ (4.5)
We dedicate Section 8 to isometric properties of this transform.
However, square integrable representations and admissible vectors do not cover
all interesting cases.
###### Example 4.6.
For the above $G=\mathrm{Aff}$ and representation (3.4), we consider the
operators $F_{\pm}:L_{p}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by:
$F_{\pm}(f)=\frac{1}{\pi\mathrm{i}}\int_{\mathbb{R}{}}\frac{f(x)\,dx}{\mathrm{i}\mp
x}.$ (4.6)
In $L_{2}{}(\mathbb{R}{})$ we note that $F_{+}(f)=\left\langle
f,c\right\rangle$, where $c(x)=\frac{1}{\pi\mathrm{i}}\frac{1}{\mathrm{i}+x}$.
Computing the Fourier transform
$\hat{c}(\lambda)=\chi_{(0,+\infty)}(\lambda)\,e^{-\lambda}$, we see that
$\bar{c}\in H_{2}{}(\mathbb{R}{})$. Moreover, $\hat{c}$ does not satisfy
admissibility condition (4.4) for representation (3.4).
Then, covariant transform (4.1) is Cauchy integral (2.1) from
$L_{p}{}(\mathbb{R}{})$ to the space of functions $\tilde{f}(a,b)$ such that
$a^{-\frac{1}{p}}\tilde{f}(a,b)$ is in the Hardy space on the upper/lower
half-plane $H_{p}{}(\mathbb{R}^{2}_{\pm}{})$ [Nikolski02a]*§ A.6.3. Due to
inadmissibility of $c(x)$, the complex analysis become decoupled from the
traditional wavelet theory.
Many important objects in harmonic analysis are generated by inadmissible
mother wavelets like (4.6). For example, the functionals
$P=\frac{1}{2}(F_{+}+F_{-})$ and $Q=\frac{1}{2\mathrm{i}}(F_{+}-F_{-})$ are
defined by kernels:
$\displaystyle p(x)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\left(\frac{1}{\mathrm{i}-x}-\frac{1}{\mathrm{i}+x}\right)=\frac{1}{\pi}\frac{1}{1+x^{2}},$
(4.7) $\displaystyle q(x)$
$\displaystyle=-\frac{1}{2\pi}\left(\frac{1}{\mathrm{i}-x}-\frac{1}{\mathrm{i}+x}\right)=-\frac{1}{\pi}\frac{x}{1+x^{2}}$
(4.8)
which are _Poisson kernel_ (2.2) and the _conjugate Poisson kernel_
[Grafakos08]*§ 4.1 [Garnett07a]*§ III.1 [Koosis98a]*Ch. 5 [Nikolski02a]*§
A.5.3, respectively. Another interesting non-admissible vector is the
_Gaussian_ $e^{-x^{2}}$.
###### Example 4.7.
A step in a different direction is a consideration of non-linear operators.
Take again the $ax+b$ group and its representation (3.4). We define $F$ to be
a homogeneous (but non-linear) functional $V\rightarrow\mathbb{R}_{+}{}$:
$F_{m}(f)=\frac{1}{2}\int\limits_{-1}^{1}\left|f(x)\right|\,dx.$ (4.9)
Covariant transform (4.1) becomes:
$[\mathcal{W}^{m}_{p}f](a,b)=F({\rho_{p}}({\textstyle\frac{1}{a},-\frac{1}{b}})f)=\frac{1}{2}\int\limits_{-1}^{1}\left|a^{\frac{1}{p}}f\left(ax+b\right)\right|\,dx=\frac{a^{\frac{1}{q}}}{2}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx,$
(4.10)
where $\frac{1}{p}+\frac{1}{q}=1$, as usual. We will see its connections with
the Hardy–Littlewood maximal functions in Example 7.6.
Since linearity has clear advantages, we may prefer to reformulate the last
example using linear covariant transforms. The idea is similar to the
representation of a convex function as an envelope of linear ones, cf.
[Garnett07a]*Ch. I, Lem. 6.1. To this end, we take a collection $\mathbf{F}$
of linear fiducial functionals and, for a given function $f$, consider the set
of all covariant transforms $\mathcal{W}_{F}f$, $F\in\mathbf{F}$.
###### Example 4.8.
Let us return to the setup of the previous Example for $G=\mathrm{Aff}$ and
its representation (3.4). Consider the unit ball $B$ in $L_{\infty}{}[-1,1]$.
Then, any $\omega\in B$ defines a bounded linear functional $F_{\omega}$ on
$L_{1}{}(\mathbb{R}{})$:
$F_{\omega}(f)=\frac{1}{2}\int\limits_{-1}^{1}f(x)\,\omega(x)\,dx=\frac{1}{2}\int_{\mathbb{R}{}}f(x)\,\omega(x)\,dx.$
(4.11)
Of course, $\sup_{\omega\in B}F_{\omega}(f)=F_{m}(f)$ with $F_{m}$ from (4.9)
and for all $f\in L_{1}{}(\mathbb{R}{})$. Then, for the non-linear covariant
transform (4.10) we have the following expression in terms of the linear
covariant transforms generated by $F_{\omega}$:
$[\mathcal{W}^{m}_{1}f](a,b)=\sup_{\omega\in
B}\,[\mathcal{W}^{\omega}_{1}f](a,b).$ (4.12)
The presence of suprimum is the price to pay for such a “linearization”.
###### Remark 4.9.
The above construction is not much different to the _grand maximal function_
[Stein93]*§ III.1.2. Although, it may look like a generalisation of covariant
transform, grand maximal function can be realised as a particular case of
Defn. 4.1. Indeed, let $M(V)$ be a subgroup of the group of all invertible
isometries of a metric space $V$. If ${\rho}$ represents a group $G$ by
isometries of $V$ then we can consider the group $\tilde{G}$ generated by all
finite products of $M(V)$ and ${\rho}(g)$, $g\in G$ with the straightforward
action ${\tilde{\rho}}$ on $V$. The grand maximal functions is produced by the
covariant transform for the representation ${\tilde{\rho}}$ of $\tilde{G}$.
###### Remark 4.10.
It is instructive to compare action (3.5) of the large
$\mathrm{SL}_{2}(\mathbb{R}{})$ group on the mother wavelet
$\frac{1}{x+\mathrm{i}}$ for the Cauchy integral and the principal case
$\omega(x)=\chi_{[-1,1]}(x)$ (the characteristic function of $[-1,1]$) for
functional (4.11). The wavelet $\frac{1}{x+\mathrm{i}}$ is an eigenvector for
all matrices $\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos t\end{pmatrix}$,
which form the one-parameter compact subgroup
$K\subset\mathrm{SL}_{2}(\mathbb{R}{})$. The respective covariant transform
(i.e., the Cauchy integral) maps functions to the homogeneous space
$\mathrm{SL}_{2}(\mathbb{R}{})/K$, which is the upper half-plane with the
Möbius (linear-fractional) transformations of complex numbers [Kisil05a]
[Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3. By contrast, the mother wavelet
$\chi_{[-1,1]}$ is an eigenvector for all matrices $\begin{pmatrix}\cosh
t&\sinh t\\\ \sinh t&\cosh t\end{pmatrix}$, which form the one-parameter
subgroup $A\in\mathrm{SL}_{2}(\mathbb{R}{})$. The covariant transform (i.e.,
the averaging) maps functions to the homogeneous space
$\mathrm{SL}_{2}(\mathbb{R}{})/A$, which can be identified with a set of
double numbers with corresponding Möbius transformations [Kisil05a]
[Kisil12a]*§ 3.3.4 [Kisil11c]*§ 3. Conformal geometry of double numbers is
suitable for real variables technique, in particular, tents [Stein93]*§ II.2
make a Möbius-invariant family.
## 5 The contravariant transform
Define the left action $\Lambda$ of a group $G$ on a space of functions over
$G$ by:
$\Lambda(g):f(h)\mapsto f(g^{-1}h).$ (5.1)
For example, in the case of the affine group it is (3.2). An object invariant
under the left action $\Lambda$ is called _left invariant_. In particular, let
$L$ and $L^{\prime}$ be two left invariant spaces of functions on $G$. We say
that a pairing $\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if
$\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime},\ g\in G.$ (5.2)
###### Remark 5.1.
1. i.
We do not require the pairing to be linear in general, in some cases it is
sufficient to have only homogeneity, see Example 5.5.
2. ii.
If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on large spaces of functions.
3. iii.
In some cases, an invariant pairing on $G$ can be obtained from an _invariant
functional_ $l$ by the formula $\left\langle
f_{1},f_{2}\right\rangle=l(f_{1}{f}_{2})$.
For a representation ${\rho}$ of $G$ in $V$ and $w_{0}\in V$, we construct a
function $w(g)={\rho}(g)w_{0}$ on $G$. We assume that the pairing can be
extended in its second component to this $V$-valued functions. For example,
such an extension can be defined in the weak sense.
###### Definition 5.2.
[Kisil09d, Kisil11c] Let $\left\langle\cdot,\cdot\right\rangle$ be a left
invariant pairing on $L\times L^{\prime}$ as above, let ${\rho}$ be a
representation of $G$ in a space $V$, we define the function
$w(g)={\rho}(g)w_{0}$ for $w_{0}\in V$ such that $w(g)\in L^{\prime}$ in a
suitable sense. The _contravariant transform_ $\mathcal{M}_{w_{0}}^{{\rho}}$
is a map $L\rightarrow V$ defined by the pairing:
$\mathcal{M}_{w_{0}}^{{\rho}}:f\mapsto\left\langle
f,w\right\rangle,\qquad\text{ where }f\in L.$ (5.3)
We can drop out sup/subscripts in $\mathcal{M}_{w_{0}}^{{\rho}}$ as we did for
$\mathcal{W}_{F}^{{\rho}}$.
###### Example 5.3 (Haar paring).
The most used example of an invariant pairing on $L_{2}{}(G,d\mu)\times
L_{2}{}(G,d\mu)$ is the integration with respect to the Haar measure:
$\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g){f}_{2}(g)\,dg.$ (5.4)
If ${\rho}$ is a square integrable representation of $G$ and $w_{0}$ is an
admissible vector, see Example 4.4, then this pairing can be extended to
$w(g)={\rho}(g)w_{0}$. The contravariant transform is known in this setup as
the _reconstruction formula_ , cf. [AliAntGaz00]*(8.19):
$\mathcal{M}_{w_{0}}f=\int_{G}f(g)\,w(g)\,dg,\qquad\text{ where
}w(g)={\rho}(g)w_{0}.$ (5.5)
It is possible to use different admissible vectors $v_{0}$ and $w_{0}$ for
wavelet transform (4.2) and reconstruction formula (5.5), respectively, cf.
Example 7.4.
Let either
* •
${\rho}$ be not a square integrable representation (even modulo a subgroup);
_or_
* •
$w_{0}$ be an inadmissible vector of a square integrable representation
${\rho}$.
A suitable invariant pairing in this case is not associated with integration
over the Haar measure on $G$. In this case we speak about a _Hardy pairing_.
The following example explains the name.
###### Example 5.4 (Hardy pairing).
Let $G$ be the $ax+b$ group and its representation ${\rho}$ (3.4) in Example
4.5. An invariant pairing on $G$, which is not generated by the Haar measure
$a^{-2}da\,db$, is:
$\left\langle f_{1},f_{2}\right\rangle_{H}=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,{f}_{2}(a,b)\,\frac{db}{a}.$ (5.6)
For this pairing, we can consider functions
$\frac{1}{\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$ or $e^{-x^{2}}$, which are not
admissible vectors in the sense of square integrable representations. For
example, for $v_{0}=\frac{1}{\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$ we obtain:
$[\mathcal{M}f](x)=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f(a,b)\,\frac{a^{-\frac{1}{p}}}{\pi\mathrm{i}(x+\mathrm{i}a-b)}\,db=-\lim_{a\rightarrow
0}\frac{a^{-\frac{1}{p}}}{\pi\mathrm{i}}\int\limits_{-\infty}^{\infty}\frac{f(a,b)\,db}{b-(x+\mathrm{i}a)}.$
In other words, it expresses the boundary values at $a=0$ of the Cauchy
integral $[-\mathcal{C}f](x+\mathrm{i}a)$.
Here is an important example of non-linear pairing.
###### Example 5.5.
Let $G=\mathrm{Aff}$ and an invariant homogeneous functional on $G$ be given
by the $L_{\infty}{}$-version of Haar functional (5.4):
$\left\langle f_{1},f_{2}\right\rangle_{\infty}=\sup_{g\in
G}\left|f_{1}(g){f}_{2}(g)\right|.$ (5.7)
Define the following two functions on $\mathbb{R}{}$:
${v}^{+}_{0}(t)=\left\\{\begin{array}[]{ll}1,&\text{ if }t=0;\\\ 0,&\text{ if
}t\neq 0,\end{array}\right.\quad\text{ and }\quad
v_{0}^{*}(t)=\left\\{\begin{array}[]{ll}1,&\text{ if }\left|t\right|\leq 1;\\\
0,&\text{ if }\left|t\right|>1.\end{array}\right.$ (5.8)
The respective contravariant transforms are generated by representation
${\rho_{\infty}}$ (3.4) are:
$\displaystyle[\mathcal{M}_{{v}^{+}_{0}}f](t)$ $\displaystyle=$ $\displaystyle
f^{+}(t)=\left\langle
f(a,b),{\rho_{\infty}}(a,b){v}_{0}^{+}(t)\right\rangle_{\infty}=\sup_{a}\left|f(a,t)\right|,$
(5.9) $\displaystyle{}[\mathcal{M}_{v^{*}_{0}}f](t)$ $\displaystyle=$
$\displaystyle f^{*}(t)=\left\langle
f(a,b),{\rho_{\infty}}(a,b)v_{0}^{*}(t)\right\rangle_{\infty}=\sup_{a>\left|b-t\right|}\left|f(a,b)\right|.$
(5.10)
Transforms (5.9) and (5.10) are the _vertical_ and _non-tangential maximal
functions_ [Koosis98a]*§ VIII.C.2, respectively.
###### Example 5.6.
Consider again $G=\mathrm{Aff}$ equipped now with an invariant linear
functional, which is a Hardy-type modification (cf. (5.6)) of
$L_{\infty}{}$-functional (5.7):
$\left\langle f_{1},f_{2}\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}={\varlimsup_{a\rightarrow
0}}\,\sup_{b\in\mathbb{R}{}}(f_{1}(a,b){f}_{2}(a,b)),$ (5.11)
where $\varlimsup$ is the upper limit. Then, the covariant transform
$\mathcal{M}^{H}$ for this pairing for functions $v^{+}$ and $v^{*}$ (5.8)
becomes:
$\displaystyle[\mathcal{M}_{{v}^{+}_{0}}^{H}f](t)$ $\displaystyle=$
$\displaystyle\left\langle
f(a,b),{\rho_{\infty}}(a,b){v}_{0}^{+}(t)\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}=\varlimsup_{a\rightarrow 0}f(a,t),$ (5.12)
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}f](t)$ $\displaystyle=$
$\displaystyle\left\langle
f(a,b),{\rho_{\infty}}(a,b)v_{0}^{*}(t)\right\rangle_{\stackrel{{\scriptstyle
H}}{{\infty}}}=\varlimsup_{\begin{subarray}{c}a\rightarrow 0\\\
\left|b-t\right|<a\end{subarray}}f(a,b).$ (5.13)
They are the _normal_ and _non-tangential_ upper limits from the upper-half
plane to the real line, respectively.
Note the obvious inequality $\left\langle
f_{1},f_{2}\right\rangle_{\infty}\geq\left\langle
f_{1},f_{2}\right\rangle_{\stackrel{{\scriptstyle H}}{{\infty}}}$ between
pairings (5.7) and (5.11), which produces the corresponding relation between
respective contravariant transforms.
There is an explicit duality between the covariant transform and the
contravariant transform. Discussion of the grand maximal function in the Rem.
4.9 shows usefulness of the covariant transform over a family of fiducial
functionals. Thus, we shall not be surprised by the contravariant transform
over a family of reconstructing vectors as well.
###### Definition 5.7.
Let $w:\mathrm{Aff}\rightarrow L_{1}{}(\mathbb{R}{})$ be a function. We define
a new function ${\rho_{1}}w$ on $\mathrm{Aff}$ with values in
$L_{1}{}(\mathbb{R}{})$ via the point-wise action
$[{\rho_{1}}w](g)={\rho_{1}}(g)w(g)$ of ${\rho_{\infty}}$ (3.4). If
$\sup_{g}\left\|w(g)\right\|_{1}<\infty$, then, for $f\in
L_{1}{}(\mathrm{Aff})$, we define the _extended contravariant transform_ by:
$[\mathcal{M}_{w}f](x)=\int_{\mathrm{Aff}}f(g)\,[{\rho_{1}}w](g)\,dg.$ (5.14)
Note, that (5.14) reduces to the contravariant transform (5.5) if we start
from the constant function $w(g)=w_{0}$.
###### Definition 5.8.
We call a function $r$ on $\mathbb{R}{}$ a _nucleus_ if:
1. i.
$r$ is supported in $[-1,1]$,
2. ii.
$\left|r\right|<\frac{1}{2}$ almost everywhere, and
3. iii.
$\int_{\mathbb{R}{}}r(x)\,dx=0$, cf. (4.5).
Clearly, for a nucleus $r$, the function $s={\rho_{1}}(a,b)r$ has the
following properties:
1. i.
$s$ is supported in a ball centred at $b$ and radius $a$,
2. ii.
$\left|s\right|<\frac{1}{2a}$ almost everywhere, and
3. iii.
$\int_{\mathbb{R}{}}s(x)\,dx=0$.
In other words, $s={\rho_{1}}(a,b)r$ is an _atom_ , cf. [Stein93]*§ III.2.2
and any atom may be obtained in this way from some nucleus and certain
$(a,b)\in\mathrm{Aff}$.
###### Example 5.9.
Let $f(g)=\sum_{j}\lambda_{j}\delta_{g_{j}}(g)$ with
$\sum_{j}\left|\lambda_{j}\right|<\infty$ be a countable sum of point masses
on $\mathrm{Aff}$. If all values of $w(g_{j})$ are nucleuses, then (5.14)
becomes:
$[\mathcal{M}_{w}f](x)=\int_{\mathrm{Aff}}f(g)\,[{\rho_{1}}w](g)\,dg=\sum_{j}\lambda_{j}s_{j},$
(5.15)
where $s_{j}={\rho_{1}}(g_{j})w(g_{j})$ are atoms. The right-hand side of
(5.15) is known as an _atomic decomposition_ of a function
$h(x)=[\mathcal{M}_{w}f](x)$, see [Stein93]*§ III.2.2.
## 6 Intertwining properties of covariant transforms
The covariant transform has obtained its name because of the following
property.
###### Theorem 6.1.
[Kisil09d, Kisil11c] Covariant transform (4.1) intertwines ${\rho}$ and the
left regular representation $\Lambda$ (5.1) on $L{}(G,U)$:
$\mathcal{W}{\rho}(g)=\Lambda(g)\mathcal{W}.$ (6.1)
###### Corollary 6.2.
The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.
The covariant transform is also a natural source of _relative convolutions_
[Kisil94e, Kisil13a], which are operators $A_{k}=\int_{G}k(g){\rho}(g)\,dg$
obtained by integration a representation ${\rho}$ of a group $G$ with a
suitable kernel $k$ on $G$. In particular, inverse wavelet transform
$\mathcal{M}_{w_{0}}f$ (5.5) can be defined from the relative convolution
$A_{f}$ as well: $\mathcal{M}_{w_{0}}f=A_{f}w_{0}$.
###### Corollary 6.3.
Covariant transform (4.1) intertwines the operator of convolution $K$ (with
kernel $k$) and the operator of relative convolution $A_{k}$, i.e.
$K\mathcal{W}=\mathcal{W}A_{k}$.
If the invariant pairing is defined by integration with respect to the Haar
measure, cf. Example 5.3, then we can show an intertwining property for the
contravariant transform as well.
###### Proposition 6.4.
[Kisil98a]*Prop. 2.9 Inverse wavelet transform $\mathcal{M}_{w_{0}}$ (5.5)
intertwines left regular representation $\Lambda$ (5.1) on $L_{2}{}(G)$ and
${\rho}$:
$\mathcal{M}_{w_{0}}\Lambda(g)={\rho}(g)\mathcal{M}_{w_{0}}.$ (6.2)
###### Corollary 6.5.
The image $\mathcal{M}_{w_{0}}(L{}(G))\subset V$ of a left invariant space
$L{}(G)$ under the inverse wavelet transform $\mathcal{M}_{w_{0}}$ is
invariant under the representation ${\rho}$.
###### Remark 6.6.
It is an important observation, that the above intertwining property is also
true for some contravariant transforms which are not based on pairing (5.4).
For example, in the case of the affine group all pairings (5.6), (5.11) and
(non-linear!) (5.7) satisfy to (6.2) for the respective representation
${\rho_{p}}$ (3.4).
There is also a simple connection between a covariant transform and right
shifts.
###### Proposition 6.7.
[Kisil10c, Kisil11c] Let $G$ be a Lie group and ${\rho}$ be a representation
of $G$ in a space $V$. Let $[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ be a
covariant transform defined by a fiducial operator $F:V\rightarrow U$. Then
the right shift $[\mathcal{W}f](gg^{\prime})$ by $g^{\prime}$ is the covariant
transform $[\mathcal{W^{\prime}}f](g)=F^{\prime}({\rho}(g^{-1})f)]$ defined by
the fiducial operator $F^{\prime}=F\circ{\rho}(g^{-1})$.
In other words the covariant transform intertwines right shifts
$R(g):f(h)\mapsto f(hg)$ on the group $G$ with the associated action
${\rho_{B}}(g):F\mapsto F\circ{\rho}(g^{-1})$ (6.3)
on fiducial operators:
$R(g)\circ\mathcal{W}_{F}=\mathcal{W}_{{\rho_{B}}(g)F},\qquad g\in G.$ (6.4)
Although the above result is obvious, its infinitesimal version has
interesting consequences. Let $G$ be a Lie group with a Lie algebra
$\mathfrak{g}$ and ${\rho}$ be a smooth representation of $G$. We denote by
$d{\rho_{B}}$ the derived representation of the associated representation
${\rho_{B}}$ (6.3) on fiducial operators.
###### Corollary 6.8.
[Kisil10c, Kisil11c] Let a fiducial operator $F$ be a null-solution, i.e.
$AF=0$, for the operator $A=\sum_{j}a_{j}d{\rho^{X_{j}}_{B}}$, where
$X_{j}\in\mathfrak{g}$ and $a_{j}$ are constants. Then the covariant transform
$[\mathcal{W}_{F}f](g)=F({\rho}(g^{-1})f)$ for any $f$ satisfies
$D(\mathcal{W}_{F}f)=0,\qquad\text{where}\quad
D=\sum_{j}\bar{a}_{j}\mathfrak{L}^{X_{j}}.$
Here, $\mathfrak{L}^{X_{j}}$ are the left invariant fields (Lie derivatives)
on $G$ corresponding to $X_{j}$.
###### Example 6.9.
Consider representation ${\rho}$ (3.4) of the $ax+b$ group with the $p=1$. Let
$\mathsf{A}$ and $\mathsf{N}$ be the basis of $\mathfrak{g}$ generating one-
parameter subgroups $A$ and $N$ (3.3), respectively. Then, the derived
representations are:
$[d{\rho^{\mathsf{A}}}f](x)=-f(x)-xf^{\prime}(x),\qquad[d{\rho^{\mathsf{N}}}f](x)=-f^{\prime}(x).$
The corresponding left invariant vector fields on $ax+b$ group are:
$\mathfrak{L}^{\mathsf{A}}=a\partial_{a},\qquad\mathfrak{L}^{\mathsf{N}}=a\partial_{b}.$
The mother wavelet $\frac{1}{x+\mathrm{i}}$ in (4.6) is a null solution of the
operator
$-d{\rho^{\mathsf{A}}}-\mathrm{i}d{\rho^{\mathsf{N}}}=I+(x+\mathrm{i})\frac{d}{dx}.$
(6.5)
Therefore, the image of the covariant transform with fiducial operator $F_{+}$
(4.6) consists of the null solutions to the operator
$-\mathfrak{L}^{\mathsf{A}}+\mathrm{i}\mathfrak{L}^{\mathsf{N}}=\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$,
that is in the essence Cauchy–Riemann operator $\partial_{\bar{z}}$ (2.3) in
the upper half-plane.
###### Example 6.10.
In the above setting, the function $p(x)=\frac{1}{\pi}\frac{1}{x^{2}+1}$ (4.7)
is a null solution of the operator:
$(d{\rho^{\mathsf{A}}})^{2}-d{\rho^{\mathsf{A}}}+(d{\rho^{\mathsf{N}}})^{2}=2I+4x\frac{d}{dx}+(1+x^{2})\frac{d^{2}}{dx^{2}}.$
The covariant transform with the mother wavelet $p(x)$ is the Poisson
integral, its values are null solutions to the operator
$(\mathfrak{L}^{\mathsf{A}})^{2}-\mathfrak{L}^{\mathsf{A}}+(\mathfrak{L}^{\mathsf{N}})^{2}=a^{2}(\partial_{b}^{2}+\partial_{a}^{2})$,
which is Laplace operator $\Delta$ (2.3).
###### Example 6.11.
Fiducial functional $F_{m}$ (4.9) is a null solution of the following
functional equation:
$\textstyle
F_{m}-F_{m}\circ{\rho_{\infty}}(\frac{1}{2},\frac{1}{2})-F_{m}\circ{\rho_{\infty}}(\frac{1}{2},-\frac{1}{2})=0.$
Consequently, the image of wavelet transform $\mathcal{W}^{m}_{p}$ (4.10)
consists of functions which solve the equation:
$\textstyle(I-R(\frac{1}{2},\frac{1}{2})-R(\frac{1}{2},-\frac{1}{2}))f=0\quad\text{
or }\quad
f(a,b)=f(\frac{1}{2}a,b+\frac{1}{2}a)+f(\frac{1}{2}a,b-\frac{1}{2}a).$
The last relation is the key to the stopping time argument [Garnett07a]*Ch.
VI, Lem. 2.2 and the dyadic squares technique, see for example [Stein93]*§
IV.3, [Garnett07a]*Ch. VII, Thm. 1.1 or the picture on the front cover of the
latter book.
The moral of the above Examples 6.9–6.11 is: there is a significant freedom in
choice of covariant transforms. However, some fiducial functionals have
special properties, which suggest the suitable technique (e.g., analytic,
harmonic, dyadic, etc.) following from this choice.
## 7 Composing the covariant and the contravariant transforms
From Props. 6.1, 6.4 and Rem. 6.6 we deduce the following
###### Corollary 7.1.
The composition $\mathcal{M}_{w}\circ\mathcal{W}_{F}$ of a covariant
$\mathcal{M}_{w}$ and contravariant $\mathcal{W}_{F}$ transforms is a map
$V\rightarrow V$, which commutes with ${\rho}$, i.e., intertwines ${\rho}$
with itself.
In particular for the affine group and representation (3.4),
$\mathcal{M}_{w}\circ\mathcal{W}_{F}$ commutes with shifts and dilations of
the real line.
Since the image space of $\mathcal{M}_{w}\circ\mathcal{W}_{F}$ is an
$\mathrm{Aff}$-invariant space, we shall be interested in the smallest
building blocks with the same property. For the Hilbert spaces, any group
invariant subspace $V$ can be decomposed into a direct integral $V=\oplus\int
V_{\mu}\,d\mu$ of _irreducible_ subspaces $V_{\mu}$, i.e. $V_{\mu}$ does not
have any non-trivial invariant subspace [Kirillov76]*§ 8.4. For
representations in Banach spaces complete reducibility may not occur and we
shall look for _primary_ subspace, i.e. space which is not a direct sum of two
invariant subspaces [Kirillov76]*§ 8.3. We already identified such subspaces
as generalized Hardy spaces in Defn. 3.2. They are also related to covariant
functional calculus [Kisil02a] [Kisil11c]*§ 6.
For irreducible Hardy spaces, we can use the following general principle,
which has several different formulations, cf. [Kirillov76]*Thm. 8.2.1:
###### Lemma 7.2 (Schur).
[AliAntGaz00]*Lem. 4.3.1 Let ${\rho}$ be a continuous unitary irreducible
representation of $G$ on the Hilbert space $H$. If a bounded operator
$T:H\rightarrow T$ commutes with ${\rho}(g)$, for all $g\in G$, then $T=kI$,
for some $\lambda\in\mathbb{C}{}$.
###### Remark 7.3.
A revision of proofs of the Schur’s Lemma, even in different formulations,
show that the result is related to the existence of joint invariant subspaces
for all operators ${\rho}(g)$, $g\in G$.
In the case of classical wavelets, the relation between wavelet transform
(4.2) and inverse wavelet transform (5.5) is suggested by their names.
###### Example 7.4.
For an irreducible square integrable representation and admissible vectors
$v_{0}$ and $w_{0}$, there is the relation [AliAntGaz00]*(8.52):
$\mathcal{M}_{w_{0}}\mathcal{W}_{v_{0}}=kI,$ (7.1)
as an immediate consequence from the Schur’s lemma. Furthermore, square
integrability condition (4.3) ensures that $k\neq 0$. The exact value of the
constant $k$ depends on $v_{0}$, $w_{0}$ and the Duflo–Moore operator
[DufloMoore] [AliAntGaz00]*§ 8.2.
It is of interest here, that two different vectors can be used as analysing
vector in (4.2) and for the reconstructing formula (5.5). Even a greater
variety can be achieved if we use additional fiducial operators and invariant
pairings.
For the affine group, recall the decomposition from Prop. 3.6 into invariant
subspaces $L_{2}{}(\mathbb{R}{})=H_{2}{}\oplus H_{2}^{\perp}{}$ and the fact,
that the restrictions ${\rho^{+}_{2}}$ and ${\rho^{-}_{2}}$ of ${\rho_{2}}$
(3.4) on $H_{2}{}$ and $H_{2}^{\perp}{}$ are not unitary equivalent. Then,
Schur’s lemma implies:
###### Corollary 7.5.
Any bounded linear operator $T:L_{2}{}(\mathbb{R}{})\rightarrow
L_{2}{}(\mathbb{R}{})$ commuting with ${\rho_{2}}$ has the form
$k_{1}I_{H_{2}{}}\oplus k_{2}I_{H_{2}^{\perp}{}}$ for some constants $k_{1}$,
$k_{2}\in\mathbb{C}{}$. Consequently, the Fourier transform maps $T$ to the
operator of multiplication by
$k_{1}\chi_{(0,+\infty)}+k_{2}\chi_{(-\infty,0)}$.
Of course, Corollary 7.5 is applicable to the composition of covariant and
contravariant transforms. In particular, the constants $k_{1}$ and $k_{2}$ may
have zero values: for example, the zero value occurs for $\mathcal{W}$ (4.2)
with an admissible vector $v_{0}$ and non-tangential limit
$\mathcal{M}_{v^{*}_{0}}^{H}$ (5.13)—because a square integrable function
$f(a,b)$ on $\mathrm{Aff}$ vanishes for $a\rightarrow 0$.
###### Example 7.6.
The composition of contravariant transform $\mathcal{M}_{v^{*}_{0}}$ (5.10)
with covariant transform $\mathcal{W}_{\infty}$ (4.10) is:
$\displaystyle[\mathcal{M}_{v^{*}_{0}}\mathcal{W}_{\infty}f](t)$
$\displaystyle=$
$\displaystyle\sup_{a>\left|b-t\right|}\left\\{\frac{1}{2a}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx\right\\}$
$\displaystyle=$
$\displaystyle\sup_{b_{1}<t<b_{2}}\left\\{\frac{1}{b_{2}-b_{1}}\int\limits^{b_{2}}_{b_{1}}\left|f\left(x\right)\right|\,dx\right\\}.$
Thus, $\mathcal{M}_{v^{*}_{0}}\mathcal{W}_{\infty}f$ coincides with
_Hardy–Littlewood maximal function_ $f^{M}$ (2.6), which contains important
information on the original function $f$ [Koosis98a]*§ VIII.B.1. Combining
Props. 6.1 and 6.4 (through Rem. 6.6), we deduce that the operator $M:f\mapsto
f^{M}$ commutes with ${\rho_{p}}$: ${\rho_{p}}M=M{\rho_{p}}$. Yet, $M$ is non-
linear and Cor. 7.5 is not applicable in this case.
###### Example 7.7.
Let the mother wavelet $v_{0}(x)=\delta(x)$ be the Dirac delta function, then
the wavelet transform $\mathcal{W}_{\delta}$ generated by ${\rho_{\infty}}$
(3.4) on $C{}(\mathbb{R}{})$ is $[\mathcal{W}_{\delta}f](a,b)=f(b)$. Take the
reconstruction vector $w_{0}(t)=(1-\chi_{[-1,1]}(t))/t/\pi$ and consider the
respective inverse wavelet transform $\mathcal{M}_{w_{0}}$ produced by Hardy
pairing (5.6). Then, the composition of both maps is:
$\displaystyle[\mathcal{M}_{w_{0}}\circ\mathcal{W}_{\delta}f](t)$
$\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{-\infty}^{\infty}f(b)\,{\rho_{\infty}}(a,b)w_{0}(t)\,\frac{db}{a}$
(7.3) $\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{-\infty}^{\infty}f(b)\,\frac{1-\chi_{[-a,a]}(t-b)}{t-b}\,{db}$
$\displaystyle=$ $\displaystyle\lim_{a\rightarrow
0}\,\frac{1}{\pi}\\!\int\limits_{\left|b\right|>a}\frac{f(b)}{t-b}\,{db}.$
The last expression is the _Hilbert transform_
$\mathcal{H}=\mathcal{M}_{w_{0}}\circ\mathcal{W}_{\delta}$, which is an
example of a _singular integral operator_ (SIO) [Stein93]*§ I.5
[MityushevRogosin00a]*§ 2.6 defined through the principal value (2.5) (in the
sense of Cauchy). By Cor. 7.5 we know that $\mathcal{H}=k_{1}I_{H_{2}{}}\oplus
k_{2}I_{H_{2}^{\perp}{}}$ for some constants $k_{1}$, $k_{2}\in\mathbb{C}{}$.
Furthermore, we can directly check that $\mathcal{H}J=-J\mathcal{H}$, for the
reflection $J$ from (3.8), thus $k_{1}=-k_{2}$. An evaluation of $\mathcal{H}$
on a simple function from $H_{2}{}$ (say, the Cauchy kernel
$\frac{1}{x+\mathrm{i}}$) gives the value of the constant $k_{1}=-\mathrm{i}$.
Thus,
$\mathcal{H}=(-\mathrm{i}I_{H_{2}{}})\oplus(\mathrm{i}I_{H_{2}^{\perp}{}})$.
In fact, the previous reasons imply the following
###### Proposition 7.8.
[Stein70a]*§ III.1.1 Any bounded linear operator on $L_{2}{}(\mathbb{R}{})$
commuting with quasi-regular representation ${\rho_{2}}$ (3.4) and
anticommuting with reflection $J$ (3.8) is a constant multiple of Hilbert
transform (7.3).
###### Example 7.9.
Consider the covariant transform $\mathcal{W}_{q}$ defined by the inadmissible
wavelet $q(t)$ (4.8), the conjugated Poisson kernel. Its composition with the
contravariant transform $\mathcal{M}_{{v}^{+}_{0}}^{H}$ (5.12) is
$[\mathcal{M}_{{v}^{+}_{0}}^{H}\circ\mathcal{W}_{q}f](t)=\varlimsup_{a\rightarrow
0}\frac{1}{\pi}\int_{\mathbb{R}{}}\frac{f(x)\,(t-x)}{(t-x)^{2}+a^{2}}\,dx$
(7.4)
We can see that this composition satisfies to Prop. 7.8, the constant factor
can again be evaluated from the Cauchy kernel $f(x)=\frac{1}{x+\mathrm{i}}$
and is equal to $1$. Of course, this is a classical result [Grafakos08]*Thm.
4.1.5 in harmonic analysis that (7.4) provides an alternative expression for
Hilbert transform (7.3).
###### Example 7.10.
Let $\mathcal{W}$ be a covariant transfrom generated either by the functional
$F_{\pm}$ (4.6) (i.e. the Cauchy integral) or $\frac{1}{2}(F_{+}-F_{-})$ (i.e.
the Poisson integral) from the Example 4.6. Then, for contravariant transform
$\mathcal{M}_{v^{+}_{0}}^{H}$ (5.9) the composition
$\mathcal{M}_{v^{+}_{0}}^{H}\mathcal{W}$ becomes the normal boundary value of
the Cauchy/Poisson integral, respectively. The similar composition
$\mathcal{M}_{v^{*}_{0}}^{H}\mathcal{W}$ for reconstructing vector $v^{*}_{0}$
(5.8) turns to be the non-tangential limit of the Cauchy/Poisson integrals.
The maximal function and SIO are often treated as elementary building blocks
of harmonic analysis. In particular, it is common to define the Hardy space as
a closed subspace of $L_{p}{}(\mathbb{R}{})$ which is mapped to
$L_{p}{}(\mathbb{R}{})$ by either the maximal operator (7.6) or by the SIO
(7.3) [Stein93]*§ III.1.2 and § III.4.3 [DziubanskiPreisner10a]. From this
perspective, the coincidence of both characterizations seems to be non-
trivial. On the contrast, we presented both the maximal operator and SIO as
compositions of certain co- and contravariant transforms. Thus, these
operators act between certain $\mathrm{Aff}$-invariant subspaces, which we
associated with generalized Hardy spaces in Defn. 3.2. For the right choice of
fiducial functionals, the coincidence of the respective invariant subspaces is
quite natural.
The potential of the group-theoretical approach is not limited to the Hilbert
space $L_{2}{}(\mathbb{R}{})$. One of possibilities is to look for a suitable
modification of Schur’s Lemma 7.2, say, to Banach spaces. However, we can
proceed with the affine group without such a generalisation. Here is an
illustration to a classical question of harmonic analysis: to identify the
class of functions on the real line such that
$\mathcal{M}_{v^{*}_{0}}^{H}\mathcal{W}$ becomes the identity operator on it.
###### Proposition 7.11.
Let $B{}$ be the space of bounded uniformly continuous functions on the real
line. Let $F:B{}\rightarrow\mathbb{R}{}$ be a fiducial functional such that:
$\lim_{a\rightarrow 0}F({\rho_{\infty}}(1/a,0)f)=0,\quad\text{ for all }f\in
B{}\text{ such that }f(0)=0$ (7.5)
and $F({\rho_{\infty}}(1,b)f)$ is a continuous function of $b\in\mathbb{R}{}$
for a given $f\in B{}$.
Then, $\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}$ is a constant multiple
of the identity operator on $B{}$.
###### Proof.
First of all we note that $\mathcal{M}_{v^{+}_{0}}^{H}\mathcal{W}_{F}$ is a
bounded operator on $B{}$. Let $v^{*}_{(a,b)}={\rho_{\infty}}(a,b)v^{*}$.
Obviously, $v^{*}_{(a,b)}(0)=v^{*}(-\frac{b}{a})$ is an eigenfunction for
operators $\Lambda(a^{\prime},0)$, $a^{\prime}\in\mathbb{R}_{+}{}$ of the left
regular representation of $\mathrm{Aff}$:
$\Lambda(a^{\prime},0)v^{*}_{(a,b)}(0)=v^{*}_{(a,b)}(0).$ (7.6)
This and the left invariance of pairing (5.2) imply that
$\mathcal{M}_{v^{*}_{0}}^{H}\circ\Lambda(1/a,0)=\mathcal{M}_{v^{*}_{0}}^{H}$
for any $(a,0)\in\mathrm{Aff}$. Then, applying intertwining properties (6.1)
we obtain that
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\Lambda(1/a,0)\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}\circ{\rho_{\infty}}(1/a,0)f](0).$
Using the limit $a\rightarrow 0$ (7.5) and the continuity of
$F\circ{\rho_{\infty}}(1,b)$ we conclude that the linear functional
$l:f\mapsto[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$ vanishes for
any $f\in\mathbb{B}{}$ such that $f(0)=0$. Take a function $f_{1}\in B{}$ such
that $f_{1}(0)=1$ and define $c=l(f_{1})$. From linearity of $l$, for any
$f\in B{}$ we have:
$l(f)=l(f-f(0)f_{1}+f(0)f_{1})=l(f-f(0)f_{1})+f(0)l(f_{1})=cf(0).$
Furthermore, using intertwining properties (6.1) and (6.2):
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](t)$
$\displaystyle=$
$\displaystyle[{\rho_{\infty}}(1,-t)\circ\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}f](0)$
$\displaystyle=$
$\displaystyle[\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}\circ{\rho_{\infty}}(1,-t)f](0)$
$\displaystyle=$ $\displaystyle l({\rho_{\infty}}(1,-t)f)$ $\displaystyle=$
$\displaystyle c[{\rho_{\infty}}(1,-t)f](0)$ $\displaystyle=$ $\displaystyle
cf(t).$
This completes the proof. ∎
To get the classical statement we need the following lemma.
###### Lemma 7.12.
For $w(t)\in L_{1}{}(\mathbb{R}{})$, define the fiducial functional on $B{}$:
$F(f)=\int_{\mathbb{R}{}}f(t)\,w(t)\,dt.$ (7.7)
Then $F$ satisfies the conditions (and thus the conclusions) of Prop. 7.11.
###### Proof.
Let $f$ be a continuous bounded function such that $f(0)=0$. For
$\varepsilon>0$ chose
* •
$\delta>0$ such that $\left|f(t)\right|<\varepsilon$ for all
$\left|t\right|<\delta$;
* •
$M>0$ such that $\int_{\left|t\right|>M}\left|w(t)\right|\,dt<\varepsilon$.
Then, for $a<\delta/M$, we have the estimation:
$\displaystyle\left|F({\rho_{\infty}}(1/a,0)f)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}{}}f\left(at\right)\,w(t)\,dt\right|$
$\displaystyle\leq$
$\displaystyle\left|\int_{\left|t\right|<M}f\left(at\right)\,w(t)\,dt\right|+\left|\int_{\left|t\right|>M}f\left(at\right)\,w(t)\,dt\right|$
$\displaystyle\leq$
$\displaystyle\varepsilon(\left\|w\right\|_{1}+\left\|f\right\|_{\infty}).$
Finally, for a uniformly continuous function $g$ for $\varepsilon>0$ there is
$\delta>0$ such that $\left|g(t+b)-g(t)\right|<\varepsilon$ for all $b<\delta$
and $t\in\mathbb{R}{}$. Then
$\left|F({\rho_{\infty}}(1,b)g)-F(g)\right|=\left|\int_{\mathbb{R}{}}(g(t+b)-g(t))\,w(t)\,dt\right|\leq\varepsilon\left\|w\right\|_{1}.$
This proves the continuity of $F({\rho_{\infty}}(1,b)g)$ at $b=0$ and, by the
group property, at any other point as well. ∎
###### Remark 7.13.
A direct evaluation shows, that the constant $c=l(f_{1})$ from the proof of
Prop. 7.11 for fiducial functional (7.7) is equal to
$c=\int_{\mathbb{R}{}}w(t)\,dt$. Of course, for non-trivial boundary values we
need $c\neq 0$. On the other hand, admissibility condition (4.5) requires
$c=0$. Moreover, the classical harmonic analysis and the traditional wavelet
construction are two “orthogonal” parts of the same covariant transform theory
in the following sense. We can present a rather general bounded function
$w=w_{a}+w_{p}$ as a sum of an admissible mother wavelet $w_{a}$ and a
suitable multiple $w_{p}$ of the Poisson kernel. An extension of this
technique to unbounded functions leads to _Calderón–Zygmund decomposition_
[Stein93]*§ I.4.
The table integral $\int_{\mathbb{R}{}}\frac{dx}{x^{2}+1}=\pi$ tells that the
“wavelet” $p(t)=\frac{1}{\pi}\frac{1}{1+t^{2}}$ (4.7) is in
$L_{1}{}(\mathbb{R}{})$ with $c=1$, the corresponding wavelet transform is the
Poisson integral. Its boundary behaviour from Prop. 7.11 is the classical
result, cf. [Garnett07a]*Ch. I, Cor. 3.2. The comparison of our arguments with
the traditional proofs, e.g. in [Garnett07a], does not reveal any significant
distinctions. We simply made an explicit usage of the relevant group
structure, which is implicitly employed in traditional texts anyway, cf.
[Burenkov12a]. Further demonstrations of this type can be found in
[Albargi13a, ElmabrokHutnik12a].
## 8 Transported norms
If the functional $F$ and the representation ${\rho}$ in (4.1) are both
linear, then the resulting covariant transform is a linear map. If
$\mathcal{W}_{F}$ is injective, e.g. due to irreducibility of ${\rho}$, then
$\mathcal{W}_{F}$ transports a norm $\left\|\cdot\right\|$ defined on $V$ to a
norm $\left\|\cdot\right\|_{F}$ defined on the image space $\mathcal{W}_{F}V$
by the simple rule:
$\left\|u\right\|_{F}:=\left\|v\right\|,\quad\text{ where the unique }v\in
V\text{ is defined by }u=\mathcal{W}_{F}v.$ (8.1)
By the very definition, we have the following
###### Proposition 8.1.
1. i.
$\mathcal{W}_{F}$ is an isometry
$(V,\left\|\cdot\right\|)\rightarrow(\mathcal{W}_{F}V,\left\|\cdot\right\|_{F})$.
2. ii.
If the representation ${\rho}$ acts on $(V,\left\|\cdot\right\|)$ by
isometries then $\left\|\cdot\right\|_{F}$ is left invariant.
A touch of non-triviality occurs if the transported norm can be naturally
expressed in the original terms of $G$.
###### Example 8.2.
It is common to consider a unitary square integrable representation ${\rho}$
and an admissible mother wavelet $f\in V$. In this case, wavelet transform
(4.2) becomes an isometry to square integrable functions on $G$ with respect
to a Haar measure [AliAntGaz00]*Thm. 8.1.3. In particular, for the affine
group and setup of Example 4.5, the wavelet transform with an admissible
vector is a multiple of an isometry map from $L_{2}{}(\mathbb{R}{})$ to the
functions on the upper half-plane, i.e., the $ax+b$ group, which are square
integrable with respect to the Haar measure $a^{-2}\,da\,db$.
A reader expects that there are other interesting examples of the transported
norms, which are not connected to the Haar integration.
###### Example 8.3.
In the setup of Example 4.6, consider the space $L_{p}{}(\mathbb{R}{})$ with
representation (3.4) of $\mathrm{Aff}$ and Poisson kernel $p(t)$ (4.7) as an
inadmissible mother wavelet. The norm transported by $\mathcal{W}_{P}$ to the
image space on $\mathrm{Aff}$ is [Nikolski02a]*§ A.6.3:
$\left\|u\right\|_{p}=\sup_{a>0}\left(\int\limits_{-\infty}^{\infty}\left|u(a,b)\right|^{p}\,\frac{db}{a}\right)^{\frac{1}{p}}.$
(8.2)
In the theory of Hardy spaces, the $L_{p}{}$-norm on the real line and
transported norm (8.2) are naturally intertwined, cf. [Nikolski02a]*Thm.
A.3.4.1(iii), and are used interchangeably.
The second possibility to transport a norm from $V$ to a function space on $G$
uses an contravariant transform $\mathcal{M}_{v}$:
$\left\|u\right\|_{v}:=\left\|\mathcal{M}_{v}u\right\|.$ (8.3)
###### Proposition 8.4.
1. i.
The contravariant transform $\mathcal{M}_{v}$ is an isometry
$(L,\left\|\cdot\right\|_{v})\rightarrow(V,\left\|\cdot\right\|)$.
2. ii.
If the composition $\mathcal{M}_{v}\circ\mathcal{W}_{F}=cI$ is a multiple of
the identity on $V$ then transported norms $\left\|\cdot\right\|_{v}$ (8.3)
and $\left\|\cdot\right\|_{F}$ (8.1) differ only by a constant multiplier.
The above result is well-known for traditional wavelets.
###### Example 8.5.
In the setup of Example 7.4, for a square integrable representation and two
admissible mother wavelets $v_{0}$ and $w_{0}$ we know that
$\mathcal{M}_{w_{0}}\mathcal{W}_{v_{0}}=kI$ (7.1), thus transported norms
(8.1) and (8.3) differ by a constant multiplier. Thus, norm (8.3) is also
provided by the integration with respect to the Haar measure on $G$.
In the theory of Hardy spaces the result is also classical.
###### Example 8.6.
For the fiducial functional $F$ with property (7.5) and the contravariant
transform $\mathcal{M}_{v^{*}_{0}}^{H}$ (5.13), Prop. 7.11 implies
$\mathcal{M}_{v^{*}_{0}}^{H}\circ\mathcal{W}_{F}=cI$. Thus, the norm
transported to $\mathrm{Aff}$ by $\mathcal{M}_{v^{*}_{0}}^{H}$ from
$L_{p}{}(\mathbb{R}{})$ up to factor coincides with (8.2). In other words, the
transition to the boundary limit on the Hardy space is an isometric operator.
This is again a classical result of the harmonic analysis, cf.
[Nikolski02a]*Thm. A.3.4.1(ii).
The co- and contravariant transforms can be used to transport norms in the
opposite direction: from a classical space of functions on $G$ to a
representation space $V$.
###### Example 8.7.
Let $V$ be the space of $\sigma$-finite signed measures of a bounded variation
on the upper half-plane. Let the $ax+b$ group acts on $V$ by the
representation adjoint to
$[{\rho_{1}}(a,b)f](x,y)=a^{-1}f(\frac{x-b}{a},\frac{y}{a})$ on
$L_{2}{}(\mathbb{R}^{2}_{+}{})$, cf. (3.2). If the mother wavelet $v_{0}$ is
the indicator function of the square $\\{0<x<1,0<y<1\\}$, then the covariant
transform of a measure $\mu$ is $\tilde{\mu}(a,b)=a^{-1}\mu(Q_{a,b})$, where
$Q_{a,b}$ is the square $\\{b<x<b+a,0<y<a\\}$. If we request that
$\tilde{\mu}(a,b)$ is a bounded function on the affine group, then $\mu$ is a
Carleson measure [Garnett07a]*§ I.5. A norm transported from
$L_{\infty}{}(\mathrm{Aff})$ to the appropriate subset of $V$ becomes the
Carleson norm of measures. Indicator function of a tent taken as a mother
wavelet will lead to an equivalent definition.
It was already mentioned in Rem. 4.9 and Example 5.9 that we may be interested
to mix several different covariant and contravariant transforms. This motivate
the following statement.
###### Proposition 8.8.
Let $(V,\left\|\cdot\right\|)$ be a normed space and ${\rho}$ be a continuous
representation of a topological locally compact group $G$ on $V$. Let two
fiducial operators $F_{1}$ and $F_{2}$ define the respective covariant
transforms $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$ to the same image space
$W=\mathcal{W}_{1}V=\mathcal{W}_{2}V$. Assume, there exists an contravariant
transform $\mathcal{M}:W\rightarrow V$ such that
$\mathcal{M}\circ\mathcal{W}_{1}=c_{1}I$ and
$\mathcal{M}\circ\mathcal{W}_{2}=c_{2}I$. Define by
$\left\|\cdot\right\|_{\mathcal{M}}$ the norm on $U$ transpordef from $V$ by
$\mathcal{M}$. Then
$\left\|\mathcal{W}_{1}v_{1}+\mathcal{W}_{2}v_{2}\right\|_{\mathcal{M}}=\left\|c_{1}v_{1}+c_{2}v_{2}\right\|,\quad\text{
for any }v_{1},v_{2}\in V.$ (8.4)
###### Proof.
Indeed:
$\begin{split}\left\|\mathcal{W}_{1}v_{1}+\mathcal{W}_{2}v_{2}\right\|_{\mathcal{M}}&=\left\|\mathcal{M}\circ\mathcal{W}_{1}v_{1}+\mathcal{M}\circ\mathcal{W}_{2}v_{2}\right\|\\\
&=\left\|c_{1}v_{1}+c_{2}v_{2}\right\|,\end{split}$
by the definition of transported norm (8.3) and the assumptions
$\mathcal{M}\circ\mathcal{W}_{i}=c_{i}I$. ∎
Although the above result is simple, it does have important consequences.
###### Corollary 8.9 (Orthogonality Relation).
Let ${\rho}$ be a square integrable representation of a group $G$ in a Hilbert
space $V$. Then, for any two admissible mother wavelets $f$ and $f^{\prime}$
there exists a constant $c$ such that:
$\int_{G}\left\langle v,{\rho}(g)f\right\rangle\,\overline{\left\langle
v^{\prime},{\rho}(g)f^{\prime}\right\rangle}\,dg=c\,\left\langle
v,v^{\prime}\right\rangle\quad\text{ for any }v_{1},v_{2}\in V.$ (8.5)
Moreover, the constant $c=c(f^{\prime},f)$ is a sesquilinear form of vectors
$f^{\prime}$ and $f$.
###### Proof.
We can derive (8.5) from (8.4) as follows. Let $\mathcal{M}_{f}$ be the
inverse wavelet transform (5.5) defined by the admissible vector $f$, then
$\mathcal{M}_{f}\circ\mathcal{W}_{f}=I$ on $V$ providing the right scaling of
$f$. Furthermore, $\mathcal{M}_{f}\circ\mathcal{W}_{f^{\prime}}=\bar{c}I$ by
(7.1) for some complex constant $c$. Thus, by (8.4):
$\left\|\mathcal{W}_{f}v+\mathcal{W}_{f^{\prime}}v^{\prime}\right\|_{\mathcal{M}}=\left\|v+\bar{c}v^{\prime}\right\|.$
Now, through the polarisation identity [KirGvi82]*Problem 476 we get the
equality (8.5) of inner products. ∎
The above result is known as the _orthogonality relation_ in the theory of
wavelets, for some further properties of the constant $c$ see
[AliAntGaz00]*Thm. 8.2.1.
Here is an application of Prop. 8.8 to harmonic analysis, cf.
[Grafakos08]*Thm. 4.1.7:
###### Corollary 8.10.
The covariant transform $\mathcal{W}_{q}$ with conjugate Poisson kernel $q$
(4.8) is a bounded map from $(L_{2}{}(\mathbb{R}{}),\left\|\cdot\right\|)$ to
$(L{}(\mathrm{Aff}),\left\|\cdot\right\|_{2})$ with norm
$\left\|\cdot\right\|_{2}$ (8.2). Moreover:
$\left\|\mathcal{W}_{q}f\right\|_{2}=\left\|f\right\|,\qquad\text{ for all
}f\in L_{2}{}(\mathbb{R}{}).$
###### Proof.
As we establish in Example 7.9 for contravariant transform
$\mathcal{M}_{{v}^{+}_{0}}^{H}$ (5.12),
$\mathcal{M}_{{v}^{+}_{0}}^{H}\circ\mathcal{W}_{q}=-\mathrm{i}I$ and
$\mathrm{i}I$ on $H_{2}{}$ and $H_{2}^{\perp}{}$, respectively. Take the
unique presentation $f=u+u^{\perp}$, for $u\in H_{2}{}$ and $u^{\perp}\in
H_{2}^{\perp}{}$. Then, by (8.4)
$\left\|\mathcal{W}_{q}f\right\|_{2}=\left\|-\mathrm{i}u+\mathrm{i}u^{\perp}\right\|=\left\|u+u^{\perp}\right\|=\left\|f\right\|.$
This completes the proof. ∎
## 9 Conclusion
We demonstrated that both, real and complex, techniques in harmonic analysis
have the same group-theoretical origin. Moreover, they are complemented by the
wavelet construction. Therefore, there is no any confrontation between these
approaches and they can be lined up as in Table 1. In other words, the binary
opposition of the real and complex methods resolves via Kant’s triad thesis-
antithesis-synthesis: complex-real-covariant.
## Acknowledgements
I am grateful to A. Albargi for careful reading of the paper and useful
comments. Prof. D. Yakubovich pointed out some recent publications in the
field. The anonymous referee made many useful suggestions which were
incorporated into the paper with gratitude.
Covariant scheme | Complex variable | Real variable
---|---|---
Covariant transform is $\mathcal{W}_{F}^{{\rho}}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v)$. In particular, the wavelet transform for the mother wavelet $v_{0}$ is $\tilde{v}(g)=\left\langle v,{\rho}(g)v_{0}\right\rangle$. | The Cauchy integral is generated by the mother wavelet $\frac{1}{2\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$. The Poisson integral is generated by the mother wavelet $\frac{1}{\pi}\frac{1}{x^{2}+1}$ | The averaging operator $\tilde{f}(b)=\frac{1}{2a}\int\limits_{b-a}^{b+a}f(t)\,dt$ is defined by the mother wavelet $\chi_{[-1,1]}(t)$, to average the modulus of $f(t)$ we use all elements of the unit ball in $L_{\infty}{}[-1,1]$.
The covariant transform maps vectors to functions on $G$ or, in the induced case, to functions on the homogeneous space $G/H$. | Functions are mapped from the real line to the upper half-plane parametrised by either the $ax+b$-group or the homogeneous space $\mathrm{SL}_{2}(\mathbb{R}{})/K$. | Functions are mapped from the real line to the upper half-plane parametrised by either the $ax+b$-group or the homogeneous space $\mathrm{SL}_{2}(\mathbb{R}{})/A$.
Annihilating action on the mother wavelet produces functional relation on the image of the covariant transform | The operator $-d{\rho^{\mathsf{A}}}-\mathrm{i}d{\rho^{\mathsf{N}}}=I+(x+\mathrm{i})\frac{d}{dx}$ annihilates the mother wavelet $\frac{1}{2\pi\mathrm{i}}\frac{1}{x+\mathrm{i}}$, thus the image of wavelet transform is in the kernel of the Cauchy–Riemann operator $-\mathfrak{L}^{\mathsf{A}}+\mathrm{i}\mathfrak{L}^{\mathsf{N}}=\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$. Similarly, for the Laplace operator. | The mother wavelet $v_{0}=\chi_{[-1,1]}$ satisfies the equality $\chi_{[-1,1]}=\chi_{[-1,0]}+\chi_{[0,1]}$, where both terms are again scaled and shifted $v_{0}$. The image of the wavelet transform is suitable for the stopping time argument and the dyadic squares technique.
An invariant pairing $\left\langle\cdot,\cdot\right\rangle$ generates the contravariant transform $[\mathcal{M}_{w_{0}}^{{\rho}}f]\left\langle f(g),{\rho}(g)w_{0}\right\rangle$ for | The contravariant transform with the invariant Hardy pairing on the $ax+b$ group produces boundary values of functions on the real line. | The covariant transform with the invariant $\sup$ pairing produces the vertical and non-tangential maximal functions.
The composition $\mathcal{M}_{v}\circ\mathcal{W}_{F}$ of the covariant and contravariant transforms is a multiple of the identity on irreducible components. | SIO is a composition of the Cauchy integral and its boundary value. | The Hardy–Littlewood maximal function is the composition of the averaging operator and the contravariant transform from the invariant $\sup$ pairing.
The Hardy space is an invariant subspace of the group representation. | The Hardy space consists of the limiting values of the Cauchy integral. SIO is bounded on this space. | The Hardy–Littlewood maximal operator is bounded on the Hardy space $H_{p}{}$ .
Table 1: The correspondence between different elements of harmonic analysis.
## References
Vladimir V. Kisil
School of Mathematics
University of Leeds
Leeds LS2 9JT
UK
E-mail: kisilv@maths.leeds.ac.uk
Web: http://www.maths.leeds.ac.uk/~kisilv/
Received: 12.09.2013
## Index
* Abel summation item Integral transforms.
* admissible
* mother wavelet Example 8.2
* admissible wavelet §4, Example 5.3
* affine
* group, see $ax+b$ group
* approximation
* identity, of the Remark 3.1
* atom §5
* atomic
* decomposition Example 5.9
* $ax+b$ group §3, Example 4.7, Example 5.4, Example 6.9
* invariant measure §3
* left regular representation §3
* representation
* co-adjoint §3
* quasi-regular §3
* calculus
* functional
* covariant §7
* Calderón–Zygmund
* decomposition Remark 7.13
* Carleson
* measure Example 8.7
* norm Example 8.7
* Cauchy
* integral item Integral transforms., Example 4.6, Table 1
* Cauchy–Riemann operator Example 6.9, Table 1
* co-adjoint
* representation of $ax+b$ group §3
* complex numbers Remark 4.10
* continuous
* wavelet transform Example 4.5
* contravariant
* transform Definition 5.2, Table 1
* convolution
* relative §6
* covariant
* functional calculus §7
* transform Definition 4.1, Table 1
* inverse, see contravariant transform
* decomposition
* atomic Example 5.9
* Calderón–Zygmund Remark 7.13
* double numbers Remark 4.10
* dyadic
* squares item Littlewood–Paley theory, Example 6.11, Table 1
* fiducial operator Definition 4.1
* formula
* reconstruction Example 5.3
* Sokhotsky–Plemelj item Boundary values and SIO.
* functional
* calculus
* covariant §7
* invariant item iii
* Gaussian Example 4.6
* grand maximal function Remark 4.9
* group
* affine, see $ax+b$ group
* $ax+b$ §3, Example 4.7, Example 5.4, Example 6.9
* invariant measure §3
* $\mathrm{SL}_{2}(\mathbb{R}{})$ Remark 1.1, Remark 3.5, Remark 4.10, Table 1
* Haar measure, see invariant measure
* Hardy
* pairing §5
* space item Hardy space., Example 3.3, Example 4.6, §7
* generalized Definition 3.2, §7, §7
* Hardy–Littlewood
* maximal functions item Hardy–Littlewood maximal function, Example 4.7, Example 7.6, Table 1
* Hilbert
* transform Example 7.7, Example 7.9
* hypercomplex numbers Remark 3.5
* identity
* approximation of the Remark 3.1
* integral
* Cauchy item Integral transforms., Example 4.6, Table 1
* Poisson, see Poisson kernel
* intertwining operator Theorem 6.1, Proposition 6.7, Example 7.6
* invariant §5
* functional item iii
* measure §3, §4, Example 5.3, Example 5.4
* pairing §5
* subspace Remark 7.3
* inverse
* covariant transform, see contravariant transform
* irreducible
* representation Definition 3.2, §7
* kernel
* Poisson item Integral transforms., Example 4.6, Table 1
* conjugated Example 4.6, Example 7.9
* Laplace
* operator Example 6.10
* left regular representation
* $ax+b$ group §3
* Littlewood–Paley theory item Littlewood–Paley theory
* maximal function
* grand Remark 4.9
* maximal functions Remark 3.1
* Hardy–Littlewood item Hardy–Littlewood maximal function, Example 4.7, Example 7.6, Table 1
* non-tangential Example 5.5, Table 1
* vertical Example 5.5, Table 1
* measure
* Carleson Example 8.7
* Haar, see invariant measure
* invariant §3, §4, Example 5.3, Example 5.4
* method
* orbit §3
* Mexican hat
* wavelet Example 4.5
* mother wavelet Example 4.4
* admissible Example 8.2
* non-tangential
* maximal functions Example 5.5, Table 1
* norm
* Carleson Example 8.7
* nucleus Definition 5.8
* numbers
* complex Remark 4.10
* double Remark 4.10
* hypercomplex Remark 3.5
* operator
* Cauchy–Riemann Example 6.9, Table 1
* fiducial Definition 4.1
* integral
* singular item Boundary values and SIO., Example 7.7, Table 1
* intertwining Theorem 6.1, Proposition 6.7, Example 7.6
* Laplace Example 6.10
* orbit
* method §3
* orthogonality
* relation §8
* pairing
* Hardy §5
* invariant §5
* Poisson kernel item Integral transforms., Example 4.6, Table 1
* conjugated Example 4.6, Example 7.9
* primary
* representation Definition 3.2
* principal
* value item Boundary values and SIO., Example 7.7
* quasi-regular
* representation of $ax+b$ group §3
* reconstruction formula Example 5.3
* relation
* orthogonality §8
* relative
* convolution §6
* representation
* $ax+b$ group
* co-adjoint §3
* quasi-regular §3
* coefficients, see wavelet transform
* irreducible Definition 3.2, §7
* left regular
* $ax+b$ group §3
* primary Definition 3.2
* square integrable §4, Example 5.3, Example 8.2
* singular
* integral operator item Boundary values and SIO., Example 7.7, Table 1
* seesingular integral operator Example 7.7
* $\mathrm{SL}_{2}(\mathbb{R}{})$ Remark 1.1, Remark 3.5, Remark 4.10, Table 1
* Sokhotsky–Plemelj formula item Boundary values and SIO.
* space
* Hardy item Hardy space., Example 3.3, Example 4.6, §7
* generalized Definition 3.2, §7, §7
* square integrable
* representation §4, Example 5.3, Example 8.2
* squares
* dyadic item Littlewood–Paley theory, Example 6.11, Table 1
* state
* vacuum, see mother wavelet
* stopping time argument Example 6.11, Table 1
* subspace
* invariant Remark 7.3
* summation
* Abel item Integral transforms.
* tent item Domain, Remark 4.10, Example 8.7
* theory
* Littlewood–Paley item Littlewood–Paley theory
* transform
* contravariant Definition 5.2, Table 1
* covariant Definition 4.1, Table 1
* inverse, see contravariant transform
* Hilbert Example 7.7, Example 7.9
* wavelet Example 4.4, Table 1
* continuous Example 4.5
* vacuum state, see mother wavelet
* value
* principal item Boundary values and SIO., Example 7.7
* vertical
* maximal functions Example 5.5, Table 1
* wavelet §4, Example 4.4
* admissible §4, Example 5.3
* Mexican hat Example 4.5
* mother Example 4.4
* admissible Example 8.2
* transform Example 4.4, Table 1
* continuous Example 4.5
|
arxiv-papers
| 2012-09-23T15:09:48 |
2024-09-04T02:49:35.447645
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/1209.5072"
}
|
1209.5091
|
# A Cheeger-type inequality on simplicial complexes
John Steenbergen, Caroline Klivans, Sayan Mukherjee Department of
Mathematics, Duke University Division of Applied Mathematics, Departments of
Computer Science and Mathematics, Brown University Departments of Statistical
Science, Mathematics and Computer Science
Institute for Genome Sciences & Policy, Duke University
###### Abstract.
In this paper, we consider a variation on Cheeger numbers related to the
coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type
inequality is proved, which is similar to a result on graphs due to Fan Chung.
This inequality is then used to study the relationship between coboundary
expanders on simplicial complexes and their corresponding eigenvalues,
complementing and extending results found by Gundert and Wagner. In
particular, we find these coboundary expanders do not satisfy natural Buser or
Cheeger inequalities.
## 1\. Introduction
### 1.1. Background
The Cheeger inequality [7, 6] is a classic result that relates the
isoperimetric constant of a manifold (with or without boundary) to the
spectral gap of the Laplace-Beltrami operator. An analog of the manifold
result was also found to hold on graphs [3, 2, 25] and is a prominent result
in spectral graph theory. Given a graph $G$ with vertex set $V$, the Cheeger
number is the following isoperimetric constant
$h:=\min_{\emptyset\subsetneq S\subsetneq V}\frac{|\delta
S|}{\min\\{|S|,|\overline{S}|\\}}$
where $\delta S$ is the set of edges connecting a vertex in $S$ with a vertex
in $\overline{S}=V\setminus S$. The Cheeger inequality on the graph relates
the Cheeger number $h$ to the algebraic connectivity $\lambda$ [14] which is
the the second eigenvalue of the graph Laplacian. It states that
$2h\geq\lambda\geq\frac{h^{2}}{2\max_{v\in V}d_{v}}$
where $d_{v}$ is the number of edges connected to vertex $v$ (also called the
degree of the vertex). For more background on the Cheeger inequality see [9].
A key motivation for studying the Cheeger inequality has been understanding
expander graphs [17] – sparse graphs with strong connectivity properties. The
edge expansion of a graph is the Cheeger number in these studies and expanders
are families of regular graphs $\mathcal{G}$ of increasing size with the
property $h(G)>\varepsilon$ for some fixed $\varepsilon>0$ and all
$G\in\mathcal{G}$. A generalization of the Cheeger number to higher dimensions
on simplicial complexes, based on ideas in [21, 24], was defined and expansion
properties studied in [11] via cochain complexes. In addition, it has long
been known [13] that the graph Laplacian generalizes to higher dimensions on
simplicial complexes. In particular one can generalize the notion of algebraic
connectivity to higher dimensions using the cochain complex and relate an
eigenvalue of the $k$-dimensional Laplacian to the $k$-dimensional Cheeger
number. This raises the question of whether the Cheeger inequality has a
higher-dimensional analog.
### 1.2. Main Results
In this paper we examine the combinatorial Laplacian which is derived from a
chain complex and a cochain complex. Precise definitions of the object studied
and the results are given in section 2. We first state our negative result –
for the cochain complex a natural Cheeger inequality does not hold. For an
$m$-dimensional simplicial complex we denote $\lambda^{m-1}$ as the analog of
the spectral gap for dimension $m-1$ on the cochain complex and we denote
$h^{m-1}$ as the $(m-1)$-dimensional coboundary Cheeger number. In addition,
let $S_{k}$ be the set of $k$-dimensional simplexes and for any $s\in S_{k}$
let $d_{s}$ be the number of $(k+1)$-simplexes incident to $s$. The following
result is an informal statement of Proposition 2.10 and implies that there
exists no Cheeger inequality of the following form for the cochain complex.
Specifically, there are no constants $p_{1},p_{2},C$ such that either of the
inequalities
$C(h^{m-1})^{p_{1}}\geq\lambda^{m-1}\text{\qquad or
\qquad}\lambda^{m-1}\geq\frac{C(h^{m-1})^{p_{2}}}{\max_{s\in S_{m-1}}d_{s}}$
hold in general for an $m$-dimensional simplicial complex $X$ with $m>1$. The
case of $h^{0}$ and $\lambda^{0}$ with $p_{1}=1$ and $p_{2}=2$ reduces to the
Cheeger inequality on the graph and the Cheeger inequality holds.
For the chain complex we obtain a positive result, there is a direct analogue
for the Cheeger inequality in certain well-behaved cases. Whereas the cochain
complex is defined using the coboundary map, the chain complex is defined
using the boundary map. Denote $\gamma_{m}$ as the analog of the spectral gap
for dimension $m$ on the chain complex and $h_{m}$ as the $m$-dimensional
Cheeger number defined using the boundary map. If the $m$-dimensional
simplicial complex $X$ is an orientable pseudomanifold or satisfies certain
more general conditions, then
$h_{m}\geq\gamma_{m}\geq\frac{h_{m}^{2}}{2(m+1)}.$
This inequality can be considered a discrete analog of the Cheeger inequality
for manifolds with Dirichlet boundary condition [7, 6].
### 1.3. Related Work
A probabilistic argument was used by Gundert and Wagner [16] to show on the
cochain complex there exists infinitely many simplicial complexes with
$h^{m-1}=0$ and $\lambda^{m-1}>c$ for some fixed constant $c>0$ – implying
that one side of the Cheeger inequality cannot hold in general. However, this
construction requires the complexes to have torsion in their integral homology
groups due to the way $h^{m-1}$ and $\lambda^{m-1}$ relate to cohomology. In
this paper we show that even for torsion-free simplicial complexes there exist
counterexamples that rule out both sides of a Cheeger inequality.
The analysis of the chain complex in our paper is related to a paper by Fan
Chung [8] which introduces a notion of a Cheeger number on graphs with the
analog of a Dirichlet boundary condition. We provide a detailed comparison on
Appendix A.
Finally, it should be mentioned that the authors in [27] prove a two-sided
Cheeger-type inequality for $\lambda^{m-1}$ using a modified higher-
dimensional Cheeger number. The modified Cheeger number used is nonzero only
if the simplicial complex has complete skeleton, and the Cheeger side of the
inequality includes an additive constant.
## 2\. Main Results
### 2.1. Simplicial Complexes
Since the concept of a Cheeger inequality is strongly associated to manifolds
we focus in this paper on abstract simplicial complexes that are analogous to
well-behaved manifolds. In particular, we will focus on simplicial complexes
that have geometric realizations homeomorphic to a Euclidean ball
$B^{m}:=\\{x\in\mathbb{R}^{m}:\lVert x\rVert_{2}\leq 1\\}$. We will call such
complexes simplicial $m$-balls
By a simplicial complex we always mean an abstract finite simplicial complex.
Simplicial complexes generalize the notion of a graph to higher dimensions.
Given a set of vertices $V$, any nonempty subset $\sigma\subseteq V$ of the
form $\sigma=\\{v_{0},v_{1},\ldots,v_{k}\\}$ is called a $k$-dimensional
simplex, or $k$-simplex. A simplicial complex $X$ is a finite collection of
simplexes of various dimensions such that $X$ is closed under inclusion, i.e.,
$\tau\subseteq\sigma$ and $\sigma\in X$ implies $\tau\in X$.
Given a simplicial complex $X$ denote the set of $k$-simplexes of $X$ as
$S_{k}:=S_{k}(X)$. We call $X$ a simplicial $m$-complex if
$S_{m}(X)\neq\emptyset$ but $S_{m+1}(X)=\emptyset$. Given two simplexes
$\sigma\in S_{k}$ and $\tau\in S_{k+1}$ such that $\sigma\subset\tau$, we call
$\sigma$ a face of $\tau$ and $\tau$ a coface of $\sigma$. Two $k$-simplexes
are lower adjacent if they share a common face and are upper adjacent if they
share a common coface.
Every simplicial complex $X$ has associated with it a geometric realization
denoted $|X|$. The simplicial $m$-complex $\Sigma^{m}$ consisting of a single
$m$-simplex and its subsets has geometric realization homeomorphic to $B^{m}$.
Thus, $\Sigma^{m}$ is an example of a simplicial $m$-ball. A subdivision of a
simplicial complex $X$ is a simplicial complex $X^{\prime}$ such that
$|X^{\prime}|=|X|$ and every simplex of $X^{\prime}$ is, in the geometric
realization, contained in a simplex of $X$. Thus, any subdivision of
$\Sigma^{m}$ is also a simplicial $m$-ball.
There is another convenient set of criteria under which a simplicial complex
is a simplicial $m$-ball. A simplicial $m$-complex $X$ is constructible if
either (1) $X=\Sigma^{m}$ or (2) $X$ can be decomposed into the union of two
constructible simplicial $m$-subcomplexes $X=X_{1}\cup X_{2}$ such that
$X_{1}\cap X_{2}$ is a constructible simplicial $(m-1)$-complex. If every
$s\in S_{m-1}$ has at most two cofaces then $X$ is said to be non-branching.
In this case, every $s\in S_{m-1}$ with exactly one coface is called a
boundary face of $X$. It is known [5] that a the geometric realization of a
non-branching constructible simplicial $m$-complex $X$ is homeomorphic to
$B^{m}$ if $X$ has at least one boundary face (otherwise it is homeomorphic to
the sphere).
### 2.2. Chain and Cochain Complexes
Given a simplicial complex $X$ and any field $F$, we can define the chain and
cochain complexes of $X$ over $F$. In this paper we consider the fields
$\mathbb{Z}_{2}$ and $\mathbb{R}$. Given a simplex
$\sigma=\\{v_{0},v_{1},\ldots,v_{k}\\}$, $\sigma$ can be ordered as a set. An
orientation, denoted by $[v_{0},v_{1},\ldots,v_{k}]$ is an equivalence class
of all even permutations of the given ordering. There are always two
orientations for $k>0$. The space of $k$-chains $C_{k}(F):=C_{k}(X;F)$ is the
vector space of linear combinations of oriented $k$-simplexes with
coefficients in $F$, with the stipulation that the two orientations of a
simplex are negatives of each other in $C_{k}(F)$. The space of $k$-cochains
$C^{k}(F):=C^{k}(X;F)$ is then defined to be the vector space dual to
$C_{k}(F)$. These spaces are isomorphic and we will make no distinction
between them. The boundary map $\partial_{k}(F):C_{k}(F)\to C_{k-1}(F)$ is
defined on the basis elements $[v_{0},\ldots,v_{k}]$ as
$\partial_{k}[v_{0},\ldots,v_{k}]=\sum_{i=0}^{k}(-1)^{i}[v_{0},\ldots,v_{i-1},v_{i+1},\ldots,v_{k}]$
The coboundary map $\delta^{k-1}(F):C^{k-1}(F)\to C^{k}(F)$ is then defined to
be the transpose of the boundary map. When there is no confusion, we will
denote the boundary and coboundary maps by $\partial$ and $\delta$. It is easy
to see that $\partial\partial=\delta\delta=0$, so that
$(C_{k}(F),\partial_{k})$ and $(C^{k}(F),\delta^{k})$ form chain and cochain
complexes. See Figures 1 and 2 for examples of $\partial$ and $\delta$ on real
and $\mathbb{Z}_{2}$ chains/cochains.
$\delta^{1}(\mathbb{R})$$\partial_{2}(\mathbb{R})$$v_{1}$$v_{2}$$v_{4}$$v_{3}$$v_{1}$$v_{2}$$v_{4}$$v_{3}$$v_{1}$$v_{2}$$v_{4}$$v_{3}$$\delta^{1}(\mathbb{R})$$\partial_{2}(\mathbb{R})$$3[v_{2},v_{1}]+4[v_{3},v_{1}]+2[v_{3},v_{2}]$$[v_{1},v_{3},v_{2}]+2[v_{2},v_{4},v_{3}]$$\
\ [v_{1},v_{3}]+[v_{2},v_{1}]+3[v_{3},v_{2}]$$\qquad\qquad\ \
+2[v_{2},v_{4}]+2[v_{4},v_{3}]$432001132212
Figure 1. An example of $\partial(\mathbb{R})$ and $\delta(\mathbb{R})$.
$v_{1}$$v_{2}$$v_{4}$$v_{3}$$v_{1}$$v_{2}$$v_{4}$$v_{3}$$v_{1}$$v_{2}$$v_{4}$$v_{3}$$\delta^{1}(\mathbb{Z}_{2})$$\partial_{2}(\mathbb{Z}_{2})$$[v_{1},v_{2},v_{3}]+[v_{2},v_{3},v_{4}]$$\delta^{1}(\mathbb{Z}_{2})$$\partial_{2}(\mathbb{Z}_{2})$$\
\ [v_{1},v_{2}]+[v_{1},v_{3}]+[v_{2},v_{3}]$$\ \
[v_{1},v_{2}]+[v_{1},v_{3}]$$\qquad\qquad\quad\
+[v_{2},v_{4}]+[v_{3},v_{4}]$110101111011
Figure 2. An example of $\partial(\mathbb{Z}_{2})$ and
$\delta(\mathbb{Z}_{2})$.
When $F=\mathbb{Z}_{2}$, positive and negative have no meaning and therefore
no distinction is made between different orientations. In particular, it is
possible to identify $C_{k}(\mathbb{Z}_{2})$ and $C_{k}(\mathbb{Z}_{2})$ with
$S_{k}$ as sets. Throughout this paper, we will identify a
$k$-chain/$k$-cochain $\phi$ over $\mathbb{Z}_{2}$ with the subset
$\phi\subset S_{k}$ of $k$-simplexes to which $\phi$ assigns the coefficient
1.
The homology and cohomology vector spaces of $X$ over $F$ are
$H_{k}(F):=H_{k}(X;F)=\frac{\operatorname{ker}\partial_{k}}{\operatorname{im}\partial_{k+1}}\text{\quad
and
\quad}H^{k}(F):=H^{k}(X;F)=\frac{\operatorname{ker}\delta^{k}}{\operatorname{im}\delta^{k-1}}.$
It is known from the universal coefficient theorem that $H^{k}(F)$ is the
vector space dual to $H_{k}(F)$.
### 2.3. Laplacians and Eigenvalues
The $k$-th Laplacian of $X$ is defined to be
$L_{k}:=L_{k}^{\text{up}}+L_{k}^{\text{down}}$
where
$L_{k}^{\text{up}}=\partial_{k+1}(\mathbb{R})\delta^{k}(\mathbb{R})\text{\quad
and
\quad}L_{k}^{\text{down}}=\delta^{k-1}(\mathbb{R})\partial_{k}(\mathbb{R}).$
By way of Rayleigh quotients, the smallest nontrivial eigenvalue of
$L_{k}^{\text{up}}$ and $L_{k}^{\text{down}}$ are given by
$\lambda^{k}=\min_{\begin{subarray}{c}f\in C^{k}(\mathbb{R})\\\
f\perp\operatorname{im}\delta\end{subarray}}\frac{\lVert\delta
f\rVert_{2}^{2}}{\lVert f\rVert_{2}^{2}}=\min_{\begin{subarray}{c}f\in
C^{k}(\mathbb{R})\\\
f\notin\operatorname{im}\delta\end{subarray}}\frac{\lVert\delta
f\rVert_{2}^{2}}{\min_{g\in\operatorname{im}\delta}\lVert f+g\rVert_{2}^{2}},$
$\lambda_{k}=\min_{\begin{subarray}{c}f\in C_{k}(\mathbb{R})\\\
f\perp\operatorname{im}\partial\end{subarray}}\frac{\lVert\partial
f\rVert_{2}^{2}}{\lVert f\rVert_{2}^{2}}=\min_{\begin{subarray}{c}f\in
C_{k}(\mathbb{R})\\\
f\notin\operatorname{im}\partial\end{subarray}}\frac{\lVert\partial
f\rVert_{2}^{2}}{\min_{g\in\operatorname{im}\partial}\lVert
f+g\rVert_{2}^{2}},$
where $\lVert\cdot\rVert_{2}$ denotes the Euclidean norm on both
$C^{k}(\mathbb{R})$ and $C_{k}(\mathbb{R})$. It is well known that the nonzero
spectrum of $L_{k}$ is the union of the nonzero spectrum of
$L_{k}^{\text{up}}$ with the nonzero spectrum of $L_{k}^{\text{down}}$. Thus,
the smallest nonzero eigenvalue of $L_{k}$ is either $\lambda^{k}$ or
$\lambda_{k}$ assuming one of them is nonzero. In addition, the nonzero
spectrum of $L_{k}^{\text{up}}$ is the same as the nonzero spectrum of
$L_{k+1}^{\text{down}}$. Thus, $\lambda^{k}=\lambda_{k+1}$ whenever
$\lambda^{k},\lambda_{k+1}$ are both nonzero.
The relationship between eigenvalues and homology/cohomology is as follows:
$\lambda_{k}=0$ | | $\lambda^{k}=0$
---|---|---
$\Updownarrow$ | and | $\Updownarrow$
$H_{k}(\mathbb{R})\neq 0$ | | $H^{k}(\mathbb{R})\neq 0$.
If we pass to the reduced cochain complex, $\lambda^{0}$ becomes the algebraic
connectivity (or Fiedler number) of a graph [14] and
$\lambda^{0}=0\Leftrightarrow\widetilde{H}^{0}(\mathbb{R})\neq 0$.
### 2.4. Cheeger Numbers
Higher-dimensional Cheeger numbers were first stated in [11] to capture a
higher-dimensional notion of expanders. They are defined via the coboundary
map as follows:
###### Definition 2.1.
Let $\lVert\cdot\rVert$ denote the Hamming norm on $C^{k}(\mathbb{Z}_{2})$.
The $k$-th (coboundary) Cheeger number of $X$ is
$h^{k}:=\min_{\begin{subarray}{c}\phi\in C^{k}(\mathbb{Z}_{2})\\\
\phi\notin\operatorname{im}\,\delta\end{subarray}}\frac{\lVert\delta\phi\rVert}{\min_{\psi\in\operatorname{im}\,\delta}\lVert\phi+\psi\rVert}.$
A similar definition can be given for the boundary map.
###### Definition 2.2.
Let $\lVert\cdot\rVert$ also denote the Hamming norm on
$C_{k}(\mathbb{Z}_{2})$. The $k$-th boundary Cheeger number of $X$ is
$h_{k}:=\min_{\begin{subarray}{c}\phi\in C_{k}(\mathbb{Z}_{2})\\\
\phi\notin\operatorname{im}\,\partial\end{subarray}}\frac{\lVert\partial\phi\rVert}{\min_{\psi\in\operatorname{im}\,\partial}\lVert\phi+\psi\rVert}.$
The relationship between Cheeger numbers and homology/cohomology is as
follows:
$h_{k}=0$ | | $h^{k}=0$
---|---|---
$\Updownarrow$ | and | $\Updownarrow$
$H_{k}(\mathbb{Z}_{2})\neq 0$ | | $H^{k}(\mathbb{Z}_{2})\neq 0$ .
If we pass to the reduced cochain complex, $h^{0}$ becomes the Cheeger number
of a graph [11] and
$h^{0}=0\Leftrightarrow\widetilde{H}^{0}(\mathbb{Z}_{2})\neq 0$.
Often, we speak of a cochain that attains the minimum in the definition of the
Cheeger number – in the graph case these are Cheeger cuts. We will say that
$\phi\in C^{k}(\mathbb{Z}_{2})$ attains $h^{k}$ if
$h^{k}=\frac{\lVert\delta\phi\rVert}{\lVert\phi\rVert}$. The same terminology
will be used for $h_{k}$.
### 2.5. Additional Notation and Preliminary Results
Here we collect some interesting results concerning Cheeger numbers which will
be needed later in section 2.6. Lemma 2.3 says that $h_{1}$ has a very simple
interpretation in terms of the diameter of the simplicial complex. Lemma 2.5
says that $h^{m-1}$ also has a very simple interpretation in terms of the
radius.
We define the diameter of a simplicial $m$-complex $X$ as follows. Given two
vertices $v_{1},v_{2}\in S_{0}$, we define the distance between them to be the
quantity
$\operatorname{dist}(v_{1},v_{2}):=\min\\{\lVert\phi\rVert:\phi\in
C_{1}(\mathbb{Z}_{2})\text{ and }\partial\phi=v_{1}+v_{2}\\}$
Any chain $\phi$ attaining the minimum is called a geodesic. Note that for any
geodesic $\phi$, $h_{1}\leq\frac{2}{\lVert\phi\rVert}$. For our purposes,
$\operatorname{dist}(v_{1},v_{2})=0$ if $v_{1}$, $v_{2}$ are not in the same
connected component. The diameter of $X$ is then defined to be
$\operatorname{diam}(X):=\max_{v_{1},v_{2}\in
S_{0}}\operatorname{dist}(v_{1},v_{2}).$
As it turns out, $h_{1}$ is strongly related to the diameter of a simplicial
complex.
###### Lemma 2.3.
Given a simplicial $m$-complex $X$ with $m\geq 1$ and satisfying
$H_{1}(\mathbb{Z}_{2})=0$, $h_{1}$ is attained by a geodesic and hence
$h_{1}=\frac{2}{\operatorname{diam}(X)}$
.
###### Proof.
Suppose that $\phi\in C_{1}(\mathbb{Z}_{2})$ attains $h_{1}$. Clearly,
$\lVert\partial\phi\rVert$ must be even and nonzero. What we will show is that
we can assume $\lVert\partial\phi\rVert=2$. Thinking of $\phi$ as a graph
(consisting of the edges in $\phi$ and their vertices), it is also clear that
every connected component $\phi_{i}$ of $\phi$ has
$\lVert\partial\phi_{i}\rVert$ even. For every pair of vertices in
$\partial\phi_{i}$, there exists a geodesic in $X$ with the given pair of
vertices as its boundary. Thus, there exist geodesics
$\psi_{1},\ldots,\psi_{q}$ such that $\partial\psi_{j}$ is a distinct pair of
vertices in $\partial\phi$ for all $j$ and
$\partial(\psi_{1}+\cdots+\psi_{q})=\partial\phi$. Since $\phi$ attains
$h_{1}$ and $H_{1}(\mathbb{Z}_{2})=0$,
$\lVert\phi\rVert=\min_{\psi\in\operatorname{im}\partial}\lVert\phi+\psi\rVert=\min_{\partial\psi=\partial\phi}\lVert\psi\rVert$
In other words, $\phi$ is a 1-chain of smallest norm with boundary
$\partial\phi$. Thus,
$\lVert\psi_{1}+\cdots+\psi_{q}\rVert\geq\lVert\phi\rVert$. Now,
$\displaystyle h_{1}$
$\displaystyle=\frac{\lVert\partial\phi\rVert}{\lVert\phi\rVert}$
$\displaystyle\geq\frac{\lVert\partial(\psi_{1}+\cdots+\psi_{q})\rVert}{\lVert\psi_{1}+\cdots+\psi_{q}\rVert}$
$\displaystyle\geq\frac{2+\cdots+2}{\lVert\psi_{1}\rVert+\cdots+\lVert\psi_{q}\rVert}$
$\displaystyle\geq\min\left\\{\frac{2}{\lVert\psi_{1}\rVert},\ldots,\frac{2}{\lVert\psi_{q}\rVert}\right\\}$
$\displaystyle\geq h_{1}$
and therefore
$h_{1}=\min\big{\\{}\frac{2}{\lVert\psi_{1}\rVert},\ldots,\frac{2}{\lVert\psi_{q}\rVert}\big{\\}}$.
Here we are using the general inequality
$\frac{a_{1}+a_{2}+\cdots+a_{k}}{b_{1}+b_{2}+\cdots+b_{k}}\geq\min_{i}\frac{a_{i}}{b_{i}}$,
valid for all $a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}>0$. Hence,
$h_{1}=\frac{2}{\lVert\psi_{j}\rVert}$ for some geodesic $\psi_{j}$. This
completes the proof. ∎
While the diameter is defined in terms of 1-chains, we define the radius in
terms of $(m-1)$-cochains as follows. Given a simplicial $m$-complex $X$, we
define the depth of an $m$-simplex $\sigma$ to be
$0pt(\sigma):=\min\\{\lVert\phi\rVert:\phi\in
C^{m-1}(\mathbb{Z}_{2}),\delta\phi=\sigma\\}.$
Any minimizing $\phi$ will be said to be a depth-attaining cochain for
$\sigma$. Note that for any such $\phi$,
$h^{m-1}\leq\frac{1}{\lVert\phi\rVert}$. All $m$-simplexes have a defined
depth when $H_{m}(\mathbb{Z}_{2})$ is trivial. In this case, we define the
radius of $X$ to be
$\operatorname{rad}(X):=\max_{\sigma\in S_{m}}0pt(\sigma).$
Depth-attaining cochains have a very predictable structure for non-branching
simplicial complexes, a fact which we will use later in proving Proposition
2.10. Roughly speaking, Lemma 2.4 says that if $\phi$ is depth-attaining for
$\sigma$, then $\phi$ is a linear non-intersecting sequence of
$(m-1)$-simplexes starting with a face of $\sigma$ and ending with a boundary
face. For the statement and proof of this Lemma we define the star
$\operatorname{st}(s)$ of a simplex $s$ to be the set of cofaces of $s$.
###### Lemma 2.4.
Let $X$ be a simplicial $m$-complex such that every $s\in S_{m-1}$ has at most
two cofaces. Suppose that $\sigma\in S_{m}$ has depth $d$ and $\phi$ is a
depth-attaining cochain for $\sigma$. Then there is a sequence
$s_{1},s_{2},\ldots,s_{d}$ of distinct $(m-1)$-simplexes and a sequence
$\sigma=\sigma_{1},\sigma_{2},\ldots,\sigma_{d}$ of distinct $m$-simplexes
satisfying
1. (1)
$\phi=\sum_{i=1}^{d}s_{i}$,
2. (2)
$\operatorname{st}(s_{i})=\\{\sigma_{i},\sigma_{i+1}\\}$ for $i<d$,
3. (3)
$\operatorname{st}(s_{d})=\\{\sigma_{d}\\}$.
###### Proof.
Assume $\phi=\sum_{i=1}^{d}s_{i}$. Clearly, at least one of the $s_{i}$ must
have $\sigma$ as a coface, so WLOG we can assume $s_{1}$ has
$\sigma=\sigma_{1}$ as a coface. If $s_{1}$ is a boundary face, we are done
and $d=1$. If not, then $s_{1}$ has another coface $\sigma_{2}$. In this case,
if there are no other $s_{i}$ with $\sigma_{2}$ as a coface then we arrive at
the contradiction that $\delta\phi$ contains $\sigma_{2}$, i.e.,
$\delta\phi\neq\sigma$. Thus, there is another $s_{i}$ with $\sigma_{2}$ as a
coface, which we can assume WLOG is $s_{2}$.
We proceed by induction. Suppose that for $k>1$ there is a sequence
$\sigma_{1},\sigma_{2},\ldots,\sigma_{k}$ of distinct $m$-simplexes such that
$\delta(s_{1}+\cdots+s_{k-1})=\sigma+\sigma_{k}$ where
$\operatorname{st}(s_{i})=\\{\sigma_{i},\sigma_{i+1}\\}$ for all $i$. Then we
can find another $s_{i}$, $i>k$, which we can assume WLOG is $s_{k}$ and which
has $\sigma_{k}$ as a coface. If no such $s_{i}$ exists then
$\delta\phi\neq\sigma$. If $s_{k}$ is a boundary face we are done and $d=k$.
If $s_{k}$ has $\sigma_{k+1}$ as a second coface and $\sigma_{k+1}=\sigma_{i}$
for some $i<k$ then $s_{i}+\ldots+s_{k}$ is a cocycle, but this means that
$\delta(\phi-s_{i}-\cdots-s_{k})=\sigma$ so $\phi$ is not depth-attaining.
Otherwise, $\sigma_{1},\sigma_{2},\ldots,\sigma_{k+1}$ is a sequence of
distinct $m$-simplexes such that
$\delta(s_{1}+\cdots+s_{k})=\sigma+\sigma_{k+1}$ where
$\operatorname{st}(s_{i})=\\{\sigma_{i},\sigma_{i+1}\\}$ for all $i$. This
leaves us back where we started. By induction, we can continue this process
until $k=d$ and $s_{d}$ is a boundary face. ∎
###### Lemma 2.5.
Let $X$ be a simplicial $m$-complex with $H^{m-1}(\mathbb{Z}_{2})=0$ and
$H_{m}(\mathbb{Z}_{2})=0$. Then $h^{m-1}$ is attained by a depth-attaining
cochain and hence
$h^{m-1}=\frac{1}{\operatorname{rad}(X)}.$
###### Proof.
Suppose $\psi$ attains $h^{m-1}$ and $\delta\psi$ is a sum of distinct
$m$-simplexes $\sigma_{1},\ldots,\sigma_{q}$ with depth-attaining cochains
$\psi_{1},\ldots,\psi_{q}$. Clearly
$\lVert\psi\rVert\leq\lVert\psi_{1}\rVert+\cdots+\lVert\psi_{q}\rVert$, so
$\displaystyle h^{m-1}$ $\displaystyle=\frac{q}{\lVert\psi\rVert}$
$\displaystyle\geq\frac{1+\cdots+1}{\lVert\psi_{1}\rVert+\cdots+\lVert\psi_{q}\rVert}$
$\displaystyle\geq\min\left\\{\frac{1}{\lVert\psi_{1}\rVert},\ldots,\frac{1}{\lVert\psi_{q}\rVert}\right\\}$
$\displaystyle\geq h^{m-1}$
and therefore
$h^{m-1}=\min\big{\\{}\frac{1}{\lVert\psi_{1}\rVert},\ldots,\frac{1}{\lVert\psi_{q}\rVert}\big{\\}}$.
Here we are using the general inequality
$\frac{a_{1}+a_{2}+\cdots+a_{k}}{b_{1}+b_{2}+\cdots+b_{k}}\geq\min_{i}\frac{a_{i}}{b_{i}}$,
valid for all $a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}>0$. Hence,
$h^{m-1}=\frac{1}{\lVert\psi_{j}\rVert}$ for some depth-attainng cochain
$\psi_{j}$. This completes the proof. ∎
An interesting result which will not be used in this paper is a Cheeger-type
inequality for the special case $X=\Sigma^{m}$.
###### Lemma 2.6.
Recall $\Sigma^{m}$ is the simplicial complex induced by an $m$-simplex. The
following holds for all $k$.
1. (1)
$h^{k}(\Sigma^{m-1})\geq\frac{m}{k+2}$
2. (2)
$h_{k}(\Sigma^{m-1})\geq\frac{m}{m-k}$.
The reason this result is Cheeger-type is because all the Laplacian
eigenvalues of all dimensions for $\Sigma^{m-1}$ are equal to $m$ (this is
easily seen from the characterization of the Laplacian in [26]). Part (1) of
this Lemma was proved by Meshulam and Wallach [24] (who, even though they did
not define the Cheeger number, still worked with its numerator and denominator
separately). Their proof can be easily modified to prove part (2) of the
Lemma.
### 2.6. Main Results
We now state the main results of this paper – there exists a Cheeger-type
inequality in the top dimension for the chain complex but not for the cochain
complex.
To state the results we need the following notion of orientational similarity.
Two oriented lower adjacent $k$-simplexes are dissimilarly oriented if they
induce the same orientation on the common face. In other words, if
$\sigma=[v_{0},\ldots,v_{k}]$ and $\tau=[w_{0},\ldots,w_{k}]$ share the face
$\\{u_{0},\ldots,u_{k-1}\\}$, then $\sigma$ and $\tau$ are dissimilarly
oriented if $\partial(\mathbb{R})\sigma$ and $\partial(\mathbb{R})\tau$ assign
the same coefficient ($+1$ or $-1$) to the oriented simplex
$[u_{0},\ldots,u_{k-1}]$. Otherwise, they are said to be similarly oriented.
If $X$ is a simplicial $m$-complex and all its $m$-simplices can be oriented
similarly, then $X$ is called orientable.
We first state the positive result – there is a Cheeger-type inequality for
the chain complex.
###### Theorem 2.7.
Let $X$ be a simplicial $m$-complex, $m>0$.
1. (1)
Let $\phi\in C_{m}(\mathbb{Z}_{2})$ minimize the quotient in
$h_{m}:=\min_{\begin{subarray}{c}\phi\in C_{m}(\mathbb{Z}_{2})\\\
\phi\notin\operatorname{im}\,\partial\end{subarray}}\frac{\lVert\partial\phi\rVert}{\min_{\psi\in\operatorname{im}\,\partial}\lVert\phi+\psi\rVert}.$
If all $m$-simplexes in $\phi$ can be similarly oriented, then
$h_{m}\geq\lambda_{m}$.
2. (2)
Assume that every $(m-1)$-dimensional simplex is incident to at most two
$m$-simplexes. Then
$\lambda_{m}\geq\frac{h_{m}^{2}}{2(m+1)}.$
The first statement is the analog of the Buser inequality for graphs. The
second statement is an analog of the Cheeger inequality for graphs, as well as
the Cheeger inequality for a manifold with Dirichlet boundary conditions. The
constraint that every $(m-1)$-simplex has at most two cofaces enforces the
boundary condition. The hypotheses required for both inequalities are always
satisfied by orientable pseudomanifolds.
The hypotheses required by the Theorem cannot be removed, as proved by the
following two examples.
###### Example 2.8 (Real Projective Plane).
Given a triangulation $X$ of $\mathbb{R}P^{2}$ (see Figure 3) we know that
$H_{2}(\mathbb{Z}_{2})\neq 0$ while $H_{2}(\mathbb{R})=0$, so that
$h_{2}=0\neq\lambda_{2}$. This is due to the nonorientability of
$\mathbb{R}P^{2}$. The chain $\phi\in C_{2}(\mathbb{Z}_{2})$ containing every
$m$-simplex has no boundary. However, the $m$-simplexes cannot all be
similarly oriented, so that there is no corresponding boundaryless chain in
$C_{2}(\mathbb{R})$. As a result, the hypothesis used in part (1) of the
Theorem cannot in general be removed.
Figure 3. The fundamental polygon of $\mathbb{R}P^{2}$.
###### Example 2.9.
Let $G_{k}$ be a graph with $2k$ vertices of degree one, half of which connect
to one end of an edge and the other half connect to the other end (see figure
4). Clearly, $h^{0}(G_{k})=\frac{1}{k+1}$ while Lemma 2.3 implies
$h_{1}=\frac{2}{3}$. By the Buser inequality for graphs,
$\lambda^{0}\leq\frac{2}{k+1}$ and since $\lambda_{1}=\lambda^{0}$, this means
that $\lambda_{1}\to 0$. As a result, we conclude that the hypothesis used in
part (2) of the Theorem cannot be removed.
$k$vertices$k$vertices
Figure 4. The family of graphs $G_{k}$.
###### Proof of Theorem 2.7.
Given the hypotheses, $\lambda_{m}$ is a linear programming relaxation of
$h_{m}$. Let $g\in C_{m}(\mathbb{R})$ be the chain which assigns a 1 to every
simplex in $\phi$ (all of them similarly oriented) and a 0 to every other
simplex. Then
$h_{m}=\frac{\lVert\partial\phi\rVert}{\lVert\phi\rVert}=\frac{\lVert\partial
g\rVert_{2}^{2}}{\lVert g\rVert_{2}^{2}}\geq\min_{\begin{subarray}{c}f\in
C_{m}(\mathbb{R})\\\ f\neq 0\end{subarray}}\frac{\lVert\partial
f\rVert_{2}^{2}}{\lVert f\rVert_{2}^{2}}=\lambda_{m}.$
∎
###### Proof of Theorem 2.7.
Let $f$ be an eigenvector of $\lambda_{m}$ and for any oriented $m$-simplex
$\sigma$ let $f(\sigma)$ denote the coefficient assigned to $\sigma$ by $f$.
Orient the $m$-simplexes of $X$ so that all the values of $f$ are non-negative
and let $S_{m}^{\text{or}}(X)$ be the set of oriented $m$-simplices of $X$. We
do not assume the $m$-simplexes are similarly oriented. Number the
$m$-simplexes from $1$ to $N:=|S_{m}^{\text{or}}(X)|$ in increasing order of
$f$:
$0\leq f(\sigma_{1})\leq f(\sigma_{2})\leq\cdots\leq f(\sigma_{N}).$
To aid us in the proof, we introduce a new simplicial $m$-complex $X^{\prime}$
which contains $X$ as a subcomplex and which is defined as follows: for every
boundary face $s=\\{v_{0},\ldots,v_{m-1}\\}$ in $X$ create a new vertex $v$
and a new $m$-simplex $\sigma=\\{v_{0},\ldots,v_{m-1},v\\}$ which includes $v$
and $s$. These new $m$-simplexes will be called border facets. Give the border
facets any orientation and let $F_{m}^{\text{or}}(X^{\prime})$ be the set of
oriented border facets. We can extend $f$ to be a function on
$S_{m}^{\text{or}}(X)\cup F_{m}^{\text{or}}(X^{\prime})$ by defining
$f(\sigma)=0$ for any $\sigma\in F_{m}^{\text{or}}(X^{\prime})$. Let
$M:=|F_{m}^{\text{or}}(X^{\prime})|$ and number the oriented border facets in
any order:
$F_{m}^{\text{or}}(X^{\prime})=\\{\sigma_{0},\sigma_{-1},\ldots,\sigma_{1-M}\\}.$
11110000
Figure 5. Making Dirichlet boundary conditions explicit.
The intuition behind introducing the border facets comes from the analogy with
the continuous Cheeger inequality for functions satisfying Dirichlet boundary
conditions (see [7]). In our case, the Dirichlet boundary condition is
implicit in the fact that $f$ is defined on $m$-simplexes (as opposed to
vertices). The border facets represent the boundary of the $m$-dimensional
part of $X$, and $f$ is in fact zero on them. See Figure 5 for a depiction. In
this analogy, $h_{m}$ plays the part of the Cheeger number defined as in [7]
for manifolds with boundary.
When two simplexes $\sigma,\tau$ are lower adjacent we write $\sigma\sim\tau$.
Now define
$C_{i}=\left\\{\\{\sigma_{j},\sigma_{k}\\}:1-M\leq j\leq i<k\leq N\text{ \ and
\ }\sigma_{j}\sim\sigma_{k}\right\\}$
and
$h[f]=\min_{0\leq i\leq N-1}\frac{|C_{i}|}{N-i}.$
Observe that $h[f]\geq h_{m}$.
We now finish the theorem. The following summations are taken over all
oriented $m$-simplexes in $S_{m}^{\text{or}}(X)\cup
F_{m}^{\text{or}}(X^{\prime})$.
(1) $\displaystyle\lambda_{m}$
$\displaystyle=\frac{\sum_{\sigma\sim\tau}(f(\sigma)\pm
f(\tau))^{2}}{\sum_{\sigma}f(\sigma)^{2}},$
$\displaystyle=\frac{\sum_{\sigma\sim\tau}(f(\sigma)\pm
f(\tau))^{2}}{\sum_{\sigma}f(\sigma)^{2}}\cdot\frac{\sum_{\sigma\sim\tau}(f(\sigma)\mp
f(\tau))^{2}}{\sum_{\sigma\sim\tau}(f(\sigma)\mp f(\tau))^{2}},$ (3)
$\displaystyle\geq\frac{\left(\sum_{\sigma\sim\tau}|f(\sigma)^{2}-f(\tau)^{2}|\right)^{2}}{\left(\sum_{\sigma}f(\sigma)^{2}\right)\cdot\left(\sum_{\sigma\sim\tau}(f(\sigma)\mp
f(t_{2}))^{2}\right)},$
$\displaystyle\geq\frac{\left(\sum_{\sigma\sim\tau}|f(\sigma)^{2}-f(\tau)^{2}|\right)^{2}}{\left(\sum_{\sigma}f(\sigma)^{2}\right)\cdot\left(2\sum_{\sigma\sim\tau}f(\sigma)^{2}+f(\tau)^{2}\right)},$
$\displaystyle=\frac{\left(\sum_{\sigma\sim\tau}|f(\sigma)^{2}-f(\tau)^{2}|\right)^{2}}{\left(\sum_{\sigma}f(\sigma)^{2}\right)\cdot
2(m+1)\cdot\left(\sum_{\sigma}f(\sigma)^{2}\right)},$ (6)
$\displaystyle=\frac{\left(\sum_{i=0}^{N-1}(f(\sigma_{i+1})^{2}-f(\sigma_{i})^{2})|C_{i}|\right)^{2}}{2(m+1)\cdot\left(\sum_{\sigma}f(\sigma)^{2}\right)^{2}},$
$\displaystyle\geq\frac{\left(\sum_{i=0}^{N-1}(f(\sigma_{i+1})^{2}-f(\sigma_{i})^{2})h[f](N-i)\right)^{2}}{2(m+1)\cdot\left(\sum_{\sigma}f(\sigma)^{2}\right)^{2}},$
$\displaystyle=\frac{h[f]^{2}}{2(m+1)}\cdot\frac{\left(\sum_{\sigma}f(\sigma)^{2}\right)^{2}}{\left(\sum_{\sigma}f(\sigma)^{2}\right)^{2}},$
$\displaystyle\geq\frac{h_{m}^{2}}{2(m+1)}.$
Step (1) follows from the Rayleigh quotient characterization of $\lambda_{m}$
and step (3) follows from the Cauchy-Schwarz inequality. We prove the
statement for step (6) below.
We want to show
$\sum_{\sigma\sim\tau}|f(\sigma)^{2}-f(\tau)^{2}|=\sum_{i=0}^{N-1}(f(\sigma_{i+1})^{2}-f(\sigma_{i})^{2})|C_{i}|.$
This can be seen by counting the number of times each $f(\sigma_{i})^{2}$
appears in each sum. In the left hand sum, each $f(\sigma_{i})^{2}$ appears a
number of times equal to
$\Delta_{i}:=\left|\\{\\{\sigma_{j},\sigma_{i}\\}:j<i\text{ \ and \
}\sigma_{j}\sim\sigma_{i}\\}|-|\\{\\{\sigma_{i},\sigma_{k}\\}:i<k\text{ \ and
\ }\sigma_{i}\sim\sigma_{k}\\}\right|.$
On the other hand, each $f(\sigma_{i})^{2}$ appears $|C_{i-1}|-|C_{i}|$ times
in the right hand sum. To see that these are the same, note that for each pair
$\\{\sigma_{j},\sigma_{k}\\}$ in $C_{i-1}$, either $k=i$ or else
$\\{\sigma_{j},\sigma_{k}\\}$ is in $C_{i}$ as well, meaning it is canceled in
the difference. Similarly, for each pair $\\{\sigma_{j},\sigma_{k}\\}$ in
$C_{i}$, either $j=i$ or else $\\{\sigma_{j},\sigma_{k}\\}$ is in $C_{i-1}$ as
well, again meaning it is canceled. Thus
$|C_{i-1}|-|C_{i}|=\Delta_{i}.$
This completes the proof. ∎
We now state the negative result – the analogous Cheeger-type inequality for
the cochain complex does not hold.
###### Proposition 2.10.
For every $m>1$, there exist families of simplicial $m$-balls $X_{k}$ and
$Y_{k}$ such that
1. (1)
for $X_{k}$, $\lambda^{m-1}(X_{k})\geq\frac{(m-1)^{2}}{2(m+1)}$ for all $k$
but $h^{m-1}(X_{k})\to 0$ as $k\to\infty$.
2. (2)
for $Y_{k}$, $\lambda^{m-1}(Y_{k})\leq\frac{1}{m^{k-1}}$ for $k>1$ but
$h^{m-1}(Y_{k})\geq\frac{1}{k}$ for all $k$.
As mentioned in the introduction, it has already been shown in [16] that there
exist infinite families of simplicial complexes for which $h^{m-1}=0$ but
$\lambda^{m-1}$ is bounded away from 0. Such a construction relies on the
presence of torsion in the integral homology groups. Indeed, any simplicial
complex with torsion can be used to show that the inequality $(h^{k})^{p}\geq
C\lambda^{k}$ need not hold in general for any $p$,$C>0$, and $k>0$. A good
example is $\mathbb{RP}^{2}$ which has $H^{1}(\mathbb{Z}_{2})\neq 0$ and
$H^{2}(\mathbb{Z}_{2})\neq 0$ but $H^{1}(\mathbb{R})=0$ and
$H^{2}(\mathbb{R})=0$. By contrast, the example presented here is a family of
orientable simplicial complexes, proving that the failure of the Cheeger
inequality to hold is not simply the result of torsion.
The fact that both families $X_{k}$ and $Y_{k}$ are simplicial $m$-balls helps
show the degree to which the Cheeger inequality fails to hold even for ‘nice’
simplicial complexes.
The proof of Proposition 2.10 puts together much of what appears earlier in
this paper. To show that $X_{k}$ is a simplicial $m$-ball we will need to
prove that it is constructible and non-branching. The $Y_{k}$ will be defined
by subdividing $\Sigma^{m}$, implying that it too is a simplicial $m$-ball. To
compute the values of $h^{m-1}$ for $X_{k}$ and $Y_{k}$ we make use of Lemmas
2.4 and 2.5. Computing $h_{m}$ will involve simple counting. By Theorem 2.7
and the fact that $\lambda_{m}=\lambda^{m-1}$, we can use our estimate of
$h_{m}$ to estimate $\lambda^{m-1}$, finishing the proof.
Now to begin the proof. We define the family $X_{k}$ recursively. To begin
with, we let $X_{1}$ be $\Sigma^{m}$, the simplicial complex induced by a
single $m$-simplex. Note that $h_{m}(X_{1})=m+1$ and $h^{m-1}(X_{1})=1$. Then,
given $X_{k}$, we define $X_{k+1}$ by gluing $m$-simplexes on to $X_{k}$ as
follows: for each boundary face $s=\\{v_{0},\ldots,v_{m-1}\\}$ in $X_{k}$ we
create a new vertex $v$ and a new $m$-simplex
$\sigma=\\{v_{0},\ldots,v_{m-1},v\\}$ which includes $v$ and $s$. A picture of
the first few iterations of $X_{k}$ for the case $m=2$ can be seen in Figure
6.
$X_{1}$$X_{2}$$X_{3}$$X_{4}$ Figure 6. The first few iterations of $X_{k}$ in
dimension 2. The 2-simplexes have been shaded according to their depth.
Clearly, $X_{1}$ is a simplicial $m$-ball. The following two lemmas prove that
indeed every $X_{k}$ is a simplicial $m$-ball.
###### Lemma 2.11.
$X_{k}$ is constructible for all $k$.
###### Proof.
The proof is by induction. We know $X_{1}$ is constructible. Assuming that
$X_{k}$ is constructible, we must prove that $X_{k+1}$ is constructible. This
reduces to proving that gluing a single $m$-simplex to $X_{k}$ along a
boundary face preserves constructibility. Let $X^{\prime}_{k}$ be the result
of taking a boundary face $s=\\{v_{0},\ldots,v_{m-1}\\}$ in $X_{k}$ and adding
a new vertex $v$ and a new $m$-simplex $\sigma=\\{v_{0},\ldots,v_{m-1},v\\}$
which includes $v$ and $s$. Then $X^{\prime}_{k}$ can be decomposed as the
union of $X_{k}$ and the simplicial subcomplex $T=\Sigma^{m}$ consisting of
$\sigma$ and its subsets, both of which are constructible $m$-complexes.
Furthermore, the intersection of $X_{k}$ and $T$ is $\Sigma^{m-1}$, which is
constructible. Therefore, $X^{\prime}_{k}$ is constructible by definition. ∎
###### Lemma 2.12.
$X_{k}$ is non-branching for all $k$.
###### Proof.
The proof is again by induction. We know that $X_{1}$ is non-branching. Assume
this is true for $X_{k}$ as well. By construction, $s\in S_{m-1}(X_{k})$ has
another coface in $S_{m}(X_{k+1})$ if and only if $s$ has only one coface in
$S_{m}(X_{k})$. The new $(m-1)$-simplexes are the boundary faces of $X_{k+1}$
and thus have exactly one coface. Thus, the total number of cofaces of every
$(m-1)$-simplex in $X_{k+1}$ is either one or two. ∎
As mentioned in the introduction, constructible non-branching simplicial
$m$-complexes are simplicial $m$-balls. Thus, every $X_{k}$ is a simplicial
$m$-ball.
To prove part (1) of Proposition 2.10, we need to keep track of how the
Cheeger numbers $h^{m-1}(X_{k})$ and $h_{m}(X_{k})$ change with $k$. This is
accomplished in the following two lemmas.
###### Lemma 2.13.
$h^{m-1}(X_{k})=\frac{1}{k}$ for all $k$.
###### Proof.
By Lemma 2.5, $h^{m-1}(X_{k})=\frac{1}{\operatorname{rad}(X_{k})}$. For $k=1$,
$\operatorname{rad}(X_{1})=1$. Now suppose that $\operatorname{rad}(X_{k})=k$.
We will prove that in passing from $X_{k}$ to $X_{k+1}$, all $m$-simplexes
originally in $X_{k}$ have their depth increased by exactly 1 (we already know
the new $m$-simplexes in $X_{k+1}$ have depth 1).
If $\tau\in S_{m}(X_{k})$ has depth $d$ and $\phi$ is a depth-attaining
cochain for $\tau$ in $X_{k}$, then $\phi$ is a sum of a sequence
$\\{s_{i}\\}_{i=1}^{d}$ of $(m-1)$-simplexes satisfying the conditions in
Lemma 2.4. All of those conditions are preserved in going from $X_{k}$ to
$X_{k+1}$, except that $s_{d}$ is no longer a boundary face. Instead, if
$s_{d}=\\{v_{0},\ldots,v_{m-1}\\}$ then a new vertex $v$ and a new $m$-simplex
$\sigma=\\{v_{0},\ldots,v_{m-1},v\\}$ are created which prevent $s_{d}$ from
being a boundary face and add $\sigma$ to the coboundary of $\phi$. However,
if we add any of the other faces of $\sigma$ to $\phi$ (which are all boundary
faces), we obtain a new cochain $\phi^{\prime}$ with
$\delta\phi^{\prime}=\tau$ and $\lVert\phi^{\prime}\rVert=d+1$. Thus, the
depth of $\tau$ in $X_{k+1}$ is at most $d+1$.
Conversely, if $\tau$ has depth $d^{\prime}$ in $X_{k+1}$ and
$\psi=\sum_{i=1}^{d^{\prime}}t_{i}$ is a depth-attaining cochain for $\tau$
with $\\{t_{i}\\}_{i=1}^{d^{\prime}}$ satisfying the conditions in Lemma 2.4,
then $\psi^{\prime}=\sum_{i=1}^{d^{\prime}-1}t_{i}$ is a cochain in $X_{k}$
with $\delta\psi^{\prime}=\tau$, so that the depth of $\sigma$ is at most
$d^{\prime}-1$. Thus, if $\tau$ has depth $d$ in $X_{k}$ then its depth in
$X_{k+1}$ must be at least $d+1$. Combined with the above result we conclude
that all $m$-simplexes originally in $X_{k}$ have their depth increased by
exactly 1 in $X_{k+1}$. ∎
###### Lemma 2.14.
$h_{m}(X_{k})\geq m-1$ for all $k$.
###### Proof.
We know that $h_{m}(X_{1})=m+1\geq m-1$. Now suppose $h_{m}(X_{k})\geq m-1$.
Any chain $\phi\in C_{m}(\mathbb{Z}_{2};X_{k+1})$ attaining $h_{m}$ can be
decomposed into a chain $\psi\in C_{m}(\mathbb{Z}_{2};X_{k})$ plus a chain
$\psi^{\prime}$ which is a sum of depth 1 simplexes in $X_{k+1}$. Then we can
write
$\lVert\partial\phi\rVert=\lVert\partial\psi\rVert+\lVert\partial\psi^{\prime}\rVert-2x$
where $x$ is the number of $(m-1)$-simplexes shared by $\partial\psi$ and
$\partial\psi^{\prime}$. Since $m$ of the $m+1$ faces of any $m$-simplex in
$\psi^{\prime}$ are boundary faces, $x\leq\lVert\psi^{\prime}\rVert$. Also, it
is clear that
$\lVert\partial\psi^{\prime}\rVert=(m+1)\lVert\psi^{\prime}\rVert$. Thus,
$\displaystyle\frac{\lVert\partial\phi\rVert}{\lVert\phi\rVert}$
$\displaystyle=\frac{\lVert\partial\psi\rVert+(m+1)\lVert\psi^{\prime}\rVert-2x}{\lVert\psi\rVert+\lVert\psi^{\prime}\rVert}$
$\displaystyle\geq\frac{\lVert\partial\psi\rVert+(m-1)\lVert\psi^{\prime}\rVert}{\lVert\psi\rVert+\lVert\psi^{\prime}\rVert}$
$\displaystyle\geq\min\left\\{\frac{\lVert\partial\psi\rVert}{\lVert\psi\rVert},m-1\right\\}$
$\displaystyle\geq m-1$
(In fact, with some effort it can be seen that
$h_{m}=\frac{(m+1)(m-1)}{(m+1)-2m^{-k+1}}$.) ∎
By Theorem 2.7,
$\lambda^{m-1}(X_{k})=\lambda_{m}(X_{k})\geq\frac{(m-1)^{2}}{2(m+1)}$. This
completes the proof of part (1) of Proposition 2.10.
In order to define the family $Y_{k}$ we need to make use of the notion of
stellar subdivision, which can be traced back to at least [1].
###### Definition 2.15 (Stellar Subdivision).
Let $Y$ be a simplicial $m$-complex and let
$\sigma=\\{v_{0},\ldots,v_{m}\\}\in S_{m}(Y)$. The stellar subdivision of $Y$
along $\sigma$, denoted by $\operatorname{sd}_{\sigma}Y$, is the simplicial
$m$-complex obtained from $Y$ by creating a new vertex $w$ and replacing
$\sigma$ with the $m$-simplexes
$\tau_{i}=\\{v_{0},\ldots,v_{i-1},w,v_{i+1},\ldots,v_{m}\\}$
where $i=0,\ldots,m$. For notational purposes, we denote the $j$-th face of
$\tau_{i}$ by $t_{i,j}:=\tau_{i}\setminus\\{v_{j}\\}$ for $i\neq j$, and
$t_{i,i}:=\tau_{i}\setminus\\{w\\}$. If $\sigma_{1},\ldots,\sigma_{n}\in
S_{m}(Y)$, then we define the stellar subdivision of $Y$ along the
$\sigma_{i}$ to be
$\operatorname{sd}_{\sigma_{1},\ldots,\sigma_{n}}Y:=\operatorname{sd}_{\sigma_{1}}\operatorname{sd}_{\sigma_{2}}\cdots\operatorname{sd}_{\sigma_{n}}Y$
We now define the $Y_{k}$ recursively. Let $\Sigma^{m}$ be the simplicial
complex induced by a single $m$-simplex $\sigma$ and let
$Y_{1}:=\operatorname{sd}_{\sigma}\Sigma^{m}$. Label the $m$-simplexes of
$Y_{1}$ as $\sigma_{0},\ldots,\sigma_{m}$ and call their common vertex (the
one created by stellar subdivision) the central vertex $v$. Now, given a
$Y_{k}$ containing the central vertex $v$, we call all $m$-simplexes
containing $v$ the inner $m$-simplexes of $Y_{k}$ and label them as
$\sigma_{0},\ldots,\sigma_{n}$. All non-inner $m$-simplexes will be referred
to as outer $m$-simplexes. We then define
$Y_{k+1}:=\operatorname{sd}_{\sigma_{0},\ldots,\sigma_{n}}Y_{k}$. Note that
$v$ and all outer $m$-simplexes (and the simplexes they contain) are preserved
unchanged in going from $Y_{k}$ to $Y_{k+1}$ while all of the inner
$m$-simplexes are subdivided. Furthermore, it is clear that all the $Y_{k}$
are subdivisions of $\Sigma^{m}$ and are thus simplicial $m$-balls. A picture
of the first few iterations of $Y_{k}$ for $m=2$ can be seen in Figure 7.
$Y_{1}$$Y_{2}$$Y_{3}$ Figure 7. The first few iterations of $Y_{k}$ in
dimension 2. The $2$-simplexes have been shaded according to their depth.
To prove part (2) of Proposition 2.10, we need to keep track of how the
Cheeger numbers $h^{m-1}(Y_{k})$ and $h_{m}(Y_{k})$ change with $k$. This is
accomplished in the following two lemmas.
###### Lemma 2.16.
$h^{m-1}(Y_{k})\geq\frac{1}{k}$ for all $k$.
###### Proof.
By Lemma 2.5, we can prove this by keeping track of the depths of all the
$m$-simplexes of $Y_{k}$. For $Y_{1}$, all the $m$-simplexes $\sigma_{i}$
contain a boundary face (using the notation of Definition 2.15 with
$\sigma_{i}=\tau_{i}$, the boundary face of $\sigma_{i}$ is $t_{i,i}$). Thus,
every $\sigma_{i}$ has depth 1 and by Lemma 2.5, $h^{m-1}(Y_{1})=1$. Note that
the cochain $\phi$ which is depth-attaining for some $\sigma_{i}$ does not
include any $(m-1)$-simplex which contains $v$.
Now suppose for induction that every outer $m$-simplex $\sigma$ of $Y_{k}$ has
depth $\leq k$ and a depth-attaining cochain $\phi\in C^{m-1}(\mathbb{Z}_{2})$
such that $\phi$ does not contain any face of any inner $m$-simplex. Then in
$Y_{k+1}$, $\phi$ remains unaltered, proving that $\sigma$ still has depth
$\leq k$ in $Y_{k+1}$.
Similarly, suppose that every inner $m$-simplex $\sigma$ of $Y_{k}$ has depth
$\leq k$ via a depth-attaining cochain $\phi$ which does not contain any
$(m-1)$-simplex containing $v$. Then in $Y_{k+1}$, $\sigma$ is removed and
replaced by new $m$-simplices. Using the notation of Definition 2.15, in
$Y_{k+1}$ the coboundary of $\phi$ becomes $\delta\phi=\tau_{m+1}$, so that
the depth of $\tau_{m+1}$ is at most $k$. Furthermore, by adding any face
$t_{(m+1),j}$ to $\phi$ ($j\neq m+1$) we obtain a cochain $\phi^{\prime}$ with
$\delta\phi^{\prime}=\tau_{j}$, proving that the depth of $\tau_{j}$ is at
most $k+1$. Since $\phi^{\prime}$ still does not contain any $(m-1)$-simplex
which contains $v$, we are back where we started. The statement now follows by
induction. ∎
###### Lemma 2.17.
$h_{m}(Y_{k})\leq\frac{1}{m^{k-1}}$ for all $k>1$.
###### Proof.
To prove this, we merely count the number of $m$-simplexes in $Y_{k}$. Note
that in going from $Y_{k}$ to $Y_{k+1}$ we replace $(m+1)m^{k-1}$ inner
$m$-simplexes with $(m+1)m^{k}$ inner $m$-simplexes. Thus, $Y_{k+1}$ has
$(m+1)m^{k}-(m+1)m^{k-1}=(m+1)(m-1)m^{k-1}$
more $m$-simplexes than $Y_{k}$. Since $|S_{m}(Y_{1})|=m+1$, this means that
$|S_{m}(Y_{k})|$ is equal to
$(m+1)+(m+1)(m-1)+(m+1)(m-1)m+\ldots\\\ +(m+1)(m-1)m^{k-2}=(m+1)m^{k-1}.$
Since $Y_{k}$ has $m+1$ boundary faces, the chain $\phi$ containing all
$m$-simplexes of $Y_{k}$ gives the upper bound on $h_{m}(Y_{k})$:
$h_{m}(Y_{k})\leq\frac{\lVert\partial\phi\rVert}{\lVert\phi\rVert}=\frac{m+1}{(m+1)m^{k-1}}=\frac{1}{m^{k-1}}.$
∎
By Theorem 2.7,
$\lambda^{m-1}(Y_{k})=\lambda_{m}(Y_{k})\leq\frac{1}{m^{k-1}}$. This completes
the proof of Proposition 2.10.
## 3\. Discussion and Open Problems
The Cheeger inequality has been relevant to a variety of algorithmic and
analysis problems in computer science and mathematics including spectral
clustering [18, 23], manifold learning [4], and the analysis of random walks
[19].
There has been interest in extending ideas from graphs to abstract simplicial
complexes including spanning trees on simplicial complexes [12], properties of
expanders on simplicial complexes [24, 11, 16], and higher-dimensional
constructions of conditional independence [22]. A motivation for our work was
to begin to develop intuition for the mathematical principles behind a higher-
dimensional notion of spectral clustering. This objective is far from being
realized.
A result of the universal coefficient theorem in algebraic topology is that
torsion will be an obstacle in relating higher-dimensional Cheeger numbers
with eigenvalues. The Cheeger inequality for graphs holds without any
assumptions since zeroth homology is never affected by torsion. For higher
dimensions either the inequality does not hold or we require assumptions that
remove torsion. The negative results for the Cheeger inequality in [16] are
for simplicial complexes with torsion. Torsion is also known to affect
algorithmic complexity. For example, the problem of finding minimal weight
cycles given a simplicial complex with weights is NP-hard if there is torsion
and is otherwise a linear program [10]. In Appendix A we use the real
projective plane to illustrate some of the issues with torsion and why they do
not appear in the graph setting.
A local Cheeger number and algebraic connectivity for graphs with Dirichlet
like boundary conditions was defined in [8] and a Cheeger inequality was
proved. There is a close relation between Theorem 1 of [8] and Theorem 2.7 in
our paper. If Theorem 1 is adapted to an unnormalized setting (see Appendix A)
then for non-branching orientable simplicial $m$-complexes Theorem 2.7 reduces
to Theorem 1. However, Theorem 2.7 covers the more general cases of non-
orientable and branching simplicial $m$-complexes.
We close with a few open problems of possible interest.
1. (1)
Intermediate values of $k$ – Given a simplicial $m$-complex, what can we say
about the relationship between $h^{k}$ and $\lambda^{k}$ or $h_{k}$ and
$\lambda_{k}$ for $1<k<m-1$? Torsion again will need to be addressed but are
there some conditions under which some Cheeger-type inequalities may hold?
2. (2)
High-order eigenvalues – In [20] the authors introduce higher-order (as
opposed to higher-dimensional) Cheeger numbers on the graph which correspond
to higher-order eigenvalues of the graph Laplacian and prove a general Cheeger
inequality for them. A natural question is how our results would extend to
higher-orders. Indeed, by analogy with the Rayleigh quotient characterization
of higher order eigenvalues, it would seem reasonable to define the
$k^{\text{th}}$ dimensional, $j^{\text{th}}$ order coboundary Cheeger numbers
to be
$h^{k,j}:=\min_{\begin{subarray}{c}\phi\in C^{k}(\mathbb{Z}_{2})\\\ \phi\notin
S_{j}\end{subarray}}\frac{\lVert\delta\phi\rVert}{\min_{\psi\in
S_{j}}\lVert\phi+\psi\rVert}$
where
$S_{j}=\operatorname{span}(\operatorname{im}\delta\cup\\{\phi_{1},\ldots,\phi_{j-1}\\})$
is the subspace of $C^{k}(\mathbb{Z}_{2})$ spanned by
$\operatorname{im}\delta$ and cochains $\phi_{1},\ldots,\phi_{j-1}$ which
attain $h^{k,1},\ldots,h^{k,j-1}$, respectively. The higher order boundary
Cheeger numbers $h_{k,j}$ could be defined similarly. One would need to prove
that this definition makes sense and then ask whether they satisfy any
inequalities with the corresponding eigenvalues.
3. (3)
Cheeger inequalities on manifolds – Ultimately, the study of higher-
dimensional Cheeger numbers on simplicial complexes should (morally speaking)
be translated back to the manifold setting if possible. A tentative definition
for the $k$-dimensional coboundary Cheeger number of a manifold $M$ might be
$h^{k}=\inf_{S}\frac{\text{Vol}_{m-k-1}(\partial S\setminus\partial
M)}{\inf_{\partial T=\partial S}\text{Vol}_{m-k}(T)}$
where $\text{Vol}_{k}$ denotes $k$-dimensional volume and the infimum is taken
over all $k$-codimensional submanifolds $S$ of $M$. Similarly, the $k$-th
boundary Cheeger number of $M$ might be
$h_{k}=\inf_{S}\frac{\text{Vol}_{k-1}(\partial S)}{\inf_{\partial T=\partial
S}\text{Vol}_{k}(T)}$
where again $\text{Vol}_{k}$ denotes $k$-dimensional volume and the infimum is
taken over all $k$-dimensional submanifolds $S$ of $M$.
## Acknowledgments
SM would like to acknowledge Shmuel Weinberger and Matt Kahle for discussions
and insight. SM is pleased to acknowledge support from grants NIH (Systems
Biology): 5P50-GM081883, AFOSR: FA9550-10-1-0436, NSF CCF-1049290, and NSF-
DMS-1209155. JS would like to acknowledge Matt Kahle, Yuan Yao, Anna Gundert,
Yuriy Mileyko, and Mikhail Belkin for discussions and insight. JS is pleased
to acknowledge support from graph NSF CCF-1209155 and a Duke Endowment
Fellowship.
## Appendix A Relation to Graphs with Dirichlet Boundaries and the Real
Projective Plane
In [8], Fan Chung defines a normalized local Dirichlet Cheeger number and
normalized local Dirichlet eigenvalue and proves an inequality between them.
If one translates Fan Chung’s result to the unnormalized case for graphs with
vertex degree upper bounded by $m+1$, it closely resembles Theorem 2.7.
Translating Theorem 1 of [8] into the unnormalized setting, it reads as
follows. Given a graph $G$ we can prescribe a certain set of vertices to be
the boundary vertices of the graph. Let $S$ be the prescribed boundary vertex
set, and let
$h_{S}:=h_{S}(G)=\min\frac{\lVert\delta\phi\rVert}{\lVert\phi\rVert}$
where the minimum is taken over all nonzero $\phi\in C^{0}(\mathbb{Z}_{2})$
such that $\phi$ does not include any boundary vertex. Similarly, let
$\lambda_{S}=\min\frac{\lVert\delta f\rVert_{2}^{2}}{\lVert f\rVert_{2}^{2}}$
where the minimum is taken over all nonzero $f\in C^{0}(\mathbb{R})$ such that
$f(s)=0$ for all $s\in S$. We can also characterize $\lambda_{S}$ as the
smallest eigenvalue of $L_{0}^{S}$, the submatrix of $L_{0}$ consisting of the
rows and columns of $L_{0}$ not indexed by vertices in $S$. In this case,
$L_{0}^{S}$ is a map on $C^{0}_{S}(\mathbb{R})$, the subspace of
$C^{0}(\mathbb{R})$ spanned by the vertices not in $S$. Then if every vertex
has degree upper bounded by $m+1$
$h_{S}\geq\lambda_{S}\geq\frac{h_{S}^{2}}{2(m+1)}.$
To relate the above inequality to the simplicial complex setting, we note that
for every non-branching simplicial $m$-complex $X$, one can construct a graph
$G$ (similar to the dual graph defined in [15]) as follows. Begin by
constructing the simplicial complex $X^{\prime}$ as in the proof of Theorem
2.7 and let $S$ be the set of border facets of $X^{\prime}$. Create a vertex
in $G$ for every $m$-simplex in $X^{\prime}$. We will use $S$ to denote both
the border facets of $X^{\prime}$ and the set of vertices in $G$ which
correspond the border facets. Connect two vertices with an edge whenever the
corresponding $m$-simplexes are lower adjacent in $X^{\prime}$. Since
$X^{\prime}$ is non-branching, the vertices of $G$ have degree upper bounded
by $m+1$. Identifying $C^{0}_{S}(G;\mathbb{R})$ with $C_{m}(X;\mathbb{R})$, we
can ask if $L_{m}:C_{m}(X;\mathbb{R})\to C_{m}(X;\mathbb{R})$ and
$L_{0}^{S}:C^{0}_{S}(G;\mathbb{R})\to C^{0}_{S}(G;\mathbb{R})$ are the same
map. They are the same if and only if $X$ is orientable (this is easy to see
from the characterization of the Laplacian in [26]). In addition, $h_{m}(X)$
and $h_{S}$ are equal regardless of orientability. Thus, for non-branching
orientable simplicial $m$-complexes, Theorem 2.7 reduces to the result proved
by Fan Chung, and the proofs are identical. The difference is that Theorem 2.7
covers the more general cases of non-orientable and branching simplicial
$m$-complexes, for which parts of the inequality may still hold.
The real projective plane provides a simple example of how orientation plays a
role in our analysis of the Cheeger inequality and why it doesn’t play a role
in [8]. In Figure 8, the first image shows the fundamental polygon that
defines $\mathbb{R}P^{2}$, the second image shows a triangulation $X$ of
$\mathbb{R}P^{2}$, and the third image is the dual graph $G$ of the
triangulation (in the second and third image, edges with similar color are
identified). In this simple example, there is no boundary ($S=\emptyset$). In
the triangulation, if one considers the 2-chain $\phi\in
C_{2}(\mathbb{Z}_{2})$ which contains every 2-simplex, then $\partial\phi=0$
and thus $h_{2}(X)=0$. However, if one considers the 2-chain $f\in
C_{2}(X;\mathbb{R})$ that assigns a 1 to every 2-simplex with the orientation
shown in the figure, the boundary of $f$ is a 1-chain which assigns a 2 to
every colored edge with the orientation shown. In particular, $\partial f\neq
0$ and in fact $\lambda_{2}\neq 0$ as a result of the nonorientability of
$\mathbb{R}P^{2}$. However, the dual graph cannot see this nonorientability,
as the 0-chain $\tilde{f}\in C_{S}^{0}(G;\mathbb{R})$ corresponding to $f$ has
empty coboundary, meaning $\lambda_{S}=0$. Thus, in this case the map $L_{2}$
is not the same as the map $L_{0}^{S}$, and Theorem 1 of [8] still holds while
part 1 of Theorem 2.7 fails.
Figure 8. The fundamental polygon of $\mathbb{R}P^{2}$, a triangulation, and
the dual graph of the triangulation.
## References
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* [19] G. Lawler and A. Sokal. Bounds on the $\ell^{2}$ spectrum of Markov chains and Markov processes: a generalization of Cheeger’s inequlity. Trans. Amer. Math. Soc., 309:557–580, 1988.
* [20] J.R. Lee, S.O. Gharan, and L. Trevisan. Multi-way spectral partitioning and higher-order cheeger inequalities. Arxiv preprint arXiv:1111.1055, 2011.
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* [23] Marina Maila and Jianbo Shi. A random walks view of spectral segmentation. In Proceedings of the Eighth International Workshop on Artificial Intelligence and Statistics, 2001.
* [24] R. Meshulam and N. Wallach. Homological connectivity of random k-dimensional complexes. Random Structures & Algorithms, 34(3):408–417, 2009.
* [25] B. Mohar. Isoperimetric numbers of graphs. Journal of Combinatorial Theory, Series B, 47(3):274–291, 1989\.
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|
arxiv-papers
| 2012-09-23T17:46:38 |
2024-09-04T02:49:35.462908
|
{
"license": "Public Domain",
"authors": "John Steenbergen and Caroline Klivans and Sayan Mukherjee",
"submitter": "John Steenbergen",
"url": "https://arxiv.org/abs/1209.5091"
}
|
1209.5184
|
# A scheme for the determination of the magnetic field in the KATRIN main
spectrometer
A. Osipowicza, U. Rauscha, A. Unrua, B. Zipfelb
a Department of Electrical Engineering and Information Technology
Corresponding author. University of Applied Sciences Marquardstr. 35 Fulda
Germany
bHF-Dept GSI Helmholtzzentrum f r Schwerionenforschung GmbH Planckstrasse 1
D-Darmstdt Germany
E-mail Alexander.Osipowicz@et.hs-fulda.de
###### Abstract
To determine the magnetic field distribution in the KATRIN main-spectrometer
with magnetic field sensors that are placed outside the main-spectrometer
vessel one can utilize the absence of magnetic rotation in main-spectrometer
volume. There a scalar magnetic potential $V(\vec{x})$ can be defined that
fulfills the Laplace equation. Large numbers of magnetic field values on an
outer surface of the main-spectrometer can be sampled by moving and fixed
magnetic field sensors. These surface samples are used as boundary values in
the relaxation of the Laplace equation for $V(\vec{x})$ and the magnetic field
components in the volume. In a simulation involving the KATRIN reference
solenoid chain, a global magnetic field and an external perturbing solenoid it
is shown that with this method the original field can be reconstructed within
2 %.
###### keywords:
Spectrometers; Detector alignment and calibration methods (lasers, sources,
particlebeams); Detector control systems (detector and experiment monitoring
and slow-control systems, architecture, hardware, algorithms, databases)
## 1 The KATRIN setup
The KArlsruhe TRItium Neutrino experiment [1] (see Fig.1) is set up at the
Karlsruher Institute of Technology (KIT), Germany. It is designed to measure
the mass of the electron anti neutrino in a direct and model-independent way
with a sensitivity of $m_{\nu}=0.2$ eV/c2 (90% confidence level) from tritium
$\beta$ decay[1]. KATRIN uses a magnetic transport field that connects the
source and detector in combination with integrating electrostatic energy
filters (MAC-E-spectrometers). Conceptual essentials of the MAC-E
spectrometer[2, 3] are the magnetic field gradients in pre - and main-
spectrometer that adiabatically convert cyclotron energy $E_{cyc}$ into energy
$E_{p}$ parallel to the magnetic field lines and vice versa.
Figure 1: Schematic view of the KATRIN experiment (total length 70 m)
consisting of calibration and monitor rear system, with the windowless gaseous
$\rm T_{2}$-source (WGTS), differential pumping (DPS) and cryo-trapping
section (SPS), the small pre-spectrometer and the large main spectrometer with
the large magnetic coil systems to compensate the earth magnetic field (EMCS)
and to shape the magnetic transport flux (LFCS) and lastly the segmented PIN-
diode detector.
At the center of the main-spectrometer (MS) in the minimal magnetic field
$B_{A}\approx 3-6$ $\mu$T, a retarding electric field allows an integral
energy analysis of $E_{p}$. The magnetic field in the analyzing volume defines
the magnetic resolution, i.e. the amount of residual cyclotron energy
$E_{cyc}$ that can not be analyzed and thus strongly influences the resolution
function. Error analysis [4] of the influence of uncertainty of the magnetic
field in the analyzing plane on the uncertainty of the neutrino mass square
$\Delta m_{v}^{2}$ leads to a relative accuracy of the magnetic field of
$\frac{\Delta B}{B}<2.4\%$. In addition, the alignment of magnetic field lines
plays a crucial role in the production of secondary electrons and electronic
background either through penning traps or inner wall contact.
Large coil systems [5] are arranged around the MS for a) global magnetic field
compensation, e.g. earth magnetic field (EMCS) and b) fine tuning of the
magnetic transport flux with a set of large circular low field coils (LFCS)
mounted coaxially with the MS (see Fig.1). However, possible influences of
residual external dipoles, magnetization in the MS environment by the high
field solenoids and/or EMCS, LFCS and the correct orientation of the
spectrometer solenoids have to be controlled. Due to the extreme MS vacuum
conditions the installation of magnetic sensors inside the MS is not possible.
Figure 2: View of the main spectrometer tank with the LFCS ring system. Right:
The mobile sensor unit with 2 sensors on the inner belt of a LFCS support
ring.
We therefore propose to determine the magnetic field inside the main
spectrometer by taking magnetic field samples at an outer surface of the main
spectrometer. The sensor network will involve fixed position magnetic sensors
and mobile magnetic field sensors [6, 7, 8] which move along the inner belts
of the LFCS support structure (see Fig.2), close to the outer MS surface but
well inside the current lines of the EMCS and LFCS . The magnetic field
samples serve as boundary values for the relaxation of the Laplace equation of
the scalar magnetic potential $V(\vec{x})$ at the interior of the KATRIN main
spectrometer.
## 2 Volume and surface considerations
For a volume $G$ with surface area $\Gamma$ Amperes equation
$\vec{\nabla}\times\vec{B}=\mu_{0}\cdot\left(\vec{J}+\frac{\epsilon_{0}\cdot\partial\vec{E}}{\partial
t}\right)$ (1)
can be simplified to the rotationally free case if the current density
$\vec{J}$ is vanishing ($J=0$) and the electric field $\vec{E}$ is constant
($\partial\vec{E}/\partial t=0$).
$\vec{\nabla}\times\vec{B}=0$ (2)
For the KATRIN MS the relevant surface $\Gamma$ (see Fig.3) has to be outside
the outer MS surface and inside the current leading elements (LFCS, EMCS,
spectrometer solenoids). As the analyzing potential distribution $U(x,y,z)$
inside the MS volume is constant during KATRIN runtime intervals (and magnetic
field sampling time intervals) the electrical fields produced are time
independent. Therefore eq. (2) can assumed to be valid for the KATRIN MS
interior.
Figure 3: View of the KATRIN main reference solenoid chain and the main
spectrometer area. The cylindrical volume $G$ enclosed by the boundary
$\Gamma$. The origin of the coordinate system is located at the geometrical
center of the MS and the symmetry point of the LFCS. To ensure that all
current leading elements are outside the boundary $\Gamma$ in $y-$ and $z-$
direction the allowed radius $R_{G}$ to $G$ is 5600 mm $<R_{G}<$ 6155 mm. The
extreme $x-$ values are -11600 mm $=-x_{n}<x<+x_{n}=11600$ mm.
Vector analysis [10] states for a scalar function $V(\vec{x})$ that:
$\vec{\nabla}\times\vec{\nabla}\cdot V(\vec{x})=0$ and one can identify
$V(\vec{x})$ with the magnetic scalar potential.
$\vec{B}=\vec{\nabla}\cdot V(\vec{x})$
Utilizing Gauss’s law for magnetism $\vec{\nabla}\cdot\vec{B}=0$ we can write
down the Laplace-equation (LPE) for $V(\vec{x})$
$\nabla^{2}V(x,y,z)=0$ (3)
The finite difference method (FDM) [11] is chosen to solve the above equation
on a 3 dimensional rectangular grid, because of its well known numerical
stability and the manageable coding effort. In the simulation the magnetic
field components at a the boundary representing the normal derivatives
$\partial V/\partial x=B_{x};\>\partial V/\partial y=B_{y};\>\partial
V/\partial z=B_{z}$ at $\Gamma$ can be exported and used in the FD-relaxation
as a von Neumann boundary values.
## 3 Simulation
The usability of the numerical approach is demonstrated in a simulation based
on magnetic field values provided by the simulation package PartOpt [9]. The
definition of a magnetic scenario (Fig. 4) at the KATRIN main spectrometer
includes: a) the energized KATRIN reference solenoid chain, b) the energized
LFCS as listed in [13], c) a magnetic field over $G$ with $B_{x}=210$ mG,
$B_{y}=35$ mG, $B_{z}=0$, d) a small disturbing magnetic dipole with central
induction $B_{c}=600$ G adjacent to the main spectrometer.
Figure 4: PartOpt view of the Simulation scenario. In volume $G$ the
effective magnetic field is composed of the KATRIN solenoid field, LFCS, an
external global field and disturbing external dipole. The perturbed magnetic
field lines magnetic field lines have been tracked starting from the center of
the WGTS. The extreme field lines indicate the boundary of the $191$ T cm2
nominal magnetic transport flux connecting source and detector.
The field values $B_{x},B_{y},B_{z}$ along the cylindrical surface of volume
$G$ with radius $R_{G}=6$ m between $x_{min}=-7.03$ m $<x<x_{max}=6.83$ m to
cover the cylindrical part of the MS are exported in ASCII format. The spacing
of the samples in $x$-direction is 0.45 m in agreement with the real
$x$-spacing of the sensor positions. In azimuthal direction a $3^{\circ}$
spacing was chosen to get $120\times 2$ samples (because 2 sensors are on
board) in 15 minutes, the time for one revolution. To simulate sensor error
the exported values are randomized according to a Gaussean distribution with a
2% relative uncertainty. This value was chosen as an upper limit according to
the sensor types used in [6] . Due to the cylindrical geometry the surface
samples points usually do not coincide with surface mesh points (cut surfaces
problem). Therefore the magnetic samples are interpolated to produce values at
the regular surface mesh points. The relaxation is performed via a basic $7$
point stencil. The resulting values for the scalar potential and the values
for the magnetic field components are generated by deriving $V(\vec{x})$
numerically.
Figure 5: View of the mesh point structure. Left: $G$ in an interval in ,
$-x_{n}<x<x_{n}$, Right: on the surface $\Gamma$, at $x=-x_{n},x_{n}$
The relaxation code is written in C. Typically 1400 iterations in 5 minutes on
a standard PC are performed to meet the terminating condition that the
difference for $V(0,0,0)$, the magnetic potential at the origin, between
successive iterations is $<0.0002$.
## 4 Simulation results
The results of the simulation is displayed as magnetic field components in
geometric planes with given coordinates within the main spectrometer. Figs.:
6,8,10 show the original PartOpt magnetic $B_{org}$ and the reconstructed
magnetic field $B_{rec}$ components for a randomly chosen $x,y$ plane at
$z=2.4994$ m. The relative differences $\Delta B$ are displayed in Fig.:
7,9,11,.
Figure 6: Left: The original magnetic field component $B_{x_{org}}$ in a in a
$x,y$ plane at $z=2.4994$ m. Right: The reconstructed magnetic field values
$B_{x_{rec}}$ in the same plane. Figure 7: Relative difference between the
original $B_{x_{org}}$ and reconstructed $B_{x_{rec}}$ magnetic field
component $\Delta B_{x}=(B_{x_{org}}-B_{x_{rec}})/B_{x_{org}}$ in a $x,y$
plane with $z=2.4994$ m. The sharp peaks at $x\approx 6$ arise numerically
from a division by zero as $B_{x_{org}}\approx 0$ in the vicinity of the
negatively charged LFCS coil towards the detector side as given in [13].
Elsewhere the difference is less than 2%. Figure 8: Left: The original
magnetic field component $B_{y_{org}}$ in a in a $x,y$ plane at $z=2.4994$ m.
Right: The reconstructed magnetic field values $B_{y_{rec}}$ in the same
plane. The sawtooth structure at the extreme y-values are due to the close
proximity of the energized LFCS Coils. Figure 9: Relative difference between
the original $B_{y_{org}}$ and reconstructed $B_{y_{rec}}$ magnetic field
component $\Delta B_{y}=(B_{y_{org}}-B_{y_{rec}})/B_{y_{org}}$ in a $x,y$
plane with $z=2,4994$ m. The sharp peaks arise numerically from a division by
zero as $B_{y_{org}}\approx 0$. Figure 10: Left: The original magnetic field
component $B_{z_{org}}$ in a in a $x,y$ plane at $z=2.4994$ m. Right: The
reconstructed magnetic field values $B_{z_{rec}}$ in the same plane. The
sawtooth structure at the extreme y-values are due to the close proximity of
the energized LFCS Coils. Figure 11: Relative difference between the original
$B_{z_{org}}$ and reconstructed $B_{z_{rec}}$ magnetic field component $\Delta
B_{z}=(B_{z_{org}}-B_{z_{rec}})/B_{z_{org}}$ in a $x,y$ plane with $z=2,4994$
m. The sharp peaks at extreme $y$ -values arise numerically from a division by
zero as $B_{z_{org}}\approx 0$.
Results with similar precision can be found in all areas of the inner
volume.Fig. 12 shows the the relative difference $\Delta B_{x}$ for for a
$y,z$-plane at $x=2.475$ m.
Figure 12: Relative difference between the original $B_{x_{org}}$ and
reconstructed $B_{x_{rec}}$ magnetic field component $\Delta
B_{x}=(B_{x_{org}}-B_{x_{rec}})/B_{x_{org}}$ in a $z,y$ plane with $x=2.475$
m. The sawtooth structure at the fringes is due to the vanishing
$B-x$-component at large radii.
## 5 Summary and Outlook
In a simulation it is shown that with a large number of magnetic field samples
taken close to the KATRIN main-spectrometer surface and inside the current
leading elements of the LFSC -, EMCS system and spectrometer solenoids it is
possible to determine the magnetic field profile inside the spectrometer at
least within a 2% precision. With better numerical techniques (e.g. stencils
involving more meshpoints, interpolation routines with more supporting points)
and longer computer relaxation times an increase in precission is possible.
Also the number and distribution of the sampling positions on the surface can
in the case of the mobile sensor units be varied to achieve better results.
As the front face (at $-x_{n}$) and the end face (at $+x_{n}$) of the
cylindric volume still intersect the KATRIN MS volume no samples can be taken
there. However, the magnetic field of these surfaces is predominantly given by
the spectrometer solenoids which can be modeled numerically to produce
calculated field values. These models can be controlled by fixed position
magnetic field sensors close to the relevant surfaces.
Unlike in a simulation, where the magnetic field components are per se given
according to the chosen coordinate system, the magnetic field sensors in
KATRIN environment have to be aligned according to the KATRIN global
coordinate system. In the case of moving sensor units moving on the inner
rails of the LFCS structure as proposed in [6] this requires information about
position and inclination along the track.
AKNOWLEDGMENTS
The authors wish to express gratitude to the group for Experimental Techniques
of the Institute for Nuclear Physics (IK) at KIT for highly efficient and
competent support. Furthermore, we wish to thank Prof. Dr. E. W. Otten, Mainz
University and Prof. Dr. Ch. Weinheimer, M nster University for helpful
discussions and support. In addition, we like to thank the University of
Applied Sciences, Fulda and the Fachbereich Elektrotechnik und
Informationstechnik, for the enduring support for this work.
This work has been funded by the German Ministry for Education and Research
under the Project codes 05A11REA, 05A08RE1.
## References
* [1] KATRIN collaboration, _KATRIN design report_ 2004, technical report, Forschungszentrum Karlsruhe,http://www-ik.fzk.de/http://www-ik.fzk.de/, Karlsruhe Germany (2004).
* [2] A. Picard et al., _A solenoid retarding spectrometer with high resolution and transmission for keV electrons_ , Nucl. Instrum. Meth. B 63 (1992) 345.
* [3] V.M. Lobashev and P.E. Spivac, _A method for measuring the anti-electron-neutrino rest mass_ , Nucl. Instrum. Meth. A 240 (1985) 305.
* [4] K. Valerius, _Elektromagnetisches Design f r das Hauptspektrometer des KATRIN Experiments_ , Diploma-Thesis, Universit t Bonn, 2004
* [5] A. Osipowicz and F. Gl ck, _Air coil design at the main spectrometer_ , KATRIN internal document, http://fuzzy.fzk.de/bscw/bscw.cgi/d443733/95-TRP-4440-D1-F.Glueck-A.Osipowicz.ppt.http://fuzzy.fzk.de/bscw/bscw.cgi/d443733/95-TRP-4440-D1-F.Glueck-A.Osipowicz.ppt.
* [6] A. Osipowicz, W. Seller, J. Letnev, P. Marte, A. M ller, A. Spengler and A. Unru
_A mobile magnetic sensor unit for the KATRIN main spectrometer_ , 2012 JINST
7 T06002; arXiv:1207.3926
* [7] A. Unru, _Elektrische und mechanische Konzipierung und prototypische Realisierung einer mobilen Sensoreinheit_ (in German), Diploma-Thesis, Univ. of Appl. Sciences, Fulda, Germany, July 2009.
* [8] J. Letnev, Systemintegration des Magnetfeldsensornetzes (in German), Master-Thesis, Univ. of Appl. Sciences, Fulda, Germany May 2011.
* [9] _The PartOpt project webpage_ , http://www.PartOpt.net/http://www.PartOpt.net/
* [10] T. M. Apostol, _Mathematical Analysis_ , Addison-Wesley, 4th ed. 1971, p. 312, Library of Congress Catalog Card No. 57-8707
* [11] H. R. Schwarz, N. Koeckler, _Numerische Mathematik_ , 6.Auflage
B.G. Teubner Verlag / GWE Fachverlage GmbH, Wiesbaden
* [12] S. Flachs, A. Osipowicz, A. Unru, _Design Document, A wiresless magnetic sensor grid for the KATRIN mainspectrometer_ , KATRIN internal document https://fuzzy.fzk.de/bscw/bscw.cgi/d698744/ https://fuzzy.fzk.de/bscw/bscw.cgi/d698744/
* [13] Fernec Glueck et al., _New pinch, detector and axisymmetric air coil design_ , KATRIN internal document, 95-TRP-4341-D1-FGlueck.ppt
|
arxiv-papers
| 2012-09-24T08:15:36 |
2024-09-04T02:49:35.475343
|
{
"license": "Public Domain",
"authors": "A. Osipowicz, U. Rausch, A. Unru, B. Zipfel",
"submitter": "Alexander Osipowicz",
"url": "https://arxiv.org/abs/1209.5184"
}
|
1209.5343
|
# Viscosity solutions to complex Hessian equations
Lu Hoang Chinh
###### Abstract.
We study viscosity solutions to complex Hessian equations. In the local case,
we consider $\Omega$ a bounded domain in $\mathbb{C}^{n},$ $\beta$ the
standard Kähler form in $\mathbb{C}^{n}$ and $1\leq m\leq n.$ Under some
suitable conditions on $F,g$, we prove that the equation
$(dd^{c}\varphi)^{m}\wedge\beta^{n-m}=F(x,\varphi)\beta^{n},\ \varphi=g$ on
$\partial\Omega$ admits a unique viscosity solution modulo the existence of
subsolution and supersolution. If moreover, the datum is Hölder continuous
then so is the solution. In the global case, let $(X,\omega)$ be a compact
Hermitian homogeneous manifold where $\omega$ is an invariant Hermitian metric
(not necessarily Kähler). We prove that the equation
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n}$ has a
unique viscosity solution under some natural conditions on $F.$
## 1\. Introduction
The complex Hessian equation has been studied intensively in recent years. In
[26], Li solved Dirichlet problems for complex Hessian equations in
$m$-pseudoconvex domains with smooth right-hand side and smooth boundary data
by using the continuity method. In [3], Blocki considered degenerate complex
Hessian equations in $\mathbb{C}^{n}$ and developed a first step of a
potential theory for this equation. Recently, Sadullaev and Abdullaev also
studied capacities and polar sets for $m$-subharmonic functions [31].
Hou [16], Jbilou [20] and Kokarev [22] began the program of solving the non-
degenerate complex Hessian equation on compact Kähler manifolds four years
ago. This is a generalization of the famous Calabi-Yau equation [35]. In [22],
this equation is solved under rather restrictive assumptions on the underlined
manifold. Hou [16] and Jbilou [20] independently solved this equation in the
case the manifold has non-negative holomorphic bisectional curvatures. The
curvature assumption served as a technical point in an a priori
$\mathcal{C}^{2}$ estimates and people wanted to remove it. Later on, Hou, Ma
and Wu [17] provided an important $\mathcal{C}^{2}$ estimate without this
hypothesis. Using this estimate and a blowing-up analysis, Dinew and Kolodziej
recently solved the equation in full generality [10].
Degenerate complex Hessian equations on compact Kähler manifold were
considered in [9] and [29]. This approach is global in nature since it relies
on some difficult integrations by parts.
The study of real Hessian equations is a classical subject which has been
developed previously in many papers, for example [6, 8, 19, 24, 25, 32, 33,
34, 36].
The viscosity method introduced in [27] (see also [7] for a survey) is purely
local and very efficient to study weak solutions to nonlinear elliptic partial
differential equations. In [12] the authors used this method to study
degenerate complex Monge-Ampère equation on compact Kähler manifolds. In the
local context, using this approach Wang [37] considered the Dirichlet problem
for complex Monge-Ampère equations where the right-hand side also depends on
the solution.
From recent developments on viscosity method applied to complex Monge-Ampère
equations it is natural to develop such a treatment for complex Hessian
equations. It is the main purpose of this paper, precisely we consider the
following complex Hessian equation:
(1) $-(dd^{c}\varphi)^{m}\wedge\beta^{n-m}+F(x,\varphi)\beta^{n}=0,$
with boundary value $\varphi=g$ on $\partial\Omega,$ where $1\leq m\leq n$,
$\Omega$ is a bounded domain in $\mathbb{C}^{n},$ $\beta$ is the standard
Kähler form in $\mathbb{C}^{n}$,
(2) $g\in\mathcal{C}(\partial\Omega),\ \text{and}$ (3) $\displaystyle F(x,t)\
\text{is a continuous function on}\
\Omega\times\mathbb{R}\rightarrow\mathbb{R}^{+}$ $\displaystyle\text{which is
non-decreasing in the second variable}.$
We say that $F^{1/m}$ is $\gamma$-Hölder continuous uniformly in $t$ if
(4) $\displaystyle\sup_{|t|\leq M}\sup_{x\neq
y\in\bar{\Omega}}\frac{|F^{1/m}(x,t)-F^{1/m}(y,t)|}{|x-y|^{\gamma}}<+\infty,\
\forall M>0.$
Equation (1) is nonlinear degenerate second order elliptic in the viscosity
sense (see [7]) when restricted to $m$-subharmonic functions. So, we can use
the concepts of subsolutions and supersolutions.
The main results are the following:
Theorem A. Let $g,F$ be functions satisfying (2) and (3) respectively. Assume
that there exist a bounded subsolution $u$ and a bounded supersolution $v$ to
(1) such that $u_{*}=v^{*}=g$ on $\partial\Omega.$ Then there exists a unique
viscosity solution to (1) with boundary value $g$. It is also the unique
potential solution.
Theorem B. With the same assumption as in Theorem A, assume moreover that
$u,v$ are $\gamma$-Hölder continuous in $\bar{\Omega}$ and $F$ satisfies (4).
Then the unique solution of (1) with boundary value $g$ is also
$\gamma$-Hölder continuous in $\bar{\Omega}.$
In Theorem A and Theorem B, to solve the equation we need to find a
subsolution and a supersolution. When the domain $\Omega$ is strictly
pseudoconvex these existences are guaranteed.
Corollary C. Assume that $g\in\mathcal{C}(\partial\Omega)$ and $F$ satisfies
(3) and (4). If $\Omega$ is strongly pseudoconvex then (1) has a unique
viscosity solution with boundary value $g.$ If, moreover, $g$ is
$(2\gamma)$-Hölder continuous and $F$ satisfies (4) with $0<\gamma\leq 1$ then
the unique solution is $\gamma$-Hölder continuous.
Remark. When $m=n$ we recover the results in [37].
We also study viscosity solutions on compact homogeneous Hermitian manifolds.
We assume that $X$ is a Hermitian manifold with a Hermitian metric $\omega$
such that the following conditions are verified:
(H1) $X=G/H$ where $G$ is a connected Lie group and $H$ is a closed subgroup.
(H2) There exists a compact subgroup $K\subset G$ which acts transitively on
$X.$
(H3) $\omega$ is invariant by $K.$
Theorem D. Assume that $(X,\omega)$ satisfies (H1), (H2) and (H3). Let
$F(x,t)$ be a continuous function which is increasing in $t$ and assume that
there exist $t_{0},t_{1}\in\mathbb{R}$ such that
(5) $F(x,t_{0})\leq 1\leq F(x,t_{1}),\ \forall x\in X.$
Then there exists a unique viscosity solution to
$-(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}+F(x,\varphi)\omega^{n}=0.$
Remark. Our proof here does not use a priori $\mathcal{C}^{2}$ estimates in
contrast with a similar result in [29] where we use potential method [29] and
existence result of [10] which relies on $\mathcal{C}^{2}$ estimate of Hou, Ma
and Wu [17]. Moreover, we do not assume that $\omega$ is closed. An example of
compact Hermitian manifold satisfying (H1), (H2), (H3) which is not Kähler was
given to us by Karl Oeljeklaus to whom we are indebted (Example 6.2).
## 2\. Preliminaries
In this section, $\Omega$ is a bounded domain and $\beta$ is the standard
Kähler form in $\mathbb{C}^{n}.$ We introduce the notion and basic properties
of $m$-subharmonic functions in the local context and one of
$(\omega,m)$-subharmonic functions on compact Kähler manifolds.
### 2.1. Elementary symmetric functions
We begin by a brief review of elementary symmetric functions (see [3], [8],
[13]). We use the notations in [3].
Let $1\leq k\leq n$ be natural numbers. The elementary symmetric function of
order $k$ is defined by
$S_{k}(\lambda)=\sum_{1\leq i_{1}<i_{2}<...<i_{k}\leq
n}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{k}},\ \
\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}.$
Let $\Gamma_{k}$ denote the closure of the connected component of
$\\{S_{k}(\lambda)>0\\}$ containing $(1,...,1).$ It is easy to show that
$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ /\
S_{k}(\lambda_{1}+t,...,\lambda_{n}+t)\geq 0,\ \forall t\geq 0\big{\\}}.$
and hence
$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ /\ S_{j}(\lambda)\geq 0,\ \
\forall 1\leq j\leq k\big{\\}}.$
We have an obvious inclusion $\Gamma_{n}\subset...\subset\Gamma_{1}.$
The set $\Gamma_{k}$ is a convex cone in $\mathbb{R}^{n}$ and $S_{k}^{1/k}$ is
concave on $\Gamma_{k}$ [13].
Let $\mathcal{H}$ denote the vector space (over $\mathbb{R}$) of complex
Hermitian matrices of dimension $n\times n.$ For $A\in\mathcal{H}$ we set
$\widetilde{S}_{k}(A)=S_{k}(\lambda(A)),$
where $\lambda(A)\in\mathbb{R}^{n}$ is the vector of eigenvalues of $A.$ The
function $\widetilde{S}_{k}$ can also be defined as the sum of all principal
minors of order $k$,
$\widetilde{S}_{k}(A)=\sum_{|I|=k}A_{II}.$
From the latter we see that $\widetilde{S}_{k}$ is a homogeneous polynomial of
order $k$ on $\mathcal{H}$ which is hyperbolic with respect to the identity
matrix $I$ (that is for every $A\in\widetilde{S}$ the equation
$\widetilde{S}_{k}(A+tI)=0$ has $n$ real roots; see [13]). As in [13] (see
also [3]), the cone
$\widetilde{\Gamma}_{k}:=\big{\\{}A\in\mathcal{H}\ /\
\widetilde{S}_{k}(A+tI)\geq 0,\forall t\geq 0\big{\\}}=\\{A\in\mathcal{H}\ /\
\lambda(A)\in\Gamma_{k}\\}$
is convex and the function $\widetilde{S}_{k}^{1/k}$ is concave on
$\widetilde{\Gamma}_{k}.$
### 2.2. m-subharmonic functions and the Hessian operator
We associate real (1,1)-forms $\alpha$ in $\mathbb{C}^{n}$ with Hermitian
matrices $[a_{j\bar{k}}]$ by
$\alpha=\frac{i}{\pi}\sum_{j,k}a_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$
Then the canonical Kähler form $\beta$ is associated with the identity matrix
I. It is easy to see that
$\binom{n}{k}\alpha^{k}\wedge\beta^{n-k}=\widetilde{S}_{k}(A)\beta^{n}.$
###### Definition 2.1.
Let $\alpha$ be a real $(1,1)$-form on $\Omega$. We say that $\alpha$ is
$m$-positive at a given point $P\in\Omega$ if at this point we have
$\alpha^{j}\wedge\beta^{n-j}\geq 0,\ \ \forall j=1,...,m.$
$\alpha$ is called $m$-positive if it is $m$-positive at any point of
$\Omega.$ If there is no confusion we also denote by $\widetilde{\Gamma}_{m}$
the set of $m$-positive (1,1)-forms.
Let $T$ be a current of bidegree $(n-k,n-k)$ with $k\leq m$. Then $T$ is
called $m$-positive if
$\alpha_{1}\wedge...\wedge\alpha_{k}\wedge T\geq 0,$
for all $m$-positive $(1,1)$-forms $\alpha_{1},...,\alpha_{k}.$
###### Definition 2.2.
A function $u:\Omega\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$m$-subharmonic if it is subharmonic and
$dd^{c}u\wedge\alpha_{1}\wedge...\wedge\alpha_{m-1}\wedge\beta^{n-m}\geq 0,$
for every $m$-positive (1,1)-forms $\alpha_{1},...,\alpha_{m-1}.$ The class of
all $m$-subharmonic functions in $\Omega$ will be denoted by
$\mathcal{P}_{m}(\Omega).$
We summarize basic properties of $m$-subharmonic functions in the following:
###### Proposition 2.3.
[3] (i) If $u$ is $\mathcal{C}^{2}$ smooth then $u$ is $m$-subharmonic if and
only if the form $dd^{c}u$ is $m$-positive at every point in $\Omega.$
(ii) If $u,v\in\mathcal{P}_{m}(\Omega)$ then $\lambda u+\mu
v\in\mathcal{P}_{m}(\Omega),\forall\lambda,\mu>0.$
(iii) If $u$ is $m$-subharmonic in $\Omega$ then the standard regularization
$u\star\chi_{\epsilon}$ is also $m$-subharmonic in
$\Omega_{\epsilon}:=\\{x\in\Omega:d(x,\partial\Omega)>\epsilon\\}$.
(iv) If $(u_{l})\subset\mathcal{P}_{m}(\Omega)$ is locally uniformly bounded
from above then $(\sup u_{l})^{\star}\in\mathcal{P}_{m}(\Omega)$, where
$v^{\star}$ is the upper semicontinuous regularization of $v$.
(v) $PSH=\mathcal{P}_{n}\subset...\subset\mathcal{P}_{1}=SH.$
(vi) Let $\emptyset\neq U\subset\Omega$ be a proper open subset such that
$\partial U\cap\Omega$ is relatively compact in $\Omega$. If
$u\in\mathcal{P}_{m}(\Omega)$, $v\in\mathcal{P}_{m}(U)$ and $\limsup_{x\to
y}v(x)\leq u(y)$ for each $y\in\partial U\cap\Omega$ then the function $w$,
defined by
$w=\left\\{{\begin{array}[]{*{20}c}{u\ \textrm{on}\ \Omega\setminus U}\\\
{\max(u,v)\ \textrm{on}\ U}\end{array}}\right.,$
is $m$-subharmonic in $\Omega.$
For locally bounded $m$-subharmonic functions $u_{1},...,u_{p}$ ($p\leq m$)
and a closed $m$-positive current $T$ we can inductively define a closed
$m$-positive current $dd^{c}u_{1}\wedge...\wedge dd^{c}u_{p}\wedge T$
(following Bedford and Taylor [2]).
###### Lemma 2.4.
Let $u_{1},...,u_{k}(k\leq m)$ be locally bounded $m$-subharmonic functions in
$\Omega$ and let $T$ be a closed $m$-positive current of bidegree $(n-p,n-p)$
($p\geq k$). Then we can define inductively a closed $m$-positive current
$dd^{c}u_{1}\wedge dd^{c}u_{2}\wedge...\wedge dd^{c}u_{k}\wedge T,$
and the product is symmetric, i.e.
$dd^{c}u_{1}\wedge dd^{c}u_{2}\wedge...\wedge dd^{c}u_{p}\wedge
T=dd^{c}u_{\sigma(1)}\wedge dd^{c}u_{\sigma(2)}\wedge...\wedge
dd^{c}u_{\sigma(k)}\wedge T,$
for every permutation $\sigma:\\{1,...,k\\}\to\\{1,...,k\\}.$
In particular, the Hessian measure of $u\in\mathcal{P}_{m}(\Omega)\cap
L^{\infty}_{loc}$ is well defined as
$H_{m}(u)=(dd^{c}u)^{m}\wedge\beta^{n-m}.$
### 2.3. $(\omega,m)$-subharmonic functions
In this section, $(X,\omega)$ is a compact Kähler manifold and $U\subset X$ is
an open subset contained in a local chart.
###### Definition 2.5.
A function $u\in L^{1}(U)$ is called weakly $\omega$-subharmonic if
$dd^{c}u\wedge\omega^{n-1}\geq 0,$
in the weak sense of currents.
Thanks to Littman [28] we have the following approximation properties.
###### Proposition 2.6.
Let $u$ be a weakly $\omega$-subharmonic function in $U$. Then there exists a
one parameter family of functions $u_{h}$ with the following properties: For
every compact subset $U^{\prime}\subset U$
a) $u_{h}$ is smooth in $U^{\prime}$ for $h$ sufficiently large,
b) $dd^{c}u_{h}\wedge\omega^{n-1}\geq 0$ in $U^{\prime},$
c) $u_{h}$ is non-increasing with increasing $h,$ and
$\lim_{h\to\infty}u_{h}(x)=u(x)$ almost everywhere in $U^{\prime},$
d) $u_{h}$ is given explicitly as $u_{h}(y)=\int_{U}K_{h}(x,y)u(x)dx,$ where
$K_{h}$ is a smooth non-negative function and $\int_{U}K_{h}(x,y)dy\to 1,$
uniformly in $x\in U^{\prime}.$
###### Definition 2.7.
A function $u$ is called $\omega$-subharmonic if it is weakly
$\omega$-subharmonic and for every $U^{\prime}\Subset U$,
$\lim_{h\to\infty}u_{h}(x)=u(x),\forall x\in U^{\prime},$ where $u_{h}$ is
constructed as in Proposition 2.6.
###### Remark 2.8.
Any continuous weakly $\omega$-subharmonic function is $\omega$-subharmonic.
If $(u_{j})$ is a sequence of continuous $\omega$-subharmonic functions
decreasing to $u\not\equiv-\infty$ then $u$ is $\omega$-subharmonic.
If $u$ is weakly $\omega$-subharmonic then the pointwise limit of $(u_{h})$ is
an $\omega$-subharmonic function.
Let $(u_{j})$ be a sequence of $\omega$-subharmonic functions and $(u_{j})$ is
uniformly bounded from above. Then $u:=(\limsup_{j}u_{j})^{\star}$ is
$\omega$-subharmonic, where for a function $v$, $v^{\star}$ denotes the upper
semicontinuous regularization of $v.$
###### Definition 2.9.
Let $\alpha$ be a real $(1,1)$-form on $X$. We say that $\alpha$ is
$(\omega,m)$-positive at a given point $P\in X$ if at this point we have
$\alpha^{k}\wedge\omega^{n-k}\geq 0,\ \ \forall k=1,...,m.$
We say that $\alpha$ is $(\omega,m)$-positive if it is $(\omega,m)$-positive
at any point of $X.$
###### Remark 2.10.
Locally at $P\in X$ with local coordinates $z_{1},...,z_{n}$, we have
$\alpha=\frac{i}{\pi}\sum_{j,k}\alpha_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}},$
and
$\omega=\frac{i}{\pi}\sum_{j,k}g_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$
Then $\alpha$ is $(\omega,m)$-positive at $P$ if and only if the vector of
eigenvalues $\lambda(g^{-1}\alpha)=(\lambda_{1},...,\lambda_{n})$ of the
matrix $\alpha_{j\bar{k}}(P)$ with respect to the matrix $g_{j\bar{k}}(P)$ is
in $\Gamma_{m}$. These eigenvalues are independent of any choice of local
coordinates.
Following Blocki [3] we can define $(\omega,m)$-subharmonicity for (non-
smooth) functions.
###### Definition 2.11.
A function $\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$(\omega,m)$-subharmonic if the following conditions hold:
(i) in any local chart $\Omega,$ given $\rho$ a local potential of $\omega$
and set $u:=\rho+\varphi$, then $u$ is $\omega$-subharmonic,
(ii) for every smooth $(\omega,m)$-positive forms $\beta_{1},...,\beta_{m-1}$
we have, in the weak sense of distributions,
$(\omega+dd^{c}\varphi)\wedge\beta_{1}\wedge...\wedge\beta_{m-1}\wedge\omega^{n-m}\geq
0.$
Let $SH_{m}(X,\omega)$ be the set of all $(\omega,m)$-subharmonic functions on
$X.$ Observe that, by definition, any $\varphi\in SH_{m}(X,\omega)$ is upper
semicontinuous.
The following properties of $(\omega,m)$-subharmonic functions are easy to
show.
###### Proposition 2.12.
(i) If $\varphi\in\mathcal{C}^{2}(X)$ then $\varphi$ is
$(\omega,m)$-subharmonic if the form $(\omega+dd^{c}\varphi)$ is
$(\omega,m)$-positive, or equivalently
$(\omega+dd^{c}\varphi)\wedge(\omega+dd^{c}u_{1})\wedge...\wedge(\omega+dd^{c}u_{m-1})\wedge\omega^{n-m}\geq
0,$
for all $\mathcal{C}^{2}$ $(\omega,m)$-subharmonic functions
$u_{1},...,u_{m-1}.$
(ii) If $\varphi,\psi\in SH_{m}(X,\omega)$ then $\max(\varphi,\psi)\in
SH_{m}(X,\omega).$
(iii) If $\varphi,\psi\in SH_{m}(X,\omega)$ and $\lambda\in[0,1]$ then
$\lambda\varphi+(1-\lambda)\psi\in SH_{m}(X,\omega).$
(iv) If $(\varphi_{j})\subset SH_{m}(X,\omega)$ is uniformly bounded from
above then
$(\limsup_{j}\varphi_{j})^{\star}\in SH_{m}(X,\omega).$
## 3\. Viscosity solutions vs. potential solutions
In this section we introduce the notion of viscosity (sub, super)-solutions to
degenerate complex Hessian equations and systematically compare them with
potential ones. We prove an important comparison principle which is the key
point in the proof of our main results. The idea of our proof is taken from
[12], [37, 5], [7].
###### Definition 3.1.
Let $u:\Omega\rightarrow\mathbb{R}\cup\\{-\infty\\}$ be a function. Let
$\varphi$ be a $\mathcal{C}^{2}$ function in a neighborhood of
$x_{0}\in\Omega.$ We say that $\varphi$ touches $u$ from above (resp. below)
at $x_{0}$ if $\varphi(x_{0})=u(x_{0})$ and $\varphi(x)\geq u(x)$ (resp.
$\varphi(x)\leq u(x)$) for every $x$ in a neighborhood of $x_{0}.$
###### Definition 3.2.
An upper semicontinuous function
$\varphi:\Omega\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is a viscosity
subsolution to
(6) $-(dd^{c}\varphi)^{m}\wedge\beta^{n-m}+F(x,\varphi)\beta^{n}=0$
if $\varphi\not\equiv-\infty$ and for any $x_{0}\in\Omega$ and any
$\mathcal{C}^{2}$ function $q$ which touches $\varphi$ from above at $x_{0}$
then
$H_{m}(q)\geq F(x,q)\beta^{n},\ \text{at}\ x_{0}.$
Here we use the notation $H_{m}(u)=(dd^{c}u)^{m}\wedge\beta^{n-m}$ for
$u\in\mathcal{C}^{2}(X).$ We also say that $H_{m}(\varphi)\geq
F(x,q)\beta^{n}$ “in the viscosity sense”.
###### Definition 3.3.
A lower semicontinuous function
$\varphi:X\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a viscosity supersolution
to (6) if $\varphi\not\equiv+\infty$ and for any $x_{0}\in X$ and any
$\mathcal{C}^{2}$ function $q$ which touches $\varphi$ from below at $x_{0}$
then
$[(dd^{c}q)^{m}\wedge\beta^{n-m}]_{+}\leq F(x,q)\beta^{n},\ \text{at}\ x_{0}.$
Here $[\alpha^{m}\wedge\beta^{n-m}]_{+}$ is defined to be itself if $\alpha$
is $m$-positive and $0$ otherwise.
###### Remark 3.4.
If $u\in\mathcal{C}^{2}(\Omega)$ then $H_{m}(\varphi)\geq
F(x,\varphi)\beta^{n}$ (or $[H_{m}(\varphi)]_{+}\leq F(x,\varphi)\beta^{n}$)
holds in the viscosity sense iff it holds in the usual sense.
###### Definition 3.5.
A function $\varphi:X\rightarrow\mathbb{R}$ is a viscosity solution to (1) if
it is both a subsolution and a supersolution. Thus, a viscosity solution is
automatically continuous.
The notion of viscosity subsolutions is stable under taking maximum. It is
also stable along monotone sequences as the following lemma shows.
###### Lemma 3.6.
Assume that $F:\Omega\times\mathbb{R}\rightarrow\mathbb{R}^{+}$ is a
continuous function. Let $(\varphi_{j})$ be a monotone sequence of viscosity
subsolutions of equation
(7) $-(dd^{c}u)^{m}\wedge\beta^{n-m}+F(x,u)\beta^{n}=0,$
If $\varphi_{j}$ is uniformly bounded from above and
$\varphi:=(\lim\varphi_{j})^{*}\not\equiv-\infty$ then $\varphi$ is also a
viscosity subsolution of (7).
###### Proof.
The proof can be found in [7]. For convenience, we reproduce it here. Observe
that if $z_{j}\to z$ then
$\limsup_{j\to+\infty}\varphi_{j}(z_{j})\leq\varphi(z).$
Fix $x_{0}\in\Omega$ and $q$ a $\mathcal{C}^{2}$ function in a neighborhood of
$x_{0}$, say $B(x_{0},r)\subset\Omega$ which touches $\varphi$ from above at
$x_{0}$. We can choose a sequence $(x_{j})\subset B=\bar{B}(x_{0},r/2)$
converging to $x_{0}$ and a subsequence of $(\varphi_{j})$ (still denoted by
$(\varphi_{j})$) such that
$\varphi_{j}(x_{j})\to\varphi(x_{0}).$
Fix $\epsilon>0.$ For each $j,$ let $y_{j}$ be the maximum point of
$\varphi_{j}-q-\epsilon|x-x_{0}|^{2}$ on $B.$ Then
(8)
$\displaystyle\varphi_{j}(x_{j})-q(x_{j})-\epsilon|x_{j}-x_{0}|^{2}\leq\varphi_{j}(y_{j})-q(y_{j})-\epsilon|y_{j}-x_{0}|^{2}.$
We claim that $y_{j}\to x_{0}.$ Indeed, assume that $y_{j}\to y\in B.$ Letting
$j\to+\infty$ in (8) and noting that
$\limsup\varphi_{j}(y_{j})\leq\varphi(y)$, we get
$0\leq\varphi(y)-q(y)-\epsilon|y-x_{0}|^{2}.$
Remember that $q$ touches $\varphi$ above in $B$ at $x_{0}$ and $y\in B.$
Thus, the above inequality implies that $y=x_{0},$ which means $y_{j}\to
x_{0}.$ Then again by (8) we deduce that
$\varphi_{j}(y_{j})\to\varphi(x_{0}).$
For $j$ large enough, the function
$q+\epsilon|x-x_{0}|^{2}+\varphi_{j}(y_{j})-q(y_{j})-\epsilon|y_{j}-x_{0}|^{2}$
touches $\varphi_{j}$ from above at $y_{j}.$ Thus
$H_{m}(q+\epsilon|x-x_{0}|^{2})(y_{j})\geq
F(y_{j},\varphi_{j}(y_{j}))\beta^{n}.$
It suffices now to let $j\to+\infty$. ∎
When $F\equiv 0,$ viscosity subsolutions of (1) are exactly $m$-subharmonic
functions.
###### Lemma 3.7.
A function $u$ is $m$-subharmonic in $\Omega$ if and only if it is a viscosity
subsolution of
(9) $-(dd^{c}u)^{m}\wedge\beta^{n-m}=0.$
###### Proof.
Assume that $u$ is $m$-subharmonic in $\Omega$ and let $u_{\epsilon}$ be its
standard smooth regularization. Then $u_{\epsilon}$ is $m$-subharmonic and
smooth, hence $u_{\epsilon}$ is a classical subsolution of (9). Thus, it
follows from Lemma 3.6 that $u$ is a viscosity subsolution of (9).
Conversely, assume that $u$ is a viscosity subsolution of (9). Fix
$\alpha_{1},...,\alpha_{m-1}$ $m$-positive (1,1)-forms with constant
coefficients such that
$\alpha_{1}\wedge...\wedge\alpha_{m-1}\wedge\beta^{n-m}$
is strictly positive. Let $x_{0}\in\Omega$ and
$q\in\mathcal{C}^{2}(V_{x_{0}})$ such that $u-q$ has a local maximum at
$x_{0}.$ Then for any $\epsilon>0$, $q+\epsilon|z-z_{0}|^{2}$ also touches $u$
from above. By the definition of viscosity subsolutions, we have
$(dd^{c}q+\epsilon\beta)^{m}\wedge\beta^{n-m}\geq 0,\forall\epsilon>0,$
which means that the Hessian matrix $\dfrac{\partial^{2}q}{\partial
z_{j}\partial\bar{z}_{k}}(x_{0})$ is $m$-positive. Hence
$L_{\alpha}q:=dd^{c}q\wedge\alpha_{1}\wedge...\wedge\alpha_{m-1}\wedge\beta^{n-m}\geq
0,$
holds at $x_{0}.$
This implies $L_{\alpha}u\geq 0$ in the viscosity sense. In appropriate
complex coordinates this constant coefficient differential operator is the
Laplace operator. Hence, [15] Proposition 3.2.10’ p. 147 implies that $u$ is
$L_{\alpha}$-subharmonic hence is $L^{1}_{\rm loc}(V_{x_{0}})$ and satisfies
$L_{\alpha}u\geq 0$ in the sense of distributions. Since
$\alpha_{1},...,\alpha_{m-1}$ were taken arbitrarily, by continuity we have
$dd^{c}u\wedge\alpha_{1}\wedge...\wedge\alpha_{m-1}\wedge\beta^{n-m}\geq 0$
in the sense of distributions for any $m$-positive (1,1)-forms $\alpha.$
Therefore, $u$ is $m$-subharmonic. ∎
###### Corollary 3.8.
Lemma 3.6 still holds if the sequence $\varphi_{j}$ is not monotone.
###### Proof.
For each $j,$ set
$u_{j}=(\sup_{k\geq j}\varphi_{k})^{*},\ \
v_{l}:=\max(\varphi_{j},...,\varphi_{j+l}).$
Since the notion of viscosity subsolution is stable under taking the maximum,
we deduce that $v_{l}$ is a viscosity subsolution of (7). Observe that
$u_{j}=(\sup_{l\geq 0}v_{l})^{*}$ and the sequence $(v_{l})$ is monotone. It
follows from what we have done before that $u_{j}$ is a viscosity subsolution
of (7). By Lemma 3.7 each $\varphi_{j}$ is $m$-subharmonic. Hence,
$u_{j}\downarrow\varphi$ and the proof is complete. ∎
For real $(1,1)$-form $\alpha$, we denote by
$S_{m}(\alpha):=\frac{\alpha^{m}\wedge\beta^{n-m}}{\beta^{n}}.$
Set
$U_{m}:=\\{\alpha\in\widetilde{\Gamma}_{m}\ \text{of constant coefficients
such that}\ S_{m}(\alpha)=1\\}.$
It is elementary to prove the following lemma:
###### Lemma 3.9.
Let $\alpha$ be a real $m$-positive $(1,1)$-form. Then the following identity
holds
$(S_{m}(\alpha))^{1/m}=\inf\Big{\\{}\frac{\alpha\wedge\alpha_{1}\wedge...\wedge\alpha_{m-1}\wedge\beta^{n-m}}{\beta^{n}}\
/\ \alpha_{j}\in U_{m},\forall j\Big{\\}}.$
Now, we compare viscosity and potential subsolutions when the right-hand side
$F(x,t)$ does not depend on $t.$
###### Proposition 3.10.
Let $\varphi$ be a bounded upper semicontinuous function in $\Omega$ and
$0\leq f$ be a continuous function.
(i) If $\varphi$ is $m$-subharmonic such that
(10) $H_{m}(\varphi)\geq f\beta^{n}$
in the potential sense then it also holds in the viscosity sense.
(ii) Conversely, if (10) holds in the viscosity sense then $\varphi$ is
$m$-subharmonic and the inequality holds in the potential sense.
###### Proof.
We follow [12].
Proof of (i): Let $x_{0}\in\Omega$ and assume that $q$ is a $\mathcal{C}^{2}$
functions which touches $\varphi$ from above at $x_{0}.$ Suppose that
$H_{m}(q(x_{0}))<f(x_{0})\beta^{n}.$ There exists $\epsilon>0$ such that
$H_{m}(q_{\epsilon})<f\beta^{n}$ in a neighborhood of $x_{0}$ since $f$ is
continuous, here $q_{\epsilon}=q+\epsilon|z-x_{0}|^{2}.$ It follows from the
proof of Lemma 3.7 that $q_{\epsilon}$ is $m$-subharmonic in a neighborhood of
$x_{0}$, say $B.$ Now, for $\delta>0$ small enough, we have
$q_{\epsilon}-\delta\geq\varphi$ on $\partial B$ but it fails at $x_{0}$ which
contradicts the potential comparison principle (see Theorem 1.14 and Corollary
1.15 in [30]).
Proof of (ii): We proceed steps by steps.
Step 1: Assume that $0<f$ is smooth. Let $x_{0}\in\Omega$ and assume that $q$
is a $\mathcal{C}^{2}$ functions which touches $\varphi$ from above at
$x_{0}.$ Fix $\alpha_{1},...,\alpha_{m-1}\in U_{m}.$
We can find $h\in\mathcal{C}^{2}(\\{x_{0}\\})$ such that
$L_{\alpha}h=f^{1/m}\beta^{n}.$ As in the proof of Lemma 3.7, we can prove
that $\varphi-h$ is $L_{\alpha}$-subharmonic, which gives
$L_{\alpha}\varphi\geq L_{\alpha}h=f^{1/m}\beta^{n}$ in the potential sense.
Consider the standard regularization $\varphi_{\epsilon}$ of $\varphi$ by
convolution with a smoothing kernel. Then
$L_{\alpha}\varphi_{\epsilon}\geq(f^{1/m})_{\epsilon}\beta^{n},$
in the potential sense and hence in the usual sense. Now, use Lemma 3.9, we
obtain
$H_{m}(\varphi_{\epsilon})\geq(f^{1/m})_{\epsilon}^{m}\beta^{n}.$
Letting $\epsilon\to 0$ and noting that the Hessian operator is continuous
under decreasing sequence, we get
$H_{m}(\varphi)\geq f\beta^{n}.$
Step 2: Assume that $0<f$ is only continuous. Note that
$f=\sup\\{h\in\mathcal{C}^{\infty}(\Omega),\ 0<h\leq f\\}.$
Now, if $H_{m}(\varphi)\geq f\beta^{n}$ in the viscosity sense then we also
have $H_{m}(\varphi)\geq h\beta^{n}$ in the viscosity sense provided that
$f\geq h$. Thus, by Step 1,
$H_{m}(\varphi)\geq h\beta^{n},$
for every $0<h\leq f\in\mathcal{C}^{\infty}(\Omega).$ This yields
$H_{m}(\varphi)\geq f\beta^{n}$
in the viscosity sense.
Step 3: $0\leq f$ is merely continuous. We consider
$\varphi_{\epsilon}=\varphi+\epsilon|z|^{2}.$ Then
$H_{m}(\varphi_{\epsilon})\geq(f+\epsilon^{m})\beta^{n}$
in the viscosity sense. By Step 2 we have
$H_{m}(\varphi_{\epsilon})\geq(f+\epsilon^{m})\beta^{n}$
in the potential sense and the result follows by letting $\epsilon$ go to $0.$
∎
###### Theorem 3.11.
Let $F:\Omega\times\mathbb{R}\to\mathbb{R}^{+}$ be a continuous function which
is non-decreasing in the second variable. Let $\varphi$ be a bounded u.s.c.
function in $\Omega.$ Then the inequality
(11) $H_{m}(\varphi)\geq F(x,\varphi)\beta^{n}$
holds in the viscosity sense if and only if $\varphi$ is $m$-subharmonic in
$\Omega$ and (11) holds in the potential sense.
###### Proof.
Let us prove the first implication. Assume that (11) holds in the viscosity
sense. Consider the sup-convolution of $\varphi$:
(12) $\varphi^{\delta}(x):=\sup\\{\varphi(y)-\frac{1}{\delta^{2}}|x-y|^{2}\ /\
y\in\Omega\\},\ x\in\Omega_{\delta},$
where $\Omega_{\delta}:=\\{x\in\Omega\ /\ d(x,\partial\Omega)>A\delta\\},$ and
the positive constant $A$ is chosen so that
$A^{2}>\text{osc}_{\Omega}\varphi.$ Then $\varphi^{\delta}\downarrow\varphi$
and as in [18] (see also [12]) it can be shown that
(13) $H_{m}(\varphi^{\delta})\geq F_{\delta}(x,\varphi^{\delta})\beta^{n},\
\text{in}\ \Omega_{\delta},$
in the viscosity sense, where $F_{\delta}(x,t)=\inf_{|y-x|\leq
A\delta}F(y,t).$
It follows from Proposition 3.10 that (13) holds in the potential sense and
the result follows by letting $\delta$ go to $0.$
Let us prove the other implication. Suppose that $\varphi$ satisfies (11) in
the potential sense. As in [12] it can be shown that
(14) $H_{m}(\varphi^{\delta})\geq F_{\delta}(x,\varphi^{\delta})\beta^{n},$
in the potential sense. Now, applying Proposition 3.10 to $\varphi^{\delta}$
we see that (14) holds in the viscosity sense. It suffices to let $\delta\to
0.$ ∎
## 4\. Local comparison principle
In this section we follow [37] (see also [CC95]) to prove a viscosity
comparison principle for equation (6).
###### Definition 4.1.
A function $u:\Omega\rightarrow\mathbb{R}$ is called semiconcave (resp.
semiconvex) if there exists $K>0$ (resp. $K<0$) such that for every
$z_{0}\in\Omega$ there exists a quadratic polynomial $P=K|z|^{2}+l$, where $l$
is an affine function, which touches $u$ from above (resp. below) at $z_{0}.$
###### Definition 4.2.
A function $u:\Omega\rightarrow\mathbb{R}$ is called punctually second order
differentiable at $z_{0}\in\Omega$ if there exists a quadratic polynomial $q$
such that
$u(z)=q(z)+o(|z-z_{0}|^{2})\ \text{as}\ z\to z_{0}.$
Note that such a $q$ is unique if it exists. We thus define
$dd^{c}u(z_{0}),D^{2}u(z_{0})$ to be $dd^{c}q(z_{0}),D^{2}q(z_{0}).$
The following result is a theorem of Alexandroff-Buselman-Feller (see [11,
Theorem 1, Section 6.4], or [23, Theorem 1, Section 1.2], or [23, Appendix
2]).
###### Theorem 4.3.
Every continuous semiconvex (or semiconcave) function is punctually second
order differentiable almost everywhere.
###### Theorem 4.4 (Local comparison principle).
Let $F$ be a continuous function which is non-decreasing in the second
variable. Let $u$ be a bounded viscosity subsolution and $v$ be a bounded
viscosity supersolution of
$-H_{m}(\varphi)+F(x,\varphi)\beta^{n}=0.$
If $u\leq v$ on $\partial\Omega$ then $u\leq v$ on $\Omega.$
###### Proof.
By considering $u-\epsilon$, $\epsilon>0$ and then letting $\epsilon\to 0$
noting that $F$ is non-decreasing in the second variable, we can assume that
$u<v$ near the boundary of $\Omega.$ Assume by contradiction that there exists
$x_{0}\in\Omega$ such that
$u(x_{0})-v(x_{0})=a>0.$
Let $u^{\epsilon},v_{\epsilon}$ be the sup-convolution and inf-convolution
(which is defined similarly as in (12)). They are semiconvex and semiconcave
functions respectively. By Dini’s Lemma
$w_{\epsilon}:=v_{\epsilon}-u_{\epsilon}\geq 0$ near the boundary
$\partial\Omega$ for $\epsilon>0$ small enough. Thus, we can fix some open
subset $U\Subset\Omega$ such that $w_{\epsilon}\geq 0$ on $\Omega\setminus U.$
Fix $\epsilon>0$ small enough. Denote by $E_{\epsilon}$ the set of all points
in $U$ where $w_{\epsilon},u^{\epsilon},v_{\epsilon}$ are punctually second
order differentiable. Then by Theorem 4.3, the Lebesgue measure of $U\setminus
E_{\epsilon}$ is $0.$ Fix some $r>0$ such that $\Omega\subset B_{r}\subset
B_{2r}.$ Define
$G_{\epsilon}(x)=\sup\\{\varphi(x)\ /\ \varphi\ \text{is convex in}\ B_{2r},\
\varphi\leq\min(w_{\epsilon},0)\ \text{in}\ \Omega\\}.$
Since $w_{\epsilon}\geq 0$ on $\partial U$ and $w_{\epsilon}(x_{0})\leq a<0$,
using Alexandroff-Bakelman-Pucci (ABP) estimate (see also [37, Lemma 4.7]) we
can find $x_{\epsilon}\in E_{\epsilon}$ such that
(i) $w_{\epsilon}(x_{\epsilon})=G_{\epsilon}(x_{\epsilon})<0,$
(ii) $G_{\epsilon}$ is punctually second order differentiable at
$x_{\epsilon}$ and
det${}_{\mathbb{R}}(D^{2}G_{\epsilon}(x_{\epsilon}))\geq\delta,$ where
$\delta>0$ depends only on $a,n$ and $diam(\Omega).$
Since $G_{\epsilon}$ is convex, we also have
det${}_{\mathbb{C}}(dd^{c}G_{\epsilon})(x_{\epsilon})\geq\delta^{1/2}.$ It
follows from Gårding’s inequality [13] that
$(dd^{c}G_{\epsilon})^{m}\wedge\beta^{n-m}(x_{\epsilon})\geq\delta_{1}\beta^{n},$
where $\delta_{1}$ does not depend on $\epsilon.$ On the other hand,
$H_{m}(u^{\epsilon})(x_{\epsilon})\geq
F_{\epsilon}(x_{\epsilon},u^{\epsilon}(x_{\epsilon}))\beta^{n},$
Moreover $G_{\epsilon}+u^{\epsilon}$ touches $v_{\epsilon}$ from below at
$x_{\epsilon}.$ Since $G_{\epsilon}+u^{\epsilon}$ is $m$-subharmonic and
punctually second order differentiable at $x_{\epsilon}$ it follows that
$H_{m}(G_{\epsilon}+u^{\epsilon})(x_{\epsilon})\leq
F^{\epsilon}(x_{\epsilon},v_{\epsilon}(x_{\epsilon}))\beta^{n}.$
Since $F$ is non-decreasing in the second variable and since
$w_{\epsilon}(x_{\epsilon})<0$, the above inequality implies that
$\delta_{2}+F_{\epsilon}(x_{\epsilon},u^{\epsilon}(x_{\epsilon}))\leq
F^{\epsilon}(x_{\epsilon},u^{\epsilon}(x_{\epsilon})),$
where $\delta_{2}>0$ is another constant which does not depend on $\epsilon.$
Letting $\epsilon\to 0$, after a subsequence if necessary we obtain a
contradiction. ∎
## 5\. Viscosity solutions on homogeneous compact Hermitian manifolds
In this section we consider viscosity solutions to
(15) $-(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}+F(x,\varphi)\omega^{n}=0,$
where $(X,\omega)$ satisfies (H1), (H2) and (H3).
The notion of viscosity subsolutions and supersolutions are defined similarly
as in the local case. We compare viscosity and potential subsolutions in the
two following theorems.
###### Proposition 5.1.
Assume that $\omega$ is Kähler and $\varphi$ is a continuous function on $X.$
Then $\varphi$ is $(\omega,m)$-subharmonic iff
(16) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}\geq 0$
in the viscosity sense.
###### Proof.
Assume that $\varphi$ is $(\omega,m)$-subharmonic and let $\varphi_{\epsilon}$
be the smooth regularizing sequence of $\varphi$ as in [29]. Then
$\varphi_{\epsilon}$ is $(\omega,m)$-subharmonic in the viscosity sense.
Fix $x_{0}\in X$, $\delta>0$ and $q$ a $\mathcal{C}^{2}$ function which
touches $\varphi$ from above at $x_{0}.$ Let $B$ be a small closed ball where
the touching appears and let $x_{\epsilon}$ be a maximum point of
$\varphi_{\epsilon}-q-\delta\rho$ in $B.$ Here $\rho=|z-x_{0}|^{2}.$ Then due
to the uniform convergence of $\varphi_{\epsilon}$ and Dini’s Lemma we have
$x_{\epsilon}\to x_{0}$ as $\epsilon\downarrow 0.$ Also, for small
$\epsilon>0$, $q+\delta\rho+\varphi_{\epsilon}(x_{\epsilon})-q(x_{\epsilon})$
touches $\varphi_{\epsilon}$ from above at $x_{\epsilon}$. This implies that
$(\omega+dd^{c}q+\delta dd^{c}\rho)^{m}\wedge\omega^{n-m}\geq 0$
holds at $x_{\epsilon}$ which, in turn, implies one implication by letting
$\epsilon\downarrow 0$ and $\delta\downarrow 0.$
Let us prove the other implication. Assume that $\varphi$ satisfies (16) in
the viscosity sense. Fix $\alpha=\alpha_{1}\wedge...\wedge\alpha_{m-1},$ where
$\alpha_{i}$ are smooth $(\omega,m)$-positive closed (1,1)-forms. By Gårding’s
inequality we see that
$(\omega+dd^{c}\varphi)\wedge\alpha\wedge\omega^{n-m}\geq 0$
in the viscosity sense. Thanks to [14, Corollary 7.20] the same arguments as
in [15, page 147] show that the above inequality also holds in the sense of
currents. Thus, $\varphi$ is $(\omega,m)$-subharmonic. ∎
###### Theorem 5.2.
Assume that $\omega$ is Kähler, $F$ is continuous on $X\times\mathbb{R}$ and
increasing in the second variable, and $\varphi\in\mathcal{C}(X).$ Then
$\varphi$ is $(\omega,m)$-subharmonic and satisfies
(17) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}\geq F(x,\varphi)\omega^{n}$
in the potential sense if and only if the above inequality holds in the
viscosity sense.
###### Proof.
Set
$f(x)=F(x,\varphi(x)),\ x\in X.$
Assume that $\varphi$ satisfies (17) in the potential sense. Let $x_{0}\in X$
and $q\in\mathcal{C}^{2}(U)$ which touches $\varphi$ from above at $x_{0}$ in
$U$, a small neighborhood of $x_{0}.$ Suppose by contradiction that
$(\omega+dd^{c}q)^{m}\wedge\omega^{n-m}<f\omega^{n}$
holds at $x_{0}.$ Then for $\epsilon$ small enough we have
$(\omega+dd^{c}q_{\epsilon})^{m}\wedge\omega^{n-m}<f\omega^{n}$
in a small ball $B$ containing $x_{0}.$ Here
$q_{\epsilon}=q+\epsilon|x-x_{0}|^{2}$ defined in a local chart near $x_{0}.$
Since $q$ touches $\varphi$ from above at $x_{0}$ in $B$, we can find
$\delta>0$ small enough such that $q_{\epsilon}-\delta\geq\varphi$ on
$\partial B$. But $q_{\epsilon}(x_{0})-\delta<\varphi(x_{0})$ which
contradicts the potential comparison principle.
Now, we prove the other implication. Assume that $\varphi$ satisfies (17) in
the viscosity sense. Then from Proposition 5.1 we see that $\varphi$ is
$(\omega,m)$-subharmonic.
We consider two cases.
Case 1: $F$ does not depend on the second variable. We denote $f(x)=F(x,0)$
for $x\in X$.
We first treat the case when $f>0$ . Fix $\tilde{f}$ a smooth function such
that $0<\tilde{f}\leq f.$ Then $\varphi$ satisfies
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}\geq\tilde{f}\omega^{n}$
in the viscosity sense.
Fix $\alpha=\alpha_{1}\wedge...\wedge\alpha_{m-1},$ where $\alpha_{i}$ are
smooth $(\omega,m)$-positive closed (1,1)-forms and set
$\alpha_{j}^{m}\wedge\omega^{n-m}=h_{j}\omega^{n},\ \ j=1...m-1.$
From Gårding’s inequality we see that
(18) $(\omega+dd^{c}\varphi)\wedge\alpha\wedge\omega^{n-m}\geq
h_{1}^{1/m}...h_{m-1}^{1/m}\tilde{f}^{1/m}\omega^{n},$
in the viscosity sense. As in the proof of Proposition 5.1 it also holds in
the potential sense. Let $\varphi_{\epsilon}$ be the smooth regularization of
$\varphi$ constructed in [29]. We claim that
$(\omega+dd^{c}\varphi_{\epsilon})\wedge\alpha\wedge\omega^{n-m}\geq
h_{1}^{1/m}...h_{m-1}^{1/m}(\tilde{f}^{1/m})_{\epsilon}\omega^{n},$
in the usual sense pointwise on $X$. Indeed, recall the definition of
$\varphi_{\epsilon}$:
$\varphi_{\epsilon}(x)=\int_{K}\mathcal{L}^{*}_{g}\varphi(x)\chi_{\epsilon}(g)dg,$
where by $\mathcal{L}_{g}$ we denote the left translation by $g$, i.e
$\mathcal{L}_{g}(x)=g.x,\ \forall x\in X.$ We compute
$\displaystyle(\omega+dd^{c}\varphi_{\epsilon})\wedge\alpha\wedge\omega^{n-m}$
$\displaystyle=$
$\displaystyle\int_{K}\mathcal{L}^{*}_{g}\Big{(}(\omega+dd^{c}\varphi)\wedge\mathcal{L}^{*}_{g^{-1}}\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle(\text{ By }\ (\ref{eq: Garding 1}))\ \ $ $\displaystyle\geq$
$\displaystyle\int_{K}\mathcal{L}^{*}_{g}\Big{(}\mathcal{L}^{*}_{g^{-1}}(h_{1}^{1/m}...h_{m-1}^{1/m})\tilde{f}^{1/m}\omega^{n}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle=$ $\displaystyle
h_{1}^{1/m}...h_{m-1}^{1/m}(\tilde{f}^{1/m})_{\epsilon}\omega^{n}.$
Thus, the claim is proved. By choosing
$\alpha_{j}=(\omega+dd^{c}\varphi_{\epsilon}),\ j=1,...,m-1$ it follows that
$(\omega+dd^{c}\varphi_{\epsilon})^{m}\wedge\omega^{n-m}\geq((\tilde{f}^{1/m})_{\epsilon})^{m}\omega^{n}.$
By letting $\epsilon\downarrow 0$ we get
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}\geq\tilde{f}\omega^{n},$
in the potential sense. Since $\tilde{f}$ was chosen arbitrarily, we deduce
that
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}\geq f\omega^{n},$
in the viscosity sense.
If $0\leq f$ is continuous we consider $\varphi_{t}:=(1-t)\varphi+t\psi$ where
$\psi$ is a smooth strictly $(\omega,m)$-subharmonic function and $0<t<1.$
Then for each fixed $t\in(0,1)$, $\varphi_{t}$ satisfies
(19) $(\omega+dd^{c}\varphi_{t})^{m}\wedge\omega^{n-m}\geq f_{t}\omega^{n},$
in the viscosity sense with $f_{t}$ continuous and strictly positive:
$f_{t}=(1-t)^{m}f+t^{m}\frac{(\omega+dd^{c}\psi)^{m}\wedge\omega^{n-m}}{\omega^{n}}.$
We then can apply what we have done above to infer that $\varphi_{t}$ verifies
(19) in the potential sense. It suffices now to let $t\downarrow 0.$
Case 2: $F$ depends on the second variable. Since $\varphi$ is continuous, the
function $f:X\rightarrow\mathbb{R}$, $f(x)=F(x,\varphi(x))$ is continuous. We
can apply Case 1 to complete the proof. ∎
### 5.1. Global Comparison Principle
Let $u,v$ be bounded viscosity subsolution and supersolution of (15).
Construct a distance $d$ on $K$ such that $d^{2}:K\times
K\rightarrow\mathbb{R}^{+}$ is smooth. Consider the sup-convolution and inf-
convolution as follows
(20) $u^{\epsilon}(x):=\sup\Big{\\{}u(g.x)-\frac{1}{\epsilon^{2}}d^{2}(g,e)\
/\ \ g\in K\Big{\\}},$
and
(21) $v_{\epsilon}(x):=\inf\Big{\\{}v(g.x)+\frac{1}{\epsilon^{2}}d^{2}(g,e)\
/\ \ g\in K\Big{\\}}.$
###### Lemma 5.3.
Fix $x_{0}\in X$ and consider local coordinates $z:\Omega\to B(0,2),$ where
$\Omega$ is a small open neighborhood of $x_{0}$ and $B$ is the ball of radius
$2$ in $\mathbb{C}^{n}.$ Then $u^{\epsilon},v_{\epsilon}$ read in this local
chart as semiconvex and semiconcave functions. In particular, they are
punctually second order differentiable almost everywhere in $B(0,1).$
###### Proof.
We only need to prove the result for $u^{\epsilon}$ since for $v_{\epsilon}$
it follows similarly. Consider a smooth section $s:\Omega\rightarrow K$ such
that $\pi\circ s(x)=x,\forall x\in\Omega,$ where $\pi$ is the projection of
$K$ onto $X.$
For simplicity we identify a point in $\Omega$ with its image in $B(0,2).$
Put
$\rho(x)=u^{\epsilon}+C|x|^{2},$
where $C>0$ is a big constant to be specified later.
We claim that for any $x\in B(0,1)$ there exists $\delta>0$ such that
$\rho(x+h)+\rho(x-h)\geq 2\rho(x),\ \forall h\in\mathbb{C}^{n},|h|\leq\delta.$
It is classical that this property implies the convexity of $\rho.$ Let us
prove the claim. Let $x_{0}\in B(0,1)$ and $y_{0}=g_{0}.x_{0}$ be such that
(22) $u^{\epsilon}(x_{0})=u(y_{0})-\frac{1}{\epsilon^{2}}d^{2}(g_{0},e).$
By considering $\epsilon>0$ small enough we can assume that $y_{0}\in
B(0,3/2).$ For $h\in\mathbb{C}^{n}$ small enough such that $x_{0}+h,x_{0}-h\in
B(0,1)$, set
$\theta(h)=g_{0}.s(x_{0}).s(x_{0}+h)^{-1}.$
Then it is easy to see that $\theta(h).(x_{0}+h)=y_{0}.$ By definition of
$u^{\epsilon}$ we thus get
(23) $u^{\epsilon}(x_{0}+h)\geq
u(y_{0})-\frac{1}{\epsilon^{2}}d^{2}(\theta(h),e),$
and
(24) $u^{\epsilon}(x_{0}-h)\geq
u(y_{0})-\frac{1}{\epsilon^{2}}d^{2}(\theta(-h),e).$
From (22), (23) and (24) we obtain
$u^{\epsilon}(x_{0}+h)+u^{\epsilon}(x_{0}-h)-2u^{\epsilon}(x_{0})\geq-\frac{1}{\epsilon^{2}}\Big{(}d^{2}(\theta(h),e)+d^{2}(\theta(-h),e)-2d^{2}(\theta(0),e)\Big{)}.$
Since $s$ is smooth and $K$ is compact we can choose $C>0$ big enough (does
not depend on $x_{0}$) such that
$u^{\epsilon}(x_{0}+h)+u^{\epsilon}(x_{0}-h)-2u^{\epsilon}(x_{0})\geq-2C|h|^{2},$
for $h\in\mathbb{C}^{n}$ small enough. This proves the claim. The last
statement follows from Alexandroff-Buselman-Feller’s theorem (Theorem 4.3). ∎
###### Lemma 5.4.
$u^{\epsilon}$ is a viscosity subsolution of
(25) $-(\omega+dd^{c}u)^{m}\wedge\omega^{n-m}+F_{\epsilon}(x,u)\omega^{n}=0,$
where
$F_{\epsilon}(x,t):=\inf\Big{\\{}F(g.x,t)\ /\ g\in
K,d(g,e)\leq\sqrt{\text{osc}(u)}\epsilon\Big{\\}}.$
Similarly, $v_{\epsilon}$ is a viscosity supersolution of
(26) $-(\omega+dd^{c}u)^{m}\wedge\omega^{n-m}+F^{\epsilon}(x,u)\omega^{n}=0,$
where
$F^{\epsilon}(x,t):=\sup\Big{\\{}F(g.x,t)\ /\ g\in
K,d(g,e)\leq\sqrt{\text{osc}(v)}\epsilon\Big{\\}}.$
###### Proof.
We only need to prove the first assertion since the second one follows
similarly. Let $q$ be a function of class $\mathcal{C}^{2}$ in a neighborhood
of $x_{0}\in X$ that touches $u^{\epsilon}$ from above at $x_{0}.$ Let
$g_{0}\in K$ be such that
$u^{\epsilon}(x_{0})=u(g_{0}.x_{0})-\frac{1}{\epsilon^{2}}d^{2}(g_{0},e).$
Consider the function $Q$ defined by
$Q(x):=q(g_{0}^{-1}x)+\frac{1}{\epsilon^{2}}d^{2}(g_{0},e).$
Then $Q$ touches $u$ from above at $g_{0}.x_{0}.$ Since $u$ is a subsolution
of (15), we have
$(\omega+dd^{c}Q)^{m}\wedge\omega^{n-m}\geq F(x,Q)\omega^{n},\ \text{at}\
g_{0}.x_{0}.$
Since $\mathcal{L}^{*}_{g_{0}}\omega=\omega$ we get
$(\omega+dd^{c}q)^{m}\wedge\omega^{n-m}\geq
F(g_{0}.x_{0},q(x_{0}))\omega^{n},\ \text{at}\ x_{0}.$
From the definition of $u^{\epsilon}$ we know that
$u^{\epsilon}(x_{0})=u(g_{0}.x_{0})-\frac{1}{\epsilon^{2}}d^{2}(g_{0},e)\geq
u(x_{0}).$ Thus $d(g_{0},e)\leq\epsilon\sqrt{\text{osc}(u)}$ and the result
follows. ∎
Now, we prove a viscosity comparison principle on homogeneous manifolds. The
fact that the metric $\omega$ is invariant under group actions allows us to
follow the proof of Theorem 4.4 in this global context.
###### Theorem 5.5.
Assume that $u,v$ are bounded viscosity subsolution and supersolution of
$-(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}+F(x,\varphi)\omega^{n}=0,$
where $0\leq F(x,t)$ is a continuous function which is increasing in the
second variable. Then we have $u\leq v$ on $X.$
###### Proof.
We consider the sup-convolution and inf-convolution of $u,v$ as in (20) and
(21). These functions read in local coordinates as semiconvex and semiconcave
functions which are punctually second order differentiable almost everywhere.
For each $\epsilon>0$ let $x_{\epsilon}$ be a maximum point of
$u^{\epsilon}-v_{\epsilon}$ on $X.$
We first treat the case when $u^{\epsilon},v_{\epsilon}$ are punctually second
order differentiable at $x_{\epsilon}.$ In this case, by the classical maximum
principle we have
$dd^{c}u^{\epsilon}\leq dd^{c}v_{\epsilon}\ \text{at}\ x_{\epsilon}.$
The form $(\omega+dd^{c}u^{\epsilon})$ is $(\omega,m)$-positive at
$x_{\epsilon}$. Thus,
$(\omega+dd^{c}u^{\epsilon})^{m}\wedge\omega^{n-m}\leq(\omega+dd^{c}v_{\epsilon})^{m}\wedge\omega^{n-m}\
\text{at}\ x_{\epsilon},$
and hence Lemma 5.4 yields
(27) $F_{\epsilon}(x_{\epsilon},u^{\epsilon})\leq
F^{\epsilon}(x_{\epsilon},v_{\epsilon}).$
We can assume that $x_{\epsilon}\to x_{0}\in X.$ By extracting a subsequence
(twice), there exists a sequence $\epsilon_{j}\downarrow 0$ such that
$F_{\epsilon_{j}}(x_{\epsilon_{j}},u^{\epsilon_{j}}(x_{\epsilon_{j}}))\
\text{and}\
F^{\epsilon_{j}}(x_{\epsilon_{j}},v_{\epsilon_{j}}(x_{\epsilon_{j}}))$
converge when $j\to+\infty.$ We thus deduce from (27) that
$F(x_{0},\liminf_{j}u^{\epsilon_{j}}(x_{\epsilon_{j}}))\leq
F(x_{0},\limsup_{j}v_{\epsilon_{j}}(x_{\epsilon_{j}})).$
Since $F$ is increasing in the second variable the latter implies that
(28) $\liminf u^{\epsilon_{j}}(x_{\epsilon_{j}})\leq\limsup
v_{\epsilon_{j}}(x_{\epsilon_{j}}).$
Since $u^{\epsilon}\downarrow u$ and $v_{\epsilon}\uparrow v$ we have
$\sup_{X}(u-v)\leq\sup_{X}(u^{\epsilon_{j}}-v_{\epsilon_{j}})=u^{\epsilon_{j}}(x_{\epsilon_{j}})-v_{\epsilon_{j}}(x_{\epsilon_{j}}).$
Then (28) implies that $\sup_{X}(u-v)\leq 0.$
Now, if $u^{\epsilon},v_{\epsilon}$ are not punctually second order
differentiable at $x_{\epsilon}$ for fixed $\epsilon,$ we proceed as in [12]
to prove that (27) still holds. Consider a local holomorphic chart centered at
$x_{\epsilon}.$ For simplicity we identify a point near $x_{\epsilon}$ with
its image in $\mathbb{C}^{n}.$ For each $k\in\mathbb{N}^{*}$, the semiconvex
function $u^{\epsilon}-v_{\epsilon}-\frac{1}{2k}\|x-x_{\epsilon}\|^{2}$
attains its strict maximum at $x_{\epsilon}.$ By Jensen’s lemma ([21]; see
also [7, Lemma A.3, page 60]), there exist sequences $(p_{k}),(y_{k})$
converging to $0$ and $x_{\epsilon}$ respectively such that the functions
$u^{\epsilon},v_{\epsilon}$ are punctually second order differentiable at
$y_{k}$ and the function
$u^{\epsilon}-v_{\epsilon}-\frac{1}{2k}\|x-x_{\epsilon}\|^{2}-\langle
p_{k},x\rangle$
attains its local maximum at $y_{k}.$ We thus get
$dd^{c}u^{\epsilon}\leq dd^{c}v_{\epsilon}+O(1/k)\omega\ \text{at}\ y_{k}.$
Since $v_{\epsilon}$ is semi-concave, and $u^{\epsilon}$ is
$(\omega,m)$-subharmonic we get
$(\omega+dd^{c}u^{\epsilon})^{m}\wedge\omega^{n-m}\leq(\omega+dd^{c}v_{\epsilon})^{m}\wedge\omega^{n-m}+O(1/k)\omega^{n}\
\text{at}\ y_{k}.$
This together with (25) and (26) yield
$F_{\epsilon}(y_{k},u^{\epsilon}(y_{k}))\leq
F^{\epsilon}(y_{k},v_{\epsilon}(y_{k}))+O(1/k).$
Now, let $k\to+\infty$ we obtain (27) which completes the proof. ∎
## 6\. Proof of the main results
### 6.1. Proof of Theorem A
Let $\mathcal{F}$ denote the family of all subsolutions $w$ of (6) such that
$u\leq w\leq v.$ It is not empty thanks to the local comparison principle. We
set
$\varphi:=\sup\\{w:w\in\mathcal{F}\\}.$
By Choquet’s lemma $\varphi^{*}=(\limsup w_{j})^{*}$ where $w_{j}$ is a
sequence in $\mathcal{F}.$ It follows from Lemma 3.6 that $\varphi^{*}$ is a
subsolution of (6).
We claim that $\varphi_{*}$ is a supersolution of (6). Indeed, assume that
$\varphi_{*}$ is not a supersolution of (6). Then there exist $x_{0}\in\Omega$
and $q\in\mathcal{C}^{2}(\\{x_{0}\\})$ such that $q$ touches $\varphi_{*}$
from below at $x_{0}$ but
$H_{m}(q)(x_{0})>F(x_{0},q(x_{0}))\beta^{n}.$
By the continuity of $F$, we can find $r>0$ small enough such that
$q\leq\varphi_{*}$ in $B(x_{0},r)$ and
$H_{m}(q)(x)>F(x,q(x))\beta^{n},\ \ \forall x\in B=B(x_{0},r).$
We then choose $0<\epsilon$ small enough and $0<\delta<<\epsilon$ so that the
function $Q=Q_{\epsilon,\delta}:=q+\delta-\epsilon|x-x_{0}|^{2}$ satisfies
$H_{m}(Q)(x)>F(x,Q(x))\beta^{n},\ \forall x\in B.$
Define $\phi$ to be $\varphi$ outside $B$ and $\phi=\max(\varphi,Q)$ in $B.$
Since $Q<\varphi$ near $\partial B$, we see that $\phi$ is upper semi
continuous and it is a subsolution of (6) in $\Omega.$
Let $(x_{j})$ be a sequence in $B$ converging to $x_{0}$ such that
$\varphi(x_{j})\to\varphi_{*}(x_{0}).$ Then $Q(x_{j})-\varphi(x_{j})\to
Q(x_{0})-\varphi_{*}(x_{0})=\delta>0.$ Thus $\phi\not\equiv\varphi,$ which
contradicts the maximality of $\varphi.$
From the above steps we know that $\varphi_{*}$ is a supersolution and
$\varphi^{*}$ is a subsolution. We also have
$g=u_{*}\leq\varphi_{*}\leq\varphi^{*}\leq v^{*}=g$ on $\partial\Omega.$ Thus
by the viscosity comparison principle $\varphi=\varphi_{*}=\varphi^{*}$ is a
continuous viscosity solution of (1) with boundary value $g.$
It remains to prove that $\varphi$ is also a potential solution of (1). From
Theorem 3.11 we know that
$H_{m}(\varphi)\geq F(x,\varphi)\beta^{n}$
in the potential sense. Let $B=B(x_{0},r)\subset\Omega$ is a small ball in
$\Omega.$ Thanks to Dinew and Kolodziej [9, Theorem 2.10] we can solve the
Dirichlet problem to find $\psi\in\mathcal{P}_{m}(B)\cap\mathcal{C}(\bar{B})$
with boundary value $\varphi$ such that
$H_{m}(\psi)=F(x,\varphi)\beta^{n},\ \text{in}\ B.$
By the potential comparison principle we have $\varphi\leq\psi$ in $\bar{B}$.
Define $\tilde{\psi}$ to be $\psi$ in $B$ and $\varphi$ in $\Omega\setminus
B$. Set $G(x)=F(x,\varphi(x)),x\in\Omega.$ It is easy to see that
$\tilde{\psi}$ is a viscosity solution of
$-(dd^{c}u)^{m}\wedge\beta^{n-m}+G\beta^{n}=0.$
By the viscosity comparison principle we deduce that $\tilde{\psi}\leq\varphi$
in $\Omega$ which implies that $\varphi=\psi$ in $B.$ The proof is thus
complete.
### 6.2. Proof of Theorem B
Let $\varphi$ be the unique viscosity solution obtained from Theorem A. Since
$u,v$ are $\gamma$-Hölder continuous in $\bar{\Omega}$ and $F$ satisfies (4),
we can find a constant $C>0$ such that
$\sup_{x,y\in\bar{\Omega}}\Big{(}|u(x)-u(y)|+|v(x)-v(y)|\Big{)}\leq
C|x-y|^{\gamma},$
and
$\sup_{|t|\leq M}\sup_{x,y\in\bar{\Omega}}|F^{1/m}(x,t)-F^{1/m}(y,t)|\leq
C|x-y|^{\gamma},$
where $M>0$ is such that $|\varphi|\leq M,$ on $\bar{\Omega}.$
Fix $R>0$ such that $\Omega\subset B(0,R).$ Define
$\psi:\bar{\Omega}\rightarrow\mathbb{R}$ by
$\psi(x):=\sup_{y\in\bar{\Omega}}\Big{\\{}\varphi(y)+C|x-y|^{\gamma}(|x|^{2}-R^{2}-1)\Big{\\}}.$
Step 1: Prove that $\psi$ is $\gamma$-Hölder continuous. Fix
$x_{1},x_{2}\in\bar{\Omega}$, and $y_{1},y_{2}$ corresponding maximum points
in $\bar{\Omega}$ as in the definition of $\psi.$ We obtain
$\psi(x_{1})-\psi(x_{2})\geq
C|x_{1}-y_{2}|^{\gamma}(|x_{1}|^{2}-R^{2}-1)-C|x_{2}-y_{2}|^{\gamma}(|x_{2}|^{2}-R^{2}-1)\\\
=C(|x_{1}|^{2}-R^{2}-1)(|x_{1}-y_{2}|^{\gamma}-|x_{2}-y_{2}|^{\gamma})+C|x_{2}-y_{2}|^{\gamma}(|x_{1}|^{2}-|x_{2}|^{2})\\\
\geq
C(|x_{1}|^{2}-R^{2}-1)|x_{1}-x_{2}|^{\gamma}+C|x_{2}-y_{2}|^{\gamma}(|x_{1}|^{2}-|x_{2}|^{2})\geq-C^{\prime}|x_{1}-x_{2}|^{\gamma},$
where $C^{\prime}>0$ depends only on $C,R.$ Similarly, we have
$\psi(x_{1})-\psi(x_{2})\leq C^{\prime}|x_{1}-x_{2}|^{\gamma}.$
The above inequalities show that $\psi$ is $\gamma$-Hölder continuous in
$\bar{\Omega}.$
Step 2: Prove that $\psi$ is a subsolution of (6). Let $x_{0}\in\Omega$ and
$q\in\mathcal{C}^{2}(\\{x_{0}\\})$ which touches $\psi$ from above at $x_{0}.$
Let $y_{0}\in\bar{\Omega}$ be such that
$\psi(x_{0})=\varphi(y_{0})+C|x_{0}-y_{0}|^{\gamma}(|x_{0}|^{2}-R^{2}-1).$
If $y_{0}\in\partial\Omega$ then $\varphi(y_{0})=u(y_{0}),$ hence
$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle
C|x_{0}-y_{0}|^{\gamma}(|x_{0}|^{2}-R^{2})=\psi(x_{0})-\varphi(y_{0})+C|x_{0}-y_{0}|^{\gamma}$
$\displaystyle\geq$ $\displaystyle
u(x_{0})-u(y_{0})+C|x_{0}-y_{0}|^{\gamma}\geq 0.$
We thus get $\varphi(x_{0})=\psi(x_{0})$ and the result follows since
$\varphi$ is a subsolution. Let us treat the case $y_{0}\in\Omega.$ The
function $Q$, defined around $y_{0}$ by
$Q(x):=q(x+x_{0}-y_{0})-C|x_{0}-y_{0}|^{\gamma}\Big{(}|x+x_{0}-y_{0}|^{2}-R^{2}-1\Big{)},$
touches $\varphi$ from above at $y_{0}.$ Since $\varphi$ is a subsolution of
(6), we have
$\widetilde{S}_{m}^{1/m}\Big{(}dd^{c}Q(y_{0})\Big{)}\geq
F^{1/m}(y_{0},Q(y_{0})).$
By the concavity of $\widetilde{S}_{m}^{1/m}$ we get
$\displaystyle\widetilde{S}_{m}^{1/m}\Big{(}dd^{c}q(x_{0})\Big{)}$
$\displaystyle\geq$ $\displaystyle
F^{1/m}(y_{0},Q(y_{0}))+C|x_{0}-y_{0}|^{\gamma}$ $\displaystyle=$
$\displaystyle F^{1/m}(y_{0},\varphi(x_{0}))+C|x_{0}-y_{0}|^{\gamma}$
$\displaystyle\geq$ $\displaystyle F^{1/m}(x_{0},\varphi(x_{0})),$
which implies that $\psi$ is a subsolution of (6).
It is easy to see that $\varphi\leq\psi$ and for any
$x\in\partial\Omega,y\in\bar{\Omega}$, we have
$\varphi(y)-C|x-y|^{\gamma}\leq v(y)-C|x-y|^{\gamma}\leq v(x)=g(x).$
This implies $\psi=g$ on $\partial\Omega.$ Hence, since $\varphi$ is maximal
we obtain $\varphi=\psi$ which, in turn, shows that $\varphi$ is
$\gamma$-Hölder continuous.
### 6.3. Proof of Corollary C
Let $h$ be the harmonic function with boundary value $g;$ it is a continuous
supersolution of (6). It follows from [2] that there exists a continuous psh
function $u$ with boundary value $g.$ Then for $A>>1$, the function $u+A\rho$,
where $\rho$ is a defining function of $\Omega,$ is a subsolution. Thus, by
Theorem A there exists a continuous viscosity solution.
Now, assume that $g$ is $(2\gamma)$-Hölder continuous in $\bar{\Omega}.$ Then
we can choose $u$ to be $\gamma$-Hölder continuous in $\bar{\Omega}$ thanks to
[2]. The same thing holds for $h.$ It suffices to apply Theorem B. The proof
is thus complete.
###### Remark 6.1.
In Corollary C it is natural to consider a strongly $m$-pseudoconvex domain
(i.e. the defining function is strongly $m$-subharmonic). The existence of
continuous subsolution and supersolution is obvious which yields the existence
of viscosity solution. However, the Hölder continuity is delicate.
### 6.4. Proof of Theorem D
It follows from (5) that $u\equiv t_{0}$ is a subsolution and $v\equiv t_{1}$
is a supersolution of (15). The global comparison principle (Theorem 5.5)
allows us to repeat the proof of Theorem A to prove Theorem D.
In the following, we give an example of compact Hermitian homogeneous manifold
satisfying our conditions (H1), (H2), (H3) which is not Kähler. It is
communicated to us by Karl Oeljeklaus to whom we are indebted.
###### Example 6.2.
Consider $G=SL(3,\mathbb{C})$, $K=SU(3)$ and
$H=\left\\{\left(\begin{array}[]{ccc}e^{w}&z_{1}&z_{2}\\\ 0&e^{iw}&z_{3}\\\
0&0&e^{-w-iw}\end{array}\right)\ \Big{/}\
w,z_{1},z_{2},z_{3}\in\mathbb{C}\right\\}.$
Then $G$ is a connected complex Lie group and $H$ is a closed complex
subgroup. The manifold $X=G/H$ is Hermitian. It is clear that $K$ acts freely
and transitively on $X.$ Taking any Hermitian metric and averaging it over the
Haar measure of $K$ we obtain a Hermitian metric $\omega$ verifying (H3). Now
we prove that $X$ is not Kähler. Since $K$ acts freely on $X$ we see that $X$
is simply connected.
Consider
$I=\left\\{\left(\begin{array}[]{ccc}\lambda_{1}&z_{1}&z_{2}\\\
0&\lambda_{2}&z_{3}\\\ 0&0&(\lambda_{1}.\lambda_{2})^{-1}\end{array}\right)\
\Big{/}\
z_{1},z_{2},z_{3}\in\mathbb{C};\lambda_{1},\lambda_{2}\in\mathbb{C}^{*}\right\\}.$
Then $H$ is a closed subgroup of $I$ and $Y=G/I$ is a rational-projective
manifold. If $X$ admits a Kähler metric then it follows from [4] (see also
[1]) that the Tits fibration
$\pi:G/H\rightarrow G/I$
is holomorphically trivial and its fiber $I/H$ is a complex compact torus.
This implies that $\pi_{1}(X)$ is non-trivial which is impossible since $X$ is
simply connected. Thus $X$ does not admit any Kähler metric.
Acknowledgement. I would like to express my deep gratitude to Ahmed Zeriahi
and Vincent Guedj for inspirational discussions and constant encouragements. I
also would like to thank Philippe Eyssidieux and Dan Popovici for valuable
discussions. I am very grateful to Nguyen Van Dong for finding an important
error in the previous proof of Lemma 5.3. I am indebted to Karl Oeljeklaus for
communicating an important example and references for it and also for very
useful discussions. Finally, I would like to thank the referee for useful
comments and corrections which improve the presentation of this paper.
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* [22] V. N. Kokarev, Mixed volume forms and a complex equation of Monge-Ampère type on Kähler manifolds of positive curvature, Izv. RAN. Ser. Mat. 74:3 (2010), 65-78.
* [23] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Translated from the Russian by P. L. Buzytsky, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. xiv+462 pp.
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* [25] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002), 1-49.
* [26] S. Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8 (2004),no. 1, 87-106.
* [27] P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Comm. Partial Differential Equations 8 (1983), no. 11, 1229-1276.
* [28] W. Littman, Generalized subharmonic functions: Monotonic approximations and an improved maximum principle, Ann. Scuola Norm. Sup. Pisa (3) 17 1963 207-222.
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* [30] N. C. Nguyen, Subsolution theorem for the complex Hessian equation, arXiv:1212.4627.
* [31] A. S. Sadullaev, B. I. Abdullaev Capacities and Hessians in a class of m-subharmonic functions, preprint arXiv:1201.6531v2 .
* [32] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164.
* [33] N. S. Trudinger, X.-J. Wang, Hessian measures II, Ann. of Math. 150 (1999), 579-604.
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LU Hoang Chinh
Université Paul Sabatier
Institut de Mathématiques de Toulouse
118 Route de Narbonne
31062 Toulouse
lu@math.univ-toulouse.fr.
|
arxiv-papers
| 2012-09-24T17:34:38 |
2024-09-04T02:49:35.487854
|
{
"license": "Public Domain",
"authors": "Lu Hoang Chinh",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1209.5343"
}
|
1209.5429
|
§ INTRODUCTION
The field of numerical optimization (see e.g., Nocedal and Wright, 1999)
is a research area with a considerable number of applications in engineering,
science, and business. Many mathematical problems involve finding the most
favorable configuration of a set of parameters that achieve an objective quantified
by a function. Numerical optimization entails the case where these parameters
can take continuous values, in contrast with combinatorial optimization,
which involves discrete variables. The mathematical formulation of a numerical
optimization problem is given by $\underset{\boldsymbol{x}}{\min\,}f(\boldsymbol{x})$,
where $\boldsymbol{x}\in\mathbb{R}^{n}$ is a real vector with $n\geq1$ components
and $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is the objective function
(also known as the fitness, loss or cost function).
In particular, we consider within numerical optimization a black-box
(or direct search) scenario where the function values of evaluated search
points are the only available information on $f$. The algorithms do
not assume any knowledge of the function $f$ regarding continuity,
the existence of derivatives, etc. A black-box optimization procedure
explores the search space by generating solutions, evaluating them, and
processing the results of this evaluation in order to generate new promising
solutions. In this context, the performance measure of the algorithms is
generally the number of function evaluations needed to reach a certain
value of $f$.
Algorithms that have been proposed to deal with this kind of optimization
problems can be classified in two groups according to the approach
followed for the generation of new solutions. On the one hand, deterministic
direct search algorithms, such as the Hooke-Jeeves [Hooke and Jeeves, 1961]
and Nelder-Mead [Nelder and Mead, 1965] methods, perform transformations
to one or more candidate solutions at each iteration. Given their
deterministic approach, these algorithms may have limited global search
capabilities and can get stuck in local optima, depending on an appropriate
selection of the initial solutions. On the other hand, randomized
optimization algorithms offer an alternative to ensure a proper global
exploration of the search space. Examples of these algorithms are
simulated annealing [Kirkpatrick et al., 1983], evolution strategies
(see e.g., Beyer and Schwefel, 2002), particle swarm optimization
[Kennedy and Eberhart, 1995], and differential evolution [Storn and Price, 1997].
In this paper, we focus on EDAs <cit.>,
which are stochastic black-box optimization algorithms characterized
by the explicit use of probabilistic models to explore the search
space. These algorithms combine ideas from genetic and evolutionary
computation, machine learning, and statistics into an optimization
procedure. The search space is explored by iteratively estimating and
sampling from a probability distribution built from promising
solutions, a characteristic that differentiates EDAs among other
randomized optimization algorithms. One key advantage of EDAs is that
the search distribution may encode probabilistic dependences between
the problem variables that represent structural properties of the
objective function, performing a more effective optimization by using
this information.
Due to its tractable properties, the normal distribution has been
commonly used to model the search distributions in EDAs for real-valued
optimization problems [Bosman and Thierens, 2006, Kern et al., 2003].
However, once a multivariate normal distribution is assumed, all the
margins are modeled with the normal density and only linear correlation
between the variables can be considered. These characteristics could
lead to the construction of incorrect models of the search space.
For instance, the multivariate normal distribution cannot represent
properly the fitness landscape of multimodal objective functions.
Also, the use of normal margins imposes limitations on the performance
when the sample of the initial solutions is generated asymmetrically
with respect to the optimum of the function (see 78
for an illustrative example of this situation).
Copula functions (see e.g.,
Joe, 1997, Nelsen, 2006)
offer a valuable alternative to tackle these problems. By means of
Sklar's Theorem [Sklar, 1959], any multivariate
distribution can be decomposed into the (possibly different)
univariate marginal distributions and a multivariate copula that
determines the dependence structure between the variables. EDAs based
on copulas inherit these properties, and consequently, can build more
flexible search distributions that may overcome the limitations of a
multivariate normal probabilistic model. The advantages of using
copula-based search distributions in EDAs extend further with the
possibility of factorizing the multivariate copula with the copula
decomposition in terms of lower-dimensional copulas. Multivariate
dependence models based on copula factorizations, such as nested
Archimedean copulas [Joe, 1997] and vines
[Joe, 1996, Bedford and Cooke, 2001, Aas et al., 2009],
provide great advantages in high dimensions. Particularly in the case
of vines, a more appropriate representation of multivariate
distributions having pairs of variables with different types of
dependence is possible.
Although various EDAs based on copulas have been proposed in the
literature, as far as we know there are no publicly available
implementations of these algorithms (see
Santana, 2011 for a comprehensive review of
EDA software). Aiming to fill this gap, the copulaedas package
[Gonzalez-Fernandez and Soto, 2014] for the R language and
environment for statistical computing [R Core Team, 2014] has been
published on the Comprehensive R Archive Network at
<http://CRAN.R-project.org/package=copulaedas>. This package
provides a modular platform where EDAs based on copulas can be
implemented and studied. It contains various EDAs based on copulas, a
group of well-known benchmark problems, and utility functions to study
EDAs. One of the most remarkable features of the framework offered by
copulaedas is that the components of the EDAs are decoupled
into separated generic functions, which promotes code factorization
and facilitates the implementation of new EDAs that can be easily
integrated into the framework.
The remainder of this paper provides a presentation of the copulaedas
package organized as follows. Section <ref> continues
with the necessary background on EDAs based on copulas. Next, the
details of the implementation of copulaedas are described in
Section <ref>, followed by an illustration of
the use of the package through examples in Section <ref>.
Finally, concluding remarks are given in Section <ref>.
§ ESTIMATION OF DISTRIBUTION ALGORITHMS BASED ON COPULAS
This section begins by describing the general procedure of an EDA,
according to the implementation in copulaedas. Then, we present
an overview of the EDAs based on copulas proposed in the literature
with emphasis on the algorithms implemented in the package.
§.§ General procedure of an EDA
The procedure of an EDA is built around the concept of performing
the repeated refinement of a probabilistic model that represents the
best solutions of the optimization problem. A typical EDA starts with
the generation of a population of initial solutions sampled from the
uniform distribution over the admissible search space of the problem.
This population is ranked according to the value of the objective
function and a subpopulation with the best solutions is selected.
The algorithm then constructs a probabilistic model to represent the
solutions in the selected population and new offspring are generated
by sampling the distribution encoded in the model. This process is
repeated until some termination criterion is satisfied (e.g., when a
sufficiently good solution is found) and each iteration of this procedure
is called a generation of the EDA. Therefore, the feedback for the
refinement of the probabilistic model comes from the best solutions
sampled from an earlier probabilistic model.
Let us illustrate the basic EDA procedure with a concrete example.
Figure <ref> shows the steps performed to minimize
the two-dimensional objective function $f(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}$
using a simple continuous EDA that assumes independence between the
problem variables. Specifically, we aim to find the global optimum
of the function $f(0,0)=0$ with a precision of two decimal places.
The algorithm starts by generating an initial population of 30
candidate solutions from a continuous uniform distribution in
$[-10,10]^{2}$. Out of this initial sampling, the best solution found
so far is $f(-2.20,-0.01)=4.85$. Next, the initial population is
ranked according to their evaluation in $f(x_{1},x_{2})$, and the best
30% of the solutions is selected to estimate the probabilistic
model. This EDA factorizes the joint probability density function
(PDF) of the best solutions as
which describes mutual independence, and where $\phi_{1}$ denotes the
univariate normal PDF of $x_{1}$ with mean $\mu_{1}$ and variance
$\sigma_{1}^{2}$, and $\phi_{1}$ denotes the univariate normal PDF of
$x_{2}$ with mean $\mu_{1}$ and variance $\sigma_{2}^{2}$. In the
first generation, the parameters of the probabilistic model are
$\mu_{1}=-0.04$, $\sigma_{1}=3.27$, $\mu_{2}=-0.66$ and
$\sigma_{2}=3.81$. The second generation starts with the simulation of
a new population from the estimated probabilistic model. Afterwards,
the same selection procedure is repeated and the resulting selected
population is used to learn a new probabilistic model. These steps are
then repeated for a third generation.
Notice how in the first three generations the refinement of the probabilistic
model that represents the best solutions is evidenced in the reduction
of the variance of the marginal distributions towards a mean value
around zero. Also, the convergence of the algorithm is reflected in
the reduction of the value of the objective function from one generation
to another. Ultimately, the simulation of the probabilistic model
estimated at the third generation produces $f(-0.02,-0.07)=0.00$,
which satisfies our requirements and the algorithm terminates.
Steps performed to minimize the function $f(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}$
using a continuous EDA that assumes independence between the variables.
The search distribution models each variable with the normal distribution
and, since mutual independence is assumed, the joint PDF is factorized
as $\phi_{1,2}(x_{1},x_{2})=\phi_{1}(x_{1},\mu_{1},\sigma_{1}^{2})\phi_{2}(x_{2},\mu_{2},\sigma_{2}^{2})$.
The simulation of the probabilistic model estimated at the third generation
produces a solution $f(-0.02,-0.07)=0.00$ that approximates the global
optimum of the function with the required precision.
In practice, EDAs include other steps in addition to the ones illustrated
in the previous basic example. The general procedure of an EDA implemented
in copulaedas is outlined in Algorithm <ref>.
In the following, we provide a description of the purpose of the main
steps of the algorithm, which are highlighted in italics in the pseudocode:
$i \leftarrow 1$
if $i = 1$ then
Generate an initial population $P_1$
using a seeding method.
Evaluate the solutions in the population $P_1$.
If required, apply a local optimization method
to the population $P_1$.
Select a population $P_i^{Selected}$ from $P_{i-1}$
according to a selection method.
Learn a probabilistic model $M_i$ from $P_i^{Selected}$
using a learning method.
Sample a new population $P_i^{Sampled}$ from $M_i$
using a sampling method.
Evaluate the solutions in the population $P_i^{Sampled}$.
If required, apply a local optimization method
to the population $P_i^{Sampled}$.
Create the population $P_i$ from $P_{i-1}$ and
$P_i^{Sampled}$ using a replacement method.
end if
If required, report progress information using a
reporting method.
$i \leftarrow i + 1$
until A criterion of the termination method is met.
Pseudocode of an EDA.
* The first step is the generation of an initial population of solutions
following a seeding method, which is usually random, but it
can use a particular heuristic when a priori information about the
characteristics of the problem is available.
* The results of global optimization algorithms such as EDAs can often
be improved if combined with local optimization methods that
look for better solutions in the neighborhood of each candidate solution.
Local optimization methods can also be used to implement repairing
methods for constrained problems where the simulated solutions may
be unfeasible and a strategy to repair these solutions is available.
* A selection method is used to determine the most promising
solutions of the population. An example selection method is truncation
selection, which creates the selected population with a percentage
of the best solutions of the current population.
* The estimation and simulation of the search distribution are the essential
steps of an EDA. These steps are implemented by learning and
sampling methods, which are tightly related. Learning methods estimate
the structure and parameters of the probabilistic model used by the algorithm
from the selected population, while sampling methods are used to generate
new solutions from the estimated probabilistic model.
* A replacement method is used to incorporate a new group of
solutions into the current population. For example, a replacement
strategy is to substitute the current population with the newly sampled
population. Other replacement strategies retain the best solutions
found so far or try to maintain the diversity of solutions.
* Reporting methods provide progress information during the execution
of the EDA. Relevant progress information can be the number of evaluations
of the objective function and the best solution found so far.
* A termination method determines when the algorithm stops according
to certain criteria; for example, when a fixed number of function
evaluations are realized or a certain value of the objective function
is reached.
Although it was possible to locate with the required precision the
optimum of the simple function presented in this section, it is not
always possible to perform a successful search by considering only the
marginal information. As we show later in this paper, the assumption
of independence between the variables constitutes a strong limitation
that may compromise the convergence of an EDA. The use of information
about the relationships between the variables allows searching efficiently
for the best solutions and it constitutes one of the main advantages of EDAs.
Among the algorithms that consider dependences between the variables,
we are particularly interested in EDAs whose learning and sampling
steps involve probabilistic models based on copulas. The next section
provides an overview of such algorithms.
§.§ Overview of EDAs based on copulas
To the best of our knowledge, the technical report [Soto et al., 2007]
and the theses [Arderí, 2007, Barba-Moreno, 2007]
constitute the first attempts to incorporate copulas into EDAs. Since then,
a considerable number of EDAs based on copula theory have been proposed
in the literature and, as evidence of its increasing popularity, the
use of copulas in EDAs has been identified as an emerging approach
for the solution of real-valued optimization problems [Hauschild and Pelikan, 2011].
The learning step of copula-based EDAs consists of two tasks: the
estimation of the marginal distributions and the estimation of the
probabilistic dependence structure. In general, these tasks have been
performed by following one of the two-step estimation procedures known
in the copula literature as the IFM <cit.> and the
semiparametric estimation method
[Genest et al., 1995]. Firstly, the marginal
distributions are estimated and the selected population is transformed
into uniformly distributed variables in $(0,1)$ by means of the
evaluation of each marginal cumulative distribution
function. Secondly, the transformed population is used to estimate a
copula-based model of the dependence structure among the
variables. Usually, a particular parametric distribution (e.g., normal
or beta) is assumed for each margin and its parameters are estimated
by maximum likelihood (see e.g.,
Soto et al., 2007, Salinas-Gutiérrez et al., 2009).
In other cases, empirical marginal distributions or kernel density
estimation have been used (see e.g.,
Soto et al., 2007, Gao, 2009, Cuesta-Infante et al., 2010).
The simulation step typically starts with the generation of a
population of uniformly distributed variables in $(0,1)$ with the
dependence structure described by the copula-based model that was
estimated in the learning step. Finally, this uniform population is
transformed to the domain of the variables through the evaluation of
the inverse of each marginal cumulative distribution function.
According to the copula model being used, EDAs based on copulas can
be classified as EDAs based on either multivariate or factorized copulas.
In the rest of this section we give an overall description of representative
algorithms belonging to each group that have been proposed in the
§.§.§ EDAs based on multivariate copulas
The research on EDAs based on multivariate copulas has focused on the
use of multivariate elliptical copulas
[Abdous et al., 2005, Fang et al., 2002] and
Archimedean copulas
[Joe, 1997, McNeil and Nešlehová, 2009]. The
algorithms described in
[Soto et al., 2007, Arderí, 2007] and
[Barba-Moreno, 2007] are both based on the
multivariate normal copula and theoretically similar, but they present
differences in the estimation of the marginal distributions and the
use of techniques such as variance scaling. Wang et al., 2009
present an EDA based on the bivariate normal copula and, since only
normal marginal distributions are used, the proposed algorithm is
equivalent to EMNA <cit.>. On the other hand,
the algorithms presented in [Wang et al., 2009] and
[Gao, 2009] use exchangeable Archimedean
copulas. Wang et al., 2009 propose two algorithms
that use Clayton and Ali-Mikhail-Haq copulas with fixed parameters,
while Gao, 2009 does not state which
particular members of the family of Archimedean copulas are used.
Two EDAs based on multivariate copulas are implemented in
copulaedas, one is based on the product or independence copula
and the other on the normal copula. The first algorithm is UMDA
(univariate marginal distribution algorithm) for continuous variables
[Larrañaga et al., 1999, Larrañaga et al., 2000],
which can be integrated into the framework of copula-based EDAs
although originally it was not defined in terms of copulas. A
consequence of Sklar's Theorem is that random variables are
independent if and only if the underlying copula is the product
copula. Thus, UMDA can be described as an EDA that models the
dependence structure between the variables using a multivariate
product copula.
The second EDA based on a multivariate copula implemented in
copulaedas is GCEDA <cit.>.
This algorithm is based on the multivariate normal copula, which
allows the construction of multivariate distributions with normal
dependence structure and non-normal margins. The dependence structure
of the multivariate normal copula is determined by a positive-definite
correlation matrix. If the marginal distributions are not normal, the
correlation matrix is estimated through the inversion of the
non-parametric estimator of Kendall's tau for each pair of variables
(see e.g., Genest and Favre, 2007, Hult and Lindskog, 2002).
If the resulting matrix is not positive-definite, the transformation
proposed by Rousseeuw and Molenberghs, 1993 is applied.
GCEDA is equivalent to EMNA when all the marginal distributions are
§.§.§ EDAs based on copula factorizations
The use of multivariate copulas to model the dependence structure
between the variables offers a considerable number of advantages over
the use of the multivariate normal distribution; nevertheless, it
presents limitations. The number of tractable copulas available when
more than two variables are involved is limited, most available
copulas are just investigated in the bivariate case. In addition, the
multivariate elliptical copulas might not be appropriate when all
pairs of variables do not have the same dependence structure. Another
limitation is that some multivariate extensions, such as exchangeable
Archimedean copulas or the multivariate $t$ copula, have only one
parameter to describe certain aspects of the overall dependence. This
characteristic can be a serious limitation when the type and strength
of the dependence is not the same for all pairs of variables. One
alternative to these limitations is to use copula factorizations that
build high-dimensional probabilistic models by using lower-dimensional
copulas as building blocks. Several EDAs based on copula
factorizations, such as nested Archimedean copulas
[Joe, 1997] and vines
[Joe, 1996, Bedford and Cooke, 2001, Aas et al., 2009],
have been proposed in the literature.
The EDA introduced in [Salinas-Gutiérrez et al., 2009] is an
extension of MIMIC (mutual information maximization for input
clustering) for continuous domains
[Larrañaga et al., 1999, Larrañaga et al., 2000] that
uses bivariate copulas in a chain structure instead of bivariate
normal distributions. Two instances of this algorithm were presented,
one uses normal copulas and the other Frank copulas. In
Section <ref>, we illustrate the implementation of
this algorithm using copulaedas.
The exchangeable Archimedean copulas employed in
[Wang et al., 2009] and
[Gao, 2009] represent highly specialized
dependence structures
[Berg and Aas, 2007, McNeil, 2008].
Within the domain of Archimedean copulas, nested Archimedean copulas
provide a more flexible alternative to build multivariate copula
distributions. In particular, hierarchically nested Archimedean
copulas present one of the most flexible solutions among the different
nesting structures that have been studied (see e.g.,
Berg and Aas, 2007 for a review).
Building from these models, Ye et al., 2010 propose an EDA that
uses a representation of hierarchically nested Archimedean copulas
based on Lévy subordinators [Hering et al., 2010].
Cuesta-Infante et al., 2010 investigate the use
of bivariate empirical copulas and a multivariate extension of Archimedean
copulas. The EDA based on bivariate empirical copulas is completely
nonparametric: it employs empirical marginal distributions and a construction
based on bivariate empirical copulas to represent the dependence between
the variables. The marginal distributions and the bivariate empirical
copulas are defined through the linear interpolation of the sample
in the selected population. The EDA based on Archimedean copulas uses
a construction similar to a fully nested Archimedean copula and uses
copulas from one of the families Frank, Clayton or HRT (i.e., heavy
right tail copula or Clayton survival copula). The parameters of the
copulas are fixed to a constant value, i.e., not estimated from the
selected population. The marginal distributions are modeled as in
the EDA based on bivariate empirical copulas.
The class of VEDAs (vine EDAs) is introduced in [Soto and Gonzalez-Fernandez, 2010]
and [Gonzalez-Fernandez, 2011]. Algorithms of this
class model the search distributions using regular vines, which are
graphical models that represent a multivariate distribution by
decomposing the corresponding multivariate density into conditional
bivariate copulas, unconditional bivariate copulas and univariate
densities. In particular, VEDAs are based on the simplified
pair-copula construction [Haff et al., 2010], which
assumes that the bivariate copulas depend on the conditioning
variables only through their arguments. Since all bivariate copulas do
not have to belong to the same family, regular vines model a rich
variety of dependences by combining bivariate copulas from different
A regular vine on $n$ variables is a set of nested trees
$T_{1},\ldots,T_{n-1}$, where the edges of tree $T_{j}$ are the nodes
of the tree $T_{j+1}$ with $j=1,\ldots,n-2$. The edges of the trees
represent the bivariate copulas in the decomposition and the nodes
their arguments. Moreover, the proximity condition requires that two
nodes in tree $T_{j+1}$ are joined by an edge only if the
corresponding edges in $T_{j}$ share a common node. C-vines (canonical
vines) and D-vines (drawable vines) are two particular types of
regular vines, each of which determines a specific decomposition of
the multivariate density. In a C-vine, each tree $T_{j}$ has a unique
root node that is connected to $n-j$ edges. In a D-vine, no node is
connected to more than two edges. Two EDAs based on regular vines are
presented in [Soto and Gonzalez-Fernandez, 2010] and
[Gonzalez-Fernandez, 2011]: CVEDA (C-vine EDA) and
DVEDA (D-vine EDA) based on C-vines and D-vines, respectively. Since
both algorithms are implemented in copulaedas, we describe them
in more detail in the rest of this section.
The general idea of the simulation and inference methods for C-vines
and D-vines was developed by Aas et al., 2009.
The simulation algorithm is based on the conditional distribution
method (see e.g., Devroye, 1986), while
the inference method should consider two main aspects: the selection
of the structure of the vines and the choice of the bivariate copulas.
In the rest of this section we describe how these aspects are performed
in the particular implementation of CVEDA and DVEDA.
The selection of the structure of C-vines and D-vines is restricted
to the selection of the bivariate dependences explicitly modeled
in the first tree. This is accomplished by using greedy heuristics,
which use the empirical Kendall's tau assigned to the edges of
the tree. In a C-vine, the node that maximizes the sum of
the weights of its edges to the other nodes is chosen as the root of
the first tree and a canonical root node is assumed for the rest of
the trees. In a D-vine, the construction of the first tree consists
of finding the maximum weighted sequence of the variables, which
can be transformed into a TSP (traveling salesman problem)
instance [Brechmann, 2010]. For efficiency reasons,
in copulaedas we find an approximate solution of the TSP by
using the cheapest insertion heuristic
[Rosenkrantz et al., 1977].
The selection of each bivariate copula in both CVEDA and DVEDA starts
with an independence test
[Genest and Rémillard, 2004, Genest et al., 2007].
The product copula is selected when there is not enough evidence
against the null hypothesis of independence at a given significance
level. Otherwise, the parameters of a group of candidate copulas are
estimated and the copula that minimizes a Cramér-von Mises statistic
of the empirical copula is selected
[Genest and Rémillard, 2008].
The cost of the construction of C-vines and D-vines increases with the
number of variables. To reduce this cost, we apply the truncation
strategy presented in [Brechmann, 2010], for which the
theoretical justification can be found in
[Joe et al., 2010]. When a vine is truncated at a
certain tree during the tree-wise estimation procedure, all the
copulas in the subsequent trees are assumed to be product copulas. A
model selection procedure based on either AIC <cit.> or BIC <cit.> is applied to detect the
required number of trees. This procedure expands the tree $T_{j+1}$ if
the value of the information criterion calculated up to the
tree $T_{j+1}$ is smaller than the value obtained up to the previous
tree; otherwise, the vine is truncated at the tree $T_{j}$. At this
point, it is important to note that the algorithm presented in
[Salinas-Gutiérrez et al., 2010] also uses a D-vine. In this
algorithm only normal copulas are fitted in the first two trees and
conditional independence is assumed in the rest of the trees, i.e.,
the D-vine is always truncated at the second tree.
The implementation of CVEDA and DVEDA included in copulaedas
uses by default the truncation procedure based on AIC and the
candidate copulas normal, $t$, Clayton, Frank and Gumbel. The
parameters of all copulas but the $t$ copula are estimated using the
method of moments. For the $t$ copula, the correlation coefficient is
computed as in the normal copula, and the degrees of freedom are
estimated by maximum likelihood with the correlation parameter fixed
[Demarta and McNeil, 2005].
§ IMPLEMENTATION IN R
According to the review presented by Santana, 2011,
the approach followed for the implementation of EDA software currently available
through the Internet can be classified into three categories:
(1) implementation of a single EDA, (2) independent implementation of
multiple EDAs, and (3) common modular implementation of multiple
EDAs. In our opinion, the third approach offers greater flexibility
for the EDA community. In these modular implementations, the EDA
components (e.g., learning and sampling methods) are independently
programmed by taking advantage of the common schema shared by most
EDAs. This modularity allows the creation and validation of new EDA
proposals that combine different components, and promotes code
factorization. Additionally, as the EDAs are grouped under the same
framework, it facilitates performing empirical studies to compare the
behavior of different algorithms. Existing members of this class are
ParadisEO [Cahon et al., 2004, DOLPHIN Project Team, 2012], LiO
[Mateo and de la Ossa, 2006, Mateo and de la Ossa, 2007], Mateda-2.0
[Santana et al., 2009, Santana et al., 2010] and now copulaedas.
The implementation of copulaedas follows an object-oriented
design inspired by the Mateda-2.0 toolbox for MATLAB
[The MathWorks, Inc., 2014]. EDAs implemented in the package are represented by
S4 classes [Chambers, 2008] with generic
functions for their main steps. The base class of EDAs in the package
is `EDA', which has two slots: name and
parameters. The name slot stores a character string with
the name of the EDA and it is used by the show method to print
the name of the algorithm when it is called with an `EDA'
instance as argument. The parameters slot stores all the
EDA parameters in a list.
In copulaedas, each step of the general procedure of an EDA
outlined in Algorithm <ref> is represented by
a generic function that expects an `EDA' instance as its first
argument. Table <ref> shows a description
of these functions and their default methods. The help page of these
generic functions in the documentation of copulaedas contains
information about their arguments, return value, and methods
already implemented in the package.
Generic function Description
edaSeed Seeding method. The default method edaSeedUniform generates
the values of each variable in the initial population from a continuous
uniform distribution.
edaOptimize Local optimization method. The use of a local optimization
method is disabled by default.
edaSelect Selection method. The default method edaSelectTruncation
implements truncation selection.
edaLearn Learning method. No default method.
edaSample Sampling method. No default method.
edaReplace Replacement method. The default method edaReplaceComplete
replaces the current population with the new population.
edaReport Reporting method. Reporting progress information is disabled
by .
edaTerminate Termination method. The default method edaTerminateMaxGen
ends the execution of the algorithm after a maximum number of generations.
Description of the generic functions that implement the steps of the
general procedure of an EDA outlined in Algorithm <ref>
and their default methods.
The generic functions and their methods that implement the steps of an
EDA look at the parameters slot of the `EDA' instance
received as first argument for the values of the parameters that
affect their behavior. Only named members of the list must be used and
reasonable default values should be assumed when a certain component
is missing. The help page of each generic function describes the
members of the list in the parameters slot interpreted by each
function and their default values.
The edaRun function implements the Algorithm <ref>
by linking together the generic functions for each step. This function
expects four arguments: the `EDA' instance, the objective function
and two vectors specifying the lower and upper bounds of the variables
of the objective function. The length of the vectors with the lower
and upper bounds should be the same, since it determines the number
of variables of the objective function. When edaRun is called,
it runs the main loop of the EDA until the call to the edaTerminate
generic function returns TRUE. Then, the function returns
an instance of the `EDAResult' class that encapsulates the results
of the algorithm. A description of the slots of this class is given
in Table <ref>.
Slot Description
eda `EDA' instance.
f Objective function.
lower Lower bounds of the variables of the objective function.
upper Upper bounds of the variables of the objective function.
numGens Total number of generations.
fEvals Total number of evaluations of the objective function.
bestSol Best solution.
bestEval Evaluation of the best solution.
cpuTime Run time of the algorithm in seconds.
Description of the slots of the `EDAResult' class.
Two subclasses of `EDA' are already defined in copulaedas:
`CEDA', that represents EDAs based on multivariate copulas;
and `VEDA', that represents vine-based EDAs. The implementation
of UMDA, GCEDA, CVEDA and DVEDA relies on the copula [Hofert et al., 2014],
vines [Gonzalez-Fernandez and Soto, 2014], mvtnorm [Genz et al., 2014, Genz and Bretz, 2009],
and truncnorm [Trautmann et al., 2014] R packages.
These packages implement the techniques for the estimation and simulation
of the probabilistic models used in these EDAs.
§ USING COPULAEDAS
In this section, we illustrate how to use copulaedas through
several examples. To begin with, we show how to run the EDAs included
in the package. Next, we continue with the implementation of a new EDA
by using the functionalities provided by the package, and finally we
show how to perform an empirical study to compare the behavior of a
group of EDAs on benchmark functions and a real-world problem.
The two well-known test problems Sphere and Summation Cancellation
are used as the benchmark functions. The functions fSphere
and fSummationCancellation implement these problems in terms
of a vector $\boldsymbol{x}=(x_{1},\ldots,x_{n})$ according to
\begin{align*}
f_{\textrm{Summation\ensuremath{\,}Cancellation}}(\boldsymbol{x})&=\frac{1}{10^{-5}+\sum_{i=1}^{n}|y_{i}|},\quad \, y_{1}=x_{1},\, y_{i}=y_{i-1}+x_{i}.
\end{align*}
Sphere is a minimization problem and Summation Cancellation is
originally a maximization problem but it is defined in the package
as a minimization problem. Sphere has its global optimum at
$\boldsymbol{x}=(0,\ldots,0)$ with evaluation zero and Summation
Cancellation at $\boldsymbol{x}=(0,\ldots,0)$ with evaluation
$-10^{5}$. For a description of the characteristics of these
functions see [Bengoetxea et al., 2002] and
[Bosman and Thierens, 2006].
The results presented in this section were obtained using R
version 3.1.0 with copulaedas version 1.4.0, copula
version 0.999-8, vines version 1.1.0, mvtnorm version 0.9-99991,
and truncnorm version 1.0-7. Computations were performed on a 64-bit
Linux machine with an Intel(R) Core(TM)2 Duo 2.00 GHz processor. In the rest
of this section, we assume copulaedas has been loaded. This
can be attained by running the following command:
R> library("copulaedas")
§.§ Running the EDAs included in the package
We begin by illustrating how to run the EDAs based on copulas
implemented in copulaedas. As an example, we execute GCEDA to
optimize Sphere in five dimensions. Before creating a new instance of
the `CEDA' class for EDAs based on multivariate copulas, we set
up the generic functions for the steps of the EDA according to the
expected behavior of GCEDA. The termination criterion is either to
find the optimum of the objective function or to reach a maximum
number of generations. That is why we set the method for the
edaTerminate generic function to a combination of the functions
edaTerminateEval and edaTerminateMaxGen through the
auxiliary function edaTerminateCombined.
R> setMethod("edaTerminate", "EDA",
+ edaTerminateCombined(edaTerminateEval, edaTerminateMaxGen))
The method for the edaReport generic function is set to edaReportSimple
to make the algorithm print progress information at each generation.
This function prints one line at each iteration of the EDA with the
minimum, mean and standard deviation of the evaluation of the solutions
in the current population.
R> setMethod("edaReport", "EDA", edaReportSimple)
Note that these methods were set for the base class `EDA' and
therefore they will be inherited by all subclasses. Generally, we find
it convenient to define methods of the generic functions that
implement the steps of the EDA for the base class, except when
different subclasses should use different methods.
The auxiliary function `CEDA' can be used to create instances of
the class with the same name. All the arguments of the function are
interpreted as parameters of the EDA to be added as members of the list
in the parameters slot of the new instance. An instance of
`CEDA' corresponding to GCEDA using empirical marginal
distributions smoothed with normal kernels can be created as follows:
R> gceda <- CEDA(copula = "normal", margin = "kernel", popSize = 200,
+ fEval = 0, fEvalTol = 1e-6, maxGen = 50)
R> gceda@name <- "Gaussian Copula Estimation of Distribution Algorithm"
The methods that implement the generic functions edaLearn
and edaSample for `CEDA' instances expect three parameters.
The copula parameter specifies the multivariate copula and
it should be set to "normal" for GCEDA. The marginal distributions
are determined by the value of margin and all EDAs implemented
in the package use this parameter for the same purpose. As margin
is set to "kernel", the algorithm will look for three functions
named fkernel, pkernel and qkernel already
defined in the package to fit the parameters of the margins and to
evaluate the distribution and quantile functions, respectively. The
fkernel function computes the bandwidth parameter of the normal
kernel according to the rule-of-thumb of Silverman, 1986
and pkernel implements the empirical cumulative distribution
function. The quantile function is evaluated following the procedure
described in [Azzalini, 1981]. The popSize
parameter determines the population size while the rest of the arguments
of CEDA are parameters of the functions that implement the
termination criterion.
Now, we can run GCEDA by calling edaRun. The lower and upper
bounds of the variables are set so that the values of the variables in
the optimum of the function are located at 25% of the interval. It
was shown in [Arderí, 2007] and
[78] that the use of empirical marginal
distributions smoothed with normal kernels improves the behavior of
GCEDA when the initial population is generated asymmetrically with
respect to the optimum of the function.
R> set.seed(12345)
R> result <- edaRun(gceda, fSphere, rep(-300, 5), rep(900, 5))
Generation Minimum Mean Std. Dev.
1 1.522570e+05 1.083606e+06 5.341601e+05
2 2.175992e+04 5.612769e+05 3.307403e+05
3 8.728486e+03 2.492247e+05 1.496334e+05
4 4.536507e+03 1.025119e+05 5.829982e+04
5 5.827775e+03 5.126260e+04 2.983622e+04
6 2.402107e+03 2.527349e+04 1.430142e+04
7 9.170485e+02 1.312806e+04 6.815822e+03
8 4.591915e+02 6.726731e+03 4.150888e+03
9 2.448265e+02 3.308515e+03 1.947486e+03
10 7.727107e+01 1.488859e+03 8.567864e+02
11 4.601731e+01 6.030030e+02 3.529036e+02
12 8.555769e+00 2.381415e+02 1.568382e+02
13 1.865639e+00 1.000919e+02 6.078611e+01
14 5.157326e+00 4.404530e+01 2.413589e+01
15 1.788793e+00 2.195864e+01 1.136284e+01
16 7.418832e-01 1.113184e+01 6.157461e+00
17 6.223596e-01 4.880880e+00 2.723950e+00
18 4.520045e-02 2.327805e+00 1.287697e+00
19 6.981399e-02 1.123582e+00 6.956201e-01
20 3.440069e-02 5.118243e-01 2.985175e-01
21 1.370064e-02 1.960786e-01 1.329600e-01
22 3.050774e-03 7.634156e-02 4.453917e-02
23 1.367716e-03 3.400907e-02 2.056747e-02
24 7.599946e-04 1.461478e-02 8.861180e-03
25 4.009605e-04 6.488932e-03 4.043431e-03
26 1.083879e-04 2.625759e-03 1.618058e-03
27 8.441887e-05 1.079075e-03 5.759307e-04
28 3.429462e-05 5.077934e-04 3.055568e-04
29 1.999004e-05 2.232605e-04 1.198675e-04
30 1.038719e-05 1.104123e-04 5.888948e-05
31 6.297005e-06 5.516721e-05 2.945027e-05
32 1.034002e-06 2.537823e-05 1.295004e-05
33 8.483830e-07 1.332463e-05 7.399488e-06
The result variable contains an instance of the
`EDAResult' class. The show method prints the results of
the execution of the algorithm.
R> show(result)
Results for Gaussian Copula Estimation of Distribution Algorithm
Best function evaluation 8.48383e-07
No. of generations 33
No. of function evaluations 6600
CPU time 7.895 seconds
Due to the stochastic nature of EDAs, it is often useful to analyze
a sequence of independent runs to ensure reliable results. The edaIndepRuns
function supports performing this task. To avoid generating lot of
unnecessary output, we first disable reporting progress information
on each generation by setting edaReport to edaReportDisabled
and then we invoke the edaIndepRuns function to perform 30
independent runs of GCEDA.
R> setMethod("edaReport", "EDA", edaReportDisabled)
R> set.seed(12345)
R> results <- edaIndepRuns(gceda, fSphere, rep(-300, 5), rep(900, 5), 30)
The return value of the edaIndepRuns function is an instance of
the `EDAResults' class. This class is simply a wrapper for a
list with instances of `EDAResult' as members that contain the
results of an execution of the EDA. A show method for
`EDAResults' instances prints a table with all the results.
R> show(results)
Generations Evaluations Best Evaluation CPU Time
Run 1 33 6600 8.483830e-07 7.583
Run 2 38 7600 4.789448e-08 8.829
Run 3 36 7200 4.798364e-07 8.333
Run 4 37 7400 9.091651e-07 8.772
Run 5 34 6800 5.554465e-07 7.830
Run 6 35 7000 3.516341e-07 8.071
Run 7 35 7000 9.325531e-07 8.106
Run 8 35 7000 6.712550e-07 8.327
Run 9 36 7200 8.725061e-07 8.283
Run 10 37 7400 2.411458e-07 8.565
Run 11 36 7200 4.291725e-07 8.337
Run 12 35 7000 7.245520e-07 8.313
Run 13 37 7400 2.351322e-07 8.538
Run 14 36 7200 8.651248e-07 8.320
Run 15 34 6800 9.422646e-07 7.821
Run 16 35 7000 9.293726e-07 8.333
Run 17 36 7200 6.007390e-07 8.274
Run 18 38 7600 3.255231e-07 8.763
Run 19 36 7200 6.012969e-07 8.353
Run 20 35 7000 5.627017e-07 8.296
Run 21 36 7200 5.890752e-07 8.259
Run 22 35 7000 9.322505e-07 8.067
Run 23 35 7000 4.822349e-07 8.084
Run 24 34 6800 7.895408e-07 7.924
Run 25 36 7200 6.970180e-07 8.519
Run 26 34 6800 3.990247e-07 7.808
Run 27 35 7000 8.876874e-07 8.055
Run 28 33 6600 8.646387e-07 7.622
Run 29 36 7200 9.072113e-07 8.519
Run 30 35 7000 9.414666e-07 8.040
Also, the summary method can be used to generate a table with a
statistical summary of the results of the 30 runs of the algorithm.
R> summary(results)
Generations Evaluations Best Evaluation CPU Time
Minimum 33.000000 6600.0000 4.789448e-08 7.5830000
Median 35.000000 7000.0000 6.841365e-07 8.2895000
Maximum 38.000000 7600.0000 9.422646e-07 8.8290000
Mean 35.433333 7086.6667 6.538616e-07 8.2314667
Std. Dev. 1.250747 250.1494 2.557100e-07 0.3181519
§.§ Implementation of a new EDA based on
In this section we illustrate how to use copulaedas to implement
a new EDA based on copulas. As an example, we consider the extension
of MIMIC for continuous domains proposed in
[Salinas-Gutiérrez et al., 2009]. Similarly to MIMIC, this
extension learns a chain dependence structure, but it uses bivariate
copulas instead of bivariate normal distributions. The chain
dependence structure is similar to a D-vine truncated at the first
tree, i.e., a D-vine where independence is assumed for all the trees
but the first. Two instances of the extension of MIMIC based on
copulas were presented in [Salinas-Gutiérrez et al., 2009],
one uses bivariate normal copulas while the other uses bivariate Frank
copulas. In this article, the algorithm will be denoted as
Copula MIMIC.
Since the algorithm in question matches the general schema of an EDA
presented in Algorithm <ref>, only the functions
corresponding to the learning and simulation steps have to be
implemented. The first step in the implementation of a new EDA is to
define a new S4 class that inherits from `EDA' to
represent the algorithm. For convenience, we also define an auxiliary
function CopulaMIMIC that can be used to create new instances
of this class.
R> setClass("CopulaMIMIC", contains = "EDA",
+ prototype = prototype(name = "Copula MIMIC"))
R> CopulaMIMIC <- function(...)
+ new("CopulaMIMIC", parameters = list(...))
Copula MIMIC models the marginal distributions with the beta distribution.
A linear transformation is used to map the sample of the variables
in the selected population into the $(0,1)$ interval to match the
domain of definition of the beta distribution. Note that, since the
copula is scale-invariant, this transformation does not affect the
dependence between the variables. To be consistent with the margins
already implemented in copulaedas, we define three functions
with the common suffix betamargin and the prefixes f,
p and q to fit the parameters of the margins and for
the evaluation of the distribution and quantile functions, respectively.
By following this convention, the algorithms already implemented in
the package can use beta marginal distributions by setting the margin
parameter to "betamargin".
R> fbetamargin <- function(x, lower, upper)
+ x <- (x - lower) / (upper - lower)
+ loglik <- function(s) sum(dbeta(x, s[1], s[2], log = TRUE))
+ s <- optim(c(1, 1), loglik, control = list(fnscale = -1))$par
+ list(lower = lower, upper = upper, a = s[1], b = s[2])
+ }
R> pbetamargin <- function(q, lower, upper, a, b) {
+ q <- (q - lower) / (upper - lower)
+ pbeta(q, a, b)
+ }
R> qbetamargin <- function(p, lower, upper, a, b) {
+ q <- qbeta(p, a, b)
+ lower + q * (upper - lower)
+ }
\end{CodeInput}
\end{CodeChunk}
The `\code{CopulaMIMIC}' class inherits methods for the generic functions
that implement all the steps of the EDA except learning and sampling.
To complete the implementation of the algorithm, we must define the
estimation and simulation of the probabilistic model as methods for
the generic functions \code{edaLearn} and \code{edaSample}, respectively.
The method for \code{edaLearn} starts with the estimation of the
parameters of the margins and the transformation of the selected
population to uniform variables in $(0,1)$. Then, the mutual
information between all pairs of variables is calculated through the
copula entropy [Davy and Doucet, 2003]. To accomplish this, the
parameters of each possible bivariate copula are estimated by the
method of maximum likelihood using the value obtained through the
method of moments as an initial approximation. To determine the chain
dependence structure, a permutation of the variables that maximizes
the pairwise mutual information must be selected but, since this is a
computationally intensive task, a greedy algorithm is used to compute
an approximate solution
[Bonet \emph{et~al.}, 1997, Larra{\~n}aga \emph{et~al.}, 1999]. Finally, the
method for \code{edaLearn} returns a list with three components that
represents the estimated probabilistic model: the parameters of the
marginal distributions, the permutation of the variables, and the
copulas in the chain dependence structure.
\begin{CodeChunk}
\begin{CodeInput}
R> edaLearnCopulaMIMIC <- function(eda, gen, previousModel,
+ selectedPop, selectedEval, lower, upper) {
+ margin <- eda@parameters$margin
+ copula <- eda@parameters$copula
+ if (is.null(margin)) margin <- "betamargin"
+ if (is.null(copula)) copula <- "normal"
+ fmargin <- get(paste("f", margin, sep = ""))
+ pmargin <- get(paste("p", margin, sep = ""))
+ copula <- switch(copula,
+ normal = normalCopula(0), frank = frankCopula(0))
+ n <- ncol(selectedPop)
+ # Estimate the parameters of the marginal distributions.
+ margins <- lapply(seq(length = n),
+ function(i) fmargin(selectedPop[, i], lower[i], upper[i]))
+ uniformPop <- sapply(seq(length = n), function(i) do.call(pmargin,
+ c(list(selectedPop[ , i]), margins[[i]])))
+ # Calculate pairwise mutual information by using copula entropy.
+ C <- matrix(list(NULL), nrow = n, ncol = n)
+ I <- matrix(0, nrow = n, ncol = n)
+ for (i in seq(from = 2, to = n)) {
+ for (j in seq(from = 1, to = i - 1)) {
+ # Estimate the parameters of the copula.
+ data <- cbind(uniformPop[, i], uniformPop[, j])
+ startCopula <- fitCopula(copula, data, method = "itau",
+ estimate.variance = FALSE)@copula
+ C[[i, j]] <- tryCatch(
+ fitCopula(startCopula, data, method = "ml",
+ start = startCopula@parameters,
+ estimate.variance = FALSE)@copula,
+ error = function(error) startCopula)
+ # Calculate mutual information.
+ if (is(C[[i, j]], "normalCopula")) {
+ I[i, j] <- -0.5 * log(1 - C[[i, j]]@parameters^2)
+ } else {
+ u <- rcopula(C[[i, j]], 100)
+ I[i, j] <- sum(log(dcopula(C[[i, j]], u))) / 100
+ }
+ C[[j, i]] <- C[[i, j]]
+ I[j, i] <- I[i, j]
+ }
+ }
+ # Select a permutation of the variables.
+ perm <- as.vector(arrayInd(which.max(I), dim(I)))
+ copulas <- C[perm[1], perm[2]]
+ I[perm, ] <- -Inf
+ for (k in seq(length = n - 2)) {
+ ik <- which.max(I[, perm[1]])
+ perm <- c(ik, perm)
+ copulas <- c(C[perm[1], perm[2]], copulas)
+ I[ik, ] <- -Inf
+ }
+ list(margins = margins, perm = perm, copulas = copulas)
+ }
R> setMethod("edaLearn", "CopulaMIMIC", edaLearnCopulaMIMIC)
\end{CodeInput}
\end{CodeChunk}
The \code{edaSample} method receives the representation of the
probabilistic model returned by \code{edaLearn} as the \code{model}
argument. The generation of a new solution with $n$ variables starts
with the simulation of an $n$-dimensional vector $U$ having uniform
marginal distributions in $(0,1)$ and the dependence described by the
copulas in the chain dependence structure. The first step is to
simulate an independent uniform variable $U_π_n$ in $(0,1)$,
where $π_n$ denotes the variable in the position $n$ of the
permutation $π$ selected by the \code{edaLearn} method. The rest of
the uniform variables are simulated conditionally on the previously
simulated variable by using the conditional copula
$C(U_π_k|U_π_k+1)$, with $k=n-1,n-2,…,1$. Finally,
the new solution is determined through the evaluation of the beta
quantile functions and the application of the inverse of the linear
\begin{CodeChunk}
\begin{CodeInput}
R> edaSampleCopulaMIMIC <- function(eda, gen, model, lower, upper) {
+ popSize <- eda@parameters$popSize
+ margin <- eda@parameters$margin
+ if (is.null(popSize)) popSize <- 100
+ if (is.null(margin)) margin <- "betamargin"
+ qmargin <- get(paste("q", margin, sep = ""))
+ n <- length(model$margins)
+ perm <- model$perm
+ copulas <- model$copulas
+ # Simulate the chain structure with the copulas.
+ uniformPop <- matrix(0, nrow = popSize, ncol = n)
+ uniformPop[, perm[n]] <- runif(popSize)
+ for (k in seq(from = n - 1, to = 1)) {
+ u <- runif(popSize)
+ v <- uniformPop[, perm[k + 1]]
+ uniformPop[, perm[k]] <- hinverse(copulas[[k]], u, v)
+ }
+ # Evaluate the inverse of the marginal distributions.
+ pop <- sapply(seq(length = n), function(i) do.call(qmargin,
+ c(list(uniformPop[, i]), model$margins[[i]])))
+ pop
+ }
R> setMethod("edaSample", "CopulaMIMIC", edaSampleCopulaMIMIC)
\end{CodeInput}
\end{CodeChunk}
The code fragments given above constitute the complete implementation
of Copula~MIMIC. As it was illustrated with GCEDA in the previous
section, the algorithm can be executed by creating an instance of
the `\code{CopulaMIMIC}' class and calling the \code{edaRun} function.
\subsection{Performing an empirical study on benchmark problems}
We now show how to use \pkg{copulaedas} to perform an empirical study
of the behavior of a group of EDAs based on copulas on benchmark problems.
The algorithms to be compared are UMDA, GCEDA, CVEDA, DVEDA and Copula~MIMIC.
The first three algorithms are included in \pkg{copulaedas} and the
fourth algorithm was implemented in Section~\ref{sec:example-new-eda}.
The two functions Sphere and Summation Cancellation described at the
beginning of Section~\ref{sec:examples} are considered as benchmark
problems in 10 dimensions.
The aim of this empirical study is to assess the behavior of these
algorithms when only linear and independence relationships are considered.
Thus, only normal and product copulas are used in these EDAs. UMDA
and GCEDA use multivariate product and normal copulas, respectively.
CVEDA and DVEDA are configured to combine bivariate product and normal
copulas in the vines. Copula~MIMIC learns a chain dependence structure
with normal copulas. All algorithms use normal marginal distributions.
Note that in this case, GCEDA corresponds to EMNA and Copula~MIMIC
is similar to MIMIC for continuous domains. In the following code
fragment, we create class instances corresponding to these algorithms.
\begin{CodeChunk}
\begin{CodeInput}
R> umda <- CEDA(copula = "indep", margin = "norm")
R> umda@name <- "UMDA"
R> gceda <- CEDA(copula = "normal", margin = "norm")
R> gceda@name <- "GCEDA"
R> cveda <- VEDA(vine = "CVine", indepTestSigLevel = 0.01,
+ copulas = c("normal"), margin = "norm")
R> cveda@name <- "CVEDA"
R> dveda <- VEDA(vine = "DVine", indepTestSigLevel = 0.01,
+ copulas = c("normal"), margin = "norm")
R> dveda@name <- "DVEDA"
R> copulamimic <- CopulaMIMIC(copula = "normal", margin = "norm")
R> copulamimic@name <- "CopulaMIMIC"
\end{CodeInput}
\end{CodeChunk}
The initial population is generated using the default \code{edaSeed}
method, therefore, it is sampled uniformly in the real interval of
each variable. The lower and upper bounds of the variables are set so
that the values of the variables in the optimum of the function are
located in the middle of the interval. We use the intervals
$[-600,600]$ in Sphere and $[-0.16,0.16]$ in Summation Cancellation.
All algorithms use the default truncation selection method with a
truncation factor of 0.3. Three termination criteria are combined
using the \code{edaTerminateCombined} function: to find the
global optimum of the function with a precision greater
than $10^-6$, to reach $300000$ function evaluations, or to loose
diversity in the population (i.e., the standard deviation of the
evaluation of the solutions in the population is less than
$10^-8$). These criteria are implemented in the functions
\code{edaTerminateEval}, \code{edaTerminateMaxEvals} and
\code{edaTerminateEvalStdDev}, respectively.
The population size of EDAs along with the truncation method determine
the sample available for the estimation of the search distribution.
An arbitrary selection of the population size could lead to misleading
conclusions of the results of the experiments. When the population
size is too small, the search distributions might not be accurately
estimated. On the other hand, the use of an excessively large population
size usually does not result in a better behavior of the algorithms
but certainly in a greater number of function evaluations. Therefore,
we advocate for the use of the critical population size when comparing
the performance of EDAs. The critical population size is the minimum
population size required by the algorithm to find the global optimum
of the function with a high success rate, e.g., to find the optimum
in 30 of 30 sequential independent runs.
An approximate value of the critical population size can be determined
empirically using a bisection method (see e.g., Pelikan, 2005
for a pseudocode of the algorithm). The bisection method begins with
an initial interval where the critical population size should be located
and discards one half of the interval at each step. This procedure
is implemented in the \code{edaCriticalPopSize} function. In the
experimental study carried out in this section, the initial interval
is set to $[50,2000]$. If the critical population size is not found
in this interval, the results of the algorithm with the population
size given by the upper bound are presented.
The complete empirical study consists of performing 30 independent
runs of every algorithm on every function using the critical population
size. We proceed with the definition of a list containing all algorithm-function
\begin{CodeChunk}
\begin{CodeInput}
R> edas <- list(umda, gceda, cveda, dveda, copulamimic)
R> fNames <- c("Sphere", "SummationCancellation")
R> experiments <- list()
R> for (eda in edas) {
+ for (fName in fNames) {
+ experiment <- list(eda = eda, fName = fName)
+ experiments <- c(experiments, list(experiment))
+ }
+ }
\end{CodeInput}
\end{CodeChunk}
Now we define a function to process the elements of the \code{experiments}
list. This function implements all the experimental setup described
before. The output of \code{edaCriticalPopSize} and \code{edaIndepRuns}
is redirected to a different plain text file for each algorithm-function
\begin{CodeChunk}
\begin{CodeInput}
R> runExperiment <- function(experiment) {
+ eda <- experiment$eda
+ fName <- experiment$fName
+ # Objective function parameters.
+ fInfo <- list(
+ Sphere = list(lower = -600, upper = 600, fEval = 0),
+ SummationCancellation = list(lower = -0.16, upper = 0.16,
+ fEval = -1e5))
+ lower <- rep(fInfo[[fName]]$lower, 10)
+ upper <- rep(fInfo[[fName]]$upper, 10)
+ f <- get(paste("f", fName, sep = ""))
+ # Configure termination criteria and disable reporting.
+ eda@parameters$fEval <- fInfo[[fName]]$fEval
+ eda@parameters$fEvalTol <- 1e-6
+ eda@parameters$fEvalStdDev <- 1e-8
+ eda@parameters$maxEvals <- 300000
+ setMethod("edaTerminate", "EDA",
+ edaTerminateCombined(edaTerminateEval, edaTerminateMaxEvals,
+ edaTerminateEvalStdDev))
+ setMethod("edaReport", "EDA", edaReportDisabled)
+ sink(paste(eda@name, "_", fName, ".txt", sep = ""))
+ # Determine the critical population size.
+ set.seed(12345)
+ results <- edaCriticalPopSize(eda, f, lower, upper,
+ eda@parameters$fEval, eda@parameters$fEvalTol, lowerPop = 50,
+ upperPop = 2000, totalRuns = 30, successRuns = 30,
+ stopPercent = 10, verbose = TRUE)
+ if (is.null(results)) {
+ # Run the experiment with the largest population size, if the
+ # critical population size was not found.
+ eda@parameters$popSize <- 2000
+ set.seed(12345)
+ edaIndepRuns(eda, f, lower, upper, runs = 30, verbose = TRUE)
+ }
+ sink(NULL)
+ }
\end{CodeInput}
\end{CodeChunk}
We can run all the experiments by calling \code{runExperiment} for
each element of the list.
\begin{CodeChunk}
\begin{CodeInput}
R> for (experiment in experiments) runExperiment(experiment)
\end{CodeInput}
\end{CodeChunk}
Running the complete empirical study sequentially is a computationally
demanding operation. If various processing units are available, it can
be speeded up significantly by running the experiments in parallel.
The \pkg{snow} package [Tierney \emph{et~al.}, 2013] offers a great platform
to achieve this purpose, since it provides a high-level interface
for using a cluster of workstations for parallel computations
in \proglang{R}. The functions \code{clusterApply} or \code{clusterApplyLB}
can be used to call \code{runExperiment} for each element of the
\code{experiments} list in parallel, with minimal modifications to
the code presented here.
A summary of the results of the algorithms with the critical population size
is shown in Table~\ref{tab:benchmark-results}. Overall, the five algorithms are
able to find the global optimum of Sphere in all the 30 independent
runs with similar function values but only GCEDA, CVEDA and DVEDA
optimize Summation Cancellation. In the rest of this section
we provide some comments about results of the algorithms on each
\begin{table}[t!]
\centering
\begin{tabular}{@{}lccc@{\hspace{15pt}}c@{\hspace{18pt}}c@{}}
\hline
Algorithm & Success & Pop. & Evaluations & Best Evaluation & CPU Time\\
\hline
\multicolumn{6}{@{}l}{\emph{Sphere:}}\\
UMDA & 30/30 & 81 & \hspace{-0.5em}3788.1~$\pm$~99.0 & $6.7$e$-07$~$\pm$~$2.0$e$-07$ & 0.2~$\pm$~0.0\\
GCEDA & 30/30 & 310 & \hspace{-0.5em}13102.6~$\pm$~180.8 & $6.8$e$-07$~$\pm$~2.0e$-07$ & 0.4~$\pm$~0.0\\
CVEDA & 30/30 & 104 & \hspace{-0.5em}4804.8~$\pm$~99.9 & 5.9e$-07$~$\pm$~1.6e$-07$ & 4.5~$\pm$~0.8\\
DVEDA & 30/30 & 111 & 5080.1~$\pm$~111.6 & 6.5e$-07$~$\pm$~2.3e$-07$ & 4.4~$\pm$~0.8\\
Copula~MIMIC & 30/30 & 172 & 7441.8~$\pm$~127.2 & 6.8e$-07$~$\pm$~1.9e$-07$ & \hspace{-0.5em}65.6~$\pm$~1.2\\
\hline
\multicolumn{6}{@{}l}{\emph{Summation Cancellation:}}\\
UMDA & \hspace{0.5em}0/30 & 2000 & \hspace{-2em}300000.0~$\pm$~0.0 & \hspace{-0.75em}$-7.2$e+02~$\pm$~4.5e+02 & \hspace{-0.5em}32.5~$\pm$~0.3\\
GCEDA & 30/30 & 325 & \hspace{-0.5em}38913.3~$\pm$~268.9 & \hspace{-0.75em}$-1.0$e+05~$\pm$~1.0e$-07$ & 4.2~$\pm$~0.1\\
CVEDA & 30/30 & 294 & 41228.6~$\pm$~1082.7 & \hspace{-0.75em}$-1.0$e+05~$\pm$~1.0e$-07$ & \hspace{-1em}279.1~$\pm$~5.1\\
DVEDA & 30/30 & 965 & \hspace{-0.5em}117022.3~$\pm$~1186.8 & \hspace{-0.75em}$-1.0$e+05~$\pm$~1.1e$-07$ & \hspace{-1em}1617.5~$\pm$~39.1\\
Copula~MIMIC & \hspace{0.5em}0/30 & 2000 & \hspace{-2em}300000.0~$\pm$~0.0 & \hspace{-0.75em}$-4.2$e+04~$\pm$~3.5e+04 & \hspace{-1em}1510.6~$\pm$~54.9\\
\hline
\end{tabular}
\caption{Summary of the results obtained in 30 independent runs of UMDA, GCEDA,
CVEDA, DVEDA and Copula~MIMIC in the 10-dimensional Sphere (top) and
Summation Cancellation (bottom). Pop.~denotes Population.\label{tab:benchmark-results}}
\end{table}
UMDA exhibits the best behavior in terms of the number of function
evaluations in Sphere. There are no strong dependences between the
variables of this function and the results suggest that considering
the marginal information is enough to find the global optimum efficiently.
The rest of the algorithms being tested require the calculation of
a greater number of parameters to represent the relationships between
the variables and hence larger populations are needed to compute them
reliably (78 illustrate this issue in
more detail with GCEDA). CVEDA and DVEDA do not assume a normal dependence
structure between the variables and for this reason are less affected
by this issue. The estimation procedure used by the vine-based algorithms
selects the product copula if there is not enough evidence of dependence.
Both UMDA and Copula~MIMIC fail to optimize Summation Cancellation.
A correct representation of the strong linear interactions between the
variables of this function seems to be essential to find the global
optimum. UMDA completely ignores this information by assuming
independence between the variables and it exhibits the worst
behavior. Copula~MIMIC reaches better fitness values than UMDA but
neither can find the optimum of the function. The probabilistic model
estimated by Copula~MIMIC cannot represent important dependences
necessary for the success of the optimization. The algorithms GCEDA,
CVEDA and DVEDA do find the global optimum of Summation Cancellation.
The results of GCEDA are slightly better than the ones of CVEDA and
these two algorithms achieve much better results than DVEDA in
terms of the number of function evaluations. The correlation matrix
estimated by GCEDA can properly represent the multivariate linear
interactions between the variables. The C-vine structure used in CVEDA,
on the other hand, provides a very good fit for the dependence structure
between the variables of Summation Cancellation, given that it is possible
to find a variable that governs the interactions in the sample
(see Gonzalez-Fernandez, 2011 for more details).
The running time of Copula~MIMIC is considerably greater than the
running time of the other algorithms for all the functions. This
result is due to the use of a numerical optimization algorithm for the
estimation of the parameters of the copulas by maximum likelihood. In
the context of EDAs, where copulas are fitted at every generation, the
computational effort required to estimate the parameters of the
copulas becomes an important issue. As was illustrated with CVEDA and
DVEDA, using a method of moments estimation is a viable alternative to
maximum likelihood that requires much less CPU time. The empirical
investigation confirms the robustness of CVEDA and DVEDA in problems
with both weak and strong interactions between the variables.
Nonetheless, the flexibility afforded by these algorithms comes with an
increased running time when compared to UMDA or GCEDA, since the
interactions between the variables have to be discovered during
the learning step.
A general result of this empirical study is that copula-based EDAs
should use copulas other than the product only when there is evidence
of dependence. Otherwise, the EDA will require larger populations
and hence a greater number of function evaluations to accurately determine
the parameters of the copulas that correspond to independence.
\subsection{Solving the molecular docking problem}
Finally, we illustrate the use of \pkg{copulaedas} for solving a
so-called real-world problem. In particular, we use CVEDA and DVEDA to
solve an instance of the molecular docking problem, which is an
important component of protein-structure-based drug design. From the
point of view of the computational procedure, it entails predicting
the geometry of a small ligand that binds to the active site of a
large macromolecular protein receptor. Protein-ligand docking remains
being a highly active area of research, since the algorithms for
exploring the conformational space and the scoring functions that have
been implemented so far have significant limitations
[Warren \emph{et~al.}, 2006].
In our docking simulations, the protein is treated as a rigid body
while the ligand is fully flexible. Thus, a candidate solution represents
only the geometry of the ligand and it is encoded as a vector of real
values that represent its position, orientation and flexible torsion
angles. The first three variables of this vector represent the ligand
position in the three-dimensional space constrained to a box enclosing
the receptor binding site. The construction of this box is based on
the minimum and maximum values of the ligand coordinates in its crystal
conformation plus a padding distance of 5{\AA{}} added to
each main direction of the space. The remainder vector variables are
three Euler angles that represent the ligand orientation as a rigid
body and take values in the intervals $[0,2\pi]$, $[-\pi/2,\pi/2]$
and $[-\pi,\pi]$, respectively; and one additional variable restricted
to $[-\pi,\pi]$ for each flexible torsion angle of the ligand.
The semiempirical free-energy scoring function implemented as part
of the suite of automated docking tools \pkg{AutoDock}~4.2 [Morris \emph{et~al.}, 2009]
is used to evaluate each candidate ligand conformation. The overall
binding energy of a given ligand molecule is expressed as the sum
of the pairwise interactions between the receptor and ligand atoms
(intermolecular interaction energy), and the pairwise interactions
between the ligand atoms (ligand intramolecular energy). The terms
of the function consider dispersion/repulsion, hydrogen bonding, electrostatics,
and desolvation effects, all scaled empirically by constants determined
through a linear regression analysis. The aim of an optimization algorithm
performing the protein-ligand docking is to minimize the overall energy
value. Further details of the energy terms and how the function is
derived can be found in [Huey \emph{et~al.}, 2007].
Specifically, we consider as an example here the docking of the 2z5u
test system, solved by \mbox{X-ray} crystallography and available as part of
the Protein Data Bank [Berman \emph{et~al.}, 2000]. The protein receptor is
lysine-specific histone demethylase~1 and the ligand is a \mbox{73-atom}
molecule (non-polar hydrogens are not counted) with 20 ligand torsions
in a box of 28{\AA{}}$\times$32{\AA{}}$\times$24{\AA{}}. In order to
make it easier for the readers of the paper to reproduce this example,
the implementation of the \pkg{AutoDock}~4.2 scoring function in
\proglang{C} was extracted from the original program and it is
included (with unnecessary dependences removed) in the supplementary
material as \code{docking.c}. During the evaluation of the scoring
function, precalculated grid maps (one for each atom type present in
the ligand being docked) are used to make the docking calculations
fast. The result of these precalculations and related metadata for the
2z5u test system are contained in the attached ASCII file
\code{2z5u.dat}.
We make use of the system command \code{R CMD SHLIB} to build a shared
object for dynamic loading from the file \code{docking.c}. Next, we
integrate the created shared object into \proglang{R} using the
function \code{dyn.load} and load the precalculated grids using the
utility \proglang{C} function \code{docking_load} as follows.
\begin{CodeChunk}
\begin{CodeInput}
R> system("R CMD SHLIB docking.c")
R> dyn.load(paste("docking", .Platform$dynlib.ext, sep = ""))
R> .C("docking_load", as.character("2z5u.dat"))
\end{CodeInput}
\end{CodeChunk}
The docking of the 2z5u test system results in a minimization problem
with a total of 26 variables. Two vectors with the lower and upper
bounds of these variables are defined using utility functions provided
in \code{docking.c} to compute the bounds of the variables that
determine the position of the ligand and the total number of
torsions. For convenience, we also define an \proglang{R} wrapper
function \code{fDocking} for the \proglang{C} scoring function
\code{docking_score} provided in the compiled code that was loaded
into \proglang{R}.
\begin{CodeChunk}
\begin{CodeInput}
R> lower <- c(.C("docking_xlo", out = as.double(0))$out,
+ .C("docking_ylo", out = as.double(0))$out,
+ .C("docking_zlo", out = as.double(0))$out, 0, -pi/2, -pi,
+ rep(-pi, .C("docking_ntor", out = as.integer(0))$out))
R> upper <- c(.C("docking_xhi", out = as.double(0))$out,
+ .C("docking_yhi", out = as.double(0))$out,
+ .C("docking_zhi", out = as.double(0))$out, 2 * pi, pi/2, pi,
+ rep(pi, .C("docking_ntor", out = as.integer(0))$out))
R> fDocking <- function(sol)
+ .C("docking_score", sol = as.double(sol), out = as.double(0))$out
\end{CodeInput}
\end{CodeChunk}
CVEDA and DVEDA are used to solve the minimization problem, since they
are the most robust algorithms among the EDAs based on copulas
implemented in \pkg{copulaedas}. The parameters of these algorithms
are set to the values reported by Soto \emph{et~al.}, 2012 in
the solution of the 2z5u test system. The
population size of CVEDA and DVEDA is set to 1400 and 1200,
respectively. Both algorithms use the implementation of the truncated
normal marginal distributions
[Johnson \emph{et~al.}, 1994] provided by
Trautmann \emph{et~al.}, 2014 to satisfy the box constraints of the
variables. The termination criterion of both CVEDA and DVEDA is to
reach a maximum of 100 generations, since an optimum value of the
scoring function is not known. Instances of the `\code{VEDA}' class
that follow the description given above are created with the following
\begin{CodeChunk}
\begin{CodeInput}
R> setMethod("edaTerminate", "EDA", edaTerminateMaxGen)
R> cveda <- VEDA(vine = "CVine", indepTestSigLevel = 0.01,
+ copulas = "normal", margin = "truncnorm", popSize = 1400, maxGen = 100)
R> dveda <- VEDA(vine = "DVine", indepTestSigLevel = 0.01,
+ copulas = "normal", margin = "truncnorm", popSize = 1200, maxGen = 100)
\end{CodeInput}
\end{CodeChunk}
Now we proceed to perform 30 independent runs of each algorithm using
the \code{edaIndepRuns} function. The arguments of this function
are the instances of the `\code{VEDA}' class corresponding to CVEDA
and DVEDA, the scoring function \code{fDocking}, and the vectors
\code{lower} and \code{upper} that determine the bounds of the variables.
\begin{CodeChunk}
\begin{CodeInput}
R> set.seed(12345)
R> cvedaResults <- edaIndepRuns(cveda, fDocking, lower, upper, runs = 30)
R> summary(cvedaResults)
\end{CodeInput}
\begin{CodeOutput}
Generations Evaluations Best Evaluation CPU Time
Minimum 100 140000 -30.560360 1421.9570
Median 100 140000 -29.387815 2409.0680
Maximum 100 140000 -23.076769 3301.5960
Mean 100 140000 -29.139028 2400.4166
Std. Dev. 0 0 1.486381 462.9442
\end{CodeOutput}
\begin{CodeInput}
R> set.seed(12345)
R> dvedaResults <- edaIndepRuns(dveda, fDocking, lower, upper, runs = 30)
R> summary(dvedaResults)
\end{CodeInput}
\begin{CodeOutput}
Generations Evaluations Best Evaluation CPU Time
Minimum 100 120000 -30.93501 1928.2030
Median 100 120000 -30.70019 3075.7960
Maximum 100 120000 -24.01502 4872.1820
Mean 100 120000 -30.01427 3053.9579
Std. Dev. 0 0 1.68583 695.0071
\end{CodeOutput}
\end{CodeChunk}
\begin{table}[!]
\centering
\begin{tabular}{@{}lccccc@{}}
\hline
{Algorithm} & {Pop.} & {Evaluations} & {Lowest Energy} & {RMSD} & {CPU Time}\\
\hline
CVEDA & 1400 & 140000.0~$\pm$~0.0 & $-$29.13~$\pm$~1.48 & 0.58~$\pm$~0.13 & 2400.4~$\pm$~462.9\\
DVEDA & 1200 & 120000.0~$\pm$~0.0 & $-$30.01~$\pm$~1.68 & 0.56~$\pm$~0.14 & 3053.9~$\pm$~695.0\\
\hline
\end{tabular}
\par
\caption{Summary of the results obtained in 30 independent runs of CVEDA and
DVEDA for the docking of the 2z5u test system. Pop.~denotes
\end{table}
\begin{figure}[t!]
\centering
\input{normal-copulas.tex}
\par
\caption{Average number of normal copulas selected at each generation
of CVEDA and DVEDA in 30 independent runs for the docking of the
2z5u test system. Since this is an optimization problem with 26
variables, the C-vines and D-vines have a total of 325
\end{figure}
The results obtained in the docking of the 2z5u test system are
summarized in Table~\ref{tab:docking-results}. In addition to the
information provided by \code{edaIndepRuns}, we present a column for
the RMSD (root mean square deviation) between the coordinates of the
atoms in the experimental crystal structure and the predicted ligand
coordinates of the best solution found at each run. These values can
be computed for a particular solution using the \code{docking_rmsd}
function included in \code{docking.c}. The RMSD values serve as a
measure of the quality of the predicted ligand conformations when an
experimentally determined solution is known.
Generally, a structure with RMSD below 2{\AA{}} can be considered as
successfully docked. Therefore, both CVEDA and DVEDA achieve good
results when solving the 2z5u test system. DVEDA exhibits slightly
lower energy and RMSD values, and it requires a smaller population
size than CVEDA. On the other hand, DVEDA uses more CPU time than
CVEDA, a fact that might be related with more dependences being
encoded in the D-vines estimated by DVEDA than in the C-vines used by
CVEDA. This situation is illustrated in
Figure~\ref{fig:normal-copulas}, which presents the average number of
normal copulas selected by CVEDA and DVEDA at each generation in the
30 independent runs. In both algorithms, the number of normal copulas
increases during the evolution, but DVEDA consistently selects more
normal copulas than CVEDA. Although the estimation procedure of the
C-vines in CVEDA intends to represent the strongest dependences in the
first tree, the constraint that only one variable can be connected to
all the others prevent strong correlations to be explicitly
encoded in the C-vines.
It is also worth noting that because of the use of the truncation
procedure based on AIC in both CVEDA and DVEDA, the number of
statistical tests required to estimate the vines was dramatically
reduced. The median number of vine trees fitted in the 30 independent
runs was 4 in CVEDA and 5 in DVEDA out of a total of 25 trees. The
interested reader is referred to [Soto \emph{et~al.}, 2012] for a deeper
study of the use of EDAs based on copulas for solving the molecular
docking problem.
\section[Concluding remarks]{Concluding remarks\label{sec:conclusions}}
We have developed \pkg{copulaedas} aiming at providing in a single
package an open-source implementation of EDAs based on copulas and
utility functions to study these algorithms. In this paper, we
illustrate how to run the copula-based EDAs implemented in the
package, how to implement new algorithms, and how to perform an
empirical study to compare copula-based EDAs on benchmark functions
and practical problems. We hope that these functionalities help the
research community to improve EDAs based on copulas by getting a
better insight of their strengths and weaknesses, and also help
practitioners to find new applications of these algorithms to
real-world problems.
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\end{document}
|
arxiv-papers
| 2012-09-24T21:24:17 |
2024-09-04T02:49:35.501089
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yasser Gonzalez-Fernandez, Marta Soto",
"submitter": "Yasser Gonzalez Fernandez",
"url": "https://arxiv.org/abs/1209.5429"
}
|
1209.5572
|
AN INTERTWINING OPERATOR FOR THE HARMONIC OSCILLATOR AND THE DIRAC OPERATOR
WITH
APPLICATION TO THE HEAT AND WAVE KERNELS
Ahmedou Yahya ould Mohameden and Mohamed Vall Ould Moustapha
Abstract. In this article an intertwining operator is constructed which
transforms the harmonic oscillator to the Dirac operator (the first order
derivative operator). We give also the explicit solutions to the heat and wave
equation associated to Dirac operator. As an application the heat and the wave
kernels of the harmonic oscillator are computed.
Key words: Intertwining Operator, Harmonic oscillator, Dirac operator, Heat
equation, Wave equation, Fourier transform, confluent hypergeometric function
1 – Introduction
The harmonic oscillators are interesting and important in their own right and
play a fundamental role in the modeling of the quantum fields and are related
to many mathematical and physical problems ($[1],[2]$, $[6]$, $[7]$, $[8]$,
$[9]$). The aim of this paper is to give an intertwining operator $T$ which
relate the harmonic oscillator:
$L^{a}=\frac{\partial^{2}}{\partial x^{2}}-a^{2}x^{2}\ \ \ \ \ \ \ a>0$ $None$
to the Dirac operator (first order derivative operator):
$D=\frac{\partial}{\partial X}$ $None$
We give also the explicit solutions to the following heat and wave equations
associated to Dirac operator:
$(HD)\qquad\left\\{\begin{array}[]{cc}\partial_{t}U(t,X)=\frac{\partial}{\partial
X}U(t,X)&(t,X)\in R^{\ast}_{+}\times I\\!\\!R\\\ U(0,X)=U_{0}(X)&U_{0}\in
C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
$(WD)\qquad\left\\{\begin{array}[]{cc}\partial_{t}^{2}V(t,X)=\frac{\partial}{\partial
X}V(t,X)&(t,X)\in R^{\ast}_{+}\times I\\!\\!R\\\
V(0,X)=0&\partial_{t}V(0,X)=V_{0}(X)\in
C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
Note that the Cauchy problem $(HD)$ for the heat equation associated to the
Dirac operator $D$ is nothing but the Cauchy problem for the transport
equation. The wave equation $(WD)$ associated to the Dirac operator is
considered in $[3]$ p.$64$ where the unequeness of solution is obtained but
there are no known explicit solutions until now.
As an application of our intertwining operator, from the heat and wave kernels
of the Dirac operator, we give the explicit solutions of the following heat
and wave equations associated to the harmonic oscillator:
$(HL^{a})\qquad\left\\{\begin{array}[]{cc}\partial_{t}u(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right)u(t,x)&(t,x)\in R^{\ast}_{+}\times I\\!\\!R\\\
u(0,x)=u_{0}(x)&u_{0}\in C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
$(WL^{a})\qquad\left\\{\begin{array}[]{cc}\partial_{t}^{2}v(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right)v(t,x)&(t,x)\in I\\!\\!R_{+}^{\ast}\times I\\!\\!R\\\
v(0,x)=0&\partial_{t}v(0,x)=v_{0}(x),v_{0}\in{\cal
S}_{a}(I\\!\\!R)\end{array}\right.$
.
with
${\cal S}_{a}(I\\!\\!R)=\left\\{\phi:{\cal
F}\left[e^{\frac{-ax^{2}}{2}}\phi\right](\xi)\in
C_{0}^{\infty}(I\\!\\!R)\right\\}$
Note that the heat kernel for the harmonic oscillator is known for long time
$[1]$:
$K_{a}(x,x^{\prime},t)=$
$\sqrt{\frac{a}{2\pi}}\frac{1}{\sqrt{sinh(2at)}}exp\left[-\frac{a}{2}(x^{2}+x^{\prime
2})coth(2at)+\frac{axx^{\prime}}{sh(2at)}\right]$ $None$
but to our knowledge the method used here and the obtained formula are new.
For the wave kernel of the harmonic oscillator $w_{a}(t,x,x^{\prime})$, the
following integral representation is given for the wave kernel at the origin
in $[2]p.358$:
$w_{1}(x,0,t)=\frac{i}{4\pi}\int_{C}\sqrt{\frac{1}{2z\sinh(z)}}e^{\frac{t^{2}}{2z}-\frac{x^{2}coth(z)}{2}}dz$
$None$
where C is a contour symmetric with respect to the x -axis going throught the
origin obtained by a smooth deformation of the circle
$C(\frac{1}{2}c,\frac{1}{2}c)$ of center $c/2$ and radius $c/2$ and for
$t<\left|x\right|$ the kernel vanishes.
Now we recall some facts about the Fourier transform:
for $f\in L^{1}(I\\!\\!R)$ the Fourier transform of $f$ and its inverse:
$({\cal
F}f)(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{-ix\xi}f(x)dx$
$None$ $({\cal
F}^{-1}f)(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{i\xi
x}f(\xi)d\xi$ $None$
and we have the following formulas:
${\cal F}f(\alpha\xi)=\frac{1}{\alpha}{\cal F}f(\frac{x}{\alpha})(\xi)$ $None$
${\cal F}^{-1}[e^{-s\xi^{2}}](x)=\frac{1}{\sqrt{2s}}e^{-x^{2}/4s};\ \ \ s>0\ \
\ $ $None$
Recall also the complementary error function $[3]$ p.272
$Erfc(z)=\int_{z}^{\infty}e^{-t^{2}}dt$ $None$
and in terms of the Tricomi confluent hypergeometric $U(a,c,z)$:
$Erfc(z)=\frac{1}{2}ze^{-z^{2}}U(1,3/2,z^{2})=\frac{1}{2}e^{-z^{2}}U(1/2,1/2,z^{2})$
$None$
2–The intertwining operator
Definition 2.1 An operator $T$ is said to be an intertwining operator if it
relates operators, $L$ and $D$, by
$TL=DT$ $None$
Lemma 2.2 For $a>0$ and $\phi\in{\cal S}_{a}(I\\!\\!R)$ we have
$\left[\frac{\partial^{2}}{\partial x^{2}}-a^{2}x^{2}\right]\phi(x)=$
$e^{\frac{ax^{2}}{2}}{\cal
F}^{-1}\left[e^{-\frac{\xi^{2}}{4a}-\frac{log|\xi|}{2}}\,(-2a\xi\frac{\partial}{\partial\xi})\,e^{\frac{\xi^{2}}{4a}+\frac{log|\xi|}{2}}{\cal
F}\left[e^{\frac{-ax^{2}}{2}}\phi\right](\xi)\right](x)$ $None$
Proof: Set
$\phi(x)=e^{\frac{ax^{2}}{2}}\psi(x)$ $None$
we get
$e^{\frac{-ax^{2}}{2}}\left[\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right]e^{\frac{ax^{2}}{2}}\psi(x)=\left[\frac{\partial^{2}}{\partial
x^{2}}+2ax\frac{\partial}{\partial x}+a\right]\psi(x)$ $None$
take the Fourier transform
${\cal F}\left[e^{\frac{-ax^{2}}{2}}\left[\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right]e^{\frac{ax^{2}}{2}}\psi(x)\right](\xi)=\left[-\xi^{2}-2a\xi\frac{\partial}{\partial\xi}-a\right]({\cal
F\psi})(\xi)$ $None$
set
$({\cal F}\psi)(\xi)=e^{-\frac{\xi^{2}}{4a}-\frac{log|\xi|}{2}}w(\xi)$ $None$
we obtain
$e^{\frac{\xi^{2}}{4a}+\frac{log|\xi|}{2}}{\cal
F}\left[e^{-\frac{ax^{2}}{2}}\left[\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right]e^{\frac{ax^{2}}{2}}\psi(x)\right](\xi)=-2a\xi\frac{\partial
w(\xi)}{\partial\xi}$ $None$
using $(2.6))$ we have
$-2a\xi\frac{\partial w(\xi)}{\partial\xi}=$
$e^{\frac{\xi^{2}}{4a}+\frac{log|\xi|}{2}}{\cal
F}\left[e^{-\frac{ax^{2}}{2}}\left[\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right]e^{\frac{ax^{2}}{2}}{\cal
F}^{-1}[e^{-\frac{\xi^{2}}{4a}-\frac{log|\xi|}{2}}w(\xi)](x)\right](\xi)$
$None$
Proposition 2.3 If $f\in{\cal S}(I\\!\\!R)$ and set
$(T\phi)(\xi)_{|\xi|=\exp{(-2aX)}}=e^{\frac{\xi^{2}}{4a}+\frac{log|\xi|}{2}}{\cal
F}\left[e^{\frac{-ax^{2}}{2}}\phi\right](\xi)_{|\xi|=\exp{(-2aX)}}$ $None$
Then the operator $T$ is an intertwining operator for the harmonic oscillator
$L^{a}$ given in $(1.1)$ and the Dirac opetrator $D$ in $(1.2)$:
$TL^{a}=DT$ $None$
Proof The formula $(2.10)$ is a consequence of the lemma $2.2$
3–Heat and wave kernels for the Dirac operator
Proposition 3.1 The Cauchy problem $(HD)$ for the heat equation associated to
the Dirac operator has the unique solution guiven by
$U(t,X)=U_{0}(t+X)$ $None$
Proof: The Cauchy problem for the heat equation associted to the Dirac
operator $D$ reduced to the Cauchy problem for the classical transport eqution
and its unique solution is given by $(3.1)$.
Proposition 3.2 The function
$W(t,X,X^{\prime})=\frac{t}{\sqrt{4\pi|X-X^{\prime}|}}\exp{\left(\frac{-t^{2}}{4|X-X^{\prime}|}\right)}U\left(1,\frac{3}{2},\frac{t^{2}}{4|X-X^{\prime}|}\right)$
$None$
satisfies the wave equation associated Dirac operator $D$.
and we have
$W_{a}(t,X,X^{\prime})=\frac{2}{\sqrt{\pi}}Erfc\left(\frac{t}{\sqrt{4|X-X^{\prime}|}}\right)$
$None$
where $Erfc(x)$ is the complementary error function in $(1.9)$.
Proof: Set $z=-\frac{t^{2}}{4(X-X^{\prime})}$ we get $\frac{\partial}{\partial
t}=-\frac{2t}{4(X-X^{\prime})}\frac{\partial}{\partial z}$,
$\frac{\partial^{2}}{\partial
t^{2}}=\frac{t^{2}}{4(X-X^{\prime})^{2}}\frac{\partial^{2}}{\partial
z^{2}}-\frac{2}{4(X-X^{\prime})}\frac{\partial}{\partial z}$,
$\frac{\partial}{\partial
X}=\frac{t^{2}}{4(X-X^{\prime})^{2}}\frac{\partial}{\partial z}$.
$\frac{\partial}{\partial X}\phi(t,X)-\frac{\partial^{2}}{\partial
t^{2}}\phi(t,X)=\frac{1}{X-X^{\prime}}\left\\{z\frac{\partial}{\partial
z^{2}}+(1/2-z)\frac{\partial}{\partial z}\right\\}=0$ $None$
the wave equation associated to the first order derivative operator is
equivalent to the equation of confluent hypergeometric type $[5]$ $p.268$
$z\varphi^{\prime\prime}(z)+\left(1/2-z\right)\varphi^{\prime}(z)=0$ $None$
and an appropriate solution $[5]$ $p.270$ is
$z^{1/2}\exp{(z)}U\left(1,3/2,-z\right)$.
Theorem 3.3 The Cauchy problem $(WD)$ for the wave equation associated to the
dirac operator has the unique solution given by:
$V(t,X)=\int_{|X-X^{\prime}|<~{}\frac{t}{2}}W\left(X,X^{\prime}\right)V_{0}(X^{\prime})dX^{\prime}$
$None$
where $W(t,X,X^{\prime})$ is given by $(3.2)$ and $(3.2)^{\prime}$.
Proof In view of the proposition 3.2 it remain to show the limit conditions
in $(WD)$. And for this set $X^{\prime}=X+\frac{t}{2}s$ we have
$V(t,X)=Ct^{3/2}\int_{-1}^{1}\frac{V_{0}(X+s\frac{t}{2})}{\sqrt{|s|}}\exp{\left(-\frac{t}{2|s|}\right)}U\left(1,3/2,\frac{t}{2|s|}\right)ds$
$None$
where $C=2^{-1/2}\frac{1}{\sqrt{4\pi}}$. By the formula giving the first
derivative of the Lommel confluent hypergeometric function $[4]$ $p.265$
$\frac{d}{dz}U(a,c,z)=-aU(a+1,c+1,z)$ $None$
we obtain the limit conditions using the following behavior of the degenerate
confluent hypergeometric function $U(a,c,z)$, $[5]$ p.$288-289$ For
$z\rightarrow 0$:
$U(a,c,z)=(\Gamma(c-1)/\Gamma(a))z^{1-c}+O(1),1<\Re c<2$ $None$
$U(a,c,z)=(\Gamma(c-1)/\Gamma(a))z^{1-c}+O(|z|^{\Re c-2}),\Re c\geq 2,c\neq 2$
$None$
4–Heat and wave kernels for the Harmonic oscillator
The Cauchy problem $(HL^{a})$ for heat equation associated to the Harmonic
oscillator for $u_{0}\in L^{2}(I\\!\\!R)$ has the unique solution $u$
belonging to $C^{0}\left([0,\infty[,L^{2}(I\\!\\!R)\right)$ see $[6]$ more
precisely there exists a semigroup $(S_{t})_{t\geq 0}$ of
$L^{2}(I\\!\\!R)$-contractions such that for all $t>0$, $u(t,.)=S_{t}u_{0}$,
the explicit solution given by
$u(t,x)=\int_{-\infty}^{+\infty}K_{a}(t,x,x^{\prime})u_{0}(x^{\prime})dx^{\prime}$
where the kernel $K_{a}(t,x,x^{\prime})$ is given by the Mehler’s formula
$(1.3)$
Theorem 4.1 If $T$ is the intertwining operator given by $(2.9)$ and if
$L^{a}$ and $D$ are the harmonic oscillator and the Dirac operator given
respectively by $(1.1)$ and $(1.2)$ then we have:
$e^{tL^{a}}u_{0}=T^{-1}\left\\{e^{tD}(Tu_{0})\right\\}$ $None$ $\frac{\sin
t\sqrt{L^{a}}}{\sqrt{L^{a}}}v_{0}=T^{-1}\left\\{\frac{\sin
t\sqrt{D}}{\sqrt{D}}(Tv_{0})\right\\}$ $None$
where $e^{tA}$ and $\frac{\sin t\sqrt{A}}{\sqrt{A}}$ are respectively the heat
and the wave kernels for the operator $A$.
Proof The formula $(4.1)$ is a consequence of the formulas $(2.10)$ and the
Cauchy problems $(HD)$ and $(HL^{a})$.
The formula $(4.2)$ is a consequence of the formulas $(2.10)$ and the Cauchy
problems $(WL^{a})$ and $(WD)$.
Theorem 4.2 The Cauchy problem $(HL^{a})$ for the heat equation associted to
the harmonic oscillator has the unique solution given by:
$u(t,x)=\int_{-\infty}^{+\infty}H_{a}(t,x,x^{\prime})u_{0}(x^{\prime})dx^{\prime}$
$None$
where
$H_{a}(t,x,x^{\prime})=$
$a\sqrt{\frac{2}{\pi}}\left(e^{2at}-e^{-2at}\right)^{-1/2}\exp\left\\{\frac{(e^{at}x-e^{-at}x^{\prime})^{2}}{e^{2at}-e^{-2at}}+\frac{a}{2}(x^{2}-x^{\prime
2})\right\\}$ $None$
Proof: Using $(4.1)$ of theorem $4.1$ we have
$u(t,x)=e^{-at}e^{ax^{2}/2}{\cal F}^{-1}[e^{-(1-e^{-4at})\xi^{2}/4a}{\cal
F}\left(e^{-ax^{2}/2}u_{0}\right)(\xi e^{-2at})](x)$ $None$
$u(t,x)=\frac{1}{\sqrt{2\pi}}e^{-at}e^{ax^{2}/2}\times$
${\cal F}^{-1}[e^{-(1-e^{-4at})\xi^{2}/4a}]\ast{\cal F}^{-1}[{\cal
F}\left(e^{-ax^{2}/2}u_{0}\right)(\xi e^{-2at})](x)$ $None$
using $(1.7)$ and $(1.8)$ we can write
$u(t,x)=\frac{e^{ax^{2}/2}}{\sqrt{\pi}}[\frac{\sqrt{a}}{\sqrt{1-e^{-4at}}}e^{-\frac{ax^{2}}{1-e^{-4at}}}]\ast
e^{2at}u_{0}(xe^{2at})e^{-(a/2)x^{2}e^{4at}}$ $None$
$=\frac{e^{at+ax^{2}/2}}{\sqrt{\pi}}\frac{\sqrt{a}}{\sqrt{1-e^{-4at}}}\int_{-\infty}^{+\infty}u_{0}(x^{\prime}e^{2at})e^{-(a/2)x^{\prime
2}e^{4at}}e^{\frac{-a(x-x^{\prime})^{2}}{1-e^{-4at}}}dx^{\prime}$ $None$
set $x^{\prime\prime}=x^{\prime}e^{2at}$ in $(4.8)$ we get the formula
$(4.3)$.
Remark 4.3: The formula in $(4.3)$ for the heat kernel associated to the
harmonic ocillator agree with that given by $(1.3)$ and the proof is left to
the reader.
Theorem 4.4 The Cauchy problem for the wave equation associated to the
harmonic oscillator $(WH)$ has the unique solution given by:
$v(t,x)=-\frac{1}{a\sqrt{\pi}}e^{ax^{2}/2}{\cal
F}^{-1}[\frac{e^{-\xi^{2}/4a}}{\sqrt{\xi}}\int_{|\ln\xi/\xi^{\prime}|<at}Erfc\left(\frac{\sqrt{a}t}{\sqrt{2|\ln\xi/\xi^{\prime}|}}\right)\times$
$\frac{e^{\xi^{\prime 2}/4a}}{\sqrt{\xi^{\prime}}}{\cal F}(e^{ax^{\prime
2}/2}v_{0})(\xi^{\prime})d\xi^{\prime}](x)$ $None$
Proof : The formula $(4.8)$ follows from the theorem $(4.1)$ and the Fubini
theorem using the formulas $(3.2)$ and $(3.2)^{\prime}$ and the fact that
$|Erfc(z)|\leq\sqrt{\pi}$
We finish this section by the folowing corollary:
Corollary 4.5 The heat and wave kernels for the cursin operator
$M=\frac{\partial^{2}}{\partial x^{2}}+x^{2}\frac{\partial^{2}}{\partial
y^{2}}$
are given respectively by:
$~{}\widetilde{H}_{a}(t,x,y,x^{\prime},y^{\prime},t)~{}=~{}\frac{1}{2\pi}\int^{+\infty}_{-\infty}e^{i(y-y^{\prime})a}H_{a}(t,x,x^{\prime})da$
$~{}\widetilde{w}_{a}(t,x,y,x^{\prime},y^{\prime},t)~{}=~{}\frac{1}{2\pi}\int^{+\infty}_{-\infty}e^{i(y-y^{\prime})a}w_{a}(t,x,x^{\prime})da$
5–Directions for further studies
We suggest here a certain number of open related problems connected to this
paper. we are intersted in the heat and wave equations for the harmonic
oscillator with an inverse square potential
$(HL^{a})^{\prime}\qquad\left\\{\begin{array}[]{cc}\partial_{t}u(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}-\frac{b^{2}}{x^{2}}\right)u(t,x)&(t,x)\in R\times
I\\!\\!R\\\ u(0,x)=u_{0}(x)&u_{0}\in
C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
$(WL^{a})^{\prime}\qquad\left\\{\begin{array}[]{cc}\partial_{t}^{2}u(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}-\frac{b^{2}}{x^{2}}\right)u(t,x)&(t,x)\in R_{+}^{\ast}\times
I\\!\\!R\\\ u(0,x)=0&\partial_{t}u(0,x)=u_{0}(x)\in
C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
.
Another possible extension is to consider the heat and wave equation
associated to the power of the harmonic oscillator
$(HL^{a})^{\prime\prime}\qquad\left\\{\begin{array}[]{cc}\partial_{t}u(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right)^{s}u(t,x)&(t,x)\in R_{+}^{\ast}\times I\\!\\!R\\\
u(0,x)=u_{0}(x)&u_{0}\in C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
$(WL^{a})^{\prime\prime}\qquad\left\\{\begin{array}[]{cc}\partial_{t}^{2}u(t,x)=\left(\frac{\partial^{2}}{\partial
x^{2}}-a^{2}x^{2}\right)^{s}u(t,x)&(t,x)\in R_{+}^{\ast}\times I\\!\\!R\\\
u(0,x)=0&\partial_{t}u(0,x)=u_{0}(x)\in
C_{0}^{\infty}(I\\!\\!R)\end{array}\right.$
.
Finally, we suggest a problems in direction of the non linear heat and wave
equations for the harmonic oscillator and to look for global solution and a
possible blow up in finite times.
References
${\bf[1]-}$ Berline, N.,Getzler, E., Vergne, M. Heat kernels and dirac
operator Springer Verlag 2004.
${\bf[2]-}$Greiner P. C. , Daniel Holocman, and Yakar Kannai, Wave kernels
related to the second order operator, Duke Math.J.vol. 114, $329-387(2002).$
[3]- Guelfand, I.M., and Chilov, C. E. Les distributions Tome 3 Théorie des
équations différentielles, Dunod 1968
${\bf[4]}$-Lebedev, N., N. ; Special Functions and their applications ; Dover
Publications INC New York 1972.
[5]-Magnus, W. Oberhittenger, F. and Soni R. P. Formulas and Theorems for the
special functions of Mathematical physics,Springer-Verlog New-York 1966.
[6]- Pazy, A. Semigroups of linear operators and applications to partial
differential equations Applied mathematical sciences $44$ Springer Verlag
$1983$.
${\bf[7]-}$Sheng-Ya Feng,A note on heat kernels of generalized Hermite
operators, Tawanese Journal of Math. Vol 15,$N^{o}.5,2035-2041(2011)$.
[8]- Read ,S. and Simon B., Methods of modern mathematical physics $II$,
Fourier analysis Self-adjointness, Academic Press, New York - London 1997.
[9]-Thangavelu Hermite and Laguerre Semigroups some recent developpent
Technical Report $N^{o}2006/7$, March 26,2006.
Université Gaston Berger de Saint-Louis B.P: 234. Sénégal.
E-mail adress: ahmeddou2011@yahoo.fr
Université de Nouakchott
Faculté des sciences et techniques
B.P: 5026, Nouakchott-Mauritanie.
E-mail adresse: khames@univ-nkc.mr
|
arxiv-papers
| 2012-09-25T10:49:55 |
2024-09-04T02:49:35.525857
|
{
"license": "Public Domain",
"authors": "Ahmedou Yahya ould Mohameden and Mohamed Vall Ould Moustapha",
"submitter": "Ould Moustapha Mohamed Vall",
"url": "https://arxiv.org/abs/1209.5572"
}
|
1209.5634
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-275 LHCb-PAPER-2012-028 October 18, 2012
Measurements of $B_{c}^{+}$ production and mass with the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay
The LHCb collaboration†††Authors are listed on the following pages.
Measurements of $B_{c}^{+}$ production and mass are performed with the decay
mode $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
using 0.37 fb-1 of data collected in $pp$ collisions at $\sqrt{s}=7$ TeV by
the LHCb experiment. The ratio of the production cross-section times branching
fraction between the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
is measured to be $(0.68\pm 0.10\,({\rm stat.})\pm 0.03\,({\rm syst.})\pm
0.05\,({\rm lifetime}))\%$ for $B_{c}^{+}$ and $B^{+}$ mesons with transverse
momenta $p_{\rm T}>4~{}$GeV/$c$ and pseudorapidities $2.5<\eta<4.5$. The
$B_{c}^{+}$ mass is directly measured to be $6273.7\pm 1.3\,({\rm stat.})\pm
1.6\,({\rm syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the
measured mass difference with respect to the $B^{+}$ meson is
$M(B_{c}^{+})-M(B^{+})=994.6\pm 1.3\,({\rm stat.})\pm 0.6\,({\rm
syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Submitted to Phys. Rev. Lett.
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B.
Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
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Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
The $B_{c}^{+}$ meson is unique in the Standard Model as it is the ground
state of a family of mesons containing two different heavy flavour quarks. At
the 7 TeV LHC centre-of-mass energy, the most probable way to produce
$B_{c}^{(*)+}$ mesons is through the $gg$-fusion process, $gg\rightarrow
B_{c}^{(*)+}+b+\bar{c}$ [1]. The production cross-section of the $B_{c}^{+}$
meson has been calculated by a complete order-$\alpha_{s}^{4}$ approach and
using the fragmentation approach [1]. It is predicted to be about 0.4 $\upmu$b
[2, 3] at $\sqrt{s}=7$ TeV including contributions from excited states. This
is one order of magnitude higher than that predicted at the Tevatron energy
$\sqrt{s}=1.96$ TeV. However, the theoretical predictions suffer from large
uncertainties, and an accurate measurement of the $B_{c}^{+}$ production
cross-section is needed to guide experimental studies at the LHC. As is the
case for heavy quarkonia, the mass of the $B_{c}^{+}$ meson can be calculated
by means of potential models and lattice QCD, and early predictions lay in the
range from $6.2-6.4\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [1]. The
inclusion of charge conjugate modes is implied throughout this Letter.
The $B_{c}^{+}$ meson was first observed in the semileptonic decay mode
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})\ell^{+}X\ (\ell=e,\mu)$ by CDF [4, *Abe:1998fb]. The
production cross-section times branching fraction for this decay relative to
that for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{+}$
was measured to be $0.132\,^{+0.041}_{-0.037}\,({\rm stat.})\,\pm 0.031\,({\rm
syst.})\,^{+0.032}_{-0.020}\,({\rm lifetime})$ for $B_{c}^{+}$ and $B^{+}$
mesons with transverse momenta $p_{\rm T}>6$ GeV/$c$ and rapidities $|y|<1$.
Measurements of the $B_{c}^{+}$ mass by CDF [6] and D0 [7] using the fully
reconstructed decay
${B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})\pi^{+}}$ gave $M(B_{c}^{+})=6275.6\pm 2.9\,({\rm
stat.})\pm 2.5\,({\rm syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$M(B_{c}^{+})=6300\pm 14\,({\rm stat.})\pm 5\,({\rm
syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. A more
precise measurement of the $B_{c}^{+}$ mass would allow for more stringent
tests of predictions from potential models and lattice QCD calculations.
In this Letter, we present a measurement of the ratio of the production cross-
section times branching fraction of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
relative to that for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ for $B_{c}^{+}$ and $B^{+}$ mesons with transverse momenta
$p_{\rm T}>4$ GeV/$c$ and pseudorapidities $2.5<\eta<4.5$, and a measurement
of the $B_{c}^{+}$ mass. These measurements are performed using
$0.37\,\mbox{\,fb}^{-1}$ of data collected in $pp$ collisions at $\sqrt{s}=7$
TeV by the LHCb experiment. The LHCb detector [8] is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The muon identification efficiency is about 97%, with a misidentification
probability $\epsilon(\pi\rightarrow\mu)\sim 3$%.
The $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
and $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
modes are topologically identical and are selected with requirements as
similar as possible to each other. Events are selected by a trigger system
consisting of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage which applies a full event
reconstruction. At the hardware trigger stage, events are selected by
requiring a single muon candidate or a pair of muon candidates with high
transverse momenta. At the software trigger stage [9, 10], events are selected
by requiring a pair of muon candidates with invariant mass within
$120\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [11], or a two- or three-
track secondary vertex with a large track $p_{\rm T}$ sum, a significant
displacement from the primary interaction, and at least one track identified
as a muon.
At the offline selection stage, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates are formed from pairs of oppositely charged tracks with transverse
momenta $p_{\rm T}>0.9\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and identified
as muons. The two muons are required to originate from a common vertex.
Candidates with a dimuon invariant mass between
$3.04{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$3.14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are combined with charged
hadrons with $p_{\rm T}>1.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to form the
$B_{c}^{+}$ and $B^{+}$ meson candidates. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass window is about seven
times larger than the mass resolution. No particle identification is used in
the selection of the hadrons. To improve the $B_{c}^{+}$ and $B^{+}$ mass
resolutions, the mass of the $\mu^{+}\mu^{-}$ pair is constrained to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [11]. The $b$-hadron
candidates are required to have $p_{\rm
T}>4\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, decay time $t>0.25\,{\rm ps}$ and
pseudorapidity in the range $2.5<\eta<4.5$. The fiducial region is chosen to
be well inside the detector acceptance to have a reasonably flat efficiency
over the phase space. To further suppress background to the $B_{c}^{+}$ decay,
the IP $\chi^{2}$ values of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and $\pi^{+}$ candidates with respect to any primary vertex (PV) in the event
are required to be larger than 4 and 25, respectively. The IP $\chi^{2}$ is
defined as the difference between the $\chi^{2}$ of the PV reconstructed with
and without the considered particle. The IP $\chi^{2}$ of the $B_{c}^{+}$
candidates with respect to at least one PV in the event is required to be less
than 25. After all selection requirements are applied, no event has more than
one candidate for the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay, and less than 1% of the events have more than one candidate for the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay.
Such multiple candidates are retained and treated the same as other
candidates; the associated systematic uncertainty is negligible.
The ratio of the production cross-section times branching fraction measured in
this analysis is
$\displaystyle\begin{aligned} R_{c/u}&=\frac{\sigma(B_{c}^{+})\,{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}{\sigma(B^{+})\,{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})}\\\
&=\frac{N\left(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\right)}{\epsilon_{\rm tot}^{c}}\frac{\epsilon_{\rm
tot}^{u}}{N\left(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\right)},\end{aligned}$ (1)
where $\sigma(B_{c}^{+})$ and $\sigma(B^{+})$ are the inclusive production
cross-sections of the $B_{c}^{+}$ and $B^{+}$ mesons in $pp$ collisions at
$\sqrt{s}=7$ TeV, ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
and ${\cal B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})$ are the branching fractions of the reconstructed decay chains,
$N\left(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\right)$ and
$N\left(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\right)$ are the yields of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
decays, and $\epsilon_{\rm tot}^{c}$, $\epsilon_{\rm tot}^{u}$ are the total
efficiencies, including geometrical acceptance, reconstruction, selection and
trigger effects.
The signal event yields are obtained from extended unbinned maximum likelihood
fits to the invariant mass distributions of the reconstructed $B_{c}^{+}$ and
$B^{+}$ candidates in the interval
$6.15<M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})<6.55{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for $B_{c}^{+}$
candidates and $5.15<M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})<5.55{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for $B^{+}$
candidates. The $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ signal mass shape is described by a double-sided Crystal Ball
function [12]. The power law behaviour toward low mass is due primarily to
final state radiation (FSR) from the bachelor hadron, whereas the high mass
tail is mainly due to FSR from the muons in combination with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass constraint. The
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
mass shape is described by the sum of two double-sided Crystal Ball functions
that share the same mean but have different resolutions. From simulated
decays, it is found that the tail parameters of the double-sided Crystal Ball
function depend mildly on the mass resolution. This functional dependence is
determined from simulation and included in the mass fit. The combinatorial
background is described by an exponential function. Background to
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ from the
Cabibbo-suppressed decay
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ is
included to improve the fit quality. The distribution is determined from the
simulated events. The ratio of the number of
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ decays
to that of the signal is fixed to
$\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})/\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=3.83\%$ [13]. The Cabibbo-suppressed decay
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ is
neglected as a source of background to the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay. The invariant mass distributions of the selected
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidates
and the fits to the data are shown in Fig. 1. The numbers of signal events are
$162\pm 18$ for $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ and $56\,243\pm 256$ for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, as
obtained from the fits. The goodness of fits is checked with a $\chi^{2}$
test, which returns a probability of 97% for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
87% for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$.
Figure 1: Invariant mass distributions of selected (a)
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
candidates and (b) $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates, used in the production measurement. The fits to the
data are superimposed.
The efficiencies, including geometrical acceptance, reconstruction, selection
and trigger effects are determined using simulated signal events. The
production of the $B^{+}$ meson is simulated using Pythia 6.4 [14] with the
configuration described in Ref. [15]. A dedicated generator BcVegPy [16] is
used to simulate the $B_{c}^{+}$ meson production. Decays of $B_{c}^{+}$,
$B^{+}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are
described by EvtGen [17] in which final state radiation is generated using
Photos [18]. The decay products are traced through the detector by the Geant4
package [19, *Agostinelli:2002hh] as described in Ref. [21]. As the
efficiencies depend on $p_{\rm T}$ and $\eta$, the efficiencies from the
simulation are binned in these variables to avoid a bias. The signal yield in
each bin is obtained from data by subtracting the background contribution
using the sPlot technique [22], where the signal and background mass shapes
are assumed to be uncorrelated with $p_{\rm T}$ and $\eta$. The efficiency-
corrected numbers of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
decays are $2470\pm 350$ and $364\,188\pm 2270$, respectively, corresponding
to a ratio of $R_{c/u}=(0.68\pm 0.10)\%$, where the uncertainties are
statistical only.
The systematic uncertainties related to the determination of the signal yields
and efficiencies are described in the following. Concerning the former,
studies of simulated events show that effects due to the fit model on the
measured ratio $R_{c/u}$ can be as much as 1%, which is taken as systematic
uncertainty. The uncertainties from the contamination due to the Cabibbo-
suppressed decays are found to be negligible.
The uncertainties on the determination of the efficiencies are dominated by
the knowledge of the $B_{c}^{+}$ lifetime, which has been measured by CDF [23]
and D0 [24] to give $\tau(B_{c}^{+})=0.453\pm 0.041\,{\rm ps}$ [11]. The
distributions of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
simulated events have been reweighted after changing the $B_{c}^{+}$ lifetime
by one standard deviation around its mean value and the efficiencies are
recomputed. The relative difference of 7.3% between the recomputed
efficiencies and the nominal values is taken as a systematic uncertainty.
Since the $B^{+}$ lifetime is known more precisely, its contribution to the
uncertainty is neglected.
The effects of the trigger requirements have been evaluated by only using the
events triggered by the lifetime unbiased (di)muon lines, which is about 85%
of the total number of events. Repeating the complete analysis, a ratio of
$R_{c/u}=(0.65\pm 0.10)\%$ is found, resulting in a systematic uncertainty of
4%.
The tracking uncertainty includes two components. The first is the difference
in track reconstruction efficiency between data and simulation, estimated with
a tag and probe method [25] of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays, which is found to be negligible. The
second is due to the 2% uncertainty on the effect from hadronic interactions
assumed in the detector simulation.
The uncertainty due to the choice of the $(p_{\rm T},\eta)$ binning is found
to be negligible. Combining all systematic uncertainties in quadrature, we
obtain $R_{c/u}=(0.68\pm 0.10\,({\rm stat.})\pm 0.03\,({\rm syst.})\pm
0.05\,({\rm lifetime}))\%$ for $B_{c}^{+}$ and $B^{+}$ mesons with transverse
momenta $p_{\rm T}>4$ GeV/$c$ and pseudorapidities $2.5<\eta<4.5$.
For the mass measurement, different selection criteria are applied. All events
are used regardless of the trigger line. The fiducial region requirement is
also removed. Only candidates with a good measured mass uncertainty ($<20$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) are used, and a loose particle
identification requirement on the pion of the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay is introduced to remove the small contamination from
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
decays.
The alignment of the tracking system and the calibration of the momentum scale
are performed using a sample of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays in periods corresponding to different
running conditions, as described in Ref. [26]. The validity of the calibrated
momentum scale has been checked using samples of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ and $\Upsilon\rightarrow\mu^{+}\mu^{-}$ decays.
In all cases, the effect of the final state radiation, which cause the fitted
masses to be underestimated, is taken into account. The difference between the
correction factors determined using the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\Upsilon$ resonances, 0.06%, is taken as the systematic
uncertainty.
The $B_{c}^{+}$ mass is determined with an extended unbinned maximum
likelihood fit to the invariant mass distribution of the selected
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
candidates. The mass difference $M(B_{c}^{+})-M(B^{+})$ is obtained by fitting
the invariant mass distributions of the selected
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidates
simultaneously. The fit model is the same as in the production cross-section
ratio measurement. Figure 2 shows the invariant mass distribution for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$. The
$B_{c}^{+}$ mass is determined to be $6273.0\pm
1.3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, with a resolution of $13.4\pm
1.1\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the mass difference
$M(B_{c}^{+})-M(B^{+})$ is $994.3\pm
1.3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The uncertainties are
statistical only.
Figure 2: Invariant mass distribution of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays, used in the mass measurement. The fit to the data is superimposed.
The mass measurement is affected by the systematic uncertainties due to the
invariant mass model, momentum scale calibration, detector description and
alignment. To evaluate the systematic uncertainty, the complete analysis,
including the track fit and the momentum scale calibration when needed, is
repeated. The parameters to which the mass measurement is sensitive are varied
within their uncertainties. The changes in the central values of the masses
obtained from the fits relative to the nominal results are then assigned as
systematic uncertainties.
Table 1 summarizes the systematic uncertainties assigned to the measured
$B_{c}^{+}$ mass and mass difference $\Delta M=M(B_{c}^{+})-M(B^{+})$. The
main source is the uncertainty in the momentum scale calibration. After the
calibration procedure a residual $\pm 0.06$% variation of the momentum scale
remains as a function of the particle pseudorapidity $\eta$. The impact of
this variation is evaluated by parameterizing the momentum scale as a function
of $\eta$. The amount of material traversed by a particle in the tracking
system is known to 10% accuracy, the magnitude of the energy loss correction
in the reconstruction is therefore varied by 10%. To quantify the effects due
to the alignment uncertainty, the horizontal and vertical slopes of the tracks
close to the interaction region, which are determined by measurements in the
vertex detector, are changed by $\pm$0.1%, corresponding to the estimated
precision of the length scale along the beam axis [27]. To test the relative
alignment of different sub-detectors, the analysis is repeated ignoring the
hits of the tracking station between the vertex detector and the magnet. Other
uncertainties arise from the signal and background line shapes. The bias due
to the final state radiation is studied using a simulation based on Photos
[18]. The mass returned by the fit model is found to be underestimated by
$0.7\pm 0.1\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the $B_{c}^{+}$
meson, and by $0.4\pm 0.1\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the
$B^{+}$ meson. The mass and mass difference are corrected accordingly, and the
uncertainties are propagated. The effects of the background shape are
evaluated by using a constant or a first-order polynomial function instead of
the nominal exponential function. The stability of the measured $B_{c}^{+}$
mass is studied by dividing the data samples according to the polarity of the
spectrometer magnet and the pion charge. The measured $B_{c}^{+}$ masses are
consistent with the nominal result within the statistical uncertainties.
Table 1: Systematic uncertainties (in ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the $B_{c}^{+}$ mass and mass difference $\Delta M=M(B_{c}^{+})-M(B^{+})$. Source of uncertainty | $M(B_{c}^{+})$ | $\Delta M$
---|---|---
Mass fitting: | |
– Signal model | $0.1$ | $0.1$
– Background model | $0.3$ | $0.2$
Momentum scale: | |
– Average momentum scale | $1.4$ | $0.5$
– $\eta$ dependence | $0.3$ | $0.1$
Detector description: | |
– Energy loss correction | $0.1$ | -
Detector alignment: | |
– Vertex detector (track slopes) | $0.1$ | -
– Tracking stations | $0.6$ | $0.3$
Quadratic sum | $1.6$ | $0.6$
In conclusion, using 0.37 fb-1 of data collected in $pp$ collisions at
$\sqrt{s}=7$ TeV by the LHCb experiment, the ratio of the production cross-
section times branching fraction of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
relative to that for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ is measured to be $R_{c/u}=(0.68\pm 0.10\,({\rm stat.})\pm
0.03\,({\rm syst.})\pm 0.05\,({\rm lifetime}))\%$ for $B_{c}^{+}$ and $B^{+}$
mesons with transverse momenta $p_{\rm T}>4$ GeV/$c$ and pseudorapidities
$2.5<\eta<4.5$. Given the large theoretical uncertainties on both production
and branching fractions of the $B_{c}^{+}$ meson, more precise theoretical
predictions are required to make a direct comparison with our result. The
$B_{c}^{+}$ mass is measured to be $6273.7\pm 1.3\,({\rm stat.})\pm 1.6\,({\rm
syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The measured mass
difference with respect to the $B^{+}$ meson is
$M(B_{c}^{+})-M(B^{+})=994.6\pm 1.3\,({\rm stat.})\pm 0.6\,({\rm
syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Taking the world average
$B^{+}$ mass [11], we obtain $M(B_{c}^{+})=6273.9\pm 1.3\,({\rm stat.})\pm
0.6\,({\rm syst.})\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which has a
smaller systematic uncertainty. The measured $B_{c}^{+}$ mass is in agreement
with previous measurements [6, 7] and a recent prediction given by the lattice
QCD calculation, $6278(6)(4){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [28].
These results represent the most precise determinations of these quantities to
date.
## 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 (The 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.
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|
arxiv-papers
| 2012-09-25T14:57:49 |
2024-09-04T02:49:35.536569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, 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, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, 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, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, O. Kochebina, V. Komarov, R.F. Koopman,\n P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps,\n G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil,\n D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R.\n Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau,\n M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K. Pal, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, 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, K. Petridis, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf,\n H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta,\n C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A.\n Satta, M. Savrie, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M.\n Schlupp, 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, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, M. Smith, K. Sobczak, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De\n Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp,\n S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah,\n S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, P. Tsopelas, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X.\n Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, H. Voss, C. Vo\\ss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Jibo He",
"url": "https://arxiv.org/abs/1209.5634"
}
|
1209.5637
|
Particle Identification with the ALICE detector at the LHC
Chiara Zampolli ALICE Collaboration
Istituto Nazionale di Fisica Nucleare, Sez. di Bologna
I-40126, ITALY
## 1 Introduction
The extreme conditions of high energy density and high temperature achieved in
Pb–Pb collisions at LHC energies are expected to realize a deconfined plasma
of quarks and gluons (the so-called Quark-Gluon Plasma, QGP [1]), from which a
phase transition to ordinary colourless hadronic matter takes place as a
consequence of subsequent expansion and cooling down. ALICE (A Large Ion
Collider Experiment, [2]) is the LHC experiment dedicated to the investigation
of the nature and the properties of the QGP using heavy-ion collisions.
Especially, it is designed and built to cope with the high track density
environment expected in Pb–Pb collisions. ALICE also can provide unique
information on low-$p_{\rm{T}}$ pp physics (thanks to the low material budget
and low magnetic field of 0.5 T), which makes the experiment complementary to
CMS and ATLAS.
ALICE addresses many observables, spanning from the global characteristics of
the events (such as multiplicity densities and rapidity distributions), to
more specific QGP signals (like direct photons, charmonium and bottomonium).
One of the basic requirement in order to carry out such measurements is an
excellent particle identification (PID) performance. Making use of all known
PID techniques, the ALICE detector is capable to identify hadrons and leptons
over a very wide momentum range covering three orders of magnitudes, from
$\sim 100$ MeV/$c$ to $\sim 100$ GeV/$c$.
In the following sections, the ALICE detector will be briefly described (see
Sec. 2). The PID techniques used by the experiment will be presented in Sec.
3. Finally, a few examples of PID applications in physics analyses will be
presented (Sec. 4). The conclusions will be drawn in Section 5.
## 2 The ALICE detector
The left panel of Figure 1 shows a schematic view of the ALICE experiment.
The detector closest to the beam pipe is the Inner Tracking System (ITS, shown
also in the small inset), which is in charge of the reconstruction of the
primary and secondary vertices. Three types of Si sensors are used for the
ITS. The two innermost layers, for which a high granularity was needed to cope
with the requirement for the position resolution of primary and secondary
vertices, consist of Silicon Pixel Detectors (SPD). They are followed by a
pair of Silicon Drift Detectors (SDD), characterized by a very good multitrack
reconstruction capability. Finally, two layers of Silicon Strip Detectors
(SSD) complete the ALICE ITS. In addition to participate to the ALICE global
tracking, the ITS is capable to perform standalone reconstruction with the
advantage to recuperate the tracks lost in the global tracking due to the
spacial acceptance and the intrinsic $p_{\rm{T}}$ cutoff of the outer
detectors, and to particle decay.
The ALICE Time Projection Chamber (TPC), following the ITS in radial
direction, is the main ALICE tracking detector. Its tracking efficiency
reaches $\sim 80\%$ in $|\eta|<0.8$ with a momentum resolution
$\sigma(p_{\rm{T}})/p_{\rm{T}}\sim 5\%$, which gets down to $\sim 2.5\%$ up to
$p_{\rm{T}}=10$ $\mathrm{GeV}/c$ (and increasing at higher transverse momenta)
when combined with the ITS. Each track in the TPC is reconstructed using up to
a maximum of 159 space points, with a resolution of 0.8 mm in the xy plane,
and 1.2 mm in the z direction.
After TPC comes the Transition Radiation Detector (TRD), mainly dedicated to
the electron identification. The TRD is followed by the Time Of Flight
detector (TOF), at a radius of 3.7 m from the interaction point.
In the central $\eta$ region, ALICE has several detectors, referred to as
single-arm detectors, which have a limited acceptance. Namely, they are a
Cherenkov RICH detector (the HMPID), a homogeneous photon spectrometer (PHOS),
and a sampling electromagnetic calorimeter (EMCAL). At forward rapidities, a
Photon Multiplicity Detector (PMD) and a muon spectrometer (MUON) are placed.
Some more detectors complete the ALICE setup, but they won’t be described in
these proceedings since they do not contribute to the particle identification
of the experiment. For more details about them and about the other ALICE
detectors in general, see [2].
|
---|---
Figure 1: Left panel: schematic view of the ALICE detector. Right panel: ITS
$dE/dx$ resolution as a function of $p_{\rm{T}}$ from data (filled markers)
and Monte Carlo simulations (hollow markers). The performance is shown for the
two different configurations, when either only three layers give a signal
(red), or all of them (blue).
## 3 ALICE PID
Out of the 16 detectors in ALICE, 6 provide particle identification
information, using all the PID techniques known nowadays, implementing them at
their state of art. Below, the different ALICE PID procedure will be
presented. It is worth to mention that on top of PID technologies, ALICE
identifies also cascades, V0 and kinks thanks to its excellent capability for
tracking and secondary vertex determination.
### 3.1 PID in the central barrel
The central barrel detectors (i.e. those with full $\phi$ coverage) perform
particle identification using the specific energy loss of a charged particle
traversing a medium, the transition radiation emitted by charged particles
when crossing the boundary between two materials, and the time of flight that
it takes to a charged particle to reach a detector’s sensitive volume from the
interaction point.
$dE/dx$ measurements are provided by the last four layers of the ITS detector,
i.e. the SDD and the SSD, thanks to their analog readout. A truncated mean is
applied to the measurements, that is, an average of the lowest two is taken if
all the four layers gave a signal, or a weighted average is taken if only
three are available. The ITS PID is performed in the low $p_{\rm{T}}$ region,
up to $\sim 1$ $\mathrm{GeV}/c$, and pions reconstructed in standalone mode
can be identified down to $\sim 100$ $\mathrm{MeV}/c$. The right panel of Fig.
1 shows the $dE/dx$ resolution achieved by the ITS detector which stays around
10-15% over the whole $p_{\rm{T}}$ range.
The ALICE TPC detector adds PID information using specific energy loss
measurements as well. Also in this case, a truncated mean is applied over the
maximum number of 159 cluster information. The performance is excellent, with
a resolution of $\sim 5\%$ calculated for isolated tracks, in the cases when
159 space points were available. In addition to the identification of charged
hadrons up to $p_{\rm{T}}\sim$ 1 – 2 $\mathrm{GeV}/c$, the TPC wide dynamic
range (up to 26 MIP) allows to identify light nuclei, as shown in the left
panel of Fig. 2. Moreover, while in the $1/\beta^{2}$ Bethe-Bloch region of
the $dE/dx$ distribution particle identification for individual tracks is
possible, in the region of relativistic rise a statistical approach is
utilized, allowing the TPC to identify charged hadrons up to $p_{\rm{T}}$ of a
few tens of $\mathrm{GeV}/c$.
Electron identification in ALICE is carried out by the TRD in the momentum
region $p>1$ $\mathrm{GeV}/c$, with a pion rejection factor of $~{}100$. The
PID relies on a 1-dimentional likelihood approach, which makes it possible to
distinguish between pions and electrons due to the different shapes of the
signals they release in the detector111The signal from electrons in a TRD
detector is characterized by a further peak at late times due to the presence
of transition radiation photons, which is absent in the case of pions..
Charged hadrons in the intermediate momentum range (i.e. up to a few
$\mathrm{GeV}/c$) are identified in ALICE by the TOF detector. In this case,
the mass (and as a consequence the identity) of a particle is obtained by
combining the measurement of its time of flight (from TOF) and its momentum
(from ITS and TPC). The reference time of the event is given by a combination
of the the event time information from the ALICE T0 detector, and the one
estimated from the particle arrival times measured by TOF. The right panel of
Fig. 2 shows the TOF resolution for the identification of pions (the most
abundant particle specie) in terms of the difference between the measured time
of flight, and the expected one calculated from the track length and momentum
assuming that the particle is a pion. As one can see, the detector performance
is outstanding, with $<90$ ps of resolution.
|
---|---
Figure 2: Left panel: Energy loss measured by the TPC as a function of
rigidity for negative tracks. The inset shows the distribution of
$m^{2}/z^{2}$ for the light nuclei candidates obtained with the TOF detector.
Right panel: Difference between the measured and the expected time of flight
in the pion hypothesis. Superimposed, the gaussian fit of the distribution.
### 3.2 PID with single-arm detectors
Hadron identification at high momenta (up to 3–5 $\mathrm{GeV}/c$ depending on
the specie) is performed by the HMPID detector. This is a single-arm proximity
focusing RICH, which determines the $\beta$ of a particle from the measurement
of the Cherenkov angle. This information is then combined with the momentum
measured by the TPC and ITS to assign an identity to the particle. The left
panel of Fig. 3 shows the Cherenkov angle measured by the HMPID as a function
of $p$. As one can see, the $\pi$, K and p bands are clearly distinguishable.
The two ALICE electromagnetic calorimeters, PHOS and EMCAL, have partial
$\eta$ and $\phi$ coverage as well. They measure $\gamma$ up to 100 and 250
$\mathrm{GeV}$ respectively. The EMCAL is also used in ALICE to help hadron
rejection when identifying electrons, thanks to the $E/p$ distribution
characteristically peaked at 1 only for electrons due to their small mass.
At forward rapidities where the multiplicities are too high to use
calorimetry, photons are identified in ALICE also using the PMD, through the
pre-shower method.
On the opposite side of the experiment, a MUON spectrometer reconstructs and
identifies muons in the momentum range $p>4$ $\mathrm{GeV}$. Hadron rejection
is possible requiring matching between the tracks reconstructed by the
tracking chambers with one track segment in the triggering chambers. Moreover,
geometrical and topological cuts are applied in order to reduce contamination
from fake tracks, and Monte Carlo simulations are used to estimate muon
contributions from hadron decays. The right panel of Fig. 3 shows the
invariant mass distribution of $\mu$ pairs ($1<p_{\rm{T}}<4$ $\mathrm{GeV}/c$)
reconstructed and identified by the MUON detector after background
subtraction. The various contributions to the spectrum are shown.
|
---|---
Figure 3: Left panel: Cherenkov angle measured by the HMPID as a function of
$p$. Right panel: Invariant mass distribution of muon pairs measured by the
MUON detector.
## 4 Physics Results with PID
Many of the ALICE physics results rely on Particle Identification. Since a
comprehensive review is not possible in these proceedings, only a few examples
will be presented. For more information about the ALICE results, see [3].
Hadron identification is one of the key elements in the femtoscopy studies.
These are aimed at the measurement of the size and shape of the emitting
source especially important in Pb–Pb collisions, by utilizing the Bose-
Einstein correlations between identical particles. When pion pairs are used,
the particle identification response of the TPC is used. Studies are also made
for charged kaons, and in this case the analysis depends on both TPC and TOF
PID. Recently, results using neutral kaons have also become available despite
the smaller statistics, In this case, the topological PID is exploited for the
K0 identification.
The knowledge of the particle composition of the low $p_{\rm{T}}$ hadrons at
mid-rapidity is important in order to understand the hadronization mechanisms.
ALICE combines the responses of its different PID detectors in order to build
the identified charged hadrons spectra as shown in the left panel of Fig. 4
for the case of positive pions, kaons and protons. Here, the ITS, TPC and TOF
PID information222The inclusion in the analysis of the HMPID is ongoing. are
used allowing to extend the PID reach of the experiment. The fit with the Lévy
function are superimposed.
|
---|---
Figure 4: Left panel: Identified pions, kaons and protons using the
combination of the information from ITS, TPC and TOF. The Lévy fit are
superimposed. Right panel: Inclusive electron spectrum measured by three ALICE
analysis using different detectors.
The right panel of Fig. 4 shows the inclusive electron spectrum measured by
ALICE in pp collisions at 7 TeV. Here, the results from three different
analysis based on the use of different combination of detectors are
superimposed. Up to 3 $\mathrm{GeV}/c$, the TPC and TOF are used for the
identification of electrons; in the range $1<p_{\rm{T}}<8$ $\mathrm{GeV}/c$,
the TRD is included in the analysis. Finally in the $p_{\rm{T}}$ range between
3 and 7 $\mathrm{GeV}/c$ the EMCAL is used together with the TPC. From the
figure one can see their remarkable agreement in the overlapping ranges, and
will infer the complementarity of the ALICE PID techniques.
## 5 Conclusions
The ALICE experiment at the LHC is endowed with many different PID detectors.
Thanks to the different momentum range that they cover, the ALICE PID
capability is extended to a broad momentum interval. The use of the most up-
to-date techniques and technologies makes ALICE PID capabilities unique. Many
analysis reckon on PID results, both in pp and Pb–Pb collisions. The
outstanding PID results demonstrates that ALICE is fully complying with its
wide and varied physics program.
## References
* [1] F. Karsch and E. Laermann, In *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 1-59 [hep-lat/0305025].
* [2] F. Carminati et al., ALICE Collaboration, Physics Performance Report Vol. I, CERN/LHCC 2003-049 and J. Phys. G30 1517 (2003); B. Alessandro et al., ALICE Collaboration, Physics Performance Report Vol. II, CERN/LHCC 2005-030 and J. Phys. G32 1295 (2006); K. Aamodt et al., ALICE Collaboration, JINST 3 (2008) S08002.
* [3] B. Abelev, ALICE Collaboration, these proceedings; P. Antonioli, ALICE Collaboration, these proceedings; L. Bianchi, ALICE Collaboration, these proceedings; D.J. Kim, ALICE Collaboration, these proceedings; P. Kuijer, ALICE Collaboration, these proceedings; C. Loizides, ALICE Collaboration, these proceedings; A. Ortiz Velasquez, ALICE Collaboration, these proceedings; Y. Pachmayer, ALICE Collaboration, these proceedings; H.D.A. Pereira Da Costa, ALICE Collaboration, these proceedings.
|
arxiv-papers
| 2012-09-25T15:03:25 |
2024-09-04T02:49:35.548243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chiara Zampolli (for the ALICE Collaboration)",
"submitter": "Chiara Zampolli",
"url": "https://arxiv.org/abs/1209.5637"
}
|
1209.5745
|
# Numerical Simulations of the Dark Universe:
State of the Art and the Next Decade
Michael Kuhlen mqk@astro.berkeley.edu Mark Vogelsberger
mvogelsb@cfa.harvard.edu Raul Angulo reangulo@slac.stanford.edu Theoretical
Astrophysics Center, University of California Berkeley, Hearst Field Annex,
Berkeley, CA 94720, USA Hubble Fellow, Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Max-Planck-Institute
for Astrophysics, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo
Park, CA 94025, USA
###### Abstract
We present a review of the current state of the art of cosmological dark
matter simulations, with particular emphasis on the implications for dark
matter detection efforts and studies of dark energy. This review is intended
both for particle physicists, who may find the cosmological simulation
literature opaque or confusing, and for astro-physicists, who may not be
familiar with the role of simulations for observational and experimental
probes of dark matter and dark energy. Our work is complementary to the
contribution by M. Baldi in this issue, which focuses on the treatment of dark
energy and cosmic acceleration in dedicated N-body simulations.
Truly massive dark matter-only simulations are being conducted on national
supercomputing centers, employing from several billion to over half a trillion
particles to simulate the formation and evolution of cosmologically
representative volumes (cosmic scale) or to zoom in on individual halos
(cluster and galactic scale). These simulations cost millions of core-hours,
require tens to hundreds of terabytes of memory, and use up to petabytes of
disk storage. Predictions from such simulations touch on almost every aspect
of dark matter and dark energy studies, and we give a comprehensive overview
of this connection. We also discuss the limitations of the cold and
collisionless DM-only approach, and describe in some detail efforts to include
different particle physics as well as baryonic physics in cosmological galaxy
formation simulations, including a discussion of recent results highlighting
how the distribution of dark matter in halos may be altered. We end with an
outlook for the next decade, presenting our view of how the field can be
expected to progress.
###### keywords:
numerical simulations , dark matter , dark energy , cosmology , structure
formation
††journal: Physics of the Dark Universe
## 1 Introduction
For almost 80 years now [1] astronomers have been accumulating evidence that
the dominant form of matter in the universe is dark and non-baryonic. Just a
bit more than a decade ago, we obtained solid observational evidence of cosmic
acceleration [2, 3], requiring yet another mysterious contribution to the
total energy budget of the universe, dark energy. The conclusion is
staggering: we live in a universe that is energetically dominated by dark
matter (DM) and dark energy (DE). For DM at least we have a well-motivated
theoretical framework, in which it is comprised of fundamental particles
predicted to arise in extensions of the standard model of particle physics
like supersymmetry [4]. There are many ongoing and proposed efforts to obtain
experimental confirmation of the hypothesis of particle DM, for example
through so called indirect (annihilation/decay) and direct (nuclear
scattering) detection. These signatures depend on the detailed distribution of
DM throughout the universe, from cosmic all the way down to sub-galactic
scales. For DE, on the other hand, we have only a very rudimentary theoretical
understanding, and are a firmly in an exploratory phase, conducting and
designing future surveys and measurements that will provide us with additional
clues to its nature.
The last 40 years have seen tremendous progress in our understanding of cosmic
structure and galaxy formation. Much of this understanding has come at the
hand of beautifully simple analytic arguments and insights. The calculation of
the Cold Dark Matter (CDM) power spectrum [5, 6], Press & Schechter theory
[7], the statistics of peaks in Gaussian random fields [8], and White & Rees’
galaxy formation model [9] are a few seminal examples. Yet, it is clear that
purely analytical approaches have arrived at the limits of their reach. Fueled
by continuing advances in numerical methods and computational capabilities,
the future of structure formation and galaxy formation theory is going to be
led by numerical simulations.
Indeed, in the last few decades Moore’s law combined with heavy infrastructure
investments has already tremendously increased available computing resources,
and the field of computational cosmology has taken full advantage. Full-box
simulations of substantial fractions of the observable Universe are now being
conducted with over half a trillion particles, and zoom-in simulations of
individual halos have exceeded the billion particle level. In this review we
present a snapshot of the current state of the art of cosmological numerical
simulations, with a particular emphasis on the DM and DE problems. Its
intended audience is both the particle physicist with an interest in DM and
DE, who may find the simulation literature to be opaque and confusing, as well
as the astro-physicist, who may not be up to speed with observational and
experimental probes of DM and DE or with the importance of simulations for
their interpretation. Our work is complementary to the contribution by M.
Baldi in this issue [10], which focuses on the treatment of DE and cosmic
acceleration in dedicated N-body simulations.
This review is structured as follows. We begin in §2 with a review of the
domain of DM simulations. What sorts of predictions do they make? How are
these predictions relevant for DM detection efforts and for probes of DE? We
cover astrophysical probes, indirect, and direct detection of DM, as well as
probes of DE in the form of baryon acoustic oscillations, redshift-space
distortions, cluster mass functions, and weak gravitational lensing. In §3 we
present a survey of the state of the art in late 2012 of cosmological DM
simulations on cosmic, cluster, and galactic scales. We give an overview of
the most commonly employed numerical techniques, then describe some of the
largest simulations run to date, and the computational resources they used. We
discuss the limits of the cold and collisionless DM-only approach, and efforts
to go beyond it by simulating alternative DM physics. We include an discussion
of hydrodynamic galaxy formation simulations, cover numerical methods,
algorithmic and technical difficulties, and highlight some recent results
regarding how the DM distribution in halos may be altered by baryonic physics.
Finally, in §4 we present our vision of this field for the next decade. What
are the important questions to tackle, and how best to do so? What
developments should be pursued in order to take advantage of technological
advances?
## 2 Dark Matter Simulations and the Dark Universe
Figure 1: $\Delta^{2}(k)\equiv 4\pi(k/2\pi)^{3}P(k)$, the linear power
spectrum of density fluctuations at $z=0$. The solid line is the canonical
cold DM model with an Eisenstein & Hu (1997) [11] transfer function. The
dashed line is a thermal relic warm DM model with $m_{\rm WDM}=8$ keV [12].
The dotted line is an atomic DM model [13]. We used WMAP7 cosmological
parameters [14], $\Omega_{m}=0.265$, $\Omega_{\Lambda}=0.735$,
$\Omega_{b}=0.0449$, $h=0.71$, $\sigma_{8}=0.801$, and $n_{s}=0.963$.
The numerical simulation discussed in this review together span an enormous
range of length scales, more than 8 orders of magnitude reaching from near
horizon scale ($\sim$ 20 Gpc) down to sub-Galactic (tens of pc). Individually
they focus on different regimes (see §3 and Table 2), but all have in common
that they evolve the growth of DM density fluctuations all the way to the
present epoch at redshift zero.111We deliberately omit from our discussion
multi-billion particle simulations that focus only on the first billion years
of cosmic evolution, for studying the epoch of reionization [15] or early
supermassive black hole growth [16].
The shape of the CDM power spectrum results in a hierarchical, bottom-up
process of structure formation, in which small and low mass objects collapse
first and over time merge to form ever more massive structures, until the
onset at $z\approx 1$ of DE induced accelerated expansion begins to halt
further collapse. In Fig. 1 we show a plot of the linear dimensionless matter
power spectrum $\Delta^{2}(k)\equiv 4\pi(k/2\pi)^{3}P(k)$ at $z=0$ versus the
wavenumber $k$ of the fluctuation. Where $\Delta\gtrsim 1$, gravitational
collapse will have proceeded to the non-linear regime and typical objects of
the corresponding mass will have collapsed. Cosmic scales, including the
Baryon Acoustic Oscillation feature discussed in §2.3.i, remain in the linear
or mildly non-linear regime, while cluster and galactic scales are strongly
non-linear. Note that computational demands grow strongly with the degree of
non-linearity resolved in the simulation.
Observational constraints from the Cosmic Microwave Background, cluster
abundances, galaxy clustering, weak lensing and the Lyman-$\alpha$ forest have
constrained the power spectrum of density fluctuations and provide a
remarkably good agreement with the predictions of $\Lambda$CDM cosmology. On
smaller scales (shaded gray in Fig. 1) we currently do not have robust
observational constraints, and here numerical simulations typically rely on
extrapolations under the assumption of CDM theory. As we discuss in §3.3,
different assumptions are plausible and are the subject of many ongoing
investigations. As an example, we show two alternative models with a
suppression of small scale power, a warm DM [12] and an atomic DM [13] model.
### 2.1 Domain
In the following we provide a brief summary of the domain of cosmological DM
simulations, roughly organized from large scales to small.222Also see [17] for
a recent review of simulation results concerning the evolution and structure
of CDM halos. This is not meant to be an exhaustive review of all current
results, but rather an overview of those results with particular relevance for
DM and DE experiments. The references that we list provide a jumping-off point
for further reading.
1. i)
Large Scale Structure
The largest scale density fluctuations in the universe never grow beyond
mildly non-linear, and even early CDM simulations predicted that the large
scale distribution of DM in the universe is not completely homogeneous,
instead exhibiting voids, walls, and filaments whose statistical description
is in remarkable agreement with the large scale distribution of galaxies [e.g.
18].
Simulations spanning an ever larger fraction of the volume of the observable
Universe at increasingly high resolution have been able to quantify the DM
density and velocity fields as well as the halo mass function together with
the full hierarchy of halo correlation functions, and their evolution with
cosmic time [19, 20, 21, 22, 23].
2. ii)
Individual isolated halo properties
On the scale of individual halos, DM-only numerical simulations have measured
halo shapes to show significant departures from sphericality, with halos
typically being prolate and increasingly so towards their centers. Major-to-
minor axis ratios of 2 or greater are not uncommon, and more massive halos
tend to be less spherical than lower mass halos [24, 25]. Shapes and
kinematics seem to be closely connected. While the spherically averaged
anisotropy profile ($\beta(r)=1-0.5\,\sigma_{t}^{2}/\sigma_{r}^{2}$) grows
from zero (isotropic) to about 0.4 (mild radial anisotropy) [26, 27], the
local $\beta$ values correlate with halo shape: positive (radial) on the major
axis and negative (tangential) on the minor axis [28, 29].
The DM mass distribution within halos is well described by a near-universal
density profile, the so-called NFW profile [30], which has the form of a
double-power-law with the logarithmic slope $\gamma\equiv{\rm
d}\\!\log\rho/{\rm d}\\!\log r$ transitioning at the scale radius $r_{s}$ from
$\gamma=-3$ at large radii to $\gamma=-1$ in the center. More recent higher
resolution simulations, however, have found a central slope shallower than
$\gamma=-1$, indicating that the density profile may be better described by a
functional form with a central slope gradually flattening to $\gamma=0$, e.g.
the Einasto profile [31, 32].
The scaling of the transition radius $r_{s}$ with halo mass, formation time,
and environment is typically described in terms of a “concentration”, defined
as the ratio of the virial radius to the scale radius, $c=R_{\rm vir}/r_{s}$.
DM simulations have quantified the concentration-mass relationship, its
scatter, and its evolution with time [33, 34, 35]. Concentrations typically
increase for lower mass halos, presumably reflecting their earlier collapse
times when the mean density of the universe was higher, although recent work
has reported an upturn of concentrations at high masses [36] presumably caused
by out-of-equilibrium systems [37].
Lastly, we mention the remarkable finding from simulations that the pseudo-
phase-space profile, the ratio of the spherically averaged DM mass density to
the cube of its spherically averaged radial velocity dispersion, is well
described by a single power-law, $Q(r)\equiv\rho(r)/\sigma_{r}(r)^{3}\sim
r^{-1.84}$, even though neither the density nor the velocity dispersion
profiles by themselves are [38, 39, 40, 32]. The power law slope is remarkably
close to analytic predictions based on spherical secondary-infall similarity
solution [41] and their generalization [42] in the inner, virialized regions
of halos [43]. Departures from a pure power-law occur around the virial
radius, close to the location of first shell crossing, where particles have
not yet fully virialized. Note also that the low velocity dispersion in
subhalos leads to large fluctuations in local estimates of the phase-space
density and thus its spherical average does not follow a single power law [32,
43].
3. iii)
Substructure
The numerical resolution achievable by state of the art DM-only simulations
has grown to the point where it has become possible to follow bound DM
structures beyond their merging with a larger halo. This has allowed studies
of DM substructure, consisting both of a population of surviving self-bound
subhalos orbiting within the potential of their hosts and the debris
associated with their tidal stripping and disruption.
These simulations have been able to probe the subhalo mass function and
$\mathbf{V_{\rm max}}$ function over $\sim 5$ of magnitude in subhalo mass
[44, 45], and have shown that they are well fit by simple power laws, ${\rm
d}N/{\rm d}M\sim M^{-1.9}$ and $N(>V_{\rm max})\sim V_{\rm max}^{-3}$. It has
even been possible to resolve up to 4 levels of the sub-substructure hierarchy
[45], but the statistics are currently not sufficient to quantify sub-
substructure scaling laws.
As with isolated halos, determining subhalo density profiles and
concentrations is of great interest [44, 45]. At current resolutions, subhalo
density profiles appear to be well fit by both NFW or Einasto profiles, and
there is some evidence for a radial scaling of subhalo concentration, with
higher concentrations for subhalos closer to the center [46, 45]. The latter
effect is likely due to a combination of stronger tidal forces and the earlier
collapse times of subhalos found close to the center [47]. The spatial
distribution of subhalos within their hosts appears to be “anti-biased” with
respect to the host’s mass distribution [26, 27, 46, 48, 45, 49], meaning that
the subhalo density normalized by the host’s mass density profile decreases
towards the center. The degree of this anti-bias depends on how the subhalo
sample is selected: a current mass-selected sample is more affected by tidal
stripping, which is stronger closer to the host’s center, resulting in a
pronounced anti-bias than a for sample that is selected by properties
unaffected by tidal stripping, like the mass before accretion [50, 51].
Velocity-space substructure, in the form of tidal streams and debris flow, is
another topic that has received attention [52, 28, 53, 54]. DM tidal streams
have low configuration space density, typically only a few per cent of the
underlying host halo density. However, since they are significantly colder
than the host halo, they have a high phase-space density contrast. Another
form of velocity-space substructure may be contributed by a so-called dark
disk [55], a component of the DM halo that is co-rotating with the stellar
disk, which may have been deposited by satellites disrupted in the plane of
the Galaxy.
On smallest scales the fine-grained phase-space structure describes the
detailed distribution of DM in configuration and velocity space. Before the
onset of nonlinear structure formation CDM was almost uniformly distributed
with particles lying on a thin three-dimensional hypersurface embedded in six-
dimensional phase-space. Due to their collisionless character DM particles
then follow the Vlasov-Poisson equations leading to stretching and folding of
this initial sheet. Therefore, at later times the velocity distribution of DM
at a given point in configuration space is a superposition of fine-grained
streams of different velocities. Furthermore, phase-space fold catastrophes
lead to caustics, where the DM configuration space density can become very
large, many order of magnitudes larger than the mean halo density [56, 57]. It
has only very recently become possible to study these effects numerically by
extending classical N-body schemes [58, 59].
4. iv)
Local DM
Lastly, numerical simulations provide expectations regarding the DM
distribution at the solar circle, $\sim 8$ kpc from the Galactic Center.
Simulations indicate that the local density of DM is likely to be quite smooth
and uniform [52, 60], since strong tidal forces disrupt most subhalos close to
the center. The flipside of this coin is that the local neighborhood should be
crossed by many DM tidal streams [52, 53], cumulatively referred to as debris
flow [54].
### 2.2 Relevance for Dark Matter Detection
| LSS | Halos | Substructure | Local
---|---|---|---|---
| voids, walls, filaments | halo mass functions | concentration-mass relation | halo shapes | density profiles | pseudo-phase-space density | mass (or Vmax) functions | density profiles | central density | spatial distribution | streams | folds & caustics | local density | tidal streams | dark disk
Astrophysical | Dwarf galaxy abundance | | | | | | | | | | | | | | |
Dwarf galaxy kinematics | | | | | | | | | | | | | | |
Stellar streams | | | | | | | | | | | | | | |
Gravitational lensing | | | | | | | | | | | | | | |
Indirect Detection | Extra-galactic DGRB | | | | | | | | | | | | | | |
Galactic DGRB | | | | | | | | | | | | | | |
Clusters | | | | | | | | | | | | | | |
Galactic Center | | | | | | | | | | | | | | |
Milky Way Dwarfs | | | | | | | | | | | | | | |
Dark Subhalos | | | | | | | | | | | | | | |
Local anti-matter | | | | | | | | | | | | | | |
Neutrinos from Earth & Sun | | | | | | | | | | | | | | |
Substructure boost | | | | | | | | | | | | | | |
Sommerfeld boost | | | | | | | | | | | | | | |
Direct | “Vanilla” $\sim\\!100$ GeV DM | | | | | | | | | | | | | | |
light / inelastic DM | | | | | | | | | | | | | | |
axions | | | | | | | | | | | | | | |
directionally sensitive experiments | | | | | | | | | | | | | | |
Table 1: A matrix showing which predictions from numerical DM simulations
affect which astrophysical probes, indirect, and direct DM detection efforts,
and vice versa.
Predictions from cosmological DM simulations affect nearly all DM detection
efforts, in a variety of ways. In the following sections we highlight some of
these inter-dependencies, which are also summarized in Table 1.
#### 2.2.1 Astrophysical Probes
1. i)
Dwarf galaxies
The abundance of dwarf satellite galaxies orbiting our Milky Way and M31 is
potentially sensitive to the nature of DM (see §3.2). Results from CDM
simulations have been used to predict how many more ultra-faint dwarf galaxies
should be detected, once surveys more sensitive than the Sloan Digital Sky
Survey (SDSS) and covering the southern hemisphere (e.g. DES, Skymapper, Pan-
STARRS, LSST) come online [61]. The kinematics of stars in the very centers of
dwarf spheroidal galaxies have been used to constrain the DM mass enclosed
within their half-light radius ($\sim 300$ pc) [62, 63] and the shape of the
DM density profile of their host halos, and these measurements have been
confronted with the predictions from CDM simulations [66, 64, 65, 67].
2. ii)
Stellar streams
The discovery in the SDSS of extended stellar streams [68], arising from the
tidal disruption of dwarf galaxies, has provided first-hand evidence for the
hierarchical build-up of the Milky Way. DM counterparts to these stellar
streams are seen in numerical simulations [28, 69], and raise expectations
that many more stellar streams remain to be discovered [70]. Additionally,
cold stellar streams associated with the disruption of globular clusters [71]
are promising probes of the clumpiness of the Milky Way’s DM halo, since
interactions with passing subhalos should produce kinks and gaps the stream
[72, 73].
3. iii)
Gravitational lensing
Gravitational lensing provides important probes of DM on cosmic, cluster, and
galactic scales that can be compared to the predictions from numerical
simulations. We can distinguish between weak lensing (see also §2.3.iv), which
can be used to tomographically map the large scale distribution of DM [74] and
to measure the total masses and shapes of individual halos through cluster and
galaxy-galaxy lensing [75, 76], and strong lensing, which can probe the
central slope of DM density profiles [77] and is sensitive, through flux ratio
anomalies [78] and potentially time-delay perturbations [79], to the amount of
DM substructure present in cluster and galaxy halos [80, 81]. Recent studies
comparing to predictions from numerical simulations tend to find that the
amount of substructure present in DM halos may be insufficient to explain the
observed occurence of flux ratio anomalies [82, 83, 84]. However, the effects
of intervening line-of-sight structures can be important [85].
#### 2.2.2 Indirect Detection
Indirect detection of DM refers to the search for the products of pair-
annihilations of DM. The direct annihilation into two photons is typically
loop-suppressed, and so the dominant annihilation channel is into quarks,
gauge (or Higgs) bosons, or directly into leptons. The hadronization of heavy
annihilation products results in gamma ray photons, electrons and positrons,
and neutrinos. All of these are potentially observable, for example with
ground based Atmospheric Cerenkov Telescopes (MAGIC, VERITAS, H.E.S.S.) and
neutrino detectors (IceCube), balloon-borne detectors (ATIC), and space-based
satellites (PAMELA, Fermi Gamma-ray Space Telescope) and experiments (AMS-02
on the International Space Station). In the following we discuss some of the
possible DM annihilation signatures.
1. i)
Extra-galactic Diffuse Gamma-ray Background
The extra-galactic diffuse gamma-ray background (DGRB) refers to the sum-total
of all gamma-ray radiation produced by DM annihilations throughout cosmic
history [86]. The amplitude of this signal depends on the large scale
distribution of DM in the universe, the evolution of the isolated halo mass
function, the concentration-mass relationship, and of course the density
profile. Clumpy substructure may also play an important role, by boosting the
annihilation luminosity of individual halos (see below). Large scale numerical
simulations (e.g. Millennium-II [87]) have been used to make predictions for
both the amplitude of such a signal and the level of its anisotropies [88],
and these have been confronted with current Fermi data [89]. The uncertainties
of these constraints are dominated by the unknown contribution of subhalos
below the simulations’ resolution limit.
2. ii)
Galactic Diffuse Gamma-ray Background
A second DGRB should arise from DM annihilations within the Milky Way’s halo,
with one component stemming from the smooth halo DM and another from clumpy
substructure. The substructure component is expected to have a shallower
angular intensity profile than the host halo component [90, 91], for two
reasons: (i) since it consists of the combined emission from a population of
subhalos, it should scale with radius as the number density of subhalos
$n_{\rm sub}(r)$, rather than as the square of the DM density $\rho_{\rm
host}(r)^{2}$, and (ii) $n_{\rm sub}(r)$ is anti-biased with respect to
$\rho_{\rm host}(r)$, with less substructure found close to the host’s center.
DM substructures introduce characteristic anisotropies in the Galactic DGRB
[92], which may allow the signal to be disentangled from an astrophysical
DGRB.
The detectability of the Galactic DGRB from DM annihilation thus depends on
the abundance of substructure, its internal structure (concentrations and
density profiles), and its spatial distribution within the host halo. If the
substructure contribution remains sub-dominant, the shape of the Milky Way’s
DM halo may determine the large-scale angular variations of the signal.
3. iii)
Galaxy Clusters
Galaxy clusters are the most massive gravitationally bound systems in the
universe, and thus have long been considered good targets for indirect
detection searches [93]. The detectability of an annihilation signal from
clusters relies on a substantial cross section boost (either from substructure
or Sommerfeld enhancement, see below) [94, 95], and the resulting emission
would likely be extended. A difficulty is that any gamma-ray signal from
annihilation has to compete with the cosmic ray induced gamma-ray emission
[96]. Nevertheless, H.E.S.S. [97] and Fermi [95] data have been able to
constrain DM parameters, and there is tentative evidence for a $\sim 130$ GeV
line signal from a subset of the most promising cluster targets [98].
4. iv)
The Galactic Center
The most prominent DM annihilation signal is thought to arise from the
Galactic Center (GC) [99], given its proximity ($\sim 8$ kpc) and the expected
high DM density there. Unfortunately, the GC is also an extraordinarily
astrophysically active place [100] hosting numerous SN remnants, pulsars,
X-ray binaries, and other high-energy sources, not to mention a super-massive
black hole. Although these astrophysical foregrounds encumber DM searches
directed towards the GC, it nevertheless has remained a popular target for
indirect detection efforts. In fact, several gamma-ray “excesses” and
anomalies from the GC have been reported [101, 102, 103, 104], including the
recent intriguing reports of a gamma-ray line at $\sim 130$ GeV [105, 106,
108, 107]. It is too early to confidently claim a detection of DM annihilation
for any of these signals, and additional data will be necessary before
statistical fluctuations, instrumental effects, or astrophysical sources can
be ruled out.
The strength of the GC DM annihilation signal depends sensitively on the shape
of the Milky Way host halo’s DM density profile at scales that are currently
not well resolved in numerical simulations. Predictions thus rely on
extrapolations of fitting function that have been calibrated at larger radii
(several hundred of pc) to a small number of high resolution simulations [44,
32, 31]. Furthermore, the gravitational potential is baryon dominated at the
GC, and one must thus account for modifications of the DM density profile due
to baryonic physics. As discussed in more detail below (§3.4), these processes
may lead to either a steepening or a flattening of the profile, and may even
displace the location of maximum DM density from the dynamical center.
5. v)
Milky Way Dwarf Galaxy Satellites
The most DM dominated objects known are the Milky Way dwarf spheroidal
satellite galaxies, which have mass-to-light ratios of up to a 1000 [109].
They are thus natural candidates for indirect detection searches [110]. Since
their distances are fairly well known, the detectability of their DM
annihilation signal is determined by the mass, concentration, and density
profile of their DM host halos. For many of these systems, stellar kinematic
data has provided tight constraints on the enclosed mass within the stellar
half-light radius [63], under assumptions of equilibrium and spherical
symmetry. Fermi data from individual and stacked dwarf galaxies have provided
some of the most stringent limits on the DM annihilation cross section,
extending to below $3\times 10^{-26}\,{\rm cm}^{3}\,{\rm s}^{-1}$ for a DM
particle mass of $\sim 40$ GeV and annihilation into pure $b\bar{b}$ [111],
but these limits assume cuspy NFW-like DM density profiles and may be
significantly weakened if baryonic processes or departures from the CDM
assumption result in a flatter profile than inferred from DM-only simulations.
6. vi)
Dark Subhalos
The vast majority of subhalos predicted in CDM cosmology are expected to be
completely dark, since their masses are too low to have allowed gas to cool
and form stars [112]. Individual dark subhalos may be promising sources for
indirect detection searches, and results from high-resolution simulations have
been used to quantify their detectability [113, 91, 114]. The Fermi point
source catalog contains hundreds of “unassociated” sources without identified
astrophysical counterparts [115], and it is possible that DM subhalos may be
lurking among them [116]. Very nearby sources could appear as faint, spatially
extended gamma-ray sources to Fermi [91], and it may even one day be possible
to measure proper motions of very nearby subhalos [117].
Once again, these results are highly uncertain due to insufficient knowledge
of the abundance, spatial distribution, and luminosity-mass relation of
subhalos on scales below the simulations’ resolution limit, as well as their
ability to survive interactions with the Galactic disk.
7. vii)
Local Anti-matter
DM annihilations in the local neighborhood would produce high energy positrons
and anti-protons, either through direct annihilation into leptons
($e^{-}e^{+},\mu^{-}\mu^{+},\tau^{-}\tau^{+}$) or via the hadronization and
decay of other annihilation products. These high energy anti-particles might
be detectable as an excess over astrophysical cosmic ray backgrounds, and have
been searched for by the Fermi [118], H.E.S.S. [119], PAMELA [120], ATIC-2
[121], and AMS-02 [122] experiments, among others. Owing to energy losses from
inverse Compton and synchrotron radiation, the propagation distance for
positrons is short ($\sim 1$ kpc), and thus any injection of positrons from DM
annihilations would have to originate from the local neighborhood. The
expected contribution from DM annihilations hence depends on the local density
of the Milky Way halo at 8 kpc. The presence of subhalos within $\sim 1$ kpc
of Earth could affect both the amplitude of this signal and its energy
spectrum [123]. Improved numerical simulations with higher resolution and
accounting for baryonic physics effects will be necessary to better
characterize the role of local DM annihilations in the high energy cosmic ray
spectrum.
8. viii)
Neutrinos from Earth & Sun
DM annihilations occurring in the center of the Sun or Earth could produce
high energy neutrinos that may be observable with neutrino observatories like
AMANDA [124] and IceCube [125]. DM particles can be captured by the Sun and
Earth through elastic scattering off of heavy nuclei [126]. Subsequent
scatterings then thermalize the population of bound DM particles, and an
equilibrium is established between annihilations and capture. The strength of
the signal depends on the local DM density, but additionally also on the speed
distribution of incident particles, since lower speed particles are easier to
capture [127]. Rates are thus especially sensitive to the presence of a “dark
disk” component [128], which can result in a marked increase in the fraction
of DM particles traveling at low speeds with respect to the solar system.
9. ix)
Substructure Boost Factors
Given that the smallest collapsed structures in WIMP CDM are expected to be
$\sim 10^{-12}-10^{-3}\,{\rm M}_{\odot}$ [129, 130], even the highest
resolution numerical simulations can only resolve a small fraction of the
expected substructure hierarchy. Since annihilation rates scale with the
square of the density $\rho$ and
$\langle\rho^{2}\rangle\geq\langle\rho\rangle^{2}$, any unresolved small-scale
clumpiness should result in a boost of the annihilation rate. Halo
annihilation rates calculated from average density profiles (like NFW or
Einasto), or even directly from the simulated particle distribution, could
well underestimate the true luminosity by several orders of magnitude. This
so-called substructure boost factor has been invoked to motivate effective
annihilation cross sections orders of magnitude larger than the thermal relic
value [e.g. 131, 132, 133, 134, 94].
Figure 2: An extrapolation of the subhalo contribution to the total luminosity
to masses far below the simulation’s resolution limit. Depending on what one
assumes for the concentration-mass relation, one can get very different total
substructure boost factors. Extrapolations from the high-mass behavior seen in
simulations (red dashed) or assuming a constant power law concentration-mass
relationship (green) are unlikely to hold at masses below $\sim 1\,{\rm
M}_{\odot}$ (visually indicated with thin faint lines).
In the following we discuss two important facts about substructure boost
factors that are perhaps not as widely appreciated as they should be:
1. (a)
There is no one single boost factor.
The expected substructure boost depends on the distance from the halo center,
with results from state of the art simulations implying very little (or no)
boost at the Galactic Center, possibly $\mathcal{O}(1)$ in the local
neighborhood, and perhaps as large as 100 - 1000 for the total luminosity of a
halo [44, 135, 91, 60, 114]. As a result a different boost factor applies to
spatially extended sources (Galactic DGRB, MW satellite galaxies, dark
subhalos) than for unresolved sources (distant halos, extra-galactic DGRB),
and similarly a gamma-ray boost factor may not be the same as those for
positron or anti-proton production [136]. Furthermore, if a significant
fraction of the mean density at a given radius is locked up in substructure,
then properly accounting for the substructure boost will actually lower the
smooth density contribution to the luminosity [114], further reducing the
contrast between the outer regions of a halo and its center. The total halo
luminosity boost likely depends on the mass of the halo, since numerical
simulations indicate a roughly equal contribution from every decade of
substructure mass, and larger mass host halos contain more decades of
substructure mass [94].
2. (b)
Substructure boosts depend sensitively on subhalo properties many orders of
magnitude below the resolution limit of state of the art simulations.
One approach to estimating the full substructure boost is to stay as close as
possible to the results from ultra-high-resolution numerical simulations like
Via Lactea II and Aquarius, by fitting the luminosity boost from all subhalos
with mass greater than $M_{\rm min}$, $B(M_{\rm min})=L(>M_{\rm min})/L_{\rm
smooth}$, to a power law of $M_{\rm min}$ over the 4-5 decades of substructure
mass that are currently resolved in the simulations, and then extrapolating
this power law down to the free-streaming cutoff scale. This approach was
taken, for example, by Springel et al. (2008) [135], who found $B(M_{\rm
min})\sim M_{\rm min}^{-0.226}$, and inferred a total boost factor for a
Milky-Way-like halo of $230$ for $M_{\rm min}=10^{-6}\,{\rm M}_{\odot}$.
Another approach is to use the numerical simulation results only to constrain
the mass function of subhalos, which is measured to be a power law,
$dn/dM_{\rm sub}\sim M_{\rm sub}^{\alpha}$ with logarithmic slope
$\alpha\simeq-1.9$ [44, 45], and to use an analytical approach to determine
the subhalo luminosity-mass relation down to the smallest mass halos [137, 90,
91]. The luminosity of a subhalo of mass $M$ is completely determined by its
concentration $c$, $L/M\sim c^{3}/f(c)$, where $f(c)$ depends on the shape of
the density profile: for an NFW profile, $L/M$ scales approximately as
$c^{2.24}$; for an Einasto profile, as $c^{2.46}$ [138]. The subhalo
annihilation luminosity-mass relation is then completely determined by the
concentration-mass relation. Again, one may choose to use a simple power law
relation, for example $c(M)\sim M^{-0.11}$, which well describes the
concentration-mass relation of Galactic scale halos [33]. Alternatively one
may choose a model in which the concentration of a halo reflects the mean
density of the universe at its typical collapse time, as in the analytical
model of Bullock et al. (2001) [33]. In this case, the concentration-mass
relation is not a simple power law, but instead rolls over at low masses, and
concentrations asymptotically become independent of mass. A comparison of the
three approaches discussed so far is shown in Fig. 2, which demonstrates how
sensitively the total halo boost factor depends on assumptions about the small
scale behavior of subhalo luminosities. Depending on what one assumes for the
concentration-mass relation, the total boost of a Milky Way halo ranges from 3
to 300 (for $M_{\rm min}=10^{-6}\,{\rm M}_{\odot}$). Note that these three
different approaches are not all equally likely to apply in reality. Simple
extrapolations from the high-mass behavior observed in simulations or assuming
a simple power law concentration-mass relation are inconsistent with
expectation from theoretical models of CDM structure formation. Microhalo
simulations find concentrations of the smallest and earliest collapsing DM
halos that are incompatible with a single power law $c(M)$ over the full
substructure hierarchy [139, 140, 141]. Consequently, substructure boost
factors much in excess of $\sim 10$ are unlikely to apply in nature.
A third approach, employed by Kamionkowski et al. (2010) [60], it to model the
volumetric probability function of density fluctuations, calibrate it to a
high resolution simulation (Via Lactea II) as a function of halo-centric
radius, and then to integrate this PDF to obtain an estimate of the boost
factor as a function of radius. This approach also results in a modest total
boost factor for a galaxy-scale halo of $\mathcal{O}(10)$.
Lastly, the annihilation luminosity can in principle also be enhanced by
caustics in the fine-grained substructure, however recent numerical studies of
this caustic boost find only percent level increases due to very efficient
mixing in phase-space [57].
10. x)
Sommerfeld Boost Factor
A second type of boost factor is of particle physics nature. When the mass of
the force carrier particle is sufficiently lighter than the DM particle, the
so-called Sommerfeld effect, a non-perturbative correction for long range
attractive forces, can lead to a velocity-dependent enhancement in the
annihilation cross section [142, 143]. Instead of the usual $\langle\sigma
v\rangle=$ constant behavior, with Sommerfeld enhancement the cross section
scales as $\langle\sigma v\rangle\sim 1/v$, or even $1/v^{2}$ at resonances
[144]. Although the effect typically saturates at small velocities ($v/c\sim
10^{-4}-10^{-5}$) owing to the finite range of the interaction, this effect
may significantly enhance the annihilation rate in subhalos compared to the
smooth host halo, given the subhalos’ lower velocity dispersions [145, 146,
147]. The details depend on the predictions of numerical simulations of the
velocity structure in the host and subhalos. On the fine-grained level of DM
the Sommerfeld effect can have interesting implications. Whereas in non-
Sommerfeld models the largest annihilation signal is expected to occur near
caustics due to their high density, this situation changes if Sommerfeld
enhancement processes are invoked. In that case, cold low-velocity dispersion
phase-space structures are enhanced compared to hotter regions. Liouville’s
theorem dictates that DM is very hot in caustics to preserve the fine-grained
phase-space density. Depending on the details of the Sommerfeld model this can
make fine-grained streams more prominent for annihilation radiation than
caustics, because streams are very cold due to their potentially low density.
This can cause the annihilation rate in streams to dominate over the rate of
the smooth mean density contribution in halos [148].
#### 2.2.3 Direct Detection
Direct detection refers to efforts to detect nuclear recoil signatures
produced in rare DM-nucleus scattering events in shielded underground
detectors. A large number of experiments are currently pursuing this goal, and
are utilizing a variety of different technologies and target materials (see
[149] for a review). The expected event rate and recoil spectrum depends on
the mass of the target nuclei and of the DM particle, on the nature of the
interaction (spin-dependent vs. spin-independent), the nuclear form factor,
the local DM density $\rho_{0}$, and the Earth frame velocity distribution
$f(\vec{v})$ of incident DM particles. Until recently, most event rate and
parameter exclusion calculations assumed a simplified model of the local DM
distribution, taking the local DM density to be $0.3-0.4$ GeV cm3 and a
Maxwellian speed distribution with a 1-D velocity dispersion of $\sigma=155$
km/s (such that the most likely speed $v_{0}$ is equal to the Galactic
rotation speed at the solar circle, $v_{0}=220$ km/s), and an escape speed of
550 km/s.
In recent years these assumptions have been directly confronted with the
predictions from high resolution simulations like Via Lactea II and Aquarius.
The large number of self-bound subhalos found to be orbiting in the Milky
Way’s potential raises the question of whether one might expect large
fluctuations in the DM density at the solar radius. If the Earth happened to
be passing through a subhalo, for example, the local density of DM might
significantly exceed the mean value at 8 kpc. Analytical calculations [150]
and direct “measurements” in simulations [52, 60] indicate that the volumetric
probability distribution of over-densities $\delta=\rho/\langle\rho\rangle$
consists of a narrow log-normal reflecting variations in the smooth halo
density and a $\mathcal{P}_{V}(\delta)\sim\delta^{-2}$ power law tail
extending over several orders of magnitude before steepening to $\delta^{-4}$
at an overdensity corresponding to the mean density of the universe at the
collapse time of the smallest halos. Barring dramatic changes in the abundance
and internal properties of subhalos below the simulations’ resolution limit,
the normalization of the power law tail at 8 kpc appears to be too low to lead
to a non-negligible chance of the Earth lying in a substantial overdensity. It
thus seems safe to use the mean value of the DM density at 8 kpc in direct
detection calculation. However, what that value is remains uncertain at least
at the factor of two level, with recent studies finding values ranging from
0.2 to 0.85 GeV cm-3 [151, 152, 153, 154].
The speed distribution is another matter. DM-only simulations have
definitively demonstrated that $f(v)$ shows clear departures from a pure
Maxwellian [27, 155, 156, 52, 53], with the typical shape instead exhibiting a
deficit near the peak and an excess at lower and higher velocities. This is a
consequence of the non-Gaussian nature of the three velocity components
aligned with the density ellipsoid, with the major axis component being
platykurtic (broader than Gaussian) and the other two leptokurtic (narrower)
[52]. In addition to these coarse departures from a Maxwellian, additional
small scale structures are often visible in the high speed tail, in the form
of broad bumps that are stable in both space and time [156, 52, 54] and
occasionally as narrow spikes at discrete speeds indicating the presence of a
tidal stream or subhalo in the sample volume [52, 53]. The presence of a
strong “dark disk” [55] would change the relative proportion of high speed and
low speed particles, which could affect scattering rates and the recoil
spectrum.
The scattering rate is proportional to $\int_{v_{\rm min}}^{\infty}f(v)/vdv$,
where $v_{\rm min}$ is the minimum speed that can result in a recoil of energy
$E_{R}$, which for elastic scattering is given by $v_{\rm
min}=\sqrt{m_{N}E_{R}/2\mu^{2}}$. $m_{N}$ is the mass of the target nucleus
and $\mu=m_{N}m_{\rm DM}/(m_{N}+m_{\rm DM})$ is the reduced mass. The smaller
$m_{N}$, the lower the speeds that are required to produce a given recoil
energy. This implies that experiments with different target materials and
different recoil energy sensitivities probe different parts of the speed
distribution. Likewise, the mass of the DM particle can affect what range of
speeds an experiment is sensitive to. For very massive particles ($m_{\rm
DM}\gg m_{N}$), the experiment becomes insensitive to $m_{\rm DM}$, but for
so-called “vanilla” WIMP DM with $m_{\rm DM}\approx 100$ GeV, the current
experiments’ $E_{R}$-thresholds of $\sim 10$ keV correspond to $v_{\rm
min}\approx 150$ km/s. The scattering rate will thus be dominated by fairly
low speed particles near the peak of $f(v)$ and below. In this case,
experiments will not be able to see effects from the interesting velocity
substructures (the bumps and occasional spikes) that occur primarily at high
speeds. A strong dark disk, on the other hand, may boost event rates.
On the other hand, with inelastic DM [157], for which the relation between
$E_{R}$ and $v_{\rm min}$ can become inverted, or light DM [158] ($m_{\rm
DM}\lesssim 10$ GeV) experiments would be sensitive only to much higher speed
particles, $v_{\rm min}\gtrsim 400-500$ km/s. In this case, the departures
from Maxwellian, both global and local ones, would perhaps be more important.
They could alter the shape and extent of current parameter exclusion curves
[53], potentially reconcile some (but not all) of the currently discrepant
results from different experiments [159, 160], change the phase and amplitude
of the annual modulation signal [53] and shift it to higher recoil energies
[54]. Directionally sensitive experiments [161] should be especially sensitive
to velocity substructure, since they typically have high recoil energy
thresholds ($\sim 50$ keV), implying a large $v_{\rm min}$. The majority of
high recoil energy particles may in fact be coming from a hotspot
significantly offset from the direction anti-parallel to Earth’s motion
through the halo [53], and debris flow particles can result in ring-like
features in the arrival direction [54, 162].
Besides generic WIMPs, axions provide another interesting DM candidate. Axions
were introduced to explain the absence in Nature of strong-CP (Charge
conjugation and Parity) violations [163]. They are expected to be extremely
weakly interacting with ordinary particles, so that they never were in thermal
equilibrium in the early Universe. This implies that axions can be very light
($\mu$eV range) and nevertheless form a cold (non-relativistic) component of
cosmic matter. The cosmological axion population formed out of equilibrium as
a zero momentum Bose condensate leading to a very small velocity dispersion.
In the absence of clustering their present day velocity dispersion would be
negligible ($\delta v\sim 10^{-17}c$ compared to $\delta v\sim 10^{-10}c$ for
generic WIMPs) making them a good CDM candidate. Axions can be detected
through their conversion to microwave photons in the presence of a strong
magnetic field [164]. Galactic axions have non-relativistic velocities
($\beta=v/c\sim 10^{-3}$) and the axion-to-photon conversion process conserves
energy, so that the frequency of converted photons can be written as:
$\nu_{a}=\nu_{a}^{0}+\Delta\nu_{a}=241.8\left(\frac{m_{a}}{1\mu{\rm
eV}/c^{2}}\right)\left(1+\frac{1}{2}\beta^{2}\right)\rm{MHz}$ (1)
where $m_{a}$ is the axion mass that lies between $10^{-6}$ eV/$c^{2}$ and
$10^{-3}$ eV/$c^{2}$. $5\mu$eV axions would therefore convert into
$\nu_{a}^{0}\cong 1200$ MHz photons with an upward spread of $\Delta\sim\cong
2$ kHz due to their kinetic energy. An advantage of axion detection compared
to WIMP searches is the fact that it is directly sensitive to the energy
rather than to the integral over the velocity distribution. Narrow velocity
streams can therefore be more easily detected and lead to spikes in the axion
detection spectrum. In that case the fine-grained structure can be relevant
for detection experiments. A low number of fine-grained streams could
potentially leave a distinct imprint in velocity-sensitive detector signals.
For example, recent simulations [57] predict that the most massive fine-
grained streams should be observable with axion detectors like ADMX.
### 2.3 Relevance for Dark Energy Studies
One of the simplest astrophysical observations, galaxy imaging and the
measurement of their redshifts and angular positions on the sky, has emerged
as a very powerful method to explore the nature of DE. For instance, from this
data baryon acoustic oscillations, redshift-space distortions, abundance of
galaxy clusters and weak gravitational lensing can be measured, all of which
can put constrains on the properties of DE. Consequently, several galaxy
surveys (e.g. DES, BOSS, BigBOSS, LSST, JPAS, Euclid) are planned or underway
to exploit this fact. DM numerical simulations have been crucial in this
process.
There are three main areas in which DM-only simulations are essential for the
cosmological exploration of DE. Firstly, DM simulations have been used to
quantify and understand the effects of various DE models on structure
formation in the Universe [e.g. 165, 166, 167]. This allows one to identify
possible ways of constraining or ruling out some DE models. Secondly, DM
simulations are the most reliable way to assess potential systematic errors on
modeling and cosmological parameter extraction from different experiments [e.g
168]. Since most of the DE probes involve complex and nonlinear processes, an
accurate modeling of the signal and related uncertainties in a given
experiment is of paramount importance in the discovery of new physics.
Thirdly, numerical simulations can be used to construct mock galaxy and
cluster catalogs, which help in the design and correct interpretation of
surveys aiming to constrain the properties of DE [e.g. 169]. Since current and
future surveys cover large solid angles and probe redshifts out to $z\approx
2$, simulations are forced to very large box sizes and high particle counts,
in order to model volumes comparable to the surveys while simultaneously
resolving structure down to the scale of individual galaxies.
In the following we briefly describe four of the most established probes
aiming to constrain the properties of DE and highlight the role of numerical
simulations in their development.
1. i)
Baryon Acoustic Oscillations
Before recombination, the coupling between free electrons and photons via
Thomson scattering resulted in a distinct pattern of oscillations in the
baryon and temperature power spectra. These BAOs were also imprinted in the
late-time total mass field (albeit with smaller amplitude) due to
gravitational interactions between baryons and DM. Thus BAOs are expected to
be present in the galaxy power spectrum, and could be used as a cosmic
standard ruler [11, 170, 171, 172]. Currently, the feature has indeed been
detected at high significance in different galaxy surveys: 2dFGRS [173], SDSS
[174], WiggleZ [175], 6dfGS [176] and BOSS [177], placing constrains on
cosmological parameters [e.g. 175, 178]. Future surveys are expected to
significantly tighten these constraints, in particular those on the DE
equation of state, which could rule out some DE candidates.
Large-scale DM simulations played an important role in this development. They
showed that BAOs survive the diffusing effects of nonlinear evolution and of
galaxy peculiar velocities, and that they should be detectable in biased
tracers of the density field [179, 180, 181, 182]. At the same time, numerical
simulations unveiled significant distortions in the shape of the acoustic
oscillations due to these effects [180, 182, 181, 183, 184, 185] which would
lead to a systematic error on measurements of the acoustic scale. However,
recently methods to remove this bias have been proposed [186, 187, 188], and
numerical simulations have been used to test their validity and performance.
In the future, specially-designed numerical simulations will help us to
understand and model better the impact of structure and galaxy formation on
the observed BAO signal in the clustering of galaxies.
2. ii)
Redshift space distortions
The redshift of a galaxy not only contains information about its cosmological
distance, but also about its peculiar velocity. This difference between
angular positions and redshifts creates an anisotropy in the observed two-
dimensional galaxy correlation function that can be used to establish the
relation between density and velocity fields in the Universe [189]. These
redshift-space distortions (RSD) have been historically used to constrain the
value of the matter density of the Universe [190, 191], but recently have also
been employed to constrain the gravity law [192, 193, 175, 194]. Current
measurements are consistent with General Relativity (GR), but future surveys
are expected to significantly improve the constraints. Hypothetical departures
from GR could be related to the DE model.
However, extracting this information is not trivial. Numerical simulations
have shown that linear perturbation applies only on extremely large scales
($>50$ Mpc/h) [195, 183, 196]. Quasi-linear corrections and the so-called
“finger-of-God” (FoG) cannot be neglected on smaller scales. In particular,
FoGs are produced by the motions of galaxies inside DM halos (whose velocity
is comparable to bulk motions produced by large-scale density perturbations),
which introduces a strong damping in the galaxy clustering along the line of
sight.
Given the increasing number of Fourier modes, it is desirable to use RSD down
to the smallest possible scales. Observations are usually modeled as a
combination of linear theory expectations plus a damping to account for FoG
[e.g. 192]. However, numerical simulations have highlighted the pitfalls of
this approach and the systematic error that it would introduce for future
surveys [197, 198, 193, 168]. This finding has fueled the development of more
accurate new estimators which can robustly use the clustering information at
smaller scales [199, 200]. This is another example of the importance of DM
numerical simulations in the optimal exploitation of observational datasets.
3. iii)
Abundance of galaxy clusters
The position of galaxies in an optical survey can also be used to identify
galaxy clusters. The number of these objects a function of their mass is of
great interest because it contains information about the underlying
probability distribution function of density perturbations in the Universe.
The evolution of the mass function on group and cluster scales has indeed been
used to measure cosmological parameters, helping to break degeneracies in the
constraints from other probes [201, 202, 203, 204, 205, 206]. In addition, the
highest mass end is sensitive to primordial non-Gaussianities and early DE,
and thus the detection of massive, high-redshift clusters has been used as
evidence of their existence [e.g. 207, 208, 209, 210, 211, 212, 213]. However,
uncertainties in the mass estimation of clusters and the respective Eddington
bias have seriously hampered these measurements [214, 215].
The most relevant aspect of DM simulations for this cosmological probe is the
halo mass function and its dependence on cosmology [e.g. 23]. This prediction
is usually parametrized in terms of the linearly extrapolated variance of the
underlying DM field [20, 21, 216, 22]. However, recently numerical simulations
have shown evidence for dependencies with other parameters [217]. This finding
could seriously limit the maximum performance of current approaches to
cosmological parameter extraction using clusters. For this reason, in the
future numerical simulations will probably play a direct role in the modeling
of the observed abundance of clusters.
Another important aspect in this probe is in the characterization of the
performance of cluster finder algorithms. DM-only simulations can quantifying
the impact of projection effects, misidentification of the cluster’s center,
and false detections [218, 219, 220], as well as the relation between cluster
mass and observed richness or weak lensing signal. Hydrodynamical simulations
have an analogous role for experiments using X-rays or thermal Sunyaev-
Zeldovich detected clusters.
4. iv)
Weak lensing
The light from high-redshift galaxies gets distorted by intervening mass
before it reaches us. Deep gravitational potentials cause large distortions
which can split the image of a galaxy into multiple lensed images, in a
phenomenon known as strong gravitational lensing. This has been discussed in
2.2.1 and can be used to probe the mass of DM halos and even the law of
gravity and the nature of DM. Smaller distortions in the shape (and size) of
background galaxies caused by the cosmic web are known as weak gravitational
lensing. These changes in the properties of galaxies can be related to
integrals of the DM mass power spectrum and thus they can be used to
reconstruct the full three-dimensional DM density field. This allows
measurements of the growth of structure, which can be used to constrain DE and
modified gravity [221].
The shear of galaxies has been detected statistically in many surveys [222,
223, 224, 225, 226, 227, 228] and has been used to place constraints on
cosmological parameters [229, 230, 231]. For these, the main ingredient is the
dependence of the nonlinear DM correlation function on cosmology, which is
normally taken from fitting formulae calibrated using predictions of DM
simulations [232, 233, 234, 235, 236]. However, weak lensing measurements are
affected by many sources of systematic errors: most importantly the intrinsic
alignments in the shape of galaxies caused by tidal forces [237], as well as
the PSF ellipticity caused by atmospheric distortions [238], among others.
Over the last few years extensive studies of these effects have been carried
out, with DM simulations helping to create synthetic data as well as
constraining the impact and magnitude of intrinsic alignments.
The next generation of surveys are expected to be able to reduce systematic
effects drastically, and thus will require high-precision predictions of the
nonlinear DM power spectrum down to small scales. Since perturbation theory
approaches can provide a reasonable description only in the mildly nonlinear
regime, the necessary predictions and modeling of data will have to rely on DM
numerical simulations, either directly or via emulators.
## 3 Current State of the Art
In this section we present a late 2012 snapshot of the state of the art of
cosmological numerical simulations, with a focus on runs with particular
relevance to the DM and DE problems. We first discuss DM-only simulations
(§3.1), which are mature, mostly computational resource limited, and have been
pushed to extremely high resolution, and then DM+hydro simulations (§3.4),
which are algorithmically more challenging, less well developed, limited to
lower resolution, and do not yet produce robust or even converged results.
### 3.1 Dissipationless Dark-Matter-only Simulations
#### 3.1.1 Numerical Techniques and Codes
Pure DM simulations take the ansatz of completely neglecting any dissipational
baryonic physics and treat all matter as collisionless DM. The density field
is sampled with discrete “N-body” particles, whose gravitational evolution is
governed by the Poisson-Vlasov equations in a coordinate system that is co-
moving with the mean expansion of the universe. The effects of DE are
generally confined to the expansion history, i.e. the translation between
cosmic time and expansion scale factor [239, 240, 241]. Many different
techniques have been developed to solve this set of equations, and we refer
the reader to [242, 243, 244, 245, 246] for extensive discussion. For the
present purpose, it suffices to briefly describe two of the major techniques
in use today.
One is the so-called tree code [247], in which the particle distribution is
organized in a hierarchical tree structure. Contributions to the gravitational
potential from distant particles are approximated by the lowest order terms in
a multipole expansion of the mass distribution at a coarse level of the tree.
If accuracy requirements demand it, the tree is “opened” to a higher level and
a more detailed particle distribution is accounted for. This method reduces
the computational complexity of the N-body problem from $\mathcal{O}(N^{2})$
to $\mathcal{O}(N\log N)$, with a well controlled error. A further improvement
to $\mathcal{O}(N)$ scaling is possible through the use of the Fast Multipole
Method (FFM) [248], in which forces are calculated between two tree nodes
rather than between individual particles and nodes. In order to avoid
unphysical hard scatterings between nearby particles (which are just tracers
of an underlying continuous density field), gravitational interactions are
“softened” on small scales [249], typically either with a Plummer or a cubic
spline kernel. The force resolution of this method is then given by the
softening length $\epsilon_{\rm soft}$, which in DM-only simulations is
usually kept constant in co-moving coordinates. Pkdgrav2 [244] is a prominent
example of a pure tree code, and it uses FFM.
The second commonly used N-body technique is the adaptive particle-mesh (PM)
method, in which the particles are deposited onto an regular mesh to produce a
density field. The mesh structure is often adaptively refined in high density
regions demanding increased force accuracy. The gravitational potential is
obtained via Fourier transform on the periodic root grid (coarsest level), and
a multi-grid relaxation technique is used to evaluated it on the refined
grids. This method also achieves $\mathcal{O}(N\log N)$ scaling, but here $N$
refers to the number of mesh cells, which is typically taken to be $2^{3}$
times the number of particles. No explicit force softening is necessary, since
particles interact with each other through a mean field not individually, and
the force resolution is effectively given by the cell spacing of the most
refined mesh. Examples of pure adaptive-PM codes are Art [242] and Ramses
[250].
One of the most widely used cosmological simulation codes is the hybrid Tree-
PM code Gadget [245], which uses the PM method to evaluate long range forces
and the tree method for short range interactions. The Gotpm code [251] is
another example of such a hybrid.
The choice of gravitational softening length in cosmological simulations with
tree codes is a contentious issue that has not been truly resolved. The
difficulty arises because there are conflicting demands on the softening [252,
253]: on the one hand it is desirable to choose as small of a softening as
allowed by computational resources (the smaller the softening, the shorter the
time steps, and the more expensive the simulation), since it represents a
distortion of the true gravitational potential and leads to overmerging [254].
On the other hand, smaller softening results in stronger unphysical two-body
relaxation effects, which can cause spurious heating as well as artificial
fragmentation [255, 256]. Some studies have advocated for softening lengths no
smaller than the mean particle separation in the initial conditions [255,
256], while others have argued that it is sufficient to choose a softening
that suppresses unphysical discreteness effects in collapsed region [257,
258]. Cosmological zoom-in simulations (see §3.1.2) generally employ
softenings significantly below the mean initial condition separation of high-
resolution particles (e.g. ranging from 0.006 to 0.02 for the zoom-ins in
Table 2). As an aside, the need to avoid two-body relaxation effects is
responsible for the slow $\approx N^{1/3}$ rate of convergence of halo density
profiles [258]. Note that due to the hierarchical nature of collapse in CDM,
no matter how many particles are used in a simulation, the first structures to
collapse are always the smallest halos that are resolved with only a small
number of particles and hence susceptible to 2-body relaxation effects.
#### 3.1.2 Simulations
Cosmological DM-only simulations can be divided into two types: (i) full-box
and (ii) zoom-in simulations. The former resolve the entire computational
domain with a single particle mass and force resolution, and typically cover a
representative volume of the universe, with box sizes ranging from $\sim 100$
Mpc to tens of Gpc. They are generally focused on resolving the large scale
structure of the universe and are most useful for statistical studies of DM
halos. We refer to this class as cosmic scale simulations.
DM-only simulations
---
Cosmic |
Name | Code | $\rm L_{box}$ | $\rm N_{p}$ | $\rm m_{p}$ | $\rm\epsilon_{soft}$ | $\rm N_{halo}^{>100p}$ | ref.
| | $\rm[h^{-1}Mpc]$ | $[10^{9}]$ | $\rm[h^{-1}\,{\rm M}_{\odot}]$ | $\rm[h^{-1}kpc]$ | $[10^{6}]$ |
DEUS FUR | Ramses-Deus | 21000 | $550$ | $1.2\times 10^{12}$ | 40.0† | $145$ | [259]
Horizon Run 3 | Gotpm | 10815 | 370 | $2.5\times 10^{11}$ | 150.0 | $\sim\\!190$ | [260]
Millennium-XXL | Gadget-3 | 3000 | $300$ | $6.2\times 10^{9}$ | 10.0 | 170 | [220]
Horizon-4$\Pi$ | Ramses | 2000 | 69 | $7.8\times 10^{9}$ | 7.6† | $\sim\\!40$ | [261]
Millennium | Gadget-2 | 500 | 10 | $8.6\times 10^{8}$ | 5.0 | 4.5 | [181]
Millennium-II | Gadget-3 | 100 | 10 | $6.9\times 10^{6}$ | 1.0 | 2.3 | [87]
MultiDark Run1 | Art | 1000 | 8.6 | $8.7\times 10^{9}$ | 7.6† | 3.3 | [36]
Bolshoi | Art | 250 | 8.6 | $1.4\times 10^{8}$ | 1.0† | 2.4 | [262]
†For AMR simulations (Ramses, Art) $\epsilon_{\rm soft}$ refers to the highest
resolution cell width.
Cluster |
Name | Code | $\rm L_{hires}$ | $\rm N_{p,hires}$ | $\rm m_{p,hires}$ | $\rm\epsilon_{soft}$ | $\rm N_{sub}^{>100p}$ | ref.
| | $\rm[h^{-1}Mpc]$ | $[10^{9}]$ | $\rm[h^{-1}\,{\rm M}_{\odot}]$ | $\rm[h^{-1}kpc]$ | $[10^{3}]$ |
Phoenix A-1 | Gadget-3 | 41.2 | 4.1 | $6.4\times 10^{5}$ | 0.15 | 60 | [263]
Galactic |
Name | Code | $\rm L_{hires}$ | $\rm N_{p,hires}$ | $\rm m_{p,hires}$ | $\rm\epsilon_{soft}$ | $\rm N_{sub}^{>100p}$ | ref.
| | $\rm[Mpc]$ | $[10^{9}]$ | $\rm[\,{\rm M}_{\odot}]$ | $\rm[pc]$ | $[10^{3}]$ |
Aquarius A-1 | Gadget-3 | 5.9 | $4.3\times 10^{9}$ | $1.7\times 10^{3}$ | 20.5 | 82 | [45]
GHalo | Pkdgrav2 | 3.89 | $2.1\times 10^{9}$ | $1.0\times 10^{3}$ | 61.0 | 43 | [32]
Via Lactea II | Pkdgrav2 | 4.86 | $1.0\times 10^{9}$ | $4.1\times 10^{3}$ | 40.0 | 13 | [44]
Table 2: Current state of the art in DM-only simulations on cosmic, cluster, and galactic scale, ordered by number of simulation particles. $\rm L_{hires}$ is a proxy for the size of the high-resolution region in zoom-in simulations, and is defined to be equal to the size of a cube at mean density enclosing all high resolution particles. $\rm N_{halo/sub}^{>100p}$ is the number of halos in the box (Cosmic) or subhalos within $r_{50}$ (Cluster and Galactic) with at least 100 particles at $z=0$. In some cases (DEUS FUR, Horizon-4$\Pi$) mass functions have not been published, and so we estimated $\rm N_{halo}^{>100p}$ from a Sheth & Tormen [19] mass function fit. Simulation | Supercomputer | Type | Center | Country | core-hours | $N_{\rm cores}$ | memory | disk space
---|---|---|---|---|---|---|---|---
| | | | | $[10^{6}]$ | | [TB] | [TB]
DEUS FUR | Curie Thin Nodes | Bullx B510 | Très Grand Centre de Calcul (TGCC) | France | 10 | 38016 | 230 | 3000
Horizon Run 3 | Tachyon II | Sun Blades B6275 | KISTI Supercomputing Center | Korea | 4 | 8240 | 21 | 400
Millennium-XXL | JuRoPa | Bull / Sun Blades | Forschungzentrum Jülich | Germany | 2.86 | 12288 | 28.5 | 100
Horizon-4$\Pi$ | Platine | Bull Novascale 3045 | Commissariat a l’Energie Atomique | France | 8 | 6144 | 14.7 | 300
Millennium | p690 | IBM Power 4 | Rechenzentrum Garching | Germany | 0.35 | 512 | 1 | 20
Millennium-II | VIP | IBM Power 6 | Rechenzentrum Garching | Germany | 1.4 | 2048 | 8 | 35
MultiDark Run1 | Pleiades | SGI Altix ICE | NASA Ames Research Center | USA | 0.4 | 4000 | 8 | 20
Bolshoi | Pleiades | SGI Altix ICE | NASA Ames Research Center | USA | 6 | 13900 | 12 | 100
Phoenix A-1 | DeepComp 7000 | HS21/x3950 Cluster | Chinese Academy of Science | China | 1.9 | 1024 | 3 | 15
Aquarius A-1 | HLRB-II | SGI Altix 4700 | Leibniz Rechenzentrum Garching | Germany | 3.5 | 1024 | 3 | 45
GHalo | Marenostrum | IBM JS21 Blades | Barcelona Supercomputing Center | Spain | 2 | 1000 | 1 | 60
Via Lactea II | Jaguar | Cray XT4 | Oak Ridge National Lab | USA | 1.5 | 3000 | 0.3 | 20
Table 3: Supercomputers and computational resources utilized for each
simulation.
In the zoom-in class, simulations forgo capturing a representative fraction of
the universe, and instead focus all available computational resources on one
halo of interest, resolving its internal structure and substructure at the
highest possible resolution. In order to achieve this goal, these simulations
make use of nested initial conditions, in which the great majority of the
computational domain is sampled only with very coarse resolution (large
particle masses and force softenings), but a small volume containing an object
of interest is resolved with much higher resolution. We distinguish between
cluster scale simulations, in which the object of interest is a single galaxy
cluster ($10^{14}->\\!10^{15}\,{\rm M}_{\odot}$), and galactic scale
simulations, which zoom in on a single galactic halo ($\lesssim{\rm few}\times
10^{12}\,{\rm M}_{\odot}$). In both cases the halo of interest is typically
identified in the $z=0$ output of a lower resolution full-box simulation. The
particles contained within some multiple (usually $3-5$) of its virial radius
are traced back to the initial conditions, and the low resolution particles in
this Lagrangian volume are replaced with a nested set of higher resolution
(lower mass) particles. The phases and amplitudes of the large wavelength
Fourier modes used to calculate the initial condition displacements of these
higher resolution particles are kept the same as in the coarse realization,
but additional random modes are introduced at smaller wavelengths. This
process ensures that the large scale matter distribution remains identical to
the coarse simulation, but with greatly enhanced power in small scale
substructure fluctuations. The technical details of this procedure are
discussed in detail in the literature [264, 265, 267, 266].
In the following we briefly go over each of these three classes of simulations
(cosmic, cluster, and galactic), highlighting both state of the art
achievements as well as limitations and shortcomings. Over the last decade
progress in this field has been driven by advances in computing technology and
available resources at national supercomputing facilities, and were aided by
the algorithmic developments discussed in the previous section. The
simulations discussed below all required multiple millions of CPU-hours on
thousands of processors, and required terabytes of memory and petabytes of
disk storage. Some of the characteristics of these simulations are summarized
in Tables 2 and 3, and visualizations for a subset are shown in Fig. 3.
1. i)
Cosmic scale
In this class the state of the art has reached $\gtrsim 10$ billion particle
simulations, with the current record holder (in terms of particle number), the
recently completed DEUS Full Universe Simulation (FUR) [259], utilizing more
than half a trillion particles in a 21 $h^{-1}$ Gpc box, which corresponds to
the entire observable universe. It was run with a modified version of the
Ramses code and took about 10 million CPU-hours using 38,016 MPI tasks on the
European supercomputer Curie. With a particle mass of $1.2\times 10^{12}\,{\rm
M}_{\odot}$ and force resolution of $\rm 40\,h^{-1}kpc$, DEUS FUR cannot
resolve individual galaxies and thus is limited to studying the large scale
distribution of matter in the universe. The main driver of this simulation has
been to quantify the imprint that DE leaves on cosmic structures (e.g. BAO),
and how the nature of DE may be inferred from observations of large scale
structure. In addition to a standard $\rm\Lambda$CDM ($w=-1$) run, two more
FUR simulations at the same resolution but with different DE models ($w=-0.87$
Ratra-Peebles quintessence and $w=-1.2$ phantom fluid) have recently been
completed.
Currently the only calculation able to simultaneously resolve scales relevant
for BAO detection as well as those DM halos and subhalos expected to host
galaxies to be seen in future surveys is the Millennium-XXL simulation. This
simulation uses slightly fewer particles (303 billion) than DEUS FUR to
represent the mass field in the Universe, but it has almost 200 times better
mass resolution due to its smaller computational domain. It was run during the
summer of 2010 at the Jülich Supercomputer Centre in Germany using 12,288 CPUs
using a memory-efficient version of the Gadget-3 code. The main goal of this
simulation is to explore the impact of galaxy formation physics on
cosmological probes, in particular for BAO detection and redshift-space
distortion tests.
On considerably smaller but still cosmic scales, two of the most prominent
simulations are the Millennium-II and the Bolshoi simulations. Millennium-II,
a Gadget-3 simulation, has 10 billion particles in a $\rm 100\,h^{-1}Mpc$ box,
for a particle mass of $6.9\times 10^{6}\,{\rm M}_{\odot}$. It cost 1.4
million CPU-hours on an IBM Power-6 supercomputer at the Max-Planck Computing
Center in Garching, Germany. Bolshoi, an Art simulation, uses 8.6 billion
particles in a $\rm 250\,h^{-1}Mpc$ box, giving a particle mass of $1.4\times
10^{8}\,{\rm M}_{\odot}$, and required 6 million CPU-hours on the Pleiades
supercomputer at NASA Ames. Both simulations have a force resolution of $\rm
1h^{-1}kpc$. Although Bolshoi has 20 times poorer mass resolution, it covers
16 times more volume than Millennium-II. One additional difference between the
two is the choice of cosmological parameters, with Millennium-II employing
values inspired by the first year WMAP results ($\Omega_{m}=0.25$,
$\Omega_{\Lambda}=0.75$, $h=0.73$, $\sigma_{8}=0.9$, and $n_{s}=1$), which for
$\sigma_{8}$ and $n_{s}$ are more than $3\sigma$ discrepant with the more
recent WMAP 5-year and 7-year results, while Bolshoi used values
($\Omega_{m}=0.27$, $\Omega_{\Lambda}=0.73$, $h=0.70$, $\sigma_{8}=0.82$, and
$n_{s}=0.95$) that are consistent with the more recent measurements.333Results
from the Millennium simulations have been rescaled to the latest set of
cosmological parameters [268, 269]. For both cases, the mass and force
resolution is sufficient to resolve some of the internal (sub-)structure of
Milky Way-like halos, while at the same time capturing a large enough sample
of such galaxies ($\sim\\!5000$ in Millennium-II, $\sim\\!90,000$ in Bolshoi)
to enable statistical studies. These simulations have provided precise and
robust results on DM halo statistics like the mass function, subhalo
abundance, mass and environment dependence of collapse times, and spatial
correlation functions, spanning a wide range of scales, from dwarf galaxy
halos to rich galaxy clusters.
Figure 3: Visualizations of state of the art simulations on cosmic
(Millennium-XXL [220], upper left), cluster (Phoenix A-1 [263], upper right),
and galactic scale (Aquarius A-1 [45], lower left, and GHalo [32], lower
right).
2. ii)
Cluster scale
About ten years ago, cluster scale DM-only simulation were leading the effort
to study the properties of individual DM halos and the abundance and
properties of their substructure [27, 270]. With around ten million high
resolution particles, these simulations resolved thousands of subhalos and
established important substructure scaling relations. DM substructure studies
then shifted focus to the Galactic scale (see below), and until very recently
cluster simulations had not pushed into the billion particle regime. The
Phoenix simulation suite [263] has now changed that, with their highest
resolution Gadget-3 simulation employing 4.1 billion particles to resolve a
$6.6\times 10^{14}\,h^{-1}\,{\rm M}_{\odot}$ cluster, and identifying a total
of almost 200,000 individual subhalos (60,000 with more than 100 particles).
This simulation was run on 1024 cores of the DeepComp 7000 supercomputer of
the Chinese Academy of Science and cost 1.9 million CPU-hours. Additional
simulations of rare and dynamically young objects like galaxy clusters will
help to clarify to what degree the internal structure of DM halos and
substructure scaling laws are universal and self-similar.
3. iii)
Galactic scale
On Galactic scales the three flag-ship simulations are Via Lactea II [44],
Aquarius A-1 [45], and GHalo [32], in chronological order. Via Lactea II (1.5
million core-hours on Oak Ridge National Lab’s Jaguar), was the first
simulation to use over a billion high resolution particles to resolve a single
halo, Aquarius A-1 (3.5 million core-hours at the Leibniz Rechenzentrum in
Garching, Germany) the first to have over a billion particles within the
virial volume of the halo, and GHalo (2 million core-hours on Marenostrum at
the Barcelona Supercomputing Center, Spain) is currently the simulation with
the highest mass resolution. With particle masses ranging from 1000 to 4100
$\,{\rm M}_{\odot}$ and force resolutions from 20 to 60 pc, these simulations
are able to resolve in unprecedented detail the formation and accretion
history of Milky Way-sized DM halos ($M\approx 10^{12}\,{\rm M}_{\odot}$),
their inner density profiles, and the properties and survival of stripped
subhalo cores, as well as tidal debris orbiting within these systems. Density
profiles have been reliably measured to $\sim\\!\\!100$ pc, and the
substructure hierarchy is resolved over five decades in mass, down to
$\sim\\!\\!10^{5}\,{\rm M}_{\odot}$ subhalos. The Aquarius project simulated
an additional five halos at somewhat lower resolution (particle mass $\approx
10^{4}\,{\rm M}_{\odot}$), which has enabled a valuable initial assessment of
halo-to-halo scatter.
Even though the simulations were run with different codes (Gadget-3 for
Aquarius, Pkdgrav2 for Via Lactea II and GHalo) and used somewhat different
cosmological parameters (most notably $(\sigma_{8},n_{s})=(0.9,1.0)$ and
$(0.74,0.95)$, respectively), the numerical results agree remarkably well with
each other when scaled by the mass of the simulated host halo. Some
disagreements persist, however, in the interpretation of these results, for
example in the assessment of the relative detectability of the Galactic DGRB
indirect detection signal and that from individual subhalos [91, 135], and in
the self-similarity of the (sub-)substructure population [44, 45].
As will have become clear from the previous sections, these state of the art
DM-only simulations on cosmic, cluster, and galactic scales require truly
massive computational efforts (see Table 3). Note that computational demands
do not scale solely with N-body particle count, but also sensitively depend on
the degree to which the simulations resolve small scale structure and non-
linear clustering, mostly because more time steps are required. Single halo
zoom-in simulations also require more communication and it is more difficult
to balance their memory and CPU requirements than for full-box single
resolution runs. For this reason a galactic-scale simulation like Via Lactea
II required about the same number of CPU-hours (about one million) as the
cosmic-scale Millennium-II run, even though the latter employed 10 times more
particles. In addition to high computational demands at run time, simulations
at this level present enormous challenges for data transfer, storage, and
analysis (see §4.3.1.ii). As detailed nicely in the DEUS FUR simulation paper
[259], analyzing the simulation often requires computing resources comparable
to running it.
### 3.2 Small scale challenges for Cold Dark Matter
Predictions from CDM simulations of the large scale distribution of DM, post-
processed to include mock galaxy populations, agree remarkably well with the
observed clustering of galaxies measured in modern surveys like the Sloan
Digital Sky Survey (SDSS) [18]. Yet at smaller scales the agreement between
CDM predictions and observations is not as good: the number of dwarf satellite
galaxies observed to be orbiting our Milky Way (and our nearest neighbor
galaxy, M31) is less than one would naively infer from the predictions of DM-
only simulations in a CDM cosmology [271, 272]. The severity of this so-called
Missing Satellites Problem has been reduced in recent years through the
discovery of more than ten new ultra-faint dwarf satellites in the SDSS [109,
and references therein], raising the possibility that hundreds more remain yet
to be discovered [61]. Nevertheless, reconciling the steep slope of the DM
subhalo mass function with the shallow faint end of the satellite luminosity
functions remains a theoretical and computational puzzle.
A second major challenge to CDM is the Cusp/Core Controversy concerning the
central slope of DM density profiles in low mass galaxies. Two-dimensional
stellar and gaseous kinematic measurements in low surface brightness field
galaxies [275, 273, 64, 274], as well as chemo-dynamical measurements in at
least two Milky Way dwarf satellite galaxies [65], imply that the slopes of
the DM density profiles are shallower than the NFW slope of $\sim-1$ predicted
by CDM simulations without baryons.444Cored DM distributions have also been
inferred for more massive spiral galaxies [e.g. 276], but given that these
systems are strongly baryon dominated, this observation is not commonly
considered a major challenge for CDM.
Lastly and possibly connecting the previous two concerns, it has recently been
pointed out [277, 278] that there may be a problem in the abundance of even
the most massive Galactic subhalos. Dubbed Too Big To Fail, this problem
refers to the inference from stellar kinematic data that the central densities
of the “classical dwarfs” (bright satellites with luminosities greater than
$10^{5}L_{\odot}$) are too low to be consistent with inhabiting the most
massive subhalos predicted in the Via Lactea II and Aquarius simulations. The
consequence being that either there exists a population of massive subhalos
orbiting within the Milky Way’s virial volume that have remained completely
dark and devoid of stars, despite the fact that less massive subhalos
manifestly were able to form galaxies, or that the Via Lactea II and Aquarius
halos are somehow not representative of our Milky Way. For example, if the
mass of the Milky Way were a factor of two less than in these simulations (but
still within the range allowed from observational constraints), then the
number of discrepant (too dense) halos may be small enough to not be a major
worry [279, 280]. Alternatively, some process not captured in the DM-only
simulations could reduce the central densities in the most massive subhalos.
### 3.3 Simulations with Departures from Collisionless Cold Dark Matter
The two assumptions that underlie all DM-only simulations described so far are
(i) that DM is “cold”, meaning that the cutoff in the density fluctuation
power spectrum occurs on scales far below what is resolved in the simulations,
and (ii) that it is collisionless, meaning that the only dynamically relevant
interactions are gravitational, i.e. that any self-scattering effects are
negligible. Although these assumptions are theoretically well motivated,
holding for example for most supersymmetric DM models as well as for axions,
they are not a priori requirements. A number of studies in the literature have
investigated whether departures from the assumptions of cold and collisionless
DM can provide solutions to the small scale challenges to CDM discussed in the
previous section.
In the Warm Dark Matter (WDM) scenario the DM particle exhibits some non-
negligible thermal velocities at high redshifts, instead of being truly cold.
In this case, free streaming in the early universe will erase small scale
density fluctuations, preventing the formation of low mass DM halos. For a WDM
particle of mass $m_{\chi}$ and temperature $T_{\chi}$, a cutoff in the power
spectrum then occurs [12, 281] at a scale of
$k_{\rm FS}\approx 50\,{\rm Mpc}^{-1}\,\left(\frac{m_{\chi}}{2\,{\rm
keV}}\right)\,\left(\frac{T_{\chi}/T_{\nu}}{0.2}\right)^{-1}.$ (2)
Here $T_{\chi}$ has been expressed in terms of the temperature of the cosmic
neutrino background $T_{\nu}$, and $(T_{\chi}/T_{\nu})$ in general is model
dependent. For thermal relic WDM particles (e.g. the gravitino [282]) it can
be related to the relic DM density via
$\Omega_{\chi}h^{2}=(T_{\chi}/T_{\nu})^{3}\,(m_{\chi}/94\,{\rm eV})$, but non-
thermally produced particles (e.g. the sterile neutrino [283]) can have a wide
range of temperatures. Constraints from the Lyman-$\alpha$ forest limit the
mass of a WDM particle to be $\gtrsim 2-8$ keV [281, 284, 285], depending on
the details of the particle physics.
The suppression of small scale power may help to explain the puzzling dearth
of Milky Way satellite galaxies [286, 287]. A secondary effect arising from
the lack of small scale structure is that the collapse times of halos above
the free-streaming cutoff are delayed. This results in lower concentrations
and reduced central densities, which may help to address the Too Big To Fail
problem [288]. Low concentration halos are also more prone to tidal
disruption, which further reduces the abundance of low mass objects in WDM
halos. Lastly, we mention that WDM halos are expected to have central density
cores, since the WDM particles’ non-zero temperature results in a finite
phase-space density in the early universe [289], which by Liouville’s theorem
cannot grow during the formation of a halo. For realistic models that are
consistent with constraints from the Lyman-$\alpha$ forest, however, it can be
shown [290, 291] that phase-space density limited cores only occur on very
small scales, $r_{\rm core}/R_{\rm vir}\lesssim 10^{-3}$, far below where
there is observational evidence for a flattening of the DM density profile.
WDM models by themselves thus do not appear to be capable of solving the
cusp/core controversy.
There are several technical difficulties associated with numerically
simulating WDM models. One is that the presence of a cutoff in the power
spectrum in the initial conditions gives rise to the formation of a large
number of spurious halos of purely numerical origin [292]. Another difficulty
is that for sufficiently light WDM particles, small box sizes, or early
simulation starting times the thermal velocities can become comparable to the
bulk flows induced by the density fluctuations in the simulation’s initial
conditions. One should then apply thermal streaming velocities to the N-body
particles, ideally by splitting each particle into $2N$ sub-particles and
applying equal and opposite velocities randomly drawn from the primordial
velocity distribution to each $N$ pairs of sub-particles [293]. In practice,
however, thermal streaming velocities can usually safely be neglected, unless
one is simulating WDM with $m_{\chi}<1$ keV (observationally ruled out) or
using a boxsize smaller than 1 Mpc.
The hypothesis of essentially collisionless DM has also been contested. This
leads to the idea of Self-Interacting Dark Matter (SIDM) [294, 295, 296, 297].
Initially SIDM models with a constant scattering cross section were quickly
abandoned since those that could solve the small-scale CDM problems seemed to
violate several astrophysical constraints, such as the observed ellipticity of
the mass distribution of galaxy clusters [298] and the survivability of
satellite halos [299]. But recently it was pointed out that some of these
earlier constraints were overstated, and small velocity-independent self-
interaction cross sections can have sizable effects on halo profiles without
violating astrophysical constraints [300]. Also simple ad hoc velocity-
dependent cross sections of the form $1/v^{\alpha}$ were explored [301],
yielding encouraging results that however lacked a proper underlying particle
physics model. More recently it was realized that self-interactions through a
Yukawa potential can resolve the challenges facing velocity-independent SIDM
models [302]. The velocity dependence of scattering through the massive
mediator of this dark force (similar to a screened Coulomb scattering in a
plasma) could make scattering important for dwarf galaxies with low velocity
dispersion, but unimportant at the much higher velocities encountered in
galaxy clusters. Such models have been explored numerically and it has been
shown that they can help to resolve some of the small-scale CDM problems
through the formation of a central density core [303]. Note that there also
exist hybrid models (e.g. Atomic DM [13], see Fig. 1), in which the DM
exhibits both self-interactions and suppression in small scale power.
### 3.4 Simulations Including Baryons Physics
As we have seen, the tension on small scales between dwarf galaxy observations
and the predictions of DM-only simulations might be an indication that the
true properties of the DM particle differ from the cold and collisionless
assumptions of these simulations. Unfortunately, however, the effects of
modified DM particle physics can be mimicked by the complicated baryonic
physics governing the formation of stars and galaxies inside DM halos. For
this reason the problem of DM is closely coupled to the problem of galaxy
formation, which of course is a worthy topic of study in its own right. A
survey of the current state of the art in numerical simulations of galaxy
formation is considerably beyond the scope of this review, and so in the
following we instead provide a limited overview of recent results with
particular pertinence to the DM and DE problems.
#### 3.4.1 Numerical Techniques and Codes
The basic equations that are solved in cosmological hydrodynamics simulations
are the Euler equations (conservation of mass, momentum, and energy) governing
the flow of an ideal gas, coupled gravitationally to the DM sector through a
source term in the energy equation and the Poisson equation. Neglecting
viscosity (ideal gas) is a good assumption on cosmological and galactic
scales, but ignoring the effects of magnetic fields and radiation less so, and
accounting for magneto- and radiation-hydrodynamic effects in cosmological
galaxy formation simulations is an active area of research [e.g. 304, 305,
306, 307, 308]. Results from such studies, however, have not yet been brought
to bear on the DM and DE problems, and so we focus here on the simpler pure
hydrodynamic case.
Unlike for the purely gravitational N-body problem, where even conceptually
quite different solvers (e.g. tree and adaptive PM codes, see §3.1.1) robustly
produce similar results, the choice of method with which to treat the
hydrodynamics can lead to marked differences in the results [309, 310, 311,
312, 313, 314, 315]. Numerous detailed discussions of the different approaches
and their relative advantages and disadvantages exist in the literature [e.g.
316, 317, 311, 313], and we only briefly summarize the essentials here.
In general one can distinguish between Eulerian and Lagrangian methods, which
discretize either space (Eulerian) or mass (Lagrangian). In Smoothed Particle
Hydrodynamics (SPH) [317], the most commonly used Lagrangian approach555Really
it is “pseudo-Lagrangian”, since shearing flows with distinct internal
properties are not followed in a truly Lagrangian way on scales below the
smoothing kernel [314]., the fluid flow is followed with particles, whose
equations of motion are derived from a discretized particle Lagrangian [318],
which ensures excellent conservation of mass, momentum, energy, entropy, and
angular momentum. Thermodynamic quantities (density, pressure, etc.) are
obtained by smoothing over neighboring particles with a particle-dependent
smoothing length. Advantages of SPH are that it is automatically adaptive,
delivering higher resolution in collapsing regions, geometrically flexible,
inherently Galilean invariant (errors don’t depend on bulk flows),
computationally inexpensive, and that it couples easily to an N-body gravity
treatment (as for the DM). It is also often simpler to implement new physics
prescriptions. However, SPH is not without its drawbacks. Its Lagrangian
nature results in less resolution in lower density environments, and its
estimates of thermodynamic quantities are noisy on the scale of the smoothing
kernel. Artificial viscosity must be added in order to inject the entropy
generated at shocks and to suppress unphysical oscillations in the states
immediately surrounding it. This broadens the discontinuity to several
smoothing lengths and has a tendency to make the method more dissipative. It
has low accuracy for contact discontinuities, and as a result suppresses some
astrophysically relevant fluid instabilities and mixing. However, not all of
these disadvantages are inherent to the SPH method, and several recently
proposed modifications have resulted in very promising improvements [319, 320,
321, 322]. SPH simulations commonly employ a gravitational force softening
length that is fixed in physical coordinates below some redshift (see [323]),
which results in poorer spatial resolution at early times compared to a
constant co-moving softening scale and may suppress early star formation. The
most widely used SPH codes in the galaxy formation field are Gadget [245] and
Gasoline [324].
With Adaptive Mesh Refinement (AMR), the most widely used Eulerian
hydrodynamics approach, the fluid flow is instead discretized in space.
Euler’s equations are solved on a regular mesh, which is adaptively refined in
regions requiring higher accuracy [325]. In finite volume methods, conserved
quantities (mass, momentum, energy) are stored on the cells of the mesh, and
their values are updated in a conservative fashion by solving for the fluxes
across cell interfaces (for a primer, see [316]). In the widely used Godunov
schemes, fluxes are calculated by considering the independent variables to be
piecewise constant across each cell, with discontinuities at the cell
boundaries, and solving the resulting Riemann problem for the characteristic
waves (shocks, rarefactions, and contact discontinuities) traveling into the
neighboring cells [326]. In practice higher order accurate methods (piecewise
linear or piecewise parabolic) are commonly employed [327, 328, 329].
Advantages of AMR are its low noise and high accuracy for shocks, contact
discontinuities, and shear waves, allowing it to capture fluid instabilities
with high fidelity, and its full control over where to place high resolution.
Disadvantages are its higher algorithmic complexity, lower numerical
stability, the fact that errors are not Galilean invariant, a tendency to
overmix fluid, and that runtime memory requirements grow with refinement. The
most commonly used AMR codes in the galaxy formation community are Hydro-Art
[242, 330, 331], Enzo [332, 333], and Ramses [250] (with the Flash code [334]
about to join the fray [335]).
Figure 4: Visualizations of three recent cosmological hydrodynamical galaxy
formation simulations. Top row: gas surface density at $z=4$ in three galaxies
(out of $\sim\\!\\!100$ in the box) simulated with the AMR code Enzo [336].
Middle row: a series of zooms onto the density field surrounding a $z=2$
galaxy, simulated with the moving mesh code Arepo [314]. Bottom: Optical and
UV composite images of Eris, a Milky Way-like galaxy simulated with the SPH
code Gasoline [337].
Both SPH and AMR techniques have weaknesses, which are related to the
numerical approach taken to solve the fluid equations. One advantage of SPH is
its pseudo-Lagrangian nature which fits very well the needs of cosmological
structure formation simulations, where adaptivity and a large spatial and
dynamical range is required. On the other hand AMR, as a finite volume scheme,
provides highly accurate results for fluid problems providing, for example,
very good resolution of shocks, discontinuities and mixing, which are
typically harder to resolve very well with SPH schemes. A natural way to
combine the advantages of SPH and AMR techniques is to allow for Moving Meshes
in the volume discretization. This idea goes back to the 1990’s, where moving
meshes were first explored in the context of astrophysical applications [338,
339]. Although the idea of having the computational mesh move with the
hydrodynamical flow seems very natural, its practical implementation turned
out to be rather difficult. Approaches relying on deformed Cartesian grids
lead to problems in handling grid deformation properly in fully astrophysical
applications [338, 339]. Only recently new moving mesh schemes have been
developed, which are able to circumvent this problem [313, 340]. These new
schemes do not use coordinate transformations like previous moving mesh codes
in cosmology, but instead employ an unstructured Voronoi tessellation of the
computational domain. The mesh-generating points of this tessellation are
allowed to move freely, offering significant flexibility for representing the
geometry of the flow. If the mesh motion is tied to the gas flow, the results
are Galilean-invariant (like in SPH), while at the same time a high accuracy
for shocks and contact discontinuities is achieved (like in Eulerian schemes).
Furthermore the mesh is free of the distortion problems inherent to previous
schemes. Using the Arepo code [313], this new computational approach has
recently been applied successfully to initial large-scale cosmological
simulations of galaxy formation [314, 315] (see Fig. 4).
An accurate treatment of the hydrodynamics is necessary, but far from
sufficient. In order to model the galaxy formation process, simulations must
go beyond adiabatic hydrodynamics and include gas cooling through radiative
energy losses, as well as heating from an externally calculated meta-galactic
UV background [341, 342]. The cooling is typically implemented through
tabulated cooling functions, which provide externally calculated (using the
Cloudy code [343, 344, 345], for example) equilibrium cooling rates as a
function of density, ionization fraction, temperature, metallicity, and
intensity of the UV background. Some codes follow the non-equilibrium
abundance of hydrogen and helium species (including molecular hydrogen in some
cases) coupled to gas cooling and heating by solving a chemical network sub-
cycled on each hydrodynamic time step. The details of how gas cooling is
implemented can make a difference in the simulation’s outcome.
In regions that have cooled and condensed to sufficiently high density, stars
will form. This process is captured with a sub-grid model, in which a fraction
of the available gas is converted to “star particles” representing an entire
stellar population, which from then on are treated as collisionless and
evolved in the same fashion as DM particles. The nature of this star formation
(SF) prescription is another important differentiating aspect of galaxy
formation simulations. Most commonly, a Schmidt law [346] is employed, whereby
the star formation rate (SFR) is proportional to the gas density divided by
the local free fall time, resulting in a SFR proportional to $\rho^{3/2}$.
However, some authors instead prefer a linear SF law corresponding to a fixed
SF time scale [330], especially with simulations that distinguish between
atomic and molecular gas phases and tie the star formation to the latter [347,
336, 348]. The SF efficiency is a free parameter in principle, but is usually
set to a few percent, motivated by observational constraints [349].
Furthermore, a SF density threshold is often employed to limit SF to
especially high density regions. Some methods additionally restrict SF to
regions with a converging flow and a cooling time shorter than the dynamical
time [350]. In order to prevent the formation of excessive numbers of star
particles, a minimum star particle mass is sometimes enforced. For any given
model, the parameters are typically tuned to reproduce macroscopic SF scaling
laws like the Kennicutt-Schmidt relation [351]. Nevertheless, with so many
different SF prescriptions and tunable parameters, it is not surprising that
there is no unique solution, and that results vary greatly, especially in
conditions far from where the prescriptions were calibrated.
Last, but by no means least, galaxy formation simulations must account for so-
called feedback processes, which attempt to capture the injection of mass,
momentum, energy, radiation, and metals (nucleosynthetic products) from
massive young stars, evolved asymptotic giant branch stars, exploding
supernovae (both type Ia and II), and accreting black holes into the
surrounding interstellar medium. Feedback appears to be a crucial ingredient
in explaining the macroscopic properties of individual galaxies [352, 353,
354, 337, 355] as well as galaxy population statistics [356, 357, 358], and
may be the key for explaining the small scale challenges CDM faces (see next
section). At the limited resolution of current simulations, feedback is
implemented using heuristic and often ad-hoc prescriptions (for a review, see
[359]). Numerical difficulties that must be overcome include preventing the
injected energy from immediately being lost owing to the high densities and
cooling rates in star forming regions, accounting for radiation pressure from
young massive stars, forming and maintaining large scale galactic outflows,
and properly accounting for the mixing of metals. Even more so than for the SF
prescription, results depend sensitively on the details of how feedback is
implemented [323], and this is probably the greatest source of uncertainty in
present day galaxy formation simulations.
For completeness, we also briefly mention here that baryonic physics effects
are also commonly accounted for through the use of Semi-Analytic Models (SAM).
In this approach gas cooling, star formation, feedback from SNe and AGNs are
all implemented in a simplified but self-consistent manner on top of halo
matter merger trees derived from analytical calculations or from DM-only
simulations. SAMs have been used to explore the connection between dark matter
structure and galaxies (for a review, see [360]) have helped to quantify how
uncertainties in galaxy formation can propagate to DE studies [183, 167].
#### 3.4.2 Baryonic effects on DM
Given the uncertainties arising from the choice of hydrodynamic method, the
sub-grid physics prescriptions, and the large number of adjustable parameters,
it is not surprising that there is not yet consensus on how baryonic physics
alters the distribution of DM in halos. In the following we report on some of
the results from recent hydrodynamic galaxy formation simulations. We caution
the reader, however, that in most cases these results should not be viewed as
definitive answers, and that the conclusions are subject to change with higher
resolution and improvements in the treatment of sub-grid physics
prescriptions.
Concerning the large scale distribution of DM, it has been shown [361, 331,
362] that the inclusion of baryonic physics substantially alters the matter
power spectrum on scales ($k\gtrsim 1h\,{\rm Mpc}^{-1}$ or $l\gtrsim 800$)
that are relevant for contemporary and future weak lensing galaxy surveys
aiming to constrain the nature of DE. The weak lensing shear signal has also
been shown to be affected by baryonic physics, especially when strong AGN
feedback in galaxy clusters is accounted for [363], and this can lead to
significant biases (tens of percent) in the inferred DE equation of state
parameter $w_{0}$. DM halo mass functions can also be affected, with baryons
causing a $10\%$ enhancement in the cumulative mass function for
$>10^{12}h^{-1}\,{\rm M}_{\odot}$ halos in one study [331], and $\sim 30$
percent deviations in the number density of $10^{14}h^{-1}\,{\rm M}_{\odot}$
halos in another [364], with the sign of the deviation (increase or decrease
in halo mass) depending on how the baryonic physics was implemented. Note that
$10-30\%$ differences are larger than the 5% statistical uncertainty in DM-
only halo mass function [22]. Gas mass fractions in galaxy clusters and halo
mass scaling relations of the thermal Sunyaev-Zel’dovich effect depend on the
treatment of baryons [365]. Together these results imply that deep galaxy
cluster surveys designed to tightly constrain cosmological parameters and the
nature of DE must account for baryonic effects.
The abundance and spatial distribution of subhalos inside galactic and cluster
scale halos is another area likely affected by baryonic physics, and here
again even the sign of the effect is unknown. Baryonic condensations within
subhalos could increase the central density and make them more resiliant to
tidal disruption. This could lead to an increase in the subhalo abundance
close to the halo center, as seen in some simulations [366, 367]. On the other
hand, if the host halo itself has a sizeable stellar disk, then disk shocking
[368], as well as interactions with individual stars [369], could lead to
enhanced subhalo destruction and lower their abundance in the inner part of
the galaxy.
From the point of view of DM detection experiments, the most significant
concern is the possibility that baryonic effects could modify the shape of the
DM density profile. This is the arena of a long standing debate between
advocates of adiabatic contraction increasing the central DM density on one
side, and proponents of processes removing DM from the halo center on the
other. If the cooling and condensation of gas proceeds slowly, with baryons
gradually sinking to the center, then DM will be dragged in adiabatically,
leading to a steepening of the DM density profile from the NFW slope of
$\sim-1$ to something closer to isothermal $-2$. This adiabatic contraction
effect was worked out analytically in [370] and its description has since been
refined [371, 372]. It is routinely observed in cosmological galaxy formation
simulations [373, 374, 375, 376, 337, 372, 377].
If, on the other hand, baryonic material is rapidly delivered to the center,
for example through cold flows [378], then adiabatic contraction may not
operate and other dissipationless processes that transfer energy from the
baryons to the DM could lower the central DM density [379]. Examples of such
cusp-to-core conversion processes are resonant interactions of the DM halo
with a stellar bar666Some authors, however, instead find that the formation of
a stellar bar actually increases the central DM density [380, 381]. [382, 383,
384], the decay of a supermassive black hole binary [385], dynamical friction
of dense stellar clumps against the smooth background DM halo [386, 387, 373,
366], and repeated and violent oscillations in the central potential due to
energy injection from active galactic nuclei [388] or supernova-driven
galactic outflows [389, 390, 391, 392]. This last process in particular has
recently attracted much attention, since galaxy formation simulations with
very efficient “blastwave” supernova feedback have for the first time resulted
in spiral galaxies with realistically low bulge-to-disk ratios [354, 337], and
also exhibit pronounced DM density cores [393]. The removal of the central DM
density cusp through baryonic processes is especially interesting in the
context of the Missing Satellites and Too Big To Fail problems of CDM (see
§3.2) [394, 395].
Lastly, we mention the possibility that baryonic physics effects could lead to
a displacement between the point of maximum DM density in a halo and its
dynamical center [396, 397], as recently identified in the Eris simulation
[337], one of the highest resolution and most realistic cosmological
hydrodynamic simulations of a Milky Way-like galaxy (see Fig. 4). In this
simulation, but not in comparable DM-only simulations, the DM offset was $\sim
340$ pc averaged over the last 8 Gyr, typically in the plane of the stellar
disk, and aligned to about 30 degrees with the orientation of the stellar bar.
Such an offset would considerably alter expectations for the indirect DM
detection signal from the Galactic Center, where Sgr A*, the compact radio
source associated with a supermassive black hole, is a good marker of the
dynamical center of the Galaxy [100]. Intriguingly, the recently reported 130
GeV gamma-ray line from the Galactic Center [105, 106, 108, 107] appears to be
offset by about 1.5 degrees (about 200 pc projected) from Sgr A*.
## 4 The Next Decade
Having rextensively reviewed the current state of the art of cosmological DM
and galaxy formation simulations, we now present our vision for this field for
the next ten years. What progress should be possible and where should
priorities be focused in order to maximize our understanding of the DM and DE
problems?
### 4.1 Dissipationless Dark-Matter-only Simulations
On the cosmic scale, there is a need for very large numbers of high resolution
simulations in large box sizes, scanning over cosmological parameters. Future
DE surveys (e.g. DES, BigBOSS, LSST) will cover enormous volumes (10000’s of
square degrees out to $z\approx 2$) and are sensitive to quite low mass
galaxies. In order to perform grid-based or Markov Chain Monte Carlo
estimations of cosmological parameters, their errors, and especially the co-
variances of their errors, it will be necessary to have highly accurate
theoretical predictions of the non-linear clustering of matter for a finely
sampled scan of cosmological parameters. At the desired percent level
accuracy, such predictions can and must come from cosmological simulations
[398] – hundreds to thousands of them [399, 400]. Each simulation must cover
volumes comparable to the surveys, and have a mass and force resolution
sufficient to resolve nonlinear clustering on galactic scales. Furthermore,
owing to non-linear mode coupling, small scales are affected by the sampling
variance of the largest modes, necessitating multiple realizations for any
given cosmology. In the mildy non-linear regime ($k<0.3\,h\,{\rm Mpc}^{-1}$),
it may be possible to greatly speed up cosmological parameter scans by
utilizing rescaling algorithms [268] or second order Lagrangian Perturbation
Theory in conjunction with time- and scale-dependent transfer functions that
provide good approximations to particle trajectories and can be derived from
cheap N-body simulations [401].
Beyond the standard $\Lambda$CDM simulations, there is considerable interest
in exploring simulations with alternative gravity laws, such as $f(R)$
theories, that explain cosmic acceleration without DE. In addition to the
N-body gravity treatment, such simulations must also solve for the non-linear
dynamics of the Chameleon scalar field that screens the gravity law
modifications from small scales. In many cases, this is accomplished with
adaptive multi-grid relaxation techniques, quite similar to how gravity is
treated in AMR hydrodynamics codes. Promising initial progress is already
being made in this area [402, 403, 404, 405].
On cluster and galactic scales, efforts will be focused on the substructure
problem, and will attack the problem from three directions:
1. i)
Higher resolution
Given that the CDM substructure hierarchy extends for 10 - 15 orders of
magnitude below current resolution limits, direct numerical simulations will
realistically not be able to resolve the full hierarchy even in the
intermediate future. Nevertheless, pushing on resolution in zoom-in
simulations of individual halos (as in the Via Lactea and Aquarius projects)
is a worthwhile goal, for the following reasons:
* (a)
Stars in the newly discovered ultra-faint dwarf galaxies of the Milky Way are
typically only found out to $\lesssim 100$ pc from the dwarf’s center. In
order to compare stellar kinematics in these systems with predictions of
central densities and profile slopes from CDM simulations, it will be
necessary to push to a mass and force resolution of $\sim 100\,{\rm
M}_{\odot}$ and $\sim 10$ pc. Note that baryonic physics modifications are
expected to be less important in the low mass ($<10^{9}\,{\rm M}_{\odot}$)
host halos of ultra-faints [390], and so DM-only simulations should still
provide valuable predictions.
* (b)
The DM annihilation boost factors from substructure depend sensitively on the
properties of the subhalo population below current simulation’s resolution
limit (§2.2.2 ix). Since the power spectrum of DM density fluctuations is not
truly scale invariant, one might expect quantitative changes in subhalo
properties at lower masses. Being able to resolve the abundance, spatial
distribution, and density profiles of $<10^{5}\,{\rm M}_{\odot}$ subhalos
would help to clarify the relevance of substructure boost factors.
* (c)
The phase-space structure in the local neighborhood is only beginning to be
resolved by current simulations. Higher resolution simulations will provide a
somewhat more fine-grained view, and will additionally resolve more tidal
streams from disrupted subhalos. This will allow a better assessment of the
importance of velocity substructure for the intepretation of DM direct
detection results. A caveat is that baryonic affects are likely to be very
important.
The “Silver River” simulation is an example of an on-going effort in this
class. With a particle mass of $100\,{\rm M}_{\odot}$ and a force softening of
27 pc, this simulation is pushing the frontier in Galactic zoom-in
simulations. The run has progressed to $z\approx 5$ and will be completed to
$z=0$ in the next 1 - 2 years, pending computational resource support from
national supercomputing centers (30 - 40 million core-hours required).
2. ii)
Different host halos
The total number of ultra-high resolution ($>10^{9}$ particles) simulations of
individual halos is still quite small: one cluster simulation (Phoenix) and
three galactic scale ones (Via Lactea II, Aquarius, and GHalo), and all four
of these simulations have been run with only two codes, Pkdgrav2 and Gadget.
The six lower resolution Aquarius simulations and Millennium-II
notwithstanding, the question of cosmic variance, of how much halo-to-halo
scatter there is in the substructure population, is not fully settled: see for
example [406], who find a considerably larger halo-to-halo scatter in the
subhalo abundance than what is seen in Aquarius and Millenium-II. Trends with
host halo mass and accretion history are equally important and in need of
further study. These questions are of particular relevance to the Too Big To
Fail problem (§3.2), where even a factor of a few variation would go a long
way towards a resolution [280]. Lastly, current simulations don’t properly
account for the immediate environment of the Milky Way, because they typically
don’t have a nearby massive M31-like companion.
The Rhapsody Cluster Resimulation Project [407] is an early effort along these
lines, consisting of 96 zoom-in simulations of cluster scale halos with $\sim
10^{7}$ particles per halo. Simulation suites of hundreds of Milky Way-like
halos with varying masses and accretion histories are also currently being
pursued. Capturing more of the Local Group structure can be achieved either
through constrained realization simulations [408] or by picking suitable halo
configurations from an ensemble of lower resolution simulations (S. Garrison-
Kimmel et al., work in progress).
3. iii)
Alternate DM models
Going beyond the cold and collisionless DM paradigm, sharpening the
predictions that WDM or SIDM models make will be another promising avenue of
exploration. In models with a cutoff in the density power spectrum it will be
interesting to study how halo formation is suppressed right around the free
streaming scale. This is numerically challenging, due to the extremely high
resolution required to push the artificial fragmentation to small enough
scales. If this challenge can be overcome, it may be feasible to conduct
ultra-high resolution Milky Way-scale simulations (like Via Lactea II or
Aquarius A-1) with one of the alternative DM models. This would be very useful
for comparing with stellar kinematic data in MW dwarf satellite galaxies and
with data from future strong lensing surveys of flux ratio anomalies. It may
also be of interest to simulate not generic WDM or SIDM, but specific well-
motivated particle physics models, for example the Atomic DM model [13] with
its baryonic-like small scale power spectrum oscillations.
### 4.2 Simulations Including Baryons Physics
We anticipate that improvements in the treatment of baryonic physics processes
will be a major focus in cosmological simulations over the next decade and
beyond. As detailed in §3.4, there are numerous conceptual and technical
challenges in properly implementing baryonic physics. Yet these are important
problems to tackle, since they have potentially far reaching consequences for
our understanding of the distribution of DM in and around galaxies, groups,
and clusters, and directly affect expectations for indirect and direct
detection experiments.
On cosmic scales, it will be crucial for numerical simulations to quantify how
baryonic physics modifies the matter power spectrum [363], how it affects the
bias between galaxies and DM halos, and how these effects impact cosmological
parameter estimations from upcoming surveys. For individual cluster
simulations, it is imperative to establish how reliable X-ray and Sunyaev-
Zeldovich measurements of halo masses are [409], and how well cluster mass-
observable relations can be calibrated. In addition to radiative cooling, star
formation, and stellar feedback, such simulations must account for magnetic
fields and anisotropic conduction, non-thermal pressure support from
turbulence and cosmic rays, as well as AGN feedback.
On galactic scales, one of the most urgent questions that numerical simulation
must strive to clarify is under what conditions adiabatic contraction steepens
the central DM density profile, and whether this effect can be overcome by
feedback processes that may redistribute large amounts of DM and result in
shallower or even cored density profiles. If both processes occur in nature,
which one dominates, and how does the answer depend on halo mass, environment,
and cosmic time? It seems clear from past work that simulations with cooling,
but little or inefficient supernova (SN) feedback, exhibit a substantial
amount of adiabatic contraction of the DM halo. On the other hand, these
simulations typically also suffer from a baryonic over-cooling problem, and
produce too many stars, that are too centrally concentrated. Some form of
stellar feedback thus appears to be a necessity. Whether the resulting
regulation of star formation is accompanied by a removal of substantial
amounts of DM from the central regions is an open question.
When efficient SN feedback is imposed “by hand”, either by artificially
turning off gas cooling, or by employing unphysically high star formation or
SN energy injection efficiencies, or by hydrodynamically decoupling wind
particles from the surrounding gas, substantial redistribution of DM has been
observed in a variety of simulations. But how realistically are these ad-hoc
implementations capturing the actual physical processes occuring in star
forming regions inside giant molecular clouds? Is the resulting DM cusp
flattening a robust outcome? The answers await more sophisticated treatments
of star formation and feedback processes. Promising directions of future
investigations along these lines include, but are not limited to:
* i)
Suppressing star formation in low mass and low metallicity systems by using
H2-regulated star formation prescriptions rather than (or in conjunction with)
SN feedback [347, 336];
* ii)
Accounting for radiation pressure from young massive stars [410], which
imparts momentum into the surrounding gas and can increase the efficiency of
subsequent SN feedback without resorting to artificial enhancement;
* iii)
Modeling non-thermal support provided by unresolved turbulence, magnetic
fields, and cosmic ray propagation [411, 412].
In general more sophisticated treatments of star formation and feedback
physics will require resolving the hydrodynamic component of the simulations
with parsec scale resolution (the DM can be treated with coarser resolution
for these purposes). This is at least a factor of ten higher than what is
currently achievable in cosmological zoom-in simulations. Full-box simulations
with this resolution will be limited to the high redshift domain, and thus it
will be challenging to obtain large samples of realistically simulated
galaxies for comparison with local observations.
It is commonly hoped that it will eventually be possible to include realistic
star formation and feedback prescriptions in low resolution, large volume,
full-box simulations by calibrating them to the results from much higher
resolution zoom-in, or even isolated, simulations, in which a more accurate
treatment of the relevant physics is possible. However, it is yet to be
demonstrated that this approach will be feasible. Recent algorithmic
developments in creating initial conditions (by generating white noise fields
hierarchically in real space [266, 220]) are now enabling simulations of
unprecedented dynamic range. The MUSIC code [266] has demonstrated high
accuracy in reproducing in nested simulations the same detailed structures as
in corresponding high uniform resolution full-box runs. This provides the
basis for the Cosmic Renaissance (CORE) Project (T. Abel et al., work in
progress), which aims to conduct a large set of self-consistent nested
simulations over a wide range of scales, from extreme zoom-ins on the
formation of the first stars in a volume the size of the observable Universe
to increasingly less focused and more coarsely resolved simulations of the
formation of galaxies, clusters, and the large scale structure of the
Universe.
Lastly, it is also necessary to understand how these numerical results depend
on which hydrodynamic technique and indeed which code was used. The Aquila
Code Comparison Project [323] was an important step in this direction, but
owing to its approach of allowing simultaneous variations in code and feedback
physics implementations, it was difficult to disentangle which factor the
observed differences can be attributed to. Nevertheless, it clearly
demonstrated that the current generation of cosmological hydrodynamic
simulations are not yet able to uniquely predict the properties of the
baryonic component of a galaxy forming in a fully specified dark matter
accretion history. The recently initiated Santa Cruz High-resolution Galaxy
Simulation Comparison
Project777https://sites.google.com/site/santacruzcomparisonproject/ is a
similar effort, focusing on $\lesssim 100$ pc resolution galaxy formation
simulations with a wide range of current state of the art codes representing
SPH, AMR, and Moving Mesh techniques.
### 4.3 Computational Trends and Algorithmic Advances
#### 4.3.1 Processing
Over the last decade there have been significant advances in High Performance
Computing (HPC) available to astrophysical research. The top 20 machines in
the June 2012 TOP500 list888http://www.top500.org/list/2012/06/100 have all
exceeded the petaflops barrier, and utilize upwards of 100,000 cores to do so
(Sequoia, the current No.1 has 1,572,864 cores). Supercomputers are expected
to reach the exaflop scale during the next decade. This will enable us to
carry out much more detailed modeling of the problems described in this
manuscript, but it also poses serious challenges for the development of
numerical codes and algorithms capable of fully exploiting new technologies.
1. i)
Scalability and performance
One characteristic of current and future supercomputers is the very large
number of computing cores. The most obvious problem concerns how to distribute
the computational load over an increasingly large computational domain, while
at the same time keeping communication time to a minimum. Preserving adequate
balance in the CPU and load requirements will require development and
improvements in all algorithms present in N-body calculations, in order to
achieve scalability to millions of compute tasks.
Another interesting aspect of supercomputing in the next decade will be the
architecture of their Central Processing Units (CPU). Modern compute nodes
contain multiple processing cores, each of which have their own sets of
instructions and cache, but have access to common memory (RAM). In the near
future we can expect nodes with hundreds of such cores, a feature that needs
to be exploited by simulation codes via mixed parallelization schemes, in
which the commonly used Message Passing Interface (MPI) model for distributed
memory is combined with shared memory parallelism, via e.g. OpenMP or Posix
threads. Some existing N-body solvers already take advantage of SIMD (Single
Instruction Multiple Data), which allows parallel vector calculations in the
CPU. Speed-ups can be factors of a few, however exploiting SIMD requires
machine-dependent code.
In recent years, Graphical Processing Units (GPU) have become a competitive
alternative to CPUs for intensive numerical tasks. Initially developed for
rapid and highly parallelized manipulation of computer graphics, they have
also found wide use for general-purpose computation. For certain algorithms
able to take advantage of data-parallelism, GPUs can result in very
significant speed-ups of factors of ten or more. HPC is already beginning to
embrace this trend, with 3 of the top 10 supercomputers (Tianhe-1A, Jaguar,
and Nebulae) taking advantage of GPUs. A downside of using GPUs is the
comparatively small amount of memory available on board, requiring a lot of
slow data transfer from CPU to GPU and back. Nevertheless, there have been
efforts to implement widely used algorithms to take advantage of GPUs, for
instance Fast Fourier Transforms, kd-Trees, as well as visualization and
analysis tools. Custom code modifications can be performed using extensions to
the C programming language, like CUDA (specific to NVIDIA GPUs) and OpenCL
(multiple architectures). As graphics cards improve in reliability and
performance, cosmological N-body codes will likely take advantage of this
technology (see e.g. [413, 414, 415]), most likely in programming models that
employ hybrid algorithms for both CPUs and GPUs.
Perhaps a point of convergence between CPUs and GPUs will be in hardware such
as the Intel MIC cards (Many Integrated Core). This product, resulting from
Intel’s Larrabee research project, is essentially a GPU-CPU hybrid and
functions as a co-processor for HPC. It is expected to become available by the
end of 2012 and could become an alternative to GPUs with a simpler programming
model.
2. ii)
Big Data
As we have seen, state of the art simulations already generate petabytes of
data. Handling this data is becoming an increasingly difficult problem and we
can only anticipate that it will get worse during the next decade. One
limiting aspect is the disk I/O speed; even with parallel distributed
filesystems (e.g. Lustre) it does not scale with the size of the
supercomputer, and thus disk I/O can take a considerable fraction of the
runtime of a simulation. Another aspect is how to manage and store extremely
large datasets, especially with collaborations commonly spread across the
world and the considerable cost of dedicated storage hardware. Simply
migrating the data from large supercomputing centers to smaller local
facilities can be difficult and time consuming.
These are serious challenges for computational cosmologists, and strategies
will need to be developed to cope with these issues. One natural development
already in practice is to merge analysis and visualization software with the
simulation code so as to reduce as much as possible the output of data during
runtime and the handling afterwards.999This is also one of the stated
development goals of the yt Project [416], http://yt-project.org/. For large
DM simulations this may mean identifying and extracting (sub)halo information,
constructing merger trees and lightcones on-the-fly, and storing only the
resulting reduced data. However, it is never possible to anticipate all the
uses of a simulation, and this motivates saving at least some full outputs.
Novel data compression techniques are a promising way forward. The idea is
that data analysis often does not require the same precision as run-time
computation. Substantial compression of the output can the be achieved either
by neglecting irrelevant bytes or by spatial averaging (a compression approach
widely used in image and video displays). For instance, the data can be stored
in a hierarchical tree fashion, where the initial bytes provide a coarse
representation of the dataset, and subsequent bytes provide refinement in
specific quantities and/or spatial regions. These would alleviate access and
reading time for postprocessing, at a cost of giving up the ability to restart
simulations and with file formats very specific to a given simulation.
The distribution of data worlwide is a serious but important problem, as
serving data products is necessary and desirable for scientific progress. In
the last few years, an increasingly popular option has been the developments
of internet databases providing reduced data products, for example from the
Millennium and MultiDark simulation databases101010http://www.mpa-
garching.mpg.de/millennium/ and http://www.multidark.org/MultiDark/. Another
alternative is to store the data itself in a distributed fashion on the
internet in the ”cloud” and using Hadoop for distributed data analysis.
#### 4.3.2 Novel Approaches
1. i)
New Gravity Solvers
Despite well-known problems with the N-body treatment of cosmological
structure formation (e.g. two body relaxation and artificial fragmentation),
there has been little progress in how to solve the collisionless Boltzmann
equation with self-gravity (Poisson-Vlasov equations). As the accuracy
demanded of cosmological simulations increases, it is likely that the short-
comings of current N-body schemes will eventually become limiting. This
motivates improvements in existing treatments, such as the introduction of
adaptive gravitational softening [417, 418] or improved density estimators
based on following the distortion of a tetrahedral tessellation of the initial
density field [419, 420], as well as radical departures from the Monte Carlo
N-body approach, such as full six-dimensional Vlasov solvers [421] or
employing kinetic theory [422].
2. ii)
New Programming Languages - the need for parallelism
Parallelism and HPC become increasingly important also for industry and
business applications, where server farms and computing cluster with tens of
thousands of cores are now used for large-scale data mining and reduction
problems. Furthermore, since the break-down of Moore’s law for the raw core
processor performance, developers in all fields, ranging from mobile devices
to high performance cluster, need to start thinking parallel. It is therefore
not surprising that new programming languages, paradigms, and models are now
emerging, which have parallelism more naturally built in.
One example is Unified Parallel C (UPC) as an extension of the ISO C 99
programming language designed for high-performance computing on large-scale
parallel systems. This includes architectures with common global address
spaces (SMP and NUMA) and also distributed memory systems, which are more
common for cosmological applications. The main idea of UPC is to present the
developer with a single shared, partioned address space, and a Single
Instruction Multiple Data (SIMD) programming model. UPC is a specific
implementation of the more general class of PGAS (partitioned global address
space) programming languages. Other languages which follow the PGAS principles
are Co-array Fortran, Titanium, Fortress, Chapel, X10 and Global Arrays. PGAS
is based on two main concepts: multiple execution contexts with separate
address spaces and access of memory locations on one execution instance by
other execution instances. It is very likely, that in the near future
simulation codes will start using some of these language.
Besides the actual simulation software, data processing software also becomes
increasingly complex and requires efficient parallelization. Large parts of
available post-processing pipelines are programmed in general-purpose,
interpreted high-level programming languages like Python, which is slowly
replacing older IDL implementations. Both languages were essentially designed
without a particular focus on parallelism, though it is possible to use them
for parallel processing. But new languages are also arising in this field,
which will likely replace Python or IDL as the de-facto standards in the near
future. For example, Julia is new a high-level, high-performance dynamic
programming language for technical computing, which provides a sophisticated
compiler, distributed parallel execution, and an extensive mathematical
function library. Altough it is a LLVM-based just-in-time (JIT) compiler based
language it reaches nearly C/C++ performance due to heavy optimization
outperforming IDL, MATLAB, R, Python, Octave by a large factor. Most
importently it has parallelism naturally build in, which makes it suitable for
large-scale data mining and reduction on huge compute clusters.
UPC, as a PGAS parallel programming language, and Julia, as a high-performance
dynamic programming language, are just two examples and it is very likely that
the programming language landscape will move significantly more towards HPC
and parallel computing in the very near future mainly driven by industry
needs.
## 5 Conclusions
Two of the greatest mysteries of contemporary physics are the nature of dark
matter and of dark energy. As we have seen over and again in this work,
experimental and observational efforts to get at the answers to these
questions are intimately tied to predictions from cosmological DM simulations.
The simulation field, however, is rapidly evolving, as computational resources
continue to grow, enabling larger, more complicated, and more realistic
simulations to be performed. As a result, it has become more difficult for
physicists not directly involved with simulations to follow the progress of
the field, and to stay up to date with the evolving implications for studies
of DM and DE. Motivated by this realization, we have in this review attempted
to provide an overview of the current state of the field of cosmological DM
simulations, with a particular emphasis on the connection to DM detection
experiments and observational probes of DE. We have summarized the successes
and accomplishments of the current generation of simulations, but also
detailed their problems and short-comings, as well as the challenges faced by
the next generation of simulations in the coming decade.
For the DE problem, the greatest need appears to be a tremendous increase in
the number of very high resolution, very large volume, DM-only simulations,
scanning over cosmological parameters in order to allow brute-force Monte-
Carlo estimates of the parameter errors and their co-variances for future DE
surveys. At the same time, cosmic scale simulations will continue to
investigate the effects of baryonic physics, non-Gaussian initial conditions,
and modified laws of gravity.
For the DM problem, on the other hand, it seems that the DM-only approach by
itself is nearing the end of its usefulness – baryonic physics effects are
simply too important to neglect on galactic scales. There is some room for
improvement in the DM-only approach, in particular for the internal structure
of subhalos, for exploring cosmic variance and host halo mass dependency, and
for departures from the cold and collisionless DM assumption. However, the
main driver of the field must be studies of baryonic effects on the
distribution of DM in halos, since these have the potential to profoundly
alter expectations for DM detection.
The next decade promises to be filled with exciting challenges and the
potential for great discoveries. It is safe to say that the field of Numerical
Simulations of the Dark Universe will not run out of things to do.
## Acknowledgments
We are very grateful to Tom Abel, Jean-Michel Alimi, Michael Boylan-Kolchin,
Francis-Yan Cyr-Racine, Liang Gao, Phil Hopkins, Anatoly Klypin, Chung-Pei Ma,
Kristin Riebe, Graziano Rossi, Uros Seljak, Kris Sigurdson, Joachim Stadel,
Romain Teyssier, and Martin White for valuable discussions and assistance with
the simulation survey. This work has made extensive use of NASA’s Astrophysics
Data System.
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|
arxiv-papers
| 2012-09-25T20:00:02 |
2024-09-04T02:49:35.563362
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Kuhlen, Mark Vogelsberger, Raul Angulo",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/1209.5745"
}
|
1209.5746
|
# A SEARCH FOR CO-EVOLVING ION AND NEUTRAL GAS SPECIES IN PRESTELLAR MOLECULAR
CLOUD CORES
Konstantinos Tassis11affiliation: Max-Planck Institut für Radioastronomie,
53121 Bonn, Germany , Talayeh Hezareh11affiliation: Max-Planck Institut für
Radioastronomie, 53121 Bonn, Germany , & Karen Willacy22affiliation: Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109,
USA
###### Abstract
Comparison of linewidths of spectral line profiles of ions and neutral
molecules have been recently used to estimate the strength of the magnetic
field in turbulent star-forming regions. However, the ion (HCO+) and neutral
(HCN) species used in such studies may not be necessarily co-evolving at every
scale and density and may thus not trace the same regions. Here, we use
coupled chemical/dynamical models of evolving prestellar molecular cloud cores
including non-equilibrium chemistry, with and without magnetic fields, to
study the spatial distribution of HCO+ and HCN, which have been used in
observations of spectral linewidth differences to date. In addition, we seek
new ion-neutral pairs that are good candidates for such observations because
they have similar evolution and are approximately co-spatial in our models. We
identify three such good candidate pairs: HCO+/NO, HCO+/CO, and NO+/NO.
###### Subject headings:
ISM: molecules – ISM: clouds – ISM: magnetic fields – magnetohydrodynamics
(MHD) – stars: formation – ISM: abundances
## 1\. Introduction
Observations of large velocity dispersions in the line profiles of molecules
(line widths wider than expected from thermal motions) in interstellar
molecular clouds is a signature of localized supersonic turbulence, while
observations of the Zeeman effect (e.g., Crutcher et al., 1999) and the
polarization of dust thermal emission (e.g., Hildebrand et al., 2009) indicate
that these clouds are permeated by magnetic fields. Thus interstellar
molecular clouds need to be studied in the framework of magnetohydrodynamics.
For typical molecular cloud conditions (e.g., $B\sim 10\mu G$, density $\sim
10^{3}{\rm cm^{-3}}$, and ionized fraction $\lesssim 10^{-6}$) magnetic
forces, which are experienced directly by ionized particles, are transmitted
to the bulk of the neutral gas molecules via frequent collisions. However the
coupling of the magnetic field with the gas is not perfect. Motions of the
bulk neutral gas with frequencies higher than the typical ion-neutral
collision frequency can not be transfered to the ions. This relative motion
between charged and neutral particles, known as ambipolar drift (Mestel &
Spitzer, 1956), results in the dissipation of hydromagnetic waves in small
scales (Braginskii, 1965; Kulsrud & Pearce, 1969; Zweibel & Josafatsson, 1983)
and can cause differences between the velocity dispersion spectra of the ions
and the neutrals, as was recognized by Houde et al. (2000a, b) and explored
further by Li & Houde (2008). Indeed, Mouschovias et al. (2011), in a
comprehensive study of magnetohydrodynamic waves in weakly-ionized gases,
showed that most hydromagnetic waves with wavelengths below a critical value
cannot propagate in the neutrals because they are damped rapidly by ambipolar
diffusion, but certain long-lived modes in the neutrals exist at all
wavelengths; in contrast, waves in the ions damp very rapidly at short
wavelengths.
Observationally, the narrower line profiles of ions compared to that of co-
existing neutral species was first noticed and investigated by Houde et al.
(2000a, b), who argued that the reason of this ion line narrowing effect was
due to the presence of magnetic fields in turbulent molecular clouds that are
on average not aligned with the local flows. To test this analysis, Lai et al.
(2003) observed DR21(OH) at a high spatial resolution in H13CO+ and H13CN and
confirmed that the ion-neutral pair is co-existent within their observing
scale ($6^{\prime\prime}$) and that the ion line widths are indeed narrower
than the neutral line widths. Li & Houde (2008) re-visited this magnetically
induced ion line narrowing effect. They mapped the M17 star forming region in
${\rm HCO^{+}}$ $(J=4\rightarrow 3)$ and HCN $(J=4\rightarrow 3)$ and fitted
the lower envelopes of the velocity dispersion spectra of the observed line
profiles of the ion and neutral pair to a Kolmogorov-type power-law. They
determined the ambipolar diffusion length scale using the fitting parameters,
and used it to calculate the strength of the plane-of-sky component of the
magnetic field. Hezareh et al. (2010) further tested the Li & Houde (2008)
technique by mapping DR21(OH) in the optically thin ${\rm H^{13}CO^{+}}$ and
${\rm H^{13}CN}$ $(J=4\rightarrow 3)$ lines and obtained the turbulent
ambipolar diffusion length and magnetic field strength in that source. Hezareh
et al. (2012) used interferometry maps of the ground transition of the latter
pair of molecules in a few massive dense cores in the Cygnus X region to
verify that the method is reproducible for both single-dish and interferometry
data.
One concern in interpreting these observations is that chemical
differentiation might also be responsible for differences in the spectral
appearance of different molecular species: species with different spatial
distributions would also exhibit different velocity profiles. Although HCN and
HCO+ have been shown to be approximately co-existent in the observed molecular
clouds on the scales probed (Houde et al., 2000a, b; Lai et al., 2003; Houde
et al., 2004), their chemical co-evolution has yet to be examined from a
theoretical perspective. Here, we use both magnetic and non-magnetic dynamical
models of core collapse coupled with, and self-consistently following, non-
equilibrium chemistry both in the gas phase and on grains (Tassis et al.,
2012, hereafter paper I) to examine whether the typically observed neutrals
and ions are indeed theoretically expected to be similarly distributed in a
star-forming core. We also repeat this exercise for other frequently observed
neutral and ionized molecules, and we propose neutral-ion pairs that,
according to our models, best maintain a co-spatial distribution.
## 2\. Models
Figure 1.— Line types and colors used to denote each of the models studied.
Solid normal-thickness red line: “reference” magnetic model; solid normal-
thickness blue line: “reference” non-magnetic model. Dotted lines: “fast”
models; dashed lines: “slow” models. Brown/purple lines: magnetic/nonmagnetic
models with temperatures differing from the “reference” models. Orange/cyan
shaded areas: variation in C/O ratio. Thin/thick solid red/blue lines:
lower/higher cosmic-ray ionization rate magnetic/non-magnetic models (see text
for details).
We are interested in the co-spatiality of various observable neutral-ion pairs
as a function of density in a star-forming region. Since the behavior of the
abundance of many molecules with density is sensitive to various model
parameters, we use all Paper I models to assess the behavior of the abundance
ratios of neutral-ion pairs with density. The best candidate molecule pairs
for studies of systematic differences of the linewidths between neutrals and
ions will be those with abundance ratios that vary weakly with density for any
set of model parameters.
The physics of the chemodynamical models we use and its effect on molecular
abundances are exhaustively discussed in Paper I. Here, we briefly review the
parameters we have varied for each class of models: temperature, C/O ratio,
cosmic ray ionization rate, and a parameter controlling the time available for
chemical evolution. The latter is the mass-to-magnetic-flux ratio for magnetic
models, and the collapse delay time for non-magnetic models (an initial time
period during which chemistry evolves but the core does not evolve
dynamically, representing an early stage of support due to turbulence which
later decays).
The initial parameter values for our “reference” magnetic model are a mass-to-
magnetic-flux ratio equal to the critical value for collapse (Mouschovias &
Spitzer, 1976), a temperature of 10 K, a C/O ratio of 0.4, and the canonical
value for the cosmic ray ionization rate $\zeta=1.3\times
10^{-17}{\rm\,s^{-1}}$. The relevant parameter values of our “reference” non-
magnetic model are a collapse delay time (hereafter “delay”) of 1 Myr, with
the other parameters being identical to the “reference” magnetic model.
For magnetic models we vary the initial mass-to-magnetic-flux ratio, examining
two additional values: 1.3 times the critical value (a faster-evolving,
magnetically supercritical model), and 0.7 of the critical value (a slower,
magnetically subcritical model). For non-magnetic models we examine two
additional values of delay: zero, and 10 Myr.
For each of these six dynamical models, the carbon-to-oxygen ratio is varied
from its “reference” value by keeping the abundance of C constant and changing
that of O. Additional values of C/O ratio we examined are 1 and 1.2. This way,
we study a “matrix” of 18 combinations (9 magnetic and 9 non-magnetic) of
available times for chemical evolution and C/O ratios.
To test the effect of temperature ($T$), we have varied each of the six basic
dynamical models by changing $T$ by a factor of $\sim 1.5$ from its reference
value of 10 K and examined models with $T=7$ K and $T=15$ K. This results in
12 additional models (6 magnetic and 6 non-magnetic).
To test the cosmic ray ionization variation, we studied four additional models
(two magnetic and two non-magnetic), which have the “reference” value for the
temperature, C/O ratio, and mass-to-flux ratio or delay (for magnetic and non-
magnetic models respectively), but for which $\zeta$ is varied by a factor of
four above ($\zeta=5.2\times 10^{-17}$ $s^{-1}$) and below ($\zeta=3.3\times
10^{-18}$ $s^{-1}$) its “reference” value, covering the range of observational
estimates (e.g., McCall et al. (2003); Hezareh et al. (2008)).
Figure 1 shows a visual representation of these suite of models and is a quick
reference guide for the line colors and types we have used to depict each
model.
## 3\. Results
### 3.1. HCO+/HCN
Figure 2.— Left column: Evolution of central abundance ratio of HCO+ and HCN
as a function of central number density. Middle and right columns: radial
profiles of the HCO+ / HCN abundance ratio, for two different evolutionary
stages as quantified by the central number density. Line types and colors are
as described in Fig. 1. Top row: Different dynamical models for fiducial
values of the C/O ratio, CR ionization rate, and temperature. Seond row:
effect of the CR ionization rate for the fiducial magnetic and non-magnetic
models. Third row: effect of the C/O ratio for each dynamical model, for
fiducial values of the CR ionization rate and temperature. Bottom row: effect
of temperature for each dynamical model for fiducial values of the CR
ionization rate and C/O ratio. Figure 3.— As in Fig. 2 but for the HCO+ / NO
pair. Figure 4.— As in Fig. 2 but for the HCO+ / CO pair. Figure 5.— As in
Fig. 2 but for the NO+ / NO pair.
Figure 2 shows the results of our aforementioned models for the evolution of
the commonly observed ion-neutral pair HCO+/HCN. We show the evolution of the
central abundance ratio with central density (left column), and radial
profiles of the abundance ratio at two different evolutionary stages (of
central density $10^{4}$ and $10^{6}$ cm-3, central and right columns,
respectively). The top row shows the effect of dynamics on the abundance
ratio, while the second, third, and bottom rows demonstrate the effect of the
CR ionization rate, C/O ratio, and temperature, respectively.
As shown in the left column, the central abundance ratio, especially in the
magnetic models, is not particularly stable with central density. The reason,
as can be seen in Figure 5 of Paper I, is that although both HCO+ and HCN
deplete with increasing central density, in our magnetic models HCO+ depletes
initially faster than HCN and therefore their ratio declines by almost three
orders of magnitude by a central density of $10^{4}$ cm-3. However, at higher
central densities, the depletion rate of HCO+ slows down while HCN keeps
depleting at its initial rate. This effect is responsible for the increase of
the abundance ratio at higher densities. In non-magnetic models, the
dependence of the abundance ratio on central density is much less pronounced,
as the behavior of each of HCO+ and HCN with central density is qualitatively
similar (Figure 5 of Paper I), with only a small rise in the abundance ratio
above $10^{6}$ cm-3 due to some flattening of the depletion rate of HCO+ at
the highest densities examined.
This behavior is also reflected in the radial profiles of this abundance
ratio, as seen in the middle and right columns of Figure 2. The magnetic
models and the slow non-magnetic models (blue, cyan, and purple dashed curves)
exhibit a remarkable variation between the outer layers of the star-forming
core where significant amounts of HCO+ still exist, and the innermost and
densest parts. The fast non-magnetic models, however, have a much milder
dependence on radius. At a more advanced evolutionary stage (n${}_{c}\sim
10^{6}$ cm-3), the minimum seen in the evolution of the central abundance
ratio with density in magnetic models also appears in the radial profile at
intermediate scales. At scales comparable to the size of the core, the radial
profiles of the HCO+/HCN abundance ratio flatten out for all models.
The total variation of the abundance ratio across models, parameters, and
densities is large (several orders of magnitude). As seen in the top row of
Fig. 2, most of the total variation has its origin in the dynamics, with the
CR ionization rate having the second most important contribution (second row).
However, different models represent different possible clouds in nature, and
for this reason the variation of the abundance ratio across models is not the
relevant concern for the Houde et al. type of analysis; instead, the quantity
of interest is the total variation of the ratio within a single model, across
space and time. Still, as seen in for example in the left column of Fig. 2,
the maximum possible variation within a single model can be as large as
several orders of magnitude.
Observationally, the intensity maps of the HCO+/HCN pair show highest degree
of correlation at large length scales, about $0.5-1.0$ pc (Lo et al., 2009),
which correspond to scales comparable with or larger than the entire core in
our models ($r/R_{0}\gtrsim 1$ in Figure 2). However, this correlation can be
reduced at smaller scales of about $0.1$ pc, i.e., resolutions achieved by
interferometric observations (Csengeri et al., 2011).
HCO+ and HCN follow independent chemical networks and thus have no reason to
be co-evolving. The two molecules evolve differently and cannot be considered
co-spatial in most of our models. HCO+ is more abundant at lower densities and
larger spatial scales. Furthermore, the critical densities of the commonly
observed rotational transitions of HCO+ are lower than these of HCN.
Consequently, HCO+ would in principle be expected to trace larger spatial
scales, regions with higher levels of turbulence, which would lead to wider
linewidths for HCO+ rather than HCN. However, this is contradictory to
observations. Houde et al. (2000a) observed the $J=3\rightarrow 2$ and
$J=4\rightarrow 3$ transitions of HCO+ and HCN in several molecular clouds
where they found these species co-existent, with the ion lines indeed more
opaque but narrower than the neutral lines. This can be explained by the
effect of the magnetic field on the dynamics of molecular ions through the
cyclotron interaction, which can in turn be interpreted as a signature of
ambipolar diffusion (Houde et al., 2004) and be used to estimate the strength
of the magnetic field. Indeed, it was later shown that clouds with stronger
magnetic fields (projected in the plane of the sky) show larger differences
between the ion and neutral spectral linewidths (Li et al., 2010; Hezareh et
al., 2012). In contrast, chemical and radiative transfer effects act in the
opposite direction causing the widths of neutral spectral lines to become
narrower than those of ions, thus underestimating the strength of the magnetic
field.
To make a more robust comparison of our models with existing observations we
must perform radiative transfer calculations for the lines observed. We will
return to this problem in the future. An observational correlation similar to
HCO+ and HCN is observed for the H13CO+ and H13CN pair but since our chemical
models do not incorporate the chemistry of the isotopes of carbon and other
elements, we do not include these isotopologue pairs in our discussion. For
the remainder of this paper, we identify additional ion and neutral species
that are good candidates for being considered as co-existent pairs.
### 3.2. HCO+/NO
Figure 3 shows the same pattern as in Figure 2, i.e., the evolution of central
abundance ratio with central density in the left column, and radial profiles
of the abundance ratio for central densities of $10^{4}$ and
$10^{6}{\rm\,cm^{-3}}$ in the middle and right columns, respectively, for the
HCO+ / NO pair. The central abundance ratio of this pair is significantly more
stable with central density within any single model than the HCO+/HCN ratio,
as HCO+ and NO evolve in a similar manner with central density, with the
exception of the high C/O ratio models (third row: yellow and cyan lines for
the magnetic and non-magnetic models, respectively). In all other cases, the
variation of the abundance ratio with central density within a single model is
$\lesssim$ an order of magnitude over the central density range we examine.
A similar behavior is reflected in the radial profiles of the HCO+ / NO
abundance ratio, with most models exhibiting little variation. The large
variation in the HCO+ / NO pair for the magnetic high C/O-ratio model (yellow
curve) only affects very small scales, which are also associated with high
densities. At these densities, however, the depletion of these molecules is
significant (Fig. 5 in Paper I), so these regions are not expected to have a
considerable contribution to the observed lines.
### 3.3. HCO+/CO
The plots for the HCO+ / CO pair is shown in Figure 4. This pair exhibits very
small variations of the central abundance ratio with central density for all
but the fastest non-magnetic models. However, even in the case of these fast
models (dotted blue, cyan, and purple curves), the variation is less than an
order of magnitude for central densities above 104 cm-3.
In the radial profiles, some variation in the abundance ratio occurs at large
radii, but it is less than about an order of magnitude, at least in early
times, and not much more than an order of magnitude in later times, even in
the most-varying, fast non-magnetic models.
It is not surprising that the abundances of the two molecules are co-evolving,
since the dominant reaction for the production of HCO+ in our models involves
CO:
${\rm CO+H_{3}^{+}\rightarrow HCO^{+}+H_{2}}\,,$ (1)
while the destruction of HCO+ proceeds primarily through dissociative
recombination with an electron:
${\rm HCO^{+}+e^{-}\rightarrow CO+H}\,,$ (2)
which leads back to creating CO, forming a production/destruction loop for the
two molecules.
### 3.4. NO+/NO
Figure 5 shows the same plots as before for the NO+ / NO pair. This is the
best example of co-evolution and chemical co-spatiality of an ion-neutral pair
that we have identified in our coupled chemical/dynamical models. The
variation of the central abundance ratio with central density in any single
model is at most a factor of a few. These findings are also reflected in the
radial profiles as well.
The reason for the co-evolution of the two molecules is that NO+ is mainly
produced through ionization of NO or charge-exchange reactions, also involving
NO:
${\rm NO+H^{+}\rightarrow NO^{+}+H}$ (3)
and
${\rm NO+C^{+}\rightarrow NO^{+}+C}\,.$ (4)
The ion is destroyed through dissociative recombination with an electron:
${\rm NO^{+}+e^{-}\rightarrow O+N}\,,$ (5)
so the abundance of NO+ is coupled to that of NO.
## 4\. Discussion
In addition to the co-spatiality of the three new ion-neutral pairs that we
have identified, their observability and critical densities for excitation of
their rotational transitions must also be considered to decide whether they
are viable candidates for studies of the ambipolar ion-neutral drift in star-
forming regions.
The most frequently observed transitions of HCO+ have critical densities111All
molecular data are taken from the Leiden Atomic and Molecular Database
(Schöier et al., 2005), http://home.strw.leidenuniv.nl/$\sim$moldata/ of the
order of 10${}^{5-6}{\rm\,cm^{-3}}$. NO generally has much lower critical
densities, so overall unless higher critical density transitions of NO can be
observed, this would pose a problem in the use of the HCO+/NO pair. The reason
is that NO would trace lower-density, higher-turbulence medium, which would
lead to wider linewidths for NO, a trend that could mimic the effect observed
in the HCO+/HCN pair. Possible higher critical-density transitions would, for
example, be the $J=7/2\rightarrow 5/2$ transition of NO at $350.6$ GHz, with a
critical density of $\simeq 10^{5}{\rm\,cm^{-3}}$. The multiplets of the first
two rotational transitions of NO, i.e., $J=3/2\rightarrow 1/2$ and
$J=5/2\rightarrow 3/2$ have been observed in both dark and massive clouds
(Gerin et al., 1992) at 150 GHz and 250 GHz with relative abundances
calculated to vary from $10^{-7}-10^{-8}$.
A similar problem arises with the HCO+/CO pair, as CO, with generally low
critical densities in usually observed transitions, traces more extended
regions. Because the abundance of CO is high, such transitions would result in
saturated and optically thick lines. To avoid this effect, one needs to use
higher CO transitions. A possibility would be the $J=6\rightarrow 5$
transition at $691$ GHz, with critical density around $10^{5}{\rm\,cm^{-3}}$.
The NO+/NO pair is in principle the best choice in terms of co-spatiality of
the neutral and the ion, but although NO+ has sub-millimeter and radio
transitions identified in the laboratory (Bowman et al., 1982), the ion has
not yet been observed in molecular clouds. This may be due to a general trend
in which nitrogen-bearing molecules, with the exception of NH3 and N2H+, have
not been targeted for observations as much as the carbon-bearing species, in
part because nitrogen chemistry in molecular clouds is even less understood
(especially at low temperatures) than that of carbon (e.g., Hily-Blant et al.,
2010).
We note that our models do not include the effect of turbulence; as a result,
the relative abundance variation that we find is an upper limit, since
turbulent mixing would lessen this variation (Xie et al., 1995). Also, our
models represent prestellar cores, and do not include the effect of feedback
sources, because our parameter studies do not feature very large variations of
the temperature, and UV ionization has not been accounted for. However,
regions with feedback sources are avoided when applying the Houde et al.
method.
We thank Harold Yorke and Helmut Wiesemeyer for insightful and constructive
comments that improved this paper. T.H. is funded by the Alexander von
Humboldt foundation. The project was supported in part by the NASA Origins of
Solar Systems program. Part of this work was carried out at the Jet Propulsion
Laboratory, California Institute of Technology under contract with the
National Aeronautics and Space Administration.
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|
arxiv-papers
| 2012-09-25T20:00:02 |
2024-09-04T02:49:35.589583
|
{
"license": "Public Domain",
"authors": "Konstantinos Tassis (MPIfR), Talayeh Hezareh (MPIfR), and Karen\n Willacy (JPL)",
"submitter": "Konstantinos Tassis",
"url": "https://arxiv.org/abs/1209.5746"
}
|
1209.5794
|
# Preliminary Analysis of WISE/NEOWISE 3-Band Cryogenic and Post-Cryogenic
Observations of Main Belt Asteroids
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:
Planetary Science Institute, Tucson, AZ 85719 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 , C. Nugent44affiliation: University of California, Los Angeles,
CA, 90095 , M. S. Cabrera11affiliation: Jet Propulsion Laboratory/California
Institute of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109,
USA, Joseph.Masiero@jpl.nasa.gov 55affiliation: California State Polytechnic
University Pomona, Pomona, CA, 91768
###### Abstract
We present preliminary diameters and albedos for 13511 MBAs that were observed
during the 3-Band Cryo phase of the WISE survey (after the outer cryogen tank
was exhausted) and as part of the NEOWISE Post-Cryo Survey (after the inner
cryogen tank was exhausted). With a reduced or complete loss of sensitivity in
the two long wavelength channels of WISE, the uncertainty in our fitted
diameters and albedos is increased to $\sim 20\%$ for diameter and $\sim 40\%$
for albedo. Diameter fits using only the $3.4$ and $4.6~{}\mu$m channels are
shown to be dependent on the literature optical $H$ absolute magnitudes. These
data allow us to increase the number of size estimates for large MBAs which
have been identified as members of dynamical families. We present thermal fits
for $14$ asteroids previously identified as the parents of a dynamical family
that were not observed during the fully cryogenic mission.
## 1 Introduction
In Masiero et al. (2011, hereafter Mas11) we presented thermal model fits for
$129,750$ Main Belt asteroids that were observed during the fully cryogenic
portion of the Wide-field Infrared Survey Explorer (WISE, Wright et al., 2010)
mission, which ran from 7 January 2010 to 6 August 2010. Sensitivity to Solar
system objects was enabled by the NEOWISE augmentation to the WISE mission
(Mainzer et al., 2011a) which provided capability for processing and archiving
of single-frame exposures and detection of previously known and new asteroids
and comets. On 6 August 2010 the hydrogen ice in the outer cryogen tank was
exhausted and the telescope began to warm up, resulting in an almost immediate
loss of the W4 ($22~{}\mu$m) channel and a decreasing sensitivity in W3
($12~{}\mu$m) beginning the 3-Band Cryo portion of the mission. On 29
September 2010 the hydrogen ice in the inner cryogen reservoir, used to cool
the detectors, was exhausted and the W3 channel was lost. From 29 September
2010 to 1 February 2011, WISE continued to survey the sky in the NEOWISE Post-
Cryo survey phase (Mainzer et al., 2012), searching for new near-Earth objects
(NEOs) and completing the survey of the largest Main Belt asteroids (MBAs)
using the two shortest bandpasses: W1 ($3.4~{}\mu$m) and W2 ($4.6~{}\mu$m).
MBAs have temperatures of $\sim 200~{}$K, depending on their distance from the
Sun and surface properties. This places the peak of their blackbody flux near
$\lambda_{peak}\sim 15~{}\mu$m. During the fully cryogenic portion of the WISE
mission the W3 bandpass straddled this peak and was the primary source of data
used for identification and analysis of the thermal emission from MBAs. For
objects detected during the 3-Band Cryo portion of the mission we used the W3
data to constrain the thermal emission, and thus the diameter, of the objects
observed. As the telescope warmed up, the integration times in W3 were
shortened to prevent saturation of the detectors from the increasing thermal
emission of the telescope (Cutri et al., 2012), resulting in a decrease in
sensitivity to sources in the bandpass. During the Post-Cryo Survey only W1
and W2 were operational: for MBAs W1 was sensitive solely to reflected light,
while W2 was a blend of reflected and emitted flux dictated by the object’s
physical and orbital parameters (e.g. distance to Sun at the time of
observation, surface temperature, albedo, etc.).
In this work, we present preliminary thermal model fits for 13511 Main Belt
asteroids observed during the 3-Band Cryo phase of the WISE survey and the
NEOWISE Post-Cryo Survey. During the fully cryogenic portion of the survey,
detectability of most minor planets was dominated by their thermal emission
and so was essentially independent of their albedo (Mainzer et al., 2011c).
However, the Post-Cryo Survey data at $3.4~{}\mu$m and $4.6~{}\mu$m are a mix
of reflected and emitted light. Thus detectability is strongly coupled to
albedo. Additionally, objects with lower temperatures will have a smaller
thermal emission component to their flux in the W2 band, resulting in a less
accurate estimate of diameter. In general, diameter fits using either the
3-Band Cryo or the Post-Cryo Survey data will typically have larger errors and
lower precision than fits from the fully cryogenic survey given in Mas11,
though they still provide useful information about the observed population of
MBAs.
One of the drivers for completing the NEOWISE survey of the inner Main Belt
after the cryogen was exhausted was to have a complete census of the largest
asteroids, particularly those that may be members of asteroid families. Having
this list allows us to constrain the mass of the pre-breakup body and more
precisely model the age of the family (Vokrouhlický et al., 2006; Masiero et
al., 2012). We present in this work preliminary albedos and diameters for
objects observed during the 3-Band Cryo and Post-Cryo Survey and discuss the
accuracy of these values because these fits use data processed with the
preliminary survey calibration values. Future work by the NEOWISE team will
include second-pass processing of the raw data with finalized calibration
values as well as extraction of sources at lower signal-to-noise that will
precede a final release of albedos and diameters.
## 2 Observations
In Mas11 we focused our analysis on data taken during the fully cryogenic
portion of the WISE mission. For this work, we analyze the 3-Band Cryo and
Post-Cryo Survey observations taken by WISE as part of the NEOWISE survey.
Observations obtained between Modified Julian Dates (MJDs) of 55414 and 55468
are available in the 3-Band Cryo Single-Exposure database, served by the
Infrared Science Archive (IRSA)111http://irsa.ipac.caltech.edu. Post-Cryo
data, spanning a MJD range of 55468 to 55593, are archived in the NEOWISE
Preliminary Post-Cryo database and also served by IRSA. Data from the 3-Band
Cryo survey were released to the public on 29 June
2012222http://wise2.ipac.caltech.edu/docs/release/3band/ and preliminary data
from the NEOWISE Post-Cryo Survey were released to the public on 31 July
2012333http://wise2.ipac.caltech.edu/docs/release/postcryo_prelim/. We note
that the Post-Cryo Survey data have only undergone first-pass processing, and
users are strongly encouraged to consult the Explanatory Supplement (Cutri et
al., 2012) associated with the database.
We follow the same method as described in Mas11 to acquire detections of MBAs
that have been vetted both by our internal WISE Moving Object Processing
System (WMOPS; Mainzer et al., 2011a) and by the Minor Planet Center (MPC).
This includes the use of the same quality flag settings from the pipeline
extraction for cleaning of detections before thermal fitting as discussed in
Mas11. Of the 14638 objects observed by WISE between MJDs 55414 and 55593,
13511 MBAs had data of sufficient quality to perform thermal model fits.
Due to the nature of WISE’s orbit and the synodic period of MBAs,
approximately half of the objects observed during the 3-Band Cryo and Post-
Cryo Survey had also been observed earlier during the fully cryogenic survey.
We use these overlap objects as standards to evaluate the accuracy of the
thermal model fits using these data (see Section 4.1). While in some cases
extremely irregularly shaped slow-rotating objects may show significant
changes in projected area between epochs and thus large variations in both
emitted and reflected flux, this is expected to be a small fraction of all
objects observed and only to add a small component of random error to the
comparison (Grav et al., 2011).
## 3 Thermal Fitting
Following the procedure discussed in Mas11, we use a faceted NEATM thermal
model to determine the diameter and albedo of the MBAs observed after the
outer cryogen tank was exhausted. In most cases we only have thermal emission
data in a single band, and so we are forced to assume a beaming parameter for
the models. We use a beaming parameter of $\eta=1.0\pm 0.2$, based on the peak
of the distribution for MBAs given in Mas11. Our measured flux in W2 is
typically dominated by thermal emission, however the reflected component of
the W2 flux will influence our models.
In order to remove the reflected component from the measured W2 flux, we need
to determine the optical geometric albedo ($p_{V}$) and assume a ratio between
the near-IR (NIR) and optical albedos. In Mas11 we were able to fit this ratio
for objects with with observations in $W3$ and/or $W4$ as well as $W1$ and
$W2$, however we cannot do this for the Post-Cryo Survey data. Following the
best-fit value from Mas11, for those objects we assume a NIR/optical
reflectance ratio of $1.4\pm 0.5$. In all cases, we also assume that the
reflectivity in W1 is identical to that in W2 ($p_{W1}=p_{W2}=p_{NIR}$). For
objects with very red spectral slopes this may not necessarily be a good
assumption (cf. Mainzer et al., 2011b; Grav et al., 2012) however without
additional data (e.g. spectral taxonomy) it is impossible to disentangle these
two values for this dataset.
To determine optical albedo we used the $H$ absolute magnitude and $G$ slope
parameter given in the Minor Planet Center’s MPCORB
file444http://www.minorplanetcenter.net/iau/MPCORB.html, and updated using
other databases following Mas11. We note that recent work has shown that these
$H$ values may be systematically offset in some magnitude ranges by up to
$0.4~{}$mags when comparing predicted and observed apparent magnitudes (Pravec
et al., 2012). This will affect the albedos that we calculate for the
asteroids presented here, which in turn will change the relative contribution
of emitted and reflected light in W2. Unlike the results presented in Mas11,
where the diameter determination is independent of the optical $H$
measurement, any future revision to the measured $H$ values will require a
refitting of the thermal models and will likely result in an change in modeled
diameter.
## 4 Discussion
### 4.1 Comparison of Overlap Objects
Mainzer et al. (2012) showed a comparison between the thermal fits performed
with the Post-Cryo Survey data and non-radiometrically determined diameters
for a range of NEOs and MBAs to derive a relative accuracy of $\sim 20\%$ on
diameter and $\sim 40\%$ on albedo. As a parallel check we have taken objects
that were observed both before and after the exhaustion of the outer cryogen
reservoir and compared the diameters and albedos found here to those values
given in Mas11. Of the fits presented here, 7222 unique objects also appeared
in the fully cryogenic observations that were presented in Mas11. Of these,
2844 were observed during the 3-Band Cryo phase of the survey and 4403 were
observed during the Post-Cryo Survey (note that 25 objects appeared in all
three phases of the survey).
For all objects seen in both the Post-Cryo Survey and in the fully cryogenic
4-band survey, we have refit the 4-band Cryo data using only the W1 and W2
measurements as a way of differentiating changes in the quality of fit due to
the loss of W3 and W4 sensitivity from changes due to the different observing
circumstances. The results of this test are shown in Figure 1. We include a
running box average of the data in order to assess the population trends,
which bins by 100 objects, in steps of 20. In general these tests follow the
expected one-to-one relationship, with the exception of the comparison of the
2-band and 4-band fits of the fully cryogenic data (Figure 1b), which deviates
at both high and low albedos, and effect that was also observed for the NEOs
in the Post-Cryo Survey data by Mainzer et al. (2012). (Mainzer et al., 2011b)
have shown that high albedo objects tend to have optical/NIR reflectance
ratios of $\sim 1.6$, while objects with low albedos tend to have reflectance
ratios of $\sim 1.0$ (though D-type objects deviate from this trend and have
very large reflectance ratios). As we use a fixed reflectance ratio of $1.4$,
low albedo objects with W1 measurements will have a final fitted $p_{V}$ below
the true value, while high albedo objects will have a $p_{V}$ slightly above,
which corresponds to the twist observed in Figure 1b.
Figure 1: Comparison of thermal fits for objects appearing in both the fully
cryogenic data set as well as the Post-Cryo Survey data. The left column shows
the comparison of the diameters ($\log~{}D$) while the right column shows the
comparison of the visual albedos ($\log~{}p_{V}$). The top row shows the
fractional difference between the 4-band fits presented in Mas11 and refits of
those data using only the W1 and W2 bandpasses, while the bottom row shows the
fractional difference between the 2-band refit of the fully cryogenic data and
the 2-band fits of the Post-Cryo Survey data. The dotted line in each case
shows a one-to-one relationship, and the solid red line shows a running box
average for each comparison.
Comparison of the 2-band refits to the results from Mas11 show the uncertainty
induced by the loss of W3 and W4 information (Figure 1a-b) results in a
$1\sigma$ scatter of $16\%$ in diameter and $32\%$ in albedo (three points
fall outside the plotted range for Figure 1(a); all other panels show all
objects considered). Comparison of the 2-band refits of the fully cryogenic
data to the Post-Cryo Survey fits (Figure 1c-d) shows the errors induced by
both changes in observing aspect as well as calibration differences between
the two data sets, which collectively result in a $1\sigma$ scatter of $13\%$
in diameter and $31\%$ in albedo as well. Combined, these two errors result in
a measured $21\%$ relative error on diameter and $45\%$ in albedo.
These total errors are in line with what was found by Mainzer et al. (2012)
when comparing the fits from Post-Cryo Survey data to literature diameters.
Our errors are also in line with the uncertainties measured for the
ExploreNEOs project which uses a similar pair of bandpasses ($3.6~{}\mu$m and
$4.5~{}\mu$m) from the Warm Spitzer mission to model diameters and albedos for
previously known NEOs (Trilling et al., 2010; Harris et al., 2011). Our
measured level of error indicates that the random error introduced by the
combined effect of irregular shape and observing geometry is below this level.
We note that the method of source extraction used for all phases of the WISE
data processing relies on the position of the object in all detected bands. In
general the W1 and W2 measurements from the 4-band data will be at lower
signal-to-noise than the data for those objects extracted from the Post-Cryo
Data, inflating the errors quoted above.
We show in Figure 2 the comparison between the fits for objects appearing in
the fully cryogenic data as well as the 3-Band Cryo or Post-Cryo Survey data.
As in Figure 1 we include a running box average using the same parameters as
above. We see no large-scale systematic shifts between datasets, however we do
confirm the increase in scatter in the fits using the latter data sets. We
note that in Figure 2d shows a behavior similar to what we observe in Figure
1b, where the fits of albedo deviate to more extreme values for both high and
low albedo objects. As discussed above, this is attributed to the use of an
assumed optical/NIR reflectance ratio that is between the values measured for
high and low albedo objects when they are considered independently.
Figure 2: Comparison of thermal fits for objects appearing both in the fully
cryogenic data set as well as in the 3-Band Cryo or Post-Cryo Survey data. The
left column shows the comparison of the diameters ($\log~{}D$) while the right
column shows the comparison of the visual albedos ($\log~{}p_{V}$). The top
row shows the fractional difference between the 3-Band Cryo fits and the
values from Mas11, while the bottom row shows the fractional difference
between the 2-band Post-Cryo Survey fits and the Mas11 values. The dotted line
in each case shows a one-to-one relationship, and the solid red line shows a
running box average for each comparison.
### 4.2 Preliminary Diameters and Albedos
We present in Table 1 the preliminary fitted diameters and albedos for all
MBAs observed during the 3-Band Cryo and Post-Cryo surveys, along with their
associated errors (note that errors do not include the systematic $\sim 20\%$
diameter error or $\sim 40\%$ albedo error discussed above). We also include
the number of detections used in each band as well as the $H$ and $G$ values
used for the fit. Objects without measured visible magnitudes have “nan”
entered for their $H$, $G$, and albedo values. The recommended method for
extracting fluxes for asteroid detections is discussed in Mainzer et al.
(2011a) and Cutri et al. (2012). Figure 3 shows the preliminary diameter and
albedo distributions for the asteroids observed during 3-Band Cryo and Post-
Cryo Surveys compared to the population presented in Mas11. With the loss of
the long wavelength channels the sensitivity to small objects was reduced and
peak of the diameter distribution moves to larger sizes. In both the 3-Band
Cryo and Post-Cryo Survey data we see a shift in the high and low branches of
the albedo distribution to more extreme values when compared to the population
from Mas11. This shift was also observed for the NEOs by Mainzer et al.
(2012), and attributed to the forced values for both beaming and NIR/optical
reflectance ratio in the fits of the Post-Cryo Survey data.
Reflected light is a much more significant component in the W1 and W2
bandpasses for MBAs than in the W3 and W4 bands used in Mas11 to perform
thermal fits. As such, results from the model fits presented here are
inherently tied to the optical measurements and cannot be considered
insensitive to albedo as was assumed in Mas11. This bias will most strongly
affect objects that are small and have low albedos. Thus, care must be taken
before extrapolating the trends observed in the these fits to the greater MBA
population.
Table 1: Thermal model fits for MBAs in the 3-Band Cryo and NEOWISE Post-Cryo
Survey. Table 1 is published in its entirety in the electronic edition of
ApJL; a portion is shown here for guidance regarding its form and content.
Name | H | G | D (km) | $p_{V}$ | nW1 | nW2 | nW3
---|---|---|---|---|---|---|---
00003 | 5.33 | 0.32 | 246.60 $\pm$ 10.59 | 0.214 $\pm$ 0.026 | 11 | 11 | 0
00005 | 6.85 | 0.15 | 106.70 $\pm$ 3.14 | 0.282 $\pm$ 0.050 | 14 | 14 | 0
00011 | 6.55 | 0.15 | 154.13 $\pm$ 3.92 | 0.178 $\pm$ 0.030 | 12 | 12 | 0
00014 | 6.30 | 0.15 | 145.68 $\pm$ 5.27 | 0.251 $\pm$ 0.041 | 9 | 9 | 0
00016 | 5.90 | 0.20 | 288.29 $\pm$ 4.63 | 0.093 $\pm$ 0.024 | 5 | 5 | 0
00017 | 7.76 | 0.15 | 69.64 $\pm$ 2.26 | 0.287 $\pm$ 0.051 | 9 | 10 | 0
00018 | 6.51 | 0.25 | 155.84 $\pm$ 5.63 | 0.181 $\pm$ 0.033 | 12 | 12 | 0
00019 | 7.13 | 0.10 | 209.81 $\pm$ 2.20 | 0.056 $\pm$ 0.012 | 11 | 11 | 0
00020 | 6.50 | 0.25 | 135.68 $\pm$ 3.67 | 0.241 $\pm$ 0.018 | 13 | 13 | 0
Figure 3: Preliminary diameter (a) and albedo (b) distributions for all MBAs
from Mas11 (black dotted), MBAs from the 3-Band Cryo data (blue solid), and
MBAs from the Post-Cryo Survey (red dashed). Note the scales are normalized:
the total number of objects presented in Mas11 is over an order of magnitude
larger than the other two populations.
### 4.3 Asteroid Family Members
One of the primary drivers of the Post-Cryo Survey was to complete the census
of large MBAs that are related to dynamically associated asteroid families.
The largest body in a family anchors both the mass estimate of the pre-breakup
body and the starting point for family age simulations. The 3-Band Cryo and
Post-Cryo Survey data contain 3319 objects identified by (Nesvorný, 2010) as
members of asteroid families that were able to have thermal models fit to
their measurements. Of these, 14 were identified as family parents and were
not observed during the WISE fully cryogenic mission, including (3) Juno, (20)
Massalia, (44) Nysa, (170) Maria, (298) Baptistina, (363) Padua, (434)
Hungaria, (490) Veritas, (569) Misa, (778) Theobalda, (1270) Datura, (1892)
Lucienne, (4652) Iannini, and (7353) Kazuya.
Of these 14 bodies that are the largest in their family, only 4 had albedos
below $p_{V}=0.1$, in contrast with the general population presented here
where $\sim 60\%$ of the MBAs had low albedos. This is due to a number of
overlapping selection biases, including the dominance of high albedo objects
in the literature family lists (cf. Mas11), preferential sensitivity to high
albedo objects in the 3-Band Cryo and Post-Cryo Survey data compared to Mas11
(meaning this data set is more likely to miss low albedo asteroids), and the
longer synodic periods of MBAs with smaller semimajor axes. The differences in
synodic period resulted in a larger fraction of objects in the inner Main Belt
that were not observed during the fully cryogenic phase of the WISE survey,
compared to the outer Main Belt. Future work in family identification will
begin to mitigate these biases.
## 5 Conclusions
We present preliminary thermal model fits for 13511 MBAs using observations
acquired by the WISE and NEOWISE surveys following the exhaustion of the outer
cryogen tank that marked the end of the fully cryogenic WISE survey. Accuracy
of these fits is degraded with respect to the results discussed in Mas11 due
to the loss of the W3 and W4 bandpasses, however fits of diameter with
relative accuracy of $\sim 20\%$ are still possible. Unlike the fits presented
in Mas11, these determinations depend strongly on the measured value of the
optical albedo (as calculated from the $H$ absolute magnitude). Thus, any
revision to the $H$ values will require a new thermal model to be fit to the
data. This dataset includes detection of 3319 members of previously identified
asteroid families, one of the main goals of the Post-Cryo Survey. Future work
by the NEOWISE team will include second-pass processing of these data sets
using updated calibration products, which is expected to improve the accuracy
of diameter and albedo determination.
## 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. We thank the anonymous referee for their helpful comments. 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.
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|
arxiv-papers
| 2012-09-25T23:35:29 |
2024-09-04T02:49:35.600592
|
{
"license": "Public Domain",
"authors": "Joseph R. Masiero, A. K. Mainzer, T. Grav, J. M. Bauer, R. M. Cutri,\n C. Nugent, M. S. Cabrera",
"submitter": "Joseph Masiero",
"url": "https://arxiv.org/abs/1209.5794"
}
|
1209.5869
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-268 LHCb-PAPER-2012-027 26 September 2012
A model-independent Dalitz
plot analysis of $B^{\pm}\rightarrow DK^{\pm}$ with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ ($h=\pi,K$) decays and constraints
on the CKM angle $\gamma$
The LHCb collaboration†††Authors are listed on the following pages.
A binned Dalitz plot analysis of $B^{\pm}\rightarrow DK^{\pm}$ decays, with
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, is performed to measure the
$C\\!P$-violating observables $x_{\pm}$ and $y_{\pm}$ which are sensitive to
the CKM angle $\gamma$. The analysis exploits 1.0 $\rm fb^{-1}$ of data
collected by the LHCb experiment. The study makes no model-based assumption on
the variation of the strong phase of the $D$ decay amplitude over the Dalitz
plot, but uses measurements of this quantity from CLEO-c as input. The values
of the parameters are found to be $x_{-}=(0.0\pm 4.3\pm 1.5\pm 0.6)\times
10^{-2}$, $y_{-}=(2.7\pm 5.2\pm 0.8\pm 2.3)\times 10^{-2}$, $x_{+}=(-10.3\pm
4.5\pm 1.8\pm 1.4)\times 10^{-2}$ and $y_{+}=(-0.9\pm 3.7\pm 0.8\pm 3.0)\times
10^{-2}$. The first, second, and third uncertainties are the statistical, the
experimental systematic, and the error associated with the precision of the
strong-phase parameters measured at CLEO-c, respectively. These results
correspond to $\gamma=(44^{\,+43}_{\,-38})^{\circ}$, with a second solution at
$\gamma\rightarrow\gamma+180^{\circ}$, and $r_{B}=0.07\pm 0.04$, where $r_{B}$
is the ratio between the suppressed and favoured $B$ decay amplitudes.
Submitted to Physics Letters B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B.
Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
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Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
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S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
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Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
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Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
A precise determination of the Unitarity Triangle angle $\gamma$ (also denoted
as $\phi_{3}$), is an important goal in flavour physics. Measurements of this
weak phase in tree-level processes involving the interference between
$b\rightarrow c\bar{u}s$ and $b\rightarrow u\bar{c}s$ transitions are expected
to be insensitive to new physics contributions, thereby providing a Standard
Model benchmark against which other observables, more likely to be affected by
new physics, can be compared. A powerful approach for measuring $\gamma$ is to
study $C\\!P$-violating observables in $B^{\pm}\rightarrow DK^{\pm}$ decays,
where $D$ designates a neutral $D$ meson reconstructed in a final state common
to both $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays.
Examples of such final states include two-body modes, where LHCb has already
presented results [1], and self $C\\!P$-conjugate three-body decays, such as
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, designated collectively as
$K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$.
The proposal to measure $\gamma$ with $B^{\pm}\rightarrow DK^{\pm}$,
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ decays was first made
in Refs. [2, 3]. The strategy relies on comparing the distribution of events
in the $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ Dalitz plot
for $B^{+}\rightarrow DK^{+}$ and $B^{-}\rightarrow DK^{-}$ decays. However,
in order to determine $\gamma$ it is necessary to know how the strong phase of
the $D$ decay varies over the Dalitz plot. One approach for solving this
problem, adopted by BaBar [4, 5, 6] and Belle [7, 8, 9], is to use an
amplitude model fitted on flavour-tagged $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ decays to provide this input. An
attractive alternative [2, 10, 11] is to make use of direct measurements of
the strong phase behaviour in bins of the Dalitz plot, which can be obtained
from quantum-correlated $D\kern 1.99997pt\overline{\kern-1.99997ptD}{}$ pairs
from $\psi(3770)$ decays and that are available from CLEO-c [12], thereby
avoiding the need to assign any model-related systematic uncertainty. A first
model-independent analysis was recently presented by Belle [13] using
$B^{\pm}\rightarrow DK^{\pm}$, $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ decays. In this Letter, $pp$ collision data at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$1.0~{}{\rm fb^{-1}}$ and accumulated by LHCb in 2011, are exploited to
perform a similar model-independent study of the decay mode
$B^{\pm}\rightarrow DK^{\pm}$ with $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$. The results are used to set constraints on the value of
$\gamma$.
## 2 Formalism and external inputs
The amplitude of the decay $B^{+}\rightarrow DK^{+}$, $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ can be written as the superposition
of the $B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}$
and $B^{+}\rightarrow D^{0}K^{+}$ contributions as
$A_{B}(m_{+}^{2},m_{-}^{2})=\overline{A}+r_{B}e^{i(\delta_{B}+\gamma)}A.$ (1)
Here $m_{+}^{2}$ and $m_{-}^{2}$ are the invariant masses squared of the
$K^{0}_{\rm\scriptscriptstyle S}h^{+}$ and $K^{0}_{\rm\scriptscriptstyle
S}h^{-}$ combinations, respectively, that define the position of the decay in
the Dalitz plot, $A=A(m_{+}^{2},m_{-}^{2})$ is the $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ amplitude, and
$\overline{A}=\overline{A}(m_{+}^{2},m_{-}^{2})$ the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ amplitude. The parameter $r_{B}$,
the ratio of the magnitudes of the $B^{+}\rightarrow D^{0}K^{+}$ and
$B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}$
amplitudes, is $\sim$0.1 [14], and $\delta_{B}$ is the strong-phase difference
between them. The equivalent expression for the charge-conjugated decay
$B^{-}\rightarrow DK^{-}$ is obtained by making the substitutions
$\gamma\rightarrow-\gamma$ and $A\leftrightarrow\overline{A}$. Neglecting
$C\\!P$ violation, which is known to be small in $D^{0}-\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing and Cabibbo-favoured $D$
meson decays [15], the conjugate amplitudes are related by
$A(m_{+}^{2},m_{-}^{2})=\overline{A}(m_{-}^{2},m_{+}^{2})$.
Following the formalism set out in Ref. [2], the Dalitz plot is partitioned
into $2N$ regions symmetric under the exchange $m_{+}^{2}\leftrightarrow
m_{-}^{2}$. The bins are labelled from $-N$ to $+N$ (excluding zero), where
the positive bins satisfy $m_{-}^{2}>m_{+}^{2}$. At each point in the Dalitz
plot, there is a strong-phase difference
$\delta_{D}(m_{+}^{2},m_{-}^{2})=\arg\overline{A}-\arg{A}$ between the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $D^{0}$ decay. The cosine of
the strong-phase difference averaged in each bin and weighted by the absolute
decay rate is termed $c_{i}$ and is given by
$c_{i}=\frac{\int_{{\cal D}_{i}}(|A||\overline{A}|\cos{\delta_{D}})\,d{\cal
D}}{\sqrt{\int_{{\cal D}_{i}}|A|^{2}\,d{\cal D}}\,\sqrt{\int_{{\cal
D}_{i}}|\overline{A}|^{2}\,d{\cal D}}},$ (2)
where the integrals are evaluated over the area ${\cal D}$ of bin $i$. An
analogous expression may be written for $s_{i}$, which is the sine of the
strong-phase difference within bin $i$, weighted by the decay rate. The values
of $c_{i}$ and $s_{i}$ can be determined by assuming a functional form for
$|A|$, $|\overline{A}|$ and $\delta_{D}$, which may be obtained from an
amplitude model fitted to flavour-tagged $D^{0}$ decays. Alternatively direct
measurements of $c_{i}$ and $s_{i}$ can be used. Such measurements have been
performed at CLEO-c, exploiting quantum-correlated $D\kern
1.99997pt\overline{\kern-1.99997ptD}{}$ pairs produced at the $\psi(3770)$
resonance. This has been done with a double-tagged method in which one $D$
meson is reconstructed in a decay to either $K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{-}$ or $K^{0}_{\rm\scriptscriptstyle L}h^{+}h^{-}$, and the other
$D$ meson is reconstructed either in a $C\\!P$ eigenstate or in a decay to
$K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$. The efficiency-corrected event
yields, combined with flavour-tag information, allow $c_{i}$ and $s_{i}$ to be
determined [2, 10, 11]. The latter approach is attractive as it avoids any
assumption about the nature of the intermediate resonances which contribute to
the $K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ final state; such an
assumption leads to a systematic uncertainty associated with the variation in
$\delta_{D}$ that is difficult to quantify. Instead, an uncertainty is
assigned that is related to the precision of the $c_{i}$ and $s_{i}$
measurements.
The population of each positive (negative) bin in the Dalitz plot arising from
$B^{+}$ decays is $N_{+i}^{+}$ ($N_{-i}^{+}$), and that from $B^{-}$ decays is
$N_{+i}^{-}$ ($N_{-i}^{-}$). From Eq. (1) it follows that
$\displaystyle N_{\pm i}^{+}$ $\displaystyle=$ $\displaystyle
h_{B^{+}}\left[K_{\mp i}+(x_{+}^{2}+y_{+}^{2})K_{\pm
i}+2\sqrt{K_{i}K_{-i}}(x_{+}c_{\pm i}\mp y_{+}s_{\pm i})\right],$
$\displaystyle N_{\pm i}^{-}$ $\displaystyle=$ $\displaystyle
h_{B^{-}}\left[K_{\pm i}+(x_{-}^{2}+y_{-}^{2})K_{\mp
i}+2\sqrt{K_{i}K_{-i}}(x_{-}c_{\pm i}\pm y_{-}s_{\pm i})\right],$ (3)
where $h_{B^{\pm}}$ are normalisation factors which can, in principle, be
different for $B^{+}$ and $B^{-}$ due to the production asymmetries, and
$K_{i}$ is the number of events in bin $i$ of the decay of a flavour tagged
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ Dalitz plot. The
sensitivity to $\gamma$ enters through the Cartesian parameters
$x_{\pm}=r_{B}\cos(\delta_{B}\pm\gamma){\rm\ and\
}\;y_{\pm}=r_{B}\sin(\delta_{B}\pm\gamma).$ (4)
In this analysis the observed distribution of candidates over the
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ Dalitz plot is used
to fit $x_{\pm}$, $y_{\pm}$ and $h_{B^{\pm}}$. The parameters $c_{i}$ and
$s_{i}$ are taken from measurements performed by CLEO-c [12]. In this manner
the analysis avoids any dependence on an amplitude model to describe the
variation of the strong phase over the Dalitz plot. A model is used, however,
to provide the input values for $K_{i}$. For the $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decay the model is taken from
Ref. [5] and for the $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$ decay the model is taken from Ref. [6]. This choice incurs no
significant systematic uncertainty as the models have been shown to describe
well the intensity distribution of flavour-tagged $D^{0}$ decay data.
The effect of $D^{0}-\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing
is ignored in the above discussion, and was neglected in the CLEO-c
measurements of $c_{i}$ and $s_{i}$ as well as in the construction of the
amplitude model used to calculate $K_{i}$. This leads to a bias of the order
of $0.2^{\circ}$ in the $\gamma$ determination [16] which is negligible for
the current analysis.
The CLEO-c study segments the $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$
Dalitz plot into $2\times 8$ bins. Several bin definitions are available. Here
the ‘optimal binning’ variant is adopted. In this scheme the bins have been
chosen to optimise the statistical sensitivity to $\gamma$ in the presence of
a low level of background, which is appropriate for this analysis. The
optimisation has been performed assuming a strong-phase difference
distribution as predicted by the BaBar model presented in Ref. [5]. The use of
a specific model in defining the bin boundaries does not bias the $c_{i}$ and
$s_{i}$ measurements. If the model is a poor description of the underlying
decay the only consequence will be to reduce the statistical sensitivity of
the $\gamma$ measurement.
For the $K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ final state $c_{i}$ and
$s_{i}$ measurements are available for the Dalitz plot partitioned into
$2\times 2$, $2\times 3$ and $2\times 4$ bins, with the guiding model being
that from the BaBar study described in Ref. [6]. The bin boundaries divide the
Dalitz plot into bins of equal size with respect to the strong-phase
difference between the $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ amplitudes. The current analysis
adopts the $2\times 2$ option, a decision driven by the size of the signal
sample. The binning choices for the two decay modes are shown in Fig. 1.
Figure 1: Binning choices for (a) $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and (b) $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$. The diagonal line separates the positive and negative bins.
## 3 The LHCb detector
The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$. The detector includes a high precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region, a large-area silicon-strip detector (VELO) located
upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and
three stations of silicon-strip detectors and straw drift-tubes placed
downstream. The combined tracking system has a momentum resolution of (0.4 –
0.6)% in the range of 5 – 100 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an
impact parameter (IP) resolution of 20 $\,\upmu\rm m$ for tracks with high
transverse momentum ($p_{\rm T}$). The dipole magnet can be operated in either
polarity and this feature is used to reduce systematic effects due to detector
asymmetries. In the data set considered in this analysis, 58% of data were
taken with one polarity and 42% with the other. Charged hadrons are identified
using two ring-imaging Cherenkov (RICH) detectors. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers.
A two-stage trigger is employed. First a hardware-based decision is taken at a
frequency up to 40 MHz. It accepts high transverse energy clusters in either
the electromagnetic calorimeter or hadron calorimeter, or a muon of high
$p_{\rm T}$. For this analysis, it is required that one of the charged final-
state tracks forming the $B^{\pm}$ candidate points at a deposit in the hadron
calorimeter, or that the hardware-trigger decision was taken independently of
these tracks. A second trigger level, implemented in software, receives 1 MHz
of events and retains $\sim$0.3% of them [18]. It searches for a track with
large $p_{\rm T}$ and large IP with respect to any $pp$ interaction point
which is called a primary vertex (PV). This track is then required to be part
of a two-, three- or four-track secondary vertex with a high $p_{\rm T}$ sum,
significantly displaced from any PV. In order to maximise efficiency at an
acceptable trigger rate, the displaced vertex is selected with a decision tree
algorithm that uses $p_{\rm T}$, impact parameter, flight distance and track
separation information. Full event reconstruction occurs offline, and a loose
preselection is applied.
Approximately three million simulated events for each of the modes
$B^{\pm}\rightarrow D(K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-})K^{\pm}$
and $B^{\pm}\rightarrow D(K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-})\pi^{\pm}$ , and one million simulated events for each of
$B^{\pm}\rightarrow D(K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-})K^{\pm}$ and
$B^{\pm}\rightarrow D(K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-})\pi^{\pm}$ are
used in the analysis, as well as a large inclusive sample of generic
$B\rightarrow DX$ decays for background studies. These samples are generated
using a version of Pythia 6.4 [19] tuned to model the $pp$ collisions [20].
EvtGen [21] encodes the particle decays in which final state radiation is
generated using Photos [22]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [23,
*Agostinelli:2002hh] as described in Ref. [25].
## 4 Event selection and invariant mass spectrum fit
Selection requirements are applied to isolate both $B^{\pm}\rightarrow
DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ candidates, with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$. Candidates selected in the
Cabibbo-favoured $B^{\pm}\rightarrow D\pi^{\pm}$ decay mode provide an
important control sample which is exploited in the analysis.
A production vertex is assigned to each $B$ candidate. This is the PV for
which the reconstructed $B$ trajectory has the smallest IP $\chi^{2}$, where
this quantity is defined as the difference in the $\chi^{2}$ fit of the PV
with and without the tracks of the considered particle. The
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are formed from two oppositely
charged tracks reconstructed in the tracking stations, either with associated
hits in the VELO detector (long $K^{0}_{\rm\scriptscriptstyle S}$ candidate)
or without (downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidate). The IP
$\chi^{2}$ with respect to the PV of each of the long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ daughters is required to be greater than 16
(4). The angle $\theta$ between the $K^{0}_{\rm\scriptscriptstyle S}$
candidate momentum and the vector between the decay vertex and the PV,
expected to be small given the high momentum of the $B$ meson, is required to
satisfy $\cos\theta>0.99$, reducing background from combinations of random
tracks.
The $D$ meson candidates are reconstructed by combining the long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ candidates with two oppositely charged
tracks for which the values of the IP $\chi^{2}$ with respect to the PV are
greater than 9 (16). In the case of the $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ a loose particle identification
(PID) requirement is placed on the kaons to reduce combinatoric backgrounds.
The IP $\chi^{2}$ of the candidate $D$ with respect to any PV is demanded to
be greater than 9 in order to suppress directly produced $D$ mesons, and the
angle $\theta$ between the $D$ candidate momentum and the vector between the
decay and PV is required to satisfy the same criterion as for the
$K^{0}_{\rm\scriptscriptstyle S}$ selection ($\cos\theta>0.99$). The invariant
mass resolution of the signal is $8.7$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($11.9$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) for $D$ mesons reconstructed with
long (downstream) $K^{0}_{\rm\scriptscriptstyle S}$ candidates, and a common
window of $\pm 25$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is imposed
around the world average $D^{0}$ mass [15]. The $K^{0}_{\rm\scriptscriptstyle
S}$ mass is determined after the addition of a constraint that the invariant
mass of the two $D$ daughter pions or kaons and the two
$K^{0}_{\rm\scriptscriptstyle S}$ daughter pions have the world average $D$
mass. The invariant mass resolution is $2.9$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($4.8$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) for long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ decays. Candidates are retained for which
the invariant mass of the two $K^{0}_{\rm\scriptscriptstyle S}$ daughters lies
within $\pm 15$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the world
average $K^{0}_{\rm\scriptscriptstyle S}$ mass [15].
The $D$ meson is combined with a candidate kaon or pion bachelor particle to
form the $B$ candidate. The IP $\chi^{2}$ of the bachelor with respect to the
PV is required to be greater than 25. In order to ensure good discrimination
between pions and kaons in the RICH system only tracks with momentum less than
$100$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are considered. The bachelor is
considered as a candidate kaon (pion) according to whether it passes (fails) a
cut placed on the output of the RICH PID algorithm. The PID information is
quantified as a difference between the logarithm of the likelihood under the
mass hypothesis of a pion or a kaon. Criteria are then imposed on the $B$
candidate: that the angle between its momentum and the vector between the
decay and the PV should have a cosine greater than 0.9999 for candidates
containing long $K^{0}_{\rm\scriptscriptstyle S}$ decays (0.99995 for
downstream $K^{0}_{\rm\scriptscriptstyle S}$ decays); that the $B$ vertex-
separation $\chi^{2}$ with respect to its PV is greater than 169; and that the
$B$ IP $\chi^{2}$ with respect to the PV is less than 9. To suppress
background from charmless $B$ decays it is required that the $D$ vertex lies
downstream of the $B$ vertex. In the events with a long
$K^{0}_{\rm\scriptscriptstyle S}$ candidate, a further background arises from
$B^{\pm}\rightarrow Dh^{\pm}$, $D\rightarrow\pi^{+}\pi^{-}h^{+}h^{-}$ decays,
where the two pions are reconstructed as a long $K^{0}_{\rm\scriptscriptstyle
S}$ candidate. This background is removed by requiring that the flight
significance between the $D$ and $K^{0}_{\rm\scriptscriptstyle S}$ vertices is
greater than 10.
In order to obtain the best possible resolution in the Dalitz plot of the $D$
decay, and to provide further background suppression, the $B$, $D$ and
$K^{0}_{\rm\scriptscriptstyle S}$ vertices are refitted with additional
constraints on the $D$ and $K^{0}_{\rm\scriptscriptstyle S}$ masses, and the
$B$ momentum is required to point back to the PV. The $\chi^{2}$ per degree of
freedom of the fit is required to be less than 5.
Less than 0.4% of the selected events are found to contain two or more
candidates. In these events only the $B$ candidate with the lowest $\chi^{2}$
per degree of freedom from the refit is retained for subsequent study. In
addition, 0.4% of the candidates are found to have been reconstructed such
that their $D$ Dalitz plot coordinates lie outside the defined bins, and these
too are discarded.
The invariant mass distributions of the selected candidates are shown in Fig.
2 for $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$, with
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays, divided
between the long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ categories.
Figure 3 shows the corresponding distributions for final states with
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, here integrated over
the two $K^{0}_{\rm\scriptscriptstyle S}$ categories. The result of an
extended, unbinned, maximum likelihood fit to these distributions is
superimposed. The fit is performed simultaneously for $B^{\pm}\rightarrow
DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$, including both $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays, allowing several parameters
to be different for long and downstream $K^{0}_{\rm\scriptscriptstyle S}$
categories. The fit range is between 5110
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 5800
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in invariant mass. At this stage in
the analysis the fit does not distinguish between the different regions of
Dalitz plot or $B$ meson charge. The purpose of this global fit is to
determine the parameters that describe the invariant mass spectrum in
preparation for the binned fit described in Sect. 5.
Figure 2: Invariant mass distributions of (a,c) $B^{\pm}\rightarrow DK^{\pm}$
and (b,d) $B^{\pm}\rightarrow D\pi^{\pm}$ candidates, with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, divided between the (a,b) long
and (c,d) downstream $K^{0}_{\rm\scriptscriptstyle S}$ categories. Fit
results, including the signal and background components, are superimposed.
Figure 3: Invariant mass distributions of (a) $B^{\pm}\rightarrow DK^{\pm}$
and (b) $B^{\pm}\rightarrow D\pi^{\pm}$ candidates, with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, shown with both
$K^{0}_{\rm\scriptscriptstyle S}$ categories combined. Fit results, including
the signal and background components, are superimposed.
The signal probability density function (PDF) is a Gaussian function with
asymmetric tails where the unnormalised form is given by
$f(m;m_{0},\alpha_{L},\alpha_{R},\sigma)=\left\\{{\exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{L}(m-m_{0})^{2})],m<m_{0};\atop\exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{R}(m-m_{0})^{2})],m>m_{0};}\right.$
(5)
where $m$ is the candidate mass, $m_{0}$ the $B$ mass and $\sigma$,
$\alpha_{L}$, and $\alpha_{R}$ are free parameters in the fit. The parameter
$m_{0}$ is taken as common for all classes of signal. The parameters
describing the asymmetric tails are fitted separately for events with long and
downstream $K^{0}_{\rm\scriptscriptstyle S}$ categories. The resolution of the
Gaussian function is left as a free parameter for the two
$K^{0}_{\rm\scriptscriptstyle S}$ categories, but the ratio between this
resolution in $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow
D\pi^{\pm}$ decays is required to be the same, independent of category. The
resolution is determined to be around 15
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $B^{\pm}\rightarrow D\pi^{\pm}$
decays of both $K^{0}_{\rm\scriptscriptstyle S}$ classes, and is smaller by a
factor $0.95\pm 0.06$ for $B^{\pm}\rightarrow DK^{\pm}$. The yield of
$B^{\pm}\rightarrow D\pi^{\pm}$ candidates in each category is determined in
the fit. Instead of fitting the yield of the $B^{\pm}\rightarrow DK^{\pm}$
candidates separately, the ratio $\mathcal{R}=N(B^{\pm}\rightarrow
DK^{\pm})$/$N(B^{\pm}\rightarrow D\pi^{\pm})$ is a free parameter and is
common across all categories.
The background has contributions from random track combinations and partially
reconstructed $B$ decays. The random track combinations are modelled by linear
PDFs, the parameters of which are floated separately for each class of decay.
Partially reconstructed backgrounds are described empirically. Studies of
simulated events show that the partially reconstructed backgrounds are
dominated by decays that involve a $D$ meson decaying to
$K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$. Therefore the same PDF is used to
describe these backgrounds as used in a similar analysis of
$B^{\pm}\rightarrow DK^{\pm}$ decays, with $D\rightarrow K^{\pm}\pi^{\mp}$,
$K^{+}K^{-}$ and $\pi^{+}\pi^{-}$[1]. In that analysis the shape was
constructed by applying the selection to a large simulated sample containing
many common backgrounds, each weighted by its production rate and branching
fraction. The invariant mass distribution for the surviving candidates was
corrected to account for small differences in resolution and PID performance
between data and simulation, and two background PDFs were extracted by kernel
estimation [26]; one for $B^{\pm}\rightarrow DK^{\pm}$ and one for
$B^{\pm}\rightarrow D\pi^{\pm}$ decays. The partially reconstructed background
PDFs are found to give a good description of both
$K^{0}_{\rm\scriptscriptstyle S}$ categories.
An additional and significant background component exists in the
$B^{\pm}\rightarrow DK^{\pm}$ sample, arising from the dominant
$B^{\pm}\rightarrow D\pi^{\pm}$ decay on those occasions where the bachelor
particle is misidentified as a kaon by the RICH system. In contrast, the
$B^{\pm}\rightarrow DK^{\pm}$ contamination in the $B^{\pm}\rightarrow
D\pi^{\pm}$ sample can be neglected. The size of this background is calculated
through knowledge of PID and misidentification efficiencies, which are
obtained from large samples of kinematically selected $D^{\ast\pm}\rightarrow
D\pi^{\pm}$, $D\rightarrow K^{\mp}\pi^{\pm}$ decays. The kinematic properties
of the particles in the calibration sample are reweighted to match those of
the bachelor particles in the $B$ decay sample, thereby ensuring that the
measured PID performance is representative of that in the $B$ decay sample.
The efficiency to identify a kaon correctly is found to be around 86%, and
that for a pion to be around 96%. The misidentification efficiencies are the
complements of these numbers. From this information and from knowledge of the
number of reconstructed $B^{\pm}\rightarrow D\pi^{\pm}$ decays, the amount of
this background surviving the $B^{\pm}\rightarrow DK^{\pm}$ selection can be
determined. The invariant mass distribution of the misidentified candidates is
described by a Crystal Ball function [27] with the tail on the high mass side,
the parameters of which are fitted in common between all the
$B^{\pm}\rightarrow DK^{\pm}$ samples.
The number of $B^{\pm}\rightarrow DK^{\pm}$ candidates in all categories is
determined by $\mathcal{R}$, and the number of $B^{\pm}\rightarrow D\pi^{\pm}$
events in the corresponding category. The ratio $\mathcal{R}$ is determined in
the fit and measured to be 0.085$\pm$0.005 (statistical uncertainty only) and
is consistent with that observed in Ref. [1]. The yields returned by the
invariant mass fit in the full fit region are scaled to the signal region,
defined as 5247–5317 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and are
presented in Tables 1 and 2 for the $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$
selections respectively. In the $B^{\pm}\rightarrow
D(K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-})K^{\pm}$ sample there are
$654\pm 28$ signal candidates, with a purity of 86%. The corresponding numbers
for the $B^{\pm}\rightarrow D(K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-})K^{\pm}$ sample are $102\pm 5$ and 88%, respectively. The
contamination in the $B^{\pm}\rightarrow DK^{\pm}$ selection receives
approximately equal contributions from misidentified $B^{\pm}\rightarrow
D\pi^{\pm}$ decays, combinatoric background and partially reconstructed
decays. The partially reconstructed component in the signal region is
dominated by decays of the type $B\rightarrow D\rho$, in which a charged pion
from the $\rho$ decay is misidentified as the bachelor kaon, and
$B^{\pm}\rightarrow D^{*}\pi^{\pm}$, again with a misidentified pion.
Table 1: Yields and statistical uncertainties in the signal region from the invariant mass fit, scaled from the full fit mass range, for candidates passing the $B^{\pm}\rightarrow Dh^{\pm}$, $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ selection. Values are shown separately for candidates containing long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and 5317 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and the full fit range is between 5110 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and 5800 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. | $B^{\pm}\rightarrow DK^{\pm}$ selection | $B^{\pm}\rightarrow D\pi^{\pm}$ selection
---|---|---
Fit component | Long | Downstream | Long | Downstream
$B^{\pm}\rightarrow DK^{\pm}$ | $213\pm 13$ | $441\pm 25$ | – | –
$B^{\pm}\rightarrow D\pi^{\pm}$ | $11\pm 3$ | $22\pm 5$ | $2809\pm 56$ | $5755\pm 82$
Combinatoric | $\phantom{0}9\pm 4$ | $29\pm 6$ | $\phantom{0}22\pm 3$ | $\phantom{0}90\pm 7$
Partially reconstructed | $11\pm 1$ | $25\pm 2$ | $\phantom{0}25\pm 1$ | $\phantom{0}55\pm 1$
Table 2: Yields and statistical uncertainties in the signal region from the invariant mass fit, scaled from the full fit mass range, for candidates passing the $B^{\pm}\rightarrow Dh^{\pm}$, $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ selection. Values are shown separately for candidates containing long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and 5317 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and the full fit range is between 5110 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ and 5800 ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. | $B^{\pm}\rightarrow DK^{\pm}$ selection | $B^{\pm}\rightarrow D\pi^{\pm}$ selection
---|---|---
Fit component | Long | Downstream | Long | Downstream
$B^{\pm}\rightarrow DK^{\pm}$ | $32\pm 2$ | $70\pm 4$ | – | –
$B^{\pm}\rightarrow D\pi^{\pm}$ | $\phantom{0}1.6\pm 1.2$ | $\phantom{0}3.4\pm 1.8$ | $417\pm 20$ | $913\pm 29$
Combinatoric | $\phantom{0}0.6\pm 0.5$ | $\phantom{0}2.5\pm 0.9$ | $\phantom{0}4.8\pm 1.4$ | $18\pm 2$
Partially reconstructed | $\phantom{0}2.2\pm 0.4$ | $\phantom{0}2.9\pm 0.5$ | $\phantom{0}3.7\pm 0.3$ | $\phantom{0}7.7\pm 0.5$
The Dalitz plots for $B^{\pm}\rightarrow DK^{\pm}$ data in the signal region
for the two $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ final
states are shown in Fig. 4. Separate plots are shown for $B^{+}$ and $B^{-}$
decays.
Figure 4: Dalitz plots of $B^{\pm}\rightarrow DK^{\pm}$ candidates in the
signal region for (a,b) $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and (c,d) $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$ decays, divided between (a,c) $B^{+}$ and (b,d) $B^{-}$. The
boundaries of the kinematically-allowed regions are also shown.
## 5 Binned Dalitz fit
The purpose of the binned Dalitz plot fit is to measure the $C\\!P$-violating
parameters $x_{\pm}$ and $y_{\pm}$, as introduced in Sect. 2. Following Eq.
(3) these parameters can be determined from the populations of each
$B^{\pm}\rightarrow DK^{\pm}$ Dalitz plot bin given the external information
that is available for the $c_{i}$, $s_{i}$ and $K_{i}$ parameters. In order to
know the signal population in each bin it is necessary both to subtract
background and to correct for acceptance losses from the trigger,
reconstruction and selection.
Although the absolute numbers of $B^{+}$ and $B^{-}$ decays integrated over
the Dalitz plot have some dependence on $x_{\pm}$ and $y_{\pm}$, the
additional sensitivity gained compared to using just the relative bin-to-bin
yields is negligible, and is therefore not used. Consequently the analysis is
insensitive to any $B$ production asymmetries, and only knowledge of the
relative acceptance is required. The relative acceptance is determined from
the control channel $B^{\pm}\rightarrow D\pi^{\pm}$. In this decay the ratio
of $b\rightarrow u\bar{c}d$ to $b\rightarrow c\bar{u}d$ amplitudes is expected
to be very small ($\sim 0.005$) and thus, to a good approximation,
interference between the transitions can be neglected. Hence the relative
population of decays expected in each $B^{\pm}\rightarrow D\pi^{\pm}$ Dalitz
plot bin can be predicted using the $K_{i}$ values calculated with the
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ model. Dividing the
background-subtracted yield observed in each bin by this prediction enables
the relative acceptance to be determined, and then applied to the
$B^{\pm}\rightarrow DK^{\pm}$ data. In order to optimise the statistical
precision of this procedure, the bins $+i$ and $-i$ are combined in the
calculation, since the efficiencies in these symmetric regions are expected to
be the same in the limit that there are no charge-dependent reconstruction
asymmetries. It is found that the variation in relative acceptance between
non-symmetric bins is at most $\sim 50\%$, with the lowest efficiency
occurring in those regions where one of the pions has low momentum.
Separate fits are performed to the $B^{+}$ and $B^{-}$ data. Each fit
simultaneously considers the two $K^{0}_{\rm\scriptscriptstyle S}$ categories,
the $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$
candidates, and the two $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{-}$ final states. In order to assess the impact of the $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ data the fit is then repeated
including only the $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ sample. The PDF parameters for both the signal and
background invariant mass distributions are fixed to the values determined in
the global fit. The yields of all the background contributions in each bin are
free parameters, apart from bins where a very low contribution is determined
from an initial fit, in which case they are fixed to zero, to facilitate the
calculation of the error matrix. The yields of signal candidates for each bin
in the $B^{\pm}\rightarrow D\pi^{\pm}$ sample are also free parameters. The
amount of signal in each bin for the $B^{\pm}\rightarrow DK^{\pm}$ sample is
determined by varying the integrated yield and the $x_{\pm}$ and $y_{\pm}$
parameters.
A large ensemble of simulated experiments are performed to validate the fit
procedure. In each experiment the number and distribution of signal and
background candidates are generated according to the expected distribution in
data, and the full fit procedure is then executed. The values for $x_{\pm}$
and $y_{\pm}$ are set close to those determined by previous measurements [14].
It is found from this exercise that the errors are well estimated. Small
biases are, however, observed in the central values returned by the fit and
these are applied as corrections to the results obtained on data. The bias is
$(0.2-0.3)\times 10^{-2}$ for most parameters but rises to $1.0\times 10^{-2}$
for $y_{+}$. This bias is due to the low yields in some of the bins and is an
inherent feature of the maximum likelihood fit. This behaviour is associated
with the size of data set being fit, since when simulated experiments are
performed with larger sample sizes the biases are observed to reduce.
The results of the fits are presented in Table 3. The systematic uncertainties
are discussed in Sect. 6. The statistical uncertainties are compatible with
those predicted by simulated experiments. The inclusion of the $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ data improves the precision on
$x_{\pm}$ by around 10%, and has little impact on $y_{\pm}$. This behaviour is
expected, as the measured values of $c_{i}$ in this mode, which multiply
$x_{\pm}$ in Eq. (4), are significantly larger than those of $s_{i}$, which
multiply $y_{\pm}$. The two sets of results are compatible within the
statistical and uncorrelated systematic uncertainties.
Table 3: Results for $x_{\pm}$ and $y_{\pm}$ from the fits to the data in the case when both $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ are considered and when only the $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ final state is included. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strong-phase parameters, respectively. The correlation coefficients are calculated including all sources of uncertainty (the values in parentheses correspond to the case where only the statistical uncertainties are considered). Parameter | All data | $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ alone
---|---|---
$x_{-}$ [$\times 10^{-2}$] | $0.0\pm 4.3\pm 1.5\pm 0.6$ | $1.6\pm 4.8\pm 1.4\pm 0.8$
$y_{-}$ [$\times 10^{-2}$] | $2.7\pm 5.2\pm 0.8\pm 2.3$ | $1.4\pm 5.4\pm 0.8\pm 2.4$
corr($x_{-}$,$y_{-}$) | $-0.10$ ($-0.11$) | $-0.12$ ($-0.12$)
$x_{+}$ [$\times 10^{-2}$] | $-10.3\pm 4.5\pm 1.8\pm 1.4$ | $-8.6\pm 5.4\pm 1.7\pm 1.6$
$y_{+}$ [$\times 10^{-2}$] | $-0.9\pm 3.7\pm 0.8\pm 3.0$ | $-0.3\pm 3.7\pm 0.9\pm 2.7$
corr($x_{+}$,$y_{+}$) | 0.22 (0.17) | $0.20$ (0.17)
The measured values of $(x_{\pm},y_{\pm})$ from the fit to all data, with
their statistical likelihood contours are shown in Fig. 5. The expected
signature for a sample that exhibits $C\\!P$-violation is that the two vectors
defined by the coordinates $(x_{-},y_{-})$ and $(x_{+},y_{+})$ should both be
non-zero in magnitude, and have different phases. The data show this
behaviour, but are also compatible with the no $C\\!P$ violation hypothesis.
Figure 5: One (solid), two (dashed) and three (dotted) standard deviation
confidence levels for $(x_{+},y_{+})$ (blue) and $(x_{-},y_{-})$ (red) as
measured in $B^{\pm}\rightarrow DK^{\pm}$ decays (statistical only). The
points represent the best fit central values.
In order to investigate whether the binned fit gives an adequate description
of the data, a study is performed to compare the observed number of signal
candidates in each bin with that expected given the fitted total yield and
values of $x_{\pm}$ and $y_{\pm}$. The number of signal candidates is
determined by fitting in each bin for the $B^{\pm}\rightarrow DK^{\pm}$
contribution for long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ decays
combined, with no assumption on how this component is distributed over the
Dalitz plot. Figure 6 shows the results in effective bin number separately for
$N_{B^{+}+B^{-}}$, the sum of $B^{+}$ and $B^{-}$ candidates, which is a
$C\\!P$-conserving observable, and for the difference $N_{B^{+}-B^{-}}$, which
is sensitive to $C\\!P$ violation. The effective bin number is equal to the
normal bin number for $B^{+}$, but is defined to be this number multiplied by
$-1$ for $B^{-}$. The expectations from the ($x_{\pm}$, $y_{\pm}$) fit are
superimposed as is, for the $N_{B^{+}-B^{-}}$ distribution, the prediction for
the case $x_{\pm}=y_{\pm}=0$. Note that the zero $C\\!P$ violation prediction
is not a horizontal line at $N_{B^{+}-B^{-}}=0$ because it is calculated using
the total $B^{+}$ and $B^{-}$ yields from the full fit, and using bin
efficiencies that are determined separately for each sample. The data and fit
expectations are compatible for both distributions yielding a $\chi^{2}$
probability of 10% for $N_{B^{+}+B^{-}}$ and 34% for $N_{B^{+}-B^{-}}$. The
results for the $N_{B^{+}-B^{-}}$ distribution are also compatible with the no
$C\\!P$-violation hypothesis ($\chi^{2}$ probability = 16% ).
Figure 6: Signal yield in effective bins compared with prediction of
$(x_{\pm},y_{\pm})$ fit (black histogram) for $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$. Figure (a) shows the sum of
$B^{+}$ and $B^{-}$ yields. Figure (b) shows the difference of $B^{+}$ and
$B^{-}$ yields. Also shown (dashed line and grey shading) is the expectation
and uncertainty for the zero $C\\!P$-violation hypothesis.
## 6 Systematic uncertainties
Systematic uncertainties are evaluated for the fits to the full data sample
and are presented in Table 4. In order to understand the impact of the CLEO-c
$(c_{i},s_{i})$ measurements the errors arising from this source are kept
separate from the other experimental uncertainties. Table 5 shows the
uncertainties for the case where only $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays are included. Each
contribution to the systematic uncertainties is now discussed in turn.
Table 4: Summary of statistical, experimental and strong-phase uncertainties on $x_{\pm}$ and $y_{\pm}$ in the case where both $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays are included in the fit. All entries are given in multiples of $10^{-2}$. Component | $\sigma(x_{-})$ | $\sigma(y_{-})$ | $\sigma(x_{+})$ | $\sigma(y_{+})$
---|---|---|---|---
Statistical | $4.3$ | $5.2$ | $4.5$ | $3.7$
Global fit shape parameters | $0.4$ | $0.4$ | $0.6$ | $0.4$
Efficiency effects | $0.3$ | $0.4$ | $0.3$ | $0.4$
$C\\!P$ violation in control mode | $1.3$ | $0.4$ | $1.5$ | $0.2$
Migration | $0.4$ | $0.2$ | $0.4$ | $0.2$
Partially reconstructed background | $0.2$ | $0.3$ | $0.2$ | $0.2$
PID efficiency | $0.1$ | $0.2$ | $0.2$ | $<0.1$
Shape of misidentified $B^{\pm}\rightarrow D\pi^{\pm}$ | $0.1$ | $0.1$ | $0.3$ | $<0.1$
Bias correction | $0.2$ | $0.3$ | $0.2$ | $0.5$
Total experimental systematic | $1.5$ | $0.9$ | $1.8$ | $0.8$
Strong-phase systematic | $0.6$ | $2.3$ | $1.4$ | $3.0$
Table 5: Summary of statistical, experimental and strong-phase uncertainties on $x_{\pm}$ and $y_{\pm}$ in the case where only $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays are included in the fit. All entries are given in multiples of $10^{-2}$. Component | $\sigma(x_{-})$ | $\sigma(y_{-})$ | $\sigma(x_{+})$ | $\sigma(y_{+})$
---|---|---|---|---
Statistical | $4.8$ | $5.4$ | $5.4$ | $3.7$
Global fit shape parameters | $0.4$ | $0.4$ | $0.6$ | $0.4$
Efficiency effects | $0.2$ | $0.2$ | $0.3$ | $0.4$
$C\\!P$ violation in control mode | $1.2$ | $0.5$ | $1.5$ | $0.2$
Migration | $0.4$ | $0.2$ | $0.4$ | $0.2$
Partially reconstructed background | $0.1$ | $0.1$ | $0.3$ | $0.2$
PID efficiency | $<0.1$ | $0.2$ | $<0.1$ | $<0.1$
Shape of misidentified $B^{\pm}\rightarrow D\pi^{\pm}$ | $0.1$ | $<0.1$ | $0.1$ | $<0.1$
Bias correction | $0.2$ | $0.3$ | $0.2$ | $0.6$
Total experimental systematic | $1.4$ | $0.8$ | $1.7$ | $0.9$
Strong-phase systematic | $0.8$ | $2.4$ | $1.6$ | $2.7$
The uncertainties on the shape parameters of the invariant mass distributions
as determined from the global fit when propagated through to the binned
analysis induce uncertainties on $x_{\pm}$ and $y_{\pm}$. In addition,
consideration is given to certain assumptions made in the fit. For example,
the slope of the combinatoric background in the data set containing
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays is fixed to be
zero on account of the limited sample size. The induced errors associated with
these assumptions are evaluated and found to be small compared to those coming
from the parameter uncertainties themselves, which vary between $0.4\times
10^{-2}$ and $0.6\times 10^{-2}$ for the fit to the full data sample.
The analysis assumes an efficiency that is flat across each Dalitz plot bin.
In reality the efficiency varies, and this leads to a potential bias in the
determination of $x_{\pm}$ and $y_{\pm}$, since the non-uniform acceptance
means that the values of $(c_{i},s_{i})$ appropriate for the analysis can
differ from those corresponding to the flat-efficiency case. The possible size
of this effect is evaluated in LHCb simulation by dividing each Dalitz plot
bin into many smaller cells, and using the BaBar amplitude model [5, 6] to
calculate the values of $c_{i}$ and $s_{i}$ within each cell. These values are
then averaged together, weighted by the population of each cell after
efficiency losses, to obtain an effective $(c_{i},s_{i})$ for the bin as a
whole, and the results compared with those determined assuming a flat
efficiency. The differences between the two sets of results are found to be
small compared with the CLEO-c measurement uncertainties. The data fit is then
rerun many times, and the input values of $(c_{i},s_{i})$ are smeared
according to the size of these differences, and the mean shifts are assigned
as a systematic uncertainty. These shifts vary between $0.2\times 10^{-2}$ and
$0.3\times 10^{-2}$.
The relative efficiency in each Dalitz plot bin is determined from the
$B^{\pm}\rightarrow D\pi^{\pm}$ control sample. Biases can enter the
measurement if there are differences in the relative acceptance over the
Dalitz plot between the control sample and that of signal $B^{\pm}\rightarrow
DK^{\pm}$ decays. Simulation studies show that the acceptance shapes are very
similar between the two decays, but small variations exist which can be
attributed to kinematic correlations induced by the different PID requirements
on the bachelor particle from the $B$ decay. When included in the data fit,
these variations induce biases that vary between $0.1\times 10^{-2}$ and
$0.3\times 10^{-2}$. In addition, a check is performed in which the control
sample is fitted without combining together bins $+i$ and $-i$ in the
efficiency calculation. As a result of this study small uncertainties of $\leq
0.3\times 10^{-2}$ are assigned for the $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ measurement to account for possible
biases induced by the difference in interaction cross-section for $K^{-}$ and
$K^{+}$ mesons interacting with the detector material. These contributions are
combined together with the uncertainty arising from efficiency variation
within a Dalitz plot bin to give the component labelled ‘Efficiency effects’
in Tables 4 and 5.
The use of the control channel to determine the relative efficiency on the
Dalitz plot assumes that the amplitude of the suppressed tree diagram is
negligible. If this is not the case then the $B^{-}$ final state will receive
a contribution from $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays,
and this will lead to the presence of $C\\!P$ violation via the same mechanism
as in $B\rightarrow DK$ decays. The size of any $C\\!P$ violation that exists
in this channel is governed by $r^{D\pi}_{B}$, $\gamma$ and
$\delta^{D\pi}_{B}$, where the parameters with superscripts are analogous to
their counterparts in $B^{\pm}\rightarrow DK^{\pm}$ decays. The naive
expectation is that $r^{D\pi}_{B}\sim 0.005$ but larger values are possible,
and the studies reported in Ref. [1] are compatible with this possibility.
Therefore simulated experiments are performed with finite $C\\!P$ violation
injected in the control channel, conservatively setting $r^{D\pi}_{B}$ to be
0.02, taking a wide variation in the value of the unknown strong-phase
difference $\delta^{D\pi}_{B}$, and choosing $\gamma=70^{\circ}$. The
experiments are fit under the no $C\\!P$ violation hypothesis and the largest
shifts observed are assigned as a systematic uncertainty. This contribution is
the largest source of experimental systematic uncertainty in the measurement,
for example contributing an error of $1.5\times 10^{-2}$ in the case of
$x_{+}$ in the full data fit.
The resolution of each decay on the Dalitz plot is approximately 0.004
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for candidates with long
$K^{0}_{\rm\scriptscriptstyle S}$ decays and 0.006
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for those containing downstream
$K^{0}_{\rm\scriptscriptstyle S}$ in the $m^{2}_{+}$ and $m_{-}^{2}$
directions. This is small compared to the typical width of a bin, nonetheless
some net migration is possible away from the more densely populated bins. At
first order this effect is accounted for by use of the control channel, but
residual effects enter because of the different distribution in the Dalitz
plot of the signal events. Once more a series of simulated experiments is
performed to assess the size of any possible bias which is found to vary
between $0.2\times 10^{-2}$ and $0.4\times 10^{-2}$.
The distribution of the partially reconstructed background is varied over the
Dalitz plot according to the uncertainty in the make-up of this background
component. From these studies an uncertainty of $(0.2-0.3)\times 10^{-2}$ is
assigned to the fit parameters in the full data fit.
Two systematic uncertainties are evaluated that are associated with the
misidentified $B^{\pm}\rightarrow D\pi^{\pm}$ background in the
$B^{\pm}\rightarrow DK^{\pm}$ sample. Firstly, there is a $0.2\times 10^{-2}$
uncertainty on the knowledge of the efficiency of the PID cut that
distinguishes pions from kaons. This is found to have only a small effect on
the measured values of $x_{\pm}$ and $y_{\pm}$. Secondly, it is possible that
the invariant mass distribution of the misidentified background is not
constant over the Dalitz plot, as is assumed in the fit. This can occur
through kinematic correlations between the reconstruction efficiency on the
Dalitz plot of the $D$ decay and the momentum of the bachelor pion from the
$B^{\pm}$ decay. Simulated experiments are performed with different shapes
input according to the Dalitz plot bin and the results of simulation studies,
and these experiments are then fitted assuming a uniform shape, as in data.
Uncertainties are assigned in the range $(0.1-0.3)\times 10^{-2}$.
An uncertainty is assigned to each parameter to accompany the correction that
is applied for the small bias which is present in the fit procedure. These
uncertainties are determined by performing sets of simulated experiments, in
each of which different values of $x_{\pm}$ and $y_{\pm}$ are input,
corresponding to a range that is wide compared to the current experimental
knowledge, and also encompassing the results of this analysis. The spread in
observed bias is taken as the systematic error, and is largest for $y_{+}$,
reaching a value of $0.5\times 10^{-2}$ in the full data fit.
Finally, several robustness checks are conducted to assess the stability of
the results. These include repeating the analysis with alternative binning
schemes for the $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$
data and performing the fits without making any distinction between
$K^{0}_{\rm\scriptscriptstyle S}$ category. These tests return results
compatible with the baseline procedure.
The total experimental systematic uncertainty from LHCb-related sources is
determined to be $1.5\times 10^{-2}$ on $x_{-}$, $0.9\times 10^{-2}$ on
$y_{-}$, $1.8\times 10^{-2}$ on $x_{+}$ and $0.8\times 10^{-2}$ on $y_{+}$.
These are all smaller than the corresponding statistical uncertainties. The
dominant contribution arises from allowing for the possibility of $C\\!P$
violation in the control channel, $B\rightarrow D\pi$. In the future, when
larger data sets are analysed, alternative analysis methods will be explored
to eliminate this potential source of bias.
The limited precision on $(c_{i},s_{i})$ coming from the CLEO-c measurement
induces uncertainties on $x_{\pm}$ and $y_{\pm}$ [12]. These uncertainties are
evaluated by rerunning the data fit many times, and smearing the input values
of $(c_{i},s_{i})$ according to their measurement errors and correlations.
Values of $(0.6-3.0)\times 10^{-2}$ are found for the fit to the full sample.
When evaluated for the $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ data set alone, the results are similar in magnitude, but
not identical, to those reported in the corresponding Belle analysis [13].
Differences are to be expected, as these uncertainties have a dependence on
the central values of the $x_{\pm}$ and $y_{\pm}$ parameters, and are sample-
dependent for small data sets. Simulation studies indicate that these
uncertainties will be reduced when larger $B^{\pm}\rightarrow DK^{\pm}$ data
sets are analysed.
After taking account of all sources of uncertainty the correlation coefficient
between $x_{-}$ and $y_{-}$ in the full fit is calculated to be $-0.10$ and
that between $x_{+}$ and $y_{+}$ to be $0.22$. The correlations between
$B^{-}$ and $B^{+}$ parameters are found to be small and can be neglected.
These correlations are summarised in Table 3, together with those coming from
the statistical uncertainties alone, and those from the fit to $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ data.
## 7 Interpretation
The results for $x_{\pm}$ and $y_{\pm}$ can be interpreted in terms of the
underlying physics parameters $\gamma$, $r_{B}$ and $\delta_{B}$. This is done
using a frequentist approach with Feldman-Cousins ordering [28], using the
same procedure as described in Ref. [13]. In this manner confidence levels are
obtained for the three physics parameters. The confidence levels for one, two
and three standard deviations are taken at 20%, 74% and 97%, which is
appropriate for a three-dimensional Gaussian distribution. The projections of
the three-dimensional surfaces bounding the one, two and three standard
deviation volumes onto the $(\gamma,r_{B})$ and $(\gamma,\delta_{B})$ planes
are shown in Fig. 7. The LHCb-related systematic uncertainties are taken as
uncorrelated and correlations of the CLEO-c and statistical uncertainties are
taken into account. The statistical and systematic uncertainties on $x$ and
$y$ are combined in quadrature.
Figure 7: Two-dimensional projections of confidence regions onto the
$(\gamma,r_{B})$ and $(\gamma,\delta_{B})$ planes showing the one (solid) and
two (dashed) standard deviations with all uncertainties included. For the
($\gamma,r_{B}$) projection the three (dotted) standard deviation contour is
also shown. The points mark the central values.
The solution for the physics parameters has a two-fold ambiguity,
$(\gamma,\delta_{B})$ and $(\gamma+180^{\circ},\delta_{B}+180^{\circ})$.
Choosing the solution that satisfies $0<\gamma<180^{\circ}$ yields
$r_{B}=0.07\pm 0.04$, $\gamma=(44^{\,+43}_{\,-38})^{\circ}$ and
$\delta_{B}=(137^{\,+35}_{\,-46})^{\circ}$. The value for $r_{B}$ is
consistent with, but lower than, the world average of results from previous
experiments [15]. This low value means that it is not possible to use the
results of this analysis, in isolation, to set strong constraints on the
values of $\gamma$ and $\delta_{B}$, as can be seen by the large uncertainties
on these parameters.
## 8 Conclusions
Approximately 800 $B^{\pm}\rightarrow DK^{\pm}$ decay candidates, with the $D$
meson decaying either to $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ or
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, have been selected from 1.0 ${\rm
fb^{-1}}$ of data collected by LHCb in 2011. These samples have been analysed
to determine the $C\\!P$-violating parameters
$x_{\pm}=r_{B}\cos(\delta_{B}\pm\gamma)$ and
$y_{\pm}=r_{B}\sin(\delta_{B}\pm\gamma)$, where $r_{B}$ is the ratio of the
absolute values of the $B^{+}\rightarrow D^{0}K^{-}$ and
$B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}$
amplitudes, $\delta_{B}$ is the strong-phase difference between them, and
$\gamma$ is the angle of the unitarity triangle. The analysis is performed in
bins of $D$ decay Dalitz space and existing measurements of the CLEO-c
experiment are used to provide input on the $D$ decay strong-phase parameters
$(c_{i},s_{i})$ [12]. Such an approach allows the analysis to be essentially
independent of any model-dependent assumptions on the strong phase variation
across Dalitz space. It is the first time this method has been applied to
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays. The following
results are obtained
$\displaystyle x_{-}$ $\displaystyle=(0.0\pm 4.3\pm 1.5\pm 0.6)\times
10^{-2},\;\,$ $\displaystyle y_{-}$ $\displaystyle=(2.7\pm 5.2\pm 0.8\pm
2.3)\times 10^{-2},$ $\displaystyle x_{+}$ $\displaystyle=(-10.3\pm 4.5\pm
1.8\pm 1.4)\times 10^{-2},\;\,$ $\displaystyle y_{+}$ $\displaystyle=(-0.9\pm
3.7\pm 0.8\pm 3.0)\times 10^{-2},$
where the first uncertainty is statistical, the second is systematic and the
third arises from the experimental knowledge of the $(c_{i},s_{i})$
parameters. These values have similar precision to those obtained in a recent
binned study by the Belle experiment [13].
When interpreting these results in terms of the underlying physics parameters
it is found that $r_{B}=0.07\pm 0.04$, $\gamma=(44^{\,+43}_{\,-38})^{\circ}$
and $\delta_{B}=(137^{\,+35}_{\,-46})^{\circ}$. These values are consistent
with the world average of results from previous measurements [15], although
the uncertainties on $\gamma$ and $\delta_{B}$ are large. This is partly
driven by the relatively low central value that is obtained for the parameter
$r_{B}$. More stringent constraints are expected when these results are
combined with other measurements from LHCb which have complementary
sensitivity to the same physics parameters.
## 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 (The 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.
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|
arxiv-papers
| 2012-09-26T08:36:01 |
2024-09-04T02:49:35.610603
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, 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, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, 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, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov,\n R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S.\n Malde, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. 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Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. 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Wiggers, G. Wilkinson, M. P. Williams,\n M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Guy Wilkinson",
"url": "https://arxiv.org/abs/1209.5869"
}
|
1209.5873
|
# Relativistic Correction to Charmonium Dissociation Temperature
Xingyu Guo Shuzhe Shi Pengfei Zhuang Physics Department, Tsinghua
University, Beijing 100084, China
###### Abstract
By solving the covariant relativistic Schrödinger equations for a pair of
heavy quarks, we obtained the wave functions for the ground and excited
quarkonium states at finite temperature. In comparison with the non-
relativistic calculation, the $J/\psi$ dissociation temperature determined by
the infinity size or zero binding energy of the system increases $7\%-13\%$,
when the central potential varies between the free energy and internal energy.
It is widely accepted that there exists a quantum chromodynamics (QCD) phase
transition from hadron gas to a new state of matter, the quark-gluon plasma
(QGP) at finite temperature and baryon density. To probe the realization of
such a phase transition in relativistic heavy ion collisions, some signatures
of the new state of matter have been discussed for decadesqgp , and among
which the quarkonium suppression is considered as a smoking gun of the
formation of QGPsatz1 .
A quarkonium is a deeply bound state of a pair of heavy quarks, its
dissociation temperature $T_{D}$ in a hot medium should be higher than the
critical temperature $T_{c}$ for the deconfinement of light hadrons. Since a
quarkonium is so heavy, one normally uses non-relativistic Schrödinger
equation to describe its dynamical evolution at finite temperature. The free
energy $F$ between a pair of heavy quarks can be extracted from the lattice
QCD simulationsdigal ; kaczmarek . Taking $F$ and the internal energy $U$ as
the two limits of the heavy quark potential $V$, which correspond respectively
to a slow and a rapid quarkonium dissociation in the hot medium, the $J/\psi$
dissociation temperature $T_{D}$ determined by the infinity size or the zero
binding energy of the system is in between $1.2T_{c}$ and $2T_{c}$digal ;
satz2 .
A nature question we ask ourselves is the relativistic correction to the
dynamical evolution of a quarkonium in the hot medium. The correction to a
bottonium is expected to be neglected safely, but for a lighter quarkonium
like $J/\psi$, the correction might be remarkable. The two-body Dirac equation
(TBDE)t1 ; t2 ; t3 ; t4 ; t5 ; long of constrained dynamics was successfully
applied to the relativistic description of light meson spectras1 ; s2 ; s3 ;
s4 ; crater2 ; crater1 in vacuum. In this Letter we take the TBDE to
calculate the charmonium wave functions at finite temperature and see the
relativistic correction to the dissociation temperature.
We can qualitatively estimate the relativistic effect on the quarkonium
dissociation before a strict calculation. Neglecting the quark spin, the
relative part of the Hamiltonian for a pair of heavy quarks can be
approximately written as a non-relativistic form
$H=\sqrt{\mu^{2}+p^{2}}-\mu+V(r)\simeq{p^{2}\over 2\mu}+V_{eff}$ (1)
with an effective potential
$V_{eff}=V-{p^{4}\over 8\mu^{3}},$ (2)
where $\mu=m_{Q}/2$ is the reduced mass with $m_{Q}$ being the heavy quark
mass. Since the relativistic correction leads to a deeper potential well,
$V_{eff}<V$, the quarkonium becomes a more deeply bound state and the
temperature needed to dissociate the quarkonium should be higher.
We now calculate the wave functions for a pair of heavy quarks in the frame of
TBDE. Taking Pauli reduction and scale transformation333 , the Dirac equation
can be expressed as a covariant relativistic Schrödinger equation for a four-
component spinorlong . Explicitly, the radial motion relative to the center of
mass is controlled by the following four equationscrater2 ,
$\displaystyle\left[-{d^{2}\over dr^{2}}+{J(J+1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-2\Phi_{SO}+\Phi_{SS}+2\Phi_{T}-2\Phi_{SOT}\right]u_{1}^{0}=b^{2}u_{1}^{0},$
(3) $\displaystyle\left[-{d^{2}\over dr^{2}}+{J(J-1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}+2(J-1)\Phi_{SO}+\Phi_{SS}+{2(J-1)\over
2J+1}(\Phi_{SOT}-\Phi_{T})\right]u_{1}^{+}$ $\displaystyle+$
$\displaystyle{2\sqrt{J(J+1)}\over
2J+1}\left(3\Phi_{T}-2(J+2)\Phi_{SOT}\right)u_{1}^{-}=b^{2}u_{1}^{+},$
$\displaystyle\left[-{d^{2}\over dr^{2}}+{(J+1)(J+2)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-2(J+2)\Phi_{SO}+\Phi_{SS}+{2(J+2)\over
2J+1}\left(\Phi_{SOT}-\Phi_{T}\right)\right]u_{1}^{-}$ $\displaystyle+$
$\displaystyle{2\sqrt{J(J+1)}\over
2J+1}\left(3\Phi_{T}+2(J-1)\Phi_{SOT}\right)u_{1}^{+}=b^{2}u_{1}^{-}$
for the spin triplet $u_{1}^{0},u_{1}^{+}$ and $u_{1}^{-}$ with quantum
numbers $n^{2s+1}L_{J}=n^{3}L_{L},\ n^{3}L_{L+1}$ and $n^{3}L_{L-1}$, and
$\left[-{d^{2}\over dr^{2}}+{J(J+1)\over
r^{2}}+2m_{w}B+B^{2}-A^{2}+2\epsilon_{w}A+\Phi_{D}-3\Phi_{SS}\right]u_{0}=b^{2}u_{0}$
(4)
for the spin singlet $u_{0}$ with quantum numbers $n^{1}L_{L}$, where
$b^{2}=(m_{m}^{2}-4m_{Q}^{2})/4$ is the energy eigenvalue in the meson rest
frame with $m_{m}$ being the meson mass, $n$ is the principal quantum number,
and $L,s$ and $J$ are the orbital, spin and total angular momentum numbers.
Note that the components $u_{1}^{+}$ with $J=L+1$ and $u_{1}^{-}$ with $J=L-1$
are coupled to each other. Following the notations in crater2 , we have
separated the central potential into two parts,
$V(r)=A(r)+B(r),$ (5)
and the abbreviations for the Darwin, spin-spin, spin-orbit and tensor terms
introduced in the dynamical equations are defined as crater2
$\displaystyle\Phi_{D}$ $\displaystyle=M+F^{\prime 2}+K^{\prime
2}-\nabla^{2}F+2K^{\prime}P-2\left(F^{\prime}+{1\over r}\right)Q,$ (6)
$\displaystyle\Phi_{T}$ $\displaystyle={1\over
3}\left[N+2F^{\prime}K^{\prime}-\nabla^{2}K+\left(3F^{\prime}-K^{\prime}+{3\over
r}\right)P+\left(F^{\prime}-3K^{\prime}+{1\over r}\right)Q\right]$
$\displaystyle\Phi_{SO}$ $\displaystyle=-{F^{\prime}\over
r}+K^{\prime}P-\left(F^{\prime}+{1\over r}\right)Q,$ $\displaystyle\Phi_{SS}$
$\displaystyle=O+{2\over 3}F^{\prime}K^{\prime}-{1\over 3}\nabla^{2}K+{2\over
3}K^{\prime}P-2\left(F^{\prime}+{1\over 3r}\right)Q,$
$\displaystyle\Phi_{SOT}$ $\displaystyle=-{K^{\prime}\over
r}+\left(F^{\prime}+{1\over r}\right)P-K^{\prime}Q,$ $\displaystyle F$
$\displaystyle={1\over 2}L-{3\over 2}G,$ $\displaystyle G$
$\displaystyle=-{1\over 2}\ln\left(1-2{A\over m_{m}}\right),$ $\displaystyle
K$ $\displaystyle={1\over 2}L+{1\over 2}G,$ $\displaystyle L$
$\displaystyle=\ln\sqrt{1+{2m_{w}B+B^{2}\over
m_{Q}^{2}\left(1-2A/m_{m}\right)}},$ $\displaystyle M$ $\displaystyle=-{1\over
2}\nabla^{2}G+{3\over 4}G^{\prime 2}+G^{\prime}F^{\prime}-K^{\prime 2},$
$\displaystyle N$ $\displaystyle={1\over 3}\left[\nabla^{2}K-{1\over
2}\nabla^{2}G+{3\over 2}{G^{\prime}-2K^{\prime}\over
r}+F^{\prime}(G^{\prime}-2K^{\prime})\right]$ $\displaystyle O$
$\displaystyle={1\over 3}\nabla^{2}(K+G)-{1\over
3}F^{\prime}(G^{\prime}+K^{\prime})-{1\over 2}G^{\prime 2},$ $\displaystyle P$
$\displaystyle={\sinh 2K\over r},$ $\displaystyle Q$ $\displaystyle={\cosh
2K-1\over r}$
and
$m_{w}=m_{Q}^{2}/m_{m},\epsilon_{w}=(m_{m}^{2}-2m_{Q}^{2})/(2m_{m}),F^{\prime}=dF/dr,G^{\prime}=dG/dr$
and $K^{\prime}=dK/dr$.
We now focus on the charmonium states. The $J/\psi$ state includes two
components $1^{3}S_{1}$ and $1^{3}D_{1}$ determined simultaneously by the two
coupled equations of (3) with energy eigenvalue
$b^{2}=(m_{J/\psi}^{2}-4m_{c}^{2})/4$. Similarly, the two components
$2^{3}S_{1}$ and $2^{3}D_{1}$ for the state $\psi^{\prime}$ are controlled by
the two coupled equations but with eigenvalue
$b^{2}=(m_{\psi^{\prime}}^{2}-4m_{c}^{2})/4$. The $\chi_{0}$ state
$1^{3}P_{0}$ is described by the last equation of (3) with disappeared
component $u_{1}^{+}$, and the $\chi_{1}$ state $1^{3}P_{1}$ is characterized
by the independent equation (4) for $u_{1}^{0}$.
In vacuum, the potential between two quarks is usually taken as the Cornell
form, including a Coulomb-like part which dominates the wave functions around
$r=0$ and a linear part which leads to the quark confinement,
$\displaystyle A(r)$ $\displaystyle=-{\alpha\over r},$ (7) $\displaystyle
B(r)$ $\displaystyle=\sigma r.$
The three parameters in the model, namely the charm quark mass $m_{c}$ in the
Schrödinger equation and the two coupling constants $\alpha$ and $\sigma$ in
the potential, can be fixed by fitting the charmonium masses in vacuum. By
taking $m_{c}=1.422$ GeV, $\alpha=0.492$ and $\sigma=0.186$ GeV2, we obtain
the charmonium masses $m_{J/\psi}=3.113$ GeV, $m_{\psi^{\prime}}=3.692$ GeV,
$m_{\chi_{0}}=3.404$ GeV and $m_{\chi_{1}}=3.504$ GeV, which are very close to
the experimental values $m_{J/\psi}=3.097$ GeV, $m_{\psi^{\prime}}=3.686$ GeV,
$m_{\chi_{0}}=3.415$ GeV and $m_{\chi_{1}}=3.511$ GeV.
The charmonium dissociation temperature in a hot medium can be determined by
solving the corresponding dynamical equation for the $c\bar{c}$ system with
potential $V$ between the two heavy quarks at finite temperature. The
potential depends on the dissociation process in the medium. At the moment, we
know only its two limits. For a rapid dissociation where there is no heat
exchange between the heavy quarks and the medium, the potential is just the
internal energy $U$, while for a slow dissociation, there is enough time for
the heavy quarks to exchange heat with the medium, the free energy $F$ which
can be extracted from the lattice calculations is taken as the
potentialshuryak . From the thermodynamic relation $F=U-T{\cal S}$ where
${\cal S}$ is the entropy, the potential well is deeper for $V=U$ and
therefore the dissociation temperature of charmonium states with potential
$V=U$ is higher than that with $V=F$. In the literatures, a number of
effective potentials in between $F$ and $U$ have been used to evaluate the
charmonium evolution in QCD mediumdigal ; satz2 ; shuryak ; wong . In non-
relativistic case, H.Satz and his collaboratorsdigal ; satz2 solved the
Schrödinger equation and found the dissociation temperatures $T_{d}/T_{c}=2.1$
and $1.26$ for $V=U$ and $V=F$ respectively, where $T_{c}=165$ MeVdigal ;
satz2 is the critical temperature for the deconfinement.
Considering the Debye screening at finite temperature, the potential $V=F=A+B$
can be written asdigal ; satz2
$\displaystyle A(r,T)$ $\displaystyle=$ $\displaystyle-{\alpha\over r}e^{-\mu
r},$ $\displaystyle B(r,T)$ $\displaystyle=$
$\displaystyle{\sigma\over\mu}\left[{\Gamma\left({1\over 4}\right)\over
2^{3\over 2}\Gamma\left({3\over 4}\right)}-{\sqrt{\mu r}\over 2^{3\over
4}\Gamma\left({3\over 4}\right)}K_{1\over
4}\left(\mu^{2}r^{2}\right)\right]-\alpha\mu,$ (8)
where $\Gamma$ is the Gamma function, $K$ is the modified Bessel function of
the second kind, and the temperature dependent parameter $\mu(T)$, namely the
screening mass or the inverse screening radius, can be extracted from fitting
the lattice simulated free energy digal ; kaczmarek . From the known free
energy $F$, one can then obtain the other limit of the potential,
$V=U=F+T{\cal S}$ by taking ${\cal S}=-\partial F/\partial T$.
We use the inverse power methodcrater3 to solve the Schrödinger equations (3)
and (4) and obtain the radial wave functions
$\Psi(r,T)={u(r,T)\over r}$ (9)
for the charmonium states $J/\psi$, $\psi^{\prime}$ and $\chi_{c}$ at finite
temperature.
Figure 1: The $S$-wave (solid lines) and $D$-wave (dashed lines) functions
for the $J/\psi$ meson at three temperatures in the limit of quark potential
$V=F$. From the top down the temperature $T$ is zero, critical temperature
$T_{c}$ and relativistic dissociation temperature $T_{d}$.
The $S-$ and $D-$wave functions for the meson $J/\psi$ at different
temperature are shown in Fig.1 in the limit of $V=F$. The wave functions in
vacuum are very similar to the results obtained in crater2 , and the component
$1^{3}S_{1}$ dominates the state at any temperature. With increasing
temperature, both the $S$\- and $D$-wave functions expand continuously. At the
critical temperature of deconfinement $T_{c}$, while the peak values of the
wave functions drop down a little, their behavior is similar to the one in
vacuum. This means that the $J/\psi$ meson is still a bound state at the
deconfinement phase transition where the light mesons start to melt in the hot
medium, and therefore the observed $J/\psi$s in the final state of heavy ion
collisions can signal the QGP formation in the early stage. However, with
further increasing temperature, the wave functions expand rapidly around
$T_{d}=1.35T_{c}$. By calculating the average distance between the $c$ and
$\bar{c}$,
$\langle r\rangle(T)={\int drr^{3}\left|\psi(r,T)\right|^{2}\over\int
drr^{2}\left|\psi(r,T)\right|^{2}},$ (10)
we found that at $T_{d}$ the average size becomes infinite, see Fig.2, which
indicates the $J/\psi$ dissociation. Therefore, $T_{d}$ is called the
relativistic dissociation temperature.
Figure 2: The average size of the $J/\psi$ meson in the quark-gluon plasma in
the limit of potential $V=F$. The temperature and average size are scaled by
their corresponding values at the critical point of the deconfinement.
The other quantity which can be used to characterize the quarkonium
dissociation is the heavy quark binding energy. For the Dirac equation or
equivalently the Schrödinger equations (3) and (4), the usual binding energy
defined as $\epsilon=V(\infty)+2m_{c}-m_{m}$ is no longer valid to describe
the quarkonium dissociation. In the limit of $r\to\infty$, the four equations
for the spin singlet and triplet degenerate and the asymptotic equation is
simplifies as
$\left(-{d^{2}\over dr^{2}}+2m_{w}V(\infty)+V^{2}(\infty)\right)u=b^{2}u$ (11)
with $V(\infty)=B(\infty)$, and the energy for the scattering state is
$2m_{w}V(\infty)+V^{2}(\infty)=b^{2},$ (12)
which leads to the binding energy for the $c\bar{c}$ bound state
$\epsilon(T)=V(\infty,T)+\sqrt{V^{2}(\infty,T)+4m_{c}^{2}}-m_{m}.$ (13)
Figure 3: The binding energy of the $J/\psi$ meson in the quark-gluon plasma
and in the limit of potential $V=F$. The temperature and binding energy are
scaled by their corresponding values at the critical point of the
deconfinement.
Fig.3 shows the temperature dependence of the binding energy in the QGP phase
and in the limit of potential $V=F$. It drops down monotonously with
increasing temperature and reaches zero at $T_{d}$. The result is consistent
with the calculation of the charmonium average size. The infinite size and
zero binding energy of the $c\bar{c}$ system define the unique dissociation
temperature. In comparison with the non-relativistic calculation, the $J/\psi$
dissociation temperature increases from $1.26T_{c}$ to $1.35T_{c}$, the
relativistic correction is $7\%$. We also calculated the $J/\psi$ wave
functions at finite temperature in the other limit of potential $V=U$ and
found that the dissociation temperature $T_{d}$ goes up from the non-
relativistic value $2.1T_{c}$ to $2.38T_{c}$. The relativistic correction is
$13\%$.
Figure 4: The wave function $u(r)=r\Psi(r)$ for the $\chi_{0}$ meson at zero
temperature and critical temperature of deconfinement $T_{c}$ in the limit of
quark potential $V=F$.
The wave function for the excited state $\chi_{0}$ is shown in Fig.4 in the
limit of potential $V=F$. Considering the fact that the radial function $\Psi$
is divergent at the origincrater2 , we plot $u=r\Psi$ directly from the
Schrödinger equation (3). In comparison with the ground state $J/\psi$,
$\chi_{0}$ wave function distributes in a wider region, the average size is
larger and the binding energy is smaller. The corresponding dissociation
temperature defined by $\langle r\rangle(T_{d})=\infty$ and
$\epsilon(T_{d})=0$ is around the critical temperature $T_{c}$ and a little
bit higher than the non-relativistic value. It is easy to understand that the
relativistic correction for the excited states should be smaller than the one
for the ground state.
In summary, we calculated the charmonium wave functions at finite temperature
by solving the covariant relativistic Schrödinger equations for the $c\bar{c}$
spin singlet and triplet states. The relativistic effect makes the quark
potential well more deep, and charmonia can survive in a more hot medium. By
considering the two limits of the central potential, the relativistic
correction to the $J/\psi$ dissociation temperature in the QGP phase is in
between $7\%$ and $13\%$.
Acknowledgement: The work is supported by the NSFC (Grant Nos. 10975084 and
11079024), RFDP (Grant No.20100002110080 ) and MOST (Grant No.2013CB922000).
## References
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* [11] H.W.Crater, R.L.Becker, C.Y.Wong and P.Van Alstine, Phys. Rev.D 46, 5117 (1992).
* [12] H.W.Crater and P.Van Alstine, Phys. Rev. D70, 034026(2004).
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* [14] J.H.Yoon, B.N.Kim, H.W.Crater and C.Y.Wong, arXiv:1110.1598 [hep-ph].
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* [17] B.Pan, ArXiv:1106.3028 [hep-ph].
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|
arxiv-papers
| 2012-09-26T08:48:56 |
2024-09-04T02:49:35.623188
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xingyu Guo, Shuzhe Shi, Pengfei Zhuang",
"submitter": "Shuzhe Shi",
"url": "https://arxiv.org/abs/1209.5873"
}
|
1209.5876
|
# Emergent gravity: From statistical point of view
Bibhas Ranjan Majhi IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus,
Pune 411 007, India bibhas@iucaa.ernet.in
###### Abstract
Near the event horizon of a black hole, the effective theory is two
dimensional conformal theory. Here we show that the holographic modes
characterising this underlying conformal symmetry and the basic definition of
entropy $S$ in statistical mechanics lead the equipartition law of energy. We
also show that $S$ is proportional to the gravity action which suggests the
emergent nature of gravity. This is further bolstered by expressing the
generalised Smarr formula as a thermodynamic relation, $S=E/2T$, where $T$ is
the Hawking temperature and $E$ is shown to be the Komar energy.
## 1 Introduction
In absence of a true quantum theory of gravity, semi-classical approaches have
become increasingly popular and are widely used, particularly in the context
of thermodynamics of gravity. It is now evident that gravity and
thermodynamics are closely connected to each other [1, 2, 3]. The repeated
failure to quantise gravity led to a parallel development [4, 5, 6] where
gravity is believed to be an emergent phenomenon just like thermodynamics and
hydrodynamics instead treating it as a fundamental force. In this context, the
fundamental role of gravity is replaced by thermodynamical interpretations
leading to similar or equivalent results without knowing the underlying
microscopic details. Nevertheless, understanding the entropic or thermodynamic
origin of gravity is far from complete since the arguments are more heuristic
than concrete and depend upon specific ansatz or assumptions.
In this writeup, based on our work [7], we readdressed some of the facts
related to the emergent nature of gravity, in the context of the black holes
in Einstein gravity.
Here using certain basic results derived by us [8, 9] and applying the
standard definition of entropy given in statistical mechanics, we are able to
provide a statistical interpretation of gravity. We first give the
equipartition law of energy and show that this leads to the identification of
entropy with the action for gravity. The immediate consequence of it is that
the Einstein equations, obtained by a variational principle involving the
action, can be equivalently obtained by an extremisation of the entropy. This
implies that the gravity can be thought of as the emergent phenomenon.
The emergent nature has been further bolstered by deriving the relation
$S=E/2T$, connecting the entropy ($S$) with the Hawking temperature ($T$) and
energy ($E$) for a black hole with stationary metric. We show that this energy
corresponds to Komar’s expression [10, 11]. Using this fact we show that the
relation $S=E/2T$ is also compatible with the generalised Smarr formula [12,
3, 13].
## 2 Previous results
Here we briefly introduce three relevant results, which were derived
previously by some of us, for our main purpose. $\bullet$ Thermodynamics of a
black hole is universally governed by its properties near the event horizon.
It is also well understood that near the event horizon the effective theory
becomes two dimensional conformal theory whose metric is given by the two
dimensional ($t-r$)- sector of the original metric. For details see [14].
$\bullet$ Using the WKB approximation, the left ($L$) and right ($R$) moving
(holomorphic) modes are obtained by solving the appropriate field equation
under the effective two dimensional metric. It has been shown that the modes
inside and outside the horizon are related by the transformations [8, 9]:
$\displaystyle\phi^{(R)}_{in}=e^{-\frac{\pi\omega}{\kappa}}\phi^{(R)}_{out};\,\,\
\phi^{(L)}_{in}=\phi^{(L)}_{out}$ (1)
where “$\omega$” is the energy of the particle as measured by an asymptotic
observer and “$\kappa$” is the surface gravity of the black hole. Furthermore,
the $L$ mode gets trapped while the $R$ mode tunnels through the horizon and
is observed at asymptotic infinity as Hawking radiation [8, 9]. The
probability of this “$R$” mode, to go outside, as measured by the outside
observer is given by
$\displaystyle
P^{(R)}=\Big{|}\phi^{(R)}_{in}\Big{|}^{2}=\Big{|}e^{-\frac{\pi\omega}{\kappa}}\phi^{(R)}_{out}\Big{|}^{2}=e^{-\frac{2\pi\omega}{\kappa}}$
(2)
where, in the second equality, (1) has been used. This is essential since the
measurement is done from outside and hence $\phi^{(R)}_{in}$ has to be
expressed in favour of $\phi^{(R)}_{out}$. $\bullet$ The effective two
dimensional curved metric can be embedded in a flat space which has exactly
two space-like coordinates. This is a consequence of a modification in the
original GEMS (globally embedding in Minkowskian space) approach of [15] and
has been elaborated by us in [16]. This tells that each $R$ mode can be
associated to two degrees of freedom. Therefore, the total number of degrees
of freedom for $n$ number of $R$ modes is $N=2n$.
## 3 Partition function
The partition function for the space-time with matter field is given by [13],
$\displaystyle{\cal{Z}}=\int~{}D[g,\Phi]~{}e^{iI[g,\Phi]}$ (3)
where $I[g,\Phi]$ is the action representing the whole system and $D[g,\Phi]$
is the measure of all field configurations ($g,\Phi$). Since we want to
confine ourself within the usual semi-classical regime, we shall neglect all
the higher order terms for the subsequent analysis. Therefore, in the semi-
classical regime the partition function is expressed as [13],
$\displaystyle{\cal{Z}}\simeq e^{iI[g_{0},\Phi_{0}]},$ (4)
where ($g_{0},\Phi_{0}$) are the background fields and the classical action
$I[g_{0},\Phi_{0}]$ leads to the Einstein equation.
## 4 Relation between different thermodynamical quantities with the action
and their implications
Here using the previous results and the above definition of partition function
combined with the standard definition of entropy in statistical mechanics we
derive three important relations among different thermodynamical quantities of
a stationary black hole. The physical implications are also being described at
the end.
In statistical mechanics the entropy is related to the partition function by
the relation: $S=\ln{\cal{Z}}+\frac{E}{T}$ and so (4) leads to,
$S=iI[g_{0},\Phi_{0}]+\frac{E}{T}$ where $E$ and $T$ are respectively the
energy and temperature of the system. Now since there are $N$ number of
degrees of freedom in which all the information is encoded, the entropy ($S$)
of the system must be proportional to $N$. Hence
$\displaystyle N=N_{0}S=N_{0}(iI[g_{0},\Phi_{0}]+\frac{E}{T}),$ (5)
where $N_{0}$ is a proportionality constant. Now the average value of the
energy, measured from outside, is calculated as,
$\displaystyle<\omega>=\frac{\int_{0}^{\infty}~{}d\omega~{}\omega~{}P^{(R)}}{\int_{0}^{\infty}~{}d\omega~{}P^{(R)}}=T$
(6)
where $T=\kappa/2\pi$ is the temperature of the black hole [9]. Therefore if
we consider that the energy $E$ of the system is encoded near the horizon and
the total number of pairs created is $n$ among which this energy is
distributed, then we must have, $E=nT$ where only the $R$ mode of the pair is
significant. Hence, since $N=2n$, we obtain the energy of the system as
$\displaystyle E=\frac{1}{2}NT.$ (7)
Noted that (7) can be interpreted as the usual law of equipartition of energy,
since it implies that if the energy $E$ is distributed equally over each
degree of freedom, then each degree of freedom should contain an energy equal
to $T/2$. The fact that the energy is equally distributed among the degrees of
freedom may be understood from the symmetry of two space-like coordinates
($z^{1}\longleftrightarrow z^{2}$) such that the metric is unchanged [16].
Now using (5) and (7) we obtain following relations:
$\displaystyle E=\frac{N_{0}}{2-N_{0}}iTI[g_{0},\Phi_{0}];\,\,\,\
S=\frac{2E}{N_{0}T}.$ (8)
In order to fix the value of “$N_{0}$” we consider the simplest example, the
Schwarzschild black hole for which the entropy, energy and temperature are
given by, $S=\frac{A}{4}=4\pi M^{2},\,\,\ E=M,\,\,\ T=\frac{1}{8\pi M}$, where
“$M$” is the mass of the black hole. Substitution of these in the second
relation of (8) leads to $N_{0}=4$. Finally, putting back $N_{0}=4$ in the
preceding relations we obtain,
$\displaystyle E=-\frac{i\kappa I[g_{0},\Phi_{0}]}{\pi};\,\,\
S=-iI[g_{0},\Phi_{0}]=\frac{E}{2T}.$ (9)
The implications of the above relations are as follows: $\bullet$ Use of the
explicit form of Einstein-Hilbert action in the first relation shows that $E$
is the Komar conserved quantity corresponding to Killing vector. For details,
see [7]. $\bullet$ First equality of the second relation signifies that the
extremisation of entropy leads to Einstein’s equations. It illustrates the
emergent nature of gravity. $\bullet$ The last equality $S=E/2T$ is the
general expression for Smarr formula. This can be checked by substituting the
relevant quantities of Kerr-Newman black hole to obtain the relation
$\frac{M}{2}=\frac{\kappa A}{8\pi}+\frac{VQ}{2}+\Omega J$, which is the
generalised Smarr formula [12, 3, 13]. This further clarifies the fact that
the gravity is an emergent phenomenon.
## 5 Conclusions
We, in the context of black hole, using the statistical definition of entropy
and the partition function for gravity, further clarified the possibility of
considering gravity as an emergent phenomenon. There are certain issues still
to be addressed or clarified. Let us tabulate some of them: $\bullet$ First of
all, we saw that the proportionality constant $N_{0}$ was not fixed from the
fundamental ground. Rather, we did it by using the parameters of a specific
black hole. An independent derivation will be very much interesting. $\bullet$
Is it possible to discuss the same for more general situation, like the
spacetime without black hole? This may be done by considering the local
Rindler frame in the spacetime which has a null surface on which the
temperature and entropy can be associated. $\bullet$ A more simpler but
interesting extension would be inclusion of higher dimensions. In this regard,
one can be interested to see if the higher dimensional gravity exhibit same
feature or in other words if the above features depend on dimension of
spacetime. Some part of this issue has already been addressed in [17]. Here we
did for the higher dimensional black hole in Einstein gravity and showed that
we again have a relation similar to $E=2ST$, but $E$ is interpreted as the
Komar conserved quantity rather than the energy itself. $\bullet$ As a final
remark we feel that although our results were derived for Einstein gravity,
the methods are general enough to include other possibilities like $f(R)$
gravity, Lancozs-Lovelock gravity etc.
References
## References
* [1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
* [2] S. W. Hawking, Nature 248, 30 (1974). S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)].
* [3] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).
* [4] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995) [arXiv:gr-qc/9504004].
* [5] T. Padmanabhan, Rept. Prog. Phys. 73, 046901 (2010) [arXiv:0911.5004 [gr-qc]] and references therein.
* [6] E. P. Verlinde, JHEP 1104, 029 (2011) [arXiv:1001.0785 [hep-th]].
* [7] R. Banerjee and B. R. Majhi, Phys. Rev. D 81, 124006 (2010) [arXiv:1003.2312 [gr-qc]].
* [8] R. Banerjee and B. R. Majhi, Phys. Rev. D 79, 064024 (2009) [arXiv:0812.0497 [hep-th]]. R. Banerjee and B. R. Majhi, Phys. Lett. B 675, 243 (2009) [arXiv:0903.0250 [hep-th]].
* [9] R. Banerjee, B. R. Majhi and E. C. Vagenas, Phys. Lett. B 686, 279 (2010) [arXiv:0907.4271 [hep-th]].
* [10] A. Komar, Phys. Rev. 113, 934 (1959).
* [11] R. M. Wald, “General Relativity,” Chicago, Usa: Univ. Pr. ( 1984) 491p.
* [12] L. Smarr, Phys. Rev. Lett. 30, 71 (1973) [Erratum-ibid. 30, 521 (1973)].
* [13] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
* [14] B. R. Majhi, arXiv:1110.6008 [gr-qc].
* [15] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159].
* [16] R. Banerjee and B. R. Majhi, Phys. Lett. B 690, 83 (2010) [arXiv:1002.0985 [gr-qc]].
* [17] R. Banerjee, B. R. Majhi, S. K. Modak and S. Samanta, Phys. Rev. D 82, 124002 (2010) [arXiv:1007.5204 [gr-qc]].
|
arxiv-papers
| 2012-09-26T09:05:22 |
2024-09-04T02:49:35.629077
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1209.5876"
}
|
1209.5878
|
The charm physics programme at the LHCb upgrade, and Atlas and CMS upgrades
Marco Gersabeck
on behalf of the LHCb collaboration
CERN, 1211 Geneva 23, Switzerland
> Charm physics has been established at the LHC based on several high-
> precision measurements. The future of charm physics at the LHC experiments
> is discussed in detail. The bulk of the charm physics programme will be
> performed by LHCb and the LHCb upgrade. In particular, the impact of the
> LHCb upgrade on mixing and $C\\!P$ violation measurements is presented.
> PRESENTED AT
>
>
>
>
> The $5^{th}$ International Workshop on Charm Physics
> Honolulu, Hawai’i, USA, 14–17 May 2012
## 1 Introduction
The LHC has performed excellently in its first years of operation and provided
large datasets at unprecedented collision energies. With the data recorded at
the four interaction points all experiments have proven their feasibility and
outstanding operational quality. Charm physics has been an integral part of
this road to success, ranging from first cross-section measurements at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [1] to the first evidence of $C\\!P$
violation in the charm system [2]. A full account of the current status of
charm physics is given in Reference [3].
## 2 The LHC upgrade schedule
The first running phase of the LHC will last until the end of 2012 for proton-
proton collisions, followed by a roughly four-week run of proton-lead
collisions at the beginning of 2013. This is followed by an approximately
18-months shutdown (LS1) for maintenance and consolidation work after which
the LHC is expected to operate close to its design collision energy of
$13$–$14\mathrm{\,Te\kern-1.00006ptV}$. In addition, also the filling scheme
is expected to go from the current $50\mbox{\,ns}$ bunch spacing to the
nominal $25\mbox{\,ns}$ bunch spacing, i.e. doubling the number of bunches.
This break also allows first work on detector upgrade installations. The
second running period is foreseen to last from late 2014 until 2017 included,
followed by another long shutdown (LS2) which will see significant work on
detector upgrades. The running period beyond 2018 will head towards the high-
intensity LHC, following another long shutdown around 2022 (LS3).
## 3 The detector upgrade plans
The Atlas collaboration plans to pursue their charm programme until LS2. The
main relevant detector upgrade is thus the installation of a fourth barrel
pixel layer in LS1. With an improved impact parameter resolution this is of
particular importance to the heavy flavour physics programme. The CMS
collaboration has no plans to pursue a charm programme after LS1.
LHCb, as a dedicated heavy flavour experiment, will continue its charm
programme throughout its running period. A major upgrade of the experiment is
foreseen to be installed in LS2. The LS1 will be used to adapt the existing
data acquisition and processing infrastructure to the changes in accelerator
conditions after the shutdown. The key challenge when operating at higher
instantaneous luminosities is the efficiency for collecting data of hadronic
decays. Based on the current LHCb layout, the hadronic trigger efficiency
reduces due to harsher cuts in the hardware trigger stage. This leads to a
saturation of the signal yield as function of instantaneous luminosity.
The solution to allow operation at higher luminosities is a complete re-design
of the trigger system. The LHCb upgrade is based on the ability to read out
the detector at the LHC clock frequency of $40\mbox{\,MHz}$. The current first
trigger level, which reduced the data rate to $1\mbox{\,MHz}$, will be
replaced by a flexible custom-electronics trigger which can be tuned to output
rates between the current $1\mbox{\,MHz}$ and the full $40\mbox{\,MHz}$. This
is followed by a software-based trigger stage which, using the full detector
information, has to reduce the output rate to $20\mbox{\,kHz}$, i.e. a factor
of four higher than the current output rate.
The upgrade task of the sub-detector system is to maintain the current high
performance while providing the possibility of the $40\mbox{\,MHz}$ readout in
the presence of a significantly increased particle rate. The LHCb upgrade is
currently in the design phase and TDR documents which finalise several
technology choices are planned for 2013. A framework TDR outlining the various
detector options as well as the financial planning and insitute’s interests
has been submitted to LHCC in June 2012 [4]. The vertex detector will be
replaced either by a silicon-strip detector with finer strip pitch or by a
silicon-pixel detector of similar layout. The remaining tracking system will
be replaced by a combination of silicon detectors and possibly scintillating
fibre or straw detectors. The RICH detectors have to be equipped with new
photon detectors which can be read out at the increased rate. The calorimeters
have to receive new readout electronics.
The data yield per year is expected to increase for several reasons. The charm
production cross-section is projected to increase by a factor of about $1.8$
when going to nominal LHC energy. This gain will take effect already after
LS1. The trigger efficiency is assumed to increase by a factor of $2$,
however, this factor may be significantly larger for multi-body decay modes.
With the annual integrated luminosity expected to increase by a factor between
$3$ and $5$, the annual signal yield is estimated to increase by about an
order of magnitude. The total integrated luminosity recorded during the LHCb
upgrade period is assumed to be $50\mbox{\,fb}^{-1}$. As an example, this
leads to an expectation of $4\times 10^{10}$ offline selected $D^{0}\\!\to
K^{-}\pi^{+}$ decays.
## 4 Charm production and spectroscopy
The LHCb collaboration has recently published a set of measurements of the
production of double-charm events, i.e. events containing double charmonium,
charmonium plus open charm, or double open charm [5]. These studies will be
continued with increased data samples. They will be extended by studies of the
production of other charmonium modes, both at Atlas and at LHCb. This will be
complemented by studies of the combined production of charmonium and jets or
vector bosons.
Another topic requiring high luminosity and hence the upgrade is the search
for doubly-heavy and triply-heavy baryons. Given their production cross-
sections which fall sharply with increased transverse momentum [6] these need
large data samples to be discovered. At the same time, it is mandatory to have
an efficient triggering to reconstruct these complex multi-body final states
without requiring large transverse momenta. Beyond the discovery of new states
it is of interest to study their properties such as lifetimes, branching
ratios, quantum numbers, and their spectrum of excited states.
## 5 Rare decay searches and analyses
Rare decay searches naturally gain in sensitivity with increasing luminosity
and hence benefit from the increased data sets planned for the detector
upgrades at the LHC. A second important component to maximise sensitivity is a
low level of background. For this reason only LHCb plans to continue rare
decay searches with their upgraded detector.
Figure 1: Current best limits on $D^{0}$ decays. The different regions
indicate flavour changing neutral current decays, lepton-flavour violating
decays (LF), lepton-number violating decays (L), and lepton and baryon-number
violating decays (BL). Reproduced from Ref. [7].
LHCb recently set the best limit for the flavour-changing neutral-current
decay $D^{0}\\!\to\mu^{-}\mu^{+}$ at $1.1\times 10^{-8}$ [8] (see Figure 1).
With the LHCb upgrade this limit is expected to be improved by about one order
of magnitude.
Another set of promising measurements are those of the dimuon invariant-mass
spectrum of $D^{\pm}\\!\to h^{\pm}\mu^{-}\mu^{+}$ decays. The LHCb acceptance
allows the study of the full spectrum down to the kinematic threshold to
search for suppressed resonances such as sGolstinos. The same decay with a
same-sign lepton pair is one example for a lepton-number violating decay. This
decay can be mediated for example by Majorana neutrinos.
Four-body decays offer several different approaches for searching for physics
beyond the standard model. An interesting example among rare decays is the
decay $D^{0}\\!\to K^{-}K^{+}\mu^{-}\mu^{+}$. Beyond the search and eventual
discovery of this mode it is of interest to study its symmetries. The
measurement of the forward-backward asymmetry of one of the muons, the search
for $C\\!P$ violation, and the measurement of T-odd correlations provide
complementary information [9]. Assuming a branching fraction of $10^{-6}$ the
LHCb upgrade should provide a sample of several hundred selected candidates.
In comparison, the decay $D^{0}\\!\to K^{-}K^{+}\pi^{-}\pi^{+}$ is expected to
yield a sensitivity to T-odd correlations of $2.5\times 10^{-4}$ at the end of
the LHCb upgrade.
## 6 Mixing measurements
There are a number of open questions in the area of charm mixing and $C\\!P$
violation. The mixing mechanism is well established through a number of
complementary measurements. However, there is still no precise determination
of the underlying parameters. The LHCb collaboration has reported first
evidence for $C\\!P$ violation in the charm system [10], but it is not clear
whether this is due to physics beyond the standard model or whether it can be
explained by hadronic uncertainties within the standard model. Beyond this
measurement there is no further evidence for $C\\!P$ violation in the charm
sector to date. In particular, indirect $C\\!P$ violation is still out of
reach.
The ultimate goal of mixing and $C\\!P$ violation measurements in the charm
sector is to reach the precision necessary to probe the standard model
predictions or better. Theoretically, precise standard model predictions are
still challenging. As a general goal, $10\%$ relative uncertainties on the
underlying parameters should be achieved. There are a number of possible ways
to measure mixing parameters. Possibly the most powerful is the observable
$y_{CP}$, measured as a ratio of effective lifetimes in the $C\\!P$
eigenstates $K^{-}K^{+}$ or $\pi^{-}\pi^{+}$ with respect to the Cabibbo-
favoured state $K^{\mp}\pi^{\pm}$. The recent LHCb measurement, based on the
small data sample recorded in 2010, is a proof of principle for such
measurements at a hadron collider. Its systematic uncertainty is expected to
scale well with the increasing sample size as more sophisticated treatments of
the contributing backgrounds become possible.
Another mixing measurement based on two-body decays is that using the wrong-
sign decay $D^{0}\\!\to K^{+}\pi^{-}$. This measurement requires external
information on the strong phase shift to yield information on the mixing
parameters themselves. The measurement of the forbidden decay rate
$D^{0}\\!\to K^{+}\mu^{-}\nu$ which is only accessible through mixing gives
access to the absolute mixing rate $R_{m}\equiv(x^{2}+y^{2})/2$. Through a
time-dependent Dalitz analysis, the decays $D^{0}\\!\to K^{0}_{\rm\scriptstyle
S}\pi^{-}\pi^{+}$ and $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ give
access to the individual mixing parameters $x$ and $y$. The expected
sensitivities are given in Table 1.
Decay | Observable | Expected sensitivity (in $10^{-3}$)
---|---|---
$D^{0}\\!\to K^{-}K^{+}$ | $y_{CP}$ | $0.04$
$D^{0}\\!\to\pi^{-}\pi^{+}$ | $y_{CP}$ | $0.08$
$D^{0}\\!\to K^{+}\pi^{-}$ | $x^{\prime 2}$, $y^{\prime}$ | $0.01$, $0.1$
$D^{0}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ | $x$, $y$ | $0.15$, $0.1$
$D^{0}\\!\to K^{+}\mu^{-}\nu$ | $x^{2}+y^{2}$ | $0.0001$
Table 1: Expected statistical sensitivities for mixing observables for
$50\mbox{\,fb}^{-1}$.
## 7 $C\\!P$ violation measurements
### 7.1 Indirect $C\\!P$ violation
Indirect $C\\!P$ violation measurements at LHCb are mostly constrained by the
observable $A_{\Gamma}$ [11], which is the asymmetry of effective lifetimes of
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $D^{0}$ decays to a
$C\\!P$ eigenstate and which can be written as
$A_{\Gamma}\approx\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\approx-
a_{C\\!P}^{\rm ind}-a_{C\\!P}^{\rm dir}y_{CP},$ (1)
where $A_{m}$ is the deviation of $|q/p|$ from $1$, $A_{d}$ is the deviation
of $|A_{f}/\overline{A}_{f}|$ from $1$, and $\phi$ is the relative weak phase
between the two fractions, following Reference [3]. The $C\\!P$ violating
parameters in this observable are multiplied by the mixing parameters $x$ and
$y$, respectively. Hence, the relative precision on the $C\\!P$ violating
parameters is limited by the relative precision of the mixing parameters.
Therefore, aiming at a relative precision below $10\%$ and taking into account
the current mixing parameter world averages, the target precision for the
mixing parameters would be $2-3\times 10^{-4}$, which can be reached with the
LHCb upgrade. With standard model indirect $C\\!P$ violation expected to be of
the order of $10^{-4}$, the direct $C\\!P$ violation parameter contributing to
$A_{\Gamma}$ has to be measured to a precision of $10^{-3}$ in order to
distinguish the two types of $C\\!P$ violation in $A_{\Gamma}$. Based in the
existing LHCb measurement [12], the ultimate statistical precision expected
for $A_{\Gamma}$ is better than $10^{-4}$ in both decays to $K^{-}$ $K^{+}$
and to $\pi^{-}$ $\pi^{+}$.
The above mentioned decays $D^{0}\\!\to K^{0}_{\rm\scriptstyle
S}\pi^{-}\pi^{+}$ and $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ give
access not only to the mixing parameters but also to the $C\\!P$-violating
parameters $|q/p|$ and $\phi$. Two different approaches exist to extract these
parameters: a model-dependent measurement of the time evolution of Dalitz-plot
parameters, and a model-independent measurement of the time evolution in
regions of different strong-phase difference. The latter approach requires
input from measurements using quantum-correlated charm meson pairs produced at
threshold. Such measurements exist from CLEOc (see e.g. [13]) and can be
performed by the BESIII collaboration in the future. It is important that the
existing measurements are further improved in precision to minimise systematic
limitations in this approach.
The interplay between indirect and direct $C\\!P$-violation parameters shows
the importance of measuring both sets of parameters in order to interpret the
result. At the same time precise theory predictions are required to identify
the source of $C\\!P$ violation. Where this is not possible it may be feasible
to constrain theoretical uncertainties through theoretically clean control
measurements. Beyond complementary $C\\!P$ violation measurements such control
measurements can also be ratios of particle lifetimes [14]. LHCb has shown
that lifetime ratio measurements involving different hadronic final states are
feasible at hadron colliders [12].
### 7.2 Direct $C\\!P$ violation
Measurements of direct $C\\!P$ violation at the LHC are linked to several
challenges. They require control of the production asymmetries present in the
proton-proton collisions. Furthermore, several sources of detection
asymmetries can arise. When measuring a $D^{0}$ decay the flavour at
production is commonly determined by the charge of the pion from a $D^{*+}$
decay which is subject to detection asymmetries. The usage of $D^{0}$ mesons
originating in semileptonic $B$ decays, where the lepton charge determines the
flavour, is another possibility and subject to similar detection asymmetries.
While some decays have pairs of oppositely charged hadrons of the same type,
for which detection asymmetries cancel provided sufficient kinematic overlap,
charged $D$ decays and some $D^{0}$ decays will have unmatched decay products
leading to inevitable detection efficiency effects.
If independent measurements of the asymmetries masking the $C\\!P$ asymmetry
are not available it is helpful to construct new observables where some
asymmetries cancel. Differences of asymmetries are a common choice as the
measured asymmetries can be expressed as sums of individual asymmetries to
first order, provided all asymmetries are small. Differences of similar final
states, e.g. two singly Cabibbo-suppressed decays, have the advantage that
potentially all unwanted asymmetries cancel, but the measured asymmetry is the
difference of two $C\\!P$ asymmetries. Access to individual asymmetries can be
obtained by using a Cabibbo-favoured decay in the difference and assuming no
$C\\!P$ violation in this decay. While this cancels the production asymmetry
it usually leaves a detection asymmetry due to the difference in the final
states. The comparison of different Cabibbo-favoured decays can give access to
detection asymmetries.
At the levels of precision anticipated for the LHCb upgrade control of these
asymmetries will be paramount for measuring time-integrated $C\\!P$
asymmetries. In addition to the challenges of measuring asymmetries to sub-
percent level precision, the non-cancellation due to second-order effects in
asymmetries will have to be taken into account. In searches for $C\\!P$
violation in the phase space of a multi-body decay production asymmetries
largely drop out as they are constant throughout the phase space. Detection
asymmetries may vary more strongly as they depend on the daughter particle
momentum which naturally varies across phase space. However, a signal for
$C\\!P$ violation in a narrow resonance should still be distinguishable from a
detection efficiency effect. The following measurements are examples for those
foreseen during the upgrade period.
Table 2 summarises several sensitivity estimates for $C\\!P$ violation
measurements. For two-body $D^{0}$ decays, the individual $C\\!P$ asymmetries
will be measured as well, however, their final precision critically depends on
the knowledge of production and detection asymmetries as described above. In
addition to the two-body and three-body $D^{+}$ decays, similar decay modes of
$D^{+}_{\mathrm{s}}$ will be analysed. For multi-body decays several types of
model-independent searches will be exploited as well as model dependent
studies.
Decay | Observable | Expected sensitivity (in $10^{-3}$)
---|---|---
$D^{0}\\!\to K^{-}K^{+}$,$D^{0}\\!\to\pi^{-}\pi^{+}$ | $\Delta A_{C\\!P}$ | $0.15$
$D^{+}\\!\to K^{0}_{\rm\scriptstyle S}K^{+}$ | $A_{C\\!P}$ | $0.1$
$D^{+}\\!\to K^{-}K^{+}\pi^{+}$ | $A_{C\\!P}$ | $0.05$
$D^{+}\\!\to\pi^{-}\pi^{+}\pi^{+}$ | $A_{C\\!P}$ | $0.08$
$D^{+}\\!\to h^{-}h^{+}\pi^{+}$ | CPV in phases | $(0.01-0.1)^{\circ}$
$D^{+}\\!\to h^{-}h^{+}\pi^{+}$ | CPV in fractions | $0.1-1.0$
Table 2: Expected statistical sensitivities for direct $C\\!P$ violation
observables for $50\mbox{\,fb}^{-1}$.
Other measurements under consideration include $C\\!P$ violation searches in
modes with neutral pions. Ongoing studies are targeted at establishing the
efficiency and purity that can be achieved in these channels. An area unique
to LHCb is the search for $C\\!P$ violation in charmed baryons which will play
an important role in the upgrade era as well.
Figure 2: Expected statistical sensitivities for $\Delta A_{C\\!P}$ and
$A_{\Gamma}$ for Belle 2 and the LHCb upgrade. The central values are fixed to
the current world average. The ellipse shows the current $1\sigma$ ellipse of
the world average. The circle marks the no $C\\!P$-violation point with a
radius of approximately $10^{-4}$.
A graphical example of the reach of the LHCb upgrade is shown in Figure 2. It
shows the comparison of the expected statistical sensitivities on $\Delta
A_{C\\!P}$ and $A_{\Gamma}$ for Belle 2 [15] and the LHCb upgrade. Belle 2
will, together with continuously improving LHCb sensitivities, make a
significant step forward in these measurements. However, in order to
positively identify indirect $C\\!P$ violation beyond the standard model level
the sensitivity of the LHCb upgrade will be required.
## 8 Conclusions
Measurements made with early LHC data have proven the feasibility of
performing charm physics at a proton-proton collider. In particular the high-
precision $C\\!P$ violation studies by LHCb show the way to a future of
probing down to standard model precision levels with the LHCb upgrade.
Significant steps in precision are already expected with the current
experiments until 2017. Atlas and LHCb will further exploit the area of
production and spectroscopy measurements.
The LHCb upgrade is mandatory to achieve the level of precision required to
distinguish effects from standard model and beyond. Through many complementary
measurements of mixing and $C\\!P$ violation observables the LHCb upgrade will
pin down the underlying theory parameters. Complementary to the area of
$C\\!P$ violation, the search for rare and forbidden decays will make a leap
forward.
## References
* [1] LHCb collaboration, “Prompt charm production in $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$.” LHCb-CONF-2010-013, Dec, 2010.
* [2] LHCb collaboration, R. Aaij et al., “Evidence for $C\\!P$ violation in time-integrated $D^{0}\\!\to h^{-}h^{+}$ decay rates,” Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938 [hep-ex].
* [3] M. Gersabeck, “Brief Review of Charm Physics,” Mod.Phys.Lett. A27 (2012) 1230026, arXiv:1207.2195 [hep-ex].
* [4] LHCb collaboration, I. Bediaga et al., “Framework TDR for the LHCb Upgrade,” Tech. Rep. CERN-LHCC-2012-007. LHCB-TDR-012, CERN, Geneva, Apr, 2012.
* [5] LHCb collaboration, R. Aaij et al., “Observation of double charm production involving open charm in pp collisions at $\sqrt{s}$=7 TeV,” JHEP 1206 (2012) 141, arXiv:1205.0975 [hep-ex].
* [6] Y.-Q. Chen and S.-Z. Wu, “Production of Triply Heavy Baryons at LHC,” JHEP 1108 (2011) 144, arXiv:1106.0193 [hep-ph].
* [7] Heavy Flavor Averaging Group, Y. Amhis et al., “Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of early 2012,” arXiv:1207.1158 [hep-ex]. Online update at http://www.slac.stanford.edu/xorg/hfag.
* [8] LHCb collaboration, “Search for the $D^{0}\to\mu^{+}\mu^{-}$ decay with $0.9\mbox{\,fb}^{-1}$ at LHCb.” LHCb-CONF-2012-005, Feb, 2012.
* [9] I. Bigi, “Could charm’s ‘third time’ be the real charm? - A manifesto,” arXiv:0902.3048 [hep-ph].
* [10] LHCb collaboration, R. Aaij et al., “Evidence for $C\\!P$ violation in time-integrated $D^{0}\rightarrow h^{-}h^{+}$ decay rates,” arXiv:1112.0938 [hep-ex].
* [11] M. Gersabeck, M. Alexander, S. Borghi, V. V. Gligorov, and C. Parkes, “On the interplay of direct and indirect $C\\!P$ violation in the charm sector,” J.Phys.G G39 (2012) 045005, arXiv:1111.6515 [hep-ex].
* [12] LHCb Collaboration, R. Aaij et al., “Measurement of mixing and CP violation parameters in two-body charm decays,” JHEP 1204 (2012) 129, arXiv:1112.4698 [hep-ex].
* [13] CLEO Collaboration, J. Libby et al., “Model-independent determination of the strong-phase difference between $D^{0}$ and $\overline{D}^{0}\to K^{0}_{S,L}h^{+}h^{-}$ ($h=\pi,K$) and its impact on the measurement of the CKM angle $\gamma/\phi_{3}$,” Phys.Rev. D82 (2010) 112006, arXiv:1010.2817 [hep-ex].
* [14] M. Bobrowski, A. Lenz, and T. Rauh, “Short distance D-Dbar mixing,” arXiv:1208.6438 [hep-ph].
* [15] A. Schwartz, “Charm Physics Program at BelleII.” these proceedings.
|
arxiv-papers
| 2012-09-26T09:09:08 |
2024-09-04T02:49:35.635574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Gersabeck (for the LHCb Collaboration)",
"submitter": "Marco Gersabeck",
"url": "https://arxiv.org/abs/1209.5878"
}
|
1209.6066
|
# Recent results about the detection of unknown boundaries and inclusions in
elastic plates ††thanks: Work supported by PRIN No. 20089PWTPS
Antonino Morassi, Edi Rosset and Sergio Vessella Dipartimento di Ingegneria
Civile e Architettura, Università degli Studi di Udine, via Cotonificio 114,
33100 Udine, Italy. E-mail: antonino.morassi@uniud.itDipartimento di
Matematica e Geoscienze, Università degli Studi di Trieste, via Valerio 12/1,
34127 Trieste, Italy. E-mail: rossedi@univ.trieste.itDipartimento di
Matematica per le Decisioni, Università degli Studi di Firenze, Via delle
Pandette 9, 50127 Firenze, Italy. E-mail: sergio.vessella@dmd.unifi.it
Abstract. In this paper we review some recent results concerning inverse
problems for thin elastic plates. The plate is assumed to be made by non-
homogeneous linearly elastic material belonging to a general class of
anisotropy. A first group of results concerns uniqueness and stability for the
determination of unknown boundaries, including the cases of cavities and rigid
inclusions. In the second group of results, we consider upper and lower
estimates of the area of unknown inclusions given in terms of the work exerted
by a couple field applied at the boundary of the plate. In particular, we
extend previous size estimates for elastic inclusions to the case of cavities
and rigid inclusions.
Mathematical Subject Classifications (2000): 35R30, 35R25, 73C02.
Key words: inverse problems, elastic plates, uniqueness, stability estimates,
size estimates, three sphere inequality, unique continuation.
## 1 Introduction
The problems we consider in the present paper belong to a more general issue
that has evolved in the last fifteen years in the field of inverse problems.
Such an issue collects the problems of determining, by a finite number of
boundary measurements, unknown boundaries and inclusions entering the boundary
value problems for partial differential equations and systems of elliptic and
parabolic type. Such problems arise in nondestructive techniques by
electrostatic measurements [Ka-Sa], [In], in thermal imaging [Vo-Mo], [Br-C],
in elasticity theory [N], [Bo-Co], [M-R3], and in many other similar
applications [Is2]. In this paper we try to enlighten the different facets of
the issue fixing our attention on the theory of thin elastic plates. In
Section 3 we give a self contained derivation of the Kirchhoff-Love plate
model on which such a theory is based.
We begin with the problem of the determination of a rigid inclusion embedded
in a thin elastic plate.
Let $\Omega$ denote the middle plane of the plate. We assume that $\Omega$ is
a bounded domain of $\mathbb{R}^{2}$ of class $C^{1,1}$. Let $h$ be its
constant thickness, $h<<\hbox{diam}(\Omega)$. The rigid inclusion $D$ is
modeled as an open simply connected domain compactly contained in $\Omega$,
with boundary of class $C^{1,1}$. The transversal displacement $w\in
H^{2}(\Omega)$ of the plate satisfies the following mixed boundary value
problem, see, for example, [Fi] and [Gu],
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w))=0,&\mathrm{in}\
\Omega\setminus\overline{D},\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w)n\cdot
n=-\widehat{M}_{n},&\mathrm{on}\ \partial\Omega,\vspace{0.25em}\\\ {\rm
div}({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\
w|_{\overline{D}}\in\mathcal{A},&\mathrm{in}\ \overline{D},\vspace{0.25em}\\\
w^{e}_{,n}=w^{i}_{,n},&\mathrm{on}\ \partial{D},\vspace{0.25em}\\\
\end{array}\right.}$
coupled with the _equilibrium conditions_ for the rigid inclusion $D$
$\int_{\partial D}\left({\rm div}({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}\right)g-(({\mathbb{P}}\nabla^{2}w)n\cdot
n)g_{,n}=0,\\\ \quad\hbox{for every }g\in\mathcal{A},$ (1.6)
where $\mathcal{A}$ denotes the space of affine functions. In the above
equations, $n$ and $\tau$ are the unit outer normal and the unit tangent
vector to $\Omega\setminus\overline{D}$, respectively, and we have defined
$w^{e}\equiv w|_{\Omega\setminus\overline{D}}$ and $w^{i}\equiv
w|_{\overline{D}}$. Moreover, $\widehat{M}_{\tau}$, $\widehat{M}_{n}$ are the
twisting and bending components of the assigned couple field $\widehat{M}$,
respectively. The plate tensor $\mathbb{P}$ is given by
$\mathbb{P}=\frac{h^{3}}{12}\mathbb{C}$, where $\mathbb{C}$ is the elasticity
tensor describing the response of the material of the plate. We assume that
$\mathbb{C}$ has cartesian components $C_{ijkl}$, $i,j,k,l=1,2$, which satisfy
the standard symmetry conditions (4.2), the regularity assumption (4.3) and
the strong convexity condition (4.4).
Given any $\widehat{M}\in H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})$,
satisfying the compatibility conditions
$\int_{\partial\Omega}\widehat{M}_{i}=0$, for $i=1,2$, problem
(LABEL:eq:1.dir-pbm-incl-rig-1)–(1.6) admits a solution $w\in H^{2}(\Omega)$,
which is uniquely determined up to addition of an affine function.
Let us denote by $\Gamma$ an open portion within $\partial\Omega$ representing
the part of the boundary where measurements are taken.
The inverse problem consists in determining $D$ from the measurement of $w$
and $w_{,n}$ on $\Gamma$. For instance, the uniqueness issue can be formulated
as follows: Given two solutions $w_{i}$ to (LABEL:eq:1.dir-pbm-incl-
rig-1)–(1.6) for $D=D_{i}$, $i=1,2$, satisfying
$w_{1}=w_{2},\hbox{ on }\Gamma,$ (1.7) $w_{1,n}=w_{2,n},\hbox{ on }\Gamma,$
(1.8)
does $D_{1}=D_{2}$ hold?
It is convenient to replace each solution $w_{i}$ introduced above with
$v_{i}=w_{i}-g_{i}$, where $g_{i}$ is the affine function which coincides with
$w_{i}$ on $\partial D_{i}$, $i=1,2$. By this approach, maintaining the same
letter to denote the solution, we rephrase the equilibrium problem
(LABEL:eq:1.dir-pbm-incl-rig-1)–(LABEL:eq:1.dir-pbm-incl-rig-5) in terms of
the following mixed boundary value problem with homogeneous Dirichlet
conditions on the boundary of the rigid inclusion
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w))=0,&\mathrm{in}\
\Omega\setminus\overline{D},\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w)n\cdot
n=-\widehat{M}_{n},&\mathrm{on}\ \partial\Omega,\vspace{0.25em}\\\ {\rm
div}({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ w=0,&\mathrm{on}\ \partial
D,\vspace{0.25em}\\\ \frac{\partial w}{\partial n}=0,&\mathrm{on}\
\partial{D},\vspace{0.25em}\\\ \end{array}\right.}$
coupled with the _equilibrium conditions_ (1.6), which has a unique solution
$w\in H^{2}(\Omega\setminus\overline{D})$. Therefore, the uniqueness question
may be rephrased as follows:
Given two solutions $w_{i}$ to (LABEL:eq:1.dir-pbm-incl-
rig-1bis)–(LABEL:eq:1-dir-pbm-incl-rig-5bis), (1.6) for $D=D_{i}$, $i=1,2$,
satisfying, for some $g\in\mathcal{A}$
$w_{1}-w_{2}=g,\quad(w_{1}-w_{2})_{,n}=g_{,n},\quad\hbox{on }\Gamma,$ (1.14)
does $D_{1}=D_{2}$ hold?
Obviously, the inverse problem above is equivalent to the determination of the
portion $\partial D$ of the boundary of $\Omega\setminus\bar{D}$ in the
boundary value problem (LABEL:eq:1.dir-pbm-incl-rig-1bis)-(LABEL:eq:1-dir-pbm-
incl-rig-5bis).
In the stability issue, instead of (1.7) and (1.8), we assume
$\min_{g\in\cal{A}}\left\\{\|w_{1}-w_{2}-g\|_{L^{2}(\Gamma)}+\left\|(w_{1}-w_{2}-g)_{,n}\right\|_{L^{2}(\Gamma)}\right\\}\leq\epsilon,$
(1.15)
for some $\epsilon>0$, and we ask for the following estimate
$d_{\cal H}(\partial D_{1},\partial D_{2})\leq\eta(\epsilon),$ (1.16)
where $\eta(\epsilon)$ is a suitable infinitesimal function.
The uniqueness for the problem above has been proved in [M-R3] under the a
priori assumption of $C^{3,1}$ regularity of $\partial D$ and with only one
nontrivial couple field. Here, by nontrivial we mean that
$(\widehat{M}_{n},\widehat{M}_{\tau,s})\not\equiv 0.$ (1.17)
Concerning the stability issue, in [M-R-Ve5] we have proved a log-log type
estimate, namely in inequality (1.16) we have
$\eta(\epsilon)=O\left((\log|\log{\epsilon}|)^{-\alpha}\right)$, where the
positive parameter $\alpha$ depends on the a priori data, see Theorem 4.3
below for a precise statement.
In [M-R-Ve2] the inverse problem of determining a cavity in an elastic plate
has been faced. We recall that in such a case conditions (1.4)-(1.5) are
replaced by homogeneous Neumann boundary conditions, which are much more
difficult to handle with respect to Dirichlet boundary conditions arising in
the case of rigid inclusions. For this reason a uniqueness result has been
established making _two_ linearly independent boundary measurements. In [M-R3]
it has also been proved a uniqueness result for a variant of the problems
considered above, that is the case of a plate whose boundary has an unknown
and inaccessible portion where $\widehat{M}=0$. In this case, thanks to the
more favorable geometric situation, one measurement suffices to detect the
unknown boundary portion. The corresponding stability results for these two
cases have not yet been proved.
The methods used to prove the above mentioned uniqueness and stability results
are based on unique continuation properties and quantitative estimates of
unique continuation for solutions to the plate equation (LABEL:eq:1.dir-pbm-
incl-rig-1). Since such properties and estimates are consequences of the three
sphere inequality for solutions to equation (LABEL:eq:1.dir-pbm-incl-rig-1),
we will discuss a while about the main features of such inequality.
The three sphere inequality for solutions to partial differential equations
and systems has a long and interesting history that intertwines with the issue
of unique continuation properties and the issue of stability estimates [Al-M],
[Ho85], [Is1], [Jo], [Lan], [Lav], [Lav-Rom-S], [L-N-W], [L-Nak-W]. In many
important cases, the three sphere inequality is the elementary tool to prove
various types of quantitative estimates of unique continuation such as, for
example, stability estimates for the Cauchy problem, smallness propagation
estimates and quantitative evaluation of the vanishing rate of solutions to
PDEs. Such questions have been intensively studied in the context of second
order equations of elliptic and parabolic type. We refer to [Al-Ro-R-Ve] and
[Ve2] where these topics are widely investigated for such types of equations.
The three sphere inequality for equation (LABEL:eq:1.dir-pbm-incl-rig-1) has
been proved in [M-R-Ve5] under the very general assumption that the elastic
material of the plate is anisotropic and obeys the so called dichotomy
condition. Roughly speaking, such a condition implies that the plate operator
at the left hand side of (LABEL:eq:1.dir-pbm-incl-rig-1) can be written as
$L_{2}L_{1}+Q$, where $L_{2}$, $L_{1}$ are second order elliptic operators
with $C^{1,1}$ coefficients and $Q$ is a third order operator with bounded
coefficients. For more details we refer to (4.5a)-(4.5b) below and [M-R-Ve5].
A simplified version of such inequality is the following one
$\int_{B_{r_{2}}(x_{0})}|\nabla^{2}w|^{2}\leq
C\left(\int_{B_{r_{1}}(x_{0})}|\nabla^{2}w|^{2}\right)^{\delta}\left(\int_{B_{r_{3}}(x_{0})}|\nabla^{2}w|^{2}\right)^{1-\delta},$
(1.18)
for every $r_{1}<r_{2}<r_{3}$, where $\delta\in(0,1)$ and $C$ depend only on
the parameters related to the regularity, ellipticity and dichotomy conditions
assumed on $\mathbb{C}$, and on the ratios $r_{1}/r_{2}$, $r_{2}/r_{3}$; in
particular, $\delta$ and $C$ do not depend on $w$.
Previously, the three sphere inequality was proved in [M-R-Ve1], for the
isotropic plate (that is
$C_{ijkl}(x)=\delta_{ij}\delta_{kl}\lambda(x)+\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)\mu(x),\quad\hbox{}{i,j,k,l=1,2}$)
and in [Ge] for the class of fourth (and higher) order elliptic equation
$\mathcal{L}u=0$ where $\mathcal{L}=L_{2}L_{1}$ and $L_{2}$, $L_{1}$ are
second order elliptic equation with $C^{1,1}$ coefficients.
The proof of the stability result, of which we give a sketch in Subsection
4.2, has essentially the same structure of the proofs of analogous stability
results in the following context:
a) Second order elliptic equations: [Al-Ro], [Be-Ve], [Ro](two variables
elliptic equations); [Al-B-R-Ve1],[Al-B-R-Ve2], [Si] (several variables
elliptic equations)
b) Second order parabolic equation: [C-R-Ve1], [C-R-Ve2], [Dc-Ro-Ve], [B-Dc-
Si-Ve], [Ve1], [Ve2]
c) Elliptic systems: [M-R3], [M-R4] (elasticity); [Ba] (Stokes fluid)
It is important to say that the stability estimates proved in the papers of
list a) and b) are of logarithmic type, that is an optimal rate of
convergence, as shown by counterexamples ([Al1], [Dc-Ro] for case a) and [Dc-
Ro-Ve], [Ve2] for case b)). The stability estimates proved in the papers of
list c) and in the case of plate equation are of log-log type, that is with a
worse rate of convergence. It seems difficult to improve such an estimate. We
believe that the main difficulty to get such an improvement is due to the lack
of quantitative estimates of strong unique continuation property at the
boundary. In order to give an idea of the crucial point which marks the
difference between these cases, let us notice that, by iterated application of
the three sphere inequality (1.18) it can be proved there exists
$\bar{\rho}>0$ such that for every $\rho\in(0,\bar{\rho})$ and every
$\bar{x}\in\partial{D_{j}}$, $j=1,2$, the following inequality holds true
$\int_{B_{\rho}(\bar{x})\cap(\Omega\setminus\overline{D_{j}})}|\nabla^{2}w_{j}|^{2}\geq
C\exp\left(-A{\rho}^{-B}\right),$ (1.19)
where $A>0$, $B>0$ and $C>0$ only depend on the a priori information, in
particular they depend by the quantity (frequency)
$\frac{\|\widehat{M}\|_{L^{2}(\partial\Omega,\mathbb{R}^{2})}}{\|\widehat{M}\|_{H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})}}.$
(1.20)
In cases a) and b), it is possible to prove a refined form of inequality
(1.19), in which the exponential term is replaced with a positive power of
$\rho$, obtaining a quantitative estimate of strong unique continuation
property at the boundary. It has been shown in [Al-B-R-Ve1] that this is a key
ingredient in proving that the stability estimate for the corresponding
inverse problem with unknown boundaries in the conductivity context is not
worse than logarithm. This mathematical tool is available for second order
elliptic, [A-E], and parabolic equations, [Es-Fe-Ve], but is not currently
available for elliptic systems and plate equation. This happens even in the
simplest case of isotropic material, and this is the reason for the presence
of a double logarithm in our stability estimate. Finally, as remarked in [M-R-
Ve4], it seems hopeless the possibility that solutions to (LABEL:eq:1.dir-pbm-
incl-rig-1) can satisfy even a strong unique continuation property in the
interior, without any a priori assumption on the anisotropy of the material,
see also [Ali]. Regarding this point, our dichotomy condition (4.5a)-(4.5b)
basically contains the same assumptions under which the unique continuation
property holds for a fourth order elliptic equation in two variables.
In the present paper we have also proved constructive upper and lower
estimates of the area of a rigid inclusion or of a cavity, $D$, in terms of an
easily expressed quantity related to work. More precisely, suppose we make the
following diagnostic test. We take a reference plate, i.e. a plate without
inclusion or cavity, and we deform it by applying a couple field $\widehat{M}$
at the boundary $\partial\Omega$. Let $W_{0}$ be the work exerted in deforming
the specimen. Next, we repeat the same experiment on a possibly defective
plate. The exerted work generally changes and assumes, say, the value $W$. We
are interested in finding constructive estimates, from above and from below,
of the area of $D$ in terms of the difference $|W-W_{0}|$. In order to prove
such estimates we proceed along the path outlined in [M-R-Ve1] and [M-R-Ve6]
in which the inclusion inside the plate is made by different elastic material.
In this introduction we illustrate such _intermediate_ case, since the scheme
of the mathematical procedure is fairly simple to describe. With regard to
this intermediate case we also want to stress that, in contrast to the extreme
cases, there are not available any kind of uniqueness result for the inverse
problem of determining inclusion $D$ from the knowledge of a finite number of
measurements on the boundary . This appears to be an extremely difficult
problem. In fact, despite the wide research developed in this field, a general
uniqueness result has not been obtained yet even in the simpler context
arising in electrical impedance tomography (which involves a second order
elliptic equation), see, for instance, [Is1] and [Al1] for an extensive
reference list.
Denoting, as above, by $w$ the transversal displacement of the plate and by
$\widehat{M}_{\tau}$, $\widehat{M}_{n}$ the twisting and bending components of
the assigned couple field $\widehat{M}$, respectively, the infinitesimal
deformation of the defective plate is governed by the fourth order Neumann
boundary value problem
${\displaystyle\left\\{\begin{array}[]{lr}\textrm{div}\,(\textrm{div}\,((\chi_{\Omega\setminus{D}}{\mathbb{P}}+\chi_{D}\widetilde{\mathbb{P}})\nabla^{2}w))=0,&\mathrm{in}\
\Omega,\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w)n\cdot
n=-\widehat{M}_{n},&\mathrm{on}\ \partial\Omega,\vspace{0.25em}\\\
\textrm{div}\,({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega.\vspace{0.25em}\\\ \end{array}\right.}$
In the above equations, $\chi_{D}$ denotes the characteristic function of $D$.
The plate tensors $\mathbb{P}$, $\widetilde{\mathbb{P}}$ are given by
$\mathbb{P}=\frac{h^{3}}{12}\mathbb{C},\quad\widetilde{\mathbb{P}}=\frac{h^{3}}{12}\widetilde{\mathbb{C}},$
(1.24)
where $\mathbb{C}$ is the elasticity tensor describing the response of the
material in the reference plate $\Omega$ and satisfies the usual symmetry
conditions (4.2), regularity condition (4.3), strong convexity condition (4.4)
and the _dichotomy condition_ , whereas $\widetilde{\mathbb{C}}$ denotes the
(unknown) corresponding tensor for the inclusion $D$.
The work exerted by the couple field $\widehat{M}$ has the expression
$W=-\int_{\partial\Omega}\widehat{M}_{\tau,s}w+\widehat{M}_{n}w,_{n}.$ (1.25)
When the inclusion $D$ is absent, the equilibrium problem
(LABEL:eq:intr.equation_with_D)-(LABEL:eq:intr.bc2_with_D) becomes
${\displaystyle\left\\{\begin{array}[]{lr}\textrm{div}\,(\textrm{div}\,({\mathbb{P}}\nabla^{2}w_{0}))=0,&\mathrm{in}\
\Omega,\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w_{0})n\cdot
n=-\widehat{M}_{n},&\mathrm{on}\ \partial\Omega,\vspace{0.25em}\\\
\textrm{div}\,({\mathbb{P}}\nabla^{2}w_{0})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{0})n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ \end{array}\right.}$
where $w_{0}$ is the transversal displacement of the reference plate. The
corresponding work exerted by $\widehat{M}$ is given by
$W_{0}=-\int_{\partial\Omega}\widehat{M}_{\tau,s}w_{0}+\widehat{M}_{n}w_{0,n}.$
(1.29)
In [M-R-Ve6] the following result has been proved. Assuming that the following
fatness-condition is satisfied
$\mathrm{area}\left(\\{x\in D|\ dist\\{x,\partial
D\\}>h_{1}\\}\right)\geq\frac{1}{2}\mathrm{area}(D),$ (1.30)
where $h_{1}$ is a given positive number, then
$C_{1}\left|\frac{W-W_{0}}{W_{0}}\right|\leq\mathrm{area}(D)\leq
C_{2}\left|\frac{W-W_{0}}{W_{0}}\right|,$ (1.31)
where the constants $C_{1}$, $C_{2}$ only depend on the a priori data. Besides
the assumptions on the plate tensor $\mathbb{C}$ given above, estimates (1.31)
are established under some suitable assumption on the jump
$\widetilde{\mathbb{C}}-\mathbb{C}$.
In the _extreme_ cases corresponding to a rigid inclusion or a cavity $D$, the
estimates are a little more involute than (1.31) and additional regularity
conditions on the boundary $\partial D$ and on the plate tensor $\mathbb{P}$
are generally required. The main difference between _extreme_ and
_intermediate_ cases lies in the estimate from below of $|D|$. Indeed, in the
former case we use regularity estimates (in the interior) for the reference
solution $w_{0}$ to equation (LABEL:eq:4.equation_without_D), whereas in the
latter we combine regularity estimates, trace and Poincaré inequalities. On
the other hand, the argument used for the estimate from above of $|D|$ is
essentially the same as in [M-R-Ve6] and involves quantitative estimates of
unique continuation in the form of three sphere inequality for the hessian
$\nabla^{2}w_{0}$. It is exactly at this point that the dichotomy condition
(4.5a)–(4.5b) on the tensor $\mathbb{C}$ is needed.
The analogous bounds in plate theory for _intermediate_ inclusions were first
obtained when the reference plate satisfies isotropic conditions, [M-R-Ve1],
extended to anisotropic materials satisfying the dichotomy conditions, [M-R-
Ve6], obtained in a weaker form for general inclusions in absence of the
fatness condition, [M-R-Ve3], and recently in the context of shallow shells in
[Dc-Li-Wa] and [Dc-Li-Ve-Wa]. The reader is referred to [K-S-Sh], [Al-R], [Al-
R-S], [Be-Fr-Ve] for size estimates of inclusions in the context of the
electrical impedance tomography and to [Ik], [Al-M-R2], [Al-M-R3], [Al-M-R-V]
for corresponding problems in two and three-dimensional linear elasticity. See
also [L-Dc-N] for an application of the size estimates approach in
thermography.
Size estimates for _extreme_ inclusions were obtained in [Al-M-R1] for
electric conductors and in [M-R1] for elastic bodies, see also [Al-M-R2].
The paper is organized as follows. In Section 2 we collect some notation. In
Section 3 we provide a derivation of the Kirchhoff-Love model of the plate. In
Section 4 we present the uniqueness and stability results concerning the
determination of rigid inclusions, and the uniqueness results for the case of
cavities and unknown boundary portions. In particular we have focused our
attention on the case of rigid inclusions and, for a better comprehension of
the arguments, we have recalled the proof of the uniqueness result, using it
as a base for a sketch of the more complex proof of the stability result.
Section 5 contains the estimates of the area of _extreme_ inclusions.
## 2 Notation
Let $P=(x_{1}(P),x_{2}(P))$ be a point of $\mathbb{R}^{2}$. We shall denote by
$B_{r}(P)$ the disk in $\mathbb{R}^{2}$ of radius $r$ and center $P$ and by
$R_{a,b}(P)$ the rectangle $R_{a,b}(P)=\\{x=(x_{1},x_{2})\ |\
|x_{1}-x_{1}(P)|<a,\ |x_{2}-x_{2}(P)|<b\\}$. To simplify the notation, we
shall denote $B_{r}=B_{r}(O)$, $R_{a,b}=R_{a,b}(O)$.
###### Definition 2.1.
(${C}^{k,1}$ regularity) Let $\Omega$ be a bounded domain in
${\mathbb{R}}^{2}$. Given $k\in\mathbb{N}$, we say that a portion $S$ of
$\partial\Omega$ is of class ${C}^{k,1}$ with constants $\rho_{0}$, $M_{0}>0$,
if, for any $P\in S$, there exists a rigid transformation of coordinates under
which we have $P=0$ and
$\Omega\cap R_{\frac{\rho_{0}}{M_{0}},\rho_{0}}=\\{x=(x_{1},x_{2})\in
R_{\frac{\rho_{0}}{M_{0}},\rho_{0}}\quad|\quad x_{2}>\psi(x_{1})\\},$
where $\psi$ is a ${C}^{k,1}$ function on
$\left(-\frac{\rho_{0}}{M_{0}},\frac{\rho_{0}}{M_{0}}\right)$ satisfying
$\psi(0)=0,\quad\psi^{\prime}(0)=0,\quad\hbox{when }k\geq 1,$
$\|\psi\|_{{C}^{k,1}\left(-\frac{\rho_{0}}{M_{0}},\frac{\rho_{0}}{M_{0}}\right)}\leq
M_{0}\rho_{0}.$
When $k=0$ we also say that $S$ is of Lipschitz class with constants
$\rho_{0}$, $M_{0}$.
###### Remark 2.2.
We use the convention to normalize all norms in such a way that their terms
are dimensionally homogeneous with the $L^{\infty}$ norm and coincide with the
standard definition when the dimensional parameter equals one, see [M-R-Ve5]
for details.
Given a bounded domain $\Omega$ in $\mathbb{R}^{2}$ such that $\partial\Omega$
is of class $C^{k,1}$, with $k\geq 1$, we consider as positive the orientation
of the boundary induced by the outer unit normal $n$ in the following sense.
Given a point $P\in\partial\Omega$, let us denote by $\tau=\tau(P)$ the unit
tangent at the boundary in $P$ obtained by applying to $n$ a counterclockwise
rotation of angle $\frac{\pi}{2}$, that is $\tau=e_{3}\times n$, where
$\times$ denotes the vector product in $\mathbb{R}^{3}$, $\\{e_{1},e_{2}\\}$
is the canonical basis in $\mathbb{R}^{2}$ and $e_{3}=e_{1}\times e_{2}$.
Given any connected component $\cal C$ of $\partial\Omega$ and fixed a point
$P\in\cal C$, let us define as positive the orientation of $\cal C$ associated
to an arclength parametrization $\varphi(s)=(x_{1}(s),x_{2}(s))$,
$s\in[0,l(\cal C)]$, such that $\varphi(0)=P$ and
$\varphi^{\prime}(s)=\tau(\varphi(s))$, where $l(\cal C)$ denotes the length
of $\cal C$.
Throughout the paper, we denote by $\partial_{i}u$, $\partial_{s}u$, and
$\partial_{n}u$ the derivatives of a function $u$ with respect to the $x_{i}$
variable, to the arclength $s$ and to the normal direction $n$, respectively,
and similarly for higher order derivatives.
We denote by $\mathbb{M}^{2}$ the space of $2\times 2$ real valued matrices
and by ${\mathcal{L}}(X,Y)$ the space of bounded linear operators between
Banach spaces $X$ and $Y$.
For every $2\times 2$ matrices $A$, $B$ and for every
$\mathbb{L}\in{\mathcal{L}}({\mathbb{M}}^{2},{\mathbb{M}}^{2})$, we use the
following notation:
$({\mathbb{L}}A)_{ij}=L_{ijkl}A_{kl},$ (2.1) $A\cdot
B=A_{ij}B_{ij},\quad|A|=(A\cdot A)^{\frac{1}{2}}.$ (2.2)
Notice that here and in the sequel summation over repeated indexes is implied.
Finally, let us introduce the linear space of the affine functions on
$\mathbb{R}^{2}$
$\mathcal{A}=\\{g(x_{1},x_{2})=ax_{1}+bx_{2}+c,\ a,b,c\in\mathbb{R}\\}.$
## 3 The Kirchhoff-Love plate model
In the last two decades different methods were used to provide new
justification of the theory of thin plates. Among these, we recall the method
of asymptotic expansion [C-D], the method of internal constraints [PG],
[L-PG], the theory of $\Gamma$-convergence in conjunction with appropriate
averages [A-B-P] or on rescaled domain and with rescaled displacements [B-C-
G-R], [Pa1], and weak convergence methods on a rescaled domain and with
rescaled displacements [C]. We refer the interested reader to [Pa2] for a
recent account of the advanced results on this topic. The present section has
a more modest aim: to show how to deduce the equations governing the statical
equilibrium of an elastic thin plate following the classical approach of the
Theory of Structures.
Let us consider a thin plate
$\Omega\times\left[-\frac{h}{2},\frac{h}{2}\right]$ with middle surface
represented by a bounded domain $\Omega$ in $\mathbb{R}^{2}$ having uniform
thickness $h$, $h<<$diam$(\Omega)$, and boundary $\partial\Omega$ of class
$C^{1,1}$. Only in this section, we adopt the convention that Greek indexes
assume the values $1,2$, whereas Latin indexes run from $1$ to $3$.
We follow the direct approach to define the infinitesimal deformation of the
plate. In particular, we restrict ourselves to the case in which the points
$x=(x_{1},x_{2})$ of the middle surface $\Omega$ are subject to transversal
displacement $w(x_{1},x_{2})e_{3}$, and any transversal material fiber
$\\{x\\}\times\left[-\frac{h}{2},\frac{h}{2}\right]$, $x\in\Omega$, undergoes
an infinitesimal rigid rotation $\omega(x)$, with $\omega(x)\cdot e_{3}=0$. In
this section we shall be concerned exclusively with regular functions on their
domain of definition. For example, the above functions $w$ and $\omega$ are
such that $w\in C^{\infty}(\overline{\Omega},\mathbb{R})$ and $\omega\in
C^{\infty}(\overline{\Omega},\mathbb{R}^{3})$. These conditions are
unnecessarily restrictive, but this choice simplifies the mechanical
formulation of the equilibrium problem. The above kinematical assumptions
imply that the displacement field present in the plate is given by the
following three-dimensional vector field:
$u(x,x_{3})=w(x)e_{3}+x_{3}\varphi(x),\quad x\in\overline{\Omega},\
|x_{3}|\leq\frac{h}{2},$ (3.1)
where
$\varphi(x)=\omega(x)\times e_{3},\quad x\in\overline{\Omega}.$ (3.2)
By (3.1) and (3.2), the associated infinitesimal strain tensor
$E[u]\in\mathbb{M}^{3}$ takes the form
$E[u](x,x_{3})\equiv(\nabla
u)^{sym}(x,x_{3})=x_{3}(\nabla_{x}\varphi(x))^{sym}+(\gamma(x)\otimes
e_{3})^{sym},$ (3.3)
where $\nabla_{x}(\cdot)=\frac{\partial}{\partial
x_{\alpha}}(\cdot)e_{\alpha}$ is the surface gradient operator,
$\nabla^{sym}(\cdot)=\frac{1}{2}(\nabla(\cdot)+\nabla^{T}(\cdot))$, and
$\gamma(x)=\varphi(x)+\nabla_{x}w(x).$ (3.4)
Within the approximation of the theory of infinitesimal deformations, $\gamma$
is the angular deviation between the transversal material fiber at $x$ and the
normal direction to the deformed middle surface of the plate at $x$. In
Kirchhoff-Love theory it is assumed that every transversal material fiber
remains normal to the deformed middle surface, e.g. $\gamma=0$ in $\Omega$.
The traditional deduction of the mechanical model of a thin plate follows
essentially from integration over the thickness of the corresponding three-
dimensional quantities. In particular, taking advantage of the infinitesimal
deformation assumption, we can refer the independent variables to the initial
undeformed configuration of the plate.
Let us introduce an arbitrary portion
$\Omega^{\prime}\times\left[-\frac{h}{2},\frac{h}{2}\right]$ of plate, where
$\Omega^{\prime}\subset\subset\Omega$ is a subdomain of $\Omega$ with regular
boundary. Consider the material fiber
$\\{x\\}\times\left[-\frac{h}{2},\frac{h}{2}\right]$ for
$x\in\partial\Omega^{\prime}$ and denote by
$t(x,x_{3},e_{\alpha})\in\mathbb{R}^{3}$, $|x_{3}|\leq\frac{h}{2}$, the
traction vector acting on a plane containing the direction of the fiber and
orthogonal to the direction $e_{\alpha}$. By Cauchy’s Lemma [T] we have
$t(x,x_{3},e_{\alpha})=T(x,x_{3})e_{\alpha}$, where
$T(x,x_{3})\in\mathbb{M}^{3}$ is the (symmetric) Cauchy stress tensor at the
point $(x,x_{3})$. Denote by $n$ the unit outer normal vector to
$\partial\Omega^{\prime}$ such that $n\cdot e_{3}=0$. To simplify the
notation, it is convenient to consider $n$ as a two-dimensional vector
belonging to the plane $x_{3}=0$ containing the middle surface $\Omega$ of the
plate. By the classical Stress Principle for plates [Vi], we postulate that
the two complementary parts $\Omega^{\prime}$ and
$\Omega\setminus\Omega^{\prime}$ interact with one another through a filed of
force vectors $R=R(x,n)\in\mathbb{R}^{3}$ and couple vectors
$M=M(x,n)\in\mathbb{R}^{3}$ assigned per unit length at
$x\in\partial\Omega^{\prime}$. Denoting by
$R(x,e_{\alpha})=\int_{-h/2}^{h/2}t(x,x_{3},e_{\alpha})dx_{3}$ (3.5)
the force vector (per unit length) acting on a direction orthogonal to
$e_{\alpha}$ and passing through $x\in\partial\Omega^{\prime}$, the contact
force $R(x,n)$ can be expressed as
$R(x,n)=T^{\Omega}(x)n,\quad x\in\partial\Omega^{\prime},$ (3.6)
where the surface force tensor $T^{\Omega}(x)\in\mathbb{M}^{3\times 2}$ is
given by
$T^{\Omega}(x)=R(x,e_{\alpha})\otimes e_{\alpha},\quad\hbox{in }\Omega.$ (3.7)
Let $P=I-e_{3}\otimes e_{3}$ be the projection of $\mathbb{R}^{3}$ along the
direction $e_{3}$. $T^{\Omega}$ is decomposed additively by $P$ in its
membranal and shearing component
$T^{\Omega}=PT^{\Omega}+(I-P)T^{\Omega}\equiv T^{\Omega(m)}+T^{\Omega(s)},$
(3.8)
where, following the standard nomenclature in plate theory, the components
$T_{\alpha\beta}^{\Omega(m)}$ ($=T_{\beta\alpha}^{\Omega(m)}$),
$\alpha,\beta=1,2$, are called the membrane forces and the components
$T_{3\beta}^{\Omega(s)}$, $\beta=1,2$, are the shear forces (also denoted as
$T_{3\beta}^{\Omega(s)}=Q_{\beta}$). The assumption of infinitesimal
deformations and the hypothesis of vanishing in-plane displacements of the
middle surface of the plate allow us to take
$T^{\Omega(m)}=0,\quad\hbox{in }\Omega.$ (3.9)
Denote by
$M(x,e_{\alpha})=\int_{-h/2}^{h/2}x_{3}e_{3}\times
t(x,x_{3},e_{\alpha})dx_{3},\quad\alpha=1,2,$ (3.10)
the contact couple acting at $x\in\partial\Omega^{\prime}$ on a direction
orthogonal to $e_{\alpha}$ passing through $x$. Note that
$M(x,e_{\alpha})\cdot e_{3}=0$ by definition, that is $M(x,e_{\alpha})$
actually is a two-dimensional couple field belonging to the middle plane of
the plate. Analogously to (3.6), we have
$M(x,n)=M^{\Omega}(x)n,\quad x\in\partial\Omega^{\prime},$ (3.11)
where the surface couple tensor $M^{\Omega}(x)\in\mathbb{M}^{2\times 2}$ has
the expression
$M^{\Omega}(x)=M(x,e_{\alpha})\otimes e_{\alpha}.$ (3.12)
A direct calculation shows that
$M(x,e_{\alpha})=e_{3}\times e_{\beta}M_{\beta\alpha}(x),$ (3.13)
where
$M_{\beta\alpha}(x)=\int_{-h/2}^{h/2}x_{3}T_{\beta\alpha}(x,x_{3})dx_{3},\quad\alpha,\beta=1,2,$
(3.14)
are the bending moments (for $\alpha=\beta$) and the twisting moments (for
$\alpha\neq\beta$) of the plate at $x$ (per unit length).
The differential equilibrium equation for the plate follows from the integral
mechanical balance equations applied to any subdomain
$\Omega^{\prime}\subset\subset\Omega$ [T]. Denote by $q(x)e_{3}$ the external
transversal force per unit area acting in $\Omega$. The statical equilibrium
of the plate is satisfied if and only if the following two equations are
simultaneously satisfied:
${\displaystyle\left\\{\begin{array}[]{lr}\int_{\partial\Omega^{\prime}}T^{\Omega}nds+\int_{\Omega^{\prime}}qe_{3}dx=0,\vspace{0.25em}\\\
\int_{\partial\Omega^{\prime}}\left((x-x_{0})\times
T^{\Omega}n+M^{\Omega}n\right)ds+\int_{\Omega^{\prime}}(x-x_{0})\times
qe_{3}dx=0,\vspace{0.25em}\\\ \end{array}\right.}$
for every subdomain $\Omega^{\prime}\subseteq\Omega$, where $x_{0}$ is a fixed
point. By applying the Divergence Theorem in $\Omega^{\prime}$ and by the
arbitrariness of $\Omega^{\prime}$ we deduce
${\displaystyle\left\\{\begin{array}[]{lr}{\rm
div}_{x}T^{\Omega(s)}+qe_{3}=0,&\mathrm{in}\ \Omega,\vspace{0.25em}\\\ {\rm
div}_{x}M^{\Omega}+(T^{\Omega(s)})^{T}e_{3}\times e_{3}=0,&\mathrm{in}\
\Omega.\vspace{0.25em}\\\ \end{array}\right.}$
Consider the case in which the boundary of the plate $\partial\Omega$ is
subjected simultaneously to a couple field $\widehat{M}$, $\widehat{M}\cdot
e_{3}=0$, and a transversal force field $\widehat{Q}e_{3}$. Local equilibrium
considerations on points of $\partial\Omega$ yield the following boundary
conditions:
${\displaystyle\left\\{\begin{array}[]{lr}M^{\Omega}n=\widehat{M},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\
T^{\Omega(s)}n=\widehat{Q}e_{3},&\mathrm{on}\
\partial\Omega.\vspace{0.25em}\\\ \end{array}\right.}$
where $n$ is the unit outer normal to $\partial\Omega$. In cartesian
components, the equilibrium equations (LABEL:eq:anto-7.3)–(LABEL:eq:anto-7.7)
take the form
${\displaystyle\left\\{\begin{array}[]{lr}M_{\alpha\beta,\beta}-Q_{\alpha}=0,&\mathrm{in}\
\Omega,\mathrm{\alpha=1,2},\vspace{0.25em}\\\
Q_{\alpha,\alpha}+q=0,&\mathrm{in}\ \Omega,\vspace{0.25em}\\\
M_{\alpha\beta}n_{\alpha}n_{\beta}=\widehat{M}_{n},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\
M_{\alpha\beta}\tau_{\alpha}n_{\beta}=-\widehat{M}_{\tau},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\
Q_{\alpha}n_{\alpha}=\widehat{Q},&\mathrm{on}\
\partial\Omega.\vspace{0.25em}\\\ \end{array}\right.}$
Here, following a standard convention in the theory of plates, we have
decomposed the boundary couple field $\widehat{M}$ in local coordinates as
$\widehat{M}=\widehat{M}_{\tau}n+\widehat{M}_{n}\tau$.
To complete the formulation of the equilibrium problem, we need to introduce
the constitutive equation of the material. We limit ourselves to the
Kirchhoff-Love theory and we choose to regard the kinematical assumptions
$E_{i3}[u]=0$, $i=1,2,3$ (see (3.3), with $\gamma=0$) as internal constraints,
that is we restrict the possible deformations of the points of the plate to
those whose infinitesimal strain tensor belongs to the set
${\cal{M}}=\\{E\in\mathbb{M}^{3\times 3}|E=E^{T},E\cdot A=0,\hbox{ for
}A=e_{i}\otimes e_{3}+e_{3}\otimes e_{i},\ i=1,2,3\\}.$ (3.26)
Then, by the Generalized Principle of Determinism [T], the Cauchy stress
tensor $T$ at any point $(x,x_{3})$ of the plate is additively decomposed in
an active (symmetric) part $T_{A}$ and in a reactive (symmetric) part $T_{R}$:
$T=T_{A}+T_{R},$ (3.27)
where $T_{R}$ does not work in any admissible motion, e.g.,
$T_{R}\in{\cal{M}}^{\perp}$. Consistently with the Principle, the active
stress $T_{A}$ belongs to ${\cal{M}}$ and, in cartesian coordinates, we have
$T_{A}=T_{A\alpha\beta}e_{\alpha}\otimes e_{\beta},\quad\alpha,\beta=1,2,$
(3.28) $T_{R}=T_{R\alpha 3}e_{\alpha}\otimes e_{3}+T_{R3\alpha}e_{3}\otimes
e_{\alpha}+T_{R33}e_{3}\otimes e_{3}.$ (3.29)
In linear theory, on assuming the reference configuration unstressed, the
active stress in a point $(x,x_{3})$ of the plate, $x\in\overline{\Omega}$ and
$|x_{3}|\leq h/2$, is given by a linear mapping from ${\cal{M}}$ into itself
by means of the fourth order elasticity tensor $\mathbb{C}_{\cal{M}}$:
$T_{A}=\mathbb{C}_{\cal{M}}E[u].$ (3.30)
Here, in view of (3.26) and (3.28), $\mathbb{C}_{\cal{M}}$ can be assumed to
belong to ${\mathcal{L}}({\mathbb{M}}^{2},{\mathbb{M}}^{2})$. Moreover, we
assume that $\mathbb{C}_{\cal{M}}$ is constant over the thickness of the plate
and satisfies the minor and major symmetry conditions expressed in cartesian
coordinates as (we drop the subscript ${\cal{M}}$)
$C_{\alpha\beta\gamma\delta}=C_{\beta\alpha\gamma\delta}=C_{\alpha\beta\delta\gamma}=C_{\gamma\delta\alpha\beta},\quad\alpha,\beta,\gamma,\delta=1,2,\quad\hbox{in
}\Omega.$ (3.31)
We refer to [PG] and [L-PG] for a representation formula of $\mathbb{C}$ based
on the maximal response symmetry of the material compatible with the internal
constraints.
Using (3.27) and recalling (3.9), we obtain the corresponding decomposition
for $T^{\Omega}$ and $M^{\Omega}$:
$T^{\Omega}=T^{\Omega(s)}_{R},\quad M^{\Omega}=M^{\Omega}_{A},$ (3.32)
that is the shear forces and the moments have reactive and active nature,
respectively. By (3.30), after integration over the thickness, the surface
couple tensor is given by
$M^{\Omega}(x)=-\frac{h^{3}}{12}{\cal{E}}\mathbb{C}(x)(\nabla_{x}^{2}w(x)),\quad\hbox{in
}\Omega,$ (3.33)
where ${\cal{E}}\in\mathbb{M}^{2}$ has cartesian components
${\cal{E}}_{11}={\cal{E}}_{11}=0$, ${\cal{E}}_{12}=-1$, ${\cal{E}}_{21}=1$.
Constitutive equation (3.33) can be written in more expressive way in terms of
the bending and twisting moments as follows:
$M_{\alpha\beta}(w)=-P_{\alpha\beta\gamma\delta}(x)w,_{\gamma\delta},\quad\alpha,\beta=1,2,$
(3.34)
where
$\mathbb{P}(x)=\frac{h^{3}}{12}\mathbb{C}(x),\quad\hbox{in }\Omega,$ (3.35)
is the plate elasticity tensor. Combining (LABEL:eq:anto-7.3) and
(LABEL:eq:anto-7.4), and by eliminating the reactive term $T^{\Omega(s)}_{R}$,
we obtain the classical partial differential equation of the Kirchhoff-Love’s
bending theory of thin elastic plates, that, written in cartesian coordinates,
takes the form
$(P_{\alpha\beta\gamma\delta}(x)w,_{\gamma\delta}(x)),_{\alpha\beta}=q,\quad\hbox{in
}\Omega.$ (3.36)
In the remaining part of this section we complete the formulation of the
equilibrium problem for a Kirchhoff-Love plate by writing the boundary
conditions corresponding to (LABEL:eq:anto-8.3)–(LABEL:eq:anto-8.5). The
determination of these boundary conditions is not a trivial issue because,
first, a constitutive equation for shear forces is not available since these
have a reactive nature (see (3.32)), and, second, because the three mechanical
boundary conditions (LABEL:eq:anto-8.3)–(LABEL:eq:anto-8.5) should reasonably
”collapse” into two independent boundary conditions for the fourth order
partial differential equation (3.36). To this aim, under the additional
assumption of $\mathbb{C}$ positive definite, we adopt a variational approach
and we impose the stationarity condition on the total potential energy
functional $J$ of the plate. Consider the space of regular kinematically
admissible displacements
${\cal{D}}=\\{v:\Omega\times(-h/2,h/2)\rightarrow\mathbb{R}^{3}|\
v(x,x_{3})=\eta(x)e_{3}-x_{3}\nabla_{x}\eta(x),\hbox{ with
}\eta:\Omega\rightarrow\mathbb{R}\\}.$ (3.37)
The energy functional $J:{\cal{D}}\rightarrow\mathbb{R}$ is defined as
$J(v)=a(v,v)-l(v),$ (3.38)
where $a(v,v)$ is interpreted as the elastic energy stored in the plate for
the displacement field $v$ and $l(v)$ is the load potential that accounts for
the energy of the system of applied loads $q$, $\widehat{M}$, $\widehat{Q}$.
The three-dimensional expression of the elastic energy in the Linear Theory of
Elasticity is given by
$a(v,v)=\frac{1}{2}\int_{-h/2}^{h/2}\int_{\Omega}\mathbb{C}E[v]\cdot
E[v]dxdx_{3},$ (3.39)
where, in view of (3.37), $E[v]=-x_{3}\nabla_{x}^{2}\eta(x)$. After
integration over the thickness, we obtain
$a(v,v)=-\frac{1}{2}\int_{\Omega}M_{\alpha\beta}(\eta)\eta,_{\alpha\beta}dx,$
(3.40)
where $M_{\alpha\beta}(\eta)$ are as in (3.34) with $w$ replaced by $\eta$.
The load functional has the expression
$l(v)=\int_{\Omega}q\eta
dx+\int_{\partial\Omega}(\widehat{Q}\eta+\widehat{M}_{2}\eta,_{2}-\widehat{M}_{1}\eta,_{1})ds.$
(3.41)
Then, the stationarity condition (in fact, minimum condition) on $J$ at $w$
yields
$\int_{\Omega}M_{\alpha\beta}(w)\eta,_{\alpha\beta}dx+\int_{\Omega}q\eta+\int_{\partial\Omega}(\widehat{Q}\eta+\widehat{M}_{2}\eta,_{2}-\widehat{M}_{1}\eta,_{1})ds=0,$
(3.42)
for every regular function $\eta$. Integrating by parts twice on the first
integral we obtain
$\int_{\Omega}(M_{\alpha\beta,\alpha\beta}(w)+q)\eta
dx+\int_{\partial\Omega}(-M_{\alpha\beta,\beta}(w)n_{\alpha}+\widehat{Q})\eta
ds+\\\
+\int_{\partial\Omega}(M_{\alpha\beta}(w)n_{\beta}\eta_{\alpha}+\widehat{M}_{2}\eta,_{2}-\widehat{M}_{1}\eta,_{1})ds=0.$
(3.43)
We elaborate the last integral $I_{1}$ of (3.43) by rewriting the first order
derivatives of $\eta$ on $\partial\Omega$ in terms of the normal and arc-
length derivative of $\eta$. We have
$I_{1}=\int_{\partial\Omega}(M_{\alpha\beta}(w)n_{\beta}n_{\alpha}-\widehat{M}_{1}n_{1}+\widehat{M}_{2}n_{2})\eta,_{n}ds+\\\
+\int_{\partial\Omega}(M_{\alpha\beta}(w)n_{\beta}\tau_{\alpha}+\widehat{M}_{1}n_{2}+\widehat{M}_{2}n_{1})\eta,_{s}ds=I_{1}^{\prime}+I_{1}^{\prime\prime}$
(3.44)
and, integrating by parts on $\partial\Omega$, we get
$I_{1}^{\prime}=(M_{\alpha\beta}(w)n_{\beta}\tau_{\alpha}+\widehat{M}_{1}n_{2}+\widehat{M}_{2}n_{1})\eta|_{s=0}^{s={{{l}}(\partial\Omega)}}-\\\
-\int_{\partial\Omega}(M_{\alpha\beta}(w)n_{\beta}\tau_{\alpha}+\widehat{M}_{1}n_{2}+\widehat{M}_{2}n_{1}),_{s}\eta
ds,$ (3.45)
where ${{l}}(\partial\Omega)$ is the length of $\partial\Omega$. Since
$\partial\Omega$ is of class $C^{1,1}$, the boundary term on the right end
side of (3.45) identically vanishes. Therefore, the stationarity condition of
$J$ at $w$ takes the final form
$\int_{\Omega}(M_{\alpha\beta,\alpha\beta}(w)+q)\eta dx+\\\
\int_{\partial\Omega}\left(-(M_{\alpha\beta}(w)n_{\beta}\tau_{\alpha}),_{s}-M_{\alpha\beta,\beta}(w)n_{\alpha}+\widehat{Q}-(\widehat{M}_{1}n_{2}+\widehat{M}_{2}n_{1}),_{s}\right)\eta
ds+\\\
+\int_{\partial\Omega}\left(M_{\alpha\beta}(w)n_{\beta}n_{\alpha}-\widehat{M}_{1}n_{1}+\widehat{M}_{2}n_{2}\right)\eta,_{n}ds=0$
(3.46)
for every $\eta\in C^{\infty}(\overline{\Omega},\mathbb{R})$. By the
arbitrariness of the function $\eta$, and of the traces of $\eta$ and
$\eta,_{n}$ on $\partial\Omega$, we determine the equilibrium equation (3.36)
and the desired Neumann boundary conditions on $\partial\Omega$:
$M_{\alpha\beta}(w)n_{\alpha}n_{\beta}=\widehat{M}_{n},$ (3.47)
$M_{\alpha\beta,\beta}(w)n_{\alpha}+(M_{\alpha\beta}(w)n_{\beta}\tau_{\alpha}),_{s}=\widehat{Q}-(\widehat{M}_{\tau}),_{s}.$
(3.48)
## 4 Uniqueness and stability of extreme inclusions and free boundaries
### 4.1 Rigid inclusions: uniqueness
In the sequel we shall assume that the plate is made of nonhomogeneous linear
elastic material with plate tensor
$\mathbb{P}=\frac{h^{3}}{12}\mathbb{C},$ (4.1)
where the elasticity tensor $\mathbb{C}(x)\in{\cal
L}({\mathbb{M}}^{2},{\mathbb{M}}^{2})$ has cartesian components $C_{ijkl}$
which satisfy the following symmetry conditions
$C_{ijkl}=C_{klij}=C_{klji}\quad i,j,k,l=1,2,\hbox{ a.e. in }\Omega,$ (4.2)
and, for simplicity, is defined in all of $\mathbb{R}^{2}$.
We make the following assumptions:
I) Regularity
$\mathbb{C}\in
C^{1,1}(\mathbb{R}^{2},{\mathcal{L}}({\mathbb{M}}^{2},{\mathbb{M}}^{2})),$
(4.3)
II) Ellipticity (strong convexity) There exists $\gamma>0$ such that
${\mathbb{C}}A\cdot A\geq\gamma|A|^{2},\qquad\hbox{in }\mathbb{R}^{2},$ (4.4)
for every $2\times 2$ symmetric matrix $A$.
III) Dichotomy condition
$\displaystyle either$ $\displaystyle{\mathcal{D}}(x)>0,\quad\hbox{for every
}x\in\mathbb{R}^{2},$ (4.5a) $\displaystyle or$
$\displaystyle{\mathcal{D}}(x)=0,\quad\hbox{for every }x\in\mathbb{R}^{2},$
(4.5b)
where
${\mathcal{D}}(x)=\frac{1}{a_{0}}|\det S(x)|,$ (4.6)
$S(x)={\left(\begin{array}[]{ccccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0&0\\\
0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0\\\ 0&0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\\
4a_{0}&3a_{1}&2a_{2}&a_{3}&0&0&0\\\ 0&4a_{0}&3a_{1}&2a_{2}&a_{3}&0&0\\\
0&0&4a_{0}&3a_{1}&2a_{2}&a_{3}&0\\\ 0&0&0&4a_{0}&3a_{1}&2a_{2}&a_{3}\\\
\end{array}\right)},$ (4.7) $a_{0}=A_{0},\ a_{1}=4C_{0},\
a_{2}=2B_{0}+4E_{0},\ a_{3}=4D_{0},\ a_{4}=F_{0}.$ (4.8)
and
${\displaystyle\left\\{\begin{array}[]{lr}C_{1111}=A_{0},\ \
C_{1122}=C_{2211}=B_{0},\vspace{0.12em}\\\
C_{1112}=C_{1121}=C_{1211}=C_{2111}=C_{0},\vspace{0.12em}\\\
C_{2212}=C_{2221}=C_{1222}=C_{2122}=D_{0},\vspace{0.12em}\\\
C_{1212}=C_{1221}=C_{2112}=C_{2121}=E_{0},\vspace{0.12em}\\\
C_{2222}=F_{0},\vspace{0.25em}\\\ \end{array}\right.}$
(4.9)
###### Remark 4.1.
Whenever (4.5a) holds we denote
$\delta_{1}=\min_{\mathbb{R}^{2}}{\mathcal{D}}.$ (4.10)
We emphasize that, in all the following statements, whenever a constant is
said to depend on $\delta_{1}$ (among other quantities) it is understood that
such dependence occurs only when (4.5a) holds.
On the assigned couple field $\widehat{M}$ let us require the following
assumptions:
$\widehat{M}\in
H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2}),\quad(\widehat{M}_{n},\widehat{M}_{\tau,s})\not\equiv
0,$ (4.11) $\int_{\partial\Omega}\widehat{M}_{i}=0,\qquad i=1,2.$ (4.12)
###### Theorem 4.2 (Unique determination of a rigid inclusion with one
measurement).
Let $\Omega$ be a simply connected domain in $\mathbb{R}^{2}$ such that
$\partial\Omega$ is of class $C^{1,1}$ and let $D_{i}$, $i=1,2$, be two simply
connected domains compactly contained in $\Omega$, such that $\partial D_{i}$
is of class $C^{3,1}$, $i=1,2$. Moreover, let $\Gamma$ be a nonempty open
portion of $\partial\Omega$, of class $C^{3,1}$. Let the plate tensor
$\mathbb{P}$ be given by (4.1), and satisfying (4.2)–(4.4) and the dichotomy
condition (4.5a) or (4.5b). Let $\widehat{M}$ be a boundary couple field
satisfying (4.11)–(4.12). Let $w_{i}$, $i=1,2$, be the solutions to the mixed
problem (LABEL:eq:1.dir-pbm-incl-rig-1bis)–(LABEL:eq:1-dir-pbm-incl-rig-5bis),
coupled with (1.6), with $D=D_{i}$.
If there exists $g\in\mathcal{A}$ such that
$w_{1}-w_{2}=g,\quad(w_{1}-w_{2})_{,n}=g_{,n},\quad\hbox{on }\Gamma,$ (4.13)
then
$D_{1}=D_{2}.$ (4.14)
###### Proof of Theorem 4.2.
Let $G$ be the connected component of $\Omega\setminus(\overline{D_{1}\cup
D_{2}})$ such that $\Gamma\subset\partial G$. Let us notice that, since
$w_{i}$ satisfies homogeneous Dirichlet conditions on the $C^{3,1}$ boundary
$\partial D_{i}$, by regularity results we have that $w_{i}\in
H^{4}(\widetilde{\Omega}\setminus D_{i})$, for every $\widetilde{\Omega}$,
$D_{i}\subset\subset{\widetilde{\Omega}}\subset\subset\Omega$, $i=1,2$ (see,
for example, [Ag]). By Sobolev embedding theorems (see, for instance, [Ad]),
we have that $w_{i}$ and $\nabla w_{i}$ are continuous up to $\partial D_{i}$,
$i=1,2$. Therefore
$w_{i}\equiv 0,\quad\nabla w_{i}^{e}\equiv 0,\quad\hbox{on }\partial D_{i}.$
(4.15)
Let $w=w_{1}-w_{2}-g$, with $g(x_{1},x_{2})=ax_{1}+bx_{2}+c$. By our
assumptions, $w$ takes homogeneous Cauchy data on $\Gamma$. From the
uniqueness of the solution to the Cauchy problem (see, for instance, Theorem
$3.8$ in [M-R-Ve4]) and also Remark 4 in [M-R-Ve2]) and from the weak unique
continuation property, we have that
$\qquad\qquad w\equiv 0,\quad\hbox{in }G.$
Let us prove, for instance, that $D_{2}\subset D_{1}$. We have
$D_{2}\setminus\overline{D_{1}}\subset\Omega\setminus(\overline{D_{1}\cup
G}),$
$\partial(\Omega\setminus(\overline{D_{1}\cup G}))=\Sigma_{1}\cup\Sigma_{2},$
where $\Sigma_{2}=\partial D_{2}\cap\partial G$,
$\Sigma_{1}=\partial(\Omega\setminus(\overline{D_{1}\cup
G}))\setminus\Sigma_{2}\subset\partial D_{1}$.
Let us distinguish two cases
i) $\partial D_{1}\cap\Sigma_{2}\neq\emptyset$,
ii) $\partial D_{1}\cap\Sigma_{2}=\emptyset$.
In case i), there exists $P_{0}\in\partial D_{1}\cap\Sigma_{2}$. Since
$w_{i}(P_{0})=0$, $w(P_{0})=0$, we have $g(P_{0})=0$.
Let $P_{n}\in G$, $P_{n}\rightarrow P_{0}$. We have
$\nabla w(P_{n})=0,$ $0=\lim_{n\rightarrow\infty}\nabla w(P_{n})=\nabla
w_{1}^{e}(P_{0})-\nabla w_{2}^{e}(P_{0})-(a,b)=-(a,b),$
so that $g\equiv c$, and, since $g(P_{0})=0$,
$g\equiv 0\quad\Rightarrow\quad w_{1}\equiv w_{2},\ \hbox{in
}G,\quad\Rightarrow\quad\left\\{\begin{array}[]{lr}w_{1}\equiv w_{2}=0,\
\hbox{on }\Sigma_{2},\vspace{0.25em}\\\ \nabla w_{1}\equiv\nabla w_{2}=0,\
\hbox{on }\Sigma_{2}.\end{array}\right.$ (4.16)
Integrating by parts equation $({\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w_{1})))w_{1}=0$, we obtain
$\int_{\Omega\setminus(\overline{D_{1}\cup
G})}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}=\int_{\Sigma_{1}\cup\Sigma_{2}}(\mathbb{P}\nabla^{2}w_{1}\nu\cdot\nu)w_{1,n}+\\\
+\int_{\Sigma_{1}\cup\Sigma_{2}}(\mathbb{P}\nabla^{2}w_{1}\nu\cdot\tau)w_{1,s}-\int_{\Sigma_{1}\cup\Sigma_{2}}(\mathrm{div}(\mathbb{P}\nabla^{2}w_{1})\cdot\nu)w_{1},$
(4.17)
where $\nu$ is the outer unit normal to $\Omega\setminus(\overline{D_{1}\cup
G})$. By
$w_{1}=0,\ \nabla w_{1}=0,\hbox{ on }\Sigma_{1},$
and by (4.16), we have
$w_{1}=w_{2}=0,\ \nabla w_{1}=\nabla w_{2}=0,\hbox{ on }\Sigma_{2},$
so that
$0=\int_{\Omega\setminus(\overline{D_{1}\cup
G})}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}\geq\gamma\int_{D_{2}\setminus\overline{D_{1}}}|\nabla^{2}w_{1}|^{2}.$
(4.18)
If $D_{2}\setminus\overline{D_{1}}\neq\emptyset$, then, by the weak unique
continuation principle, $w_{1}$ coincides with an affine function in
$\Omega\setminus\overline{D_{1}}$, contradicting the choice of the nontrivial
Neumann data $\widehat{M}$ on $\partial\Omega$. Therefore
$D_{2}\subset\overline{D_{1}}$ and, by the regularity of $D_{i}$, $i=1,2$,
$D_{2}\subset D_{1}$.
In case ii), either $\overline{D_{1}}\cap\overline{D_{2}}=\emptyset$ or
$\overline{D_{1}}\subset D_{2}$.
Let us consider for instance the first case, the proof of the second case
being similar. Integrating by parts, we have
$\int_{D_{2}}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}=\int_{D_{2}}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}(w_{1}-g)=\\\
\int_{\partial D_{2}}(\mathbb{P}\nabla^{2}w_{1}\nu\cdot\nu)(w_{1}-g),_{n}+\\\
+\int_{\partial
D_{2}}(\mathbb{P}\nabla^{2}w_{1}\nu\cdot\tau)(w_{1}-g),_{s}-\int_{\partial
D_{2}}(\mathrm{div}(\mathbb{P}\nabla^{2}w_{1})\cdot\nu)(w_{1}-g).$
By the regularity of $\partial D_{2}$, we may rewrite it as
$\int_{D_{2}}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}=\int_{\partial
D_{2}}(\mathbb{P}\nabla^{2}w_{1}\nu\cdot\nu)(w_{1}-g),_{n}+\\\ -\int_{\partial
D_{2}}\left((\mathbb{P}\nabla^{2}w_{1}\nu\cdot\tau),_{s}+\mathrm{div}(\mathbb{P}\nabla^{2}w_{1})\cdot\nu\right)(w_{1}-g),$
Recalling that $w_{1}-g=w_{2}=0$, $(w_{1}-g),_{n}=w_{2,n}=0$ on $\partial
D_{2}$, we have that
$\int_{D_{2}}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}=0.$
If $D_{2}\neq\emptyset$, then $w_{1}$ coincides with an affine function in
$D_{2}$, and, by the weak unique continuation principle, also in
$\Omega\setminus\overline{D_{1}}$, contradicting the choice of a nontrivial
$\widehat{M}$. Therefore $D_{2}=\emptyset$. Symmetrically, we obtain that
$D_{1}=\emptyset$, that is $D_{1}=D_{2}$.
∎
### 4.2 Rigid inclusions: stability
In order to prove the stability estimates, we need the following further
quantitative assumptions.
Given $\rho_{0}$, $M_{0}$, $M_{1}>0$, we assume that
$|\Omega|\leq M_{1}\rho_{0}^{2},$ (4.19)
$\hbox{dist}(D,\partial\Omega)\geq\rho_{0},$ (4.20) $\partial\Omega\hbox{ is
of }class\ C^{2,1}\ with\ constants\ \rho_{0},M_{0},$ (4.21) $\Gamma\hbox{ is
of }class\ C^{3,1}\ with\ constants\ \rho_{0},M_{0},$ (4.22) $\partial D\hbox{
is of }class\ C^{3,1}\ with\ constants\ \rho_{0},M_{0},$ (4.23)
where $|\Omega|$ denotes the Lebesgue measure of $\Omega$. Moreover, we assume
that for some $P_{0}\in\Sigma$ and some $\delta_{0}$, $0<\delta_{0}<1$,
$\partial\Omega\cap R_{\frac{\rho_{0}}{M_{0}},\rho_{0}}(P_{0})\subset\Gamma,$
(4.24)
and that
$|\Gamma|\leq(1-\delta_{0})|\partial\Omega|.$ (4.25)
On the Neumann data $\widehat{M}$ we assume that
$\widehat{M}\in
L^{2}(\partial\Omega,\mathbb{R}^{2}),\quad(\widehat{M}_{n},(\widehat{M}_{\tau}),_{s})\not\equiv
0,$ (4.26) $\int_{\partial\Omega}\widehat{M}_{i}=0,\quad i=1,2,$ (4.27)
$\hbox{supp}(\widehat{M})\subset\subset\Gamma,$ (4.28)
and that, for a given constant $F>0$,
$\frac{\|\widehat{M}\|_{L^{2}(\partial\Omega,\mathbb{R}^{2})}}{\|\widehat{M}\|_{H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})}}\leq
F.$ (4.29)
On the elasticity tensor $\mathbb{C}$, we assume the same a priori information
made in Subsection 4.1, and we introduce a parameter $M>0$ such that
$\sum_{i,j,k,l=1}^{2}\sum_{m=0}^{2}\rho_{0}^{m}\|\nabla^{m}C_{ijkl}\|_{L^{\infty}(\mathbb{R}^{2})}\leq
M.$ (4.30)
We shall refer to the set of constants $M_{0}$, $M_{1}$, $\delta_{0}$, $F$,
$\gamma$, $M$, $\delta_{1}$ as the _a priori data_. The scale parameter
$\rho_{0}$ will appear explicitly in all formulas, whereas the dependence on
the thickness parameter $h$ will be omitted.
###### Theorem 4.3 (Stability result).
Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$ satisfying (4.19) and
(4.21). Let $D_{i}$, $i=1,2$, be two simply connected open subsets of $\Omega$
satisfying (4.20) and (4.23). Moreover, let $\Gamma$ be an open portion of
$\partial\Omega$ satisfying (4.22), (4.24) and (4.25). Let $\widehat{M}\in
L^{2}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.26)–(4.29) and let the plate
tensor $\mathbb{P}$ given by (4.1) satisfy (4.2), (4.30), (4.4) and the
dichotomy condition. Let $w_{i}\in H^{2}(\Omega\setminus\overline{D_{i}})$ be
the solution to (LABEL:eq:1.dir-pbm-incl-rig-1bis)–(LABEL:eq:1-dir-pbm-incl-
rig-5bis), coupled with (1.6), when $D=D_{i}$, $i=1,2$. If, given
$\epsilon>0$, we have
$\min_{g\in\cal{A}}\left\\{\|w_{1}-w_{2}-g\|_{L^{2}(\Sigma)}+\rho_{0}\left\|(w_{1}-w_{2}-g)_{,n}\right\|_{L^{2}(\Sigma)}\right\\}\leq\epsilon,$
(4.31)
then we have
$d_{\cal H}(\partial D_{1},\partial D_{2})\leq
C\rho_{0}(\log|\log\widetilde{\epsilon}|)^{-\eta},\quad
0<\widetilde{\epsilon}<e^{-1},$ (4.32)
and
$d_{\cal H}(\overline{D_{1}},\overline{D_{2}})\leq
C\rho_{0}(\log|\log\widetilde{\epsilon}|)^{-\eta},\quad\
0<\widetilde{\epsilon}<e^{-1},$ (4.33)
where
$\widetilde{\epsilon}=\frac{\epsilon}{\rho_{0}^{2}\|\widehat{M}\|_{H^{-\frac{1}{2}}}}$
and $C$, $\eta$, $C>0$, $0<\eta\leq 1$, are constants only depending on the a
priori data.
###### Proof of Theorem 4.3.
Let us rough-out a sketch of the proof, referring the interested reader to
Section 3 of [M-R-Ve5]. Retracing the proof of the uniqueness theorem, the
basic idea is that of deriving the quantitative version in the stability
context of the vanishing of the integral
$\int_{\Omega\setminus(\overline{D_{i}\cup
G})}\mathbb{P}\nabla^{2}w_{i}\cdot\nabla^{2}w_{i}$, for $i=1,2$, that is a
control with some small term emerging from the bound (4.31) on the Cauchy
data. To this aim, we need stability estimates of continuation from Cauchy
data and propagation of smallness estimates. In particular, the propagation of
smallness of $|\nabla^{2}w_{i}|$, for $i=1,2$, from a neighboorhood of
$\Gamma$ towards $\partial(\Omega\setminus(\overline{D_{i}\cup G}))$ is
performed through iterated application of the three sphere inequality (1.18)
over suitable chains of disks. A strong hindrance which occurs in this step is
related to the difficulty of getting arbitrarily closer to the boundary of the
set $\Omega\setminus(\overline{D_{i}\cup G})$, due to the absence, in our
general setting, of any a priori information on the reciprocal position of
$D_{1}$ and $D_{2}$. For this reason, as a preparatory step, we derive the
following rough estimate
$\max\left\\{\int_{D_{2}\setminus\overline{D_{1}}}|\nabla^{2}w_{1}|^{2},\int_{D_{1}\setminus\overline{D_{2}}}|\nabla^{2}w_{2}|^{2}\right\\}\leq\rho_{0}^{2}\|\widehat{M}\|_{H^{-\frac{1}{2}}}^{2}(\log|\log\widetilde{\epsilon}|)^{-\frac{1}{2}},$
(4.34)
which holds for every $\widetilde{\epsilon}<e^{-1}$, with
$\widetilde{\epsilon}=\frac{\epsilon}{\rho_{0}^{2}\|\widehat{M}\|_{H^{-\frac{1}{2}}}}$,
and where $C>0$ depends only on $\gamma$, $M$, $\delta_{1}$, $M_{0}$, $M_{1}$
and $\delta_{0}$.
Since a pointwise lower bound for $|\nabla^{2}w_{i}|^{2}$ cannot hold in
general, the next crucial step consists in the following Claim.
Claim. If
$\max\left\\{\int_{D_{2}\setminus\overline{D_{1}}}|\nabla^{2}w_{1}|^{2},\int_{D_{1}\setminus\overline{D_{2}}}|\nabla^{2}w_{2}|^{2}\right\\}\leq\
\frac{\eta}{\rho_{0}^{2}},$ (4.35)
then
$d_{\cal H}(\partial D_{1},\partial D_{2})\leq
C\rho_{0}\left[\log\left(\frac{C\rho_{0}^{4}\|\widehat{M}\|_{H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{n})}^{2}}{\eta}\right)\right]^{-\frac{1}{B}},$
(4.36)
where $B>0$ and $C>0$ only depend on $\gamma$, $M$, $\delta_{1}$, $M_{0}$,
$M_{1}$, $\delta_{0}$ and $F$.
###### Proof of the Claim.
Denoting for simplicity $d=d_{\cal H}(\partial D_{1},\partial D_{2})$, we may
assume, with no loss of generality, that there exists $x_{0}\in\partial D_{1}$
such that $\hbox{dist}(x_{0},\partial D_{2})=d$. Let us distinguish two cases:
i) $B_{d}(x_{0})\subset D_{2}$;
ii) $B_{d}(x_{0})\cap D_{2}=\emptyset$.
In case i), by the regularity assumptions made on $\partial D_{1}$, there
exists $x_{1}\in D_{2}\setminus D_{1}$ such that $B_{td}(x_{1})\subset
D_{2}\setminus D_{1}$, with $t=\frac{1}{1+\sqrt{1+M_{0}^{2}}}$.
In [M-R-Ve6], by iterated application of the three sphere inequality (1.18),
we have obtained the following _Lipschitz propagation of smallness_ estimate:
there exists $s>1$, only depending on $\gamma$, $M$, $\delta_{1}$, $M_{0}$ and
$\delta_{0}$, such that for every $\rho>0$ and every
$\bar{x}\in(\Omega\setminus\overline{D})_{s\rho}$, we have
$\int_{B_{\rho}(\bar{x})}|\nabla^{2}w|^{2}\geq\frac{C\rho_{0}^{2}}{\exp\left[A\left(\frac{\rho_{0}}{\rho}\right)^{B}\right]}\|\widehat{M}\|_{H^{-\frac{1}{2}}}^{2},$
(4.37)
where $A>0$, $B>0$ and $C>0$ only depend on $\gamma$, $M$, $\delta_{1}$,
$M_{0}$, $M_{1}$, $\delta_{0}$ and $F$. By (4.35) and by applying (4.37) with
$\rho=\frac{td}{s}$, we have
$\eta\geq\frac{C\rho_{0}^{4}}{\exp{\left[A\left(\frac{s\rho_{0}}{td}\right)^{B}\right]}}\|\widehat{M}\|_{H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})}^{2},$
(4.38)
where $A>0$, $B>0$ and $C>0$ only depend on $\gamma$, $M$, $\delta_{1}$,
$M_{0}$, $M_{1}$, $\delta_{0}$ and $F$.
By (4.38) we easily find (4.36).
Case ii) can be treated similarly by substituting $w_{1}$ with $w_{2}$. ∎
By applying the Claim to (4.34), we obtain a first stability estimates of log-
log-log type. At this stage, a tool which turns out to be very useful is a
geometrical result, firstly stated in [Al-B-R-Ve1], which ensures that there
exists $\epsilon_{0}>0$, only depending on $\gamma$, $M$, $\delta_{1}$,
$M_{0}$, $M_{1}$, $\delta_{0}$ and $F$, such that if
$\epsilon\leq\epsilon_{0}$ then $\partial G$ is of Lipschitz class with
constants $\widetilde{\rho_{0}}$, $L$, with $L$ and
$\frac{\widetilde{\rho_{0}}}{\rho_{0}}$ only depending on $M_{0}$. Lipschitz
regularity prevents the occurrence of uncontrollable narrowings or cuspidal
points in $G$ and allows to refine the geometrical constructions of the chains
of disks to which we apply the three sphere inequality, obtaining the better
estimate
$\max\left\\{\int_{D_{2}\setminus\overline{D_{1}}}|\nabla^{2}w_{1}|^{2},\int_{D_{1}\setminus\overline{D_{2}}}|\nabla^{2}w_{2}|^{2}\right\\}\leq\rho_{0}^{2}\|\widehat{M}\|_{H^{-\frac{1}{2}}}^{2}|\log\widetilde{\epsilon}|^{-\sigma},$
(4.39)
which holds for every $\widetilde{\epsilon}<1$, where $C>0$ and $\sigma>0$
depend only on $\gamma$, $M$, $\delta_{1}$, $M_{0}$, $M_{1}$, $\delta_{0}$,
$L$ and $\frac{\widetilde{\rho_{0}}}{\rho_{0}}$. Again, by applying the Claim,
the desired estimates follow. ∎
### 4.3 Cavities and unknown boundary portions: uniqueness
In this subsection we consider the inverse problems of determining unknown
boundaries of the following two kinds: i) the boundary of a cavity, ii) an
unknown boundary portion of $\partial\Omega$. In both cases, we have
homogeneous Neumann conditions on the unknown boundary. Neumann boundary
conditions lead to further complications in the arguments involving
integration by parts. To give an idea of the differences, given any connected
component $F$ of $D_{2}\setminus\overline{D_{1}}$, whose boundary is made of
two arcs $\tau\subset\partial D_{1}$ and $\gamma\subset\partial D_{2}$, having
common endpoints $P_{1}$ and $P_{2}$, the analogue of (4.17) becomes
$\int_{F}\mathbb{P}\nabla^{2}w_{1}\cdot\nabla^{2}w_{1}=\int_{\tau}(\mathbb{P}\nabla^{2}w_{1}n^{1}\cdot\tau^{1}w_{1})_{,s}-\int_{\gamma}(\mathbb{P}\nabla^{2}w_{2}n^{2}\cdot\tau^{2}w_{2})_{,s}=\\\
=\left[(\mathbb{P}\nabla^{2}w_{1}n^{1}\cdot\tau^{1})(P_{1})-(\mathbb{P}\nabla^{2}w_{1}n^{2}\cdot\tau^{2})(P_{1})\right](w_{1}(P_{1})-w_{1}(P_{2})),$
(4.40)
where $n^{i}$, $\tau^{i}$ denotes the unit normal and tangent vector to
$D_{i}$, $i=1,2$.
Since, in general, the boundaries of $D_{1}$ and $D_{2}$ intersect
nontangentially, the above expression does not vanish and the contradiction
arguments fails. For this reason, we need two boundary measurements to prove
uniqueness, as stated in Theorem 4.4. Instead, uniqueness with one measurement
can be restored in the problem of the determination of an unknown boundary
portion, by taking advantage of the fact that the two plates have a common
regular boundary portion, say $\Gamma$, see Theorem 4.5.
Let $D\subset\subset\Omega$ be a domain of class $C^{1,1}$ representing an
unknown cavity inside the plate $\Omega$. Under the same assumptions made in
Subsection 4.1, the transversal displacement $w$ satisfies the following
Neumann problem
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w))=0,&\mathrm{in}\
\Omega\setminus\overline{D},\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w)n\cdot
n=-\widehat{M}_{n},&\mathrm{on}\ \partial\Omega,\vspace{0.25em}\\\ {\rm
div}({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w)n\cdot
n=0,&\mathrm{on}\ \partial D,\vspace{0.25em}\\\ {\rm
div}({\mathbb{P}}\nabla^{2}w)\cdot
n+(({\mathbb{P}}\nabla^{2}w)n\cdot\tau),_{s}=0,&\mathrm{on}\ \partial
D,\vspace{0.25em}\\\ \end{array}\right.}$
which admits a solution $w\in H^{2}(\Omega\setminus\overline{\Omega})$, which
is uniquely determined up to addition of an affine function.
Concerning the inverse problem of the determination of the cavity $D$ inside
the plate, let us recall the following result.
###### Theorem 4.4 (Uniqueness with two boundary measurements).
Let $\Omega$ be a simply connected domain in $\mathbb{R}^{2}$ such that
$\partial\Omega$ is of class $C^{1,1}$ and let $D_{i}$, $i=1,2$, be two simply
connected domains compactly contained in $\Omega$, such that $\partial D_{i}$
is of class $C^{4,1}$, $i=1,2$. Moreover, let $\Gamma$ be a nonempty open
portion of $\partial\Omega$, of class $C^{3,1}$. Let the plate tensor
$\mathbb{P}$ be given by (4.1), and satisfying (4.3) (4.4) and the dichotomy
condition (4.5a) or (4.5b). Let $\widehat{M}$, $\widehat{M}^{*}$ be two
boundary couple fields both satisfying (4.11)–(4.12) and such that
$(\widehat{M}_{n},\widehat{M}_{\tau,s})$ and
$(\widehat{M}_{n}^{*},\widehat{M}_{\tau,s}^{*})$ are linearly independent in
$H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})\times
H^{-\frac{3}{2}}(\partial\Omega,\mathbb{R}^{2})$. Let $w_{i}$, $w_{i}^{*}$,
$i=1,2$, be solutions to the Neumann problem (LABEL:eq:dir-pbm-unkn-
bdry-1)–(LABEL:eq:dir-pbm-unkn-bdry-5), with $D=D_{i}$, corresponding to
boundary data $\widehat{M}$, $\widehat{M}^{*}$ respectively. If
$w_{1}=w_{2},\qquad w_{1,n}=w_{2,n},\qquad\hbox{on}\ \Gamma,$ (4.46)
$w_{1}^{*}=w_{2}^{*},\qquad w_{1,n}^{*}=w_{2,n}^{*},\qquad\hbox{on}\ \Gamma,$
(4.47)
then
$D_{1}=D_{2}.$ (4.48)
Next, let us consider the case of a plate whose boundary is composed by an
accessible portion $\Gamma$ and by an unknown inaccessible portion $I$, to be
determined. More precisely, let $\Gamma$, $I$ be two closed, nonempty sub-arcs
of the boundary $\partial\Omega$ such that
$\Gamma\cup I=\partial\Omega,\quad\Gamma\cap I=\\{Q,R\\},$ (4.49)
where $Q$, $R$ are two distinct points of $\partial\Omega$. On the assigned
couple field $\widehat{M}$ let us require the following assumptions:
$\widehat{M}\in
L^{2}(\Gamma,\mathbb{R}^{2}),\quad(\widehat{M}_{n},\widehat{M}_{\tau,s})\not\equiv
0,$ (4.50) $\int_{\Gamma}\widehat{M}_{i}=0,\qquad i=1,2.$ (4.51)
Under the same assumptions made in Subsection 4.1 for the plate tensor and the
domain $\Omega$, the transversal displacement $w\in H^{2}(\Omega)$ satisfies
the following Neumann problem
${\displaystyle\left\\{\begin{array}[]{lr}M_{\alpha\beta,\alpha\beta}=0,&\mathrm{in}\
\Omega,\vspace{0.25em}\\\
M_{\alpha\beta}n_{\alpha}n_{\beta}=\widehat{M}_{n},&\mathrm{on}\
\Gamma,\vspace{0.25em}\\\
M_{\alpha\beta,\beta}n_{\alpha}+(M_{\alpha\beta}n_{\beta}\tau_{\alpha}),_{s}=-(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\Gamma,\vspace{0.25em}\\\ M_{\alpha\beta}n_{\alpha}n_{\beta}=0,&\mathrm{on}\
I,\vspace{0.25em}\\\
M_{\alpha\beta,\beta}n_{\alpha}+(M_{\alpha\beta}n_{\beta}\tau_{\alpha}),_{s}=0,&\mathrm{on}\
I.\vspace{0.25em}\\\ \end{array}\right.}$
Concerning the inverse problem of the determination of the unknown boundary
portion $I$, in [M-R3] we have proved the following result.
###### Theorem 4.5 (Unique determination of unknown boundaries with one
measurement).
Let $\Omega_{1}$, $\Omega_{2}$ be two simply connected bounded domains in
$\mathbb{R}^{2}$ such that $\partial\Omega_{i}$, $i=1,2$, are of class
$C^{4,1}$. Let $\partial\Omega_{i}=I_{i}\cup\Gamma$, $i=1,2$, where $I_{i}$
and $\Gamma$ are the inaccessible and the accessible parts of the boundaries
$\partial\Omega_{i}$, respectively. Let us assume that $\Omega_{1}$ and
$\Omega_{2}$ lie on the same side of $\Gamma$ and that conditions (4.49) are
satisfied by both pairs $\\{I_{1},\Gamma\\}$ and $\\{I_{2},\Gamma\\}$. Let the
plate tensor $\mathbb{P}$ of class $C^{2,1}(\mathbb{R}^{2})$ be given by
(4.1), and satisfying (4.3) (4.4) and the dichotomy condition (4.5a) or
(4.5b). Let $\widehat{M}\in L^{2}(\Gamma,\mathbb{R}^{2})$ be a boundary couple
field satisfying conditions (4.50), (4.12). Let $w_{i}\in H^{2}(\Omega_{i})$
be a solution to the Neumann problem (LABEL:eq:3.compact_equation-bdry)–(4.51)
in $\Omega=\Omega_{i}$, $i=1,2$. If
$w_{1}=w_{2},\qquad w_{1,n}=w_{2,n},\qquad\textrm{ on }\Gamma,$ (4.57)
then
$\Omega_{1}=\Omega_{2}.$ (4.58)
## 5 Size estimates for extreme inclusions
### 5.1 Formulation of the problem and main results
Let us assume that the middle plane of the plate $\Omega$ is a bounded domain
in $\mathbb{R}^{2}$ of class $C^{1,1}$ with constants $\rho_{0}$, $M_{0}$. In
the present section we shall derive constructive upper and lower bounds of the
area of either a rigid inclusion or a cavity in an elastic plate from a single
boundary measurement. These extreme inclusions will be represented by an open
subset $D$ of $\Omega$ such that $\Omega\setminus\overline{D}$ is connected
and $D$ is compactly contained in $\Omega$; that is, there is a number
$d_{0}>0$ such that
$\textrm{dist}(D,\partial\Omega)\geq d_{0}\rho_{0}.$ (5.1)
In addition, in proving the lower bound for the area of $D$, we shall
introduce the following a priori information, which is a way of requiring that
$D$ is not ”too thin”.
###### Definition 5.1.
(Scale Invariant Fatness Condition) Given a domain $D$ having Lipschitz
boundary with constants $r\rho_{0}$ and $L$, where $r>0$, we shall say that it
satisfies the Scale Invariant Fatness Condition with constant $Q>0$ if
$\textrm{diam}(D)\leq Qr\rho_{0}.$ (5.2)
###### Remark 5.2.
It is evident that if $D$ satisfies Definition 5.1, then we have the trivial
upper and lower estimates
$\frac{\omega_{2}}{(1+\sqrt{1+L^{2}})^{2}}r^{2}\rho_{0}^{2}\leq\textrm{area}(D)\leq\omega_{2}Q^{2}r^{2}\rho_{0}^{2},$
(5.3)
where $\omega_{2}$ denotes the measure of the unit disk in $\mathbb{R}^{2}$.
Since we are interested in obtaining upper and lower bounds of the area of $D$
when $D$ is unknown, it will be necessary to consider also the number $r$ as
an unknown parameter, and all our estimates will not depend on $r$.
Conversely, the parameters $L$ and $Q$, which are invariant under scaling,
will be considered as a priori information on the unknown inclusion $D$.
For reader’s convenience and in order to introduce some useful notation, we
briefly recall the formulation of the equilibrium problem when the plate
contains either a rigid inclusion or a cavity, and when the inclusion is
absent.
Let us assume that the plate tensor $\mathbb{P}\in
L^{\infty}(\mathbb{R}^{2},{\mathcal{L}}({\mathbb{M}}^{2},{\mathbb{M}}^{2}))$
given by (4.1) satisfies the symmetry conditions (4.2) and the strong
convexity condition (4.4). Moreover, let $\widehat{M}\in
H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.27).
If an inclusion $D$ made by rigid material is present, with boundary $\partial
D$ of class $C^{1,1}$, then the transversal displacement $w_{R}$ corresponding
to the assigned couple field $\widehat{M}$ is given as the weak solution
$w_{R}\in H^{2}(\Omega\setminus\overline{D})$ of the boundary value problem
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w_{R}))=0,&\mathrm{in}\
\Omega\setminus\overline{D},\vspace{0.25em}\\\
({\mathbb{P}}\nabla^{2}w_{R})n\cdot n=-\widehat{M}_{n},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ {\rm div}({\mathbb{P}}\nabla^{2}w_{R})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{R})n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\
w_{R}|_{\overline{D}}\in\mathcal{A},&\mathrm{in}\
\overline{D},\vspace{0.25em}\\\ \frac{\partial w_{R}^{e}}{\partial
n}=\frac{\partial w_{R}^{i}}{\partial n},&\mathrm{on}\
\partial{D},\vspace{0.25em}\\\ \end{array}\right.}$
coupled with the equilibrium conditions for the rigid inclusion $D$
$\int_{\partial D}\left({\rm div}({\mathbb{P}}\nabla^{2}w_{R}^{e})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{R}^{e})n\cdot\tau),_{s}\right)g-(({\mathbb{P}}\nabla^{2}w_{R}^{e})n\cdot
n)g_{,n}=0,\\\ \quad\hbox{for every }g\in\mathcal{A},$ (5.9)
where we recall that we have defined $w_{R}^{e}\equiv
w|_{\Omega\setminus\overline{D}}$ and $w_{R}^{i}\equiv w|_{\overline{D}}$.
When a cavity is present, then the transversal displacement in
$\Omega\setminus\overline{D}$ is given as the weak solution $w_{V}\in
H^{2}(\Omega\setminus\overline{D})$ to the boundary value problem
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w_{V}))=0,&\mathrm{in}\
\Omega\setminus\overline{D},\vspace{0.25em}\\\
({\mathbb{P}}\nabla^{2}w_{V})n\cdot n=-\widehat{M}_{n},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ {\rm div}({\mathbb{P}}\nabla^{2}w_{V})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{V})n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ ({\mathbb{P}}\nabla^{2}w_{V})n\cdot
n=0,&\mathrm{on}\ \partial D,\vspace{0.25em}\\\ {\rm
div}({\mathbb{P}}\nabla^{2}w_{V})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{V})n\cdot\tau),_{s}=0,&\mathrm{on}\ \partial
D.\vspace{0.25em}\\\ \end{array}\right.}$
Finally, when the inclusion is absent, we shall denote by $w_{0}\in
H^{2}(\Omega)$ the corresponding transversal displacement of the plate which
will be given as the weak solution of the Neumann problem
${\displaystyle\left\\{\begin{array}[]{lr}{\rm div}({\rm
div}({\mathbb{P}}\nabla^{2}w_{0}))=0,&\mathrm{in}\ \Omega,\vspace{0.25em}\\\
({\mathbb{P}}\nabla^{2}w_{0})n\cdot n=-\widehat{M}_{n},&\mathrm{on}\
\partial\Omega,\vspace{0.25em}\\\ {\rm div}({\mathbb{P}}\nabla^{2}w_{0})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{0})n\cdot\tau),_{s}=(\widehat{M}_{\tau}),_{s},&\mathrm{on}\
\partial\Omega.\vspace{0.25em}\\\ \end{array}\right.}$
We shall denote by $W_{R}$, $W_{V}$, $W_{0}$ the work exerted by the couple
field $\widehat{M}$ acting on $\partial\Omega$ when $D$ is a rigid inclusion,
it is a cavity, or it is absent, respectively. By the weak formulation of the
corresponding equilibrium problem it turns out that
$W_{R}=-\int_{\partial\Omega}\widehat{M}_{\tau,s}w_{R}+\widehat{M}_{n}w_{R},_{n}=\int_{\Omega\setminus\overline{D}}\mathbb{P}\nabla^{2}w_{R}\cdot\nabla^{2}w_{R},$
(5.18)
$W_{V}=-\int_{\partial\Omega}\widehat{M}_{\tau,s}w_{V}+\widehat{M}_{n}w_{V},_{n}=\int_{\Omega\setminus\overline{D}}\mathbb{P}\nabla^{2}w_{V}\cdot\nabla^{2}w_{V},$
(5.19)
$W_{0}=-\int_{\partial\Omega}\widehat{M}_{\tau,s}w_{0}+\widehat{M}_{n}w_{0},_{n}=\int_{\Omega}\mathbb{P}\nabla^{2}w_{0}\cdot\nabla^{2}w_{0}.$
(5.20)
Note that the works $W_{R}$, $W_{V}$ and $W_{0}$ are well defined since they
are invariant with respect to the addition of any affine function to the
displacement fields $w_{R}$, $w_{V}$ and $w_{0}$, respectively. In the
following, the solutions $w_{V}$ and $w_{0}$ will be uniquely determined by
imposing the normalization conditions
$\int_{\partial D}w_{V}=0,\quad\int_{\partial D}\nabla w_{V}=0,$ (5.21)
$\int_{\Omega}w_{0}=0,\quad\int_{\Omega}\nabla w_{0}=0.$ (5.22)
Concerning the solution $w_{R}$, we found convenient to normalize it by
requiring that
$w_{R}=0,\quad\mathrm{in}\ \overline{D}.$ (5.23)
For a given positive number $h_{1}$, we denote by $D_{h_{1}\rho_{0}}$ the set
$D_{h_{1}\rho_{0}}=\\{x\in D|\ \mathrm{dist}(x,\partial D)>h_{1}\rho_{0}\\}.$
(5.24)
We are now in position to state our size estimates. In the case of a rigid
inclusion we have the following two theorems.
###### Theorem 5.3.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$, such
that $\partial\Omega$ is of class $C^{2,1}$ with constants $\rho_{0}$,
$M_{0}$, and satisfying (4.19). Let $D$ be a simply connected open subset of
$\Omega$ with boundary $\partial D$ of class $C^{1,1}$, satisfying (5.1), such
that $\Omega\setminus\overline{D}$ is connected and
$\mathrm{area}(D_{h_{1}\rho_{0}})\geq\frac{1}{2}\mathrm{area}(D),$ (5.25)
for a given positive number $h_{1}$. Let the plate tensor $\mathbb{P}$ given
by (4.1) satisfy (4.2), (4.4), (4.30) and the dichotomy condition. Let
$\widehat{M}\in L^{2}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.26)–(4.28),
with $\Gamma$ satisfying (4.25). The following inequality holds
$\mathrm{area}(D)\leq K\rho_{0}^{2}\frac{W_{0}-W_{R}}{W_{0}},$ (5.26)
where the constant $K>0$ only depends on the quantities $M_{0}$, $M_{1}$,
$d_{0}$, $h_{1}$, $\gamma$, $\delta_{1}$, $M$, $\delta_{0}$ and $F$.
###### Theorem 5.4.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$, such
that $\partial\Omega$ is of class $C^{2,1}$ with constants $\rho_{0}$,
$M_{0}$, and satisfying (4.19). Let $D$ be a simply connected domain
satisfying (5.1), (5.2), such that $\Omega\setminus\overline{D}$ is connected
and the boundary $\partial D$ is of class $C^{3,1}$ with constants
$r\rho_{0}$, $L$. Let the plate tensor $\mathbb{P}$ given by (4.1) satisfy
(4.2), (4.4) and (4.30). Let $\widehat{M}\in
L^{2}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.27). The following inequality
holds
$C\rho_{0}^{2}\Phi\left(\frac{W_{0}-W_{R}}{W_{0}}\right)\leq\mathrm{area}(D),$
(5.27)
where the function $\Phi$ is given by
$[0,1)\ni t\mapsto\Phi(t)=\frac{t^{2}}{1-t},$ (5.28)
and $C>0$ is a constant only depending on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M$.
When $D$ is a cavity, the following bounds hold.
###### Theorem 5.5.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$, such
that $\partial\Omega$ is of class $C^{2,1}$ with constants $\rho_{0}$,
$M_{0}$, and satisfying (4.19). Let $D$ be a simply connected open subset of
$\Omega$ with boundary $\partial D$ of class $C^{1,1}$, satisfying (5.1), such
that $\Omega\setminus\overline{D}$ is connected and
$\mathrm{area}(D_{h_{1}\rho_{0}})\geq\frac{1}{2}\mathrm{area}(D),$ (5.29)
for a given positive number $h_{1}$. Let the plate tensor $\mathbb{P}$ given
by (4.1) satisfy (4.2), (4.4), (4.30) and the dichotomy condition. Let
$\widehat{M}\in L^{2}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.26)–(4.28),
with $\Gamma$ satisfying (4.25). The following inequality holds
$\mathrm{area}(D)\leq K\rho_{0}^{2}\frac{W_{V}-W_{0}}{W_{0}},$ (5.30)
where the constant $K>0$ only depends on the quantities $M_{0}$, $M_{1}$,
$d_{0}$, $h_{1}$, $\gamma$, $\delta_{1}$, $M$, $\delta_{0}$ and $F$.
###### Theorem 5.6.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$, such
that $\partial\Omega$ is of class $C^{1,1}$ with constants $\rho_{0}$,
$M_{0}$, and satisfying (4.19). Let $D$ be a simply connected domain
satisfying (5.1), (5.2), such that $\Omega\setminus\overline{D}$ is connected
and the boundary $\partial D$ is of class $C^{1,1}$ with constants
$r\rho_{0}$, $L$. Let the plate tensor $\mathbb{P}$ given by (4.1) satisfy
(4.2) and (4.4), and such that $\|\mathbb{P}\|_{C^{2,1}(\mathbb{R}^{2})}\leq
M^{\prime}$, where $M^{\prime}$ is a positive parameter. Let $\widehat{M}\in
L^{2}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.27). The following inequality
holds
$C\rho_{0}^{2}\Psi\left(\frac{W_{V}-W_{0}}{W_{0}}\right)\leq\mathrm{area}(D),$
(5.31)
where the function $\Psi$ is given by
$[0,+\infty)\ni t\mapsto\Psi(t)=\frac{t^{2}}{1+t},$ (5.32)
and $C>0$ is a constant only depending on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M^{\prime}$.
### 5.2 Proof of Theorems 5.3 and 5.4
The starting point of the upper and lower estimates of the area of a rigid
inclusion is the following energy estimate, in which the works $W_{0}$ and
$W_{R}$ are compared.
###### Lemma 5.7.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$ with
boundary $\partial\Omega$ of class $C^{1,1}$. Assume that $D$ is a simply
connected open set compactly contained in $\Omega$, with boundary $\partial D$
of class $C^{1,1}$ and such that $\Omega\setminus\overline{D}$ is connected.
Let the plate tensor $\mathbb{P}\in
L^{\infty}(\Omega,\mathcal{L}(\mathbb{M}^{2},\mathbb{M}^{2}))$ given by (4.1)
satisfy (4.2) and (4.4). Let $\widehat{M}\in
H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.27). Let $w_{R}\in
H^{2}(\Omega\setminus\overline{D})$, $w_{0}\in H^{2}(\Omega)$ be the solutions
to problems (LABEL:eq:extreme-3.1a)–(5.9) and
(LABEL:eq:extreme-5.2a)–(LABEL:eq:extreme-5.2c), normalized as above. We have
$\int_{D}\mathbb{P}\nabla^{2}w_{0}\cdot\nabla^{2}w_{0}\leq
W_{0}-W_{R}=\int_{\partial D}M_{n}(w_{R})w_{0,n}+V(w_{R})w_{0},$ (5.33)
where we have denoted by
$M_{n}(w_{R})=-(\mathbb{P}\nabla^{2}w_{R})n\cdot n,$ (5.34) $V(w_{R})={\rm
div}({\mathbb{P}}\nabla^{2}w_{R})\cdot
n+(({\mathbb{P}}\nabla^{2}w_{R})n\cdot\tau),_{s}$ (5.35)
the bending moment and the Kirchhoff shear on $\partial D$ associated to
$w_{R}$, respectively. Here, $n$ denotes the exterior unit normal to
$\Omega\setminus\overline{D}$.
###### Proof.
The proof is based on the weak formulation of problems
(LABEL:eq:extreme-3.1a)–(5.9) and
(LABEL:eq:extreme-5.2a)–(LABEL:eq:extreme-5.2c), and can be obtained by
adapting the proof of the corresponding result for a rigid inclusion in an
elastic body derived in ([M-R1], Lemma 3.1). ∎
###### Proof of Theorem 5.3.
By (5.33) and from the strong convexity condition (4.4) we have
$\int_{D}|\nabla^{2}w_{0}|^{2}\leq\gamma^{-1}(W_{0}-W_{R}).$ (5.36)
Estimate (5.26) can be obtained from the following lower bound for the elastic
energy associated to $w_{0}$ in $D$
$\int_{D}|\nabla^{2}w_{0}|^{2}\geq
C\frac{\mathrm{area}(D)}{\rho_{0}^{2}}W_{0},$ (5.37)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $h_{1}$,
$\gamma$, $\delta_{1}$, $M$, $\delta_{0}$ and $F$. The above estimate was
derived in ([M-R-Ve6], Theorem 3.1) for inclusions $D$ satisfying the fatness
condition (5.25) and its proof is based on a three sphere inequality for
solutions to the plate equation (LABEL:eq:extreme-5.2a) with anisotropic
elastic coefficients obeying to the dichotomy condition. ∎
In order to prove Theorem 5.4 we need the following Poincaré inequalities of
constructive type.
For a given positive number $r>0$, we denote by $D^{r\rho_{0}}$ the following
set
$D^{r\rho_{0}}=\\{x\in\mathbb{R}^{2}|\ 0<\mathrm{dist}(x,D)<r\rho_{0}\\}.$
(5.38)
For $D$ with Lispchitz boundary and $u\in H^{1}(D)$ we define
$u_{D}=\frac{1}{|D|}\int_{D}u,\quad u_{\partial D}=\frac{1}{|\partial
D|}\int_{\partial D}u.$ (5.39)
###### Proposition 5.8.
Let $D$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$, of Lipschitz class
with constants $r\rho_{0}$, $L$, satisfying condition (5.2) with constant
$Q>0$. For every $u\in H^{1}(D)$ we have
$\int_{D}|u-u_{D}|^{2}\leq C_{1}r^{2}\rho_{0}^{2}\int_{D}|\nabla u|^{2},$
(5.40) $\int_{\partial D}|u-u_{\partial D}|^{2}\leq
C_{2}r\rho_{0}\int_{D}|\nabla u|^{2},$ (5.41)
where $C_{1}>0$, $C_{2}>0$ only depend on $L$ and $Q$.
If $u\in H^{1}(D^{r\rho_{0}})$, then
$\int_{\partial D}|u-u_{\partial D}|^{2}\leq
C_{3}r\rho_{0}\int_{D^{r\rho_{0}}}|\nabla u|^{2},$ (5.42)
where $C_{3}>0$ only depends on $L$ and $Q$. Moreover, if $u\in
H^{1}(D^{r\rho_{0}})$ and $u=0$ on $\partial D$, then we have
$\int_{D^{r\rho_{0}}}u^{2}\leq
C_{4}r^{2}\rho_{0}^{2}\int_{D^{r\rho_{0}}}|\nabla u|^{2},$ (5.43)
where $C_{4}>0$ only depends on $L$ and $Q$.
###### Proof.
We refer to [Al-M-R1] for a proof of the inequalities (5.40)–(5.42) and for a
precise evaluation of the constants $C_{1}$, $C_{2}$, $C_{3}$ in terms of the
scale invariant bounds $L$, $Q$ regarding the regularity and shape of $D$.
Inequality (5.43) is a consequence of the following result. Let $\Omega$ be a
bounded domain in $\mathbb{R}^{n}$ with Lipschitz boundary with constants
$r\rho_{0}$, $L$, and such that $\mathrm{diam}(\Omega)\leq Qr\rho_{0}$, $Q>0$.
Let $E$ be any measurable subset of $\Omega$ with positive Lebesgue measure
$|E|>0$. For every $v\in H^{1}(\Omega)$ we have
$\int_{\Omega}|v-v_{E}|^{2}\leq\left(1+\sqrt{\frac{|\Omega|}{|E|}}\right)^{2}Cr^{2}\rho_{0}^{2}\int_{\Omega}|\nabla
v|^{2},$ (5.44)
where the constant $C>0$ only depends on $L$ and $Q$. The above inequality
follows from Lemma 2.1 of [Al-M-R4] and from (5.40) applied to the function
$v$, see also inequality (3.8) of [Al-M-R4]. Let us extend the function $u\in
H^{1}(D^{r\rho_{0}})$ into the interior of $D$ by taking $u\equiv 0$ in $D$,
and let us continue to denote by $u$ this extended function, with $u\in
H^{1}(D\cup D^{r\rho_{0}})$. Inequality (5.43) follows from (5.44) by taking
$v=u$, $\Omega=D\cup D^{r\rho_{0}}$ and $E=D$. ∎
###### Proof of Theorem 5.4.
Let $g=a+bx+cy$ be an affine function such that the function
$\widetilde{w}_{0}=w_{0}+g$ satisfies
$\int_{\partial D}\widetilde{w}_{0}=0,\quad\int_{\partial
D}\nabla\widetilde{w}_{0}=0.$ (5.45)
The function $\widetilde{w}_{0}\in H^{2}(\Omega)$ is a solution of
(LABEL:eq:extreme-5.2a)–(LABEL:eq:extreme-5.2c), and by (5.9), from the right-
hand side of (5.33) and by applying Hölder’s inequality, we have
$W_{0}-W_{R}\leq\left(\int_{\partial
D}|M_{n}(w_{R})|^{2}\right)^{\frac{1}{2}}\left(\int_{\partial
D}|\widetilde{w}_{0,n}|^{2}\right)^{\frac{1}{2}}+\\\ +\left(\int_{\partial
D}|V(w_{R})|^{2}\right)^{\frac{1}{2}}\left(\int_{\partial
D}|\widetilde{w}_{0}|^{2}\right)^{\frac{1}{2}}=I_{1}+I_{2}.$ (5.46)
We start by estimating $I_{1}$. By (5.41) and by the definition of
$\widetilde{w}_{0}$ we have
$\int_{\partial D}|\widetilde{w}_{0,n}|^{2}\leq
Cr\rho_{0}\int_{D}|\nabla^{2}w_{0}|^{2}\leq
Cr\rho_{0}\|\nabla^{2}w_{0}\|_{L^{\infty}(D)}^{2}\mathrm{area}(D),$ (5.47)
where the constant $C>0$ only depends on $L$ and $Q$. By the Sobolev embedding
theorem (see, for instance, [Ad]), by standard interior regularity estimates
(see, for example, Theorem 8.3 in [M-R-Ve1]), by Proposition 5.8, by (4.4) and
by (5.20) we have
$\|\nabla^{2}w_{0}\|_{L^{\infty}(D)}\leq\frac{C}{\rho_{0}^{2}}\|w_{0}\|_{H^{2}(\Omega)}\leq\frac{C}{\rho_{0}}\left(\int_{\Omega}|\nabla^{2}w_{0}|^{2}\right)^{\frac{1}{2}}\leq\frac{C}{\rho_{0}}W_{0}^{\frac{1}{2}},$
(5.48)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $\gamma$
and $M$.
The integral $\int_{\partial D}|M_{n}(w_{R})|^{2}$ can be estimated by using a
trace inequality (see, for instance, [L-M]) and a $H^{3}$-regularity estimate
up to the boundary $\partial D$ for $w_{R}$ (see, for example, Lemma 4.2 and
Theorem 5.2 of [M-R-Ve5]):
$\int_{\partial
D}|M_{n}(w_{R})|^{2}\leq\frac{C}{r^{5}\rho_{0}^{5}}\sum_{i=0}^{3}(r\rho_{0})^{2i}\int_{D^{\frac{r\rho_{0}}{2}}}|\nabla^{i}w_{R}|^{2}\leq\frac{C}{r^{5}\rho_{0}^{5}}\sum_{i=0}^{2}(r\rho_{0})^{2i}\int_{D^{r\rho_{0}}}|\nabla^{i}w_{R}|^{2},$
(5.49)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $L$, $Q$, $\gamma$
and $M$. The function $w_{R}$ belongs to $H^{2}(D^{r\rho_{0}})$ and, by
(5.23), $w_{R}=0$ and $\nabla w_{R}=0$ on $\partial D$. Therefore, by applying
twice (5.43), by (4.4) and by (5.18) we have
$\sum_{i=0}^{2}(r\rho_{0})^{2i}\int_{D^{r\rho_{0}}}|\nabla^{i}w_{R}|^{2}\leq
Cr^{4}\rho_{0}^{4}\int_{D^{r\rho_{0}}}|\nabla^{2}w_{R}|^{2}\leq
Cr^{4}\rho_{0}^{4}W_{R},$ (5.50)
where the constant $C>0$ only depends on $L$, $Q$ and $\gamma$. By (5.47),
(5.48), (5.49), (5.50) we have
$I_{1}\leq\frac{C}{\rho_{0}}W_{0}^{\frac{1}{2}}W_{R}^{\frac{1}{2}}(\mathrm{area}(D))^{\frac{1}{2}},$
(5.51)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M$.
The control of the term $I_{2}$ can be obtained similarly. By a standard
Poincaré inequality, by (5.41) and by (5.48) we have
$\int_{\partial D}|\widetilde{w}_{0}|^{2}\leq Cr^{2}\rho_{0}^{2}\int_{\partial
D}|\widetilde{w}_{0,s}|^{2}\leq
Cr^{3}\rho_{0}^{3}\int_{D}|\nabla^{2}w_{0}|^{2}\leq\\\ \leq
Cr^{3}\rho_{0}^{3}\|\nabla^{2}w_{0}\|_{L^{\infty}(D)}^{2}\mathrm{area}(D)\leq
Cr^{3}\rho_{0}W_{0}\mathrm{area}(D),$ (5.52)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M$. Concerning the integral $\int_{\partial D}|V(w_{R})|^{2}$,
by using a trace inequality, a $H^{4}$-regularity estimate up to the boundary
$\partial D$ for $w_{R}$ (see, for example, Lemma 4.3 and Theorem 5.3 in [M-R-
Ve5]) and by (5.50), we have
$\int_{\partial
D}|V(w_{R})|^{2}\leq\frac{C}{r^{7}\rho_{0}^{7}}\sum_{i=0}^{4}(r\rho_{0})^{2i}\int_{D^{\frac{r\rho_{0}}{2}}}|\nabla^{i}w_{R}|^{2}\leq\\\
\leq\frac{C}{r^{3}\rho_{0}^{3}}\int_{D^{r\rho_{0}}}|\nabla^{2}w_{R}|^{2}\leq\frac{C}{r^{3}\rho_{0}^{3}}W_{R},$
(5.53)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $L$, $Q$, $\gamma$
and $M$. Then, by (5.52) and (5.53) we have
$I_{2}\leq\frac{C}{\rho_{0}}W_{0}^{\frac{1}{2}}W_{R}^{\frac{1}{2}}(\mathrm{area}(D))^{\frac{1}{2}},$
(5.54)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M$. By (5.51) and (5.54), the inequality (5.27) follows. ∎
### 5.3 Proof of Theorems 5.5 and 5.6
As in the proof of the upper and lower estimates of the area of a rigid
inclusion, we need to compare the works $W_{0}$ and $W_{V}$. The analogue of
Lemma 5.7 is the following result.
###### Lemma 5.9.
Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{2}$ with
boundary $\partial\Omega$ of class $C^{1,1}$. Assume that $D$ is a simply
connected open set compactly contained in $\Omega$, with boundary $\partial D$
of class $C^{1,1}$ and such that $\Omega\setminus\overline{D}$ is connected.
Let the plate tensor $\mathbb{P}\in
L^{\infty}(\Omega,\mathcal{L}(\mathbb{M}^{2},\mathbb{M}^{2}))$ given by (4.1)
satisfy (4.2) and (4.4). Let $\widehat{M}\in
H^{-\frac{1}{2}}(\partial\Omega,\mathbb{R}^{2})$ satisfy (4.27). Let $w_{V}\in
H^{2}(\Omega\setminus\overline{D})$, $w_{0}\in H^{2}(\Omega)$ be the solutions
to problems (LABEL:eq:extreme-4.1a)–(LABEL:eq:extreme-4.1e) and
(LABEL:eq:extreme-5.2a)–(LABEL:eq:extreme-5.2c), normalized as above. We have
$\int_{D}\mathbb{P}\nabla^{2}w_{0}\cdot\nabla^{2}w_{0}\leq
W_{V}-W_{0}=\int_{\partial D}M_{n}(w_{0})w_{V,n}+V(w_{0})w_{V}\ ,$ (5.55)
where the functions $M_{n}(w_{0})$, $V(w_{0})$ are defined as in (5.34) and
(5.35), and $n$ denotes the exterior unit normal to
$\Omega\setminus\overline{D}$.
###### Proof.
As for Lemma 5.7, the proof is based on the weak formulation of problems
(LABEL:eq:extreme-4.1a)–(LABEL:eq:extreme-4.1e) and
(LABEL:eq:extreme-5.2a)–(LABEL:eq:extreme-5.2c), and can be obtained following
the same guidelines of the corresponding result for a cavity in an elastic
body derived in ([M-R1], Lemma 3.5). However, here we simplify the approach
presented in [M-R1] without extending the function $w_{V}$ in the interior of
$D$. ∎
###### Proof of Theorem 5.5.
The proof follows from the left-hand side of (5.55) by using the same
arguments as in the proof of Theorem 5.3. ∎
###### Proof of Theorem 5.6.
From the right-hand side of (5.55) and by applying Hölder’s inequality, we
have
$W_{V}-W_{0}\leq\left(\int_{\partial
D}|M_{n}(w_{0})|^{2}\right)^{\frac{1}{2}}\left(\int_{\partial
D}|{w}_{V,n}|^{2}\right)^{\frac{1}{2}}+\\\ +\left(\int_{\partial
D}|V(w_{0})|^{2}\right)^{\frac{1}{2}}\left(\int_{\partial
D}|{w}_{V}|^{2}\right)^{\frac{1}{2}}=J_{1}+J_{2}.$ (5.56)
Let us estimate the integral $J_{2}$. By the Sobolev embedding theorem (see,
for instance, [Ad]), by standard interior regularity estimates for $w_{0}$
(see, for example, Lemma 1 in [M-R-Ve2]), by (5.40), by (4.4) and by (5.20) we
have
$\int_{\partial D}|V(w_{0})|^{2}\leq
C\|\nabla^{3}w_{0}\|_{L^{\infty}(D)}^{2}|\partial
D|\leq\frac{C}{\rho_{0}^{6}}\|w_{0}\|_{H^{2}(\Omega)}^{2}|\partial D|\leq\\\
\leq\frac{C}{\rho_{0}^{4}}\int_{\Omega}|\nabla^{2}w_{0}|^{2}|\partial
D|\leq\frac{C}{\rho_{0}^{4}}W_{0}|\partial D|,$ (5.57)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $\gamma$
and $M^{\prime}$. By ([Al-R], Lemma 2.8) we have
$|\partial D|\leq C\frac{\mathrm{area}(D)}{r\rho_{0}},$ (5.58)
where the constant $C>0$ only depends on $L$.
To control the integral $\int_{\partial D}|{w}_{V}|^{2}$ we use a standard
Poincaré inequality on $\partial D$, Proposition 5.8, inequality (5.42), the
strong convexity condition (4.4) for $\mathbb{P}$ and the definition of
$W_{V}$, that is
$\int_{\partial D}|{w}_{V}|^{2}\leq Cr^{2}\rho_{0}^{2}\int_{\partial
D}|{w}_{V,s}|^{2}\leq
Cr^{3}\rho_{0}^{3}\int_{D^{r\rho_{0}}}|\nabla^{2}w_{V}|^{2}\leq\\\ \leq
Cr^{3}\rho_{0}^{3}\int_{\Omega\setminus\overline{D}}\mathbb{P}\nabla^{2}w_{V}\cdot\nabla^{2}w_{V}=Cr^{3}\rho_{0}^{3}W_{V},$
(5.59)
where the constant $C>0$ only depends on $L$, $Q$ and $\gamma$. Then, by
(5.57), (5.58) and (5.59), and since, trivially, $r\leq K$, where $K>0$ is a
constant only depending on $L$ and $M_{1}$, we have
$J_{2}\leq\frac{C}{\rho_{0}}W_{0}^{\frac{1}{2}}W_{V}^{\frac{1}{2}}(\mathrm{area}(D))^{\frac{1}{2}},$
(5.60)
where the constant $C>0$ only depends on $M_{0}$, $M_{1}$, $d_{0}$, $L$, $Q$,
$\gamma$ and $M^{\prime}$. By using similar arguments we can also find the
analogous bound for $J_{1}$, and the thesis follows. ∎
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|
arxiv-papers
| 2012-09-26T20:23:54 |
2024-09-04T02:49:35.647718
|
{
"license": "Public Domain",
"authors": "Antonino Morassi, Edi Rosset, Sergio Vessella",
"submitter": "Sergio Vessella",
"url": "https://arxiv.org/abs/1209.6066"
}
|
1209.6081
|
# Spectroscopic Pulsational Frequency Identification and Mode Determination of
$\gamma$ Doradus Star HD 12901 ††thanks: This paper includes data taken at the
Mount John University Observatory of the University of Canterbury (New
Zealand), the McDonald Observatory of the University of Texas at Austin
(Texas, USA), and the European Southern Observatory at La Silla (Chile).
E. Brunsden1, K. R. Pollard1, P. L. Cottrell1, D. J. Wright2, P. De Cat3
1Department of Physics and Astronomy, University of Canterbury, Private Bag
4800, Christchurch, New Zealand
2Department of Astrophysics, University of New South Wales, Sydney, NSW 2052
Australia
3Royal Observatory of Belgium, Ringlaan 3, 1180 Brussel, Belgium E-mail:
emily.brunsden@gmail.com
###### Abstract
Using multi-site spectroscopic data collected from three sites, the
frequencies and pulsational modes of the $\gamma$ Doradus star HD 12901 were
identified. A total of six frequencies in the range 1-2 d-1 were observed,
their identifications supported by multiple line-profile measurement
techniques and previously-published photometry. Five frequencies were of
sufficient signal-to-noise for mode identification and all five displayed
similar three-bump standard deviation profiles which were fitted well with
($l$,$m$)=(1,1) modes. These fits had reduced $\chi^{2}$ values of less than
18. We propose that this star is an excellent candidate to test models of non-
radially pulsating $\gamma$ Doradus stars as a result of the presence of
multiple (1,1) modes.
###### keywords:
line: profiles, techniques: spectroscopic, HD12901, stars: variables: general,
stars: oscillations
††pagerange: Spectroscopic Pulsational Frequency Identification and Mode
Determination of $\gamma$ Doradus Star HD 12901 ††thanks: This paper includes
data taken at the Mount John University Observatory of the University of
Canterbury (New Zealand), the McDonald Observatory of the University of Texas
at Austin (Texas, USA), and the European Southern Observatory at La Silla
(Chile).–8††pubyear: 2012
## 1 Introduction
The identification of the geometry of a $\gamma$ Doradus type pulsation
remains one of the more difficult spectroscopic fields. However, it is a
powerful way to characterise the interior structure of a star and improve
stellar pulsation models. The pulsations of $\gamma$ Doradus stars are
particularly of interest as they propagate through the deep layers of a star.
The flux-blocking mechanism at the base of the surface convective zone (Guzik
et al., 2000) is held accountable for the origin of the pulsations which are
g-modes, where the restoring force is gravity.
Briefly $\gamma$ Doradus stars are slightly evolved late-A to late-F stars at
the cooler end of the classical instability strip that display high-order
g-mode pulsations with frequencies on the order of 1 cycle-per-day. For a full
definition see Kaye et al. (1999) and recent reviews of the field are in Kaye
(2007) and Pollard (2009). Currently there are less than 100 bright bona fide
$\gamma$ Doradus stars known (see list in Henry et al. 2011) and a handful of
$\gamma$ Doradus/$\delta$ Scuti hybrid stars (Henry & Fekel, 2005;
Uytterhoeven et al., 2008), with a further 100 $\gamma$ Doradus and 171 hybrid
stars thus far reported by the Kepler mission (Grigahcène et al., 2010;
Uytterhoeven et al., 2011).
Now is truly the age of satellite photometry and no ground-based methods can
compete with the long uninterrupted datasets that satellites such as CoRoT and
Kepler produce. The Fourier frequency spectra of such studies show the high-
levels of precision only obtainable from space (e.g. at least 840 frequencies
found in HD 49434 using the CoRoT satellite; Chapellier et al. 2011). However,
to further our understanding of g-mode pulsations we need more data than
frequencies alone. Several successful techniques using ground-based multi-
colour photometry can be employed to determine the number of nodal lines ($l$)
in a star. The full mode-identification of a star, that is finding the $l$ and
also the number of nodal lines passing through the pole of a star $m$, remains
the sole domain of spectroscopy. It is hoped that modelling of the frequencies
and modes of a star will allow determination of the $n$, number of shells
interior to the stellar surface.
The spectroscopic study of $\gamma$ Doradus stars relies on the collection of
a large amount of high-resolution and preferably multi-site data of sufficient
signal-to-noise for classification of the pulsations. Improvement is gained by
the use of data from multiple sites to reduce daily aliasing patterns in the
Fourier spectra. Such observational campaigns take many months to years in
order to collect a sufficient number of spectra.
Table 1: Observation log of the multi-site data for HD 12901 Observatory | Telescope | Spectrograph | Observation range | $\Delta$T (d) | $\\#$ spectra
---|---|---|---|---|---
MJUO | 1.0m | HERCULES | Feb 2009-Sep 2011 | 935 | 478
La Silla | 1.2m | CORALIE | Nov 1998-Nov 2002 | 1166 | 53
McDonald | 2.1m | SES | Sep 2009-Oct 2009 | 8 | 60
When such datasets are compiled for $\gamma$ Doradus stars, the pulsational
frequencies and modes can be examined and compared to those from photometry.
There are still only a handful of $\gamma$ Doradus stars with full mode
identifications and our immediate goal is to classify as many as we can. The
results of these mode identifications can be used to improve pulsational
models (such as those of Grigahcène et al. 2012; Townsend 2003) by providing
information about the amplitudes of excited modes and also to start to
identify patterns within the class.
This paper focuses on the $\gamma$ Doradus star HD 12901 outlining the
observations made and reduction procedure in Section 2. The spectroscopic
frequency analysis for the individual observatories and the combined data
results are described in Section 3 with each method tested. Section 4
describes the reanalysis of white-light and seven-colour photometry taken of
this star. The mode identification of the five identified frequencies follows
in Section 5. Finally Section 6 discusses the findings and their implications
for future work in this field.
## 2 Observations and data treatment
The major findings of this research are the identification of the frequencies
and modes using spectra obtained at three observing sites (Mt John University
Observatory (MJUO), La Silla Observatory and McDonald Observatory) and are
summarised in Table 1.
Spectra were reduced according to the standard local spectrograph software
then they were normalised, continuum fitted and order-merged by the authors
using a semi-automated MATLAB routine. The full spectra were cross-correlated
for each site using a scaled delta function routine (Wright 2008, Wright et
al. 2007). This produces the line profiles used for pulsational analysis. The
signal-to-noise of a cross-correlated line profile is much higher than a
single spectral line and thus pulsations are easier to extract. The line
profiles for each site were combined (scaled, shifted in velocity space and
continuum fitted) to produce a single dataset of consistent line profiles.
This was done post-cross correlation to maximise the number of available lines
used for each spectrum, as different spectrographs operate over different
wavelength ranges.
The line profiles were analysed in FAMIAS, a pulsational frequency mode-
identification toolbox (Zima, 2008), using techniques applied from Zima et al.
(2006). The Fourier spectra for the Pixel-by-Pixel (PbP) technique and that of
the 0th-3rd moments (Balona, 1986; Briquet & Aerts, 2003) were used to
identify frequencies. These frequencies were then analysed to determine mode
identifications using the Fourier Parameter Fit method (Zima, 2009).
Additionally, the software package SIGSPEC (Reegen, 2007) was used to compare
the frequency selection method of FAMIAS. SIGSPEC performs a Fourier analysis
of a two-dimensional dataset and selects frequencies based on their spectral
significance. The frequencies obtained with SIGSPEC can be regarded as more
precise than that of the highest-peak direct selection as, after producing the
Fourier frequency spectrum, the peak with the highest spectral significance is
selected. Spectral significance includes the analysis of the false-alarm
probability to remove frequency peaks caused by irregular data sampling or
noise in the data.
Both FAMIAS and SIGSPEC were used to re-analyse photometric data, originally
published in Handler (1999), Eyer & Aerts (2000) and Aerts et al. (2004), as
further insight into the frequencies of the pulsations. To identify the $l$
values of the modes, the frequency amplitude ratio and phase differences
method, based on pre-computed grids of models, were used (Balona & Stobie,
1979; Watson, 1988; Cugier et al., 1994; Daszyńska-Daszkiewicz et al., 2002).
This was done using the photometric analysis toolbox in FAMIAS (Zima, 2008).
## 3 Spectroscopic Frequency Analysis
In total $591$ spectra from the three sites were of sufficient quality to be
analysed. The data spanned a total of 4667 days, just over 12 years. Each
dataset was analysed independently for frequencies and they were then
combined. By doing this analysis we can comment on the limitations of single-
site data and the extent to which multi-site data reduces aliasing.
### 3.1 Frequency Identification For Each Dataset
#### 3.1.1 MJUO
This is the largest single-site dataset and was analysed most extensively.
First the cross-correlated line profiles were analysed in FAMIAS. The
frequency list is in Table 2. The PbP technique found eight possible frequency
peaks including a one-cycle-per-day frequency ($f_{m5}$), likely an alias. The
zeroth moment was too noisy to extract any frequencies. The first and third
moment Fourier peaks each only showed one frequency clearly, $1.186$ d-1 and
$1.184$ d-1 respectively, which is the same frequency given a conservative
uncertainty estimate of $\pm 0.001$ (uncertainties are dealt with more
formally in Section 3.3). The second moment yielded $0.144$ d-1 as the only
viable frequency. Though many more frequencies may be evident in the data,
further extraction of frequencies above the noise was difficult with any
measure of certainty.
The same moments were then tested in SIGSPEC for further analysis. SIGSPEC
found 20 frequencies above a spectral significance level of 5. It is likely
that noise in the data is causing many misidentifications rather than the
possibility that these could all be real, given the noise level in the Fourier
spectra. The frequencies with spectral significance greater than 15 are
reported for moments one to three in Table 2. From the table it is clear that
$f_{m2}$ to $f_{m8}$ (excluding $f_{m5}$) are viable frequencies found using
multiple methods. The addition of further datasets from other observation
sites should improve the signal-to-noise and reduce any aliasing, particularly
1-day aliasing that occurs in the data.
Table 2: Frequencies of HD 12901 found in the MJUO data. Pixel-by-Pixel (PbP) frequencies found using FAMIAS and moments and significances (Sig) found using SIGSPEC are listed. The amplitudes of the PbP frequencies are scaled to the first identified amplitude. Note some frequencies have been identified as $1-f$ aliases of the PbP frequency. The frequency $f_{m5}$ shows identifications near 1 d-1, likely from the data sampling. | PbP | Moments
---|---|---
| Freq. | Amp. | $1^{st}$ | Sig | $2^{nd}$ | Sig | $3^{rd}$ | Sig
| (d-1) | (rel. to $f_{m1}$) | (d-1) | | (d-1) | | (d-1) |
$f_{m1}$ | 1.271 | 1 | | | | | |
$f_{m2}$ | 1.186 | 1.8 | 1.183 | 24 | | | 1.183 | 23
$f_{m3}$ | 1.681 | 1.5 | 0.678 | 26 | | | 0.678 | 28
$f_{m4}$ | 1.396 | 1.7 | 0.403 | 20 | 1.396 | 20 | |
$f_{m5}$ | 1.001 | 1.3 | | | 1.001 | 26 | |
$f_{m6}$ | 1.560 | 1.6 | 0.460 | 16 | 1.560 | 17 | 1.560 | 19
| | | | | | | 0.541 | 18
$f_{m7}$ | 1.216 | 1.6 | 2.214 | 18 | 0.213 | 23 | 2.214 | 20
$f_{m8}$ | 0.244 | 1.1 | 0.756 | 21 | | | |
$f_{m9}$ | | | | | 0.104 | 16 | 0.899 | 19
$f_{m10}$ | | | | | | | 2.352 | 20
$f_{m11}$ | | | | | | | 0.836 | 17
#### 3.1.2 La Silla
The spectra taken on the CORALIE spectrograph have previously been analysed
and published by Aerts et al. (2004) and De Cat et al. (2006). Frequencies of
$1.04\pm 0.28$ d-1 and $1.30\pm 0.30$ d-1 were reported.
Although the small size of the dataset made it difficult to analyse, frequency
peaks were identified at $0.005$ d-1 and $1.26$ d-1 in the PbP measure,
$0.997$ d-1 and $3.01$ d-1 in the first moment and $0.997$ d-1 and $1.999$ d-1
in the third moment. Peaks that occur very close to integer values are less
reliable as they are likely 1-day alias patterns from the observation times
artificially amplified by data sampling effects in the window function. No
identifiable peaks occurred in the zeroth and second moments. Only the $1.26$
d-1 can be identified with any confidence. It is notable that a peak was
apparent in all the above Fourier spectra at $0.23$ d-1.
SIGSPEC was better able to distinguish the 1-day aliases and identified
frequencies of $0.26$ d-1 in the zeroth moment and $0.23$ d-1 in the first and
third moments. These had significances of $3.3$, $3.5$ and $2.7$ respectively,
which are usually regarded as too small to be significant.
#### 3.1.3 McDonald
The McDonald data set comprised of 60 observations taken in September and
October 2009. The data were considered to be particularly useful as having
time intervals that overlap some of the MJUO observations, providing some
independent confirmation of the line profile variation. As a stand-alone data
set the shorter wavelength range of the spectrograph, and the larger regions
of telluric lines, reduced the number of stellar lines that could be cross-
correlated from several thousand to around 100 lines. This has a significant
impact on the signal-to-noise of the final line profile. This, and the low
number of observations, meant frequencies detected in this dataset alone were
unconvincing. Despite this, it is noted that frequencies near $1.38$ d-1 and
$1.24$ d-1 are present in the PbP analysis and a peak near $1.67$ d-1 is
visible in the first moment Fourier spectrum.
Figure 1: Scaled line profiles of the three datasets. McDonald (red), La Silla
(blue) and MJUO (black) observations are shown with the mean line profile
(white). All datasets scale well to show a consistent line profile. The data
are provided as an electronic file (see Supporting Information).
### 3.2 Frequency Identification Of Combined Data For Each Method
Figure 2: Effect of the multi-site datasets on the Pixel-by-Pixel Fourier
spectra, the MJUO data on the left and the combined data on the right. Each
series shows the window function for the data at the top then successive pre-
whitening of frequencies $f_{p1}$ to $f_{p5}$. The Fourier spectra for the
Pixel-by-Pixel analysis of the combined data continues to $f_{p10}$. The
frequencies are listed in Table 3.
The cross-correlated line profiles of the three datasets were combined post
cross-correlation to form a single dataset. The line profiles are displayed in
Figure 1. Combining the line profiles instead of the individual spectra was
the best way to preserve the signal-to-noise in the MJUO and La Silla datasets
which had many more lines than the McDonald spectra. There was a small shift
(6 km s-1) in the systemic velocity between the McDonald and the other two
datasets. The McDonald spectra were shifted to ensure consistent data. The
line profiles were scaled to have the same equivalent widths, a parameter
shown to remain constant by the lack of variation in the zeroth moment in the
MJUO data. The Fourier spectra of the MJUO data alone and the combined
datasets are compared in Figure 2.
The spectral windows for the MJUO data alone and all three data sets are very
similar, but, as expected, the addition of the multi-site data reduces the
amplitude of the secondary peaks of the identified frequency. From Figure 2 it
is clear that the addition of the extra data increases the amplitude of the
Fourier peaks. This is an effect of not only the reinforced signal found in
the other sites spectra, but also an artificial amplification due to the
increased base noise level. The frequency peaks remain in nearly the same
positions and show approximately the same repeating pattern. The one-day
aliasing, although still present, has been reduced to make the true peaks more
evident. In general, the frequency identification becomes clearer with the
addition of the multi-site data, despite it being from small sets compared to
the MJUO data.
#### 3.2.1 Pixel-by-Pixel (PbP)
The line profiles showed many periodic variations in each pixel. The ten
frequencies found in the combined dataset are listed in Table 3 and the
Fourier spectra displayed in Figure 2. The first two frequencies found were
the $f_{2}$ and $f_{3}$ from Aerts et al. (2004) from photometry and their
$f_{1}$ shows up as our $f_{p4}$. Further frequencies $f_{p3}$ and $f_{p5}$
have credible three bump variations in their line profiles. It is likely that
$f_{p6}$ is a one-day alias frequency arising from the dominance of one single
site (MJUO contributes about $80\%$ of the observations). The frequencies
$f_{p7}$, $f_{p8}$ and $f_{p10}$ showed clear three bump standard deviation
profiles each with smooth phase changes. This increases the likelihood that
they are real frequencies, although the possibility remains that they are from
residual power from a previous poorly-defined frequency. Finally, $f_{p9}$
showed a clear four bump standard deviation profile at $3.14$ d-1 and $4.14$
d-1 frequencies when tested. As none of the previous frequencies showed a
four-bump pattern in the variation it is more likely this is a real frequency
rather than a misidentification of an earlier frequency.
A check on the independence of the frequencies was done by prewhitening the
spectra by 1-day aliases and 1-$f$ aliases of the first five identified
frequencies. Only $f_{p3}\pm 1$ had clear standard deviation profile
variations resembling pulsation and none of the alternate identifications
altered the identification of following frequencies, indicating that they are
all independent.
A second check of the variability of the line profile with different
frequencies involves phasing the data to the proposed frequency and examining
the structure. To see the changes most clearly this was done using the
residuals of the line profiles after subtraction of the mean. The frequencies
$f_{p1}$ to $f_{p5}$ in Figure 3 the residuals on the left side of each plot
show the ’braided rope’ structure typical of non-radial pulsation, but for the
other frequencies this was less clear. If we require pulsations to produce
regular changes in both red and blue halves of the line profile then we must
discard $f_{p6}$, $f_{p7}$ and $f_{p9}$ as candidate frequencies leaving just
$f_{p8}$ and $f_{p10}$ to investigate further.
An investigation into the phase coverage of the data for each frequency can
also be used as an indicator of the reliability of the frequency. All of the
frequencies $f_{p1}-f_{p10}$ were well covered in phase space except $f_{p6}$,
which was poorly sampled between phases 0.2-0.3 and 0.5-0.8 due to the
dominance of a single site in the data.
Table 3: Frequencies from the Pixel-by-Pixel analysis of the combined dataset. The uncertainty estimate for the frequencies is $\pm 0.0002$d-1 as described in Section 3.3. Frequencies with a strike-though were discarded as described in the text. ID | Freq. | Period | Amp. | Phase | Variation
---|---|---|---|---|---
| (d-1) | (d) | (rel. to $f_{p1}$) | | Explained
$f_{p1}$ | 1.3959 | 0.7164 | 1 | 0.286 | $13\%$
$f_{p2}$ | 1.1863 | 0.8430 | 0.72 | 0.205 | $21\%$
$f_{p3}$ | 1.6812 | 0.5948 | 0.93 | 0.267 | $30\%$
$f_{p4}$ | 1.2157 | 0.8226 | 0.93 | 0.265 | $39\%$
$f_{p5}$ | 1.5596 | 0.6412 | 0.80 | 0.230 | $45\%$
$f_{p6}$ | 1.0004 | 0.9996 | 0.55 | 0.158 | $48\%$
$f_{p7}$ | 1.2465 | 0.8022 | 0.50 | 0.143 | $50\%$
$f_{p8}$ | 1.2743 | 0.7848 | 0.53 | 0.152 | $53\%$
$f_{p9}$ | 3.1392 | 0.3186 | 0.36 | 0.102 | $54\%$
$f_{p10}$ | 2.5803 | 0.3876 | 0.37 | 0.106 | $55\%$
#### 3.2.2 Moment Analysis
The zeroth moment analysis shows few peaks that are similar to those from
other methods. There is a cyclic variation around 1-cycle per day but with the
$1.0$ d-1 peak missing due to the combination of multi-site data. The small
amplitude of variations in the second moment indicates we have only small
periodic temperature variations to account for in the line profiles.
The first moment Fourier spectra showed clear peak frequencies similar to the
PbP method. These are shown in the first section of Table 4. The amount of
variation of the first moment was moderately well explained by the selection
of the highest six peaks in the Fourier spectrum. Beyond $f_{f6}$ the
improvements to the explanation of variation are too small to be conclusive.
When the PbP frequencies were extracted from the first moment data (in
approximately the same order as the highest peak frequencies) a better fit to
the variation was found. With the six frequencies, $f_{p1}-f_{p5}$ and
$f_{p8}$, $76\%$ of the variation was removed. The frequencies found using
SIGSPEC closely matched the first four frequencies found in the highest peak
method then reproduced PbP frequencies $f_{p3}$ and $f_{p7}$ as $f_{f9}$ and
$f_{f10}$.
The Fourier spectra of the second moment also showed a few promising
frequencies once the first peak at $0.0006$ d-1 was removed. The frequencies
$f_{s2}$ to $f_{s4}$ match to, or match to one-day aliases of, frequencies
found in the PbP dataset. It is likely that $f_{s6}$ is a misidentification of
$f_{p2}$. The second section of Table 4 catalogues the frequencies found. Both
the highest peak and PbP frequencies accounted for around the same amount of
variation as they identified nearly all the same frequencies.
Frequency peaks found in the third moment are presented in the final section
of Table 4. The first few frequencies found using this method were generally
similar to those found in the PbP method except the double identification of
$f_{t2}$. This is usually the result of small errors in the identification of
a strong frequency, leaving residual power in the Fourier spectrum. It is
likely that this then impacts the following sequence $f_{t5}$ to $f_{t10}$
which all seem close frequencies to other identifications, and possibly one-
day aliases. The PbP frequencies identified explained more variation,
accounting for a total of $83\%$ of the line profile variation.
Overall the PbP variations have been shown to remove the variation from the
data more accurately, demonstrated by the higher percentage of variation
removed for each moment. This leads us to conclude that the frequencies found
using the highest peaks in the moment methods are less reliable. It is notable
that the SIGSPEC identified frequencies are more reliable than the highest
peak selection.
As an additional test of the robustness of the derived frequencies, a
synthetic dataset, modelled on the time spacings and modes identified in
Section 5, was created to compare with the PbP derived frequencies. The
synthetic line profiles were phased and plotted to compare with the real data
in Figure 3. The synthetic profiles strongly match the observed profiles,
strengthening our confidence in the frequency identification.
Table 4: Resulting frequencies from the First, Second and Third Moments found
using peaks in the Fourier spectra and in SIGSPEC
.
First Moment
---
| Highest Peak | SIGSPEC
ID | Freq. | Var. | Freq. | Sig.
$f_{f1}$ | 1.1835 | $16\%$ | 1.18624 | 26
$f_{f2}$ | 1.6100 | $32\%$ | |
$f_{f3}$ | 1.4017 | $42\%$ | 1.39591 | 26
$f_{f4}$ | 1.2247 | $52\%$ | 1.2156 | 28
| 1.2154 | | 2.2273 | 21
$f_{f5}$ | 0.6490 | $57\%$ | | 22
$f_{f6}$ | 1.5627 | $63\%$ | 1.5596 | 20
$f_{f7}$ | 0.7559 | $66\%$ | |
$f_{f8}$ | 1.5179 | $69\%$ | |
$f_{f9}$ | | | 1.6812 | 22
$f_{f10}$ | | | 0.2436 | 22
$f_{f11}$ | | | 2.1362 | 17
$f_{f12}$ | | | 0.0002 | 17
$f_{f13}$ | | | 2.7789 | 15
$f_{f14}$ | | | 0.6736 | 15
Second Moment
| Highest Peak | SIGSPEC
ID | Freq. | Var. | Freq. | Sig.
$f_{s1}$ | 0.0006 | $7\%$ | |
$f_{s2}$ | 0.2159 | $22\%$ | |
$f_{s3}$ | 0.3767 | $38\%$ | |
| 1.3962 | | 1.3959 | 19
$f_{s4}$ | 1.5596 | $46\%$ | 1.5596 | 20
$f_{s6}$ | 2.1786 | $57\%$ | |
$f_{s7}$ | | | 0.1622 | 25
$f_{s8}$ | | | 0.9449 | 23
$f_{s9}$ | | | 0.9061 | 19
$f_{s10}$ | | | 0.7382 | 19
$f_{s11}$ | | | 2.6809 | 17
Third Moment
| Highest Peak | SIGSPEC
ID | Freq. | Var. | Freq. | Sig.
$f_{t1}$ | 1.3987 | $17\%$ | 1.3956 | 31
$f_{t2}$ | 1.1864 | $40\%$ | 1.1862 | 25
| 2.1833 | | |
$f_{t3}$ | 1.6810 | $49\%$ | 1.6809 | 23
$f_{t4}$ | 1.3489 | $58\%$ | |
$f_{t5}$ | 1.1661 | $64\%$ | |
$f_{t6}$ | 2.5624 | $32\%$ | 0.5597 | 21
$f_{t7}$ | 2.7952 | $41\%$ | |
$f_{t8}$ | 1.1716 | $49\%$ | |
$f_{t9}$ | 3.5169 | $58\%$ | |
$f_{t10}$ | 1.2658 | $64\%$ | |
$f_{t11}$ | | | 1.2156 | 27
$f_{t12}$ | | | 2.1923 | 20
$f_{t13}$ | | | 0.2437 | 18
Figure 3: Line profile residuals phased on the frequencies identified $f_{p1}$
(top) to $f_{p5}$ (bottom) compared with synthetic profiles. The left panel
shows the observed profiles and the right is a synthetic model of the line
profile using the modes identified in Section 5. All are (1,1) modes.
### 3.3 Frequency Results
The frequencies chosen to proceed with in the analysis were those originally
identified in the PbP results, $f_{p1}$ to $f_{p5}$ and $f_{p8}$ as noted in
Table 3. This choice was made based on the prevalence of these frequencies
through the other techniques used, the decreased sensitivity of the PbP
technique to asymmetric variations and generally observed higher signal-to-
noise of the PbP frequencies.
It is clear from the above sections that, due to the inclusion of multi-site
data, 1-day aliasing was not a big problem when identifying frequencies
present in these data.
Identifying the uncertainty range on the frequencies found is a complex task.
When using SIGSPEC we can use the relation of Kallinger et al. (2008):
$\sigma(f)=\frac{1}{T*\sqrt{sig(a)}},$ (1)
where T is the time base of observations in days and $sig(a)$ is the spectral
significance of the frequency. For the frequencies $f_{p1}$ to $f_{p10}$
identified in the first moment method this gives an uncertainty estimate of
$\pm 0.0002$. Additionally we used the estimation from Montgomery & Odonoghue
(1999) who propose
$\sigma(f)=\sqrt{\frac{6}{N}}\frac{1}{\pi{T}}\frac{\sigma(m)}{A}.$ (2)
Here N is the number of observations. The value $\sigma(f)$ is a one-sigma
uncertainty with an amplitude (A) root-mean-square deviation $\sigma(m)$. This
method was used to derive (an underestimate) of the uncertainties of the
pixel-by-pixel frequencies. The results for the frequencies $f_{p1}$ to
$f_{p5}$ ranged from $\pm 0.00013$ to $\pm 0.00018$. Given the observed
differences between the frequencies identified in the PbP and the moment
methods we view $\pm 0.0002$ to be a good estimate of the uncertainty.
The data were also analysed for possible frequency combinations. It was found
that $f_{p5}(1.559)+f_{p3}(1.681)=3.240$d-1 which is possibly the same as
$f_{p7}$(1.246) and strengthens our earlier removal of this frequency.
It is clear from the above sections that many frequencies extracted from the
data are robust as they appear in multiple methods with clear line profile
variations. The first five frequencies in the PbP analysis appear in almost
every analysis method. The high signal-to-noise in the line profiles for these
frequencies makes them suitable for mode identification. The frequency
$f_{p8}$, although showing evidence for being an independent frequency, showed
a misshapen standard deviation profile, which meant there was not enough
certainty in this frequency identification to proceed with a mode
identification. The final PbP frequency $f_{p10}$ showed no evidence of
appearing in other analysis methods and was rejected for mode identification.
Even with all these frequencies identified, variations in the data remain. The
PbP method only removed $50\%$ of the variation and the moment methods up to
$80\%$. Some of the remaining variation is due to the noise in the data and
possible slight misidentifications of the frequencies, but it is likely that
there remain multiple unidentified frequencies below the detection threshold
of the present data. The presence of these further frequencies is expected for
$\gamma$ Doradus stars as they have dozens and sometimes even hundreds of
frequencies identifiable in photometry. The large number of similar amplitude
frequencies make this star challenging to study. However already we are able
to see more frequencies than previously identified in spectroscopy, which
makes this a promising star for further study.
Table 5: Frequencies found in the HIPPARCOS white-light photometry. The first three frequencies match well with $f_{p4}$, $f_{p5}$ and $f_{p7}$. Frequency (d-1) | Uncertainty | Significance
---|---|---
1.2158 | $\pm$0.0003 | 7
1.5525 | $\pm$0.0003 | 6
2.2449 | $\pm$0.0003 | 6
2.5665 | $\pm$0.0004 | 5
Table 6: Frequencies (d-1) found in each of the Geneva photometry filters and significances using SIGSPEC. | B | B1 | B2 | G | U | V | V1
---|---|---|---|---|---|---|---
| f. | sig. | f. | sig. | f. | sig. | f. | sig. | f. | sig. | f. | sig. | f. | sig.
$f_{1}$ | 1.215 | 13 | 1.215 | 13 | 1.215 | 12 | 2.218 | 11 | 1.215 | 13 | 1.215 | 11 | 1.215 | 12
$f_{2}$ | 0.09 | 8 | 0.09 | 8 | 0.09 | 8 | 0.393 | 6 | 0.07 | 5 | 0.09 | 7 | 0.095 | 8
$f_{3}$ | 1.396 | 8 | 1.396 | 8 | 1.396 | 8 | 0.071 | 6 | | | 1.396 | 7 | 1.396 | 6
$f_{4}$ | 2.567 | 5 | 1.565 | 5 | 3.56 | 4 | 3.32 | 5 | | | 4.32 | 4 | 3.32 | 6
$f_{5}$ | | | | | | | 3.19 | 4 | | | | | 2.95 | 4
## 4 Photometric Frequency and Mode Identification
Two photometric datasets were analysed for frequencies. The first was a time
series of white light observations from the satellite HIPPARCOS (Perryman &
ESA, 1997). Specifically these were taken in the Hp filter. The data span a
period of 1166 days from November 1989 to February 1993 during which 122
measurements were taken. The second dataset was Geneva photometry taken on the
0.7m Swiss telescope at La Silla over a period of 25 years from 1973 to 1997.
A total of 174 observations for each filter in this time. This dataset has
been extensively analysed in Aerts et al. (2004). We present our re-analysis
to complement the spectroscopic results above.
### 4.1 HIPPARCOS Photometric Frequencies
The Fourier analysis of the HIPPARCOS data measurements have a range of 0.09
Hp magnitudes with an average uncertainty of 0.1 Hp magnitudes. It was
difficult to identify clear frequencies from the Fourier spectra, but the
highest peak was at 1.2701 d-1. Using SIGSPEC we were able to identify the
frequencies and their significances as shown in Table 5.
### 4.2 Geneva Photometric Frequencies
The seven filters of the Geneva photometry provide us with enough information
to extract frequencies and identify the $l$ modes. Each of the seven filters
showed very similar variation and this was reflected in the individual Fourier
spectra. Frequency peaks were observed at 1.218 d-1, 1.396 d-1 and 0.0907 d-1
(or one-day aliases) in most filters. To formalise the frequency
identification we used SIGSPEC to identify the frequencies and their
significances as shown in Table 6. These frequencies match well to those
previously found in the same data as is discussed further in Section 6.
### 4.3 Geneva Photometry Modes
The identification of the $l$ value of the spectroscopic frequencies $f_{p1}$
to $f_{p5}$ was attempted using the seven filters in the Geneva photometric
data. The identification was done using the amplitude ratio method in FAMIAS
with modes from $l=1$ to $l=3$ tested in all filters. All five frequencies
were found to be solely consistent with $l=1$ modes distinguished primarily by
the first amplitude ratio. The results are largely identical to Aerts et al.
(2004), Their Figure 6 shows the unique fit to the $l=1$ mode. This method,
although not as powerful as spectroscopic mode identification, gives us
independent support to the $l=1$ mode fits of the star.
## 5 Mode Identification of Spectroscopic Line Profiles
Figure 4: The $95\%$ confidence limit (solid line) of the zero-point fit to
the $v$sin$i$. The dashed lines show the limits at the confidence level and
the dotted line indicates the minimum and hence best fit value.
The mode identification was performed using the PbP frequencies $f_{p1}$ to
$f_{p5}$, as chosen in Section 3.3 ($f_{p8}$ not being considered due to the
distorted line profile variations). The modes of the individual frequencies
were first identified and then a best fit, including all five frequencies, was
computed. Initially the zero-point profile was fitted to determine the basic
line parameters. The best fit is shown in Figure 5(a). The $v$sin$i$ was found
to converge at $63.9$ km s-1 with a $\chi^{2}$ of 103. Figure 4 shows the 95%
confidence limit on the determination of $v$sin$i$ which gives a range of
$63.9\pm 0.5$ km s-1. The zero-point fit set the initial line and stellar
parameters of the space searched. The details of the parameters are given in
Table 8 and the resulting best fit modes are given in Table 9. The unusually
high values for mass and radius are discussed in Section 5.1.
The best fit models found for each individual frequency are plotted in Figures
5(b) \- 5(f). The results in Table 9 and Figures 6(a) to 6(e) show that the
mode identifications are well determined as there is no ambiguity in choosing
the model with the lowest $\chi^{2}$. The inclination appears to lie in the
region $20\degree-45\degree$, with the simultaneous fit giving a value of
$27\degree$ for a $\chi^{2}$ of 16.59. Formally the 95% confidence limit of
this parameter gives $27\degree$ ${}^{+52}_{-12}$, so it is poorly constrained
by the mode identification. Given the $v$sin$i$ of the star, a value near
$30\degree$ is plausible as described further in Section 5.1. The $v$sin$i$
value determined varies slightly between the fits as it is modified to change
the width of the standard deviation profile. The measured $v$sin$i$ of the
star is best determined from the fit to the zero-point profile given above.
The detection of five independent frequencies and the occurrence of a large
number of (1,1) modes makes this star an excellent candidate for further
asteroseismic analysis. The prevalence of the (1,1) modes may indicate
sequencing of the $n$-values, or the number of interior shells. It also is
possible that the (1,1) modes may be linked to the rapid rotation of the star.
This is discussed in the context of all $\gamma$ Doradus stars in Section 6.
### 5.1 Rotation and Pulsation Parameters
The results of a preliminary mode identification were tested using the
rotation and pulsation parameter tools in FAMIAS. Using mass = 1.5 $M_{\odot}$
and radius = 1.7 $R_{\odot}$ (typical for a $\gamma$ Doradus star), $v$sin$i$
= 64 km s-1 and various inclinations indicates the rotational parameters of
the star. Shown in Table 7 are the results for $i=30\degree$, which indicates
that this star is not approaching critical rotational velocity. This is the
case for all tested values of $i$ in the range $i=10\degree-90\degree$. The
rotational frequency of the star is dependent on the inclination, ranging from
4.28 d-1 at $i=10\degree$ to 0.74d-1 at $i=90\degree$. The rotational
frequency for a $\gamma$ Doradus star is expected to be on the same order as
the pulsation frequency and these values fall within this range.
Table 7: Rotational parameters and critical limits for a pulsating star with mass = 1.5 $M_{\odot}$,radius = 1.7 $R_{\odot}$, $v$sin$i$ = 64 km s-1 observed at an inclination of $i=30\degree$. vrot | 128 km s-1
---|---
Trot | 0.67 d
$f_{rot}$ | 1.49 d-1
vcrit | 410 km s-1
$v$sin$i_{crit}$ | 205 km s-1
$i_{crit}$ | 9$\degree$
The pulsation parameter tool can be used to get an indication of the
horizontal-to-vertical pulsation amplitude parameter ($\kappa$) and whether
the pulsation and rotation frequencies fall within the mode determination
limits of FAMIAS. The $\kappa$ value falls between the values 50-800 for
frequencies $f_{1}-f_{2}$, which is above 1.0 as expected for $\gamma$ Doradus
stars. These values then show the natural pulsational frequencies to lie
between 0.2 d-1 and 0.7 d-1. Taking the ratio $f_{rot}$/$f_{co-rot}$ gives
values in the range 1-5 for the determined frequencies and modes (all $m=1$).
FAMIAS has an operational range of $f_{rot}$/$f_{co-rot}\leq 0.5$, being
designed to deal with p-mode pulsations with much larger vertical pulsation
components. The very high values of $\kappa$ for the g-mode pulsations in
$\gamma$ Doradus stars and the high rotation rate of this star mean we are
operating beyond this limit. To obtain a reasonable mode identification the
mass and radius were extended to non-physical values for a $\gamma$ Doradus
star to provide observable amplitudes of pulsation (see Table 8). However the
mass and radius are solely used for the calculation of the ratio $\kappa$, and
do not affect other aspects of the fit. Making these changes allowed us to
consider pulsations with $f_{rot}$/$f_{co-rot}$ values of 0.3-0.5 for the
multi-frequency fit in Table 9.
Including more physics to more accurately describe the effects of rotation of
g-modes in current pulsation models would ultimately solve such problems.
Townsend (2003) investigated the effect of higher rotation on g-modes with
high $n$ and low $l$ by increasing the Coriolis forces. It was seen that the
identification of prograde modes (defined here as $m>0$, in Townsend 2003 as
$m<0$), have smaller differences in the rotating scenarios than retrograde
modes. Smaller values of $m$ are also less affected. There is also a
discussion of this in Wright et al. (2011) who estimate the true limits of the
$f_{rot}$/$f_{co-rot}$ ratio in FAMIAS for various $m$ for a $\gamma$ Doradus
range frequency and found FAMIAS could identify modes up to $f_{rot}$/$f_{co-
rot}=1$ for a $m=1$ mode. These results indicate that the mode identification
is unlikely to be affected by the rotation of this star. We can be confident
that using non-physical values for mass and radius affects only the observed
amplitude of the pulsation and not the mode, itself.
A convincing mode identification of the frequencies $f_{1}-f_{5}$ was
obtained.
Table 8: Stellar parameters used in the mode identification. Fixed values for Teff, [M/H] and $\log g$ were taken from Bruntt et al. (2008) and $v$sin$i$ values from the ranges in Bruntt et al. (2008); De Cat et al. (2006); Dupret et al. (2005); Aerts et al. (2004); Eyer & Aerts (2000). Parameter | Fixed | Min | Max | Step
---|---|---|---|---
| Value | | |
Radius (solar units) | | 1 | 10 | 0.1
Mass (solar units) | | 0.5 | 50 | 0.01
Temperature (K) | 7193 | | |
Metallicity [M/H] | 0.13 | | |
log $g$ | 4.18 | | |
Inclination ($\degree$) | | 0 | 90 | 1
$v$sin$i$ (km s-1) | | 60 | 75 | 0.1
Table 9: Results of mode identification of all four frequencies individually after a least-squares fit is applied (lsf), and all four frequencies simultaneously (sim). | Mode | $\chi^{2}$ | Inc. | Amp | Ph.
---|---|---|---|---|---
| ID | | ($\degree$) | (km s-1) |
$f_{1}$ lsf | (1,1) | 7.65 | 43.15 | 0.82 | 0.32
$f_{2}$ lsf | (1,1) | 7.00 | 35.79 | 0.74 | 0.80
$f_{3}$ lsf | (1,1) | 7.57 | 22.40 | 1.50 | 0.18
$f_{4}$ lsf | (1,1) | 10.43 | 33.11 | 1.50 | 0.72
$f_{5}$ lsf | (1,1) | 5.49 | 33.11 | 0.50 | 0.61
$f_{1}$ sim | (1,1) | 16.59 | 29.67 | 2.06 | 0.31
$f_{2}$ sim | (1,1) | | | 1.07 | 0.81
$f_{3}$ sim | (1,1) | | | 2.90 | 0.18
$f_{4}$ sim | (1,1) | | | 1.07 | 0.71
$f_{5}$ sim | (1,1) | | | 2.17 | 0.60
(a) Fit of the zero-point profile ($\chi^{2}=104$).
(b) Mode identification of $f_{p1}=1.3959$ d-1 ($\chi^{2}=7.7$).
(c) Mode identification of $f_{p2}=1.1863$ d-1 ($\chi^{2}=7.0$).
(d) Mode identification of $f_{p3}=1.6812$ d-1 ($\chi^{2}=7.6$).
(e) Mode identification of $f_{p4}=1.2157$ d-1 ($\chi^{2}=10.4$).
(f) Mode identification of $f_{p5}=1.5596$ d-1 ($\chi^{2}=5.5$).
Figure 5: The fit (dashed) of the mode identification to the mean line
profile, variation and phase (solid) of the five identified frequencies. All
have been identified as (1,1) modes with the labelled $\chi^{2}$ for the best
fit. An indication of the maximum uncertainty is given on each plot.
## 6 Discussion
This study allowed a direct comparison between single-site and multi-site
data. From this we can judge the usefulness of large single-site datasets. The
findings show that although the addition of further sites increased the
amplitudes of the frequencies, it also elevated the base noise level.
Additionally it is clear from the window function spectra in Figure 2 that the
1-day aliasing pattern is reduced but not entirely eliminated. The above leads
us to conclude that the addition of multi-site data is useful, but not
required to extract frequencies from sufficiently large datasets. We must also
require that the data from any one site have a sufficient number of
observations to produce a balanced (well sampled in phase space) mean line
profile in order to combine the cross-correlated line profiles with the
highest precision.
The frequencies found in the PbP method were the most reliable and were
consistent with all the other analysis methods. Past papers have analysed the
photometric data from HIPPARCOS and also some multi-coloured photometry. In
Handler (1999), the author finds a frequencies of 2.18 d-1 and 1.2155 d-1
($f_{p4}$). In this paper we found frequencies $f_{p4}$, $f_{p5}$ and (2x)
$f_{p8}$ in the HIPPARCOS photometry.
The multi-colour photometry was previously published by Aerts et al. (2004),
who found frequencies equivalent to $f_{p4}$, $f_{p1}$ and a 1-day alias of
$f_{p3}$. In the same data we find $f_{p4}$, $f_{p1}$ and $f_{p5}$. The mode
identification showed the frequencies $f_{p4}$, $f_{p1}$ and $f_{p3}$ to best
fit $l=1$ modes, the same as found in this analysis. The same result was found
by Dupret et al. (2005) who showed that the three frequencies from Aerts et
al. (2004) are $l=1$ modes when using time-dependent convection models.
Aerts et al. (2004) also describe the spectroscopic dataset taken with CORALIE
that was also used in this analysis. The authors did not find any frequencies
from the spectroscopic data alone (it is, as the authors note, too small a
dataset for spectroscopic analysis). The imposition of the $f_{p4}$, $f_{p1}$
frequencies found in their photometric analysis did provide some harmonic fits
with low amplitudes.
(a) $\chi^{2}$ values for $l=0-3$ for $f_{p1}=1.3959$d-1.
Best fit value is for ($l,m$) = (1,1).
(b) $\chi^{2}$ values for $l=0-3$ for $f_{p2}=1.1863$d-1.
Best fit value is for ($l,m$) = (1,1).
(c) $\chi^{2}$ values for $l=0-4$ for $f_{p3}=1.6812$d-1.
Best fit value is for ($l,m$) = (1,1).
(d) $\chi^{2}$ values for $l=0-3$ for $f_{p4}=1.2157$d-1.
Best fit value is for ($l,m$) = (1,1).
(e) $\chi^{2}$ values for $l=0-3$ for $f_{p5}=1.5596$d-1.
Best fit value is for ($l,m$) = (1,1).
Figure 6: Lowest $\chi^{2}$ values for each possible ($l,m$) combination for
the final identified frequencies. The best fit $\chi^{2}$ is identified in
bold.
A study applying the frequency ratio method to the frequencies found in Aerts
et al. (2004) was done by Moya et al. (2005), who found three models
consistent with $l=1$ modes. The models have a $T$= $6760K$, log$g=3.88-4.12$
and stellar ages around $2-3$ Myr.
Considering all the prior studies, all of the candidate frequencies we confirm
($f_{p1}$ to $f_{p5}$ and $f_{p8}$) are well supported. The mode
identification is also partially confirmed by photometry. There is no
suggestion of different frequencies observed in photometry and spectroscopy as
there are from some $\gamma$ Doradus stars (see Maisonneuve et al. 2011;
Uytterhoeven et al. 2008).
A prevalence of (1,1) modes in this star, and in $\gamma$ Doradus stars in
general, is beginning to emerge. This includes two modes in HD 135825
(Brunsden et al., 2012), two in $\gamma$ Doradus (Balona et al., 1996; Dupret
et al., 2005), one in HD 40745 (Maisonneuve et al., 2011), one in HR8799
(Wright et al., 2011), two in HD 189631 (Davie, 2011) and two in HD 65526
(Greenwood, 2012). This is possibly due to the large surface area covered in
each segment of a (1,1) mode, meaning pulsations have larger amplitudes and
large changes in amplitude across the stellar surface. This indicates an
observational selection effect. Additionally Balona et al. (2011) discuss the
prevalence of $m=\pm 1$ modes in a sample of $\gamma$ Doradus candidates from
Kepler photometry. These authors assume the dominant frequency to be the
rotational frequency and they then show further frequencies to be close to
this value. This constrains the light maxima to once per rotation cycle,
requiring a $m=\pm 1$ mode. The occurrence of five of these such modes in this
star suggests some physical linking between them. The identification of
several (1,1) modes in this star led to an investigation into the period
spacings of the six identified frequencies (Table 3). The asymptotics of
oscillation theory (Tassoul, 1980) predicts a characteristic period spacing
for high-order g-modes of the same low degree ($l$) for sequential values of
$n$. An investigation into the period spacings of the PbP identified
frequencies shows that the spacing between $f_{p1}$-$f_{p2}$ and
$f_{p1}$-$f_{p3}$ to be close ($0.1266$d and $0.1215$d respectively). This
suggests they could be subsequent values of $n$ if we allow our frequencies to
vary by $\pm 0.003$ d-1. The frequencies $f_{p4}$, $f_{p5}$ and $f_{p8}$
however do not fit with this spacing and additionally it may be that the close
spacing of $f_{p2}$ and $f_{p4}$ is inconsistent with current theoretical
models. The identification of the frequencies could be further improved by
photometric studies to confirm this. Ultimately the sequencing of $n$-values
could provide us with direct information about the stellar interior.
For the reasons above we propose this star as an excellent candidate with
which to test asteroseismic models and potentially give us insight into the
pulsational behaviour of $\gamma$ Doradus stars.
## 7 Acknowledgements
This work was supported by the Marsden Fund administered by the Royal Society
of New Zealand.
The authors acknowledge the assistance of staff at Mt John University
Observatory, a research station of the University of Canterbury.
We appreciate the time allocated at other facilities for multi-site campaigns,
particularly McDonald Observatory and La Silla (European Southern
Observatory).
Gratitude must be extended to the numerous observers who make acquisition of
large datasets possible. We thank P. M. Kilmartin at MJUO and all the
observers at La Silla and the HIPPARCOS team for their dedication to acquiring
precise data.
This research has made use of the SIMBAD astronomical database operated at the
CDS in Strasbourg, France.
Mode identification results obtained with the software package FAMIAS
developed in the framework of the FP6 European Coordination action HELAS
(http://www.helas-eu.org/).
We thank our reviewer Gerald Handler for his helpful comments that improved
this paper.
## 8 Supporting Information
Additional Supporting information may be found in the online version of this
article:
Data file. The line profile data including Modified Julian Date, velocity (on
a relative scale) and the intensity at each of the 120 velocity sampling
points for each profile. Line 1 is the axis scale, lines 2-478 are MJUO
observations, lines 479-526 are La Silla observations and lines 527-586 are
McDonald observations.
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|
arxiv-papers
| 2012-09-26T21:20:18 |
2024-09-04T02:49:35.662607
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Brunsden, K. R. Pollard, P. L. Cottrell, D. J. Wright, P. De Cat",
"submitter": "Emily Brunsden",
"url": "https://arxiv.org/abs/1209.6081"
}
|
1209.6199
|
# Determination of the number of $\psi^{\prime}$ events at BESIII
M. Ablikim1, M. N. Achasov5, O. Albayrak3, D. J. Ambrose39, F. F. An1, Q.
An40, J. Z. Bai1, Y. Ban27, J. Becker2, J. V. Bennett17, M. Bertani18A, J. M.
Bian38, E. Boger20,a, O. Bondarenko21, I. Boyko20, R. A. Briere3, V. Bytev20,
X. Cai1, O. Cakir35A, A. Calcaterra18A, G. F. Cao1, S. A. Cetin35B, J. F.
Chang1, G. Chelkov20,a, G. Chen1, H. S. Chen1, J. C. Chen1, M. L. Chen1, S. J.
Chen25, X. Chen27, Y. B. Chen1, H. P. Cheng14, Y. P. Chu1, D. Cronin-
Hennessy38, H. L. Dai1, J. P. Dai1, D. Dedovich20, Z. Y. Deng1, A. Denig19, I.
Denysenko20,b, M. Destefanis43A,43C, W. M. Ding29, Y. Ding23, L. Y. Dong1, M.
Y. Dong1, S. X. Du46, J. Fang1, S. S. Fang1, L. Fava43B,43C, F. Feldbauer2, C.
Q. Feng40, R. B. Ferroli18A, C. D. Fu1, J. L. Fu25, Y. Gao34, C. Geng40, K.
Goetzen7, W. X. Gong1, W. Gradl19, M. Greco43A,43C, M. H. Gu1, Y. T. Gu9, Y.
H. Guan6, A. Q. Guo26, L. B. Guo24, Y. P. Guo26, Y. L. Han1, F. A. Harris37,
K. L. He1, M. He1, Z. Y. He26, T. Held2, Y. K. Heng1, Z. L. Hou1, H. M. Hu1,
J. F. Hu36, T. Hu1, G. M. Huang15, G. S. Huang40, J. S. Huang12, X. T.
Huang29, Y. P. Huang1, T. Hussain42, C. S. Ji40, Q. Ji1, Q. P. Ji26,c, X. B.
Ji1, X. L. Ji1, L. L. Jiang1, X. S. Jiang1, J. B. Jiao29, Z. Jiao14, D. P.
Jin1, S. Jin1, F. F. Jing34, N. Kalantar-Nayestanaki21, M. Kavatsyuk21, W.
Kuehn36, W. Lai1, J. S. Lange36, C. H. Li1, Cheng Li40, Cui Li40, D. M. Li46,
F. Li1, G. Li1, H. B. Li1, J. C. Li1, K. Li10, Lei Li1, Q. J. Li1, S. L. Li1,
W. D. Li1, W. G. Li1, X. L. Li29, X. N. Li1, X. Q. Li26, X. R. Li28, Z. B.
Li33, H. Liang40, Y. F. Liang31, Y. T. Liang36, G. R. Liao34, X. T. Liao1, B.
J. Liu1, C. L. Liu3, C. X. Liu1, C. Y. Liu1, F. H. Liu30, Fang Liu1, Feng
Liu15, H. Liu1, H. H. Liu13, H. M. Liu1, H. W. Liu1, J. P. Liu44, K. Y. Liu23,
Kai Liu6, P. L. Liu29, Q. Liu6, S. B. Liu40, X. Liu22, Y. B. Liu26, Z. A.
Liu1, Zhiqiang Liu1, Zhiqing Liu1, H. Loehner21, G. R. Lu12, H. J. Lu14, J. G.
Lu1, Q. W. Lu30, X. R. Lu6, Y. P. Lu1, C. L. Luo24, M. X. Luo45, T. Luo37, X.
L. Luo1, M. Lv1, C. L. Ma6, F. C. Ma23, H. L. Ma1, Q. M. Ma1, S. Ma1, T. Ma1,
X. Y. Ma1, Y. Ma11, F. E. Maas11, M. Maggiora43A,43C, Q. A. Malik42, Y. J.
Mao27, Z. P. Mao1, J. G. Messchendorp21, J. Min1, T. J. Min1, R. E.
Mitchell17, X. H. Mo1, C. Morales Morales11, C. Motzko2, N. Yu. Muchnoi5, H.
Muramatsu39, Y. Nefedov20, C. Nicholson6, I. B. Nikolaev5, Z. Ning1, S. L.
Olsen28, Q. Ouyang1, S. Pacetti18B, J. W. Park28, M. Pelizaeus2, H. P. Peng40,
K. Peters7, J. L. Ping24, R. G. Ping1, R. Poling38, E. Prencipe19, M. Qi25, S.
Qian1, C. F. Qiao6, X. S. Qin1, Y. Qin27, Z. H. Qin1, J. F. Qiu1, K. H.
Rashid42, G. Rong1, X. D. Ruan9, A. Sarantsev20,d, B. D. Schaefer17, J.
Schulze2, M. Shao40, C. P. Shen37,e, X. Y. Shen1, H. Y. Sheng1, M. R.
Shepherd17, X. Y. Song1, S. Spataro43A,43C, B. Spruck36, D. H. Sun1, G. X.
Sun1, J. F. Sun12, S. S. Sun1, Y. J. Sun40, Y. Z. Sun1, Z. J. Sun1, Z. T.
Sun40, C. J. Tang31, X. Tang1, I. Tapan35C, E. H. Thorndike39, D. Toth38, M.
Ullrich36, G. S. Varner37, B. Wang9, B. Q. Wang27, D. Wang27, D. Y. Wang27, K.
Wang1, L. L. Wang1, L. S. Wang1, M. Wang29, P. Wang1, P. L. Wang1, Q. Wang1,
Q. J. Wang1, S. G. Wang27, X. L. Wang40, Y. D. Wang40, Y. F. Wang1, Y. Q.
Wang29, Z. Wang1, Z. G. Wang1, Z. Y. Wang1, D. H. Wei8, J. B. Wei27, P.
Weidenkaff19, Q. G. Wen40, S. P. Wen1, M. Werner36, U. Wiedner2, L. H. Wu1, N.
Wu1, S. X. Wu40, W. Wu26, Z. Wu1, L. G. Xia34, Z. J. Xiao24, Y. G. Xie1, Q. L.
Xiu1, G. F. Xu1, G. M. Xu27, H. Xu1, Q. J. Xu10, X. P. Xu32, Z. R. Xu40, F.
Xue15, Z. Xue1, L. Yan40, W. B. Yan40, Y. H. Yan16, H. X. Yang1, Y. Yang15, Y.
X. Yang8, H. Ye1, M. Ye1, M. H. Ye4, B. X. Yu1, C. X. Yu26, H. W. Yu27, J. S.
Yu22, S. P. Yu29, C. Z. Yuan1, Y. Yuan1, A. A. Zafar42, A. Zallo18A, Y.
Zeng16, B. X. Zhang1, B. Y. Zhang1, C. Zhang25, C. C. Zhang1, D. H. Zhang1, H.
H. Zhang33, H. Y. Zhang1, J. Q. Zhang1, J. W. Zhang1, J. Y. Zhang1, J. Z.
Zhang1, S. H. Zhang1, X. J. Zhang1, X. Y. Zhang29, Y. Zhang1, Y. H. Zhang1, Y.
S. Zhang9, Z. P. Zhang40, Z. Y. Zhang44, G. Zhao1, H. S. Zhao1, J. W. Zhao1,
K. X. Zhao24, Lei Zhao40, Ling Zhao1, M. G. Zhao26, Q. Zhao1, Q. Z. Zhao9,f,
S. J. Zhao46, T. C. Zhao1, X. H. Zhao25, Y. B. Zhao1, Z. G. Zhao40, A.
Zhemchugov20,a, B. Zheng41, J. P. Zheng1, Y. H. Zheng6, B. Zhong1, J. Zhong2,
Z. Zhong9,f, L. Zhou1, X. K. Zhou6, X. R. Zhou40, C. Zhu1, K. Zhu1, K. J.
Zhu1, S. H. Zhu1, X. L. Zhu34, Y. C. Zhu40, Y. M. Zhu26, Y. S. Zhu1, Z. A.
Zhu1, J. Zhuang1, B. S. Zou1, J. H. Zou1 (BESIII Collaboration) 1 Institute of
High Energy Physics, Beijing 100049, P. R. China
2 Bochum Ruhr-University, 44780 Bochum, Germany
3 Carnegie Mellon University, Pittsburgh, PA 15213, USA
4 China Center of Advanced Science and Technology, Beijing 100190, P. R. China
5 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Graduate University of Chinese Academy of Sciences, Beijing 100049, P. R.
China
7 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
8 Guangxi Normal University, Guilin 541004, P. R. China
9 GuangXi University, Nanning 530004,P.R.China
10 Hangzhou Normal University, Hangzhou 310036, P. R. China
11 Helmholtz Institute Mainz, J.J. Becherweg 45,D 55099 Mainz,Germany
12 Henan Normal University, Xinxiang 453007, P. R. China
13 Henan University of Science and Technology, Luoyang 471003, P. R. China
14 Huangshan College, Huangshan 245000, P. R. China
15 Huazhong Normal University, Wuhan 430079, P. R. China
16 Hunan University, Changsha 410082, P. R. China
17 Indiana University, Bloomington, Indiana 47405, USA
18 (A)INFN Laboratori Nazionali di Frascati, Frascati, Italy; (B)INFN and
University of Perugia, I-06100, Perugia, Italy
19 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, 55099
Mainz, Germany
20 Joint Institute for Nuclear Research, 141980 Dubna, Russia
21 KVI/University of Groningen, 9747 AA Groningen, The Netherlands
22 Lanzhou University, Lanzhou 730000, P. R. China
23 Liaoning University, Shenyang 110036, P. R. China
24 Nanjing Normal University, Nanjing 210046, P. R. China
25 Nanjing University, Nanjing 210093, P. R. China
26 Nankai University, Tianjin 300071, P. R. China
27 Peking University, Beijing 100871, P. R. China
28 Seoul National University, Seoul, 151-747 Korea
29 Shandong University, Jinan 250100, P. R. China
30 Shanxi University, Taiyuan 030006, P. R. China
31 Sichuan University, Chengdu 610064, P. R. China
32 Soochow University, Suzhou 215006, China
33 Sun Yat-Sen University, Guangzhou 510275, P. R. China
34 Tsinghua University, Beijing 100084, P. R. China
35 (A)Ankara University, Ankara, Turkey; (B)Dogus University, Istanbul,
Turkey; (C)Uludag University, Bursa, Turkey
36 Universitaet Giessen, 35392 Giessen, Germany
37 University of Hawaii, Honolulu, Hawaii 96822, USA
38 University of Minnesota, Minneapolis, MN 55455, USA
39 University of Rochester, Rochester, New York 14627, USA
40 University of Science and Technology of China, Hefei 230026, P. R. China
41 University of South China, Hengyang 421001, P. R. China
42 University of the Punjab, Lahore-54590, Pakistan
43 (A)University of Turin, Turin, Italy; (B)University of Eastern Piedmont,
Alessandria, Italy; (C)INFN, Turin, Italy
44 Wuhan University, Wuhan 430072, P. R. China
45 Zhejiang University, Hangzhou 310027, P. R. China
46 Zhengzhou University, Zhengzhou 450001, P. R. China
a also at the Moscow Institute of Physics and Technology, Moscow, Russia
b on leave from the Bogolyubov Institute for Theoretical Physics, Kiev,
Ukraine
c Nankai University, Tianjin,300071,China
d also at the PNPI, Gatchina, Russia
e now at Nagoya University, Nagoya, Japan
f Guangxi University,Nanning,530004,China
###### Abstract
The number of $\psi^{\prime}$ events accumulated by the BESIII experiment from
March 3 through April 14, 2009, is determined by counting inclusive hadronic
events. The result is $106.41\times(1.00\pm 0.81\%)\times 10^{6}$. The error
is systematic only; the statistical error is negligible.
$\psi^{\prime}$, inclusive, hadron, Bhabha
###### pacs:
13.25.Gv, 13.66.Bc, 13.20.Gd
††preprint: APS/123-QED
## I Introduction
In 2009, the world’s largest $\psi^{\prime}$ sample to date was collected at
BESIII, allowing more extensive and precise studies of $\psi^{\prime}$ decays.
The number of $\psi^{\prime}$ events, $N_{\psi^{\prime}}$, is important in all
$\psi^{\prime}$ analyses, including studies both of the direct decays of the
$\psi^{\prime}$, as well as its daughters, $\chi_{cJ}$, $h_{c}$, and
$\eta_{c}$. The precision of $N_{\psi^{\prime}}$ will directly affect the
precision of all these measurements.
In this paper, we determine $N_{\psi^{\prime}}$ with $\psi^{\prime}\rightarrow
inclusive~{}hadrons$, whose branching ratio is known rather precisely,
$(97.85\pm 0.13)$% PDG . Also, a large off-resonance continuum data sample at
$E_{cm}=3.650$ GeV with an integrated luminosity of 44 pb-1 was collected.
These events are very similar to the continuum background under the
$\psi^{\prime}$ peak. Since the energy difference is very small, we can use
the off-resonance data to estimate this background.
BEPCII is a double-ring $e^{+}e^{-}$ collider designed to provide $e^{+}e^{-}$
interactions with a peak luminosity of $10^{33}~{}\rm{cm}^{-2}\rm{s}^{-1}$ at
a beam current of 0.93 A. The cylindrical core of the BESIII detector consists
of a helium-based main drift chamber (MDC), a plastic scintillator time-of-
flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which
are all enclosed in a superconducting solenoidal magnet providing a 1.0 T
magnetic field. The solenoid is supported by an octagonal flux-return yoke
with resistive plate counter muon identifier modules interleaved with steel.
The acceptance for charged particles and photons is 93% over 4$\pi$ stereo
angle, and the charged-particle momentum and photon energy resolutions at 1
GeV are 0.5% and 2.5%, respectively.
The BESIII detector is modeled with a Monte Carlo (MC) simulation based on
geant4 geant4 ; geant42 . For the simulation of inclusive $\psi^{\prime}$
decays, we use the evtgen generator evtgen . Known $\psi^{\prime}$ decay
channels are generated according to branching ratios in the PDG PDG ; the
remaining unknown decays are generated by the lundcharm model lund .
## II Event selection
There are many types of events in the data collected at the $\psi^{\prime}$
energy point, including $\psi^{\prime}\rightarrow$ hadrons and lepton pairs
($e^{+}e^{-},~{}\mu^{+}\mu^{-}$, and $\tau^{+}\tau^{-}$), radiative returns to
the $J/\psi$, and $J/\psi$ decays from the extended tail of the $J/\psi$
Breit-Wigner distribution. In addition, there are non-resonance (QED)
processes, which make up the continuum background, including
$e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow$ hadrons, lepton pairs, and
$e^{+}e^{-}\rightarrow e^{+}e^{-}$ +X (X=hadrons, lepton pairs). Non-collision
events include cosmic rays, beam-associated background, and electronic noise.
The signal channel is the process $\psi^{\prime}\rightarrow$ hadrons. The data
collected at the off-resonance energy include all of the above except
$\psi^{\prime}\rightarrow$ hadrons and lepton pairs.
Event selection includes track level selection and event level selection. At
the track level, good charged tracks are required to pass within 1 cm of the
beam line in the plane perpendicular to the beam and within $\pm$15 cm from
the Interaction Point (IP) in the beam direction. Photon candidate showers
reconstructed from the EMC barrel region ($|\cos\theta|<0.8$) must have a
minimum energy of 25 MeV, while those in the end-caps
($0.86<|\cos\theta|<0.92$) must have at least 50 MeV. The showers in the
angular range between the barrel and end-cap are poorly reconstructed and
excluded from the analysis. Requirements on the EMC cluster timing are applied
to suppress electronic noise and energy deposits unrelated to the event.
At the event level, at least one good charged track is required. If the number
of good charged tracks is larger than 2, i.e. $N_{good}>2$, no additional
selection is needed. If $N_{good}=2$, where the Bhabha and dimuon events are
dominant backgrounds, the momentum of each track is required to be less than
1.7 GeV/$c$, and the opening angle between the two tracks is required to be
less than $176^{\circ}$ to suppress these backgrounds. Figures 1 and 2 show
scatter plots of the momentum of one track versus the momentum of the second
track for MC simulated Bhabha events and inclusive MC events with two charged
tracks, respectively. Figures 3 and 4 show the opening angle distributions of
MC simulated Bhabha events and inclusive MC events with two charged tracks,
respectively. In addition, $E_{visible}/E_{cm}>0.4$ is required to suppress
low energy background (LEB), comprised mostly of $e^{+}e^{-}\rightarrow
e^{+}e^{-}+X$ and double ISR events
($e^{+}e^{-}\to\gamma_{ISR}\gamma_{ISR}X$). Here, $E_{visible}$ denotes the
visible energy which is defined as the energy sum of all charged tracks
(calculated with the track momentum and assuming a $\pi^{\pm}$ mass) and
neutral showers, and $E_{cm}$ denotes the center-of-mass energy. Figure 5
shows the $E_{visible}$ distribution for data and inclusive MC events with two
charged tracks. The excess in data at low energy is from the LEB.
If $N_{good}=1$, at least two additional photons are required in an event.
From all photon pair combinations, the combination whose invariant mass,
$M_{\gamma\gamma}$, is closest to the $\pi^{0}$ mass is selected, and
$|M_{\gamma\gamma}-M_{\pi^{0}}|<0.015$ GeV/$c^{2}$ is required.
$E_{visible}/E_{cm}>0.4$ is also required to suppress the LEB. Figure 6 shows
the $M_{\gamma\gamma}$ distributions in the $\pi^{0}$ mass region for data and
MC simulation. Figure 7 shows the $E_{visible}$ distribution for data and
inclusive MC events. The excess in data at low energy is from LEB.
Figure 1: The distribution of $P_{2}$ versus $P_{1}$ from MC simulated Bhabha
events. The horizontal and vertical lines show the selection requirements to
remove Bhabha and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ events. Figure 2: The
distribution of $P_{2}$ versus $P_{1}$ from inclusive MC events with two
charged tracks. The horizontal and vertical lines show the selection
requirements to remove Bhabha and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ events. Figure
3: The distribution of angle between tracks for MC simulated Bhabha events.
The arrow shows the angle requirement used to remove most Bhabha events.
Figure 4: The distribution of angle between tracks for inclusive MC events
with two charged tracks. The arrow shows the angle requirement used to remove
most Bhabha events. Figure 5: The $E_{visible}/E_{cm}$ distribution for
$N_{good}=2$ events. Dots with error bars are data; the histogram is MC
simulation, normalized to $E_{visible}/E_{cm}>0.4$. Figure 6: The
$\gamma\gamma$ invariant mass ($M_{\gamma\gamma}$) distribution in the
$\pi^{0}$ mass region for $N_{good}=1$ events. Dots with error bars are data;
the histogram is MC simulation. Figure 7: The $E_{visible}/E_{cm}$
distribution for $N_{good}=1$ events. Dots with error bars are data; the
histogram is MC simulation, normalized to $E_{visible}/E_{cm}>0.3$.
The average Z-direction vertex for an event is defined as
$\bar{V}_{Z}=\frac{\sum\limits^{N_{good}}_{i=1}V_{Z}^{i}}{N_{good}},$
where $V_{Z}$ is the distance along the beam direction of the point of closest
approach of a track to the IP. Figure 8 shows the $\bar{V}_{Z}$ distribution
for $\psi^{\prime}$ data after the above selection. Events satisfying
$|\bar{V}_{Z}|<4.0$ cm are taken as signal, while events in the sideband
region 6.0 cm$<|\bar{V}_{Z}|<10.0$ cm are taken as non-collision background
events. The number of observed hadronic events ($N^{obs.}$) is determined by
$N^{obs.}=N_{signal}-N_{sideband}.$ (1)
Another method to determine the number of hadronic events (described below) is
to fit the average Z-vertex with a double Gaussian to describe the signal and
a polynomial to describe the non-collision events.
Figure 8: The average $Z$ vertex ($\bar{V}_{Z}$) distribution of hadronic
events in $\psi^{\prime}$ data. The curves are a double Gaussian to describe
the signal and a polynomial to describe the non-collision events. Figure 9:
The average $Z$ vertex ($\bar{V}_{Z}$) distribution of hadronic events in off-
resonance data. The curves are a double Gaussian to describe the signal and a
polynomial to describe the non-collision events.
## III Background subtraction
In principle, the number of QED events can be estimated from:
$N^{QED}={\cal L}\cdot\sigma\cdot\epsilon,$ (2)
where ${\cal L}$ is the luminosity, and $\sigma$ and $\epsilon$ are the cross-
section and efficiency, respectively. $\sigma$ is usually obtained from
theoretical prediction, and $\epsilon$ is determined from MC simulation.
However in this analysis, we use the large sample of off-resonance data
collected at 3.65 GeV to estimate the continuum background. The events
remaining, after imposing the same selection criteria in the off-resonance
data, also form a peak in the $\bar{V}_{Z}$ distribution, as shown in Figure
9. The same signal and sideband regions are used as for the $\psi^{\prime}$
data to determine the collision and non collision events. With this method,
the continuum background subtraction is independent of MC simulation, and
little systematic bias is introduced.
The contributions from radiative returns to $J/\psi$ and $J/\psi$ decays from
the extended tail of the Breit-Wigner are very similar at the $\psi^{\prime}$
peak and off-resonance energy due to the small energy difference. They are
estimated to be 1.11 and 1.03 $nb$ at the $\psi^{\prime}$ peak and the off-
resonance energy point, respectively, and according to MC simulation, the
efficiencies for the known continuum processes at the two energy points are
also similar. Therefore, the off-resonance data can be employed to subtract
both the continuum QED and $J/\psi$ decay backgrounds using a scaling factor,
$f$, determined from the integrated luminosity multiplied by a factor of $1/s$
($s=E_{cm}^{2}$) to account for the energy dependence of the cross-section:
$f=\frac{{\cal L}_{\psi^{\prime}}}{{\cal
L}_{3.65}}\cdot\frac{3.65^{2}}{3.686^{2}}=3.677,$ (3)
where, ${\cal L}_{\psi^{\prime}}$ and ${\cal L}_{3.65}$ are the integrated
luminosities for $\psi^{\prime}$ data and 3.65 GeV data, respectively.
The luminosities at the two different energy points are determined from
$e^{+}e^{-}\rightarrow\gamma\gamma$ events using the same track and event
level selection criteria. At the track level, no good charged tracks and at
least two showers are required. The energy for the most energetic shower
should be higher than $0.7\times E_{beam}$ while the second most energetic
shower should be larger than $0.4\times E_{beam}$, where $E_{beam}$ is the
beam energy. At the event level, the two most energetic showers in the
$\psi^{\prime}$ rest frame should be back to back, and their phi angles must
satisfy $178^{\circ}<|\phi_{1}-\phi_{2}|<182.0^{\circ}$. The luminosity
systematic errors nearly cancel in calculating the scaling factor due to small
energy difference between these two energy points. The $f$ factor can also be
obtained using luminosities determined with Bhabha events. It is found to be
3.685.
Also of concern is the LEB remaining in the $\psi^{\prime}$ events after the
$E_{visible}/E_{cm}$ requirement. In order to test if the continuum background
subtraction is also valid for these events, candidate LEB events are selected
by requiring $E_{visible}/E_{cm}<0.35$ where there are few QED events
expected. Figures 10 and 11 show the comparison of $E_{visible}/E_{cm}$
between peak and off-resonance data for $N_{good}=1$ and $N_{good}=2$ events,
respectively. The agreement between the two energy points is good for these
events. The ratios of the numbers of peak and off-resonance events for
$N_{good}=1$ and $N_{good}=2$ are 3.3752 and 3.652, respectively. Compared
with the scaling factor obtained from luminosity normalization in Eq.( 3), a
difference of about 10% is found for $N_{good}=1$ while there is almost no
difference for $N_{good}=2$ events. These differences will be taken as
systematic errors.
The small numbers of events from $\psi^{\prime}\rightarrow
e^{+}e^{-},~{}\mu^{+}\mu^{-}$, and $\tau^{+}\tau^{-}$ in data that pass our
selection do not need to be explicitly subtracted since $\psi^{\prime}\to
lepton$ events are included in the inclusive MC and those passing the
selection criteria will contribute to the MC determined efficiency, so that
their contribution cancels.
Figure 10: Comparison of LEB events between $\psi^{\prime}$ peak and off-
resonance data for $N_{good}=1$ events. Dots with error bars are
$\psi^{\prime}$ data, and the histogram is off-resonance data. Figure 11:
Comparison of LEB events between $\psi^{\prime}$ peak and off-resonance for
$N_{good}=2$ events. Dots with error bars denote $\psi^{\prime}$ data, and the
histogram denotes off-resonance data.
Table 1 shows the number of observed hadronic events for different
multiplicity requirements for $\psi^{\prime}$ and off-resonance data. Figures
12, 13, and 14 show the $\cos\theta$, $E_{visible}$, and charged-track
multiplicity distributions after subtracting background.
Table 1: $N^{obs.}$ for peak and off-resonance data ($\times 10^{6}$), and the detection efficiency for inclusive $\psi^{\prime}$ decay events determined with $106\times 10^{6}$ $\psi^{\prime}\rightarrow~{}inclusive$ MC events. | $N_{good}\geq 1$ | $N_{good}\geq 2$ | $N_{good}\geq 3$ | $N_{good}\geq 4$
---|---|---|---|---
$\psi^{\prime}$ data | 106.928 | 102.791 | 81.158 | 63.063
off-resonance data | 2.192 | 1.98 | 0.704 | 0.433
$\epsilon$(%) | 92.912 | 89.860 | 74.624 | 58.188
Figure 12: The $\cos\theta$ distribution for charged tracks. Dots with error
bars are data; the histogram is MC simulation. Figure 13: The visible energy
distribution. Dots with error bars are data; the histogram is MC simulation.
Figure 14: The charged-track multiplicity distribution. Dots with error bars
are data; the histogram is MC simulation.
## IV Numerical result
The number of $\psi^{\prime}$ events is determined from
$N_{\psi^{\prime}}=\frac{N_{peak}^{obs.}-f\cdot N_{off-
resonance}^{obs.}}{\epsilon},$ (4)
where, $N_{peak}^{obs.}$ is the number of hadronic events observed at the
$\psi^{\prime}$ peak from Eq. (1), $N_{off-resonance}^{obs.}$ is the number of
hadronic events observed at the off-resonance energy point, $E_{cm}=3.650$
GeV, with the same selection criteria as those for peak data, and $\epsilon$
is the selection efficiency obtained from the inclusive $\psi^{\prime}$ MC
sample, the branching fraction of
$\psi^{\prime}\rightarrow~{}inclusive~{}hadron$ is included in the efficiency.
The relevant numbers are listed in Table 1 for different $N_{good}$ selection
requirements. The factor $f$ is the scaling factor which has been introduced
in Eq. (2). With these numbers, we obtain the numerical result for
$N_{\psi^{\prime}}$ listed in Table 2 for different choices of $N_{good}$. We
take the result for $N_{good}\geq 1$ as the central value of our final result.
Table 2: $N_{\psi^{\prime}}$ ($\times 10^{6}$) for different charged-track multiplicity requirements. | $N_{good}\geq 1$ | $N_{good}\geq 2$ | $N_{good}\geq 3$ | $N_{good}\geq 4$
---|---|---|---|---
$N_{\psi^{\prime}}$ | 106.414 | 106.279 | 105.289 | 105.643
## V Systematic Uncertainties
The systematic uncertainties include the uncertainties caused by tracking, the
event start time ($T_{0}$), trigger efficiency, background contamination, the
selection of the signal and sideband regions, etc.
### V.1 Tracking
Generally, the tracking efficiency for MC events is higher than that of data
according to various studies track . Assuming the average efficiency
difference between data and MC is 1% per track, the effect can be measured by
randomly tossing out 1% of MC simulated tracks. Only a difference of 0.03% on
$N_{\psi^{\prime}}$ is found for $N_{good}\geq 1$ events with and without this
tracking efficiency change; $N_{\psi^{\prime}}$ is not sensitive to the
tracking efficiency.
### V.2 Charged-track multiplicity
Figure 14 shows that the MC does not simulate the charged-track multiplicity
very well. The error due to charged-track multiplicity simulation can be
estimated by an unfolding method, which is described as follows. The generated
true charged multiplicity in MC simulation is even, i.e., 0, 2, 4, 6, 8,
$\cdots$. The observed MC multiplicity distribution is obtained after
simulation and event selection. For example, if the generated true
multiplicity is 4, the observed multiplicities are 0, 1, 2, 3, or 4 with
different probabilities. Therefore, an efficiency matrix, $\epsilon_{ij}$,
which describes the efficiency of an event generated with $j$ charged tracks
to be reconstructed with $i$ charged tracks, is obtained from MC simulation.
The distribution of the number of observed charged-track events in data,
$N_{i}^{obs.}$, is known. The true multiplicity distribution in data can be
estimated from the observed multiplicity distribution in data and the
efficiency matrix by minimizing the $\chi^{2}$. The $\chi^{2}$ is defined as
$\chi^{2}=\sum\limits^{10}_{i=1}\frac{(N_{i}^{obs.}-\sum\limits_{j=0}^{10}\epsilon_{ij}\cdot
N_{j})^{2}}{N_{i}^{obs.}},$ (5)
where the $N_{j}~{}(j=0,~{}2,~{}4,~{}6,~{}8,~{}10)$ describe the true
multiplicity distribution in data and are taken as floating parameters in the
fit. The simulation is only done up to a true multiplicity of 10, since there
are few events at high multiplicity. The total true number of events in data
can be obtained by summing all fitted $N_{j}$; it is $105.96\times 10^{6}$
which is lower than the nominal value by 0.4%. We take this difference as the
error due to the charged-track multiplicity distribution.
### V.3 Momentum and opening angle
For $N_{good}=2$ events, momentum and opening angle requirements are used to
remove the huge number of Bhabha events. When the momentum requirement is
changed from $P<1.7$ GeV/$c$ to $P<1.55$ GeV/$c$, the corresponding $N^{obs.}$
for peak and resonance data, as well as the efficiency change, but the change
in $N_{\psi^{\prime}}$ is only 0.05%. When the angle requirement is changed
from $\theta<176^{\circ}$ to $\theta<160^{\circ}$, the change in
$N_{\psi^{\prime}}$ is 0.01%. Therefore, the total uncertainty due to momentum
and opening angle requirements is 0.05%. Figures 15 and 16 show comparisons
between data and MC simulations for momentum and opening angle distributions
after background subtraction, respectively.
Figure 15: The distribution of total momentum for $N_{good}=2$ events. Dots
with error bars are data; the histogram is MC simulation. Figure 16: The
distribution of opening angle between tracks for $N_{good}=2$ events. Dots
with error bars are data; the histogram is MC simulation.
### V.4 LEB background contamination
$N_{\psi^{\prime}}$ is insensitive to the visible energy requirement. The
difference between a tight requirement, $E_{visible}/E_{cm}>0.45$, and no
requirement is only 0.1%. Conservatively, an error of 0.1% is assigned due to
the background contamination.
### V.5 Determination of number of hadronic events
Two methods are used to obtain $N^{obs.}$. The first is to directly count the
numbers of events in the signal and sideband regions; the second method is to
fit the $\bar{V}_{Z}$ distribution with a double Gaussian for the signal and a
polynomial for the background, as shown in Figs. 8 and 9. A difference of
0.28% is found between these two methods which is taken as the error due to
the uncertainty from the $N^{obs.}$ determination.
### V.6 Vertex limit
When $V_{r}<1$ cm is changed to $V_{r}<2$ cm, $N_{\psi^{\prime}}$ changes by
0.35%, while if $|\bar{V}_{Z}|<10$ cm is changed to $|\bar{V}_{Z}|<15$ cm,
there is almost no change. Therefore, the difference of 0.35% is taken as the
error from the vertex requirement.
### V.7 Scaling factor
The scaling factor can be obtained for two different QED processes,
$e^{+}e^{-}\rightarrow\gamma\gamma$ and $e^{+}e^{-}\rightarrow e^{+}e^{-}$.
The corresponding results are 3.677 and 3.685. The difference on
$N_{\psi^{\prime}}$ due to the $f$ factor can be calculated by $\Delta f\cdot
N^{obs.}_{N_{good}\geq 1}(3.650$ GeV$)/N_{\psi^{\prime}}=(3.685-3.677)\cdot
3.1808/106.32=0.023\%$. The slight difference indicates the uncertainty caused
by the normalization factor is negligibly small.
### V.8 Choice of sideband region
We take $|\bar{V}_{Z}|<4.0$ cm as the signal region and $6<|\bar{V}_{Z}|<10$
cm as the sideband region. A difference of 0.45% in $N_{\psi^{\prime}}$ is
found by shifting the sideband region outward by 1.0 cm, which is about
1$\sigma$ of the $\bar{V}_{Z}$ resolution, i.e., the sideband region is
changed from 6 cm$<|\bar{V}_{Z}|<10$ cm to 7cm $<|\bar{V}_{Z}|<11$ cm. We take
this difference as the error due to the uncertainty caused by choice of the
sideband region.
### V.9 $\pi^{0}$ mass requirement
This requirement is only used for $N_{good}=1$ events. $N_{\psi^{\prime}}$ has
a slight change of 0.11% when the mass window requirement is changed from
$|M_{\gamma\gamma}-M_{\pi^{0}}|<0.015$ GeV/$c^{2}$ to
$|M_{\gamma\gamma}-M_{\pi^{0}}|<0.025$ GeV/$c^{2}$; this difference is taken
as the uncertainty due to $\pi^{0}$ mass requirement.
### V.10 The cross section of $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$
Since the off-resonance energy point is not very far from $\tau\tau$
threshold, $\sigma({e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}})$ does not vary as
$1/s$ between the off-resonance energy and the $\psi^{\prime}$ peak, as other
QED processes. The difference between the observed and the cross section
assuming a $1/s$ dependence causes a change of 0.17% in $N_{\psi^{\prime}}$.
This change is taken as a systematic error.
### V.11 $B(\psi^{\prime}\rightarrow X+J/\psi)$
The $\psi^{\prime}$ MC assumes $B(\psi^{\prime}\rightarrow X+J/\psi)\approx
57\%$ from the PDG PDG , while the CLEO experiment determined a branching
ratio of 62% cleo . Using CLEO’s result, a new inclusive MC sample was
generated. The corresponding efficiencies are 92.912%, 89.761%, 74.838% and
58.528% for $N_{good}\geq 1,~{}2,~{}3$ and 4, respectively. Compared with
numbers in Table 1, the efficiency differences between these two MC samples
are negligible.
### V.12 Event start time determination
The Event Start Time (EST) algorithm is used to determine the common start
time of the recorded tracks in an event. The efficiency of the EST
determination affects the resolution of tracks from the tracking algorithm.
These efficiencies for different charged tracks, $e$, $\mu$, $\pi$, $K$, and
$p$, and photons are studied with different control samples for both data and
inclusive MC events, for example, $J/\psi\rightarrow\pi^{+}\pi^{-}\pi^{0}$,
$\pi^{+}\pi^{-}p\bar{p}$, and
$\psi^{\prime}\rightarrow\pi^{+}\pi^{-}J/\psi,J/\psi\to l^{+}l^{-}$, etc. All
comparisons indicate that the efficiencies of the EST determination are high
for both track and event level $(>98\%)$ selection, and the difference between
data and MC simulated events is quite small $(\sim 0.1\%)$. We take this
difference as the uncertainty caused by the EST determination.
### V.13 Trigger efficiency
The fraction of events with $N_{ngood}\geq 2$ is about 97%. The trigger
efficiency for these events is close to 100.0% according to a study of the
trigger efficiency nik . For $N_{good}=1$ events, an extra $\pi^{0}$ is
required, and the hadron trigger efficiency for this channel is 98.7% nik .
Since the fraction of $N_{good}=1$ events is only about 3%, the uncertainty
caused by the trigger is negligible.
### V.14 The missing 0-prong hadronic events
A detailed topology analysis is performed for $N_{good}=0$ events in the
inclusive MC sample. Most of these events come from known decay channels, such
as $\psi^{\prime}\rightarrow
X+J/\psi~{}(X=\eta,~{}\pi^{0}\pi^{0},~{}$and$~{}\pi^{+}\pi^{-})$,
$\psi^{\prime}\rightarrow\gamma\chi_{cJ}$, and $\psi^{\prime}\rightarrow
e^{+}e^{-}$, $\mu^{+}\mu^{-}$. The fraction of pure neutral events is less
than 1.0%. For the known charged decay modes, the MC simulation works well
according to many comparisons between data and MC simulation in Section 3. To
investigate the pure neutral channels, the same selection criteria at the
track level are used. The criteria at the event level include $N_{good}=0$ and
$N_{\gamma}>3$. The latter requirement is used to suppress
$e^{+}e^{-}\rightarrow\gamma\gamma$ and beam-associated background events. The
same selection criteria are imposed on the off-resonance data. Figures 17 and
18 show the distribution of total energy in the EMC for data and inclusive MC
events. The peaking events correspond to the pure neutral candidates, and the
number of events is extracted by fitting. The difference in the number of
fitted events between data and inclusive MC events is found to be 17%.
Therefore, the uncertainty due to the pure neutral events should be less than
$17\%\times 1\%=0.17\%$, and this is taken as the systematic error on the
missing 0-prong events.
Figure 17: The distribution of total energy in the EMC with $N_{good}=0$ for
data. The dot-dashed line denotes the signal shape of
$\psi^{\prime}\rightarrow~{}neutral~{}channel$, the dashed line denotes the
background shape from QED processes, and the shaded region is the background
shape from $\psi^{\prime}$ decay. Figure 18: The distribution of total energy
in the EMC with $N_{good}=0$ for inclusive MC events. The dashed line denotes
the signal shape of $\psi^{\prime}\rightarrow~{}neutral~{}channel$, and the
shaded region is the background shape from $\psi^{\prime}$ decay.
### V.15 $B(\psi^{\prime}\rightarrow hadrons)$
The uncertainty of $B(\psi^{\prime}\rightarrow hadrons)$ is very small
according to the PDG PDG , 0.13%, which is taken as the error due to
uncertainty of $\psi^{\prime}$ decays to hadronic events.
### V.16 Total error
Table 3 lists all systematic errors. The total systematic error is determined
by the quadratic sum of all errors.
Table 3: The systematic error (%) Source | Error
---|---
Background contamination | 0.10
$N^{obs.}$ determination | 0.28
Choice of sideband region | 0.45
Vertex selection | 0.35
Momentum and opening angle | 0.05
Scaling factor ($f$) | 0.02
0-prong events | 0.17
Tracking | 0.03
Charged-track multiplicity | 0.40
$\sigma(e^{+}e^{-}\rightarrow\tau^{+}\tau^{-})$ | 0.17
$B(\psi^{\prime}\rightarrow X+J/\psi)$ | 0.00
$\pi^{0}$ mass requirement | 0.11
EST determination | 0.10
Trigger efficiency | negligible
$B(\psi^{\prime}\rightarrow hadron)$ | 0.13
Total | 0.81
## VI Summary
The number of $\psi^{\prime}$ events is determined using
$\psi^{\prime}\rightarrow~{}hadrons$. The large off-resonance data sample at
$E_{cm}=3.65$ GeV is used to estimate the background under the $\psi^{\prime}$
peak. The number of $\psi^{\prime}$ events taken in 2009 is measured to be
$(106.41\pm 0.86)\times 10^{6}$, where the error is systematic only and the
statistical error is negligible.
## VII Acknowledgment
The BESIII collaboration thanks the staff of BEPCII and the computing center
for their hard efforts. This work is supported in part by the Ministry of
Science and Technology of China under Contract No. 2009CB825200; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 10625524,
10821063, 10825524, 10835001, 10935007, 11125525, 10975143, 10979058; Joint
Funds of the National Natural Science Foundation of China under Contracts Nos.
11079008, 11179007; the Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; CAS under Contracts Nos. KJCX2-YW-N29,
KJCX2-YW-N45; 100 Talents Program of CAS; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; U. S. Department of Energy under Contracts Nos. DE-
FG02-04ER41291, DE-FG02-91ER40682, DE-FG02-94ER40823; U.S. National Science
Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research
Foundation of Korea under Contract No. R32-2008-000-10155-0.
## References
* (1)
* (2) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
* (3) S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
* (4) J. Allison et al., IEEE Trans. Nucl. Sci. 53, 270 (2006).
* (5) R. G. Ping et al., Chinese Physics C 32, 599 (2008).
* (6) J. C. Chen et al., Phys. Rev. D 70, 011102(R) (2005).
* (7) M. Ablikim et al. (BES Collaboration), Phys. Lett. B 677, 239 (2009).
* (8) M. Ablikim et al. (BESIII Collaboration), Phys. Lett. B 710, 594 (2012).
* (9) H. Mendez et al. (CLEO Collaboration), Phys. Rev. D 78, 011102(R) (2008).
* (10) N. Berger et al., Chinese Physics C 34, 1779 (2010).
|
arxiv-papers
| 2012-09-27T11:51:18 |
2024-09-04T02:49:35.675211
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.Ablikim, M. N. Achasov, D.J. Ambrose",
"submitter": "Zhiyong Wang",
"url": "https://arxiv.org/abs/1209.6199"
}
|
1209.6224
|
# Interaction of a graphene sheet with a ferromagnetic metal plate
Anh D. Phan1, N. A. Viet2, Nikolai A. Poklonski3, Lilia M. Woods1, Chi H. Le4
1Department of Physics, University of South Florida, Tampa, Florida 33620, USA
2Institute of Physics, 10 Daotan, Badinh, Hanoi, Vietnam 3Physics Department,
Belarusian State University, Minsk 220030, Belarus 4School of Engineering,
University of Greenwich, Medway, United Kingdom
###### Abstract
Nanoscale surface forces such as Casimir and the van der Waals forces can have
a significant influence on fabrication, handling and assembly processes as
well as the performance of micro and nano devices. In this paper, the
investigation and the calculation of the Casimir force between a graphene
sheet and a ferromagnetic metal substrate in a vacuum are presented. The
reflection coefficients of graphene are graphene-conductivity dependent, and
the conductivity of graphene is described by the Kubo formalism. There is an
effect of magnetic properties of the metal on the Casimir interaction. The
magnetic effect plays a significant role at low temperatures or high value of
chemical potential. The numerical results also demonstrate that the thickness
of a metal slab has a minor influence on the Casimir force. The investigation
and findings about the Casimir force in this study would lead to useful
information and effective solutions for design and manufacturing of micro and
nano devices, especially in the areas of micro and nano machining,
fabrication, manipulation, assembly and metrology.
###### pacs:
78.20.Ls, 31.30.jh, 78.67.Wj, 12.20.Fv
## I Introduction
Micro and nano technologies are potential economic engines and are considered
as enabling technologies with exceptional economic capabilities. There have
been increasingly more products in which the integrated micro and nano-
electromechanical systems (MEMS/NEMS) are crucial in today’s market because
they allow for improved functionality, lower costs and higher quality. With
the decrease in the size of MEMS and NEMS, additional nanoscale surface
forces, such as the Casimir force and the van der Waals force, should be
considered 1 ; 2 , especially in the areas of micro and nano machining,
fabrication, manipulation, assembly and metrology. Therefore, a fundamental
understanding of the Casimir force has recently been under intense discussion
in the research and technological development (RTD) communities.
The Casimir force can cause the small elements in a device to stick together.
At small scales, the nanoscale surface forces may overcome elastic restoring
actions in the device and lead to the plates’ sticking during the fabrication
process 1 . It has been recently shown that Casimir forces may hamper the
functioning of MEMS and NEMS devices by providing a pull-in instability 36 .
The Casimir force between two objects can induce adhesion and stiction leading
to failures in devices 2 ; 3 ; 32 ; 36 . Particularly, when the size of a
material is under the threshold length, the influence of the loading
interaction on design and manufacturing of micro and nanodevices becomes much
more prominent.
The Casimir interaction originates from the quantum electromagnetic
fluctuations between two objects 2 ; 3 ; 40 . It provides a fundamental
understanding about nanoscience and has a significant influence on the high
performance of devices. Over six decades, the Lifshitz theory has been
employed to investigate this force in metal-metal 4 ; 5 , semiconductor-
semiconductor 6 , metal-superconductor 7 ; 28 , metamaterials-metamaterials 8
; 9 and graphene-graphene systems 10 ; 11 ; 12 ; 34 . Results of some
configurations have been experimentally examined and have good agreement with
theoretical calculations 3 ; 13 .
The interaction between a non-magnetic metal with a magneto-dielectric
material has also recently been discussed 29 ; 14 ; 15 . These studies
illustrate that the magnetic properties can significantly influence in the
Casimir pressure. Theoretical and experimental studies also show that it is
possible to obtain the repulsive Casimir force in such systems 2 ; 15 ; 16 ,
and the results could provide possible solutions to handle the stiction and
adhesion problems found in MEMS/NEMS devices. In terms of theory, the
calculations depend substantially on the model that describes the dielectric
function of metals.
Graphene, an atomically thin layered material with novel properties, has
gained a great deal of attention since its discovery 30 . Researchers from a
wide range of scientific fields have spent considerable effort in exploring
its nature and its potential for practical applications 17 ; 18 ; 19 ; 20 .
Recently obtained results demonstrate that graphene can be a promising
candidate for next-generation electronic devices 31 . The functions of
graphene have been applied to diverse areas from biology to material science.
Figure 1: (Color online) Schematics of a two-dimensional layer (graphene) and
a substrate (FM metal) with thickness $D$.
In this paper, we calculate the Casimir force between a graphene layer and a
ferromagnetic (FM) substrate shown in Fig.1. The FM material is taken to be Fe
described via a dielectric and magnetic response properties. We utilize the
Lifshitz theory to examine the role of the separation, temperature, and
thickness of the substrate. We also investigate the influence of the magnetic
response of the FM and show that in most cases its contribution is relatively
small compared to the one from the dielectric response.
This paper is organized as follows: Sec. II presents the detailed expressions
and calculations of the thermal Casimir interactions. The results and
discussions are mentioned in Sec. III. Finally, Sec. IV presents the
conclusions.
## II CASIMIR INTERACTION BETWEEN GRAPHENE AND A METAL SUBSTRATE
Here we calculate the Casimir force between the graphene sheet and a substrate
in vacuum at a separation $a$. The force per unit area at temperature $T$ is
given as 3 ; 21 ; 34 ; 24
$\displaystyle
F(a,T)=-\frac{k_{B}T}{\pi}\sum_{l=0}^{\infty}\left(1-\frac{1}{2}\delta_{l0}\right)\int_{0}^{\infty}q_{l}k_{\perp}dk_{\perp}$
$\displaystyle\times\left(\frac{r_{TE}^{(1)}r_{TE}^{(2)}}{e^{2q_{l}a}-r_{TE}^{(1)}r_{TE}^{(2)}}+\dfrac{r_{TM}^{(1)}r_{TM}^{(2)}}{e^{2q_{l}a}-r_{TM}^{(1)}r_{TM}^{(2)}}\right),$
(1)
where $k_{B}$ is the Plank constant, $r_{TM}^{(1,2)}$ and $r_{TE}^{(1,2)}$ are
the reflection coefficients corresponding to the transverse magnetic (TM) and
transverse electric (TE) field modes.
The reflection coefficients for the substrate are given as 2 ; 14 ; 15
$\displaystyle r_{TE}^{(1)}\equiv
r_{TE}^{(1)}(i\xi_{l},k_{\perp})=\frac{\mu_{1}(i\xi_{l})q_{l}-k_{1}}{\mu_{1}(i\xi_{l})q_{l}+k_{1}},$
$\displaystyle r_{TM}^{(1)}\equiv
r_{TM}^{(1)}(i\xi_{l},k_{\perp})=\frac{\varepsilon_{1}(i\xi_{l})q_{l}-k_{1}}{\varepsilon_{1}(i\xi_{l})q_{l}+k_{1}},$
(2)
where
$\displaystyle q_{l}\equiv
q_{l}(i\xi_{l},k_{\perp})=\sqrt{k_{\perp}^{2}+\frac{\xi_{l}^{2}}{c^{2}}},$
$\displaystyle k_{1}\equiv
k_{1}(i\xi_{l},k_{\perp})=\sqrt{k_{\perp}^{2}+\mu_{1}(i\xi_{l})\varepsilon_{1}(i\xi_{l})\frac{\xi_{l}^{2}}{c^{2}}}.$
(3)
It is important to note that $k_{\perp}$ is the wave vector component
perpendicular to the plate, $c$ is the speed of light, $\xi_{l}=2\pi
k_{B}Tl/\hbar$ is the Matsubara frequencies. The response properties of the
metal are characterized by the dielectric function $\varepsilon_{1}(i\xi_{l})$
and the permeability function $\mu_{1}(i\xi_{l})$ which are frequency
dependent along the imaginary axis ($\omega_{l}=i\xi_{l}$). We use the Drude
model $\varepsilon_{1D}$ and the plasma model $\varepsilon_{1P}$ to describe
the dielectric function of metals 3 ; 14 ; 29
$\displaystyle\varepsilon_{1D}(i\xi)=1+\frac{\omega_{p}^{2}}{\xi(\xi+\gamma_{p})},$
$\displaystyle\varepsilon_{1P}(i\xi)=1+\frac{\omega_{p}^{2}}{\xi^{2}},$ (4)
where $\omega_{p}$ is the plasma frequency and the damping parameter is
$\gamma_{p}$. At room temperature, the $l=0$ term of the Casimir-Lifshitz
force in the metallic system is dominant. This reason allows us to consider
the contribution of the static magnetic permeability $\mu(0)\gg 1$ on the
Casimir force. For higher orders of $l$, one should put $\mu(\xi_{l})=1$ 29 .
Both models lead to $r_{TM}^{(1)}(0)=1$. However, we can obtain two different
expressions of $r_{TE}^{(1)}(0)$ when applying these two models 29
$\displaystyle r_{TE,D}^{(1)}(0,k_{\perp})=\frac{\mu(0)-1}{\mu(0)+1},$
$\displaystyle
r_{TE,P}^{(1)}(0,k_{\perp})=\frac{\mu(0)ck_{\perp}-\sqrt{c^{2}k_{\perp}^{2}+\mu(0)\omega_{p}^{2}}}{\mu(0)ck_{\perp}+\sqrt{c^{2}k_{\perp}^{2}+\mu(0)\omega_{p}^{2}}},$
(5)
here $r_{TE,D}^{(1)}(0,k_{\perp})$ and $r_{TE,P}^{(1)}(0,k_{\perp})$ are the
TE reflection coefficients corresponding to the Drude and plasma model,
respectively.
For graphene, the reflection coefficients are found to be 11 ; 16
$\displaystyle r_{TE}^{(2)}\equiv
r_{TE}^{(2)}(i\xi_{l},k_{\perp})=-\frac{2\pi\xi_{l}\sigma/q_{l}c^{2}}{1+2\pi\xi_{l}\sigma/q_{l}c^{2}},$
$\displaystyle r_{TM}^{(2)}\equiv
r_{TM}^{(2)}(i\xi_{l},k_{\perp})=\frac{2\pi\sigma q_{l}/\xi_{l}}{1+2\pi\sigma
q_{l}/\xi_{l}},$ (6)
where $\sigma\equiv\sigma(i\xi)$ is the 2D conductivity described via the the
Kubo formalism 11
$\displaystyle\sigma(i\xi)=\frac{2e^{2}k_{B}T\ln(2)}{\pi\hbar^{2}\xi}+\frac{e^{2}\xi}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh(x)dx}{x^{2}+(\frac{\hbar\xi}{4k_{B}T})^{2}}.$
(7)
The first term in Eq.(7) corresponds to the intraband contribution, while the
second term corresponds to the interband contribution. At low temperatures,
the graphene conductivity $\sigma(i\xi_{l})$ approaches to the universal value
$\sigma_{0}=e^{2}/4\hbar$.
Figure 2: (Color online) The normalized graphene conductivity
$\sigma(i\xi)/\sigma_{0}$ vs frequency at given temperatures.
Using Eq.(6) and Eq.(7) the TM reflection coefficient of graphene is given by
$\displaystyle
r_{TM}^{(2)}=\frac{\frac{2e^{2}k_{B}T}{\pi\hbar^{2}}+\frac{e^{2}\xi_{l}^{3}}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh
xdx}{x^{2}+(\hbar\xi_{l}/4k_{B}T)^{2}}}{\frac{\xi_{l}^{2}}{2\pi
q_{l}}+\frac{2e^{2}k_{B}T}{\pi\hbar^{2}}+\frac{e^{2}\xi_{l}^{3}}{8\pi
k_{B}T}\int_{0}^{\infty}\frac{\tanh xdx}{x^{2}+(\hbar\xi_{l}/4k_{B}T)^{2}}}.$
(8)
The equation suggests $r_{TM}^{(2)}(0,k_{\perp})=1$. In the same way, it can
be found the expression of $r_{TE}^{(2)}(0,k_{\perp})$
$\displaystyle
r_{TE}^{(2)}(0,k_{\perp})=-\frac{(2\pi/c)8\ln(2)\sigma_{0}}{\pi\lambda_{T}k_{\perp}+(2\pi/c)8\ln(2)\sigma_{0}},$
(9)
where $\lambda_{T}=\hbar c/k_{B}T$ is the thermal wavelength in most
materials. The explicit expression of $r_{TE}^{(2)}(0,k_{\perp})$ shows that
it is susceptible to temperature. $r_{TE}^{(2)}(0,k_{\perp})$ is nonzero only
because of the intraband component of the graphene conductivity.
The first Matsubara frequencies $\xi_{1}$ are approximately 0.027, 0.162, and
0.325 $eV$ corresponding to the temperature 50, 300 and 600 $K$, respectively.
As seen in Fig. 2, for $l\geq 2$, it is possible to substitute $\sigma_{0}$
for $\sigma(i\xi_{l})$ in Eq.(6) to calculate the reflection coefficients and
the higher order terms in the Casimir-Lifshitz force formula.
## III RESULTS AND DISCUSSIONS
In this section, we consider the Casimir interactions between graphene and a
Fe metal plate. The parameters of Fe are $\omega_{p}=4.09$ $eV$,
$\gamma_{p}=0.018$ $eV$, and $\mu(0)=10^{4}$ 4 . As discussed in previous part
$r_{TM}^{(1)}(0,k_{\perp})=r_{TM}^{(2)}(0,k_{\perp})=1$, the TM contribution
to the $l=0$ term of the Casimir force is given
$\displaystyle
F^{(0)}_{TM}(a,T)=-\frac{k_{B}T}{2\pi}\int_{0}^{\infty}\frac{k_{\perp}^{2}dk_{\perp}}{e^{2k_{\perp}a}-1}=-\frac{k_{B}T\zeta(3)}{8\pi
a^{3}}.$ (10)
The expression above is identical to the term with $l=0$ in the Casimir force
between two metals. The sum of the $l\geq 1$ is much smaller than that of the
metallic system due to the presence of the low graphene conductivity 11 . As a
result, the $l=0$ term is the dominating term. Now, to calculate the
contribution of the TE mode $F^{(0)}_{TE}(a,T)$ with $l=0$, it is necessary to
choose a good model for Fe. Both the plasma 14 and Drude model 22 , however,
has been widely used to compute the Casimir interaction and fit with
experimental data. Determining an accuracy of two models as compared to
measurement data has been a controversial issue. In the following work, we
utilize the plasma model to calculate the interactions. The expression of
$F^{(0)}_{TE}(a,T)$ is expressed by
$\displaystyle
F^{(0)}_{TE}(a,T)=-\frac{k_{B}T}{2\pi}\int_{0}^{\infty}\frac{k_{\perp}^{2}dk_{\perp}}{\frac{e^{2k_{\perp}a}}{r_{TE}^{(1)}(0,k_{\perp})r_{TE}^{(2)}(0,k_{\perp})}-1}.$
(11)
Note that $F^{(0)}_{TE}(a,T)$ is a unique term affected by the magnetic
property of the ferromagnetic substrate. In Fig.3, the ratio
$F^{(0)}_{TE}/F^{(0)}_{TM}$ is less than 0.75 $\%$ for distances $a$ in the
range from 0 to 1 $\mu m$ at given temperatures. It indicates that the effect
of $\mu(0)$ on the Casimir force can be neglectable. For this reason, Fe can
be treated as a regular non-magnetic metal $\mu(i\xi)=1$.
For a finite-thickness iron slab $D$, the reflection coefficients of the metal
material are modified as follows 15 ; 23
$\displaystyle
R_{TE,TM}(i\xi_{l},k_{\perp})=r_{TE,TM}^{(1)}\frac{1-e^{-2k_{1}D}}{1-(r_{TE,TM}^{(1)})^{2}e^{-2k_{1}D}}.$
(12)
Figure 3: (Color online) The ratios of $F^{(0)}_{TE}(a,T)/F^{(0)}_{TM}(a,T)$
and $F^{(2)}(a,T)/F^{(0)}_{TM}(a,T)$ at 50, 300, and 600 $K$ as a function of
separation distance $a$.
To study the influence of thickness on the Casimir force, we rewrite Eq.(1) in
the following way
$\displaystyle F(a,T)=F^{(0)}_{TM}(a,T)+F^{(0)}_{TE}(a,T)+F^{(1)}(a,T),$ (13)
here $F^{(1)}(a,T)$ is the sum of all $l\geq 1$ terms and is thickness-
dependent. Because $r_{TM}^{(1)}(0,k_{\perp})=1$, so
$R_{TM}^{(1)}(0,k_{\perp})=1$. From this, the expression of
$F^{(0)}_{TM}(a,T)$ is in Eq.(10) and is independent of $D$. The component
force $F^{(0)}_{TE}(a,T)$ is weaker than that for the thick plate and one can
consider $F^{(0)}_{TE}(a,T)\approx 0$. Therefore, only $F^{(1)}(a,T)$ is
sensitive to a variation of subtrate thickness.
Figure 4: (Color online) The Casimir pressures corresponding to the
approximate and full expression at different temperatures. Here
$F_{0}(a)=-\pi^{2}\hbar c/240a^{4}$.
Figure 3 shows that the thickness of the Fe slab does not influence much the
Casimir forces. At the same temperature, the separation between the two curves
corresponding to the finite plate and the semi-infinite plate is small. In
bulk materials, a reduction of the thickness gives rise to a small decrease of
the Casimir force. The thermal wavelength of graphene is
$\lambda_{T}/200\approx 38$ $nm$ at room temperature. At distances beyond the
thermal wavelength of a system, the contribution of the sum of $l\geq 1$ terms
can be ignorable. In the investigated range, the influence of $F^{(1)}(a,T)$
is not significant. This finding suggest that since graphene is a two
dimensional monolayer material, it significantly interacts only with the
nearest interface layers of the bottom substrate. Graphene’s effect on layers
separated by a sufficient number of layers is negligable. The Fe slab with
$D=20$ nm has more layers than that which are meaningful. Another interesting
special feature is that at large distance limit ($a\geq 0.8\mu m$) the
contribution of $F^{(1)}(a,T)$ to the total Casimir force is minor. It can be
explained that The $l\geq 2$ terms provide weak force components due to the
presence of the small universal conductivity in the reflection coefficients.
It is important to note that a universal graphene conductivity is a main
reason for less than 2 $\%$ difference as one compares the dispersion force
between two graphene sheets with $F_{0}(a)$ 11 . It demonstrates that the
approximate formula for the Casimir interaction equivalent to the $l=0$ term
of the Casimir-Lifshitz expression in this regime is
$F_{app}(a,T)=-k_{B}T\zeta(3)/8\pi a^{3}$. In Fig.4, we plot the normalized
Casimir force with the full expression (see Eq.(1)) and the approximate
expression. The $l=0$ term is dominant and can replace the full expression of
the Casimir interaction when $a\geq 0.8$ $\mu m$ for 300 $K$.
The critical $T_{c}$ of Fe is 1043 $K$. At $T>T_{c}$, the magnetic properties
of the ferromagnetic material nearly vanish. Spins in the Fe slab are
rearranged so that $\mu(i\xi)=1$ at all frequencies. Nevertheless, the
dielectric function is not affected by the directions of spins. Therefore, the
plasma frequency and damping parameter are unchanged by temperature increases
at $T\geq T_{c}$. If we consider the Casimir interaction between Fe and
another metal, the phase transition leads to a significant change in the
mutual interaction 14 ; 29 . The fact that graphene is quite transparent with
the presence of the magnetic properties, the dispersion force may not be
modified.
A notable point is that the formula in Eq.(7) was used for pure graphene with
the chemical potential $\mu_{C}=0$. If $\mu_{C}\neq 0$, the graphene
conductivity has to be rewritten in the form of 25
$\displaystyle\sigma(\mu_{C},i\xi)=\frac{e^{2}k_{B}T\ln(2)}{\pi\hbar^{2}\xi}+\frac{e^{2}k_{B}T\ln(1+\cosh(\mu_{C}/k_{B}T))}{\pi\hbar^{2}\xi}$
$\displaystyle+\frac{e^{2}\xi}{\pi}\int_{0}^{\infty}\frac{\sinh(E/k_{B}T)}{\cosh(E/k_{B}T)+\cosh(\mu_{C}/k_{B}T)}\frac{dE}{(\hbar\xi)^{2}+4E^{2}}.$
(14)
The first two terms are known as the intraband conductivity, another is the
interband component. Eq.(14) shows that the intraband contribution of
$\sigma(\mu_{C},i\xi)$ still induces a non-zero value of
$r_{TE}^{(2)}(0,k_{\perp})$. Obviously, the graphene conductivity is strongly
susceptible to the chemical potential. The chemical potential of a graphene
sheet can be controlled by using an applied electric field $E_{d}$ 26 ; 27
$\displaystyle\frac{\pi\varepsilon_{0}\hbar^{2}v_{F}^{2}}{e}E_{d}=\int_{0}^{\infty}E\left(f(E)-f(E+2\mu_{C})\right)dE,$
(15)
here $f(E)$ is the Fermi distribution function, $v_{F}=c/300$ is the Fermi
velocity.
Other ways to modulate $\mu_{C}$ are an applied magnetic field 27 and
chemical doping. Using a magnetic field also varys the expression of graphene
conductivity and creates Landau energy levels. In the limit of extremely low
chemical potential, $r_{TE}^{(2)}(0,k_{\perp})$ is represented as Eq.(9). For
$\mu_{C}\gg k_{B}T$, the expression of the TE reflection coefficient with
$l=0$ is written by
$\displaystyle
r_{TE}^{(2)}(0,k_{\perp})\approx-\frac{4\sigma_{0}\mu_{C}/(\pi\hbar
c)}{k_{\perp}+4\sigma_{0}\mu_{C}/(\pi\hbar c)}.$ (16)
Figure 5: (Color online) The Casimir forces at the case with and without
taking into account the magnetic properties at $\mu_{C}=1$ $eV$.
In Fig.5, the Casimir interactions are calculated at low temperatures ($\leq
50$ $K$) and $\mu_{C}=1$ $eV$ with and without the magnetic properties. Unlike
the case of a pristine graphene sheet ($\mu_{C}=0$ $eV$), the influence of
permeability of Fe on the dispersion force is noticeable at 10 $K$. The
properties, however, have much less effect on the Casimir force at
temperatures greater than 50 $K$. The majority of $F(a,T)$ comes from the
interband conductivity of graphene. At 10 $K$, the contribution of the
intraband on the Casimir force can be considerable to that of the interband
conductivity. However, as temperature increases the interband contribution
increases substantially, and causing a decrease in the contribution from
intraband. For this reason, the magnetic properties nearly disappear at $T\geq
50$ $K$.
To investigate further the Casimir force dependence on chemical potential, we
consider the fluctuation interactions at $a=100$ $nm$ and various temperatures
versus $\mu_{C}$ shown in Fig.6. $F(a,T)/F_{0}$ changes from 0.018 to 0.034 at
$T=10$ $K$ and from 0.045 to 0.064 at $T=300$ $K$ when $\mu_{C}$ varys in the
regime below $1.2$ $eV$. In addition, at higher temperatures, the curves of
$F(a,T)/F_{0}$ is of the linear form.
Figure 6: (Color online) The Casimir pressures as a function of chemical
potential.
It is remarkable that Fe is not a superconducting material, so iron has no
temperature phase transition as temperature decreases to 10 $K$.
Alternatively, the universal value of conductivity $\sigma_{0}$ also
demonstrates that even if temperature is 0 $K$, graphene still does not change
phase and properties.
## IV Conclusions
When design and manufacturing of micro and nano devices and components, it is
important and necessary to understand the loading and effects of related
forces on the systems and its components, especially the nanoscale surface
forces such as Casimir and the van der Waals. In this study, a comprehensive
investigation and discussion of the Casimir interaction between graphene and
FM materials is presented. We discussed about the Casimir force which is a
function of the graphene conductivity and the permeability constant of $Fe$.
It was shown that the conductivity of graphene heavily depends on the chemical
potential which again has a large impact on the dispersion force and causes a
difference between non-magnetic and magnetic calculations at the ultra-low
temperature. In other cases, there is no effect of the magnetic properties on
the Casimir force. One can apply a bias electric field to a graphene sheet to
tailor the chemical potential of graphene in order to control the magnitude of
the Casimir force. According to our numerical results, the force between a
graphene layer and the semi-infinite metal is smaller than the interaction
between two metals in 29 because the conductivity of metals is higher than
that of graphene. The numerical results also demonstrated that the thickness
of a metal slab had a minor influence on the Casimir force. The investigation
and above findings about the Casimir force in this study would lead to useful
information and effective solutions for design and manufacturing of micro and
nano devices where graphene and FM metal materials are utilised, especially in
the areas of micro and nano machining, fabrication, manipulation, assembly and
metrology.
###### Acknowledgements.
We give thanks to Dr. David Drosdoff for discussions. This research was
supported by the Nafosted Grant No. 103.06-2011.51. Lilia M. Woods
acknowledges the Department of Energy under contract DE-FG02-06ER46297.
Nikolai A. Poklonski acknowledges the financial support from Belarusian
Republican Foundation for Fundamental Research Grant No. F11V-001.
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|
arxiv-papers
| 2012-09-27T13:36:22 |
2024-09-04T02:49:35.683254
|
{
"license": "Public Domain",
"authors": "Anh D. Phan, N. A. Viet, Nikolai A. Poklonski, Lilia M. Woods, Chi H.\n Le",
"submitter": "Anh Phan Mr.",
"url": "https://arxiv.org/abs/1209.6224"
}
|
1209.6282
|
# Open heavy flavor production via semi-leptonic decayed muons in Pb+Pb
collisions at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV with the ATLAS detector at the
LHC
ATLAS Collaboration
###### Abstract
Measurements of heavy quark production and suppression in ultra-relativistic
nuclear collisions probe the interactions of heavy quarks with the hot, dense
medium created in the collisions. ATLAS has measured heavy quark production in
$\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb collisions via semi-leptonic decays of open
heavy flavour hadrons to muons. Results obtained from an integrated luminosity
of approximately $7~{}\mu\mathrm{b}^{-1}$ collected in 2010 are presented for
the per-event muon yield as a function of muon transverse momentum,
$p_{\mathrm{T}}$, over the range of $4<p_{\mathrm{T}}<14$ GeV. Over that
momentum range single muon production results largely from heavy quark decays.
The centrality dependence of the muon yields is characterized by the “central
to peripheral” ratio, $R_{\rm CP}$. Using this measure, muon production from
heavy quark decays is found to be suppressed by a centrality-dependent factor
that increases smoothly from peripheral to central collisions. Muon production
is suppressed by approximately a factor of two in central collisions relative
to peripheral collisions. Within the experimental errors, the observed
suppression is independent of muon $p_{\mathrm{T}}$ for all centralities.
###### keywords:
Heavy Ion , Heavy flavor suppression , $R_{\rm CP}$
## 1 Introduction
Collisions between lead ions at the LHC are thought to produce strongly
interacting matter at temperatures well above the QCD critical temperature. At
such temperatures, strongly interacting matter is expected to take the form of
“quark-gluon plasma.” High-$p_{\mathrm{T}}$ quarks and gluons generated in
hard-scattering processes during the initial stages of the nuclear collisions
are thought to lose energy in the quark-gluon plasma resulting in “jet
quenching” [Majumder:2010qh] . Since the energy loss results from the
interaction of a quark or gluon with the medium, jet quenching is thought to
provide a valuable tool for probing the properties of the quark-gluon plasma
[Wang:1994fx, Baier:1998yf, Gyulassy:2000fs]. Measurements of heavy quarks are
an important complement to studies of light quark and gluon quenching. The
contributions from radiative [Djordjevic:2003zk] and collisional
[Wicks:2005gt, Djordjevic:2006tw] energy loss in weakly coupled calculations
are expected to depend on the heavy quark mass. In particular, the mass of
heavy quarks is expected to reduce radiative energy loss for quark transverse
momenta less than or comparable to the quark mass ($m$),
$p_{\mathrm{T}}\lesssim m$, through the dead-cone effect [Dokshitzer:2001zm].
However, measurements of heavy quark production at RHIC via semi-leptonic
decays to electrons showed a combined charm and bottom suppression in Au+Au
collisions comparable to that observed for inclusive hadron production
[Adare:2006nq, Abelev:2006db, Adare:2010de]. There is disagreement in the
theoretical literature regarding the interpretation of the RHIC heavy quark
suppression measurements [Wicks:2005gt, Djordjevic:2011tm, Gossiaux:2008jv,
Uphoff:2011ad] particularly regarding the role of non-perturbative effects
[Adil:2006ra, vanHees:2007me, Horowitz:2008ig]. It is clear that measurements
of heavy quark quenching at the LHC are an essential complement to inclusive
jet [Aad:2010bu, Chatrchyan:1327643] or single hadron measurements
[CMS:2012aa, Aamodt:2010jd].
This paper presents ATLAS measurements of inclusive muon production in Pb+Pb
collisions at $\mbox{$\sqrt{s}$}=2.76$ TeV from an integrated luminosity of
approximately $7~{}{\rm\mu b^{-1}}$ collected in 2010\. The measurements were
performed for several intervals of collision centrality over the muon
transverse momentum range $4<p_{\mathrm{T}}<14$ GeV and the yields compared to
those in a peripheral bin using $R_{\rm CP}$:
$\mbox{$R_{\rm
CP}$}\equiv\frac{\mbox{$N_{\mathrm{coll}}$}^{\mathrm{periph}}}{\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}}\frac{\mathrm{d}n^{\mathrm{cent}}}{\mathrm{d}n^{\mathrm{periph}}},$
(1)
where $\mathrm{d}n^{\mathrm{cent}}$ and $\mathrm{d}n^{\mathrm{periph}}$
represent the per-event differential rate for the same hard-scattering
observable in central and peripheral collisions, respectively.
$\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}$ and
$\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}$ represent $N_{\mathrm{coll}}$
values calculated for the corresponding centralities (see Sec. 3). Over this
$p_{\mathrm{T}}$ range, muon production in _p_ +_p_ collisions results
predominantly from a combination of charm and bottom quark semi-leptonic
decays; all other sources including $J/\psi$ decays contribute less than 1% of
the prompt muon yield [Aad:2011rr]. The results presented here complement
previous measurements by the CMS Collaboration [Chatrchyan:2012np] of bottom
quark suppression performed using measurements of non-prompt $J/\psi$
production.
## 2 Experimental setup
Muons were detected by combining independent measurements of the muon
trajectories from the inner detector (ID) and the muon spectormeter (MS),
which covers $|\eta|<2.5$ and $|\eta|<2.7$ in pseudorapidity respectively.
Two forward calorimeters [Aad:2008zzm] placed symmetrically with respect to
$z=0$ and covering $3.2<|\eta|<4.9$ are used in this analysis to characterize
Pb+Pb collision centrality. They are composed of tungsten and copper absorbers
with liquid argon as the active medium; each calorimeter has a total thickness
of about 10 interaction lengths.
Minimum bias Pb+Pb collisions were identified using measurements from the zero
degree calorimeters (ZDCs) and the minimum-bias trigger scintillator (MBTS)
counters [Aad:2008zzm]. The ZDCs are located symmetrically at $z=\pm 140$ m
and cover $|\eta|>8.3$. In Pb+Pb collisions the ZDCs measure primarily
“spectator” neutrons, which originate from the incident nuclei and do not
interact hadronically. The MBTS detects charged particles over
$2.1<|\eta|<3.9$ using two counters placed at $z=\pm 3.6$ m. Each counter is
azimuthally divided into 16 sections, and the MBTS provides measurements of
both the pulse heights and arrival times of ionization energy deposits in each
section.
Events used in this analysis were selected for recording by the data
acquisition system using a logical OR of ZDC and MBTS coincidence triggers.
The MBTS coincidence required at least one hit in both sides of the detector,
and the ZDC coincidence trigger required the summed pulse height from each
calorimeter to be above a threshold set below the single neutron peak.
## 3 Event selection and centrality
In the offline analysis, Pb+Pb collisions were required to have a primary
vertex reconstructed from charged particle tracks with $p_{\mathrm{T}}>500$
MeV. The tracks were reconstructed from hits in the inner detector using the
standard track reconstruction algorithm [Cornelissen:2008zzc] with settings
optimized for the high hit density in heavy ion collisions [ATLAS:2011ah].
Additional requirements of at least one hit in each MBTS counter, a time
difference between the two MBTS detectors of less than 3 ns, and a ZDC
coincidence were imposed, yielding a total of 53 million minimum-bias Pb+Pb
events. The combination of the MBTS, ZDC, and the primary vertex requirements
eliminates efficiently both beam-gas interactions and photo-nuclear collisions
[Djuvsland:2010qs]. Previous studies [ATLAS:2011ah] indicate that this
combination of ZDC trigger and offline requirements select minimum-bias
hadronic Pb+Pb collisions with an efficiency of $98\pm 2\%$. The inefficiency
is concentrated in the very most peripheral collisions not used in this
analysis.
The centrality of Pb+Pb collisions was characterized by $\Sigma
E_{\mathrm{T}}^{\mathrm{FCal}}$, the total transverse energy measured in the
forward calorimeters (FCal). For the results presented in this paper, the
minimum-bias $\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution was divided
into centrality intervals according to the following percentiles of the
$\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution ordered from the most
central to the most peripheral collisions: 0-10%, 10-20%, 20-40%, 40-60%, and
60-80%. The centrality intervals were determined after accounting for very
peripheral events lost due to the minimum-bias trigger efficiency. A standard
Glauber Monte-Carlo analysis [Alver:2008aq, Miller:2007ri] was used to
estimate the average number of participating nucleons,
$\langle\mbox{$N_{\mathrm{part}}$}\rangle$, and the average number of nucleon-
nucleon collisions, $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$, for Pb+Pb
collisions in each of the centrality bins. The results are shown in Table 1.
The $R_{\rm CP}$ measurements presented in the paper use the 60-80% centrality
bin as a common peripheral reference.
The $R_{\rm CP}$ calculation uses the ratio
$\mbox{$R_{\mathrm{coll}}$}\equiv\langle N_{\mathrm{coll}}\rangle/\langle
N_{\mathrm{coll}}^{\mathrm{60-80}}\rangle$, where $\langle
N_{\mathrm{coll}}^{\mathrm{60-80}}\rangle$ is the average number of collisions
in the 60-80% centrality bin. The $R_{\mathrm{coll}}$ uncertainties have been
directly calculated by evaluating the changes in $R_{\mathrm{coll}}$ under
variations of the minimum-bias trigger efficiency, variations of the
parameters of the Glauber calculation such as the nucleon-nucleon cross-
section, the nuclear radius, and nuclear thickness, and variations in the
modeling of the $\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution
[ATLAS:2011ah]. The $R_{\mathrm{coll}}$ values and uncertainties are also
reported in Table 1.
Table 1: Results of Glauber model evaluation of $\langle\mbox{$N_{\mathrm{part}}$}\rangle$, $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$, and the $N_{\mathrm{coll}}$ ratio, $R_{\mathrm{coll}}$for each centrality. The quoted uncertainties represent the systematic uncertainties in the Glauber calculation. Centrality | $\langle\mbox{$N_{\mathrm{part}}$}\rangle$ | $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$ | $R_{\mathrm{coll}}$
---|---|---|---
0–10% | $356\pm 2$ | $1498\pm 133$ | $56.7\pm 6.2$
10–20% | $261\pm 4$ | $923\pm 83$ | $34.9\pm 3.8$
20–40% | $158\pm 4$ | $441\pm 37$ | $16.7\pm 1.5$
40–60% | $69.3\pm 3.5$ | $130\pm 10$ | $4.9\pm 0.2$
60–80% | $22.6\pm 2.3$ | $26.5\pm 2$ | $1$
## 4 Muon reconstruction
Muons used in this analysis were obtained by combining separate measurements
of the muon trajectory in the ID and the MS using the same algorithms that
were developed for _p_ +_p_ collisions [Aad:2011rr]. The ID tracks were
required to have a hit in the first pixel layer, at least 7 hits in the
microstrip detector, no missing hits in the microstrip detector in active
sensors, and a maximum of one missing hit in the pixel detector. The tracks
were also required to point to within 5 mm of the primary vertex in both the
transverse and longitudinal directions.
The reconstructed muon momenta were obtained from a combined fit to the muon
trajectories in the ID and MS. The results presented here use muons having
$4<p_{\mathrm{T}}<14$ GeV and $|\eta|<1.05$. The lower limit of the
$p_{\mathrm{T}}$ range was chosen to be near the plateau of the muon
efficiency curve. The upper $p_{\mathrm{T}}$ limit results both from the
limited statistics of the data and the desire to limit this measurement to a
muon $p_{\mathrm{T}}$ range where the contribution of $W$ decays to the muon
spectrum is less than 1% [Aad:2011rr, ATLAS-CONF-2011-078]. The muon $\eta$
interval was chosen to avoid transitional regions in calorimeter coverage
where the muon performance is degraded. The momentum resolution is 5% over the
measured $p_{\mathrm{T}}$ range. The number of reconstructed muons passing the
described cuts in each $p_{\mathrm{T}}$ and centrality bin are listed in Table
2.
Table 2: Approximate number of reconstructed muon candidates in the
pseudorapidity interval $|\eta|<1.05$ in each $p_{\mathrm{T}}$ and centrality
bin.
Centrality [%] $p_{\mathrm{T}}$ [GeV] 0–10 10–20 20–40 40–60 60–80 4–5 49k 36k
43k 17k 4.7k 5–6 19k 15k 18k 7.4k 2.0k 6–7 8.6k 6.5k 8.2k 3.4k 0.93k 7–8 4.1k
3.1k 3.9k 1.6k 0.46k 8–9 2.2k 1.7k 2.1k 0.78k 0.21k 9–10 1.2k 0.93k 1.1k 0.46k
0.13k 10–14 1.7k 1.31k 1.5k 0.62k 0.15k
The performance of the ATLAS detector and offline analysis in measuring muons
in Pb+Pb collisions was evaluated using a Monte Carlo (MC) data set [atlassim]
originally prepared for studies of jet performance. The sample also provides
enough events containing high-$p_{\mathrm{T}}$ muons from heavy flavour decays
and background sources for the purposes of this analysis. The MC data set was
obtained by overlaying GEANT4-simulated [Agostinelli:2002hh]
$\mbox{$\sqrt{s}$}=2.76$ TeV _p_ +_p_ dijet events on to 1 million
GEANT4-simulated minimum-bias Pb+Pb events obtained from version 1.38b of the
HIJING event generator [Wang:1991hta]. HIJING was run with default parameters,
except for the disabling of jet quenching. To simulate the effects of elliptic
flow [ATLAS:2011ah] in Pb+Pb collisions, a parameterized $\cos{2\phi}$
modulation that varies with centrality, $\eta$ and $p_{\mathrm{T}}$ as
determined by previous ATLAS measurements [ATLAS:2011ah] was imposed on the
particles after generation [Masera:2009zz]. The _p_ +_p_ events were obtained
from the ATLAS MC09 tune [:2010ir] of the PYTHIA event generator
[Sjostrand:2006za]. One million PYTHIA hard scattering events were generated
for each of five intervals of $\hat{p}_{\mathrm{T}}$, the transverse momentum
of outgoing partons in the $2\rightarrow 2$ hard scattering, with boundaries
$17,35,70,140,280$ and 560 GeV. The _p_ +_p_ events for each
$\hat{p}_{\mathrm{T}}$ interval were overlaid on the same sample of HIJING
events.
Figure 1: Efficiency ($\varepsilon$) for reconstructing heavy flavour semi-
leptonic decay muons as a function of muon $p_{\mathrm{T}}$ for three
collision centrality bins. The efficiency is shown over a wider
$p_{\mathrm{T}}$ range than that used in the analysis. The dotted line
indicates the plateau efficiency value, $\mbox{$\varepsilon$}=0.8$.
The efficiency for reconstructing muons associated with heavy flavour decays
was evaluated using the MC sample described above for different bins in
collision centrality. Figure 1 shows the fraction of muons from semi-leptonic
charm and bottom decays reconstructed as a function of muon $p_{\mathrm{T}}$,
$\mbox{$\varepsilon$}(p_{\mathrm{T}})$, for a subset of the centrality bins:
0-10%, 20-40%, and 60-80%. For $p_{\mathrm{T}}\gtrsim 5$ GeV, the
reconstruction efficiency is found to be constant within the statistical
uncertainties. The $\mbox{$\varepsilon$}(p_{\mathrm{T}})$ values for each
centrality bin were fit to a constant over the range $5<p_{\mathrm{T}}<14$ GeV
to estimate the “plateau” efficiency. The results for all centrality bins were
consistent with $\mbox{$\varepsilon$}_{\mathrm{plat}}=0.80\pm 0.02$. For
$4<p_{\mathrm{T}}<5$ GeV, a statistically significant variation of
$\varepsilon$ with centrality is observed and the efficiencies were taken
directly from the results shown in Fig. 1. Those vary between $0.67\pm 0.02$
in peripheral collisions (60-80%) to $0.64\pm 0.02$ in central collisions
(0-10%).
Figure 2: Simulated $C$ distributions, $\mathrm{d}P/\mathrm{d}\mbox{$C$}$,
for signal and background muons for the full (0-80%) centrality range and for
the 0-10% and 60-80% centrality bins for muons in the momentum range of 5 – 6
GeV.
## 5 Muon signal extraction
The measured muons consist of both “signal” muons and background muons. The
signal muons are those that originate directly from the Pb+Pb collision, from
vector meson decays, and from heavy quark decays. The background muons arise
from pion and kaon decays, muons produced in hadronic showers in the
calorimeters, and mis-associations of MS and ID tracks. Previous studies have
shown that the signal and background contributions to the reconstructed muon
sample can be discriminated statistically [Aad:2011rr, ATLAS-CONF-2011-003].
For this analysis a weighted combination of two discriminant quantities has
been used [ATLAS-CONF-2011-003]. The fractional momentum imbalance,
$\mbox{$\Delta p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$, quantifies the
difference between the ID and MS measurements of the muon momentum after
accounting for the energy loss of the muon in the calorimeters [ATLAS-
CONF-2010-075]. It is defined as:
$\dfrac{\mbox{$\Delta
p_{\mathrm{loss}}$}}{\mbox{$p_{\mathrm{ID}}$}}=\dfrac{\mbox{$p_{\mathrm{ID}}$}-\mbox{$p_{\mathrm{MS}}$}-\mbox{$\Delta
p_{\mathrm{calo}}$}(p,\eta,\phi)}{\mbox{$p_{\mathrm{ID}}$}},$ (2)
where $p_{\mathrm{ID}}$ and $p_{\mathrm{MS}}$ represent the reconstructed muon
momenta from the ID and MS, respectively and $\Delta p_{\mathrm{calo}}$
represents a momentum and angle-dependent average energy loss of muons in the
calorimeter obtained from MC simulations. The “scattering significance”, $S$,
characterizes deflections in the trajectory resulting from (e.g.) decays in
flight. For each measurement along the muon trajectory, $i$, the deviation in
azimuthal angle of the measured hit position from the fitted trajectory,
$\Delta\phi_{i}$, was calculated and compared to the expected deviation from
multiple scattering, $\phi^{\mathrm{msc}}$, to obtain the signed quantity
$\mbox{$s_{i}$}\equiv q\Delta\phi_{i}/\mbox{$\phi^{\mathrm{msc}}$}$, where $q$
represents the track charge. The TRT measurements were not used when
calculating $s_{i}$ due to the high occupancy of the TRT in Pb+Pb collisions.
Then, the scattering significance at the $k{\rm th}$ layer, $S(k)$, was
calculated according to:
$\displaystyle\mbox{$S(k)$}=\frac{1}{\sqrt{n}}\left(\sum_{i=1}^{k}s_{i}-\sum_{j=k+1}^{n}s_{j}\right).$
(3)
Here $n$ represents the total number of measurements of the muon trajectory;
typically $n=16$. $S(k)$ yields a statistically significant value when the
muon is deflected between measurements $k$ and $k+1$. An overall scattering
significance, $S$, was obtained from the maximum value of $S(k)$:
$\displaystyle S=\max{\left\\{|\mbox{$S(k)$}|,k=1,2,...\right\\}}.$ (4)
The effectiveness of $\mbox{$\Delta
p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$ and $S$ in discriminating
between signal and background muons varies with momentum in a complementary
manner. $S$ is more effective at lower muon momenta where the deflections
resulting from (e.g.) in-flight decays are greater. $\mbox{$\Delta
p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$ is more effective at larger muon
momenta where in-flight decays are less important and a larger fraction of the
muons are generated in hadronic showers. To take advantage of the
complementarity of the two discriminants, a “composite” discriminant was
formed [ATLAS-CONF-2011-003] according to:
$\mbox{$C$}=\left|\dfrac{\mbox{$\Delta
p_{\mathrm{loss}}$}}{\mbox{$p_{\mathrm{ID}}$}}\right|+rS$ (5)
where $r$ is a parameter that controls the relative contribution of the two
discriminants to $C$. MC studies indicate that $r=0.07$ provides optimal
separation between signal and background muons over the muon momentum range
studied here [ATLAS-CONF-2011-003].
Distributions for $C$ were obtained from the PYTHIA$+$HIJING Monte Carlo
sample for the different centrality and muon $p_{\mathrm{T}}$ bins used in
this analysis. The $C$ distributions were obtained separately for signal muons
from charm and bottom decays and for background muons generated by pions and
kaons. Figure 2 shows the resulting $C$ distributions for muons with
$5<p_{\mathrm{T}}<6$ GeV in the 0-10% and 60-80% centrality bins. As
demonstrated by the figure, the signal and background $C$ distributions were
found to vary only weakly with collision centrality. As a result, for the
template fitting procedure described below, the analysis was carried out using
centrality-integrated (0-80%) signal and background $C$ distributions which
are also shown in Fig. 2.
Figure 3: Examples of the template fits to measured $C$ distributions (see
text) in 0-10% (left) and 60-80% (right) centrality bins for two
$p_{\mathrm{T}}$ intervals: $5<p_{\mathrm{T}}<6$ GeV (top) and
$10<p_{\mathrm{T}}<14$ GeV (bottom). The curves show the contributions of
signal and background sources based on the corresponding $C$ distributions
obtained from MC including the shift, re-scaling, and smearing modifications
(see text).
The MC signal and background $C$ distributions were used to perform a template
fit to the $C$ distributions in the data. Specifically, a hypothetical
probability density distribution for $C$, $\mathrm{d}P/\mathrm{d}\mbox{$C$}$
is formed assuming that a fraction, ${f_{\mathrm{S}}}$, of the measured muons
is signal,
$\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}=\left.\mbox{${f_{\mathrm{S}}}$}\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}\right|_{\mathrm{S}}+(1-\mbox{${f_{\mathrm{S}}}$})\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}\right|_{\mathrm{B}},$
(6)
where $\mathrm{d}P/\mathrm{d}\mbox{$C$}|_{\mathrm{S}}$ and
$\mathrm{d}P/\mathrm{d}\mbox{$C$}|_{\mathrm{B}}$ represent the Monte Carlo $C$
distributions from signal and background sources, respectively.
To account for possible differences in the $C$ distribution between data and
MC due to (e.g.) inaccuracies in the MC description of multiple scattering,
the fits allow for limited adaptation of the MC templates to the data. In
particular, the fit allows a shift and re-scaling of the $C$ distribution,
$\mbox{$C$}^{\prime}=a+\mbox{$\langle
C\rangle$}+b\left(\mbox{$C$}-\mbox{$\langle C\rangle$}\right)$, where $\langle
C\rangle$ is the usual mean of the $C$ distribution. In addition, a Gaussian
smearing of the MC $C$ distribution was included such that the data $C$
distribution was fit to
$\dfrac{\mathrm{d}P^{\prime}}{\mathrm{d}\mbox{$C$}^{\prime}}\equiv\left(\mbox{${f_{\mathrm{S}}}$}\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}^{\prime}}\right|_{\mathrm{S}}+(1-\mbox{${f_{\mathrm{S}}}$})\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}^{\prime}}\right|_{\mathrm{B}}\right)\otimes\dfrac{e^{-{\mathcal{C}}^{\prime
2}/2\sigma^{2}}}{\sqrt{2\pi}\sigma}$ (7)
A kernel estimation method [Cranmer:2000du] was used to produce the unbinned
probability density distribution, $dP^{\prime}/d\mbox{$C$}^{\prime}$ from the
MC signal and background samples. The signal fraction, ${f_{\mathrm{S}}}$, was
then evaluated for each centrality and muon $p_{\mathrm{T}}$ bin using
unbinned maximum likelihood fits with four free parameters,
${f_{\mathrm{S}}}$, $a$, $b$, and $\sigma$. The fits were performed using
MINUIT [James:1975dr] as implemented in RooFit[Verkerke:2003ir]. Examples of
the resulting template fits are shown in Fig. 3 for the 0-10% and 60-80%
centrality bins and for two muon $p_{\mathrm{T}}$ intervals. Typical fitted
values for the $a,b,$ and $\sigma$ parameters in Eq. 7 are, $a\sim 0.02$,
$b\sim 0.95-1.05$, and $\sigma\sim 0.002$. The combination of the MC signal
and background $C$ distributions are found to describe well the measured $C$
distributions. The description remains good even if the adaptation of the
template described in Eq. 7 is removed, though the fit results may change (see
below).
The signal muon fractions extracted from the template fits for all
$p_{\mathrm{T}}$ and centrality bins are shown in Fig. 4. The statistical
uncertainties on ${f_{\mathrm{S}}}$ from the fits represent $1\sigma$
confidence intervals that account for the limited statistics in the data $C$
distributions and correlations between the fit parameters. The uncertainties
in the fit results due to the finite MC statistics were evaluated using a
pseudo-experiment technique in which new pseudo-MC $C$ distributions with the
same number of counts as the original MC $C$ distributions were obtained by
statistically sampling the MC distributions. The resulting pseudo-MC
distributions were then used to perform template fits. The procedure was
repeated eight times for each $p_{\mathrm{T}}$ and centrality bin and the
standard deviation of the resulting ${f_{\mathrm{S}}}$ values in each bin was
combined in quadrature with the statistical uncertainty from the original fit
to produce a combined statistical uncertainty on ${f_{\mathrm{S}}}$. The
fractional uncertainties on ${f_{\mathrm{S}}}$ due to MC statistics are found
to be below $2\%$ and are typically much smaller than the uncertainties from
the template fits.
Figure 4: Fractions of signal muons, ${f_{\mathrm{S}}}$, in the measured muon
yields as a function of muon $p_{\mathrm{T}}$ in different bins of collision
centrality. The points are shown at the mean transverse momentum of the muons
in the given $p_{\mathrm{T}}$ bin. The shaded boxes indicate systematic errors
(see text) due to possible data-MC template matches, hadron composition, the
template fitting (i.e. combination of $dP/dC$, $K/\pi$ and “Fit” entries in
Table 3), which can vary from point-to-point. The error bars show combined
statistical and systematic uncertainties.
The sensitivity of the measured ${f_{\mathrm{S}}}$ values to the adaptation of
the MC $C$ distributions to the data was evaluated by performing the fits
fixing the $a$ and $\sigma$ parameters to zero and $b$ to one. Those fits
yielded systematically lower ${f_{\mathrm{S}}}$ values than the values
obtained from the default fits. The reductions varied from as much as 18% for
the 60-80%, 4–5 GeV bin to about 3% for the highest $p_{\mathrm{T}}$ bins. The
differences between the two values for each centrality and $p_{\mathrm{T}}$
bin were used to estimate systematic uncertainties on the fit due to data-MC
matching of the $C$ distributions. Typically, those errors are largest for the
lowest $p_{\mathrm{T}}$ bin and decrease with increasing $p_{\mathrm{T}}$, but
the 6–7 GeV and 7–8 GeV bins also show enhanced sensitivity to the template
adaptation. That is particularly true for the 60-80% 7–8 GeV $p_{\mathrm{T}}$
bin which suffers from low statistics.
The use of the centrality-integrated MC $C$ distributions in the template
fitting provides another potential source of data-MC mismatch. To evaluate the
impact of this choice, the analysis was performed using centrality-selected MC
$C$ distributions. The reduction in template statistics worsened the
statistical accuracy of the fits, particularly for the most peripheral bin.
However, in all cases the differences from the default results were within the
uncertainty estimates described in the preceding paragraph. Based on that
observation and the judgment that the adaptation of the template distribution
can likely account for the small centrality differences observed in the $C$
distributions, no additional systematic error is assigned to account for the
use of the centrality-integrated templates.
Figure 5: Invariant differential muon per-event yields calculated according
to Eq. 9 as a function of muon $p_{\mathrm{T}}$ for different bins in
collision centrality. The points are shown at the mean transverse momentum of
the muons in the given $p_{\mathrm{T}}$ bin. The statistical errors are
smaller than the sizes of the points in all bins and are not shown. The error
bars indicate combined statistical and systematic errors (see text).
The MC background $C$ distributions depend on the primary hadron composition
particularly the relative proportions of kaons and pions ($K/\pi$) which may
differ from that in the data. To test the sensitivity of the results to such a
difference, new MC background $C$ distributions were obtained by separately
doubling the $\pi$ and $K$ contributions. The new $\mbox{$C$}_{\mathrm{B}}$
distributions were used in the template fits and the resulting differences in
extracted ${f_{\mathrm{S}}}$ used to estimate the $K/\pi$ systematic
uncertainties on the ${f_{\mathrm{S}}}$ values. The resulting (fractional)
uncertainties are typically of order 1%, but they can be as large as 5% in the
lowest $p_{\mathrm{T}}$ bin.
Potential systematic uncertainties resulting from the template fitting
procedure were evaluated using a simple cut procedure applied to the $C$
distributions. For a given centrality and $p_{\mathrm{T}}$ bin, a cut was
applied at a chosen value of $C$, ${C_{\mathrm{cut}}}$, and the fraction of
muons in the data below the cut, $f^{<}$ was calculated. For an appropriate
choice of cut, that fraction represents most of the signal muons with a modest
contamination of background muons. The MC signal and background $C$
distributions were used to evaluate the fraction of the signal (background)
muons below the cut, $f^{<}_{\mathrm{S}}$ ($f^{<}_{\mathrm{B}}$). Then, the
fraction of signal muons evaluated using this cut method,
$f_{\mathrm{S}}(\mbox{${C_{\mathrm{cut}}}$})$, was estimated:
$f_{\mathrm{S}}(\mbox{${C_{\mathrm{cut}}}$})=\dfrac{f^{<}-f^{<}_{\mathrm{B}}}{f^{<}_{\mathrm{S}}-f^{<}_{\mathrm{B}}}.$
(8)
$f_{\mathrm{S}}^{\mathrm{cut}}$ values were obtained using
${C_{\mathrm{cut}}}$ values of 0.15, 0.2, and 0.25. The results from the three
${C_{\mathrm{cut}}}$ values agree typically to within a few %. The averages of
the three $f_{\mathrm{S}}^{\mathrm{cut}}$ values for each $p_{\mathrm{T}}$ and
centrality bin were compared to the results of the template fit where the
adaptation was disabled. These two different estimates of the signal fraction
agreed typically to better than 1%, but differences as large as 5% were
observed for the 60-80% centrality bin. These differences were used to
establish systematic uncertainties associated with the template fitting
procedure.
Table 3 summarizes the contributions to the systematic error in the
determination of the signal fraction, ${f_{\mathrm{S}}}$, for three sample
$p_{\mathrm{T}}$ bins in 0-10% and 60-80% centrality bins.
Table 3: Systematic uncertainty contributions from different sources for sample $p_{\mathrm{T}}$ bins in 0-10% and 60-80% centrality bins. “$dP/dC$” represents the systematic uncertainties due to potential data-MC mismatches in the template, “Fit” represents the estimated systematic on the fitting procedure evaluated using the cut method. “$K/\pi$” represents the uncertainty due to the hadron composition and “$\varepsilon$” represents the uncertainty on the efficiency estimation. Centrality | $p_{\mathrm{T}}$ | Uncertainty [$\%$]
---|---|---
[%] | [GeV] | $dP/dC$ | Fit | $K/\pi$ | $\varepsilon$ | Total
0-10 | 4–5 | 4 | 0 | 5 | 3 | 7
7–8 | 5 | 0.5 | 0.5 | 2 | 5.5
10–14 | 4 | 1 | 1 | 2 | 5
60-80 | 4–5 | 18 | 1 | 5 | 3 | 19
7–8 | 14 | 5 | 0.5 | 2 | 15
10–14 | 4 | 4 | 2 | 2 | 6
## 6 Results
The invariant differential per-event muon yields for a given centrality bin
were obtained according to
$\left.\dfrac{1}{\mbox{$N_{\mathrm{evt}}$}}\dfrac{d^{2}N}{p_{\mathrm{T}}dp_{\mathrm{T}}d\eta}\right|_{\mathrm{cent}}=\dfrac{1}{N_{\mathrm{evt}}^{\mathrm{cent}}}\dfrac{N_{\mathrm{S}}^{\mathrm{cent}}(p_{\mathrm{T}})}{\varepsilon^{\mathrm{cent}}(p_{\mathrm{T}})\;p_{\mathrm{T}}\Delta
p_{\mathrm{T}}\Delta\eta}$ (9)
where $N_{\mathrm{evt}}$ is the number of sampled Pb+Pb collisions in the
centrality interval, $N_{\mathrm{S}}^{\mathrm{cent}}$ is the number of signal
muons in the centrality bin and $\varepsilon^{\mathrm{cent}}$ is the muon
reconstruction efficiency discussed in the previous section. Figure 5 shows
the resulting distributions for the five centrality bins included in this
analysis. The yields increase by more than an order of magnitude between
peripheral and central collisions as expected from the geometric enhancement
of hard scattering rates.
The muon $R_{\rm CP}$ is calculated from the ratio of the invariant
differential yields between a given centrality bin and the 60-80% bin
multiplied by $1/\mbox{$R_{\mathrm{coll}}$}$. Since the kinematic and phase
space factors cancel between the numerator and denominator, the $R_{\rm CP}$
calculation reduces to
$\left.\mbox{$R_{\rm
CP}$}(p_{\mathrm{T}})\right|_{\mathrm{cent}}=\dfrac{1}{\displaystyle
R_{\mathrm{coll}}^{\mathrm{cent}}}\left(\dfrac{\dfrac{1}{N_{\mathrm{evt}}^{\mathrm{cent}}}\dfrac{N_{\mathrm{S}}^{\mathrm{cent}}}{\varepsilon^{\mathrm{cent}}}}{\dfrac{1}{N_{\mathrm{evt}}^{60-80}}\dfrac{N_{\mathrm{S}}^{60-80}}{\varepsilon^{60-80}}}\right).$
(10)
The resulting $R_{\rm CP}$ values are shown in Fig. 6. The $R_{\rm CP}$ values
vary weakly with $p_{\mathrm{T}}$ and the points for each centrality interval
are consistent with a $p_{\mathrm{T}}$-independent $R_{\rm CP}$ within the
uncertainties on the points. The $R_{\rm CP}$ varies strongly with centrality,
increasing from about 0.4 in the 0-10% centrality bin to about 0.85 in the
40-60% bin.
Figure 6: Muon $R_{\rm CP}$ as a function of $p_{\mathrm{T}}$ calculated
according to Eq. 10 for different centrality bins. The points are shown at the
mean transverse momentum of the muons in the given $p_{\mathrm{T}}$ bin. The
error bars include both statistical and systematic uncertainties. The
contribution of the systematic uncertainties from $R_{\mathrm{coll}}$ and
$\varepsilon$, which are fully correlated between $p_{\mathrm{T}}$ bins, are
indicated by the shaded boxes.
One disadvantage of $R_{\rm CP}$ is the poor statistics in peripheral bins,
which have small geometric enhancement of hard scattering processes compared
to central collisions. Statistical errors in the reference peripheral
$p_{\mathrm{T}}$ bin then influence the $R_{\rm CP}$ results for that bin for
all centralities. Such an effect can be observed in Fig. 6 for the 7–8 GeV
$p_{\mathrm{T}}$ bin. Alternatively, the centrality variation can be
characterized by comparing different centrality bins to the most central bin
which has the highest statistics producing a peripheral-to-central ratio,
$R_{\rm PC}$. The $R_{\rm PC}$ is defined analogously to Eq. 10 but with the
60-80% reference replaced by the 0-10% centrality bin. Figure 7 shows the
resulting $R_{\rm PC}$ values as a function of $p_{\mathrm{T}}$ for different
collision centralities. The figure provides similar insights as Fig. 6, but
the $R_{\rm PC}$ allows a stronger statement about the $p_{\mathrm{T}}$
dependence of the suppression. Namely, the shape of the muon spectra in the
0-10%, 10-20%, 20-40% and 40-60% centrality bins remains the same, within
uncertainties, while over those centrality bins the suppression is changing by
a factor of nearly 2.
The variation of $R_{\rm CP}$ with centrality as characterized by the average
number of participants, $\langle N_{\mathrm{part}}\rangle$, is shown in Fig. 8
for the different $p_{\mathrm{T}}$ bins included in the analysis. The $R_{\rm
CP}$ decreases smoothly from peripheral to central collisions. The centrality
dependence is observed to be the same for all $p_{\mathrm{T}}$ bins within the
experimental uncertainties.
Figure 7: Muon peripheral to central ratio, $R_{\rm PC}$, as a function of
$p_{\mathrm{T}}$ calculated according to Eq. 10 for different centrality bins.
The points are shown at the mean transverse momentum of the muons in the given
$p_{\mathrm{T}}$ bin. The error bars include both statistical and systematic
uncertainties. The contribution of the systematic uncertainties from
$R_{\mathrm{coll}}$ and $\varepsilon$, which are fully correlated between
$p_{\mathrm{T}}$ bins, are indicated by the shaded boxes.
## 7 Discussion
CMS has previously reported measurements of $R_{\rm AA}$ for non-prompt
$J/\psi$ produced in $b$ decays in $\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb
collisions [Chatrchyan:2012np]. The $R_{\rm AA}$ in the 0-20% centrality bin
($\mbox{$\langle N_{\mathrm{part}}\rangle$}=308$) is consistent with the 0-10%
($\mbox{$\langle N_{\mathrm{part}}\rangle$}=356$) muon $R_{\rm CP}$ presented
here in the $10<p_{\mathrm{T}}<14$ GeV bin, for which the muon yield is
expected to be dominated by $b$ decay contributions. However, the $R_{\rm AA}$
reported by CMS in the 20-100% bin ($\mbox{$\langle
N_{\mathrm{part}}\rangle$}=64$) falls well below the $R_{\rm CP}$ measurements
presented here. Since the CMS results are in terms of $R_{\rm AA}$ and this
analysis uses centrality 60-80% as the peripheral reference, we have repeated
the analysis using a 70-90% bin as reference and find no evidence that our
choice of peripheral reference can account for this difference.
Figure 8: Muon $R_{\rm CP}$ as a function of $\langle
N_{\mathrm{part}}\rangle$ calculated according to Eq. 10 for different bins in
muon $p_{\mathrm{T}}$. The error bars show combined statistical and systematic
uncertainties. The sets of points for the different $p_{\mathrm{T}}$ bins are
successively displaced by $\Delta\mbox{$N_{\mathrm{part}}$}=6$ for clarity of
presentation.
The measurements presented in this paper of the suppression of muons from
heavy flavour decay differ appreciably from measurements of single hadron
suppression at the LHC at comparable transverse momenta [Aamodt:2010jd,
CMS:2012aa, ATLAS-CONF-2011-079]. The single hadron $R_{\rm CP}$ reported by
CMS [CMS:2012aa] in the 0-5% centrality bin relative to the 50-90% centrality
interval is a factor of approximately two smaller than the 0-10% muon $R_{\rm
CP}$ value reported here, while that in the 5-10% centrality bin is lower by
about 50%. The single hadron $R_{\rm CP}$ is also observed to vary much more
rapidly with $p_{\mathrm{T}}$ than the muon $R_{\rm CP}$ shown in Fig. 6.
Interpretation of this difference may be complicated by the indirect
relationship between the transverse momentum of the observed particle and the
parent parton for both the hadrons and the muons. Nonetheless, the clear
difference between semi-leptonic muon and charged hadron suppression at the
LHC should be contrasted with the situation at RHIC where the experimental
data do not indicate such a difference [Abelev:2006db].
PHENIX has reported a semi-leptonic electron $R_{\rm AA}$ for
$p_{\mathrm{T}}>4$ GeV in the 0-10% centrality bin of $0.30\pm
0.02(\mathrm{stat})\pm 0.04(\mathrm{syst})$ [Adare:2010de]. STAR has reported
similar results for $\mbox{$R_{\rm AA}$}(p_{\mathrm{T}})$ [Abelev:2006db]. The
RHIC semi-leptonic electron results show more suppression than the
measurements reported here, while the natural expectation would be that the
suppression would be greater in Pb+Pb collisions at the LHC due to the higher
initial energy densities [Horowitz:2011gd].
## 8 Conclusions
This paper has presented ATLAS measurements of muon production and suppression
in $\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb collisions in a transverse momentum
range dominated by heavy flavor decays, $4<p_{\mathrm{T}}<14$ GeV, and over
the pseudo-rapidity range $|\eta|<1.05$. The fraction of prompt muons was
estimated using template fits to the distribution of a quantity capable of
distinguishing statistically between signal and background. The
$p_{\mathrm{T}}$ spectra of signal muons were evaluated in five centrality
bins: 0-10%, 10-20%, 20-40%, 40-60%, and 60-80%. The centrality dependence of
muon production was characterized using the central-to-peripheral ratio,
$R_{\rm CP}$, calculated using the 60-80% centrality bin as a peripheral
reference. The results for $R_{\rm CP}$ indicate a factor of about 2.5
suppression in the yield of muons in the most central (0-10%) collisions
compared to the most peripheral collisions included in the analysis (60-80%).
No significant variation of $R_{\rm CP}$ with muon $p_{\mathrm{T}}$ is
observed. The $R_{\rm CP}$ decreases smoothly from peripheral to central
collisions. The central muon $R_{\rm CP}$ for $10<p_{\mathrm{T}}<14$ GeV
agrees with the central (0-10%) non-prompt $J/\psi$ $R_{\rm AA}$ measured by
CMS. The central $R_{\rm CP}$ is 1.5-2 times larger than that measured for
charged hadrons in a comparable $p_{\mathrm{T}}$ range and using comparable,
though not identical, centrality ranges. The central $R_{\rm CP}$ indicates
weaker suppression than observed in the semi-leptonic electron $R_{\rm AA}$
measured over the same $p_{\mathrm{T}}$ range in $\mbox{$\sqrt{s}$}=200$ GeV
Au+Au collisions at RHIC.
|
arxiv-papers
| 2012-09-27T16:47:59 |
2024-09-04T02:49:35.691817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yujiao Chen (for the ATLAS Collaboration)",
"submitter": "Yujiao Chen",
"url": "https://arxiv.org/abs/1209.6282"
}
|
1209.6415
|
eurm10 msam10
# Viscous boundary layer properties in turbulent thermal convection in a
cylindrical cell: the effect of cell tilting
PINGWEIANDKE-QINGXIA Department of Physics, The Chinese University of Hong
Kong, Shatin, China
(?; revised ?; accepted ?. - To be entered by editorial office)
###### Abstract
We report an experimental study of the properties of the velocity boundary
layer in turbulent Rayleigh-Bénard convection in a cylindrical cell. The
measurements were made at Rayleigh numbers $Ra$ in the range $2.8\times
10^{8}<Ra<5.6\times 10^{9}$ and were conducted with the convection cell tilted
with an angle $\theta$ relative to gravity, at $\theta=0.5^{o}$, $1.0^{o}$,
$2.0^{o}$, and $3.4^{o}$, respectively. The fluid was water with Prandtl
number $Pr=5.3$.
It is found that at small tilt angles ($\theta\leq 1^{o}$), the measured
viscous boundary layer thickness $\delta_{v}$ scales with the Reynolds number
$Re$ with an exponent close to that for a Prandtl-Blasius laminar boundary
layer, i.e. $\delta_{v}\sim Re^{-0.46\pm 0.03}$. For larger tilt angles, the
scaling exponent of $\delta_{v}$ with $Re$ decreases with $\theta$. The
normalized mean horizontal velocity profiles measured at the same tilt angle
but with different $Ra$ are found to have an invariant shape. But for
different tilt angles, the shape of the normalized profiles is different.
It is also found that the Reynolds number $Re$ based on the maximum mean
horizontal velocity scales with $Ra$ as $Re\sim Ra^{0.43}$ and the Reynolds
number $Re_{\sigma}$ based on the maximum rms velocity scales with $Ra$ as
$Re_{\sigma}\sim Ra^{0.55}$, with both exponents do not seem to depend on the
tilt angle $\theta$.
Several wall quantities are also measured directly and their dependency on
$Re$ are found to agree well with those predicted for a classical laminar
boundary layer. These are the wall shear stress $\tau$ ($\sim Re^{1.46}$), the
viscous sublayer $\delta_{w}$ ($\sim Re^{0.75}$), the friction velocity
$u_{\tau}$ ($\sim Re^{-0.86}$) and the skin friction coefficient $c_{f}$
($\sim Re^{-0.46}$). Again, all these near-wall quantities do not seem to
depend on the tilt angle.
We also examined the dynamical scaling method proposed bys Zhou and Xia [Phys.
Rev. Lett. 104, 104301 (2010)] and found that in both the laboratory and the
dynamical frames the mean velocity profiles show deviations from the
theoretical Prandtl-Blasius profile, with the deviations increase with $Ra$.
But profiles obtained from dynamical scaling in general have better agreement
with the theoretical profile. It is also found that the effectiveness of this
method appears to be independent of $Ra$.
## 1 Introduction
### 1.1 Rayleigh-Bénard convection
Rayleigh-Bénard (RB) convection, which is a fluid layer heated from below and
cooled from the top, is an idealized model to study turbulent flows involving
heat transport and has attracted much attention during the past few decades
(Siggia, 1994; Kadanoff, 2001; Ahlers, Grossmann & Lohse, 2009; Lohse & Xia,
2010). The system is characterized by two control parameters: the Rayleigh
number $Ra$, Prandtl number $Pr$, which are defined as
$Ra=\frac{\alpha g\Delta TH^{3}}{\nu\kappa},$ (1)
and
$Pr=\frac{\nu}{\kappa},$ (2)
respectively. Here $\alpha$ is the thermal expansion coefficient, $g$ the
gravitational acceleration, $\Delta T$ the temperature difference between the
bottom and the top plates, $H$ the height of the fluid layer between the
plates, $\nu$ the kinematic viscosity, and $\kappa$ the thermal diffusivity of
the convecting fluid. In addition, the aspect ratio $\Gamma=D/H$ ($D$ is the
lateral dimension of the system) also plays an important role in the
structures and dynamics of the flow.
In a fully developed Rayleigh-Bénard turbulent flow, most of the imposed
temperature difference is localized in the thermal boundary layers near the
surface of the top and bottom plates, within which heat is transported via
conduction (Wu & Libcharber, 1991; Belmonte, Tilgner & Libchaber, 1994; Lui &
Xia, 1998). The velocity field has the same character: velocity gradient is
localized in a thin layer near the plates, which is called viscous boundary
layer. Turbulent flow in the central region of the RB cell is approximately
homogenous and isotropic(Zhou, Sun & Xia, 2008; Ni, Huang & Xia, 2011a, 2012).
As the top and bottom boundary layers contribute the main resistance to heat
transfer through the cell and thus dominantly determine the Nusselt number,
they deserve special attention. Indeed, nearly all theories in RB convection
are in essence boundary-layer (BL) theories. For example, a turbulent BL was
assumed in the early marginal stability theory (Malkus, 1954) and also in the
models by Shraiman & Siggia (1990) and Siggia (1994) and by Dubrulle (2001,
2002). On the other hand, a Prandtl-Blasius (PB) type laminar BL was
assumption in the Grossmann & Lohse (GL) theory (Grossmann & Lohse, 2000,
2001, 2002, 2004). Therefore, direct characterization of the BL properties is
essential for testing and differentiating the various theoretical models, and
will also provide insight into the physical nature of turbulent heat transfer.
### 1.2 Boundary layer measurements in turbulent thermal convection
One of the earlier measurements of temperature and also velocity profiles in
turbulent RB convection was taken by Tilgner, Belmonte & Libchaber (1993) in
water ($Pr=6.6$) at the fixed $Ra=1.1\times 10^{9}$ and at a fixed lateral
position. Belmonte, Tilgner & Libchaber (1993) extended these measurements
over the range $5\times 10^{5}\leq Ra\leq 10^{11}$ in compressed gas (air) at
room temperature (Pr=0.7), but still at fixed lateral position. Lui & Xia
(1998) measured the mean temperature profiles at various horizontal positions
on the lower plate of a cylindrical convection cell, the result shows that the
thermal layer thickness $\delta_{th}$ varied over the plate for the same $Ra$,
and the thinnest BL is closed to the center of the plate. Wang & Xia (2003)
found similar results for a cubic cell. du Puits et al. (2007b) measured high-
resolution temperature profiles in Rayleigh-Bénard convection near the top
plate of a cylindrical container with air ($Pr=0.7$) as the working fluid.
Their result shows that the thermal BL thickness $\delta_{th}\sim Ra^{-0.25}$
in the cell with $\Gamma=1.13$. Sun, Cheung & Xia (2008) found that the
thermal boundary layer thickness scales with $Ra^{-0.32}$ in a rectangular
cell, at $Pr=4.3$ and $Ra$ ranging from $10^{8}$ to $10^{10}$.
For the velocity measurement, the methods for determining the velocity
profiles near the solid walls of the cell are developed in recent years. Since
strong temperature fluctuations exist in Rayleigh-Bénard convection, the well-
established hot-wire anemometry could not be applied to this system. For the
viscous boundary layer, the large temperature fluctuations make conventional
laser Doppler velocimetry ineffective because the temperature fluctuations
cause fluctuations in the refractive index of the fluid that in turn make it
difficult to steadily focus two laser beams to cross each other in the fluid
(Xia, Xin & Tong, 1995). Tilgner et al. (1993) introduced an electrochemical
labeling method and measured the velocity profile and boundary layer thickness
near the top plate of a cubic cell filled with water, but only at a single
value of $Ra$. In a later study, Belmonte, Tilgner & Libchaber (1993, 1994)
developed an indirect method — the correspondence between the peak position of
the cutoff frequency profile of the temperature power spectrum and the peak
position of the velocity — to infer the viscous boundary boundary layer
thickness in gaseous convection. This method has subsequently been used to
infer the viscous layer in thermal convection in mercury (Naert, Segawa &
Sano, 1997). It may just be that the method works in certain situations, but
there is no theoretical basis for it. Xin, Xia & Tong (1996), using a novel
light-scattering technique developed by Xia et al. (1995), conducted the first
direct systematic measurement of velocity profiles in RB convection in a
cylindrical cell as a function of $Ra$. They found $\delta_{\upsilon}\sim
Ra^{-0.16}$ from the velocity profile above the center of the lower plate. Qiu
& Xia (1998a, b) extended these measurements to convection in cubic cells,
finding the same scaling exponent $-0.16$ at the bottom plate, but at the
sidewall a different result $\delta_{\upsilon}\sim Ra^{-0.26}$. Using various
organic liquids, Lam et al. (2002) explored the $Pr$ dependence, finding
$\delta_{\upsilon}\sim Pr^{0.24}Ra^{-0.16}$. With the measured $Ra-Re$ scaling
relationship $Re\sim Ra^{0.5}$ obtained in these studies (in fact, it was the
Peclet number $Pe=\upsilon L/\kappa$ rather than $Re$ in some of these
studies; please see Sun & Xia (2005) for more detailed discussions), the above
results imply a scaling relation $\delta_{\upsilon}\sim Re^{-0.32}$. In recent
years, the technique of particle image velocimetry (PIV) has been introduced
to the experimental study of thermal convection (Xia, Sun & Zhou, 2003; Sun,
Xi & Xia, 2005a, b; Sun & Xia, 2005). Sun et al. (2008) further applied the
PIV technique to study the viscous BL in a rectangular cell. Their results
show that $\delta_{\upsilon}\sim Ra^{-0.27}$ and $\delta_{\upsilon}\sim
Re^{-0.5}$, which showed that the viscous BL in thermal turbulence has the
same $Re$-scaling as a Prandtl-Blasius laminar BL. This result validates the
laminar BL assumption made in the GL model in a scaling sense. Thus it appears
there is a discrepancy in the measured scaling exponent of $\delta_{\upsilon}$
with respect to $Ra$ ($Re$) between those obtained in cylindrical and cubic
cells and that obtained in rectangular cells. In Sun et al. (2008) it was
argued that because of the more complicated flow dynamics of the large-scale
circulation (LSC), such as the azimuthal motion in the cylindrical cell (Sun
et al., 2005b; Brown et al., 2005; Xi et al., 2006) and the secondary flows in
the cubic cell (Qiu & Xia, 1998a), the shear flow near the plates is less
steady as compared to that in the rectangular cell which is more close to
quasi-two-dimension (quasi-2D). As the viscous boundary is created by the
shear of the LSC, this may plausibly change the BL properties, resulting in a
different exponent. However, the above argument has not been substantiated
experimentally. Part of the motivation of the present work is to determine how
the three-dimensional LSC dynamics will affect the BL properties. It is known
that titling the cell by a small angle will “lock” the LSC in a fixed
azimuthal plane in the sense that it will limit the range of the LSC’s
azimuthal meandering (Sun et al., 2005a) and reduce its azimuthal oscillation
amplitude near the top and bottom plates of the cell (Ahlers, Brown &
Nikolaenko, 2006). In the present work, we present measurements of BL
properties in a cylindrical cell with the cell titled over a range of angles.
For small titling angles, we measure a boundary layer under a more steady
shear comparing to the “leveled” case when the LSC can freely meander in the
azimuthal direction but presumably the BL is otherwise unperturbed under such
a small titling angle ($\leq 1^{o}$). We also examine how the BL scaling
exponent and other BL properties behave when the titling angle becomes not so
small ($\geq 1^{o}$), which would amount to a perturbation to the BLs.
Boundary layers play such an important role in turbulent thermal convection,
it is therefore important to examine how BLs respond to external
perturbations. Understanding the stability or instability of BLs is also
relevant to the search for the so-called ultimate state of thermal convection,
as the transition from the “classical state” to the ultimate one is
essentially an instability transition of the BL from being laminar to being
turbulent.
In addition to the scaling of the BL thickness, the shape of the velocity
profile near the top and bottom plates has attracted a lot of attention
recently. Although the BL has been found scaling wise to be of Prandtl-Blasius
type (at least in the quasi-2D case), the time-averaged velocity profiles are
found to differ with the theoretically predicted one (du Puits et al., 2007a;
Sun et al., 2008), especially for the region around the thermal BL. Recently,
Zhou & Xia (2010) have proposed a dynamic scaling method that shows that the
mean velocity profile measured in the laboratory frame can be brought into
coincidence with the theoretical Prandtl-Blasius laminar BL profile, if it is
resampled relative to the time-dependent frame that fluctuates with the
instantaneous BL thickness. This method was tested initially for the case of
velocity profile in turbulent convection in a quasi-2D rectangular cell with
water as working fluid ($Pr=4.3$). In a follow-up study using two-dimensional
DNS data, Zhou et al. (2010) found that the method is also valid for thermal
boundary layers and for the case of $Pr=0.7$ as well. More recently, these
authors further shown, again using numerical data, that the method works also
in other positions in the horizontal plate other than the central axis (Zhou
et al., 2011) and in three-dimension (3D) cylindrical cell for moderate values
of $Ra$ (Stevens et al., 2012). However, Scheel, Kim & White (2012) and Shi,
Emran & Schumacher (2012), both using numerical approaches, have found that
dynamic scaling works less well in the 3D cylindrical geometry than in the
quasi-2D case. However, the method has not been tested experimentally so far
in a 3D system. Here we would like to examine the dynamical scaling method
using the experimentally obtained instantaneous velocity profiles in our
three-dimensional cylindrical cell.
### 1.3 Organization of the paper
The remainder of this paper is organized as follows. We give detailed
descriptions of the experimental setup and measurement instrumentation in §2
and present and analyze experimental results in §3, which are divided into six
subsections. In §3.1 we present the measured temperature profiles and
corresponding position-dependent fluid properties, which will be used to
calculate the viscous and Reynolds stresses. In §3.2, the measured velocity
profiles and their characterizations are presented. In §3.3, the scaling
properties, with both $Ra$ and $Re$, of the thickness $\delta_{v}$ obtained
from the mean velocity profiles and $\delta_{\sigma}$ obtained form r.m.s.
velocity profiles, are presented and discussed. We also discuss the influence
of the cell tilting angle $\theta$ on the boundary layer scaling. In §3.4
statistical properties (r.m.s. and skewness) of the velocity field in the
boundary layer region are discussed. In §3.5 we present results of the viscous
and Reynolds shear stresses distributions in the boundary layer, and discuss
the scaling of the wall quantities. In §3.6 we test the dynamic scaling method
with respect to the measured instantaneous velocity profiles. We summarize our
findings and conclude in §4.
## 2 Experimental apparatus
### 2.1 Convection cell
The measurements were made in a cylindrical Rayleigh-Bénard convection cell,
which has been described in detail previously (Zhou & Xia, 2002; Sun et al.,
2005b; Ni et al., 2011b). Here we give only its essential features. The top
and bottom conducting plates are made of pure copper with a thin layer of
nickel to avoid oxidation. The sidewall is made of Plexiglas. To avoid
distortions in the images viewed by the camera, a square-shaped jacket is
fitted around the sidewall of the convection cell. As shown in figure 1(b),
the jacket is filled with water. The diameter and height of the cell is
$D=19.6$ cm and $H=18.6$ cm, respectively. The aspect ratio $\Gamma=D/H$ is
thus close to $1$. Two (three) thermistors are embedded in the top (bottom)
plate. The top plate temperature is maintained constant by a refrigerated
circulator (Polyscience Model 9702) that has a temperature stability of
$0.01^{o}C$. A NiChrome wire (26 Gauge, Aerocon Systems) surrounded by
fiberglass sleeving and Teflon tape is distributed inside the grooves carved
under the bottom plate. The wire is connected with five DC power supplies (GE
Model GPS-3030) in series to provide constant and uniform heating. During the
measurement, the whole cell is placed in a homemade thermostat box that is
kept at the same temperature ($30^{o}C$) as that of the fluid at the centre of
the cell. During the experiment the cell was tilted by an angle $\theta$ such
that the circulation plane of the LSC was parallel to the image plane of the
camera (the $x$-$z$ plane, see figure 1).
### 2.2 PIV measurement
The application of PIV to thermal turbulence has been described in detail in
several previous publications (Xia et al., 2003; Sun et al., 2005b, 2008).
Here we only provide details concerning the particular features of the present
experiment. The PIV system consists of one CCD camera with $2048\times 2048$
pixels, a dual pulse Nd-YAG laser with $135$ mJ per pulse, a synchronizer and
software. As the cell was titled, both the CCD and the laser light-sheet were
titled accordingly with the same angle. A 105 mm focal-length macro lens was
attached to the CCD to achieve a measuring area with size varying from
$18\times 18$ mm2 to $30\times 30$ mm2. Each 2D velocity vector is calculated
from a subwindow (32 pixels $\times$ 32 pixels) that has 50$\%$ overlap with
its neighboring subwindows, so each vector corresponds to a region of 16
pixels $\times$ 16 pixels and each velocity map contains $127\times 127$
velocity vectors in the $x$-$z$ plane (see figure 1). This corresponds to
spatial resolutions of about $0.135\times 0.135$ mm2 to $0.236\times 0.236$
mm2 for velocities $u$ and $w$ measured in the horizontal $x$ and vertical $z$
directions, respectively. For the measurement at $\theta=3.4^{o}$ particles
with diameter $2~{}\mu$m were used, while particles with diameter of
$10~{}\mu$m were used for measurements with other three tilted angles. For
each run, typically about 25200 image pairs were acquired with frame rate of
$2$ Hz.
Figure 1: Sketch of the convection cell and the Cartesian coordinates used in
temperature and velocity measurements. (a) side view of the setup, and (b) the
top view.
Figure 2: (a) A profile of mean temperature $\langle T\rangle$ measured at
$Ra=6.8\times 10^{8}$ and $\theta=3.4^{o}$. (b) A profile of dynamic viscosity
$\mu$ obtained from the mean temperature profile in (a).
## 3 Results and discussion
PIV measurements were made at four values of the titling angle $\theta$
$=0.5^{o}$, $1^{o}$, $2^{o}$, and $3.4^{o}$. For each $\theta$, measurements
over a range of $Ra$ were made. Table 1 lists the parameters ($\theta$, $Ra$
and $Pr$) of each measurement, which typically lasted for about 3.5 hours. As
already mentioned, titling the cell by a small angle has the effect of
“locking” the LSC’s circulation plane at a fixed azimuthal angle (in reality
it restricts the angular range of the LSC’s azimuthal meandering). Thus,
measurements made with small $\theta$ are aimed at studying BL properties
under more steady shear, but the BL itself is assumed to be unperturbed
otherwise. For large values of $\theta$ we wish to examine how the BL responds
to relatively large perturbations.
$\theta$ | Ra | Pr | $U_{max}$ | $\delta_{v}$ | $\theta$ | Ra | Pr | $U_{max}$ | $\delta_{v}$
---|---|---|---|---|---|---|---|---|---
$(deg.)$ | | | $(mm/s)$ | $(mm)$ | $(deg.)$ | | | $(mm/s)$ | $(mm)$
$0.5$ | $5.77\times 10^{8}$ | 5.45 | 4.77 | 2.80 | $2.0$ | $1.34\times 10^{9}$ | 5.42 | 7.47 | 2.57
| $2.79\times 10^{8}$ | 5.41 | 3.57 | 3.40 | | $5.85\times 10^{8}$ | 5.42 | 5.21 | 3.25
| $1.55\times 10^{9}$ | 5.39 | 7.38 | 2.44 | | $5.53\times 10^{9}$ | 5.32 | 13.49 | 1.97
| $2.93\times 10^{9}$ | 5.39 | 9.11 | 2.13 | $3.4$ | $2.74\times 10^{8}$ | 5.67 | 4.02 | 4.54
| $4.26\times 10^{9}$ | 5.38 | 10.88 | 2.06 | | $4.19\times 10^{8}$ | 5.66 | 4.70 | 4.23
$1.0$ | $1.68\times 10^{9}$ | 5.45 | 7.53 | 3.46 | | $6.69\times 10^{8}$ | 5.66 | 6.00 | 3.58
| $3.19\times 10^{9}$ | 5.39 | 9.96 | 2.87 | | $1.38\times 10^{9}$ | 5.68 | 7.59 | 2.70
| $5.54\times 10^{9}$ | 5.42 | 11.60 | 2.66 | | $2.19\times 10^{9}$ | 5.64 | 9.40 | 2.50
| $9.46\times 10^{8}$ | 5.45 | 5.72 | 3.82 | | $2.66\times 10^{9}$ | 5.66 | 10.20 | 2.11
| $2.40\times 10^{8}$ | 5.42 | 3.06 | 5.07 | | $9.89\times 10^{8}$ | 5.66 | 6.47 | 3.00
| $6.00\times 10^{8}$ | 5.43 | 4.71 | 3.94 | | $3.20\times 10^{9}$ | 5.65 | 10.84 | 2.14
$2.0$ | $2.76\times 10^{8}$ | 5.41 | 3.76 | 4.12 | | $4.28\times 10^{9}$ | 5.63 | 12.65 | 1.89
| $2.78\times 10^{9}$ | 5.42 | 10.09 | 2.15 | | $5.19\times 10^{9}$ | 5.56 | 13.49 | 1.84
Table 1: Control parameters of the experiment: cell tilt angle $\theta$, the
Rayleigh number $Ra$ and the Prandtl number $Pr$; and the measured maximum
horizontal velocity $U_{max}$ and viscous boundary layer thickness
$\delta_{v}$. The data are listed in chronological order.
### 3.1 Temperature profile and fluid properties
The local values of fluid properties are needed in calculating the viscous and
Reynolds shear stresses, which requires measurement of the local temperature.
Temperature profiles for the leveled case have been measured systematically by
Lui & Xia (1998) in a similar cylindrical cell. To check whether titling the
cell by a relatively large angle will change the temperature profile, we
measured one mean temperature along the central axis ($x=y=0$) of the cell at
a titling angle $\theta=3.4^{o}$ ($Ra=6.8\times 10^{8}$). The result is shown
in figure 2(a) and the dynamic viscosity corresponding to the local
temperature is shown in figure 2(b). As these results are similar to those
obtained in previous studies by Lui & Xia (1998) and Sun et al. (2008), we
will use results from those studies at similar $Ra$ in the calculations of
Reynolds stress (Sec. 3.5) and other wall quantities that require position
dependent viscosity (density).
### 3.2 Velocity profiles and the Reynolds number scaling
Figure 3: Coarse-grained vector maps of the instantaneous (a) and time-
averaged (b) velocity field measured near the center of the bottom plate
($Ra=4.2\times 10^{8}$ with $\theta=3.4^{o}$), the velocity scale bar is in
unit of $mm/s$.
Figure 4: Time-averaged horizontal velocity profiles measured at tilt angles
$\theta=0.5^{o}$ (a), $1.0^{o}$ (b), $2.0^{o}$ (c) and $3.4^{o}$ (d). In each
plot the corresponding value of $Ra$ decreases from top to bottom (see Table 1
for exact values).
Figure 5: Profiles normalized by their respective maximum velocity
$U_{max}(Ra)$ and the corresponding viscous boundary layer thickness
$\delta_{v}(Ra)$ with tilt angles $\theta=0.5^{o}$ (a), $1.0^{o}$ (b),
$2.0^{o}$ (c) and $3.4^{o}$ (d). The solid line in each plot represents the
theoretical Prandtl-Blasius profile.
Figure 6: Normalized profiles measured at different tilt angles $\theta$ but
with approximately the same value of $Ra$. Profiles in (a) have a nominal
value of $Ra=5\times 10^{8}$ and in (b) have a nominal value of $Ra=1.5\times
10^{9}$. In both figures the symbols are: inverted triangles
($\theta=0.5^{o}$); squares ($\theta=1.0^{o}$); triangles ($\theta=2.0^{o}$);
and circles ($\theta=3.4^{o}$). The solid line in each plot represents the
theoretical Prandtl-Blasius profile.
Figure 7: (a) $Re$ based on the maximum horizontal velocity $U_{max}$, and
(b) $Re_{\sigma}$ based on the maximum velocity fluctuation $\sigma_{max}$ as
a function of $Ra$ for different tilting angles. Inverted triangles:
$\theta=0.5^{o}$; squares: $1^{o}$; triangles: $2^{o}$; circles: $3.4^{o}$.
The dashed lines in (a) represent power-law fits to the individual data sets
all with a scaling exponent $-0.43$ (see text for the fitting results). The
solid line in (b) is a power law fit to all data sets in the plot, which gives
$Re_{\sigma}=0.007Ra^{0.55\pm 0.01}$.
Figure 3 (a) shows an example of measured instantaneous velocity map and (b)
time-averaged velocity field taken over a period of $3.5$ h (corresponding to
$25200$ velocity frames), with the cell tilted at $\theta=3.4^{o}$ and at
$Ra=4.2\times 10^{8}$. In the present measurement, $x$ spans from $-8.75$ mm
to $8.75$ mm, and $z$ spans from $0$ to $17.5$ mm. From the velocity scale in
figure 3 (a) and (b), it is seen that there exist velocity bursts with values
much larger than the maximum velocity in the time-averaged velocity field. It
is found that velocity maps measured at other tilt angles have similar
features. As the mean velocity and the velocity fluctuations do not exhibit
any obvious dependence on the horizontal position $x$ over the small range of
the measurement, the quantities presented below are based on values averaged
along the $x$-direction over the width of the measuring area.
Figure 4 plots the velocity profiles for different tilt angles and various
values of $Ra$, which shows that the shapes of the profiles are rather similar
at this level of detail. Figure 5 plots normalized profiles in which $U(z)$ is
normalized by the maximum horizontal velocity $U_{max}(Ra)$ (for ease of
reference the values of $U_{max}$ are also listed in Table 1) and the distance
$z$ from the wall by the viscous boundary layer thickness $\delta_{v}(Ra)$ (to
be defined below). The figure shows that up to $2\delta_{v}$ profiles for
different $Ra$ and for the same tilt angle collapse on to a single curve quite
well (except perhaps those correspond to the largest $Ra$ for $\theta=2.0$ and
$3.4^{o}$). Note that $z\simeq 2\delta_{v}$ is around where $U$ reaches its
maximum value and beyond this position it decays toward cell centre. So this
position may be taken as the separation between the boundary layer region and
the bulk. The above results suggest that for the same tilt angle the profiles
in the boundary layer region have an invariant shape with respect to different
values of $Ra$. This result is consistent with the finding by Sun et al.
(2008). In figure 5 we also plot the theoretical Prandtl-Blasius profile. It
is seen that within the BL ($z\leq\delta_{v}$) the profiles match the
theoretical solution very well, while in the region just outside the boundary
layer where plume emissions occur, all measured profiles are generally less
steep than the Prandtl-Blasius profile. This feature is also similar to that
observed by (Zhou & Xia, 2010) and will be further discussed in Sec. 3.6. On
the other hand, it is seen from figure 5 that profiles obtained at different
$\theta$ seem to have different degrees of deviation from the Prandtl-Blasius
profile. This can be seen more clearly in figure 6 where we show two examples
in which profiles for different $\theta$ but with values of $Ra$ close to each
other are plotted together along with the theoretical PB profile. This result
suggests that the shape of the velocity profile near the plume-emission region
is modified by the tilting angle. It is also noted that the profiles measured
with $\theta=1.0^{o}$ show strong deviations from the linear dependence with
zero interception. We shall come back to this when discussing boundary layer
scalings in the next section.
Taking $U_{max}$ as the characteristic velocity of LSC, we define the Reynolds
number $Re=U_{max}H/\nu$ and plot $Re$ as a function of $Ra$ and for different
$\theta$ in figure 7(a). When fitting a power-law to the data for different
$\theta$ separately, they all produce an exponent close to 0.43. To better
compare the amplitude of $Re$ for different $\theta$, we fix the scaling
exponent at $0.43$ and fit power laws to the different data sets again. This
gives $Re=(0.185\pm 0.002,0.182\pm 0.003,0.206\pm 0.001,0.203\pm 0.001)\times
Ra^{0.43}$, where the amplitudes in the brackets are for $\theta=0.5^{o}$,
$1.0^{o}$, $2.0^{o}$, and $3.4^{o}$, respectively. These results show that in
general the values of $Re$ with larger $\theta$ are larger than those with
smaller $\theta$. In an earlier study of the effect of cell titling, Ahlers et
al. (2006) have found that $Re$ obtained indirectly from temperature
measurement increases with the tilted angle, which is consistent with the
trend observed here. We note also that the value of the scaling exponent of
$Re$ obtained from many previous studies, and sometimes under nominally
similar conditions, varies over a rather wide range from $0.43$ to $0.55$ (see
for example, Xin et al. (1996); Xin & Xia (1997); Qiu & Xia (1998a, b);
Ashkenazi & Steinberg (1999); Lam et al. (2002); Brown et al. (2007); Sun et
al. (2008); Xie et al. (2012)). The reason for such variations is not
completely clear at present. A detailed study on this issue is beyond the
scope of this paper. For interested readers, we refer to Sun & Xia (2005) who
offered an explanation that can account some of these dispersions in the
exponent.
From the measured profile of the RMS velocity (see figure 8), we can define
another Reynolds number $Re_{\sigma}=\sigma_{max}H/\nu$, which is shown in
figure 7(b) as a function of Ra in a log-log scale for the four tilt angles.
Here it is seen that $Re_{\sigma}$ does not seem to have an obvious dependence
on $\theta$. We therefore fitted a single power law to all four data sets on
the plot, which gave $Re_{\sigma}=0.007Ra^{0.55\pm 0.01}$. The value of the
exponent is somewhat larger than $0.5$ that was obtained from several previous
studies (Xin et al., 1996; Xin & Xia, 1997; Qiu & Xia, 1998a, b; Sun et al.,
2008). But given the uncertainties in the experimental measurements, it is
hard for one to attach too much significance to this difference.
### 3.3 The viscous boundary layer and its scaling with $Ra$ and $Re$
Figure 8: Determination of the viscous boundary layer thickness $\delta_{v}$
through the slope-method from the mean horizontal velocity profile $U(z)$
(circles) and the thickness $\delta_{\sigma}$ from the standard deviation
profile $\sigma_{U}$ (crosses). The measurement was made near the bottom plate
with tilt angle $\theta=3.4^{o}$ and at $Ra=4.2\times 10^{8}$.
We define the thickness $\delta_{v}$ of the viscous boundary layer through the
“slope-method” as shown in figure 8 where a mean velocity $U(z)$ (circles)
profile and the corresponding standard deviation profile $\sigma_{u}(z)$
(crosses) are shown, which are measured at $Ra=4.2\times 10^{8}$ with
$\theta=3.4^{o}$. It is seen that $\delta_{v}$ is defined as the distance at
which the extrapolation of the linear part of $U(z)$ equals its maximum value
$U_{max}$, i.e. $\delta_{v}=U_{max}[dU/dz|_{z=0}]^{-1}$. A length scale
$\delta_{\sigma}$ can also be defined from the profile of $\sigma_{u}(z)$
where $\sigma_{u}$ reaches its maximum value. For the present example, the
values for the two boundary layer length scales $\delta_{v}$ and
$\delta_{\sigma}$ are found to be $4.20$ and $6.05$ mm, respectively. For ease
of reference, the values of $\delta_{v}$ are listed in Table 1.
Figure 9: Measured viscous boundary layer thickness $\delta_{v}$ (a)versus
$Ra$ and (b) versus $Re$ for the four tilt angles: $\theta=0.5^{o}$ (inverted
triangles); $1.0^{o}$ (squares); $2.0^{o}$ (triangles); and $3.4^{o}$
(circles). The dashed lines are power-law fits
$\delta_{v}/H=A_{1}Ra^{\beta_{1}}$ and $\delta_{v}/H=A_{2}Re^{\beta_{2}}$ to
the respective data sets, with the fitting results listed in Table 2.
Figure 10: The scaling exponent $\beta_{2}$ versus cell tilt angle $\theta$,
where $\beta_{2}$ is obtained from the power law fit $\delta_{v}/H\sim
Re^{\beta_{2}}$. The dashed line indicates $\beta_{2}=-0.5$ for a Pradtl-
Blasius laminar boundary layer.
We now examine the scalings of the boundary layer thickness with both the
Rayleigh number $Ra$ and the Reynolds number $Re$. In figures 9(a) and (b) we
plot the measured viscous boundary layer thickness $\delta_{v}$ vs,
respectively, $Ra$ and $Re$ for the four tilt angles. The lines in the figures
represent the best power-law fits $\delta_{v}/H=A_{1}Ra^{\beta_{1}}$ and
$\delta_{v}/H=A_{2}Re^{\beta_{2}}$ to the respective data sets and the
obtained fitting parameters are listed in Table 2. Also shown in the Table for
comparison are results obtained in cells with different geometries and using
different methods. It is seen from the table that for small tilt angles
($\theta=0.5$ and $1.0^{o}$), the exponents are essentially the same and
within the experimental uncertainties the $Re$-scaling exponent may be taken
as the same as that predicted for a Prandtl-Blasius boundary layer, i.e.
$\delta_{v}\sim Re^{-1/2}$. For larger titling angles, there appears to be a
trend for both $\beta_{1}$ and $\beta_{2}$ to decrease (absolute value
increases) with increasing $\theta$. It thus appears that titling the cell by
over $1^{o}$ is a rather strong perturbation to the BL, at least for its
scaling. The situation for the amplitude of viscous boundary layer thickness
$\delta_{v}$ is a bit more complicated. From both figures 9(a) and (b) it
seems that at lower values of $Ra$ ($Re$) the BL thickness increases with
increasing tilting angle, except for $\theta=1^{o}$. For this latter titling
angle, $\delta_{v}$ appears to have an overall upward shift from the rest data
sets. While we do not know the exact reason(s) for this, we note from figure 6
that the profiles for this tilt angle seem to have a nonzero intercept on the
horizontal axis. This appears to suggest that the origin of the $z$-axis for
this was somehow shifted. But even if this is the case, the relatively small
“shift” cannot account for the large “deviation” of this $\delta_{v}$ from the
rest data sets (assuming there is indeed something “wrong” with this data
set). Aside from the amplitude, the behavior of the $Ra$\- and $Re$-scaling
exponents may be summarized as follows. For small tilting angle ($\theta\leq
1^{o}$), the effect of tilting is to lock the azimuthal plane of the LSC (or
restrict its azimuthal meandering range) but the BL is otherwise not strongly
perturbed and scaling wise the BL is approximately Prandtl-Blasius type. For
relatively large titling angle ($\theta>1^{o}$), the BL appears to be strongly
perturbed as far as scaling is concerned and the magnitude of the scaling
exponent increases with titling angle, i.e. the BL thickness $\delta_{v}$
decays with increasing $Ra$ ($Re$) with a steeper slope. The situation is
illustrated in figure 10 where $\beta_{2}$ is plotted as a function of the
tilt angle $\theta$.
Quantity | $Ra$ | Pr | Geometry | $A_{1}$ | $-\beta_{1}$ | $A_{2}$ | $-\beta_{2}$ | $\theta(^{0})$ | Source
---|---|---|---|---|---|---|---|---|---
$\delta_{v}/H$ | $10^{8}\sim 10^{10}$ | $\sim 7$ | cylin. | 0.51 | $0.16\pm 0.02$ | | $0.32$ | 0 | a
| $10^{8}\sim 10^{10}$ | $6\sim 1027$ | cylin. | $0.65Pr^{0.24}$ | $0.16\pm 0.02$ | | $0.32$ | 0 | b
| $10^{8}\sim 10^{10}$ | $\sim 7$ | cubic | 0.69 | $0.18\pm 0.04$ | | $0.36$ | 0 | c
| | | | 3.6 | $0.26\pm 0.03$ | | $0.52$ | 0 | d
| $10^{9}\sim 10^{10}$ | $4.3$ | rectan. | 4.95 | $0.27\pm 0.01$ | $0.64$ | $0.50$ | 0 | e
| $10^{8}\sim 10^{9}$ | $5.4$ | cylin. | 0.745 | $0.19\pm 0.01$ | 0.369 | $0.45\pm 0.04$ | $0.5$ | f
| | | | 1.41 | $0.20\pm 0.01$ | 0.564 | $0.46\pm 0.03$ | $1.0$ | f
| | | | 5.86 | $0.29\pm 0.01$ | 1.41 | $0.61\pm 0.04$ | $2.0$ | f
| | | | 13.3 | $0.32\pm 0.01$ | 6.26 | $0.81\pm 0.01$ | $3.4$ | f
| | | | $A_{3}$ | $-\beta_{3}$ | $A_{4}$ | $-\beta_{4}$ | |
$\delta_{\sigma}/H$ | $10^{7}\sim 10^{11}$ | $\sim 7$ | cylin. | 1.02 | $0.25\pm 0.02$ | | $0.5$ | 0 | a
| $10^{8}\sim 10^{10}$ | $\sim 7$ | cubic | 0.95 | $0.25\pm 0.04$ | | $0.5$ | 0 | c
| | | | 43 | $0.38\pm 0.03$ | | $1.0$ | 0 | d
| $10^{9}\sim 10^{10}$ | $4.3$ | rectan. | 16.5 | $0.37\pm 0.10$ | $0.69$ | $0.72\pm 0.14$ | 0 | e
| $10^{8}\sim 10^{9}$ | $5.4$ | cylin. | 0.58 | $0.15\pm 0.02$ | $0.14$ | $0.26\pm 0.03$ | $0.5$ | f
| | | | 1.77 | $0.20\pm 0.02$ | $0.27$ | $0.37\pm 0.04$ | $1.0$ | f
| | | | 2.68 | $0.23\pm 0.02$ | $0.32$ | $0.41\pm 0.04$ | $2.0$ | f
| | | | 9.9 | $0.29\pm 0.02$ | $0.75$ | $0.54\pm 0.04$ | $3.4$ | f
Table 2: Fitting results for the normalized viscous boundary layer thickness
$\delta_{v}/H$ determined from the mean horizontal velocity profile and
$\delta_{\sigma}/H$ determined from the rms horizontal velocity profile. The
fitting parameters $A_{i}$ and $\beta_{i}$ ($i=1,2,3,4$) are defined through
the power laws: $\delta_{v}/H=A_{1}Ra^{\beta_{1}}$,
$\delta_{v}/H=A_{2}Re^{\beta_{2}}$, $\delta_{\sigma}/H=A_{3}Ra^{\beta_{3}}$,
and $\delta_{\sigma}/H=A_{4}Re_{\sigma}^{\beta_{4}}$. The control parameters
$Ra$ and $Pr$ and cell geometry of measurements are also listed. Also shown in
the table are results from some previous experiments. The sources are: a. Xin
et al. (1996); b. Lam et al. (2002); c. Qiu & Xia (1998a) (bottom); d. Qiu &
Xia (1998b) (side wall); e. Sun et al. (2008); and f. present work. (Note: the
cell tilt angle $\theta$ is indicated as $0$ when it was not mentioned in the
respective papers and we assume the cell was nominally leveled in those
cases.)
Figure 11: Scalings of the boundary layer scale $\delta_{\sigma}$ determined
from the measured rms velocity profiles: (a) versus $Ra$ and (b) versus
$Re_{\sigma}$. The symbols represent: $\theta=0.5^{o}$ (inverted triangles),
$1.0^{o}$ (squares), $2.0^{o}$ (triangles), and $3.4^{o}$ (circles). The lines
are power law fits $\delta_{\sigma}/H=A_{3}Ra^{\beta_{3}}$ and
$\delta_{\sigma}/H=A_{4}Re_{\sigma}^{\beta_{4}}$ to the respective data sets,
with the fitting results listed in Table 2.
In addition to the boundary layer thickness $\delta_{v}$ determined from the
mean horizontal velocity profile, another length scale can also be defined
based on the profile of the horizontal r.m.s. velocity $\sigma_{u}$, which may
be called the r.m.s. velocity boundary layer thickness, as defined in figure
8. In figures 11(a) and (b) we plot $\delta_{\sigma}$ versus $Ra$ and
$Re_{\sigma}$ respectively. The $Ra$-scaling exponent varies from $-0.15$ to
$-0.29$, which appears to follow similar trend as that of $\delta_{v}$, i.e.
its absolute value increases with increasing $\theta$. But it and that of
$Re_{\sigma}$-scaling exponent show significant difference with those obtained
in previous studies. Table 2 shows the fitting results of
$\delta_{\sigma}=A_{3}Ra^{\beta_{3}}$ and
$\delta_{\sigma}=A_{4}Re_{\sigma}^{\beta_{4}}$.
Now we compare our result with previous experimental results obtained in the
cells with different geometries. As shown in Table 2, the value of $\beta_{1}$
obtained in both cylindrical and cubic geometries and measured near the bottom
plate of the cell is $-0.16$. In all these previous measurements, the Reynolds
number based on the maximum horizontal velocity near the plate was also
obtained and they gave a scaling exponent $\gamma=0.5$ via $Re\sim
Ra^{\gamma}$. From this we obtain $\beta_{2}=-0.32$. In these studies, the
convection cell was nominally leveled, i.e. not intensionally tilted. In the
present study, for the small tilting angle cases, where we assume the BL is
not strongly perturbed, the measured $\beta_{1}\simeq-0.19$ when combined with
combined with $\gamma=0.43$ give a $\beta_{2}\simeq-0.45\pm 0.04$ (note that
the actual value of $\beta_{2}$ are obtained from fitting the $\delta_{v}$ vs
$Re$ data, not from the relationship between the exponents). If we take these
values to be close to the Prandtl-Blasius result, then scaling wise the
viscous BL in a cylindrical geometry is also of a Prandtl-Blasius type, as was
already found in a rectangular cell (Sun et al., 2008). For the relatively
large deviations found in the untilted case, it may be attributed to the
random azimuthal motion of the LSC.
Finally we remark that as far as the scaling of the viscous BL is concerned,
there is no theoretical prediction for the dependence of $\delta_{v}$ on $Ra$,
only that on $Re$ (for example, $\delta_{v}\sim Re^{-1/2}$ for the Prandtl-
Blasius BL). In the literature, it is sometimes stated that $\delta_{v}$
should scale as $Ra^{-1/4}$ for the Prandtl-Blasius BL. This is based on the
assumption that $Re\sim Ra^{1/2}$. From above we have seen that the scaling
exponent of $Re$ with $Ra$ varies over a rather wide range. It is therefore
more meaningful to talk about the scaling of $\delta_{v}$ with $Re$, rather
than with $Ra$. We further note that in Sun et al. (2008) it was found that
$\delta_{v}\sim Ra^{-0.27}$ and $Re\sim Ra^{0.55}$, which together give
$\delta_{v}\sim Re^{-0.50}$. In the present case, we have $\delta_{v}\sim
Ra^{-0.2}$ and $Re\sim Ra^{0.43}$, which together give $\delta_{v}\sim
Re^{-0.46\pm 0.03}$. Whether this is fortuitous or there is something deep
here remains remains to be explored.
### 3.4 Fluctuations and statistical properties of the velocity field in the
boundary layer
Figure 12: Time traces of horizontal $u(t)$ (left panels) and vertical $w(t)$
(right panels) velocity components measured at $Ra=2.4\times 10^{8}$ and
$\theta=1^{o}$, at $x=0$ and different distances $z$ from the bottom plate.
Figure 13: Histograms of (a) the horizontal velocity $u(t)$ and (b) vertical
velocity $w(t)$ measured at various distances from the bottom plate with
$Ra=2.4\times 10^{8}$ and $\theta=1^{o}$.
Figure 14: Profiles of (a) the normalized rms velocity $\sigma_{u}$
($\sigma_{w}$) and of (b) the skewness $S_{u}$ ($S_{w}$) measured at
$Ra=2.4\times 10^{8}$ and $\theta=1^{o}$. The vertical distance $z$ is
normalized by the velocity boundary layer thickness $\delta_{v}$. In both
plots the circles represent those for the horizontal velocity component $u$
and the crosses represent those for the vertical velocity component $w$.
In previous BL measurements in the cylindrical cell, owing to the nature of
the dual-beam incoherent cross-correlation technique employed (Xin et al.,
1996; Lam et al., 2002), only time-averaged velocity profiles are measured and
no time-dependent quantities are obtained. It is therefore interesting to
examine these quantities and compare them with similar quantities obtained in
other type of turbulent flows. Figure 12 shows the time series of both the
horizontal component $u(t)$ (left panel) and the vertical component $w(t)$
(right panel) of the velocity, measured at various positions from the plate.
The corresponding velocity histograms are shown in figure 13. The measurements
were made at $Ra=2.4\times 10^{8}$ and $\theta=1^{o}$. We show the velocity
trace at several typical positions: (i) inside the thermal boundary layer,
(ii) around the thermal boundary layer, (iii) around the viscous boundary
layer; (iv) at the position of the maximum velocity; and (v) far away from the
boundary layers. The figures show that the absolute horizontal velocity is
much higher than vertical velocity at each position. One general feature we
observed is that velocity time series and histograms look similar for
different tilting angles. For this reason, we show here results for only one
tilting angle.
At $Ra=2.4\times 10^{8}$, the viscous BL thickness is $\delta_{v}=5.07$ mm. It
is seen from figures 12(a) and 12(b) that at positions inside the BL, the
horizontal velocity $u(t)$ skews toward the positive side, i.e. the velocity
is skewed toward the mean flow direction. This may be understood by the fact
that close to the viscous sublayer the flow speed is very close to zero and a
fluctuation smaller than the mean would mean a flow reversal, which is a
rather rare event. Once outside of the BL, one observes more symmetric
fluctuations around the mean velocity. For the vertical velocity $w(t)$, its
mean velocity is very small at most positions. But the fluctuation increases
significantly when the position is outside of the BL, which are signatures of
plume emissions at these positions. These properties can also be seen from the
velocity histograms shown in figure 13. A notable difference of the present
results from those observed in Sun et al. (2008) is that for positions outside
of the BL the horizontal velocity fluctuates more or less symmetrically around
the mean, rather than skewed toward the negative as seen in the rectangular
cell.
The statistical properties of the velocity may be characterized more
quantitatively by its root-mean-square (r.m.s.) value and its skewness, which
are shown in figure 14. Figure 14(a) plots the velocity r.m.s $\sigma_{u}$ and
$\sigma_{w}$ normalized by maximum horizontal velocity $U_{max}$ versus the
normalized distance $z/\delta_{v}$. Figure 14(b) shows the skewness profiles
$S_{u}=\langle(u-\langle u\rangle)^{3}\rangle/(\langle(u-\langle
u\rangle)^{2}\rangle)^{3/2}$ and $S_{w}=\langle(w-\langle
w\rangle)^{3}\rangle/(\langle(w-\langle w\rangle)^{2}\rangle)^{3/2}$ for the
horizontal and vertical velocities, respectively. Similar to Sun et al.
(2008), our result could not tell whether $\sigma_{w}$ favors a power law or a
logarithmic scaling with the distance $z$, even though our measurement had a
much higher spatial resolution. This is partly due to the limited size of the
measurement area.
### 3.5 Properties of shear stresses and near-wall quantities
Figure 15: Viscous stress $\tau_{v}$ (solid squares), the Reynolds stress
$\tau_{R}$ (open circles) and the total stress $\tau$ (open squares) as
functions of the normalized distance from the plate for (a) $Ra=4.2\times
10^{8}$ and (b) $Ra=9.9\times 10^{8}$, both with $\theta=3.4^{o}$.
Figure 16: The Ra-dependence of (a) the wall shear stress $\tau_{w}$, (b) the
friction velocity $u_{\tau}$, (c) the wall thickness (viscous sublayer)
$\delta_{w}$, and (d) the friction coefficient $c_{f}$, with power law fits
shown as solid lines. The symbols represent data for different tilting angles:
$\theta=0.5^{o}$ (inverted triangles), $1^{o}$ (squares), $2^{o}$ (triangles),
and $3.4^{o}$ (circles).
Figure 17: The Re-dependence of (a) the wall shear stress $\tau_{w}$, (b) the
friction velocity $u_{\tau}$, (c) the wall thickness (viscous sublayer)
$\delta_{w}$, and (d) the friction coefficient $c_{f}$. The symbols represent
data for different tilting angles: $\theta=0.5^{o}$ (inverted triangles),
$1^{o}$ (squares), $2^{o}$ (triangles), and $3.4^{o}$ (circles). Power law
fits are indicated in the figure.
Figure 18: Measured horizontal velocity profiles normalized by wall units for
four different Ra all taken at tilting angle $\theta=3.4^{o}$.
One of the advantages of PIV measurement is that it enables one to measure the
horizontal and vertical velocities at the same time, so that one can calculate
the Reynolds shear stress $\tau_{R}=-\rho(z)\langle
u^{\prime}w^{\prime}\rangle$. Here $u^{\prime}$ and $w^{\prime}$ are the
fluctuations of the horizontal and vertical velocity components respectively,
$\rho(z)$ is the z-dependent fluid density. Viscous shear stress is defined as
$\tau_{v}=\mu(z)du/dz$, where $\mu(z)$ is the dynamic viscosity dependent on
position $z$. The Reynolds stress represents the transport of momentum by
turbulent fluctuations, whereas the viscous stress describes the momentum
transfer by viscosity. The total shear stress is then
$\tau=\tau_{v}+\tau_{R}$.
Figure 15 plots the profiles of the viscous shear stress, Reynolds stress and
total stress for (a) $Ra=4.2\times 10^{8}$, and (b) $9.9\times 10^{8}$. It is
seen that both Ra have the same qualitative features. Here the examples are
for $\beta=3.4^{o}$, and results for other tilting angles are similar. Near
the plate, it is seen that the Reynolds stress $\tau_{R}$ is close to zero,
while the viscous shear stress $\tau_{v}$ is maximum because of the large
velocity gradient $du/dz$ at the wall. So the total stress at the wall
$\tau_{w}(=\tau(0))$ comes almost entirely from the contribution of the
viscous shear stress. Moving away from the plate, the velocity gradient
becomes smaller and the viscous shear stress decreases to zero. The Reynolds
stress $\tau_{R}$ increases and attains its maximum at $z\approx
1.5\delta_{v}$. It then decreases to around zero $z\approx 2\delta_{v}$ and
becomes negative in the bulk flow. It is also seen clearly from the figure
that $\tau_{R}$ and $\tau_{v}$ cross at $z\approx 1.5\delta_{v}$, where
$\tau_{R}$ is close to its maximum value. This suggests that the momentum
transfer in the outer region is dominated by turbulent fluctuations. But in
the viscous boundary layer, the momentum transfer is still dominated by the
viscous diffusion, which implies that the viscous boundary layer is still
laminar in this range of $Ra$.
With the measured near-wall high-resolution velocity field, we are now in a
position to check the dynamic wall properties in turbulent thermal convection.
We first consider the scaling of four basic wall quantities with both $Ra$ and
$Re$. These are the wall shear stress $\tau_{w}$, the skin-friction velocity
$u_{\tau}=(\tau/\rho_{0})^{1/2}$, the viscous sublayer length scale
$\delta_{w}=\nu_{0}/u_{\tau}$, and the skin-friction coefficient
$c_{f}=\tau/\rho_{0}U_{max}^{2}$. Here $\rho_{0}\equiv\rho(z=0)$ and
$\nu_{0}\equiv\nu(z=0)$. Figure 16 shows the scaling of these quantities with
$Ra$. It is seen that within experimental uncertainties there is no difference
between data with different $\theta$. This suggests that tilting the cell does
not have any appreciable effect on BL properties near the wall. Without
differentiating the different data sets, power law fits to all data yield
$\tau_{w}\sim Ra^{0.63}$, $u_{\tau}\sim Ra^{0.32}$, $\delta_{w}\sim
Ra^{-0.37}$ and $c_{f}\sim Ra^{-0.19}$. In a rectangular cell, Sun et al
(2008) found for the same quantities the fitted power law exponents $0.86$,
$0.44$, $-0.50$, and $-0.28$ respectively. It is seen that the absolute values
of these exponents are all larger than those obtained in the present
experiment. There is no theoretical prediction for the Ra-scaling of these
quantities in turbulent thermal convection, so we do not know what the
difference means.
It will be more useful perhaps to examine the scaling of these quantities with
the Reynolds number $Re$, since theoretical predictions exist for such
scalings for wall-bounded shear flows (Schlichting & Gersten, 2000). Figure 17
plots these quantities as a function of Re, the symbols are the same as in
figure 16. For the quantities $\tau_{w}$, $u_{\tau}$, $\delta_{w}$, and
$c_{f}$ our results give the exponents $1.46$, $0.75$, $-0.86$, and $-0.46$.
For a laminar boundary layer over a flat plate, the theoretically predicted
‘classical’ exponents for these quantities are $3/2$, $3/4$, $-1$, and $-1/2$
respectively. One sees that within the experimental uncertainties there is an
excellent agreement between the present experiment and the theoretical
predictions for all the wall quantities except for $\delta_{w}$, which is a
bit smaller. For reference, the previous measurement in rectangular cell gives
$1.55$, $0.8$, $-0.91$, and $-0.34$ for the corresponding quantities (Sun et
al., 2008).
To further compare the present system with classical boundary layers, we
examine velocity profiles in terms of the wall units. Figure 18 shows the
normalized mean horizontal velocity profiles for four different values of $Ra$
taken at $\theta=3.4^{o}$ in a semi-log plot, here $u^{+}=u(z)/u_{\tau}$ and
$z^{+}=z/\delta_{w}$. The linear scaling of $u^{+}$ over $z^{+}$ in the
viscous sublayer below $z^{+}<5$ is reflected quite well by the measured
profiles confirming that the boundary layer is not turbulent in the present
range of $Ra$ and $Pr$. The velocity normalized by wall unit decrease after
reaching the maximum value in $z^{+}\sim 10$. Comparing to the same quantity
measured in the rectangular cell (Sun et al., 2008), however, our result shows
some deviation from the theoretical profile. This is a reflection of the fact
that in the cylindrical cell it is more difficult to measure the profile
accurately very close to the wall.
### 3.6 Dynamical scaling and the shape of velocity profiles in the boundary
layer
Figure 19: Comparison between profiles obtained in the dynamical frame
($u^{*}(z^{*})$, red circles) and the laboratory frame ($u(z)$, blue squares),
measured at different tilt angles $\theta$ but with comparable values of $Ra$.
Also shown for comparison is the theoretical Prandtl-Blasius laminar velocity
profile (solid line). (a) $\theta=0.5^{o}$, $Ra=5.77\times 10^{8}$; (b)
$\theta=1.0^{o}$, $Ra=6.00\times 10^{8}$; (c) $\theta=2.0^{o}$, $Ra=5.85\times
10^{8}$; and (d) $\theta=3.4^{o}$, $Ra=6.69\times 10^{8}$.
Figure 20: The shape factor $H=\delta_{d}/\delta_{m}$ of profiles
$u^{*}(z^{*})$ (red circles) obtained in the dynamical frame and of $u(z)$
(blue squares) obtained in the laboratory frame as a function of $Ra$ and for
different titling angles. (a) $\theta=0.5^{o}$, (b) $1.0^{o}$, (c) $2.0^{o}$,
and (d) $3.4^{o}$. The dashed line represents the value of $2.59$ for the
theoretical Prandtl-Blasius laminar BL.
Figure 21: Examples of instantaneous horizontal velocity profiles scaled by
the instantaneous kinematic BL thickness and the instantaneous maximum
velocity (measured at $Ra=6.00\times 10^{8}$ and $\theta=1^{o}$). The
corresponding instantaneous shape factor is also indicated on the plot. The
solid curves are the Prandtl-Blasius velocity profiles.
Figure 22: PDFs of the shape-factor difference $\delta H$ between those of
the rescaled instantaneous profiles and that of the Prandtl-Blasius profile,
measured at (a) $\theta=0.5^{o}$, (b) $1.0^{o}$, (c) $2.0^{o}$, and (d)
$3.4^{o}$.
The dynamic scaling method of Zhou & Xia (2010) has been found to work well
when tested in quasi-2D experiment and 2D numerical simulations (Zhou et al.,
2010, 2011). But it has not been examined in 3D experiments. Here we
investigate how it works in our cylindrical geometry. As the method has been
well documented elsewhere (Zhou & Xia, 2010; Zhou et al., 2010, 2011), we will
only give a brief description of it here. From the measured instantaneous
velocity profile $u(z,t)$ one can obtain an instantaneous viscous boundary
layer thickness $\delta_{v}(t)$ using the same ‘slope’ method as used for the
mean velocity profiles. A local dynamical BL frame can then be constructed by
defining the time-dependent rescaled distance $z^{*}(t)$ from the plate as
$z^{*}(t)\equiv z/\delta_{v}(t).$ (3)
The dynamically time averaged mean velocity profile $u^{*}(z^{*})$ in the
dynamical BL frame is then obtained by averaging over all values of $u(z,t)$
that were measured at different discrete times $t$ but at the same relative
position $z^{*}$, i.e.
$u^{*}(z^{*})\equiv\langle u(z,t)|z=z^{*}\delta_{v}(t)\rangle.$ (4)
Figure 19 shows the mean velocity profiles measured in the laboratory and the
dynamical frames respectively at the four tilting angles and for comparable
values of $Ra$ (as indicated in the figure caption). These results show that
the dynamical scaling method appears to be more effective for larger values of
$\theta$. This may be understood based on the fact that a larger tilt angle
will place stronger restriction on the azimuthal meandering of the LSC so that
it has less fluctuations in the horizontal direction perpendicular to the mean
flow. We note, however, regardless of the tilt angle, the method works less
effectively than it is in quasi-2D experiment and 2D simulations.
A more quantitative approach to characterize the shape of the mean velocity
profiles is to investigate their shape factor $H=\delta_{v}/\delta_{m}$
defined as the ratio between the displacement thickness $\delta_{v}$ and the
momentum thickness $\delta_{m}$, where
$\delta_{d}=\int_{0}^{\infty}[1-\frac{u(z)}{u_{max}}]dz,~{}{and}~{}\delta_{m}=\int_{0}^{\infty}[1-\frac{u(z)}{u_{max}}]\frac{u(z)}{u_{max}}dz.$
(5)
Since $u(z)$ decays after reaching its maximum value, the above integrations
are evaluated only over the range from $z=0$ to where $u(z)=u_{max}$. For our
profiles the obtained shape factors range between $1.9$ to $2.3$, which are
smaller than $H=2.59$, the value for a laminar Prandtl-Blasius boundary layer.
A shape factor smaller than the theoretical value means the corresponding
profile will approach its asymptotic value (the maximum velocity) slower than
the theoretical profile does.
In figure 20 we show the shape factor $H$ for mean velocity profiles obtained
in the laboratory and dynamical frames respectively for the four tilting
angles and for all $Ra$ measured. The dashed lines in the figure indicate the
Prandtl-Blasius value of 2.59. It is seen that, despite the data scatter,
there is a general trend that for both lab- and dynamical-frame profiles the
deviation from the Prandtl-Blasius profile increases with $Ra$. This is no
surprise, since, as the convective flow above the BL becomes more turbulent
with increasing $Ra$, the BL itself will experience stronger fluctuations and
hence larger deviations from the laminar case. This finding that the dynamical
rescaling method works better for smaller Ra than larger ones is consistent
with those found in DNS studies in the same geometry by Stevens et al. (2012)
for the temperature profile and by Shi et al. (2012) and by Scheel et al.
(2012) for the velocity profile. The second feature is that for all $\theta$
and $Ra$ the profiles obtained in the dynamical frame in general show some
degree of improvement towards that of Prandtl-Blasius value as compared to
those obtained in the laboratory frame. We also note that the “degree of
improvement does not seem to have an obvious dependence on $Ra$, which is also
consistent with the findings of Zhou & Xia (2010); Zhou et al. (2010).
Some insight can be obtained by examining the rescaled instantaneous velocity
profiles. Figure 21 show examples of rescaled instantaneous velocity profiles,
where the distance from the plate has been normalized by the instantaneous BL
thickness corresponding to that moment and the velocity has been normalized by
the instantaneous maximum horizontal velocity. It is seen that there are quite
few cases where the rescaled instantaneous velocity profile is rather close to
the theoretical Prandtl-Blasius profile (up to the point of the maximum
velocity) and deviations of the instantaneous shape are likely caused by
distubances such as plume emissions. Also shown in the figure are the shape
factor $H(t)$ of these instantaneous profiles. To quantify how the
instantaneous profiles are distributed with respect to the Prandtl-Blasius
profile, we examine the PDF of the shape factor difference $\delta
H(t)=H(t)-H^{PB}$ where $H^{PB}=2.59$. Figure 22 plots the PDFs of $\delta
H(t)$ for the 4 tilting angles and for all measured $Ra$ respectively. Despite
the seemingly large variations among them, these PDFs show the general trend
that the rescaled instantaneous profiles measured at lower values of $Ra$
($\lesssim 1\times 10^{9}$) are more of the time having a shape closer to that
of the Prandtl-Blasius profile and that for higher values of $Ra$ the peak of
the PDFs shift to smaller values of $H$. This indicates that with increasing
$Ra$ the profiles around the BL thickness becomes more rounded, i.e. the
approach to the maximum velocity becomes slower and slower. We further note
that these general trends are true across all tilt angles. Another feature
observed in the present 3D case is that we did not find any strong correlation
between the instantaneous BL thickness $\delta_{v}$ and the velocity $u(t)$
just above the BL. This is in contrast to the finding in the quasi-2D
experiment where $\delta_{v}$ and $u(t)$ are found to have a strong negative
correlation, i.e. a large velocity above would exert a stronger shear and
therefore thins the BL thickness (Zhou & Xia, 2010). This result suggest that
in certain aspect the BLs in the 3D and in the 2D/quais-2D cases are
dynamically different.
## 4 Summary and conclusions
We have conducted an experimental study of velocity boundary layer properties
in turbulent thermal convection. High-resolution two-dimensional velocity
field was measured using the particle image velocimetry (PIV) technique in a
cylindrical cell of height $H=18.6$ cm and aspect ratio close to unity, with
the Rayleigh number $Ra$ varying from $10^{8}$ to $6\times 10^{9}$ and the
Prandtl number $Pr$ fixed at $\sim 5.4$, with the convection cell tilted with
respect to gravity at angles $\theta=0.5^{o}$, $1^{o}$, $2^{o}$, and
$3.4^{o}$, respectively. Measurements made with small $\theta$ are aimed at
studying BL properties under more steady shear, but the BL itself is assumed
to be unperturbed otherwise. For large values of $\theta$ we wish to examine
how the BL responds to relatively large perturbations. We also examined
effectiveness of the dynamical BL scaling method in a three-dimensional
system.
It is found that the Reynolds number $Re$ ($=U_{max}H/\nu$) based on the
maximum mean horizontal velocity scales with $Ra$ as $Re\sim Ra^{0.43}$ and
the Reynolds number $Re_{\sigma}$ ($=\sigma_{max}H/\nu$) based on the maximum
rms velocity scales with $Ra$ as $Re_{\sigma}\sim Ra^{0.55}$. Both exponents
do not seem to have an apparent dependence on the tilt angle. On the other
hand, the amplitude of $Re$ seem to show a weak increasing trend with
$\theta$.
With the measured horizontal velocity, we obtain two length scales, i.e. the
viscous BL thickness $\delta_{v}$ based on the mean horizontal velocity
profile and the length scale $\delta_{\sigma}$ based on the rms horizontal
velocity profile. It is found that as far as scaling with the Reynolds number
$Re$ is concerned, the behavior of $\delta_{v}$ can be divided into two
regimes according to the tilting angle of the cell. For $\theta\leq 1^{o}$, it
is found that $\delta_{v}\sim Re^{-0.46\pm 0.03}$, which within experimental
uncertainty may be considered to be consistent with that of the Prandtl-
Blasius BL. It thus appears that the main effect of tilting the cell is to
restrict the azimuthal meandering of the large-scale circulation but the BL is
otherwise not strongly perturbed. For $\theta\geq 1^{o}$, the absolute value
of the exponent is found to increase with $\theta$ and in this case the BL may
be considered to be strongly perturbed. It is found that the scaling exponent
of $\delta_{\sigma}$ with respect to $Ra$ ($Re$) does not have a strong
dependence on $\theta$ as $\delta_{v}$ does. But similar to $\delta_{v}$, the
absolute values of these exponents increase with increasing $\theta$.
It is also found that tilting the cell modifies the velocity profile in the BL
region, i.e. for different tilt angles the shape of profiles is different. But
for the same tilting angle the velocity profiles measured at different $Ra$
can be brought to collapse on a single curve when the mean velocity is
normalized by the maximum velocity $U_{max}$ and the distance from the plate
by the viscous BL thickness $\delta_{v}$.
With simultaneously measured horizontal and vertical velocity components, we
also obtain the Reynolds stress $\tau_{R}$ in the velocity boundary layer. It
is found that $\tau_{R}$ is stronger in the mixing zone comparing with the
rectangular cell. The wall quantities such as the wall shear stress$\tau_{w}$,
the viscous sublayer$\delta_{w}$, the friction velocity $u_{\tau}$ are also
measured. Their scaling exponents with the Reynolds number are very close to
those predicted for classical laminar boundary layers, which is also
consistent with the measurement in rectangular cell.
Regarding the dynamical scaling method, we found that the method in general
works better when the cell is tilted at larger angle $\theta$ than it does at
smaller angles, but the effect is somewhat marginal. With respect to the
influence of $Ra$, it is found that in general the mean velocity profile
sampled in both the laboratory and dynamical frames are more closer to the
Prandtl-Blasius profile at smaller values of $Ra$ than they are at larger
$Ra$, which is consistent with findings from previous DNS studies. Moreover,
it is found that for smaller values of $Ra$ ($\lesssim 1\times 10^{9}$) the
PDF’s of the shape factor $H$ for the rescaled instantaneous profiles exhibit
a peak close to that for the Prandtl-Blasius profile, whereas for larger
values of $Ra$ the peaks shift to smaller values of $H$, indicating the
profile’s approach to the maximum velocity becomes slower and slower with
increasing $Ra$. Another finding is that the effectiveness of the dynamical
scaling method, in terms of its ability of bringing the mean velocity profile
closer to that of Prandtl-Blasius profile, does not have any apparent
dependence on $Ra$. Our general conclusion is that as far as the effectiveness
of the dynamical scaling method is concerned the influence of titling angle is
much smaller than that of the Rayleigh number $Ra$. We note that the Prandtl-
Blasius boundary layer theory is a 2D model, so it is perhaps no surprise that
the dynamic method works less well in 3D than in 2D.
We would like to thank X.-D. Shang and S.-Q. Zhou for kindly making their PIV
facility available to us and L. Qu, Y.-C. Xie and S.-D. Huang for helping with
the experiment. This work was supported in part by the Hong Kong Research
Grants Council under Project Nos. CUHK404409 and CUHK403811.
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|
arxiv-papers
| 2012-09-28T03:08:19 |
2024-09-04T02:49:35.711839
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ping Wei and Ke-Qing Xia",
"submitter": "Ping Wei",
"url": "https://arxiv.org/abs/1209.6415"
}
|
1209.6424
|
119–126
# 22GHz water maser survey of Xinjiang Astronomical Observatory
Jian-jun. Zhou1,2 Jarken. Esimbek1,2 Gang. Wu1,2 1Xinjiang Astronomical
Observatory, CAS
150 Science 1 street Urumqi, Xinjiang 830011, China
email: zhoujj@xao.ac.cn
2Key Laboratory of Radio Astronomy, CAS,
150 Science 1 street Urumqi, Xinjiang 830011, China
(2012)
###### Abstract
Water masers are good tracers of high-mass star-forming regions. Water maser
VLBI observations provide a good probe to study high-mass star formation and
the galactic structure. We plan to make a blind survey toward the northern
Galactic plane in future years using 25m radio telescope of Xinjiang
Astronomical Observatory. We will select some water maser sources discovered
in the survey and make high resolution observations and study the gas
kinematics close to the high-mass protostar.
###### keywords:
water maser, survey, star formation.
††volume: 287††journal: Cosmic masers-from OH to H∘††editors: Roy. Booth, Liz.
Hmphries & W.H.T. Vlemmings, eds.
## 1 Introduction
High-mass star forming regions are usually at far distances, heavy obscuration
makes it difficult to observe them. The water masers are good probes of
physical conditions and dynamics of the star forming regions. Maser VLBI
observations are the unique mean by which one can explore the gas kinematics
close (within tens or hundreds of AU) to the forming high-mass protostar
([Moscadelli et al. 2011, Moscadelli et al. 2011]). Measure trigonometric
parallaxes and proper motions of water masers found in high-mass star-forming
regions by VLBI reference method can provide very accurate distance of them.
Combining positions, distances, proper motions and radial velocities yields
complete 3-dimensional kinematic information of the Galaxy ([Xu et al. 2006,
Xu et al. 2006]; [Reid et al. 2009, Reid et al. 2009]). Water masers are very
rich in the Galaxy, they are reliable tracers of high-mass star-forming
regions ([Caswell et al. 2011, Caswell et al. 2011]). Therefore, it is
valuable to discover more water masers associated with high-mass star-forming
regions.
Earlier water maser searches have chiefly been made to targeted sources, many
masers may not be discovered. There are only a few unbiased water maser
surveys ([Breen et al. 2007, Breen et al. 2007]; [Walsh et al. 2008, Walsh et
al. 2008]; [Caswell & Breen 2010, Caswell & Breen 2010]). Recently, one much
larger blind survey toward 100 square degree of southern Galactic plane has
been completed successfully ([Walsh et al. 2011, Walsh et al. 2011]). However,
no large blind water maser survey has been done toward northern Galactic
plane. We will make a blind survey toward 90 square degree of the northern
galactic plane using our 25m radio telescope. We hope to discover a large
sample of water masers and high-mass star-forming regions at earlier stages,
and study high-mass star formation and galactic structure.
## 2 25m radio telescope of Xinjiang Astronomical Observatory
Nanshan 25m radio telescope of Xinjiang Astronomical Observatory was built in
1992 as a station for the Chinese very long baseline interferometry network.
It is located at Nanshan mountains west of Urumqi city at an altitude of
2080m. Its front-end receiver system includes several receivers working at 18,
13, 6, 3.6 and 1.3cm. At 1.3 cm, one dual-polarization cryogenic receiver has
been installed on the telescope recently, the noise temperature of the
receiver is better than 20K. When weather is good, the system temperature is
better than 50K. We built a molecular spectrum observing system in 1997\. One
digital filter bank (DFB) system is employed as the spectrometer, it is is
capable of processing up to 1 GHz of bandwidth with 8192 channels. Our
telescope now can observe several molecules such as OH, H2O, NH3, H2CO and
H110α.
## 3 Our plan
We will make a large scale blind survey toward Northern Galactic plane. For
that most water masers concentrated in the region along the galactic plane
($|b|<0.5^{\circ}$). We plan to survey 90 square degrees of the northern
galactic plane, it covers the region between l=30∘ and l=120∘,and
$|b|<0.5^{\circ}$. In order to complete the project in reasonable time, scan
observation mode (on the fly) will be used in our observation, and final
sensitivity of the survey is about 1.4Jy.
On the other hand, many surveys at millimeter, submillimeter, infrared
wavelengths discovered a large sample of possible star-forming regions, e.g.
Bolocam, Planck, Glimpse and MIPS. These sources provide us good candidates
for searching water masers. We also can select some sources and make targeted
survey.
###### Acknowledgements.
This work was funded by The National Natural Science Foundation of China under
Grant $10778703$, China Ministry of Science and Technology under State Key
Development Program for Basic Research (2012CB821800) and The National Natural
Science Foundation of China under Grant $10873025$.
## References
* [Breen et al. 2007] Breen, S. L., Ellingsen, S. P., Johnston-Hollitt, M., et al. 2007, MNRAS, 377, 491
* [Caswell & Breen 2010] Caswell, J. L., & Breen, S. L. 2010, MNRAS, 407, 2599
* [Caswell et al. 2011] Caswell, J. L., Breen, S. L., & Ellingsen, S. P. 2011, MNRAS, 410, 1283
* [Moscadelli et al. 2011] Moscadelli, L., Sanna, A. & Goddi, C. 2011, A&A, 536, 38
* [Reid et al. 2009] Reid, M. J., Menten, K. M., Zheng, X. W., Brunthaler, A., et al. 2009, ApJ, 700, 137
* [Walsh et al. 2008] Walsh, A. J., Lo, N., Burton, M. G., et al. 2008, PASA, 25, 105
* [Walsh et al. 2011] Walsh, A. J., Breen, S. L., Britton, T., et al. 2011, MNRAS, 416, 1764
* [Xu et al. 2006] Xu, Y., Reid, M.J., Zheng, X. W., & Menten, K. M. 2006, Science, 311, 54
|
arxiv-papers
| 2012-09-28T04:55:21 |
2024-09-04T02:49:35.722799
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-jun Zhou, Jarken Esimbek, Gang Wu",
"submitter": "Jian-Jun Zhou",
"url": "https://arxiv.org/abs/1209.6424"
}
|
1209.6594
|
# Azimuthally fluctuating magnetic field and its impacts on observables in
heavy-ion collisions
John Bloczynski Physics Department and Center for Exploration of Energy and
Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN
47408, USA. Xu-Guang Huang Physics Department and Center for Exploration of
Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane,
Bloomington, IN 47408, USA. Xilin Zhang Physics Department and Center for
Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson
Lane, Bloomington, IN 47408, USA. Jinfeng Liao Physics Department and Center
for Exploration of Energy and Matter, Indiana University, 2401 N Milo B.
Sampson Lane, Bloomington, IN 47408, USA. RIKEN BNL Research Center, Bldg.
510A, Brookhaven National Laboratory, Upton, NY 11973, USA.
###### Abstract
The heavy-ion collisions can produce extremely strong transient magnetic and
electric fields. We study the azimuthal fluctuation of these fields and their
correlations with the also fluctuating matter geometry (characterized by the
participant plane harmonics) using event-by-event simulations. A sizable
suppression of the angular correlations between the magnetic field and the
$2$nd and $4$th harmonic participant planes is found in very central and very
peripheral collisions, while the magnitudes of these correlations peak around
impact parameter $b\sim 8-10\rm fm$ for RHIC collisions. This can lead to
notable impacts on a number of observables related to various magnetic field
induced effects, and our finding suggests that the optimal event class for
measuring them should be that corresponding to $b\sim 8-10$ fm.
###### pacs:
12.38.Mh, 25.75.-q, 25.75.Ag, 24.10.Nz
## I Introduction
Ultra-relativistic heavy-ion collisions create not only a domain of extremely
high energy density where a new state of matter — the deconfined quark-gluon
plasma (QGP) may form, but also extremely strong (electro)magnetic fields due
to the relativistic motion of the colliding heavy ions carrying large positive
electric charge (Rafelski:1975rf, ). Previous computations showed that the
magnetic fields generated in Au + Au collision at RHIC ($\sqrt{s}=200\rm GeV$)
can reach about $eB\sim m_{\pi}^{2}\sim 10^{18}$ G (arXiv:0711.0950, ;
arXiv:0907.1396, ; arXiv:1003.2436, ; arXiv:1103.4239, ; arXiv:1107.3192, ;
arXiv:1111.1949, ; arXiv:1201.5108, ), which is $10^{13}$ times larger than
the strongest man-made steady magnetic field in the laboratory. The magnetic
field generated at LHC energy can be an order of magnitude larger than that at
RHIC (arXiv:0907.1396, ; arXiv:1201.5108, ), according to a simple scaling law
recently found in (arXiv:1201.5108, ) for the event-averaged magnetic field:
$\langle eB_{y}\rangle\propto Zb\sqrt{s}$ for $b\lesssim 2R_{A}$ , where $Z$
is the charge number of the ions, $b$ is the impact parameter, $R_{A}$ is the
radius of nucleus, and the $y$-axis is perpendicular to the reaction plane.
Thus, heavy-ion collisions provide a unique terrestrial environment with
ultra-strong magnetic fields.
There have been very strong interests and intensive efforts recently in
studying various possible physical effects induced by the presence of strong
magnetic field. Receiving particular enthusiasm is the set of ideas to look
for experimental manifestation of QCD effects stemming from topology and
anomaly and aided by the external magnetic field. These include, e.g., the so-
called Chiral Magnetic Effect (CME) (Kharzeev:2004ey, ; Kharzeev:2007tn, ;
arXiv:0711.0950, ; Fukushima:2008xe, ) in which a nonzero axial charge density
in the matter (presumably from topological transitions via sphalerons)
together with the magnetic field ${\bf B}$ will induce a dipole charge
separation along the ${\bf B}$ direction. A lot of works have been done in the
past few years to experimentally measure this effect by analyzing the charged
particle correlations (Voloshin:2004vk, ; Abelev:2009ac, ;
Selyuzhenkov:2011xq, ) and to understand the interpretation and background
effects related to these measurements (Schlichting:2010qia, ; Pratt:2010zn, ;
Bzdak:2009fc, ; Bzdak:2010fd, ; Liao:2010nv, ; Wang:2009kd, ), as recently
reviewed in Ref. (Bzdak:2012ia, ). There is also the so-called Chiral
Separation Effect (CSE) (Son:2004tq, ), from its interplay with the CME there
arises a gapless collective excitation called the Chiral Magnetic Wave (CMW)
(Kharzeev:2010gd, ). An observable effect of CMW was proposed in
(Burnier:2011bf, ): the CMW with the presence of nonzero vector charge density
and the magnetic field transports the charges in QGP towards an electric
quadrupole distribution with more positive charges near the poles of the
produced fireball (pointing outside of the reaction plane) while more negative
charges near the equator (in the reaction plane), and this leads to a
measurable splitting of negative and positive pions’ elliptic flow. Recently
STAR collaboration reported the first measurement of the charged pion flow
spitting versus the charge asymmetry which appears in agreement with the
predictions from CMW (Burnier:2011bf, ; Burnier:2012ae, ). Yet one more
interesting effect is possible soft photon production through the QCD
conformal anomaly in the external magnetic field as suggested in
(Basar:2012bp, ). (The CME could also possibly cause anisotropic soft photon
production, see Fukushima:2012fg .) In this mechanism the photons are emitted
perpendicular to the ${\bf B}$ direction and there is an appreciable azimuthal
anisotropy of the emitted photons which might partially account for the
unusually large $v_{2}$ of direct photons reported by PHENIX collaboration
(Adare:2011zr, ). In addition to the above anomaly phenomena in magnetic
field, there are also discussions of other novel effects in strong magnetic
fields (arXiv:1008.1055, ; arXiv:1102.3819, ; arXiv:1108.4394, ;
arXiv:1108.0602, ), e.g., the spontaneous electromagnetic superconductivity of
QCD vacuum, the possible enhancement of elliptic flow of charged particles,
possible anisotropic electromagnetic radiation from the QGP, the energy loss
due to the synchrotron radiation of quarks, and the emergence of anisotropic
viscosities.
In most of the phenomenological studies of these effects, the magnetic field
${\bf B}$ in heavy-ion collisions represents the largest source of uncertainty
in their quantitative calculations, and a precise knowledge of the magnetic
field is urgently needed. This is particularly so in the context of the
realization and intensive investigations in the last two or three years that
there are very strong fluctuations in the initial conditions of heavy-ion
collisions. Such fluctuations have been shown to lead to strong observable
effects both in the bulk collective expansions (as “harmonic flows”)
(harmonic_flow, ) and in the anisotropy of penetrating hard probe (as
“harmonic tomography”) (harmonic_tomography, ). Since the magnetic field
directly relies on the initial distributions of protons in both nuclei, a
realistic computation has to take into account such strong initial
fluctuations. A first step has been made in (arXiv:1111.1949, ;
arXiv:1201.5108, ) to compute the magnitude of ${\bf B}$ with event-by-event
fluctuations, which was indeed found to be remarkably modified from the naive
“optical” estimates.
In this paper, we focus on the strong fluctuations in the azimuthal
orientation of the magnetic field ${\bf B}$, and particularly investigate its
angular correlation with the underlying matter geometry (specified by
participant planes) bearing concurrent fluctuations on an event-by-event
basis. This correlation is a very important link between the experimental
measurements and any of the magnetic field induced effects in heavy-ion
collisions. Most (if not all) previous studies rely on the assumption that the
magnetic field is pointing in the out-of-plane direction which would be true
without fluctuations. In the real world, however, both the ${\bf B}$ and the
“planes” (event planes or participant planes) bear strong fluctuations and as
we will show their orientations are never fully “locked” and only strongly
correlated in some circumstances. We will study these correlations in great
details and examine the consequences for a number of observables related to
some aforementioned effects.
The rest of this paper is organized as follows. In Sec. II we show how the
initial fluctuations modify the magnetic and electric fields generation in
heavy-ion collisions. We study the azimuthal fluctuation of the magnetic field
in Sec. III. We discuss the physical implications of these results in Sec. IV.
For completeness we also show results for the electric field in Sec. V. In
sec. VI a new class of charge dependent measurements partly motivated by our
results will be discussed. Finally we summarize in Sec. VII. The natural unit
$\hbar=c=k_{B}=1$ will be used throughout this article.
## II Event-by-event calculation of the electromagnetic field
The aim of this section is to give an event-by-event calculation of the
electromagnetic fields in heavy-ion collisions. We focus on the fields at the
initial time $t=0$, that is, the moment when the two colliding nuclei overlap
completely. Our starting point is the Lorentz boosted Coulomb formulas which
are equivalent to the Liénard-Wiechert potentials for constantly moving
charges:
$\displaystyle e{\bf B}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}({\bf
R}_{n})\frac{1-v_{n}^{2}}{[R_{n}^{2}-({\bf R}_{n}\times{\bf
v}_{n})^{2}]^{3/2}}{\bf v}_{n}\times{\bf R}_{n},$ $\displaystyle e{\bf
E}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}({\bf
R}_{n})\frac{1-v_{n}^{2}}{[R_{n}^{2}-({\bf R}_{n}\times{\bf
v}_{n})^{2}]^{3/2}}{\bf R}_{n},$ (1)
where ${\bf R}_{n}={\mathbf{r}}-{\mathbf{r}}_{n}(t)$ is the relative position
of the field point ${\mathbf{r}}$ to the $n$th proton at time $t$,
${\mathbf{r}}_{n}(t)$, and ${\bf v}_{n}$ is the velocity of the $n$th proton.
The summations run over all protons in the projectile and target nuclei.
Equations (1)contain singularities at $R_{n}=0$ if we treat protons as point
charges. In practical calculation, to avoid such singularities we treat
protons as uniformly charged spheres with radius $R_{p}$. The charge number
factor $Z_{n}(\bf{R}_{n})$ in Eqs. (1) is introduced to encode this aspect:
when the field point locates outside the $n$th proton (in the rest frame of
the proton) $Z_{n}=1$, otherwise $Z_{n}<1$ depends on ${\bf R}_{n}$. The in-
medium charge radius $R_{p}$ of proton is unknown (the most recent measurement
of the rms charge radius of proton gives $R_{p}=0.84184(67)$ fm in vacuum
Pohl:2010zza ), we choose $R_{p}=0.7$ fm in our numerical simulations. We
checked that varying $R_{p}$ from $0.6$ fm to 0.9 fm will shift the numerical
results within $15\%$ but no qualitative conclusion is altered. The nucleons
in one nucleus move at constant velocity along the beam direction (we choose
it as $z$-direction), with the nucleons in another nucleus moving oppositely
at the same speed. The energy for each nucleon is set to be $\sqrt{s}/2$ in
the center-of-mass frame, therefore the value of the velocity of each nucleon
is given by $v_{n}^{2}=1-(2m_{N}/\sqrt{s})^{2}$, where $m_{N}$ is the mass of
the nucleon. We set the $x$-axis along the impact parameter vector so that the
reaction plane is the $x$-$z$ plane. Finally, the positions of nucleons in the
rest frame of a nucleus are sampled according to the Woods-Saxon distribution.
Figure 1: (Color online) The event-averaged $(eB)^{2}$ and $(eE)^{2}$ (in unit
of $m^{4}_{\pi}$) at $t=0$ and four different points on the transverse plane
as functions of the impact parameter $b$.
Figure 2: (Color online) The event-by-event histograms of $\Psi_{\bf
B}-\Psi_{2}$ at impact parameters $b=0,5,10,12$ fm for Au + Au collision at
RHIC energy. Here $\Psi_{\bf B}$ is the azimuthal direction of ${\bf B}$ field
(at $t=0$ and ${\mathbf{r}}=(0,0,0)$) and $\Psi_{2}$ is the second harmonic
participant plane.
Figure 3: The scatter plots on $\Psi_{B}$-$\Psi_{2}$ plane at impact
parameters $b=0,5,10,12$ fm for Au + Au collision at RHIC energy. Here
$\Psi_{\bf B}$ is the azimuthal direction of ${\bf B}$ field (at $t=0$ and
${\mathbf{r}}=(0,0,0)$) and $\Psi_{2}$ is the second harmonic participant
plane.
In this computation we have obtained in each event all the components of both
the ${\bf B}$ and ${\bf E}$ fields at four different points,
${\mathbf{r}}=(0,0,0),{\mathbf{r}}=(3\,{\rm fm},0,0),{\mathbf{r}}=(0,3\,{\rm
fm},0)$, and ${\mathbf{r}}=(3\,{\rm fm},3\,{\rm fm},0)$, on the transverse
plane for a wide range of impact parameter $b$ in Au + Au collision at RHIC
energy $\sqrt{s}=200$ GeV. We have found that all the $z$ components are
negligibly small, while the transverse components are strong, with their
centrality trends in agreement with previous results in (arXiv:1111.1949, ;
arXiv:1201.5108, ). The absolute magnitudes of these fields are somewhat
smaller than those reported in (arXiv:1111.1949, ; arXiv:1201.5108, ), due to
the important difference that we treat each proton as a charged sphere with
radius $R_{p}$ rather than a point charge, which is both more physical and
mathematically less of singularity problem. As also noted in (arXiv:1111.1949,
; arXiv:1201.5108, ) and confirmed in our computation, although the
$x$-component of the magnetic field as well as the $x$\- and $y$-components of
the electric field vanish after averaging over many events, their magnitudes
in each event can be huge and comparable to the $y$-component of the magnetic
field due to the fluctuations. This already implies that both the magnitude
and the direction of the electromagnetic field (albeit in the unmeasurable
$x$-$y$ plane) fluctuate strongly. In Fig. 1 we show $\langle(eB)^{2}\rangle$
and $\langle(eE)^{2}\rangle$ as functions of $b$ at different points: clearly
the electromagnetic fields bear considerable inhomogeneity on the transverse
plane. We also notice that the electric field can be very strong particularly
for more central collisions. The $\langle(eB)^{2}\rangle$ is particularly
interesting because the signal strengths of several magnetic-field-induced
effects are proportional to $(eB)^{2}$, as we will discuss more in Sec. IV.
## III Azimuthal correlations between magnetic field and matter geometry
As already mentioned before, on the event-by-event basis the electromagnetic
field fluctuates strongly both in magnitude and in azimuthal direction. The
direction of the ${\bf B}$-field is very important as the ${\bf B}$-field
induced effects inherit this information and occur either along (e.g. CME,
CMW) or perpendicular (e.g. photon emission) to ${\bf B}$.
Even more important from the measurement perspective is the question of the
${\bf B}$ orientation with respect to a frame that could be identified
experimentally rather than to the ideal world reaction plane. In reality what
could be determined is the final-state hadrons’ distribution geometry in
momentum space, in particular the second harmonic $v_{2}$ event plane (EP)
which is relatively more tightly correlated with the initial matter
distribution geometry that can be specified primarily by the second harmonic
$\epsilon_{2}$ participant plane (PP). Therefore the really useful information
is the ${\bf B}$ orientation with respect to the matter geometry per event.
Here we make a first detailed examination of such a kind, studying the
azimuthal correlation between the magnetic field and the participant planes.
In order to do that, we determine the participant planes for various harmonics
from the Monte Carlo Glauber simulations of the initial condition and analyze
the angular correlations between the ${\bf B}$ and the participant plane
orientations from the same event. The $n$th harmonic participant plane angle
$\Psi_{n}$ and eccentricity $\epsilon_{n}$ are calculated from participant
density $\rho({\mathbf{r}})$ as in the literature (e.g. (harmonic_flow, )):
$\epsilon_{1}e^{i\Psi_{1}}=-(\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{3}e^{i\phi})/({\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{3}}),$ and
$\epsilon_{n}e^{in\Psi_{n}}=-({\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{n}e^{in\phi})/(\int
d^{2}{\mathbf{r}}_{\perp}\rho({\mathbf{r}}_{\perp})r_{\perp}^{n}})$ for $n>1$.
In this Section we focus on the 2nd harmonic participant plane $\Psi_{2}$ as
it is the most prominent anisotropy from both geometry and fluctuations.
Correlations of ${\bf B}$ with other harmonics will be discussed in Sec. IV.
By determining ${\bf B}$ and $\Psi_{2}$ in each event we can examine the
distribution of their relative angle over many events: in Fig. 2 we plot the
event-by-event histograms of $\Psi_{\bf B}-\Psi_{2}$ at $b=0,5,10$ and $12$ fm
for Au + Au collisions at $\sqrt{s}=200$ GeV where $\Psi_{\bf B}$ is the
azimuthal direction of the magnetic field at $t=0$ and ${\mathbf{r}}=(0,0,0)$.
Strikingly at $b=0$ the histogram of $\Psi_{\bf B}-\Psi_{2}$ is almost uniform
indicating that $\Psi_{\bf B}$ and $\Psi_{2}$ are basically uncorrelated. For
$b=5,10$ and $12$ fm, the histograms have Gaussian shapes peaking at $\pi/2$
with the corresponding widths not small at all. The width first decreases when
$b$ increased from $5$ to $10$ fm and then increases again toward $b=12$ fm.
So although the ${\bf B}$ field indeed points at $\pi/2$ with respect to
$\Psi_{2}$ plane on average, it fluctuates significantly in each event. The
correlation between $\Psi_{\bf B}$ and $\Psi_{2}$ is the strongest in middle-
centrality while weakens much in the most central and most peripheral
collisions.
To further reveal the correlation pattern between $\Psi_{\bf B}$ and
$\Psi_{2}$ and to understand better the non-monotonous centrality trend of the
widths of the histograms in Fig. 2, we show the scatter plots from all events
at given $b$ on the $\Psi_{\bf B}$-$\Psi_{2}$ plane in Fig. 3 which visualize
the 2D probability distribution density. Again for $b=0$, the events are
almost uniformly distributed indicating negligible correlation between
$\Psi_{\bf B}$ and $\Psi_{2}$. For $b=5,10$, and $12$ fm, the event
distributions evidently concentrate around $(\Psi_{\bf B},\Psi_{2})=(\pi/2,0)$
indicating a correlation between the two. Going from $b=5$ to $10$ and to $12$
fm, the spread in $\Psi_{\bf B}$ keeps shrinking while the spread in
$\Psi_{2}$ clear grows with larger $b$. This is because for non-central
collisions with increasing $b$, the ${\bf B}$ is mostly from the spectators
whose number increases and bears less fluctuations while the $\Psi_{2}$ is
determined by participants whose number decreases and fluctuates more. This
explains the non-monotonic trend of the widths in the histograms in Fig. 2. In
short, we have found that the event-by-event fluctuations of the initial
condition bring azimuthal fluctuations in both $\Psi_{\bf B}$ and $\Psi_{2}$,
and the angular correlation between them is smeared out significantly in the
very central and very peripheral collisions while stays strong for middle-
centrality collisions. This observation certainly influences the
interpretation of observables related with ${\bf B}$-induced effects, as will
be discussed in the next section.
## IV Impact on various observables
We now discuss the impacts of the azimuthal fluctuations of the magnetic field
with respect to matter geometry (the participant planes) on a number of
pertinent observables recently measured in heavy-ion collisions. We recap a
few points already discussed: the ${\bf B}$-induced effects are sensitive to
the azimuthal direction of the ${\bf B}$ field, which however is not
experimentally known; in the past when observables are proposed and
interpreted for measuring certain $\bf B$-related effects, it is often assumed
that the $\bf B$ direction is perpendicular to the reaction plane; as already
shown in last section, this assumption is not true, and the azimuthal
orientation between $\Psi_{\bf B}$ of $\bf B$ and $\Psi_{2}$ fluctuates with
sizable spread in their relative angle $(\Psi_{\bf B}-\Psi_{2})$. In what
follows we evaluate the impacts of such fluctuations on the experimentally
measured quantities.
First, let us consider the pair correlation
$\gamma=\langle\cos(\phi_{1}+\phi_{2}-2\Psi_{2})\rangle$ with $\phi_{1,2}$ the
azimuthal angles of the particle 1 and 2 where the average is taken over
events. The measurements of $\gamma$ are motivated by the search for the
Chiral Magnetic Effect (arXiv:0711.0950, ). (The $\gamma$ is actually
extracted through three particle correlations
$\langle\cos(\phi_{1}+\phi_{2}-2\phi_{3})\rangle$ divided by elliptic flow
$v_{2}$ with the third particle serving as a “projector” toward the Event
Plane Voloshin:2004vk . ) Now let us focus on the same-charge pairs, and
suppose the CME indeed gives rise to the same-side azimuthal correlations
along the $\bf B$-field direction for the same-charge pairs. The two-particle
density then receives the following contribution (with $A_{++}$ the signal
strength)
$\displaystyle f_{++}=A_{++}\cos(\phi_{1}-\Psi_{\bf
B})\,\cos(\phi_{2}-\Psi_{\bf B}).$ (2)
This translates into the following form after re-defining angles
$\bar{\phi}_{i}=\phi_{i}-\Psi_{2}$ and $\bar{\Psi}_{\bf B}=\Psi_{\bf
B}-\Psi_{2}$:
$\displaystyle f_{++}$ $\displaystyle=$
$\displaystyle\frac{A_{++}}{2}\cos(\bar{\phi}_{1}-\bar{\phi}_{2})+\frac{A_{++}}{2}[\cos(2\bar{\Psi}_{\bf
B})]\,\cos(\bar{\phi}_{1}+\bar{\phi}_{2})$ (3)
$\displaystyle+\frac{A_{++}}{2}[\sin(2\bar{\Psi}_{\bf
B})]\,\sin(\bar{\phi}_{1}+\bar{\phi}_{2}).$
We therefore see that the CME’s contribution to the $\gamma_{++}$:
$\displaystyle\gamma_{++}\sim\frac{\langle A_{++}\cos(2\bar{\Psi}_{\bf
B})\rangle}{2}.$ (4)
If the $\bf B$-direction were to be always perfectly aligned with the out-of-
plane, i.e. $\bar{\Psi}_{\bf B}=\pi/2$, then we simply have
$\gamma_{++}\to-\langle A_{++}\rangle/2$. But the fluctuations in magnetic
field as well as in matter geometry will blur the angular relation between the
two and modify the signal by the factor $\sim\langle\cos(2\bar{\Psi}_{\bf
B})\rangle$. Similarly, if one measures the charge separation with respect to
higher harmonic participant plane, e.g. the fourth harmonic plane $\Psi_{4}$,
the corresponding correlation will be
$\langle\cos[2(\phi_{1}+\phi_{2}-2\Psi_{4})]\rangle$ (as recently proposed
with the hope to disentangle the collective-flow and CME contributions to
$\gamma$ Voloshin:2011mx ). The azimuthal fluctuation of ${\bf B}$ field with
respect to $\Psi_{4}$ will again contribute a modification factor
$\sim\langle\cos[4(\Psi_{\bf B}-\Psi_{4})]\rangle$ in the above correlation.
Figure 4: (Color online) The correlations $\langle\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle$ as functions of impact parameter for $n=1,2,3,4$ at four
different positions on the transverse plane: (from left to right)
${\mathbf{r}}=(0,0,0)$ fm; ${\mathbf{r}}=(3,0,0)$ fm; ${\mathbf{r}}=(0,3,0)$
fm; ${\mathbf{r}}=(3,3,0)$ fm. Figure 5: (Color online) The $(e{\bf
B})^{2}$-weighted correlations $\langle(e{\bf B})^{2}\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle/\langle(e{\bf B})^{2}\rangle$ as functions of impact
parameter for $n=1,2,3,4$ at ${\mathbf{r}}=(0,0,0)$ fm.
Recently it has been suggested that a splitting between the elliptic flow of
the $\pi^{-}$ and $\pi^{+}$ could occur in heavy-ion collisions due to the
Chiral Magnetic Wave (Burnier:2011bf, ; Burnier:2012ae, ), as briefly
discussed in Sec. I. Specifically the CMW will induce an electric quadruple in
the net charge distribution along the $\bf B$ field:
$\displaystyle\rho_{e}(\phi)\sim 2r_{e}\cos[2(\phi-\Psi_{\bf
B})]=2r_{e}\cos[2(\bar{\phi}-\bar{\Psi}_{\bf B})],$ (5)
where $r_{e}\propto\int d\phi\rho_{e}(\phi)\cos[2(\phi-\Psi_{\bf B})]$
quantifies the quadruple from CMW. This modifies the final charged hadron
distribution by an amount proportional to $r_{e}$:
$\displaystyle\frac{dN_{\pm}(\phi)}{d\phi}\sim
N_{\pm}\left\\{1+2v_{2}\cos(2\bar{\phi})\pm\frac{r_{e}}{N}\cos[2(\bar{\phi}-\bar{\Psi}_{\bf
B})]\right\\},\quad$ (6)
where $N=N_{+}+N_{-}$ is the charge multiplicity. Thus one obtains a splitting
of the charged pions $v_{2}$
$\displaystyle
v_{2}^{\pi^{-}}-v_{2}^{\pi^{+}}=-\,\langle\frac{r_{e}}{N}\cos(2\bar{\Psi}_{\bf
B})\rangle.$ (7)
Again, we see the modification factor $\sim\langle\cos(2\Psi_{\bf B})\rangle$
arising from relative orientation between $\bf B$ and $\Psi_{2}$. If
$\bar{\Psi}_{\bf B}$ were to be simply $\pi/2$ (i.e. $\bf B$ always out-of-
plane) then the sign of splitting is always $v_{2}^{\pi^{-}}>v_{2}^{\pi^{+}}$.
However due to the fluctuation, $\bar{\Psi}_{\bf B}$ deviates from $\pi/2$ and
leads to a smaller splitting.
Finally, we consider the example of recently suggested soft photon production
from conformal anomaly in the presence of an external ${\bf B}$ field
(Basar:2012bp, ). The photon emitted via this mechanism carries azimuthal
information of the external magnetic field. If one assumes a homogeneous
external $\bf B$ field, then the produced photon spectrum goes like
$q_{0}dN/d^{3}{\bm{q}}\sim(\bm{q}\times\bm{B})^{2}$ leading to the following
azimuthal distribution:
$\displaystyle
dN_{\gamma}/d\phi\propto\left\\{1-\cos[2(\phi-\Psi_{B})]\right\\}=\left\\{1-\cos[2(\bar{\phi}-\bar{\Psi}_{B})]\right\\}\quad$
(8)
which will contribute the following to the elliptic flow of these photons
(with $V_{\gamma}$ being the signal strength)
$\displaystyle v_{2}^{\gamma}=-\langle V_{\gamma}\cos(2\bar{\Psi}_{\bf
B})\rangle.$ (9)
Once again we see the same modification factor
$\sim\langle\cos(2\bar{\Psi}_{\bf B})\rangle$ due to the mismatch between $\bf
B$ field and the out-of-plane direction from event to event. If
$\bar{\Psi}_{\bf B}$ were to be simply $\pi/2$ (i.e. $\bf B$ always out-of-
plane) then indeed these photons will have positive and sizable elliptic flow.
However the fluctuation may bring $\bar{\Psi}_{\bf B}$ to depart from $\pi/2$
considerably.
Figure 6: (Color online) The correlations $\langle\cos[n(\Psi_{\bf
E}-\Psi_{n})]\rangle$ as functions of impact parameter for $n=1,2,3,4$ at four
different positions on the transverse plane: (from left to right)
${\mathbf{r}}=(0,0,0)$ fm; ${\mathbf{r}}=(3,0,0)$ fm; ${\mathbf{r}}=(0,3,0)$
fm; ${\mathbf{r}}=(3,3,0)$ fm.
In all three examples, the fluctuating $\Psi_{\bf B}-\Psi_{2}$ brings in a
reduction to the intrinsic strength of the signal by
$\displaystyle R_{1}=\langle\cos(2\bar{\Psi}_{\bf B})\rangle.$ (10)
It is therefore worth a close examination of this factor: see Fig. 4 for the
computed average values of $\langle\cos[n(\Psi_{\bf B}-\Psi_{n})]\rangle$ for
varied centralities from event-by-event determination of the $\bf B$-field
direction $\Psi_{\bf B}$ (at several different spatial points) and the
participants harmonics, $\Psi_{n}$, $n=1,2,3,4$. The plots suggest that the
correlations between $\Psi_{\bf B}$ and the odd harmonics $\Psi_{1},\Psi_{3}$
are practically zero (in accord with parity invariance), while the the
correlations of $\Psi_{\bf B}$ with even harmonics $\Psi_{2},\Psi_{4}$ are
nonzero but get suppressed as compared with the results in optical geometry
limit (without fluctuations) e.g. $\langle\cos[2(\Psi_{\bf
B}-\Psi_{2})]\rangle_{\rm opt}=-1$. The centrality dependence of
$\langle\cos[2(\Psi_{\bf B}-\Psi_{2})]\rangle$ is in perfect agreement with
the patterns seen in the histograms Fig. 2 and scatter plots Fig. 3: it is
significantly suppressed in the most central and most peripheral cases
(indicating no correlations) while is maximized around $b=8\sim 10$ fm with
peak values $-0.6\sim 0.7$. Similar behavior is observed for
$\langle\cos[4(\Psi_{\bf B}-\Psi_{4})]\rangle$ too albeit with a weaker
correlation strengths, e.g. with the peak values $\sim 0.2$ or so. Although
suppressed, the correlation between $\Psi_{\bf B}$ and $\Psi_{4}$ appears
still sizable for moderate values of $b$. This may imply complications and
caveats that must be seriously addressed for the proposal in Ref.
Voloshin:2011mx to use the two-particle correlation with respect to the
fourth harmonic event plane to disentangle the CME contribution and the flow
contribution.
The plots in Fig.4 for four different field points demonstrate the spatial
dependence of the correlations $\langle\cos[n(\Psi_{\bf B}-\Psi_{n})]\rangle$
on the transverse plane. We first see minor differences when the field point
deviates from the origin and such differences become sizable for the outer
most point ${\mathbf{r}}=(3,3,0)$ fm. The comparison suggests that further
away from the origin the correlations between $\Psi_{\bf B}$ and $\Psi_{n}$
become even weaker. However, near the central overlapping region, the
correlations are almost homogeneous over, for example, the distance of the
typical size of a sphaleron, $(\alpha_{s}T)^{-1}\lesssim 1$ fm for RHIC and
LHC McLerran:1990de .
The last issue we want to address is the possible correlation between the
${\bf B}$ orientation and its own strength. It is conceivable that there could
be correlation (on an event-by-event basis) between the signal strength and
the $\bf B$ field direction. Typically the signal strength of these $\bf
B$-field induced effects will scale as $\sim\bf B^{2}$. For example, recalling
that the CME current is proportional to ${\bf B}$ and the signal strength
$A_{++}$ as mentioned earlier would be nearly proportional to ${\bf B}^{2}$.
The quadrupole strength $r_{e}$ through CMW also scales similarly
$r_{e}\propto{\bf B}^{2}$. So does the signal strength $V_{\gamma}$ in the
case of soft photon emissions. If the field strength and the orientation of
${\bf B}$ do correlate with each other, then one may not be able to factorize
the signal strength and the field orientation factor $\cos(2\bar{\Psi}_{\bf
B})$ when taking the event average. We therefore further examine the
correlations between the two by evaluating the factor
$\displaystyle R_{2}=\frac{\langle(e{\bf B})^{2}\cos(2\bar{\Psi}_{\bf
B})\rangle}{\langle(e{\bf B})^{2}\rangle}.$ (11)
In Fig. 5 we show the impact parameter dependence of the $(e{\bf
B})^{2}$-weighted correlations $\langle(e{\bf B})^{2}\cos[n(\Psi_{\bf
B}-\Psi_{n})]\rangle/\langle(e{\bf B})^{2}\rangle$ for $n=1,2,3,4$. We find
little difference between the two factors $R_{1}$ in Fig.4(the most left
panel) and $R_{2}$ in Fig.5 at all centralities for all $n$ values. While only
results for field point ${\mathbf{r}}=(0,0,0)$ are shown in Fig.5, we have
checked all other three points and the observation is the same. Therefore, we
conclude that the magnitude of the magnetic field has no noticeable
correlation to its azimuthal direction with respect to the matter geometry.
## V Azimuthal correlation between electric field and matter geometry
In this section, we briefly discuss how the event-by-event fluctuation affects
the correlation between the orientation of ${\bf E}$-field and the
participants harmonic planes, $\Psi_{n}$. The motivation is that from Fig. 1
we notice that the electric field can be as strong as the magnetic one, and
thus they may lead to observable effects too. An obvious example is possible
multiple charge distributions induced by the strong electric field. Similarly
to the magnetic field case, possible ${\bf E}$-induced effects would be
affected by the azimuthal correlations between ${\bf E}$ and matter geometry.
It is therefore interesting to also study these correlations. In Fig. 6, we
show the correlations $\langle\cos[n(\Psi_{\bf E}-\Psi_{n})]\rangle$,
$n=1,2,3,4$, as functions of the impact parameter at the four field points as
before. There are a number of interesting features that differ from the
magnetic field case: 1) there is a clear negative correlation (i.e. back-to-
back) between $\Psi_{\bf E}$ and $\Psi_{1}$ and it is most strong in the more
central collisions and at the field point near the origin — this is
understandable as the pole of $\Psi_{1}$ with more matter will concurrently
have more positive charges from protons generating ${\bf E}$ pointing in the
opposite; 2) at field points away from the origin the $\Psi_{\bf E}$ is
strongly correlated to $\Psi_{2}$ (and also weakly correlated to $\Psi_{4}$) —
the direction of ${\bf E}$ is more in-plane for the field point on $x$-axis
while more out-of-plane for points away from $x$-axis; 3) there is also a weak
correlation between $\Psi_{\bf E}$ and $\Psi_{3}$. Finally we have also
checked the $(eE)^{2}$-weighted correlations $\langle(e{\bf
E})^{2}\cos[n(\Psi_{\bf E}-\Psi_{n})]\rangle/\langle(e{\bf E})^{2}\rangle$ and
find no visible correlation between the ${\bf E}$ magnitude and orientation.
## VI Discussion on charge-dependent measurements
The studies of correlations between ${\bf B}$ and ${\bf E}$ fields and matter
geometry suggest in general that there may be nontrivial charge distributions,
particularly in azimuthal angles (e.g. charged dipole and quadrupole), induced
by varied ${\bf B}$\- and ${\bf E}-$induced effects. There are also other
effects not related the initial ${\bf E}$ and ${\bf B}$ fields, e.g., the
local charge conservation effect (Schlichting:2010qia, ) that can lead to
nontrivial charge azimuthal correlations when coupled with various harmonic
flows. It is therefore tempting to think about possible measurements that may
fully extract information for the azimuthal charge distributions. In parallel
to the measurements of (charge-inclusive) global azimuthal particle
distributions that can be subsequently Fourier-decomposed into various
harmonic components, here we suggest a class of observables, the charged
multiple vectors $\hat{Q}^{c}_{n}$. Consider the measured charged hadrons in
an event we can construct the charged multiple vector $\hat{Q}_{n}^{c}$ with
magnitude $Q^{c}_{n}$ and azimuthal angle $\Psi^{c}_{n}$:
$\displaystyle Q^{c}_{n}\,e^{in\Psi^{c}_{n}}=\sum_{i}\,q_{i}\,e^{in\phi_{i}}$
(12)
where the summation runs over all particles with $q_{i}$ and $\phi_{i}$ the
electric charge and azimuthal angle of the $i$-th particle. This idea
generalizes the earlier proposal of the charged dipole vector analysis
suggested in (Liao:2010nv, ) in the context of CME observables. One may
introduce properly $p_{t}$-weighed definition. One may also think of sub-event
version of this analysis or possible multi-particle correlation improved
version. We emphasize that these are different and independent from the
existing $\hat{Q}_{n}$ vectors related to flow measurements, i.e.
$Q_{n}\,e^{in\Psi_{n}}=\sum_{i}\,e^{in\phi_{i}}$. The $\hat{Q}_{n}$ is charge
blind and includes all charges similarly while the $\hat{Q}_{n}^{c}$ takes the
difference between positive and negative charges therefore yields information
on the charge distribution. The technical difficulty of $\hat{Q}_{n}^{c}$
measurements should be at the same level as previous $\hat{Q}_{n}$ analysis
and quite feasible, while clearly the $\hat{Q}_{n}^{c}$ analysis provides
orthogonal and unique information on the charge distributions. With a joint
$\hat{Q}_{n}$ and $\hat{Q}_{n}^{c}$ analysis one can study the strength and
azimuthal correlations among all harmonics and charged multipoles. Therefore
it is very valuable to do a systematic charged multiple vector analysis,
leading toward a quantitative “charge landscape survey” in heavy-ion
collisions.
## VII Summary
In summary, we have performed a detailed study of the event-by-event
fluctuations of both the azimuthal orientation of the magnetic and electric
fields as well as the matter geometry (which is specified by the participant
planes of a series of harmonics) in the initial condition of heavy-ion
collisions. Such fluctuations suppress the azimuthal correlations between the
magnetic field and the second harmonic participant plane particularly in the
very central and very peripheral collisions, while leaving a window around
$b=8\sim 10$ fm with still sizable correlation between the two. We have
further studied similar azimuthal correlations between ${\bf B}$ and other
harmonic participant planes, and found similar yet weaker correlation with
$\Psi_{4}$ while no correlation with $\Psi_{1}$ and $\Psi_{3}$. The
correlation between $\Psi_{\bf B}$ and $\Psi_{4}$ may indicate that the CME
can also contribute to the charge-pair correlation
$\langle\cos[2(\phi_{1}+\phi_{2}-2\Psi_{4})]\rangle$ as opposed to the
assumption in Ref. Voloshin:2011mx . Examination of these correlations at
different field points shows notable dependence on spatial positions. For
completeness we have presented similar studies for electric field ${\bf E}$
azimuthal correlations with matter geometry which show quite different
patterns from that for the ${\bf B}$ field. We have evaluated the impact of
such azimuthal fluctuations and correlations on a number of observables
related to magnetic-field induced effects in heavy-ion collisions: the charged
pair azimuthal correlations, the charge dependent elliptic flow of pions, as
well as the azimuthal anisotropy of soft photons due to conformal anomaly.
Specifically we have quantified the modification factors
$R_{1}=\langle\cos(2\bar{\Psi}_{\bf B})\rangle$ and the $(e{\bf
B})^{2}$-weighted reduction factor $R_{2}=\langle(e{\bf
B})^{2}\cos(2\bar{\Psi}_{\bf B})\rangle/\langle(e{\bf B})^{2}\rangle$ at
different centralities (see Figs. 4 and 5) and found sizable reduction in both
very central and very peripheral collisions. From these results we conclude
that the optimal centrality class for observing the above mentioned ${\bf
B}$-induced effects corresponds to impact parameter range $b\sim 8-10$ fm. The
qualitative conclusion should hold also for Pb + Pb collisions at LHC.
We end with a few pertinent remarks:
(1) In our computations (as well as all the previous computations
arXiv:0711.0950 ; arXiv:0907.1396 ; arXiv:1003.2436 ; arXiv:1103.4239 ;
arXiv:1107.3192 ; arXiv:1111.1949 ; arXiv:1201.5108 ), the classical Liénard-
Wiechert potentials have been used. One may worry about quantum corrections to
the field equations as the magnitude of the electromagnetic field is much
larger than the electron and light quark masses. In principle a calculation
including all relevant QED processes is needed, but a one-loop Euler-
Heisenberg effective lagrangian (see Ref. Dunne:2004nc for review) may give a
good indication on the size of such corrections. At strong-field limit the
field equation derived from this lagrangian can be regarded as linear Maxwell
equations with the renormalized charge
$e^{2}\rightarrow\tilde{e}^{2}=e^{2}/\left[1-\frac{e^{2}}{24\pi^{2}}\ln\frac{e^{2}|F|^{2}}{m_{e}^{4}}\right]$.
An estimate implies that the computed electromagnetic field may be amended
only by a few percent even for $e|F|\sim 100m_{\pi}^{2}$.
(2) As shown in Refs. arXiv:1103.4239 ; arXiv:1201.5108 , the magnetic field
magnitude bears strong time dependence. Furthermore, the electromagnetic
response of QGP could dramatically modify the time evolution of the ${\bf B}$
and ${\bf E}$ fields. It would be important to incorporate these into
realistic modeling of effects induced by these fields, as recently attempted
in Toneev:2012zx to incorporate the time evolution into the event-by-event
estimates of the CME by using the parton-hadron-string-dynamics approach.
(3) As suggested in Ref. Voloshin:2010ut , in order to disentangle the effects
of the magnetic field and the collective flow, it is very useful to study the
U + U collisions where the geometry becomes a nontrivial level arm. It will be
interesting to study in the future the various correlations among the field
strengths, field azimuthal orientations with the matter geometric
eccentricities as well as the harmonic participant planes.
Acknowledgments: We are grateful to Y. Burnier, A. Bzdak, U. Heinz, D.
Kharzeev, V. Koch, V. Skokov, H. Yee, and Z. Qiu for helpful communications
and discussions. We also thank the RIKEN BNL Rsearch Center for partial
support.
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|
arxiv-papers
| 2012-09-28T18:27:33 |
2024-09-04T02:49:35.732966
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "John Bloczynski, Xu-Guang Huang, Xilin Zhang, and Jinfeng Liao",
"submitter": "Xu-Guang Huang",
"url": "https://arxiv.org/abs/1209.6594"
}
|
1210.0039
|
Generalizations and specializations of generating functions for Jacobi,
Gegenbauer, Chebyshev and Legendre polynomials with definite integrals
Howard S. Cohl∗ and Connor MacKenzie†
∗ Applied and Computational Mathematics Division, National Institute of
Standards and Technology, Gaithersburg, Maryland, U.S.A.
† Department of Mathematics, Westminster College, 319 South Market Street, New
Wilmington, Pennsylvania, U.S.A.
###### Abstract
In this paper we generalize and specialize generating functions for classical
orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre
polynomials. We derive a generalization of the generating function for
Gegenbauer polynomials through extension a two element sequence of generating
functions for Jacobi polynomials. Specializations of generating functions are
accomplished through the re-expression of Gauss hypergeometric functions in
terms of less general functions. Definite integrals which correspond to the
presented orthogonal polynomial series expansions are also given.
## 1 Introduction
This paper concerns itself with analysis of generating functions for Jacobi,
Gegenbauer, Chebyshev and Legendre polynomials. The analysis involves
generalization and specialization by re-expression of Gauss hypergeometric
generating functions for these orthogonal polynomials. The generalizations
that we present here are for two of the most important generating functions
for Jacobi polynomials, namely [4, (4.3.1-2)].111The interesting question of
which orthogonal polynomial generating functions are important is addressed in
[4, Section 4.3] (see also R. A. Askey’s MathSciNet review of Srivastava &
Manocha (1984) [12]). In fact, these are the first two generating functions
which appear in Section 4.3 of [4]. As we will show, these two generating
functions, traditionally expressed in terms of Gauss hypergeometric functions,
can be re-expressed in terms of associated Legendre functions (and also in
terms of Ferrers functions, associated Legendre functions on the real segment
$(-1,1)$). Our Jacobi polynomial generating function generalizations, Theorem
1, Corollary 1 and Corollary 2, generalize the generating function for
Gegenbauer polynomials. The presented proofs of these generalizations rely
upon the series re-arrangment technique. The motivation for the proofs of our
generalizations was purely intuitive. Examination of the two Jacobi polynomial
generating functions which we generalize,222As well as their companion
identities, see Section 2. indicate that these generating functions represent
two elements of an infinite sequence of eigenfunction expansions. The
resulting formal proofs of our generalizations are then simple consequences.
Our generalized expansions and hypergeometric orthogonal polynomial generating
functions are given in terms of Gauss hypergeometric functions. The Gauss
hypergeometric Jacobi polynomial generating functions which we generalize, as
well as their eigenfunction expansion generalizations, are all re-expressible
in terms of associated Legendre functions. Associated Legendre functions [9,
Chapter 14] are given in terms of Gauss hypergeometric functions which satisfy
a quadratic transformation of variable. These have an abundance of
applications in Physics, Engineering and Applied Mathematics for solving
partial differential equations in a variety of contexts. Recently, efficient
numerical evaluation of these functions has been investigated in [11].
Associated Legendre functions are more elementary than Gauss hypergeometric
functions because Gauss hypergeometric functions have three free parameters,
whereas associated Legendre functions have only two. One can make this
argument of elementarity with all of the functions which can be expressed in
terms of Gauss hypergeometric functions, namely (inverse) trigonometric,
(inverse) hyperbolic, exponential, logarithmic, Jacobi, Gegenbauer, and
Chebyshev polynomials, and complete elliptic integrals. We summarize how
associated Legendre functions, Gegenbauer, Chebyshev and Legendre polynomials
and complete elliptic integrals of the first kind are interrelated. In a one-
step process, we obtain definite integrals from our orthogonal polynomial
expansions and generating functions.
To the best of our knowledge our generalizations, re-expressions of Gauss
hypergeometric generating functions for orthogonal polynomials and definite
integrals are new and have not previously appeared in the literature.
Furthermore, the generating functions presented in this paper are some of the
most important generating functions for these hypergeometric orthogonal
polynomials and any specializations and generalizations will be similarly
important.
This paper is organized as follows. In Sections 2, 3, 4, 5, 6, we present
generalized and simplified expansions for Jacobi, Gegenbauer, Chebyshev of the
second kind, Legendre, and Chebyshev of the first kind polynomials
respectively. In A we present definite integrals which correspond to the
derived hypergeometric orthgonal polynomial expansions. Unless stated
otherwise the domains of convergence given in this paper are those of the
original generating function and/or its corresponding definite integral.
Throughout this paper we rely on the following definitions. Let
$a_{1},a_{2},a_{3},\ldots\in{\mathbf{C}}$. If $i,j\in{\mathbf{Z}}$ and $j<i,$
then $\sum_{n=i}^{j}a_{n}=0$ and $\prod_{n=i}^{j}a_{n}=1$. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$ Let
${\mathbf{D}}:=\\{z\in{\mathbf{C}}:|z|<1\\}$ be the open unit disk.
## 2 Expansions over Jacobi polynomials
The Jacobi polynomials $P_{n}^{(\alpha,\beta)}:{\mathbf{C}}\to{\mathbf{C}}$
can be defined in terms of the terminating Gauss hypergeometric series as
follows ([9, (18.5.7)])
$P_{n}^{(\alpha,\beta)}(z):=\frac{(\alpha+1)_{n}}{n!}\,{}_{2}F_{1}\left(\begin{array}[]{c}-n,n+\alpha+\beta+1\\\\[2.84544pt]
\alpha+1\end{array};\frac{1-z}{2}\right),$
for $n\in{\mathbf{N}}_{0}$, and $\alpha,\beta>-1$ such that if
$\alpha,\beta\in(-1,0)$ then $\alpha+\beta+1\neq 0$. The Gauss hypergeometric
function
${{}_{2}}F_{1}:{\mathbf{C}}^{2}\times({\mathbf{C}}\setminus{\mathbf{N}}_{0})\times{\mathbf{D}}\to{\mathbf{C}}$
(see Chapter 15 in [9]) is defined as
${{}_{2}}F_{1}\left(\begin{array}[]{c}a,b\\\\[5.69046pt]
c\end{array};z\right):=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},$
where the Pochhammer symbol (rising factorial)
$(\cdot)_{n}:{\mathbf{C}}\to{\mathbf{C}}$ [9, (5.2.4)] is defined by
$(z)_{n}:=\prod_{i=1}^{n}(z+i-1),$
where $n\in{\mathbf{N}}_{0}$. Note that the Gauss hypergeometric function can
be analytically continued through for instance the Euler’s integral
representation for $z\in{\mathbf{C}}\setminus(1,\infty)$ (see for instance [2,
Theorem 2.2.1]).
Consider the generating function for Gegenbauer polynomials (see §3 for their
definition) given by [9, (18.12.4)], namely
$\frac{1}{(1+\rho^{2}-2\rho
x)^{\nu}}=\sum_{n=0}^{\infty}\rho^{n}C_{n}^{\nu}(x).$ (1)
We attempt to generalize this expansion using the representation of Gegenbauer
polynomials in terms of Jacobi polynomials given by [9, (18.7.1)], namely
$C_{n}^{\nu}(x)=\frac{(2\nu)_{n}}{\left(\nu+\frac{1}{2}\right)_{n}}P_{n}^{(\nu-1/2,\nu-1/2)}(x).$
(2)
By making the replacement $\nu-1/2$ to $\alpha$ and $\beta$ in (1) using (2),
we see that there are two possibilities for generalizing the generating
function for Gegenbauer polynomials to a generating function for Jacobi
polynomials. These two possibilities are given below, namely (6), (13). The
first possibility is given for $\rho\in{\mathbf{D}}\setminus(-1,0]$ by [9,
(18.12.3)]
$\displaystyle\frac{1}{(1+\rho)^{\alpha+\beta+1}}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+1}{2},\frac{\alpha+\beta+2}{2}\\\\[2.84544pt]
\beta+1\end{array};\frac{2\rho(1+x)}{(1+\rho)^{2}}\right)$ (5)
$\displaystyle=\left(\frac{2}{\rho(1+x)}\right)^{\beta/2}\frac{\Gamma(\beta+1)}{{\rm
R}^{\alpha+1}}P_{\alpha}^{-\beta}(\zeta_{+})=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_{n}}{(\beta+1)_{n}}\rho^{n}P_{n}^{(\alpha,\beta)}(x),$
(6)
where we have used the definitions
${\rm R}={\rm R}(\rho,x):=\sqrt{1+\rho^{2}-2\rho
x},\quad\zeta_{\pm}=\zeta_{\pm}(\rho,x):=\frac{1\pm\rho}{\sqrt{1+\rho^{2}-2\rho
x}}.$
Note that the restriction given by $\rho\in{\mathbf{D}}\setminus(-1,0]$ is so
that the values of $\rho$ are ensured to remain in the domain of
$P_{\nu}^{\mu}$, but may otherwise be relaxed to ${\mathbf{D}}$ by analytic
continuation if one uses the Gauss hypergeometric representation. The Ferrers
function of the first kind representation given below provides the analytic
continuation to the segment $(-1,0]$. Here
$P_{\nu}^{\mu}:{\mathbf{C}}\setminus(-\infty,1]\to{\mathbf{C}}$ is the
associated Legendre function of the first kind (see Chapter 14 in [9]), which
can be defined in terms of the Gauss hypergeometric function as follows [9,
(14.3.6), (15.2.2), §14.21(i)]
$P_{\nu}^{\mu}(z):=\frac{1}{\Gamma(1-\mu)}\left(\frac{z+1}{z-1}\right)^{\mu/2}\,{}_{2}F_{1}\left(\begin{array}[]{c}-\nu,\nu+1\\\\[2.84544pt]
1-\mu\end{array};\frac{1-z}{2}\right).$ (7)
The associated Legendre function of the first kind can also be expressed in
terms of the Gauss hypergeometric function as (see [9, (14.3.18),
§14.21(iii)]), namely
$P_{\nu}^{\mu}\left(z\right)=\frac{2^{\mu}z^{\nu+\mu}}{\Gamma\left(1-\mu\right)\left(z^{2}-1\right)^{\mu/2}}~{}{{}_{2}}F_{1}\left(\begin{array}[]{c}\frac{-\nu-\mu}{2},\frac{-\nu-\mu+1}{2}\\\\[5.69046pt]
1-\mu\end{array};1-\frac{1}{z^{2}}\right),$ (8)
where $|\arg(z-1)|<\pi$. We have used (8) to re-express the generating
function (6). We will refer to a companion identity as one which is produced
by applying the map $x\mapsto-x$ to an expansion over Jacobi, Gegenbauer,
Chebyshev, or Legendre polynomials with argument $x$, in conjunction with the
parity relations for those orthogonal polynomials.
In our first possibility for generalizing the generating function for
Gegenbauer polynomials to a generating function for Jacobi polynomials, namely
(6), we use the parity relation for Jacobi polynomials (see for instance [9,
Table 18.6.1])
$P_{n}^{(\alpha,\beta)}(-x)=(-1)^{n}P_{n}^{(\beta,\alpha)}(x),$ (9)
and the replacement $\alpha,\beta\mapsto\beta,\alpha$. This produces a
companion identity which is the second possibility for generalizing the
generating function for Gegenbauer polynomials to a generating function for
Jacobi polynomials for $\rho\in(0,1)$ by
$\displaystyle\frac{1}{(1-\rho)^{\alpha+\beta+1}}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+1}{2},\frac{\alpha+\beta+2}{2}\\\\[2.84544pt]
\alpha+1\end{array};\frac{-2\rho(1-x)}{(1-\rho)^{2}}\right)$ (12)
$\displaystyle=\left(\frac{2}{\rho(1-x)}\right)^{\alpha/2}\frac{\Gamma(\alpha+1)}{{\rm
R}^{\beta+1}}{\rm
P}_{\beta}^{-\alpha}(\zeta_{-})=\sum_{n=0}^{\infty}\frac{(\alpha+\beta+1)_{n}}{(\alpha+1)_{n}}\rho^{n}P_{n}^{(\alpha,\beta)}(x).$
(13)
Note that the restriction given by $\rho\in(0,1)$ is so that the values of
$\rho$ are ensured to remain in the domain of ${\rm P}_{\nu}^{\mu}$, but may
otherwise be relaxed to ${\mathbf{D}}$ by analytic continuation if one uses
the Gauss hypergeometric representation. Here
$\mathrm{P}_{\nu}^{\mu}:(-1,1)\to{\mathbf{C}}$ is the Ferrers function of the
first kind (associated Legendre function of the first kind on the cut) through
[9, (14.3.1)], defined as
$\mathrm{P}_{\nu}^{\mu}(x):=\frac{1}{\Gamma(1-\mu)}\left(\frac{1+x}{1-x}\right)^{\mu/2}{}_{2}F_{1}\left(\begin{array}[]{c}-\nu,\nu+1\\\\[2.84544pt]
1-\mu\end{array};\frac{1-x}{2}\right).$ (14)
The Ferrers function of the first kind can also be expressed in terms of the
Gauss hypergeometric function as (see [7, p. 167]), namely
${\mathrm{P}}_{\nu}^{\mu}(x)=\frac{2^{\mu}x^{\nu+\mu}}{\Gamma(1-\mu)(1-x^{2})^{\mu/2}}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{-\nu-\mu}{2},\frac{-\nu-\mu+1}{2}\\\\[2.84544pt]
1-\mu\end{array};1-\frac{1}{x^{2}}\right),$ (15)
for $x\in(0,1)$. We have used (15) to express (13) in terms of the Ferrers
function of the first kind. One can easily see that (6) and (13) are
generalizations of the generating function for Gegenbauer polynomials by
taking $\alpha=\beta=\nu-1/2$. The right-hand sides easily follow using the
identification (2) and the left-hand sides follow using [1, (8.6.16-17)].
There exist natural extensions of (6), (13) in the literature (see [4,
(4.3.2)]). The extension corresponding to (13) is given for $\rho\in(0,1)$ by
$\displaystyle\frac{(\alpha+\beta+1)(1+\rho)}{(1-\rho)^{\alpha+\beta+2}}{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+2}{2},\frac{\alpha+\beta+3}{2}\\\\[2.84544pt]
\alpha+1\end{array};\frac{-2\rho(1-x)}{(1-\rho)^{2}}\right)$ (17)
$\displaystyle\hskip
14.22636pt=\left(\frac{2}{\rho(1-x)}\right)^{\alpha/2}\frac{(\alpha+\beta+1)(1+\rho)\Gamma(\alpha+1)}{{\rm
R}^{\beta+2}}{\rm P}_{\beta+1}^{-\alpha}(\zeta_{-})$ $\displaystyle\hskip
56.9055pt=\sum_{n=0}^{\infty}(2n+\alpha+\beta+1)\frac{(\alpha+\beta+1)_{n}}{(\alpha+1)_{n}}\rho^{n}P_{n}^{(\alpha,\beta)}(x),$
(18)
and its companion identity corresponding to (6), for
$\rho\in{\mathbf{D}}\setminus(-1,0]$ is
$\displaystyle\frac{(\alpha+\beta+1)(1-\rho)}{(1+\rho)^{\alpha+\beta+2}}{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+2}{2},\frac{\alpha+\beta+3}{2}\\\\[2.84544pt]
\beta+1\end{array};\frac{2\rho(1+x)}{(1+\rho)^{2}}\right)$ (21)
$\displaystyle\hskip
14.22636pt=\left(\frac{2}{\rho(1+x)}\right)^{\beta/2}\frac{(\alpha+\beta+1)(1-\rho)\Gamma(\beta+1)}{{\rm
R}^{\alpha+2}}P_{\alpha+1}^{-\beta}(\zeta_{+})$ $\displaystyle\hskip
56.9055pt=\sum_{n=0}^{\infty}(2n+\alpha+\beta+1)\frac{(\alpha+\beta+1)_{n}}{(\beta+1)_{n}}\rho^{n}P_{n}^{(\alpha,\beta)}(x).$
(22)
We have used (8), (15) to re-express these Gauss hypergeometric function
generating functions as associated Legendre functions. We have not seen the
companion identity (22) in the literature, but it is an obvious consequence of
[4, (4.3.2)] using parity. On the other hand, we have not seen the associated
Legendre function representations of (18), (22) in the literature.
Upon examination of these two sets of generating functions, we suspected that
these were just two examples of an infinite sequence of such expansions. This
led us to the proof of the following theorem, which is a Jacobi polynomial
expansion which generalizes the generating function for Gegenbauer polynomials
(1). According to Ismail (2005) [4, (4.3.2)], the generating functions (18),
(22), their generalizations Theorem 1, Corollary 1 and their corresponding
definite integrals (A), (A), are closely related to the Poisson kernel for
Jacobi polynomials, so our new generalizations will have corresponding
applications.
###### Theorem 1
Let $m\in{\mathbf{N}}_{0}$, $\alpha,\beta>-1$ such that if
$\alpha,\beta\in(-1,0)$ then $\alpha+\beta+1\neq 0$, $x\in[-1,1],$
$\rho\in{\mathbf{D}}\setminus(-1,0].$ Then
$\frac{(1+x)^{-\beta/2}}{{\rm
R}^{\alpha+m+1}}P_{\alpha+m}^{-\beta}(\zeta_{+})=\sum_{n=0}^{\infty}a_{n,m}^{(\alpha,\beta)}(\rho)P_{n}^{(\alpha,\beta)}(x),$
(23)
where $a_{n,m}^{(\alpha,\beta)}:{\mathbf{D}}\setminus(-1,0]\to{\mathbf{C}}$ is
defined by
$\displaystyle
a_{n,m}^{(\alpha,\beta)}(\rho):=\frac{(2n+\alpha+\beta+1)\Gamma(\alpha+\beta+n+1)(\alpha+\beta+m+1)_{2n}}{2^{\beta/2}\Gamma(\beta+n+1)}$
$\displaystyle\hskip
142.26378pt\times\frac{1}{\rho^{(\alpha+1)/2}(1-\rho)^{m}}P_{-m}^{-\alpha-\beta-2n-1}\left(\frac{1+\rho}{1-\rho}\right).$
Proof. Let $\rho\in(0,\epsilon)$ with $\epsilon$ sufficiently small. Then
using the definition of the following Gauss hypergeometric function
$\displaystyle{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+m+1}{2},\frac{\alpha+\beta+m+2}{2}\\\\[2.84544pt]
\beta+1\end{array};\frac{2\rho(1+x)}{(1+\rho)^{2}}\right)$ (26)
$\displaystyle\hskip
56.9055pt=\sum_{n=0}^{\infty}\frac{\left(\frac{\alpha+\beta+m+1}{2}\right)_{n}\left(\frac{\alpha+\beta+m+2}{2}\right)_{n}(2\rho)^{n}(1+x)^{n}}{n!(\alpha+1)_{n}(1+\rho)^{2n}},$
(27)
the expansion of $(1+x)^{n}$ in terms of Jacobi polynomials is given by
$(1+x)^{n}=2^{n}(\beta+1)_{n}\sum_{k=0}^{n}\frac{(-1)^{k}(-n)_{k}(\alpha+\beta+2k+1)(\alpha+\beta+1)_{k}}{(\alpha+\beta+1)_{n+k+1}(\beta+1)_{k}}P_{k}^{(\alpha,\beta)}(x),$
(28)
whose coefficients can be determined using orthogonality of Jacobi polynomials
(see Appendix) combined with the Mellin transform given in [9, (18.17.36)]. By
inserting (28) in the right-hand side of (27), we obtain an expansion of the
Gauss hypergeometric function on the left-hand side of (27) in terms of Jacobi
polynomials. By interchanging the two sums (with justification by absolute
convergence), shifting the $n$-index by $k$, and taking advantage of standard
properties such as
$\displaystyle(-n-k)_{k}=\frac{(-1)^{k}(n+k)!}{n!},$
$\displaystyle(a)_{n+k}=(a)_{k}(a+k)_{n},$
$\displaystyle\left(\frac{a}{2}\right)_{n}\left(\frac{a+1}{2}\right)_{n}=\frac{1}{2^{2n}}\left(a\right)_{2n},$
$n,k\in{\mathbf{N}}_{0}$, $a\in{\mathbf{C}}$, produces a Gauss hypergeometric
function as the coefficient of the Jacobi polynomial expansion. The resulting
expansion is
${}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+m+1}{2},\frac{\alpha+\beta+m+2}{2}\\\\[2.84544pt]
\beta+1\end{array};\frac{2\rho(1+x)}{(1+\rho)^{2}}\right)=\sum_{n=0}^{\infty}f_{n,m}^{(\alpha,\beta)}(\rho)P_{n}^{(\alpha,\beta)}(x),$
(29)
where $f_{n,m}^{(\alpha,\beta)}:(0,\epsilon)\to{\mathbf{R}}$ is defined by
$\displaystyle
f_{n,m}^{(\alpha,\beta)}(\rho):=\frac{(2n+\alpha+\beta+1)(\alpha+\beta+1)_{n}(\alpha+\beta+m+1)_{2n}\,\rho^{n}}{(\beta+1)_{n}(\alpha+\beta+1)_{2k+1}(1+\rho)^{2n}}$
$\displaystyle\hskip
85.35826pt\times\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+m+2n+1}{2},\frac{\alpha+\beta+m+2n+2}{2}\\\\[2.84544pt]
\alpha+\beta+2n+2\end{array};\frac{4\rho}{(1+\rho)^{2}}\right).$
The above expansion is actually analytic on ${\mathbf{D}}$. However, if we
express it in terms of associated Legendre functions, then we must necessarily
subdivide it into two regions. The Gauss hypergeometric function coefficient
of this expansion, as well as the Gauss hypergeometric function on the left-
hand of (29) are realized to be associated Legendre functions of the first
kind through (8). Both sides of the resulting Jacobi polynomial expansion are
analytic functions on $\rho\in{\mathbf{D}}\setminus(-1,0].$ Since we know that
(23) is valid for $\rho\in(0,\epsilon),$ then by the identity theorem for
analytic functions, the equation holds on this domain. This completes the
proof. $\hfill\blacksquare$
Note that the left-hand side of Theorem 1 can be rewritten as
$\frac{(\rho/2)^{\beta/2}}{\Gamma(\beta+1)(1+\rho)^{\alpha+\beta+m+1}}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{\alpha+\beta+m+1}{2},\frac{\alpha+\beta+m+2}{2}\\\\[5.69046pt]
\beta+1\end{array};\frac{2\rho(1+x)}{(1+\rho)^{2}}\right).$
We have also derived the companion identity to (23), which we give in the
following corollary.
###### Corollary 1
Let $m\in{\mathbf{N}}_{0}$, $\alpha,\beta>-1$ such that if
$\alpha,\beta\in(-1,0)$ then $\alpha+\beta+1\neq 0$, $x\in[-1,1]$,
$\rho\in(0,1)$. Then
$\frac{(1-x)^{-\alpha/2}}{{\rm
R}^{\beta+m+1}}{\mathrm{P}}_{\beta+m}^{-\alpha}(\zeta_{-})=\sum_{n=0}^{\infty}b_{n,m}^{(\alpha,\beta)}(\rho)P_{n}^{(\alpha,\beta)}(x),$
(31)
where $b_{n,m}^{(\alpha,\beta)}:(0,1)\to{\mathbf{R}}$ is defined by
$\displaystyle
b_{n,m}^{(\alpha,\beta)}(\rho):=\frac{(2n+\alpha+\beta+1)\Gamma(\alpha+\beta+n+1)(\alpha+\beta+m+1)_{2n}}{2^{\alpha/2}\Gamma(\alpha+n+1)}$
$\displaystyle\times\frac{1}{\rho^{(\beta+1)/2}(1+\rho)^{m}}\mathrm{P}_{-m}^{-\alpha-\beta-2n-1}\left(\frac{1-\rho}{1+\rho}\right).$
Proof. We start with (29) and apply the parity relation for Jacobi polynomials
(9). Let $\rho\in(0,1)$. The Gauss hypergeometric function coefficient of the
Jacobi expansion is seen to be a Ferrers function of the first kind (14).
After the application of the parity relation, the left-hand side also reduces
to a Ferrers function of the first kind through (15). This completes the
proof. $\hfill\blacksquare$
Theorem 1 generalizes (6), (22), while Corollary 1 generalizes (13), (18).
Both Theorem 1 and Corollary 1 generalize the generating function for
Gegenbauer polynomials (1), which is its own companion identity.
### 2.1 Expansions and definite integrals from the Szegő transformation
If one applies on the complex plane, the Szegő transformation (conformal map)
$z=\frac{1+\rho^{2}}{2\rho},$ (32)
(which maps a circle with radius less than unity to an ellipse with the foci
at $\pm 1$) to the expansion in Theorem 1, then one obtains a new expansion.
By [13, Theorem 12.7.3], this new Jacobi polynomial expansion is convergent
for all $x\in{\mathbf{C}}$ within the interior of this ellipse. Applying (32)
to (23) yields the following corollary.
###### Corollary 2
Let $m\in{\mathbf{N}}_{0}$, $\alpha,\beta>-1$ such that if
$\alpha,\beta\in(-1,0)$ then $\alpha+\beta+1\neq 0$, $x,z\in{\mathbf{C}}$,
with $z\in{\mathbf{C}}\setminus(-\infty,1]$ on any ellipse with the foci at
$\pm 1$ and $x$ in the interior of that ellipse. Then
$\frac{(1+x)^{-\beta/2}}{(z-x)^{(\alpha+m+1)/2}}P_{\alpha+m}^{-\beta}\left(\frac{1+z-\sqrt{z^{2}-1}}{\sqrt{2(z-\sqrt{z^{2}-1})(z-x)}}\right)=\sum_{n=0}^{\infty}c_{n,m}^{(\alpha,\beta)}(z)P_{n}^{(\alpha,\beta)}(x),$
(33)
where
$c_{n,m}^{(\alpha,\beta)}:{\mathbf{C}}\setminus(-\infty,1]\to{\mathbf{C}}$ is
defined by
$\displaystyle
c_{n,m}^{(\alpha,\beta)}(z):=\frac{(2n+\alpha+\beta+1)\Gamma(\alpha+\beta+n+1)(\alpha+\beta+m+1)_{2n}}{2^{(\beta-\alpha-m-1)/2}\Gamma(\beta+n+1)}$
$\displaystyle\hskip
113.81102pt\times\frac{(z-\sqrt{z^{2}-1})^{m/2}}{\left(1-z+\sqrt{z^{2}-1}\right)^{m}}P_{-m}^{-\alpha-\beta-2n-1}\left(\sqrt{\frac{z+1}{z-1}}\right).$
We would just like to briefly note that one may use the Szegő transformation
(32) to obtain new expansion formulae and corresponding definite integrals
from all the Jacobi, Gegenbauer, Legendre and Chebyshev polynomial expansions
used in this paper. For the sake of brevity, we leave this to the reader.
## 3 Expansions over Gegenbauer polynomials
The Gegenbauer polynomials $C_{n}^{\mu}:{\mathbf{C}}\to{\mathbf{C}}$ can be
defined in terms of the terminating Gauss hypergeometric series as follows
([9, (18.5.9)])
$C_{n}^{\mu}(z):=\frac{(2\mu)_{n}}{n!}\,{}_{2}F_{1}\left(\begin{array}[]{c}-n,n+2\mu\\\\[2.84544pt]
\mu+\frac{1}{2}\end{array};\frac{1-z}{2}\right),$ (34)
for $n\in{\mathbf{N}}_{0}$ and $\mu\in(-1/2,\infty)\setminus\\{0\\}.$ The
Gegenbauer polynomials (34) are defined for
$\mu\in(-1/2,\infty)\setminus\\{0\\}.$ However many of the formulae listed
below actually make sense in the limit as $\mu\to 0$. In this case, one should
take the limit of the expression as $\mu\to 0$ with the interpretation of
obtaining Chebyshev polynomials of the first kind (see §6 for the details of
this limiting procedure).
###### Corollary 3
Let $m\in{\mathbf{N}}_{0},$ $\mu\in(-1/2,\infty)\setminus\\{0\\},$
$x\in[-1,1]$. If $\rho\in{\mathbf{D}}\setminus(-1,0],$ then
$\frac{1}{{\rm
R}^{2\mu+m}}C_{m}^{\mu}(\zeta_{+})=\frac{2\Gamma(2\mu+m)}{m!\rho^{\mu}(1-\rho)^{m}}\sum_{n=0}^{\infty}(n+\mu)(2\mu+m)_{2n}P_{-m}^{-2\mu-2n}\left(\frac{1+\rho}{1-\rho}\right)C_{n}^{\mu}(x),$
(35)
and if $\rho\in(0,1)$ then
$\frac{1}{{\rm
R}^{2\mu+m}}C_{m}^{\mu}(\zeta_{-})=\frac{2\Gamma(2\mu+m)}{m!\rho^{\mu}(1+\rho)^{m}}\sum_{n=0}^{\infty}(n+\mu)(2\mu+m)_{2n}{\mathrm{P}}_{-m}^{-2\mu-2n}\left(\frac{1-\rho}{1+\rho}\right)C_{n}^{\mu}(x).$
(36)
Proof. Using (23), substitute $\alpha=\beta=\mu-1/2$ along with (2) and [9,
(14.3.22)], namely
$P_{n+\mu-1/2}^{1/2-\mu}(z)=\frac{2^{\mu-1/2}\Gamma(\mu)n!}{\sqrt{\pi}\,\Gamma(2\mu+n)}(z^{2}-1)^{\mu/2-1/4}\,C_{n}^{\mu}(z).$
Through (7), we see that the Gauss hypergeometric function in the definition
of the associated Legendre function of the first kind on the right-hand side
is terminating and therefore defines an analytic function for
$\rho\in{\mathbf{D}}$. The analytic continuation to the segment $\rho\in(0,1]$
is provided by replacing the associated Legendre function of the first kind
with the Ferrers function of the first kind with argument $(1-\rho)/(1+\rho)$.
$\hfill\blacksquare$
As an example for re-expression using the elementarity of associated Legendre
functions which was mentioned in the introduction, we now apply to two
generating function results of Koekoek et al. (2010) [6, (9.8.32)] and
Rainville (1960) [10, (144.8)].
###### Theorem 2
Let $\lambda\in{\mathbf{C}},$ $\mu\in(-1/2,\infty)\setminus\\{0\\},$
$\rho\in(0,1),$ $x\in[-1,1]$. Then
$\displaystyle(1-x^{2})^{1/4-\mu/2}P_{\mu-\lambda-1/2}^{1/2-\mu}\left({\rm
R}+\rho\right)\,{\mathrm{P}}_{\mu-\lambda-1/2}^{1/2-\mu}\left({\rm
R}-\rho\right)$ $\displaystyle\hskip
71.13188pt=\frac{2^{1/2-\mu}}{\Gamma(\mu+\frac{1}{2})}\sum_{n=0}^{\infty}\frac{(\lambda)_{n}\,(2\mu-\lambda)_{n}}{(2\mu)_{n}\,\Gamma(\mu+\frac{1}{2}+n)}\rho^{\mu-1/2+n}C_{n}^{\mu}(x).$
(37)
Proof. [6, (9.8.32)] give a generating function for Gegenbauer polynomials,
namely
$\displaystyle{{}_{2}}F_{1}\left(\begin{array}[]{c}\lambda,2\mu-\lambda\\\\[5.69046pt]
\mu+\frac{1}{2}\end{array};\frac{1-{\rm
R}-\rho}{2}\right){{}_{2}}F_{1}\left(\begin{array}[]{c}\lambda,2\mu-\lambda\\\\[5.69046pt]
\mu+\frac{1}{2}\end{array};\frac{1-{\rm R}+\rho}{2}\right)$ (42)
$\displaystyle\hskip
162.18062pt=\sum_{n=0}^{\infty}\frac{(\lambda)_{n}\,(2\mu-\lambda)_{n}}{(2\mu)_{n}\,(\mu+\frac{1}{2})_{n}}\rho^{n}C_{n}^{\mu}(x).$
Using [9, (15.8.17)] to do a quadratic transformation on the Gauss
hypergeometric functions and then using (8), with degree and order given by
$\mu-\lambda-1/2,$ $1/2-\mu,$ respectively and either $z={\rm R}+\rho$ or
$z={\rm R}-\rho$. Simplification completes the proof. $\hfill\blacksquare$
###### Theorem 3
Let $\alpha\in{\mathbf{C}},$ $\mu\in(-1/2,\infty)\setminus\\{0\\},$
$\rho\in(0,1),$ $x\in[-1,1]$. Then
$\displaystyle\frac{\left(1-x^{2}\right)^{1/4-\mu/2}}{{\rm
R}^{1/2+\alpha-\mu}}{\mathrm{P}}_{\mu-\alpha-1/2}^{1/2-\mu}\left(\frac{1-\rho
x}{{\rm
R}}\right)=\frac{(\rho/2)^{\mu-1/2}}{\Gamma(\mu+\frac{1}{2})}\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{(2\mu)_{n}}\rho^{n}C_{n}^{\mu}(x).$
(43)
Proof. On p. 279 of [10, (144.8)] there is a generating function for
Gegenbauer polynomials, namely
$(1-\rho
x)^{-\alpha}\,{{}_{2}}F_{1}\left(\begin{array}[]{c}\frac{\alpha}{2},\frac{\alpha+1}{2}\\\\[5.69046pt]
\mu+\frac{1}{2}\end{array};\frac{-\rho^{2}(1-x^{2})}{(1-\rho
x)^{2}}\right)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{(2\mu)_{n}}\rho^{n}C_{n}^{\mu}(x).$
Using (8) to rewrite the Gauss hypergeometric function on the left-hand side
of the above equation completes the proof. $\hfill\blacksquare$
## 4 Expansions over Chebyshev polynomials of the second kind
The Chebyshev polynomials of the second kind can be obtained from the
Gegenbauer polynomials using [9, (18.7.4)], namely
$U_{n}(z)=C_{n}^{1}(z),$ (44)
for $n\in{\mathbf{N}}_{0}$. Hence and through (34), the Chebyshev polynomials
of the second kind $U_{n}:{\mathbf{C}}\to{\mathbf{C}}$ can be defined in terms
of the terminating Gauss hypergeometric series as follows
$U_{n}(z):=(n+1)\,{}_{2}F_{1}\left(\begin{array}[]{c}-n,n+2\\\\[2.84544pt]
\frac{3}{2}\end{array};\frac{1-z}{2}\right).$ (45)
###### Corollary 4
Let $m\in{\mathbf{N}}_{0},$ $x\in[-1,1].$ If
$\rho\in{\mathbf{D}}\setminus(-1,0]$ then
$\displaystyle\frac{1}{{\rm
R}^{m+2}}U_{m}(\zeta_{+})=\frac{2(m+1)}{\rho(1-\rho)^{m}}\sum_{n=0}^{\infty}(n+1)(m+2)_{2n}P_{-m}^{-2n-2}\left(\frac{1+\rho}{1-\rho}\right)U_{n}(x),$
(46)
and if $\rho\in(0,1)$ then
$\displaystyle\frac{1}{{\rm
R}^{m+2}}U_{m}(\zeta_{-})=\frac{2(m+1)}{\rho(1+\rho)^{m}}\sum_{n=0}^{\infty}(n+1)(m+2)_{2n}{\mathrm{P}}_{-m}^{-2n-2}\left(\frac{1-\rho}{1+\rho}\right)U_{n}(x).$
(47)
Proof. Using (35), with (44) and
$U_{m}(z)=\sqrt{\frac{\pi}{2}}\frac{m+1}{(z^{2}-1)^{1/4}}P_{m+1/2}^{-1/2}(z),$
(48)
which follows from (7), (45), and [9, (15.8.1)]. Analytically continuing to
the segment $\rho\in(0,1)$ completes the proof. $\hfill\blacksquare$
Note that using [1, (8.6.9)], namely
$P_{\nu}^{-1/2}\left(z\right)=\sqrt{\frac{2}{\pi}}\frac{\left(z^{2}-1\right)^{-1/4}}{\left(2\nu+1\right)}\left[\left(z+\sqrt{z^{2}-1}\right)^{\nu+1/2}-\left(z+\sqrt{z^{2}-1}\right)^{-\nu-1/2}\right],$
and (48) one can derive the elementary function representation for the
Chebyshev polynomials of the second kind [8, (1.52)].
###### Corollary 5
Let $\rho\in{\mathbf{D}},$ $x\in[-1,1]$. Then
$\displaystyle(1-x^{2})^{-1/4}P_{1/2-\lambda}^{-1/2}\left({\rm
R}+\rho\right){\mathrm{P}}_{1/2-\lambda}^{-1/2}\left({\rm R}-\rho\right)$
$\displaystyle\hskip
88.2037pt=\frac{2^{5/2}\sqrt{\rho}}{\pi}\sum_{n=0}^{\infty}\frac{(\lambda)_{n}\,(2-\lambda)_{n}\,2^{2n}\rho^{n}}{(2n+2)!}U_{n}(x).$
(49)
Proof. Substituting $\mu=1$ into (37), and using (44) with simplification,
completes the proof. $\hfill\blacksquare$
###### Corollary 6
Let $\alpha\in{\mathbf{C}},$ $\rho\in(0,1),$ $x\in[-1,1]$. Then
$\frac{{\rm
R}^{1/2-\alpha}}{(1-x^{2})^{1/4}}{\mathrm{P}}_{1/2-\alpha}^{-1/2}\left(\frac{1-\rho
x}{{\rm
R}}\right)=\sqrt{\frac{2\rho}{\pi}}\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{(n+1)!}\rho^{n}U_{n}(x).$
(50)
Proof. Using (44), and substituting $\mu=1$ into (43) with simplification,
produces this generating function for Chebyshev polynomials of the second
kind. $\hfill\blacksquare$
## 5 Expansions over Legendre polynomials
Legendre polynomials can be obtained from the Gegenbauer polynomials using [9,
(18.7.9)], namely
$P_{n}(z)=C_{n}^{1/2}(z),$ (51)
for $n\in{\mathbf{N}}_{0}$. Hence and through (34), the Legendre polynomials
of the second kind $P_{n}:{\mathbf{C}}\to{\mathbf{C}}$ can be defined in terms
of the terminating Gauss hypergeometric series as follows
$P_{n}(z):={}_{2}F_{1}\left(\begin{array}[]{c}-n,n+1\\\\[2.84544pt]
1\end{array};\frac{1-z}{2}\right).$ (52)
Using (51) we can write the previous expansions over Gegenbauer polynomials in
terms of expansions over Legendre polynomials.
###### Corollary 7
Let $m\in{\mathbf{N}}_{0},$ $x\in[-1,1]$. If
$\rho\in{\mathbf{D}}\setminus(-1,0]$ then
$\displaystyle\frac{1}{{\rm
R}^{m+1}}P_{m}(\zeta_{+})=\frac{(1-\rho)^{-m}}{\sqrt{\rho}}\sum_{n=0}^{\infty}(2n+1)(m+1)_{2n}P_{-m}^{-2n-1}\left(\frac{1+\rho}{1-\rho}\right)P_{n}(x),$
(53)
and if $\rho\in(0,1)$ then
$\displaystyle\frac{1}{{\rm
R}^{m+1}}P_{m}(\zeta_{-})=\frac{(1+\rho)^{-m}}{\sqrt{\rho}}\sum_{n=0}^{\infty}(2n+1)(m+1)_{2n}{\mathrm{P}}_{-m}^{-2n-1}\left(\frac{1-\rho}{1+\rho}\right)P_{n}(x).$
(54)
Proof. Using (35), substitute $\mu=1/2$ with (51) and $P_{m}(z)=P_{m}^{0}(z),$
which follows from (7), (52). Analytic continuation to $\rho\in(0,1)$
completes the proof. $\hfill\blacksquare$
###### Corollary 8
Let $\lambda\in{\mathbf{C}},$ $\rho\in\\{x\in{\mathbf{C}}:|z|<1\\},$
$x\in[-1,1]$. Then
$P_{-\lambda}\left({\rm R}+\rho\right){\mathrm{P}}_{-\lambda}\left({\rm
R}-\rho\right)=\sum_{n=0}^{\infty}\frac{(\lambda)_{n}\,(1-\lambda)_{n}}{(n!)^{2}}\rho^{n}P_{n}(x).$
(55)
Proof. Substituting $\mu=1/2$ into (37) and using (51) with simplification
completes the proof. $\hfill\blacksquare$
Note that Corollary 55 is just a restatement of [14, Theorem A], and therefore
Theorem 2 is a generalization of Brafman’s theorem.
###### Corollary 9
Let $\alpha\in{\mathbf{C}},$ $\rho\in{\mathbf{D}},$ $x\in[-1,1].$ Then
${\rm R}^{-\alpha}\,{\mathrm{P}}_{\alpha-1}\left(\frac{1-\rho x}{{\rm
R}}\right)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}\rho^{n}P_{n}(x).$ (56)
Proof. Substituting $\mu=1/2$ in (43) with simplification completes the proof.
$\hfill\blacksquare$
As a further example of the elementarity of associated Legendre functions
mentioned in the introduction, we apply to the recent generating function
results of Wan & Zudelin (2012) [14].
###### Theorem 4
Let $x,y$ be in a neighborhood of 1. Then
$\displaystyle\hskip
2.84544pt\frac{\pi^{2}}{2}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_{n}^{2}}{(n!)^{2}}P_{2n}\left(\frac{(x+y)(1-xy)}{(x-y)(1+xy)}\right)\left(\frac{x-y}{1+xy}\right)^{2n}$
$\displaystyle\hskip
28.45274pt=\left\\{\begin{array}[]{ll}\displaystyle\frac{\pi^{2}}{2}&\mathrm{if}\
x=y=1,\\\\[11.38092pt]
\displaystyle\frac{1+xy}{xy}\,K\left(\frac{\sqrt{x^{2}-1}}{x}\right)K\left(\frac{\sqrt{y^{2}-1}}{y}\right)&\mathrm{if}\
x,y\geq 1,\\\\[17.07182pt]
\displaystyle\frac{1+xy}{x}\,K\left(\frac{\sqrt{x^{2}-1}}{x}\right)K\left(\sqrt{1-y^{2}}\right)&\mathrm{if}\
x\geq 1{\rm\ and\ }y\leq 1,\\\\[17.07182pt]
\displaystyle\frac{1+xy}{y}\,K\left(\sqrt{1-x^{2}}\right)K\left(\frac{\sqrt{y^{2}-1}}{y}\right)&\mathrm{if}\
x\leq 1{\rm\ and\ }y\geq 1,\\\\[17.07182pt]
\displaystyle(1+xy)\,K\left(\sqrt{1-x^{2}}\right)K\left(\sqrt{1-y^{2}}\right)&\mathrm{if}\
x,y\leq 1.\end{array}\right.$
Proof. If we start with (10) from [14], namely
$\displaystyle\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}^{2}}{(n!)^{2}}P_{2n}\left(\frac{(x+y)(1-xy)}{(x-y)(1+xy)}\right)\left(\frac{x-y}{1+xy}\right)^{2n}$
$\displaystyle\hskip
113.81102pt=\frac{1+xy}{2}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{1}{2},\frac{1}{2}\\\\[5.69046pt]
1\end{array};1-x^{2}\right){}_{2}F_{1}\left(\begin{array}[]{c}\frac{1}{2},\frac{1}{2}\\\\[5.69046pt]
1\end{array};1-y^{2}\right),$
and use [9, (15.9.21)] we can express the Gauss hypergeometric functions as
Legendre functions. For instance
${}_{2}F_{1}\left(\begin{array}[]{c}\frac{1}{2},\frac{1}{2}\\\\[5.69046pt]
1\end{array};1-x^{2}\right)=P_{-1/2}(2x^{2}-1),$
with $x\in{\mathbf{C}}\setminus(-\infty,0]$. This domain is because the
Legendre function of the first kind $P_{\nu}$ and the Ferrers function of the
first kind ${\rm P}_{\nu}$, both with order $\mu=0,$ are given by the same
Gauss hypergeometric function and are continuous across argument unity (cf.
(7), (14)). So there is no distinction between these two functions, except
that the Ferrers function has argument on the real line with modulus less than
unity and the Legendre function is defined on
${\mathbf{C}}\setminus(-\infty,1)$ (both being well defined with argument
unity). (Hence there really is no need to use two different symbols to denote
this function.) The proof is completed by noting the two formulae
$P_{-1/2}(z)=\frac{2}{\pi}\sqrt{\frac{2}{z+1}}K\left(\sqrt{\frac{z-1}{z+1}}\right),$
${\rm P}_{-1/2}(x)=\frac{2}{\pi}K\left(\sqrt{\frac{1-x}{2}}\right)$
[1, (8.13.1), (8.13.8)], where $K:[0,1)\to[\pi/2,\infty)$ is the complete
elliptic integral of the first kind defined by [9, (19.2.8)]
$K(k):=\frac{\pi}{2}\,{}_{2}F_{1}\left(\begin{array}[]{c}\frac{1}{2},\frac{1}{2}\\\\[5.69046pt]
1\end{array};k^{2}\right).$
$\hfill\blacksquare$
###### Theorem 5
Let $x,y$ be in a neighborhood of 1. Then
$\displaystyle=3\sum_{n=0}^{\infty}\frac{\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}}{(n!)^{2}}P_{3n}\left(\frac{x+y-2x^{2}y^{2}}{(x-y)\sqrt{1+4xy(x+y)}}\right)\left(\frac{x-y}{\sqrt{1+4xy(x+y)}}\right)^{3n}.$
$\displaystyle=\sqrt{1+4xy(x+y)}\left\\{\begin{array}[]{ll}1&\mathrm{if}\
x=y=1\\\\[5.69046pt]
P_{-1/3}\left(2x^{3}-1\right)P_{-1/3}\left(2y^{3}-1\right)&\mathrm{if}\
x,y>1\\\\[5.69046pt] {\rm
P}_{-1/3}\left(2x^{3}-1\right)P_{-1/3}\left(2y^{3}-1\right)&\mathrm{if}\
x<1{\rm\ and\ }y>1\\\\[5.69046pt] P_{-1/3}\left(2x^{3}-1\right){\rm
P}_{-1/3}\left(2y^{3}-1\right)&\mathrm{if}\ x>1{\rm\ and\ }y<1\\\\[5.69046pt]
{\rm P}_{-1/3}\left(2x^{3}-1\right){\rm
P}_{-1/3}\left(2y^{3}-1\right)&\mathrm{if}\ x,y<1\end{array}\right.$
Proof. This follows by [14, (11)] and [9, (15.9.21)]. $\hfill\blacksquare$
## 6 Expansions over Chebyshev polynomials of the first kind
The Chebyshev polynomials of the first kind
$T_{n}:{\mathbf{C}}\to{\mathbf{C}}$ can be defined in terms of the terminating
Gauss hypergeometric series as follows ( [7, p. 257])
$T_{n}(z):={}_{2}F_{1}\left(\begin{array}[]{c}-n,n\\\\[2.84544pt]
\frac{1}{2}\end{array};\frac{1-z}{2}\right),$ (60)
for $n\in{\mathbf{N}}_{0}$. The Chebyshev polynomials of the first kind can be
obtained from the Gegenbauer polynomials using [2, (6.4.13)], namely
$T_{n}(z)=\frac{1}{\epsilon_{n}}\lim_{\mu\to
0}\frac{n+\mu}{\mu}C_{n}^{\mu}(z),$ (61)
where the Neumann factor $\epsilon_{n}\in\\{1,2\\}$, commonly seen in Fourier
cosine series, is defined as $\epsilon_{n}:=2-\delta_{n,0}.$
###### Corollary 10
Let $m\in{\mathbf{N}}_{0},$ $x\in[-1,1]$. If
$\rho\in{\mathbf{D}}\setminus(-1,0]$ then
$\frac{1}{{\rm
R}^{m}}T_{m}(\zeta_{+})=\frac{1}{(1-\rho)^{m}}\sum_{n=0}^{\infty}\epsilon_{n}(m)_{2n}P_{-m}^{-2n}\left(\frac{1+\rho}{1-\rho}\right)T_{n}(x),$
(62)
and if $\rho\in(0,1)$ then
$\frac{1}{{\rm
R}^{m}}T_{m}(\zeta_{-})=\frac{1}{(1+\rho)^{m}}\sum_{n=0}^{\infty}\epsilon_{n}(m)_{2n}{\mathrm{P}}_{-m}^{-2n}\left(\frac{1-\rho}{1+\rho}\right)T_{n}(x).$
(63)
Proof. Using (35), (61), and
$T_{m}(z)=\sqrt{\frac{\pi}{2}}(z^{2}-1)^{1/4}P_{m-1/2}^{1/2}(z),$ (64)
which follows from (7), (60), [9, (15.8.1)]. Analytic continuation to
$\rho\in(0,1)$ completes the proof. $\hfill\blacksquare$
Note that using [1, (8.6.8)], namely
$P_{\nu}^{1/2}\left(z\right)=\frac{1}{\sqrt{2\pi}}\left(z^{2}-1\right)^{-1/4}\left[\left(z+\sqrt{z^{2}-1}\right)^{\nu+1/2}+\left(z+\sqrt{z^{2}-1}\right)^{-\nu-1/2}\right],$
and (64) one can derive the elementary function representation for the
Chebyshev polynomials of the first kind [5, p. 177].
## Appendix A Definite integrals
As a consequence of the series expansions given above, one may generate
corresponding definite integrals (in a one-step procedure) as an application
of the orthogonality relation for these hypergeometric orthogonal polynomials.
We now describe this correspondence. Given an expansion over a set of
orthogonal polynomials $p_{n}$ such that
$f(x)=\sum_{n=0}^{\infty}a_{n}p_{n}(x),$ (65)
and the orthogonality relation
$\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)dx=c_{n}\delta_{n,m},$ (66)
where $w:(-1,1)\to[0,\infty),$ then using (66) one has
$\int_{-1}^{1}f(x)p_{n}(x)w(x)dx=\sum_{m=0}^{\infty}a_{m}\int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)dx=a_{n}c_{n},$
and therefore
$a_{n}=\frac{1}{c_{n}}\int_{-1}^{1}f(x)p_{n}(x)w(x)dx.$ (67)
The definite integral expression (67) for the coefficient $a_{n}$ is of equal
importance to the expansion (65), since one may use it to derive the other.
Integrals of such sort are always of interest since they are very likely to
find applications in applied mathematics and theoretical physics and could be
included in tables of integrals such as [3].
We now give the orthogonality relations for the orthogonal polynomials used in
the main text. For Jacobi, Gegenbauer, Chebyshev of the second kind, Legendre,
and Chebyshev of the first kind polynomials, the orthogonality relations can
be found in [9, (18.2.1), (18.2.5), Table 18.3.1]. Using the above procedure,
we obtain the following definite integrals for products of special functions
with Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. Let
$m,n\in{\mathbf{N}}_{0}$, $\alpha,\beta>-1$ such that if
$\alpha,\beta\in(-1,0)$ then $\alpha+\beta+1\neq 0$,
$\rho\in\left\\{z\in{\mathbf{C}}:0<|z|<1\right\\}\setminus(-1,0]$. Then
$\displaystyle\int_{-1}^{1}\frac{(1-x)^{\alpha}(1+x)^{\beta/2}}{{\rm
R}^{\alpha+m+1}}P_{\alpha+m}^{-\beta}(\zeta_{+})P_{n}^{(\alpha,\beta)}(x)dx$
$\displaystyle\hskip
31.2982pt=\frac{2^{\alpha+\beta/2+1}\Gamma(\alpha+n+1)(\alpha+\beta+m+1)_{2n}}{n!\rho^{(\alpha+1)/2}(1-\rho)^{m}}P_{-m}^{-\alpha-\beta-2n-1}\left(\frac{1+\rho}{1-\rho}\right).$
Let $\rho\in(0,1).$ Then
$\displaystyle\int_{-1}^{1}\frac{(1-x)^{\alpha/2}(1+x)^{\beta}}{{\rm
R}^{\beta+m+1}}{\mathrm{P}}_{\beta+m}^{-\alpha}(\zeta_{-})P_{n}^{(\alpha,\beta)}(x)dx$
$\displaystyle\hskip
31.2982pt=\frac{2^{\alpha/2+\beta+1}\Gamma(\beta+n+1)(\alpha+\beta+m+1)_{2n}}{n!\rho^{(\beta+1)/2}(1+\rho)^{m}}{\mathrm{P}}_{-m}^{-\alpha-\beta-2n-1}\left(\frac{1-\rho}{1+\rho}\right).$
Let $\mu\in(-1/2,\infty)\setminus\\{0\\},$
$\rho\in\\{z\in{\mathbf{C}}:0<|z|<1\\}\setminus(-1,0].$ Then
$\displaystyle\int_{-1}^{1}\frac{(1-x^{2})^{\mu-1/2}}{{\rm
R}^{2\mu+m}}C_{m}^{\mu}(\zeta_{+})C_{n}^{\mu}(x)dx$ $\displaystyle\hskip
71.13188pt=\frac{2^{2-2\mu}\pi\Gamma(2\mu+n)\Gamma(2\mu+2n+m)}{m!n!\Gamma^{2}(\mu)\rho^{\mu}(1-\rho)^{m}}P_{-m}^{-2n-2\mu}\left(\frac{1+\rho}{1-\rho}\right).$
A similar integral on $\rho\in(0,1)$ can be obtained using (36).
It should be noted that by using (44), (51), (61), the previous definite
integral over Gegenbauer polynomials can be written as an integral over
Chebyshev polynomials of the first and second kind, and well as Legendre
polynomials. Let $\lambda\in{\mathbf{C}}$, $\rho\in{\mathbf{D}}.$ Then
$\displaystyle\int_{-1}^{1}(1-x^{2})^{\mu/2-1/4}P_{\mu-1/2-\lambda}^{1/2-\mu}\left({\rm
R}+\rho\right){\mathrm{P}}_{\mu-1/2-\lambda}^{1/2-\mu}\left({\rm
R}-\rho\right)C_{n}^{\mu}(x)dx$ $\displaystyle\hskip
142.26378pt=\frac{(\lambda)_{n}\,(2\mu-\lambda)_{n}\,\rho^{n+\mu-1/2}\,2^{\mu-1/2}}{(n+\mu)\Gamma(2\mu)(\mu+1/2)_{n}\,n!}.$
Let $\rho\in(0,1)$. Then
$\displaystyle\int_{-1}^{1}\frac{(1-x^{2})^{1/4-\mu/2}}{{\rm
R}^{1/2-\mu+\lambda}}{\mathrm{P}}_{\mu-\lambda-1/2}^{1/2-\mu}\left(\frac{1-\rho
x}{{\rm R}}\right)C_{n}^{\mu}(x)dx$ $\displaystyle\hskip
113.81102pt=\frac{(\lambda)_{n}\,\sqrt{\pi}\rho^{n+\mu-1/2}\,2^{1/2-\mu}}{(n+\mu)\Gamma(\mu)n!}.$
The previous two definite integrals over Gegenbauer polynomials can also be
written as integrals over Chebyshev polynomials of the second kind and
Legendre polynomials using (44), (51).
Acknowledgements. The authors thank Hans Volkmer for valuable discussions. C.
MacKenzie would like to thank the Summer Undergraduate Research Fellowship
program at the National Institute of Standards and Technology for financial
support while this research was carried out. We also would like to acknowledge
two anonymous referees whose comments greatly improved the paper. This work
was conducted while H. S. Cohl was a National Research Council Research
Postdoctoral Associate in the Applied and Computational Mathematics Division
at the National Institute of Standards and Technology, Gaithersburg, Maryland,
U.S.A.
## References
* [1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C., 1972.
* [2] G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
* [3] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007.
* [4] M. E. H. Ismail. Classical and quantum orthogonal polynomials in one variable, volume 98 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche, With a foreword by Richard A. Askey.
* [5] A. Jakimovski, A. Sharma, and J. Szabados. Walsh equiconvergence of complex interpolating polynomials. Springer Monographs in Mathematics. Springer, Dordrecht, 2006.
* [6] R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric orthogonal polynomials and their $q$-analogues. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder.
* [7] W. Magnus, F. Oberhettinger, and R. P. Soni. Formulas and theorems for the special functions of mathematical physics. Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer-Verlag New York, Inc., New York, 1966.
* [8] J. C. Mason and D. C. Handscomb. Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton, FL, 2003.
* [9] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. Cambridge University Press, Cambridge, 2010.
* [10] E. D. Rainville. Special functions. The Macmillan Co., New York, 1960.
* [11] B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat. A new Fortran 90 program to compute regular and irregular associated Legendre functions. Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, 181(12):2091–2097, 2010.
* [12] H. M. Srivastava and H. L. Manocha. A treatise on generating functions. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester, 1984.
* [13] G. Szegő. Orthogonal polynomials. American Mathematical Society Colloquium Publications, Vol. 23. Revised ed. American Mathematical Society, Providence, R.I., 1959.
* [14] J. Wan and W. Zudilin. Generating functions of Legendre polynomials: a tribute to Fred Brafman. Journal of Approximation Theory, 164(4):488–503, 2012.
|
arxiv-papers
| 2012-09-28T21:21:27 |
2024-09-04T02:49:35.748483
|
{
"license": "Public Domain",
"authors": "Howard Cohl and Connor MacKenzie",
"submitter": "Howard Cohl",
"url": "https://arxiv.org/abs/1210.0039"
}
|
1210.0071
|
###### Abstract
Content distribution networks (CDNs) which serve to deliver web objects (e.g.,
documents, applications, music and video, etc.) have seen tremendous growth
since its emergence. To minimize the retrieving delay experienced by a user
with a request for a web object, caching strategies are often applied -
contents are replicated at edges of the network which is closer to the user
such that the network distance between the user and the object is reduced. In
this literature survey, evolution of caching is studied. A recent research
paper [15] in the field of large-scale caching for CDN was chosen to be the
anchor paper which serves as a guide to the topic. Research studies after and
relevant to the anchor paper are also analyzed to better evaluate the
statements and results of the anchor paper and more importantly, to obtain an
unbiased view of the large scale collaborate caching systems as a whole.
###### Index Terms:
Cooperative caching, Content distribution networks.
A Literature Survey of Cooperative Caching in Content Distribution Networks
Jing Zhang
Electrical and Computer Engineering
Drexel University
Email: jz334@drexel.edu
May 10, 2012
## I Introduction
Decades ago when the World Wide Web was new, making an index of all the
webpages was just like executing an Unix egrep command over 110,000 documents
[1] . Within 7 years of its existence, the web had grown to compete with long
existed information services such as television and telephone networks [5].
After around 20 years, in the month of May 2010 alone, U.S. Internet users
watched nearly 34 billion videos [2]. Nowadays, omnipresent data access and
data sharing for a large number of end-users are enabled by content
distribution systems, such as on-demand video services [15], file-sharing
networks [21], and content clouds [20].
To follow the exponential growth in demand of web service [8], caching was
introduced to reduce the retrieve latency. Caching, however, was not a new
idea at all. The first web browsers, such as Mosaic [9], were capable of
caching web objects for later reference and thus reduce bandwidth for web
traffic and latency to the users. Caching rapidly developed from being local
for a single browser to being shared serving number of clients within a
certain institution [8].
This literature survey is organized as follows: the evolution of caching is
described in section II. In section III, a chosen anchor paper in the field of
distributed caching is analyzed. Further research results in the field are
presented in section IV. Conclusion and anticipated future work are discussed
in section V.
## II Evolution of Caching
In 1995, Jacobson[3] proposed that caching could deal with the exponential
growth of the internet. According to Jacobson, data has to find _local_
sources near consumers rather than always coming from the place it was
originally produced. To match with the exponential growth of the internet,
cache has to grow as fast as the internet - exponentially. The question
remains as to how should we architect _lots_ of caches.
The taxonomy of caching was identified in 1982. Dowdy et al. proposed that an
acceptable File Assignment Problem (FAP) solution assigns files to the nodes
in some “optimal” fashion. “optimality” was identified to be measured by cost
and performance [4].
### II-A Single Cache and Multiple Caches
In 1999, Shim et al. introduced a single cache algorithm in [5]. Albeit
relatively simple with single cache proxy architecture, the LNC-R-W3-U
algorithm took into consideration both cache replacement and consistency
maintenance for web proxies. In 2000, Fan et al. described a multiple caching
protocol - “summary cache”. In the summary cache protocol, every proxy keeps a
summary of the cache directory of each participating proxy and checks the
summaries for potential hits before sending any queries [6]. Fan et al.
recognized that to fully benefit from caching, caches should cooperate and
serve each other’s misses to reduce the total traffic through the bottleneck.
### II-B Hierarchical Caching
To increase the hit rate of web cache, large scale caching structure in which
caches cooperate with each other were brought into sight - “hierarchical” and
“distributed” caching. In hierarchical caching, caches are placed in different
network levels [8]. As early as 1993, polynomial time algorithm for
hierarchical placement problem was presented [12]. Harvest structure was
considered to be the first hierarchical caching structure[8]. It was named to
illustrate its initial focus on reaping the growing crop of Internet
information [7]. Figure 1 is the overall Harvest architecture in which a
Harvest Gatherer collects indexing information from across the internet, while
a Broker collects information from many gatherers and provide query interface
to the gathered information. Brokers can also collect information from other
brokers to cascade the views from others [7].
Figure 1: Overall Harvest Architecture
Advantages with hierarchical cache are reduced network distance to hit a
document and reduced administrative concerns comparing to distributed
architecture whereas some problems associated with hierarchical caching
include: every level introduces latency [10], high level caches may experience
long queuing delays and file placement redundancy that same document gets
stored in different levels [8].
### II-C Distributed Caching
Another approach to implement large-scale cache is distributed caching. In
distributed caching, only caches at the edge of the network cooperate to serve
each other’s misses. In the year of 2001, it was realized that distributed
caching were becoming popular with the emerging of new applications that allow
distributions of web pages, images, and music since distributed caching has
lower transmission times than distributed caching due to the fact of most
traffic flows through less congested lower network levels[8]. Distributed
cache also creates fair share of loads for the system and does not generate
hot spots with high load. Nonetheless, distributed caching has longer
connection times comparing to hierarchical caching [8]. Rodriguez et al [8]
proposed that a hybrid scheme with optimal number of cooperating caches at
each level could improve performance of hierarchical and distributed caching
with reduced latency, load and bandwidth cost.
### II-D Other Caching Structures
A new “transparent” structure developed around 2002 was called _en-route
caching_. An en-route cache intercepts a client request that passes through
it. If the requested object is in the cache, object will be sent to the client
and the request will no longer be propagating further along the path.
Otherwise, the node will forward the request along the regular routing path
[11].
En-route caching has several advantages: 1) it is transparent to both content
server and clients 2) no request is detoured off the regular path which
minimizes network delay for cache miss and eliminates extra overhead such as
broadcast queries [11].
In more recent research, more specific problems of distributed caching were
studied such as adaptive distributed cache update algorithm for mobile ad-hoc
networks [13] and distributed selfish caching [14] which more realistically
models content delivery networks (CDNs), and peer-to-peer(P2P) systems. In
Laoutaris et al’s study [14], traditional grouping of distributed resources to
ensure scalability and efficiency is deemed to be common strategic goal. New
classes of network applications (e.g., overlay networks and P2P) are more “ad
hoc” in the sense that it is not as strictly dictated by organizational
boundaries or “strategic goals” described above. Individual nodes were
described to be autonomous in the sense that membership within the group is
solely motivated by benefiting from the group and thus a fluid nature of group
membership was identified to be expected. Focusing on the susceptibility of
nodes being mistreated (i.e. mistreatment due to interactions between group
members and use of common scheme for cache management for all members of the
group), [14] analyzed causes of mistreatments and suggested approach for an
individual node to decide autonomously whether to stay or leave a group. This
study carries great significance since it models realistic scenarios and
suggested decisions to take while encountering the problem.
## III Anchor Paper
Published in 2010, the anchor paper focuses on minimizing the bandwidth cost
instead of focusing on minimizing retrieving latency in previous studies such
as [8],[13]. Borst et al. chose to focus on bandwidth minimization seeing the
great momentum fueled by popularity of Video on Demand libraries. For example,
Youtube is estimated to attract tens of millions of viewers a day generating
around 2000 TB of traffic [15]. Borst et al. reasoned that for high-definition
videos with sizes of few GBs and hours-long durations, minimize bandwidth
usage is of far more significance than reducing initial play-out delay by
several hundred of milliseconds. Objectives of the research were defined as
designing light-weight cooperative content placement algorithms to maximize
the traffic volume served from cache and minimizing the bandwidth cost. Cache
replication strategies were proposed as linear programming formulations and
studied numerically. Performance of the light-weight cooperative cache
algorithm was also guaranteed to be within a constant factor from global
optimal performance in [15].
In this study, instead of focusing on general network topologies, Borst et al.
opted to focus on specific topologies motivated by real system deployments.
Figure 2 is the cache cluster model considered in the anchor paper. The cache
cluster model need not be stand-alone but rather be a part of a larger
hierarchical tree network as in Figure 3. The structure is mostly hierarchical
which also has some degree of connectivity among nodes of the same level. This
structure should fit into the category of hybrid cache structure described in
section II since hierarchy and distribution caching coexist in it.
Figure 2: Graphical Illustration of Cache Cluster Figure 3: Cache Cluster
Embedded in Hierchical Tree Network
Assuming equal bandwidth cost and cache sizes which is usually the way IPTV is
configured [15], the bandwidth minimalization $P_{min}$ or cache maximum
utilization $P_{max}$ could be modeled as
$max\sum_{n=1}^{N}s_{n}d_{n}(c^{```}p_{n}+c^{`}(M-1)q_{n}+Mc_{0}x_{0n})$ (1)
$sub\sum_{n=1}^{N}s_{n}x_{0n}\leq B_{0}$ (2)
$\sum_{n=1}^{N}s_{n}(p_{n}+(M-1)q_{n})\leq MB$ (3) $p_{n}+x_{0}n\leq
1,n=1,...,N$ (4) $q_{n}+x_{0}n\leq 1,n=1,...,N$ (5)
where $M$ denotes the number of ‘leaf’ nodes, $s_{1},...,s_{N}$ denotes the
collection of $N$ content items, $d_{in}$ denotes the demand for $n$-th
content item in node $i$, $c_{ij}$ denotes the cost associated with
tranferring content from node $i$ to node $j$, $x_{ijn}$ is a boolean
indicates whether requests for the $n$-th item at node $i$ are served from
node $j$ or not.
The problem was simplified to a linear programming problem - _knapsack-type
problem_. The problem was further analyzed in two scenarios - intra level
cache cooperation and inter level cache cooperation. In intra-level cache
cooperation case, the knapsack-type problem was further simplified to a
knapsack of size $MB$ and $2N$ items of sizes
$a_{n}=s_{n},a_{N+n}=(M-1)s_{n},n=1..N$. Based on the above insight, a _Local
- Greedy_ algorithm was proposed. Borst et al stated that under symmetric
demands, the worst case performance of Local - Greedy is $3/4$ of the global
optimal performance. In the inter-level cache cooperation scenario, the
solution can be even more simplified to a simple _Greedy_ algorithm.
The demand was estimated by a Zipf-Mandelbrot distribution with shape
parameter $\alpha$ and shift parameter $q$. Using the global optimum solution,
the bandwidth costs were observed to be decreasing with increasing values of
$\alpha$ which reflects the fact that cache performance improves as popularity
function gets steeper as in Figure 4. However, it could be observed that
neither full nor zero replication performs well across range of all $\alpha$
values. The author proposed that adjusting the degree of replication should be
done according to the particular distribution and the _Local - Greedy
algorithm_ does just that. Therefore, the _Local - Greedy algorithm_ is not
only good in a way that it has low computing complexity but also that it
combats the unknown popularity distribution problem. Further simulations on
the _Local - Greedy algorithm_ also showed that it performs close to the
global optimum with a constant margin. When aging is slow, the algorithm
converges fast (in around 2500 requests)to the global optimum.
From the performance analysis, we could see that _Local - Greedy algorithm_
performs fairly well comparing to the global optimum solution. The worst-case
performance ratio with respect to global optimum performance was guaranteed.
The most fascinating feature of the algorithm is that instead of requiring
prior knowledge of popularity distribution as needed by numerical analysis,
the algorithm combats the problem of unknown popularity distribution in its
nature of making locally optimal choices in every stage.
Figure 4: Cache Cluster Embedded in Hierchical Tree Network
## IV Further Researches
The rapid increase of content delivery over the Internet has led to the
proliferation of content distribution networks (CDNs) [18] which complied with
Borst et al’s prediction in [15] and thus confirmed the significance of
studying large scale cooperative caching system. Several research studies
relevant to the anchor paper published after [15] are analyzed in this
section.
Wireless access technology including Wi-Fi, HSPA+, 4G LTE have made broadband
wireless connections (e.g., peak downlink rate of 100 Mbps in LTE) a near-term
reality [16]. This advancement enabled user to stream videos on their hand-
held devices. According to Dai et el, cache servers of Wireless Service
Providers (WSPs) are typically deployed at Mobile Switching Centers (MSC) to
locate video content closer to end users in order to maximize bandwidth
efficiency. However, the dynamics of mobile users made resources provisioning
at cache servers a great challenge. [16] was the first to suggest a
collaborative caching mechanism between multiple WSPs while guaranteeing both
fairness and truthfulness. The collaborative caching problem was formulated as
a Vickrey-Clarke-Groves( VCG) auctions problem in game theory which encourages
cache servers to cooperate for trading their bandwidth as commodities in
auctions. Dai et al. suggested that the VCG auction from game theory fits for
designing the mechanism for several reasons: buyers have to _pay_ for each
successful trade which serves as incentives to contribute, _truthfulness_ can
be ensured by the payment method in VCG auction in which buyers are willing to
truthfully reveal their bidding information in an auction [16]. Simulation
results shows that with maximizing social welfare in VCG auction problem the
video streaming quality can be tremendously improved through the collaboration
of caching among various WSPs.
One of the significances of [16] is that by acknowledging the benefits from
collaborative caching in [15], it takes one step further considering how
collaboration between several WSPs could improve the problem caused by dynamic
nature of mobile users.
Similar to [16] in the consideration that nodes can make autonomous selfish
decisions, [17] proposed a distributed algorithm which accounts for the
selfishness of autonomous nodes and the churn phenomena (i.e. random “join”
and “leave” events of nodes in the group). Although the algorithm proposed in
[17] might not converge at equilibrium, it is still highly desired for several
reasons: it decreases access cost for all nodes compared to greedy local or
churn-unaware strategy, and it provides a fairer treatment to nodes according
to their reliability while churn-unaware strategy can have higher overheads
accessing unreliable nodes [17]. This study also went further beyond the study
of the anchor paper [15] in a realistic consideration that nodes are becoming
more dynamic recently with the advancement of mobile technology.
In [18], policies for request routing, content placement and content eviction
for CDN were designed with the goal of small user delays. While [15] focuses
on minimizing bandwidth usage considering video streaming scenarios, [18]
focuses on achieving small user delay considering smart-phone applications,
music and video files and concluded that low delays would be desired in every
case. Although content in a cache that is close to a user is likely to
experience shorter delay, placing massive contents at the nearest cache might
be counterproductive since link capacity between cache and end-user is not
infinite [18]. Figure 5 from [18] is an abstract switch model of a CDN
network.
Figure 5: A Content Distribution Network. (a) Control Plane: Requests arrive
at frontend servers (S), and must be routed to one of (possibly) several
backend caches (D) that have the content. Caches can only host a finite number
of content files (C), and the caches may be refreshed by placement and
eviction of content, (b) Data Plane: Content is served to end-users across a
network consisting of finite capacity links.
For a CDN as in Figure 5, each query could be potentially served from multiple
backend caches and each frontend takes a decision on picking a backend cache.
The constraint of the system is that the network connecting the backend cache
has finite capacity, each backend cache host finite amount of content and
refreshing content in the caches risks to incur a cost. Objective of [18] is
to develop algorithms for jointly solving the request routing and content
caching problems. Four algorithms satisfy the objectives were proposed.
#### IV-1 Periodic Max-Weight algorithm with random eviction
The algorithm tries to stabilize a system using a Lyapunov function that is
quadratic in queue lengths. [18] showed that algorithm is throughput optimal,
and has bounded queue lengths which is desirable
#### IV-2 Periodic Max-Weight Scheduling with Min-Weight Eviction policy
Queue size is related to the drift of Lyapunov function that larger negative
drift of Lyapunov function implies shorter queue length which is the insight
of the algorithm. This algorithm is also throughput optimal and has low
computational complexity.
#### IV-3 Iterative Periodic Max-Weight algorithm
Both 1) and 2) are throughput optimal but inefficient in link capacity usage.
This algorithm attempts to maximize link capacity utilization. Amble et al
showed that this is also a throughput algorithm and likely to have shorter
queue length than PMW.
#### IV-4 IPMW Scheduling with Min-Weight algorithm
The algorithm is an IPMW version of algorithm 2 that is also throughput
optimal and has average queue length at most that of PMW with min-weight
evictions.
The performance of the four algorithms proved that algorithms generating
large, negative Lyapunov drift in a system are desirable since it indicates
short average queue delays [18]. This study took a more “network” perspective
model of content distribution networks by modeling it as a switch structure.
The anchor paper [15] provides general analysis of large scale content
distribution network while [18] takes a closer view to the network recognizing
that capacity links are not infinite and tries to reduce content retrieving
latency by minimize queuing delay of the abstract CDN model.
Bj¨orkqvist et al. proposed two Peer Aware Content Caching(PACC) for CDN
networks in [19]. Similar to the anchor paper [15], objectives of [19] were
recognized as minimize total content retrieval cost and optimize bandwidth
consumption. As opposed to a specific topology setup in [15], [19] aimed on
_generic_ three layer CDN systems where both vertical and horizontal retrieval
is enabled. The two PACC policies proposed are PACC-AR and PACC-CL. PACC-AR is
implemented distributively whereas PACC-CL collaborates with peer nodes in
caching process. Simulation result shows that both policies come close to
achieving the optimum diffusion and low retrieval costs especially the
collaborated PACC-CL. Bj¨orkqvist et al argued that [15] is not applicable
while evaluating from a system perspective(i.e., dimensions of the system size
and storage capacity) and further stated that the framework in [19] is capable
of evaluating the content retrieval costs. Further work suggested by [19] is
to conduct experimental analysis.
## V Conclusion
In this literature survey, the evolution of caching was studied as background
information to better understand the current collaborative caching structures
such as Content Distributed Networks (CDN). An anchor paper [15] focusing on
large scale collaborating caching was analyzed. In the anchor paper, the total
bandwidth minimization problem was reduced to a linear programming knapsack-
type problem and further simplified to a _Local-Greedy algorithm_. The anchor
paper successfully stressed the fact that collaborated caching improves
bandwidth usage. In later on studies, [16] suggested a collaborative caching
mechanism between multiple Wireless Service Providers (WSPs) to optimize
caching performance for mobile users. [17] focuses on developing a distributed
caching algorithm that takes selfishness and churn phenomena(i.e., random
“join” and “leave” in the group) into consideration also due to the realistic
fact that nodes are becoming more dynamic lately. [18] modeled CDN as a switch
model and take into account finite link capacity between the cache and end-
user which provides a more “network” perspective to the CDN than [15]. [19]
proposed two Peer Aware Content Caching (PACC) policies for CDN networks also
taken into account dimension of the system size and storage capacity as one
step further from [15]. All the later on work suggested that nodes are being
more and more mobile and more specific system models considering the system
dimension should be developed. [19] suggested that experimental results should
be done in its future work which is also an anticipation for all the work done
currently in the field. With all the numerical analysis and simulation
experiment work done in the cooperative caching field, it is time to examine
the theory results by practical experiments.
## VI References
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(2007). Distributed selfish caching. IEEE Transactions on Parallel and
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[16]J. Dai, B. Li, F.-M. Liu, B. Li and J.-C. Liu (2012). ”Collaborative
Caching in Wireless Video Streaming Through Resource Auctions,” accepted and
to appear in IEEE Journal on Selected Areas in Communications, Special Issue
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content replication in networks with autonomous nodes. Comput. Commun. 35, 5,
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|
arxiv-papers
| 2012-09-29T02:35:49 |
2024-09-04T02:49:35.758690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jing Zhang",
"submitter": "Jing Zhang",
"url": "https://arxiv.org/abs/1210.0071"
}
|
1210.0098
|
# Infinitely Many Periodic Solutions for Some N-Body
Type Problems with Fixed Energies††thanks: Supported partially by NSF of
China.
Pengfei Yuan and Shiqing Zhang
Department of Mathematics, Sichuan University, Chengdu 610064, China
###### Abstract
In this paper, we apply the Ljusternik-Schnirelman theory with local Palais-
Smale condition to study a class of N-body problems with strong force
potentials and fixed energies. Under suitable conditions on the potential $V$,
we prove the existence of infinitely many non-constant and non-collision
symmetrical periodic solutions .
Key Words: Periodic solutions, N-body problems, Ljusternik-Schnirelman theory,
Local Palais-Smale condition.
2000 Mathematicals Subject Classification: 34C15, 34C25, 58F
## 1 Introduction and Main Results
The N-body problem is an old and very difficult problem. Many mathematicians
studied this problem using many different methods, here we only concern
variational methods. In 1975 and 1977, Gordon $([15],[16])$ firstly used
variational methods to study the periodic solutions of 2-body problems. In
$[16]$, he put forward to the strong force condition (SF for short) and got
the following Lemma:
###### Lemma 1.1.
(Gordon$[16]$) Suppose that $V(x)$ satisfies the so called Gordon’s Strong
Force condition: There exists a neighborhood $\mathcal{N}$ of 0 and a function
$U\in C^{1}(\mathcal{N}\setminus\\{0\\},\mathbb{R})$ such that:
$(i).\>\underset{|x|\rightarrow 0}{\lim}U(x)=-\infty;$
$(ii).\>-V(x)\geq|\nabla U(x)|^{2}$ for every
$x\in\mathcal{N}\,\backslash\,\\{0\\}.$
Let
$\displaystyle\Lambda$ $\displaystyle\triangleq\\{x|\,x(t)\in
H^{1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}),\,x(t)\neq
0,\forall\,t\,\in[0,1]\\}$ $\displaystyle\partial\Lambda$
$\displaystyle\triangleq\\{x|\,x(t)\in
H^{1}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}),\,\exists\,t_{0}\in[0,1]\,s.t.\,x(t_{0})=0\\}$
Then we have
$\int_{0}^{1}V(x_{n})\,\mathrm{d}t\rightarrow-\infty,\quad\forall\>x_{n}\rightharpoonup
x\in\partial\Lambda.$
After Gordon, many researchers used variational methods to study N ($N\geq 3$)
body problems $([1]-[8],[10]-[19],[22]-[27],etc.)$.
In $[18]$, P.Majer used Ljusternik-Schnirelman theory (LS for short) with
local Palais-Smale condition to seek T-periodic solutions of the following
second order Hamiltonian system:
$\ddot{x}+ax+\nabla_{x}W(t,x)=0$ $None$
where $W$ is singular at $x=0,W(t+T,x)=W(t,x).$
He got the following theorem :
###### Theorem 1.2.
Suppose
$\quad(P_{1}).a<(\dfrac{\pi}{T})^{2};$
$\quad(P_{2}).W\in
C^{1}(S_{T}^{1}\times(\mathbb{R}^{n}\setminus\\{0\\}),\mathbb{R})$ satisfies
$(SF)$;
$\quad(P_{3}).\exists\,c,\,\theta<2,\,r>0$, such that $\forall\,|x|\geq
r,\forall\,t\in S_{T}^{1}$
$W(t,x)\leq c|x|^{\theta},\quad\langle\nabla W_{x}(t,x),x\rangle-2W(t,x)\leq
c|x|^{\theta}$
$\quad(P_{4}).W(t,x)\leq b.$
Then the system $(1.1)$ has infinitely many T-periodic non-collision solutions
.
After Majer , Zhang-Zhou $([27])$ studied a class of N-Body problems. They
considered the following system :
$m_{i}\ddot{x}(t)+\nabla_{x_{i}}V(t,x_{1}(t),\cdots,x_{N}(t))=0,\,x_{i}\in\mathbb{R}^{n},\,i=1,\cdots,N$
$None$
where $m_{i}>0$ for all $i$, and $V$ satisfies the following conditions :
$\quad(V1).V(t,x_{1},\cdots,x_{N})=\dfrac{1}{2}\underset{1\leq i\neq j\leq
N}{\sum}V_{ij}(t,x_{i}-x_{j});$
$\quad(V2).V_{ij}\in
C^{2}(\mathbb{R}\times(\mathbb{R}^{n}\setminus\\{0\\});\mathbb{R}),$ for all
$1\leq i\neq j\leq N$;
$\quad(V3).V_{ij}(t,\xi)\rightarrow-\infty$ uniformly on $t$ as
$|\xi|\rightarrow 0$, for all $1\leq i\neq j\leq N$;
$\quad(V4).V_{ij}(t,\xi)\leq 0$, for all
$t\in\mathbb{R},\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}$;
$\quad(V5).V_{ij}$ satisfies Gordon’s strong force condition;
$\quad(V6)$. There exists an element $g$ of finite order $s$ in $SO(k)$ which
has no fixed point other than origin (i.e. 1 is not an eigenvalue of g), such
that
$V(t,x_{1},\cdots,x_{N})=V(t+T/s,gx_{1},\cdots,gx_{N}),\,\forall\,t\in[0,1],\,x_{i}\in\mathbb{R}^{n}.$
If the potential $V$ satisfies $(V1)-(V6)$, and
$x(t)=(x_{1}(t),\cdots,x_{N}(t))$ is a T-periodic non-collision solution of
system $(1.2)$ and satisfies
$x(t+T/s)=(x_{1}(t+T/s),\cdots,x_{N}(T+t/s))=(gx_{1}(t),\cdots,gx_{N}(t))=gx(t)$,
we say that $x(t)$ is a g-symmetric T-periodic non-collision solutions (we
also call $(g,T)$ is a non-collision solution for short).
Zhang-Zhou obtained the following theorem:
###### Theorem 1.3.
Suppose the potential $V$ satisfies $(V1)-(V6)$ and $T$ is any positive real
number. Then system $(1.2)$ has infinitely many $(g,T)$ non-collision
solutions.
Motivated by P.Majer, Zhang S.Q.- Zhou Q. ’s work , we consider the following
system:
$\left\\{\begin{array}[]{cc}&m_{i}\ddot{u}_{i}+\nabla_{u_{i}}V(u_{1},\ldots,u_{N})=0,\quad(1\leq
i\leq N),\quad(Ph.1)\\\ &\dfrac{1}{2}\sum
m_{i}|\dot{u}_{i}(t)|^{2}+V\left(u_{1}(t),\ldots,u_{N}(t)\right)=h.\quad\qquad(Ph.2)\end{array}\right.$
$None$
We have the following theorem :
###### Theorem 1.4.
Suppose $V(u_{1},\cdots,u_{N})=\dfrac{1}{2}\underset{1\leq i\neq j\leq
N}{\sum}V_{ij}(u_{i}-u_{j}),\,\,V_{ij}(\xi)\in
C^{1}(\mathbb{R}^{n}\setminus\\{0\\},\mathbb{R})$, and satisfies
$\quad(V5^{\prime}).V_{ij}(\xi)\leq
0,\forall\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}$;
$\quad(V6^{\prime}).\exists\,\alpha>2$ and $r_{1}>0$ such that $\langle\nabla
V_{ij}(\xi),\xi\rangle\geq-\alpha V_{ij}(\xi),\forall 0<|\xi|\leq r_{1};$
$\quad(V7^{\prime}).\exists\,c\geq 0,-2<\theta<0,r_{2}>r_{1}$, such that when
$|\xi|\geq r_{2}$, there holds
$\langle\nabla_{\xi}V_{ij}(\xi),\xi\rangle\leq c|\xi|^{\theta};$
$\quad(V8^{\prime}).V_{ij}(\xi)=V_{ji}(\xi),\forall\,\xi\in\mathbb{R}^{n}\setminus\\{0\\}.$
Then for any $h>0$, the system $(Ph)$ has infinitely many non-constant and
non-collision periodic solutions .
###### Example 1.5.
We take
$V_{ij}(\xi)=\left\\{\begin{array}[]{cc}-\dfrac{1}{|\xi|^{\alpha}},&\ \ {\rm
if}\ \ 0<|\xi|\leq r_{1},\\\ smooth\,\,connecting,&\ \ {\rm if}\ \
r_{1}<|\xi|<r_{2},\\\ -|\xi|^{\theta},&\ \ {\rm if}\ \ |\xi|\geq
r_{2}>r_{1}.\end{array}\right.$
## 2 Some Lemmas
In this section, we collect some known lemmas which are necessary for the
proof of Theorem 1.4.
Let us introduce the following notations:
$\displaystyle m^{*}=\min{\\{m_{1},\cdots,m_{N}\\}}\>;\qquad
H^{1}=W^{1,2}(\mathbb{R}/\mathbb{Z},\mathbb{R}^{n}).$ $\displaystyle
E=\\{u=(u_{1},\ldots,u_{N})\mid\ u_{i}\in
H^{1},\,u_{i}(t+\dfrac{1}{2})=-u_{i}(t)\\}.$ $\displaystyle\Lambda_{0}=\\{u\in
E\mid u_{i}(t)\neq u_{j}(t),\>\forall\,t,\,\forall\,i\neq j\\}.$
$\displaystyle\partial\Lambda_{0}=\\{u\in E\mid\exists\,t_{0},1\leq i_{0}\neq
j_{0}\leq N\,s.t.\,u_{i_{0}}(t_{0})=u_{j_{0}}(t_{0})\\}.$ $\displaystyle
p(u)=\underset{1\leq i\neq j\leq N,\,t\in[0,1]}{\min{}}|u_{i}(t)-u_{j}((t)|.$
$\displaystyle\\{f\leq c\\}=\\{u\in\Lambda_{0},\,f(u)\leq c\\}.$
###### Lemma 2.1.
$([1]-[4])$ Let
$f(u)=\dfrac{1}{2}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|u_{i}|^{2}\,\mathrm{d}t\int_{0}^{1}(h-V(u))\,\mathrm{d}t$
and $\tilde{u}\in\Lambda_{0}$ satisfy $f^{\prime}\left(\tilde{u}\right)=0$ and
$f\left(\tilde{u}\right)>0$. Set
$\dfrac{1}{T^{2}}=\dfrac{\int_{0}^{1}\left(h-V\left(\tilde{u}\right)\right)\,\mathrm{d}t}{\dfrac{1}{2}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{\tilde{u}}_{i}|^{2}\,\mathrm{d}t}.$
$None$
Then $\tilde{q}(t)=\tilde{u}\left(t/T\right)$ is a non-constant $T$-periodic
solution for (Ph).
###### Lemma 2.2.
$(Palais[20])$ Let $\sigma$ be an orthogonal representation of a finite or
compact group $G$ in the real Hilbert space $H$ such that for
$\forall\>\sigma\in\,G$,
$f(\sigma\cdot x)=f(x),$
where $f\in C^{1}(H,\mathbb{R})$.
Let $S=\\{x\in H\mid\sigma x=x,\>\forall\>\sigma\ in\ G\\}$, then the critical
point of $f$ in $S$ is also a critical point of $f$ in $H$.
By Lemma 2.1-2.2, we have
###### Lemma 2.3.
$([1]-[4])$ Assume $V_{ij}\in
C^{1}(\mathbb{R}^{n}\setminus\\{0\\},\mathbb{R})$ satisfies $(V8^{\prime}).$
If $\bar{u}\in\Lambda_{0}$ is a critical point of $f(u)$ and $f(\bar{u})>0$,
then $\bar{q}(t)=\bar{u}(t/T)$ is a non-constant $T$-periodic solution of
(Ph).
###### Lemma 2.4.
($[28]$) Let $q\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{n})$ and
$\int_{0}^{T}q(t)\,\mathrm{d}t=0$, then we have
$(i)$. Poincare-Wirtinger’s inequality:
$\int_{0}^{T}|\dot{q}(t)|^{2}\,\mathrm{d}t\geq{\Big{(}\frac{2\pi}{T}\Big{)}}^{2}\int_{0}^{T}|{q}(t)|^{2}\,\mathrm{d}t.$
$None$
$(ii)$. Sobolev’s inequality:
$\underset{0\leq t\leq
T}{\max}|q(t)|=|q|_{\infty}\leq\sqrt{\frac{T}{12}}\big{(}\int_{0}^{T}|\dot{q}(t)|^{2}\mathrm{d}t\big{)}^{1/2}.$
$None$
By the definition of $\Lambda_{0}$ and Lemma 2.3, for
$\forall\,u\in\Lambda_{0},(\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t)^{1/2}$
is equivalent to the $(H^{1})^{N}=H^{1}\times\cdots H^{1}$ norm:
$\parallel
u\parallel_{(H^{1})^{N}}=\left(\int_{0}^{1}|\dot{u}|^{2}\,\mathrm{d}t\right)^{1/2}+\left(\int_{0}^{1}|u|^{2}\,\mathrm{d}t\right)^{1/2}.$
So we take norm $\parallel
u\parallel=(\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t)^{1/2}$.
###### Lemma 2.5.
(Coti Zelati$\,[11]$) Let
$X=(x_{1},\ldots,x_{N})\in\mathbb{R}^{n}\times\cdots\mathbb{R}^{n},x_{i}\neq
x_{j},\,\forall\,1\leq i\neq j\leq N.$ Then
$\sum\limits_{1\leq i<j\leq N}\dfrac{m_{i}m_{i}}{|x_{i}-x_{j}|^{\alpha}}\geq
C_{\alpha}(m_{1},\ldots,m_{N})(\sum\limits_{i=1}^{N}m_{i}|x_{i}|^{2})^{-\alpha/2},$
where
$C_{\alpha}(m_{1},\ldots,m_{N})\overset{\triangle}{=}C_{\alpha}=(\sum\limits_{i=1}^{N}m_{i})^{-\alpha/2}(\sum\limits_{1\leq
i<j\leq N}m_{i}m_{j})^{\frac{2+\alpha}{2}}.$
###### Lemma 2.6.
$([18])$ Let $X$ be a Banach space with norm $\parallel\cdot\parallel$,
$\Lambda$ be an open subset of $X$, and suppose a functional
$f:\Lambda\rightarrow\mathbb{R}$ is given such that the following conditions
hold:
$\quad(i).Cat_{\Lambda}(\Lambda)=+\infty;$
$\quad(ii).f\in C^{1}(\Lambda)$ and $\forall
u_{n}\rightharpoonup\partial\Lambda,f(u_{n})\rightarrow+\infty;$
$\quad(iii).\forall\lambda\in\mathbb{R},Cat_{\Lambda}(\\{f\leq\lambda\\})<+\infty;$
Suppose in addition that there exist $g\in C^{1}(\Lambda),\beta\in(0,1)$ and
$\lambda_{0}\in\mathbb{R}$ such that
$\quad(iv).Cat_{\Lambda}(\\{f\leq g\\})<+\infty$;
$\quad(v).$ the PS condition holds in the set $\\{f\geq g\\};$
$\quad(vi).\beta\parallel f^{\prime}(u)\parallel\geq\parallel
g^{\prime}(u)\parallel,\forall u\in\\{f=g\geq\lambda_{0}\\}.$
Then $f$ has a sequence $\\{u_{n}\\}\subset\Lambda$ of critical points such
that $f(u_{n})\rightarrow+\infty$ and $f(u_{n})\geq g(u_{n})-1.$
## 3 The Proof of Theorem 1.4
###### Lemma 3.1.
Let $\\{u_{n}\\}\subset\Lambda_{0}$ and $u_{n}\rightharpoonup
u\in\partial\Lambda_{0}.$ Then $f(u_{n})\rightarrow+\infty.$
###### Proof.
First of all , we recall that
$f(u_{n})=\dfrac{1}{2}\int_{0}^{1}\sum_{i=1}^{N}m_{i}|\dot{u}_{n}^{i}|^{2}dt\int_{0}^{1}(h-V(u_{n}))dt$
(1).If $u=$ constant, we can deduce $u=0$ by
$u_{i}(t+\dfrac{1}{2})=-u_{i}(t)$. By Sobolev’s embedding theorem, we obtain
$|u_{n}|_{\infty}\rightarrow 0,\quad n\rightarrow\infty$ $None$
So when $n$ is large enough, $0\leq|u_{n}^{i}-u_{n}^{j}|\leq r_{1}$. By
$(V6^{\prime})$, there exists an $A>0$ such that
$V_{ij}(\xi)\leq-\dfrac{A}{|\xi|^{\alpha}},\quad\forall 0<|\xi|\leq r_{1}$
$None$
By $h>0$, Sobolev’s inequality $(2.3)$ , $(3.2)$ and Lemma 2.5, we have
$\displaystyle f(u_{n})$ $\displaystyle=\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}(h-\sum_{1\leq i<j\leq
N}V_{ij}(u_{n}^{i}-u_{n}^{j}))dt$ $\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\\{-\sum_{1\leq i<j\leq
N}V_{ij}(u_{n}^{i}-u_{n}^{j})\\}dt$ $\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\sum_{1\leq i<j\leq
N}\dfrac{A}{|u_{n}^{i}-u_{n}^{j}|^{\alpha}}dt$
$\displaystyle\geq\dfrac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}A[\dfrac{N(N-1)}{2}]^{\frac{2+\alpha}{2}}N^{\frac{-\alpha}{2}}|u_{n}|^{-\alpha}dt$
$\displaystyle\geq
3m^{*}A2^{-\frac{\alpha}{2}}N(N-1)^{1+\frac{\alpha}{2}}|u_{n}|_{\infty}^{2-\alpha}$
Then by $(3.1)$ and $\alpha>2$, we can deduce
$f(u_{n})\rightarrow+\infty,\quad n\rightarrow\infty$
(2). If $u\not\equiv$ constant, by the weakly lower-semi-continuity property
for norm, we have
$\displaystyle\underset{n\rightarrow\infty}{\liminf}\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{n}^{i}|^{2}\,\mathrm{d}t$
$\displaystyle\geq\int_{0}^{1}\sum\limits_{i=1}^{N}m_{i}|\dot{u}_{i}|^{2}\,\mathrm{d}t>0.$
(3.3)
Since $u\in\partial\Lambda_{0}$, there exist $t_{0},1\leq i_{0}\neq j_{0}\leq
N$ s.t. $u_{i_{0}}(t_{0})=u_{j_{0}}(t_{0}).$ Set
$\displaystyle\xi_{n}(t)=u_{n}^{i_{0}}(t)-u_{n}^{j_{0}}(t)$
$\displaystyle\xi(t)=u_{i_{0}}(t)-u_{j_{0}}(t)$
By $u_{n}\rightharpoonup u$, we have $\xi_{n}(t)\rightharpoonup\xi(t)$. Then
by $(V6^{\prime})$ and Lemma 1.1, we have
$\int_{0}^{1}V_{i_{0}j_{0}}(u_{n}^{i_{0}}-u_{n}^{j_{0}})\,\mathrm{d}t\rightarrow-\infty.$
Recalling that
$V(u_{n})=\sum\limits_{i<j}V_{ij}(u_{n}^{i}-u_{n}^{j}).$
So we have
$f(u_{n})\rightarrow+\infty,\quad n\rightarrow\infty.$
∎
###### Lemma 3.2.
For every $\lambda\in\mathbb{R}$ there exists a constant $k=k(\lambda)$ such
that
$\parallel u\parallel\leq k(\lambda)p(u),\quad\forall u\in\\{f\leq\lambda\\}$
$None$
###### Proof.
Recall that
$f(u)=\dfrac{1}{2}\parallel u\parallel^{2}\int_{0}^{1}(h-V(u))dt$
Since $V_{ij}\leq 0$, we have $f(u)\geq\dfrac{1}{2}h\parallel u\parallel^{2}$.
Since $u\in\\{f\leq\lambda\\}$, we can deduce that
$\parallel u\parallel\leq C.$ $None$
If $(3.4)$ is not true, then there exists a sequence $\\{u_{n}\\}$ such that
$\parallel u_{n}\parallel\geq np(u_{n})$. By $(3.5)$, we have
$0<p(u_{n})\leq\dfrac{C}{n}.$ $None$
Let $n\rightarrow\infty$ in (3.6), we have $p(u_{n})\rightarrow 0.$ Then there
exists a subsequence $\\{u_{n}\\}\rightharpoonup u\in\partial\Lambda_{0}$, by
Lemma 3.1, $f(u_{n})\rightarrow+\infty$, which is a contradiction since
$\\{u_{n}\\}\subset\\{f\leq\lambda\\}.$
∎
###### Lemma 3.3.
For every $\lambda\in\mathbb{R}$, the set
$\Lambda_{\lambda}=\\{u\in\Lambda_{0}:\dfrac{\parallel
u\parallel}{p(u)}\leq\lambda\\}$ is of finite category in $\Lambda_{0}$.
###### Proof.
The proof is almost the same as the proof of Lemma 4.3 of [27]. For the
covenience of the readers, we write the complete proof.
It is sufficient that we give a homotopy
$h:[0,1]\times\Lambda_{c}\subset\subset\Lambda_{0}.$
Take $\delta\in(0,1)$ s.t. $2\dfrac{1}{m^{*}}\sqrt{\delta}c<1$, and define
$\phi(t)=\dfrac{p(u}{\delta}$, if $t\in[0,\delta]$,otherwise $\phi(t)=0.$
Define
$h(u,\lambda)=(1-\lambda)u(t)+\lambda\dfrac{(u\ast\phi)(t)}{p(u)},\quad\forall
u\in\Lambda_{c},\,0\leq\lambda\leq 1$
where the convolution
$(u\ast\phi)(t)=(\int_{0}^{1}u_{1}\phi(s)ds,\cdots,\int_{0}^{1}u_{N}\phi(s)ds)$,
then $h(u,0)$ is an inclusion and $h_{1}(\Lambda_{c})$ is paracompact since
$h(u,1)$ is a convolution operator , so it’s a compact operator. We need to
prove $h(\Lambda_{c}\times I)\subset\Lambda_{0}.$
Suppose this is not true , then
$\exists\lambda_{0}\in(0,1],\,u\in\Lambda_{c},\,1\leq i_{0}\neq j_{0}\leq
N,\,t_{0}\in[0,1]$, such that
$(1-\lambda_{0})(u_{i_{0}}-u_{j_{0}})+\lambda_{0}\dfrac{1}{p(u)}((u_{i_{0}}-u_{j_{0}})\ast\phi)(t_{0})$
Then
$(u_{i_{0}}-u_{j_{0})}\ast\phi(t_{0})=p(u)(1-\dfrac{1}{\lambda_{0}})(u_{i_{0}}(t_{0})-u_{j_{0}}(t_{0}))$
So we have
$\displaystyle|p(u)(u_{i_{0}}(t_{0})-u_{j_{0}}(t_{0})-((u_{i_{0}}-u_{j_{0}}\ast\phi))(t_{0}))|$
$\displaystyle=\dfrac{p(u)}{\lambda_{0}}\geq p(u)p(u)=p^{2}(u).$
On the other hand , $\forall u=(u_{1},\cdots,u_{N})\in\Lambda_{c},\,i\neq
j,\,t\in[0,1]$, we have
$\displaystyle|p(u)(u_{i}(t)-u_{j}(t))-((u_{i}-u_{j})\ast\phi)(t)|$
$\displaystyle=|\dfrac{p(u)}{\delta}\int_{0}^{\delta}(u_{i}(t)-u_{j}(t))ds-\dfrac{p(u)}{\delta}(u_{i}(t-s)-u_{j}(t-s))ds|$
$\displaystyle\leq\dfrac{p(u)}{\delta}\int_{0}^{\delta}|u_{i}(t)-u_{j}(t)-(u_{i}(t-s)-u_{j}(t-s))|ds$
$\displaystyle\leq
p(u)\cdot\underset{|s|\leq\delta}{\sup{}}|u_{i}(t)-u_{j}(t)-(u_{i}(t-s)-u_{j}(t-s))|$
$\displaystyle\leq
p(u)\sqrt{\delta}\parallel\dot{u}_{i}-\dot{u}_{j}\parallel_{L^{2}}\leq
p(u)\sqrt{\delta}(\parallel\dot{u}_{i}\parallel_{L^{2}}+\parallel\dot{u}_{j}\parallel_{L^{2}})$
$\displaystyle\leq p(u)2\sqrt{\delta}\parallel\dot{u}\parallel_{L^{2}}\leq
p(u)\cdot(2\dfrac{1}{m^{*}}\sqrt{\delta}c)\cdot p(u)<p^{2}(u)$
Which is a contradiction. ∎
###### Lemma 3.4.
The functional $f$ verifies the Palais-Smale condition on $\Lambda_{0}$.
###### Proof.
Let $\\{u_{n}\\}\subset\Lambda_{0}$ be a P.S. sequence, then up to a
subsequence, it converges weakly in $(H^{1})^{N}$ and uniformly in
$|u|_{\infty}$ to an element $u\in\Lambda_{0}$ by Lemma 3.1. Hence
$\langle\nabla_{\xi}V_{ij}(u_{i}^{n}-u_{j}^{n}),(u_{i}-u_{i}^{n}-u_{j}-u_{j}^{n})\rangle$
converges uniformly to zero. Since $f^{\prime}(u_{n})\rightarrow 0$, and
$u-u_{n}$ is $(H^{1})^{N}$-bounded, and
$\langle f^{\prime}(u),v\rangle=\int_{0}^{1}\sum_{1\leq i\leq
N}m_{i}\langle\dot{u_{i}},\dot{v_{i}}\rangle
dt\int_{0}^{1}(h-V(u))dt-\dfrac{1}{2}\parallel
u\parallel^{2}\int_{0}^{1}\langle\nabla_{u}V(u),v\rangle dt$
So we have
$\displaystyle\parallel
u\parallel^{2}-\underset{n\rightarrow\infty}{\lim{}}\parallel
u_{n}\parallel^{2}$
$\displaystyle=\underset{n\rightarrow\infty}{\lim{}}\int_{0}^{1}m_{i}\langle\dot{u}_{n},(\dot{u}-\dot{u_{n}})\rangle
dt$ $\displaystyle=\underset{n\rightarrow\infty}{\lim}{}\dfrac{\langle
f^{\prime}(u_{n}),u-u_{n}\rangle}{\int_{0}^{1}(h-V(u_{n}))dt}+\underset{n\rightarrow\infty}{\lim}{}\dfrac{\frac{1}{2}\parallel
u_{n}\parallel^{2}\int_{0}^{1}\langle\nabla_{u}V(u_{n}),(u-u_{n})\rangle
dt}{\int_{0}^{1}(h-V(u_{n}))dt}=0$
∎
###### Lemma 3.5.
$Cat_{\Lambda_{0}}(\Lambda_{0})=+\infty$
###### Proof.
See the Corollary 3.4 of [27]. ∎
###### Lemma 3.6.
There exist a functional $g\in C^{1}(\Lambda_{0}),\beta\in(0,1)$ and
$\lambda_{0}\in\mathbb{R}$ such that:
(iv)$Cat_{\Lambda_{0}}\\{f\leq g\\}<+\infty;$
(v) the P.S. condition holds in the set $\\{f\leq g\\};$
(vi)$\beta\parallel f^{\prime}(u)\parallel\geq\parallel
g^{\prime}(u)\parallel,\quad\forall u\in\\{f=g\geq\lambda_{0}\\}.$
###### Proof.
By Sobolev’s embedding theorem, we know there is a $k_{\infty}>0$ s.t.
$\parallel u\parallel_{\infty}\leq k_{\infty}\parallel
u\parallel,\forall\,u\in\Lambda_{0}.$
Take $\beta$ such that $\beta>\dfrac{\theta+2}{2}$, take $\gamma>0$ such that
$\beta\cdot[2\gamma-N(N-1)c2^{\theta-2}k_{\infty}^{\theta}]>\gamma(\theta+2).$
Let $g(u)=\gamma\parallel u\parallel^{\theta+2}$, we shall show that $\\{f\leq
g\\}$ is a set of finite category. We take $0<\varepsilon<\dfrac{h}{2}$ and
$M>0$ such that $\forall s\in\mathbb{R},\gamma|s|^{\theta+2}\leq\varepsilon
s^{2}+M$.
Define
$f_{\varepsilon}(u)=f(u)-\varepsilon\parallel u\parallel^{2}$
Then
$\\{f\leq g\\}\subset\\{f_{\varepsilon}(u)\leq M\\}$
We can use the similar proof of Lemma 3.2 to show that there exists
$k_{1}\in\mathbb{R}$ such that
$\parallel u\parallel\leq k_{1}p(u),\forall u\in\\{f_{\varepsilon}\leq M\\}$
then by Lemma 3.3,
$Cat_{\Lambda_{0}}\\{f\leq g\\}\leq Cat_{\Lambda_{0}}\\{f_{\varepsilon}\leq
M\\}<+\infty$
From Lemma 3.4, we know that $(v)$ is satisfied.
Since $\parallel u\parallel\leq k_{1}p(u)$, we take
$\lambda_{0}=:\gamma(k_{1}r_{2})^{\theta+2}$. Then if
$u\in\\{f=g\geq\lambda_{0}\\}$, so
$\parallel
u\parallel\geq[\dfrac{\lambda_{0}}{\gamma}]^{\frac{1}{\theta+2}}=k_{1}r_{2}$
So we can obtain :
$r_{2}\leq p(u)\leq|u_{i}(t)-u_{j}(t)|\leq 2|u|_{\infty}\leq
2k_{\infty}\parallel u\parallel$
$\displaystyle\int_{0}^{1}\langle\nabla_{u}V(u),u\rangle dt$
$\displaystyle=\int_{0}^{1}\underset{1\leq i<j\leq N}{\sum}\langle\nabla
V_{ij}(u_{i}-u_{j}),(u_{i}-u_{j})\rangle dt$
$\displaystyle\leq\int_{0}^{1}\sum_{1\leq i<j\leq N}c|u_{i}-u_{j}|^{\theta}dt$
$\displaystyle\leq c\dfrac{N(N-1)}{2}(2k_{\infty}\parallel
u\parallel)^{\theta}$ (3.7)
Since
$\displaystyle\parallel f^{\prime}(u)\parallel\parallel u\parallel$
$\displaystyle\geq\langle f^{\prime}(u),u\rangle$
$\displaystyle=2f(u)-\dfrac{1}{2}\parallel
u\parallel^{2}\int_{0}^{1}\langle\nabla_{u}(u),u\rangle dt$ (3.8)
Then by $(3.7)$ and $(3.8)$, we have
$\parallel
f^{\prime}(u)\parallel\geq[\,2\gamma-N(N-1)c2^{\theta-2}k_{\infty}^{\theta}\,]\parallel
u\parallel^{\theta+1}$ $None$
Since $\parallel g^{\prime}(u)\parallel=\gamma(\theta+2)\parallel
u\parallel^{\theta+1}$, from our choice of $\gamma$, we have
$\beta\parallel f^{\prime}(u)\parallel-\parallel g^{\prime}(u)\parallel\geq
0,\forall u\in\\{f=g\geq\lambda_{0}\\}$ $None$
That is, $(vi)$ holds. ∎
## References
* [1] A.Ambrosetti and V.Coti Zelati, Closed orbits of fixed energy for a class of N-body problems, Ann. Inst. H.Poincare, Analyse Non Lineaire 9(1992), 187-200, Addendum, Ann. Inst. H.Poincare, Analyse Non Lineaire 9(1992), 337-338.
* [2] A.Ambrosetti and V.Coti Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rat. Mech. Anal. 112(1990), 339-362.
* [3] A.Ambrosetti and V.Coti Zelati, Non-collision periodic solutions for a class of symmetric 3-body type problems.Topol. Methods Nonlinear Anal.3(1994),197-207.
* [4] A.Ambrosetti and V.Coti Zelati, Periodic solutions for singualr Lagrangian systems, Springer, 1993.
* [5] A.Ambrosetti and P.H.Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14(1973), 349-381.
* [6] A.Ambrosetti and K.Tanaka and E.Vitillaro, Periodic solutions with prescribed energy for some Keplerian N-body problems, Ann.Inst.H.P.Poincar Anal.Nonlinaire 11(1994),613-632.
* [7] A.Bahri and P.H.Rabinowitz, Periodic solutions of Hamiltonian systems of three body type. Ann. Inst. H. Poincare Anal. Non Lineaire 8(1991), 561-649.
* [8] U.Bessi and V.Coti Zelati, Symmetries and noncollision closed orbits for planar N-body-type problems, Nonlinear Anal. 16(1991), 587-598.
* [9] G.Buttazzo and M.Giaquinta and S.Hildebrandt, One-dimensional variational problems, Oxford University Press, 1998.
* [10] A.Chenciner and R.Montgomery, A remarkable periodic solutions of the three-body problem in the case of equal masses, Annals of Math, 152(2000), 881-901
* [11] V.Coti Zelati, The periodic solutions of N-body type problems, Ann. Inst. H.Poincare Anal. Nonlineaire 7(1990), 477-492.
* [12] M.Degiovanni and F.Giannoni, Dynamical systems with Newtonian type potentials, Ann. Sc. Norm. Sup. Pisa 15(1989), 467-494.
* [13] G.F.Dell’s Antonio, Classical solutions of a perturbed N-body system, In Top. Nonlinear Anal., M.Matzeu etc. ed., Birkhauser, 1997, 1-86.
* [14] E.Fadell and S.Husseini, A note on the categoty of free loop space, Pro.Am.Math.Soc., 107(1989),527- 536.
* [15] W.B.Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99(1977), 961-971.
* [16] W.B.Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204(1975), 113-135.
* [17] Y.M.Long and S.Q.Zhang, Geometric characterizations for variational minimization solutions of the 3-body problems, Act Math. Sinica 16(2000), 579-592.
* [18] P.Majer, Ljusterni-Schnirelman theory with local Palais-Smale condition and singular dynamical systems, Ann. Inst. Henri Poincare Anal. Non Lineaire 8(1991), 459-476.
* [19] P.Majer and S.Terracini, Periodic solutions to some N-body type problems:the fixed energy case. Duke Math.J.69(1993), 683-697.
* [20] R.Palais, The principle of symmetric criticality, Comm. Math. Phys. 69(1979),19-30.
* [21] J.P.Serre, Homologic singular des espaces fibers, Ann.Math. 54(1951),425-505.
* [22] E.Serra and S.Terracini , Collisionless periodic solutions to some 3-body problems, Arch. Rational. Mech. Anal. 120(1992), 305-325.
* [23] S.Terracini, Multiplicity of periodic solution with prescribed energy to singular dynamical systems.Ann.Inst.H.Poincar Anal.Nonlinaire 9(1992),597-641.
* [24] A.Venturelli, Une caracterisation variationnelle des solutions de Lagrange du problem plan des trois corps, C.R. Acad. Sci. Paris 332(2001), 641-644.
* [25] S.Q.Zhang and Q.Zhou, A minimizing property of Lagrangian solutions, Acta Math. Sinica 17(2001), 497-500.
* [26] S.Q.Zhang and Q.Zhou, Variational methods for the choregraphy solution to the three-body problem, Sci.China 45(2002), 594-597.
* [27] S.Q.Zhang and Q.Zhou, Symmetric periodic noncollision solutions for N-Body-type problems, Acta Math.Sinica 11(1995),37-43.
* [28] W.P.Ziemer, Weakly differentiable functions, Springer, 1989.
|
arxiv-papers
| 2012-09-29T12:31:06 |
2024-09-04T02:49:35.768134
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pengfei Yuan and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1210.0098"
}
|
1210.0268
|
# Two Species Evolutionary Game Model of User and Moderator Dynamics
Christopher Griffin and Douglas Mercer Applied Resarch Laboratory and
Department of Mathematics
Penn State University
University Park, PA 16802
E-mail: griffinch@ieee.org James Fan Dept. of Supply Chain and Logistics
Penn State University
University Park, PA 16802
E-mail: juf187@psu.edu Anna Squicciarini College of Info. Sci. and Tech.
Penn State University
University Park, PA 16802
E-mail: asquicciarini@psu.edu
###### Abstract
We construct a two species evolutionary game model of an online society
consisting of ordinary users and behavior enforcers (moderators). Among
themselves, moderators play a coordination game choosing between being
“positive” or ”negative” (or harsh) while ordinary users play prisoner’s
dilemma. When interacting, moderators motivate good behavior (cooperation)
among the users through punitive actions while the moderators themselves are
encouraged or discouraged in their strategic choice by these interactions. We
show the following results: (i) We show that the $\omega$-limit set of the
proposed system is sensitive both to the degree of punishment and the
proportion of moderators in closed form. (ii) We demonstrate that the basin of
attraction for the Pareto optimal strategy
$(\text{Cooperate},\text{Positive})$ can be computed exactly. (iii) We
demonstrate that for certain initial conditions the system is self-regulating.
These results partially explain the stability of many online users communities
such as Reddit. We illustrate our results with examples from this online
system.
## I Introduction
Online social networks (OSNs) allow individuals to share not only information
about themselves, but also information about their world. While traditional
OSNs, such as Facebook, Myspace, and Google+, are designed for sharing
personal information, new OSNs with a focus on sharing all interesting
information, personal or not, have become popular. These OSNs, such as Reddit,
Digg, and Funnyjunk, still allow users to create online connections through
shared interests and experiences. Much like traditional OSNs, these new OSNs
are susceptible to user deception.
Reddit in particular rewards users for posting information with a form of
social capital called karma. Shared information, in the form of posts, is
upvoted or downvoted by the community. The user in turn accumulates karma
based on how popular his or her posts are. While karma has no intrinsic value,
it appears to be a desirable commodity within the Reddit community. Karma is a
unique and concrete measure of social capital, a great tool to analyze online
social behavior. One interesting component of Reddit is the communal agreement
that honesty is highly preferred to dishonesty within the system, should be
rewarded, and moderators actively enforce this policy through a variety of
methods. In this paper, we seek to develop dynamics that lead to the unique
properties found on Reddit: that is a self-stabilizing society in which
moderators (or authority figures) act in as kind a way as possible with users
committed to cooperation through honesty at the potential expense of karma.
### I-A An Overview of Reddit
From the perspective of users, lying without malice does not necessarily
affect the holistic well-being of the Reddit social network. Instead, users
identify “trolls” as problematic. A troll is a member of an online community
whose contributions are intended to enrage or offend as many people as
possible, as significantly as possible. In essence, trolls seek out
opportunities to defend or propose indefensible, reprehensible positions in
order to receive a negative reaction from other community members.
Many Reddit users will ignore trolls. However, in cases where a troll is not
ignored, online altercations may occur leading to communal breakdown within
Reddit. Beyond the use of karmic voting and individual interactions, Reddit
enforces community standards by utilizing two layers of moderators.
Specifically, subreddit111A subreddit is a specific discussion forum within
Reddit specific moderators enforce Reddit and subreddit guidelines by deleting
posts and comments that were not already buried by downvotes. These are the
moderators upon which the paper focuses. If these moderators are not strict
enough, subreddits sympathetic to trolling can harbor and encourage trolling
behavior, disrupting the community.
In this paper we model negative behavior within Reddit (or a similar online
community) as the defect behavior in a classic prisoner’s dilemma. We justify
this assumption by noting that trolling is analogous to defecting, in the
sense that defection is characterized by an action that goes against what is
best for societal wellbeing for selfish reasons. In Section IV, where we
explictly define our user-user payoff matrix, we will detail our justification
for this assumption in further detail.
### I-B Paper Summary
In this paper, we develop a simple evolutionary game that attempts to model
the behavior of Reddit users and moderators and illustrates how the Reddit
equilibrium can be reached; that is, an equilibrium in which most users agree
to cooperatively share information and in which moderators beneficially
interact with the system. We assume that ordinary users interact with each
other playing a prisoner’s dilemma style game, while moderators interact with
each other playing a coordination game. The moderators’ strategy space
consists of the strategies, “Positive” and “Negative,” which attempts to
capture their view of users, in particular when they are engaged in negative
behavior. When a moderator and a user interact, moderators may or may not
derive benefit from the interaction depending on the strategy of the user. A
similar statement holds for the user.
Specifically, given our evolutionary game system discussed in the sequel, we
show the following results: (i) We show that the $\omega$-limit set of the
proposed system is sensitive both to the degree of punishment and the
proportion of moderators in closed form. (ii) We demonstrate that the basin of
attraction for the Pareto optimal strategy
$(\text{Cooperate},\text{Positive})$ can be computed exactly. (ii) We
demonstrate that for certain initial conditions the system is a regulating.
These results partially explain the stability of many online users communities
such as Reddit.
The remainder of this paper is organized as follows: In Section II, we provide
a brief literature review of deviance (or negative behavior) in online social
environments and a review of the impact moderators have in this situation. In
Section III we layout our modeling approach and contrast it to the one in [1].
In Section IV we present our basic model of the system. In Section V we
present our results on the dynamical system under consideration. We present
future directions and conclusions in Section VI.
## II Literature Review
Online deviance (defection, in our model) has been studied from the
perspective of both social science and computer science. To date these two
perspectives have not been adequately integrated. Using classical labeling and
identity theories, several social researchers have focused on Internet users’
behavior [2, 3, 4, 5, 6]. Labeling theory holds that being labeled as a
“deviant” leads a person to engage in deviant behavior, and explains why
people’s behavior clashes with social norms [2]. However, social sciences
research has not yet developed a normative definition of cyber communication
and the online subculture. A significant study of online behavior that will
inform our model is the “Palace Study” in which Suler and Philips [4] classify
deviant behavior (strategies) into several types and provide a taxonomy of
possible counter-strategies, ranging from mild, premeditative actions (e.g.,
warnings) to preventative systems. The study confirms that existing prevention
and remediation of aversive online behavior techniques have been found to be
difficult and expensive. For instance, reputation systems were found to be
useful for this purpose, but are unreliable due to the lack of identity
validation and control [7]. Recent socio-computational studies focus mostly on
single short text analysis to automatically identify spam/deviant comments in
user-contributed sites [8, 9, 10, 11, 12]. While content-based methods have
shown encouraging results, they are limited to single-post analysis, and they
do not target specific users’ behaviors or follow traces.
Several tools exist to help moderators identify bots and vandalism (e.g. [13,
14]). Automated bots (e.g., Cluebot), filters (e.g., abusefilter), and editing
assistants (e.g., Huggle and Twinkle) all aim to locate acts of vandalism.
Such tools work via regular expressions and manually-authored rule sets. In
addition, a notable effort is from West and colleagues [15], who adopted
classifiers to detect vandalism on Wikipedia. At the core of the West’s
solution is a lightweight classifier capable of identifying vandalism. The
classifier exploits temporal and spatial features, extracted from revision
metadata of articles.
Our work also parallels the body of work on free-riding in peer-to-peer
systems [16]. Peer-to-peer systems are designed to allow users to connect with
others and share resources. Similar to online communities, users are free to
access and contribute as much as desired, and few controls are in place. As
for online communities, punishments, although applied, are shown not to be
truly effective, most likely because users can abandon the system. To tackle
these issues, the common solution is to implement incentive-based mechanisms.
Incentives are applied in certain online forums, whereby end users are given
special roles and privileges as a result of their good-standing (see [17] for
a discussion on the community enforcement mechanisms of E-Bay). In this paper,
we assume that users cannot easily abandon the system (i.e., there are few
competitors and a barrier to change, as there is with Facebook) and study the
case when moderators focus on punishments rather than incentives. (See Section
IV.) We discuss how to vary the model to study the incentives based case in
Section VI.
## III Relevant Previous Work in Evolutionary Games
Consider a two-player bimatrix game. That is, the payoff matrix for the row
player is $\mathbf{A}\in\mathbb{R}^{n\times n}$ ($n\in\mathbb{Z}_{+}$) and for
the column player it is $\mathbf{B}$. In an evolutionary game, let
$\boldsymbol{\zeta}(t)\in\mathbb{R}^{n\times 1}$ be a vector whose
$i^{\text{th}}$ component $\boldsymbol{\zeta}_{i}(t)$ yields the proportion of
the population of row players that chooses pure strategy $i$ at time $t$. We
will likewise define $\boldsymbol{\chi}(t)\in\mathbb{R}^{n\times 1}$ for the
column players. Hofbauer [1] proposes the following replicator dynamics for
this case:
$\displaystyle\dot{\boldsymbol{\zeta}}_{i}=\boldsymbol{\zeta}_{i}\left(\left(\mathbf{A}\boldsymbol{\chi}\right)_{i}-\boldsymbol{\zeta}^{T}\mathbf{A}\boldsymbol{\chi}\right)\quad
i=1,\dots,n$ (1)
$\displaystyle\dot{\boldsymbol{\chi}}_{i}=\boldsymbol{\chi}_{j}\left(\left(\boldsymbol{\zeta}^{T}\mathbf{B}\right)_{j}-\boldsymbol{\zeta}^{T}\mathbf{B}\boldsymbol{\chi}\right)\quad
j=1,\dots,n$ (2)
This is a simple generalization of the replicator dynamics from a zero-sum
game to a general sum game.
For games in which a population is to play a symmetric bimatrix game, we
propose the following simplified dynamics. Let
$\boldsymbol{\zeta}(t)\in\mathbb{R}^{n\times n}$ simply be the vector whose
$i^{\text{th}}$ component is the proportion of the population that is playing
pure strategy $i$ at time $t$. Then for an individual playing strategy $i$,
the expected payoff value is nothing more than
$(\mathbf{A}\boldsymbol{\zeta})_{i}$, regardless of whether this individual is
a row or column player since
$\boldsymbol{\zeta}^{T}\mathbf{A^{T}}=\mathbf{A}\boldsymbol{\zeta}$ by
symmetry. Care must be taken, however, when computing the population average.
In this case, the population average is not
$\boldsymbol{\zeta}^{T}\mathbf{A}\boldsymbol{\zeta}$ as it is in the case of
the classical replicator dynamics [18]. Instead, the population average is
given by:
$\bar{u}=\frac{1}{2}\boldsymbol{\zeta}^{T}\left(\mathbf{A}+\mathbf{A}^{T}\right)\boldsymbol{\zeta}$
(3)
To see this, assume that (as expected) half the time a player meets a
competitor she will play the role of the row player and the other half of the
time she will play the roll of the column player. Then the population average
can be computed as:
$\bar{u}=\frac{\boldsymbol{\zeta}^{T}\mathbf{A}\boldsymbol{\zeta}+\boldsymbol{\zeta}^{T}\mathbf{A}^{T}\boldsymbol{\zeta}}{2}$
(4)
which is identical to Equation 3. This leads to a simplified replicator
dynamic in the case of a symmetric game:
$\dot{\boldsymbol{\zeta}}_{i}=\boldsymbol{\zeta}_{i}\left(\left(\mathbf{A}\boldsymbol{\zeta}\right)_{i}-\frac{1}{2}\boldsymbol{\zeta}^{T}\left(\mathbf{A}+\mathbf{A}^{T}\right)\boldsymbol{\zeta}\right)$
(5)
We will use this dynamic for the evolution of strategy within our
subpopulations of ordinary users and moderators while we will use the
formulation of Hofbauer in our inter-population strategy evolution dynamics.
## IV Model
Let $x(t)\in[0,1]$ for all $t\in\mathbb{R}_{+}$ be the proportion of ordinary
users who choose to cooperate (e.g., behave appropriately, act honestly, etc.)
while $y(t)\in[0,1]$ for all $t\in\mathbb{R}_{+}$ is the proportion of users
who choose to defect (e.g., behave negatively, deceive, etc.). Naturally our
dynamics will require $x(t)+y(t)=1$ for all $t\in\mathbb{R}_{+}$. Likewise,
let $z(t)$ be the proportion of moderators who choose to be positive and let
$w(t)$ be the proportion of moderators who choose to be negative so that
$z(t)+w(t)=1$ for all $t\in\mathbb{R}_{+}$ as well.
When interacting, each subpopulation plays a symmetric general sum game:
prisoner’s dilemma ([19], Page 67) and a coordination game respectively. In
Prisoner’s Dilemma, users who cooperate gain a benefit, but not so much as a
user who defects from a cooperating user. Two defecting users reap a smaller
benefit than they would if they cooperate.
For ordinary users, assume they have the following prisoner’s dilemma payoff
matrix:
$\mathbf{A}=\begin{bmatrix}\frac{r}{2}&-r\\\ r&\frac{r}{4}\end{bmatrix}$ (6)
This assumption follows naturally from the following obersations. Consider an
interaction between two OSN users. If both choose to cooperate, we would
expect their interaction to be beneficial to the community as a whole (e.g.,
produce an insightful, genuine conversation). However, if one user chooses to
defect while the other opts to abide by community standards, the defecting
player will derive his satisfaction at the expense of the unsuspecting,
cooperating user (e.g., the troll reaping the reward of anger from the
legitimate user). If both users defect, each will derive some satisfaction
from the experience (e.g., a shared joke between two like-minded defectors),
though not as much as if they had cooperated. Thus, the dominant strategy is,
as in the case of the textbook prisoner’s dilemma problem, to defect, despite
the social optimum strategy being user cooperation.
In a coordination game, users who play the same strategy are rewarded, while
users who do not are penalized. The moderators have the following coordination
game payoff matrix:
$\mathbf{F}=\begin{bmatrix}v&-v\\\ -v&v\end{bmatrix}$ (7)
For simplicity in this paper, we will assume that $r=v=1$ and leave results on
the more general case to subsequent work. When the two populations interact,
they play a bimatrix with payoff matrices given as:
$\displaystyle\mathbf{B}=\begin{bmatrix}\frac{a}{2}&0\\\
-\frac{a}{2}&-a\end{bmatrix}$ (8)
$\displaystyle\mathbf{C}=\begin{bmatrix}\frac{s}{2}&0\\\
\frac{s}{4}&s\end{bmatrix}$ (9)
Here $\mathbf{B}$ is the payoff matrix for ordinary users (as the row players)
and $\mathbf{C}$ is the payoff matrix for moderators (as the column players).
It is easy to see for $a,s>0$ that ordinary users benefit from meeting a
positive moderator when they are cooperating and gain nothing when they meet a
negative moderator. When an ordinary user is defecting he is penalized when he
meets any member of the moderator subpopulation, but more so when he meets a
negative moderator. Likewise, moderators acting positively benefit when they
meet any player, but less so when they meet a defector (presumably it makes
them unhappy to consider a user engaged in negative behavior, but they are
able to moderate behavior, thus “improving society”). Moderators who are
negative derive no pleasure from meeting a cooperating user, but substantial
pleasure from punishing (or expelling) a defecting user. For the sake of
simplicity, we will assume that $s=1$ for the remainder of this paper. While
these are specific payoff matrices, we assert that the qualitative behavior we
observe will be largely the same no matter how we assign values, even in the
presence of a more complex game structure. Essentially, as long as the users
are playing prisoner’s dilemma, the moderators are playing a coordination
game, and there is a penalty when a defector meets a moderator, then the
qualitative behaviors we observe will be present.
Assume that in a population of players a proportion $n_{p}\in(0,1)$ are
ordinary users and $n_{c}\in(0,1)$ are moderators where $n_{p}+n_{c}=1$. Thus,
$n_{p}$ is the proportion of the population that is an ordinary user. Let:
$\boldsymbol{\xi}(t)=\begin{bmatrix}x(t)\\\
y(t)\end{bmatrix}\quad\boldsymbol{\eta}(t)=\begin{bmatrix}z(t)\\\
w(t)\end{bmatrix}$ (10)
Combining the dynamics given in Equation 5 with Hofbauer’s dynamics (Equations
1 and 2), we obtain the following dynamics for the player:
$\dot{\boldsymbol{\xi}}_{i}=\boldsymbol{\xi}_{i}\left(n_{p}\left(\left(\mathbf{A}\boldsymbol{\xi}\right)_{i}-\frac{1}{2}\boldsymbol{\xi}^{T}\left(\mathbf{A}+\mathbf{A}^{T}\right)\boldsymbol{\xi}\right)+\right.\\\
\left.n_{c}\left(\left(\mathbf{B}\boldsymbol{\eta}\right)_{i}-\boldsymbol{\xi}^{T}\mathbf{B}\boldsymbol{\eta}\right)\right)\quad
i=1,2$ (11)
$\dot{\boldsymbol{\eta}}_{j}=\boldsymbol{\eta}_{j}\left(n_{c}\left(\left(\mathbf{F}\boldsymbol{\eta}\right)_{j}-\frac{1}{2}\boldsymbol{\eta}^{T}\left(\mathbf{F}+\mathbf{F}^{T}\right)\boldsymbol{\eta}\right)+\right.\\\
\left.n_{p}\left(\left(\boldsymbol{\xi}^{T}\mathbf{C}\right)_{j}-\boldsymbol{\xi}^{T}\mathbf{C}\boldsymbol{\eta}\right)\right)\quad
j=1,2$ (12)
In the sequel, we will explore the dynamics of stability for varying values of
$n_{p}$ and $a$, the relative punitive value of defecting when playing against
a member of the moderators subpopulation.
We first state a theorem that will simplify our analysis of these dynamics.
Essentially, it simply asserts we can solve these differential equations
completely on the subspace $x(t)+y(t)=1$ and $z(t)+w(t)=1$.
###### Theorem IV.1.
On the subspace defined by the equalities $x(t)+y(t)=1$ and $z(t)+w(t)=1$, the
dynamical system given in Equations 11 and 12 is equivalent to the two-
variable differential system:
$\displaystyle\dot{x}=\frac{1}{4}\,x\left(-1+x\right)\left(-3{\it
n_{p}}\,x+5\,{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a\right)$ (13)
$\displaystyle\dot{z}=-\frac{1}{4}z\left(-1+z\right)\left(-16{\it
n_{p}}\,z+16\,z+5\,{\it n_{p}}-8+5{\it n_{p}}\,x\right)$ (14)
## V Theoretical Results
The following theorem on the equilibria of the dynamical system given by
Equations 11 and 12 is easily verified by substitution. We note there are nine
equilibria that can be identified by finding the roots of the right hand sides
of Equations 13 and 14.
###### Theorem V.1.
For the dynamical system given by Equations 13 and 14, there are always 9
equilibria, (some possibly spurious):
$x=0,z=0$
$x=0,z=1$
$x=1,z=0$
$x=1,z=1$
$x=0,z=\tfrac{1}{16}{\tfrac{5{\it n_{p}}-8}{-1+{\it n_{p}}}}$
$x=\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4{\it n_{p}}\,a-4\,a}{{\it n_{p}}}},z=0$
$x=1,z=\tfrac{1}{8}{\tfrac{5{\it n_{p}}-4}{-1+{\it n_{p}}}}$
$x=\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it n_{p}}}},z=1$
$x=\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it
n_{p}}}},z=\tfrac{1}{12}{\tfrac{10\,{\it n_{p}}-6+5\,{\it
n_{p}}\,a-5\,a}{-1+{\it n_{p}}}}$
It is worthwhile noting that these equilibria may not always be valid for our
equations. It may be that the equilibrium points fall outside the solution
space $x\in[0,1]$ and $z\in[0,1]$. In this case, these stationary points are
spurious. Of interest is the ninth equilibrium point, because it is an
interior equilibrium point:
###### Corollary V.2.
Assume $n_{p}\in(0,1)$, then there is a non-trivial, non-spurious equilibrium
point for which $x,z\in(0,1)$ (and thus $y,w\in(0,1)$) just in case:
$\begin{cases}\frac{1}{2}\,{\frac{{\it n_{p}}}{1-{\it
np}}}<a<\frac{5}{4}\,{\frac{{\it n_{p}}}{1-{\it
n_{p}}}}&\text{if}\,\,n_{p}<\frac{8}{11}\\\ \frac{1}{2}\,{\frac{{\it
n_{p}}}{1-{\it np}}}<a<\frac{2}{5}\,{\frac{3-{\it n_{p}}}{1-{\it
np}}}&\text{if}\,\,\frac{8}{11}\leq n_{p}<\frac{4}{5}\\\
\frac{2}{5}\,{\frac{5\,{\it n_{p}}-3}{1-{\it
np}}}<a<\frac{2}{5}\,{\frac{3-{\it n_{p}}}{1-{\it np}}}&\frac{4}{5}\leq
n_{p}\end{cases}$
A more interesting question revolves around the stability of the various
equilibrium points. It would be nice to know that the presence of the
moderator subpopulation causes the cooperate strategy to become stable within
the ordinary users, even though in general prisoner’s dilemma it is not in any
player’s interest to cooperate. Moreover, we would also like a society in
which the moderators play the positive strategy, since there’s no point in
living in a society where users behave because they are terrified of their
system of justice. (Presumably, the online society would fall apart.) We can
explore this problem by computing the eigenvalues of the Jacobian matrix of
the dynamical system. That is, by studying the characteristics of the non-
linear dynamical system described in Equations 11 and 12 by linearizing about
a stable point of interest. In the following lemma, we linearize about the
utopian equilibrium point $x=1,z=1$ and use the eigenvalues of the Jacobian to
determine when this point is stable.
###### Lemma V.3.
The Jacobian matrix $\mathbf{H}$ of the dynamical system described by
Equations 13 and 14 about $x=1,z=1$ (and $y=0$ and $w=0$) is specified by:
$\mathbf{H}=\left[\begin{array}[]{cc}1/2\,{\it n_{p}}-a+{\it n_{p}}\,a&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&-2+3/2\,{\it
n_{p}}\end{array}\right]$ (15)
with eigenvalues:
$\left[\begin{array}[]{c}-2+3/2\,{\it n_{p}}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr 1/2\,{\it n_{p}}-a+{\it n_{p}}\,a\end{array}\right]$
From the previous lemma and Theorem 3.2 of [20] the following theorem is
immediate:
###### Theorem V.4.
Assume $n_{p}\in(0,1)$. If:
$\frac{n_{p}}{2(1-n_{p})}<a$
then $x=1,z=1$ (and $y=0$ and $w=0$) is a stable equilibrium point.
The resulting relationship between the two variables is illustrated in Figure
1.
Figure 1: The relationship between $n_{p}$ and $a$ when we ensure that
$[x=1,y=0,z=1,w=0]$ is a stable equilibrium point.
This figure makes a great deal of sense. As the proportion of the population
becomes overwhelmingly dominated by ordinary users, the probability of
encountering a member of the moderators subpopulation drops. Therefore, to
ensure proper behavior, stricter and stricter punitive action is required.
By a similar process, we can also explore the case of the dystopian society in
which $[x=0,y=1,z=0,w=1]$ is stable.
###### Theorem V.5.
Assume $n_{p}\in(0,1)$. If:
$\frac{5n_{p}}{4(1-n_{p})}>a$
then $x=0,z=0$ (and $y=1$ and $w=1$) is a stable equilibrium point.
###### Corollary V.6.
There is at least one pair of values for $a$ and $n_{p}$ so that both
$[x=0,y=1,z=0,w=1]$ and $[x=1,y=0,z=1,w=0]$ are stable.
Corollary V.6 tells us that in our online system, it is possible to “descend
into chaos” in the sense that all users are actively defecting (deceiving,
scamming etc.) and all moderators are engaged in highly punitive activities.
Corollary V.6 also suggests that an investigation of the basins of attraction
for the two attracting points could lead to a complete characterization of the
behavior of the dynamical system in light of the following theorem, which
follows from Theorem 3.1 of [20]:
###### Theorem V.7.
Under no conditions is the interior equilibrium point
$\displaystyle x=\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it
n_{p}}}}$ $\displaystyle z=\tfrac{1}{12}{\tfrac{10\,{\it n_{p}}-6+5\,{\it
n_{p}}\,a-5\,a}{-1+{\it n_{p}}}}$
ever stable.
###### Proof.
Analysis of Jacobian matrix and eigenvalues shows that for this point to be
stable, we must have:
$\displaystyle a<\frac{1}{2}\,{\frac{{\it n_{p}}}{1-{\it n_{p}}}}\text{ or
}\frac{5}{4}\,{\frac{{\it n_{p}}}{1-{\it n_{p}}}}<a\text{ and }$
$\displaystyle a<\frac{2}{5}\,{\frac{5\,{\it n_{p}}-3}{1-{\it n_{p}}}}\text{
or }\frac{2}{5}\,{\frac{3-{\it n_{p}}}{1-{\it n_{p}}}}<a$
From Theorem V.2, for the equilibrium to be non-spurious (i.e., in
$[0,1]\times[0,1]$) it must be the case that:
$a<\frac{1}{2}\,{\frac{{\it n_{p}}}{1-{\it n_{p}}}}\text{ and
}\frac{2}{5}\,{\frac{3-{\it n_{p}}}{1-{\it n_{p}}}}<a$
If these intervals overlap, then there is a point at which:
$\frac{n_{p}}{2}=\frac{2}{5}(3-n_{p})$
which only occurs if $n_{p}=\tfrac{4}{3}$, but we know $n_{p}\in(0,1)$. Thus,
the interior equilibrium is always unstable. ∎
What the preceding theorem means is that for many online systems that obey the
dynamics discussed will either converge to a final state in which all users
are behaving cooperatively and moderators who are positive or it will descend
into chaos. This is illustrated in the figures below and proved explicitly for
certain $n_{p}$.
To illustrate the nature of the equilibria, we consider the case when
$n_{p}=0.9$ and $a=7$. (In this case, 90% of the population is composed of
ordinary users.) From Theorem V.1, we can see that the non-trivial interior
point equilibrium is present with values: $x=\tfrac{17}{27}$ and
$z=\tfrac{5}{12}$ as are 6 other equilibrium points. The system phase portrait
is shown in Figure 2:
Figure 2: Phase portrait of the dynamical system when $a=7$ and $n_{p}=0.9$
In a case like this, we can see that the space half-space $x<\tfrac{17}{27}$,
$z\in(0,1)$ is the basin of attraction for the point for $x=0,z=0$ (except for
a set of measure zero) while the half-space $x>\tfrac{17}{27}$, $z\in(0,1)$ is
the basin of attraction for $x=1,z=1$ (again except for a set of measure
zero).
The behavior of the population varies substantially with the value of $a$ and
the previously illustrated behavior is not the only possible outcome for this
online society. An interesting situation arises when we set $n_{p}=0.9$ and
$a=12$. In this case, there is no interior equilibrium point and the basin of
attraction for $x=1,z=1$ is almost the entire region $[0,1]\times[0,1]$. This
is shown in Figure 3.
Figure 3: Phase portrait of the dynamical system when $a=12$ and $n_{p}=0.9$
What’s interesting about this case is the self-regulating nature of the
system. Note the trajectory beginning at $x=\tfrac{3}{100}$ and
$z=\tfrac{99}{100}$. We see that the moderators, while starting with the
positive strategy, quickly change to the negative strategy, which drives the
ordinary users to move from the defect strategy to the cooperate strategy.
This, in turn, drives the moderators to move from the negative strategy to the
positive strategy, arriving in the utopian scenario. We now provide a result
on the basin of attraction in scenarios like the previous example. This result
shows that the behavior illustrated in Figure 2 is somewhat typical of this
system and describes how one online site can become successful and (mostly)
stable like Reddit, while other sites might descend into chaos and fail.
Suppose that there is a non-trivial equilibrium solution for which
$x^{*}\in(0,1)$, that is:
$x^{*}=\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it n_{p}}}}$
(16)
Consider $x^{*}-\epsilon$ where $\epsilon\in(0,x^{*})$. If we evaluate
$\dot{x}=\frac{1}{4}\,x\left(-1+x\right)\left(-3{\it n_{p}}\,x+5\,{\it
n_{p}}+4\,{\it n_{p}}\,a-4\,a\right)$
at $x^{*}-\epsilon$, then we obtain:
$\frac{1}{12}\frac{r\epsilon s}{n_{p}}$ (17)
where
$\displaystyle r=\left(5\,{\it n_{p}}-4\,a+4\,{\it n_{p}}\,a-3\,\epsilon\,{\it
n_{p}}\right)$ $\displaystyle s=\left(4\,{\it n_{p}}\,a-4\,a+2\,{\it
n_{p}}-3\,\epsilon\,{\it n_{p}}\right)$
Expression 17 is cubic in $\epsilon$ and it has three roots:
$\left\\{0,\frac{2}{3}\,{\frac{2\,{\it n_{p}}\,a-2\,a+{\it n_{p}}}{{\it
n_{p}}}},\frac{1}{3}\,{\frac{5\,{\it n_{p}}-4\,a+4\,{\it n_{p}}\,a}{{\it
n_{p}}}}\right\\}$
We can see at once that:
$\frac{1}{3}\,{\frac{5\,{\it n_{p}}-4\,a+4\,{\it n_{p}}\,a}{{\it
n_{p}}}}-\frac{2}{3}\,{\frac{2\,{\it n_{p}}\,a-2\,a+{\it n_{p}}}{{\it
n_{p}}}}=1$ (18)
and, by our assumption in Equation 16, the Expression 17 is either always
positive or always negative on the interval:
$\left[0,\frac{1}{3}\,{\frac{5\,{\it n_{p}}-4\,a+4\,{\it n_{p}}\,a}{{\it
n_{p}}}}\right]$
since the right endpoint of this interval is positive by assumption. To
determine the sign of the function, we can can compute the critical points of
the derivative of the cubic equation as:
$\frac{\left({\frac{7}{9}}\,{\it n_{p}}+{\frac{8}{9}}\,{\it
n_{p}}\,a-{\frac{8}{9}}\,a\pm 1/9\,\sqrt{v}\right)}{{{\it n_{p}}}}$ (19)
where:
$v=19\,{{\it n_{p}}}^{2}+28\,a{{\it n_{p}}}^{2}-28\,{\it n_{p}}\,a+16\,{{\it
n_{p}}}^{2}{a}^{2}-\\\ 32\,{\it n_{p}}\,{a}^{2}+16\,{a}^{2}$
The positive root is clear and the second derivative of the cubic equation in
$\epsilon$ evaluated at this root is:
$6\,{\it np}\,\sqrt{v}>0$ (20)
meaning that the positive root corresponds to a minimum and thus, for all
appropriately chosen values of $n_{p}$ and $a$, we know that $\dot{x}<0$ when
$\epsilon>0$ and $x$ must decrease toward $0$. By a similar argument, we can
show that if $\epsilon<0$, then $\dot{x}>0$ and $x$ must increase toward $1$.
Thus we have proved:
###### Lemma V.8.
Assume a non-spurious, non-trivial interior equilibrium exists in the game;
i.e., the ninth equilibrium point from Theorem V.1 is contained in
$(0,1)\times(0,1)$. If:
$x(0)<\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it n_{p}}}}$
(21)
then $\lim_{t\rightarrow\infty}x(t)=0$. Otherwise, if $x(0)$ is greater than
this value, $\lim_{t\rightarrow\infty}x(t)=1$.
To complete the characterization of the limiting behavior of the differential
equation, we analyze the eigenvalues of the Jacobian matrix at $x=0,z=1$ and
$x=1,z=0$. Our last lemma follows from Theorem 3.1 of [20]:
###### Lemma V.9.
The point $x=0,z=1$ is stable if and only if:
$n_{p}<\frac{8}{11}\quad\text{and}\quad a<\frac{5}{4}\,{\frac{{\it
n_{p}}}{1-{\it n_{p}}}}$ (22)
Furthermore, the point $x=1,z=0$ is stable if and only if:
$n_{p}<\frac{4}{5}\quad\text{and}\quad\frac{1}{2}\,{\frac{{\it n_{p}}}{-1+{\it
n_{p}}}}<a$ (23)
From these lemmas and Corollary V.2, we have the following theorem:
###### Theorem V.10.
Suppose $n_{p}>\tfrac{4}{5}$ and
$\frac{2}{5}\,{\frac{5\,{\it n_{p}}-3}{1-{\it
np}}}<a<\frac{2}{5}\,{\frac{3-{\it n_{p}}}{1-{\it np}}}$
The basin of attraction for $x=1,z=1$ is the set of $(x,z)$ pairs so that:
$\displaystyle\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a}{{\it
n_{p}}}}<x\leq 1$ $\displaystyle 0<z<1$
and the basin of attraction for $x=0,z=0$ is:
$\displaystyle 0\leq<x<\tfrac{1}{3}{\tfrac{5{\it n_{p}}+4\,{\it
n_{p}}\,a-4\,a}{{\it n_{p}}}}$ $\displaystyle 0<z<1$
Theorem V.10 is illustrated in Figure 2. The more complex behaviors this
system is able to exhibit yield more complex basins of attraction. However,
since we anticipate $n_{p}>0.9$, we have focused on this case explicitly in
Theorem V.10.
This tells us that under certain conditions the final state of an (online)
society (governed by these simple dynamics) can depend substantially on the
initial conditions of the system. That is, if Reddit had been governed in this
fashion, but the initial user group was slightly less interested in the
posting of honest information (but the moderators were), then Reddit could
have easily descended into a more chaotic state.
## VI Future Directions
We discuss three future directions for research: an incentives based model, an
optimal control model in which the penalty is assigned dynamically and a model
in which $n_{p}$ is not fixed and determined by epidemic dynamics.
### VI-A Incentives Based Model
As noted, in the literature review, for online communities, punishments,
although applied, are shown not to be truly effective [21]. In this instance,
incentives are applied to encourage good behavior. We can modify the payoff
matrix $\mathbf{B}$ as:
$\mathbf{B}=\begin{bmatrix}a&\frac{a}{2}\\\ 0&-1\end{bmatrix}$ (24)
In this scenario, we can vary $a$ to adjust the payoff received by a user
engaged in cooperative behavior. The following result is immediate:
###### Theorem VI.1.
When $\mathbf{B}$ is given by Equation 24 and the remaining payoff matrices
are held constant, the new set of differential equations has four equilibria:
$x=0$, $z=0$, $x=1$, $z=1$, $x=1$, $z=0$ and $x=0$, $z=1$. Furthermore, $x=1$,
$z=1$ is stable just in case:
$\frac{1}{2}\frac{n_{p}}{1-n_{p}}<a$ (25)
and $x=0$,$z=0$ is stable just in case:
$a<\frac{1}{2}\frac{9n_{p}-4}{1-n_{p}}$ (26)
Example dynamics for the incentivizing game are shown in Figure 4
Figure 4: Dynamics for the incentives game in which $a=15$ and $n_{p}=0.9$.
Under these dynamics, the basin of attraction for the utopian solution
$(x=1,z=1)$ is difficult to identify in closed form, however you will observe
he have dynamics similar to those shown in Figure 3. Additional work on this
problem might yield interesting conditions on the incentive structures for
encouraging stable and beneficial social networks.
### VI-B An Optimal Control Problem
In most Social Networking sites, the number of moderators is static (that is,
both $n_{p}$ and $n_{c}$ are fixed). For the remainder of this discussion, we
will assume our original dynamics, rather than the incentivizing behavior
describe above. If we can measure $x(0)$ (and $y(0)$), we would like to
identify a time varying optimal value for $a$ (the penalty) so that
$x^{*}=1,z^{*}=1$ is an attractor that is reachable from $x(0)$. However, an
$a$ that is too large may cripple the social network (in a way not captured by
the dynamics in this paper). We can phrase this problem as a finite (or
infinite) time horizon optimal control problem:
$\displaystyle\min$
$\displaystyle\int_{0}^{T}(1-x(t))^{2}+(1-z(t))^{2}+a(t)^{2}dt$ (27)
$\displaystyle s.t.$
$\displaystyle\dot{x}=\frac{1}{4}\,x\left(-1+x\right)\left(-3{\it
n_{p}}\,x+5\,{\it n_{p}}+4\,{\it n_{p}}\,a-4\,a\right)$
$\displaystyle\dot{z}=-\frac{1}{4}z\left(-1+z\right)\left(-16{\it
n_{p}}\,z+16\,z+5\,{\it n_{p}}-8+5{\it n_{p}}\,x\right)$ $\displaystyle
x(0)=x_{0},z(0)=z_{0}$ $\displaystyle a(t)\in[0,\infty)$
This problem will have a Hamiltonian [22] that is quadratic in $a$ and thus
may admit a non bang-bang solution. Study of this problem is reserved for
future work.
### VI-C Time Varying Population Proportions
As a second generalization of this problem, consider the case where $n_{p}$ is
not static. We can model this scenario using epidemic dynamics in which users
become moderators in a manner consistent with an infection:
$\displaystyle\dot{n}_{p}=\lambda+\rho n_{c}-\beta n_{p}n_{c}-\mu n_{p}$
$\displaystyle\dot{n}_{c}=\beta n_{p}n_{c}-\rho n_{c}-\mu n_{c}$
If we assume a stable population, then $\lambda=\mu$. Here $\beta$ is the
infection rate, while $\rho$ is a recovery rate that leads back to a
susceptible state. Using these differential equations with the equations from
(14 \- 14) yields a more realistic dynamic. We can also define a more complex
optimal control problem in which we attempt to find values for $\beta$ and $a$
that minimize the objective functional of Expression 27.
## VII Conclusions
In this paper we constructed a simple model of a self-regulating system that
describes (to some degree) the behavior of a moderated online community. We
showed that we need only two differential equations to describe a complex two-
population evolutionary game, rather than the six that would be used following
the analytical techniques described in [1]. We completely described the nature
of the evolutionary dynamics of the proposed system and illustrated how the
level of incentive (or penalty) associated with meeting a moderator can effect
the limiting behavior of the system. These results are related to the online
system Reddit. We also discuss future work in which the population structure
is varied and we posed an optimal control problem that is relevant to the
mechanism design for social networks.
## References
* [1] J. Hofbauer, “Evolutionary dynamics for bimatrix games: A Hamiltonian system?” _J. Math. Bio_ , vol. 34, pp. 675–688, 1996.
* [2] N. J. Davis, “Labeling theory in deviance research. a critique and reconsideration,” _The Sociological Quarterly_ , vol. 13, no. 4, pp. 447–474, 1972.
* [3] P. Shaver, J. Schwartz, D. Kirson, and C. O’Connor, “Emotion knowledge further exploration of a prototype approach,” _Journal of Personality and Social Psychology_ , vol. 52, no. 6, 1987.
* [4] J. R. Suler and W. L. Phillips, “The bad boys of cyberspace: Deviant behavior in a multimedia chat community,” _CyberPsychology & Behavior_, vol. 1, pp. 275–294, 1998.
* [5] A. Bruckman, C. Danis, C. Lampe, J. Sternberg, and C. Waldron, “Managing deviant behavior in online communities,” in _CHI Extended Abstracts_ , 2006, pp. 21–24.
* [6] J. Denegri-Knott and J. Taylor, “The labeling game: a conceptual exploration of deviance on the internet,” _Soc. Sci. Comput. Rev._ , vol. 23, no. 1, pp. 93–107, Mar. 2005. [Online]. Available: http://dx.doi.org/10.1177/0894439304271541
* [7] S. David and T. Pinch, “Six degrees of reputation: The use and abuse of online review and recommendation systems,” _First Monday_ , vol. 11, no. 3, 2006\.
* [8] A. H. Wang, “Detecting spam bots in online social networking sites: a machine learning approach,” in _Proceedings of the 24th annual IFIP WG 11.3 working conference on Data and applications security and privacy_ , ser. DBSec’10. Berlin, Heidelberg: Springer-Verlag, 2010, pp. 335–342.
* [9] A. Sureka, “Mining user comment activity for detecting forum spammers in youtube,” _CoRR_ , vol. abs/1103.5044, 2011.
* [10] T. Moh and A. Murmann, “Can you judge a man by his friends?-enhancing spammer detection on the twitter microblogging platform using friends and followers,” _Information Systems, Technology and Management_ , pp. 210–220, 2010.
* [11] A. Antonucci and C. de Campos, “Decision making by credal nets,” in _Intelligent Human-Machine Systems and Cybernetics (IHMSC), 2011 International Conference on_ , vol. 1, aug. 2011, pp. 201 –204.
* [12] K. Lee, J. Caverlee, and S. Webb, “Uncovering social spammers: social honeypots + machine learning,” in _Proceedings of the 33rd international ACM SIGIR conference on Research and development in information retrieval_ , ser. SIGIR ’10. New York, NY, USA: ACM, 2010, pp. 435–442. [Online]. Available: http://doi.acm.org/10.1145/1835449.1835522
* [13] Fassim, “Fassim: a forum spam prevention plugin,” http://www.fassim.com/about/.
* [14] S. F. SPam, 2012, http://www.stopforumspam.com.
* [15] A. G. West, S. Kannan, and I. Lee, “Stiki: an anti-vandalism tool for wikipedia using spatio-temporal analysis of revision metadata,” in _6th International Symposium on Wikis and Open Collaboration_ , ser. WikiSym ’10. New York, NY, USA: ACM, 2010, pp. 32:1–32:2. [Online]. Available: http://doi.acm.org/10.1145/1832772.1832814
* [16] M. Feldman, “Free-riding and whitewashing in peer-to-peer systems,” _IEEE Journal on Selected Areas in Communication_ , vol. 24, no. 5, pp. 1010 – 1019, 2006.
* [17] J. Boyd, “In community we trust: Online security communication at ebay,” _Journal of Computer-Mediated Communication_ , vol. 7, no. 3, pp. 0–0, 2002\. [Online]. Available: http://dx.doi.org/10.1111/j.1083-6101.2002.tb00147.x
* [18] J. W. Weibull, _Evolutionary Game Theory_. MIT Press, 1997.
* [19] C. Griffin, “Graph theory: Penn state math 485 lecture notes (v 0.9.8),” http://www.personal.psu.edu/cxg286/Math485.pdf, 2011-2012.
* [20] F. Verhulst, _Nonlinear Differential Equations and Dynamical Systems_ , 2nd ed. Springer, 2006.
* [21] G. P. Anna Squicciarini, William Mcgill and S. Huang, “Early detection of policies violations in a social media site: A bayesian belief network approach,” in _IEEE Symposium on Policies for Distributed Systems & Networks_, 2012.
* [22] D. E. Kirk, _Optimal Control Theory: An Introduction_. Dover Press, 2004.
|
arxiv-papers
| 2012-10-01T00:46:29 |
2024-09-04T02:49:35.779734
|
{
"license": "Public Domain",
"authors": "Christopher Griffin and Douglas Mercer and James Fan and Anna\n Squicciarini",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1210.0268"
}
|
1210.0433
|
# A geometric characterization of
invertible quantum measurement maps
Kan He Faculty of Mathematics, Institute of Mathematics, Taiyuan University
of Technology, Taiyuan, 030024, P. R. China kanhemath@yahoo.com.cn , Jin-
Chuan Hou Faculty of Mathematics, Institute of Mathematics, Taiyuan
University of Technology, Taiyuan, 030024, P. R. China
jinchuanhou@yahoo.com.cn and Chi-Kwong Li Department of Mathematics, College
of William Mary, Williamsburg, VA 23187-8795, USA; Faculty of Mathematics,
Institute of Mathematics, Taiyuan University of Technology, Taiyuan, 030024,
P. R. China ckli@math.wm.edu
###### Abstract.
A geometric characterization is given for invertible quantum measurement maps.
Denote by ${\mathcal{S}}(H)$ the convex set of all states (i.e., trace-1
positive operators) on Hilbert space $H$ with dim$H\leq\infty$, and
$[\rho_{1},\rho_{2}]$ the line segment joining two elements
$\rho_{1},\rho_{2}$ in ${\mathcal{S}}(H)$. It is shown that a bijective map
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ satisfies
$\phi([\rho_{1},\rho_{2}])\subseteq[\phi(\rho_{1}),\phi(\rho_{2})]$ for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}$ if and only if $\phi$ has one of the
following forms
$\rho\mapsto\frac{M\rho M^{*}}{{\rm tr}(M\rho M^{*})}\quad\hbox{ or
}\quad\rho\mapsto\frac{M\rho^{T}M^{*}}{{\rm tr}(M\rho^{T}M^{*})},$
where $M$ is an invertible bounded linear operator and $\rho^{T}$ is the
transpose of $\rho$ with respect to an arbitrarily fixed orthonormal basis.
2002 Mathematical Subject Classification. 47B49, 47L07, 47N50
Key words and phrases. Quantum states, Quantum measurement, Segment preserving
maps
This work is partially supported by National Natural Science Foundation of
China (11171249, 11271217, 11201329), a grant to International Cooperating
Research from Shanxi (2011081039). Li was also supported by a USA NSF grant
and a HK RCG grant.
## 1\. Introduction and the main result
In the mathematical framework of the theory of quantum information, a state is
a positive operator of trace 1 acting on a complex Hilbert space $H$. Denote
by ${\mathcal{S}}(H)$ the set of all states on $H$, that is, of all positive
operators with trace 1. It is clear that ${\mathcal{S}}(H)$ is a closed convex
subset of ${\mathcal{T}}(H)$, the Banach space of all trace-class operators on
$H$ endowed with the trace-norm $\|\cdot\|_{\rm Tr}$. In quantum information
science and quantum computing, it is important to understand, characterize,
and construct different classes of maps on states. For instance, all quantum
channels and quantum operations are completely positive linear maps; in
quantum error correction, one has to construct the recovery map for a given
channel; to study the entanglement of states, one constructs entanglement
witnesses, which are special types of positive maps; see [11]. In this
connection, it is helpful to know the characterizations of maps leaving
invariant some important subsets or quantum properties. Such questions have
attracted the attention of many researchers; for example, see [1, 2, 4, 6, 8,
9, 10].
In this paper, we characterize invertible maps
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ that satisfies
$\phi([\rho_{1},\rho_{2}])\subseteq[\phi(\rho_{1}),\phi(\rho_{2})]\quad\hbox{
for any }\rho_{1},\rho_{2}\in{\mathcal{S}}(H),$
where $[\rho_{1},\rho_{2}]=\\{t\rho_{1}+(1-t)\rho_{2}:t\in[0,1]\\}$ denotes
the closed line segment joining two states $\rho_{1},\rho_{2}$. In other
words, we characterize maps on states such that for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$ and $0\leq t\leq 1$, there is some $s$
with $0\leq s\leq 1$ such that
$\phi(t\rho_{1}+(1-t)\rho_{2})=s\phi(\rho_{1})+(1-s)\phi(\rho_{2}).$
This question is motivated by the study of affine isomorphisms on
${\mathcal{S}}(H)$; see [2]. Recall that an affine isomorphism on
${\mathcal{S}}(H)$ is a bijective map
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ satisfying
$\phi(t\rho_{1}+(1-t)\rho_{2})=t\phi(\rho_{1})+(1-t)\phi(\rho_{2})\quad\hbox{
for all }t\in[0,1]\hbox{ and }\rho_{1},\rho_{2}\in{\mathcal{S}}(H).$
Evidently, we have the implications (c) $\Rightarrow$ (b) $\Rightarrow$ (a)
for a bijective map $\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ for the
following conditions.
* (a)
$\phi([\rho_{1},\rho_{2}])\subseteq[\phi(\rho_{1}),\phi(\rho_{2})]$ for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$.
* (b)
$\phi([\rho_{1},\rho_{2}])=[\phi(\rho_{1}),\phi(\rho_{2})]$ for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$.
* (c)
$\phi$ is an affine isomorphism.
It was shown in [2] that an affine isomorphism
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ has the form
$None$ $\rho\mapsto U\rho U^{*}\quad\hbox{ or }\quad\rho\mapsto
U\rho^{T}U^{*},$
where $U$ is a unitary operator and $\rho^{T}$ is the transpose of $\rho$ with
respect to a certain orthonormal basis for $H$. Note that unitary similarity
transforms correspond to evolutions of quantum systems, and many maps that
leave invariant subsets or quantum properties of the states have the form
described in (1.1). One may be tempted to conjecture that maps on states
satisfying (a) or (b) above also have the forms described in (1.1). However,
this is not true as shown by our results. It turns out that the maps
satisfying condition (a) and (b) are closely related to quantum measurements.
Recall that in quantum mechanics a fine-grained quantum measurement is
described by a collection $\\{M_{m}\\}$ of measurement operators acting on the
state space $H$ satisfying $\sum_{m}M_{m}^{*}M_{m}=I$. Let $M_{j}$ be a
measurement operator. If the state of the quantum system is
$\rho\in{\mathcal{S}}(H)$ before the measurement, then the state after the
measurement is $\frac{M_{j}\rho M_{j}^{*}}{{\rm tr}(M_{j}\rho M_{j}^{*})}$
whenever $M_{j}\rho M_{j}^{*}\not=0$. If $M_{j}$ is fixed, we get a
measurement map $\phi_{j}$ defined by $\phi_{j}(\rho)=\frac{M_{j}\rho
M_{j}^{*}}{{\rm tr}(M_{j}\rho M_{j}^{*})}$ from the convex subset
${\mathcal{S}}_{M}(H)=\\{\rho:M_{j}\rho M_{j}^{*}\not=0\\}$ of the (convex)
set ${\mathcal{S}}(H)$ of states into ${\mathcal{S}}(H)$. If $M_{j}$ is
invertible, then $\phi_{j}:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ is
bijective and will be called an invertible measurement map. Observe that a
measurement map $\phi_{j}$ satisfies (a), (b), and is not of the standard form
(1.1) in general.
In this paper, we show that, up to the transpose, bijective maps on states
satisfying (a) or (b) are precisely invertible measurement maps. The following
is our main result.
Theorem 1. Let ${\mathcal{S}}(H)$ be the convex set of all states on Hilbert
space $H$ with $2\leq\dim H\leq\infty$. The following statements are
equivalent for a bijective map
$\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$.
* (a)
$\phi([\rho_{1},\rho_{2}])\subseteq[\phi(\rho_{1}),\phi(\rho_{2})]$ for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$.
* (b)
$\phi([\rho_{1},\rho_{2}])=[\phi(\rho_{1}),\phi(\rho_{2})]$ for any
$\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$.
* (c)
There is an invertible bounded linear operator $M\in{\mathcal{B}}(H)$ such
that $\phi$ has the form
$\rho\mapsto\frac{M\rho M^{*}}{{\rm tr}(M\rho M)}\quad{\rm
or}\quad\rho\mapsto\frac{M\rho^{T}M^{*}}{{\rm tr}(M\rho^{T}M^{*})},$
where $\rho^{T}$ is the transpose of $\rho$ with respect to an orthonormal
basis.
It is interesting to note that condition (a) is much weaker than condition
(b). For example, condition (a) does not even ensure that
$\phi([\rho_{1},\rho_{2}])$ is a convex (connected) subset of
$[\phi(\rho_{1}),\phi(\rho_{2})]$. It turns out that the two conditions (a)
and (b) are equivalent for a bijective map, and the map must be a measurement
map or the composition of the transpose map with a measurement map.
The proof of Theorem 1 is done in the next few sections. In Section 2, we will
establish the equivalence of (a) and (b) using a result of P$\breve{a}$les
[12]. Then we verify the equivalence of (b) and (c). We treat the finite
dimensional case in Section 3. Using the result in Section 3, we complete the
proof for the infinite dimensional case in Section 4.
## 2\. The equivalence of the first two conditions
The implication of (b) $\Rightarrow$ (a) is clear. We consider the implication
(a)$\Rightarrow$(b).
Assume (a) holds. We will prove that
$\phi([\rho,\sigma])=[\phi(\rho),\phi(\sigma)]$ for any quantum states
$\rho,\sigma$. If $\rho=\sigma$, it is trivial. Suppose $\rho\neq\sigma$.
Note that $\rho,\sigma\in{\mathcal{S}}(H)$ are linearly dependent if and only
if $\rho=\sigma$. So, if $\rho,\sigma$ are linearly independent, then
$\phi(\rho),\phi(\sigma)$ are linearly independent as
$\phi(\rho)\neq\phi(\sigma)$ by the injectivity of $\phi$. Let
${\mathcal{HT}}(H)$ be the real linear space of all self-adjoint trace-class
operators on $H$. As $\phi$ is injective, we must have
$\phi(]\rho,\sigma[)\subset]\phi(\rho),\psi(\sigma)[$ for any
$\rho,\sigma\in{\mathcal{S}}(H)$, where
$]\rho,\sigma_{2}[=[\rho,\sigma]\setminus\\{\rho,\sigma\\}$ is the open line
segment joining $\rho,\sigma$. So by P$\breve{a}$les’ result [12, Theorem 2],
there exists a real linear map
$\psi:{\mathcal{HT}}(H)\rightarrow{\mathcal{HT}}(H)$, a real linear functional
$f:{\mathcal{HT}}(H)\rightarrow{\mathbb{R}}$, an operator
$B\in{\mathcal{H}T}(H)$ and a real number $c$ such that
$None$ $\phi(\rho)=\frac{\psi(\rho)+B}{f(\rho)+c}\quad\mbox{\rm and}\quad
f(\rho)+c>0$
hold for all $\rho\in{\mathcal{S}}(H)$. Thus, for any
$\rho,\sigma\in{\mathcal{S}}(H)$ with $\rho\not=\sigma$ and any $t\in[0,1]$,
there exists $s\in[0,1]$ such that
$\phi(t\rho+(1-t)\sigma)=s\phi(\rho)+(1-s)\phi(\sigma)=s\frac{\psi(\rho)+B}{f(\rho)+c}+(1-s)\frac{\psi(\sigma)+B}{f(\sigma)+c}.$
On the other hand, by the linearity of $\psi$ and $f$, we have
$\displaystyle\phi(t\rho+(1-t)\sigma)$ $\displaystyle=$
$\displaystyle\frac{\psi(t\rho+(1-t)\sigma)+B}{f(t\rho+(1-t)\sigma)+c}$
$\displaystyle=$ $\displaystyle
t\frac{\psi(\rho)+B}{f(t\rho+(1-t)\sigma)+c}+(1-t)\frac{\psi(\sigma)+B}{f(t\rho+(1-t)\sigma)+c}.$
Write $\lambda_{t,\rho,\sigma}=f(t\rho+(1-t)\sigma)+c$, we get
$(\frac{s}{f(\rho)+c}-\frac{t}{\lambda_{t,\rho,\sigma}})(\psi(\rho)+B)+(\frac{1-s}{f(\sigma)+c}-\frac{1-t}{\lambda_{t,\rho,\sigma}})(\psi(\sigma)+B)=0.$
As $\rho\neq\sigma$, $\phi(\rho)$ and $\phi(\sigma)$ are linearly independent.
This implies that $\psi(\rho)+B$ and $\psi(\sigma)+B$ are linearly
independent, too. It follows that
$\frac{t}{f(t\rho+(1-t)\sigma)+c}=\frac{s}{f(\rho)+c}\ {\rm and}\
\frac{1-t}{f(t\rho+(1-t)\sigma)}=\frac{1-s}{f(\sigma)+c}.$
Clearly, $s$ is continuously dependent of $t$ such that $\lim_{t\rightarrow
0}s=0$ and $\lim_{t\rightarrow 1}s=1$. Hence we must have
$\phi([\rho,\sigma])=[\phi(\rho),\phi(\sigma)]$. Thus, condition (b) holds. ∎
Denote by ${\mathcal{P}ur}(H)=\\{x\otimes x:x\in H,\|x\|=1\\}$ the set of pure
states in ${\mathcal{S}}(H)$. The following lemma is useful for our future
discussion.
Lemma 2.1. If condition (b) of Theorem 1 holds, then $\phi$ preserves pure
states in both directions, that is,
$\phi({\mathcal{P}ur}(H))={\mathcal{P}ur}(H)$ .
Proof It is clear that ${\mathcal{S}}(H)$ is a convex set and its extreme
point set is the set ${\mathcal{P}ur}(H)$ of all pure states (rank-1
projections). For any $P\in{\mathcal{P}ur}(H)$, if
$\phi^{-1}(P)\not\in{\mathcal{P}ur}(H)$, then there are two states
$Q,R\in{\mathcal{S}}(H)$ such that $Q\not=R$ and $\phi^{-1}(P)=tQ+(1-t)R$. As
$\phi([\rho,\sigma])\subseteq[\phi(\rho),\phi(\sigma)]$ for any $\rho,\sigma$,
there is some $s\in[0,1]$ such that
$P=\phi(\phi^{-1}(P))=\Phi(tQ+(1-t)R)=s\phi(Q)+(1-s)\phi(R)$. Since
$\phi(Q)\neq\phi(R)$, this contradicts the fact that $P$ is extreme point. So
$\phi^{-1}$ sends pure states to pure states. Similarly, since
$\phi([\rho,\sigma])\supseteq[\phi(\rho),\phi(\sigma)]$ for any states
$\rho,\sigma$, one can show that $\phi$ maps pure states into pure states. ∎
## 3\. Proof of Theorem 1: finite dimensional case
In this section we assume that $\dim H=n<\infty$. In such a case, we may
regard ${\mathcal{HT}}(H)$ the same as ${\bf H}_{n}$, the real linear space of
$n\times n$ Hermitian matrices. Since the implication (c) $\Rightarrow$ (b),
we needs only prove the implication (b) $\Rightarrow$ (c). We divide the proof
of this implication into several assertions. Assume (b) holds.
Assertion 3.1. $\phi(\frac{I}{n})$ is invertible.
Let $\phi(\frac{I}{n})=T$. In order to prove $T$ is invertible, we show that
$\phi$ maps invertible states to invertible states. Note that $\phi$ has the
form of Eq.(2.1), that is, for any $\rho\in{\mathcal{S}}(H)$,
$\phi(\rho)=\frac{\psi(\rho)+B}{f(\rho)+c}$. Since ${\bf H}_{n}$ is finite
dimensional, the linear map $\psi$ and the linear functional $f$ are bounded.
So $\phi$ is continuous. $\phi^{-1}$ is also continuous as $\phi$ preserves
line segment and hence has the form of Eq.(2.1). Thus $\phi$ maps open sets to
open sets. Denote by $G({\mathcal{S}}(H))$ the subset of all invertible
states. $G({\mathcal{S}}(H))$ is an open subset of ${\mathcal{S}}(H)$. In
fact, $G({\mathcal{S}}(H))$ is the maximal open set of all interior points of
${\mathcal{S}}(H)$. To see this, assume that a state $\rho$ is not invertible;
then there are mutually orthogonal rank-one projections $P_{i}$ ($i=1,2,\ldots
n$), an integer $1\leq k<n$ and scalars $t_{i}>0$ with $\sum_{i=1}^{k}t_{i}=1$
such that $\rho=\Sigma^{k}_{i=1}t_{i}P_{i}$. For any $\varepsilon>0$ small
enough so that $\frac{\varepsilon}{2k}<\min\\{t_{1},t_{2}\ldots,t_{k}\\}$, let
$\rho_{\varepsilon}=\Sigma^{k}_{i=1}(t_{i}-\frac{\varepsilon}{2k})P_{i}+\Sigma^{n}_{j=k+1}(\frac{\varepsilon}{2(n-k)})P_{j}.$
Then $\rho_{\varepsilon}$ is an invertible state and
$\|\rho-\rho_{\varepsilon}\|_{{\rm
tr}}\leq\Sigma^{k}_{i=1}\frac{\varepsilon}{2k}+\Sigma^{n}_{j=k+1}\frac{\varepsilon}{2(n-k)}=\varepsilon.$
It follows that for any state $\rho$ and any $\varepsilon>0$, there is an
invertible state $\sigma$ such that
$\rho\in\\{\tau\in{\mathcal{S}}(H):\|\tau-\sigma\|_{\rm Tr}<\varepsilon\\}$.
So the trace norm closure of $G({\mathcal{S}}(H))$ equals ${\mathcal{S}}(H)$.
Thus $G({\mathcal{S}}(H))$ is the set of all interior points of
${\mathcal{S}}(H)$. Since $\phi$ preserves the open sets, we have
$\phi(G({\mathcal{S}}(H)))\subseteq G({\mathcal{S}}(H))$. So $\phi$ preserves
the invertible states. In particular, $\phi(\frac{I}{n})$ is invertible. ∎
By Assertion 1, there is an invertible operator $R\in{\mathcal{B}}(H)$ such
that $\phi(\frac{I}{n})=RR^{*}$. Let $S=R^{-1}$; then the map
$\tilde{\phi}:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ defined by
$\rho\mapsto\frac{S\phi(\rho)S^{*}}{{\rm tr}(S\phi(\rho)S^{*})}$
is bijective, sends line segments to line segments in both directions, i.e.,
$\tilde{\phi}([\rho,\sigma])=[\tilde{\phi}(\rho),\tilde{\phi}(\sigma)]$, and
satisfies $\tilde{\phi}(\frac{I}{n})=\frac{I}{n}$.
Assertion 3.2. $\tilde{\phi}$ maps orthogonal rank one projections to
orthogonal rank one projections.
If $\\{P_{1},\dots,P_{n}\\}$ is an orthogonal set of rank one projections
satisfying $P_{1}+\cdots+P_{n}=I$, then there are $t_{i}\in[0,1]$
$(i=1,...,n)$ with $\Sigma_{i=1}^{n}t_{i}=1$ such that
$\frac{I}{n}=\tilde{\phi}(\frac{I}{n})=\tilde{\phi}(\frac{(P_{1}+\cdots+P_{n})}{n})=t_{1}\tilde{\phi}(P_{1})+\cdots+t_{n}\tilde{\phi}(P_{n})\geq
t_{i}\tilde{\phi}(P_{i})$
for each $i=1,\dots,n$. Because $\tilde{\phi}(P_{i})$ is a rank one orthogonal
projection and $I/n-t_{i}\tilde{\phi}(P_{i})$ is positive semidefinite, we see
that $1/n\geq t_{i}$ for $i=1,\dots,n$. Taking trace, we have
$1={\rm tr}(I/n)=\sum_{i=1}^{n}t_{i}.$
Thus, $t_{1}=\cdots=t_{n}=1/n$. So, $I=\sum_{i=1}^{n}\tilde{\phi}(P_{i})$.
This implies that $\\{\tilde{\phi}(P_{1}),\ldots,\tilde{\phi}(P_{n})\\}$ is an
orthogonal set of rank one projections. Hence, $\tilde{\phi}$ sends orthogonal
rank one projections to orthogonal rank one projections. ∎
By [12, Theorem 2] again, $\tilde{\phi}$ has the form of Eq.(2.1), that is,
$None$ $\tilde{\phi}(\rho)=\frac{\psi(\rho)+B}{f(\rho)+c}$
holds for any $\rho\in{\mathcal{S}}(H)$, where
$\psi:{\mathbf{H}}_{n}({\mathbb{C}})\rightarrow{\mathbf{H}}_{n}({\mathbb{C}})$
is a real linear map, ${\mathbf{H}}_{n}({\mathbb{C}})$ is the real linear
space of all $n\times n$ hermitian matrices,
$B\in{\mathbf{H}}_{n}({\mathbb{C}})$,
$f:{\mathbf{H}}_{n}({\mathbb{C}})\rightarrow{\mathbb{R}}$ is a real linear
functional and $c$ is a real constant with $f(\rho)+c>0$ for all
$\rho\in{\mathcal{S}}(H)$.
Next we consider the two cases of dim$H>2$ and dim$H=2$ respectively.
Assertion 3.3. Assume dim$H>2$. The functional $f$ in Eq.(3.1) is a constant
on ${\mathcal{S}}(H)$, that is, there is a real number $a$ such that
$f(\rho)=a$ for all $\rho\in{\mathcal{S}}(H)$.
For any normalized orthogonal basis $\\{e_{i}\\}^{n}_{i=1}$, let
$P_{i}=e_{i}\otimes e_{i}$. We first claim that $f(e_{i}\otimes
e_{i})=f(e_{j}\otimes e_{j})$ for any $i$ and $j$. Since $\tilde{\phi}$
preserves the rank one projections in both directions, there is a rank one
projection $Q_{i}=x_{i}\otimes x_{i}$ such that
$x_{i}\otimes x_{i}=Q_{i}=\tilde{\phi}(P_{i})=\frac{\psi(e_{i}\otimes
e_{i})+B}{f(e_{i}\otimes e_{i})+c}.$
So
$\psi(e_{i}\otimes e_{i})+B=(f(e_{i}\otimes e_{i})+c)(x_{i}\otimes x_{i}).$
As $\tilde{\phi}(\frac{I}{n})=\frac{I}{n}$ and
$\frac{I}{n}=\frac{1}{n}\sum^{n}_{i=1}e_{i}\otimes e_{i}$, we have
$\frac{I}{n}=\tilde{\phi}(\frac{1}{n}\sum^{n}_{i=1}e_{i}\otimes
e_{i})=\frac{\psi(\sum^{n}_{i=1}\frac{1}{n}e_{i}\otimes
e_{i})+B}{f(\sum^{n}_{i=1}\frac{1}{n}e_{i}\otimes
e_{i})+c}=\frac{\sum^{n}_{i=1}\frac{1}{n}\psi(e_{i}\otimes
e_{i})+n\frac{1}{n}B}{\sum^{n}_{i=1}\frac{1}{n}f(e_{i}\otimes
e_{i})+n\frac{1}{n}c}.$
Then
$None$ $\frac{I}{n}=\frac{\frac{1}{n}(\sum^{n}_{i=1}\psi(e_{i}\otimes
e_{i})+B)}{\frac{1}{n}(\sum^{n}_{i=1}f(e_{i}\otimes
e_{i})+c)}=\frac{\sum^{n}_{i=1}(\psi(e_{i}\otimes
e_{i})+B)}{\sum^{n}_{i=1}(f(e_{i}\otimes e_{i})+c)}.$
On the other hand, by Assertion 3.2, we have
$\frac{I}{n}=\tilde{\phi}(\frac{I}{n})=\frac{1}{n}\sum^{n}_{i=1}\tilde{\phi}(e_{i}\otimes
e_{i})=\frac{1}{n}\sum^{n}_{i=1}\frac{\psi(e_{i}\otimes
e_{i})+B}{f(e_{i}\otimes e_{i})+c}.$
Thus we get
$None$ $I=\sum^{n}_{i=1}\frac{\psi(e_{i}\otimes e_{i})+B}{f(e_{i}\otimes
e_{i})+c}.$
Let $A_{i}=\psi(e_{i}\otimes e_{i})+B$ and $a_{i}=f(e_{i}\otimes e_{i})+c$.
Then Eq.(3.2) and Eq.(3.3) imply that
$I=n(\frac{A_{1}+A_{2}+\ldots+A_{n}}{a_{1}+a_{2}+\ldots+a_{n}})=\frac{A_{1}}{a_{1}}+\frac{A_{2}}{a_{2}}+\ldots+\frac{A_{n}}{a_{n}}.$
Note that $A_{i}=a_{i}Q_{i}$, where $Q_{i}=\tilde{\phi}(e_{i}\otimes
e_{i})=x_{i}\otimes x_{i}$. Therefore, we get that
$I=n(\frac{a_{1}Q_{1}+a_{2}Q_{2}+\ldots+a_{n}Q_{n}}{a_{1}+a_{2}+\ldots+a_{n}})=\frac{a_{1}Q_{1}}{a_{1}}+\frac{a_{2}Q_{2}}{a_{2}}+\ldots+\frac{a_{n}Q_{n}}{a_{n}}.$
It follows that
$n(\frac{a_{1}Q_{1}+a_{2}Q_{2}+\ldots+a_{n}Q_{n}}{a_{1}+a_{2}+\ldots+a_{n}})=Q_{1}+Q_{2}+\ldots+Q_{n}.$
Since $\\{Q_{i}\\}_{i=1}^{n}$ is an orthogonal set of rank one projections, we
see that
$\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}=a_{1}=a_{2}=\ldots=a_{n}.$
This implies that there is some scalar $a$ such that $f(e_{i}\otimes e_{i})=a$
holds for all $i$. Now for arbitrary unit vectors $x,y\in H$, as dim$H>2$,
there is a unit vector $z\in H$ such that $z\in[x,y]^{\perp}$. It follows from
the above argument that $f(x\otimes x)=f(z\otimes z)=f(y\otimes y)$. So
$f(x\otimes x)=a$ for all unit vectors $x\in H$. Since each state is a convex
combination of pure states, by the linearity of $f$, we get that $f(\rho)=a$
holds for every state $\rho$. ∎
Assertion 3.4. Assume dim$H>2$. $\phi$ has the form stated in Theorem 1 (c).
Every state is a convex combination of some pure states, i.e. convex
combination of some rank one projections. Therefore, by Assertion 3.3, we have
$\tilde{\phi}(\rho)=\frac{\psi(\rho)+B}{\alpha+c}$
holds for all $\rho$. Then by the linearity of $\psi$, it is clear that
$\tilde{\phi}$ is an affine isomorphism, i.e., for any states $\rho,\sigma$
and scalar $\lambda$ with $0\leq\lambda\leq 1$,
$\tilde{\phi}(\lambda\rho+(1-\lambda)\sigma)=\lambda\tilde{\phi}(\rho)+(1-\lambda)\tilde{\phi}(\sigma).$
By a result due to Kadison (Ref. [2, Theorem 8.1]), $\tilde{\phi}$ has the
standard form, that is, there exists a unitary operator $U\in{\mathcal{B}}(H)$
such that $\tilde{\phi}$ has the form
$\tilde{\phi}(\rho)=U\rho U^{*}\ \mbox{\rm for all }\rho\quad\hbox{ or
}\quad\rho\mapsto U\rho^{T}U^{*}\ \mbox{for all }\ \rho.$
Now recalled that $\tilde{\phi}$ is defined by
$\tilde{\phi}(\rho)=S\phi(\rho)S^{*}/{\rm tr}(S\phi(\rho)S^{*})$. If
$\tilde{\phi}$ takes the first form, then we have
$\phi(\rho)={\rm
tr}(S\phi(\rho)S^{*})S^{-1}\tilde{\phi}(\rho)(S^{*})^{-1}={\rm
tr}(S\phi(\rho)S^{*})S^{-1}U\rho U^{*}(S^{*})^{-1}.$
As $1={\rm tr}(\phi(\rho))={\rm tr}(S\phi(\rho)S^{*}){\rm tr}(S^{-1}U\rho
U^{*}(S^{*})^{-1})$, so
${\rm tr}(S\phi(\rho)S^{*})=\frac{1}{{\rm tr}(S^{-1}U\rho
U^{*}(S^{*})^{-1})}.$
Letting $M=S^{-1}U$, we get $\phi(\rho)=\frac{M\rho M^{*}}{{\rm tr}(M\rho
M^{*})}$ for all $\rho$, that is, $\phi$ has the first form stated in (c) of
Theorem 1.
Similarly, if $\tilde{\phi}$ takes the second form, then $\phi$ takes the
second form stated in (c) of Theorem 1. ∎
Assertion 3.5. Condition (c) of Theorem 1 holds for the case of dim$H=2$.
Assume that dim$H=2$. Denote by ${\mathcal{S}}_{2}=\mathcal{S}(H)$ the convex
set of $2\times 2$ positive matrices with the trace 1. Then the map
$\tilde{\phi}:{\mathcal{S}}_{2}\rightarrow{\mathcal{S}}_{2}$ is a bijective
map preserving segment in both directions satisfying
$\tilde{\phi}(\frac{1}{2}I_{2})=\frac{1}{2}I_{2}$. Let us identify
${\mathcal{S}}_{2}$ with the unit ball
$({\mathbb{R}}^{3})_{1}=\\{(x,y,z)^{T}\in{\mathbb{R}}^{3}:x^{2}+y^{2}+z^{2}\leq
1\\}$ of ${\mathbb{R}}^{3}$ by the following way. Let
$\pi:({\mathbb{R}}^{3})_{1}\rightarrow{\mathcal{S}}_{2}$ be the map defined by
$(x,y,z)^{T}\mapsto\frac{1}{2}I_{2}+\frac{1}{2}\left(\begin{array}[]{ccccccccccccccc}z&x-iy\\\
x+iy&-z\\\ \end{array}\right).$
$\pi$ is a bijective affine isomorphism. Note that $v=(x,y,z)^{T}$ satisfies
$x^{2}+y^{2}+z^{2}=1$ if and only if the corresponding matrix $\pi(v)$ is a
rank one projection, and $0=(0,0,0)^{T}$ if and only if the corresponding
matrix is $\pi(0)=\frac{1}{2}I$. The map
$\tilde{\phi}:{\mathcal{S}}_{2}\rightarrow{\mathcal{S}}_{2}$ induces a map
$\hat{\phi}:({\mathbb{R}}^{3})_{1}\rightarrow({\mathbb{R}}^{3})_{1}$ by the
following equation
$\tilde{\phi}(\rho)=\frac{1}{2}I+\pi(\hat{\phi}(\pi^{-1}(\rho))).$
Since $\tilde{\phi}$ is a segment preserving bijective map and $\pi$ is an
affine isomorphism, the map $\hat{\phi}$ is a bijective map preserving segment
in both directions, that is, $\hat{\phi}([u,v])=[\hat{\phi}(u),\hat{\phi}(v)]$
for $u,v\in({\mathbb{R}}^{3})_{1}$. So $\hat{\phi}$ maps the surface of
$({\mathbb{R}}^{3})_{1}$ onto the surface of $({\mathbb{R}}^{3})_{1}$. Since
$\tilde{\phi}(\frac{1}{2}I)=\frac{1}{2}I$, we have that
$\hat{\phi}((0,0,0)^{T})=(0,0,0)^{T}$.
Applying the P$\breve{a}$les’ result [12, Theorem 2] to $\hat{\phi}$, there
exists a linear transformation
$L:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{3}$, a linear functional
$f:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$, a vector
$u_{0}\in{\mathbb{R}}^{3}$ and a scalar $r\in{\mathbb{R}}$ such that
$f((x,y,z)^{T})+r>0$ and
$\hat{\phi}((x,y,z)^{T})=\frac{L((x,y,z)^{T})+u_{0}}{f((x,y,z)^{T})+r}$
for each $(x,y,z)^{T}\in({\mathbb{R}}^{3})_{1}$. Since
$\hat{\phi}((0,0,0)^{T})=(0,0,0)^{T}$, we have $u_{0}=0$ and $r>0$.
Furthermore, the linearity of $f$ implies that there are real scalars
$r_{1},r_{2},r_{3}$ such that $f((x,y,z)^{T})=r_{1}x+r_{2}y+r_{3}z$. We claim
that $r_{1}=r_{2}=r_{3}=0$ and hence $f=0$. If not, then there is a vector
$(x_{0},y_{0},z_{0})^{T}$ satisfying $x^{2}_{0}+y^{2}_{0}+z^{2}_{0}=1$ such
that $f((x_{0},y_{0},z_{0})^{T})=r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}\neq 0$. It
follows that
$1=\|\hat{\phi}((x_{0},y_{0},z_{0})^{T})\|=\|\frac{L((x_{0},y_{0},z_{0})^{T})}{r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+r}\|,$
and thus
$\|L((x_{0},y_{0},z_{0})^{T})\|=r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+r.$
Similarly
$\|L((-x_{0},-y_{0},-z_{0})^{T})\|=-r_{1}x_{0}-r_{2}y_{0}-r_{3}z_{0}+r.$
By the linearity of $L$ we have
$r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}+r=-r_{1}x_{0}-r_{2}y_{0}-r_{3}z_{0}+r$.
Hence $r_{1}x_{0}+r_{2}y_{0}+r_{3}z_{0}=0$, a contradiction. So, we have
$f=0$, and thus $\hat{\phi}=\frac{L}{r}$ is linear. Now it is clear that
$\tilde{\phi}$ is an affine isomorphism as $\pi$ is an affine isomorphism.
Applying a similar argument to the proof of Assertion 3.4 and the Kadison’s
result, one sees that $\tilde{\phi}$ has the standard form. Thus, Theorem 1
(c) holds. ∎
By Assertions 3.4 and 3.5, we get the proof of Theorem 1 for finite-
dimensional case.
## 4\. Proof: infinite dimensional case
In this section we give a proof of our main result for infinite dimensional
case. Similar to the previous section, we need only establish the implication
(b) $\Rightarrow$ (c). We begin with two lemmas.
Let $V_{1},V_{2}$ be linear spaces on a field ${\mathbb{F}}$,
$\upsilon:{\mathbb{F}}\rightarrow{\mathbb{F}}$ a nonzero ring automorphism. A
map $A:V_{1}\rightarrow V_{2}$ is called a $\upsilon$-linear operator if
$A(\lambda x)=\upsilon(\lambda)Ax$ for all $x\in V_{1}$. The following lemma
is similar to [7, Lemma 2.3.1].
Lemma 4.1 Let $V_{1},V_{2}$ be linear spaces on a field ${\mathbb{F}}$,
$\tau,\upsilon:{\mathbb{F}}\rightarrow{\mathbb{F}}$ nonzero ring auto-
isomorphisms. Suppose $A:V_{1}\rightarrow V_{2}$ is a $\tau$-linear
transformation, $B:V_{1}\rightarrow V_{2}$ is a $\upsilon$-linear
transformation, and $\dim{\rm span}({\rm ran}(B))\geq 2$. If $\ker
B\subseteq\ker A$ and $Ax$ and $Bx$ are linearly dependent for all $x\in V$,
then $\tau=\upsilon$ and $A=\lambda B$ for some scalar $\lambda$.
Proof As $\ker B\subseteq\ker A$, for every $x\in V_{1}$, there is some scalar
$\lambda_{x}$ such that $Ax=\lambda_{x}Bx$. If $Bx\not=0$, then there exists
$y\in V_{1}$ such that $Bx,By$ are linearly independent. Then
$\lambda_{x+y}(Bx+By)=A(x+y)=\lambda_{x}Bx+\lambda_{y}By$. This implies that
$\lambda_{x}=\lambda_{x+y}=\lambda_{y}$. Moreover, for any
$\alpha\in{\mathbb{F}}$, we have $\lambda_{\alpha x}=\lambda_{x}$. If $Bx=0$,
then $Ax=0$. Thus it follows that there exists a scalar $\lambda$ such that
$Ax=\lambda Bx$ holds for all $x\in V_{1}$. So, $A=\lambda B$ and
$\tau=\upsilon$. $\Box$
Lemma 4.2 Let ${\mathcal{S}}(H)$ be the set of all states on Hilbert space $H$
with $\dim H=\infty$, and $\phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(H)$ a
bijective map. If $\phi$ satisfies that, for any $t\in[0,1]$ and
$\rho,\sigma\in{\mathcal{S}}(H)$, there is $s\in[0,1]$ such that
$\phi(t\rho+(1-t)\sigma)=s\phi(\rho)+(1-s)\phi(\sigma),$
then, $\phi$ is continuous and there is an invertible bounded linear or
conjugate linear operator $T$ such that
$\phi(x\otimes x)=\frac{Tx\otimes Tx}{\|Tx\|^{2}}{\ for\ all\ unit\ vectors\
}x\in H.$
Proof We complete the proof by checking several assertions. First we restate
Lemma 2.1 as:
Assertion 4.1. $\phi$ preserves pure states (rank one projections) in both
directions.
Assertion 4.2. For any $x_{i}\otimes x_{i}\in{\mathcal{P}ur}(H)$ with
$\\{x_{1},x_{2}\ldots,x_{n}\\}$ linearly independent, let
$F(x_{1},\ldots,x_{n})=C(x_{1},\ldots,x_{n})\cup F_{0}(x_{1},\ldots,x_{n}),$
where $C(x_{1},\ldots,x_{n})={\rm cov}\\{x_{i}\otimes x_{i}:i=1,2\ldots,n\\}$
is the convex hull of $\\{x_{i}\otimes x_{i}\\}_{i=1}^{n}$,
$\begin{array}[]{rl}F_{0}(x_{1},\ldots,x_{n})=&\\{Z\in{\mathcal{S}}(H)\setminus
C(x_{1},\ldots,x_{n}):\mbox{\rm there exists some }\\\
&W\in{\mathcal{S}}(H)\setminus C(x_{1},\ldots,x_{n})\ \mbox{\rm such that
}[Z,W]\cap C(x_{1},\ldots,x_{n})\not=\emptyset\\}.\end{array}$
Let $H_{0}={\rm span}\\{x_{1},\ldots,x_{n}\\}$. Then we have
$None$ $F(x_{1},\ldots,x_{n})={\mathcal{S}}(H_{0})\oplus\\{0\\}.$
Obviously, $C(x_{1},\ldots,x_{n})\subset{\mathcal{S}}(H_{0})\oplus\\{0\\}$. If
$Z\in F_{0}(x_{1},\ldots,x_{n})$, then there exists some
$W\in{\mathcal{S}}(H)\setminus C(x_{1},\ldots,x_{n})$, $t_{i}>0$ with
$\sum_{i=1}^{n}t_{i}=1$ and $t\in(0,1)$ such that
$\sum_{i=1}^{n}t_{i}x_{i}\otimes x_{i}=tZ+(1-t)W.$
Let $P_{0}\in{\mathcal{B}}(H)$ be the projection from $H$ onto $H_{0}$. As
$\sum_{i=1}^{n}t_{i}x_{i}\otimes x_{i}-tZ=(1-t)W\geq 0$ and
$(I-P_{0})\sum_{i=1}^{n}t_{i}x_{i}\otimes
x_{i}=\sum_{i=1}^{n}t_{i}x_{i}\otimes x_{i}(I-P_{0})=0$, we see that
$(I-P_{0})Z=Z(I-P_{0})=0$, which implies that $P_{0}ZP_{0}=Z$ and hence
$Z\in{\mathcal{S}}(H_{0})\oplus\\{0\\}$.
Conversely, assume that $Z\in{\mathcal{S}}(H_{0})\oplus\\{0\\}$. Since
$C(x_{1},\ldots,x_{n})\subset{\mathcal{S}}(H_{0})\oplus\\{0\\}$, we may assume
that $Z$ is not a convex combination of $\\{x_{i}\otimes x_{i}\\}_{i=1}^{n}$.
Because $\\{x_{i}\\}_{i=1}^{n}$ is a linearly independent set, there exists an
operator $S\in{\mathcal{B}}(H_{0})$ such that $\\{e_{i}=Sx_{i}\\}_{i=1}^{n}$
is an orthonormal basis of $H_{0}$. Then, consider
$S(\sum_{i=1}^{n}a_{i}x_{i}\otimes
x_{i}-Z)S^{*}=\sum_{i=1}^{n}a_{i}Sx_{i}\otimes
Sx_{i}-SZS^{*}=\sum_{i=1}^{n}a_{i}e_{i}\otimes e_{i}-SZS^{*}.$
It is clear that for sufficient large $a_{i}>0$,
$\sum_{i=1}^{n}a_{i}e_{i}\otimes e_{i}-SZS^{*}\geq 0$, and hence,
$W=\sum_{i=1}^{n}a_{i}x_{i}\otimes x_{i}-Z\geq 0$. This entails that
$\frac{\sum_{i=1}^{n}a_{i}x_{i}\otimes
x_{i}}{\sum_{i=1}^{n}a_{i}}=\frac{1}{\sum_{i=1}^{n}a_{i}}Z+\frac{{\rm
tr}(W)}{\sum_{i=1}^{n}a_{i}}(\frac{W}{{\rm tr}(W)}),$
that is, $Z\in F_{0}(x_{1},\ldots,x_{n})\subset F(x_{1},\ldots,x_{n})$. This
finishes the proof of Eq.(4.1). ∎
Assertion 4.3. For any finite-dimensional subspace $H_{0}\subset H$, there
exists a subspace $H_{1}$ with $\dim H_{1}=\dim H_{0}$ such that
$\phi({\mathcal{S}}(H_{0})\oplus\\{0\\})={\mathcal{S}}(H_{1})\oplus\\{0\\}.$
Assume that $\dim H_{0}=n$. Choose an orthonormal basis
$\\{x_{i}\\}_{i=1}^{n}$ of $H_{0}$. Then by Assertion 4.1, there are unit
vectors $u_{i}\in H$ such that $\phi(x_{i}\otimes x_{i})=u_{i}\otimes u_{i}$.
It is clear that $\\{u_{i}\\}_{i=1}^{n}$ is a linearly independent set. Let
$H_{1}={\rm span}\\{u_{i}\\}_{i=1}^{n}$. Then $\dim H_{1}=n$, and by Eq.(4.1)
in Assertion 4.2, we have
$F(x_{1},\ldots,x_{n})={\mathcal{S}}(H_{0})\oplus\\{0\\}$,
$F(u_{1},\ldots,u_{n})={\mathcal{S}}(H_{1})\oplus\\{0\\}$. Since the bijection
$\phi$ preserves segments and pure states in both directions, it is easily
checked that $\phi(F(x_{1},\ldots,x_{n}))=F(u_{1},\ldots,u_{n})$, and the
conclusion of Assertion 4.3 follows. ∎
Assertion 4.4. For any finite dimensional subspace $\Lambda\subset H$, there
exists a subspace $H_{\Lambda}\subset H$ with $\dim H_{\Lambda}=\dim\Lambda$
and an invertible linear or conjugate linear operator
$M_{\Lambda}:\Lambda\rightarrow H_{\Lambda}$ such that
$\phi(P_{\Lambda}\rho P_{\Lambda})=\frac{Q_{\Lambda}M_{\Lambda}\rho
M_{\Lambda}^{*}Q_{\Lambda}}{{\rm tr}(M_{\Lambda}\rho M_{\Lambda}^{*})}$
for all $\rho\in{\mathcal{S}}(\Lambda)$, where $P_{\Lambda}$ and $Q_{\Lambda}$
are respectively the projections onto $\Lambda$ and $H_{\Lambda}$. Moreover,
the $M_{\Lambda}$ can be chosen so that
$M_{\Lambda_{1}}=M_{\Lambda_{2}}|_{\Lambda_{1}}$ whenever
$\Lambda_{1}\subseteq\Lambda_{2}$.
Let $H_{0}$ be a finite dimensional subspace of $H$ and let
$\\{e_{1},e_{2},\ldots,e_{n}\\}$ be an orthonormal basis of $H_{0}$. By
Assertion 4.1 there exist unit vectors $\\{u_{1},u_{2},\ldots,u_{n}\\}$ such
that $\phi(e_{i}\otimes e_{i})=u_{i}\otimes u_{i}$. Let $H_{1}={\rm
span}\\{u_{1},u_{2},\ldots,u_{n}\\}$. By Assertion 4.1 again, $\dim
H_{1}=n=\dim H_{0}$. It follows from Assertion 4.3 that, for any
$\rho\in{\mathcal{S}}(H)$, $P_{0}\rho P_{0}=\rho$ implies that
$P_{1}\phi(\rho)P_{1}=\phi(\rho)$. Thus $\phi$ induces a bijective map
$\phi_{0}:{\mathcal{S}}(H_{0})\rightarrow{\mathcal{S}}(H_{1})$ by
$\phi_{0}(\rho)=\phi(P_{0}\rho P_{0})|_{H_{1}}$. Applying Theorem 1 for finite
dimensional case just proved in Section 2, we obtain that there is an
invertible bounded linear operator $M:H_{0}\rightarrow H_{1}$ such that
$\phi_{0}$ has the form
$\rho\mapsto\frac{M\rho M^{*}}{{\rm tr}(M^{*}M\rho)}\quad{\rm
or}\quad\rho\mapsto\frac{M\rho^{T}M^{*}}{{\rm tr}(M^{*}M\rho^{T})},$
where $\rho^{T}$ is the transpose of $\rho$ with respect to the orthonormal
basis $\\{e_{1},e_{2},\ldots,e_{n}\\}$. In the last case, we let
$J:H_{0}\rightarrow H_{0}$ be the conjugate linear operator defined by
$J(\sum_{i=1}^{n}\xi_{i}e_{i})=\sum_{i=1}^{n}\bar{\xi_{i}}e_{i}$, and let
$M^{\prime}=MJ$. Then, $M^{\prime}:H_{0}\rightarrow H_{1}$ is invertible
conjugate linear and
$\phi_{0}(\rho)=\frac{M^{\prime}\rho{M^{\prime}}^{*}}{{\rm
tr}({M^{\prime}}^{*}M^{\prime}\rho)}$ for all $\rho\in{\mathcal{S}}(H_{0})$.
Therefore, the first part of the Assertion 4.4 is true.
Let $\Lambda_{i}$, $i=1,2$, are finite dimensional subspaces of $H$ and
$M_{i}$s are associated operators as that obtained above way. If
$\Lambda_{1}\subseteq\Lambda_{2}$, then, for any unit vector
$x\in\Lambda_{1}$, we have $\frac{M_{1}x\otimes
M_{1}x}{\|M_{1}x\|^{2}}=\phi(x\otimes x)=\frac{M_{2}x\otimes
M_{2}x}{\|M_{2}x\|^{2}}$. It follows that $M_{1}x$ and $M_{2}x$ are linearly
dependent. By Lemma 4.2 we see that $M_{2}|_{\Lambda_{1}}=\lambda M_{1}$ for
some scalar $\lambda$. As $\frac{(\lambda M)\rho(\lambda M)^{*}}{{\rm
tr}((\lambda M)^{*}(\lambda M)\rho)}=\frac{M\rho M^{*}}{{\rm
tr}(M^{*}M\rho)}$, we may choose $M_{2}$ so that $M_{2}|_{\Lambda_{1}}=M_{1}$.
∎
Assertion 4.5. There exists a linear or conjugate linear bijective
transformation $T:H\rightarrow H$ such that
$\phi(x\otimes x)=\frac{Tx\otimes Tx}{\|Tx\|^{2}}$
for every unit vector $x\in H$ and $T|_{\Lambda}=M_{\Lambda}$ for every finite
dimensional subspace $\Lambda$ of $H$.
For any $x\in H$, there is finite dimensional subspace $\Lambda$ such that
$x\in\Lambda$. Let $Tx=M_{\Lambda}x$. Then, by Assertion 4.4, $T:H\rightarrow
H$ is well defined, linear or conjugate linear. And by Assertion 4.1, $T$ is
bijective.
Note that P$\breve{a}$les’ result (Theorem 2 in [12]) holds true for the
infinite dimensional case. Since $\phi$ preserves segment, by [12, Theorems
1-2], there exists a linear operator
$\Gamma:{\mathcal{HT}}(H)\rightarrow{\mathcal{HT}}(H)$, a linear functional
$g:{\mathcal{HT}}(H)\rightarrow{\mathbb{R}}$, a scalar $b\in{\mathbb{R}}$ and
some operator $B\in\mathcal{HT}(B)$ such that
$None$ $\phi(\rho)=\frac{\Gamma\rho+B}{g(\rho)+b}$
for all $\rho\in{\mathcal{S}}(H)$, where ${\mathcal{HT}}(H)$ denotes the set
of all self-adjoint Trace-class operators in ${\mathcal{B}}(H)$ and
$g(\rho)+b>0$ for all $\rho\in{\mathcal{S}}(H)$. ∎
Assertion 4.6. The functions $g$, $\Gamma$ in Eq.(4.2) are bounded and hence
$\phi$ is continuous.
Note that, for any $\rho_{1},\rho_{2}\in{\mathcal{S}}(H)$ and any $t\in(0,1)$,
there exists some $s(t)\in(0,1)$ such that
$\phi(t\rho_{1}+(1-t)\rho_{2})=s(t)\phi(\rho_{1})+(1-s(t))\phi(\rho_{2}).$
Combining this with Eq.(4.2), one gets
$None$
$\frac{t\Gamma\rho_{1}+(1-t)\Gamma\rho_{2}+B}{tg(\rho_{1})+(1-t)g(\rho_{2})+b}=s(t)\frac{\Gamma\rho_{1}+B}{g(\rho_{1})+b}+(1-s(t))\frac{\Gamma\rho_{2}+B}{g(\rho_{2})+b}.$
Note that different states are linearly independent. Comparing the
coefficients of $\Gamma\rho_{1}$ in Eq.(4.3), one sees that
$None$ $s(t)=\frac{t(g(\rho_{1})+b)}{tg(\rho_{1})+(1-t)g(\rho_{2})+b}.$
It follows that $s(t)\rightarrow 1$ when $t\rightarrow 1$. If
$\rho=\sum_{i=1}^{n}t_{i}\rho_{i}\in{\mathcal{S}}(H)$ with
$\rho_{i}\in{\mathcal{S}}(H)$, one can get some $p_{i}$ so that
$\phi(\rho)=\phi(\sum_{i=1}^{n}t_{i}\rho_{i})=\sum_{i=1}^{n}p_{i}\phi(\rho_{i})$,
where $\sum_{i=1}^{n}t_{i}=\sum_{i=1}^{n}p_{i}=1$. Similarly we can check that
$None$ $p_{i}=\frac{t_{i}(g(\rho_{i})+b)}{\sum_{i=1}^{n}t_{i}g(\rho_{i})+b}.$
Suppose that $\rho,\rho_{i}\in{\mathcal{S}}(H)$ with
$\rho=\sum_{i=1}^{\infty}t_{i}\rho_{i}$, where $t_{i}>0$ and
$\sum_{i=1}^{\infty}t_{i}=1$. Then
$None$
$\begin{array}[]{rl}\phi(\rho)=&\phi(\sum_{i=1}^{\infty}t_{i}\rho_{i})\\\
=&\phi((\sum_{j=1}^{k}t_{j})\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i}+(1-\sum_{j=1}^{k}t_{j})\sum_{i=k+1}^{\infty}(\frac{t_{i}}{1-\sum_{j=1}^{k}t_{j}})\rho_{i})\\\
=&s_{k}\phi(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i})+(1-s_{k})\phi(\sum_{i=k+1}^{\infty}(\frac{t_{i}}{1-\sum_{j=1}^{k}t_{j}})\rho_{i}).\end{array}$
Thus there exist scalars $q_{i}^{(k)}>0$ with $\sum_{i=1}^{k}q_{i}^{(k)}=1$
such that
$s_{k}\phi(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i})=\sum_{i=1}^{k}s_{k}q_{i}^{(k)}\phi(\rho_{i}).$
According to Eq.(4.4), Eq.(4.5), and keeping in mind that $g$ is a linear
functional, a simple calculation reveals that
$None$
$\begin{array}[]{rl}s_{k}=&\frac{(\sum_{j=1}^{k}t_{j})(g(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i})+b)}{(\sum_{j=1}^{k}t_{j})(g(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i}))+(1-\sum_{j=1}^{k}t_{j})g(\sum_{i=k+1}^{\infty}(\frac{t_{i}}{1-\sum_{j=1}^{k}t_{j}})\rho_{i}))+b}\\\
=&\frac{(\sum_{j=1}^{k}t_{j})(g(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i})+b)}{g(\rho)+b},\end{array}$
$None$
$q_{i}^{(k)}=\frac{(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})(g(\rho_{i})+b)}{\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})g(\rho_{i})+b}=\frac{(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})(g(\rho_{i})+b)}{g(\sum_{i=1}^{k}(\frac{t_{i}}{\sum_{j=1}^{k}t_{j}})\rho_{i})+b},$
and
$None$ $s_{k}q_{i}^{(k)}=\frac{t_{i}(g(\rho_{i})+b)}{g(\rho)+b}.$
Observe that $s_{k}q_{i}^{(k)}$ is independent to $k$. Since
$\sum_{i=1}^{k}t_{i}\rightarrow 1$ as $k\rightarrow\infty$, we must have
$s_{k}\rightarrow 1$ as $k\rightarrow\infty$. Eqs.(4.6)-(4.9) imply that
$\sum_{i=1}^{\infty}\frac{t_{i}(g(\rho_{i})+b)}{g(\rho)+b}=1$
and
$None$
$\phi(\sum_{i=1}^{\infty}t_{i}\rho_{i})=\sum_{i=1}^{\infty}(\frac{t_{i}(g(\rho_{i})+b)}{g(\rho)+b})\phi(\rho_{i}).$
In particular, we have
$None$
$g(\sum_{i=1}^{\infty}t_{i}\rho_{i})=\sum_{i=1}^{\infty}t_{i}g(\rho_{i}).$
We assert that $\sup\\{g(\rho):\rho\in{\mathcal{S}}(H)\\}<\infty$. Assume that
$\sup\\{g(\rho):\rho\in{\mathcal{S}}(H)\\}=\infty$. Then, for any positive
integer $i$, there exists $\rho_{i}\in{\mathcal{S}}(H)$ satisfying that
$g(\rho_{i})>2^{i}$. Let
$\rho_{0}=\sum_{i=1}^{\infty}\frac{1}{2^{i}}\rho_{i}$,
$\sigma_{k}=\sum_{i=1}^{k}\frac{1}{2^{i}}\rho_{i}$, then
$\sigma_{k}\rightarrow\rho_{0}$, and
$g(\sigma_{k})=\sum_{i=1}^{k}\frac{1}{2^{i}}g(\rho_{i})\geq\sum_{i=1}^{k}1=k.$
Since $g(\rho_{i})\geq 0$, by Eq.(4.11), we have $g(\rho_{0})\geq
g(\sigma_{k})\geq k$ for every $k$, contradicting to the fact that
$g(\rho_{0})<\infty$. Now the fact $g(\rho)+b>0$ for all $\rho$ entails that
there exists a positive number $c$ such that
$\sup\\{|g(\rho)|:\rho\in{\mathcal{S}}(H)\\}=c$. Thus $g$ is continuous on
${\mathcal{HT}}(H)$ and
$None$ $\|g\|=c<\infty.$
Since
$\|\Gamma\rho\|\leq\|\Gamma\rho+B\|+\|B\|\leq\|\Gamma\rho+B\|_{\rm
Tr}+\|B\|=g(\rho)+b+\|B\|\leq c+|b|+\|B\|$
holds for all $\rho\in{\mathcal{S}}(H)$, it follows that $\Gamma$ is
$\|\cdot\|_{\rm tr}$-$\|\cdot\|$ continuous from ${\mathcal{HT}}(H)$ into
itself. Hence, if $\rho_{n},\rho\in{\mathcal{S}}(H)$ and $\|\cdot\|_{\rm
tr}$-$\lim_{n\rightarrow\infty}\rho_{n}=\rho$, then
$\|\cdot\|$-$\lim_{n\rightarrow\infty}\phi(\rho_{n})=\phi(\rho)$. However,
convergence under trace-norm topology and convergence under uniform-norm
topology are the same for states [15]. Hence we have $\|\cdot\|_{\rm
tr}$-$\lim_{n\rightarrow\infty}\phi(\rho_{n})=\phi(\rho)$, i.e., $\phi$ is
continuous under the trace-norm topology. ∎
Assertion 4.7. The operator $T$ in Assertion 4.5 is bounded.
For any finite dimensional subspace $\Lambda\subset H$, let $M_{\Lambda}$ be
the invertible linear or conjugate linear operator stated in Assertion 4.4.
Then for any $\rho\in{\mathcal{S}}(H)$ with range in $\Lambda$, we have
$\frac{\Gamma\rho+B}{g(\rho)+b}=\frac{Q_{\Lambda}M_{\Lambda}\rho
M_{\Lambda}^{*}Q_{\Lambda}}{{\rm tr}(M_{\Lambda}\rho M_{\Lambda}^{*})}$. Thus
$\Gamma\rho+B=\lambda_{\rho}Q_{\Lambda}M_{\Lambda}\rho
M_{\Lambda}^{*}Q_{\Lambda},$
where
$\lambda_{\rho}=\frac{g(\rho)+b}{{\rm tr}(M_{\Lambda}\rho M_{\Lambda}^{*})}.$
For any $\sigma\in{\mathcal{S}}(H)$ with range in $\Lambda$ and
$\sigma\not=\rho$, and for any $0<t<1$, by considering $t\rho+(1-t)\sigma$ one
gets
$\lambda_{\rho}=\lambda_{t\rho+(1-t)\sigma}=\lambda_{\sigma}.$
This implies that there exists a scalar $d>0$ such that $\lambda_{\rho}=d$ for
all $\rho$ with range in $\Lambda$. Use Assertion 4.4 again, it is clear that
$d$ is not dependent to $\Lambda$. Thus, the equation
${\rm tr}(M_{\Lambda}\rho M_{\Lambda}^{*})=d^{-1}(g(\rho)+b)$
holds for all finite rank $\rho\in{\mathcal{S}}(H)$. In particular, for any
unit vector $x\in\Lambda$, by Assertion 4.6, $\|g\|<\infty$ and we have
$\|M_{\Lambda}x\|^{2}=d^{-1}(g(x\otimes x)+b)\leq d^{-1}(\|g\|+|b|)<\infty,$
which implies that $\|M_{\Lambda}\|\leq\sqrt{d^{-1}(\|g\|+|b|)}.$ It follows
that, for any unit vector $x\in H$, we have
$\|Tx\|\leq\sqrt{d^{-1}(\|g\|+|b|)}$ and hence
$\|T\|\leq\sqrt{d^{-1}(\|g\|+|b|)}$.
The proof is finished.∎
Now we are in a position to give a proof of the main theorem for infinite
dimensional case.
Proof of Theorem 1: infinite dimensional case. Similar to the finite
dimensional case, we need only to show (b) $\Rightarrow$ (c).
Assume (b). By Lemma 4.2, there is a bounded invertible linear or conjugate
linear operator $T$ such that $\phi(x\otimes x)=\frac{Tx\otimes
Tx}{\|Tx\|^{2}}=\frac{Tx\otimes xT^{*}}{\|Tx\|^{2}}$ for all unit vectors
$x\in H$. Let $\rho$ be any finite rank state. Then there exists a finite
dimensional subspace $\Lambda$ of $H$ such that the range of $\rho$ is
contained in $\Lambda$. By Assertion 4.4 in the proof of Lemma 4.3, we have
$\phi(\rho)=\frac{(Q_{\Lambda}M_{\Lambda})\rho(Q_{\Lambda}M_{\Lambda})^{*}}{{\rm
tr}((Q_{\Lambda}M_{\Lambda})\rho(Q_{\Lambda}M_{\Lambda})^{*})}=\frac{T\rho
T^{*}}{{\rm tr}(T\rho T^{*})}$. Since the set of finite-rank states is dense
in ${\mathcal{S}}(H)$ and, by Lemma 4.3, $\phi$ is continuous, we get that
$\phi(\rho)=\frac{T\rho T^{*}}{{\rm tr}(T^{*}T\rho)}$ for all states $\rho$ as
desired, completing the proof.∎
Acknowldegement
We thank Wen-ling Huang for helpful discussions on the segment preserving
bijections $\hat{\phi}$ in Assertion 3.5.
## References
* [1] E. Alfsen, F. Shultz, Unique decompositions, faces, and automorphisms of separable states, Journal of Mathematical Physics, 51(2010), 052201.
* [2] I. Bengtsson, K. Zyczkowski, Geometry of quantum states: an introduction to quantum entanglement, Cambridge University Press, Cambridge, 2006.
* [3] C.-A. Faure, An elementary proof of the fundamental theorem of projective geometry, Geom. Dedicata, 90(2002), 145-151.
* [4] S. Friedland, C.-K. Li Y.-T. Poon and N.-S. Sze, The automorphism group of separable states in quantum information theory, Journal of Mathematical Physics, 52(2011), 042203.
* [5] S. Gudder, A structure for quantum measurements, Reports on Mathematical Physics, 55(2005) 2, 249-267.
* [6] J. C. Hou, A characterization of positive linear maps and criteria of entanglement for quantum states, J. Phys. A: Math. Theor., 43(2010) 385201.
* [7] J. C. Hou, J. L. Cui, Introduction to linear maps on operator algebras, Scince Press in China, Beijing, 2004.
* [8] L. Moln$\acute{a}$r, Characterizations of the automorphisms of Hilbert space effect algebras, Commun. Math. Phys., 223(2001), 437-450.
* [9] L. Moln$\acute{a}$r, On some automorphisms of the set of effects on Hilbert space. Lett. Math. Phys., 51(2000), 37-45
* [10] L. Moln$\acute{a}$r, W. Timmermann, Mixture preserving maps on von Neumann algebra effects, Lett. Math. Phys., 79(2007), 295-302
* [11] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
* [12] Z. P$\breve{a}$les, Characterization of segment and convexity preserving maps, preprint.
* [13] U. Uhlhorn, Representation of symmetry transformations in quantum mechanics, Ark. Fysik, 23(1963), 307-340.
* [14] E. P. Wigner, Group theory: And its application to the quantum mechanics of atomic spectra, Academic Press, 1959.
* [15] S. Zhu, Z.-H. Ma, Topologies on quantum states, Phys. Lett. A, 374(2010), 1336-1341.
|
arxiv-papers
| 2012-10-01T15:01:10 |
2024-09-04T02:49:35.797577
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kan He, Jin-Chuan Hou and Chi-Kwong Li",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1210.0433"
}
|
1210.0530
|
# Best Practices for Scientific Computing
Greg Wilson Software Carpentry / gvwilson@software-carpentry.org D.A. Aruliah
University of Ontario Institute of Technology / Dhavide.Aruliah@uoit.ca C.
Titus Brown Michigan State University / ctb@msu.edu Neil P. Chue Hong
Software Sustainability Institute / N.ChueHong@epcc.ed.ac.uk Matt Davis
Space Telescope Science Institute / mrdavis@stsci.edu Richard T. Guy
University of Toronto / guy@cs.utoronto.ca Steven H.D. Haddock Monterey Bay
Aquarium Research Institute / steve@practicalcomputing.org Kathryn D. Huff
University of California / huff@berkeley.edu Ian M. Mitchell University of
British Columbia / mitchell@cs.ubc.ca Mark D. Plumbley Queen Mary University
of London / mark.plumbley@eecs.qmul.ac.uk Ben Waugh University College
London / b.waugh@ucl.ac.uk Ethan P. White Utah State University /
ethan@weecology.org Paul Wilson University of Wisconsin /
wilsonp@engr.wisc.edu
###### Abstract
Scientists spend an increasing amount of time building and using software.
However, most scientists are never taught how to do this efficiently. As a
result, many are unaware of tools and practices that would allow them to write
more reliable and maintainable code with less effort. We describe a set of
best practices for scientific software development that have solid foundations
in research and experience, and that improve scientists’ productivity and the
reliability of their software.
Software is as important to modern scientific research as telescopes and test
tubes. From groups that work exclusively on computational problems, to
traditional laboratory and field scientists, more and more of the daily
operation of science revolves around developing new algorithms, managing and
analyzing the large amounts of data that are generated in single research
projects, and combining disparate datasets to assess synthetic problems, and
other computational tasks.
Scientists typically develop their own software for these purposes because
doing so requires substantial domain-specific knowledge. As a result, recent
studies have found that scientists typically spend 30% or more of their time
developing software [19, 53]. However, 90% or more of them are primarily self-
taught [19, 53], and therefore lack exposure to basic software development
practices such as writing maintainable code, using version control and issue
trackers, code reviews, unit testing, and task automation.
We believe that software is just another kind of experimental apparatus [63]
and should be built, checked, and used as carefully as any physical apparatus.
However, while most scientists are careful to validate their laboratory and
field equipment, most do not know how reliable their software is [21, 20].
This can lead to serious errors impacting the central conclusions of published
research [43]: recent high-profile retractions, technical comments, and
corrections because of errors in computational methods include papers in
_Science_ [7, 14], _PNAS_ [39], the _Journal of Molecular Biology_ [6],
_Ecology Letters_ [37, 9], the _Journal of Mammalogy_ [34], _Journal of the
American College of Cardiology_ [1], _Hypertension_ [27] and _The American
Economic Review_ [22].
In addition, because software is often used for more than a single project,
and is often reused by other scientists, computing errors can have
disproportionate impacts on the scientific process. This type of cascading
impact caused several prominent retractions when an error from another group’s
code was not discovered until after publication [43]. As with bench
experiments, not everything must be done to the most exacting standards;
however, scientists need to be aware of best practices both to improve their
own approaches and for reviewing computational work by others.
This paper describes a set of practices that are easy to adopt and have proven
effective in many research settings. Our recommendations are based on several
decades of collective experience both building scientific software and
teaching computing to scientists [2, 65], reports from many other groups [23,
30, 31, 35, 41, 51, 52], guidelines for commercial and open source software
development [62, 15], and on empirical studies of scientific computing [5, 32,
60, 58] and software development in general (summarized in [48]). None of
these practices will guarantee efficient, error-free software development, but
used in concert they will reduce the number of errors in scientific software,
make it easier to reuse, and save the authors of the software time and effort
that can used for focusing on the underlying scientific questions.
For reasons of space, we do not discuss the equally important (but
independent) issues of reproducible research, publication and citation of code
and data, and open science. We do believe, however, that all of these will be
much easier to implement if scientists have the skills we describe.
### Acknowledgments
We are grateful to Joel Adamson, Aron Ahmadia, Roscoe Bartlett, Erik Bray,
Steven Crouch, Michael Jackson, Justin Kitzes, Adam Obeng, Karthik Ram, Yoav
Ram, and Tracy Teal for feedback on this paper.
Neil Chue Hong was supported by the UK Engineering and Physical Sciences
Research Council (EPSRC) Grant EP/H043160/1 for the UK Software Sustainability
Institute.
Ian M. Mitchell was supported by NSERC Discovery Grant #298211.
Mark Plumbley was supported by EPSRC through a Leadership Fellowship
(EP/G007144/1) and a grant (EP/H043101/1) for SoundSoftware.ac.uk.
Ethan White was supported by a CAREER grant from the US National Science
Foundation (DEB 0953694).
Greg Wilson was supported by a grant from the Sloan Foundation.
## 1 Write programs for people, not computers.
Scientists writing software need to write code that both executes correctly
and can be easily read and understood by other programmers (especially the
author’s future self). If software cannot be easily read and understood, it is
much more difficult to know that it is actually doing what it is intended to
do. To be productive, software developers must therefore take several aspects
of human cognition into account: in particular, that human working memory is
limited, human pattern matching abilities are finely tuned, and human
attention span is short [3, 24, 38, 4, 56].
First, _a program should not require its readers to hold more than a handful
of facts in memory at once (1.1)_. Human working memory can hold only a
handful of items at a time, where each item is either a single fact or a
“chunk” aggregating several facts [3, 24], so programs should limit the total
number of items to be remembered to accomplish a task. The primary way to
accomplish this is to break programs up into easily understood functions, each
of which conducts a single, easily understood, task. This serves to make each
piece of the program easier to understand in the same way that breaking up a
scientific paper using sections and paragraphs makes it easier to read. For
example, a function to calculate the area of a rectangle can be written to
take four separate coordinates:
def rect_area(x1, y1, x2, y2):
...calculation...
or to take two points:
def rect_area(point1, point2):
...calculation...
The latter function is significantly easier for people to read and remember,
while the former is likely to lead to errors, not least because it is possible
to call the original with values in the wrong order:
surface = rect_area(x1, x2, y1, y2)
Second, scientists should _make names consistent, distinctive, and meaningful
(1.2)_. For example, using non-descriptive names, like a and foo, or names
that are very similar, like results and results2, is likely to cause
confusion.
Third, scientists should _make code style and formatting consistent (1.3)_. If
different parts of a scientific paper used different formatting and
capitalization, it would make that paper more difficult to read. Likewise, if
different parts of a program are indented differently, or if programmers mix
CamelCaseNaming and pothole_case_naming, code takes longer to read and readers
make more mistakes [38, 4].
## 2 Let the computer do the work.
Science often involves repetition of computational tasks such as processing
large numbers of data files in the same way or regenerating figures each time
new data is added to an existing analysis. Computers were invented to do these
kinds of repetitive tasks but, even today, many scientists type the same
commands in over and over again or click the same buttons repeatedly [2]. In
addition to wasting time, sooner or later even the most careful researcher
will lose focus while doing this and make mistakes.
Scientists should therefore _make the computer repeat tasks (2.1)_ and _save
recent commands in a file for re-use (2.2)_. For example, most command-line
tools have a “history” option that lets users display and re-execute recent
commands, with minor edits to filenames or parameters. This is often cited as
one reason command-line interfaces remain popular [55, 18]: “do this again”
saves time and reduces errors.
A file containing commands for an interactive system is often called a
_script_ , though there is real no difference between this and a program. When
these scripts are repeatedly used in the same way, or in combination, a
workflow management tool can be used. The paradigmatic example is compiling
and linking programs in languages such as Fortran, C++, Java, and C# [12]. The
most widely used tool for this task is probably
Make111http://www.gnu.org/software/make, although many alternatives are now
available [61]. All of these allow people to express dependencies between
files, i.e., to say that if A or B has changed, then C needs to be updated
using a specific set of commands. These tools have been successfully adopted
for scientific workflows as well [16].
To avoid errors and inefficiencies from repeating commands manually, we
recommend that scientists _use a build tool to automate workflows (2.3)_ ,
e.g., specify the ways in which intermediate data files and final results
depend on each other, and on the programs that create them, so that a single
command will regenerate anything that needs to be regenerated.
In order to maximize reproducibility, everything needed to re-create the
output should be recorded automatically in a format that other programs can
read. (Borrowing a term from archaeology and forensics, this is often called
the _provenance_ of data.) There have been some initiatives to automate the
collection of this information, and standardize its format [47], but it is
already possible to record the following without additional tools:
* •
unique identifiers and version numbers for raw data records (which scientists
may need to create themselves);
* •
unique identifiers and version numbers for programs and libraries;
* •
the values of parameters used to generate any given output; and
* •
the names and version numbers of programs (however small) used to generate
those outputs.
## 3 Make incremental changes.
Unlike traditional commercial software developers, but very much like
developers in open source projects or startups, scientific programmers usually
don’t get their requirements from customers, and their requirements are rarely
frozen [58, 59]. In fact, scientists often _can’t_ know what their programs
should do next until the current version has produced some results. This
challenges design approaches that rely on specifying requirements in advance.
Programmers are most productive when they _work in small steps with frequent
feedback and course correction (3.1)_ rather than trying to plan months or
years of work in advance. While the details vary from team to team, these
developers typically work in steps that are sized to be about an hour long,
and these steps are often grouped in iterations that last roughly one week.
This accommodates the cognitive constraints discussed in Section 1, and
acknowledges the reality that real-world requirements are constantly changing.
The goal is to produce working (if incomplete) code after each iteration.
While these practices have been around for decades, they gained prominence
starting in the late 1990s under the banner of _agile development_ [40, 36].
Two of the biggest challenges scientists and other programmers face when
working with code and data are keeping track of changes (and being able to
revert them if things go wrong), and collaborating on a program or dataset
[41]. Typical “solutions” are to email software to colleagues or to copy
successive versions of it to a shared folder, e.g.,
Dropbox222http://www.dropbox.com. However, both approaches are fragile and can
lead to confusion and lost work when important changes are overwritten or out-
of-date files are used. It’s also difficult to find out which changes are in
which versions or to say exactly how particular results were computed at a
later date.
The standard solution in both industry and open source is to _use a version
control system (3.2)_ (VCS) [42, 15]. A VCS stores snapshots of a project’s
files in a _repository_ (or a set of repositories). Programmers can modify
their working copy of the project at will, then _commit_ changes to the
repository when they are satisfied with the results to share them with
colleagues.
Crucially, if several people have edited files simultaneously, the VCS
highlights the differences and requires them to resolve any conflicts before
accepting the changes. The VCS also stores the entire history of those files,
allowing arbitrary versions to be retrieved and compared, together with
metadata such as comments on what was changed and the author of the changes.
All of this information can be extracted to provide provenance for both code
and data.
Many good VCSes are open source and freely available, including
Subversion333http://subversion.apache.org, Git444http://git-scm.com, and
Mercurial555http://mercurial.selenic.com. Many free hosting services are
available as well (SourceForge666http://sourceforge.net, Google
Code777http://code.google.com, GitHub888https://github.com, and
BitBucket999https://bitbucket.org being the most popular). As with coding
style, the best one to use is almost always whatever your colleagues are
already using [15].
Reproducibility is maximized when scientists _put everything that has been
created manually in version control (3.3)_ , including programs, original
field observations, and the source files for papers. Automated output and
intermediate files can be regenerated at need. Binary files (e.g., images and
audio clips) may be stored in version control, but it is often more sensible
to use an archiving system for them, and store the metadata describing their
contents in version control instead [45].
## 4 Don’t repeat yourself (or others).
Anything that is repeated in two or more places is more difficult to maintain.
Every time a change or correction is made, multiple locations must be updated,
which increases the chance of errors and inconsistencies. To avoid this,
programmers follow the DRY Principle [26], for “don’t repeat yourself”, which
applies to both data and code.
For data, this maxim holds that _every piece of data must have a single
authoritative representation in the system (4.1)_. Physical constants ought to
be defined exactly once to ensure that the entire program is using the same
value; raw data files should have a single canonical version, every geographic
location from which data has been collected should be given an ID that can be
used to look up its latitude and longitude, and so on.
The DRY Principle applies to code at two scales. At small scales, _modularize
code rather than copying and pasting (4.2)_. Avoiding “code clones” has been
shown to reduce error rates [29]: when a change is made or a bug is fixed,
that change or fix takes effect everywhere, and people’s mental model of the
program (i.e., their belief that “this one’s been fixed”) remains accurate. As
a side effect, modularizing code allows people to remember its functionality
as a single mental chunk, which in turn makes code easier to understand.
Modularized code can also be more easily repurposed for other projects.
At larger scales, it is vital that scientific programmers _re-use code instead
of rewriting it (4.3)_. Tens of millions of lines of high-quality open source
software are freely available on the web, and at least as much is available
commercially. It is typically better to find an established library or package
that solves a problem than to attempt to write one’s own routines for well
established problems (e.g., numerical integration, matrix inversions, etc.).
## 5 Plan for mistakes.
Mistakes are inevitable, so verifying and maintaining the validity of code
over time is immensely challenging [17]. While no single practice has been
shown to catch or prevent all mistakes, several are very effective when used
in combination [42, 11, 57].
The first line of defense is _defensive programming_. Experienced programmers
_add assertions to programs to check their operation (5.1)_ because experience
has taught them that everyone (including their future self) makes mistakes. An
_assertion_ is simply a statement that something holds true at a particular
point in a program; as the example below shows, assertions can be used to
ensure that inputs are valid, outputs are consistent, and so
on101010Assertions do not require language support: it is common in languages
such as Fortran for programmers to create their own test-and-fail functions
for this purpose..
def bradford_transfer(grid, point, smoothing):
assert grid.contains(point),
’Point is not located in grid’
assert grid.is_local_maximum(point),
’Point is not a local maximum in grid’
assert len(smoothing) > FILTER_LENGTH,
’Not enough smoothing parameters’
...do calculations...
assert 0.0 < result <= 1.0,
’Bradford transfer value out of legal range’
return result
Assertions can make up a sizeable fraction of the code in well-written
applications, just as tools for calibrating scientific instruments can make up
a sizeable fraction of the equipment in a lab. These assertions serve two
purposes. First, they ensure that if something does go wrong, the program will
halt immediately, which simplifies debugging. Second, assertions are
_executable documentation_ , i.e., they explain the program as well as
checking its behavior. This makes them more useful in many cases than comments
since the reader can be sure that they are accurate and up to date.
The second layer of defense is _automated testing_. Automated tests can check
to make sure that a single unit of code is returning correct results (_unit
tests_), that pieces of code work correctly when combined (_integration
tests_), and that the behavior of a program doesn’t change when the details
are modified (_regression tests_). These tests are conducted by the computer,
so that they are easy to rerun every time the program is modified. Creating
and managing tests is easier if programmers _use an off-the-shelf unit testing
library (5.2)_ to initialize inputs, run tests, and report their results in a
uniform way. These libraries are available for all major programming languages
including those commonly used in scientific computing [68, 44, 49].
Tests check to see whether the code matches the researcher’s expectations of
its behavior, which depends on the researcher’s understanding of the problem
at hand [25, 33, 46]. For example, in scientific computing, tests are often
conducted by comparing output to simplified cases, experimental data, or the
results of earlier programs that are trusted. Another approach for generating
tests is to _turn bugs into test cases (5.3)_ by writing tests that trigger a
bug that has been found in the code and (once fixed) will prevent the bug from
reappearing unnoticed. In combination these kinds of testing can improve our
confidence that scientific code is operating properly and that the results it
produces are valid. An additional benefit of testing is that it encourages
programmers to design and build code that is testable (i.e., self-contained
functions and classes that can run more or less independently of one another).
Code that is designed this way is also easier to understand (Section 1) and
more reusable (Section 4).
Now matter how good ones computational practice is, reasonably complex code
will always initially contain bugs. Fixing bugs that have been identified is
often easier if you _use a symbolic debugger (5.4)_ to track them down. A
better name for this kind of tool would be “interactive program inspector”
since a debugger allows users to pause a program at any line (or when some
condition is true), inspect the values of variables, and walk up and down
active function calls to figure out why things are behaving the way they are.
Debuggers are usually more productive than adding and removing print
statements or scrolling through hundreds of lines of log output [69], because
they allow the user to see exactly how the code is executing rather than just
snapshots of state of the program at a few moments in time. In other words,
the debugger allows the scientist to witness what is going wrong directly,
rather than having to anticipate the error or infer the problem using indirect
evidence.
## 6 Optimize software only after it works correctly.
Today’s computers and software are so complex that even experts find it hard
to predict which parts of any particular program will be performance
bottlenecks [28]. The most productive way to make code fast is therefore to
make it work correctly, determine whether it’s actually worth speeding it up,
and—in those cases where it is—to _use a profiler to identify bottlenecks
(6.1)_.
This strategy also has interesting implications for choice of programming
language. Research has confirmed that most programmers write roughly the same
number of lines of code per unit time regardless of the language they use
[54]. Since faster, lower level, languages require more lines of code to
accomplish the same task, scientists are most productive when they _write code
in the highest-level language possible (6.2)_ , and shift to low-level
languages like C and Fortran only when they are sure the performance boost is
needed111111Using higher-level languages also helps program comprehensibility,
since such languages have, in a sense, “pre-chunked” the facts that
programmers need to have in short-term memory. Taking this approach allows
more code to be written (and tested) in the same amount of time. Even when it
is known before coding begins that a low-level language will ultimately be
necessary, rapid prototyping in a high-level language helps programmers make
and evaluate design decisions quickly. Programmers can also use a high-level
prototype as a test oracle for a high-performance low-level reimplementation,
i.e., compare the output of the optimized (and usually more complex) program
against the output from its unoptimized (but usually simpler) predecessor in
order to check its correctness.
## 7 Document design and purpose, not mechanics.
In the same way that a well documented experimental protocol makes research
methods easier to reproduce, good documentation helps people understand code.
This makes the code more reusable and lowers maintenance costs [42]. As a
result, code that is well documented makes it easier to transition when the
graduate students and postdocs who have been writing code in a lab transition
to the next career phase. Reference documentation and descriptions of design
decisions are key for improving the understandability of code. However, inline
documentation that recapitulates code is _not_ useful. Therefore we recommend
that scientific programmers _document interfaces and reasons, not
implementations (7.1)_. For example, a clear description like this at the
beginning of a function that describes what it does and its inputs and outputs
is useful:
def scan(op, values, seed=None):
# Apply a binary operator cumulatively to the values given
# from lowest to highest, returning a list of results.
# For example, if ’op’ is ’add’ and ’values’ is ’[1, 3, 5]’,
# the result is ’[1, 4, 9]’ (i.e., the running total of the
# given values). The result always has the same length as
# the input.
# If ’seed’ is given, the result is initialized with that
# value instead of with the first item in ’values’, and
# the final item is omitted from the result.
# Ex: scan(add, [1, 3, 5], seed=10) => [10, 11, 14]
...implementation...
In contrast, the comment in the code fragment below does nothing to aid
comprehension:
i = i + 1 # Increment the variable ’i’ by one.
If a substantial description of the implementation of a piece of software is
needed, it is better to _refactor code in preference to explaining how it
works (7.2)_ , i.e., rather than write a paragraph to explain a complex piece
of code, reorganize the code itself so that it doesn’t need such an
explanation. This may not always be possible—some pieces of code simply are
intrinsically difficult—but the onus should always be on the author to
convince his or her peers of that.
The best way to create and maintain reference documentation is to _embed the
documentation for a piece of software in that software (7.3)_. Doing this
increases the probability that when programmers change the code, they will
update the documentation at the same time.
Embedded documentation usually takes the form of specially-formatted and
placed comments. Typically, a _documentation generator_ such as Javadoc,
Doxygen, or
Sphinx121212http://en.wikipedia.org/wiki/Comparison_of_documentation_generators
extracts these comments and generates well-formatted web pages and other
human-friendly documents. Alternatively, code can be embedded in a larger
document that includes information about what the code is doing (i.e.,
literate programming). Common approaches to this include this use of knitr
[67] and IPython Notebooks [50].
## 8 Collaborate.
In the same way that having manuscripts reviewed by other scientists can
reduce errors and make research easier to understand, reviews of source code
can eliminate bugs and improve readability. A large body of research has shown
that _code reviews_ are the most cost-effective way of finding bugs in code
[13, 8]. They are also a good way to spread knowledge and good practices
around a team. In projects with shifting membership, such as most academic
labs, code reviews help ensure that critical knowledge isn’t lost when a
student or postdoc leaves the lab.
Code can be reviewed either before or after it has been committed to a shared
version control repository. Experience shows that if reviews don’t have to be
done in order to get code into the repository, they will soon not be done at
all [15]. We therefore recommend that projects _use pre-merge code reviews
(8.1)_.
An extreme form of code review is _pair programming_ , in which two developers
sit together while writing code. One (the driver) actually writes the code;
the other (the navigator) provides real-time feedback and is free to track
larger issues of design and consistency. Several studies have found that pair
programming improves productivity [64], but many programmers find it
intrusive. We therefore recommend that teams _use pair programming when
bringing someone new up to speed and when tackling particularly tricky
problems (8.2)_.
Once a team grows beyond a certain size, it becomes difficult to keep track of
what needs to be reviewed, or of who’s doing what. Teams can avoid a lot of
duplicated effort and dropped balls if they _use an issue tracking tool (8.3)_
to maintain a list of tasks to be performed and bugs to be fixed [10]. This
helps avoid duplicated work and makes it easier for tasks to be transferred to
different people. Free repository hosting services like GitHub include issue
tracking tools, and many good standalone tools exist as well, such as
Trac131313http://trac.edgewall.org.
## 9 Conclusion
We have outlined a series of recommended best practices for scientific
computing based on extensive research, as well as our collective experience.
These practices can be applied to individual work as readily as group work;
separately and together, they improve the productivity of scientific
programming and the reliability of the resulting code, and therefore the speed
with which we produce results and our confidence in them. They are also, we
believe, prerequisites for reproducible computational research: if software is
not version controlled, readable, and tested, the chances of its authors (much
less anyone else) being able to re-create results are remote.
Our 25 recommendations are a beginning, not an end. Individuals and groups who
have incorporated them into their work will find links to more advanced
practices on the Software Carpentry website141414http://software-
carpentry.org/biblio.html.
Research suggests that the time cost of implementing these kinds of tools and
approaches in scientific computing is almost immediately offset by the gains
in productivity of the programmers involved [2]. Even so, the recommendations
described above may seem intimidating to implement. Fortunately, the different
practices reinforce and support one another, so the effort required is less
than the sum of adding each component separately. Nevertheless, we do not
recommend that research groups attempt to implement all of these
recommendations at once, but instead suggest that these tools be introduced
incrementally over a period of time.
How to implement the recommended practices can be learned from many excellent
tutorials available online or through workshops and classes organized by
groups like Software Carpentry151515http://software-carpentry.org. This type
of training has proven effective at driving adoption of these tools in
scientific settings [2, 66].
For computing to achieve the level of rigor that is expected throughout other
parts of science, it is necessary for scientists to begin to adopt the tools
and approaches that are known to improve both the quality of software and the
efficiency with which it is produced. To facilitate this adoption,
universities and funding agencies need to support the training of scientists
in the use of these tools and the investment of time and money in building
better scientific software. Investment in these approaches by both individuals
and institutions will improve our confidence in the results of computational
science and will allow us to make more rapid progress on important scientific
questions than would otherwise be possible.
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## Reference
1. 1.
Write programs for people, not computers.
1. (a)
A program should not require its readers to hold more than a handful of facts
in memory at once.
2. (b)
Make names consistent, distinctive, and meaningful.
3. (c)
Make code style and formatting consistent.
2. 2.
Let the computer do the work.
1. (a)
Make the computer repeat tasks.
2. (b)
Save recent commands in a file for re-use.
3. (c)
Use a build tool to automate workflows.
3. 3.
Make incremental changes.
1. (a)
Work in small steps with frequent feedback and course correction.
2. (b)
Use a version control system.
3. (c)
Put everything that has been created manually in version control.
4. 4.
Don’t repeat yourself (or others).
1. (a)
Every piece of data must have a single authoritative representation in the
system.
2. (b)
Modularize code rather than copying and pasting.
3. (c)
Re-use code instead of rewriting it.
5. 5.
Plan for mistakes.
1. (a)
Add assertions to programs to check their operation.
2. (b)
Use an off-the-shelf unit testing library.
3. (c)
Turn bugs into test cases.
4. (d)
Use a symbolic debugger.
6. 6.
Optimize software only after it works correctly.
1. (a)
Use a profiler to identify bottlenecks.
2. (b)
Write code in the highest-level language possible.
7. 7.
Document design and purpose, not mechanics.
1. (a)
Document interfaces and reasons, not implementations.
2. (b)
Refactor code in preference to explaining how it works.
3. (c)
Embed the documentation for a piece of software in that software.
8. 8.
Collaborate.
1. (a)
Use pre-merge code reviews.
2. (b)
Use pair programming when bringing someone new up to speed and when tackling
particularly tricky problems.
3. (c)
Use an issue tracking tool.
|
arxiv-papers
| 2012-10-01T01:04:04 |
2024-09-04T02:49:35.809856
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Greg Wilson, D. A. Aruliah, C. Titus Brown, Neil P. Chue Hong, Matt\n Davis, Richard T. Guy, Steven H. D. Haddock, Katy Huff, Ian M. Mitchell, Mark\n Plumbley, Ben Waugh, Ethan P. White, Paul Wilson",
"submitter": "Greg Wilson",
"url": "https://arxiv.org/abs/1210.0530"
}
|
1210.0627
|
FERMILAB-PUB-12-541-E
The D0 Collaboration111with visitors from aAugustana College, Sioux Falls, SD,
USA, bThe University of Liverpool, Liverpool, UK, cUPIITA-IPN, Mexico City,
Mexico, dDESY, Hamburg, Germany, eSLAC, Menlo Park, CA, USA, fUniversity
College London, London, UK, gCentro de Investigacion en Computacion - IPN,
Mexico City, Mexico, hECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico
and iUniversidade Estadual Paulista, São Paulo, Brazil.
# Measurement of the $\bm{p\bar{p}\to W+b+X}$ production cross section at
$\bm{\sqrt{s}=1.96}$ TeV
V.M. Abazov Joint Institute for Nuclear Research, Dubna, Russia B. Abbott
University of Oklahoma, Norman, Oklahoma 73019, USA B.S. Acharya Tata
Institute of Fundamental Research, Mumbai, India M. Adams University of
Illinois at Chicago, Chicago, Illinois 60607, USA T. Adams Florida State
University, Tallahassee, Florida 32306, USA G.D. Alexeev Joint Institute for
Nuclear Research, Dubna, Russia G. Alkhazov Petersburg Nuclear Physics
Institute, St. Petersburg, Russia A. Altona University of Michigan, Ann
Arbor, Michigan 48109, USA A. Askew Florida State University, Tallahassee,
Florida 32306, USA S. Atkins Louisiana Tech University, Ruston, Louisiana
71272, USA K. Augsten Czech Technical University in Prague, Prague, Czech
Republic C. Avila Universidad de los Andes, Bogotá, Colombia F. Badaud
LPC, Université Blaise Pascal, CNRS/IN2P3, Clermont, France L. Bagby Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA B. Baldin
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA D.V.
Bandurin Florida State University, Tallahassee, Florida 32306, USA S.
Banerjee Tata Institute of Fundamental Research, Mumbai, India E. Barberis
Northeastern University, Boston, Massachusetts 02115, USA P. Baringer
University of Kansas, Lawrence, Kansas 66045, USA J.F. Bartlett Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA U. Bassler
CEA, Irfu, SPP, Saclay, France V. Bazterra University of Illinois at
Chicago, Chicago, Illinois 60607, USA A. Bean University of Kansas,
Lawrence, Kansas 66045, USA M. Begalli Universidade do Estado do Rio de
Janeiro, Rio de Janeiro, Brazil L. Bellantoni Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA S.B. Beri Panjab University,
Chandigarh, India G. Bernardi LPNHE, Universités Paris VI and VII,
CNRS/IN2P3, Paris, France R. Bernhard Physikalisches Institut, Universität
Freiburg, Freiburg, Germany I. Bertram Lancaster University, Lancaster LA1
4YB, United Kingdom M. Besançon CEA, Irfu, SPP, Saclay, France R.
Beuselinck Imperial College London, London SW7 2AZ, United Kingdom P.C. Bhat
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA S. Bhatia
University of Mississippi, University, Mississippi 38677, USA V. Bhatnagar
Panjab University, Chandigarh, India G. Blazey Northern Illinois University,
DeKalb, Illinois 60115, USA S. Blessing Florida State University,
Tallahassee, Florida 32306, USA K. Bloom University of Nebraska, Lincoln,
Nebraska 68588, USA A. Boehnlein Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA D. Boline State University of New York, Stony
Brook, New York 11794, USA E.E. Boos Moscow State University, Moscow, Russia
G. Borissov Lancaster University, Lancaster LA1 4YB, United Kingdom A.
Brandt University of Texas, Arlington, Texas 76019, USA O. Brandt II.
Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen,
Germany R. Brock Michigan State University, East Lansing, Michigan 48824,
USA A. Bross Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA D. Brown LPNHE, Universités Paris VI and VII, CNRS/IN2P3, Paris, France
J. Brown LPNHE, Universités Paris VI and VII, CNRS/IN2P3, Paris, France X.B.
Bu Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA M.
Buehler Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
V. Buescher Institut für Physik, Universität Mainz, Mainz, Germany V.
Bunichev Moscow State University, Moscow, Russia S. Burdinb Lancaster
University, Lancaster LA1 4YB, United Kingdom C.P. Buszello Uppsala
University, Uppsala, Sweden E. Camacho-Pérez CINVESTAV, Mexico City, Mexico
B.C.K. Casey Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA H. Castilla-Valdez CINVESTAV, Mexico City, Mexico S. Caughron Michigan
State University, East Lansing, Michigan 48824, USA S. Chakrabarti State
University of New York, Stony Brook, New York 11794, USA D. Chakraborty
Northern Illinois University, DeKalb, Illinois 60115, USA K.M. Chan
University of Notre Dame, Notre Dame, Indiana 46556, USA A. Chandra Rice
University, Houston, Texas 77005, USA E. Chapon CEA, Irfu, SPP, Saclay,
France G. Chen University of Kansas, Lawrence, Kansas 66045, USA S.
Chevalier-Théry CEA, Irfu, SPP, Saclay, France S.W. Cho Korea Detector
Laboratory, Korea University, Seoul, Korea S. Choi Korea Detector
Laboratory, Korea University, Seoul, Korea B. Choudhary Delhi University,
Delhi, India S. Cihangir Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA D. Claes University of Nebraska, Lincoln, Nebraska
68588, USA J. Clutter University of Kansas, Lawrence, Kansas 66045, USA M.
Cooke Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
W.E. Cooper Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA M. Corcoran Rice University, Houston, Texas 77005, USA F. Couderc CEA,
Irfu, SPP, Saclay, France M.-C. Cousinou CPPM, Aix-Marseille Université,
CNRS/IN2P3, Marseille, France A. Croc CEA, Irfu, SPP, Saclay, France D.
Cutts Brown University, Providence, Rhode Island 02912, USA A. Das
University of Arizona, Tucson, Arizona 85721, USA G. Davies Imperial College
London, London SW7 2AZ, United Kingdom S.J. de Jong Nikhef, Science Park,
Amsterdam, the Netherlands Radboud University Nijmegen, Nijmegen, the
Netherlands E. De La Cruz-Burelo CINVESTAV, Mexico City, Mexico F. Déliot
CEA, Irfu, SPP, Saclay, France R. Demina University of Rochester, Rochester,
New York 14627, USA D. Denisov Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA S.P. Denisov Institute for High Energy Physics,
Protvino, Russia S. Desai Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA C. Deterre CEA, Irfu, SPP, Saclay, France K. DeVaughan
University of Nebraska, Lincoln, Nebraska 68588, USA H.T. Diehl Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA M. Diesburg
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA P.F. Ding
The University of Manchester, Manchester M13 9PL, United Kingdom A. Dominguez
University of Nebraska, Lincoln, Nebraska 68588, USA A. Dubey Delhi
University, Delhi, India L.V. Dudko Moscow State University, Moscow, Russia
D. Duggan Rutgers University, Piscataway, New Jersey 08855, USA A. Duperrin
CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France S. Dutt Panjab
University, Chandigarh, India A. Dyshkant Northern Illinois University,
DeKalb, Illinois 60115, USA M. Eads University of Nebraska, Lincoln,
Nebraska 68588, USA D. Edmunds Michigan State University, East Lansing,
Michigan 48824, USA J. Ellison University of California Riverside,
Riverside, California 92521, USA V.D. Elvira Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA Y. Enari LPNHE, Universités Paris
VI and VII, CNRS/IN2P3, Paris, France H. Evans Indiana University,
Bloomington, Indiana 47405, USA A. Evdokimov Brookhaven National Laboratory,
Upton, New York 11973, USA V.N. Evdokimov Institute for High Energy Physics,
Protvino, Russia G. Facini Northeastern University, Boston, Massachusetts
02115, USA L. Feng Northern Illinois University, DeKalb, Illinois 60115, USA
T. Ferbel University of Rochester, Rochester, New York 14627, USA F. Fiedler
Institut für Physik, Universität Mainz, Mainz, Germany F. Filthaut Nikhef,
Science Park, Amsterdam, the Netherlands Radboud University Nijmegen,
Nijmegen, the Netherlands W. Fisher Michigan State University, East Lansing,
Michigan 48824, USA H.E. Fisk Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA M. Fortner Northern Illinois University,
DeKalb, Illinois 60115, USA H. Fox Lancaster University, Lancaster LA1 4YB,
United Kingdom S. Fuess Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA A. Garcia-Bellido University of Rochester, Rochester,
New York 14627, USA J.A. García-González CINVESTAV, Mexico City, Mexico
G.A. García-Guerrac CINVESTAV, Mexico City, Mexico V. Gavrilov Institute for
Theoretical and Experimental Physics, Moscow, Russia P. Gay LPC, Université
Blaise Pascal, CNRS/IN2P3, Clermont, France W. Geng CPPM, Aix-Marseille
Université, CNRS/IN2P3, Marseille, France Michigan State University, East
Lansing, Michigan 48824, USA D. Gerbaudo Princeton University, Princeton,
New Jersey 08544, USA C.E. Gerber University of Illinois at Chicago,
Chicago, Illinois 60607, USA Y. Gershtein Rutgers University, Piscataway,
New Jersey 08855, USA G. Ginther Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA University of Rochester, Rochester, New York
14627, USA G. Golovanov Joint Institute for Nuclear Research, Dubna, Russia
A. Goussiou University of Washington, Seattle, Washington 98195, USA P.D.
Grannis State University of New York, Stony Brook, New York 11794, USA S.
Greder IPHC, Université de Strasbourg, CNRS/IN2P3, Strasbourg, France H.
Greenlee Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
G. Grenier IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and
Université de Lyon, Lyon, France Ph. Gris LPC, Université Blaise Pascal,
CNRS/IN2P3, Clermont, France J.-F. Grivaz LAL, Université Paris-Sud,
CNRS/IN2P3, Orsay, France A. Grohsjeand CEA, Irfu, SPP, Saclay, France S.
Grünendahl Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA M.W. Grünewald University College Dublin, Dublin, Ireland T. Guillemin
LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France G. Gutierrez Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA P. Gutierrez
University of Oklahoma, Norman, Oklahoma 73019, USA J. Haley Northeastern
University, Boston, Massachusetts 02115, USA L. Han University of Science
and Technology of China, Hefei, People’s Republic of China K. Harder The
University of Manchester, Manchester M13 9PL, United Kingdom A. Harel
University of Rochester, Rochester, New York 14627, USA J.M. Hauptman Iowa
State University, Ames, Iowa 50011, USA J. Hays Imperial College London,
London SW7 2AZ, United Kingdom T. Head The University of Manchester,
Manchester M13 9PL, United Kingdom T. Hebbeker III. Physikalisches Institut
A, RWTH Aachen University, Aachen, Germany D. Hedin Northern Illinois
University, DeKalb, Illinois 60115, USA H. Hegab Oklahoma State University,
Stillwater, Oklahoma 74078, USA A.P. Heinson University of California
Riverside, Riverside, California 92521, USA U. Heintz Brown University,
Providence, Rhode Island 02912, USA C. Hensel II. Physikalisches Institut,
Georg-August-Universität Göttingen, Göttingen, Germany I. Heredia-De La Cruz
CINVESTAV, Mexico City, Mexico K. Herner University of Michigan, Ann Arbor,
Michigan 48109, USA G. Heskethf The University of Manchester, Manchester M13
9PL, United Kingdom M.D. Hildreth University of Notre Dame, Notre Dame,
Indiana 46556, USA R. Hirosky University of Virginia, Charlottesville,
Virginia 22904, USA T. Hoang Florida State University, Tallahassee, Florida
32306, USA J.D. Hobbs State University of New York, Stony Brook, New York
11794, USA B. Hoeneisen Universidad San Francisco de Quito, Quito, Ecuador
J. Hogan Rice University, Houston, Texas 77005, USA M. Hohlfeld Institut
für Physik, Universität Mainz, Mainz, Germany I. Howley University of Texas,
Arlington, Texas 76019, USA Z. Hubacek Czech Technical University in Prague,
Prague, Czech Republic CEA, Irfu, SPP, Saclay, France V. Hynek Czech
Technical University in Prague, Prague, Czech Republic I. Iashvili State
University of New York, Buffalo, New York 14260, USA Y. Ilchenko Southern
Methodist University, Dallas, Texas 75275, USA R. Illingworth Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A.S. Ito Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA S. Jabeen Brown
University, Providence, Rhode Island 02912, USA M. Jaffré LAL, Université
Paris-Sud, CNRS/IN2P3, Orsay, France A. Jayasinghe University of Oklahoma,
Norman, Oklahoma 73019, USA M.S. Jeong Korea Detector Laboratory, Korea
University, Seoul, Korea R. Jesik Imperial College London, London SW7 2AZ,
United Kingdom P. Jiang University of Science and Technology of China,
Hefei, People’s Republic of China K. Johns University of Arizona, Tucson,
Arizona 85721, USA E. Johnson Michigan State University, East Lansing,
Michigan 48824, USA M. Johnson Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA A. Jonckheere Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA P. Jonsson Imperial College London,
London SW7 2AZ, United Kingdom J. Joshi University of California Riverside,
Riverside, California 92521, USA A.W. Jung Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A. Juste Institució Catalana de
Recerca i Estudis Avançats (ICREA) and Institut de Física d’Altes Energies
(IFAE), Barcelona, Spain E. Kajfasz CPPM, Aix-Marseille Université,
CNRS/IN2P3, Marseille, France D. Karmanov Moscow State University, Moscow,
Russia P.A. Kasper Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA I. Katsanos University of Nebraska, Lincoln, Nebraska 68588, USA
R. Kehoe Southern Methodist University, Dallas, Texas 75275, USA S. Kermiche
CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France N. Khalatyan
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA A. Khanov
Oklahoma State University, Stillwater, Oklahoma 74078, USA A. Kharchilava
State University of New York, Buffalo, New York 14260, USA Y.N. Kharzheev
Joint Institute for Nuclear Research, Dubna, Russia I. Kiselevich Institute
for Theoretical and Experimental Physics, Moscow, Russia J.M. Kohli Panjab
University, Chandigarh, India A.V. Kozelov Institute for High Energy
Physics, Protvino, Russia J. Kraus University of Mississippi, University,
Mississippi 38677, USA A. Kumar State University of New York, Buffalo, New
York 14260, USA A. Kupco Center for Particle Physics, Institute of Physics,
Academy of Sciences of the Czech Republic, Prague, Czech Republic T. Kurča
IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and Université de
Lyon, Lyon, France V.A. Kuzmin Moscow State University, Moscow, Russia S.
Lammers Indiana University, Bloomington, Indiana 47405, USA G. Landsberg
Brown University, Providence, Rhode Island 02912, USA P. Lebrun IPNL,
Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and Université de Lyon,
Lyon, France H.S. Lee Korea Detector Laboratory, Korea University, Seoul,
Korea S.W. Lee Iowa State University, Ames, Iowa 50011, USA W.M. Lee Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA X. Lei
University of Arizona, Tucson, Arizona 85721, USA J. Lellouch LPNHE,
Universités Paris VI and VII, CNRS/IN2P3, Paris, France D. Li LPNHE,
Universités Paris VI and VII, CNRS/IN2P3, Paris, France H. Li LPSC,
Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National
Polytechnique de Grenoble, Grenoble, France L. Li University of California
Riverside, Riverside, California 92521, USA Q.Z. Li Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J.K. Lim Korea Detector
Laboratory, Korea University, Seoul, Korea D. Lincoln Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Linnemann Michigan
State University, East Lansing, Michigan 48824, USA V.V. Lipaev Institute
for High Energy Physics, Protvino, Russia R. Lipton Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA H. Liu Southern
Methodist University, Dallas, Texas 75275, USA Y. Liu University of Science
and Technology of China, Hefei, People’s Republic of China A. Lobodenko
Petersburg Nuclear Physics Institute, St. Petersburg, Russia M. Lokajicek
Center for Particle Physics, Institute of Physics, Academy of Sciences of the
Czech Republic, Prague, Czech Republic R. Lopes de Sa State University of
New York, Stony Brook, New York 11794, USA H.J. Lubatti University of
Washington, Seattle, Washington 98195, USA R. Luna-Garciag CINVESTAV, Mexico
City, Mexico A.L. Lyon Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA A.K.A. Maciel LAFEX, Centro Brasileiro de Pesquisas
Físicas, Rio de Janeiro, Brazil R. Madar Physikalisches Institut,
Universität Freiburg, Freiburg, Germany R. Magaña-Villalba CINVESTAV, Mexico
City, Mexico S. Malik University of Nebraska, Lincoln, Nebraska 68588, USA
V.L. Malyshev Joint Institute for Nuclear Research, Dubna, Russia Y. Maravin
Kansas State University, Manhattan, Kansas 66506, USA J. Martínez-Ortega
CINVESTAV, Mexico City, Mexico R. McCarthy State University of New York,
Stony Brook, New York 11794, USA C.L. McGivern The University of Manchester,
Manchester M13 9PL, United Kingdom M.M. Meijer Nikhef, Science Park,
Amsterdam, the Netherlands Radboud University Nijmegen, Nijmegen, the
Netherlands A. Melnitchouk Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA D. Menezes Northern Illinois University, DeKalb,
Illinois 60115, USA P.G. Mercadante Universidade Federal do ABC, Santo
André, Brazil M. Merkin Moscow State University, Moscow, Russia A. Meyer
III. Physikalisches Institut A, RWTH Aachen University, Aachen, Germany J.
Meyer II. Physikalisches Institut, Georg-August-Universität Göttingen,
Göttingen, Germany F. Miconi IPHC, Université de Strasbourg, CNRS/IN2P3,
Strasbourg, France N.K. Mondal Tata Institute of Fundamental Research,
Mumbai, India M. Mulhearn University of Virginia, Charlottesville, Virginia
22904, USA E. Nagy CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille,
France M. Naimuddin Delhi University, Delhi, India M. Narain Brown
University, Providence, Rhode Island 02912, USA R. Nayyar University of
Arizona, Tucson, Arizona 85721, USA H.A. Neal University of Michigan, Ann
Arbor, Michigan 48109, USA J.P. Negret Universidad de los Andes, Bogotá,
Colombia P. Neustroev Petersburg Nuclear Physics Institute, St. Petersburg,
Russia H.T. Nguyen University of Virginia, Charlottesville, Virginia 22904,
USA T. Nunnemann Ludwig-Maximilians-Universität München, München, Germany
J. Orduna Rice University, Houston, Texas 77005, USA N. Osman CPPM, Aix-
Marseille Université, CNRS/IN2P3, Marseille, France J. Osta University of
Notre Dame, Notre Dame, Indiana 46556, USA M. Padilla University of
California Riverside, Riverside, California 92521, USA A. Pal University of
Texas, Arlington, Texas 76019, USA N. Parashar Purdue University Calumet,
Hammond, Indiana 46323, USA V. Parihar Brown University, Providence, Rhode
Island 02912, USA S.K. Park Korea Detector Laboratory, Korea University,
Seoul, Korea R. Partridgee Brown University, Providence, Rhode Island 02912,
USA N. Parua Indiana University, Bloomington, Indiana 47405, USA A. Patwa
Brookhaven National Laboratory, Upton, New York 11973, USA B. Penning Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA M. Perfilov
Moscow State University, Moscow, Russia Y. Peters II. Physikalisches
Institut, Georg-August-Universität Göttingen, Göttingen, Germany K. Petridis
The University of Manchester, Manchester M13 9PL, United Kingdom G. Petrillo
University of Rochester, Rochester, New York 14627, USA P. Pétroff LAL,
Université Paris-Sud, CNRS/IN2P3, Orsay, France M.-A. Pleier Brookhaven
National Laboratory, Upton, New York 11973, USA P.L.M. Podesta-Lermah
CINVESTAV, Mexico City, Mexico V.M. Podstavkov Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA A.V. Popov Institute for High
Energy Physics, Protvino, Russia M. Prewitt Rice University, Houston, Texas
77005, USA D. Price Indiana University, Bloomington, Indiana 47405, USA N.
Prokopenko Institute for High Energy Physics, Protvino, Russia J. Qian
University of Michigan, Ann Arbor, Michigan 48109, USA A. Quadt II.
Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen,
Germany B. Quinn University of Mississippi, University, Mississippi 38677,
USA M.S. Rangel LAFEX, Centro Brasileiro de Pesquisas Físicas, Rio de
Janeiro, Brazil K. Ranjan Delhi University, Delhi, India P.N. Ratoff
Lancaster University, Lancaster LA1 4YB, United Kingdom I. Razumov Institute
for High Energy Physics, Protvino, Russia P. Renkel Southern Methodist
University, Dallas, Texas 75275, USA I. Ripp-Baudot IPHC, Université de
Strasbourg, CNRS/IN2P3, Strasbourg, France F. Rizatdinova Oklahoma State
University, Stillwater, Oklahoma 74078, USA M. Rominsky Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA A. Ross Lancaster
University, Lancaster LA1 4YB, United Kingdom C. Royon CEA, Irfu, SPP,
Saclay, France P. Rubinov Fermi National Accelerator Laboratory, Batavia,
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de Pesquisas Físicas, Rio de Janeiro, Brazil G. Savage Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA L. Sawyer Louisiana
Tech University, Ruston, Louisiana 71272, USA T. Scanlon Imperial College
London, London SW7 2AZ, United Kingdom R.D. Schamberger State University of
New York, Stony Brook, New York 11794, USA Y. Scheglov Petersburg Nuclear
Physics Institute, St. Petersburg, Russia H. Schellman Northwestern
University, Evanston, Illinois 60208, USA C. Schwanenberger The University
of Manchester, Manchester M13 9PL, United Kingdom R. Schwienhorst Michigan
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48824, USA A.A. Shchukin Institute for High Energy Physics, Protvino, Russia
R.K. Shivpuri Delhi University, Delhi, India V. Simak Czech Technical
University in Prague, Prague, Czech Republic P. Skubic University of
Oklahoma, Norman, Oklahoma 73019, USA P. Slattery University of Rochester,
Rochester, New York 14627, USA D. Smirnov University of Notre Dame, Notre
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68588, USA J. Snow Langston University, Langston, Oklahoma 73050, USA S.
Snyder Brookhaven National Laboratory, Upton, New York 11973, USA S.
Söldner-Rembold The University of Manchester, Manchester M13 9PL, United
Kingdom L. Sonnenschein III. Physikalisches Institut A, RWTH Aachen
University, Aachen, Germany K. Soustruznik Charles University, Faculty of
Mathematics and Physics, Center for Particle Physics, Prague, Czech Republic
J. Stark LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut
National Polytechnique de Grenoble, Grenoble, France D.A. Stoyanova
Institute for High Energy Physics, Protvino, Russia M. Strauss University of
Oklahoma, Norman, Oklahoma 73019, USA L. Suter The University of Manchester,
Manchester M13 9PL, United Kingdom P. Svoisky University of Oklahoma,
Norman, Oklahoma 73019, USA M. Titov CEA, Irfu, SPP, Saclay, France V.V.
Tokmenin Joint Institute for Nuclear Research, Dubna, Russia Y.-T. Tsai
University of Rochester, Rochester, New York 14627, USA K. Tschann-Grimm
State University of New York, Stony Brook, New York 11794, USA D. Tsybychev
State University of New York, Stony Brook, New York 11794, USA B. Tuchming
CEA, Irfu, SPP, Saclay, France C. Tully Princeton University, Princeton, New
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Petersburg, Russia S. Uvarov Petersburg Nuclear Physics Institute, St.
Petersburg, Russia S. Uzunyan Northern Illinois University, DeKalb, Illinois
60115, USA R. Van Kooten Indiana University, Bloomington, Indiana 47405, USA
W.M. van Leeuwen Nikhef, Science Park, Amsterdam, the Netherlands N. Varelas
University of Illinois at Chicago, Chicago, Illinois 60607, USA E.W. Varnes
University of Arizona, Tucson, Arizona 85721, USA I.A. Vasilyev Institute
for High Energy Physics, Protvino, Russia P. Verdier IPNL, Université Lyon
1, CNRS/IN2P3, Villeurbanne, France and Université de Lyon, Lyon, France A.Y.
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Vertogradov Joint Institute for Nuclear Research, Dubna, Russia M. Verzocchi
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA M.
Vesterinen The University of Manchester, Manchester M13 9PL, United Kingdom
D. Vilanova CEA, Irfu, SPP, Saclay, France P. Vokac Czech Technical
University in Prague, Prague, Czech Republic H.D. Wahl Florida State
University, Tallahassee, Florida 32306, USA M.H.L.S. Wang Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA J. Warchol University
of Notre Dame, Notre Dame, Indiana 46556, USA G. Watts University of
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Dame, Notre Dame, Indiana 46556, USA J. Weichert Institut für Physik,
Universität Mainz, Mainz, Germany L. Welty-Rieger Northwestern University,
Evanston, Illinois 60208, USA A. White University of Texas, Arlington, Texas
76019, USA D. Wicke Fachbereich Physik, Bergische Universität Wuppertal,
Wuppertal, Germany M.R.J. Williams Lancaster University, Lancaster LA1 4YB,
United Kingdom G.W. Wilson University of Kansas, Lawrence, Kansas 66045, USA
M. Wobisch Louisiana Tech University, Ruston, Louisiana 71272, USA D.R. Wood
Northeastern University, Boston, Massachusetts 02115, USA T.R. Wyatt The
University of Manchester, Manchester M13 9PL, United Kingdom Y. Xie Fermi
National Accelerator Laboratory, Batavia, Illinois 60510, USA R. Yamada
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA S. Yang
University of Science and Technology of China, Hefei, People’s Republic of
China T. Yasuda Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA Y.A. Yatsunenko Joint Institute for Nuclear Research, Dubna,
Russia W. Ye State University of New York, Stony Brook, New York 11794, USA
Z. Ye Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA H.
Yin Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA K.
Yip Brookhaven National Laboratory, Upton, New York 11973, USA S.W. Youn
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA J.M. Yu
University of Michigan, Ann Arbor, Michigan 48109, USA J. Zennamo State
University of New York, Buffalo, New York 14260, USA T. Zhao University of
Washington, Seattle, Washington 98195, USA T.G. Zhao The University of
Manchester, Manchester M13 9PL, United Kingdom B. Zhou University of
Michigan, Ann Arbor, Michigan 48109, USA J. Zhu University of Michigan, Ann
Arbor, Michigan 48109, USA M. Zielinski University of Rochester, Rochester,
New York 14627, USA D. Zieminska Indiana University, Bloomington, Indiana
47405, USA L. Zivkovic Brown University, Providence, Rhode Island 02912, USA
###### Abstract
We present a measurement of the cross section for $W$ boson production in
association with at least one $b$-quark jet in proton-antiproton collisions.
The measurement is made using data corresponding to an integrated luminosity
of $6.1$ fb-1 recorded with the D0 detector at the Fermilab Tevatron
$p\bar{p}$ Collider at $\sqrt{s}=1.96$ TeV. We measure an inclusive cross
section of $\sigma(W~{}(\to\mu\nu)+b+X)=1.04~{}\pm~{}0.05\thinspace$(stat.)
$\pm~{}0.12\thinspace$(syst.) pb and $\sigma(W(\to e\nu)+b+X)=1.00$
$\pm~{}0.04\thinspace$(stat.) $\pm~{}0.12\thinspace$(syst.) pb in the phase
space defined by $p_{T}^{\nu}>25$ GeV, $p_{T}^{\text{$b$-jet}}>20$ GeV,
$|\eta^{\text{$b$-jet}}|<1.1$, and a muon (electron) with $p_{T}^{\ell}>20$
GeV and $|\eta^{\mu}|<1.7$ ($|\eta^{e}|<1.1$ or $1.5<|\eta^{e}|<2.5$). The
combined result per lepton family is $\sigma(W(\to\ell\nu)+b+X)=1.05$
$\pm~{}0.12\thinspace$(stat.+syst.) for $|\eta^{\ell}|<1.7$. The results are
in agreement with predictions from next-to-leading order QCD calculations
using mcfm, $\sigma(W+b)\cdot{\cal
B}(W\to\ell\nu)=1.34~{}^{+0.41}_{-0.34}\thinspace(\textrm{syst.})$, and also
with predictions from the sherpa and madgraph Monte Carlo event generators.
###### pacs:
12.38.Qk, 13.85.Qk, 14.65.Fy, 14.70.Fm
The measurement of the production cross section of a $W$ boson in association
with a $b$-quark jet provides a stringent test of quantum chromodynamics
(QCD). Processes involving $W/Z$ bosons in association with $b$ quarks are
also the largest backgrounds in studies of the standard model (SM) Higgs boson
decaying to two $b$ quarks, in measurements of top quark properties in both
single and pair production, and in numerous searches for physics beyond the
SM. The cross section for the process $p\bar{p}\to W+b+X$ has been calculated
with next-to-leading order (NLO) precision assoc_wb1 ; assoc_wb . Subprocesses
at NLO include $q\bar{q}~{}\to~{}Wb\bar{b},~{}q\bar{q}~{}\to~{}Wb\bar{b}g$,
and $qg~{}\to~{}Wb\bar{b}q^{\prime}$. An additional small contribution comes
from sea $b$ quarks in the incoming proton or antiproton, $bq\to
Wbq^{\prime}$.
In this letter we describe a measurement of the cross section for $W$ boson
production in association with $b$-quark jets in $p\bar{p}$ interactions,
where a $W$ boson is identified via its electronic or muonic decay modes. A
measurement of $W+b$ production cross section with up to two jets at
$\sqrt{s}=1.96$ TeV has been published by the CDF Collaboration wbb_cdf and
an inclusive measurement has been published by the ATLAS Collaboration
wbb_atlas at $\sqrt{s}=7$ TeV. The measured production cross section reported
by CDF is $\sigma\cdot{\cal B}(W\to\ell\nu)=2.74\pm
0.27\thinspace(\text{stat.})\pm 0.42\thinspace(\text{syst.})$ pb
($\ell=e,~{}\nu$), while the theoretical expectation for this quantity based
on NLO calculations is $1.22\pm 0.14\thinspace\text{(syst.)}$ pb wbb_cdf .
With the CDF measurement of $W+b$ production exceeding significantly the NLO
prediction, while the ATLAS result is in agreement with the expectation, an
independent measurement is important to understand the production of $W$
bosons in association with $b$ jets at hadron colliders.
The data used in this analysis were collected between July 2006 and December
2010 using the D0 detector at the Fermilab Tevatron Collider at
$\sqrt{s}~{}=~{}1.96$ TeV, and correspond to an integrated luminosity of 6.1
$\mathrm{fb}^{-1}$. We first briefly describe the main components of the D0
Run II detector run2det relevant to this analysis. The D0 detector has a
central tracking system consisting of a silicon microstrip tracker (SMT)
layer0 and a central fiber tracker (CFT), both located within a 2 T
superconducting solenoidal magnet, with designs optimized for tracking and
vertexing at pseudorapidities $|\eta|<3$ and $|\eta|<2.5$, respectively coord
. A liquid argon and uranium calorimeter has a central section (CC) covering
pseudorapidities $|\eta|\lesssim 1.1$, and two end calorimeters (EC) that
extend coverage to $|\eta|\approx 4.2$, with all three housed in separate
cryostats calopaper . An outer muon system, at $|\eta|<2$, consists of a layer
of tracking detectors and scintillation trigger counters in front of 1.8 T
toroids, followed by two similar layers after the toroids. Luminosity is
measured using plastic scintillator arrays located in front of the EC
cryostats. The trigger and data acquisition systems are designed to
accommodate the high instantaneous luminosities of Run II.
The $W+b$ candidates are selected by triggering on single lepton or lepton-
plus-jet signatures with a three-level trigger system. The trigger
efficiencies are approximately $70\%$ for the muon channel and $95\%$ for the
electron channel.
$W$ boson candidates are identified in the $\mu+\nu$ and $e+\nu$ decay
channels whereas a small fraction of selected events arises from leptonical
decaying tau leptons. Offline event selection requires a reconstructed primary
$p\bar{p}$ interaction primary vertex (PV) that has at least three associated
tracks and is located within $60$ cm of the center of the detector along the
beam direction. The vertex selection for $W+b$ events is about $97\%$
efficient as measured in simulations.
Electrons are identified using calorimeter and tracking information. The
selection requires exactly one electron with transverse momentum
$p_{T}^{e}>20$ GeV identified by an electromagnetic (EM) shower in the central
($|\eta^{e}|<1.1$) or endcap ($1.5<|\eta^{e}|<2.5$) calorimeter by comparing
the longitudinal and transverse shower profiles to those of simulated
electrons. The showers must be spatially isolated from other energetic
particles, deposit most of their energy in the EM part of the calorimeter, and
pass a likelihood criterion that includes a spatial track match. In the
central detector region, an $E/p$ requirement is applied, where $E$ is the
energy of the calorimeter cluster and $p$ is the momentum of the track. The
transverse momentum measurement of electrons is based on calorimeter energy
information.
The muon selection requires the candidate to be reconstructed from hits in the
muon system and matched to a reconstructed track in the central tracker. The
transverse momentum of the muon must exceed $p_{T}^{\mu}>20$ GeV, with
$|\eta^{\mu}|<1.7$. Muons are required to be spatially isolated from other
energetic particles using information from the central tracking detectors and
calorimeter diboson_prd . Muons from cosmic rays are rejected by applying a
timing criterion on the hits in the scintillator layers and by applying
restrictions on the displacement of the muon track with respect to the
selected PV.
Candidate $W+\textrm{jets}$ events are then selected by requiring at least one
reconstructed jet with $|\eta^{\text{jet}}|<1.1$ and $p_{T}^{\text{jet}}>20$
GeV. Jets are reconstructed from energy deposits in the calorimeter using the
iterative midpoint cone algorithm jet_algo and a cone of radius $\Delta
R=0.5$ in $y$-$\varphi$ space coord . The energies of jets are corrected for
detector response, the presence of noise and multiple $p\bar{p}$ interactions,
and for energy deposited outside of the jet reconstruction cone. To enrich the
sample with $W$ bosons, events are required to have missing transverse energy
$\mbox{$E\kern-6.00006pt/$ }_{T}\\!$ $>25$ GeV due to the neutrino escaping
detection.
Background processes for this analysis are electroweak
$W+\textrm{jets}/\gamma$ production, $Z/\gamma^{*}$ production, $t{\bar{t}}$
and single top quark production, diboson production, and multijet events with
jets misidentified as leptons. The $W+b$ signal and SM background processes
are simulated using a combination of pythia v6.409 pythia and alpgen v2.3
alpgen with pythia providing parton showering and hadronization. We use
pythia Tune A with CTEQ6L1 pdf_cteq6M parton distribution functions (PDFs)
and perform a detailed geant-based geant simulation of the D0 detector. The
$V{\rm+jets}$ ($V=W/Z$) processes are normalized to the inclusive $W$ and
$Z$-boson cross sections calculated at NNLO hamberg . The $Z$-boson $p_{T}$
distribution is modeled to match the distribution observed in data zpt_xsec ,
taking into account the dependence on the number of reconstructed jets. To
reproduce the $W$-boson $p_{T}$ distribution in simulated events, the product
of the measured $Z$-boson $p_{T}$ spectrum and the ratio of $W$ to $Z$-boson
$p_{T}$ distributions at NLO is used as correction. NLO+NNLL (next-to-next-to-
leading log) calculations are used to normalize $t{\bar{t}}~{}$ production
mochuwer , while single top quark production is normalized to NNLO single_top
. The NLO $WW$, $WZ$, and $ZZ$ production cross section values are obtained
with mcfm program mcfm . For the $W$+heavy-flavor jet ($b$ or $c$ quark)
events, the ratio of the alpgen prediction to the NLO prediction for
$W+b\bar{b}$ and $W+c\bar{c}$ is obtained from mcfm mcfm and applied as a
correction factor. The simulation is also corrected for the trigger
efficiencies measured in data.
Instrumental backgrounds and those from semileptonic decays of hadrons,
referred to as “multijet” background, are estimated from data. The
instrumental background is important for the electron channel, where a jet
with a high electromagnetic fraction can pass electron identification
criteria, or a photon can be misidentified as an electron. In the muon
channel, the multijet background is less significant and arises mainly from
the semileptonic decay of heavy quarks in which the muon satisfies the
isolation requirements. We require that the $W$ boson candidates have a
transverse mass $M_{T}$ mtw satisfying
$40~{}\text{GeV}+~{}\frac{1}{2}\mbox{$\mbox{$E\kern-6.00006pt/$ }_{T}\\!$
}<M_{T}<120$ GeV to suppress multijet background and mis-reconstructed events.
The average efficiency determined in simulation for a $W+b$ signal to pass
these requirements is about $82\%$.
Identification of $b$ jets is crucial for this measurement. Once the inclusive
$W+{\rm jets}$ sample is defined, the jets considered for $b$ tagging are
subject to a requirement called taggability. This requirement is imposed to
decouple the performance of the $b$-jet identification from detector effects.
For a jet to be taggable, it must contain at least two tracks with at least
one hit in the SMT, $p_{T}>1$ GeV for the highest-$p_{T}$ track and
$p_{T}>0.5$ GeV for the next-to-highest $p_{T}$ track. The efficiency for a
jet to be taggable is about $90\%$ in the selected phase space.
The D0 $b$-tagging algorithm for identifying heavy flavor jets is based on a
combination of variables sensitive to the presence secondary vertices (SV) or
tracks displaced from the PV. This analysis uses an updated $b$ tagger
utilizing a multivariate analysis (MVA) tmva ; gamma_b that provides improved
performance over the previous neural network based algorithm bid_nim . The
most sensitive input variables to the MVA are the number of reconstructed
secondary vertices in the jet, the invariant mass of charged particles
associated with the SV ($M_{\text{SV}}$), the number of tracks used to
reconstruct the SV, the two-dimensional decay length significance of the SV in
the plane transverse to the beam, a weighted combination of the tracks’
transverse impact parameter significances, and the probability that the tracks
from the jet originate from the PV, which is referred to as the jet lifetime
probability (JLIP). The MVA provides a continuous output value that tends
towards one for $b$ jets and zero for non-$b$ jets. Events are considered in
which at least one jet passes a tight MVA requirement corresponding to an
efficiency of $\approx 50\%$ for $b$ jets. The likelihood for a light jet
($u$, $d$, $s$ quarks and gluons) to be misidentified for the corresponding
MVA selection is about $0.5$%. Simulated events are corrected to have the same
efficiencies for taggability and $b$-tagging requirements as found in data.
These corrections are derived in a flavor dependent manner bid_nim , using
independent $QCD$ enriched data samples and simulated events with enriched
light and heavy jet contributions. Jets containing $b$ quarks have a different
energy response and receive an additional energy correction of about $6\%$ as
determined from simulation. Figure 1 shows the transverse mass of the
candidate events before and after applying $b$-jet identification.
In addition to the MVA output, we perform further selections using
$M_{\text{SV}}$ and JLIP variables. $M_{\text{SV}}$ provides good
discrimination between $b$, $c$, and light quark jets due to their different
masses gamma_b . The two variables together take into account the kinematics
of the event and, in order to further improve the separation power, they are
combined in a single variable ${\cal
D}_{\text{MJL}}~{}=\frac{1}{2}~{}\left(M_{\text{SV}}/(5~{}\text{GeV})-\ln(\text{JLIP})/20\right)$
zbzlf .A loose criterion for an event to pass at least ${\cal
D}_{\text{MJL}}>0.1$ is applied to remove poorly reconstructed events. The
efficiency for signal events to pass this selection is about $97\%$.
The numbers of expected and observed events before and after applying the
$b$-jet identification in data and simulation are listed in Table 1. The
$b$-tagging column includes the selection requirement on ${\cal
D}_{\text{MJL}}$.
Process | No $b$-tag | $b$-tag
---|---|---
$V+$heavy flavor | 41093 | $\pm$ | 8924 | 5068 | $\pm$ | 1124
$V+$light flavor | 516661 | $\pm$ | 56734 | 5718 | $\pm$ | 678
Diboson | 4728 | $\pm$ | 519 | 222 | $\pm$ | 26
Top | 5431 | $\pm$ | 536 | 1602 | $\pm$ | 181
Multijet | 20527 | $\pm$ | 4458 | 794 | $\pm$ | 180
Expected events | 588440 | $\pm$ | 57610 | 13405 | $\pm$ | 1338
Data | 586289 | 12793
Table 1: Numbers of events for data and contributing processes before and
after applying $b$-jet identification. Uncertainties include statistical and
systematic contributions. The contribution of $Z+\textrm{jets}$ events to the
$V+\textrm{jets}$ samples is $\approx 5\%$ for heavy and light flavor jets
before and after $b$-tagging.
Figure 1: [color online] Transverse mass of the $\ell\nu$ system (a) before
and (b) after $b$-jet identification. The data are shown by black markers,
simulated background processes are shown by filled histograms. The data
uncertainties are statistical only. An estimate of the systematic uncertainty
on the simulated background processes is shown by the shaded bands
We measure the fraction of $W+b+X$ events in the final selected sample by
performing a binned maximum likelihood fit to the observed data distribution
of the ${\cal D}_{\text{MJL}}$ discriminant in our sample shown in Fig. 2. The
templates for $W$+light flavor, $W+b$, and $W+c$ jets shown in Fig. 2 are
taken from the efficiency-corrected simulation. Expected contributions from
$Z$+jets, single top quark, $t\bar{t}$, diboson, and multijet production are
subtracted from the data. After performing the fits, we obtain the number of
events with different jet flavors listed in Table 2.
The measured cross sections are presented at the particle level by correcting
for detector acceptance, selection-efficiencies, and $b$-jet identification.
We quote our result as a cross section in a restricted phase space: at least
one $b$-jet with $p_{T}^{\text{$b$-jet}}>20$ GeV,
$|\eta^{\text{$b$-jet}}|<1.1$ and a muon with $p_{T}^{\mu}~{}>~{}20$ GeV and
$|\eta^{\mu}|<1.7$ or an electron with $p_{T}^{e}>20$ GeV and $|\eta^{e}|<1.1$
or $1.5<|\eta^{e}|<2.5$. For the neutrino momentum we require $p_{T}^{\nu}>25$
GeV.
| $W\to\mu\nu$ | $W\to e\nu$
---|---|---
Processxxxx | Events | Fraction | Events | Fraction
$W+b$ | $1306\pm 166$ | $0.3\pm 0.04$ | $1676\pm 212$ | $0.27\pm 0.03$
$W+c$ | $664\pm 97$ | $0.1\pm 0.02$ | $1096\pm 159$ | $0.18\pm 0.03$
$W+\textrm{l.f.}$ | $2152\pm 265$ | $0.5\pm 0.07$ | $3479\pm 425$ | $0.56\pm 0.07$
Data$-$Bkgd | $4127\pm 150$ | | $6255\pm 168$ |
Table 2: Estimated numbers of $W+\text{jet}$ events from fitting the flavor-
specific processes, along with the expected background of $W$ boson processes
and the data after subtracting $Z$+jets, single top quark, $t\bar{t}$, and
diboson background processes. $l.f.$ stands for light flavor jets.
Uncertainties include statistical and systematic contributions.
Systematic uncertainties are determined by varying experimental parameters and
efficiency/acceptance corrections by one standard deviation and propagating
the effect on ${\cal D}_{\text{MJL}}$. The systematic uncertainties are
dominated by effects related to the measurement of jets. The contributions
from jet energy resolution, jet modeling, and detector effects are about
$2.5\%,~{}3\%$, and $4\%$, respectively. Uncertainties on $b$-jet
identification are determined in data and simulations by using $b$-jet-
enriched samples and are about $2\%-5\%$ per jet. The uncertainties due to
lepton identification are about $2\%$. The integrated luminosity is known to a
precision of $6.1\%$ lumi . The uncertainty of the template fit is estimated
by varying the normalization and shape from the data corrections of the $W$
boson processes and the fit parameters (about $6\%$). By summing the
uncertainties in quadrature we obtain a final total systematic uncertainty on
the cross section measurements of approximately $12\%$.
The cross section times branching fraction is calculated by dividing the
number of signal events measured by integrated luminosity ($\mathcal{L}$),
acceptance ($\mathcal{A}$), and efficiencies ($\epsilon$) of the selection
requirements:
Figure 2: [color online] Contributions of the various jet flavors normalized
to the measured cross section obtained from a fit in the $W\to\mu\nu$ channel
on both (a) linear and (b) logarithmic scales. The various $W+\textrm{jets}$
processes are shown as filled histograms and data, after the subtraction of
contributions from Drell-Yan, diboson, and top quark production, are
represented with black markers. The uncertainties include both statistical and
systematic contributions.
$\sigma(W+b)\cdot{\cal B}(W\to\ell\nu)=\frac{N_{W+b}}{{\cal L}\cdot{\cal
A}\cdot\epsilon},$ (1)
where $\epsilon$ is given by the product of the trigger, object
reconstruction, and selection efficiencies.
We first present results separately for the muon channel and electron channel
because they are performed in slightly different requirements on the phase
space of the lepton and then combine using a common phase space. We measure
from the cross section in the muon channel where $W\to\mu\nu$ in a visible
phase space defined by $p_{T}^{\mu}>20$ GeV, $|\eta^{\mu}|<1.7$ with at least
one $b$-jet limited to $p_{T}^{\text{$b$-jet}}>20$ GeV and
$|\eta^{\text{$b$-jet}}|<1.1$ as,
$\begin{split}\sigma(W+b)\cdot{\cal B}(W\to\mu\nu)=\phantom{xxxxxx}\\\ 1.04\pm
0.05\thinspace\textrm{(stat.)}\pm
0.12\thinspace\textrm{(syst.)~{}pb.}\end{split}$ (2)
We perform an NLO QCD prediction using mcfm v6.1, based on CTEQ6M PDF
pdf_cteq6M and a central scale of $M_{W}+2m_{b}$, where $m_{b}=4.7$ GeV is
the mass of the $b$ quark. Uncertainties are estimated by varying
renormalization and factorization scales by a factor of two in each direction,
varying $m_{b}$ between $4.2$ and $5$ GeV, and by using an alternative PDF
set. The mcfm calculation predicts $\sigma(W+b)\cdot{\cal
B}(W\to\mu\nu)=1.34~{}^{+0.40}_{-0.33}~{}(\textrm{scale})\pm
0.06~{}(\textrm{PDF})~{}^{+0.09}_{-0.05}~{}(m_{b})$ pb. Predictions obtained
using sherpa v1.4 and CTEQ6.6 PDFs pdf_cteq6M lead to a value $1.21\pm
0.03\thinspace(\textrm{stat.})$ pb. Using madgraph5 mg5 with CTEQ6L1 PDFs, we
obtain $1.52\pm 0.02\thinspace(\textrm{stat.})$ pb. Uncertainties for scale
variations, PDFs, and the $b$-quark mass are on the order of about 30%.
In the electron channel, we measure the cross section times branching fraction
by selecting $p_{T}^{e}>20$ GeV, $|\eta^{e}|<1.1$ or $1.5<|\eta^{e}|<2.5$, at
least one $b$-jet as above and obtain
$\begin{split}\sigma(W+b)\cdot{\cal B}(W\to e\nu)=\phantom{xxxxxx}\\\ 1.00\pm
0.04\thinspace\textrm{(stat.)}\pm
0.12\thinspace\textrm{(syst.)~{}pb.}\end{split}$ (3)
The mcfm calculated cross section for this channel is $\sigma(W+b)\cdot{\cal
B}(W\to e\nu)=1.28~{}^{+0.40}_{-0.33}~{}(\textrm{scale})\pm
0.06~{}(\textrm{PDF})~{}^{+0.09}_{-0.05}~{}(m_{b})$ pb. The sherpa prediction
is $1.08\pm 0.03\thinspace(\textrm{stat.})$ pb, while the madgraph5 prediction
is $1.44\pm 0.02\thinspace(\textrm{stat.})$ pb. The combined systematic effect
scale, PDF and $m_{b}$ variations is also around 30%.
Using the mcfm prediction we extrapolate the measurement in the electron final
state to the same selection requirements as the muon final state to allow for
a consistent combination. Combining the results in $W\to\mu\nu$ and $W\to
e\nu$ decays we obtain
$\begin{split}\sigma(W+b)\cdot{\cal B}(W\to\ell\nu)=\phantom{xxxxxx}\\\
1.05\pm 0.03\thinspace\textrm{(stat.)}\pm
0.12\thinspace\textrm{(syst.)~{}pb.}\end{split}$ (4)
The small experimental uncertainty should allow to further constrain
theoretical predictions. In summary, we have performed a measurement of the
inclusive cross section for $W$ boson production in association with at least
one $b$-jet at $\sqrt{s}=1.96$ TeV, considering final states with $W\to\mu\nu$
($W\to e\nu$) events in a restricted phase space of $p_{T}^{\ell}>20$ GeV,
$|\eta^{\mu}|<1.7$ ($|\eta^{e}|<1.1$ or $1.5<|\eta^{e}|<2.5$), with $b$ jets
limited to $p_{T}^{\text{$b$-jet}}>20$ GeV and $|\eta^{\text{$b$-jet}}|<1.1$.
The measured cross sections agree within uncertainties with NLO QCD
calculations and predictions obtained using the sherpa and madgraph
generators.
We thank the staffs at Fermilab and collaborating institutions, and
acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France);
MON, NRC KI and RFBR (Russia); CNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil);
DAE and DST (India); Colciencias (Colombia); CONACyT (Mexico); NRF (Korea);
FOM (The Netherlands); STFC and the Royal Society (United Kingdom); MSMT and
GACR (Czech Republic); BMBF and DFG (Germany); SFI (Ireland); The Swedish
Research Council (Sweden); and CAS and CNSF (China). Special thanks as well to
John Campbell for his support with mcfm and Patrick Fox with madgraph.
## References
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|
arxiv-papers
| 2012-10-02T01:14:25 |
2024-09-04T02:49:35.822989
|
{
"license": "Public Domain",
"authors": "D0 Collaboration",
"submitter": "Bj\\\"orn Penning",
"url": "https://arxiv.org/abs/1210.0627"
}
|
1210.0785
|
††thanks: Corresponding authors: J. K. S. Poon and A. Joushaghani
# Low-voltage broadband hybrid plasmonic-vanadium dioxide switches
Arash Joushaghani1 Brett A. Kruger1 Suzanne Paradis2 David Alain2 J. Stewart
Aitchison1 Joyce K. S. Poon1 1Department of Electrical and Computer
Engineering and Institute for Optical Sciences, University of Toronto, 10
King’s College Road, Toronto, Ontario, M5S 3G4, Canada 2Defence Research and
Development Canada - Valcartier, 2459 Pie-XI Blvd. North, Quebec, Quebec G3J
1X5, Canada
Surface plasmon polaritons can substantially reduce the sizes of optical
devices, since they can concentrate light to (sub)wavelength scales Ozbay
(2006); MacDonald and Zheludev (2010); Schuller _et al._ (2010). However,
(sub)wavelength-scale electro-optic plasmonic switches or modulators with high
efficiency, low insertion loss, and high extinction ratios remain a challenge
due to their small active volumes Dionne _et al._ (2009); Min and Veronis
(2009); Cai _et al._ (2009). Here, we use the insulator-metal phase
transition of a correlated-electron material, vanadium dioxide, to overcome
this limitation and demonstrate compact, broadband, and efficient plasmonic
switches with integrated electrical control. The devices are micron-scale in
length and operate near a wavelength of 1550 nm. The switching bandwidths
exceed 100 nm and applied voltages of only 400 mV are sufficient to attain
extinction ratios in excess of 20 dB. Our results illustrate the potential of
using phase transition materials for highly efficient and ultra-compact
plasmonic switches and modulators.
The miniaturization of optical devices, whether using surface plasmon
polaritons (SPPs) at metal-dielectric interfaces or strongly confined
electromagnetic modes in high-index-contrast dielectric waveguides, presents
an opportunity to reduce the sizes of electro-optic switches and modulators.
As the volume of the active region shrinks, if the optical confinement is
maintained, the amount of energy required to activate the material for
modulation decreases. However, small active regions and low power consumption
often come at the cost of a reduced extinction ratio. In dielectric devices,
this trade-off is often overcome by recirculating light in high-$Q$
microcavities, which restrict the operation wavelengths to narrowband cavity
resonances Xu _et al._ (2005). Plasmonic devices can be significantly more
broadband, but the losses of SPPs typically prevent high-$Q$ microcavities
from being formed and limit the devices to micron or sub-micron lengths. These
constraints have led to large switching voltages $>10$ V Dicken _et al._
(2008); Melikyan _et al._ (2011) and extinction ratios that are at best $\sim
10$ dB Sorger Volker _et al._ (2012); Dionne _et al._ (2009) for short
plasmonic switches that are no more than several microns long.
Figure 1: VO2 properties and the hybrid SPP-VO2 switch. (a) Measured
temperature-dependent resistivity, $\rho$, of the VO2 films. The phase
transition occurred at $T_{c}\approx 340$ K. (b) The real, $n$, and imaginary,
$k$, components of the refractive index of the deposited VO2 film measured by
ellipsometry. (c) SEM image of the device cross-section with the thicknesses
of various layers labeled. (d) SEM image of the hybrid SPP-VO2 switch with an
integrated heater. The simulated electric field intensities of the
electromagnetic modes of the SPP-VO2 waveguide when the temperature of VO2
layer is (e) $T<T_{c}$ and (f) $T>T_{c}$.
Efficient and compact electro-optic plasmonic switches require materials that
exhibit exceptionally large changes in their refractive indices when subjected
to electrical control signals. A promising material for infrared wavelengths
is the correlated-electron material, vanadium dioxide (VO2). The electronic
distribution and lattice of VO2 can reversibly reconfigure to produce an
insulator-metal phase transition with a resistivity change of up to three
orders of magnitude. The phase transition can be initiated by temperature
changes Berglund and Guggenheim (1969), optical Cavalleri _et al._ (2001) or
terahertz Liu _et al._ (2012) pulses, electric fields Stefanovich _et al._
(2000), surface charge accumulations Nakano _et al._ (2012), or mechanical
strain Cao _et al._ (2009).
In this work, we demonstrate plasmonic switches with low switching voltages
and record high extinction ratios using a hybrid SPP-VO2 geometry and the
thermally-induced VO2 phase transition. In contrast to previous proposals and
demonstrations Sorger Volker _et al._ (2012); Dionne _et al._ (2009);
Randhawa _et al._ (2012); Papaioannou _et al._ (2012); Briggs _et al._
(2010); Sweatlock and Diest (2012), our devices are highly compact (between 5
to 15 $\mathrm{\mu m}$ long) and have integrated electrical control. To show
the potential for integration with silicon waveguides, polycrystalline VO2 was
first deposited on a silicon-on-insulator (SOI) substrate using radio-
frequency (RF) magnetron sputtering at a substrate temperature of 773 K
(details described in Methods). The thicknesses of the VO2, Si, and buried
oxide (BOX) layers were 280 nm, 250 nm, and 3 $\mathrm{\mu m}$, respectively.
The temperature-dependent resistivity of this film, shown in Fig. 1(a),
indicates that the critical temperature for the insulator-metal phase
transition was about $T_{c}\approx 340$ K, typical for VO2.
Accompanied with the change in the resistivity in the phase transition is a
dramatic change in the real, $n$, and imaginary, $k$, parts of the refractive
index in the infrared wavelength range as shown in Fig. 1(b) Berglund and
Guggenheim (1969). The measured refractive indices for our VO2 film at
$\lambda=1550$ nm were $2.9+0.4i$ at $T=298$ K $<T_{c}$ and $2.0+3.0i$ at
$T=373$ K $>T_{c}$. Even though VO2 exhibits a large index change near
$T_{c}$, it is not an ideal waveguide material by itself; the imaginary part
of the refractive index is too high in the insulator phase to support a low-
loss optical mode, but it is not high enough in the metal phase to support a
low-loss SPP. Therefore, to form a switch, we incorporated VO2 in a hybrid
geometry with a plasmonic waveguide Kruger _et al._ (2012); Sweatlock and
Diest (2012).
The devices were formed in a 300 nm thick silver (Ag) film on a 820 nm thick
SiO2 spacer on the VO2 using a lift-off step. Figures 1(c)-(d) respectively
show scanning electron microscope (SEM) images of the material stack and
device. The Ag features and thickness of SiO2 spacer were designed using
combined optical, thermal, and electrical modelling to optimize the extinction
ratio (ER), insertion loss (IL), and switching voltage, $V$, for the given
thicknesses of VO2 and SOI. The grating couplers (10 periods with a period of
1040 nm) coupled light into and out of the SPP-VO2 waveguide. A 5 $\mathrm{\mu
m}$ wide strip perpendicular to the waveguide acted as a thin film heater to
heat the local volume of VO2 above $T_{c}$ when a current passed through it.
The heater widened to the contact pads.
The SPP-VO2 waveguides were $w_{g}=8$ $\mathrm{\mu m}$ wide to facilitate the
measurements and were varied in length, $L$. To achieve a high extinction
ratio, the SPP-VO2 waveguides should switch between a pair of hybrid modes
Oulton _et al._ (2008); Alam _et al._ (2010) depending on the phase of the
VO2. To ensure that the switching is due to propagation through the SPP-VO2
waveguides and not a change in the input/output coupling losses between the
waveguide and the gratings, this pair of hybrid modes should have different
propagation losses but similar effective indices. At $T<T_{c}$, our designed
SPP-VO2 waveguide supports a low-loss, transverse magnetic (TM) polarized mode
shown in Fig. 1(e), which is similar to the SPP mode of a single Ag-SiO2
interface Kruger _et al._ (2012). This mode has a calculated effective index
of $n_{\mathrm{eff}}=1.45$ and a propagation loss of 0.4 dB/$\mathrm{\mu}$m.
At $T>T_{c}$, the waveguide supports a TM polarized metal-oxide-metal-like
mode that is mainly confined between the VO2 and Ag layers as shown in Fig.
1(f). By tracing the modes while increasing thickness of the SiO2 spacer
layer, the hybrid mode of Fig. 1(f) is found to be of a different order
compared to the low-loss mode of Fig. 1(e). This lossy mode has a calculated
effective index of $n_{\mathrm{eff}}=1.50$ and a propagation loss of 3.1
dB/$\mathrm{\mu}$m. Therefore, using the VO2 phase transition, the excitation
of the two different types of modes leads to a theoretical ER-per-length of
2.7 dB/$\mathrm{\mu}$m.
Figure 2: The temperature distribution of the device and the associated
refractive index profile in the VO2. (a) The simulated temperature profile
along $y$ at the mid-line of the SPP-VO2 waveguide under an applied current of
$I_{0}=180$ mA. (b) The refractive index contours of VO2 film near the edge of
the hybrid SPP-VO2 waveguide computed using the temperature profile of 2(a).
The regions with insulating and metallic VO2 phases are marked with arrows.
The fully metallic VO2 region is directly under the heater. (c) The change in
the surface reflectivity, $\Delta R$, measured in the vicinity of the heater
and SPP-VO2 waveguide (estimated position marked by white lines). The heater
is along $x$, and the waveguide is along $y$.
We designed the devices such that the heating would minimally affect
input/output coupling losses, so our results can be generalized to cases with
integrated input/output waveguides. Figure 2(a) shows the temperature
distribution along the mid-line of the device computed from a three-
dimensional thermal simulation with an applied current of $I=180$ mA. The heat
is localized under the SPP-VO2 waveguide and, as a result, mostly changes the
phase of the VO2 directly under the heater. Figure 2(b) shows the contours of
the refractive index in the VO2 film computed from the temperature profile of
Fig. 2(a). The VO2 under the heater experiences the largest index change and
is completely metallic, while the VO2 under the gratings remain predominantly
in the insulating phase. Since the spot size of the optical input in the
experiment spanned $\sim$ 6 grating periods and was positioned near the center
of the grating-coupler, for the currents tested, the input grating-coupler
mainly excited the low-loss SPP-VO2 mode of Fig. 1(e). This mode propagated
through the transition region of Fig. 2(b) and would in turn be converted to
the high-loss mode of Fig. 1(f) when the VO2 under the waveguide region was in
its metallic phase. Electromagnetic simulations show that the theoretical net
input and output coupling losses due to the grating-couplers is about 14.5 dB.
Using the temperature profile of Fig. 2(a), the net change in the coupling
losses, resulting from both the change in grating coupling losses and the
mismatch between the two modes of Fig. 1(e)-(f), is estimated to be $<3$ dB.
To experimentally confirm that the heating was localized, we uniformly
illuminated the fabricated devices from the top at a wavelength of
$\lambda=1550$ nm to image the change in the reflectivity, defined as
$\Delta R=\frac{R_{I=I_{0}}-R_{I=0}}{R_{I=0}},$ (1)
where $R_{I}$ is the surface reflectivity when a current of $I$ passes through
the heater. Figure 2(c) shows $\Delta R$ at $I_{0}=180$ mA. $\Delta R$ is
localized near the edges of the heater and SPP-VO2 waveguide, away from the
grating-couplers. $\Delta R>0$ is in agreement with the reflection spectra
extracted from the ellipsometry data of the VO2 film on SOI.
Figure 3: The static transmission and power consumption characteristics. (a)
The normalized transmission of devices with varying lengths of SPP-VO2
waveguides ($L$) as a function of the input current. An ER of 24.1 dB is
reached when $L=12$ $\mathrm{\mu m}$. For $L\geq 15$ $\mathrm{\mu m}$ , the
measured ER was limited by the background noise. (b) The measured IL and ER of
the SPP-VO2 waveguides as a function of $L$. Linear fits of IL and ER are
included. (c) The spectral dependence of IL and ER of the $L=7$
$\mathrm{\mu}$m device. (d) The current and electrical power required for the
reported ERs as a function of $L$. Figure 4: The dynamic switching
characteristics of the $\bm{L=10}$ $\mathbf{\mu m}$ device. (a) (right axis)
The time-dependent transmission when a (left axis) triangular voltage pulse
with a ramp duration of $\tau=2.5$ ms was applied at the heater. (b)
Transmission-voltage hysteresis of (a). (c) The optical transmission when the
ramp duration of the drive voltage is reduced. (d) The change in transmission,
$\Delta\mathrm{Trans}$, as a function of $1/\tau$.
To investigate the device operation, we measured the static and dynamic
characteristics of a set of devices with identical grating couplers and
integrated heaters, but different SPP-VO2 waveguide lengths between $L=5$
$\mathrm{\mu m}$ and $L=15$ $\mathrm{\mu m}$. Figure 3(a) shows that when the
applied current was increased, the normalized transmission at $\lambda=1550$
nm dropped abruptly as $T_{c}$ was reached. As shown in Fig. 3(b), the ER and
IL increased with the VO2-SPP waveguide length. An ER of 10.3 dB was achieved
when $L=5$ $\mathrm{\mu}$m and increased to a maximum of 24.1 dB when $L=12$
$\mathrm{\mu}$m. For $L\geq 15$ $\mathrm{\mu}$m, the optical output power
became too low to measure the ER accurately. The ER of the $L=15$
$\mathrm{\mu}$m device in Fig. 3(a) was limited by background noise.
Linear fits of the data show that the ER-per-length was $1.9\pm 0.2$
dB/$\mathrm{\mu}$m and the IL-per-length was $0.9\pm 0.1$ dB/$\mathrm{\mu}$m.
This ER-per-length, to our knowledge, is the highest reported to date amongst
plasmonic switches (Table 1). The intercept of the IL linear fit indicates the
losses due to the grating-couplers were $17.6\pm 1.1$ dB. The intercept of the
ER linear fit of $1.8\pm 2.2$ dB represents the difference between the
grating-coupler losses when the VO2 was in the insulator and metal phases and
the mode conversion losses at the transition between the metallic and
insulating VO2 regions. This small offset supports that the switching was
mainly due to the propagation through the SPP-VO2 waveguide and not lumped
coupling losses. The measured IL, ER, and grating coupling losses are in good
agreement with the theoretical values from the simulations.
The broadband nature of the devices is evident by the spectra of the
transmission below $T_{c}$ and the ER of the $L=7$ $\mathrm{\mu m}$ device in
Fig. 3(c). The measured device bandwidth was limited by the grating couplers,
which had a 3 dB bandwidth of about $117\pm 15$ nm. The calculated dispersion
relation of the VO2-SPP waveguide suggests the theoretical 3 dB bandwidth of
the ER is about 160 nm Kruger _et al._ (2012).
We estimate the power consumed by the switch by subtracting the potential
difference at the contacts from the net potential difference across the
probes. Figure 3(d) shows the current and power consumption for different
waveguide lengths. From the linear fit, the total power dissipated in regions
of the heater away from the SPP-VO2 waveguide was about $31.2\pm 3$ mW, and
the power used to switch the SPP-VO2 waveguide was $5.0\pm 0.3$
mW/$\mathrm{\mu m}$. About 28 mW of power was required to switch the $L=5$
$\mathrm{\mu m}$ SPP-VO2 waveguide to attain an ER of 10.3 dB. The applied
current was about 148 mA and the voltage was about 400 mV.
Since the switching was reversible and repeatable, we investigated the
modulation dynamics by driving the heater with a periodic train of triangular
voltage pulses with various amplitudes and ramp times, $\tau$. The delay
between the pulses was 100 ms. Figure 4(a) shows the time-dependent optical
transmission of the $L=10$ $\mathrm{\mu m}$ SPP-VO2 switch under an applied
voltage pulse with $\tau=2.5$ ms. The voltage was sufficiently high for
complete switching. The transmission remained at a minimum for about 1 ms
after the peak of the voltage pulse because of the intrinsic hysteresis of the
VO2 insulator-metal transition (Fig. 1(a)) and the finite time required for
thermal dissipation. This hysteresis is evident in Fig. 4(b), where the
transmission is plotted against the applied voltage.
Next, we changed $\tau$ of the voltage pulses to measure the frequency
response of the $L=10$ $\mathrm{\mu m}$ device. Figure 4(c) shows the
transmission at several values of $\tau$, while the voltage amplitude was kept
to $V_{0}=450$ mV. As $\tau$ shortened, the ER diminished because the
temperature change could no longer track the voltage pulse. Figure 4(d) shows
the change in the transmission of the switch,
$\Delta\mathrm{Trans}=\left|\frac{\mathrm{Trans}_{V=V_{0}}-\mathrm{Trans}_{V=0}}{\mathrm{Trans}_{V=0}}\right|,$
(2)
as a function of $1/\tau$. The 50% roll-off frequency was about 25 kHz,
typical of thermally activated devices Murzina _et al._ (2012). Increasing
the applied voltage amplitude increased the roll-off frequency, but could
damage the Ag strip.
Finally, we compare the hybrid SPP-VO2 switch against state-of-the-art
plasmonic Sorger Volker _et al._ (2012); Dionne _et al._ (2009); Randhawa
_et al._ (2012); Papaioannou _et al._ (2012) and Si-based Liu _et al._
(2008); Watts _et al._ (2011) optical switches in Table 1 to illustrate its
unique advantages. The $L=7$ $\mathrm{\mu m}$ SPP-VO2 switch is included in
the table. The hybrid SPP-VO2 switch is one of the most compact switches
demonstrated to date, yet it simultaneously has the highest ER-per-length and
among the lowest switching voltages. Its ER (16.4 dB) is superior to other
plasmonic switches and similar to the double Si microdisk switch Watts _et
al._ (2011). Its switching power is of the same order of magnitude as other
thermo-optic switches. The power can be improved by shortening the heater,
narrowing the waveguide, and removing the Si underneath the VO2. We expect
that the power consumption can be reduced to 1 - 10 mW. The Si microdisk
switch Watts _et al._ (2011) did not have thermal tuners to bias the
wavelength, which would have added about 10-20 mW of power Li _et al._
(2011). The GeSi electro-absorption switch Liu _et al._ (2008), though
significantly larger in size, had a low power consumption because it operated
by the field-induced Franz-Keldysh effect rather than a thermo-optic effect.
The comparison shows that hybrid SPP-VO2 switches have the unparalleled
capability to exhibit large ERs at short device lengths while maintaining a
broad operation bandwidth.
In summary, we have demonstrated the first hybrid plasmonic switches that use
a transition metal oxide, VO2. Our results show that through the choice of
materials and design, electrically-controlled plasmonic switches can have
performance characteristics competitive and superior to their dielectric
counterparts. This work opens the avenue toward using VO2 for (sub)wavelength
size-scale yet efficient opto-electronic devices. The ability of the VO2 phase
transition to be initiated at sub-picosecond timescales by electric fields
Stefanovich _et al._ (2000); Nakano _et al._ (2012) are promising for ultra-
high-speed and low-power modulation.
Methods:The devices were designed using a commercial finite element solver
(COMSOL Multiphysics). The optical design was carried out using the RF module,
and the electrical and thermal designs were carried out using coupled heat
transport and DC current modules. The optical properties of silver and SiO2
were taken from Palik and Ghosh (1998). The thermal properties of VO2 were
derived from Berglund and Guggenheim (1969); Oh _et al._ (2010).
The VO2 film was deposited on SOI substrates using RF magnetron sputtering at
100 W RF power and a pressure of 7 mT. The substrate was kept at a temperature
of 773 K. The sputtering used a 2 inch diameter 99.7 % vanadium target along
with a gas mixture of argon (Ar) and oxygen (O2) with a flow rate of 83 sccm
for Ar and 4.15 sccm for O2, resulting in a O2 concentration of 5.77%. The
resistivity of the VO2 film on SOI was measured using a four-point-probe setup
(Four-Dimensions Six-Point-Probe 101C). The refractive indices were measured
using ellipsometry with the sample mounted on a temperature controlled stage
(Horiba Jobin Yvon UVISEL spectroscopic ellipsometer) Crunteanu _et al._
(2010).
The fabrication of the devices used electron-beam lithography and lift-off.
First, a 200 nm planarizing spin-on-glass (HSQ 6%) was spin-coated and cured
on top of the VO2 film. A 620 nm SiO2 layer was then deposited using a plasma-
enhanced chemical vapor deposition system (Oxford Plasmalab). The SiO2
planarized the VO2 to reduce the scattering losses and set the waveguide
dispersion. The features were defined using electron-beam lithography (Vistec
EBPG 5000+) on a resist (ZEP-520A), followed by electron-beam evaporation of
$\approx$ 0.3 nm thick chromium adhesion layer and silver. To enhance the
adhesion of the electron-beam resist, a thin layer of HMDS was spin-coated on
the SiO2 prior to the spin-coating of ZEP-520A. Finally, the features were
formed using a single lift-off step.
To measure the static characteristics of each device, a 20X infinity-corrected
objective lens mounted over the chip was used to focus light from a tunable
laser (Agilent 81682A) on the input grating and collect the optical signal
from the output grating. A half-wave plate was used to adjust the polarization
of the input light. The output was separated from the input light using a
beam-splitter, spatially filtered, and fiber-coupled to a power meter (Agilent
81633A). DC electrical probes contacted the pads. Current was ramped up and
down using a sourcemeter (Keithley 2636A) while the optical transmission and
the voltage across the electrical probes were monitored. The chip was mounted
on a temperature controlled stage maintained at 298 K. By measuring the
resistance of calibration heaters of different lengths, we found the contact
resistance of the probes was about $2.5\pm 1$ $\mathrm{\Omega}$.
To measure the dynamic response, periodic triangular voltage pulses from a
function generator (Tektronix AFG3102) were applied across electrical
contacts. The same method as above was used to couple light into and out of
the SPP-VO2 waveguides; however, the input light was pre-amplified using an
erbium doped optical amplifier before the input grating. The output light was
then coupled into a nanosecond detector (New Focus 1623). The voltage output
of the photodetector was captured by an 200 MHz oscilloscope (Tektronix TDS
2022B), which also simultaneously monitored the voltage across the electrical
probes.
Acknowledgments:A.J., B.A., J.S.A., and J.K.S.P. thank the University of
Toronto Emerging Communications Technology Institute for access to the
cleanroom facilities. They are also grateful for the financial support of the
Natural Sciences and Engineering Research Council of Canada.
## References
* Ozbay (2006) E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
* MacDonald and Zheludev (2010) K. F. MacDonald and N. I. Zheludev, “Active plasmonics: current status,” Laser & Photonics Reviews 4, 562–567 (2010).
* Schuller _et al._ (2010) J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat Mater 9, 193–204 (2010).
* Dionne _et al._ (2009) J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “Plasmostor: A metal-oxide-Si field effect plasmonic modulator,” Nano Letters 9, 897–902 (2009).
* Min and Veronis (2009) C. H. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009).
* Cai _et al._ (2009) W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Letters 9, 4403–4411 (2009).
* Xu _et al._ (2005) Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
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* Dicken _et al._ (2008) M. J. Dicken, L. A. Sweatlock, D. Pacifici, H. J. Lezec, K. Bhattacharya, and H. A. Atwater, “Electrooptic modulation in thin film barium titanate plasmonic interferometers,” Nano Letters 8, 4048–4052 (2008).
* Sorger Volker _et al._ (2012) J. Sorger Volker, D. Lanzillotti-Kimura Norberto, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17 (2012).
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* Randhawa _et al._ (2012) S. Randhawa, S. Lachèze, J. Renger, A. Bouhelier, R. E. de Lamaestre, A. Dereux, and R. Quidant, “Performance of electro-optical plasmonic ring resonators at telecom wavelengths,” Opt. Express 20, 2354–2362 (2012).
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* Watts _et al._ (2011) M. R. Watts, W. A. Zortman, D. C. Trotter, R. W. Young, and A. L. Lentine, “Vertical junction silicon microdisk modulators and switches,” Opt. Express 19, 21989–22003 (2011).
* Li _et al._ (2011) G. Li, X. Zheng, J. Yao, H. Thacker, I. Shubin, Y. Luo, K. Raj, J. E. Cunningham, and A. V. Krishnamoorthy, “25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning,” Opt. Express 19, 20435–20443 (2011).
* Palik and Ghosh (1998) Edward D. Palik and Gorachand Ghosh, _Handbook of optical constants of solids_ (Academic Press, San Diego, 1998).
* Oh _et al._ (2010) D.-W. Oh, C. Ko, S. Ramanathan, and D. G. Cahill, “Thermal conductivity and dynamic heat capacity across the metal-insulator transition in thin film VO2,” Applied Physics Letters 96, 151906–3 (2010).
* Crunteanu _et al._ (2010) A. Crunteanu, J. Givernaud, P. Blondy, J.-C. Orlianges, C. Champeaux, and A. Catherinot, “Exploiting the semiconductor-metal phase transition of VO2 materials: a novel direction towards tuneable devices and systems for RF-microwave applications,” in _Advanced Microwave and Millimeter Wave Technologies Semiconductor Devices Circuits and Systems_ , edited by M. Mukherjee (Intech, 2010) pp. 35–66.
|
arxiv-papers
| 2012-10-02T14:38:31 |
2024-09-04T02:49:35.835964
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arash Joushaghani, Brett A. Kruger, Suzanne Paradis, David Alain, J.\n Stewart Aitchison, and Joyce K. S. Poon",
"submitter": "Joyce Poon",
"url": "https://arxiv.org/abs/1210.0785"
}
|
1210.0840
|
# ADS Labs - Supporting Information Discovery in Science Education
Edwin A. Henneken Smithsonian Astrophysical Observatory, 60 Garden Street, MA
02138 Cambridge
###### Abstract
The SAO/NASA Astrophysics Data System (ADS) is an open access digital library
portal for researchers in astronomy and physics, operated by the Smithsonian
Astrophysical Observatory (SAO) under a NASA grant, successfully serving the
professional science community for two decades. Currently there are about
55,000 frequent users (100+ queries per year), and up to 10 million infrequent
users per year. Access by the general public now accounts for about half of
all ADS use, demonstrating the vast reach of the content in our databases. The
visibility and use of content in the ADS can be measured by the fact that
there are over 17,000 links from Wikipedia pages to ADS content, a figure
comparable to the number of links that Wikipedia has to OCLC’s WorldCat
catalog. The ADS, through its holdings and innovative techniques available in
ADS Labs (http://adslabs.org), offers an environment for information discovery
that is unlike any other service currently available to the astrophysics
community. Literature discovery and review are important components of science
education, aiding the process of preparing for a class, project, or
presentation. The ADS has been recognized as a rich source of information for
the science education community in astronomy, thanks to its collaborations
within the astronomy community, publishers and projects like ComPADRE. One
element that makes the ADS uniquely relevant for the science education
community is the availability of powerful tools to explore aspects of the
astronomy literature as well as the relationship between topics, people,
observations and scientific papers. The other element is the extensive
repository of scanned literature, a significant fraction of which consists of
historical literature.
## 1 Introduction
Information discovery is central to many activities in life, from finding
restaurants while attending a conference to making strategic decisions in a
big company. In science, information discovery is, for example, used to stay
up-to-date, doing literature research, and it is crucial in the process of
scientific research itself. In a world where online information is generated
24 hours a day, 7 days a week, this journey of discovery can easily become a
daunting task. We need powerful discovery tools to help us on this journey.
General search engines support a low level of information retrieval,
sufficient to get a general idea, but when you are looking for technical
material in a rich metadata environment, you need specialized digital
libraries. The SAO/NASA Astrophysics Data System (ADS) is such a digital
library (Kurtz et al. (2000), Henneken et al. (2011)). It has very
successfully served the astronomy and physics community for almost 20 years,
free of charge. In order to support a richer and more efficient information
discovery experience, we created the ADS Labs environment, in which we expose
our users to new search paradigms and tools, to better support our community’s
research needs. In the following we will argue that the ADS Labs environment
will prove to be a useful tool for people involved in science education and
outreach.
## 2 Using ADS Labs
General search engines return so many results that it quickly consumes
excessive amounts of time to parse all these results and determine if any of
these are relevant in a science education environment. As a result,
instructors often return to resources they have been using time and time
again, while missing out on a wealth of new material continuously being added
through a broad spectrum of resources. A specialized digital library is a
highly efficient tool to locate these resources. Through its contents and
functionality, ADS Labs will prove to be such tool.
In addition to publications relevant for scientific research, the ADS
repository also contains a set of journals that are directly relevant to
people involved in science education. Examples of such journals are: Astronomy
Education Review, American Journal of Physics; The Science Teacher; Journal of
Science Education and Technology; Journal of Science Teacher Education;
International Journal of Science Education; Research in Science Education;
Science & Education; Spark, the AAS Education Newsletter. A large portion of
the astronomical research of the 19th and early 20th centuries was reported in
publications written and published by individual observatories. Many of these
collections were not widely distributed and complete sets of these volumes are
now, at best, difficult to locate. This material is often requested by amateur
astronomers and researchers, because the observatory report is the only
published record of the research and observations. This makes these
publications a great resource for classes that have a component dealing with
the history of astronomy. We have a large number of historical publications
(mostly from microfilm) in our repository of scanned literature (Thompson et
al. (2007)).
The functionality that makes ADS Labs such a powerful tool is a combination of
being able to specify ahead of time what kind of results you are interested
in, and the ability to efficiently filter the results afterwards, using
facets. We will illustrate this using an example. Imagine you are looking for
publications describing or on discussing extrasolar planets in a classroom
environment. Figure 1 shows how you could start this search, using the
“streamlined search” of ADS Labs.
Figure 1.: Streamlined Search. This query example will find publication that
have the terms extrasolar planets classroom in their title, sorted by
“relevancy”. The “AST” next to the search button refers to the fact that the
entire database is searched (not just astronomy, for example).
In the search box we specify “extrasolar planets classroom”, as search scope
we select “ALL” (next to the search button), to search the entire ADS
repository, and we specify “most relevant” for sorting. This sorts on a
combination of several indicators, including date, position of the query words
in the document, position of the author in the author list, citation
statistics and usage statistics (this is how many popular search engines
rank).
Figure 2.: Results page. The results page consists of a list of publications
with a panel of facets on the left. Via snippets the user is given a view
inside the abstract or full text to see matches of the query terms.
This example query generates the results list shown in figure 2. The results
page consists of a list of publications with a panel of facets on the left,
which offer an efficient way of further filtering the results. Besides serving
as a filter, the author facet summarizes the people active in the field
defined by the query. The author facet also shows all the spelling variants by
which author names occur in the ADS repositories (by clicking on the author
name), which can assist in filtering out different authors with the same
initials (where e.g. the first name is spelled out). The diagram below the
facets shows the number of publications as a function of year, providing a
measure for the activity in a field. In this way the facets, besides serving
as filters, also provide you with valuable information. The “Data” facet
provides you with an overview of available data products, if available. Every
entry in the results list has a potential option to look inside the
publication. If an abstract is available, the “Matches in Abstract” link will
open a snippet showing the matches in the abstract of the query terms. The
link “Matches in fulltext” will show these matches in the full text version of
the publication (which could be the preprint version from arXiv). The results
page also provides the menu “More” (in the upper right corner) that contains
tools to further explore the results. The “author network” allows one to
visualize collaborations between authors and further filter the search results
by selecting within the nodes in the network. The paper network allows one to
visualize the relationships between papers and further filter the search
results by selecting within the nodes in the network. If the publications in
the results list have astronomical objects identified in them, you can
visualize these using the “Sky Map” option. This uses Google Sky. The sky map
can also be used for further filtering. The “Word Cloud” visualizes the most
relevant terms found in the list of results and further filter the list by
selecting interesting terms from the cloud. The word cloud is only indirectly
based on word frequencies: the word frequencies in the astronomy corpus are
subtracted, so that relatively frequent words stand out. The “Metrics” page
provides an overview of bibliometric indicators, based on the publications in
the results list. This overview can be exported in Excel format.
Figure 3.: Abstract page. This page provides the user with a concentrated
description of an article.
The next level is the abstract page (see figure 3). The principal function of
this page is to provide the user with a concentrated description of an
article, sufficient for the user to decide whether or not to download and read
it. This page provides a view on the basic metadata (authors, affiliations,
title, abstract). It also provides links to the full text (including open
access versions) and opportunities to share the abstract on various social
media sites, or save it in an ADS Private Library (available for users with an
ADS account, see below). When sufficient data is available, a set of
“Suggested Articles” is provided, which is generated by a recommender system
(Henneken et al. (2011)). Depending on availability, the abstract view
provides access to the bibliography of the publication (those papers cited by
the publication, for which there is a record in the ADS), an overview of
publications citing the publication, a list of papers most frequently read in
conjunction with this publication (“Co-reads”) and a list of papers that are
similar to this publication, based on word similarity (“Similar Articles”).
Figure 1 has a area on the right with two tabs (“myADS articles” and “Recently
viewed articles”). These relate to the existence of user accounts in the ADS.
The ADS offers the option to create a login, providing its users with the
possibility to personalize the service. This includes access to a powerful
alert service called “myADS” (see Henneken et al. (2007)), the creation of
“Private Libraries” (which are essentially baskets in which users can store,
and annotate, links to publications) and a way to specify a “Library Link
Server”, allowing access to full text using institutional subscriptions. This
makes the ADS portable, because it provides you with online access to the full
text of articles from anywhere in the world. When a user is logged in to their
account, visiting the streamlined search will result in displaying the most
recent content for their “daily myADS” service and an overview of their most
recently viewed records.
As an illustration of finding historical material, consider the following
example: you are interested in the history and use of an instrument called the
“mural circle”. When you run the query “m̈ural circle1̈800-1850” in the
streamlined search, you will find about two dozen results. These results show
that in this period there were such instruments at the Madras Observatory,
Royal Greenwich Observatory, Armagh Observatory, U.S. Naval Observatory and
Cape Observatory. Most publications in this results list are available in PDF
format.
## 3 Concluding Remarks
Through the publications in its holdings and the user-friendly, intuitive
streamlined search, the ADS is a useful instrument in the tool box of search
engines for professionals involved in science education and outreach. We do
realize that this is a group of users with requirements and needs that, in
some aspects, differ significantly from those that have traditionally been
using the ADS. We would love to get feedback and suggestions to help us
optimize the search experience for all our users. Our users are a big part of
our curation efforts, so if you encounter material in our database relevant
for science education, but not flagged as such, we would love to hear from you
as well. Feedback should be sent to ads@cfa.harvard.edu.
### Acknowledgments
The ADS is funded by NASA grant NNX12AG54G.
## References
* Henneken et al. (2007) Henneken, E., Kurtz, M. J., Eichhorn, G., et al. 2007, Library and Information Services in Astronomy V, 377, 106
* Henneken et al. (2011) Henneken, E. A., Kurtz, M. J., & Accomazzi, A. 2011, arXiv:1106.5644
* Henneken et al. (2011) Henneken, E. A., Kurtz, M. J., Accomazzi, A., et al. 2011, Astrophysics and Space Science Proceedings, 1, 125
* Kurtz et al. (2000) Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Murray, S. S.,Watson, J. M. (2000). Astronomy and Astrophysics Supplement Series 143, 41-59.
* Thompson et al. (2007) Thompson, D. M., Accomazzi, A., Eichhorn, G., et al. 2007, Library and Information Services in Astronomy V, 377, 102
|
arxiv-papers
| 2012-10-02T16:51:35 |
2024-09-04T02:49:35.850543
|
{
"license": "Public Domain",
"authors": "Edwin A. Henneken, Donna Thompson",
"submitter": "Edwin Henneken",
"url": "https://arxiv.org/abs/1210.0840"
}
|
1210.0847
|
# Identities involving $q$-Genocchi numbers and polynomials
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com;
saraci88@yahoo.com.tr , Mehmet Acikgoz University of Gaziantep, Faculty of
Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY
acikgoz@gantep.edu.tr , Hassan Jolany School of Mathematics, Statistics and
Computer Science, University of Tehran, Iran hassan.jolany@khayam.ut.ac.ir;
hassan.jolany@mail.com and Yuan He Department of Mathematics, Kunming
University of Science and Technology, Kunming, Yuannan 650500, People’
Republic of China hyyhe@yahoo.com.cn
###### Abstract.
In this paper, we focus on the $q$-Genocchi numbers and polynomials. We shall
introduce new identities of the q-Genocchi numbers and polynomials by using
the fermionic $p$-adic integral on $\mathbb{Z}_{p}$ which are very important
in the study of Frobenius-Genocchi numbers and polynomials. Also, we give
Cauchy-integral formula for the $q$-Genocchi polynomials and moreover by using
measure theory on $p$-adic integral we derive the distribution formula
$q$-Genocchi polynomials. Finally, we present a new definition of $q$-Zeta-
type function by using Mellin transformation which is the interpolation
function of the $q$-Genocchi polynomials at negative integers.
###### Key words and phrases:
Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials,
$p$-adic $q$-integral on $\mathbb{Z}_{p}$, Mellin transformation, $q$-Zeta
function.
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. PRELIMINARIES
Recently, Kim and Lee have given some properties for the $q$-Euler numbers and
polynomials in [7]. Actually, this type numbers and polynomials and their
$q$-extensions or variously generalizations have been studied in several
different ways for a long time (for details, see [1-21]).
By using $p$-adic $q$-integral on $\mathbb{Z}_{p}$, Kim defined many new
generating functions of the $q$-Bernoulli poynomials, $q$-Euler polynomials
and $q$-Genocchi polynomials in his arithmetic works (for details, see
[6-19]). The works of Kim have been benefited for further works of many
mathematicians in Analytic numbers theory. Actually, we motivated from his
inspiring works to write this paper.
Assume that $p$ be a fixed odd prime number. Throughout this work, we require
the definitions of the some notations such that let $\mathbb{Q}_{p}$ be the
field $p$-adic rational numbers and let $\mathbb{C}_{p}$ be the completion of
algebraic closure of $\mathbb{Q}_{p}$. That is,
$\boldsymbol{\mathbb{Q}}_{p}=\left\\{x=\sum_{n=-k}^{\infty}a_{n}p^{n}:0\leq
a_{n}\leq p-1\right\\}\text{.}$
Then $\mathbb{Z}_{p}$ is integral domain which is defined by
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x=\sum_{n=0}^{\infty}a_{n}p^{n}:0\leq
a_{n}\leq p-1\right\\}$
or
$\boldsymbol{\mathbb{Z}}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq
1\right\\}\text{.}$
We assume that $q\in\mathbb{C}_{p}$ with $\left|1-q\right|_{p}<1$ as an
indeterminate. The $p$-adic absolute value $\left|.\right|_{p}$, is normally
given by
$\left|x\right|_{p}=p^{-r}$
where $x=p^{r}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and $r\in\mathbb{Q}$.
The $q$-extension of $x$ with the display notation of $\left[x\right]_{q}$ is
introduced by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{.}$
We want to note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ (see[1-22]).
Also, we use notation $\mathbb{N}^{\ast}$ for the union of zero and natural
numbers.
We consider that $\eta$ is a uniformly differentiable function at a point
$a\in\mathbb{Z}_{p}$, if the difference quotient
$\Phi_{\eta}\left(x,y\right)=\frac{\eta\left(x\right)-\eta\left(y\right)}{x-y},$
have a limit $\eta{\acute{}}\left(a\right)$ as
$\left(x,y\right)\rightarrow\left(a,a\right)$ and denote this by $\eta\in
UD\left(\mathbb{Z}_{p}\right)$.
Then, for $\eta\in UD\left(\mathbb{Z}_{p}\right)$, we can discuss the
following
$\frac{1}{\left[p^{n}\right]_{q}}\sum_{0\leq\xi<p^{n}}\eta\left(\xi\right)q^{\xi}=\sum_{0\leq\xi<p^{n}}\eta\left(\xi\right)\mu_{q}\left(\xi+p^{n}\mathbb{Z}_{p}\right)\text{,}$
which represents as a $p$-adic $q$-analogue of Riemann sums for $\eta$. The
integral of $\eta$ on $\mathbb{Z}_{p}$ will be defined as the limit
$\left(n\rightarrow\infty\right)$ of these sums, when it exists. The $p$-adic
$q$-integral of function $\eta\in UD\left(\mathbb{Z}_{p}\right)$ is defined by
T. Kim in [6], [10], [15] by
(1.1)
$I_{q}\left(\eta\right)=\int_{\mathbb{Z}_{p}}\eta\left(\xi\right)d\mu_{q}\left(\xi\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q}}\sum_{\xi=0}^{p^{n}-1}\eta\left(\xi\right)q^{\xi}\text{.}$
The bosonic integral is considered as a bosonic limit $q\rightarrow 1,$
$I_{1}\left(\eta\right)=\lim_{q\rightarrow 1}I_{q}\left(\eta\right)$.
Similarly, the fermionic $p$-adic integral on $\mathbb{Z}_{p}$ is introduced
by T. Kim as follows:
(1.2)
$I_{-q}\left(\eta\right)=\int_{\mathbb{Z}_{p}}\eta\left(\xi\right)d\mu_{-q}\left(\xi\right)$
(for more details, see [16-18]). Obviously that
(1.3) $\lim_{q\rightarrow
1}I_{-q}\left(\eta\right)=I_{-1}\left(\eta\right)=\int_{\mathbb{Z}_{p}}\eta\left(\xi\right)d\mu_{-1}\left(\xi\right)=\lim_{n\rightarrow\infty}\sum_{\xi=0}^{p^{n}-1}\eta\left(\xi\right)\left(-1\right)^{\xi}\text{.}$
From (1.3), it is well-known as the useful property for the fermionic $p$-adic
$q$-integral on $\mathbb{Z}_{p}$:
(1.4)
$I_{-1}\left(\eta_{1}\right)+I_{-1}\left(\eta\right)=2\eta\left(0\right)\text{,}$
where $\eta_{1}\left(x\right)=\eta\left(x+1\right)$ (for details, see[2-4,
11-14, 16-22]).
The $q$-Genocchi polynomials can be introduced as follows:
(1.5)
$G_{n,q}\left(x\right)=n\int_{\mathbb{Z}_{p}}q^{\xi}\left(x+\xi\right)^{n-1}d\mu_{-1}\left(\xi\right)\text{.}$
From (1.5), we have
$G_{n,q}\left(x\right)=\sum_{l=0}^{n}\binom{n}{l}x^{l}G_{n-l,q}$
where $G_{n,q}(0):=G_{n,q}$ are called $q$-Genocchi numbers. Then,
$q$-Genocchi numbers can be given by
$G_{0,q}=0\text{ and
}q\left(G_{q}+1\right)^{n}+G_{n,q}=\left\\{\begin{array}[]{cc}2,&\text{if
}n=1\\\ 0,&\text{if\ }n\geq 1\end{array}\right.$
with the usual convention about replacing $\left(G_{q}\right)^{n}$ by
$G_{n,q}$ is used (for details, see [4]).
Our objective in the present paper is to derive not only new but also novel
and interesting properties of the $q$-Genocchi numbers and polynomials. Our
applications for the $q$-Genocchi polynomials seem to be useful in mathematics
for engineerings (on this subject, see [23]).
## 2\. SOME PROPERTIES ON THE $q$-GENOCCHI NUMBERS AND POLYNOMIALS
Let $\eta\left(x\right)=q^{x}e^{t\left(x+\xi\right)}$. Then, by using (1.4),
we see that
$t\int_{\mathbb{Z}_{p}}q^{\xi}e^{t\left(x+\xi\right)}d\mu_{-1}\left(\xi\right)=\frac{2t}{qe^{t}+1}e^{xt}\text{.}$
From the last equality and (1.5), we easily derive the following generating
function of the $q$-Genocchi polynomials:
(2.1)
$t\int_{\mathbb{Z}_{p}}q^{\xi}e^{t\left(x+\xi\right)}d\mu_{-1}\left(\xi\right)=\sum_{n=0}^{\infty}G_{n,q}\left(x\right)\frac{t^{n}}{n!}=\frac{2t}{qe^{t}+1}e^{xt}\text{,
}\left|\log q+t\right|<\pi\text{.}$
Substituting $x\rightarrow x+y$ into (2.1), then we write
$\displaystyle\sum_{n=0}^{\infty}G_{n,q}\left(x+y\right)\frac{t^{n}}{n!}=\frac{2t}{qe^{t}+1}e^{\left(x+y\right)t}$
$\displaystyle=\left(\sum_{n=0}^{\infty}G_{n,q}\left(x\right)\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}y^{n}\frac{t^{n}}{n!}\right)$
$\displaystyle=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\binom{n}{k}G_{k,q}\left(x\right)y^{n-k}\right)\frac{t^{n}}{n!}$
From the above, we easily express the following theorem:
###### Theorem 1.
The following holds:
(2.2)
$G_{n,q}\left(x+y\right)=\sum_{k=0}^{n}\binom{n}{k}G_{k,q}\left(x\right)y^{n-k}\text{.}$
By (2.2), we consider the following
$G_{n,q}\left(x+y\right)=\frac{2}{\left[2\right]_{q}}ny^{n-1}+\sum_{k=2}^{n}\binom{n}{k}G_{k,q}\left(x\right)y^{n-k}\text{.}$
From this, it follows that
$G_{n,q}\left(x+y\right)-\frac{2}{\left[2\right]_{q}}ny^{n-1}=\sum_{k=2}^{n}\binom{n}{k}G_{k,q}\left(x\right)y^{n-k}$
can be derived and so we reach the following theorem:
###### Theorem 2.
For $n\in\mathbb{N}^{\ast}$, one has
(2.3)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{n}{k}}{\left(k+2\right)\left(k+1\right)}G_{k+2,q}\left(x\right)y^{n-k}$
$\displaystyle=\frac{G_{n+2,q}\left(x+y\right)-\frac{2}{\left[2\right]_{q}}\left(n+2\right)y^{n+1}}{\left(n+2\right)\left(n+1\right)}\text{.}$
Replacing $y$ by $-y$ into (2.3), then we get
(2.4)
$\displaystyle\frac{G_{n+2,q}\left(x-y\right)-\frac{2}{\left[2\right]_{q}}\left(n+2\right)\left(-1\right)^{n+1}y^{n+1}}{\left(n+2\right)\left(n+1\right)}$
$\displaystyle=\sum_{k=0}^{n}\frac{\binom{n}{k}\left(-1\right)^{n-k}}{\left(k+2\right)\left(k+1\right)}G_{k+2,q}\left(x\right)y^{n-k}\text{.}$
By (2.4), it follows that
(2.5)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{n}{k}\left(-1\right)^{k}}{\left(k+2\right)\left(k+1\right)}G_{k+2,q}\left(x\right)y^{n-k}$
$\displaystyle=\frac{\left(-1\right)^{n}G_{n+2,q}\left(x-y\right)+\frac{2}{\left[2\right]_{q}}\left(n+2\right)y^{n+1}}{\left(n+1\right)\left(n+2\right)}\text{.}$
Therefore, from the expressions of (2.3) and (2.5), we procure the following
theorem:
###### Theorem 3.
The following holds true:
(2.6)
$\displaystyle\sum_{k=0}^{\left[\frac{n}{2}\right]}\frac{\binom{n}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)y^{n-2k}$
$\displaystyle=\frac{\left(-1\right)^{n}G_{n+2,q}\left(x-y\right)+G_{n+2,q}\left(x+y\right)}{\left(n+1\right)\left(n+2\right)}$
where $\left[.\right]$ is Gauss’ symbol.
By (2.4), we have the following identity
(2.7)
$\displaystyle\sum_{k=2}^{n}\frac{\binom{n}{k}\left(-1\right)^{k}}{k\left(k-1\right)}G_{k,q}\left(x\right)y^{n-k}$
$\displaystyle=\frac{\left(-1\right)^{n}G_{n,q}\left(x-y\right)+\frac{2}{\left[2\right]_{q}}ny^{n-1}}{n\left(n-1\right)}\text{.}$
By (2.4), (2.5) and (2.7), then we have the following theorem:
###### Theorem 4.
For $n\in\mathbb{N}^{\ast}$, we get
(2.8)
$\displaystyle\frac{\left(-1\right)^{n}G_{n+2,q}\left(x-y\right)+G_{n+2,q}\left(x+y\right)}{\left(n+2\right)\left(n+1\right)}$
$\displaystyle=\sum_{k=0}^{\left[\frac{n+1}{2}\right]}\frac{\binom{n}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)y^{n-2k}\text{.}$
Taking $y=1$ into (2.3), it follows that
(2.9)
$q\sum_{k=0}^{n}\frac{\binom{n}{k}}{\left(k+2\right)\left(k+1\right)}G_{k+2,q}\left(x\right)=\frac{qG_{n+2,q}\left(x+1\right)}{\left(n+1\right)\left(n+2\right)}-\frac{2q}{\left[2\right]_{q}\left(n+1\right)}\text{.}$
We need the following for sequel of this paper:
$2e^{tx}=\frac{1}{t}\left(q\frac{2t}{qe^{t}+1}e^{\left(x+1\right)t}+\frac{2t}{qe^{t}+1}e^{xt}\right)\text{.}$
From the above, we easily develop the following:
(2.10)
$qG_{n+1,q}\left(x+1\right)+G_{n+1,q}\left(x\right)=\left(n+1\right)2x^{n}\text{.}$
By (2.9) and (2.10), we state the following theorem:
###### Theorem 5.
The following holds:
(2.11)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{n}{k}}{\left(k+2\right)\left(k+1\right)}G_{k+2,q}\left(x\right)$
$\displaystyle=\frac{2x^{n+1}}{qn+q}-\frac{G_{n+2,q}\left(x\right)}{\left(qn+q\right)\left(n+2\right)}-\frac{2}{\left[2\right]_{q}\left(n+1\right)}\text{.}$
Thanks to equality of $\lim_{q\rightarrow
1}G_{n,q}\left(x\right)=G_{n,1}\left(x\right):=G_{n}\left(x\right)$, where
$G_{n}\left(x\right)$ are known as Genocchi polynomials which is given via the
exponential generating function, as follows:
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2t}{e^{t}+1}e^{xt}\text{
}\left(\left|t\right|<\pi\right),$
(see [1-4, 12, 13, 21]). From the above, as $q\rightarrow 1$ in (2.11), we
discover the following corollary:
###### Corollary 1.
The following
$\displaystyle\sum_{k=0}^{n}\frac{\binom{n}{k}}{\left(k+2\right)\left(k+1\right)}G_{k+2}\left(x\right)$
$\displaystyle=\frac{2x^{n+1}}{n+1}-\frac{G_{n+2}\left(x\right)}{\left(n+1\right)\left(n+2\right)}-\frac{1}{n+1}$
is true.
Let us take $y=1$ and $n\rightarrow 2n$ into (2.6), becomes
(2.12)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)$
$\displaystyle=\frac{G_{2n+2,q}\left(x-1\right)+G_{2n+2,q}\left(x+1\right)}{\left(2n+1\right)\left(2n+2\right)}$
$\displaystyle=\frac{\frac{1}{q}\left(qG_{2n+2,q}\left(x+1\right)+G_{2n+2,q}\left(x\right)\right)+qG_{2n+2,q}\left(x\right)+G_{2n+2,q}\left(x-1\right)}{\left(2n+1\right)\left(2n+2\right)}$
$\displaystyle-\frac{G_{2n+2,q}\left(x\right)}{q\left(2n+1\right)\left(2n+2\right)}-\frac{qG_{2n+2,q}\left(x\right)}{\left(2n+1\right)\left(2n+2\right)}$
$\displaystyle=\frac{2\left(n+2\right)x^{n+1}}{\left(2n+1\right)\left(2n+2\right)}+\frac{2\left(n+2\right)\left(x-1\right)^{n+1}}{\left(2n+1\right)\left(2n+2\right)}-\frac{G_{2n+2,q}\left(x\right)}{q\left(2n+1\right)\left(2n+2\right)}-\frac{qG_{2n+2,q}\left(x\right)}{\left(2n+1\right)\left(2n+2\right)}$
After these applications, we conclude with the following theorem:
###### Theorem 6.
The following identity
(2.13)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)$
$\displaystyle=\frac{\left(n+2\right)x^{n+1}}{\left(2n+1\right)\left(n+1\right)}+\frac{\left(n+2\right)\left(x-1\right)^{n+1}}{\left(2n+1\right)\left(n+1\right)}-\frac{G_{2n+2,q}\left(x\right)}{q\left(2n+1\right)\left(2n+2\right)}-\frac{qG_{2n+2,q}\left(x\right)}{\left(2n+1\right)\left(2n+2\right)}$
is true.
Now, we analyse as $q\rightarrow 1$ for the equation (2.13) and so we readily
state the following corollary which seems to be interesting property for the
Genocchi polynomials.
###### Corollary 2.
The following equality holds:
(2.14)
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2}\left(x\right)$
$\displaystyle=\frac{2\left(n+2\right)x^{n+1}}{\left(2n+1\right)\left(2n+2\right)}+\frac{2\left(n+2\right)\left(x-1\right)^{n+1}}{\left(2n+1\right)\left(2n+2\right)}-\frac{G_{2n+2}\left(x\right)}{\left(2n+1\right)\left(2n+2\right)}-\frac{G_{2n+2}\left(x\right)}{\left(2n+1\right)\left(2n+2\right)}\text{.}$
Substituting $n\rightarrow 2n+1$ and $y=1$ into (2.8), we compute
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n+1}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)$
$\displaystyle=\frac{G_{2n+3,q}\left(x+1\right)-G_{2n+3,q}\left(x-1\right)}{\left(2n+3\right)\left(2n+2\right)}$
$\displaystyle=\frac{\frac{1}{q}\left(qG_{2n+3,q}\left(x+1\right)+G_{2n+3,q}\left(x\right)\right)-\left(G_{2n+3,q}\left(x\right)+G_{2n+3,q}\left(x-1\right)\right)}{\left(2n+3\right)\left(2n+2\right)}$
$\displaystyle+\left(\frac{q-1}{q}\right)\frac{G_{2n+3,q}\left(x\right)}{\left(2n+3\right)\left(2n+2\right)}$
$\displaystyle=\frac{x^{2n+2}}{q\left(n+1\right)}-\frac{\left(x-1\right)^{2n+2}}{\left(n+1\right)}+\left(\frac{q-1}{q}\right)\frac{G_{2n+3,q}\left(x\right)}{\left(2n+3\right)\left(2n+2\right)}\text{.}$
Therefore, we obtain the following theorem:
###### Theorem 7.
The following equality holds:
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n+1}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)$
$\displaystyle=\frac{x^{2n+2}}{q\left(n+1\right)}-\frac{\left(x-1\right)^{2n+2}}{\left(n+1\right)}+\left(\frac{q-1}{q}\right)\frac{G_{2n+3,q}\left(x\right)}{\left(2n+3\right)\left(n+1\right)}\text{.}$
As $q\rightarrow 1$ in the above theorem, then we easily derive the following
corollary:
###### Corollary 3.
For $n\in\mathbb{N}^{\ast}$, then we have
$\displaystyle\sum_{k=0}^{n}\frac{\binom{2n+1}{2k}}{\left(k+1\right)\left(2k+1\right)}G_{2k+2,q}\left(x\right)$
$\displaystyle=\frac{2x^{2n+2}}{\left(2n+2\right)}-\frac{2\left(x-1\right)^{2n+2}}{\left(2n+2\right)}\text{.}$
## 3\. CONCLUSION
In this final section, we recall the generating function of the $q$-Genocchi
polynomials, as follows:
(3.1)
$\mathcal{F}_{q}\left(x,t\right)=\frac{2t}{qe^{t}+1}e^{xt}=\sum_{n=0}^{\infty}G_{n,q}\left(x\right)\frac{t^{n}}{n!}\text{.}$
Using the definition of the $k$-th derivative as $\frac{d^{k}}{dt^{k}}$ to
(3.1), then we easily see that
(3.2)
$\frac{d^{k}}{dt^{k}}\left(\frac{2t}{qe^{t}+1}e^{xt}\right)=\frac{d^{k}}{dt^{k}}\left(\sum_{n=0}^{\infty}G_{n,q}\left(x\right)\frac{t^{n}}{n!}\right)\text{.}$
By applying $\lim_{t\rightarrow 0}$ on the both sides in (3.2), then we
conclude with the following theorem:
###### Theorem 8.
The following identity
(3.3) $G_{k,q}\left(x\right)=\lim_{t\rightarrow
0}\left[\frac{d^{k}}{dt^{k}}\left(\frac{2t}{qe^{t}+1}e^{xt}\right)\right]$
is true.
We now consider Cauchy-integral formula of the $q$-Genocchi polynomials which
is a vital and important in complex analysis, is an important statement about
line integrals for holomorphic functions in the complex plane. So, by using
the equation of (3.3), we can state the following theorem:
###### Theorem 9.
The following Cauchy-integral holds true:
$G_{n,q}\left(x\right)=\frac{n!}{2\pi
i}\int_{C}\mathcal{F}_{q}\left(x,t\right)\frac{dt}{t^{n+1}}$
where $C$ is a loop which starts at $-\infty$, encircles the origin once in
the positive direction, and the returns $-\infty$.
Distribution formula for the $q$-Genocchi polynomials is important to study
regarding $p$-adic Measure theory. That is,
$\displaystyle\int_{\mathbb{Z}_{p}}q^{y}\left(x+y\right)^{n}d\mu_{-1}\left(y\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\sum_{\xi=0}^{dp^{n}-1}\left(-1\right)^{\xi}\left(x+\xi\right)^{n}q^{\xi}$
$\displaystyle=$ $\displaystyle
d^{n}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{a}\left(\lim_{n\rightarrow\infty}\sum_{\xi=0}^{p^{n}-1}\left(-1\right)^{\xi}\left(\frac{x+a}{d}+\xi\right)^{n}q^{d\xi}\right)$
$\displaystyle=$ $\displaystyle
d^{n}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{a}\frac{G_{n+1,q}\left(\frac{x+a}{d}\right)}{n+1}.$
After the above applications, we procure the following theorem.
###### Theorem 10.
For $n\in\mathbb{N}^{\ast}$, then we have
$G_{n,q}\left(dx\right)=d^{n-1}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{a}G_{n,q}\left(x+\frac{a}{d}\right)\text{.}$
By utilizing from the definition of the geometric series in (3.1), we easily
see that
$\sum_{m=0}^{\infty}G_{m,q}\left(x\right)\frac{t^{m}}{m!}=\sum_{m=0}^{\infty}\left(2\left(m+1\right)\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}n^{m}\right)\frac{t^{m+1}}{\left(m+1\right)!},$
by comparing the coefficients on the both sides, then we have, for
$m\in\mathbb{N}$
(3.4)
$\frac{G_{m+1,q}\left(x\right)}{m+1}=\left[2\right]_{q}\sum_{n=1}^{\infty}\left(-1\right)^{n}q^{n}n^{m}\text{.}$
Now also, we develop the following applications by using Mellin transformation
to the generating function of the $q$-Genocchi polynomaials: For
$s\in\mathbb{C}$ and $\Re e\left(s\right)>1$,
$\displaystyle\zeta\left(s,x:q\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-2}\left\\{-\mathcal{F}_{q}\left(x,-t\right)\right\\}dt$
$\displaystyle=$ $\displaystyle
2\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}\left(\int_{0}^{\infty}t^{s-1}e^{-nt}dt\right)$
$\displaystyle=$ $\displaystyle
2\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}q^{n}}{n^{s}}$
Thus, we can state the definition of the $q$-Zeta-type function as follows:
(3.6)
$\zeta\left(s,x:q\right)=2\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}q^{n}}{n^{s}}\text{.}$
As $q\rightarrow 1$ in (3.6), turns into
$\lim_{q\rightarrow
1}\zeta\left(s,x:q\right)=\zeta\left(s,x:1\right):=\zeta\left(s,x\right)=2\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{n^{s}}$
which is well-known as Euler-Zeta function (see [6]). By (3.4) and (3.6), we
get
$\zeta\left(-n,x:q\right)=\frac{G_{n+1,q}\left(x\right)}{n+1}\text{.}$
## References
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|
arxiv-papers
| 2012-10-02T17:15:26 |
2024-09-04T02:49:35.858310
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz, Hassan Jolany, and Yuan He",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1210.0847"
}
|
1210.1068
|
# Novel method of fractal approximation
K. Igudesman and G. Shabernev
###### Abstract
We introduce new method of optimization for finding free parameters of affine
iterated function systems (IFS), which are used for fractal approximation. We
provide the comparison of effectiveness of fractal and quadratic types of
approximation, which are based on a similar optimization scheme, on the
various types of data: polynomial function, DNA primary sequence, price graph
and graph of random walking.
## 1 Introduction
It is well known that approximation is a crucial method for making complicated
data easier to describe and operate. In many cases we have to deal with
irregular forms, which can’t be approximate with desired precision. Fractal
approximation become a suitable tool for that purpose. Ideas for interpolation
and approximation with the help of fractals appeared in works of M. Barnsley
[2] and was developed by P. Massopust [6] and C. Bandt and A. Kravchenko [1].
Today we can apply fractals to approximate such interesting and
interdisciplinary data as graphs of DNA primary sequences of different species
and interbeat heart intervals [7], price waves and many others.
Section 2 of this work is devoted to the construction of fractal interpolation
functions. Necessary condition on free parameters $d_{i}$ of affine iterated
function systems is shown. One graphical example is given.
In section 3 we give the common scheme of approximation of general function
$g\in L^{2}[a,b]$ and obtain the equation for direct calculation of free
parameters $d_{i}$.
In section 4 we illustrate the results on concrete examples.
## 2 Fractal Interpolation Functions
There are two methods for constructing fractal interpolation functions. In
1986 M. Barnsley [2] defined such functions, as attractors of some specific
iterated function systems. In this work we use common approach, which was
developed by P. Massopust [6].
Let $[a,b]\subset\mathbb{R}$ be a nonempty interval, $1<N\in\mathbb{N}$ and
$\\{(x_{i},y_{i})\in[a,b]\times\mathbb{R}\mid
a=x_{0}<x_{1}<\cdots<x_{N-1}<x_{N}=b\\}$ — are points of interpolation. For
all $i=\overline{1,N}$ consider affine transformations of the plane
$A_{i}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\quad
A_{i}\left(\begin{array}[]{c}x\\\ y\\\
\end{array}\right):=\left(\begin{array}[]{cc}a_{i}&0\\\ c_{i}&d_{i}\\\
\end{array}\right)\left(\begin{array}[]{c}x\\\ y\\\
\end{array}\right)+\left(\begin{array}[]{c}e_{i}\\\ f_{i}\\\
\end{array}\right).$
We require following two conditions hold true for all $i$:
$A_{i}(x_{0},y_{0})=(x_{i-1},y_{i-1}),\quad A_{i}(x_{N},y_{N})=(x_{i},y_{i}).$
In this case
$\begin{array}[]{ll}\displaystyle
a_{i}=\frac{x_{i}-x_{i-1}}{b-a},&\displaystyle
c_{i}=\frac{y_{i}-y_{i-1}-d_{i}(y_{N}-y_{0})}{b-a},\\\ \displaystyle
e_{i}=\frac{bx_{i-1}-ax_{i}}{b-a},&\displaystyle
f_{i}=\frac{by_{i-1}-ay_{i}-d_{i}(by_{0}-ay_{N})}{b-a},\end{array}$ (1)
there $\\{d_{i}\\}_{i=1}^{N}$ act like family of parameters. Notice, that for
all $i$ operator $A_{i}$ takes the line segment between $(x_{0},y_{0})$ and
$(x_{N},y_{N})$ to the line segment passes through points of interpolation
$(x_{i-1},y_{i-1})$ and $(x_{i},y_{i})$.
Let $\mathcal{K}$ be a space of nonempty compact subsets $\mathbb{R}^{2}$ with
Hausdorff metric. Define the Hutchinson operator [5]
$\Phi:\mathcal{K}\rightarrow\mathcal{K},\qquad\Phi(E)=\bigcup_{i=1}^{N}A_{i}(E).$
It is easily seen [2], that the Hutchinson operator $\Phi$ take a graph of any
continuous function on a segment $[a,b]$ to a graph of a continuous function
on the same segment. Thus, $\Phi$ can be treated as operator on the space of
continuous functions $C[a,b]$.
For all $i=\overline{1,N}$ denote
$\begin{array}[]{l}u_{i}:[a,b]\rightarrow[x_{i-1},x_{i}],\quad
u_{i}(x):=a_{i}x+e_{i},\\\ p_{i}:[a,b]\rightarrow\mathbb{R},\quad
p_{i}(x):=c_{i}x+f_{i}.\end{array}$ (2)
Massopust [6] has shown, that $\Phi$ acts on $C[a,b]$ according to the rule
$(\Phi g)(x)=\sum_{i=1}^{N}\left((p_{i}\circ u_{i}^{-1})(x)+d_{i}(g\circ
u_{i}^{-1})(x)\right)\chi_{[x_{i-1},x_{i}]}(x).$ (3)
Moreover, if $|d_{i}|<1$ for all $i=\overline{1,N}$, then operator $\Phi$ is
contractive on the Banach space $(C[a,b],\|\ \|_{\infty})$ with contractive
constant $d\leq\max\\{|d_{i}|\mid i=\overline{1,N}\\}$. By the fixed-point
theorem there exists unique function $g^{\star}\in C[a,b]$, such that $\Phi
g^{\star}=g^{\star}$ and for all $g\in C[a,b]$ we have
$\lim_{n\to\infty}\|\Phi^{n}(g)-g^{\star}\|_{\infty}=0.$
We will call $g^{\star}$ fractal interpolation function. It is clear, that if
$g\in C[a,b]$, $g(x_{0})=y_{0}$ and $g(x_{N})=y_{N}$, then $\Phi(g)$ passes
through points of interpolation. In this case we will call $\Phi^{n}(g)$ pre-
fractal interpolation functions of order $n$.
###### Example 1
Picture shows fractal interpolation function, which was constructed on points
of interpolation $(0,0)$, $(0.5,0.5)$ $(1,0)$ with parameters
$d_{1}=d_{2}=0.5$.
Figure 1: Fractal interpolation function.
## 3 Approximation
From now on we assume, that $|d_{i}|<1$ for all $i=\overline{1,N}$. We try to
approximate function $g\in C[a,b]$ by the fractal interpolation function
$g^{\star}$, which is constructed on points of interpolation
$\\{(x_{i},y_{i})\\}_{i=0}^{N}$. Thus, it is sufficient to fit parameters
$d_{i}\in(-1,1)$ to minimize the distance between $g$ and $g^{\star}$.
We use methods that have been developed for fractal image compression [3].
Notice, that from (3), (2) and (1) follows, that for all $g,h\in L^{2}[a,b]$
$\begin{split}\|\Phi g-\Phi h\|_{2}=\sqrt{\int_{a}^{b}(\Phi g-\Phi
h)^{2}\,\mathrm{d}x}=\sqrt{\sum_{i=1}^{N}d_{i}^{2}\int_{x_{i-1}}^{x_{i}}(g\circ
u_{i}^{-1}(x)-h\circ u_{i}^{-1}(x))^{2}\,\mathrm{d}x}\\\
\leq\max_{i=\overline{1,N}}\\{|d_{i}|\\}\cdot\sqrt{\sum_{i=1}^{N}a_{i}\int_{a}^{b}(g-h)^{2}\,\mathrm{d}x}=\max_{i=\overline{1,N}}\\{|d_{i}|\\}\cdot\|g-h\|_{2}.\end{split}$
Thus, $\Phi:L^{2}[a,b]\rightarrow L^{2}[a,b]$ is contractive operator with a
fixed point $g^{\star}$.
Furthermore, instead of minimization of $\|g-g^{\star}\|_{2}$ we will minimize
$\|g-\Phi g\|_{2}$, that makes the problem of optimization much easier. The
collage theorem provides validity of such approach.
###### Theorem 1
Let $(X,d)$ be a non-empty complete metric space. Let $T:X\to X$ be a
contraction mapping on $X$ with contractivity factor $c<1$. Then for all $x\in
X$
$d(x,x^{\star})\leq\frac{d(x,T(x))}{1-c}$
where $x^{\star}$ is the fixed point of $T$.
$\blacktriangleright$ For all integer $n$ we have
$\begin{split}d(x,x^{\star})\leq
d(x,T(x))+d(T(x),T^{2}(x))+\cdots+d(T^{n-1}(x),T^{n}(x))+d(T^{n}(x),x^{\star})\\\
\leq d(x,T(x))(1+c+c^{2}+\cdots+c^{n-1})+d(T^{n}(x),x^{\star}).\end{split}$
Letting $n\rightarrow\infty$ we establish the formula. $\blacktriangleleft$
Considering (1) and (2), we rewrite (3):
$(\Phi
g)(x)=\sum_{i=1}^{N}\Big{(}\alpha_{i}(x)-d_{i}\big{(}\beta_{i}(x)-g\circ\gamma_{i}(x)\big{)}\Big{)}\chi_{[x_{i-1},x_{i}]}(x),$
(4)
where
$\alpha_{i}(x)=\frac{(y_{i}-y_{i-1})x+(x_{i}y_{i-1}-x_{i-1}y_{i})}{x_{i}-x_{i-1}},$
$\beta_{i}(x)=\frac{(y_{N}-y_{0})x+(x_{i}y_{0}-x_{i-1}y_{N})}{x_{i}-x_{i-1}},$
(5) $\gamma_{i}(x)=\frac{(b-a)x+(x_{i}a-x_{i-1}b)}{x_{i}-x_{i-1}}.$
Thus, we have to minimize functional
$(\|g-\Phi
g\|_{2})^{2}=\sum_{i=1}^{N}\int_{x_{i-1}}^{x_{i}}\Big{(}g(x)-\alpha_{i}(x)+d_{i}\big{(}\beta_{i}(x)-g\circ\gamma_{i}(x)\big{)}\Big{)}^{2}\,\mathrm{d}x.$
Setting partial derivatives with respect to $d_{i}$ to zero we obtain
$d_{i}=\frac{\int_{x_{i-1}}^{x_{i}}\big{(}\alpha_{i}(x)-g(x)\big{)}\big{(}\beta_{i}(x)-g\circ\gamma_{i}(x)\big{)}\,\mathrm{d}x}{\int_{x_{i-1}}^{x_{i}}\big{(}\beta_{i}(x)-g\circ\gamma_{i}(x)\big{)}^{2}\,\mathrm{d}x},\qquad
i=1,\ldots,N.$ (6)
## 4 Discretization and results
In this section we will approximate discrete data
$Z=\\{(z_{m},w_{m})\\}_{m=1}^{M}$, $a=z_{1}<z_{2}<\cdots<z_{M}=b$ by the
fractal interpolation function $g^{\star}$, which is constructed on points of
interpolation $X=\\{(x_{i},y_{i})\\}_{i=0}^{N}$, $N\ll M$. Taking $X\subset
Z$, $(x_{0},y_{0})=(z_{1},w_{1})$ and $(x_{N},y_{N})=(z_{M},w_{M})$ we fit
parameters $d_{i}\in(-1,1)$ to minimize
$\sum_{m=1}^{M}(w_{m}-g^{\star}(z_{m}))^{2}.$
Let us approximate $Z$ by the piecewise constant function
$g:[a,b]\to\mathbb{R}$. More precisely $g(z)=w_{m}$, where $(z_{m},w_{m})\in
Z$ and $z_{m}$ is a nearest neighbor of $z$. From (6) we obtain the discrete
formulas for $d_{i}$:
$d_{i}=\frac{\sum\limits_{z_{m}\in[x_{i},x_{i+1}]}\big{(}\alpha_{i}(z_{m})-w_{m}\big{)}\big{(}\beta_{i}(z_{m})-g\circ\gamma_{i}(z_{m})\big{)}}{\sum\limits_{z_{m}\in[x_{i},x_{i+1}]}\big{(}\beta_{i}(z_{m})-g\circ\gamma_{i}(z_{m})\big{)}^{2}\,},\qquad
i=1,\ldots,N-1.$ (7)
After finding $d_{i}$ we obtain formulas for affine transformations $A_{i}$
and we are able to construct fractal interpolation function $g^{\star}$ for
$g$.
Our aim is to compare fractal approximation with a piecewise quadratic
approximation function which is based on the same discretization. On each
segment $[x_{i-1},x_{i}]$ approximating function has the quadratic form
$q_{i}(x)=k_{i}x^{2}+r_{i}x+l_{i}$. To get a continuous function we claim that
$q_{i}(x_{i-1})=g(x_{i-1})$ and $q_{i}(x_{i})=g(x_{i})$. From this we find
coefficients $k_{i}$ and $l_{i}$. To find free parameter $r_{i}$ we minimize
functional
$\sum\limits_{z_{m}\in[x_{i-1},x_{i}]}(w_{m}-q_{i}(z_{m}))^{2}$
with respect to $r_{i}$ on each segment $[x_{i-1},x_{i}],i=\overline{1,N}$.
The approximating function $q(x)$ will have following form:
$q(x)=\left\\{\begin{array}[]{ll}q_{1}(x)=k_{1}x^{2}+r_{1}x+l_{1},&x\in[a=x_{0},x_{1}];\\\
q_{2}(x)=k_{2}x^{2}+r_{2}x+l_{2},&x\in[x_{1},x_{2}];\\\ \qquad\vdots\\\
q_{N}(x)=k_{N}x^{2}+r_{N}x+l_{N},&x\in[x_{N-1},x_{N}=b].\end{array}\right.$
Since there is one free parameter $r_{i}$ in each function $q_{i}(x)$ and one
parameter $d_{i}$ for each affine transformation $A_{i}$ it makes the
comparison correct.
To compare fractal and quadratic approximations we consider four types of
data.
1. 1.
Polynomial function.
2. 2.
DNA sequence.
3. 3.
Price graph.
4. 4.
Random walking graph.
For all types of data $M=10000$, $z_{m}=m$, $[a,b]=[1,M]$,
$\\{w_{m}\\}_{m=1}^{M}$ are normalized sequences, that is $E(\\{w_{m}\\})=0$
and $E(\\{w_{m}^{2}\\})=1$. For all cases we choose $(x_{0},y_{0})=(1,w_{1})$,
$(x_{N},y_{N})=(M,w_{M})$ and other interpolation points $(x_{i},y_{i})$,
$i=\overline{1,N-1}$ are local extremums of the given data.
###### Example 2
Let $f(x)=-6x+5x^{2}+5x^{3}-5x^{4}+x^{5},\,x\in[-1,2.5]$. As we work with the
segment $[1,M]$ we map $[-1,2.5]$ to it. Consider sequence
$v_{m}=f\left(\frac{7(m-1)}{2(M-1)}-1\right)$, $m=\overline{1,M}$. Set
$w_{m}=(v_{m}-s_{1})/s_{2}$, where $s_{1}$ and $s_{2}$ are mean and deviation
of $\\{v_{m}\\}_{m=1}^{M}$. Figure 2 shows the normalized sequence
$\\{w_{m}\\}$.
Figure 2: The graph of original function $g$.
Choose five interpolation points $x_{0}=1$, $x_{1}=500$, $x_{2}=4000$,
$x_{3}=7500$, $x_{4}=10000$. Applying (7) we obtain $d_{1}=0.066$,
$d_{2}=0.155$, $d_{3}=0.033$, $d_{4}=0.096$. The small values of $|d_{i}|$
mean that on segments $[x_{i-1},x_{i}]$ fractal approximation function looks
as a straight line. Figure 3 shows the graphs of fractal and quadratic
approximating functions.
Figure 3: Fractal and quadratic interpolations of the polynomial function.
###### Example 3
A DNA sequence can be identified with a word over an alphabet
$\mathcal{N}=\\{A,C,G,T\\}$. Here we have the sequence of 10000 nucleotides of
Edwardsiella tarda. The graph represented by the formula
$v_{1}=0,\,v_{m}=v_{m-1}+\left\\{\begin{array}[]{ll}+1,&\mbox{if}\,\,m^{th}\mbox{nucleotide
belongs to (A,G)};\\\ -1,&\mbox{if}\,\,m^{th}\mbox{nucleotide belongs to
(C,T)}.\end{array}\right.$
For full description of representation of DNA primary sequences see [4].
Figure 4 shows the sequence $\\{w_{m}\\}$ after normalization of $\\{v_{m}\\}$
according to the formula in the previous example.
Figure 4: Picture shows DNA Graph of 10000 nucleotides of Edwardsiella tarda.
Interpolation points are $x_{0}=1$, $x_{1}=1000$, $x_{2}=2500$, $x_{3}=3000$,
$x_{4}=3500$,$x_{5}=5000$, $x_{6}=6500$, $x_{7}=7000$, $x_{8}=8000$,
$x_{9}=9000$, $x_{10}=10000$. Applying (7) we obtain $d_{1}=-0.001$,
$d_{2}=0.274$, $d_{3}=0.31$, $d_{4}=0.24$, $d_{5}=-0.057$, $d_{6}=0.211$,
$d_{7}=-0.42$, $d_{8}=-0.121$, $d_{9}=0.215$, $d_{10}=0.158$. Figure 5 shows
the graphs of fractal and quadratic approximating functions.
Figure 5: Fractal and quadratic interpolations of the DNA Graph.
###### Example 4
We take price wave of 10000 prices $v_{m},\,m=\overline{1,M}$ of one day
period for EUR/USD, then normalize it (Figure 6).
Figure 6: Picture shows Price Graph for EUR/USD.
Interpolation points are $x_{0}=0$, $x_{1}=500$, $x_{2}=1500$, $x_{3}=2000$,
$x_{4}=2500$,$x_{5}=3000$,$x_{6}=4000$, $x_{7}=5000$, $x_{8}=6000$,
$x_{9}=8000$, $x_{10}=10000$. Applying (7) we obtain $d_{1}=-0.334$,
$d_{2}=-0.004$, $d_{3}=0.315$, $d_{4}=0.307$, $d_{5}=0.333$, $d_{6}=-0.28$,
$d_{7}=-0.067$, $d_{8}=0.027$, $d_{9}=0.047$, $d_{10}=-0.33$. Figure 7 shows
the graphs of fractal and quadratic approximating functions.
Figure 7: Fractal and quadratic interpolations of the Price Graph.
###### Example 5
Picture shows Random Walking Graph. It represented by the formula
$v_{0}=0,\,v_{i}=v_{i-1}+\xi_{i}$, where $\xi_{i}$ is a random value with
normal distribution.
Figure 8: Normalized Random Walking graph.
Interpolation points are $x_{0}=0$, $x_{1}=1500$, $x_{2}=2000$, $x_{3}=3000$,
$x_{4}=4000$, $x_{5}=5500$, $x_{6}=6300$, $x_{7}=7600$, $x_{8}=8000$,
$x_{9}=9000$, $x_{10}=10000$. Applying (7) we obtain $d_{1}=-0.237$,
$d_{2}=0.14$, $d_{3}=-0.020$, $d_{4}=-0.105$, $d_{5}=0.105$, $d_{6}=0.0545$,
$d_{7}=-0.184$, $d_{8}=-0.368$, $d_{9}=0.081$, $d_{10}=-0.111$. Figure 9 shows
the graphs of fractal and quadratic approximating functions.
Figure 9: Fractal and quadratic interpolations of the Random Walking graph.
To compare the results we calculate approximation errors for each type of
data. Let $h(x)$ be the approximating function for data
$\\{w_{m}\\}_{m=1}^{M}$. Then approximation error is
$\sqrt{\sum\limits_{m=1}^{M}\frac{(h(x_{m})-w_{m}))^{2}}{M}}.$
Here we represent the table of approximation errors for each type
$\begin{array}[]{ccc}&Fractal&Quadratic\\\
PolynomialFunction&0.0359037&0.0245094\\\
DNA\,Primary\,Sequence&0.0692072&0.0624714\\\
Price\,Graph&0.0501345&0.0533686\\\
Random\,Walking&0.1015339&0.101438\end{array}$
From it we see, that fractal approximation is better for price graph and
nearly equal for random walking, but much worse for smooth function and
slightly for DNA sequence. Different results were appearing during
calculations of errors. We assume that some conditions could give us more
exact approximation results from fractal interpolation function and for that
extra observations should be established.
## References
* [1] C. Bandt, A. Kravchenko. Differentiability of fractal curves. Nonlinearity, 24(10):2717–2728, 2011.
* [2] M. F. Barnsley. Fractals Everywhere. Academic Press Inc., MA, 1988.
* [3] M. F. Barnsley and L. P. Hurd. Fractal Image Compression. Wellesley, MA:AK Peters, 1993.
* [4] Feng-lan Bai, Ying-zhao Liu, Tian-ming Wang. A representation of DNA primary sequences by random walk. Mathematical Biosciences 209 (2007) 282–291.
* [5] J. Hutchinson. Fractals and self similarity. Indiana Univ. Math. J., 30:713–747, 1981.
* [6] P. Massopust. Interpolation and approximation with splines and fractals. Oxford University Press, Oxford, 2010.
* [7] H. E. Stanley, S. V. Buldyrev, A. L. Goldbergerb, J. M. Hausdorff, S. Havlin, J. Mietusb, C. K. Peng, F. Sciortino and M. Simons. Fractal landscapes in biological systems: Long-range correlations in DNA and interbeat heart intervals. Physica A 191 (1992) 1–12 North-Holland
|
arxiv-papers
| 2012-10-03T11:38:57 |
2024-09-04T02:49:35.879170
|
{
"license": "Public Domain",
"authors": "K. Igudesman and G. Shabernev",
"submitter": "Konstantin Igudesman",
"url": "https://arxiv.org/abs/1210.1068"
}
|
1210.1089
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-286 LHCb-PAPER-2012-025 January 8, 2013
First evidence for the annihilation decay mode $B^{+}\rightarrow
D_{s}^{+}\phi$
The LHCb collaboration†††Authors are listed on the following pages.
Evidence for the hadronic annihilation decay mode $B^{+}\rightarrow
D_{s}^{+}\phi$ is found with greater than $3\sigma$ significance. The
branching fraction and $C\\!P$ asymmetry are measured to be
$\displaystyle\mathcal{B}(B^{+}\rightarrow D_{s}^{+}\phi)$ $\displaystyle=$
$\displaystyle\left(1.87^{\,+1.25}_{\,-0.73}\,({\rm stat})\pm 0.19\,({\rm
syst})\pm 0.32\,({\rm norm})\right)\times 10^{-6},$
$\displaystyle\mathcal{A}_{CP}(B^{+}\rightarrow D_{s}^{+}\phi)$
$\displaystyle=$ $\displaystyle-0.01\pm 0.41\,({\rm stat})\pm 0.03\,({\rm
syst}).$
The last uncertainty on $\mathcal{B}(B^{+}\rightarrow D_{s}^{+}\phi)$ is from
the branching fractions of the $B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ normalization mode and
intermediate resonance decays. Upper limits are also set for the branching
fractions of the related decay modes $B^{+}_{(c)}\rightarrow
D^{+}_{(s)}K^{*0}$, ${B^{+}_{(c)}\rightarrow D^{+}_{(s)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}}$ and ${B_{c}^{+}\rightarrow
D^{+}_{s}\phi}$, including the result ${\mathcal{B}(B^{+}\rightarrow
D^{+}K^{*0})}<1.8\times 10^{-6}$ at the 90% credibility level.
Submitted to the Journal of High Energy Physics
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
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Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
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Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The decays111Throughout this paper, charge conjugation is implied.
Furthermore, $K^{*0}$ and $\phi$ denote the $K^{*0}(892)$ and $\phi(1020)$
resonances, respectively. $B^{+}\rightarrow
D_{s}^{+}\phi,\;D^{+}K^{*0},\;D_{s}^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ occur in the Standard Model (SM)
via annihilation of the quarks forming the $B^{+}$ meson into a virtual
$W^{+}$ boson (Fig. 1). There is currently strong interest in annihilation-
type decays of $B^{+}$ mesons due, in part, to the roughly $2\sigma$ deviation
above the SM prediction observed in the branching fraction of
$B^{+}\rightarrow\tau^{+}\nu$ [1, 2]. Annihilation diagrams of $B^{+}$ mesons
are highly suppressed in the SM; no hadronic annihilation-type decays of the
$B^{+}$ meson have been observed to-date. Branching fraction predictions
(neglecting rescattering) for $B^{+}\\!\rightarrow D_{s}^{+}\phi$ and
$B^{+}\\!\rightarrow D^{+}K^{*0}$ are $(1-7)\times 10^{-7}$ in the SM [3, 4,
5, 6], where the precision of the calculations is limited by hadronic
uncertainties. The branching fraction for the $B^{+}\\!\rightarrow
D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay mode is
expected to be about 20 times smaller due to the CKM quark-mixing matrix
elements involved. The current upper limits on the branching fractions of
these decay modes are $\mathcal{B}(B^{+}\\!\rightarrow
D_{s}^{+}\phi)<1.9\times 10^{-6}$ [7], ${\cal B}(B^{+}\\!\rightarrow
D^{+}K^{*0})<3.0\times 10^{-6}$ [8] and ${\cal B}(B^{+}\\!\rightarrow
D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<4.0\times 10^{-4}$
[9], all at the 90% confidence level.
Contributions from physics beyond the SM (BSM) could greatly enhance these
branching fractions and/or produce a large $C\\!P$ asymmetry [4, 5]. For
example, a charged Higgs ($H^{+}$) boson mediates the annihilation process.
Interference between the $W^{+}$ and $H^{+}$ amplitudes could result in a
$C\\!P$ asymmetry if the two amplitudes are of comparable size and have both
strong and weak phase differences different from zero. An $H^{+}$ contribution
to the amplitude could also significantly increase the branching fraction.
In this paper, first evidence for the decay mode $B^{+}\\!\rightarrow
D_{s}^{+}\phi$ is presented using 1.0 fb-1 of data collected by LHCb in 2011
from $pp$ collisions at a center-of-mass energy of 7 TeV. The branching
fraction and $C\\!P$ asymmetry are measured. Limits are set on the branching
fraction of the decay modes $B^{+}\\!\rightarrow D^{+}K^{*0}$ and
$B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, along with the highly suppressed
decay modes $B^{+}\\!\rightarrow D^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{+}\\!\rightarrow
D^{+}_{s}K^{*0}$. Limits are also set on the product of the production rate
and branching fraction for $B_{c}^{+}$ decays to the final states
$D_{s}^{+}\phi$, $D_{(s)}^{+}K^{*0}$ and $D_{(s)}^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$.
Figure 1: Feynman diagrams for $B^{+}\\!\rightarrow D_{s}^{+}\phi$,
$B^{+}\\!\rightarrow D^{+}K^{*0}$ and $B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays.
## 2 The LHCb experiment
The LHCb detector [10] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Discrimination between different types of charged particles is provided by two
ring-imaging Cherenkov detectors [11]. Photon, electron and hadron candidates
are identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a muon system composed of alternating
layers of iron and multiwire proportional chambers.
The LHCb trigger [12] consists of a hardware stage, based on information from
the calorimeter and muon systems, followed by a software stage which applies a
partial event reconstruction (only tracks with $\mbox{$p_{\rm
T}$}>0.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are used). The software stage
of the LHCb trigger builds two-, three- and four-track partial $b$-hadron
candidates that are required to be significantly displaced from the primary
interaction and have a large sum of $p_{\rm T}$ in their tracks. At least one
of the tracks used to form the trigger candidate must have $\mbox{$p_{\rm
T}$}>1.7$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter
$\chi^{2}$ with respect to the primary interaction $\chi^{2}_{\rm IP}>16$. The
$\chi^{2}_{\rm IP}$ is defined as the difference between the $\chi^{2}$ of the
primary interaction vertex reconstructed with and without the considered
track. A boosted decision tree (BDT) [13, 14] is used to distinguish between
trigger candidates originating from $b$-hadron decays and those that originate
from prompt $c$-hadrons or combinatorial background. The BDT provides a pure
sample of $b\bar{b}$ events for offline analysis.
For the simulation, $pp$ collisions are generated using Pythia 6.4 [15] with a
specific LHCb configuration [16]. Decays of hadronic particles are described
by EvtGen [17] in which final state radiation is generated using Photos [18].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [19, *Agostinelli:2002hh] as
described in Ref. [21].
## 3 Event selection
Candidates of the decays searched for are formed from tracks that are required
to have $\mbox{$p_{\rm T}$}>0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\chi^{2}_{\rm IP}>4$ and $p>1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For the
$\phi$ and $K^{*0}$ decay products the momentum requirement is increased to
$p>2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. These momentum requirements are
100% efficient on simulated signal events. The $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$, ${D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}}$,
${\phi\rightarrow K^{+}K^{-}}$ and $K^{*0}\rightarrow K^{+}\pi^{-}$ candidates
are required to have invariant masses within 25, 25, 20 and
50${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their respective world-average
(PDG) values [22]. The mass resolutions for $D_{s}^{+}\rightarrow
K^{+}K^{-}\pi^{+}$ and ${D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}}$ are about
7${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
8${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The decay chain is
fit constraining the $D^{+}_{(s)}$ candidate mass to its PDG value. The
$D^{+}_{(s)}$ vertex is required to be downstream of the $B^{+}$ vertex and
the $p$-value formed from $\chi^{2}_{\rm IP}+\chi^{2}_{\rm vertex}$ of the
$B^{+}$ candidate is required to be greater than 0.1%. Backgrounds from
charmless decays are suppressed by requiring significant separation between
the $D^{+}_{(s)}$ and $B^{+}$ decay vertices. This requirement reduces
contributions from charmless backgrounds by a factor of about 15 while
retaining 87% of the signal.
Cross-feed between $D^{+}$ and $D_{s}^{+}$ candidates can occur if one of the
child tracks is misidentified. If a $D_{s}^{+}\\!\rightarrow
K^{+}K^{-}\pi^{+}$ candidate can also form a $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ candidate that falls within
25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the PDG $D^{+}$ mass, then it
is rejected unless either ${|m_{KK}-m_{\phi}^{\rm
PDG}|<10}$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or the ambiguous child
track satisfies a stringent kaon particle identification (PID) requirement.
This reduces the $D^{+}\rightarrow D_{s}^{+}$ cross-feed by a factor of about
200 at the expense of only 4% of the signal. For decay modes that contain a
$D^{+}$ meson, a $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ candidate that can
also form a $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ candidate whose mass is
within 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the PDG $D^{+}_{s}$
mass is rejected if either ${|m_{KK}-m_{\phi}^{\rm
PDG}|<10}$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or the ambiguous child
track fails a stringent pion PID requirement. For all modes,
$\Lambda_{c}^{+}\rightarrow D^{+}_{(s)}$ cross-feed (from the
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ decay mode) is suppressed using
similar requirements.
When a pseudoscalar particle decays into a pseudoscalar and a vector, $V$, the
spin of the vector particle (in this case a $\phi$ or $K^{*0}$) must be
orthogonal to its momentum to conserve angular momentum; i.e., the vector
particle must be longitudinally polarized. For a longitudinally-polarized
$\phi$ ($K^{*0}$) decaying into the $K^{+}K^{-}(K^{+}\pi^{-}$) final state,
the angular distribution of the $K^{+}$ meson in the $V$ rest frame is
proportional to $\cos^{2}{\theta_{K}}$, where $\theta_{K}$ is the angle
between the momenta of the $K^{+}$ and $B^{+}$ in the $V$ rest frame. The
requirement $|\cos{\theta_{K}}|>0.4$, which is 93% efficient on signal and
rejects about 40% of the background, is applied in this analysis.
Four BDTs that identify $D_{s}^{+}\\!\rightarrow K^{+}K^{-}\pi^{+}$,
${D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}}$, $\phi\rightarrow K^{+}K^{-}$ and
${K^{*0}\rightarrow K^{+}\pi^{-}}$ candidates originating from $b$-hadron
decays are used to suppress the backgrounds. The BDTs are trained using large
clean $D_{(s)}^{+}$, $\phi$ and $K^{*0}$ samples obtained from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D_{(s)}^{+}\pi^{-}$,
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ data,
respectively, where the backgrounds are subtracted using the sPlot technique
[23]. Background samples for the training are taken from the $D_{(s)}^{+}$,
$\phi$ and $K^{*0}$ sidebands in the same data samples. The BDTs take
advantage of the kinematic similarity of all $b$-hadron decays and avoid using
any topology-dependent information. The BDTs use kinematic, track quality,
vertex and PID information to obtain a high level of background suppression.
In total, 23 properties per child track and five properties from the parent
$D_{(s)}^{+}$, $\phi$ or $K^{*0}$ meson are used in each BDT. The boosting
method used is known as bagging [24], which produces BDT response values in
the unit interval.
A requirement is made on the product of the BDT responses of the $D_{(s)}^{+}$
and $\phi$ or $K^{*0}$ candidates. Tests on several $B^{0}_{(s)}\rightarrow
DD^{\prime}$ decay modes show that this provides the best performance [25].
The efficiencies of these cuts are obtained using large $\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{(s)}^{0}\rightarrow
D_{(s)}^{+}\pi^{-}$,
$B_{s}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ data
samples that are not used in the BDT training. The efficiency calculation
takes into account the kinematic differences between the signal and training
decay modes using additional input from simulated data. Correlations between
the properties of the $D_{(s)}^{+}$ and $\phi$ or $K^{*0}$ mesons in a given
$B^{+}$ candidate are also accounted for.
The optimal BDT requirements are chosen such that the signal significance is
maximized for the central value of the available SM branching fraction
predictions. The signal efficiency of the optimal BDT requirement is 51%, 69%
and 51% for $B^{+}\rightarrow D^{+}_{s}\phi$, $B^{+}\rightarrow D^{+}K^{*0}$
and $B^{+}\rightarrow D_{s}^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay modes, respectively. The
final sample contains no events with multiple candidates. Finally, no
consideration is given to contributions where the $K^{+}K^{-}(K^{+}\pi^{-})$
is in an $S$-wave state or from the tails of higher $\phi(K^{*0})$ resonances.
Such contributions are neglected as they are expected to be much smaller than
the statistical uncertainties.
## 4 Branching fraction for the $B^{+}\rightarrow D_{s}^{+}\phi$ decay
Table 1: Summary of fit regions for $B^{+}\rightarrow D^{+}_{s}\phi$. About 89% of the signal is expected to be in region A. | $|m_{K\kern-0.70004pt{K}}-m_{\phi}|$ (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
---|---
$|\cos\theta_{K}|$ | $<20$ | $(20,40)$
$>0.4$ | A | B
$<0.4$ | C | D
The $B^{+}\\!\rightarrow D_{s}^{+}\phi$ yield is determined by performing an
unbinned maximum likelihood fit to the invariant mass spectra of $B^{+}$
candidates. Candidates failing the $\cos{\theta_{K}}$ and/or $m_{KK}$
selection criteria that are within
40${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of $m_{\phi}^{\rm PDG}$ are used
in the fit to help constrain the background probability density function
(PDF). The data set is comprised of the four subsamples given in Table 1. They
are fit simultaneously to a PDF with the following components:
* •
$B^{+}\\!\rightarrow D_{s}^{+}\phi$: A Gaussian function whose parameters are
taken from simulated data and fixed in the fit is used for the signal shape.
The fraction of signal events in each of the subsamples is also fixed from
simulation to be as follows: (A) 89%; (B) 4%; (C) 7% and (D) no signal
expected. Thus, almost all signal events are expected to be found in region A,
while region D should contain only background. A 5% systematic uncertainty is
assigned to the branching fraction determination due to the shape of the
signal PDF. This value is obtained by considering the effect on the branching
fraction for many variations of the signal PDFs for $B^{+}\\!\rightarrow
D_{s}^{+}\phi$ and the normalization decay mode.
* •
$B^{+}\rightarrow D_{s}^{*+}\phi$: The $\phi$ in this decay mode does not need
to be longitudinally polarized. When the photon from the $D_{s}^{*+}$ decay is
not reconstructed, the polarization affects both the invariant mass
distribution and the fraction of events in each of the subsamples. Studies
using a wide range of polarization fractions, with shapes taken from
simulation, show that the uncertainties in this PDF have a negligible impact
on the signal yield.
* •
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{(*)+}K^{-}K^{*0}$: These decay modes, which arise as backgrounds to
$B^{+}\\!\rightarrow D_{s}^{+}\phi$ when the pion from the $K^{*0}$ decay is
not reconstructed, have not yet been observed; however, they are expected to
have similar branching fractions to the decay modes $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{(*)+}K^{-}K^{*0}$.
The ratio ${\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{*+}K^{-}K^{*0})}/{\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}K^{*0})}$ is fixed to be the same as the value of
$\mathcal{B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{*+}K^{-}K^{*0})/{\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}K^{*0})}$
[26]. The fraction of events in each subsample is constrained by simulation.
Removing these constraints results in a 1% change in the signal yield.
* •
Combinatorial background: An exponential shape is used for this component. The
exponent is fixed to be the same in all four subsamples. This component is
assumed to be uniformly distributed in $\cos{\theta_{K}}$. Removing these
constraints produces shifts in the signal yield of up to 5%; thus, a 5%
systematic uncertainty is assigned to the branching fraction measurement.
To summarize, the parameters allowed to vary in the fit are the signal yield,
the yield and longitudinal polarization fraction of $B^{+}\rightarrow
D_{s}^{*+}\phi$, the yield of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{(*)+}K^{-}K^{*0}$ in each subsample, the combinatorial background yield
in each subsample and the combinatorial exponent.
Figure 2 shows the $B^{+}$ candidate invariant mass spectra for each of the
four subsamples, along with the various components of the PDF. The signal
yield is found to be $6.7^{\,+4.5}_{\,-2.6}$, where the confidence interval
includes all values of the signal yield for which $\log{(\mathcal{L}_{\rm
max}/\mathcal{L})}<0.5$. The statistical significance of the signal is found
using Wilks Theorem [27] to be $3.6\sigma$. A simulation study consisting of
an ensemble of $10^{5}$ data sets confirms the significance and also the
accuracy of the coverage to within a few percent. All of the variations in the
PDFs discussed above result in significances above $3\sigma$; thus, evidence
for $B^{+}\\!\rightarrow D_{s}^{+}\phi$ is found at greater than $3\sigma$
significance including systematics.
Figure 2: Fit results for $B^{+}\\!\rightarrow D_{s}^{+}\phi$. The fit
regions, as given in Table 1, are labelled on the panels. The PDF components
are as given in the legend.
The $B^{+}\\!\rightarrow D_{s}^{+}\phi$ branching fraction is normalized to
$\mathcal{B}(B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$. The selection for the
normalization mode, which is similar to that used here for
$B^{+}\\!\rightarrow D_{s}^{+}\phi$, is described in detail in Ref. [25]. The
ratio of the efficiency of the product of the geometric, trigger,
reconstruction and selection (excluding the charmless background suppression
and BDT) requirements of the signal mode to the normalization mode is found
from simulation to be $0.93\pm 0.05$. The ratio of BDT efficiencies, which
include all usage of PID information, is determined from data (see Sect. 3) to
be $0.52\pm 0.02$. The large branching fraction of the normalization mode
permits using a BDT requirement that is nearly 100% efficient. For the
charmless background suppression requirement, the efficiency ratio is
determined from simulation to be $1.15\pm 0.01$. The difference is mostly due
to the fact that the normalization mode has two charmed mesons, while the
signal mode only has one. The branching fraction is measured as
$\displaystyle\mathcal{B}(B^{+}\\!\rightarrow D_{s}^{+}\phi)$ $\displaystyle=$
$\displaystyle\frac{\epsilon(B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})}{\epsilon(B^{+}\\!\rightarrow
D_{s}^{+}\phi)}\frac{\mathcal{B}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{-}\pi^{+})}{\mathcal{B}(\phi\rightarrow
K^{+}K^{-})}\frac{N(B^{+}\\!\rightarrow D_{s}^{+}\phi)}{N(B^{+}\\!\rightarrow
D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})}\,\mathcal{B}(B^{+}\\!\rightarrow
D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ $\displaystyle=$
$\displaystyle\left(1.87^{\,+1.25}_{\,-0.73}\,({\rm stat})\pm 0.19\,({\rm
syst})\pm 0.32\,({\rm norm})\right)\times 10^{-6},$
where $\epsilon$ denotes efficiency. The normalization uncertainty includes
contributions from $\mathcal{B}(B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})=(1.0\pm 0.17)\%$,
$\mathcal{B}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{-}\pi^{+})=(3.88\pm 0.05)\%$ and ${\mathcal{B}(\phi\rightarrow
K^{+}K^{-})}={(48.9\pm 0.5)\%}$ [22]. The systematic uncertainties are
summarized in Table 2. The value obtained for $\mathcal{B}(B^{+}\\!\rightarrow
D_{s}^{+}\phi)$ is consistent with the SM calculations given the large
uncertainties on both the theoretical and experimental values.
Table 2: Systematic uncertainties contributing to $\mathcal{B}(B^{+}\\!\rightarrow D_{s}^{+}\phi)/\mathcal{B}(B^{+}\\!\rightarrow D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0})$. Source | Uncertainty (%)
---|---
Selection | $\phantom{1}7$
Signal PDF | $\phantom{1}5$
Background PDF | $\phantom{1}5$
Normalization | 17
## 5 Branching fractions for the decays ${B^{+}\rightarrow
D_{(s)}^{+}K^{*0}}$ and ${B^{+}\rightarrow D_{(s)}^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}}$
The SM predicts the branching fraction ratios $\mathcal{B}(B^{+}\rightarrow
D^{+}K^{*0})/\mathcal{B}(B^{+}\rightarrow D_{s}^{+}\phi)\sim 1$ and
${\mathcal{B}(B^{+}\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}/{\mathcal{B}(B^{+}\rightarrow
D_{s}^{+}\phi)}\sim|V_{cd}/V_{cs}|^{2}$ [3]. The partially reconstructed
backgrounds are expected to be much larger in these channels compared to
$B^{+}\\!\rightarrow D_{s}^{+}\phi$ mainly due to the large $K^{*0}$ mass
window. Producing an exhaustive list of decay modes that contribute to each of
these backgrounds is not feasible; thus, reliable PDFs for the backgrounds are
not available. Instead, data in the sidebands around the signal region are
used to estimate the expected background yield in the signal region. The
signal region is chosen to be $\pm 2\sigma$ around the $B^{+}$ mass, where
$\sigma=13.8$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is determined from
simulation.
Our prior knowledge about the background can be stated as the following three
assumptions: (1) the slope is negative, which will be true provided $b$-baryon
background contributions are not too large; (2) it does not peak or form a
shoulder222No evidence of peaking backgrounds is found in either the
$D_{(s)}^{+}$ or $K^{*0}$ sidebands. If peaking backgrounds do make
significant contributions, then the limits set in this paper are conservative.
and (3) the background yield is non-negative. These background properties are
assumed to hold throughout the signal and sideband regions. To convert these
assumptions into background expectations, ensembles of background-only data
sets are generated using the observed data in the sidebands and assuming
Poisson distributed yields. For each simulated data set, all interpolations
into the signal region that satisfy our prior assumptions are assigned equal
probability. These probabilities are summed over all data sets to produce
background yield PDFs, all of which are well described by Gaussian lineshapes
(truncated at zero) with the parameters $\mu_{\rm bkgd}$ and $\sigma_{\rm
bkgd}$ given in Table 3. The $B^{+}$ candidate invariant mass distributions,
along with the background expectations, are shown in Fig. 3. The results of
spline interpolation using data in the sideband bins, along with the 68%
confidence intervals obtained by propagating the Poisson uncertainties in the
sidebands to the splines, are shown for comparison. As expected, the spline
interpolation results, which involve a stronger set of assumptions, have less
statistical uncertainty.
Figure 3: Invariant mass distributions for (a) $B^{+}\\!\rightarrow
D^{+}K^{*0}$, (b) $B^{+}\\!\rightarrow D^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, (c) ${B^{+}\rightarrow
D^{+}_{s}K^{*0}}$ and (d) $B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. The bins are each $4\sigma$
wide, where ${\sigma=13.8}$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is the
expected width of the signal peaks (the middle bin is centred at the expected
$B^{+}$ mass). The shaded regions are the $\mu_{\rm bkgd}\pm\sigma_{\rm bkgd}$
intervals (see Table 3) used for the limit calculations; they are taken from
the truncated-Gaussian priors as discussed in the text. Spline interpolation
results (solid blue line and hashed blue areas) are shown for comparison.
A Bayesian approach [28] is used to set the upper limits. Poisson
distributions are assumed for the observed candidate counts and uniform, non-
negative prior PDFs for the signal branching fractions. The systematic
uncertainties in the efficiency and $B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ normalization are encoded in log-
normal priors, while the background prior PDFs are the truncated Gaussian
lineshapes discussed above. The posterior PDF, $p(\mathcal{B}|n_{\rm obs})$,
where $n_{\rm obs}$ is the number of candidates observed in the signal region,
is computed by integrating over the background, efficiency and normalization.
The 90% credibility level (CL) upper limit, $\mathcal{B}_{90}$, is the value
of the branching fraction for which
$\int_{0}^{\mathcal{B}_{90}}p(\mathcal{B}|n_{\rm obs}){\rm
d}\mathcal{B}=0.9\int_{0}^{\infty}p(\mathcal{B}|n_{\rm obs}){\rm
d}\mathcal{B}$. The upper limits are given in Table 3. The limit on
$B^{+}\\!\rightarrow D^{+}K^{*0}$ is 1.7 times lower than any previous limit,
while the $B^{+}\\!\rightarrow D^{+}_{s}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ limit is 91 times lower. For the
highly suppressed decay modes $B^{+}\\!\rightarrow D^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{+}\\!\rightarrow
D^{+}_{s}K^{*0}$ these are the first limits to be set.
Table 3: Upper limits on ${\cal B}(B^{\pm}\rightarrow D_{(s)}^{\pm}K^{*0})$, where $n_{\rm obs}$ is the number of events observed in each of the signal regions, while $\mu_{\rm bkgd}$ and $\sigma_{\rm bkgd}$ are the Gaussian parameters used in the background prior PDFs. Decay | $n_{\rm obs}$ | $\mu_{\mathrm{bkgd}}$ | $\sigma_{\mathrm{bkgd}}$ | Upper Limit at 90% CL
---|---|---|---|---
$B^{+}\\!\rightarrow D^{+}K^{*0}$ | 08 | 02.2 | 3.4 | $1.8\times 10^{-6}$
$B^{+}\\!\rightarrow D^{+}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | 08 | 07.1 | 3.6 | $1.4\times 10^{-6}$
$B^{+}\\!\rightarrow D^{+}_{s}K^{*0}$ | 19 | 20.0 | 4.2 | $3.5\times 10^{-6}$
$B^{+}\\!\rightarrow D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | 16 | 14.8 | 5.6 | $4.4\times 10^{-6}$
The posterior PDF for the $B^{+}\\!\rightarrow D^{+}K^{*0}$ decay excludes the
no-signal hypothesis at the 89% CL and gives a branching fraction measurement
of $\mathcal{B}(B^{+}\\!\rightarrow D^{+}K^{*0})=(0.8^{\,+0.6}_{\,-0.5})\times
10^{-6}$, where the uncertainty includes statistics and systematics. This
result is consistent with both the SM expectation and, within the large
uncertainties, with the value obtained above for
$\mathcal{B}(B^{+}\\!\rightarrow D_{s}^{+}\phi)$. If processes beyond the SM
are producing an enhancement in $\mathcal{B}(B^{+}\\!\rightarrow
D_{s}^{+}\phi)$, then a similar effect would also be expected in
$B^{+}\\!\rightarrow D^{+}K^{*0}$. While an enhancement cannot be ruled out by
the data, the combined $\mathcal{B}(B^{+}\\!\rightarrow D_{s}^{+}\phi)$ and
$\mathcal{B}(B^{+}\\!\rightarrow D^{+}K^{*0})$ result is consistent with the
SM interpretation.
## 6 Limits on branching fractions of $B_{c}^{+}$ decay modes
Annihilation amplitudes are expected to be much larger for $B_{c}^{+}$ decays
due to the large ratio of $|V_{cb}/V_{ub}|$. In addition, the
$B_{c}^{+}\rightarrow D_{s}^{+}\phi,\;D^{+}K^{*0},\;D_{s}^{+}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay modes can also proceed via
penguin-type diagrams. However, due to the fact that $B_{c}^{+}$ mesons are
produced much more rarely than $B^{+}$ mesons in
7$\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions (the ratio of $B_{c}^{+}$ to
$B^{+}$ mesons produced is denoted by $f_{c}/f_{u}$), no signal events are
expected to be observed in any of these $B_{c}^{+}$ channels. The Bayesian
approach is again used to set the limits. A different choice is made here for
the background prior PDFs because the background levels are so low. The
background prior PDFs are now taken to be Poisson distributions, where the
observed background counts are obtained using regions of equal size to the
signal regions in the high-mass sidebands. Only the high-mass sidebands are
used to avoid possible contamination from partially reconstructed $B_{c}^{+}$
backgrounds. In none of the decay modes is more than a single candidate seen
across the combined signal and background regions. The limits obtained, which
are set on the product of $f_{c}/f_{u}$ and the branching fractions (see Table
4), are four orders of magnitude better than any previous limit set for a
$B_{c}^{+}$ decay mode that does not contain charmonium. As expected given the
small numbers of candidates observed, the limits have some dependence on the
choice made for the signal prior PDF. As a cross check, the limits were also
computed using various frequentist methods. The largest difference found is
20%.
Table 4: Upper limits on $f_{c}/f_{u}\cdot\mathcal{B}(B_{c}\rightarrow X)$, where $n_{\rm obs}$ and $n_{\rm bkgd}$ are the number of events observed in the signal and background (sideband) regions, respectively. Decay | $n_{\rm obs}$ | $n_{\rm bkgd}$ | Upper Limit at 90% CL
---|---|---|---
$B_{c}^{+}\rightarrow D_{s}^{+}\phi$ | 0 | 0 | $0.8\times 10^{-6}$
$B_{c}^{+}\rightarrow D^{+}K^{*0}$ | 1 | 0 | $0.5\times 10^{-6}$
$B_{c}^{+}\rightarrow D^{+}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | 0 | 0 | $0.4\times 10^{-6}$
$B_{c}^{+}\rightarrow D^{+}_{s}K^{*0}$ | 0 | 0 | $0.7\times 10^{-6}$
$B_{c}^{+}\rightarrow D^{+}_{s}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | 1 | 0 | $1.1\times 10^{-6}$
## 7 $C\\!P$ asymmetry for the decay $B^{+}\\!\rightarrow D_{s}^{+}\phi$
To measure the $C\\!P$ asymmetry, $\mathcal{A}_{CP}$, in $B^{+}\\!\rightarrow
D_{s}^{+}\phi$, only candidates in region (a) and in a $\pm 2\sigma$ window
($\pm 26.4$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) around the $B^{+}$ mass
are considered. The number of $B^{+}$ candidates is $n_{+}=3$, while the
number of $B^{-}$ candidates is $n_{-}=3$. The integral of the background PDF
from the fit described in detail in Sect. 4 in the signal region is $n_{\rm
bkgd}=0.75$ (the background is assumed to be charge symmetric). The observed
charge asymmetry is $\mathcal{A}_{\rm obs}=(n_{-}-n_{+})/(n_{-}+n_{+}-n_{\rm
bkgd})=0.00\pm 0.41$, where the 68% confidence interval is obtained using the
Feldman-Cousins method [29].
To obtain $\mathcal{A}_{CP}$, the production, $\mathcal{A}_{\rm prod}$,
reconstruction, $\mathcal{A}_{\rm reco}$, and selection, $\mathcal{A}_{\rm
sel}$, asymmetries must also be accounted for. The $D^{+}_{s}\phi$ final state
is charge symmetric except for the pion from the $D_{s}^{+}$ decay. The
observed charge asymmetry in the decay modes
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$,
along with the interaction asymmetry of charged kaons [30] and the pion-
detection asymmetry [31] in LHCb are used to obtain the estimate
$\mathcal{A}_{\rm prod}+\mathcal{A}_{\rm reco}=(-1\pm 1)\%$. The large $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
sample used to determine the BDT efficiency is employed to estimate the
selection charge asymmetry yielding $\mathcal{A}_{\rm sel}=(2\pm 3)\%$, where
the precision is limited by the sample size. Finally, the $C\\!P$ asymmetry is
found to be
$\mathcal{A}_{CP}(B^{+}\\!\rightarrow D_{s}^{+}\phi)=\mathcal{A}_{\rm
obs}-\mathcal{A}_{\rm prod}-\mathcal{A}_{\rm reco}-\mathcal{A}_{\rm
sel}=-0.01\pm 0.41\,({\rm stat})\pm 0.03\,({\rm syst}),$
which is consistent with the SM expectation of no observable $C\\!P$
violation.
## 8 Summary
The decay mode $B^{+}\\!\rightarrow D_{s}^{+}\phi$ is seen with greater than
$3\sigma$ significance. This is the first evidence found for a hadronic
annihilation-type decay of a $B^{+}$ meson. The branching fraction and $C\\!P$
asymmetry for $B^{+}\\!\rightarrow D_{s}^{+}\phi$ are consistent with the SM
predictions. Limits have also been set for the branching fractions of the
decay modes $B^{+}_{(c)}\rightarrow D^{+}_{(s)}K^{*0}$,
$B^{+}_{(c)}\rightarrow D^{+}_{(s)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and ${B_{c}^{+}\rightarrow
D^{+}_{s}\phi}$. These limits are the best set to-date.
## 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 (The 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.
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|
arxiv-papers
| 2012-10-03T12:42:32 |
2024-09-04T02:49:35.887808
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, 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, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, 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, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov,\n R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S.\n Malde, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, 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. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, H. Voss, C. Vo{\\ss}, R. Waldi, R. Wallace,\n S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale,\n M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams,\n M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Sam Hall",
"url": "https://arxiv.org/abs/1210.1089"
}
|
1210.1099
|
# Simply-connected minimal surfaces with finite total curvature in
$\mathbb{H}^{2}\times\mathbb{R}$
Juncheol Pyo and M. Magdalena Rodríguez Research partially supported by the
CEI BioTIC GENIL project (CEB09-0010) and the Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry
of Education, Science and Technology (2012-0007728).Research partially
supported by the MEC-FEDER Grant no. MTM2011-22547 and the Regional J.
Andalucía Grant no. P09-FQM-5088.
###### Abstract
Laurent Hauswirth and Harold Rosenberg developed in [4] the theory of minimal
surfaces with finite total curvature in $\mathbb{H}^{2}\times\mathbb{R}$. They
showed that the total curvature of one such a surface must be a non-negative
integer multiple of $-2\pi$. The first examples appearing in this context are
vertical geodesic planes and Scherk minimal graphs over ideal polygonal
domains. Other non simply-connected examples have been constructed recently in
[6, 11, 14].
In the present paper, we show that the only complete minimal surfaces in
$\mathbb{H}^{2}\times\mathbb{R}$ of total curvature $-2\pi$ are Scherk minimal
graphs over ideal quadrilaterals. We also construct properly embedded simply-
connected minimal surfaces with total curvature $-4k\pi$, for any integer
$k\geq 1$, which are not Scherk minimal graphs over ideal polygonal domains.
Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42
## 1 Introduction
In the classical theory of minimal surfaces in $\mathbb{R}^{3}$, the ones
better known are those with finite total curvature. We recall that the total
curvature of a surface $M$ is defined as $C(M)=\int_{M}K$, where $K$ denotes
the Gauss curvature of $M$. If a minimal surface $M$ of $\mathbb{R}^{3}$ has
finite total curvature (i.e. $|C(M)|<+\infty$) then either $M$ is a plane or
it must be $C(M)=-4\pi k$, for some integer $k\geq 1$, and the equality only
holds for $M$ being the catenoid or Enneper’s surface (see [13, Theorems 9.2
and 9.4]).
In the last decade, the geometry of minimal surfaces in
$\mathbb{H}^{2}\times\mathbb{R}$ has been actively studied, and many examples
have been constructed (see for instance [1, 3, 9, 10, 12, 15]). Hauswirth and
Rosenberg started in [4] the study of complete minimal surfaces of finite
total curvature in $\mathbb{H}^{2}\times\mathbb{R}$. The only known examples
at that moment were the Scherk minimal graphs over ideal polygonal domains
with an even number of edges, with boundary values $\pm\infty$ disposed
alternately. Morabito and the authors constructed in [11, 14] non simply-
connected properly embedded minimal surfaces with finite total curvature and
genus zero. Quite recently, in a joint work with Martín and Mazzeo, the second
author [6] has constructed properly embedded minimal surfaces with finite
total curvature and positive genus.
The classification of minimal surfaces of finite total curvature in
$\mathbb{H}^{2}\times\mathbb{R}$ arises very naturally. The first result of
classification appearing in this theory was that the only complete minimal
surfaces with vanishing total curvature are the vertical geodesic planes (see
[5, Corollary 5]). Quite recently, Hauswirth, Nelli, Sa Earp and Toubiana have
proved in [7] that a complete minimal surface in
$\mathbb{H}^{2}\times\mathbb{R}$ with finite total curvature and two ends,
each one asymptotic to a vertical geodesic plane, must be one of the
horizontal catenoids constructed in [11, 14]. In this paper, we show that the
Scherk minimal graphs over ideal quadrilaterals (i.e. ideal polygonal domains
bounded by four ideal geodesics) are the only complete minimal surfaces of
total curvature $-2\pi$.
It was expected that each end of a minimal surface with finite curvature in
$\mathbb{H}^{2}\times\mathbb{R}$ were asymptotic to either a vertical geodesic
plane or a Scherk graph over an ideal polygonal domain. We construct new
simply-connected examples, that we call twisted Scherk examples, that
highlight this is not the case. They all have total curvature an integer
multiple of $-4\pi$, so we cannot expect a classification result for Scherk
graphs over ideal polygonal domains bounded by $4k+2$ edges as the only
simply-connected complete minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$
with total curvature $-4k\pi$.
## 2 Preliminaires
We consider the Poincaré disk model of $\mathbb{H}^{2}$; i.e.
$\mathbb{H}^{2}=\\{z\in\mathbb{C}\ |\ |z|<1\\}$, with the hyperbolic metric
$g_{-1}=\frac{4}{(1-|z|^{2})^{2}}|dz|^{2}$. We denote by
$\partial_{\infty}\mathbb{H}^{2}$ the infinite boundary of $\mathbb{H}^{2}$
(i.e. $\partial_{\infty}\mathbb{H}^{2}=\\{z\in\mathbb{C}\ |\ |z|=1\\}$) and by
$\mathbf{0}$ the origin of $\mathbb{H}^{2}$. Also $t$ will denote the
coordinate in $\mathbb{R}$.
Let $M$ be a complete orientable minimal surface immersed in
$\mathbb{H}^{2}\times\mathbb{R}$. We define the total curvature of $M$ as
$C(M)=\int_{M}K$, where $K\leq 0$ denotes the Gaussian curvature of $M$. We
say that $M$ has finite total curvature when $|C(M)|<+\infty$.
In this section we summarize the geometric properties of minimal surfaces in
$\mathbb{H}^{2}\times\mathbb{R}$ with finite total curvature given by
Hauswirth and Rosenberg in [4].
We call the height function of $M$ the horizontal projection
$h:M\to\mathbb{R}$, and we denote by $F$ the vertical projection of $M$ over
$\mathbb{H}^{2}$. It is well-known that $h$ is a real harmonic function on $M$
and that $F$ is an harmonic map from $M$ to $\mathbb{H}^{2}$. Given a
conformal parameter $w$ on $M$, Sa Earp and Toubiana [15] proved that
$(h_{w})^{2}=-Q$, where $Q$ is the Hopf differential associated to $F$. Then
the zeroes of $Q$ are of even order and, up to a sign (which corresponds to a
reflection symmetry with respect to $\mathbb{H}^{2}\times\mathbb{R}$),
$h=\Re\left(-2i\int\sqrt{Q}\right),$ (1)
see equation (3) in [4].
We fix a unit normal vector field $N$ on $M$. We now state the main theorem in
[4].
###### Theorem 1.
[4] Let $M$ be a complete, orientable, minimal surface immersed in
$\mathbb{H}^{2}\times\mathbb{R}$ with finite total curvature. Then:
1. 1.
$M$ is conformally a closed Riemann surface $\mathbb{M}$ punctured in a finite
number of points $p_{1},\cdots,p_{n}$, called ends of $\Sigma$.
2. 2.
$Q$ is holomorphic on $M$ and extends meromorphically to its ends $p_{i}$. If
we parameterize conformally a neighborhood of $p_{i}$ in $M$ by
$\Omega=\mathbb{C}\setminus D_{0}$, where $D_{0}$ is the open unit disk in
$\mathbb{C}$ centered at the origin, then
$Q(z)=z^{2m_{i}}(dz)^{2},$
for some integer $m_{i}\geq-1$.
3. 3.
$N_{3}=\langle N,\partial_{t}\rangle$ converges uniformly to zero on each end
$p_{i}$.
4. 4.
The total curvature of $M$ is given by
$\int_{M}K=2\pi\left(2-2g-2n-\sum_{i=1}^{n}m_{i}\right).$ (2)
###### Remark 2.
Suppose $p_{i}$ is an end of $M$ for which $m_{i}=-1$. If we want to close
periods in equation (1), then we have to choose $Q(z)=-z^{-2}(dz)^{2}$,
$z\in\Omega$.
###### Assertion 3.
In the second item of Theorem 1, $m_{i}$ cannot equal $-1$.
###### Proof.
Suppose $M$ (in the setting of Theorem 1) has an end $p_{1}$ for which
$m_{1}=-1$. We know that a neighborhood $\mathcal{E}$ of $p_{1}$ can be
conformally parameterized on $\Omega=\\{z\in\mathbb{C}\ |\ |z|\geq 1\\}$,
where $Q(z)=-z^{-2}dz^{2}$ (see Remark 2). From (1) we then get
$h(z)=2\Re\left(\int_{M}\frac{dz}{z}\right)=2\ln|z|$. Therefore, $\mathcal{E}$
is a vertical annulus whose intersection with each horizontal slice
$\mathbb{H}^{2}\times\\{t\\}$, $t\geq 0$, is a compact curve.
The boundary of $\mathcal{E}$ (which corresponds to $\\{|z|=1\\}$) consists of
a horizontal compact curve $\Gamma$ at height zero. Consider $R>0$ big enough
so that the disc $D\subset\mathbb{H}^{2}$ of radius $R$ centered at the origin
contains $\Gamma$ in its interior. And let $\mathcal{C}$ be the complete
vertical rotational catenoid constructed by Nelli and Rosenberg in [12] whose
neck is $\partial D$. Since $\mathcal{E}$ intersects each horizontal slice in
a compact curve, we deduce using the Maximum Principle with vertically
translated copies of $\mathcal{C}$ that $\mathcal{E}$ must be contained in
$D\times\mathbb{R}$. But this is not possible: If we translate $\mathcal{C}$
vertically up a distance $\pi$, we reach a contradiction by applying the
Maximum Principle with the family of shrunk catenoids going from $\mathcal{C}$
to the 2-sheeted covering of the punctured slice
$(\mathbb{H}^{2}-\\{\mathbf{0}\\})\times\\{\pi\\}$. ∎
We finish this section by describing the asymptotic behavior of a complete,
orientable, minimal surface immersed in $\mathbb{H}^{2}\times\mathbb{R}$ with
finite total curvature.
###### Lemma 4.
[4] Let $M$ be a minimal surface in the hypothesis of Theorem 1, and $p_{i}$
an end of $M$. If $m_{i}\geq 0$ is the integer associated to $p_{i}$, as
defined in Theorem 1, then $p_{i}$ corresponds to $m_{i}+1$ geodesics
$\gamma_{1},\ldots,\gamma_{m_{i}+1}\subset\mathbb{H}^{2}\times\\{+\infty\\}$,
$m_{i}+1$ geodesics
$\Gamma_{1},\ldots,\Gamma_{m_{i}+1}\subset\mathbb{H}^{2}\times\\{-\infty\\}$,
and $2(m_{i}+1)$ vertical straight lines (possibly some of them coincide) in
$\partial_{\infty}\mathbb{H}^{2}\times\mathbb{R}$, each one joining an
endpoint of some $\gamma_{j}$ to an endpoint of some $\Gamma_{j}$.
## 3 Minimal examples with finite total curvature
Given any two points
$p,q\in\mathbb{H}^{2}\cup\partial_{\infty}\mathbb{H}^{2}$, we will denote by
$\overline{pq}$ the geodesic arc joining $p,q$.
We consider an even number of different points
$p_{1},\cdots,p_{2k}\in\partial_{\infty}\mathbb{H}^{2}$ (cyclically ordered),
with $k\geq 2$, and we call $A_{i}=\overline{p_{2i-1}p_{2i}}$,
$B_{i}=\overline{p_{2i}p_{2i+1}}$, for any $1\leq i\leq k$, where we consider
the cyclic notation $p_{2k+1}=p_{1}$. Let $\Omega$ be the ideal polygonal
domain bounded by $A_{1},B_{1},\cdots,A_{k},B_{k}$. We call Scherk minimal
graph over $\Omega$ to a minimal graph over $\Omega$ with boundary values
$+\infty$ over the $A_{i}$ edges and $-\infty$ over the $B_{i}$ edges (in [1,
12] it is proved that it exists and it is unique up to a vertical
translation). In [1, 4] it is proved that such a graph has total curvature
$2\pi(1-k)$. Scherk graphs over ideal polygonal domains, together with the
vertical geodesic planes, where the first known examples of minimal surfaces
with finite total curvature.
In [11, 14] other non-simply-connected examples where presented, called
minimal $k$-noids. We briefly explain their construction: Consider an even
number of points $p_{1},\cdots,p_{2k}$ (cyclically ordered) such that
$p_{2i-1}\in\mathbb{H}^{2}$ and $p_{2i}\in\partial_{\infty}\mathbb{H}^{2}$. We
call $A_{i}=\overline{p_{2i-1}p_{2i}}$ and $B_{i}=\overline{p_{2i}p_{2i+1}}$.
Consider the minimal graph $\Sigma$ over the polygonal domain bounded by
$A_{1},B_{1},\cdots,A_{k},B_{k}$ with boundary values $+\infty$ over the
$A_{i}$ edges and $-\infty$ over the $B_{i}$ edges (it exists and is unique up
to a vertical translation, by [1, 9]), which has total curvature $2\pi(1-k)$
(see [1]). The conjugate minimal surface $\Sigma^{*}$ of $\Sigma$ is a minimal
graph contained in $\mathbb{H}^{2}\times\\{t\geq 0\\}$, whose boundary
consists of $k$ geodesic curvature lines in $\mathbb{H}^{2}\times\\{0\\}$.
(The conjugation for minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ was
introduced by Daniel [2] and by Hauswirth, Sa Earp and Toubiana [5].) If we
reflect $\Sigma^{*}$ with respect to $\mathbb{H}^{2}\times\\{0\\}$, we get a
properly embedded minimal surface of genus zero, $k$ ends asymptotic to
vertical geodesic planes and total curvature $4\pi(1-k)$. For $k=2$, the
obtained examples are usually called horizontal catenoids, and have been
recently classified by Hauswirth, Nelli, Sa Earp and Toubiana as the only
complete minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ with finite
total curvature and two ends, each one asymptotic to a vertical geodesic
plane.
Using a gluing method, the second author has recently constructed in a joint
work with Martín and Mazzeo a wide range of properly embedded minimal surfaces
with finite total curvature and finite topology (with possibly positive
genus).
We wondered if Scherk minimal graphs were, together with the vertical geodesic
planes, the only complete, embedded, simply-connected examples of finite total
curvature. In this section we explain the simple construction of other
different complete, embedded, simply-connected examples, that we will call
twisted Scherk examples.
### 3.1 Twisted Scherk examples
Let us first construct an example with total curvature $-4\pi$. Let
$p_{1},p_{2}$ be two points in $\partial_{\infty}\mathbb{H}^{2}$. Up to an
isometry of $\mathbb{H}^{2}$, we can assume $p_{1}=1$ and $p_{2}=e^{i\theta}$,
for some fixed $\theta\in(0,\pi/2]$ (see Figure 1). We call
$A_{1}=\overline{\mathbf{0}p_{1}}$, $B_{1}=\overline{p_{1}p_{2}}$ and
$C_{1}=\overline{\mathbf{0}p_{2}}$. Let $\Delta$ be the geodesic triangle
bounded by $A_{1}\cup B_{1}\cup C_{1}$. By the triangle inequality at infinity
(see [1, Lemma 3]), we get that $\Delta$ satisfies the Jenkins-Serrin
condition for the existence of a minimal graph $u$ over $\Delta$ with boundary
values $+\infty$ on $A_{1}$, $-\infty$ on $B_{1}$ and $0$ on $C_{1}$ (see [1,
Theorem 3] and [9, Theorem 3.3]).
Now let us see that the graph surface $\Sigma(u)$ of $u$ has finite total
curvature: For any positive integer $n$, we denote $r=1-1/(n+1)$ and
$p_{1,n}=r$, $p_{2,n}=re^{i\theta}$. By Theorem 3 in [12], there exists a
minimal graph $u_{r}(n)$ over the geodesic triangle of vertices
$\mathbf{0},p_{1,n},p_{2,n}$ taking boundary values $+n$ on
$\overline{\mathbf{0}p_{1,n}}$, $-n$ on $\overline{p_{1,n}p_{2,n}}$ and $0$ on
$\overline{\mathbf{0}p_{2,n}}$. By the Gauss-Bonnet formula, the graph surface
of $u_{r}(n)$ has total curvature $\pi$. Since $u_{r}(n)$ converges uniformly
on compact sets of $\Delta$ to $u$ as $n\rightarrow\infty$, the total
curvature of $\Sigma(u)$ is at most $\pi$, and then finite.
By rotating $\Sigma(u)$ an angle $\pi$ about the horizontal geodesic ray
$\overline{\mathbf{0}p_{2}}$ contained in its boundary, we obtain a minimal
graph whose boundary consists of the vertical geodesic
$\\{\mathbf{0}\\}\times\mathbb{R}$. We extend such a graph by rotation of
angle $\pi$ about its boundary, and we get a properly embedded simply-
connected minimal surface $\Sigma_{1}$.
Figure 1: Left: The minimal graph over the triangle region with these
prescrived values is the fundamental piece of a twisted Scherk example
$\Sigma_{1}$ with total curvature $-4\pi$. Right: Vertical projection of
$\Sigma_{1}$.
Since $\Sigma_{1}$ consists of four copies of $\Sigma(u)$, then it has finite
total curvature. Then equation (2) applies. In our case, $g=0$, $n=1$ and
$m_{1}=2$ ($m_{1}=2$ follows from the fact that the intersection of $M$ with a
horizontal slice $\mathbb{H}^{2}\times\\{t\\}$, for $t>0$ large enough,
consists of three divergent curves, see Figure 1). Thus
$\int_{\Sigma_{1}}K=-4\pi$.
Now, let us consider $k\geq 2$. Let $\Omega$ be a polygonal domain whose
vertices are $\mathbf{0}$ and $2k-1$ different ideal points
$p_{1},\cdots,p_{2k-1}\in\partial_{\infty}\mathbb{H}^{2}$. Assume that
$\Omega$ satisfies the Jenkins-Serrin condition of Theorem 3 in [1] or Theorem
3.3 in [9]. The example below proves that there exist such domains. We call
$\Sigma$ the minimal graph over $\Omega$ with boundary values $+\infty$ on
$\overline{\mathbf{0}p_{1}}$ and on $\overline{p_{2i}p_{2i+1}}$, for $1\leq
i\leq k-1$; and $-\infty$ on $\overline{p_{2i-1}p_{2i}}$, for $1\leq i\leq
k-1$, and zero on $\overline{\mathbf{0}p_{2k-1}}$. By rotating $\Sigma$ an
angle $\pi$ about the vertical geodesic line
$\\{\mathbf{0}\\}\times\mathbb{R}$ in its boundary, we obtain a properly
embedded simply-connected minimal surface $\Sigma_{k}$. Arguing similarly as
for $\Sigma_{1}$, we can prove that $\int_{\Sigma_{k}}K=-4k\pi$. Then we have
proved the following theorem.
###### Theorem 5.
For any integer $k\geq 1$, there exists a properly embedded simply-connected
minimal surface $\Sigma_{k}$ of finite total curvature $-4k\pi$ which is not a
minimal (vertical) graph.
Now let us construct a polygonal domain $\Omega$ in the above setting. For any
$\theta\in(0,\frac{\pi}{2k})$, let $\Omega_{\theta}$ be the polygonal domain
with vertices $\mathbf{0}$, $\widetilde{p}_{1}=1$, and
$p_{n}=e^{i(n-1)\theta},\quad 2\leq n\leq k+1.$
We mark by $+\infty$ the edge $\overline{\mathbf{0},\widetilde{p}_{1}}$ and
those of the form $\overline{p_{2i}p_{2i+1}}$; by $-\infty$ the edges of the
form $\overline{p_{2i-1}p_{2i}}$; and by $0$ the edge
$\overline{\mathbf{0}p_{k+1}}$. It is clear that $\Omega_{\theta}$ does not
satisfy the Jenkins-Serrin condition (see Theorem 3 in [1] or Theorem 3.3 in
[9]), as we can consider the inscribed polygonal domain
$\mathcal{P}\subset\Omega$ with vertices
$\mathbf{0},\widetilde{p}_{1},p_{2},p_{3}$ and any choice of disjoints
horocycles $H_{1},H_{2},H_{3}$ at $\widetilde{p}_{1},p_{2},p_{3}$
respectively, for which
$\mbox{dist}_{\mathbb{H}^{2}}(\mathbf{0},H_{1})+\mbox{dist}_{\mathbb{H}^{2}}(H_{2},H_{3})=\mbox{dist}_{\mathbb{H}^{2}}(\mathbf{0},H_{3})+\mbox{dist}_{\mathbb{H}^{2}}(H_{1},H_{2})$.
To solve this problem, we consider a small perturbation of
$\widetilde{p}_{1}$: Let $\Omega_{\theta,\beta}$ be the polygonal domain with
vertices $p_{1}=e^{-i\beta}$, for $\beta\in(0,\frac{\pi}{2}-k\theta]$ small,
and $p_{n}$ defined as above, for $2\leq n\leq k+1$. This domain
$\Omega_{\theta,\beta}$ satisfies the Jenkins-Serrin condition if we label by
$+\infty$ the edge $\overline{\mathbf{0},p_{1}}$ and those of the form
$\overline{p_{2i}p_{2i+1}}$; by $-\infty$ the edges of the form
$\overline{p_{2i-1}p_{2i}}$; and by $0$ the edge
$\overline{\mathbf{0}p_{k+1}}$.
Let $R$ be the reflection with respect to the geodesic containing
$\overline{\mathbf{0}p_{k+1}}$. Then $\Omega=\Omega_{\theta,\beta}\cup
R(\Omega_{\theta,\beta})$ is in the desired conditions. See Figure 2.
Figure 2: Left: The fundamental piece of a twisted Scherk example $\Sigma_{2}$
with total curvature $-8\pi$. Right: Vertical projection of $\Sigma_{2}$.
## 4 Uniqueness of Scherk minimal graphs
###### Theorem 6.
If $M$ is a complete minimal surface of total curvature $-2\pi$ in
$\mathbb{H}^{2}\times\mathbb{R}$, then $M$ is the Scherk minimal graph over an
ideal quadrilateral.
###### Proof.
Since the total curvature of $M$ is $-2\pi$, we have by equation (2) in
Theorem 1 that
$-2\pi=2\pi\left(2-2g-2n-\sum_{i=1}^{n}m_{i}\right).$
We already know that $m_{i}\geq 0$, by Assertion 3. And $n\geq 1$, since a
complete minimal surface in $\mathbb{H}^{2}\times\mathbb{R}$ cannot be
compact. So the only possibility is $g=0$, $n=1$ (hence the complete minimal
surface $M$ is simply-connected) and $m_{1}=1$.
As $m_{1}=1$, we know by Lemma 4 that there are four points
$p_{1},p_{2},p_{3},p_{4}\in\partial_{\infty}\mathbb{H}^{2}$, with $p_{i}\neq
p_{i+1}$ for any $i$, such that the end of $M$ corresponds to
$(\overline{p_{1}p_{2}}\times\\{+\infty\\})\cup(\overline{p_{2}p_{3}}\times\\{-\infty\\})\cup(\overline{p_{3}p_{4}}\times\\{+\infty\\})\cup(\overline{p_{4}p_{1}}\times\\{-\infty\\}),$
together with the complete vertical geodesics $\\{p_{i}\\}\times\mathbb{R}$ in
the ideal cylinder $\partial_{\infty}\mathbb{H}^{2}\times\mathbb{R}$ joining
their endpoints.
Let us now prove that the four points $p_{i}$ are all different. By the
maximum principle using vertical geodesic planes, we know that at least three
of them are different as $M$ cannot be a vertical plane. Suppose $p_{1}=p_{3}$
(the case $p_{2}=p_{4}$ follows similarly). Also using the maximum principle
with vertical geodesic planes, we get that the vertical projection $\pi(M)$ of
$M$ is contained in the ideal geodesic triangle of vertices
$p_{1},p_{2},p_{4}$. Even more, $\pi(M)$ is contained in a domain ${\cal
T}\subset\mathbb{H}^{2}$ bounded by $\overline{p_{1}p_{2}}$,
$\overline{p_{1}p_{4}}$ and a strictly concave (with respect to ${\cal T}$)
curve $\alpha$. We observe that the points in $M$ projecting onto $\alpha$
have horizontal normal vector. Suppose that the vertical projection of the
limit normal vector of $M$ (that we also call $N$) along
$\overline{p_{1}p_{2}}\times\\{+\infty\\}$ points to ${\cal T}$. We observe
that the horizontal curves in $M$ with endpoint in
$\\{p_{2}\\}\times\mathbb{R}$ arrive orthogonally to
$\partial_{\infty}\mathbb{H}^{2}\times\mathbb{R}$. In particular, $N$ is
constant along the vertical asymptotic line $\\{p_{2}\\}\times\mathbb{R}$. On
one hand that implies, looking at the behavior of $N$ along the asymptotic
boundary of $M$ (corresponding to the end) that the vertical projection of $N$
along $\overline{p_{1}p_{2}}\times\\{-\infty\\}$ also points to ${\cal T}$,
and its projection along $\overline{p_{1}p_{4}}\times\\{\pm\infty\\}$ goes out
from ${\cal T}$. On the other hand, if we follow the projection of $N$ along
$\alpha$, we obtain that it points to ${\cal T}$ along
$\overline{p_{1}p_{4}}\times\\{\pm\infty\\}$, a contradiction.
We now claim that $p_{1},p_{2},p_{3},p_{4}$ are cyclically ordered. We define
the solid cylinder $C_{r,T}=\\{(z,t):|z|\leq r,|t|\leq T\\}$, for $r<1$ close
to one and $T$ large, and consider $M_{r,T}=M\cap C_{r,T}$, which is a compact
minimal surface bounded by two horizontal compact curves contained in
$\\{t=T\\}$ close to $\overline{p_{1}p_{2}}\times\\{T\\}$ and
$\overline{p_{3}p_{4}}\times\\{T\\}$, two curves on $\\{t=-T\\}$ close to
$\overline{p_{2}p_{3}}\times\\{-T\\}$ and
$\overline{p_{4}p_{1}}\times\\{-T\\}$, and four curves on $\\{|z|=r\\}$ close
to vertical lines. By the flux formula with respect to the Killing vector
field $\partial_{t}$ (see [8, Proposition 3]), we have
$\int_{\partial M_{r,T}}\langle\nu,\partial_{t}\rangle=0,$ (3)
where $\nu$ is the outward-pointing unit conormal to $M_{r,T}$ along $\partial
M_{r,T}$. We get from (3), taking limits as $r\rightarrow 1$ and
$T\to+\infty$, that
$|\overline{p_{1}p_{2}}|+|\overline{p_{3}p_{4}}|=|\overline{p_{2}p_{3}}|+|\overline{p_{4}p_{1}}|$,
where $|\bullet|$ denotes (as in [1]) the hyperbolic length of the curve
$\bullet$ outside some disjoint horocycles at the ideal points $p_{i}$,
identifying $\mathbb{H}^{2}$ with the corresponding horizontal slice. By the
triangle inequality at infinity [1, Lemma 3] we get that
$p_{1},p_{2},p_{3},p_{4}$ must be cyclically ordered.
We call $\Omega$ the ideal quadrilateral with vertices
$p_{1},p_{2},p_{3},p_{4}$. By the maximum principle using vertical geodesic
planes, we get that $\pi({M})\subset\Omega$. On the other hand, the geometry
of the end of $M$ says that a neighborhood of $\partial\Omega$ is contained in
$\pi(M)$. Since $M$ is complete and simply-connected, we conclude
$\pi({M})=\Omega$.
Now let us show that the normal vector of $M$ is never horizontal. Suppose
there exists a point $P\in M$ such that $N_{3}(P)=0$. Let
$\Gamma\times\mathbb{R}$ be the vertical geodesic plane tangent to $M$ at $P$.
Since $M$ and $\Gamma\times\mathbb{R}$ have first contact order at $P$, their
intersection consists of $k$ curves meeting at equals angles at $P$, with
$k\geq 2$. Thus, there are at least four branches of
$M\cap(\Gamma\times\mathbb{R})$ leaving $P$ (see Figure 3, left). Since $M$ is
simply-connected, we deduce using the maximum principle with vertical planes
that there cannot exists a compact cycle in $M\cap(\Gamma\times\mathbb{R})$.
Hence $\Gamma$ cannot intersect two edges of $\Omega$, so it must have some
$p_{i}$ as an endpoint. Denote by $\gamma=\gamma(t)$, $t\in\mathbb{R}$, the
arc-length parameterized geodesic of $\mathbb{H}^{2}$ orthogonal to $\Gamma$
such that $\gamma(0)=\pi(P)$; and by $\Gamma_{t}$ the geodesic of
$\mathbb{H}^{2}$ passing through $\gamma(t)$ orthogonally (in particular,
$\Gamma_{0}=\Gamma$). For $\varepsilon>0$ small, $\Gamma_{\varepsilon}$
intersects two edges of $\Omega$, say $\overline{p_{1}p_{2}}$ and
$\overline{p_{2}p_{3}}$, and the number of intersection curves between the
vertical plane $\Gamma_{\varepsilon}\times\mathbb{R}$ and $M$ is at least two
(see Figure 3, right). But only one branch of the intersection curves can
arrive to $\overline{p_{1}p_{2}}\times\\{+\infty\\}$ (resp.
$\overline{p_{2}p_{3}}\times\\{-\infty\\}$), the other branch should be a
compact loop, a contradiction.
Figure 3: Left: The nodal domains between $M$ and $\Gamma\times\mathbb{R}$ at
a point with horizontal normal vector. Right: The intersection curves between
$M$ and $\Gamma_{\varepsilon}\times\mathbb{R}$.
We have prove then that, for any point $q\in\Omega$, the intersection of
$\\{q\\}\times\mathbb{R}$ with $M$ is transverse. So the number of
intersection points does not depend on $q$. For $q$ near an edge of $\Omega$
this number is one. We conclude that $M$ is a graph over $\Omega$. ∎
## References
* [1] P. Collin and H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math., 172 (2010), 1879-1906. DOI:10.4007/annals.2010.172.1879, arXiv:math/0701547.
* [2] B. Daniel, Isometric immersions into $\mathbb{S}^{n}\times\mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$ and applications to minimal surfaces, Trans. Amer. Math. Soc., 361 (2009), 6255–6282. MR2538594, Zbl pre05638191.
* [3] L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds, Pacific J. Math. 224 (2006), 91-117. arXiv:math/0507187, MR2231653, Zbl 1108.49031.
* [4] L. Hauswirth and H. Rosenberg, Minimal surfaces of finite total curvature in $\mathbb{H}^{2}\times\mathbb{R}$, Mat. Contemp., 31 (2006), 65-80. MR2385437, Zbl 1144.53323.
* [5] L. Hauswirth, R. Sa Earp and E. Toubiana, Associate and conjugate minimal immersions in $M\times\mathbb{R}$, Tohoku Math. J. (2) 60 (2008), 267-286.
* [6] F. Martín, Rafe Mazzeo and M.M. Rodríguez, Minimal surfaces with positive genus and finite total curvature in $\mathbb{H}^{2}\times\mathbb{R}$, arXiv:1208.5253.
* [7] L. Hauswirth, B. Nelli, R. Sa Earp and E. Toubiana, Minimal ends in $\mathbb{H}^{2}\times\mathbb{R}$ with finite total curvature and a Schoen type theorem, prepint, arXiv:1111.0851.
* [8] D. Hoffman,, J. Lira and H. Rosenberg, Constant Mean Curvature Surfaces in $\mathbb{M}^{2}\times\mathbb{R}$, Trans. Amer. Math. Soc., 358:2 (2006), 491-507.
* [9] L. Mazet, M.M. Rodríguez and H. Rosenberg, The Dirichlet problem for the minimal surface equation with possible infinite boundary data over domains in a Riemannian surface, Proc. London Math. Soc., 102:3 (2011), 985-1023. DOI:10.1112/plms/pdq032. arXiv:0806.0498.
* [10] F. Morabito, A Costa-Hoffman-Meeks type surface in $\mathbb{H}^{2}\times\mathbb{R}$, Trans. Amer. Math. Soc., 363:1 (2010), 1-36.
* [11] F. Morabito and M.M. Rodríguez, Saddle Towers and minimal $k$-noids in $\mathbb{H}^{2}\times\mathbb{R}$, J. Inst. Math. Jussieu, 11(2): 333-349 (2012). DOI:10.1017/S1474748011000107. arXiv:0910.5676.
* [12] B. Nelli and H. Rosenberg, Minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, Bull. Braz. Math. Soc., 33:2 (2002), 263-292.
* [13] R. Osserman, A survey of minimal surfaces, Dover Publications, New York (1986). MR0852409, Zbl 0209.52901.
* [14] J. Pyo, New complete embedded minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, Ann. Global Anal. Geom. 40:2 (2011) 167-176. arXiv:math/0911.5577.
* [15] R. Sa Earp and E. Toubiana, Screw motion surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and $\mathbb{S}^{2}\times\mathbb{R}$, Illinois J. Math., 49:4 (2005), 1323-1362.
Juncheol Pyo
Department of Mathematics
Pusan National University
Busan 609-735, Korea
e-mail: jcpyo@pusan.ac.kr
M. Magdalena Rodríguez
Departamento de Geometría y Topología
Universidad de Granada
Fuentenueva, 18071, Granada, Spain
e-mail: magdarp@ugr.es
|
arxiv-papers
| 2012-10-03T13:14:37 |
2024-09-04T02:49:35.900242
|
{
"license": "Public Domain",
"authors": "Juncheol Pyo, Magdalena Rodriguez",
"submitter": "Juncheol Pyo",
"url": "https://arxiv.org/abs/1210.1099"
}
|
1210.1204
|
# Early Warning Signals and the Prosecutor’s Fallacy
Carl Boettiger cboettig@ucdavis.edu Alan Hastings Center for Population
Biology, 1 Shields Avenue, University of California, Davis, CA, 95616 United
States. Department of Environmental Science and Policy, University of
California, Davis
###### Abstract
Early warning signals have been proposed to forecast the possibility of a
critical transition, such as the eutrophication of a lake, the collapse of a
coral reef, or the end of a glacial period. Because such transitions often
unfold on temporal and spatial scales that can be difficult to approach by
experimental manipulation, research has often relied on historical
observations as a source of natural experiments. Here we examine a critical
difference between selecting systems for study based on the fact that we have
observed a critical transition and those systems for which we wish to forecast
the approach of a transition. This difference arises by conditionally
selecting systems known to experience a transition of some sort and failing to
account for the bias this introduces – a statistical error often known as the
Prosecutor’s Fallacy. By analysing simulated systems that have experienced
transitions purely by chance, we reveal an elevated rate of false positives in
common warning signal statistics. We further demonstrate a model-based
approach that is less subject to this bias than these more commonly used
summary statistics. We note that experimental studies with replicates avoid
this pitfall entirely.
###### keywords:
early warning signals , tipping point , alternative stable states , likelihood
methods
††journal: Proceedings of the Royal Society B
## 1 Introduction
> _Mathematics …while assisting the trier of fact in the search of truth, must
> not cast a spell over him._ – California Supreme court, 1968.
In the case of _People v. Collins_ 1968, California Supreme Court considered
the evidence of an expert witness described by the court as “an instructor of
mathematics at a state college”, which concluded that the probability that a
randomly selected individual would match the description given by the victim
would be less than 1 in 12 million (Supreme Court, 1968). The prosecution had
produced an individual matching the prosecutor’s detailed description, and
convinced by the mathematics, the lower courts had found him guilty.
The prosecution has only observed that the probability of seeing the evidence
($E$) they produced given a random innocent individual ($I$), $P(E|I)$ is very
small. From this one cannot conclude that the individual is indeed guilty,
that is, that the probability the individual is innocent given the evidence
$P(I|E)$ is also very small. In a city with millions of people, there might be
several individuals who match the description of the evidence. Mathematically
$P(E|I)$ need not equal $P(I|E)$, instead, these expressions are related by
Bayes theorem,
$P(E|I)=P(I|E)\frac{P(E)}{P(I)},$ (1)
$P(E)\ll 1$ and $P(I)\approx 1$ , so $P(E|I)\approx P(I|E)P(E)$, and
consequently we cannot conclude that $P(I|E)\ll 1$ from $P(E|I)\ll 1$.
Realizing this mistake, the California Supreme Court reversed the decision,
and the case became a widely recognized example of the Prosecutor’s Fallacy
(Thompson and Schumann, 1987). Here we explore how a similar misconception can
arise from the use of historical data to evaluate methods for detecting early
warning signals of critical transitions.
Catastrophic transitions or tipping points, where a complex system shifts
suddenly from one state to another, have been implicated in a wide array of
ecological and global climate systems such as lake ecosystems (Carpenter,
2011), coral reefs (Mumby et al., 2007), savannah (Kéfi et al., 2007),
fisheries (Berkes et al., 2006), and tropical forests (Hirota et al., 2011).
Recent research has begun to identify statistical patterns commonly associated
with these sudden catastrophic transitions which could be used as an _early
warning sign_ to identify an approaching tipping point, which might provide
managers time to react to and avert an undesirable state shift (Scheffer et
al., 2009; Lenton, 2011). An array of statistical patterns associated with
tipping point phenomena has been suggested for the detection of early warning
signals associated with such sudden transitions. Two of the most commonly used
are a pattern of increasing variance (Carpenter and Brock, 2006) and a pattern
of increasing autocorrelation (van Nes and Scheffer, 2007), which have been
tested in both experimental manipulation (Drake and Griffen, 2010; Carpenter,
2011; Veraart et al., 2011; Dai et al., 2012) and historical observations
(Livina and Lenton, 2007; Dakos et al., 2008; Lenton et al., 2012; Ditlevsen
and Johnsen, 2010; Guttal and Jayaprakash, 2008; Thompson and Sieber, 2010).
### Testing patterns on historical data
Historical examples of sudden transitions taken from the paleo-climate record
provide an important way to test and evaluate potential leading indicator
methods, and have been widely used for this purpose (Livina and Lenton, 2007;
Dakos et al., 2008; Lenton et al., 2012; Ditlevsen and Johnsen, 2010; Guttal
and Jayaprakash, 2008; Thompson and Sieber, 2010). Similarly, it has been
suggested that data gathered from ecological systems such as lakes that were
monitored before they experienced sudden eutrophication, or grasslands
subjected to overgrazing, could contain data that could help reveal when
similar systems are approaching a tipping point (Carpenter, 2011).
However, testing methods for early warning signals against historical examples
of transitions is susceptible to statistical mistakes that arise from
selecting data conditional on that data having already exhibited a sudden
transition. A central tenant of early warning theory is that the system in
question is slowly approaching a tipping point that lies some unknown distance
away. If nothing is done to remedy the situation, this slow change will
inevitably carry the system beyond the tipping point, which introduces a
sudden, rapid transition into an undesirable state (Scheffer et al., 2009).
This process can be described mathematically as a _bifurcation_ , in which a
slowly changing parameter reaches a critical value that causes the system
stability to change.
Not all sudden transitions are caused by some “guilty” process slowly driving
the system over a tipping point – the kind of process that early warning
signals are designed to detect. Some systems may experience such transitions
purely by chance, leaving a stable state on an extremely unlikely excursion
that happens to stray to far from the stable attractor (_e.g._ Ditlevsen and
Johnsen, 2010; Lenton, 2011, consider this possibility in transitions that
arise from analyzing historical climate record). Like the evidence presented
before the California Supreme Court in 1968, the chance of observing such an
“innocent” transition a priori may be very small, but when selected from a
historical record of many possible transitions, this possibility can no longer
be ignored.
Figure 1 shows a schematic illustrating critical transitions under each of
these scenarios. In the left panel, the system experiences a bifurcation and
should contain an early warning signal. In the right panel, a similar-looking
trajectory emerges from a simulation of a stable system which should not
contain a warning signal. While the simulation of the bifurcation scenario
shown on the left produces a similar transition every time, the transition
shown on the right is somewhat less likely, occurring in only 1% of
simulations.
Figure 1: The Prosecutor’s Fallacy. (a) Plot of the model functions shown in
Eq (2) with parameters $a=180$, $K=500$, $e=.5$, and $h=200$. When the death
rate is higher than the birth rate, the system dynamics drive the state
(population size) to smaller values. When birth rate is higher, the system
moves right, as indicated by the arrows.(b) The potential energy is given by
the negative integral of $b(n)-d(n)$, shown in the lower plot. The potential
function gives an intuitive picture of the stability of a system by imagining
the curve as a surface on which a ball is free to bounce across – wells
correspond to stable points and peaks to unstable points. While most
trajectories remain near the stable well, some transition out merely by
chance. An example of such a trajectory is shown in the top panel, in which
time increases along the vertical axis. Though initially oscillating around
the stable state, a chance excursion carries it beyond the Allee threshold
(vertical dotted line). Such chance trajectories can produce the statistical
patterns as observed in true critical transitions seen in panel (c): Early
warning signals are aimed at detecting systems which are slowly moving towards
a tipping point or bifurcation, illustrated in the successive curves
(deteriorating and critical). Top panel: An example trajectory from a
simulation under this process shows the state of the system as the potential
moves towards the bifurcation point. The original position of the Allee
threshold is shown by the vertical dotted line (though it moves slightly as
the parameter changes).
## 2 Methods and Results
To investigate if early warning signals are vulnerable to this fallacy, we
simulate a system that is not driven towards a bifurcation such as in Fig
reffig:1(b). This simulation approach allows us to determine whether examining
historical events is a valid way to test the utility of these indicators. We
simulated 20,000 replicates of a stochastic individual-based birth-death
process with an Allee threshold (Courchamp et al., 2008), which arises from
positive fitness effects at low densities. Above the Allee threshold the
population returns to a positive equilibrium size, whereas below the threshold
the population decreases to zero. The model can be represented as a continuous
time birth-death process where births and deaths are Poisson events which
depend on the current density with rates given by
$\displaystyle b(n)$ $\displaystyle=\frac{Kn^{2}}{n^{2}+h^{2}},$ (2)
$\displaystyle d(n)$ $\displaystyle=en+a,$ (3)
a model with a linear death rate and density-dependent birth rate that drives
the Allee effect at low densities and limits growth at high densities. In this
model $n$ indicates the discrete number of individuals in the population, $K$
indicates a carrying capacity as set by a limiting resource, $e$ a per-capita
death rate (the $e$ scaling term in the birth equation allows the carrying
capacity $K$ to correspond to a positive equilibrium point), $a$ an additional
mortality imposed on the population such as harvest, $h$ is a parameter
controlling at what population size the addition of more individuals switches
from conferring a positive benefit on growth from Allee interactions $n<h$ to
a negative impact on growth due to increased competition, $n>h$. The key
feature of this model is the alternate stable states introduced by this
effect; other functional forms for Eq. (2) could serve equally well for these
simulations (see _e.g._ Scheffer et al., 2001). Though this system can be
forced through a bifurcation by increasing the death rate, in these
simulations all parameters are held constant and no bifurcation occurs.
Consequently we do not anticipate an early warning signal of an approaching
bifurcation.
The simulation starts from the positive equilibrium population size. Though
the chance of a transition across the Allee threshold in any given time step
is small, given enough time this system will eventually experience such a rare
event driving the population extinct. We ran each replicate over 50,000 time
units, sampling the system every 50 time units. In this time window 266 of the
1,000 replicates experience population collapse. To keep the examples of
comparable sample size, we focus on a section of the data 500 time points
prior to the system approaching the transition.
To test whether selecting systems that have experienced spontaneous
transitions could bias the analysis towards false positive detection of early
warning signals, (the Prosecutor’s Fallacy) we selected replicates conditional
on having collapsed in the simulations. We then selected a window around each
system that ended just before the collapse, while the population values were
still above the Allee threshold. For each replicate, we calculated the most
common early warning indicators, variance and autocorrelation (_e.g._
Carpenter and Brock, 2006; Dakos et al., 2008; Scheffer et al., 2009), around
a moving window equal to half the length of that time series.
To test for the presence of a warning signal in these indicators we computed
values of Kendall’s $\tau$ for both indicators for each of the 266 replicates.
Kendall’s $\tau$ is a non-parametric measure of rank correlation frequently
used to identify an increasing trend ($\tau>0)$ in early warning signals
(Dakos et al., 2008, 2011), defined as $\tau{\frac{1}{2}n(n-1)}$ in $n$
observations. 111A pair of observations $(x_{i},y_{i})$ and $(x_{j},y_{j})$
are concordant if $x_{i}>x_{j}$ and $y_{i}>y_{j}$ or $x_{i}<x_{j}$ and
$y_{i}<y_{j}$ and discordant otherwise; equalities excepted. $\tau$ takes
values in $(-1,1)$. The distribution of $\tau$ values observed across these
replicates is shown in Figure 2. We compare the distribution of $\tau$ from
all the simulations to the distribution conditioned on experiencing a chance
transition to the alternative stable state. To avoid an effect of sample size
the time series are all chosen to be the same length.
To demonstrate the effect we observe is not unique to models with Allee
effects, we provide an example of the effect arising in a discrete-time model
with two non-zero stable states adapted from (May, 1977),
$X_{t+1}=X_{t}\exp\left(r\left(1-\frac{X_{t}}{K}\right)-\frac{a*X_{t}^{Q-1}}{X_{t}^{Q}+H^{Q}}\right).$
(4)
which combines a logistic growth model with a saturating predator response
(See May (1977) for detailed discussion), shown in Figure 3. Code to replicate
the analysis can be found at
`https://github.com/cboettig/earlywarning/tree/prosecutor/`.
Figure 2: The distribution of the correlation statistic $\tau$ for two early
warning indicators (variance, autocorrelation) on replicates conditionally
selected for having collapsed by chance in simulations is shown in grey bars.
Solid lines indicate the estimated density of the statistic from a random
sample of the simulations (not conditional on observing a transition).
Positive values of $\tau$ correspond to a pattern of an indicator increasing
with time; typically taken as evidence that a system is approaching a critical
transition. In these simulations, the pattern arises instead from the
Prosecutor’s fallacy of conditional selection. Figure 3: The identical
analysis from Figure 2 is shown for the model in Eq (4) using parameters
$r=0.75$, $K=10$, $a=1.7$, $Q=3$, and $H=1$. A similar statistical bias,
particularly towards positive values of $tau$ occurs in this model as well.
For each of these replicates we also take a model-based approach, estimating
parameters for an approximate linear model of the system approaching a saddle
node bifurcation, as described by Boettiger and Hastings (2012),
$\mathrm{d}X=\sqrt{r_{t}}(\phi(r_{t})-X_{t})\mathrm{d}t+\sigma\sqrt{\phi(r_{t})}\mathrm{d}B_{t}$
(5)
In this model, the parameter $m$ describes the approach towards the saddle-
node bifurcation. Estimates $m<0$ are expected in systems approaching a
bifurcation, while for stable systems $m$ should be approximately zero. None
of the estimates across the 266 simulations differed from zero in our study,
hence the model-based estimation shows no evidence of bias on data that has
been selected conditional on collapse.
## 3 Discussion
The attempts to detect early warning signs for critical transitions are based
on the concept of a deteriorating environment as embodied in a changing
parameter Scheffer et al. (2009), which is a different kind of transition than
one which is driven instead by stochasticity in an environment which is
otherwise constant and exhibiting no directional change. When trying to use
historical data to understand critical transitions we often do not know which
category, changing environment or simply chance, an observed large change
falls into.
We have shown here that systems which undergo rare sudden transitions due to
chance look statistically different from their counterparts that do not, even
though they are driven by the same stochastic process. In particular, such
conditionally selected examples are more likely to show signs associated with
an early warning of an approaching tipping point, such as increasing variance
or increasing autocorrelation, as measured by Kendall’s $\tau$. This increases
the risk of false positives – cases in which a warning signal being tested
appears to have successfully detected an underlying change in the system
leading to a tipping point, when in fact the example comes instead from a
stable system with no underlying change in parameters. Figure 2 shows that
many of the chance crashes show values of $\tau$ that are significantly larger
than those observed in the otherwise identical replicates that did not
experience a chance transition, thus “detecting” an underlying change in the
system dynamics that is not in fact present.
### 3.1 Chance transitions are false positives for early warning signals
It seems tempting to argue that this bias towards positive detection in
historical examples is not problematic – each of these systems did indeed
collapse, so the increased probability of exhibiting warning signals could be
taken as a successful detection. Unfortunately this is not the case. At the
moment the forecast is made, these systems are not likely to transition, since
they experience a strong pull towards the original stable state. A closer look
at the patterns involved shows why common indicators such as autocorrelation
and variance can be misleading.
As the system gets farther from its stable point, it it more likely to draw a
random step that returns it towards the stable point. Despite this, there is
always some probability that it will move further still, so systems that do
cross the tipping point must do so rather quickly by a string of events. This
pattern, clearly visible before the crashes in each of the examples in Figure
1, produces a string of observations that appear more highly autocorrelated
(if we are sampling the system frequently enough to catch the excursion at
all) than we observe in the rest of the fluctuations around the equilibrium.
Yet this autocorrelation comes from a chance trajectory moving quickly _away_
from the stable state, not from the critical slowing down pattern in the
return times to the stable state which precede a saddle-node bifurcation and
motivate the early warning signal.
This longer than expected excursion results in a higher than expected variance
in that window as well. Both variance and autocorrelation are calculated using
a moving window over the time-series, which allows the method to pick out a
pattern of change as the window moves along the sequence. If this chance
excursion that precedes the crash happens to fill a significant part of the
moving window, the resulting pattern will tend to show an increase in
autocorrelation or variance. If the chance excursion is relatively rapid
compared to the frequency at which the system is observed (spacing of the
data) or the width of the moving window, the excursion may not significantly
alter the general pattern. In this way, some of the events in which a crash is
observed will appear to present these statistical patterns of increased
variance or autocorrelation without being harbingers of approaching critical
transitions.
### 3.2 The truncation of observations
If we had a complete knowledge of the system dynamics, then we could eliminate
the bias we observe here since the bias arises from the transient branch of
the trajectory that crosses the threshold, and if the system were truncated at
the minimum of the potential then the effect we emphasize here would not
appear. But, it is not possible to truncate the system in any practical
application. The precise location of the minimum of the potential which is the
location of the deterministic equilibrium is unknown. Moreover, under the
hypothesis that the system is approaching a critical transition, the location
of the minimum potential moves so it cannot easily be estimated by previous
observations, (see Figure 1c where the equilibrium point moves in the
direction of the transition). Thus it is neither practical nor desirable to
suggest that historical time series can be used by following a simple
truncation rule that avoids the branch of a trajectory crossing the threshold
to another basin of attraction. Exactly where a particular study will choose
to truncate such a trajectory will necessarily be arbitrary without an
underlying model of the process. Frequently this is done by removing the very
steep, monotonic branch of the trajectory expected once the system crosses the
unstable threshold. Such an approach corresponds with our choice of
termination and produces the bias we discuss here.
The examples of Figure 1, though only single replicates, may be useful in
illustrating these issues. Figure 1c, top panel shows a sample trajectory of a
system with a parameter shift, while 1b shows a trajectory without a shift.
Both trajectories become more highly autocorrelated and higher variance near
the end of the time series (time increases on the y axis in Figure 1). The
part of the time series following the critical transition shows a fast and
monotonic trajectory to the unstable trajectory, and would usually be excluded
by an analysis for warning signals in advance of the transition. No such clear
pattern exists prior to the transition in Figure 1b. An alternative proposal
to terminate the trajectory in panel B earlier would also risk decreasing the
signal seen in panel c, and would be inconsistent with the application of
warning signals in the forecasting context, where there would be no such
truncation.
### 3.3 Comparing to the model-based method
In our numerical experiment, the model-based estimate of early warning signals
appears more robust than the summary statistics, producing the same estimates
on both the conditionally selected replicates as on a random sample of the
replicates. This is a consequence of the more rigid specifications that come
with a model-based approach – the pattern expected is less general than any
increase in variance or autocorrelation, but instead must be one that matches
its approximation of the saddle-node bifurcation. This observation highlights
the difference between the pattern driving the false positive trends in
increasing variance and increasing autocorrelation and the pattern anticipated
in the saddle-node model. This should not however be taken as evidence that
the model-based approach is immune to the bias of the Prosecutor’s Fallacy.
### 3.4 Importance of experimental approaches
The problem we highlight ultimately stems from the difficulty of having only a
single realization with which to examine a complex problem. The only way to
deal with this problem embodied is through replication, as can be done in an
experimental system in laboratory manipulations such as Drake and Griffen
(2010); Veraart et al. (2011); Dai et al. (2012) and at the scale of whole
lake ecosystems in Carpenter (2011). Experimental procedures avoid the hazard
of the Prosecutor’s fallacy by generating a complete sample of replicates,
rather than selecting a subset of cases from some larger historical sample.
## 4 Acknowledgments
This research was supported by funding from NSF Grant EF 0742674 to AH and a
Computational Sciences Graduate Fellowship from the Department of Energy grant
DE-FG02-97ER25308 and NERSC Supercomputing grant DE-AC02-05CH11231 to CB. The
authors thank M. Baskett, T.A. Perkins and N. Ross for helpful comments on
earlier drafts of the manuscript, and also P. Ditlevsen and an anonymous
reviewer for their comments.
## References
* Berkes et al. (2006) Berkes, F., Hughes, T. P., Steneck, R. S., Wilson, J. A., Bellwood, D. R., Crona, B., Folke, C., Gunderson, L. H., Leslie, H. M., Norberg, J., Nyström, M., Olsson, P., Osterblom, H., Scheffer, M., Worm, B., Mar. 2006\. Ecology. Globalization, roving bandits, and marine resources. Science (New York, N.Y.) 311 (5767), 1557–8.
* Boettiger and Hastings (2012) Boettiger, C., Hastings, A., May 2012. Quantifying limits to detection of early warning for critical transitions. Journal of The Royal Society Interface 9 (75), 2527 – 2539.
* Carpenter (2011) Carpenter, J., Feb. 2011. May the Best Analyst Win. Science 331 (6018), 698–699.
* Carpenter and Brock (2006) Carpenter, S. R., Brock, W. A., 2006. Rising variance: a leading indicator of ecological transition. Ecology letters 9 (3), 311–8.
* Courchamp et al. (2008) Courchamp, F., Berec, L., Gascoigne, J., 2008. Allee Effects in Ecology and Conservation. Oxford University Press, USA.
* Dai et al. (2012) Dai, L., Vorselen, D., Korolev, K. S., Gore, J., May 2012. Generic Indicators for Loss of Resilience Before a Tipping Point Leading to Population Collapse. Science 336 (6085), 1175–1177.
* Dakos et al. (2011) Dakos, V., Kéfi, S., Rietkerk, M., Nes, E. H. V., Scheffer, M., 2011. Slowing Down in Spatially Patterned Ecosystems at the Brink of Collapse. The American Naturalist.
* Dakos et al. (2008) Dakos, V., Scheffer, M., van Nes, E. H., Brovkin, V., Petoukhov, V., Held, H., Sep. 2008. Slowing down as an early warning signal for abrupt climate change. Proceedings of the National Academy of Sciences 105 (38), 14308–12.
* Ditlevsen and Johnsen (2010) Ditlevsen, P. D., Johnsen, S. J., Oct. 2010. Tipping points: Early warning and wishful thinking. Geophysical Research Letters 37 (19), 2–5.
* Drake and Griffen (2010) Drake, J. M., Griffen, B. D., Sep. 2010. Early warning signals of extinction in deteriorating environments. Nature 467 (7314), 456–459.
* Guttal and Jayaprakash (2008) Guttal, V., Jayaprakash, C., Dec. 2008. Spatial variance and spatial skewness: leading indicators of regime shifts in spatial ecological systems. Theoretical Ecology 2 (1), 3–12.
* Hirota et al. (2011) Hirota, M., Holmgren, M., Van Nes, E. H., Scheffer, M., Oct. 2011. Global Resilience of Tropical Forest and Savanna to Critical Transitions. Science 334 (6053), 232–235.
* Kéfi et al. (2007) Kéfi, S., Rietkerk, M., Alados, C. L., Pueyo, Y., Papanastasis, V. P., Elaich, A., de Ruiter, P. C., Sep. 2007. Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems. Nature 449 (7159), 213–7.
* Lenton (2011) Lenton, T. M., Jun. 2011. Early warning of climate tipping points. Nature Climate Change 1 (4), 201–209.
* Lenton et al. (2012) Lenton, T. M., Livina, V., Dakos, V., 2012. Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness. Philosophical Transactions of The Royal Society A (in press).
* Livina and Lenton (2007) Livina, V. N., Lenton, T. M., Feb. 2007. A modified method for detecting incipient bifurcations in a dynamical system. Geophysical Research Letters 34 (3), 1–5.
* May (1977) May, R. M., Oct. 1977. Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269 (5628), 471–477.
* Mumby et al. (2007) Mumby, P. J., Hastings, A., Edwards, H. J., Nov. 2007. Thresholds and the resilience of Caribbean coral reefs. Nature 450 (7166), 98–101.
* Scheffer et al. (2009) Scheffer, M., Bascompte, J., Brock, W. A., Brovkin, V., Carpenter, S. R., Dakos, V., Held, H., van Nes, E. H., Rietkerk, M., Sugihara, G., 2009. Early-warning signals for critical transitions. Nature 461 (7260), 53–9.
* Scheffer et al. (2001) Scheffer, M., Carpenter, S. R., Foley, J. A., Folke, C., Walker, B., Oct. 2001. Catastrophic shifts in ecosystems. Nature 413 (6856), 591–6.
* Supreme Court (1968) Supreme Court, C., 1968. People v. Collins.
* Thompson and Sieber (2010) Thompson, J. M. T., Sieber, J., Dec. 2010. Climate tipping as a noisy bifurcation: a predictive technique. IMA Journal of Applied Mathematics 76 (1), 27–46.
* Thompson and Schumann (1987) Thompson, W. C., Schumann, E. L., 1987. Interpretation of statistical evidence in criminal trials: The prosecutor’s fallacy and the defense attorney’s fallacy. Law and Human Behavior 11 (3), 167–187.
* van Nes and Scheffer (2007) van Nes, E. H., Scheffer, M., Jun. 2007. Slow recovery from perturbations as a generic indicator of a nearby catastrophic shift. The American naturalist 169 (6), 738–47.
* Veraart et al. (2011) Veraart, A. J., Faassen, E. J., Dakos, V., van Nes, E. H., Lürling, M., Scheffer, M., Dec. 2011. Recovery rates reflect distance to a tipping point in a living system. Nature, 2–5.
|
arxiv-papers
| 2012-10-03T19:56:10 |
2024-09-04T02:49:35.915243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carl Boettiger and Alan Hastings",
"submitter": "Carl Boettiger",
"url": "https://arxiv.org/abs/1210.1204"
}
|
1210.1265
|
# Note on an integral expression for the
average lifetime of the bound state 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
Recently, an exact Green’s function of the diffusion equation for a pair of
spherical interacting particles in two dimensions subject to a backreaction
boundary condition was used to derive an exact expression for the average
lifetime of the bound state. Here, we show that the corresponding divergent
integral may be considered as the formal limit of a Stieltjes transform. Upon
analytically calculating the Stieltjes transform one can obtain an exact
expression for the finite part of the divergent integral and hence for the
average lifetime.
11footnotetext: Email: prustelt@niaid.nih.gov, mms@niaid.nih.gov
## 1 Introduction
In Ref. [3] an exact analytical expression for the average lifetime of the
bound state and hence the off-rate in 2D was derived, based on an exact
Green’s function of the reversible diffusion-influenced reaction for an
isolated pair in two dimensions [4]. Furthermore, it was shown that the
associated integral has to be regularized and that its numerical evaluation
suggests the relation
$\frac{1}{k_{\text{off}}}=\frac{1}{\kappa_{d}}+\frac{\ln 2-\gamma}{2\pi
D}\frac{\kappa_{a}}{\kappa_{d}},$ (1.1)
where $\kappa_{a}$, $\kappa_{d}$, $D$ and $\gamma$ denote the intrinsic
association and dissociation constants, the diffusion constant and Euler’s
number $\gamma=0.5772156649\ldots$ [1], respectively. More precisely, up to a
constant, the (regularized) off-rate is given by
$k_{\text{off}}^{-1}\propto\int^{\infty}_{1}\frac{f(x)}{x}dx+\int^{1}_{0}\frac{f(x)-f(0)}{x}dx,$
(1.2)
where $f(x)$ is defined by
$f(x):=\frac{P^{2}(x,1)}{x^{2}}$ (1.3)
and $P(x,1)$ is the function
$\displaystyle P(x,1)$ $\displaystyle:=$
$\displaystyle\frac{\frac{2}{\pi}\tilde{h}}{[\tilde{\alpha}(x)^{2}+\tilde{\beta}(x)^{2}]^{1/2}},$
(1.4) $\displaystyle\tilde{\alpha}(x)$ $\displaystyle:=$
$\displaystyle(x^{2}-\tilde{\kappa}_{D})J_{1}(x)+\tilde{h}xJ_{0}(x),$ (1.5)
$\displaystyle\tilde{\beta}(x)$ $\displaystyle:=$
$\displaystyle(x^{2}-\tilde{\kappa}_{D})Y_{1}(x)+\tilde{h}xY_{0}(x).$ (1.6)
$J_{0},J_{1},Y_{0},Y_{1}$ denote the Bessel functions of first and second kind
and of zeroth and first order, respectively [1]. Furthermore, the
dimensionless constants $\tilde{h},\tilde{\kappa_{d}}$ are related to the
intrinsic association and dissociation constants $\kappa_{a}$ and $\kappa_{d}$
by
$\displaystyle\tilde{h}:=ha:=\frac{\kappa_{a}}{2\pi D},$ (1.7)
$\displaystyle\tilde{\kappa_{D}}:=\kappa_{D}a^{2}:=\frac{\kappa_{d}a^{2}}{D}.$
(1.8)
Here, $a$ refers to the encounter radius.
## 2 Stieltjes transform
Instead of approaching the finite integrals in Eq. (1.2) directly, we will
consider the full divergent integral as the limiting case of a Stieltjes
transform. The Stieltjes transform itself can be expressed in terms of
modified Bessel functions. Then, their limiting behavior for small arguments
opens the possibility to separate the finite and divergent contributions. In
this way, we will derive that the finite part gives indeed Eq. (1.1).
Starting point is the observation made in Ref. [2] that a twofold Laplace
transform yields a Stieltjes transform
$\int^{\infty}_{0}e^{-xu}\int^{\infty}_{0}e^{-u\xi}g(\xi)dud\xi=\int^{\infty}_{0}\frac{g(\xi)}{x+\xi}d\xi,$
(2.1)
where $g(t)$ is an arbitrary sufficiently ”well-behaved” function. This
observation was used to show that
$\frac{K_{\nu}(\sqrt{x})}{\sqrt{x}K_{\nu+1}(\sqrt{x})}=\frac{2}{\pi^{2}}\int^{\infty}_{0}\frac{\xi^{-1}}{x+\xi}\\{J^{2}_{\nu+1}(\sqrt{\xi})+Y^{2}_{\nu+1}(\sqrt{\xi})\\}^{-1}d\xi,$
(2.2)
for $\nu\geq-1$ and $x>0$, based on the relation
$\mathcal{L}\bigg{\\{}\frac{2}{\pi^{2}}\xi^{-1}[J^{2}_{\nu+1}(\sqrt{\xi})+Y^{2}_{\nu+1}(\sqrt{\xi})]^{-1}\bigg{\\}}=\mathcal{L}^{-1}\bigg{\\{}\frac{K_{\nu}(\sqrt{\xi})}{\sqrt{\xi}K_{\nu+1}(\sqrt{\xi})}\bigg{\\}},$
(2.3)
cp. Ref. [2] and references given therein. Here,
$\mathcal{L},\mathcal{L}^{-1}$ denote the Laplace and inverse Laplace
transform, respectively and $K_{\nu}$ refers to the modified Bessel function
of second kind and $\nu$th order [1].
Inspired by these results it is tempting to consider the full divergent
integral [3]
$\int^{\infty}_{0}\frac{f(x)}{x}dx$ (2.4)
which gives the off-rate as the limit of a Stieltjes transform. Indeed, one
finds [5]
$\frac{\tilde{h}}{\tilde{\kappa_{D}}}\frac{1}{x}-\frac{\tilde{h}K_{1}(\sqrt{x})}{x[(x+\tilde{\kappa_{D}})K_{1}(\sqrt{x})+\tilde{h}\sqrt{x}K_{0}(\sqrt{x})]}=\frac{2}{\pi^{2}}\tilde{h}^{2}\int^{\infty}_{0}\frac{1}{\xi(\xi+x)}\frac{d\xi}{\tilde{\alpha}(\sqrt{\xi})^{2}+\tilde{\beta}(\sqrt{\xi})^{2}}.$
(2.5)
Now, upon changing the dummy variable $\xi\rightarrow\varphi^{2}$, one notes
that the divergent integral Eq. (2.4) is formally the limiting case of the
obtained Stieltjes transform
$\int^{\infty}_{0}\frac{f(\varphi)}{\varphi}d\varphi=\lim_{x\rightarrow
0}\frac{4}{\pi^{2}}\tilde{h}^{2}\int^{\infty}_{0}\frac{1}{\varphi(\varphi^{2}+x)}\frac{d\varphi}{\tilde{\alpha}(\varphi)^{2}+\tilde{\beta}(\varphi)^{2}}.$
(2.6)
We emphasize again that both the expression on the lhs and on the rhs diverge.
However, we can invoke Eq. (2.5) to study the limit $x\rightarrow 0$ and to
extract the exact expression for the finite contribution of the divergent
integral. To this end, we employ the expansion of the modified Bessel function
suitable for small arguments [1] and arrive for small $x$ at (note that
$\ln(C):=\ln(\frac{1}{2})+\gamma$)
lhs of Eq. (2.5) $\displaystyle=$
$\displaystyle\frac{\tilde{h}}{\tilde{\kappa_{D}}}\frac{1}{x}-\frac{\tilde{h}}{\tilde{\kappa_{D}}x}\frac{\frac{1}{2}\ln(C\sqrt{x})\sqrt{x}+\frac{1}{\sqrt{x}}+\ldots}{\frac{\sqrt{x}}{\tilde{\kappa_{D}}}+\frac{1}{2}\ln(C\sqrt{x})\sqrt{x}+\frac{1}{\sqrt{x}}-\frac{\tilde{h}}{\tilde{\kappa_{D}}}\ln(C\sqrt{x})\sqrt{x}+\ldots}$
(2.7) $\displaystyle=$
$\displaystyle\frac{\tilde{h}}{\tilde{\kappa_{D}}}\frac{1}{x}-\frac{\tilde{h}}{\tilde{\kappa_{D}}x}\bigg{[}1-\frac{x}{\tilde{\kappa_{D}}}+\frac{\tilde{h}}{\tilde{\kappa_{D}}}x[\ln(C)+\ln(\sqrt{x})]+\ldots\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{\tilde{h}^{2}}{\tilde{\kappa_{D}}^{2}}(\ln(2)-\gamma)+\frac{\tilde{h}}{\tilde{\kappa_{D}}^{2}}-\frac{\tilde{h}^{2}}{\tilde{\kappa_{D}}^{2}}\ln(\sqrt{x})+\ldots.$
We see that apart from the expected logarithmic divergence we get a finite
contribution which exactly yields upon multiplication with the appropriate
factor Eq. (1.1), cp. [3, Eq. (2.11)]. We will elaborate on the whole issue of
the 2D off-rate in a forthcoming publication [5].
### 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
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] M.E.H. Ismail. Ann. Probab., 5:582, 1977.
* [3] T. Prüstel and M. Meier-Schellersheim. arXiv:1112.4010v1 [math-ph], 2011.
* [4] T. Prüstel and M. Meier-Schellersheim. J. Chem. Phys., 137:054104, 2012.
* [5] T. Prüstel and M. Meier-Schellersheim. In preparation, 2012.
|
arxiv-papers
| 2012-10-04T00:02:33 |
2024-09-04T02:49:35.925500
|
{
"license": "Public Domain",
"authors": "Thorsten Prustel and Martin Meier-Schellersheim",
"submitter": "Thorsten Pr\\\"ustel",
"url": "https://arxiv.org/abs/1210.1265"
}
|
1210.1338
|
# No-go theorem and optimization of dynamical decoupling against noise with
soft cutoff
Zhen-Yu Wang Ren-Bao Liu rbliu@phy.cuhk.edu.hk Department of Physics and
Center for Quantum Coherence, The Chinese University of Hong Kong, Shatin, N.
T., Hong Kong, China
###### Abstract
We study the performance of dynamical decoupling in suppressing decoherence
caused by soft-cutoff Gaussian noise, using short-time expansion of the noise
correlations and numerical optimization. For the noise with soft cutoff at
high frequencies, there exists no dynamical decoupling scheme to eliminate the
decoherence to arbitrary orders of the short time, regardless of the timing or
pulse shaping of the control under the population conserving condition. We
formulate the equations for optimizing pulse sequences that minimizes
decoherence up to the highest possible order of the short time for the noise
correlations with odd power terms in the short-time expansion. In particular,
we show that the Carr-Purcell-Meiboom-Gill sequence is optimal in short-time
limit for the noise correlations with a linear order term in the time
expansion.
###### pacs:
03.67.Pp, 03.65.Yz, 03.67.Lx, 82.56.Jn
## I Introduction
Quantum information processing Nielsen and Chuang (2000) relies on the
coherence of quantum systems. Unavoidable interactions between a quantum
system and its environment (bath) introduce noise on the system and lead to
error evolutions (decoherence) of the quantum system. Various methods have
been proposed to combat the decoherence, including decoherence-free subspaces
Duan and Guo (1997); Zanardi (1998); Lidar et al. (1998), error-correction
codes Shor (1995); Steane (1996), and dynamical decoupling (DD) Viola and
Lloyd (1998); Ban (1998); Zanardi (1999); Viola et al. (1999); Yang et al.
(2011). In particular, the DD scheme uses rapid unitary control pulses acting
only on the systems to suppress the effects of the noise from the
environments. DD has the advantages of suppressing decoherence without
measurement, feedback, or redundant encoding Viola et al. (1999). DD
originated from the seminal spin echo experiment Hahn (1950), in which the
effect of a static random magnetic field (inhomogeneous broadening) is
canceled. And more complex DD pulse sequences, such as the Carr-Purcell-
Meiboom-Gill (CPMG) sequence Carr and Purcell (1954); Meiboom and Gill (1958),
were designed to prolong the spin coherence time Mehring (1983).
The early DD schemes only eliminate low-order errors, i.e., the errors of
quantum evolutions up to some low order in the Magnus expansion. By unitary
symmetrization procedure Zanardi (1999); Viola et al. (1999), DD cancels the
first order (i.e., leading order) errors. To eliminate errors to the second
order in short time, mirror-symmetric arrangement of two DD sequences can be
used Viola et al. (1999). The first explicit arbitrary $M$th order DD scheme,
which suppresses errors to $\mathcal{O}(T^{M+1})$ for short evolution time
$T$, is the concatenated DD (CDD) Khodjasteh and Lidar (2005, 2007) proposed
by Khodjasteh and Lidar. CDD sequences against pure dephasing were
investigated for electron spin qubits in realistic solid-state systems with
nuclear spins as baths Yao et al. (2007); Witzel and Das Sarma (2007); Zhang
et al. (2007). Experiments Peng et al. (2011); Álvarez et al. (2010);
Tyryshkin et al. ; Wang et al. (2012); Barthel et al. (2010) have tested the
performance of CDD. CDD works for generic quantum systems coupled to a finite
bath Santos and Viola (2008); Wang and Liu (2011a). However, since CDD uses
recursively constructed pulse sequences to suppress decoherence, the number of
pulses increases exponentially with the decoupling order. As pulse errors are
inevitably introduced in each control pulse in experiments, finding efficient
DD schemes with fewer control pulses is desirable. A remarkable advance is the
Uhrig DD (UDD) Uhrig (2007); Lee et al. (2008); Uhrig (2008); Yang and Liu
(2008). UDD is optimal in the short-time limit in the sense that it suppresses
the pure dephasing of a qubit coupled to a finite bath to the $M$th order
using only $M$ qubit flips. The performance bounds for UDD against pure
dephasing were established Uhrig and Lidar (2010). Shaped pulses Pasini et al.
(2008); Fauseweh et al. (2012) of finite amplitude can be incorporated into
UDD Pasini et al. (2008). Many recent experimental studies Biercuk et al.
(2009a); Du et al. (2009); Biercuk et al. (2009b); Uys et al. (2009); Jenista
et al. (2009); Álvarez et al. (2010); Barthel et al. (2010) demonstrated the
performance of UDD.
It is important to find efficient schemes to suppress general decoherence
(including pure dephasing and population relaxation). Yang and Liu extended
UDD to the suppression of population relaxation Yang and Liu (2008). This
inspired efficient ways to suppress the general decoherence of single qubits,
including concatenation of UDD sequences (CUDD) Uhrig (2009) and a much more
efficient one called quadratic DD (QDD) West et al. (2010); Wang and Liu
(2011a); Quiroz and Lidar (2011); Kuo and Lidar (2011); Jiang and Imambekov
(2011) discovered by West _et al_ West et al. (2010). Based on the proof in
Yang and Liu (2008), Mukhtar _et al_ generalized UDD to protect arbitrary
multilevel systems with full prior knowledge of the initial states Mukhtar et
al. (2010). One can actually preserve the coherence of arbitrary multi-qubit
systems by protecting a mutually orthogonal operation set (MOOS) Wang and Liu
(2011a). By nesting UDD sequences for protecting the elements in the MOOS, the
nested UDD (NUDD) Wang and Liu (2011a); Mukhtar et al. (2010); Jiang and
Imambekov (2011) requires only a polynomially increasing number of pulses in
the decoupling order. These universal DD schemes also work for analytically
time-dependent baths Wang and Liu (2011b).
The above-mentioned variations of UDD, however, rely on the finiteness of the
baths, i.e., the existence of hard high-frequency cutoff in the noise spectra.
Legitimate questions are: For quantum systems coupled to an infinite quantum
bath or affected by soft-cutoff noise, can any DD be designed to eliminate the
decoherence to arbitrary orders of precision in the short-time limit? And if
yes, how can such DD be designed? Such questions have been previously
addressed in some specific noise models. Comparing the efficiency of various
DD sequences in suppressing pure dephasing of a qubit due to classical noise,
Cywiński _et al_ observed that if the noise spectrum cutoff is not reached,
CPMG sequences Carr and Purcell (1954); Meiboom and Gill (1958) actually
performs better than CDD and UDD sequences Cywiński et al. (2008). With the
consideration of minimum pulse separations in physical systems, Viola _et al_
observed that low-order DD sequences provide better performance than high-
order DD when the rate of pulses is not faster than the correlation time of
the noise Hodgson et al. (2010); Khodjasteh et al. (2011). It was confirmed by
experiments that for ${}^{13}\text{C}$ spin qubits in a ${}^{1}\text{H}$ spin
bath of which the high frequency cutoff was not reached by the DD sequences,
CPMG outperforms UDD Ajoy et al. (2011). Also, Pasini and Uhrig derived the
equations for minimizing decoherence for power-law spectra, and found that the
numerically optimized sequences resemble CPMG Pasini and Uhrig (2010). Chen
and Liu proved that for telegraphlike noise the CPMG sequences are the most
efficient scheme in protecting the qubit coherence in the short-time limit and
the decoherence can be suppressed at most to the third order of short
evolution time by DD Chen and Liu (2010). These results suggest that for noise
with soft cutoff in the spectrum, there are certain constraints on the optimal
order and decoupling scaling of DD. Ref. Gordon et al. (2008) presented
numerical optimization of bounded-strength DD for specific noise spectra.
However, no conclusion has been drawn on the performance of DD with arbitrary
timing and shaping for the general cases of soft-cutoff noise.
In this paper, we address the general question of the performance of DD
against soft-cutoff noise based on the general modulation function induced by
arbitrary DD with bounded-strength or pulsed control. We show that for the
noise spectrum with a power-law asymptote at high frequencies, there exists no
modulation function to eliminate the decoherence to an arbitrary order of the
short time, regardless of the timing and shaping of the DD control under the
population conserving condition. Although the decohernce can be suppressed to
be arbitrarily small by DD with a sufficiently large number of pulses, the
existence of the largest achievable decoupling order shows that DD against
soft-cutoff noise does not have the order-by-order decoupling efficiency,
which is possible for hard-cutoff noise. Since for soft-cutoff noise the
decoherence cannot be eliminated at a certain order of short time, we derive a
set of equations to minimize the leading-order term in the short-time
expansion while eliminating the lower orders. These equations are numerically
solved for optimal solutions. In particular, for noise correlations with a
linear order term in time, we prove that the CPMG sequences are optimal. For
other noise correlations with odd-order terms, the minimum pulse interval of
the optimized sequences is larger than in UDD sequences. This feature is
important in realistic experiments when there is a minimum pulse switching
time Khodjasteh et al. (2011).
This paper is organized as follows: In Sec. II, we analyze the performance of
DD against soft-cutoff noise, and we give the condition under which
decoherence suppression to an arbitrary order of short-time scaling is
impossible. The relation between the high-frequency cutoff and the short-time
expansion of correlations is also discussed. In Sec. III, we derive the
equations for sequence optimization and obtain optimal DD for noise
correlations with odd-power expansion terms. Finally, the conclusions are
drawn in Sec. IV.
## II No-go theorem on dynamical decoupling against noise with soft cutoff
We consider the pure-dephasing Hamiltonian for a single spin (qubit)
$H=\frac{1}{2}\sigma_{z}[\omega_{a}+\beta(t)],$ (1)
where $\sigma_{z}=|+\rangle\langle+|-|-\rangle\langle-|$ is the Pauli operator
of the qubit, $\omega_{a}$ is the energy splitting of the qubit, and
$\beta(t)$ describes random noise with average $\overline{\beta(t)}=0$. Here
the over bar denotes averaging over the noise realizations. We assume that the
statistics of the noise fluctuations are Gaussian.
After a duration of free evolution time $T$, the noise induces between the two
states $|\pm\rangle$ a random phase shift $\int_{0}^{T}\beta(t)$ that destroys
the quantum coherence. We can suppress the decoherence by DD control on the
qubit. There is only one noise source $\beta(t)$ in the model Eq. (1), and to
suppress the decoherence we need to protect a MOOS which consists of a Pauli
operator $\sigma_{x}$ (more generally $\sigma_{x}\cos\phi+\sigma_{y}\sin\phi$
with $\phi$ being real) Wang and Liu (2011a). We will prove later that DD can
suppress the decoherence (i.e., the protection of the MOOS $\\{\sigma_{x}\\}$)
only to a certain order of short evolution time for noise correlations that
have odd-power expansion terms in time. We expect that the proof also applies
to other quantum systems (e.g., multi-qubit systems) when the noise
correlations have odd-power terms, since in those systems there are more noise
sources and more system operators (e.g., a MOOS consisting of $L>1$ Pauli
operators) should be protected.
When we apply a sequence of instantaneous unitary operations $\sigma_{x}$ at
the moments $T_{1}$, $T_{2}$, $\ldots$, $T_{N}$, the controlled evolution
operator reads
$\displaystyle U(T)$ $\displaystyle=$
$\displaystyle(\sigma_{x})^{N}U(T_{N+1},T_{N})\sigma_{x}U(T_{N},T_{N-1})\cdots$
(2) $\displaystyle\times\sigma_{x}U(T_{2},T_{1})\sigma_{x}U(T_{1},T_{0}),$
where $T_{0}=0$, $T_{N+1}=T$, and the free evolution operator
$U(T_{j+1},T_{j})=e^{-i\frac{\sigma_{z}}{2}\int_{T_{j}}^{T_{j+1}}\left[\omega_{a}+\beta(t)\right]dt}.$
(3)
Note that when $N$ is odd, we may apply an additional $\sigma_{x}$ pulse at
the end of the sequence for the identity evolution. Using
$\sigma_{x}U(T_{j+1},T_{j})\sigma_{x}=e^{-i\frac{\sigma_{z}}{2}\int_{T_{j}}^{T_{j+1}}\left[-\omega_{a}-\beta(t)\right]dt},$
(4)
we write the evolution operator as
$U(T)=e^{-i\frac{\sigma_{z}}{2}\int_{0}^{T}\omega_{a}F_{\pi}(t/T)dt}e^{-i\frac{\sigma_{z}}{2}\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt},$
(5)
where we have defined the modulation function for instantaneous $\pi$-pulse
sequences Uhrig (2007); Cywiński et al. (2008)
$F_{\pi}(t/T)=\begin{cases}(-1)^{j}&\text{for }t\in(T_{j},T_{j+1}]\\\
0&\text{for }t>T,\mbox{\text{ or }}t\leq 0\end{cases}.$ (6)
The DD control is parametrized by the relative pulse locations $T_{j}/T$. At
the moment $T$, the off-diagonal density matrix element of an ensemble is
$\rho_{\uparrow\downarrow}(T)=\rho_{\uparrow\downarrow}(0)e^{-i\int_{0}^{T}\omega_{a}F_{\pi}(t/T)dt}\overline{e^{-i\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt}}.$
(7)
The coherence is characterized by the ensemble-averaged phase factor
$W(T)\equiv\overline{e^{-i\int_{0}^{T}\beta(t)F_{\pi}(t/T)dt}}.$ (8)
For Gaussian noise, the ensemble-averaged phase factor $W(T)$ is determined by
the two-point correlation function $\overline{\beta(t_{1})\beta(t_{2})}$ and
$W(T)$ becomes Anderson (1954); Kubo (1954); Cywiński et al. (2008)
$W(T)=e^{-\chi_{\pi}(T)},$ (9)
where the phase correlation for instantaneous pulses
$\chi_{\pi}(T)=\frac{1}{2}\int_{0}^{T}dt_{1}\int_{0}^{T}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F_{\pi}\left(\frac{t_{1}}{T}\right)F_{\pi}\left(\frac{t_{2}}{T}\right).$
(10)
can be written as the overlap between the noise power spectrum and a filter
function determined by the Fourier transform of the modulation function
Cywiński et al. (2008).
Under DD control, the qubit is flipped at different moments, and the random
field $\beta(t)$ is modulated by the modulation function $F_{\pi}(t/T)$. For
multilevel systems, the modulation functions resulting from instantaneous
$\pi$-pulse sequences may have values not restricted to $\\{\pm 1\\}$ for
$t\in(0,T]$ (see Appendix A). In Ref. Uhrig and Pasini (2010), it is shown
that for DD composed of specially engineered finite-duration pulses, the
effective modulation functions can take values from $\\{+1,-1,0\\}$
alternatively. We may also encounter effective modulation functions which are
triangle wave functions during the time of system evolution Wang and Liu . For
a more general analysis, we assume that the control conserves the populations
and the phase modulation function $F_{\pi}(t/T)$ has a general form as
$\displaystyle F\left(\frac{t}{T}\right)$ $\displaystyle=\begin{cases}\text{a
bounded function}&\text{for }t\in(0,T]\\\ 0&\text{otherwise}\end{cases},$ (11)
which has a finite number of discontinuities. The more general phase
correlation considered in this paper reads
$\displaystyle\chi(T)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int_{0}^{T}dt_{1}\int_{0}^{T}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F^{*}\left(\frac{t_{1}}{T}\right)F\left(\frac{t_{2}}{T}\right)$
(12a) $\displaystyle=$
$\displaystyle\Re\int_{0}^{T}dt_{1}\int_{0}^{t_{1}}dt_{2}\overline{\beta(t_{1})\beta(t_{2})}F^{*}\left(\frac{t_{1}}{T}\right)F\left(\frac{t_{2}}{T}\right),$
(12b)
where we have used
$\overline{\beta(t_{1})\beta(t_{2})}=\overline{\beta(t_{2})\beta(t_{1})}$ to
derive Eq. (12b) [For quantum noise, this may not be true. But in Eq. (12a),
$t_{1}$ and $t_{2}$ can be exchanged without changing the integration. So the
noise correlation can always be symmetrized]. It was shown that under suitable
approximation, Eq. (12) with a complex $F(t/T)=e^{-i\int_{0}^{t/T}V(s)ds}$
describes the average dephasing of a qubit under bounded-strength control with
the amplitude $V(s)$ Gordon et al. (2008, 2006); Gordon and Kurizki (2007);
Gordon et al. (2007). In Ref. Gordon et al. (2008) based on minimization of
$\chi(T)$, some optimal control fields $V(s)$ were obtained for some specific
noise spectra. Therefore we analyse and minimize $\chi(T)$ given by Eq. (12)
for DD design. For the special case of instantaneous $\pi$-pulse sequences for
the dephasing of a qubit, $F(t/T)=F_{\pi}(t/T)$ and $\chi(T)=\chi_{\pi}(T)$.
We assume the noise is stationary, i.e., of time translation symmetry,
$\overline{\beta(t_{1})\beta(t_{2})}=\overline{\beta(t_{1}-t_{2})\beta(0)}$.
Another symmetry is $\overline{\beta(t)\beta(0)}=\overline{\beta(0)\beta(t)}$.
These symmetries indicate that the noise correlation is an even function of
time, i.e.,
$\overline{\beta(t)\beta(0)}\equiv C_{\text{corr}}(t)=C_{\text{corr}}(|t|).$
(13)
The noise correlation can be transformed from the noise power spectrum
$S(\omega)$ as
$\overline{\beta(t)\beta(0)}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}S(\omega)e^{-i\omega
t}.$ (14)
Note that both $\overline{\beta(t)\beta(0)}$ and $S(\omega)$ are real even
functions.
The general filter function is defined as the Fourier transform of the general
modulation function,
$\displaystyle\tilde{f}(\omega T)$
$\displaystyle\equiv\frac{1}{T}\int_{-\infty}^{\infty}F\left(\frac{t}{T}\right)e^{i\omega
t}dt$ (15a) $\displaystyle=\int_{0}^{1}F(s)e^{i\omega Ts}ds,$ (15b)
which has the power expansion
$\displaystyle\tilde{f}(\omega T)=\sum_{m=0}^{\infty}\frac{(i\omega
T)^{m}}{m!}\lambda_{m},$ (16a)
$\displaystyle\lambda_{m}\equiv\int_{0}^{1}F(s)s^{m}ds.$ (16b)
Eq. (15b) shows that $\tilde{f}(\omega T)$ is bounded by $|\tilde{f}(\omega
T)|\leq\int_{0}^{1}|F(s)|ds.$
As $F(s)$ has a finite number of discontinuities, we use integration by parts
and get
$\tilde{f}(u)=\frac{1}{iu}\sum_{j}\left[\left.F(s)e^{ius}\right|_{s_{j}}^{s_{j+1}}-I_{j}(u)\right],$
(17)
where $s_{j}$ are the discontinuous points and
$|I_{j}(u)|=|\int_{s_{j}}^{s_{j+1}}F^{\prime}(s)e^{ius}ds|\leq\int_{s_{j}}^{s_{j+1}}|F^{\prime}(s)|ds\equiv\tilde{I}_{j}$
is finite. Therefore we have
$|\tilde{f}(u)|\leq a_{\tilde{f}}/u,$ (18)
where the coefficient
$a_{\tilde{f}}=\sum_{j}\left[|F(s_{j})|+|F(s_{j+1})|+\tilde{I}_{j}\right]$ is
bounded.
For a sequence of $N$ instantaneous $\pi$ pulses, $\lambda_{m}$ reads
$\lambda_{m}^{(\pi)}=\sum_{j=0}^{N}\frac{(-1)^{j}}{(m+1)}\left[\left(\frac{T_{j+1}}{T}\right)^{m+1}-\left(\frac{T_{j}}{T}\right)^{m+1}\right].$
(19)
Using Eqs. (14) and (15b), Eq. (12a) can be written as the overlap of the
noise spectrum and filter function
$\displaystyle\chi(T)$
$\displaystyle=T^{2}\int_{0}^{\infty}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2}.$ (20)
It should be stressed that in Eq. (20) the filter function $\tilde{f}(\omega
T)$ is general and not limited to the case of instantaneous pulse sequences.
### II.1 Scaling of decoupling orders
We separate the noise spectrum into two parts by a frequency $\Omega$. As the
noise spectrum $S(\omega)$ in Eq. (20) induces decoherence linearly,
$\chi(T)=\chi_{[0,\Omega]}(T)+\chi_{[\Omega,\infty]}(T),$ (21)
where
$\displaystyle\chi_{[0,\Omega]}(T)=T^{2}\int_{0}^{\Omega}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2},$ (22a)
$\displaystyle\chi_{[\Omega,\infty]}(T)=T^{2}\int_{\Omega}^{\infty}\frac{d\omega}{\pi}S(\omega)|\tilde{f}(\omega
T)|^{2}.$ (22b)
$\chi_{[0,\Omega]}(T)$ and $\chi_{[\Omega,\infty]}(T)$ account for the effects
of the low- and high-frequency noise, respectively. Both
$\chi_{[0,\Omega]}(T)$ and $\chi_{[\Omega,\infty]}(T)$ cause decoherence as
$S(\omega)\geq 0$.
#### II.1.1 Effects of low-frequency noise
For the spectrum $S(\omega)=\mathcal{O}(1/\omega^{P})$ with $P<1$ when
$\omega\rightarrow 0$, Eq. (14) gives the noise correlation of the noise with
the frequencies $\omega<\Omega$,
$\displaystyle\overline{\beta(t)\beta(0)}_{[0,\Omega]}$
$\displaystyle=\int_{0}^{\Omega}\frac{d\omega}{\pi}S(\omega)\cos\left(\omega
t\right)$ (23a) $\displaystyle=\sum_{m=0}^{\infty}C_{2m}\Omega^{2m+1}t^{2m}.$
(23b)
Here the coefficients
$C_{2m}=(-1)^{2m}\int_{0}^{1}\frac{du}{\pi}S(u\Omega)\frac{u^{2m}}{(2m)!},$
(24)
which depend on the noise spectrum $S(\omega)$ and $\Omega$, converge at low
frequencies $\omega\rightarrow 0$.
Eq. (12b) gives
$\chi_{[0,\Omega]}(T)=\sum_{m=0}^{\infty}C_{2m}\phi_{2m}\Omega^{2m+1}T^{2m+2},$
(25)
where the decoherence functions
$\phi_{k}\equiv\Re\int_{0}^{1}ds_{1}\int_{0}^{s_{1}}ds_{2}(s_{1}-s_{2})^{k}F^{*}\left(s_{1}\right)F\left(s_{2}\right),$
(26)
are modified by the modulation function of DD. The even-order decoherence
functions $\phi_{2k}$ control the effects of the low-frequency noise.
Therefore if the modulation function $F(t/T)$ is designed to make
$\\{\phi_{2m}=0\\}_{m=0}^{M-1}\equiv\\{\phi_{0}=\phi_{2}=\cdots=\phi_{2M-2}=0\\},$
(27)
the decoherence from low-frequency noise is eliminated to
$\chi_{[0,\Omega]}(T)=\mathcal{O}(T^{2M+2})$ (the prefactor of the scaling
depends on the noise spectrum and $\Omega$). Note that
$e^{-i\int_{0}^{T}\omega_{a}F(t/T)dt}=1$ in Eq. (7) when $\phi_{0}=0$.
In Appendix B, we simplify the even-order $\phi_{2m}$ as
$\displaystyle\phi_{2m}$
$\displaystyle=\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!}.$
(28)
From Eq. (28), we find that the following two sets of equations are equivalent
$\\{\phi_{2m}=0\\}_{m=0}^{M-1}\Leftrightarrow\\{\lambda_{m}=0\\}_{m=0}^{M-1}.$
(29)
For instantaneous $\pi$-pulse sequences, the optimal solution of the equation
set $\\{\lambda_{m}=\lambda_{m}^{(\pi)}=0\\}_{m=0}^{N-1}$ is
$T_{j}^{\text{UDD}}=T\sin^{2}\left[\frac{\pi
j}{2N+2}\right],\>(j=1,2,\ldots,N),$ (30)
which is the timing of UDD sequences Uhrig (2007). The conditions Eqs. (29)
and (16b) are more general than the one that leads to UDD and may lead to more
general designs of optimal DD.
For the power-law spectrum $S(\omega)\approx\alpha/\omega^{P}$ with $P\geq 1$
at low frequencies,
$\chi_{[0,\Omega]}(T)=T^{2}\int_{0}^{\Omega}\frac{d\omega}{\pi}\frac{\alpha}{\omega^{P}}|\tilde{f}(\omega
T)|^{2}$ in general diverges. For the modulation function $F_{\pi}(t/T)$, it
was shown that the divergence of the integral can be eliminated by high-order
DD sequences Pasini and Uhrig (2010). For the general modulation function
$F(t/T)$ under the conditions $\\{\lambda_{m}=0\\}_{m=0}^{M-1}$, using Eq.
(16a) we have $|\tilde{f}(u)|^{2}\sim\mathcal{O}(u^{2M})$ and
$\chi_{[0,\Omega]}(T)=\mathcal{O}(\Omega^{2M+1-P}T^{2M+2})$ when $M>(P-1)/2$.
When the noise with the frequency $\omega>\Omega$ is negligible (i.e., a hard
cutoff frequency $\omega_{c}=\Omega$ in the noise spectrum and
$\chi_{[\Omega,\infty]}\approx 0$), the decoherence can be suppressed order by
order, and $\chi(T)\approx\chi_{[0,\Omega]}(T)$ has the scaling
$\chi(T)\approx\mathcal{O}(T^{2M+2})$ in short-time limit.
#### II.1.2 Effects of high-frequency noise
Consider the noise spectrum with a power-law asymptote at high frequencies
(i.e., a soft high-frequency cutoff),
$S(\omega)\approx\frac{\alpha}{\omega^{P}},\text{ for
}\Omega\leq\omega\leq\Omega_{\infty},$ (31)
where $\Omega_{\infty}\gg 1/T$ and $\Omega\lesssim 1/T$. We assume that the
decay of the noise at the frequencies $\omega>\Omega_{\infty}$ is not slower
than $\alpha/\omega^{P}$ and the contribution is negligible. We set
$\Omega_{\infty}=\infty$. The high-frequency contribution Eq. (22b) reads
$\chi_{[\Omega,\infty]}(T)\approx\chi_{P}(T)\equiv T^{P+1}\int_{\Omega
T}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}.$ (32)
Since $\tilde{f}(u)=\mathcal{O}\left(1/u\right)$ when $u\rightarrow\infty$
[Eq. (18)], $\chi_{[\Omega,\infty]}(T)$ converges when $P>-1$.
Using $|\tilde{f}(0)|^{2}=|\lambda_{0}|^{2}$, we get $\lim_{T\rightarrow
0}\chi_{[\Omega,\infty]}(T)=0$ even though the integral $\lim_{T\rightarrow
0}\int_{\Omega
T}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}$ may diverge
when $T\rightarrow 0$. We have shown in Sec. II.1.1 that under the conditions
$\\{\lambda_{m}=0\\}_{m=0}^{M-1}$ with $2M+1-P>0$, $\int_{0}^{\Omega
T}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}=\mathcal{O}(\Omega^{2M+1-P}T^{2M+1-P})$.
Therefore when $M>(P-1)/2$, in Eq. (32) we extend the bounds of integration to
$(0,\infty)$,
$\displaystyle\chi_{[\Omega,\infty]}(T)$
$\displaystyle=C_{P}^{\text{soft}}T^{P+1}-\mathcal{O}(T^{2M+1-P})=\mathcal{O}(T^{P+1}),$
(33)
where
$C_{P}^{\text{soft}}=\int_{0}^{\infty}\frac{du}{\pi}\frac{\alpha}{u^{P}}|\tilde{f}(u)|^{2}$
is bounded when $\\{\lambda_{m}=0\\}_{m=0}^{M-1}$ with $2M+1-P>0$ and $P>-1$.
The scaling $\chi_{[\Omega,\infty]}(T)=\mathcal{O}(T^{P+1})$ is the largest
order of decoupling that can be achieved for the noise with soft cutoff. We
have the following theorem.
###### Theorem 1.
For the noise spectrum with a power-law asymptote $\alpha/\omega^{P}$ with
$P>-1$ at high frequencies, the largest achievable decoupling order of DD with
a general modulation function given by Eq. (11) is
$\chi(T)=\mathcal{O}(T^{P+1})$ in short-time limit $T\rightarrow 0$.
This theorem holds for arbitrary non-zero modulation function $F(t/T)$, and it
shows that one cannot suppress the decoherence to arbitrary order when the
noise has a soft cutoff in the spectrum.
For example, the $1/f$ noise and the Lorentz spectrum
$\alpha/(\Omega^{2}+\omega^{2})$ correspond to the cases of $P=1$ and $P=2$,
respectively. After eliminating the effect of low-frequency noise by high-
order DD which satisfy $\lambda_{0}=0$, we achieve the largest decoupling
order $\chi_{P=1}(T)=\mathcal{O}\left(T^{2}\right)$ and
$\chi_{P=2}(T)=\mathcal{O}\left(T^{3}\right)$.
Note that Theorem 1 applies to the order of short-time scaling and it does not
mean that DD can not protect the coherence to arbitrarily high precision. The
decoherence can be suppressed further by reducing the prefactor
$C_{P}^{\text{soft}}$, i.e., the the overlap of the filter function
$|\tilde{f}(\omega T)|^{2}$ and noise spectrum $S(\omega)$. We will discuss
the optimization of pulse sequences based by minimizing the overlap in Sec.
III.
The result in Ref. Chen and Liu (2010) that DD can suppress decoherence at
most to the third order of short evolution time for telegraphlike noise is
general for noise with arbitrary statistics. In deriving Theorem 1, we have
made the assumption that the statistics of the noise are Gaussian and the
decoherence is determined by the two-point noise correlation. It would be
interesting to generalize the theorem to non-Gaussian noise.
The perturbative regime $T\lesssim 1/\Omega$ is limited by the technology of
experiments. For power-law noise (e.g., $1/f$ noise), $\Omega\rightarrow 0$
and the conclusion applies to arbitrary duration of $T$ when $M>(P-1)/2$.
### II.2 Noise correlation expansion and high-frequency cutoff
The correlation in the short-time limit, which is due to high-frequency noise,
can be written as
$C_{\Omega,P}(t)=\Re\int_{\Omega}^{\infty}\frac{d\omega}{\pi}\frac{\alpha}{\omega^{P}}e^{-i\omega
t}.$ (34)
As $C_{\Omega,P}(t)$ is an even function, we just calculate the integral for
the case of $t>0$. For $P>1$ and $t>0$, we have
$\displaystyle C_{\Omega,P}(t)$
$\displaystyle=\frac{\Re}{\pi}\left[\int_{c_{\Omega}}+\int_{c_{i}}+\int_{c_{\infty}}\right]\frac{\alpha}{z^{P}}e^{izt}dz,$
where the paths $c_{\Omega}$, $c_{i}$, and $c_{\infty}$ are shown in Fig. (1).
Figure 1: The paths for the integral of $\alpha/z^{P}$.
Since the maximum of $\alpha/z^{P}\rightarrow 0$ as $|z|\rightarrow\infty$ in
the upper half-plane, the contribution
$\int_{c_{\infty}}\frac{\alpha}{z^{P}}e^{izt}dz=0$. The contribution
$\frac{\Re}{\pi}\int_{c_{i}}\frac{\alpha}{z^{P}}e^{izt}dz=\frac{\alpha}{\pi}\Re\int_{\Omega}^{\infty}i^{1-P}y^{-P}e^{-yt}dy$
(35)
vanishes for even $P$. And
$\displaystyle\frac{\Re}{\pi}\int_{c_{\Omega}}\frac{\alpha}{z^{P}}e^{izt}dz$
$\displaystyle=\frac{\Re}{\pi}\int_{0}^{\frac{\pi}{2}}\frac{\alpha}{\Omega^{P}e^{iP\theta}}e^{i\Omega
te^{i\theta}}(i\Omega)e^{i\theta}d\theta$
$\displaystyle=I_{c_{\Omega}}^{(1)}+I_{c_{\Omega}}^{(2)},$ (36)
where
$\displaystyle I_{c_{\Omega}}^{(1)}$
$\displaystyle=\frac{\Re}{\pi}\sum_{r=0,r\neq
P-1}^{\infty}\frac{\alpha}{r!}\frac{\Omega^{1-P}(i\Omega
t)^{r}}{(r+1-P)}\left(i^{r+1-P}-1\right),$ (37a) $\displaystyle
I_{c_{\Omega}}^{(2)}$ $\displaystyle=\begin{cases}\frac{1}{2}\alpha\Re
i^{P}t^{P-1}/(P-1)!,&\text{if }r\text{ is an integer,}\\\
0,&\text{otherwise.}\end{cases}$ (37b)
For even $P$, $I_{c_{\Omega}}^{(1)}$ is an expansion of even powers of $t$
$I_{c_{\Omega}}^{(1)}=\sum_{k=0}^{\infty}C_{2k}t^{2k},C_{2k}=\frac{\alpha(-1)^{k+1}\Omega^{1-P+2k}}{\pi(2k)!(2k+1-P)}\neq
0,$ (38)
and there is only one odd-order expansion term, which is
$I_{c_{\Omega}}^{(2)}=\frac{(-1)^{P/2}\alpha}{2(P-1)!}|t|^{P-1}.$ (39)
The existence of an odd-order term means that the correlation function is non-
analytical, which indicates that the noise source cannot be a finite quantum
bath. The noise with non-analytical correlation functions must come from the
fluctuations of an infinite bath, since otherwise the unitary evolution of a
finite quantum system will always lead to analytical correlation functions.
For example, the noise correlation
$\overline{\beta(t)\beta(0)}=e^{-|t|/t_{c}}$ has odd-order terms in the time
expansion, and the noise has a Lorentz spectrum, which has a power-law
decrease at high frequencies. This kind of noise can be caused by Markovian
(or instantaneous) collisions in the bath Berman and Brewer (1985).
As an example, we consider the following noise spectrum,
$S_{2K}^{\prime}(\omega)\equiv\frac{\alpha}{\Omega_{c}^{2K}+\omega^{2K}},$
(40)
for $K\in\\{1,2,\ldots\\}$. $S_{2K}^{\prime}(\omega)\approx\alpha/\omega^{2K}$
when $\omega\gg\Omega_{c}$. For example, the measured ambient noise for ions
in a Penning trap has an approximate $1/\omega^{4}$ spectrum at high
frequencies and a flat dependence at low frequencies Biercuk et al. (2009a),
which approximately corresponds to the noise spectrum Eq. (40) with $K=2$. The
corresponding correlation function of Eq. (40) is obtained by the inverse
transform
$\overline{\beta(t)\beta(0)}_{2K}=\int_{0}^{\infty}\frac{d\omega}{\pi}\frac{\alpha
e^{-i\omega t}}{\Omega_{c}^{2K}+\omega^{2K}}.$ (41)
Using the residue theorem, we have
$\overline{\beta(t)\beta(0)}_{2K}=\frac{i\alpha}{2K}\Omega_{c}^{1-2K}\sum_{n=0}^{K-1}\frac{\exp[-ie^{i\frac{\pi}{2K}(2n+1)}|\Omega_{c}t|]}{\exp[i\frac{\pi}{2K}(2n+1)(2K-1)]}.$
(42)
Expanding the right-hand side in powers of $t$, we get
$\displaystyle\overline{\beta(t)\beta(0)}_{2K}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(-i|\Omega_{c}t|)^{m}}{m!}\frac{e^{i\frac{(m-2K+1)\pi}{2K}}[(-1)^{m}+1]}{e^{i(m+1)\pi/K}-1}$
$\displaystyle\times\frac{-i\alpha}{2K}\Omega_{c}^{1-2K}.$ (43)
In Eq. (43), the leading odd-order term of the time expansion is
$\frac{(-1)^{K}\alpha}{2(2K-1)!}|t|^{2K-1}$, as predicted by Eq. (39).
For simplicity, let us consider the noise correlations that have the power
expansion
$\overline{\beta(t)\beta(0)}\equiv
C_{\text{corr}}(t)=\sum_{k=0}^{\infty}C_{k}\left|t\right|^{k},$ (44)
where the expansion coefficients
$C_{k}\equiv\left.\frac{1}{k!}\frac{d^{k}C_{\text{corr}}(t)}{dt^{k}}\right|_{t\rightarrow
0^{+}}$ (45)
are finite real numbers with
$\displaystyle C_{2k-1}$ $\displaystyle=0,\text{ for }k<K,$ (46a)
$\displaystyle C_{2K-1}$ $\displaystyle\neq 0.$ (46b)
The leading odd-order term in the short-time expansion of $C_{\text{corr}}(t)$
is $C_{2K-1}|t|^{2K-1}.$ An example is $C_{\text{corr}}(t)=e^{-|t|^{3}}$ with
$C_{1}=0$ and $C_{3}=-1$. We assume that the noise correlation decreases
smoothly at long correlation times, that is,
$\lim_{t\rightarrow\infty}\frac{d^{k}}{dt^{k}}C_{\text{corr}}(t)=0,\text{ for
}k=0,1,\ldots,$ (47)
and
$\displaystyle I_{L}(\omega)$ $\displaystyle\equiv\int_{0}^{\infty}e^{i\omega
t}\frac{d^{L}}{dt^{L}}C_{\text{corr}}(t)dt,$ (48)
vanishes at large $\omega$ for $L=0,1,\ldots$ Sun and Liu (1996). For the
noise correlations that decay in the correlation time smoothly without fast
oscillation, $I_{L}(\omega)$ vanishes at large frequency $\omega$. For
example, the noise correlation $e^{-|t|}$ has
$I_{L}(\omega)=i(-1)^{L}/(\omega+i)\rightarrow 0$ when
$\omega\rightarrow\infty$.
We consider the high frequency behavior of the noise spectrum,
$S_{2K}(\omega)=\int_{-\infty}^{\infty}C_{\text{corr}}(t)e^{i\omega t}dt.$
Integration by parts $L\geq(2K+1)$ times gives
$S_{2K}(\omega)=2\Re\left[\sum_{r=1}^{L}\frac{(r-1)!}{(-i\omega)^{r}}C_{r-1}+\frac{I_{L}(\omega)}{(-i\omega)^{L}}\right],$
(49)
where we have used Eqs. (45) and (47).
Using Eqs. (46) and (48), we obtain for large $\omega$,
$S_{2K}(\omega)\approx
2\frac{(2K-1)!}{(i\omega)^{2K}}C_{2K-1}+\mathcal{O}\left(\frac{1}{\omega^{2K+1}}\right),$
(50)
which is a power-law decrease at high frequencies.
When the noise correlation expansion contains only even-order terms, from Eq.
(49) we have the noise spectrum $S_{\text{even}}(\omega)=2\Re
I_{L}(\omega)/(-i\omega)^{L}$ for an arbitrarily large $L$. From the
assumption $\lim_{\omega\rightarrow\infty}I_{L}(\omega)=0$, we have
$\lim_{\omega\rightarrow\infty}S_{\text{even}}(\omega)\omega^{L}=0$ for an
arbitrarily large $L$ and therefore the noise spectrum has a hard high-
frequency cutoff. One example is the correlation function $e^{-t^{2}}$, which
has the noise spectrum of exponential form $\sim\exp(-\frac{\omega^{2}}{4})$,
and obviously the UDD sequence can suppress the noise effect order by order.
The large $\omega$ behavior of other correlation functions of the form
$\exp(-\sum_{j=1}^{p}\alpha_{2j}t^{2j})$ can be calculated by the saddle point
integration method, which gives a result of an exponential decrease at high
frequencies (i.e., hard cutoff). For example, when $\omega$ is very large,
$\int_{-\infty}^{\infty}e^{-t^{4}}e^{i\omega
t}dt\simeq\frac{1}{2}\Im\sqrt{\frac{2\pi}{a(\omega)}}e^{g(\omega)}$, where
$g(\omega)=3(\frac{\omega}{4})^{\frac{4}{3}}e^{-i2\pi/3}$ and
$a(\omega)=12(\frac{\omega}{4})^{2/3}e^{i2\pi/3}$.
## III SEQUENCE OPTIMIZATION IN SHORT-TIME LIMIT
In this section, we optimize DD for the noise correlations that have the power
expansion given by Eq. (44),
$\overline{\beta(t)\beta(0)}\equiv\sum_{k=0}^{\infty}C_{k}\left|t\right|^{k}$.
The performance of DD in the short-time limit is directly derived from the
time-domain expansion. The time-domain expansion of noise correlations has the
advantage to avoid the divergence of the decoherence $\chi(T)$. Using the
expansion Eq. (44), we write
$\chi(T)=\sum_{k=0}^{\infty}C_{k}\phi_{k}T^{k+2},$ (51)
where the decoherence functions $\phi_{k}$ is given by Eq. (26). It seems that
we can find DD schemes to suppress the decoherence to an arbitrary order
$\chi(T)=\mathcal{O}(T^{M+2})$ by solving the equations $\phi_{k}=0$ with
$C_{k}\neq 0$ for $k<M$. However, we have shown in Sec. II.1.2 by Theorem 1
that for soft-cutoff noise, there is a largest decoupling order.
The even-order functions $\phi_{2m}$ are given by Eq. (28). Using Eq. (82), we
have the odd-order functions $\phi_{2M-1}$,
$\displaystyle\phi_{2M-1}=-\frac{(2M-1)!}{2}\sum_{r=0}^{2M-1}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2M-1-r}}{(2M-1-r)!}$
$\displaystyle+(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2M}}\right.$
$\displaystyle\left.-\sum_{r=0}^{2M-1}\sum_{n=0}^{2M-1-r}\frac{(-1)^{r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$
(52)
The condition Eq. (29) gives
$\displaystyle\phi_{2M-1}=(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2M}}\right.$
$\displaystyle\left.-\sum_{r=M}^{2M-1}\sum_{n=M}^{2M-1-r}\frac{(-1)^{r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$
(53)
Notice in the summation $2M-1-r<M$. We obtain
$\phi_{2M-1}=(-1)^{M}(2M-1)!\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{|\tilde{f}(\omega)|^{2}}{\omega^{2M}}.$
(54)
In Eq. (54), the integrand $|\tilde{f}(\omega)|^{2}/\omega^{2M}\geq 0$ and it
cannot vanish for all $\omega$ from $-\infty$ to $\infty$. Thus we have the
following theorem.
###### Theorem 2.
There is no non-zero modulation function $F(t/T)$ to eliminate the errors so
that the equations $\\{\phi_{2m}=0\\}_{m=0}^{M-1}$ and $\phi_{2M-1}=0$ satisfy
simultaneously.
For example, for the noise with the correlation function $e^{-|t/t_{c}|}$, one
cannot simultaneously eliminate the two leading decoherence terms
$C_{0}\phi_{0}$ and $C_{1}\phi_{1}$, and the error induced by the noise is at
least $\mathcal{O}(T^{3})$. The result is consistent with Theorem 1, since the
noise correlation $e^{-|t/t_{c}|}$ implies a spectrum with a soft cutoff at
high frequencies $S(\omega)=\frac{t_{c}}{1+(\omega t_{c})}$.
### III.1 Sequence optimization
In this paper, we optimize the DD performance in the short-time limit. As
indicated in Eq. (54), a smaller $\tilde{f}(\omega T)$ at low frequencies will
give a smaller $\phi_{2M-1}$. Here we focus on DD with ideal instantaneous
$\pi$ pulses. We use more pulses to construct a more efficient modulation
function $F(t/T)=F_{\pi}(t/T)$ to minimize $\phi_{2M-1}$, with the conditions
$\\{\phi_{2m}=0\\}_{m=0}^{M-1}$ [Eq. (29)]. Using the method of Lagrange
multipliers, we need to solve a set of nonlinear equations as
$\displaystyle\nabla_{\\{y,T\\}}G_{M}=0,$ (55a) $\displaystyle
G_{M}\equiv\sum_{j=0}^{M-1}y_{j}\lambda_{j}^{(\pi)}+\phi_{2M-1},$ (55b)
$\displaystyle\nabla_{\\{y,T\\}}\equiv(\frac{\partial}{\partial
T_{1}},...,\frac{\partial}{\partial T_{N}},\frac{\partial}{\partial
y_{0}},...,\frac{\partial}{\partial y_{M-1}}).$ (55c)
The introduced variables $\\{y_{j}\\}$ are the Lagrange multipliers. Note that
the sequence optimization in Ref. Pasini and Uhrig (2010) also used the
constraints $\\{\lambda_{m}^{(\pi)}=0\\}$, but the constraints were used there
to guarantee the convergence of the calculation of $\chi(T)$. Here, the
constraints eliminate the lowest orders of errors
($\\{\phi_{2m}=0\\}_{m=0}^{M-1}$) [see Eq. (29)] in short-time limit. In
particular, the decoherence from inhomogeneous broadening is eliminated when
$\phi_{0}=0$.
We calculate Eq. (26) by separating the domain of integration according to the
value of $F_{\pi}(t_{1}/T)F_{\pi}(t_{2}/T)$. For $k\geq 0$, we obtain
$\displaystyle\phi_{k}$
$\displaystyle=\frac{-1}{T^{k+2}(k+1)(k+2)}\left[4\sum_{j=2}^{N}\sum_{i=1}^{j-1}(T_{j}-T_{i})^{k+2}(-1)^{i+j}\right.$
$\displaystyle+(T_{N+1}-T_{0})^{k+2}(-1)^{N+1}+2\sum_{j=1}^{N}(T_{j}-T_{0})^{k+2}(-1)^{j}$
$\displaystyle\left.+2\sum_{i=1}^{N}(T_{N+1}-T_{i})^{k+2}(-1)^{N+1+i}\right].$
(56)
Then we have
$\displaystyle\frac{\partial\phi_{2M-1}}{\partial(T_{k}/T)}=\frac{(-1)^{k}}{T^{2M}M}\left[2\sum_{j=k+1}^{N}(T_{j}-T_{k})^{2M}(-1)^{j}-(T_{k}-T_{0})^{2M}\right.$
$\displaystyle\left.+(T_{N+1}-T_{k})^{2M}(-1)^{N+1}-2\sum_{j=1}^{k-1}(T_{k}-T_{j})^{2M}(-1)^{j}\right].$
(57)
For the special case of $M=1$, $G_{1}=y_{0}\lambda_{0}^{(\pi)}+\phi_{1}$, we
find that the CPMG sequences are solutions to Eq. (55). The timing of an
$N$-pulse CPMG sequence reads
$T_{j}^{\text{CPMG}}=\frac{2j-1}{2N}T,\text{ for }j=1,...,N.$ (58)
The CPMG sequences obviously satisfy the constraint $\lambda_{0}^{(\pi)}=0$
[see Eq. (19)], which is the so-called echo condition that eliminates the
effect of static inhomogeneous broadening. Eqs. (57) and (19) give
$\displaystyle\left.\frac{\partial\phi_{1}}{\partial(T_{k}/T)}\right|_{\text{CPMG}}=(-1)^{k+1}\frac{1}{4N^{2}}\left[1+(-1)^{N}\right],$
(59a)
$\displaystyle\left.y_{0}\frac{\partial\lambda_{0}^{(\pi)}}{\partial(T_{k}/T)}\right|_{\text{CPMG}}=2(-1)^{k+1}y_{0}.$
(59b)
Thus the CPMG sequences also satisfy Eq. (55) with
$y_{0}=-\frac{1}{8N^{2}}\left[1+(-1)^{N}\right]$, so they are at least the
locally optimal pulse sequences. It has been proved that CPMG sequences are
the most efficient pulse sequences in protecting the qubit coherence against
telegraph-like noise in the short-time limit Chen and Liu (2010). With
numerical evidence, we conjecture that it is also true that the CPMG sequences
are globally optimal in the short-time limit when the time expansion of the
noise correlation function has the two leading terms $C_{0}$ and $C_{1}|t|$.
For other cases of minimizing $\phi_{2M-1}$ with the condition
$\\{\lambda_{m}^{(\pi)}=0\\}_{m=0}^{M-1}$, one can see that the short-time
optimized DD (ODD) coincides with UDD for pulse number $N\leq M$. And as $N$
increases, the ODD sequences gradually approach the CPMG sequences. For
example, ODD for the noise correlation
$\overline{\beta(t)\beta(0)}=C_{0}+C_{2}t^{2}+C_{3}|t|^{3}+\mathcal{O}(t^{4}),$
(60)
is shown in Fig. 2 in comparison with UDD and CPMG. The ODD sequences resemble
the CPMG sequences when $N$ is large.
Figure 2: (Color online). Comparison of the ODD, UDD and CPMG sequences for
different pulse number $N$. Squares (blue), triangles (green), and circles
(red) correspond to UDD, CPMG, and ODD. The ODD sequences are optimized to
minimize $\phi_{3}$ under the constraints $\phi_{0}=\phi_{2}=0$.
We show in Fig. 3(a) the performance of three DD schemes against the noise
described by Eq. (60). A comparison is also shown in Fig. 3(b) by considering
a hard high-frequency cutoff $\omega_{c}=40$. In Fig. 3(a), we can see that
ODD sequences give better performance than UDD and CPMG sequences. These ODD
sequences are optimal for a wide range of noise which has the noise
correlation given by Eq. (60). When we introduce a high-frequency cutoff in
the noise spectrum, as the case in Fig. 3(b), the ODD is the best initially
when the number of pulses $N\lesssim\omega_{c}T/2\approx 10$, and the UDD
sequences become better and suppress the decoherence order by order when $N$
is large and the hard cutoff is reached. In Fig. 3(b), for large $N$ UDD is
better than ODD, since the ODD sequences are optimized for soft-cutoff noise
rather than hard-cutoff noise. In Fig. 3(a) the decreasing of $\chi(T)$ is a
linear decrease in the double-logarithm plot, but in Fig. 3(b) it is much
faster. This confirms that DD is not so efficient against soft-cutoff noise.
Figure 3: (Color online). The decoherence function $\chi(T)$ as a function of
pulse number $N$ (a) without hard cutoff and (b) with a hard high-frequency
cutoff $\omega_{c}=40$. Here the noise spectrum
$S(\omega)=\frac{10^{5}}{1+\omega^{4}}$ and $T=0.5$. Squares (blue), triangles
(green), and circles (red) correspond to UDD, CPMG, and ODD. The ODD sequences
are the same as those shown in Fig. 2.
## IV Summary and conclusions
We have studied the dynamical decoupling control of decoherence caused by
Gaussian noise with soft cutoff in a general modulation formalism. We have
proved Theorem 1 which shows that, for the soft cutoff with the power-law
asymptote $~{}\alpha/\omega^{P}$ at high frequencies, DD can only suppress
decoherence to $\mathcal{O}(T^{P+1})$, where $P$ does not need to be an
integer. When the short-time expansion of noise correlation has the $(2K-1)$th
odd expansion term, DD can only suppress decoherence to
$\mathcal{O}(T^{2K+1})$ (Theorem 2). The existence of odd-order terms in the
short-time expansion corresponds to a soft high-frequency cutoff
($\sim\alpha/\omega^{2K}$) in the noise spectrum. For these noise spectra, we
have derived the equations for pulse sequence optimization, which minimizes
the leading odd-order decoherence function and eliminates even-order
decoherence functions of lower orders. The ODD sequences obtained by this
method coincide with the UDD sequences when the pulse number $N$ is small, and
they resemble CPMG sequences when $N$ is large. For the special case that the
short-time correlation function expansion has a linear term in time (i.e., a
soft cutoff $\sim\alpha/\omega^{2}$), the ODD sequences are exactly the CPMG
sequences.
Although we derived the theorems from a pure dephasing model, we expect that
the results of the existence of the largest decoupling order in short-time
limit can be extended to the general decoherence model (including both
dephasing and relaxation) of quantum systems. It is desirable to study the DD
in suppressing the general decoherence of quantum systems in the soft-cutoff
noise in the future.
###### Acknowledgements.
We thank Yi-Fan Luo and Bobo Wei for discussions. This work was supported by
the Hong Kong GRF CUHK402209, the CUHK Focused Investments Scheme, and the
National Natural Science Foundation of China Project No. 11028510.
## Appendix A Modulation Functions in Multiqubit Systems
Consider an $L$-qubit (or $2^{L}$-level) system suffering pure dephasing
described by the Hamiltonian
$H=\sum_{m=0}^{2^{L}-1}|m\rangle\langle m|[\omega_{m}+\beta_{m}(t)],$ (61)
where $\omega_{m}$ is the energy and $\beta_{m}(t)$ is the fluctuation on the
state $|m\rangle$. Here $m=(m_{L}\cdots m_{2}m_{1})$ is a binary code with
$m_{l}=0$ or $1$ for the $l$th qubit.
The Pauli operator for the $l$th qubit is
$\sigma_{x}^{(l)}=\sum_{m_{l}=0}\left(|m+2^{l-1}\rangle\langle
m|+\text{H.c.}\right),$ (62)
which exchanges two basis states $|m\rangle$ and $|m^{\prime}\rangle$ if $m$
and $m^{\prime}$ differ at and only at the $l$th bit.
After a sequence of $\sigma_{x}^{(l)}$ pulses and a final pulse
$\sigma_{\text{add}}=\sigma_{x}^{(l_{1})}\cdots\sigma_{x}^{(l_{N})}$, the
evolution operator is
$\displaystyle U(T)$
$\displaystyle=\sigma_{\text{add}}U(T_{N+1},T_{N})\sigma_{x}^{(l_{N})}U(T_{N},T_{N-1})\cdots$
$\displaystyle\times\sigma_{x}^{(l_{2})}U(T_{2},T_{1})\sigma_{x}^{(l_{1})}U(T_{1},T_{0})\sigma_{x}^{(l_{0})},$
(63)
where $T_{0}=0$, $T_{N+1}\equiv T$, and $\sigma_{x}^{(l_{0})}\equiv I$. The
free evolution operator
$U(T_{j+1},T_{j})=\exp\left[-i\int_{T_{j}}^{T_{j+1}}\sum_{m=0}^{2^{L}-1}|m\rangle\langle
m|[\omega_{m}+\beta_{m}(t)]dt\right].$ (64)
We write the evolution operator as
$U(T)=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}\sum_{m=0}^{2^{L}-1}\Xi_{m,j}[\omega_{m}+\beta_{m}(t)]dt},$
(65)
with
$\Xi_{m,j}\equiv\sigma_{x}^{(l_{0})}\cdots\sigma_{x}^{(l_{j})}|m\rangle\langle
m|\sigma_{x}^{(l_{j})}\cdots\sigma_{x}^{(l_{0})}$. The phase factor between
the states $|p\rangle$ and $|q\rangle$ changed during the evolution time $T$
is
$\varphi_{pq}(T)=\langle p|U(T)|p\rangle\langle q|U^{\dagger}(T)|q\rangle,$
(66)
where
$\langle p|U(T)|p\rangle=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}\langle
p|\sum_{m=0}^{2^{L}-1}\Xi_{m,j}|p\rangle[\omega_{m}+\beta_{m}(t)]dt}.$ (67)
Note that
$\sigma_{x}^{(l_{j})}\cdots\sigma_{x}^{(l_{0})}|p\rangle=|p\oplus[l_{0}\cdots
l_{j}]\rangle$ with
$p\oplus[l_{0}\cdots l_{j}]\equiv p\oplus 2^{l_{1}}\oplus
2^{l_{2}}\cdots\oplus 2^{l_{j}},$ (68)
for $j>0$ and $p\oplus[l_{0}]\equiv p$. Here $\oplus$ denotes addition on
binary digits, that is, $p\oplus[l_{0}\cdots l_{j}]$ is obtained by flipping
the $l_{1}$th,$\ldots$,$l_{j}$th bits of $p=(p_{L}\cdots p_{2}p_{1})$ in
binary code. We obtain
$\langle
p|U(T)|p\rangle=e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}[\omega_{p\oplus[l_{0}\cdots
l_{j}]}+\beta_{p\oplus[l_{0}\cdots l_{j}]}(t)]dt}.$ (69)
Therefore the coherence between the states $|p\rangle$ and $|q\rangle$
decreases by the average of the random phase
$\overline{e^{-i\sum_{j=0}^{N}\int_{T_{j}}^{T_{j+1}}[\beta_{p\oplus[l_{0}\cdots
l_{j}]}(t)-\beta_{q\oplus[l_{0}\cdots l_{j}]}(t)]dt}}.$ (70)
When each of the qubits feels the same noise,
$\beta_{m}(t)=(\sum_{k=1}^{L}m_{k})\beta(t)$ for the $m$th level, and
$\overline{\varphi_{pq}(T)}=e^{-i\int_{0}^{T}F_{pq}(t)\omega
dt}\overline{e^{-i\int_{0}^{T}F_{pq}(t)\beta(t)dt}},$ (71)
where the the modulation function is defined as
$F_{pq}(t)=(\sum_{k=1}^{L}\tilde{p}_{k})-(\sum_{k=1}^{L}\tilde{q}_{k}),$ (72)
with $\tilde{p}=p\oplus[l_{0}\cdots l_{j}]$ and $\tilde{q}=q\oplus[l_{0}\cdots
l_{j}]$ for $t\in(T_{j},T_{j+1}]$. For example, when $L=2$,
$F_{pq}(t)\in\\{0,\pm 1,\pm 2\\}$.
## Appendix B Decoherence Functions $\phi_{k}$
As $F(t/T)=0$ for $t\notin(0,T]$, we extend the bounds of integration for $t$
to infinity and transform Eq. (26) to
$\displaystyle\phi_{k}$ $\displaystyle=$
$\displaystyle\Re\frac{\partial^{k}}{\partial(i\eta)^{k}}\int_{-\infty}^{\infty}\frac{d\omega_{1}}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega_{2}}{2\pi}\int_{-\infty}^{\infty}dt_{1}\int_{0}^{t_{1}}dt_{2}$
(73)
$\displaystyle\times\tilde{f}^{*}(\omega_{1})\tilde{f}(\omega_{2})e^{i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}e^{i\eta(t_{1}-t_{2})},$
where we set $\eta\rightarrow 0$ after differentiation. Integrations over
$t_{2}$, $t_{1}$ and $\omega_{1}$ give
$\phi_{k}=\Re\frac{\partial^{k}}{\partial(i\eta)^{k}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\tilde{f}(\omega)}{i(\omega+\eta)}[\tilde{f}^{*}(-\eta)-\tilde{f}^{*}(\omega)].$
(74)
Using the formulas
$\frac{\partial^{k}}{\partial(i\eta)^{k}}\left[u(\eta)v(\eta)\right]=\sum_{r=0}^{k}\frac{k!}{r!(k-r)!}\left[\frac{\partial^{k-r}u(\eta)}{\partial(i\eta)^{k-r}}\right]\left[\frac{\partial^{r}v(\eta)}{\partial(i\eta)^{r}}\right],$
(75)
$\left.\frac{\partial^{k-r}}{\partial(i\eta)^{k-r}}\frac{1}{i(\omega+\eta)}\right|_{\eta=0}=\frac{-(k-r)!}{(-i\omega)^{k-r+1}},\text{
for }r\geq 0,$ (76)
and
$\left.\frac{\partial^{r}}{\partial(i\eta)^{r}}\tilde{f}^{*}(-\eta)\right|_{\eta=0}=\lambda_{r}^{*},\text{
for }r\geq 0,$ (77)
we have
$\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}-\sum_{r=0}^{k}\frac{\tilde{f}(\omega)}{(-i\omega)^{k-r+1}}\frac{\lambda_{r}^{*}}{r!}].$
(78)
Changing the summation index, we obtain
$\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}-\sum_{r=0}^{k}\frac{\tilde{f}(\omega)}{(-i\omega)^{k-r+1}}\frac{\lambda_{r}^{*}}{r!}\right].$
(79)
Note that the summation over $r$ and the integration over frequency cannot be
exchanged when the integration does not converge for each individual term.
Using Eq. (16a), we have
$\displaystyle\phi_{k}$ $\displaystyle=$ $\displaystyle
k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}+\sum_{r=0}^{k}\sum_{n=0}^{k-r}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right.$
(80) $\displaystyle+$
$\displaystyle\left.\sum_{r=0}^{k}\sum_{n=k-r+1}^{\infty}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right].$
We simplify the last line by using Eq. (16b) and the equality
$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\sum_{n=r}^{\infty}\frac{(\pm
i\omega)^{n-r}}{n!}t^{n}=\frac{1}{2}\frac{t^{r-1}}{(r-1)!},\text{ for }r\geq
1,t\geq 0,$ (81)
which is proved in Appendix C. We obtain
$\displaystyle\phi_{k}=k!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{k+1}}+\sum_{r=0}^{k}\sum_{n=0}^{k-r}\frac{(-1)^{k-r}}{(i\omega)^{k-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right]$
$\displaystyle+\frac{k!}{2}\Re\sum_{r=0}^{k}(-1)^{k-r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{k-r}}{(k-r)!}.$
(82)
For even number $2m$, $\frac{|\tilde{f}(\omega)|^{2}}{(-i\omega)^{2m+1}}$ is
an odd function and its integral vanishes. Eq. (82) gives
$\displaystyle\phi_{2m}=(2m)!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{r}}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right]$
$\displaystyle+\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!},$
(83)
which is decomposed as (with the changes of summation order and indices)
$\displaystyle\phi_{2m}$ $\displaystyle=$
$\displaystyle(2m)!\Re\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left[\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{r}/2}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{n}}{n!}\right.$
(84)
$\displaystyle+\left.\sum_{r=0}^{2m}\sum_{n=0}^{2m-r}\frac{(-1)^{n}/2}{(i\omega)^{2m-r-n+1}}\frac{\lambda_{r}}{r!}\frac{\lambda_{n}^{*}}{n!}\right]$
$\displaystyle+\frac{(2m)!}{2}\Re\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!},.$
The integrals of the terms with odd-power of $\omega$ vanish. And for even
functions of $\omega$, the sum $n+r$ is an odd number, so
$(-1)^{r}+(-1)^{n}=0$. Thusthe real part of the integral in Eq. (84) vanishes.
From the last line of Eq. (84), we obtain Eq. (28), i.e.,
$\displaystyle\phi_{2m}$
$\displaystyle=\frac{(2m)!}{2}\sum_{r=0}^{2m}(-1)^{r}\frac{\lambda_{r}^{*}}{r!}\frac{\lambda_{2m-r}}{(2m-r)!}.$
(85)
## Appendix C Proof of Equation (81)
To prove Eq. (81), we just need to prove
$\lim_{R\rightarrow\infty}\int_{-R}^{R}dx\sum_{n=r}^{\infty}\frac{(\pm
ix)^{n-r}}{n!}=\frac{\pi}{(r-1)!},\text{ for }r\geq 1,$ (86)
where the bounds in the integral guarantee that the modulation function $F(t)$
is a real function. Using
$\displaystyle\int_{-R}^{R}dx\sum_{n=r}^{\infty}\frac{(\pm
ix)^{n-r}}{n!}=\sum_{n=1}^{\infty}\frac{R^{n}}{(n+r-1)!n}(i^{n-1}+\text{c.c.}),$
(87a) $\displaystyle\frac{1}{(r-1)!}\int_{-R}^{R}\frac{\sin
x}{x}dx=\sum_{n=1}^{\infty}\frac{R^{n}}{n!n}\frac{1}{(r-1)!}(i^{n-1}+\text{c.c.}),$
(87b)
and $\lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{\sin x}{x}dx=\pi$, we just
need to prove
$\lim_{R\rightarrow\infty}\sum_{n=1}^{\infty}\left[\frac{R^{n}}{(n+r-1)!n}-\frac{R^{n}}{n!n(r-1)!}\right](i^{n-1}+\text{c.c.})=0.$
(88)
For $r=1$, it obviously holds. For $r\geq 2$, we can show the difference
$\Delta\equiv\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!n}\left[(r-1)!-\prod_{j=1}^{r-1}(n+j)\right]=\mathcal{O}\left(\frac{1}{R}\right),$
(89)
so $\text{lim}_{R\rightarrow\infty}\Delta=0$. By expanding the terms in the
square brackets of Eq. (89), we have
$\Delta=\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!}\left(\sum_{k=0}^{r-2}a_{k}n^{k}\right),$
(90)
where $a_{k}$ is a number independent of $n$. We arrange the terms in the
square brackets and get
$\Delta=\sum_{n=1}^{\infty}\frac{R^{n}(i^{n-1}+\text{c.c.})}{(n+r-1)!}\left[\sum_{k=1}^{r-2}b_{k}\prod_{j=1}^{k}[(n+r-j)]+b_{0}\right],$
(91)
with $b_{j}$ independent of $n$. After some simplification it becomes for
$r\geq 2$
$\sum_{k=1}^{r-1}\frac{b_{r-k-1}}{i^{k+1}R^{k}}\left(e^{iR}-\sum_{n=0}^{k-1}\frac{R^{n}}{n!}i^{n}\right)+\text{c.c.}=\mathcal{O}\left(\frac{1}{R}\right).$
(92)
Hence $\Delta=\mathcal{O}(\frac{1}{R})$, and Eq. (81) is proved.
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|
arxiv-papers
| 2012-10-04T08:50:19 |
2024-09-04T02:49:35.934080
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhen-Yu Wang and Ren-Bao Liu",
"submitter": "Zhen-Yu Wang",
"url": "https://arxiv.org/abs/1210.1338"
}
|
1210.1516
|
arxiv-papers
| 2012-10-01T16:15:34 |
2024-09-04T02:49:35.953017
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Asia Furones",
"submitter": "Asia Furones",
"url": "https://arxiv.org/abs/1210.1516"
}
|
|
1210.1581
|
# Noise Induced Pattern Switching in Randomly Distributed Delayed Swarm
Patterns
Brandon Lindley and Luis Mier-y-Teran-Romero and Ira B. Schwartz This work was
supported by the Office of Naval Research and the National institutes of
Health.Brandon Lindley is an NRC postodctoral fellow at the US Naval Research
Labooratory, Code 6792, Washington, DC 20375 USA,
brandon.lindley.ctr@nrl.navy.milL. Mier-y-Teran-Romero is an NIH post doctoral
fellow at the Naval Research Laboratory. luis.miery@nrl.navy.milI. B. Schwartz
is at the US Naval Research Labooratory, Code 6792, Washington, DC 20375 USA
Ira.schwartz@nrl.navy.mil
###### Abstract
We study the effects of noise on the dynamics of a system of coupled self-
propelling particles in the case where the coupling is time-delayed, and the
delays are discrete and randomly generated. Previous work has demonstrated
that the stability of a class of emerging patterns depends upon all moments of
the time delay distribution, and predicts their bifurcation parameter ranges.
Near the bifurcations of these patterns, noise may induce a transition from
one type of pattern to another. We study the onset of these noise-induced
swarm re-organizations by numerically simulating the system over a range of
noise intensities and for various distributions of the delays. Interestingly,
there is a critical noise threshold above which the system is forced to
transition from a less organized state to a more organized one. We explore
this phenomenon by quantifying this critical noise threshold, and note that
transition time between states varies as a function of both the noise
intensity and delay distribution.
## I Introduction
The dynamics of interacting multi-agent or swarming systems in various
biological and engineering fields is actively being studied. These systems are
remarkable for their ability to self-organize into very diverse, complex
spatio-temporal patterns. These studies have numerous biological applications
with widely varying spatial and temporal scales. Among them are bacterial
colonies, schooling fish, flocking birds, swarming locusts, ants, and
pedestrians [1, 2, 3, 4, 5, 6, 7, 8]. In engineering, these studies have
investigated systems of communicating robots [9, 10, 11, 12] and mobile sensor
networks [13].
A fundamental problem for the engineering of systems of autonomous,
communicating agents is the design of agent-interaction protocols to achieve
robotic space-time path planning, consensus and cooperative functions, and
other forms of spatio-temporal organization. A fruitful approach has resulted
from applying the tools developed in the study of swarms in various biological
and physical contexts to aid in the design of algorithms for systems of
communicating robots. This has led to the successful use of a combination of
inter-agent and external potentials to obtain agent organization and
cooperation; however, it must be ensured that the results from these methods
are scalable with respect to the number of agents. Important applications
comprise the following: obstacle avoidance [11], boundary tracking [14, 15],
environmental sensing [13, 16], decentralized target tracking [17],
environmental consensus estimation [13, 18] and task allocation [19].
An important aspect that must be accounted for in the design of algorithms for
the spatio-temporal organization of communicating robotic systems is that of
time delay. Time delay arises in latent communication between agents,
information processing times, hardware malfunction, as well as actuation lag
times due to inertia. Time delays in robotic systems are important in the
areas of consensus estimation [18] and task allocation, where, for example,
there is a time delay as a consequence of the time required to switch between
different tasks [19]. Previous work has shown the big impact that time delays
may have in the dynamics of a system, such as destabilization and
synchronization [20, 21]. Moreover, time delays have been used with success
for control purposes [22]. The initial studies considered at most a few
discrete time delays that are constant in time. Recent studies have extended
the aforementioned investigations to consider randomly selected time delays
[23, 24, 25] and distributed time delays, i.e., when the time evolution of the
system is affected by its history over an extended time interval in its past,
instead of at a discrete instants [26, 27, 28].
Robust algorithms for task planning with inter-agent and environmental
interactions need to account for the presence of noise at all levels in the
system. Noise in the swarm’s dynamics introduces higher complexity in the
behavior and may produce transitions from one coherent pattern to another,
something that may be detrimental to the algorithm’s purposes or, to the
contrary, that may be exploited to escape unwanted states [29, 30, 31, 32].
Here, we investigate a swarming model where the coupling between agents occurs
with randomly distributed time delays. We show that the attractive coupling,
non-uniform, random time delays and external noise intensity combine to
produce transitions between different coherent patterns. Remarkably, we show
that under certain conditions, noise produces transitions that increase the
phase space coherence of the particles.
## II Swarm Model
We study the spatio-temporal dynamics of a two dimensional system of $N$
agents under the effects of two forces: self-propulsion and mutual attraction.
We consider that the attraction between agents occurs in a time delayed
fashion due to finite communication speeds and processing times. The dynamics
of the particles is described by the following governing dimensionless
equations:
$\displaystyle\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau_{ij}))+\boldsymbol{\eta}_{i}(t),$
(1)
for $i=1,2\ldots,N$. The 2D position and velocity vectors of particle $i$ at
time $t$ are denoted by $\mathbf{r}_{i}(t)$ and $\dot{\mathbf{r}}_{i}(t)$,
respectively. The self-propulsion of agent $i$ is modeled by the term
$\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}$. The quantity
$a$ is called the coupling constant and measures the strength of attraction
between agents. At time $t$, agent $i$ is attracted to the position of agent
$j$ at the past time $t-\tau_{ij}$. The $N(N-1)$ different time delays
$0<\tau_{ij}$ are distributed according to a distribution $\rho(\tau)$ whose
mean is $\mu_{\tau}$ and whose standard deviation is $\sigma_{\tau}$. In
contrast to some of our recent work, here we allow that
$\tau_{ij}\neq\tau_{ji}$; i.e., time delays are not symmetric among pairs of
agents [33, 34]. The form of our model is based on the normal form for
particles near a supercritical bifurcation corresponding to the onset of
coherent motion [35]. In addition, the functional form of the attractive terms
may be thought of as representing the first term in a Taylor series around a
stable equilibrium point of a more general time-delayed potential. Various
models of this form have been extensively used to study the motion of swarms
[36, 30, 37, 31, 35, 32]. Lastly, the term
$\boldsymbol{\eta}_{i}(t)=(\eta_{i}^{(1)},\eta_{i}^{(2)})$ is a two-
dimensional vector of stochastic white noise with intensity equal to $D$ and
such that $\langle\eta_{i}^{(\ell)}(t)\rangle=0$ and
$\langle\eta_{i}^{(\ell)}(t)\eta_{j}^{(k)}(t^{\prime})\rangle=2D\delta(t-t^{\prime})\delta_{ij}\delta_{\ell
k}$ for $i,j=1,2,\ldots N$ and $\ell,k=1,2$.
Figure 1: A selection of the bifurcation curves of the mean field model Eq.
(2) in the parameter space of coupling strength $a$ and mean time delay
$\mu_{\tau}$ for different values of the time delay standard deviation
$\sigma_{\tau}=0.2$ (a) and 0.6 (b). In region A (a) the stationary state of
the mean field is stable; this state corresponds to the ring state (see text)
of the full swarm system Eq. (1). The swarm moves uniformly in region C (a),
this ‘translating state’ disappears by merging with the stationary state along
the curve $a\mu_{\tau}=1$ (red). The stationary state undergoes a first Hopf
bifurcation along the curve above region A (a), making the mean field adopt a
‘rotating state’ in region B (a). The stationary state has an infinite series
of higher order Hopf bifurcations; the first few members are shown in dashed
curves (green, cyan and magenta). Comparing (a) and (b), we see how sensitive
these additional Hopf curves are with respect to $\sigma_{\tau}$. The marker
at $a=2$ and $\mu_{\tau}=2$ denotes the parameter values used for our
numerical investigations. (Color online).
A mean field approximation of the swarm dynamics may be obtained by using
coordinates relative to the center of mass
$\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i}$, for $i=1,2\ldots,N$, where
$\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t)$. Following [33], we
use the following distributed delay equation to describe the mean field of the
swarm:
$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{\
R}(t)-\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau\right).$ (2)
For Eq. (2) to be accurate, we require that $N$ be sufficiently large so that
$\frac{1}{N(N-1)}\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\mathbf{R}(t-\tau_{ij})\approx\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau$
and that the swarm particles remain relatively close together. However, since
the proximity of the particles is not controlled directly by any parameter,
one must rely on numerical simulations of finite swarm populations to
establish the parameter regimes where the approximation holds.
## III Results
In the absence of noise, Eqs. (1) have been shown to possess a rich
bifurcation structure, elucidated by the use of the mean field approximation
Eq. (2) (Figure 1) [33]. In contrast to the mean field, the full system
displays bistability of several coherent patterns. In particular, in wide
portions of region B in Fig. 1a, Eqs. (1) possesses bistability of two
different patterns: (_i_) a ‘ring state’, where the center of mass of the
swarm is at rest and the agents rotate both clockwise and counterclockwise at
a constant speed and radius $1/\sqrt{a}$; (_ii_) a ‘rotating state’, where the
particles form a fairly dense clump and move along a circular arc at constant
speed, with all velocity vectors approximately aligned. The initial alignment
of the particle velocities and the width of the time delay distribution are
instrumental in determining what pattern is adopted after the decay of
transients. Specifically, decreasing initial velocity alignment and increasing
width of the delay distribution results in a higher liklihood of convergence
to the ring state.
Figure 2: Snapshots of the swarm agents at different times, illustrating the
transition from the ring state to the rotating state. Here $\sigma_{\tau}$=0.2
and $D=$0.23. The circle in the different panels is simply shown as a guide to
the eye.
When noise is introduced, the combination of the coupling strength, the time
delay and noise produces interesting pattern transitions due to the
fluctuations of the agent’s alignment. Specifically, we find that when there
is bistability of the ring and rotating state, there is a critical value of
the noise intensity, $D_{crit}$, above which the swarm transitions from the
ring state to the rotating state. In contrast, noise intensities below that
critical value do not produce such a transition; that is, no such transition
has been observed within the limits of our long-time numerical simulations.
Figure 2 shows snapshots of the swarm, illustrating this transition. Here, and
the rest of the simulations discussed below, the initial state of the
particles is considered to be at rest and the particles are randomly
distributed on the unit square. The new state of the swarm at each time step
is found by updating the stochastic system (1) using Heun’s Method. In all
simulations we assume the delays $\tau$ are uniformly distributed with a mean
delay of $\mu_{\tau}$ and a standard deviation of $\sigma_{\tau}$. The noise
is assumed to be Gaussian with intensity $D$. For all of our numerical
studies, we use the values $N=50$, $a=2$ and $\mu_{\tau}=2$.
A remarkable fact is that noise intensities $D>D_{crit}$ produce a transition
from a less coherent state into another with higher coherence. This is because
the ring state is a disorganized state with both position and velocity vectors
adopting wide probability distributions. In contrast, the rotating state is
highly coherent, with particles having nearby locations (high density swarms)
and almost perfect velocity alignment.
Figure 3: Time series showing the mean particle alignment as a function of
time, for different parameter values. The top row has time delay standard
deviation $\sigma_{\tau}=$0.2 and noise intensities $D=$0.25 (a) and 0.4 (b).
The bottom row has time delay standard deviation $\sigma_{\tau}=$0.6 and noise
intensities $D=$0.25 (c) and 0.4 (d). Please note the different scales on the
abcissa axis.
In order to properly quantify the time required by the swarm to make this
transition, we use a quantity called the mean alignment, which has been used
in the past to describe the pattern adopted by the swarm as a whole [32]. This
quantity is defined as follows. If the velocity of particle $j$ makes an angle
$\theta_{j}$ with the velocity of the center of mass, then the mean alignment
is simply the ensemble average of the cosines of all of the angles
$\theta_{j}$, for $j=1,2\ldots N$. That is,
$\displaystyle\textrm{Mean swarm
alignment}=\frac{1}{N}\sum_{j=1}^{N}\cos\theta_{j}=\frac{1}{N}\sum_{j=1}^{N}\frac{\dot{\mathbf{r}}_{j}\cdot\dot{\mathbf{R}}}{|\dot{\mathbf{r}}_{j}||\dot{\mathbf{R}}|},$
(3)
which ranges from -1 to 1. When all particles have perfectly aligned
velocities, the mean alignment is equal to 1, regardless of their location in
space or the magnitude of the individual particles’ speeds.
Figure 3 shows the mean particle alignment as a function of time, for
different values of the noise intensity and the time delay standard deviation.
The four panels show how the swarm starts with very low values of the mean
alignment, since it initially adopts the ring state. After some dwell time in
the ring state, the noise causes the swarm depart from that state and converge
to the rotating state, where the mean alignment is almost 1.0. Once the agents
begin to depart from the ring state, the transition time required to complete
the transition is very short compared to the dwell time. For the simulations
shown, the dwell time decreases with increasing noise intensity (Fig. 3a to 3b
and Fig. 3c to 3d) and increases with increasing time delay standard deviation
(Fig. 3a to 3c and Fig. 3b to 3d).
To better understand the onset of alignment due to noise, we probe different
noise intensities for various choices of $\sigma_{\tau}$, with $\mu_{\tau}$
fixed, and then plot their asymptoptic mean alignment Figure 4. Each of the
numerical experiments in Figure 4 was started in an initial state that,
outside of the presence of noise, will stay indefinitely in the ring formation
(unaligned). The simulation was run out to $t=2500$, in order to allow
transients to pass, and to ensure we capture any transition between the two
states. As reported in [32], we do see a critical value of the noise intensity
$D$ at which the noise drives the particles into the highly aligned rotating
state. Interestingly, as $\sigma_{\tau}$ is increased, the location of the
$D_{crit}$ shifts so that a larger noise intensity is required to observe this
transition. Thus, we observe a dependence of the critical noise threshold on
the distribution width of the delays.
Figure 4: Asymptotic mean particle alignment as a function of the noise
intensity parameter, $D$, and for different values of the time delay standard
deviation $\sigma_{\tau}$. Each curve (labeling different values of the
standard deviation of the time delay) has an extremely sharp transition that
occurs at a critical value of the noise intensity $D_{crit}$
($0.2<D_{crit}<0.25$). Below the transition point the swarm converges to the
ring state and remains there for times as long as our simulations permit. In
contrast, above the critical noise value, the swarm transitions to the more
coherent, rotating sate.
Further, as observed in Fig. 3, increasing $\sigma_{\tau}$ also increases the
amount of time the particles remain in the ring state before finally
transitioning to the rotating state (assuming $D$ is large enough for such a
transition to occur). To better understand this observation, we do single runs
for various values of $\sigma_{\tau}$ and $D$. For each simulation, we monitor
when a threshold value of the mean alignment is reached, and then record that
time as the ‘dwell time’. While these dwell time values, shown in figure 5,
record only single events, they do indicate that, generally speaking, below
the threshold value $D<D_{crit}(\sigma_{\tau})$, we do not observe a
transition out of the ring state, but for $D\geq D_{crit}(\sigma_{\tau})$, we
observe that the dwell time is highest near the threshold, and then rapidly
decreases as $D$ is increased. As one continues to increase $D$ even further,
the rotating state will gradually become more disorganized until noise
dominates the entire system.
Figure 5: Dwelling time in the ring state, as a function of noise intensity,
$D$, and the time delay standard deviation, $\sigma_{\tau}$. Below the
$\sigma_{\tau}$-dependent, critical value of noise intensity, $D_{crit}$, the
transition times diverge from the perspective of our long-time numerical
simulations (white region). Above the value $D_{crit}$, the transition times
become finite and accessible to computation from numerical simulations.
## IV Discussion
We have considered the general problem of multi-agent swarms of particles
where the communication is governed by a delay coupled potential field. In
particular, we have considered the case where the delay coupling is fixed in
time but randomly distributed from a chosen probability density. This
corresponds to the fact that in many cases the delays in signal transmission
and/or reception are caused by finite transmission times, processing and
control delays, and the probability of dropped packets. Modeling the swarm as
a globally coupled delay system, we have considered the role of temporal noise
due to fluctuations from external forces which occur when the swarm is
operating in a random field. In particular, we have considered parameters of
the distributed delay system in which there exists bi-stability; i.e., there
co-exists both a ring state and a rotating state.
For a given noise density and delay distribution, we have characterized the
observation that a specific range of noise intensities forces a swarm from a
disordered ring state, to a more ordered rotating state. By probing the
effects of noise intensity along with the delay distribution width, we see a
two parameter set which describes fluctuations which cause switching between
disordered and ordered states. In particular, fluctuations due to sufficient
noise intensities are observed to produce highly coherent and compact
structures, which is clearly a non-intuitive result. We note once more that
the patterns and the transitions between them, do not fundamentally change
with the addition of small, local repulsive forces between particles. Stronger
repulsion can, however, destabilize the coherent structures.
Understanding these types noise-induced transitions is key to preventing
coherence collapse in delay-coupled autonomous systems, as well as formulating
control strategies. In particular, the idea of a critical noise threshold
which can serve to facilitate transitions between different dynamic patterns
is very interesting and powerful, and understanding its dependence on the
structure of delay-coupled systems is an area of ongoing interest.
We note that in the future, more work is required to understand the role of
fluctuations on delay coupled systems. Towards this end, a general theory of
switching for non-Gaussian noise is needed. In addition, more systematic
numerical simulations will allow sufficient averaging to refine our
understanding of the transitions discussed here. In addition, realistic
networks of communication need to be considered beyond the globally coupled
system modeled here.
## V Acknowledgements
The authors gratefully acknowledge the Office of Naval Research for their
support through ONR contract no. N0001412WX2003 and the Naval Research
Laboratory 6.1 program contract no. N0001412WX30002. LMR and IBS are supported
by Award Number R01GM090204 from the National Institute Of General Medical
Sciences. The content is solely the responsibility of the authors and does not
necessarily represent the official views of the National Institute Of General
Medical Sciences or the National Institutes of Health. BSL is currently
supported by a National Research Council fellowship.
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|
arxiv-papers
| 2012-10-04T20:02:34 |
2024-09-04T02:49:35.960384
|
{
"license": "Public Domain",
"authors": "Brandon Lindley and Luis Mier-y-Teran-Romero and Ira B. Schwartz",
"submitter": "Ira Schwartz",
"url": "https://arxiv.org/abs/1210.1581"
}
|
1210.1611
|
# Efficient Tabling of Structured Data with Enhanced Hash-Consing
Neng-Fa Zhou
CUNY Brooklyn College & Graduate Center
zhou@sci.brooklyn.cuny.edu Christian Theil Have
Roskilde University
cth@ruc.dk
(2002)
###### Abstract
Current tabling systems suffer from an increase in space complexity, time
complexity or both when dealing with sequences due to the use of data
structures for tabled subgoals and answers and the need to copy terms into and
from the table area. This symptom can be seen in not only B-Prolog, which uses
hash tables, but also systems that use tries such as XSB and YAP. In this
paper, we apply hash-consing to tabling structured data in B-Prolog. While
hash-consing can reduce the space consumption when sharing is effective, it
does not change the time complexity. We enhance hash-consing with two
techniques, called input sharing and hash code memoization, for reducing the
time complexity by avoiding computing hash codes for certain terms. The
improved system is able to eliminate the extra linear factor in the old system
for processing sequences, thus significantly enhancing the scalability of
applications such as language parsing and bio-sequence analysis applications.
We confirm this improvement with experimental results.
††volume: 10 (3):
## 1 Introduction
Tabling, as provided in logic programming systems such as B-Prolog [Zhou et
al. (2008)], XSB [Swift and Warren (2012)], YAP [Santos Costa et al. (2012)],
and Mercury [Somogyi and Sagonas (2006)], has been shown to be a viable
declarative language construct for describing dynamic programming solutions
for various kinds of real-world applications, ranging from program analysis,
parsing, deductive databases, theorem proving, model checking, to logic-based
probabilistic learning. The main idea of tabling is to memorize the answers to
subgoals in a table area and use the answers to resolve their variant or
subsumed descendants. This idea of caching previously calculated solutions,
called memoization, was first used to speed up the evaluation of functions
[Michie (1968)]. Tabling can get rid of not only infinite loops for bounded-
term-size programs but also redundant computations in the execution of
recursive programs. While Datalog programs require tabling only subgoals with
atomic arguments, many other programs such as those dealing with complex
language corpora or bio-sequences require tabling structured data.
Unfortunately, none of the current tabling systems can process structured data
satisfactorily. Consider, for example, the predicate is_list/2:
:-table is_list/1.
is_list([]).
is_list([_|L]):-is_list(L).
For the subgoal is_list([1,2,...,N]), the current tabled Prolog systems
demonstrate a higher complexity than linear in N: B-Prolog (version 7.6 and
older) consumes linear space but quadratic time; YAP, with a global trie for
all tabled structured terms [Raimundo and Rocha (2011)], consumes linear space
but quadratic time; XSB is quadratic in both time and space. The nonlinear
complexity is due to the data structure used to represent tabled subgoals and
answers and the need to copy terms into and from the table area.
The inefficiency of early versions of B-Prolog in handling large sequences has
been reported and a program transformation method has been proposed to index
ground structured data to work around the problem [Have and Christiansen
(2012)]. In old versions of B-Prolog, tabled subgoals and answers were
organized as hash tables, and input sharing was exploited to allow a tabled
subgoal to share its ground structured arguments with its answers and its
descendant subgoals. Input sharing enabled B-Prolog to consume only linear
space for the tabled subgoal is_list([1,2,...,N]). Nevertheless, since the
hash code was based on the first three elements of a list, the time complexity
for a query like is_list([1,1,...,1]) was quadratic in the length of the list.
B-Prolog didn’t support output sharing, i.e. letting different answers share
structured data. Therefore, on the tabled version of the permutation program
that generates all permutations through backtracking, B-Prolog would create
$n\times n!$ cons cells where $n$ is the length of the given list.
This problem with tabling structured data has been noticed before and several
remedies have been attempted. One well known technique used in parsing is to
represent sentences as position indexed facts rather than lists. XSB provides
tabled grammar predicates that convert list representation to position
representation by redefining the built-in predicate ’C’/3.111Personal
communication with David S. Warren, 2011. The position representation is also
used for PCFG parsing in PRISM [Sato and Kameya (2008)]. A program
transformation method has been proposed to index ground structured data to
work around the quadratic time complexity of B-Prolog’s tabling system [Have
and Christiansen (2012)]. Nevertheless, these remedies have their limitations:
the position representation disallows natural declarative modeling of
sequences and the program transformation incurs considerable overhead. Have
and Christiansen advocate for native support of data sharing in tabled Prolog
systems for better scalability of their bio-sequence analysis application
[Have and Christiansen (2012)].
We have implemented full data sharing in B-Prolog in response to the
manifesto. In the new version of B-Prolog, both input sharing and output
sharing are exploited to allow tabled subgoals and answers to share ground
structured data. Hash-consing [Ershov (1959)], a technique originally used in
functional programming to share values that are structurally equal [Goto
(1974), Appel and de Rezende Goncalves (2003)], is adopted to memorize
structured data in the table area. This technique avoids storing the same
ground term more than once in the table area. While hash-consing can reduce
the space consumption when sharing is effective, it does not change the time
complexity. To avoid the extra linear time factor in dealing with sequences,
we enhance hash-consing with input sharing and hash code memoization. For each
compound term, an extra cell is used to store its hash code.
Our main contribution in this paper is to apply hash-consing to tabling and
enhance it with techniques to make it time efficient. The resulting system
demonstrates linear complexity in terms of both space and time on the query
is_list(L) for any kind of ground list L. As another contribution, we also
compare tries with hash consing in the tabling context. As long as sequences
are concerned, a trie allows for sharing of prefixes while hash-consing allows
for sharing of ground suffixes. While we can build examples that arbitrarily
favor one over the other, for recursively defined predicates such as is_list,
it is more common for subgoals to share suffixes than prefixes. The enhanced
hash-consing greatly improves the scalability of PRISM on sequence analysis
applications. Our experimental results on a simulator of a hidden Markov model
show that PRISM with enhanced hash-consing is asymptotically better than the
previous version that supports no hash-consing.
The remainder of the paper is structured as follows: Section 2 defines the
primitive operations on the table area used in a typical tabling system;
Section 3 presents the hash tables for subgoals and answers, and describes the
copy algorithm for copying data from the stack/heap to the table area; Section
4 modifies the copy algorithm to accommodate hash-consing; Section 5 describes
the techniques for speeding up computation of hash codes; Section 6 evaluates
the new tabling system with enhanced hash-consing; Section 7 gives a survey of
related work; and Section 8 concludes the paper.
## 2 Operations on the Table Area
A tabling system uses a data area, called table area, to store tabled subgoals
and their answers. A tabling system, whether it is suspension-based SLG [Chen
and Warren (1996)] or iteration-based linear tabling [Zhou et al. (2008)],
relies on the following three primitive operations to access and update the
table area.222The interpretation of these operations may vary depending on
implementations.
Subgoal lookup and registration:
This operation is used when a tabled subgoal is encountered in execution. It
looks up the subgoal table to see if there is a variant of the subgoal. If
not, it inserts the subgoal (termed a pioneer or generator) into the subgoal
table. It also allocates an answer table for the subgoal and its variants.
Initially, the answer table is empty. If the lookup finds that there already
is a variant of the subgoal in the table, then the record stored in the table
is used for the subgoal (called a consumer). Generators and consumers are
dealt with differently. In linear tabling, for example, a generator is
resolved using clauses and a consumer is resolved using answers; a generator
is iterated until the fixed point is reached and a consumer fails after it
exhausts all the existing answers.
Answer lookup and registration:
This operation is executed when a clause succeeds in generating an answer for
a tabled subgoal. If a variant of the answer already exists in the table, it
does nothing; otherwise, it inserts the answer into the answer table for the
subgoal. When the lazy consumption strategy (also called local strategy) is
used, a failure occurs no matter whether the answer is in the table or not,
which drives the system to produce the next answer.
Answer return:
When a consumer is encountered, an answer is returned immediately if any. On
backtracking, the next answer is returned. A generator starts consuming its
answers after it has exhausted all its clauses. Under the lazy consumption
strategy, a top-most looping generator does not return any answer until it is
complete.
## 3 Hash Tables for Subgoals and Answers
The data structures used for the table area are orthogonal to the tabling
mechanism, whether it is suspension-based or iteration-based; they can be hash
tables, tries, or some other data structures. In this section, we consider
hash tables and the operations for the table area without data sharing.
A hash table, called a subgoal table, is used for all tabled subgoals. For
each tabled subgoal and its variants, there is a record in the subgoal table,
which includes, amongst others, the following fields:
AnswerTable: | Pointer to the answer table for the subgoal
---|---
sym: | The functor of the subgoal
A1...An: | The arguments the subgoal
|
When a tabled predicate is invoked by a subgoal, the subgoal table is looked
up to see if a variant of the subgoal exists. If not, a record is allocated
and the arguments are copied from the stack/heap to the table area. The copy
of the subgoal shares no structured terms with the original subgoal and all of
its variables are numbered so that they have different identities from those
in the original subgoal.
The record of a subgoal in the subgoal table includes a pointer to another
hash table, called an answer table, for storing answers produced for the
subgoal. For each answer and its variants, there is a record in the answer
table, which stores amongst others a pointer to a copy of the answer. When an
answer is produced for a subgoal, the subgoal’s answer table is looked up to
see if a variant of the answer exists. If not, a record is allocated and the
answer is copied from the stack/heap to the table area. The answers in a
subgoal’s answer table are connected from the oldest one to the newest one
such that they can be consumed by the subgoal one by one through backtracking.
In the implementation, a hash table is represented as an array. To add an item
into a hash table, the system computes the hash code of the item and uses the
hash code modulo the size of the array to determine a slot for the item. All
items hashed to the same slot are connected as a linked list, called a hash
chain. A hash table is expanded when the number of records in it exceeds the
size of the array.
The WAM representation [Warren (1983)] is used to represent both terms on the
heap and terms in the table area except that variables in tabled terms are
numbered. A term is represented by a word containing a value and a tag. The
tag distinguishes the type of the term. It may be REF denoting a reference,
ATM an atomic value, STR a structure, LST a cons, or NUMVAR a numbered
variable. A STR-tagged reference to a structure $f(t_{1},\ldots,t_{n})$ points
to a block of $n+1$ consecutive words where the first word points to the
functor $f/n$ in the symbol table and the remaining $n$ words store the $n$
components of the structure. An LST-tagged reference to a list cons $[H|T]$
points to a block of two consecutive words where the first word stores the car
$H$ and the second word stores the cdr $T$.
Figure 1 gives the definition of the function copy_term that copies a numbered
term from the stack/heap to the table area. The hash function is designed in
such a way that the hash code of a non-ground term is always 0. The function
call seq_hcode(code1,code2) gives the combined hash code of the two hash codes
from two components:
int seq_hcode(int code1, int code2){
if (code1==0) return 0;
if (code2==0) return 0;
return code1+31*code2+1;
}
If either code is 0, then the resulting code is 0 too.333Note that this way of
combing hash codes is for hash consing terms. For the subgoal and answer
tables, hash codes are combined in a different way.
It is assumed that all the variables in a subgoal have been numbered before
the arguments are copied. In the real implementation, variables are numbered
inside the function copy_term. The function call
copy_subgoal_args(src,des,arity) copies the arguments of a numbered subgoal to
the table area where (src-i) points to the ith argument on the stack and
(des+i) is the destination in the table area where the argument is copied to.
In the TOAM architecture [Zhou (2012)] on which B-Prolog is based, arguments
are passed through the stack and the stack grows downward from high addresses
to low ones. That is why (src-1) points to the first argument and (src-arity)
points to the last argument of the subgoal. A similar function is used to copy
answers to the table area.
int copy_subgoal_args(TermPtr src, TermPtr des, int arity){
---
| hcsum = 0;
| for (i=1;i<=arity;i++){
| | hcode = copy_term(*(src-i), des+i);
| | hc_sum = seq_hcode(hc_sum,hcode);
| }
| return hc_sum;
}
int copy_term(Term t, TermPtr des){
| deref(t);
| switch (tag(t)){
| case NUMVAR:
| | *des = t;
| | return 0;
| case ATM:
| | *des = t;
| | return atomic_hcode(t);
| case LST:
| | p1 = untag(t);
| | p2 = allocate_from_table(2);
| | car_code = copy_term(*p1, p2);
| | cdr_code = copy_term(*(p1+1), p2+1);
| | hcode = seq_hcode(car_code,cdr_code);
| | t1 = add_tag(p2,LST);
| | *des = t1;
| | return hcode;
| case STR:
| | p1 = untag(t);
| | sym = *p1;
| | arity = get_arity(sym);
| | p2 = allocate_from_table(arity+1);
| | hcode = *p2 = sym;
| | for (i=1;i<=arity;i++)
| | | hcode = seq_hcode(hcode, copy_term(*(p1+i), p2+i));
| | t1 = add_tag(p2,STR);
| | *des = t1;
| | return hcode;
| } /* end switch */
} /* end copy_term */
Figure 1: Copy data to the table area with no sharing.
The function copy_term is not tail recursive and can easily cause the native C
stack to overflow when copying large lists. In the real implementation, an
iterative version is used to copy a list and compute its hash code. For a
cons, the function needs to compute the hash codes of the car and the cdr
before computing its hash code. The function does this in two passes: in the
first pass it reverses the list and in the second pass it computes the hash
codes while reversing the list back.
The function copy_term exploits no sharing of data. Consider, for example, the
following program and the query is_list([1,2]). After completion of the query,
the subgoal table contains three tabled subgoals, is_list([1,2]),
is_list([2]), and is_list([]), and each subgoal’s answer table contains an
answer that is just a copy of the subgoal itself. No data are shared among the
copies of the terms. So there are two separate copies of [1,2] and two
separate copies of [2] in the table area. In the WAM representation of lists,
a cons requires two words to store, so 12 words are used in total. In general,
the query is_list([1,2,...,N]) consumes $O({\mathtt{N}}^{2})$ space in the
table area.
## 4 Hash-Consing of Ground Compound Terms
Hash-consing, like tabling, is a memoization technique which uses a hash table
to memorize values that have been created. Before creating a new value, it
looks up the table to see if the value exists. If so, it reuses the existing
value, otherwise, it inserts the value into the table. The concept of hash-
consing originates from implementations of Lisp that attempt to reuse cons
cells that have been constructed before [Goto (1974)]. This technique has also
been suggested for Prolog (e.g., for sharing answers of findall/3 [O’Keefe
(2001)]), but its use in Prolog implementations is unknown, not to mention its
use in tabling.
Let’s call the hash table used for all ground terms terms-table. Figure 2
gives an updated version of copy_term that performs hash-consing. If the term
is a list or a structure, the function copies it into the table area first. If
the term is ground, it then calls the function hash_consing(t1,hcode) to look
up the terms-table to see if a copy of t1 already exists in the table. If so,
hash_consing(t1,hcode) returns the copy; otherwise, it inserts t1 into the
terms-table and returns t1 itself. If an old copy in the terms-table is
returned (`t1 != t2`), the function deallocates the memory space allocated for
the current copy.
With hash-consing, the query ?-is_list([1,2]) only creates one copy of [1,2]
in the table area and the list is shared by the subgoals and the answers. As
[2] is the cdr of [1,2], no separate copy is stored for it. So, only 4 words
are used in total for the list. The number of words used for hashing the two
lists varies, depending on if there is a collision. If no collision occurs,
two slots in the terms-table are used; otherwise, one slot in the terms-table
is used and one node with two words is used to chain the two lists. So in the
worst case, 7 words are needed in total.
int copy_term(Term t, TermPtr des){
---
| deref(t);
| switch (tag(t)){
| case NUMVAR:
| | *des = t;
| | return 0;
| case ATM:
| | *des = t;
| | return atomic_hcode(t);
| case LST:
| | p1 = untag(t);
| | p2 = allocate_from_table(2);
| | car_code = copy_term(*p1, p2);
| | cdr_code = copy_term(*(p1+1), p2+1);
| | hcode = seq_hcode(car_code,cdr_code);
| | t1 = add_tag(p2,LST);
| | if (is_ground_hcode(hcode)){
| | | t2 = hash_consing(t1,hcode);
| | | if (t1 != t2){
| | | | deallocate_to_table(2);
| | | | t1 = t2;
| | | }
| | }
| | *des = t1;
| | return hcode;
| case STR:
| | p1 = untag(t);
| | sym = *p1;
| | arity = get_arity(sym);
| | p2 = allocate_from_table(arity+1);
| | hcode = *p2 = sym;
| | for (i=1;i<=arity;i++)
| | | hcode = seq_hcode(hcode, copy_term(*(p1+i), p2+i));
| | t1 = add_tag(p2,STR);
| | if (is_ground_hcode(hcode)){
| | | t2 = hash_consing(t1,hcode);
| | | if (t1 != t2){
| | | | deallocate_to_table(arity+1);
| | | | t1 = t2;
| | | }
| | }
| | *des = t1;
| | return hcode;
| } /* end switch */
} /* end copy_term */
Figure 2: Copy data with hash-consing.
## 5 Enhanced Hash-Consing
With hash-consing, the tabled subgoal is_list([1,...,N]) consumes only linear
table space now. Nevertheless, its time complexity remains quadratic in N.
This is because for each descendant subgoal is_list([K,...,N]) (K$>$1) the
hash code of the list [K,...,N] has to be computed and the terms-table has to
be looked up. We enhance hash-consing with two techniques to lower the time
complexity of is_list([1,...,N]) to linear.444The worst case time complexity
is still quadratic in theory if a poorly designed hash function is used.
### 5.1 Hash code memoization
The first technique is to table hash codes of structured terms in the table
area. For each structure or a list cons in the table area, we use an extra
word to store its hash code. The WAM representation of terms is not changed.
The word for the hash code of a compound term is located right before the
term. So assume p is the untagged reference to a structure or a list cons,
then p-1 references the hash code.
Figure 3 gives a new version of copy_term that tables hash codes. Tabled hash
codes are used for two purposes. Firstly, when searching for the term t1 in
the hash chain, the function hash_consing(t1,hcode) always compares the hash
codes first and only when the codes are equal will it compare the terms.
Secondly, the system reuses the tabled hash codes of terms when it expands a
hash table and rehashes the terms into the new hash table.
With tabled hash codes, the subgoal is_list([1,...,N]) still takes quadratic
time since the list [1,...,N] resides on the heap and for each descendant
subgoal, the hash code of the argument is not available and hence has to be
computed. To avoid this computation, we introduce input sharing.
int copy_term(Term t, TermPtr des){
---
| deref(t);
| switch (tag(t)){
| case NUMVAR:
| | *des = t;
| | return 0;
| case ATM:
| | *des = t;
| | return atomic_hcode(t);
| case LST:
| | p1 = untag(t);
| | if (!is_heap_reference(p1)){
| | | *des = t;
| | | return *(p1-1); /* return the tabled hash code */
| | }
| | p2 = allocate_from_table( 3);
| | p2++;
| | car_code = copy_term(*p1, p2);
| | cdr_code = copy_term(*(p1+1), p2+1);
| | hcode = seq_hcode(car_code,cdr_code);
| | *(p2-1) = hcode;
| | t1 = add_tag(p2,LST);
| | if (is_ground_hcode(hcode)){
| | | t2 = hash_consing(t1,hcode);
| | | if (t1 != t2){
| | | | deallocate_to_table( 3);
| | | | t1 = t2;
| | | }
| | }
| | *des = t1;
| | return hcode;
| case STR:
| | p1 = untag(t);
| | if (!is_heap_reference(p1)){
| | | *des = t;
| | | return *(p1-1); /* return the tabled hash code */
| | }
| | sym = *p1;
| | arity = get_arity(sym);
| | p2 = allocate_from_table( arity+2);
| | p2++;
| | hcode = *p2 = sym;
| | for (i=1;i<=arity;i++)
| | | hcode = seq_hcode(hcode, copy_term(*(p1+i), p2+i));
| | *(p2-1) = hcode;
| | t1 = add_tag(p2,STR);
| | if (is_ground_hcode(hcode)){
| | | t2 = hash_consing(t1,hcode);
| | | if (t1 != t2){
| | | | deallocate_to_table( arity+2);
| | | | t1 = t2;
| | | }
| | }
| | *des = t1;
| | return hcode;
| } /* end switch */
} /* end copy_term */
Figure 3: Tabling hash codes while copying with hash-consing.
### 5.2 Input Sharing
Input sharing amounts to letting a subgoal share its ground terms with its
answers and descendant subgoals. Consider the tabled subgoal is_list([1,2,3]).
The answer is the same as the subgoal, so it shares the term [1,2,3] with the
subgoal in the table area. The direct descendant subgoal is is_list([2,3]).
Since the list [2,3] is a suffix of [1,2,3], the descendant subgoal should
share it with the original subgoal in the table area.
To implement input sharing, we let the copying procedure set the frame slot of
an argument of a tabled subgoal to the address of the copied argument in the
table area if the argument is a ground structured term. So for the tabled
subgoal is_list([1,2,3]), the frame slot of the argument initially references
the list [1,2,3] on the heap. After the subgoal is copied to the table area,
the frame slot is set to reference the copy of the list in the table area. In
this way, the list will be shared by answers and the descendant subgoals. For
programs that do not use destructive assignments, which is the case for tabled
programs, updating frame slots this way causes no problem.
The function copy_subgoal_args shown in Figure 4 implements input sharing.
When an argument is found to be ground, the function lets the stack slot of
the argument reference its copy in the table area. The function copy_term (in
Figure 3) tests the reference to a compound term to see if the term needs to
be copied. If it is not a heap reference, then the referenced term must reside
in the table area and thus can be reused.
Note that our input sharing scheme has its limitation in the sense that it
fails to facilitate sharing of ground components in non-ground arguments.
Consider, for example, the subgoal is_list([X,2,3]). The suffix [2,3] will not
be shared through input sharing in our implementation since the argument is
not ground. It will eventually be shared through hash-consing, but its hash
code needs to be computed again when it occurs in a descendant subgoal or an
answer.
int copy_subgoal_args(TermPtr src, TermPtr des, int arity){
---
| hcsum = 0;
| for (i=1;i<=arity;i++){
| | hcode = copy_term(*(src-i), des+i);
| | if (is_ground_hcode(hcode)) *(src-i) = *(des+1);
| | hc_sum = seq_hcode(hc_sum,hcode);
| }
| return hc_sum;
}
Figure 4: Input sharing by updating frame slots.
## 6 Evaluation
The improved tabling system described in this paper has been implemented and
made available with B-Prolog version 7.7 (BP7.7). We evaluate the proposed
approach by comparing BP7.7 with YAP (version 6.3.2) and XSB (version 3.3.6),
and also the previous version of B-Prolog, version 7.6 (BP7.6), which did not
have enhanced hash-consing. We also compare it with indexed programs produced
by the transformation proposed in [Have and Christiansen (2012)] running on
B-Prolog 7.6 (indexed). We use the is_list/1 predicate, the
edit_distance/3555The source code is available in [Have and Christiansen
(2012)]. program, and a PRISM program to show the effectiveness of the
proposed techniques. We also test on a program that favors prefix sharing with
tries more than suffix sharing with hash-consing. In addition, we also show
results for the CHAT suite and the ATR parser, the traditional benchmarks used
to evaluate tabling systems.
The results are obtained on a Linux machine with 16 2.4 GHz, 64 bit Intel
Xeon(R) E7340 processor cores and 64 GB of memory. For this evaluation, only a
single processor core is utilized. CPU times (in seconds) and table space (in
kilobytes) consumptions are measured using the statistics/1 built-in for BP
and XSB, and table_statistics/1 for YAP.
Table 1 shows the results on the query is_list([1,1,...,1]) where N is the
number of 1s in the list. All the systems except for BP7.6 demonstrate a
close-to-linear complexity. The higher time complexity of BP7.6 is due to that
fact that BP7.6 only uses the first three elements of a list as the key and
hashing degenerates into linear search for the query because of hash
collision. The difference in time among BP7.7, YAP and XSB is at least a large
constant factor. As mentioned above, a trie allows for sharing of prefixes
while hash-consing allows for sharing of suffixes as long as lists are
concerned. For a list that contains repeated data, there are an equal number
of prefixes and suffixes, and hence both types of sharing are equally favored.
The difference between BP7.7 and indexed is only a small constant factor.
Table 1: Results on is_list([1,1,...,1]) | BP7.7 | BP7.6 | indexed | YAP | XSB
---|---|---|---|---|---
N | time | space | time | space | time | space | time | space | time | space
500 | 0.000 | 33 | 0.098 | 43 | 0.001 | 39 | 0.007 | 90 | 0.003 | 399
1000 | 0.001 | 66 | 0.776 | 86 | 0.003 | 78 | 0.033 | 180 | 0.010 | 567
1500 | 0.001 | 99 | 2.608 | 128 | 0.004 | 117 | 0.073 | 269 | 0.019 | 735
2000 | 0.002 | 131 | 6.169 | 171 | 0.005 | 156 | 0.134 | 359 | 0.037 | 903
2500 | 0.001 | 164 | 12.034 | 214 | 0.006 | 195 | 0.186 | 449 | 0.058 | 1071
3000 | 0.002 | 197 | 20.777 | 257 | 0.008 | 234 | 0.282 | 539 | 0.078 | 1239
3500 | 0.002 | 229 | 32.975 | 300 | 0.009 | 273 | 0.384 | 629 | 0.108 | 1407
4000 | 0.003 | 264 | 49.204 | 343 | 0.011 | 312 | 0.498 | 719 | 0.139 | 1575
4500 | 0.003 | 297 | 70.048 | 386 | 0.011 | 351 | 0.571 | 809 | 0.177 | 1743
5000 | 0.003 | 330 | 96.112 | 429 | 0.013 | 390 | 0.729 | 898 | 0.217 | 1911
Table 2 shows the results on the query is_list(L) where L is a list of random
constants.666A random number generator is used to generate the lists. For each
size, the same list was used for all the systems. BP consumes linear space and
linear time; YAP consumes linear space thanks to the global trie for terms but
takes quadratic time; XSB is quadratic in both time and space. For random
lists, suffix sharing with hash consing is clearly more effective than prefix
sharing with tries.
Table 2: Results on is_list(L) where L contains random data. | BP7.7 | BP7.6 | indexed | YAP | XSB
---|---|---|---|---|---
N | time | space | time | space | time | space | time | space | time | space
500 | 0.000 | 33 | 0.000 | 43 | 0.002 | 39 | 0.008 | 90 | 0.024 | 9990
1000 | 0.001 | 66 | 0.001 | 86 | 0.002 | 78 | 0.032 | 180 | 0.063 | 39236
1500 | 0.001 | 99 | 0.001 | 128 | 0.004 | 117 | 0.082 | 270 | 0.142 | 87991
2000 | 0.001 | 132 | 0.002 | 171 | 0.005 | 156 | 0.134 | 360 | 0.252 | 156269
2500 | 0.001 | 164 | 0.003 | 214 | 0.007 | 195 | 0.218 | 450 | 0.387 | 244071
3000 | 0.002 | 197 | 0.003 | 257 | 0.008 | 234 | 0.341 | 540 | 0.559 | 351401
3500 | 0.002 | 229 | 0.004 | 300 | 0.010 | 273 | 0.401 | 630 | 0.766 | 478260
4000 | 0.003 | 264 | 0.005 | 343 | 0.011 | 312 | 0.537 | 719 | 0.978 | 624640
4500 | 0.003 | 297 | 0.006 | 386 | 0.012 | 351 | 0.703 | 809 | 1.244 | 790555
5000 | 0.004 | 330 | 0.008 | 429 | 0.013 | 390 | 0.894 | 899 | 1.504 | 975990
Tables 3 and 4 show the results on the edit_distance program with repeated
data and random data, respectively. The main predicate edit(L1,L2,D) in the
program computes the distance between L1 and L2, i.e., the number of
substitutions, insertions and deletions needed to transform L1 to L2. The
tabled version finds all solutions. BP7.7 is significantly faster than BP7.6
on the type of queries that use repeated data. BP7.7 also outperforms YAP and
XSB in both time and space on both types of queries. Similar to the is_list
benchmark, enhanced hash-consing is asymptotically more effective than tries
on random data.
Table 3: Results on edit([1,1,...,1],[1,1,...,1],D). | BP7.7 | BP7.6 | indexed | YAP | XSB
---|---|---|---|---|---
N | time | space | time | space | time | space | time | space | time | space
30 | 0.000 | 60 | 0.026 | 97 | 0.003 | 90 | 0.005 | 213 | 0.006 | 1273
60 | 0.003 | 233 | 0.726 | 378 | 0.016 | 348 | 0.034 | 819 | 0.057 | 4341
90 | 0.007 | 519 | 5.189 | 841 | 0.036 | 776 | 0.107 | 1820 | 0.235 | 9435
120 | 0.015 | 917 | 21.216 | 1487 | 0.064 | 1372 | 0.266 | 3214 | 0.736 | 16554
150 | 0.022 | 1427 | 63.536 | 2316 | 0.102 | 2137 | 0.517 | 5002 | 1.635 | 25698
180 | 0.031 | 2051 | 156.072 | 3328 | 0.142 | 3071 | 0.942 | 7183 | 3.041 | 36868
210 | 0.047 | 2786 | 334.190 | 4523 | 0.208 | 4173 | 1.533 | 9759 | 5.035 | 50064
240 | 0.060 | 3634 | 646.550 | 5900 | 0.267 | 5445 | 2.367 | 12728 | 7.662 | 65285
270 | 0.074 | 4595 | 1159.182 | 7460 | 0.339 | 6885 | 3.081 | 16090 | 11.327 | 82531
300 | 0.095 | 5668 | 1955.331 | 9204 | 0.448 | 8493 | 4.401 | 19847 | 15.664 | 101803
Table 4: Results on edit(L1,L2,D) where L1 and L2 contain random data. | BP7.7 | BP7.6 | indexed | YAP | XSB
---|---|---|---|---|---
N | time | space | time | space | time | space | time | space | time | space
30 | 0.001 | 61 | 0.000 | 97 | 0.004 | 90 | 0.005 | 214 | 0.011 | 4148
60 | 0.003 | 234 | 0.006 | 378 | 0.020 | 348 | 0.045 | 822 | 0.099 | 27706
90 | 0.010 | 521 | 0.016 | 841 | 0.038 | 776 | 0.118 | 1823 | 0.313 | 89645
120 | 0.017 | 919 | 0.033 | 1487 | 0.067 | 1372 | 0.298 | 3218 | 0.759 | 209183
150 | 0.027 | 1430 | 0.057 | 2316 | 0.105 | 2137 | 0.591 | 5007 | 1.501 | 404752
180 | 0.038 | 2054 | 0.094 | 3328 | 0.148 | 3071 | 1.058 | 7190 | 2.771 | 695363
210 | 0.056 | 2790 | 0.156 | 4523 | 0.217 | 4173 | 1.695 | 9766 | 4.271 | 1099906
240 | 0.073 | 3639 | 0.219 | 5900 | 0.282 | 5445 | 2.687 | 12736 | 6.247 | 1637354
270 | 0.092 | 4600 | 0.297 | 7460 | 0.352 | 6885 | 3.782 | 16100 | 8.787 | 2327276
300 | 0.114 | 5674 | 0.435 | 9204 | 0.466 | 8493 | 5.248 | 19857 | 11.954 | 3187340
Table 5 compares BP7.7 and BP7.6 on the PRISM program that simulates a two-
state hidden Markov model [Sato et al. (2010)]. For our benchmarking purpose,
the training data of the form hmm([a,b,a,b,...]) are used, and only the time
and space required to find all the explanations are measured. While BP7.7
consumes slightly more space than BP7.6 due to the overhead of hash-consing,
it outperforms BP7.6 in time by a linear factor.
Table 5: Results on the PRISM program HMM. | BP7.7 | BP7.6
---|---|---
N | time | space | time | space
2000 | 0.002 | 222 | 1.164 | 179
3000 | 0.005 | 333 | 3.911 | 269
4000 | 0.006 | 444 | 9.249 | 359
5000 | 0.008 | 555 | 18.044 | 449
6000 | 0.010 | 666 | 31.150 | 539
7000 | 0.011 | 776 | 49.441 | 628
8000 | 0.013 | 889 | 73.774 | 718
9000 | 0.015 | 1000 | 105.049 | 808
10000 | 0.018 | 1111 | 144.140 | 898
Although it is more common for subgoals of recursive programs to share
suffixes than prefixes, it is possible to find programs on which prefix
sharing with tries is more effective than suffix sharing with hash-consing.
The following gives such a program:
:-table create_list/2.
create_list(N,L):-
between(1,N,I),
range(1,I,L).
The query create_list(N,L) creates N lists [1], [1,2], …, and [1,2,...,N] that
have only common prefixes. As shown in Table 6, XSB consumes linear space,
while BP and YAP consume quadratic space. YAP tables all suffixes into the
global trie for terms and there are $O({\mathtt{N}}^{2})$ suffixes. BP7.7
consumes more table space than BP7.6 since all the terms are hash-consed but
none is shared. BP7.6 is slower than BP7.7 since the hash function used in
BP7.6, which is based on the first three elements of a list, results in more
collisions than BP7.7.
Table 6: Results on create_list(N,L). | BP7.7 | BP7.6 | YAP | XSB
---|---|---|---|---
N | time | space | time | space | time | space | time | space
500 | 0.035 | 2417 | 0.107 | 990 | 0.039 | 3965 | 0.023 | 290
1000 | 0.201 | 9564 | 0.827 | 3937 | 0.201 | 15742 | 0.043 | 348
1500 | 0.654 | 21635 | 2.989 | 8831 | 0.523 | 35332 | 0.095 | 407
2000 | 0.969 | 37926 | 7.245 | 15679 | 0.962 | 62734 | 0.169 | 465
2500 | 2.151 | 60082 | 14.130 | 24480 | 1.699 | 97949 | 0.264 | 524
3000 | 2.660 | 85890 | 24.343 | 35249 | 2.630 | 140976 | 0.378 | 583
3500 | 3.276 | 116011 | 38.397 | 47956 | 3.739 | 191816 | 0.517 | 641
4000 | 4.011 | 150192 | 57.217 | 62616 | 5.071 | 250468 | 0.675 | 700
4500 | 7.319 | 194310 | 80.994 | 79229 | 6.978 | 316933 | 0.853 | 758
5000 | 8.316 | 238885 | 110.631 | 97796 | 9.267 | 391211 | 1.051 | 817
Table 7 compares the systems on the CHAT benchmark suite and the ATR parser.
There is almost no difference between BP7.7 and BP7.6 in time and the space
overhead incurred by hash-consing is noticeable. Hash-consing has no positive
effect on these programs because the sequences used in the programs are very
short.
Table 7: Results on the CHAT benchmarks and the ATR parser. | BP 7.7 | BP 7.6 | YAP | XSB
---|---|---|---|---
Benchmark | time | space | time | space | time | space | time | space
cs_o | 0.015 | 198 | 0.0129 | 11 | 0.009 | 26 | 0.011 | 285
cs_r | 0.025 | 332 | 0.026 | 11 | 0.019 | 27 | 0.022 | 286
disj | 0.008 | 108 | 0.009 | 11 | 0.005 | 23 | 0.007 | 277
gabriel | 0.011 | 111 | 0.012 | 9 | 0.006 | 20 | 0.008 | 272
kalah | 0.008 | 90 | 0.008 | 15 | 0.006 | 35 | 0.008 | 304
pg | 0.006 | 69 | 0.006 | 7 | 0.004 | 15 | 0.006 | 263
read | 0.057 | 987 | 0.058 | 23 | 0.099 | 46 | 0.030 | 327
atr | 0.509 | 15111 | 0.543 | 5947 | 0.325 | 52520 | 0.280 | 45400
## 7 Related Work
Since structure sharing [Boyer and Moore (1972)] was discarded and the Warren
Abstract Machine (WAM) [Warren (1983)] triumphed as the implementation model
of Prolog, there has been little attention paid to exploiting data sharing in
Prolog implementations.777A lot of work has been done on indexing Prolog
terms, but indexing is a different kind of sharing since it does not consider
reuse of terms from different sources. In his Diploma thesis [Neumerkel
(1989)], Ulrich Neumerkel gave several example Prolog programs that would
consume an-order-of-magnitude less space with data sharing than without
sharing. He proposed applying hash-consing and DFA-minimization to sharing
terms including cyclic ones. The proposed approach would incur considerable
overhead if every compound term is hash-consed when created, and hence it is
infeasible to incorporate the approach into the WAM. Following Appel and
Goncalves’s hash-consing garbage collector for SML/NJ [Appel and de Rezende
Goncalves (2003)], Nguyen and Demoen recently built a similar garbage
collector for hProlog [Nguyen and Demoen (2012)]. The garbage collector hash-
conses compound terms on the heap in one phase and performs absorption in
another phase such that for the replications of a compound term only one copy
is kept and all the others are garbage collected. Their experiment basically
confirms the disappointing result reported in Appel and Goncalves’s paper: the
overhead outweighs the gain except for special programs.
Hash-consing can be applied to the built-in predicate findall/3, as suggested
by O’Keefe [O’Keefe (2001)], to avoid repeatedly copying the same term in
different answers. Currently, B-Prolog is the only Prolog system that supports
hash-consing for findall/3. It employs a hash table for ground terms in the
findall area. The algorithm and memory manager developed for the table area is
reused for the findall area. With hash-consing, the system copies a ground
term only once when copying answers from the findall area to the heap. Input
sharing is exploited in the same way as for tabled subgoals. For a findall
call, the compiler converts it into a call to a temporary predicate such that
each argument of the generator occupies one slot in the stack frame. At
runtime, the system first copies the arguments of the generator from the
stack/heap to the findall area before the generator is executed. When an
argument of the generator is found to be a ground compound term, its frame
slot is set to reference the copy in the findall area. In this way, the
argument and its subterms can be reused by the answers and the descendant
calls. Nguyen and Demoen’s implementation of input sharing for findall/3
[Nguyen and Demoen (2012)] distinguishes between old terms that are created
before the generator and new terms that are generated by the generator, and
have answers share the old terms. Their scheme can exploit sharing of not only
ground arguments but also ground terms in non-ground arguments. Their scheme
may not be suited for tabled data since, unlike data in the findall area which
live and die with the generator, tabled data are permanent. Also, their
implementation does not exploit output sharing.
A trie has been a popular data structure for organizing tabled subgoals and
answers [Ramakrishnan et al. (1998)]. It is adopted by all the tabled Prolog
systems except B-Prolog. As far as lists are concerned, a trie facilitates
sharing of the prefixes while hash-consing allows for sharing of the suffixes.
So for the two lists [1,2] and [1,2,3], the former shares the same path as the
latter in the trie, but they are treated as separate lists when hash-consed;
for the two lists [2,3] and [1,2,3], however, a trie allows for no sharing
while hash-consing allows for complete sharing.
Another advantage of tries is that they can be used to perform both variant
testing and subsumption testing, and thus can be used in both variant-based
and subsumption-based tabling systems. Hash-consing, on the other hand, can be
used to perform equivalence testing only and thus cannot directly be used for
subsumption-based tabling.
Terms stored in a trie have a different representation from terms on the heap.
For example, in the YAP system, tries are represented as trie instructions
[Santos Costa et al. (2012)]. For this reason, when an answer is returned, it
must be copied from its trie in the table area to the heap even if it is
ground. In our system, structured ground terms in the table area have exactly
the same representation as on the heap, so when they occur in an answer they
do not need to be copied when the answer is returned.
In the original implementation of XSB and YAP, one trie is used for all tabled
subgoals, and for each subgoal one trie is used for the answer table. To
enhance sharing, Raimundo and Rocha propose using a global trie for all tabled
structured terms [Raimundo and Rocha (2011)]. Due to the necessity of copying
answers from the table area to the heap, the time complexity remains the same
even when the space complexity drops.
To some extent, the idea of representing sentences as position indexed facts
[Have and Christiansen (2012), Swift et al. (2009)] is similar to hash-consing
in the sense that a hash-consed term always is associated with a hash code.
The translation from a program that deals with sequences represented as lists
into one that uses position representation is not trivial. When difference
lists are involved, the translation is even more complicated. The program
obtained after translation may lose sharing opportunities. Therefore, hash-
consing is a more practical solution to sharing than program transformation.
As far as we know, our implementation is the first attempt to apply hash-
consing to tabling. Our implementation enhances hash-consing with input
sharing and hash code memoization to speed-up computation of hash codes. The
extra cell used to store the hash code of a compound term is overhead if the
term is never shared. Nevertheless, while the increase of space is always a
constant factor, the gain in speed can be linear in the size of the data.
## 8 Conclusion
We have presented an implementation of hash-consing for tabling structured
data. Hash-consing facilitates sharing of structured data and can eliminate
the extra linear factor of space complexity commonly seen in early tabling
systems when dealing with sequences. Hash-consing alone does not change the
time complexity. We have enhanced it with input sharing and hash code
memoization to eliminate the extra linear factor of time complexity in dealing
with sequences. The resulting tabling system significantly improves the
scalability of language parsing and bio-sequence analysis applications.
Our work will shed some light on the discussion on what data structure to use
for tabled data. A trie is suitable for sharing prefixes and hash-consing is
suitable for sharing suffixes of sequences. Although it is possible to find
programs that make prefix sharing arbitrarily better than suffix sharing, it
is more common for subgoals of recursive programs to share suffixes than
prefixes. Therefore, hash-consing is in general a better choice than tries as
a data structure for representing tabled data. Hash-consing as it is in our
implementation is not suitable for subsumption-based tabling. It is future
work to adapt hash-consing to subsumption testing.
## Acknowledgements
The PRISM system has been the motivation for this project and we thank Taisuke
Sato and Yoshitaka Kameya for their discussion. We also thank the anonymous
referees for their detailed comments on the presentation. Neng-Fa Zhou was
supported in part by NSF (No.1018006) and Christian Theil Have was supported
by the project Logic-statistic modelling and analysis of biological sequence
data funded by the NABIIT program under the Danish Strategic Research Council.
## References
* Appel and de Rezende Goncalves (2003) Appel, A. W. and de Rezende Goncalves, M. J. 2003\. Hash-consing garbage collection. Technical Report TR 74-03, Princeton University.
* Boyer and Moore (1972) Boyer, R. S. and Moore, J. S. 1972\. A sharing of structure in theorem proving programs. Machine Intelligence 7, 101–116.
* Chen and Warren (1996) Chen, W. and Warren, D. S. 1996\. Tabled evaluation with delaying for general logic programs. Journal of the ACM 43, 1, 20–74.
* Ershov (1959) Ershov, A. 1959\. On programming of arithmetic operations. Communications of the ACM 1, 8, 3–6.
* Goto (1974) Goto, E. 1974\. Monocopy and associative algorithms in extended Lisp. Technical Report TR 74-03, University of Tokyo.
* Have and Christiansen (2012) Have, C. T. and Christiansen, H. 2012\. Efficient tabling of structured data using indexing and program transformation. In PADL. LNCS 7149, 93–107.
* Michie (1968) Michie, D. 1968\. “memo” functions and machine learning. Nature, 19–22.
* Neumerkel (1989) Neumerkel, U. 1989\. Garbage collection in Prolog systems (in German). Ph.D. thesis, Thesis, Technical University of Vienna.
* Nguyen and Demoen (2012) Nguyen, P.-L. and Demoen, B. 2012\. Representation sharing for Prolog. TPLP.
* O’Keefe (2001) O’Keefe, R. A. 2001\. O(1) reversible tree navigation without cycle. TPLP 1, 5, 617–630.
* Raimundo and Rocha (2011) Raimundo, J. and Rocha, R. 2011\. Global trie for subterms. In CICLOPS.
* Ramakrishnan et al. (1998) Ramakrishnan, I., Rao, P., Sagonas, K., Swift, T., and Warren, D. 1998\. Efficient access mechanisms for tabled logic programs. Journal of Logic Programming 38, 31–54.
* Santos Costa et al. (2012) Santos Costa, V., Rocha, R., and Damas, L. 2012\. The YAP Prolog system. TPLP, Special Issue on Prolog Systems 12, 1-2, 5–34.
* Sato and Kameya (2008) Sato, T. and Kameya, Y. 2008\. New advances in logic-based probabilistic modeling by PRISM. In Probabilistic Inductive Logic Programming. 118–155.
* Sato et al. (2010) Sato, T., Zhou, N.-F., Kameya, Y., and Yizumi, Y. 2010\. The PRISM user’s manual. http://www.mi.cs.titech.ac.jp/prism/.
* Somogyi and Sagonas (2006) Somogyi, Z. and Sagonas, K. 2006\. Tabling in Mercury: Design and implementation. In PADL. LNCS 3819, 150–167.
* Swift and Warren (2012) Swift, T. and Warren, D. S. 2012\. XSB: Extending Prolog with tabled logic programming. TPLP, Special issue on Prolog systems 12, 1-2, 157–187.
* Swift et al. (2009) Swift, T., Warren, D. S., et al. 2009\. The XSB Programmer’s Manual: vols. 1 and 2. http://xsb.sf.net.
* Warren (1983) Warren, D. H. D. 1983\. An abstract Prolog instruction set. Technical note 309, SRI International.
* Zhou (2012) Zhou, N.-F. 2012\. The language features and architecture of B-Prolog. TPLP, Special Issue on Prolog Systems 12, 1-2, 189–218.
* Zhou et al. (2008) Zhou, N.-F., Sato, T., and Shen, Y.-D. 2008\. Linear tabling strategies and optimizations. TPLP 8, 1, 81–109.
|
arxiv-papers
| 2012-10-04T23:19:16 |
2024-09-04T02:49:35.970190
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Neng-Fa Zhou and Christian Theil Have",
"submitter": "Neng-Fa Zhou",
"url": "https://arxiv.org/abs/1210.1611"
}
|
1210.1627
|
# The perturbation of the group inverse under the stable perturbation in a
unital ring
Fapeng Du
School of Mathematical & Physical Sciences, Xuzhou Institute of Technology
Xuzhou 221008, Jiangsu Province, P.R. China
E-mail: jsdfp@163.com Yifeng Xue
Department of mathematics and Research Center for Operator Algebras
East China Normal University, Shanghai 200241, P.R. China Corresponding
author, E-mail: yfxue@math.ecnu.edu.cn
###### Abstract
Let $\mathfrak{R}$ be a ring with unit $1$ and
$a\in\mathfrak{R},\,\bar{a}=a+\delta a\in\mathfrak{R}$ such that $a^{\\#}$
exists. In this paper, we mainly investigate the perturbation of the group
inverse $a^{\\#}$ on $\mathfrak{R}$. Under the stable perturbation, we obtain
the explicit expressions of $\bar{a}^{\\#}$. The results extend the main
results in [23, 24] and some related results in [22]. As an application, we
give the representation of the group inverse of the matrix
$\begin{bmatrix}d&b\\\ c&0\end{bmatrix}$ on the ring $\mathfrak{R}$ for
certain $d,\,b,\,c\in\mathfrak{R}$.
## 1 Introduction
Let $\mathfrak{R}$ be a ring with unit and $a\in\mathfrak{R}$. we consider an
element $b\in\mathfrak{R}$ and the following equations:
$(1)$ $aba=a$, $(2)$ $bab=b$, $(3)$ $a^{k}ba=a^{k}$, $(4)$ $ab=ba$.
If $b$ satisfies $(1)$, then $b$ is called a pseudo–inverse or $1$–inverse of
$a$. In this case, $a$ is called regular. The set of all $1$–inverse of $a$ is
denoted by $a^{\\{1\\}}$; If $b$ satisfies $(2)$, then $b$ is called a
$2$–inverse of $a$, and $a$ is called anti–regular. The set of all $2$–inverse
of $a$ is denoted by $a^{\\{2\\}}$; If $b$ satisfies $(1)$ and $(2)$, then $b$
is called the generalized inverse of $a$, denoted by $a^{+}$; If $b$ satisfies
$(2)$, $(3)$ and $(4)$, then $b$ is called the Drazin inverse of $a$, denoted
by $a^{D}$. The smallest integer $k$ is called the index of $a$, denoted by
$ind(a)$. If $ind(a)=1$, we say $a$ is group invertible and $b$ is the group
inverse of $a$, denoted by $a^{\\#}$.
The notation so–called stable perturbation of an operator on Hilbert spaces
and Banach spaces is introduced by G. Chen and Y. Xue in [4, 6]. Later the
notation is generalized to Banach Algebra by Y. Xue in [23] and to Hilbert
$C^{*}$–modules by Xu, Wei and Gu in [21]. The stable perturbation of linear
operator was widely investigated by many authors. For examples, in [5], G.
Chen and Y. Xue study the perturbation for Moore–Penrose inverse of an
operator on Hilbert spaces; Q. Xu, C. Song and Y. Wei studied the stable
perturbation of the Drazin inverse of the square matrices when
$I-A^{\pi}-B^{\pi}$ is nonsingular in [20] and Q. Huang and W. Zhai worked
over the perturbation of closed operators in [16, 17], etc.. Some further
results can be found in [10, 11, 12, 13].
The Drazin inverse has many applications in matrix theory, difference
equations, differential equations and iterative methods. In 1979, Campbell and
Meyer proposed an open problem: how to find an explicit expression for the
Drazin inverse of the matrix $\begin{bmatrix}A&B\\\ C&D\end{bmatrix}$ in terms
of its sub-block in [1]? The representation of the Drazin inverse of a
triangular matrix $\begin{bmatrix}A&B\\\ 0&D\end{bmatrix}$ has been given in
[3, 9, 15]. In [8], Deng and Wei studied the Drazin inverse of the anti-
triangular matrix $\begin{bmatrix}A&B\\\ C&0\end{bmatrix}$ and given its
representation under some conditions.
In this paper, we investigate the stable perturbation for the group inverse of
an element in a ring. Assume that $1-a^{\pi}-\bar{a}^{\pi}$ is invertible, we
present the expression of $\bar{a}^{\\#}$ and $\bar{a}^{D}$. This extends the
related results in [22, 24]. As an applications, we study the representation
for the group inverse of the anti–triangular matrix $\begin{bmatrix}d&b\\\
c&0\end{bmatrix}$ on the ring.
## 2 Some Lemmas
Throughout the paper, $\mathfrak{R}$ is always a ring with the unit $1$. In
this section, we give some lemmas:
###### Lemma 2.1.
Let $a,\,b\in\mathfrak{R}$. Then $1+ab$ is invertible if and only if $1+ba$ is
invertible. In this case, $(1+ab)^{-1}=1-a(1+ba)^{-1}b$ and
$(1+ab)^{-1}a=a(1+ba)^{-1},\ b(1+ab)^{-1}=(1+ba)^{-1}b.$
###### Lemma 2.2.
Let $a,\,b\in\mathfrak{R}$. If $1+ab$ is left invertible, then so is $1+ba$.
Proof. Let $c\in\mathfrak{R}$ such that $c(1+ab)=1$. Then
$1+ba=1+bc(1+ab)a=1+bca(1+ba).$
Therefore, $(1-bca)(1+ba)=1$.
###### Lemma 2.3.
Let $a$ be a nonzero element in $\mathfrak{R}$ such that $a^{+}$ exists. If
$s=a^{+}a+aa^{+}-1$ is invertible in $\mathfrak{R}$, then $a^{\\#}$ exists and
$a^{\\#}=a^{+}s^{-1}+(1-a^{+}a)s^{-1}a^{+}s^{-1}$.
Proof. According to [18] or [22, Theorem 4.5.9], $a^{\\#}$ exists. We now give
the expression of $a^{\\#}$ as follows.
Put $p=a^{+}a$, $q=aa^{+}$. Then we have
$ps=pq=sq,\ qs=qp=sp,\ sa=a^{+}a^{2}.$ (2.1)
Set $y=a^{+}s^{-1}$. Then by (2.1),
$\displaystyle yp$ $\displaystyle=a^{+}s^{-1}a^{+}a=a^{+}aa^{+}s^{-1}=y=py,$
$\displaystyle pay$ $\displaystyle=a^{+}aaa^{+}s^{-1}=pqs^{-1}=p,$
$\displaystyle ypa$ $\displaystyle=a^{+}s^{-1}a^{+}aa=a^{+}a=p.$
Put $a_{1}=pap=pa$, $a_{2}=(1-p)ap=(1-p)a$. Then $a=a_{1}+a_{2}$ and it is
easy to check that $a^{\\#}=y+a_{2}y^{2}$. Using (2.1), we can get that
$a^{\\#}=a^{+}s^{-1}+(1-a^{+}a)a(a^{+}s^{-1})^{2}=a^{+}s^{-1}+(1-a^{+}a)s^{-1}a^{+}s^{-1}.$
Let $M_{2}(\mathfrak{R})$ denote the matrix ring of all $2\times 2$ matrices
over $\mathfrak{R}$ and let $1_{2}$ denote the unit of $M_{2}(\mathfrak{R})$.
###### Corollary 2.4.
Let $b,\,c\in\mathfrak{R}$ have group inverse $b^{\\#}$ and $c^{\\#}$
respectively. Assume that $k=b^{\\#}b+c^{\\#}c-1$ is invertible in
$\mathfrak{R}$. Then ${\begin{bmatrix}0&b\\\ c&0\end{bmatrix}}^{\\#}$ exists
with ${\begin{bmatrix}0&b\\\
c&0\end{bmatrix}}^{\\#}=\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&0\end{bmatrix}.$
In particular, when $b^{\\#}bc^{\\#}c=b^{\\#}b$ and
$c^{\\#}cb^{\\#}b=c^{\\#}c$, ${\begin{bmatrix}0&b\\\
c&0\end{bmatrix}}^{\\#}=\begin{bmatrix}0&b^{\\#}bc^{\\#}\\\
c^{\\#}cb^{\\#}&0\end{bmatrix}$.
Proof. Set $a=\begin{bmatrix}0&b\\\ c&0\end{bmatrix}$. Then
$a^{+}=\begin{bmatrix}0&c^{\\#}\\\ b^{\\#}&0\end{bmatrix}$ and
$a^{+}a+aa^{+}-1_{2}=\begin{bmatrix}b^{\\#}b+c^{\\#}c-1&0\\\
0&b^{\\#}b+c^{\\#}c-1\end{bmatrix}=\begin{bmatrix}k\\\ \ &k\end{bmatrix}$
is invertible in $M_{2}(\mathfrak{R})$. Noting that
$bb^{\\#}k^{-1}=k^{-1}cc^{\\#}$. Thus, by Lemma 2.3,
$\displaystyle a^{\\#}$ $\displaystyle=a^{+}\begin{bmatrix}k^{-1}\\\ \
&k^{-1}\end{bmatrix}+(1_{2}-a^{+}a)\begin{bmatrix}k^{-1}\\\ \
&k^{-1}\end{bmatrix}a^{+}\begin{bmatrix}k^{-1}\\\ \ &k^{-1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&c^{\\#}k^{-1}+(1-c^{\\#}c)k^{-1}c^{\\#}k^{-1}\\\
b^{\\#}k^{-1}+(1-b^{\\#}b)k^{-1}b^{\\#}k^{-1}&0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&0\end{bmatrix}.$
When $b^{\\#}bc^{\\#}c=b^{\\#}b$ and $c^{\\#}cb^{\\#}b=c^{\\#}c$, $k^{-1}=k$.
In this case, $\begin{bmatrix}0&b\\\
c&0\end{bmatrix}^{\\#}=\begin{bmatrix}0&b^{\\#}bc^{\\#}\\\
c^{\\#}cb^{\\#}&0\end{bmatrix}.$
###### Lemma 2.5.
Let $a,\,b\in\mathfrak{R}$ and $p$ be a non–trivial idempotent element in
$\mathfrak{R}$, i.e., $p\not=0,1$. Put $x=pap+pb(1-p)$.
1. $(1)$
If $pap$ is group invertible and $(pap)(pap)^{\\#}b(1-p)=pb(1-p)$, then $x$ is
group invertible too and $x^{\\#}=(pap)^{\\#}+[(pap)^{\\#}]^{2}pb(1-p)$.
2. $(2)$
If $x$ is group invertible, then so is the $pap$ and
$(pap)(pap)^{\\#}b(1-p)=pb(1-p)$.
Proof. (1) It is easy to check that $p(pap)^{\\#}=(pap)^{\\#}p=(pap)^{\\#}$.
Put $y=(pap)^{\\#}+[(pap)^{\\#}]^{2}pb(1-p)$. Then $xyx=x$, $yxy=y$ and
$xy=yx$, i.e., $y=x^{\\#}$.
(2) Set $y_{1}=px^{\\#}p$, $y_{2}=px^{\\#}(1-p)$, $y_{3}=(1-p)x^{\\#}p$ and
$y_{4}=(1-p)x^{\\#}(1-p)$. Then $x^{\\#}=y_{1}+y_{2}+y_{3}+y_{4}$. From
$xx^{\\#}x=x$, $x^{\\#}xx^{\\#}=x^{\\#}$ and $xx^{\\#}=x^{\\#}x$, we can
obtain that $y_{3}=y_{4}=0$ and
$(pap)y_{1}(pap)=pap,\quad y_{1}(pap)y_{1}=y_{1},\quad
y_{1}(pap)=(pap)y_{1},\quad y_{1}(pap)pb(1-p)=pb(1-p),$
that is, $(pap)^{\\#}=y_{1}$ and $(pap)(pap)^{\\#}b(1-p)=pb(1-p)$.
At the end of this section, we will introduce the notation of stable
perturbation of an element in a ring.
Let $\mathcal{A}$ be a unital Banach algebra and $a\in\mathcal{A}$ such that
$a^{+}$ exists. Let $\bar{a}=a+\delta a\in\mathcal{A}$. Recall from [23] that
$\bar{a}$ is a stable perturbation of $a$ if
$\bar{a}\mathcal{A}\cap(1-aa^{+})\mathcal{A}=\\{0\\}$. This notation can be
easily extended to the case of ring as follows.
###### Definition 2.6.
Let $a\in\mathfrak{R}$ such that $a^{+}$ exists and let $\bar{a}=a+\delta a\in
R$. We say $\bar{a}$ is a stable perturbation of $a$ if
$\bar{a}\mathfrak{R}\cap(1-aa^{+})\mathfrak{R}=\\{0\\}$.
Using the same methods as appeared in the proofs of [23, Proposition 2.2] and
[22, Theorem 2.4.7], we can obtain:
###### Proposition 2.7.
Let $a\in\mathfrak{R}$ and $\bar{a}=a+\delta a\in\mathfrak{R}$ such that
$a^{+}$ exists and $1+a^{+}\delta a$ is invertible in $\mathfrak{R}$. Then the
following statements are equivalent:
1. $(1)$
$\bar{a}^{+}$ exists and $\bar{a}^{+}=(1+a^{+}\delta a)^{-1}a^{+}$.
2. $(2)$
$\bar{a}\mathfrak{R}\cap(1-aa^{+})\mathfrak{R}=\\{0\\}$ $($ that is, $\bar{a}$
is a stable perturbation if $a)$.
3. $(3)$
$\bar{a}(1+a^{+}\delta a)^{-1}(1-a^{+}a)=0$.
4. $(4)$
$(1-aa^{+})(1+\delta aa^{+})^{-1}\bar{a}=0$.
5. $(5)$
$(1-aa^{+})\delta a(1-a^{+}a)=(1-aa^{+})\delta a(1+a^{+}\delta
a)^{-1}a^{+}\delta a(1-a^{+}a)$.
6. $(6)$
$\mathfrak{R}\bar{a}\cap\mathfrak{R}(1-a^{+}a)=\\{0\\}$.
## 3 Main results
In this section, we investigate the stable perturbation for group inverse and
Drazin inverse of an element $a$ in $\mathfrak{R}$.
Let $a\in\mathfrak{R}$ and $\bar{a}=a+\delta a\in\mathfrak{R}$ such that
$a^{\\#}$ exists and $1+a^{\\#}\delta a$ is invertible in $\mathfrak{R}$. Put
$a^{\pi}=(1-a^{\\#}a)$, $\Phi(a)=1+\delta aa^{\pi}\delta a[(1+a^{\\#}\delta
a)^{-1}a^{\\#}]^{2}$ and $B=\Phi(a)(1+\delta aa^{\\#}),\ C(a)=a^{\pi}\delta
a(1+a^{\\#}\delta a)^{-1}a^{\\#}.$ These symbols will be used frequently in
this section.
###### Lemma 3.1.
Let $a\in\mathfrak{R}$ and $\bar{a}=a+\delta a\in\mathfrak{R}$ such that
$a^{\\#}$ exists and $1+a^{\\#}\delta a$ is invertible in $\mathfrak{R}$.
Suppose that $\Phi(a)$ is invertible, then
$(Ba)^{\\#}=Baa^{\\#}B^{-1}a^{\\#}B^{-1}.$
Proof. Put $P=aa^{\\#}$. Noting that $\Phi(a)(1-P)=1-P$, we have
$P\Phi(a)P=P\Phi(a)$, $\Phi^{-1}(a)(1-P)=(1-P)$ and $PBP=PB$,
$B^{-1}(1-P)=(1+\delta aa^{\\#})^{-1}(1-P)$, $a^{\\#}B^{-1}(1-P)=0$, i.e.,
$a^{\\#}B^{-1}=a^{\\#}B^{-1}P$. Thus, $BPB^{-1}Ba=Ba$ and
$\displaystyle(Ba)(Baa^{\\#}B^{-1}a^{\\#}B^{-1})$
$\displaystyle=BPB^{-1}=(Baa^{\\#}B^{-1}a^{\\#}B^{-1})(Ba),$
$\displaystyle(Baa^{\\#}B^{-1}a^{\\#}B^{-1})(BPB^{-1})$
$\displaystyle=Baa^{\\#}B^{-1}a^{\\#}B^{-1}.$
These indicate $(Ba)^{\\#}=Baa^{\\#}B^{-1}a^{\\#}B^{-1}.$
###### Theorem 3.2.
Let $a\in\mathfrak{R}$ such that $a^{\\#}$ exists. Let $\bar{a}=a+\delta
a\in\mathfrak{R}$ with $1+a^{\\#}\delta a$ invertible in $\mathfrak{R}$.
Suppose that $\Phi(a)$ is invertible and
$\bar{a}\mathfrak{R}\cap(1-aa^{\\#})\mathfrak{R}=\\{0\\}$. Put
$D(a)=(1+a^{\\#}\delta a)^{-1}a^{\\#}\Phi^{-1}(a)$. Then $\bar{a}^{\\#}$
exists with
$\bar{a}^{\\#}=(1+C(a))(D(a)+D^{2}(a)\delta aa^{\pi})(1-C(a)).$
Proof. Put $P=aa^{\\#}$. By Proposition 2.7 (3), we have $a^{\pi}(1+\delta
aa^{\\#})^{-1}\bar{a}=0$ and
$P\bar{a}(1+a^{\\#}\delta a)^{-1}=a(aa^{\\#}+a^{\\#}\delta a)(1+a^{\\#}\delta
a)^{-1}=a(1+a^{\\#}\delta a-a^{\pi})(1+a^{\\#}\delta a)^{-1}=a.$
Thus, we have
$\displaystyle(1-C(a))\bar{a}(1+C(a))$ $\displaystyle=[P+a^{\pi}(1+\delta
aa^{\\#})^{-1}]\bar{a}[1+a^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}]$
$\displaystyle=P\bar{a}[1+a^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}]$
$\displaystyle=P\bar{a}+P\bar{a}a^{\pi}\delta a(1+a^{\\#}\delta
a)^{-1}a^{\\#}$ $\displaystyle=P\bar{a}+P\delta aa^{\pi}\delta
a(1+a^{\\#}\delta a)^{-1}a^{\\#}$ $\displaystyle=P\delta a+P[a+\delta
aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}]$ $\displaystyle=P\delta
a(1-P)+P\delta aP+P[a+\delta aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}]$
$\displaystyle=P\delta a(1-P)+P[\delta a+a+\delta aa^{\pi}\delta
a(1+a^{\\#}\delta a)^{-1}a^{\\#}]P$ $\displaystyle=P\delta
a(1-P)+P[\bar{a}+\delta aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}]P$
$\displaystyle=P\delta a(1-P)+P[\bar{a}(1+a^{\\#}\delta a)^{-1}+\delta
aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}(1+a^{\\#}\delta
a)^{-1}](1+a^{\\#}\delta a)P$ $\displaystyle=P\delta a(1-P)+P[a+\delta
aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}(1+a^{\\#}\delta
a)^{-1}](1+a^{\\#}\delta a)P$ $\displaystyle=P\delta a(1-P)+P[a+\delta
aa^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}(1+a^{\\#}\delta
a)^{-1}]a^{\\#}(1+\delta aa^{\\#})a$ $\displaystyle=P\delta
a(1-P)+P[aa^{\\#}+\delta aa^{\pi}\delta a(1+a^{\\#}\delta
a)^{-1}a^{\\#}(1+a^{\\#}\delta a)^{-1}a^{\\#}](1+\delta aa^{\\#})a$
$\displaystyle=P\delta a(1-P)+P[1+\delta aa^{\pi}\delta a((1+a^{\\#}\delta
a)^{-1}a^{\\#})^{2}](1+\delta aa^{\\#})a$ $\displaystyle=P\delta
a(1-P)+P\Phi(a)(1+\delta aa^{\\#})aP.$
By Lemma 3.1, we have
$P(Ba)^{\\#}P=PBaa^{\\#}B^{-1}a^{\\#}B^{-1}P=PBPB^{-1}a^{\\#}B^{-1}P=a^{\\#}B^{-1}=P(Ba)^{\\#}$
and $PBa(Ba)^{\\#}\delta a=P\delta a$. So $P(Ba)^{\\#}P(Ba)P=P(Ba)^{\\#}(Ba)$
and
$\displaystyle P(Ba)PP(Ba)^{\\#}P$
$\displaystyle=P(Ba)(Ba)^{\\#}P=P(Ba)^{\\#}(Ba)P=P(Ba)^{\\#}P(Ba)P$
$\displaystyle P(Ba)^{\\#}P(Ba)P(Ba)^{\\#}P$
$\displaystyle=P(Ba)^{\\#}(Ba)(Ba)^{\\#}P=P(Ba)^{\\#}P,$ $\displaystyle
P(Ba)P(Ba)^{\\#}P(Ba)P$ $\displaystyle=P(Ba)P(Ba)^{\\#}(Ba)P=P(Ba)P,$
i.e., $(P(Ba)P)^{\\#}=P(Ba)^{\\#}=a^{\\#}B^{-1}$. So $P(Ba)P(P(Ba)P)^{\\#}=P$
and hence, we have by Lemma 2.5 (1),
$[(1-C(a))\bar{a}(1+C(a))]^{\\#}=a^{\\#}B^{-1}+[a^{\\#}B^{-1}]^{2}\delta
a(1-P).$
Therefore,
$\displaystyle\bar{a}^{\\#}$
$\displaystyle=(1+C(a))[(1-C(a))\bar{a}(1+C(a))]^{\\#}(1-C(a))$
$\displaystyle=(1+C(a))(D(a)+D^{2}(a)\delta aa^{\pi})(1-C(a))$
$\displaystyle=(1+a^{\pi}\delta a(1+a^{\\#}\delta
a)^{-1}a^{\\#})(1+a^{\\#}\delta a)^{-1}a^{\\#}\big{[}1+\delta aa^{\pi}\delta
a[(1+a^{\\#}\delta a)^{-1}a^{\\#}]^{2}\big{]}^{-1}$ $\displaystyle\
\times\big{[}1+(1+a^{\\#}\delta a)^{-1}a^{\\#}\big{[}1+\delta aa^{\pi}\delta
a[(1+a^{\\#}\delta a)^{-1}a^{\\#}]^{2}\big{]}^{-1}\delta
aa^{\pi}\big{]}(1-a^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}).\qed$
Now we consider the case when $a\in\mathfrak{R}$ and $\bar{a}=a+\delta
a\in\mathfrak{R}$ such that $a^{\\#}$, $\bar{a}^{\\#}$ exist. Firstly, we have
###### Proposition 3.3.
Let $a\in\mathfrak{R}$, $\bar{a}=a+\delta a\in\mathfrak{R}$ such that
$a^{\\#}$, $\bar{a}^{\\#}$ exist. Then the following statements are
equivalent:
1. $(1)$
$\mathfrak{R}=\bar{a}\mathfrak{R}\dotplus(1-aa^{\\#})\mathfrak{R}=a\mathfrak{R}\dotplus(1-\bar{a}\bar{a}^{\\#})\mathfrak{R}=\mathfrak{R}\bar{a}\dotplus\mathfrak{R}(1-aa^{\\#})=\mathfrak{R}a\dotplus\mathfrak{R}(1-\bar{a}\bar{a}^{\\#})$.
2. $(2)$
$K=K(a,\bar{a})=\bar{a}\bar{a}^{\\#}+aa^{\\#}-1$ is invertible.
3. $(3)$
$\bar{a}\mathfrak{R}\cap(1-aa^{\\#})\mathfrak{R}=\\{0\\}$,
$\mathfrak{R}\bar{a}\cap\mathfrak{R}(1-aa^{\\#})=\\{0\\}$ and $1+\delta
aa^{\\#}$ is invertible.
Proof. $(1)\Rightarrow(2):$ Since
$\mathfrak{R}=\bar{a}\mathfrak{R}\dotplus(1-aa^{\\#})\mathfrak{R}=a\mathfrak{R}\dotplus(1-\bar{a}\bar{a}^{\\#})\mathfrak{R}$,
we have for any $y\in\mathfrak{R}$, there are
$y_{1}\in\mathfrak{R},\,y_{2}\in\mathfrak{R}$ such that
$(1-\bar{a}\bar{a}^{\\#})y=(1-\bar{a}\bar{a}^{\\#})(1-aa^{\\#})y_{1},\
\bar{a}\bar{a}^{\\#}y=\bar{a}\bar{a}^{\\#}aa^{\\#}y_{2}.$
Put $z=aa^{\\#}y_{2}-(1-aa^{\\#})y_{1}$. Then
$K(a,\bar{a})z=(\bar{a}\bar{a}^{\\#}+aa^{\\#}-1)(aa^{\\#}y_{2}-(1-aa^{\\#})y_{1})=y.$
Since
$\mathfrak{R}=\mathfrak{R}\bar{a}\dotplus\mathfrak{R}(1-aa^{\\#})=\mathfrak{R}a\dotplus\mathfrak{R}(1-\bar{a}\bar{a}^{\\#})$,
we have for any $y\in\mathfrak{R}$, there are $y_{1},y_{2}\in\mathfrak{R}$
such that
$y(1-\bar{a}\bar{a}^{\\#})=y_{1}(1-aa^{\\#})(1-\bar{a}\bar{a}^{\\#}),y\bar{a}\bar{a}^{\\#}=y_{2}aa^{\\#}\bar{a}\bar{a}^{\\#}.$
Put $z=y_{2}aa^{\\#}-y_{1}(1-aa^{\\#})$. Then
$zK(a,\bar{a})=(y_{2}aa^{\\#}-y_{1}(1-aa^{\\#}))(\bar{a}\bar{a}^{\\#}+aa^{\\#}-1)=y.$
The above indicates $K(a,\bar{a})$ is invertible when we take $y=1$.
$(2)\Rightarrow(3):$ Let
$y\in\bar{a}\mathfrak{R}\cap(1-aa^{\\#})\mathfrak{R}$. Then
$\bar{a}\bar{a}^{\\#}y=y,\,a^{\\#}y=0$. Thus $K(a,\bar{a})y=0$ and hence
$y=0$, that is, $\bar{a}\mathfrak{R}\cap(1-aa^{\\#})\mathfrak{R}=\\{0\\}$.
Similarly, we have $\mathfrak{R}\bar{a}\cap\mathfrak{R}(1-aa^{\\#})=\\{0\\}$.
Let $T=aK^{-1}\bar{a}^{\\#}-a^{\pi}$. Since $\bar{a}a^{\\#}aK^{-1}=\bar{a}$,
we have $(1+\delta aa^{\\#})T=K$, that is, $(1+\delta aa^{\\#})$ has right
inverse $TK^{-1}$.
Since $K^{-1}aa^{\\#}\bar{a}=\bar{a}$, we have
$(\bar{a}^{\\#}K^{-1}a-a^{\pi})(1+a^{\\#}\delta a)=K$, that is,
$1+a^{\\#}\delta a$ has left inverse $K^{-1}(\bar{a}^{\\#}K^{-1}a-a^{\pi})$.
This indicates that $1+\delta aa^{\\#}$ has left inverse $1-\delta
aK^{-1}(\bar{a}^{\\#}K^{-1}a-a^{\pi})a^{\\#}$ by Lemma 2.2. Finally, $1+\delta
aa^{\\#}$ is invertible.
$(3)\Rightarrow(1):$ By Lemma 2.1, $1+a^{\\#}\delta a$ is also invertible. So
from
$1+\delta aa^{\\#}=\bar{a}a^{\\#}+(1-aa^{\\#}),\ 1+a^{\\#}\delta
a=a^{\\#}\bar{a}+(1-aa^{\\#})$
and Lemma 2.7, we get that
$\mathfrak{R}=\bar{a}\mathfrak{R}\dotplus(1-aa^{\\#})\mathfrak{R}=\mathfrak{R}\bar{a}\dotplus\mathfrak{R}(1-aa^{\\#}).$
We now prove that
$\mathfrak{R}=a\mathfrak{R}+(1-\bar{a}\bar{a}^{\\#})\mathfrak{R}=\mathfrak{R}a+\mathfrak{R}(1-\bar{a}\bar{a}^{\\#}),\
a\mathfrak{R}\cap(1-\bar{a}\bar{a}^{\\#})\mathfrak{R}=\mathfrak{R}a\cap\mathfrak{R}(1-\bar{a}\bar{a}^{\\#})=\\{0\\}.$
For any $y\in a\mathfrak{R}\cap(1-\bar{a}\bar{a}^{\\#})\mathfrak{R}$, we have
$aa^{\\#}y=y,\,\bar{a}y=0$. So $(1+a^{\\#}\delta a)y$ $=(1-a^{\\#}a)y=0$ and
hence $y=0$.
Similarly, we have
$\mathfrak{R}a\cap\mathfrak{R}(1-\bar{a}\bar{a}^{\\#})=\\{0\\}$.
By Lemma 2.7, $\bar{a}^{+}=(1+a^{\\#}\delta a)^{-1}a^{\\#}$ and
$\bar{a}^{+}\bar{a}=(1+a^{\\#}\delta a)^{-1}a^{\\#}a(1+a^{\\#}\delta a)$. So
$(1-\bar{a}^{+}\bar{a})\mathfrak{R}=(1+a^{\\#}\delta
a)^{-1}(1-a^{\\#}a)\mathfrak{R}$. From
$(1-\bar{a}^{\\#}\bar{a})(1-\bar{a}^{+}\bar{a})=1-\bar{a}^{+}\bar{a}$, we get
that
$(1-\bar{a}^{+}\bar{a})\mathfrak{R}\subset(1-\bar{a}^{\\#}\bar{a})\mathfrak{R}$.
Note that $a\mathfrak{R}=a^{\\#}\mathfrak{R}$ and $(1+a^{\\#}\delta
a)a\mathfrak{R}=a^{\\#}(1+\delta aa^{\\#})\mathfrak{R}=a^{\\#}a\mathfrak{R}$.
So
$\mathfrak{R}\supset
a\mathfrak{R}+(1-\bar{a}^{\\#}\bar{a})\mathfrak{R}\supset(1+a^{\\#}\delta
a)^{-1}a^{\\#}a\mathfrak{R}+(1+a^{\\#}\delta
a)^{-1}(1-a^{\\#}a)\mathfrak{R}=\mathfrak{R}.$
Similarly, we can get
$\mathfrak{R}a+\mathfrak{R}(1-\bar{a}^{\\#}\bar{a})=\mathfrak{R}$.
Now we present a theorem which can be viewed as the inverse of Theorem 3.2 as
follows:
###### Theorem 3.4.
Let $a\in\mathfrak{R}$ and $\bar{a}=a+\delta a\in\mathfrak{R}$ such that
$a^{\\#}$, $\bar{a}^{\\#}$ exist. If $K(a,\bar{a})$ is invertible, then
$\Phi(a)$ is invertible.
Proof. Since $K(a,\bar{a})$ is invertible, we have
$\bar{a}\mathfrak{R}\cap(1-aa^{\\#})\mathfrak{R}=\\{0\\}$ and $1+\delta
aa^{\\#}$ is invertible in $\mathfrak{R}$ by Propositon 3.3. Thus, from the
proof of Theorem 3.2, we have
$\displaystyle(1-C(a))\bar{a}(1+C(a))$ $\displaystyle=P\delta
a(1-P)+P\Phi(a)(1+\delta aa^{\\#})aP$ $\displaystyle=PBaP+P\delta a(1-P).$
Since $(1-C(a))\bar{a}(1+C(a))$ is group invertible, it follows from Lemma 2.5
(2) that $PBaP$ is group invertible and $PBa(PBa)^{\\#}\delta a(I-P)=P\delta
a(I-P).$ Consequently,
$\displaystyle[(1-C(a))\bar{a}(1+C(a))]^{\\#}$
$\displaystyle=(1-C(a))\bar{a}^{\\#}(1+C(a))$
$\displaystyle=(PBa)^{\\#}P+((PBa)^{\\#})^{2}\delta a(1-P).$
Thus,
$\displaystyle(1-C(a))\bar{a}\bar{a}^{\\#}(1+C(a))$
$\displaystyle=[PBaP+P\delta a(1-P)][(PBa)^{\\#}P+((PBa)^{\\#})^{2}\delta
a(1-P)]$ $\displaystyle=PBa(PBa)^{\\#}P+(PBa)^{\\#}\delta a(1-P)$
$\displaystyle(1-C(a))K(a,\bar{a})(1+C(a))$
$\displaystyle=(1-C(a))\bar{a}\bar{a}^{\\#}(1+C(a))-(1-C(a))a^{\pi}(1+C(a))$
$\displaystyle=(1-C(a))\bar{a}\bar{a}^{\\#}(1+C(a))-a^{\pi}(1+C(a))$
$\displaystyle=PBa(PBa)^{\\#}P+P(PBa)^{\\#}\delta a(1-P)-(1-P)C(a)P-(1-P)$
Since $(1-C(a))K(a,\bar{a})(1+C(a))$ is invertible, we get that
$\rho(a)=(PBa)^{\\#}PBa-(PBa)^{\\#}\delta
aC(a)=PBa[(PBa)^{\\#}]^{2}(PBa-\delta aC(a))$
is invertible in $P\mathfrak{R}P$. So we have $P=PBa[(Ba)^{\\#}]^{2}(Ba-\delta
aC(a))\rho^{-1}(a)$ and that $\Phi(a)$ has right inverse.
Set $E(a)=a^{\\#}(1+\delta aa^{\\#})^{-1}\delta aa^{\pi}$. Then
$1-E(a)=P+(1+a^{\\#}\delta a)^{-1}a^{\pi}$ and $(1-E(a))^{-1}=1+E(a)$. From
Lemma 2.7, we have $\bar{a}(1+a^{\\#}\delta a)^{-1}a^{\pi}=0$ and
$\displaystyle a^{\\#}(1+\delta aa^{\\#})^{-1}\bar{a}$
$\displaystyle=(1+a^{\\#}\delta a)^{-1}a^{\\#}\bar{a}=(1+a^{\\#}\delta
a)^{-1}(1+a^{\\#}\delta a-a^{\pi})$ $\displaystyle=1-(1+a^{\\#}\delta
a)^{-1}a^{\pi}.$
Put $\psi(a)=1+[(1+a^{\\#}\delta a)^{-1}a^{\\#}]^{2}\delta aa^{\pi}\delta a$
and $R=(1+a^{\\#}\delta a)\psi(a)$. Then $(1-P)\psi(a)=1-P$,
$P\psi(a)P=\psi(a)P$ and
$\displaystyle(1+E(a))\bar{a}(1-E(a))$ $\displaystyle=[1+a^{\\#}(1+\delta
aa^{\\#})^{-1}\delta aa^{\pi}]\bar{a}[P+(1+a^{\\#}\delta a)^{-1}a^{\pi}]$
$\displaystyle=[1+a^{\\#}(1+\delta aa^{\\#})^{-1}\delta aa^{\pi}]\bar{a}P$
$\displaystyle=\bar{a}P+a^{\\#}(1+\delta aa^{\\#})^{-1}\delta
aa^{\pi}\bar{a}P$ $\displaystyle=\bar{a}P+a^{\\#}(1+\delta
aa^{\\#})^{-1}\delta aa^{\pi}\delta aP$ $\displaystyle=aP+\delta
aP+a^{\\#}(1+\delta aa^{\\#})^{-1}\delta aa^{\pi}\delta aP$
$\displaystyle=(1-P)\delta aP+P\delta aP+aP+a^{\\#}(1+\delta
aa^{\\#})^{-1}\delta aa^{\pi}\delta aP$ $\displaystyle=(1-P)\delta
aP+P[\bar{a}+a^{\\#}(1+\delta aa^{\\#})^{-1}\delta aa^{\pi}\delta a]P$
$\displaystyle=(1-P)\delta aP+P(1+\delta aa^{\\#})[(1+\delta
aa^{\\#})^{-1}\bar{a}+(1+\delta aa^{\\#})^{-1}a^{\\#}(1+\delta
aa^{\\#})^{-1}\delta aa^{\pi}\delta a]P$ $\displaystyle=(1-P)\delta
aP+Pa(1+a^{\\#}\delta a)[a^{\\#}(1+\delta
aa^{\\#})^{-1}\bar{a}+a^{\\#}(1+\delta aa^{\\#})^{-1}a^{\\#}(1+\delta
aa^{\\#})^{-1}\delta aa^{\pi}\delta a]P$ $\displaystyle=(1-P)\delta
aP+Pa(1+a^{\\#}\delta a)[a^{\\#}(1+\delta
aa^{\\#})^{-1}\bar{a}+[(1+a^{\\#}\delta a)^{-1}a^{\\#}]^{2}\delta
aa^{\pi}\delta a]P$ $\displaystyle=(1-P)\delta aP+Pa(1+a^{\\#}\delta
a)[1+[(1+a^{\\#}\delta a)^{-1}a^{\\#}]^{2}\delta aa^{\pi}\delta a]P$
$\displaystyle=(1-P)\delta aP+PaRP.$
Since $(1+E(a))\bar{a}(1-E(a))$ is group invertible, we can deduce that $aRP$
is group invertible and
$\displaystyle[(1+E(a))\bar{a}(1-E(a))]^{\\#}$
$\displaystyle=(1+E(a))\bar{a}^{\\#}(1-E(a))$
$\displaystyle=P(aRP)^{\\#}+(1-P)\delta a((aRP)^{\\#})^{2}$
and
$\displaystyle(1+E(a))\bar{a}\bar{a}^{\\#}(1-E(a))$
$\displaystyle=[(1-P)\delta aP+PaRP][P(aRP)^{\\#}+(1-P)\delta
a((aRP)^{\\#})^{2}]$ $\displaystyle=PaR(aRP)^{\\#}+(1-P)\delta a(aRP)^{\\#}.$
Thus, from the invertibility of $K(a,\bar{a})$, we get that
$\displaystyle(1+E(a))$ $\displaystyle K(a,\bar{a})(1-E(a))$
$\displaystyle=(1+E(a))\bar{a}\bar{a}^{\\#}(1-E(a))-(1+E(a))a^{\pi}(1-E(a))$
$\displaystyle=PaRP(aRP)^{\\#}+(1-P)\delta aP(aRP)^{\\#}-(1+E(a))a^{\pi}$
$\displaystyle=PaRP(aRP)^{\\#}+(1-P)\delta a(aRP)^{\\#}-PE(a)(1-P)-(1-P)$
is invertible in $\mathfrak{R}$ and hence
$\displaystyle\eta(a)$ $\displaystyle=aRP(aRP)^{\\#}-E(a)\delta
a(aRP)^{\\#}=[aRP-E(a)\delta a][(aRP)^{\\#}]^{2}aRP$
$\displaystyle=[aRP-E(a)\delta a][(aRP)^{\\#}]^{2}a(1+a^{\\#}\delta
a)\psi(a)P$
is invertible in $P\mathfrak{R}P$. So $P\psi(a)P$ is left invertible and
$\psi(a)$ is left invertible and hence $\Phi(a)$ is left invertible by Lemma
2.2. Therefore, $\Phi(a)$ is invertible.
Let $a\in\mathfrak{R}$ such that $a^{D}$ exists and $ind(a)=s$. As we know if
$a^{D}$ exists, then $a^{l}$ has group inverse $(a^{l})^{\\#}$ and
$a^{D}=(a^{l})^{\\#}a^{l-1}$ for any $l\geq s$.
From Theorem 3.2 and Theorem 3.4, we have the following corollary:
###### Corollary 3.5.
Let $a$ and $b$ be nonzero elements in $R$ such that $a^{D}$ and $b^{D}$
exist. Put $s=ind(a)$ and $t=ind(b)$. Suppose that $K(a,b)=bb^{D}+aa^{D}-1$ is
invertible in $\mathfrak{R}$. Then for any $l\geq s$ and $k\geq t$, we have
1. $(1)$
$1+(a^{D})^{l}(b^{k}-a^{l})$ is invertible in $\mathfrak{R}$ and
$b^{k}\mathfrak{R}\cap(1-a^{D}a)\mathfrak{R}=\\{0\\}$.
2. $(2)$
$W_{k,l}=1+E_{k,l}Z_{k,l}(1+(a^{D})^{l}E_{k,l})^{-1}(a^{D})^{l}$ is invertible
in $\mathfrak{R}$, here $E_{k,l}=b^{k}-a^{l}$ and
$Z_{k,l}=a^{\pi}E_{k,l}(a^{D})^{l}(1+E_{k,l}(a^{D})^{l})^{-1}.$
3. $(3)$
$b^{D}=(1+Z_{k,l})[H_{k,l}+H^{2}_{k,l}E_{k,l}a^{\pi}](1-Z_{k,l})b^{k-1}$,
where $H_{k,l}=(1+(a^{D})^{l}E_{k,l})^{-1}(a^{D})^{l}W^{-1}_{k,l}.$
Proof. Noting that $(a^{D})^{l}=(a^{l})^{\\#},\ aa^{D}=a^{l}(a^{l})^{\\#},\
bb^{D}=b^{k}(b^{k})^{\\#},\ l\geq s,\ k\geq t,$ we have
$K(a,b)=b^{k}(b^{k})^{\\#}+a^{l}(a^{l})^{\\#}-1,\quad
1+(a^{D})^{l}(b^{k}-a^{l})=1+(a^{l})^{\\#}(b^{k}-a^{l}).$
Applying Theorem 3.2 and Theorem 3.4 to $b^{k}$ and $a^{l}$, we can get the
assertions.
## 4 The representation of the group inverse of certain matrix on
$\mathfrak{R}$
As an application of Theorem 3.2 and Theorem 3.4, we study the representation
of the group inverse of $\begin{bmatrix}d&b\\\ c&0\end{bmatrix}$ on the ring
$\mathfrak{R}$.
###### Proposition 4.1.
Let $b,\,c,\,d\in\mathfrak{R}$. Suppose that $b^{\\#}$ and $c^{\\#}$ exist and
$k=b^{\\#}b+c^{\\#}c-1$ is invertible. If $b^{\pi}d=0$ or $dc^{\pi}=0$, then
$\begin{bmatrix}d&b\\\ c&0\end{bmatrix}^{\\#}$ exists and
$\begin{bmatrix}d&b\\\
c&0\end{bmatrix}^{\\#}=\begin{bmatrix}-k^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1}&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}(1+dk^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1})&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}$
if $b^{\pi}d=0$. When $dc^{\pi}=0$, we have
$\begin{bmatrix}d&b\\\
c&0\end{bmatrix}^{\\#}=\begin{bmatrix}-k^{-1}b^{\pi}dk^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}&(1+k^{-1}b^{\pi}dk^{-1}c^{\\#}k^{-1}b^{\\#}d)k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}.$
Proof. Set $a=\begin{bmatrix}0&b\\\ c&0\end{bmatrix}$, $\delta
a=\begin{bmatrix}d&0\\\ 0&0\end{bmatrix}$ and $\bar{a}=\begin{bmatrix}d&b\\\
c&0\end{bmatrix}$. Since $b^{\\#}bk=kc^{\\#}c$, $c^{\\#}ck=kb^{\\#}b$, it
follows from Corollary 2.4 that
$\displaystyle 1_{2}+a^{\\#}\delta a$
$\displaystyle=1_{2}+\begin{bmatrix}0&b^{\\#}bk^{-1}c^{\\#}k^{-1}\\\
c^{\\#}ck^{-1}b^{\\#}k^{-1}&0\end{bmatrix}\begin{bmatrix}d&0\\\
0&0\end{bmatrix}=\begin{bmatrix}1&0\\\ k^{-1}b^{\\#}k^{-1}d&1\end{bmatrix}$
$\displaystyle(1_{2}+a^{\\#}\delta a)^{-1}a^{\\#}$
$\displaystyle=\begin{bmatrix}1&0\\\
-k^{-1}b^{\\#}k^{-1}d&1\end{bmatrix}\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}$
$\displaystyle aa^{\\#}$ $\displaystyle=\begin{bmatrix}0&b\\\
c&0\end{bmatrix}\begin{bmatrix}0&b^{\\#}bk^{-1}c^{\\#}k^{-1}\\\
c^{\\#}ck^{-1}b^{\\#}k^{-1}&0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}bc^{\\#}ck^{-1}b^{\\#}k^{-1}&0\\\
0&cb^{\\#}bk^{-1}c^{\\#}k^{-1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}bb^{\\#}k^{-1}&0\\\
0&cc^{\\#}k^{-1}\end{bmatrix}$ $\displaystyle a^{\pi}$
$\displaystyle=1-aa^{\\#}=\begin{bmatrix}-c^{\pi}k^{-1}&0\\\
0&-b^{\pi}k^{-1}\end{bmatrix}$ $\displaystyle\bar{a}(1_{2}+a^{\\#}\delta
a)^{-1}a^{\pi}$ $\displaystyle=\begin{bmatrix}d&b\\\
c&0\end{bmatrix}\begin{bmatrix}1&0\\\
-c^{\\#}ck^{-1}b^{\\#}k^{-1}d&1\end{bmatrix}\begin{bmatrix}-c^{\pi}k^{-1}&0\\\
0&-b^{\pi}k^{-1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}d-bc^{\\#}ck^{-1}b^{\\#}k^{-1}d&b\\\
c&0\end{bmatrix}\begin{bmatrix}-c^{\pi}k^{-1}&0\\\
0&-b^{\pi}k^{-1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-c^{\pi}k^{-1}d&b\\\
c&0\end{bmatrix}\begin{bmatrix}-c^{\pi}k^{-1}&0\\\
0&-b^{\pi}k^{-1}\end{bmatrix}=\begin{bmatrix}k^{-1}b^{\pi}dc^{\pi}k^{-1}&0\\\
0&0\end{bmatrix}$
and
$\displaystyle\delta aa^{\pi}\delta a$ $\displaystyle=\begin{bmatrix}d&0\\\
0&0\end{bmatrix}\begin{bmatrix}-c^{\pi}k^{-1}&0\\\
0&-b^{\pi}k^{-1}\end{bmatrix}\begin{bmatrix}d&0\\\
0&0\end{bmatrix}=\begin{bmatrix}-dc^{\pi}k^{-1}d&0\\\ 0&0\end{bmatrix}$
$\displaystyle a^{\pi}\delta a$
$\displaystyle=\begin{bmatrix}-k^{-1}b^{\pi}d&0\\\
0&0\end{bmatrix},\quad\delta aa^{\pi}=\begin{bmatrix}-dc^{\pi}k^{-1}&0\\\
0&0\end{bmatrix}.$
If $b^{\pi}d=0$ or $dc^{\pi}=0$, then $\bar{a}(1+a^{\\#}\delta
a)^{-1}a^{\pi}=0$ and $\delta aa^{\pi}\delta a=0$. Thus, $\Phi(a)=1_{2}$ and
$D(a)=(1+a^{\\#}\delta a)^{-1}a^{\\#}\Phi^{-1}(a)=(1+a^{\\#}\delta
a)^{-1}a^{\\#}$.
When $b^{\pi}d=0$, $C(a)=a^{\pi}\delta a(1+a^{\\#}\delta a)^{-1}a^{\\#}=0$ and
$\displaystyle D(a)\delta aa^{\pi}$
$\displaystyle=\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}\begin{bmatrix}-dc^{\pi}k^{-1}&0\\\
0&0\end{bmatrix}$ $\displaystyle=\begin{bmatrix}0&0\\\
-k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1}&0\end{bmatrix}.$
By Theorem 3.2, we have
$\displaystyle\bar{a}^{\\#}$ $\displaystyle=(1_{2}+C(a))(D(a)+D^{2}(a)\delta
aa^{\pi})(1_{2}-C(a))$ $\displaystyle=D(a)(1_{2}+D(a)\delta aa^{\pi})$
$\displaystyle=\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}\begin{bmatrix}1&0\\\
-k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1}&1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}a_{1}&k^{-1}c^{\\#}k^{-1}\\\
a_{2}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix},$
where $a_{1}=-k^{-1}c^{\\#}k^{-2}b^{\\#}k^{-1}dc^{\pi}k^{-1}$,
$a_{2}=k^{-1}b^{\\#}k^{-1}+k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-2}b^{\\#}k^{-1}dc^{\pi}k^{-1}.$
Since $cc^{\\#}b^{\\#}=kb^{\\#}$, it follows that
$c^{\\#}k^{-2}b^{\\#}=c^{\\#}(c^{\\#}c)k^{-1}k^{-1}b^{\\#}=c^{\\#}k^{-1}k^{-1}c^{\\#}cb^{\\#}=c^{\\#}k^{-1}b^{\\#}.$
So $a_{1}=-k^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1}$,
$a_{2}=k^{-1}b^{\\#}k^{-1}(1+dk^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}dc^{\pi}k^{-1})$.
When $dc^{\pi}=0$, we have by Theorem 3.2,
$\displaystyle\bar{a}^{\\#}$
$\displaystyle=(1_{2}+C(a))D(a)(1_{2}-C(a))=(1_{2}+C(a))D(a)$
$\displaystyle=\begin{bmatrix}1&-k^{-1}b^{\pi}dk^{-1}c^{\\#}k^{-1}\\\
0&1\end{bmatrix}\begin{bmatrix}0&k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-k^{-1}b^{\pi}dk^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}&(1+k^{-1}b^{\pi}dk^{-1}c^{\\#}k^{-1}b^{\\#}k^{-1}d)k^{-1}c^{\\#}k^{-1}\\\
k^{-1}b^{\\#}k^{-1}&-k^{-1}b^{\\#}k^{-1}dk^{-1}c^{\\#}k^{-1}\end{bmatrix}.\qed$
Combining Proposition 4.1 with Corollary 2.4, we have
###### Corollary 4.2.
Let $b,\,c,\,d\in\mathfrak{R}$. Assume that $b^{\\#}$ and $c^{\\#}$ exist and
satisfy conditions: $b^{\\#}bc^{\\#}c=b^{\\#}b$, $c^{\\#}cb^{\\#}b=c^{\\#}c$.
1. $(1)$
If $b^{\pi}d=0$, then $\begin{bmatrix}d&b\\\
c&0\end{bmatrix}^{\\#}=\begin{bmatrix}b^{\\#}bc^{\\#}b^{\\#}db^{\pi}&b^{\\#}bc^{\\#}\\\
c^{\\#}cb^{\\#}(1-db^{\\#}bc^{\\#}b^{\\#}db^{\pi})&-c^{\\#}cb^{\\#}db^{\\#}bc^{\\#}\end{bmatrix}.$
2. $(2)$
If $dc^{\pi}=0$, then $\begin{bmatrix}d&b\\\
c&0\end{bmatrix}^{\\#}=\begin{bmatrix}b^{\pi}db^{\\#}bc^{\\#}b^{\\#}&(1-b^{\pi}db^{\\#}bc^{\\#}b^{\\#}d)b^{\\#}bc^{\\#}\\\
c^{\\#}cb^{\\#}&-c^{\\#}cb^{\\#}db^{\\#}bc^{\\#}\end{bmatrix}.$
Recall from [14] that an involution * on $\mathfrak{R}$ is an involutory
anti–automorphism, that is,
$(a^{*})^{*}=a,\ (a+b)^{*}=a^{*}+b^{*},\ (ab)^{*}=b^{*}a^{*},\ a^{*}=0\
\text{if and only if}\ a=0$
and $\mathfrak{R}$ is called the $*$–ring if $\mathfrak{R}$ has an involution.
###### Corollary 4.3.
Let $\mathfrak{R}$ be a $*$–ring with unit $1$ and let $p$ be a nonzero
idempotent element in $\mathfrak{R}$. Then
$\begin{bmatrix}pp^{*}&p\\\
p&0\end{bmatrix}^{\\#}=\begin{bmatrix}pp^{*}(1-p)&p\\\
p-(pp^{*})^{2}(1-p)&-pp^{*}p\end{bmatrix},\quad\begin{bmatrix}p^{*}p&p\\\
p&0\end{bmatrix}^{\\#}=\begin{bmatrix}(1-p)p^{*}p&p-(1-p)(p^{*}p)^{2}\\\
p&-pp^{*}p\end{bmatrix}.$
Proof. Since $p^{\\#}=p$, we can get the assertions easily by using Corollary
4.2.
###### Remark 4.4.
(1) If $\mathfrak{R}$ is a skew field and $b=c$ in Proposition 4.1, the
conclusion of Proposition 4.1 is contained in [26].
(2) Let $p$ be an idempotent matrix. The group inverse of
$\begin{bmatrix}a&b\\\ c&0\end{bmatrix}$ is given in [2] for some
$a,\,b,\,c\in\\{pp^{*},\,p,\,p^{*}\\}$. The group inverse of this type of
matrices is also discussed in [7].
Acknowledgement. The authors thank to the referee for his (or her) helpful
comments and suggestions.
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|
arxiv-papers
| 2012-10-05T02:08:38 |
2024-09-04T02:49:35.982021
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fapeng Du and Yifeng Xue",
"submitter": "Yifeng Xue",
"url": "https://arxiv.org/abs/1210.1627"
}
|
1210.1633
|
# Insensitivity of User Distribution in
Multicell Networks under General Mobility
and Session Patterns
Wei Bao, and Ben Liang, A preliminary version of this work has appeared as
[infocom_version].
###### Abstract
The location of active users is an important factor in the performance
analysis of mobile multicell networks, but it is difficult to quantify due to
the wide variety of user mobility and session patterns. In this work, we study
the stationary distribution of users by modeling the system as a multi-route
queueing network with Poisson inputs. We consider arbitrary routing and
arbitrary joint probability distributions for the channel holding times in
each route. Through a decomposition-composition approach, we derive a closed-
form expression for the joint stationary distribution for the number of users
in all cells. The stationary user distribution (1) is insensitive to the user
movement patterns, (2) is insensitive to general and dependently distributed
channel holding times, (3) depends only on the average arrival rate and
average channel holding time at each cell, and (4) is completely characterized
by an open network with $M/M/\infty$ queues. We use the Dartmouth trace to
validate our analysis, which shows that the analytical model is accurate when
new session arrivals are Poisson and remains useful when non-Poisson session
arrivals are also included in the data set. Our results suggest that accurate
calculation of the user distribution, and other associated metrics such as the
system workload, can be achieved with much lower complexity than previously
expected.
###### Index Terms:
Mobility modeling, multicell network, user distribution, insensitivity,
dependent channel holding times.
## I Introduction
In designing ever more efficient and capable mobile access networks, the
accurate modeling of how user mobility and session connectivity patterns
affect network performance is of paramount interest. However, compared with
wired networks, the analytical modeling of mobile networks is burdened with
many additional technical challenges. Some of the most difficult factors are
the following:
* •
The movement of users may be individually arbitrary, without following any
common mobility pattern [Mobility-Survey].
* •
The session durations may have a general probability distribution, supporting
diverse data and multimedia applications [Mobility-Parameter].
* •
The channel holding times at different cells are correlated, dependent on the
speed or trajectory of different users [zahran:TMC08].
To facilitate tractable analysis, existing studies often adopt simplified
models. For example, classical models assume exponentially distributed cell
dwell times and session durations, resulting in independent and memoryless
channel holding times [Guerin1987, Hong1986, Lin1997]. Although more general
mobility and session models have been considered in the past literature
[FanChLi97, Mobility-Parameter, Wong2000, Fang2005, Marsan2004, zahran:TMC08],
to the best of our knowledge, none addresses all of the challenges above.
In this paper, we study the joint stationary distribution for the number of
users in all cells in a multicell network, which has important utilization in
network management and planning. Prior studies have proposed several
analytical models to estimate the user distribution with various degrees of
detail and generality [Mobility-RWP-Analytical, Mobility-Core2, Mobility-
Core5, Mobility-Core4, Mobility-Core1]. Instead, we consider general mobility
and session patterns, only requiring that the new session arrivals form a
Poisson process, which is well supported by experimental data [Iversen:ITU01,
Heegaard:LNCS07, Mobility-Core1]. We model the user mobility with a general
system with multiple routes, each representing one type of users with a
specific movement pattern. A general probability distribution is used to
represent the session durations. As a consequence, the channel holding times
at different cell sites are no longer independent.
Through a decomposition-composition approach, we derive a closed-form
expression for the joint stationary distribution for the number of users in
all cells. We observe five important conclusions on the stationary user
distribution: _first_ , it is insensitive to how the users move through the
system; _second_ , it is insensitive to the general distribution of channel
holding times; _third_ , it is insensitive to the correlation among the
channel holding times; _fourth_ , it depends only on the average arrival rate
and average channel holding time at each cell; and _fifth_ , it is perfectly
captured by an open Jackson network with $M/M/\infty$ queues.
We confirm our theoretical analysis through experimental validation using the
Dartmouth user mobility traces [Trace-Dartmouth-Data03]. These traces provide
a large data set, with 152 APs and more than 5000 users, to support an
accurate real-life measurement of the joint user distribution. They also
contain a large amount of handoff traffic among APs to create strong
dependency between channel holding times, which tests our claim of
insensitivity. The experimental results show that the proposed analysis
accurately predicts real-world user distributions.
The conclusion of this work has important consequence to performance analysis
and practical system design. It suggests that accurate calculation of the user
distribution, and other associated metrics such as the system workload, can be
achieved with much lower requirement for system parameter estimation than
previously expected. Furthermore, the simplicity of the resultant product-form
user distribution enables further analytical endeavors in system optimization.
The rest of the paper is organized as follows. In Section II, we discuss the
relation between our work and prior works. In Section III, we present the
system model. In Sections IV and V, we derive the analytical stationary
distributions for single-route and multiple-route networks, respectively. In
Section LABEL:section_experiment, we validate our analysis with experimental
results from the Dartmouth traces. Finally, concluding remarks are given in
Section LABEL:section_conclusion.
## II Related Work
In the following subsections, we present the related prior work in user
distribution modeling and general results in the insensitivity of queueing
networks.
### II-A User Distribution in Mobile Networks
The user distribution is an important factor in the management and planning of
mobile networks. However, relatively few analytical models are available in
the literature. The uniform user distribution has been widely adopted for
mathematical convenience, but it does not account for non-homogeneity in the
physical topology and is incorrect in some cases. For example, it is well
known that the user distribution is non-uniform under the random waypoint
model [Mobility-RWP-Analytical].
Other previous works have proposed analytical models using stochastic queueing
networks to derive the user distribution in different environments, including
wireless multimedia networks [Mobility-Core2], vehicular ad-hoc networks
[Mobility-Core5], and WLANs [Mobility-Core4, Mobility-Core1]. However, they do
not allow arbitrary mobility or arbitrary session patterns. In terms of user
movement, [Mobility-Core2], [Mobility-Core5], and [Mobility-Core1] assume that
users move from one cell to another probabilistically and memorylessly, while
[Mobility-Core4] focuses on scattered single cells, so that user movement
among multiple cells is not discussed. None of them consider the arbitrary
user movement patterns. In terms of channel holding times, [Mobility-Core2]
uses the sum of hyper-exponentials or the Coxian distribution to approximate
arbitrary distributions; [Mobility-Core4] assumes generally distributed
channel holding times but concerns only a single cell; and [Mobility-Core5]
and [Mobility-Core1] consider generally but independently distributed channel
holding times in different cells. None of the above works consider the
dependence among channel holding times.
Note that the authors of [Mobility-Core1] have also observed a surprising
match between analysis and real-life user mobility traces from the Dartmouth
study [Trace-Dartmouth-Data03], even though their analysis assumes simple
$M/G/\infty$ mobility and session models without considering arbitrary user
movement patterns or dependent channel holding times. No analytical
explanation is given in [Mobility-Core1] for this observation. In contrast,
our work provides theoretical support for it, since we show that the
stationary user distribution is also insensitive to arbitrary user movement
patterns and dependent channel holding times.
### II-B Insensitivity Property
The insensitivity of queueing networks indicates the situation where the
stationary distribution remains unchanged while the distribution of service
times takes arbitrary forms. When the service times are assumed independent
among different queues, there are several well known conditions for
insensitivity. For example, networks with symmetric queues are insensitive
[Book-Reversibility]. In [Insensitivity1] and [Insensitivity2], the partial
balance of probability flows is shown to be a sufficient condition for
insensitivity. In [Insensitivity4], partial reversibility is shown to be a
necessary and sufficient condition for insensitivity. However, none of these
known results consider the case where the service times between different
queues are dependent. For example, the queueing network closely related to
ours is one with $M/G/\infty$ queues. It is known to be insensitive when the
service times are independent [Book-Reversibility], but to the best of our
knowledge, there is no further general result for dependent service times.
### II-C Preliminary Version
A preliminary version of this work has appeared as [infocom_version]. This
full version includes the following extensions: First, we provide more
detailed derivation and discussion in the analysis of the single-route network
in Sections IV. Second, we fully expand the analysis of the multiple-route
network by proving the theorem of insensitivity and deriving the stationary
user distribution of multicell networks in Section V. Third, we include new
experimental studies in Section LABEL:section_experiment.
## III System Model
Consider a cellular network with $C$ cells. There are $L$ unique _routes_ ,
each defined as a finite ordered sequence of cells. The $j$th stage on the
$l$th route corresponds to the $j$th cell in the sequence, which is denoted as
$c(l,j)$. Let $N_{l}$ be the number of stages on the $l$th route. Each user of
the $l$th route starts a new session in cell $c(l,1)$; then it moves along the
route through cells $c(l,1),c(l,2)\ldots c(l,N_{l})$, as long as the session
remains active. The user is considered to have departed the network when its
session terminates or when it exits cell $c(l,N_{l})$. We allow an arbitrary
number of arbitrary routes to cover all possible movement patterns.
For each route, we assume the arrivals of _new_ sessions to form a Poisson
process. Note that although the arrivals of packets in the Internet may not
form Poisson processes [Poisson-against1], the arrivals of new sessions are at
a much larger time scale and are well justified as Poisson [Iversen:ITU01,
Heegaard:LNCS07]. Furthermore, in [Mobility-Core1] and later in Section
LABEL:section_experiment, experimental data show that new sessions in the type
of mobile networks under consideration are indeed Poisson barring some extreme
cases. We emphasize that only the new session arrivals are Poisson, while the
handoff arrivals at each cell have general statistics with complicated
dependencies.
The session duration of a user on the $l$th route is modeled as an arbitrarily
distributed random variable $T_{l}$. Let $\lambda_{l0}$ be the new session
arrival rate at the $l$th route. After a new session arrival, let $\tau_{l1}$
denote the residual cell dwell time of the user in the $1$st stage on the
$l$th route, which is arbitrarily distributed. Let $\tau_{lj}$, $2\leq j\leq
N_{l}$, denote the cell dwell time of the user in the $j$th stage on the $l$th
route, which are also arbitrarily distributed. Then, the channel holding time
of the $j$th stage on the $l$th route, $t_{lj}$, if it exists, can be
represented as follows:
$t_{lj}=\begin{cases}\min\\{T_{l},\tau_{l1}\\},&\textrm{ if }j=1,\\\
\displaystyle\min\\{T_{l}-\sum_{i=1}^{j-1}\tau_{li},\tau_{lj}\\},&\textrm{ if
}T_{l}>\sum_{i=1}^{j-1}\tau_{li},2\leq j\leq N_{l}.\end{cases}$ (1)
Fig. 1 shows an example network with $3$ routes. Route $1$ starts from cell
$1$ and passes cell $3$, $4$ and $6$ (i.e., $c(1,1)=1$, $c(1,2)=3$, $c(1,3)=4$
and $c(1,4)=6$). A user starts a session in cell $1$, and the session is
terminated in cell $4$. The corresponding
$T_{1},\tau_{11},\tau_{12},\tau_{13},t_{11},t_{12}$, and $t_{13}$ are labeled
in the figure.
handoff$T_{1}$$\tau_{12}$$\tau_{13}$$\tau_{11}$$t_{11}$$t_{12}$$t_{13}$$T_{1}$Cell
3Cell 4Cell 6Cell 2Route 3Cell 5Route
1$\tau_{11}=t_{11}$$\tau_{12}=t_{12}$$t_{13}$$\tau_{13}$Route 2startCell
1handoffterminate
Figure 1: System model. TABLE I: Selected Definition of Variables Name | Definition
---|---
$x_{lj}$, | Number of sessions in the $j$th stage on the $l$th route,
$\mathbf{x}$ | $\mathbf{x}=[\\{x_{lj}\\}]^{T}$.
$\widetilde{x}_{lkij}$ | On the $l$th route, number of sessions lasting $k$ stages,
$\widetilde{\mathbf{x}}$ | in the $i$th realization, in the $j$th stage, $\widetilde{\mathbf{x}}=[\\{\widetilde{x}_{lkij}\\}]^{T}$.
$y_{n}$, $\mathbf{y}$ | The number of sessions in the $n$th cell, $\mathbf{y}=[\\{y_{n}\\}]^{T}.$
$t_{lj}$ | Random variable: on the $l$th route,
| channel holding time at the $j$th stage.
$\widehat{t}_{lkj}$ | Random variable: on the $l$th route, channel holding
| time at the $j$th stage given that there are $k$ stages.
$\overline{t}_{lj}$ | On the $l$th route, the average value of $t_{lj}$,
| given that the number of stages $\geq j$.
$\mathbf{\widehat{t}_{lk}}$ | Random vector: $\\{\widehat{t}_{lk1},\ldots,\widehat{t}_{lkk}\\}$.
$\widetilde{t}_{lkij}$ | Constant: on the $l$th route, $i$th realization of channel
| holding time at $j$th stage, when session lasts $k$ stages.
$\mathbf{\widetilde{t}_{lki}}$ | Constant vector: $\\{\widetilde{t}_{lki1},\ldots,\widetilde{t}_{lkik}\\}$.
$p_{lk}$ | On the $l$th route, probability
| that a session lasts $k$ stages.
$q_{lki}$ | On the $l$th route, probability of the $i$th realization of
| a session, given that a session lasts for $k$ stages.
$P_{lki}$ | On the $l$th route, probability that a session lasts $k$
| stages and is in the $i$th realization, $P_{lki}=p_{lk}q_{lki}$.
$\lambda_{l0}$ | Arrival rate of the $l$th route.
$\lambda_{lj}$ | $1/\overline{t}_{lj}$.
$\widetilde{\lambda}_{lki0}$ | Arrival rate of sessions lasting $k$ stages, in the $i$th
| realization, on the $l$th route,
$\widetilde{\lambda}_{lki0}=P_{lki}\lambda_{l0}$.
$\widetilde{\lambda}_{lkij}$ | $1/\widetilde{t}_{lkij}$.
$w^{\prime}_{lj}$ | Invariant measure of memoryless network.
$w_{lj}$ | $w^{\prime}_{lj}/\lambda_{lj}$.
$\widetilde{w}^{\prime}_{lkij}$ | Invariant measure of decoupled memoryless network.
$\widetilde{w}_{lkij}$ | $\widetilde{w}^{\prime}_{lkij}/\widetilde{\lambda}_{lkij}$.
$\pi_{0}(\mathbf{x})$ | Stationary distribution of memoryless network w.r.t $\mathbf{x}$.
$\pi_{D}(\widetilde{\mathbf{x}})$ | Stationary user distribution of decoupled network w.r.t $\mathbf{\widetilde{\mathbf{x}}}$.
$\pi(\mathbf{x})$ | Stationary user distribution of the original network w.r.t $\mathbf{x}$.
$\pi_{1}(\mathbf{y})$ | Stationary user distribution of multicell network w.r.t $\mathbf{y}$.
$\overline{\lambda}_{n}$ | Average arrival rate of the $n$th cell.
$\overline{\mathfrak{t}}_{n}$ | Average channel holding time of the $n$th cell.
Since there is a one-to-one mapping between active users and sessions, we do
not distinguish the two. Note that given a route $l$, we know the sessions
start at cell $c(l,1)$ but the cell where they end is random. Furthermore, we
do not assume independence between $T_{l}$ and $\tau_{lj}$, and the channel
holding times $t_{lj}$ are not independent either. Finally, each route defines
the user movement trace and the distribution of channel holding times, which
implicitly characterizes the speed of users on this route.
Let $x_{lj}$, $1\leq l\leq L$, $1\leq j\leq N_{l}$, denote the number of
active users in the $j$th stage on the $l$th route; let $y_{n}$, $1\leq n\leq
C$, denote the number of active users in the $n$th cell. Let
$\mathbf{x}=[\\{x_{lj}:1\leq l\leq L,1\leq j\leq N_{l}\\}]^{T}$ and
$\mathbf{y}=[y_{1},y_{2},\ldots,y_{C}]^{T}$. We aim to derive
$\pi(\mathbf{x})$ and $\pi_{1}(\mathbf{y})$, the joint stationary user
distributions for $\mathbf{x}$ and $\mathbf{y}$, respectively. Note that since
$\pi(\mathbf{x})$ and $\pi_{1}(\mathbf{y})$ are defined in the steady state,
we explicitly ignore any temporal fluctuation in these distributions.
A partial list of nomenclature is given in Table I.
## IV Stationary User Distribution in Single-Route Network
We first derive the stationary user distribution on a single route. We
construct a reference single-route memoryless network, where all the channel
holding times are independently and exponentially distributed. We prove
insensitivity by showing an equivalence between the original network and the
memoryless network in terms of stationary user distribution.
### IV-A Queueing Network Model for Single-Route Network
$\vdots$$\ldots$(a) Single-route
network.$\lambda_{l0}$$t_{l1}$$t_{l2}$$t_{l3}$$t_{lN_{l}}$$0$(b) Reference
single-route memoryless
network.$\ldots$$\lambda_{l0}$$\frac{p_{l1}}{\sum_{j=1}^{N_{l}}p_{lj}}\lambda_{l1}$$\frac{p_{l2}}{\sum_{j=2}^{N_{l}}p_{lj}}\lambda_{l2}$$0$$\lambda_{lN_{l}}$$\frac{p_{l(N_{l})}}{p_{l(N_{l}-1)}+p_{lN_{l}}}\lambda_{l(N_{l}-1)}$$\frac{\sum_{j=2}^{N_{l}}p_{lj}}{\sum_{j=1}^{N_{l}}p_{lj}}\lambda_{l1}$$\frac{\sum_{j=3}^{N_{l}}p_{lj}}{\sum_{j=2}^{N_{l}}p_{lj}}\lambda_{l2}$$\frac{p_{l(N_{l}-1)}}{p_{l(N_{l}-1)}+p_{lN_{l}}}\lambda_{l(N_{l}-1)}$$\vdots$
Figure 2: Single-route network.
Consider exclusively the $l$th route in the network. Throughout this section,
we will carry the route index $l$ in most symbols, since they will be re-used
in the analysis of multiple-route networks.
As shown in Fig. 2(a), we model the route as a tandem-liked queueing network,
except with early exists. The node labeled with $0$ represents the exogenous
world. The $j$th queue, $1\leq j\leq N_{l}$, represents the $j$th stage of the
route, and units in this queue represent sessions in the $j$th stage. Each
queue has infinite servers, since the sessions are served in parallel with no
waiting111Users move into and out of each cell in parallel. Therefore, when
considering the channel holding time as the service time of a queue that
models mobility, this is equivalent to all users being served at the same time
by its own dedicated server, which is the same as having infinite servers. In
terms of active user sessions, this model is accurate for communication
systems with no admission control (e.g., WiFi) and gives reasonable
approximation to systems with many available channels..
The channel holding time of a session in the $j$th stage, $t_{lj}$, is
equivalent to the service time of the $j$th queue. The handoff of a session
from the $j$th stage to the $(j+1)$th stage is equivalent to a unit movement
from the $j$th queue to the $(j+1)$th queue. The termination of a session is
equivalent to the movement from a queue to node $0$.
Let $p_{lk}$ denote the probability that a session lasts for $k$ stages. It is
given by
$p_{lk}=P\Big{[}\sum_{j=1}^{k-1}\tau_{lj}<T_{l}\leq\sum_{j=1}^{k}\tau_{lj}\Big{]},\text{
for }2\leq k\leq N_{l}-1,$
with $p_{l1}=P\left[T_{l}\leq\tau_{l1}\right]$ and
$p_{lN_{l}}=P\left[\sum_{j=1}^{N_{l}-1}\tau_{lj}<T_{l}\right]$. Note that we
have $\sum_{k=1}^{N_{l}}p_{lk}=1$. Given a session in the $k$th stage, it
enters the $(k+1)$th stage with probability
$\frac{\sum_{j=k+1}^{N_{l}}p_{lj}}{\sum_{j=k}^{N_{l}}p_{lj}}$ and terminates
with probability $\frac{p_{lk}}{\sum_{j=k}^{N_{l}}p_{lj}}$.
### IV-B Reference Single-Route Memoryless Network
We define a reference _single-route memoryless network_ , as a Jackson network
with the same topology as the original single-route network, where each queue
has infinitely many independent and exponential servers. An illustration is
shown in Fig. 2(b). By matching the mean service times in this memoryless
network with those of the original network, we see that its external arrival
rate is $\lambda_{l0}$, the service rate of the $j$th queue is
$\lambda_{lj}=\frac{1}{\overline{t}_{lj}}$. The routing probability from the
$k$th queue to the $(k+1)$th queue is the probability that a session enters
the $(k+1)$th stage conditioned on it is in the $k$th stage,
$\frac{\sum_{j=k+1}^{N_{l}}p_{lj}}{\sum_{j=k}^{N_{l}}p_{lj}}$. The routing
probability from the $k$th queue to node $0$ is
$\frac{p_{lk}}{\sum_{j=k}^{N_{l}}p_{lj}}$. Thus, the service rate from the
$k$th queue to the $(k+1)$th queue is
$\frac{\sum_{j=k+1}^{N_{l}}p_{lj}}{\sum_{j=k}^{N_{l}}p_{lj}}\lambda_{lk}$, and
the service rate from the $k$th queue to node $0$ is
$\frac{p_{lk}}{\sum_{j=k}^{N_{l}}p_{lj}}\lambda_{lk}$.
Let $w_{lj}^{\prime}$ denote the positive invariant measure of the $j$th queue
that satisfies the routing balance equations of the single-route memoryless
network. $w_{0}^{\prime}$ is the positive invariant measure of the node $0$.
We adopt the convention that $w_{0}^{\prime}=1$. It can be derived from the
topology of Fig. 2(b) that
$\displaystyle w_{0}^{\prime}=$ $\displaystyle\lambda_{l0}w_{l1}^{\prime},$
(2)
$\displaystyle\frac{\sum_{n=j}^{N_{l}}p_{ln}}{\sum_{n=j-1}^{N_{l}}p_{ln}}w_{lj-1}^{\prime}=$
$\displaystyle w_{lj}^{\prime},\quad 2\leq j\leq N_{l},$ (3)
which leads to
$\displaystyle w_{l1}^{\prime}=$ $\displaystyle\lambda_{l0},$ (4)
$\displaystyle w_{lj}^{\prime}=$
$\displaystyle\lambda_{l0}(1-\sum_{n=1}^{j-1}p_{ln}),\quad 2\leq j\leq N_{l}.$
(5)
Because each queue has infinite servers, the departure intensity at the $j$th
queue is $\lambda_{lj}x_{lj}$ when there are $x_{lj}$ users in it. Let
$w_{lj}=\frac{w_{lj}^{\prime}}{\lambda_{lj}}$. Then the stationary user
distribution w.r.t. $\mathbf{x}$ of this network is [Book-StochasticNetwork]
$\pi_{0}(\mathbf{x})=\prod_{j=1}^{N_{l}}e^{-w_{lj}}w_{lj}^{x_{lj}}\frac{1}{x_{lj}!}.$
(6)
### IV-C Insensitivity of Single-Route Network
$\widetilde{t}_{lN_{l}21}$$\widetilde{\lambda}_{l110}$$p_{l1}\lambda_{l0}$$\widetilde{t}_{l2M_{l2}1}$$\widetilde{t}_{l2M_{l2}2}$$\vdots$$\vdots$$\widetilde{t}_{l221}$$\widetilde{t}_{l222}$$\widetilde{\lambda}_{l220}$$\widetilde{t}_{l211}$$\widetilde{t}_{l212}$$p_{l2}\lambda_{l0}$$\vdots$$\widetilde{\lambda}_{l120}$$\widetilde{\lambda}_{l210}$$\widetilde{\lambda}_{l1M_{l1}0}$$\widetilde{t}_{l1M_{l1}1}$$\widetilde{t}_{l111}$$\widetilde{t}_{l121}$$0$$\ldots$$\widetilde{t}_{lN_{l}M_{lN_{l}}1}$$\widetilde{t}_{lN_{l}2N_{l}}$$\vdots$$\widetilde{t}_{lN_{l}M_{lN_{l}}2}$$\widetilde{t}_{lN_{l}M_{lN_{l}}3}$$\widetilde{t}_{lN_{l}22}$$\widetilde{t}_{lN_{l}23}$$\widetilde{t}_{lN_{l}M_{lN_{l}}N_{l}}$$\widetilde{t}_{lN_{l}1N_{l}}$$\vdots$$\vdots$$\vdots$$\ldots$$\widetilde{t}_{lN_{l}12}$$\widetilde{t}_{lN_{l}13}$$\ldots$$\vdots$$\ldots$$p_{lN_{l}}\lambda_{l0}$$\widetilde{\lambda}_{lN_{l}10}$$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}0}$$\widetilde{\lambda}_{lN_{l}20}$$\widetilde{\lambda}_{l2M_{l2}0}$$\widetilde{t}_{lN_{l}11}$
Figure 3: Decoupled network.
$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}N_{l}}$$0$$\widetilde{\lambda}_{l110}$$p_{l1}\lambda_{l0}$$\widetilde{\lambda}_{l2M_{l2}1}$$\widetilde{\lambda}_{l2M_{l2}2}$$\vdots$$\vdots$$\widetilde{\lambda}_{l221}$$\widetilde{\lambda}_{l222}$$\widetilde{\lambda}_{l220}$$\widetilde{\lambda}_{l211}$$\widetilde{\lambda}_{l212}$$p_{l2}\lambda_{l0}$$\vdots$$\widetilde{\lambda}_{l120}$$\widetilde{\lambda}_{l210}$$\widetilde{\lambda}_{l1M_{l1}0}$$\widetilde{\lambda}_{l1M_{l1}1}$$\widetilde{\lambda}_{l111}$$\widetilde{\lambda}_{l121}$$\ldots$$\vdots$$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}3}$$\widetilde{\lambda}_{lN_{l}22}$$\widetilde{\lambda}_{lN_{l}23}$$\widetilde{\lambda}_{lN_{l}1N_{l}}$$\vdots$$\vdots$$\vdots$$\ldots$$\widetilde{\lambda}_{lN_{l}12}$$\widetilde{\lambda}_{lN_{l}13}$$\ldots$$\vdots$$\ldots$$p_{lN_{l}}\lambda_{l0}$$\widetilde{\lambda}_{lN_{l}10}$$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}0}$$\widetilde{\lambda}_{lN_{l}20}$$\widetilde{\lambda}_{l2M_{l2}0}$$\widetilde{\lambda}_{lN_{l}11}$$\widetilde{\lambda}_{lN_{l}2N_{l}}$$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}2}$$\widetilde{\lambda}_{lN_{l}21}$$\widetilde{\lambda}_{lN_{l}M_{lN_{l}}1}$
Figure 4: Reference memoryless decoupled network.
For the original single route network, we employ a decomposition-composition
approach to derive its stationary user distribution.
Given that one session lasts for $k$ stages, we denote the channel holding
times as a $k$-dimensional random vector
$\mathbf{\widehat{t}_{lk}}=\\{\widehat{t}_{lk1},\ldots\widehat{t}_{lkj},\ldots,\widehat{t}_{lkk}\\}$,
where $\widehat{t}_{lkj}$ is the channel holding time at the $j$th stage. We
assume that $\mathbf{\widehat{t}_{lk}}$ is an arbitrarily distributed discrete
random vector with $M_{lk}$ possible realizations222For a vector of continuous
channel holding times, we can use a sequence of discrete distributions with
decreasing granularity to approach its distribution.. For any $i$, $1\leq
i\leq M_{lk}$, we define a $k$-dimensional deterministic vector
$\mathbf{\widetilde{t}_{lki}}=[\widetilde{t}_{lki1},\ldots,\widetilde{t}_{lkij},\ldots,\widetilde{t}_{lkik}]^{T}$
corresponding to the $i$th realization of $\mathbf{\widehat{t}_{lk}}$. Let
$q_{lki}$ be the probability of the $i$th realization given that the session
lasts for $k$ stages. Also, let $P_{lki}=p_{lk}q_{lki}$ denote the probability
that a session lasts for $k$ stages and it is in the $i$th realization.
By doing so, we decompose the original network into a multiple-branch queueing
network as shown in Fig. 3, which is referred to as the _decoupled network_.
In this network, there are $N_{l}$ main branches, where the $k$th main branch
represents the event that a session lasts for $k$ stages. The $k$th main
branch contains $M_{lk}$ sub-branches, where the $i$th sub-branch represents
the realization where
$\mathbf{\widehat{t}_{lk}}=\mathbf{\widetilde{t}_{lki}}$. Furthermore, the
$j$th queue in the $i$th sub-branch of the $k$th main branch represents the
$j$th stage of the $i$th realization of the sessions that last for $k$ stages.
Hence, each queue of the decoupled network has infinite servers with
_deterministic_ service time, $\widetilde{t}_{lkij}$, for the $j$th stage of
the $i$th sub-branch of the $k$th main branch. Furthermore, the arrival rate
of the $i$th sub-branch of the $k$th main branch is
$\widetilde{\lambda}_{lki0}=P_{lki}\lambda_{l0}$. Let
$\widetilde{\mathbf{x}}=[\\{\widetilde{x}_{lkij}:1\leq k\leq N_{l},1\leq j\leq
k,1\leq i\leq M_{lk}\\}]^{T}$ be the vector of number of sessions in the $j$th
stage of the $i$th sub-branch of the $k$th main branch. Denote by
$\pi_{D}(\widetilde{\mathbf{x}})$ the stationary user distribution of the
decoupled network.
Note that the stationary distribution of a Jackson network with infinite
servers at each queue is insensitive with respect to the distribution of the
service times [Insensitivity1]. Therefore, $\pi_{D}(\widetilde{\mathbf{x}})$
remains unchanged if we create a reference Jackson network by replacing each
queue with _deterministic service time_ in the decoupled network with a queue
that has _exponential distributed memoryless service time_ with the same mean
(e.g., the service rate at the $j$th queue of the $i$th sub-branch of the
$k$th main branch
$\widetilde{\lambda}_{lkij}=\frac{1}{\widetilde{t}_{lkij}}$), as shown in Fig.
4, which is referred to as the _reference memoryless decoupled network_.
Let $\widetilde{w}_{lkij}^{\prime}$ be the positive invariant measure of the
$j$th queue of the $i$th sub-branch of the $k$th main branch of the reference
memoryless decoupled network, which satisfies the routing balance equations
with the convention that at node $0$, $w_{0}^{\prime}=1$. Since each sub-
branch is a chain network, we have
$\displaystyle\widetilde{w}_{lkij}^{\prime}=P_{lki}\lambda_{l0}.$ (7)
Let
$\widetilde{w}_{lkij}=\frac{\widetilde{w}_{lkij}^{\prime}}{\widetilde{\lambda}_{lkij}}$.
Then the stationary user distribution of the decoupled network as well as the
reference memoryless decoupled network is
$\displaystyle\pi_{D}(\widetilde{\mathbf{x}})=\prod_{j=1}^{N_{l}}\prod_{k=j}^{N_{l}}\prod_{i=1}^{M_{lk}}e^{-\widetilde{w}_{lkij}}\widetilde{w}_{lkij}^{\widetilde{x}_{lkij}}\frac{1}{\widetilde{x}_{lkij}!}.$
(8)
The stationary user distribution of the original single route network,
$\pi(\mathbf{x})$, is the sum of $\pi_{D}(\widetilde{\mathbf{x}})$ satisfying
$x_{lj}=\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}\widetilde{x}_{lkij}$, $\forall
j$. To derive $\pi(\mathbf{x})$, we first introduce the following lemma.
###### Lemma 1
Consider a stationary open Jackson network with $N$ queues each with an
infinite number of servers. Let $x_{j}$ be the number of units in the $j$th
queue and $\mathbf{x}=[x_{1},\ldots x_{N}]^{T}$. Suppose
$\\{\mathcal{J}_{1},\mathcal{J}_{2},\ldots\mathcal{J}_{M}\\}$ is a set of
mutually exclusive subsets of $\\{1,2,\ldots,N\\}$. Let
$z_{i}=\sum_{j\in\mathcal{J}_{i}}x_{j}$, $i=1,2,\ldots,M$, denoting the sum of
units in the queues inside $\mathcal{J}_{i}$. Then, the distribution of
$\mathbf{z}=[z_{1},\ldots z_{M}]^{T}$ is
$\displaystyle\pi(\mathbf{z})=\prod_{i=1}^{M}e^{-v_{i}}v_{i}^{z_{i}}\frac{1}{z_{i}!},$
(9)
where $v_{i}=\sum_{j\in\mathcal{J}_{i}}w_{j}$, and $w_{j}$ is the expected
number of units in the $j$th queue.
###### Proof:
For a Jackson network with infinite servers at each queue, the stationary
queue lengths are independent Poisson random variables with mean $w_{j}$ for
the $j$th queue. Hence, $z_{i}$ is Poisson with mean
$v_{i}=\sum_{j\in\mathcal{J}_{i}}w_{j}$ for all $i$. Furthermore, since
$\\{\mathcal{J}_{i}\\}$ are mutually exclusive, $\\{z_{i}\\}$ are independent.
∎
Next, we note that the expected service time spent in the $j$th stage given
that the $j$th stage exists, i.e., $j\leq k$ for the $k$th main branch, can be
computed as
$\displaystyle\overline{t}_{lj}$
$\displaystyle=\frac{\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}P_{lki}\widetilde{t}_{lkij}}{\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}P_{lki}}$
$\displaystyle=\frac{\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}P_{lki}\widetilde{t}_{lkij}}{1-\sum_{n=1}^{j-1}p_{ln}}.$
(10)
Combining this with (7), we have
$\displaystyle\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}\widetilde{w}_{lkij}$
$\displaystyle=\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}\frac{\lambda_{l0}P_{lki}}{\widetilde{\lambda}_{lkij}}$
$\displaystyle=\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}\lambda_{l0}P_{lki}\widetilde{t}_{lkij}$
$\displaystyle=\lambda_{l0}(1-\sum_{n=1}^{j-1}p_{ln})\overline{t}_{lj}$
$\displaystyle=\frac{\lambda_{l0}}{\lambda_{lj}}(1-\sum_{n=1}^{j-1}p_{ln})$
$\displaystyle=w_{lj}.$ (11)
Therefore, by Lemma 1, we have
$\displaystyle\pi(\mathbf{x})$
$\displaystyle=\sum_{\widetilde{\mathbf{x}}:x_{lj}=\sum_{k=j}^{N_{l}}\sum_{i=1}^{M_{lk}}\widetilde{x}_{lkij},\forall
j}\pi_{D}(\widetilde{\mathbf{x}})$
$\displaystyle=\prod_{j=1}^{N_{l}}e^{-w_{lj}}\frac{w_{lj}^{x_{lj}}}{x_{lj}!},$
(12)
which is restated as the following theorem:
###### Theorem 1
The single-route network has the same stationary user distribution as that of
the corresponding single-route memoryless network:
$\pi(\mathbf{x})=\pi_{0}(\mathbf{x})$.
## V Stationary User Distribution in Multiple-Route Network
$t_{LN_{L}}$$\ldots$$\lambda_{10}$$\lambda_{L0}$$\lambda_{20}$$\vdots$$\cdots$$\vdots$$\vdots$$t_{11}$$t_{1N_{1}}$$t_{13}$$t_{12}$$\ldots$$0$$t_{L1}$$t_{L2}$$t_{L3}$
Figure 5: Multiple-route network.
In this section, we study the general case with multiple routes. We first
extend the results from the previous section to show
$\pi(\mathbf{x})=\pi_{0}(\mathbf{x})$ in a multiple-route network. We then
derive the stationary user distribution $\pi_{1}(\mathbf{y})$ with respect to
cells and show its insensitivity.
### V-A Queueing Network Model for Multiple-Route Network
Since the $L$ routes are independent, we model the multiple-route network as a
paralleling of $L$ single-route networks, as shown in Fig. 5. Similar to
Section IV, we consider a reference multiple-route memoryless network, which
is a paralleling of $L$ corresponding single-route memoryless networks. Then,
we construct the decoupled multiple-route network, which is a paralleling of
$L$ corresponding single-route decoupled networks.
### V-B Insensitivity of $\pi(\mathbf{x})$
|
arxiv-papers
| 2012-10-05T03:13:53 |
2024-09-04T02:49:35.991625
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Bao and Ben Liang",
"submitter": "Wei Bao",
"url": "https://arxiv.org/abs/1210.1633"
}
|
1210.1653
|
# An Improved Proof-Theoretic Compilation of Logic Programs
Iliano Cervesato
Department of Computer Science
Carnegie Mellon University
E-mail: iliano@cmu.edu
(TBA)
###### Abstract
In prior work, we showed that logic programming compilation can be given a
proof-theoretic justification for generic abstract logic programming
languages, and demonstrated this technique in the case of hereditary Harrop
formulas and their linear variant. Compiled clauses were themselves logic
formulas except for the presence of a second-order abstraction over the atomic
goals matching their head. In this paper, we revisit our previous results into
a more detailed and fully logical justification that does away with this
spurious abstraction. We then refine the resulting technique to support well-
moded programs efficiently.
_To appear in Theory and Practice of Logic Programming._
###### keywords:
Compilation, Abstract Logic Programming, Hereditary Harrop Formulas, Well-
Moded Logic Programs.
††volume: 10 (3):
## 1 Introduction
In [Cervesato (1998)], we presented a general methodology for developing a
compiler and associated intermediate language for any abstract logic
programming language (ALPL) [Miller et al. (1991)] that satisfies some basic
proof-theoretic properties. We applied it abstractly to the language of
hereditary Harrop formulas and its linear variant, and also based the concrete
implementations of the Twelf [Pfenning and Schürmann (1999)] and LLF
[Cervesato and Pfenning (2002)] systems directly on it. This methodology
identified right sequent rules that behave like the left rules that can appear
in a uniform proof and used the corresponding connectives as the compilation
targets of the constructs in program clauses. The intermediate language was
therefore just another ALPL and its abstract machine relied on proof-search,
like the source ALPL. Because the transformation was based on the proof-
theoretic duality between left and right rules, proving the correctness of the
compilation process amounted to a simple induction. Finally, for Horn clauses
the connectives in the target ALPL corresponded to key instructions in the
Warren Abstract Machine (WAM) [Warren (1983)]. The WAM is an essential
component of commercial Prolog systems since many compiled programs run over
an order of magnitude faster than when interpreted.
Up to then, the notoriously procedural instruction set of the WAM was regarded
as a wondrous piece of engineering without any logical status, in sharp
contrast with the deep logical roots of Prolog. In the words of [Börger and
Rosenzweig (1995)] “[the WAM] resembles an intricate puzzle, whose many pieces
fit tightly together in a miraculous way”. As a result, understanding it was
complex in spite of the availability of excellent tutorials [Aït-Kaci (1991)],
proving its correctness was a formidable task [Börger and Rosenzweig (1995),
Russinoff (1992)], and adapting it to other logic programming languages a
major endeavor — it was done for _CLP $({\mathcal{}R})$_ [Jaffar et al.
(1992)] and _$\lambda$ Prolog_ [Nadathur and Mitchell (1999)]. By contrast,
the methodology in [Cervesato (1998)] is simple, (mostly) logic-based, easily
verifiable, and of general applicability.
The technique in [Cervesato (1998)] had however one blemish: it made use of
equality over atomic formulas together with a second-order binder over atomic
goals, which lacked logical status. In this paper, we remedy this drawback by
carefully massaging the head of clauses. This allows us to replace those
constructs with term-level equality and regular universal quantifications over
the arguments of a clause head. The result is an improved proof-theoretic
account of compilation for logic programs that sits squarely within logic. It
also opens the doors to specializing the compilation process to well-moded
programs, which brings out the potential of doing away with unification in
favor of matching, a more efficient operation in many languages. We present
these results for the language of hereditary Harrop formulas and only at the
highest level of abstraction. Just like [Cervesato (1998)], they are however
general, both in terms of the source ALPL and of the level of the abstraction
considered. We are indeed in the process of using them to implement a compiler
for CLF [Watkins et al. (2003), Cervesato et al. (2003)], a higher-order
concurrent linear logic programming language that combines backward and
forward chaining.
The paper is organized as follows: Section 2 recalls the compilation process
of [Cervesato (1998)]. In Section 3, we present our improved compilation
process. In Section 4, we refine it to support moded programs. We lay out
future developments in Sections 5 and 6.
## 2 Background and Recap
In this section, we recall the compilation process presented in [Cervesato
(1998)]. For succinctness, we focus on a smaller source language — it
corresponds to the language underlying the Twelf system [Pfenning and
Schürmann (1999)], on which this technique was first used. We will comment on
larger languages, including those examined in [Cervesato (1998)], in Section
5.
### 2.1 Source Language
We take the language freely generated from atomic propositions ($a$),
intuitionistic implication ($\supset$) and universal quantification
($\forall$) as our source language. We expand the open-ended atomic
propositions of [Cervesato (1998)], into a _predicate symbol_ $p$ followed by
zero or more terms $t$. A program is a sequence of closed formulas. This
language, which we call $\mathcal{L}^{s}$, is given by the following grammar:
$\begin{array}[t]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Formulas:}}\hskip
8.00003pt&A&::=&a\hskip 5.0pt|\hskip 5.0ptA_{1}\supset A_{2}\hskip
5.0pt|\hskip 5.0pt\forall x.\,A\\\ \mbox{\emph{Atoms:}}\hskip
8.00003pt&a&::=&p\hskip 5.0pt|\hskip 5.0pta\>t\end{array}$
$\begin{array}[t]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Programs:}}\hskip
8.00003pt&\Gamma&::=&\cdot\hskip 5.0pt|\hskip 5.0pt\Gamma,A\end{array}$
As in [Cervesato (1998)], we leave the language of terms open, but require
that it be predicative (substituting a term for a variable cannot alter the
outer structure of a formula). We will often write an atom $a$ as
$p\;\underline{t}$, where $p$ is its predicate symbol and $\underline{t}$ is
the sequence of terms it is applied to. We implicitly assume that a predicate
symbol is consistently applied to the same number of terms throughout a
program — its arity. We write $[t^{\prime}/x]t$ (resp. $[t^{\prime}/x]A$) for
the capture-avoiding substitution of term $t^{\prime}$ for all free
occurrences of variable $x$ in term $t$ (resp. in formula $A$). Simultaneous
substitution is denoted $[\underline{t^{\prime}}/\underline{x}]t$ and
$[\underline{t^{\prime}}/\underline{x}]A$.
$\mathcal{L}^{s}$ is an abstract logic programming language [Miller et al.
(1991)] and, for appropriate choices of the term language, has indeed the same
expressive power as $\lambda$Prolog [Miller and Nadathur (1986)] or Twelf
[Pfenning and Schürmann (1999)]. It differs from the first language discussed
in [Cervesato (1998)] for the omission of conjunction and truth (see Section
5).
The operational semantics of $\mathcal{L}^{s}$ is given by the two judgments
$\begin{array}[]{l@{\hspace{1.5em}}p{16em}}\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A\hfil\hskip 15.00002pt&\emph{$A$ is uniformly
provable from $\Gamma$}\\\ \Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A\,\gg\,a\hfil\hskip 15.00002pt&\emph{$a$ is
immediately entailed by $A$ in $\Gamma$}\end{array}$
Their defining rules, given in Figure 1, produce uniform proofs [Miller et al.
(1991)]: the uniform provability judgment includes the right sequent rules for
$\mathcal{L}^{s}$ and, once the goal is atomic, rule $\mathbf{u\\_atm}$ calls
the immediate entailment judgment, which focuses on a program formula $A$ and
decomposes it as prescribed by the left sequent rules. This strategy is
complete with respect to the traditional sequent rules of this logic [Miller
et al. (1991)]. From a logic programming perspective, the connectives
appearing in the goal — handled by right rules — are search directives, while
the left rules carry out a run-time preparatory phase.
${\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
\lx@intercol\mathbf{\scriptstyle}Uniform\;provability\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma,A,\Gamma^{\prime}\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A\,\gg\,a}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}u\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma,A,\Gamma^{\prime}\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,a}$}}$\hfil\cr}}\hskip
25.82562pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma,A_{1}\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A_{2}}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}u\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A_{1}\supset A_{2}}$}}$\hfil\cr}}\hskip
25.53119pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{c\;\mbox{\em``new''}\quad\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,[c/x]A}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}u\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,\forall x.\,A}$}}$\hfil\cr}}\hskip
21.60454pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Immediate\;entailment\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}i\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,a\,\gg\,a}$}}$\hfil\cr}}\hskip
24.23001pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A_{1}\,\gg\,a\quad\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A_{2}}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}i\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A_{2}\supset A_{1}\,\gg\,a}$}}$\hfil\cr}}\hskip
23.93558pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,[t/x]A\,\gg\,a}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}i\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,\forall x.\,A\,\gg\,a}$}}$\hfil\cr}}\hskip
20.00893pt}}}\\\\[-3.01389pt] \end{array}$
Figure 1: Uniform Deduction System for $\mathcal{L}^{s}$.
### 2.2 Target Language
In [Cervesato (1998)], the target language of the compilation process
distinguished compiled goals ($G$) from compiled clauses ($C$). A compiled
goal was either an atomic proposition, or a hypothetical goal (a goal to be
solved in the presence of an additional clause) or a universal goal (a goal to
be solved in the presence of a new constant). A compiled clause had the form
$\Lambda\alpha.\,C$, where the second-order variable $\alpha$ stood for the
atomic goal to be resolved against the present clause, while $C$ could either
match $\alpha$ with the head $a$ of this clause
($a\stackrel{{\scriptstyle.}}{{=}}\alpha$), invoke a goal ($C\,\land\,G$), or
request that a variable $x$ be instantiated with a term ($\exists x.\,C$). A
compiled program $\Psi$ was then a sequence of compiled clauses. The grammar
for the resulting language, which we call $\mathcal{L}^{c}_{0}$, is as
follows:
$\begin{array}[t]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Goals:}}\hskip
8.00003pt&G&::=&a\hskip 5.0pt|\hskip 5.0pt(\Lambda\alpha.\,C)\supset G\hskip
5.0pt|\hskip 5.0pt\forall x.\,G\\\ \mbox{\emph{Clauses:}}\hskip
8.00003pt&C&::=&\makebox[30.00005pt]{$a\stackrel{{\scriptstyle.}}{{=}}\alpha$}\hskip
5.0pt|\hskip 5.0ptC\,\land\,G\hskip 5.0pt|\hskip 5.0pt\exists
x.\,C\end{array}$ $\begin{array}[t]{@{}r@{\hspace{0.8em}} cr
l@{}}\mbox{\emph{Programs:}}\hskip 8.00003pt&\Psi&::=&\cdot\hskip 5.0pt|\hskip
5.0pt\Psi,\Lambda\alpha.\,C\end{array}$
The operational semantics of a compiled program, as given by the above
grammar, is defined on the basis of the following two judgments:
$\begin{array}[]{l@{\hspace{1.5em}}p{16em}}\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,G\hfil\hskip 15.00002pt&\emph{$G$ is uniformly
provable from $\Psi$}\\\ \Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,\tilde{C}\hfil\hskip 15.00002pt&\emph{$\tilde{C}$
is uniformly provable from $\Psi$}\end{array}$
Here, clause instances $\tilde{C}$ are $C$’s whose variable $\alpha$ has been
instantiated with an atomic formula $a^{\prime}$. The operational semantics of
$\mathcal{L}^{c}_{0}$ is shown in Figure 2. Observe that, with the partial
exception of $\mathbf{g0\\_atm}$, it consists solely of right rules. This
means that every connective is seen as a search directive: the dynamic clause
preparations embodied by the left rules has now been turned into right search
rules through a static compilation phase.
${\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
\lx@intercol\mathbf{\scriptstyle}Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,\Lambda\alpha.\,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,[a/\alpha]C}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi,\Lambda\alpha.\,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,a}$}}$\hfil\cr}}\hskip
28.90834pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,\Lambda\alpha.\,C\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,(\Lambda\alpha.\,C)\supset G}$}}$\hfil\cr}}\hskip
28.6139pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{c\;\mbox{\em``new''}\quad\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,[c/x]G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,\forall x.\,G}$}}$\hfil\cr}}\hskip
24.68726pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Clause\>instances\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r0\\_eq$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,a\stackrel{{\scriptstyle.}}{{=}}a}$}}$\hfil\cr}}\hskip
22.9324pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,\tilde{C}\quad\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r0\\_and$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,\tilde{C}\,\land\,G}$}}$\hfil\cr}}\hskip
27.84215pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,[t/x]\tilde{C}}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r0\\_exists$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{0}}}{{\longrightarrow}}\,\exists x.\,\tilde{C}}$}}$\hfil\cr}}\hskip
35.05893pt}}}\\\\[-3.01389pt] \end{array}$
Figure 2: Search Semantics of $\mathcal{L}^{c}_{0}$.
### 2.3 Compilation
Compilation, the process that transforms a logic program in $\mathcal{L}^{s}$
into a compiled program in $\mathcal{L}^{c}_{0}$, is expressed by means of the
following three judgments:
$\begin{array}[]{l@{\hspace{1.5em}}p{16em}}\Gamma\,\gg\,\Psi\hfil\hskip
15.00002pt&\emph{Program $\Gamma$ is compiled to $\Psi$}\\\
A\,\gg\,\alpha\,\backslash\,C\hfil\hskip 15.00002pt&\emph{Clause $A$ with
$\alpha$ is compiled to $C$ }\\\ A\,\gg\,G\hfil\hskip 15.00002pt&\emph{Goal
$A$ is compiled to $G$}\end{array}$
These judgments are defined by the rules in Figure 3 — see [Cervesato (1998)]
for details.
As our ongoing example, consider the following two clauses, taken from a type
checking specification for a Church-style simply typed $\lambda$-calculus. For
clarity, we write program clauses Prolog-style, using the reverse implication
$\subset$ instead of $\supset$ in positive formulas.
1. 1.
$\begin{array}[t]{@{}l@{\hspace{3.5em}}c@{\hspace{2em}}l@{}}\begin{array}[t]{@{}ll@{}}\\\
\lx@intercol\forall E_{1}.\,\forall E_{2}.\,\forall T_{1}.\,\forall
T_{2}.\hfil\\\ &\mathsf{of}\;(\mathsf{app}\;E_{1}\;E_{2})\;T_{2}\\\
\subset&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\\\
\subset&\mathsf{of}\;E_{2}\;T_{1}\end{array}\hfil\hskip
31.50005pt&\raisebox{-27.1249pt}{ \ $\,\gg\,$ \ }\hfil\hskip
18.00003pt&\begin{array}[t]{@{}ll@{}}\lx@intercol\Lambda\alpha.\hfil\\\
\lx@intercol\exists E_{1}.\,\exists E_{2}.\,\exists T_{1}.\,\exists
T_{2}.\hfil\\\
&(\mathsf{of}\;(\mathsf{app}\;E_{1}\;E_{2})\;T_{2})\stackrel{{\scriptstyle.}}{{=}}\alpha\\\
\,\land&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\\\
\,\land&\mathsf{of}\;E_{2}\;T_{1}\end{array}\end{array}$
2. 2.
$\begin{array}[t]{@{}lcl@{}}\begin{array}[t]{@{}ll@{}}\\\ \lx@intercol\forall
E.\,\forall T_{1}.\,\forall T_{2}.\hfil\\\
&\mathsf{of}\;(\mathsf{lam}\;T_{1}\;E)\;(\mathsf{arr}\;T_{1}\;T_{2})\\\
\subset&(\forall x.\,\mathsf{of}\;x\;T_{1}\\\ &\hskip
4.5pt\supset\mathsf{of}\;(E\;x)\;T_{2})\end{array}&\raisebox{-23.24991pt}{$\,\gg\,$}&\begin{array}[t]{@{}ll@{}}\lx@intercol\Lambda\alpha.\hfil\\\
\lx@intercol\exists E.\,\exists T_{1}.\,\exists T_{2}.\hfil\\\
&(\mathsf{of}\;(\mathsf{lam}\;T_{1}\;E)\;(\mathsf{arr}\;T_{1}\;T_{2}))\stackrel{{\scriptstyle.}}{{=}}\alpha\\\
\,\land&(\forall
x.\,\begin{array}[t]{@{}l@{\;}l@{}}&\Lambda\beta.\,((\mathsf{of}\;x\;T_{1})\stackrel{{\scriptstyle.}}{{=}}\beta)\\\
\supset&\mathsf{of}\;(E\;x)\;T_{2})\end{array}\end{array}\end{array}$
The compiled language $\mathcal{L}^{c}_{0}$ is sound and complete for
$\mathcal{L}^{s}$. See [Cervesato (1998)] for the formal statements. The proof
of both directions proceeds by straightforward induction, which contrasts
greatly with the complex proofs of soundness and correctness previously
devised for the WAM [Börger and Rosenzweig (1995), Russinoff (1992)].
${{\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
\lx@intercol\mathbf{\scriptstyle}Programs\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}p0c\\_empty$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\cdot\,\gg\,\cdot}$}}$\hfil\cr}}\hskip
38.63383pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma\,\gg\,\Psi\quad
A\,\gg\,\alpha\,\backslash\,C}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}p0c\\_clause$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma,A\,\gg\,\Psi,\Lambda\alpha.\,C}$}}$\hfil\cr}}\hskip
38.99922pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Clauses\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c0c\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut
a\,\gg\,\alpha\,\backslash\,a\stackrel{{\scriptstyle.}}{{=}}\alpha}$}}$\hfil\cr}}\hskip
31.37698pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{B\,\gg\,\alpha\,\backslash\,C\quad
A\,\gg\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c0c\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut A\supset
B\,\gg\,\alpha\,\backslash\,C\,\land\,G}$}}$\hfil\cr}}\hskip
31.08255pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,\alpha\,\backslash\,C}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c0c\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\forall
x.\,A\,\gg\,\alpha\,\backslash\,\exists x.\,C}$}}$\hfil\cr}}\hskip
27.1559pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0c\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut
a\,\gg\,a}$}}$\hfil\cr}}\hskip
31.93762pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,\alpha\,\backslash\,C\quad
B\,\gg\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0c\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut A\supset
B\,\gg\,(\Lambda\alpha.\,C)\supset G}$}}$\hfil\cr}}\hskip
31.64319pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,C}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g0c\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\forall
x.\,A\,\gg\,\forall x.\,C}$}}$\hfil\cr}}\hskip 27.71654pt}}}\\\\[-3.01389pt]
\end{array}$
Figure 3: Compilation of $\mathcal{L}^{s}$ into $\mathcal{L}^{c}_{0}$.
## 3 Fully Logical Compilation
Because clauses are compiled to expressions of the form $\Lambda\alpha.\,C$,
the language $\mathcal{L}^{c}_{0}$ is not fully logical. In this section we
consider a different compilation target, the language $\mathcal{L}^{c}_{1}$,
which lies entirely within logic.
In the previous section, a generic Horn clause of the form
$\forall\underline{y}.\,(p\>\underline{t}\subset a_{1}\subset\ldots\subset
a_{n})$ (1)
was compiled into
$\Lambda\alpha.\,\exists\underline{y}.\,(p\>\underline{t}\stackrel{{\scriptstyle.}}{{=}}\alpha\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n}).$
During execution, rule $\mathbf{c0\\_atm}$ reduced the current atomic goal $a$
to the clause instance
$\exists\underline{y}.\,(p\>\underline{t}\stackrel{{\scriptstyle.}}{{=}}a\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n})$.
Note that $\underline{t}$ may depend on $\underline{y}$, but $a$ does not. We
will now compile that Horn clause into
$\forall\underline{x}.\,(p\;\underline{x}\subset\exists\underline{y}.\,(\underline{x}\stackrel{{\scriptstyle.}}{{=}}\underline{t}\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n}))$
(2)
where $\underline{x}$ is a sequence of fresh variables, all distinct from each
other, and equal in number to the arity of $p$, and
$\underline{x}\stackrel{{\scriptstyle.}}{{=}}\underline{t}$ stands for a
conjunction of equalities between each variable $x_{i}$ in $\underline{x}$ and
the term $t_{i}$ in $\underline{t}$ in the corresponding position (or $\top$
if the arity of $p$ is zero). Notice that the non-logical second-order binder
“$\Lambda\alpha.\,\\!$” is gone. At run time, formula (2) will resolve an
atomic goal $p\;\underline{t^{\prime}}$ into the clause
$p\;\underline{t^{\prime}}\subset\exists\underline{y}.\,(\underline{t^{\prime}}\stackrel{{\scriptstyle.}}{{=}}\underline{t}\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n})$,
which immediately reduces to
$\exists\underline{y}.\,(\underline{t^{\prime}}\stackrel{{\scriptstyle.}}{{=}}\underline{t}\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n})$.
Like earlier, $\underline{t}$ may depend on $\underline{y}$, but
$\underline{t^{\prime}}$ does not. The variables $\underline{x}$ correspond
directly to the “argument registers” (`A`$n$) of the WAM [Aït-Kaci (1991)],
while the $\underline{y}$’s are closely related to its “permanent variables”
(`Y`$n$).
Formula (2) can be understood as an uncurried form of (1): outer implications
are transformed into conjunctions and universals into existentials. Doing so
literally would yield the formula
$p\;\underline{t}\subset\exists\underline{y}.\,(a_{1}\,\land\,\ldots\,\land\,a_{n})$,
which is incorrect because occurrences of variables in $\underline{y}$ within
$\underline{t}$ have escaped their scope. Instead, formula (2) installs fresh
variables $\underline{x}$ as the arguments of the head predicate $p$ and adds
the equality constraints
$\underline{x}\stackrel{{\scriptstyle.}}{{=}}\underline{t}$ in the body.
### 3.1 Target Language
We now generalize the above intuition to any formula in $\mathcal{L}^{s}$, not
just Horn clauses. Our second target language, $\mathcal{L}^{c}_{1}$, is given
by the following grammar.
$\begin{array}[t]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Goals:}}\hskip
8.00003pt&G&::=&a\hskip 5.0pt|\hskip 5.0ptC\supset G\hskip 5.0pt|\hskip
5.0pt\forall x.\,G\\\ \mbox{\emph{Clauses:}}\hskip 8.00003pt&C&::=&R\supset
p\>\underline{x}\hskip 5.0pt|\hskip 5.0pt\forall x.\,C\\\
\mbox{\emph{Residuals:}}\hskip
8.00003pt&R&::=&x\stackrel{{\scriptstyle.}}{{=}}t\hskip 5.0pt|\hskip
5.0pt\top\hskip 5.0pt|\hskip 5.0ptR\,\land\,G\hskip 5.0pt|\hskip 5.0pt\exists
x.\,R\end{array}$ $\begin{array}[t]{@{}r@{\hspace{0.8em}} cr
l@{}}\mbox{\emph{Programs:}}\hskip 8.00003pt&\Psi&::=&\cdot\hskip 5.0pt|\hskip
5.0pt\Psi,C\end{array}$
Compiled goals ($G$) are just like in Section 2.2: atoms, hypothetical goals,
or universal goals. Compiled clauses ($C$) have the form
$\forall\underline{x}.\,(R\supset p\>\underline{x})$, i.e., a (possibly empty)
outer layer of universal quantifiers enclosing an implication $R\supset
p\>\underline{x}$ whose head $p\>\underline{x}$ always consists of a predicate
name ($p$) applied to a (possibly empty) sequence of distinct variables
($\underline{x}$). Its body is a _residual_ ($R$). A residual can be either an
equality constraint ($x\stackrel{{\scriptstyle.}}{{=}}t$), the trivial
constraint $\top$ (logical truth), or like in Section 2.2 a goal invocation or
an instantiation request. Notice that $C$ is now the full result of compiling
a clause.
${{\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
\lx@intercol\mathbf{\scriptstyle}Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,C\,\gg\,a}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,a}$}}$\hfil\cr}}\hskip
28.90834pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,C\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,C\supset G}$}}$\hfil\cr}}\hskip
28.6139pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{c\;\mbox{\em``new''}\quad\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,[c/x]G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\forall x.\,G}$}}$\hfil\cr}}\hskip
24.68726pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Clauses\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\tilde{R}}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c1\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\tilde{R}\supset a\,\gg\,a}$}}$\hfil\cr}}\hskip
28.05327pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,[t/x]\tilde{C}\,\gg\,a}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c1\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\forall x.\,\tilde{C}\,\gg\,a}$}}$\hfil\cr}}\hskip
24.12662pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Residuals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r1\\_eq$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,t\stackrel{{\scriptstyle.}}{{=}}t}$}}$\hfil\cr}}\hskip
22.9324pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r1\\_true$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\top}$}}$\hfil\cr}}\hskip
29.44391pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\tilde{R}\quad\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r1\\_and$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\tilde{R}\,\land\,G}$}}$\hfil\cr}}\hskip
27.84215pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,[t/x]\tilde{R}}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r1\\_exists$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\exists x.\,\tilde{R}}$}}$\hfil\cr}}\hskip
35.05893pt}}}\\\\[-3.01389pt] \end{array}$
Figure 4: Search Semantics of $\mathcal{L}^{c}_{1}$.
The operational semantics of $\mathcal{L}^{c}_{1}$ is specified by the
following three judgments:
$\begin{array}[]{l@{\hspace{1.5em}}p{18em}}\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,G\hfil\hskip 15.00002pt&\emph{$G$ is uniformly
provable from $\Psi$}\\\ \Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\tilde{C}\,\gg\,a\hfil\hskip 15.00002pt&\emph{$a$
is immediately entailed by $\tilde{C}$ in $\Psi$}\\\
\Psi\stackrel{{\scriptstyle c_{1}}}{{\longrightarrow}}\,\tilde{R}\hfil\hskip
15.00002pt&\emph{$\tilde{R}$ is uniformly provable from $\Psi$}\end{array}$
where $\tilde{C}$ and $\tilde{R}$ differ from $C$ and $R$ by the instantiation
of some variables in a clause head and on the left-hand side of equalities,
respectively.
Their operational semantics is given in Figure 4. Goals are handled exactly in
the same way as uniform provability in $\mathcal{L}^{s}$ (top part of Figure
1). The operational reading of compiled clauses is an instance of that of
immediate entailment: rule $\mathbf{c1\\_imp}$ is a special case of
$\mathbf{i\\_imp}$ while $\mathbf{c1\\_all}$ is isomorphic to
$\mathbf{i\\_all}$. Note that rule $\mathbf{c1\\_imp}$ reduces immediately to
the residual $R$ if the head of the clause matches the atomic goal $a$ being
proved. The rules for residuals correspond closely to the rules for clause
instances for our original target language at the bottom of Figure 2: rule
$\mathbf{r1\\_eq}$ requires that the two sides of an equality be indeed equal
and rule $\mathbf{r1\\_true}$ is always satisfied.
The rules in Figure 4 build uniform proofs [Miller et al. (1991)],
characteristic of abstract logic programming languages: the operational
semantics decomposes a goal to an atomic formula (top segment of Figure 4),
then selects a clause and focuses on it until it finds a matching head (middle
segment) and then decomposes its body (bottom segment), which may eventually
expose some goals, and the cycle repeats. In particular, once an atomic goal
$p\>\underline{t}$ has been exposed, a successful derivation will necessarily
contain an instance of rule $\mathbf{g1\\_atm}$ that picks a clause $C$ with
head $p\>\underline{x}$, as many instances of rule $\mathbf{c1\\_all}$ as the
arity of $p$, and an instance of rule $\mathbf{c1\\_imp}$. This necessary
sequence of steps is captured by the following derived “macro-rule” (the
_backchaining_ rule):
$\displaystyle{\Psi,\forall\underline{x}.\,(R\supset
p\>\underline{x}),\Psi^{\prime}\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,[\underline{t}/\underline{x}]R}$
$\displaystyle{\mathstrut\Psi,\forall\underline{x}.\,(R\supset
p\>\underline{x}),\Psi^{\prime}\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,p\>\underline{t}}$
Replacing rules $\mathbf{g1\\_atm}$, $\mathbf{c1\\_all}$ and
$\mathbf{c1\\_imp}$ with rule $\mathbf{g1\\_atm^{\prime}}$ yields a system
that is equivalent to that in Figure 4. Taking it as primitive amounts to
replacing the construction for compiled clauses,
$\forall\underline{x}.\,(R\supset p\>\underline{x})$, with a synthetic
connective, call it $\Lambda_{p}\underline{x}.\,R$. Therefore, by accounting
for the structure of atomic propositions and proper quantification patterns,
$\mathcal{L}^{c}_{1}$ provides a fully logical justification for clause
compilation that $\mathcal{L}^{c}_{0}$’s $\Lambda\alpha.\,C$ lacked.
### 3.2 Compilation
Compilation transforms logic programs in $\mathcal{L}^{s}$ into compiled logic
programs in $\mathcal{L}^{c}_{1}$. In order to define it, the auxiliary notion
of pseudo clause will come handy:
$\begin{array}[]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Pseudo
Clauses:}}\hskip 8.00003pt&\mathcal{C}&::=&\Box\supset p\>\underline{x}\hskip
5.0pt|\hskip 5.0pt\forall x.\,\mathcal{C}\end{array}$
A pseudo clause retains the outer structure of a clause, but has a hole
($\Box$) in place of the residual $R$. In general, a pseudo clause
$\mathcal{C}$ has the form $\forall\underline{x}.\,\Box\supset
p\>\underline{x^{\prime}}$. In a fully compiled clause, variables
$\underline{x}$ will coincide with $\underline{x^{\prime}}$.
Pseudo clauses are generated while processing the head of a clause. The hole
then needs to be replaced with the compiled body, a residual. We write this
operation, pseudo clause instantiation, as $\mathcal{C}[R]$. It is formally
defined as follows:
$\left\\{\begin{array}[]{lcl}(\Box\supset p\>\underline{x})[R]&=&R\supset
p\>\underline{x}\\\ (\forall x.\,\mathcal{C})[R]&=&\forall
x.\,(\mathcal{C}[R])\end{array}\right.$
As is often the case with such contextual operations, pseudo clause
instantiation can, and generally will, lead to variable capture: in
$(\forall\underline{x}.\,\Box\supset p\>\underline{x})[R]$, there may be free
occurrences of variables in $\underline{x}$ within $R$. In the result, these
occurrences are bound by the outer quantifiers.
Compilation is expressed by means of the following four judgments
$\begin{array}[]{l@{\hspace{1.5em}}p{17em}}\Gamma\,\gg\,\Psi\hfil\hskip
15.00002pt&\emph{Program $\Gamma$ is compiled to $\Psi$}\\\
\underline{x}\vdash a\,\gg\,\mathcal{C}\,\backslash\,E\hfil\hskip
15.00002pt&\emph{Head $a$ with $\underline{x}$ is compiled to $\mathcal{C}$
and $E$}\\\ A\,\gg\,\mathcal{C}\,\backslash\,R\hfil\hskip
15.00002pt&\emph{Clause $A$ is compiled to $\mathcal{C}$ and $R$}\\\
A\,\gg\,G\hfil\hskip 15.00002pt&\emph{Goal $A$ is compiled to $G$}\end{array}$
and defined by the rules in Figure 5, where we wrote $E$ for conjunctions of
equalities. The judgment $A\,\gg\,\mathcal{C}\,\backslash\,R$ compiles an
$\mathcal{L}^{s}$ clause $A$ into a pseudo clause $\mathcal{C}$ and a residual
$R$. They are assembled into an $\mathcal{L}^{c}_{1}$ clause in rules
$\mathbf{p1c\\_clause}$ and $\mathbf{g1c\\_imp}$. Programs and goals are
otherwise compiled just as for $\mathcal{L}^{c}_{0}$ in Figure 3. Clause heads
are handled differently: rule $\mathbf{c1c\\_atm}$ invokes the auxiliary head
compilation judgment to compile the goal $p\>\underline{t}$ into a pseudo
clause $\forall\underline{x}.\,\Box\supset p\>\underline{x}$ and the
equalities $\underline{x}\stackrel{{\scriptstyle.}}{{=}}\underline{t}$, which
will form the seed of the clause’s residual.
Consider the first example clause in Section 2.3. Its head
($\mathsf{of}\;(\mathsf{app}\;E_{1}\;E_{2})\;T_{2}$) is compiled into the
pseudo clause $\forall x_{1}.\,\forall
x_{2}.\,(\Box\supset\mathsf{of}\;x_{1}\;x_{2})$ and the equality constraints
$\top\,\land\,(x_{1}\stackrel{{\scriptstyle.}}{{=}}\mathsf{app}\;E_{1}\;E_{2})\,\land\,(x_{2}\stackrel{{\scriptstyle.}}{{=}}T_{2})$,
where $x_{1}$ and $x_{2}$ are new variables. These core equalities are then
extended with the compiled body of that clause,
$(\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2}))\,\land\,(\mathsf{of}\;E_{2}\;T_{1})$,
and existential quantifications over the original variables of the clause,
$E_{1}$, $E_{2}$, $T_{1}$ and $T_{2}$, are finally wrapped around the result
before embedding it in the hole of the pseudo clause. The resulting
$\mathcal{L}^{c}_{1}$ clause is displayed in the top part of Figure 6.
${{{\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-9.04166pt]
\lx@intercol\mathbf{\scriptstyle}Programs\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}p1c\\_empty$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\cdot\,\gg\,\cdot}$}}$\hfil\cr}}\hskip
38.63383pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Gamma\,\gg\,\Psi\quad
A\,\gg\,\mathcal{C}\,\backslash\,R}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}p1c\\_clause$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Gamma,A\,\gg\,\Psi,\mathcal{C}[R]}$}}$\hfil\cr}}\hskip
38.99922pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Heads\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}h1c\\_p$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\underline{x}\vdash
p\,\gg\,\Box\supset p\>\underline{x}\,\backslash\,\top}$}}$\hfil\cr}}\hskip
23.5287pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{x\>\underline{x}\vdash
a\,\gg\,\mathcal{C}\,\backslash\,E\quad x\>\mbox{\em``new''}}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}h1c\\_pt$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\underline{x}\vdash
a\;t\,\gg\,\forall
x.\,\mathcal{C}\,\backslash\,E\,\land\,x\stackrel{{\scriptstyle.}}{{=}}t}$}}$\hfil\cr}}\hskip
26.05649pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Clauses\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[0.0pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\cdot\vdash
a\,\gg\,\mathcal{C}\,\backslash\,E}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c1c\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut
a\,\gg\,\mathcal{C}\,\backslash\,E}$}}$\hfil\cr}}\hskip
31.37698pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,G\quad
B\,\gg\,\mathcal{C}\,\backslash\,R}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c1c\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut A\supset
B\,\gg\,\mathcal{C}\,\backslash\,R\,\land\,G}$}}$\hfil\cr}}\hskip
31.08255pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,\mathcal{C}\,\backslash\,R}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c1c\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\forall
x.\,A\,\gg\,\mathcal{C}\,\backslash\,\exists x.\,R}$}}$\hfil\cr}}\hskip
27.1559pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1c\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut
a\,\gg\,a}$}}$\hfil\cr}}\hskip
31.93762pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,\mathcal{C}\,\backslash\,R\quad
B\,\gg\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1c\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut A\supset
B\,\gg\,\mathcal{C}[R]\supset G}$}}$\hfil\cr}}\hskip
31.64319pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{A\,\gg\,C}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g1c\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\forall
x.\,A\,\gg\,\forall x.\,C}$}}$\hfil\cr}}\hskip 27.71654pt}}}\\\\[-3.01389pt]
\end{array}$
Figure 5: Compilation of $\mathcal{L}^{s}$ into $\mathcal{L}^{c}_{1}$.
The target language $\mathcal{L}^{c}_{1}$ is sound and complete with respect
to $\mathcal{L}^{s}$. In order to show it, we need the following auxiliary
results. The first statement is proved by induction on the structure of $a$.
The second by induction on the given derivation.
###### Lemma 3.1
* •
If $\underline{x}\vdash a\,\gg\,\mathcal{C}\,\backslash\,E$, then for all
$\underline{t}$ of the same length as $\underline{x}$ and all $\Psi$ we have
$\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,[\underline{t}/\underline{x}](\mathcal{C}[E])\,\gg\,a\>\underline{t}$.
* •
If $\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\mathcal{C}[R]\,\gg\,a$, then
$\Psi\stackrel{{\scriptstyle c_{1}}}{{\longrightarrow}}\,R$.
The statements of soundness and completeness are as follows. For each of them,
the proof proceeds by mutual induction on the first derivation in the
antecedent.
###### Theorem 3.2 (Soundness of the compilation to $\mathcal{L}^{c}_{1}$)
* •
If $\Gamma\stackrel{{\scriptstyle u}}{{\longrightarrow}}\,A$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,G$, then $\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,G$.
* •
If $\Gamma\stackrel{{\scriptstyle u}}{{\longrightarrow}}\,A\,\gg\,a$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,\mathcal{C}\,\backslash\,R$, then
$\Psi\stackrel{{\scriptstyle
c_{1}}}{{\longrightarrow}}\,\mathcal{C}[R]\,\gg\,a$.
###### Theorem 3.3 (Completeness of the compilation to $\mathcal{L}^{c}_{1}$)
* •
If $\Psi\stackrel{{\scriptstyle c_{1}}}{{\longrightarrow}}\,G$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,G$, then $\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A$.
* •
If $\Psi\stackrel{{\scriptstyle c_{1}}}{{\longrightarrow}}\,C\,\gg\,a$,
$\Gamma\,\gg\,\Psi$, $C=\mathcal{C}[R]$ and
$A\,\gg\,\mathcal{C}\,\backslash\,R$, then $\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A\,\gg\,a$.
1. $\begin{array}[t]{@{}l@{\hspace{3.5em}}c@{\hspace{2em}}l@{}}\begin{array}[t]{@{}ll@{}}\lx@intercol\forall E_{1}.\,\forall E_{2}.\,\forall T_{1}.\,\forall T_{2}.\hfil\\\ &\mathsf{of}\;(\mathsf{app}\;E_{1}\;E_{2})\;T_{2}\\\ \\\ \\\ \\\ \subset&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\\\ \subset&\mathsf{of}\;E_{2}\;T_{1}\end{array}\hfil\hskip 31.50005pt&\raisebox{-38.74985pt}{ \ $\,\gg\,$ \ }\hfil\hskip 18.00003pt&\begin{array}[t]{@{}ll@{}l@{\;}l@{}}\lx@intercol\forall x_{1}.\,\forall x_{2}.\hfil\\\ &\lx@intercol\mathsf{of}\;x_{1}\;x_{2}\hfil\\\ \subset&(&\lx@intercol\exists E_{1}.\,\exists E_{2}.\,\exists T_{1}.\,\exists T_{2}.\,\top\hfil\\\ &&\,\land&x_{1}\stackrel{{\scriptstyle.}}{{=}}\mathsf{app}\;E_{1}\;E_{2}\\\ &&\,\land&x_{2}\stackrel{{\scriptstyle.}}{{=}}T_{2}\\\ &&\,\land&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\\\ &&\,\land&\mathsf{of}\;E_{2}\;T_{1})\end{array}\end{array}$ 2. $\begin{array}[t]{@{}lcl@{}}\begin{array}[t]{@{}ll@{}}\lx@intercol\forall E.\,\forall T_{1}.\,\forall T_{2}.\hfil\\\ &\mathsf{of}\;(\mathsf{lam}\;T_{1}\;E)\;(\mathsf{arr}\;T_{1}\;T_{2})\\\ \\\ \\\ \\\ \subset&(\forall x.\\\ \\\ \\\ &\hskip 18.00003pt\mathsf{of}\;x\;T_{1}\\\ &\hskip 4.5pt\supset\mathsf{of}\;(E\;x)\;T_{2})\end{array}&\raisebox{-56.18727pt}{$\,\gg\,$}&\begin{array}[t]{@{}ll@{}l@{\;}l@{}}\lx@intercol\forall x_{1}.\,\forall x_{2}.\hfil\\\ &\lx@intercol\mathsf{of}\;x_{1}\;x_{2}\hfil\\\ \subset&(&\lx@intercol\exists E.\,\exists T_{1}.\,\exists T_{2}.\,\top\hfil\\\ &&\,\land&x_{1}\stackrel{{\scriptstyle.}}{{=}}\mathsf{lam}\;T_{1}\;E\\\ &&\,\land&x_{2}\stackrel{{\scriptstyle.}}{{=}}\mathsf{arr}\;T_{1}\;T_{2}\\\ &&\,\land&(\forall x.\,\begin{array}[t]{@{}l@{\;}l@{}}&\begin{array}[t]{@{}l@{\;}l@{}}\lx@intercol\forall x_{1}^{\prime}.\,\forall x_{2}^{\prime}.\,\top\hfil\\\ \,\land&x_{1}^{\prime}\stackrel{{\scriptstyle.}}{{=}}x\\\ \,\land&x_{2}^{\prime}\stackrel{{\scriptstyle.}}{{=}}T_{1}\\\ \,\land&\mathsf{of}\;x_{1}^{\prime}\;x_{2}^{\prime})\end{array}\\\ \supset&\mathsf{of}\;(E\;x)\;T_{2})\end{array}\end{array}\end{array}$
Figure 6: $\mathcal{L}^{c}_{1}$ Compilation Example
We conclude this section by showing in Figure 6 the output of our compilation
procedure for the two examples seen in Section 2.3. We stretch the source
clauses (left) to align corresponding atoms. As can be gleaned from these
clauses, there are ample opportunities for optimizations in our compilation
process. In particular, a constraint $x\stackrel{{\scriptstyle.}}{{=}}y$
mentioning variables on both sides can often be eliminated by replacing the
existential variable $y$ with the universal variable $x$ in the rest of the
clause (and removing the existential quantifier) — the exception is when there
are multiple constraints of this form for the same $y$. The leading logical
constant $\top$ makes for a succinct presentation of the compilation process,
but plays no actual role: it can also be eliminated.
It is interesting to rewrite these clauses using the synthetic connective
$\Lambda_{p}$ discussed earlier (we have omitted occurrences of $\top$ for
readability):
$\begin{array}[]{lll}\Lambda_{\mathsf{of}}\;x_{1}\>x_{2}.&\lx@intercol\exists
E_{1}.\,\exists E_{2}.\,\exists T_{1}.\,\exists T_{2}.\hfil\lx@intercol\\\
&&x_{1}\stackrel{{\scriptstyle.}}{{=}}\mathsf{app}\;E_{1}\;E_{2}\,\land\,\;x_{2}\stackrel{{\scriptstyle.}}{{=}}T_{2}\\\
&\,\land&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\;\,\land\,\;\mathsf{of}\;E_{2}\;T_{1}\\\\[4.30554pt]
\Lambda_{\mathsf{of}}\;x_{1}\>x_{2}.&\lx@intercol\exists E.\,\exists
T_{1}.\,\exists T_{2}.\hfil\lx@intercol\\\
&&x_{1}\stackrel{{\scriptstyle.}}{{=}}\mathsf{lam}\;T_{1}\;E\;\,\land\,\;x_{2}\stackrel{{\scriptstyle.}}{{=}}\mathsf{arr}\;T_{1}\;T_{2}\\\
&\,\land&\forall
x.\,(\Lambda_{\mathsf{of}}\;x_{1}^{\prime}\>x_{2}^{\prime}.\;\;x_{1}^{\prime}\stackrel{{\scriptstyle.}}{{=}}x\>\,\land\,\>x_{2}^{\prime}\stackrel{{\scriptstyle.}}{{=}}T_{1})\;\supset\;\mathsf{of}\;(E\;x)\;T_{2}\end{array}$
## 4 Support for Moded Programs
In this section, we will specialize the compilation process just outlined to
the case where the source program is well-moded. In a well-model program, the
argument positions of each predicate symbol are designated as either input or
output. Input arguments are guaranteed to be ground terms at the time a goal
is called. Dually, output arguments are guaranteed to have been made ground by
the time the call returns.
There are operational benefits to working with well-moded programs: while an
interpreter for a generic program must implement term-level unification, well-
moded programs can be executed by relying uniquely on pattern matching and
variable instantiation. This is desirable because matching often behaves
better than general unification. For example, it is more efficient for first-
order term languages were it only because it does away with the occurs-check,
and it is decidable for higher-order term languages while general unification
is not [Stirling (2009)].
The development in this section is motivated by well-moding, but is sound
independently of whether a program is well-moded or not. Statically enforcing
well-moding brings the operational advantages just discussed, but the results
in this section do not depend on it.
### 4.1 Source Language
In this section, we assume that each predicate symbol in $\mathcal{L}^{s}$
comes with a _mode_ which declares each of its arguments as input, written
$\check{\;}$, or output, written $\hat{\;}$. For simplicity of exposition, we
decorate the actual arguments of all atomic propositions with these symbols,
so that a term $t$ in input position in an atomic proposition is written
$\check{t}$ (read “in $t$”). Similarly $t$ in output position is written
$\hat{t}$ (pronounced “out $t$”). This amounts to revising the grammar of
atomic propositions as follows:
$\begin{array}[]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Atoms:}}\hskip
8.00003pt&a&::=&p\hskip 5.0pt|\hskip 5.0pta\>\check{t}\hskip 5.0pt|\hskip
5.0pta\>\hat{t}\end{array}$
Just like we assume that the arity of a predicate symbol $p$ remains constant
in a program, we require that all atomic propositions for $p$ have their
input/output marks in the same positions. This pattern is the mode of $p$ — an
actual language would rely on explicit mode declarations.
For typographic convenience and without loss of generality, our examples
assume that input positions precede output positions so that an atomic formula
$a$ can be written as $p\;\underline{\check{t}}\;\underline{\hat{t}}$ where
$\underline{\check{t}}$ and $\underline{\hat{t}}$ are the (possibly empty)
sequences of terms in input (resp. output) positions for $p$. To avoid
notational proliferation, we use the markers $\check{\;}$ and $\hat{\;}$ both
as mode designators and as symbol decorations (like primes and subscripts)
when working with generic terms. Therefore, $\check{t}$ and $\hat{t}$ indicate
possibly different terms in $p\>\check{t}\>\hat{t}$, and similarly for term
sequences, as in $p\;\underline{\check{t}}\;\underline{\hat{t}}$ above.
At our level of abstraction, the rules in Figure 1 capture the operational
semantics of this variant of $\mathcal{L}^{s}$: mode annotations are simply
ignored. However, moded execution requires that two of the operational choices
left open by those rules be resolved using some algorithmic strategy: the
order in which rule $\mathbf{i\\_imp}$ searches for derivations of its two
premises, and the substitution term that rule $\mathbf{i\\_all}$ picks. For
both, we will assume the same strategy as Prolog: implement rule
$\mathbf{i\\_imp}$ left to right and implement rule $\mathbf{i\\_all}$ lazily
by replacing each variable $x$ with a “logical variable” $X$ which is
instantiated incrementally through unification. This allows us to view an
atomic goal as a (non-deterministic) procedure call. In a well-moded program
[Debray and Warren (1988)], terms in input position are seen as the actual
arguments of this procedure, and terms in output position yield return values.
In this section, we will not formalize the notion of well-modedness — see
[Debray and Warren (1988)] for Prolog and [Sarnat (2010)] for Twelf — nor
refine our operational semantics to make goal evaluation order and unification
explicit — see [Pientka (2003)]. We will instead refine our compilation
process to account for mode information and produce compiled programs that, if
well-moded, can be executed without appealing to unification.
### 4.2 Target Language
In $\mathcal{L}^{c}_{1}$, a (well-moded) Horn clause
$\forall\underline{y}.\,p\>\underline{\check{t}}\>\underline{\hat{t}}\subset
a_{1}\subset\ldots\subset a_{n}$ was compiled into
$\forall\underline{\check{x}}\;\underline{\hat{x}}.\,(p\>\underline{\check{x}}\>\underline{\hat{x}}\subset\exists\underline{y}.\,(\underline{\check{x}}\stackrel{{\scriptstyle.}}{{=}}\underline{\check{t}}\,\land\,\underline{\hat{x}}\stackrel{{\scriptstyle.}}{{=}}\underline{\hat{t}}\,\land\,a_{1}\,\land\,\ldots\,\land\,a_{n}))$.
Here, the left-to-right execution order forces us to guess the final values of
the output variables $\underline{\hat{x}}$ before the goals in its body have
been fully executed. In $\mathcal{L}^{c}_{2}$, we will move the equality
$\underline{\hat{x}}\stackrel{{\scriptstyle.}}{{=}}\underline{\hat{t}}$ after
the last goal $a_{n}$. Since $\underline{\hat{x}}$ appear nowhere else in the
residual, this equality is no more than an assignment of the computed instance
of $\underline{\hat{t}}$ to $\underline{\hat{x}}$. Accordingly, we will write
it as $\underline{\hat{x}}:=\underline{\hat{t}}$. Furthermore, in a well-moded
program, this clause will be invoked with ground terms in input position, so
that $\underline{\check{x}}$ will be bound to ground terms. Then, the input
equality
$\underline{\check{x}}\stackrel{{\scriptstyle.}}{{=}}\underline{\check{t}}$
will match the variables in $\underline{\check{t}}$ with appropriate subterms.
For this reason, we will write it as
$\underline{\check{x}}=:\underline{\check{t}}$. Expanding each goal $a_{i}$
into $q_{i}\;\underline{\check{t}}_{i}\;\underline{\hat{t}}_{i}$, the above
clause will be compiled (almost) as follows, where the arrows represent the
data flow of a well-moded execution (note that it parallels the control flow):
| | | | | | | | | | | | | | | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\textstyle{\forall\underline{\check{x}}\>\underline{\hat{x}}.\,(}$$\textstyle{p}$$\textstyle{\underline{\check{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\hat{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
8.00003pt\subset\hskip
8.00003pt}$$\textstyle{(\exists\underline{y}.\,}$$\textstyle{\underline{\check{x}}}$$\textstyle{=:}$$\textstyle{\underline{\check{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\and}$$\textstyle{q_{1}}$$\textstyle{\underline{\check{t}}_{1}}$$\textstyle{\underline{\hat{t}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\and}$$\textstyle{\mathord{.}\;}$$\textstyle{\\!\\!\mathord{.}\;\\!\\!}$$\textstyle{\mathord{.}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\and}$$\textstyle{q_{n}}$$\textstyle{\underline{\check{t}}_{n}}$$\textstyle{\underline{\hat{t}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\and}$$\textstyle{\underline{\hat{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{:=}$$\textstyle{\underline{\hat{t}}}$$\textstyle{))}$
When executing an atomic goal, it is desirable to separate the call from the
verification that the output terms returned by the caller match the expected
output terms in this goal. We will do so by rewriting any atomic goal
$q\>\underline{\check{t}}\>\underline{\hat{t}}$ in a compiled clause into the
formula
$\exists\underline{z}.\,(q\>\underline{\check{t}}\>\underline{z}\,\land\,\underline{z}=:\underline{\hat{t}})$
for fresh variables $\underline{z}$. This transformation preserves the left-
to-right control and data flow. No special provision needs to be made for the
input arguments of $q$ as variables in it will have been instantiated to
ground terms at the moment the call is made.
Next, we again generalize this intuition to any formula in $\mathcal{L}^{s}$,
not just Horn clauses. Our third target language, $\mathcal{L}^{c}_{2}$, is
defined by the following grammar.
$\begin{array}[t]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Goal
Matches:}}\hskip 8.00003pt&M&::=&\top\hskip 5.0pt|\hskip
5.0ptM\,\land\,z=:\hat{t}\\\ \mbox{\emph{Atomic Goals:}}\hskip
8.00003pt&F&::=&p\;\underline{\check{t}}\;\underline{\hat{z}}\,\land\,M\hskip
5.0pt|\hskip 5.0pt\exists z.\,F\\\ \mbox{\emph{Goals:}}\hskip
8.00003pt&G&::=&F\hskip 5.0pt|\hskip 5.0ptC\supset G\hskip 5.0pt|\hskip
5.0pt\forall x.\,G\\\\[4.30554pt] \mbox{\emph{Clauses:}}\hskip
8.00003pt&C&::=&R\supset p\;\underline{\check{x}}\;\underline{\hat{x}}\hskip
5.0pt|\hskip 5.0pt\forall x.\,C\\\ \mbox{\emph{Residuals:}}\hskip
8.00003pt&R&::=&\check{x}=:t\hskip 5.0pt|\hskip 5.0pt\hat{x}:=t\hskip
5.0pt|\hskip 5.0pt\top\hskip 5.0pt|\hskip 5.0ptR\,\land\,G\hskip 5.0pt|\hskip
5.0pt\exists x.\,R\end{array}\begin{array}[t]{@{}r@{\hspace{0.8em}} cr
l@{}}\mbox{\emph{Programs:}}\hskip 8.00003pt&\Psi&::=&\cdot\hskip 5.0pt|\hskip
5.0pt\Psi,C\end{array}$
Residuals ($R$) refine the equality predicate
$x\stackrel{{\scriptstyle.}}{{=}}t$ of $\mathcal{L}^{c}_{1}$ into a matching
predicate $x=:t$ and an assignment predicate $x:=t$. At our level of
abstraction, they behave just like equality. During well-moded execution, the
match predicate will have the form $t_{g}=:t_{v}$ where $t_{g}$ is a ground
term while $t_{v}$ may contain variables. It will bind these variables to
ground subterms of $t_{g}$, thereby realizing matching. However, presented
with programs that are not well-moded, the terms $t_{g}$ cannot be assumed to
be ground and $=:$ performs unification. The assignment predicate will be
called as $x:=t$ where $x$ is a variable and $t$ a term — a ground term for
well-moded programs. It simply binds $x$ to $t$. Compiled clauses and programs
are just like in $\mathcal{L}^{c}_{1}$.
Following the motivations above, an atomic goal
$p\;\underline{\check{t}}\;\underline{\hat{t}}$ is not compiled any more to
itself as in $\mathcal{L}^{c}_{1}$, but to a formula $F$ of the form
$\exists\underline{z}.\,(q\>\underline{\check{t}}\>\underline{\hat{z}}\,\land\,\underline{\check{z}}=:\underline{\hat{t}})$.
In the grammar above, we isolated the match predicates
$\underline{\check{z}}=:\underline{\hat{t}}$ as the non-terminal $M$.
${{{{\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
\lx@intercol\mathbf{\scriptstyle}Goals\;Matches\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
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4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\top}$}}$\hfil\cr}}\hskip
32.23744pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,M}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}m2\\_mtch$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,M\,\land\,t=:t}$}}$\hfil\cr}}\hskip
34.82678pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Atomic\;Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
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height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}a2\\_atm$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{2f}}}{{\longrightarrow}}\,p\;\underline{\check{t}}\;\underline{\hat{t}}\,\land\,M}$}}$\hfil\cr}}\hskip
29.01854pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2f}}}{{\longrightarrow}}\,[t/z]R}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}a2\\_exists$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2f}}}{{\longrightarrow}}\,\exists z.\,R}$}}$\hfil\cr}}\hskip
35.4065pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Goals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-1.50694pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
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to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g2\\_f$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,F}$}}$\hfil\cr}}\hskip
20.71492pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi,C\stackrel{{\scriptstyle
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to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g2\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
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28.6139pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{c\;\mbox{\em``new''}\quad\Psi\stackrel{{\scriptstyle
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height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}g2\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\forall x.\,G}$}}$\hfil\cr}}\hskip
24.68726pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt\vskip-12.0pt\vskip 1.5pt}{\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Clauses\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,R}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c2\\_imp$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,R\supset a\,\gg\,a}$}}$\hfil\cr}}\hskip
28.05327pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,[t/x]C\,\gg\,a}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}c2\\_all$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\forall x.\,C\,\gg\,a}$}}$\hfil\cr}}\hskip
24.12662pt}}}\\\\[-1.50694pt] \cr\vskip 6.0pt\hrule height=0.5pt\vskip
6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Residuals\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
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4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,t=:t}$}}$\hfil\cr}}\hskip
32.03325pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
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4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,t:=t}$}}$\hfil\cr}}\hskip
30.14952pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{}$\hfil\cr\vskip
4.0pt\cr\hrule height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
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4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\top}$}}$\hfil\cr}}\hskip
29.44391pt}}}\\\\[5.425pt]
{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,R\quad\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,G}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r2\\_and$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,R\,\land\,G}$}}$\hfil\cr}}\hskip
27.84215pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,[t/x]R}$\hfil\cr\vskip 4.0pt\cr\hrule
height=0.0pt\cr\hfil$\displaystyle{\vbox to0.0pt{\vss\hbox
to0.0pt{\hbox{$\;\mbox{$\mathbf{\scriptstyle}r2\\_exists$}$}\hss}\vss}}$\cr\hrule\cr\vskip
4.0pt\cr\hfil$\displaystyle{\hbox{$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\exists x.\,R}$}}$\hfil\cr}}\hskip
35.05893pt}}}\\\\[-3.01389pt] \end{array}$
Figure 7: Search Semantics of $\mathcal{L}^{c}_{2}$.
We specify the operational semantics of $\mathcal{L}^{c}_{2}$ by means of the
following five judgments:
$\begin{array}[]{l@{\hspace{1.5em}}p{18em}}\phantom{\Psi}\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,M\hfil\hskip 15.00002pt&\emph{$M$ is provable}\\\
\Psi\stackrel{{\scriptstyle c_{2f}}}{{\longrightarrow}}\,F\hfil\hskip
15.00002pt&\emph{$F$ is uniformly provable from $\Psi$}\\\
\Psi\stackrel{{\scriptstyle c_{2}}}{{\longrightarrow}}\,G\hfil\hskip
15.00002pt&\emph{$G$ is uniformly provable from $\Psi$}\\\
\Psi\stackrel{{\scriptstyle c_{2}}}{{\longrightarrow}}\,C\,\gg\,a\hfil\hskip
15.00002pt&\emph{$a$ is immediately entailed by $C$ in $\Psi$}\\\
\Psi\stackrel{{\scriptstyle c_{2}}}{{\longrightarrow}}\,R\hfil\hskip
15.00002pt&\emph{$R$ is uniformly provable from $\Psi$}\end{array}$
which parallel the grammar just presented. The resulting operational semantics
is shown in Figure 7. The rules for clauses are unchanged with respect to
$\mathcal{L}^{c}_{1}$ while that language’s residual rule for equality has
been duplicated into isomorphic rules for matching and assignment. The rules
for compiled goals have instead proliferated due to our handling of terms in
output position in atomic goals. Observe that rule $\mathbf{a2\\_atm}$ is
essentially a combination of rule $\mathbf{g1\\_atm}$ in $\mathcal{L}^{c}_{1}$
and the rule for conjunction. Rules $\mathbf{a2\\_exists}$ and
$\mathbf{m2\\_true}$ are just the standard rules for existential
quantification and truth. Rule $\mathbf{m2\\_mtch}$ combines the rules for
conjunction and matching.
Just like in the case of $\mathcal{L}^{c}_{1}$, the rules in Figure 7
construct proofs that are uniform [Miller et al. (1991)], which makes
$\mathcal{L}^{c}_{2}$ an abstract logic programming language. In a successful
derivation, this operational semantics decomposes a goal to formulas of the
form
$F=\exists\underline{z}.\,(p\>\underline{\check{t}}\>\underline{\hat{z}}\,\land\,\underline{\check{z}}=:\underline{\hat{t}})$
(rules in the “Goals” segment). Then, rules $\mathbf{a2\\_exists}$,
$\mathbf{m2\\_mtch}$ and $\mathbf{m2\\_true}$ necessarily reduce it in a few
steps into the atomic formula $p\>\underline{\check{t}}\>\underline{\hat{t}}$.
Similarly to $\mathcal{L}^{c}_{1}$, the left premise of rule
$\mathbf{a2\\_atm}$ selects a clause and focuses on it until it finds a
potentially matching head (“Clauses” segment). It then proceeds to decomposing
its body (“Residuals” segment) and the cycle repeats with whatever goals it
finds in there.
As just noticed, any atomic goal $F$ of the form
$\exists\underline{\hat{z}}.\,(p\>\underline{\check{t}}\>\underline{\hat{z}}\,\land\,\underline{\hat{z}}=:\underline{\hat{t}})$
is necessarily reduced to $p\>\underline{\check{t}}\>\underline{\hat{t}}$ by
as many applications of rule $\mathbf{a2\\_exists}$ as there are variables in
$\underline{\hat{z}}$, a pass-through instance of $\mathbf{a2\\_atm}$ via its
right branch, and a similar number of uses of rules $\mathbf{m2\\_mtch}$ and
$\mathbf{m2\\_true}$ respectively. This entails that the macro-rule
$\mathbf{a2\\_atm^{\prime}}$, on the left-hand side of the following display,
is derivable:
$\displaystyle{\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,p\>\check{t}\>\hat{t}}$
$\displaystyle{\mathstrut\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\exists\underline{z}.\,(p\>\underline{\check{t}}\>\underline{z}\,\land\,\underline{z}=:\underline{\hat{t}})}$
$\displaystyle{\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,C\,\gg\,p\;\underline{\check{t}}\;\underline{\hat{t}}}$
$\displaystyle{\mathstrut\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{2f}}}{{\longrightarrow}}\,p\;\underline{\check{t}}\;\underline{\hat{t}}}$
Having factored rule $\mathbf{a2\\_atm^{\prime}}$ out, the work performed by
$\mathbf{a2\\_atm}$ degenerates to rule $\mathbf{a2\\_atm^{\prime\prime}}$ on
the right-hand side of the above display, which is akin to $\mathbf{u\\_atm}$.
The system obtained by replacing the $\mathbf{m2\\_*}$ and $\mathbf{a2\\_*}$
rules as well as $\mathbf{g2\\_f}$ with rules $\mathbf{a2\\_atm^{\prime}}$ and
$\mathbf{a2\\_atm^{\prime\prime}}$ is indeed equivalent to the rule set in
Figure 7.
Rule $\mathbf{a2\\_atm^{\prime}}$ entices us to interpret the compiled formula
$\exists\underline{z}.\,(p\>\underline{\check{t}}\>\underline{z}\,\land\,\underline{z}=:\underline{\hat{t}})$
for an atomic goal $p\>\check{t}\>\hat{t}$ as a synthetic operator
$\mathsf{call}\>p\>\underline{\check{t}}=:\underline{\hat{t}}$ which invokes a
clause for $p$ with its (ground) input arguments $\underline{\check{t}}$ and
matches the returned values against its terms $\underline{\hat{t}}$ in output
position.
Having recovered atomic goals $p\>\underline{\check{t}}\>\underline{\hat{t}}$
through rules $\mathbf{a2\\_atm^{\prime}}$ and
$\mathbf{a2\\_atm^{\prime\prime}}$, we can carry out a sequence of reasoning
steps similar to what led us to the backchaining rule for
$\mathcal{L}^{c}_{1}$. Exposing the trailing assignments, a generic compiled
clause $C$ has the form
$\forall\underline{\check{x}}\>\underline{\hat{x}}.\,(\exists\underline{y}.\,R\,\land\,\underline{\hat{x}}:=\underline{\hat{s}})\supset
p\>\underline{\check{x}}\>\underline{\hat{x}}$. In a successful derivation,
all rule $\mathbf{a2\\_atm^{\prime\prime}}$ does is to pick such a clause.
Then, applications of rule $\mathbf{c2\\_all}$ will instantiate variables
$\underline{\check{x}}\>\underline{\hat{x}}$ with the terms
$\underline{\check{t}}\>\underline{\hat{t}}$, and next rule
$\mathbf{c2\\_imp}$ will invoke the instantiated residual
$[\underline{\check{t}}/\underline{\check{x}},\underline{\hat{t}}/\underline{\hat{x}}](\exists\underline{y}.\,R\,\land\,\underline{\hat{x}}:=\underline{\hat{s}})$.
Now, because $\underline{\hat{x}}$ does not occur in $R$ and
$\underline{\check{x}}\>\underline{\hat{x}}$ cannot appear in
$\underline{\hat{s}}$, this formula reduces to
$\exists\underline{y}.\,([\underline{\check{t}}/\underline{\check{x}}]R\,\land\,\underline{\hat{t}}:=\underline{\hat{s}})$
by pushing the substitution in. Rule $\mathbf{r2\\_exists}$ will then
instantiate the variables $\underline{y}$ with terms $\underline{u}$ (which
cannot mention variables $\underline{\check{x}}\>\underline{\hat{x}}$).
Pushing this substitution in yields the formula
$[\underline{\check{t}}/\underline{\check{x}},\underline{u}/\underline{y}]R\,\land\,\underline{\hat{t}}:=[\underline{u}/\underline{y}]\underline{\hat{s}}$
since variables in $\underline{y}$ can occur in neither
$\underline{\check{t}}$ nor $\underline{\hat{t}}$. Finally, by rule
$\mathbf{r2\\_assg}$, $\underline{\hat{t}}$ and
$[\underline{u}/\underline{y}]\underline{\hat{s}}$ must be equal in a
successful derivation. This necessary sequence of steps is captured by the
following derived backchaining macro-rule,
$\displaystyle{\Psi,C,\Psi^{\prime}\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,[\underline{\check{t}}/\underline{\check{x}},\underline{u}/\underline{y}]R}$
$\displaystyle{\mathstrut\Psi,\underbrace{\forall\underline{\check{x}}\>\underline{\hat{x}}.\,(\exists\underline{y}.\,R\,\land\,\underline{\hat{x}}:=\underline{\hat{s}})\supset
p\>\underline{\check{x}}\>\underline{\hat{x}}}_{C},\Psi^{\prime}\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,p\;\underline{\check{t}}\;[\underline{u}/\underline{y}]\underline{\hat{s}}}$
where we have carried out the assignment
$\underline{\hat{t}}:=[\underline{u}/\underline{y}]\underline{\hat{s}}$ in the
conclusion. This rule can be seen as a refinement of
$\mathbf{g1\\_atm^{\prime}}$ in $\mathcal{L}^{c}_{1}$ that makes use of the
trailing assignment in the compiled clauses of $\mathcal{L}^{c}_{2}$. With
this derived inference, rules $\mathbf{a2\\_atm^{\prime\prime}}$,
$\mathbf{c2\\_imp}$ and $\mathbf{c2\\_all}$ become unnecessary: the system
consisting of rules $\mathbf{a2\\_atm^{\prime}}$,
$\mathbf{g2\\_atm^{\prime}}$, the goal rules for implication and universal
quantification, and the residual rules is equivalent to that in Figure 7.
Taking rule $\mathbf{g2\\_atm^{\prime}}$ as primitive amounts to replacing
compiled clauses with the following synthetic connective, which refines
$\mathcal{L}^{c}_{1}$’s $\Lambda_{p}\underline{x}.\,R$.
$\begin{array}[]{cccl}\underbrace{\forall\underline{\check{x}}\>\underline{\hat{x}}.\,p\>\underline{\check{x}}\>\underline{\hat{x}}\subset}&\exists\underline{y}.\,(R&\,\land&\underbrace{\underline{\hat{x}}:=\underline{\hat{t}}})\\\
\Lambda_{p}\underline{\check{x}}.&\exists\underline{y}.\,(R&;&\mathsf{return}\>\underline{\hat{t}})\end{array}$
The variables $\underline{y}$ are then interpreted as local variables for the
execution of this clause. In this, they are akin to the `Y`$n$ permanent
variables of the WAM [Aït-Kaci (1991)].
In a valid proof in this system, an occurrence of $\mathbf{a2\\_atm^{\prime}}$
is always immediately followed by an instance of $\mathbf{g2\\_atm^{\prime}}$:
the conclusion of the latter must match the premise of the former. This fact
realizes the requirement that, upon returning from a call, the output terms,
here $[\underline{u}/\underline{y}]\underline{\hat{s}}$, must be checked
against the terms in output position of the caller.
### 4.3 Compilation
Compilation transforms logic programs in $\mathcal{L}^{s}$ to compiled
programs in $\mathcal{L}^{c}_{2}$. The input does not have to be well-moded at
the level of detail considered here, but this would be operationally
advantageous in a refinement of the semantics in Figure 7 that handles
quantifiers lazily. We will make use of two auxiliary notions in this section:
pseudo clauses that we encountered already in Section 3.2 and the analogous
notion of pseudo atomic goal. They are defined as follows:
$\begin{array}[]{@{}r@{\hspace{0.8em}} cr l@{}}\mbox{\emph{Pseudo
Clauses:}}\hskip 8.00003pt&\mathcal{C}&::=&\Box\supset p\>\underline{x}\hskip
5.0pt|\hskip 5.0pt\forall x.\,\mathcal{C}\\\ \mbox{\emph{Pseudo Atomic
Goals:}}\hskip
8.00003pt&\mathcal{F}&::=&p\>\underline{\check{t}}\>\underline{\hat{z}}\,\land\,\Box\hskip
5.0pt|\hskip 5.0pt\exists z.\,\mathcal{F}\end{array}$
Just like pseudo clauses retain the outer structure of a clause replacing the
embedded residual with a hole ($\Box$), pseudo atomic goals have a hole in
place of their trailing matches. The general form of pseudo clauses and pseudo
atomic formulas, accounting for input and output positions, are
$\forall\underline{\check{x}}\>\underline{\hat{x}}.\,\Box\supset
p\>\underline{\check{x^{\prime}}}\>\underline{\hat{x^{\prime}}}$ and
$\exists\underline{\hat{z}}.\,(p\>\underline{\check{t}}\>\underline{\hat{z^{\prime}}}\,\land\,\Box)$.
In Section 3.2, wrote $\mathcal{C}[R]$ for the replacement of the hole of
$\mathcal{C}$ with the residual $R$ and noted that variable capture could (and
generally will) occur. Similarly, we write $\mathcal{F}[M]$ for replacement of
the hole of $\mathcal{F}$ with matches $M$.
${{{{\begin{array}[]{@{\hfill}c@{\hfill}}\makebox[360.0pt]{}\\\\[-7.53471pt]
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6.0pt}\\\\[-13.56248pt]
\lx@intercol\mathbf{\scriptstyle}Heads\rule[-4.52083pt]{0.0pt}{0.0pt}\hfil\lx@intercol\\\\[-3.01389pt]
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26.62006pt}}}\hfill{\hbox{\raise-9.60004pt\hbox{\vbox{\halign{#\cr\hfil$\displaystyle{x\;\underline{x}\vdash
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x.\,A\,\gg\,\mathcal{C}\,\backslash\,\exists x.\,R\nnwarrow
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height=0.5pt\vskip 6.0pt}\\\\[-13.56248pt]
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\end{array}$
Figure 8: Compilation of $\mathcal{L}^{s}$ into $\mathcal{L}^{c}_{2}$.
The compilation process is modeled by the following five judgments, which are
reminiscent of the compilation judgments $\mathcal{L}^{c}_{1}$. They are more
complex because clause compilation now needs to handle both matching and
assignment as opposed to a generic equality. Furthermore, a new judgment is
needed to compile atomic goals.
$\begin{array}[]{l@{\hspace{1.5em}}p{20em}}\Gamma\,\gg\,\Psi\hfil\hskip
15.00002pt&\emph{Program $\Gamma$ is compiled to $\Psi$}\\\
\underline{x}\vdash a\,\gg\,\mathcal{C}\ssearrow I\nnwarrow O\hfil\hskip
15.00002pt&\emph{Head $a$ with $\underline{x}$ is compiled to $\mathcal{C}$,
$I$ and $O$}\\\ A\,\gg\,\mathcal{C}\,\backslash\,R\nnwarrow O\hfil\hskip
15.00002pt&\emph{Clause $A$ is compiled to $\mathcal{C}$, $R$ and $O$}\\\
\underline{t}\vdash a\,\gg\,\mathcal{F}\,\backslash\,M\hfil\hskip
15.00002pt&\emph{Atomic goal $a$ with $\underline{t}$ is compiled to
$\mathcal{F}$ and $M$}\\\ A\,\gg\,G\hfil\hskip 15.00002pt&\emph{Goal $A$ is
compiled to $G$}\end{array}$
We write $I$ and $O$ for a conjunction of matches (compilation of terms in
input position) and assignments (compilation of output terms), respectively,
in the body of a compiled clause. In compiled atomic goals, we write $M$ for a
conjunction of matches.
The rules for compilation, which define these judgments, are shown in Figure
8. Compiling a clause $A$, modeled by the judgment
$A\,\gg\,\mathcal{C}\,\backslash\,R\nnwarrow O$, returns a pseudo clause
$\mathcal{C}$, the residual $R$ (inclusive of input matches) and the output
assignments $O$ that will fill its hole. The rules in the “Clauses” segment
build up this residual starting with the compilation of its head, which is
displayed in the “Heads” segment. The rules therein differ from the similar
inference for $\mathcal{L}^{c}_{1}$ by the fact that they dispatch terms in
input and output positions in the $I$ and $O$ zones of the judgment as matches
and assignments respectively. Residuals and assignments are plugged in the
hole of the pseudo clause once this clause has been fully compiled, as can be
seen in the “Programs” segment and in rule $\mathbf{g2c\\_imp}$.
The compilation of goals differs from $\mathcal{L}^{c}_{1}$ for the treatment
of atomic formulas: upon encountering an atom $a$, the compilation appeals to
the new judgment $\cdot\vdash a\,\gg\,\mathcal{F}\,\backslash\,M$. It
generates a pseudo atomic formula $\mathcal{F}$ and matches $M$, which are
integrated in rule $\mathbf{g2c\\_atm}$. The zone to the left of the turnstile
serves as an accumulator, very much like when compiling heads.
Target language, $\mathcal{L}^{c}_{2}$, is sound and complete with respect to
$\mathcal{L}^{s}$. The following lemma collects some auxiliary results needed
to prove this property. The first two statements are proved by induction on
the structure of $a$; the third by induction on the given derivation.
###### Lemma 4.1
* •
If $\underline{x}\vdash a\,\gg\,\mathcal{C}\ssearrow I\nnwarrow O$, then for
any term sequence $\underline{t}$ of the same length as $\underline{x}$ and
program $\Psi$ we have $\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,[\underline{t}/\underline{x}](\mathcal{C}[I\,\land\,O])\,\gg\,a\>\underline{t}$.
* •
If $\underline{t}\vdash a\,\gg\,\mathcal{F}\,\backslash\,M$, then for all
$\Psi$ we have $\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\mathcal{F}[M]\,\gg\,a\>\underline{t}$.
* •
If $\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\mathcal{C}[R]\,\gg\,a$, then
$\Psi\stackrel{{\scriptstyle c_{1}}}{{\longrightarrow}}\,R$.
We have the following soundness and completeness theorems for
$\mathcal{L}^{c}_{2}$. In both cases, the proof proceeds by mutual induction
over the first derivation in the antecedent.
###### Theorem 4.2 (Soundness of the compilation to $\mathcal{L}^{c}_{2}$)
* •
If $\Gamma\stackrel{{\scriptstyle u}}{{\longrightarrow}}\,A$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,G$, then $\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,G$.
* •
If $\Gamma\stackrel{{\scriptstyle u}}{{\longrightarrow}}\,A\,\gg\,a$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,\mathcal{C}\,\backslash\,R\nnwarrow O$, then
$\Psi\stackrel{{\scriptstyle
c_{2}}}{{\longrightarrow}}\,\mathcal{C}[R\,\land\,O]\,\gg\,a$.
###### Theorem 4.3 (Completeness of the compilation to $\mathcal{L}^{c}_{2}$)
* •
If $\Psi\stackrel{{\scriptstyle c_{2}}}{{\longrightarrow}}\,G$,
$\Gamma\,\gg\,\Psi$ and $A\,\gg\,G$, then $\Gamma\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\,A$.
* •
If $\Psi\stackrel{{\scriptstyle c_{2}}}{{\longrightarrow}}\,C\,\gg\,a$,
$\Gamma\,\gg\,\Psi$, $C=\mathcal{C}[R\,\land\,O]$ and
$A\,\gg\,\mathcal{C}\,\backslash\,R\nnwarrow O$, then
$\Gamma\stackrel{{\scriptstyle u}}{{\longrightarrow}}\,A\,\gg\,a$.
1. $\begin{array}[t]{@{}l@{\hspace{3.5em}}c@{\hspace{2em}}l@{}}\begin{array}[t]{@{}ll@{}}\lx@intercol\forall E_{1}.\,\forall E_{2}.\,\forall T_{1}.\,\forall T_{2}.\hfil\\\ &\mathsf{of}\;(\mathsf{app}\;E_{1}\;E_{2})\;T_{2}\\\ \\\ \\\ \subset&\mathsf{of}\;E_{1}\;(\mathsf{arr}\;T_{1}\;T_{2})\\\ \subset&\mathsf{of}\;E_{2}\;T_{1}\end{array}\hfil\hskip 31.50005pt&\raisebox{-38.74985pt}{ \ $\,\gg\,$ }\hfil\hskip 18.00003pt&\begin{array}[t]{@{}ll@{}l@{\;}l@{}}\lx@intercol\forall x_{1}.\,\forall x_{2}.\hfil\\\ &\lx@intercol\mathsf{of}\;x_{1}\;x_{2}\hfil\\\ \subset&(&\lx@intercol\exists E_{1}.\,\exists E_{2}.\,\exists T_{1}.\,\exists T_{2}.\,\top\hfil\\\ &&\,\land&x_{1}=:\mathsf{app}\;E_{1}\;E_{2}\\\ &&\,\land&\exists z_{1}.\,(\mathsf{of}\;E_{1}\;z_{1}\,\land\,z_{1}=:\mathsf{arr}\;T_{1}\;T_{2}\,\land\,\top)\\\ &&\,\land&\exists z_{2}.\,(\mathsf{of}\;E_{2}\;z_{2}\,\land\,z_{2}=:T_{1}\,\land\,\top)\\\ &&\,\land&x_{2}:=T_{2}\,\land\,\top)\end{array}\end{array}$ 2. $\begin{array}[t]{@{}lcl@{}}\begin{array}[t]{@{}ll@{}}\lx@intercol\forall E.\,\forall T_{1}.\,\forall T_{2}.\hfil\\\ &\mathsf{of}\;(\mathsf{lam}\;T_{1}\;E)\;(\mathsf{arr}\;T_{1}\;T_{2})\\\ \\\ \\\ \subset&(\forall x.\\\ \\\ &\hskip 18.00003pt\mathsf{of}\;x\;T_{1}\\\ \\\ &\hskip 4.5pt\supset\mathsf{of}\;(E\;x)\;T_{2})\end{array}&\raisebox{-56.18727pt}{$\,\gg\,$}&\begin{array}[t]{@{}ll@{}l@{\;}l@{}}\lx@intercol\forall x_{1}.\,\forall x_{2}.\hfil\\\ &\lx@intercol\mathsf{of}\;x_{1}\;x_{2}\hfil\\\ \subset&(&\lx@intercol\exists E.\,\exists T_{1}.\,\exists T_{2}.\,\top\hfil\\\ &&\,\land&x_{1}=:\mathsf{lam}\;T_{1}\;E\\\ &&\,\land&\exists z.\,((\forall x.\,(\begin{array}[t]{@{}l@{\;}l@{}}&\begin{array}[t]{@{}l@{\;}l@{}}\lx@intercol\forall x_{1}^{\prime}.\,\forall x_{2}^{\prime}.\,\top\hfil\\\ \,\land&x_{1}^{\prime}=:x\\\ \,\land&\mathsf{of}\;x_{1}^{\prime}\;x_{2}^{\prime}\\\ \,\land&x_{2}^{\prime}:=T_{1}\,\land\,\top)\end{array}\\\ \supset&\mathsf{of}\;(E\;x)\;z)\end{array}\\\ &&&\ \ \ \ \ \,\land\,z=:T_{2}\,\land\,\top)\\\ &&\,\land&x_{2}:=\mathsf{arr}\;T_{1}\;T_{2}\,\land\,\top)\end{array}\end{array}$
Figure 9: $\mathcal{L}^{c}_{2}$ Compilation Example
To conclude this section, we revisit our ongoing examples. Here, we assume
that the mode of the predicate $\mathsf{of}$ is
$\mathsf{of}\;\check{\>}\;\hat{\>}$ — the first argument is input and the
second output. The result of compiling our two familiar clauses into
$\mathcal{L}^{c}_{2}$ is shown in Figure 9. As in Section 3.2, the moded
compilation process offers ample opportunities for optimization: matches and
assignments with variables on both side and the corresponding existential
quantification can often be elided, and all occurrences of $\top$ can be
optimized away.
It is instructive to rewrite these clauses with the two synthetic connectives
introduced earlier for $\mathcal{L}^{c}_{2}$, again omitting $\top$ for
readability:
$\begin{array}[]{lcl}\Lambda_{\mathsf{of}}\;x_{1}.&\lx@intercol\exists
E_{1}.\,\exists E_{2}.\,\exists T_{1}.\,\exists
T_{2}.\,\;\;x_{1}=:\mathsf{app}\;E_{1}\;E_{2}\hfil\lx@intercol\\\
&\,\land&\mathsf{call}\>(\mathsf{of}\;E_{1})\;=:\;(\mathsf{arr}\;T_{1}\;T_{2})\;\,\land\,\;\mathsf{call}\>(\mathsf{of}\;E_{2})\;=:\;T_{1};\\\
&&\mathsf{return}\>T_{2}\\\\[4.30554pt]
\Lambda_{\mathsf{of}}\;x_{1}.&\lx@intercol\exists E.\,\exists T_{1}.\,\exists
T_{2}.\,\;\;x_{1}=:\mathsf{lam}\;T_{1}\;E\hfil\lx@intercol\\\ &\,\land&\forall
x.\,(\Lambda_{\mathsf{of}}\;x_{1}^{\prime}.\;\;x_{1}^{\prime}=:x\;;\mathsf{return}\>T_{1})\;\supset\;\mathsf{call}\>(\mathsf{of}\;(E\;x))\;=:\;T_{2};\\\
&&\mathsf{return}\>(\mathsf{arr}\;T_{1}\;T_{2})\end{array}$
## 5 Larger Source Languages
In [Cervesato (1998)], we illustrated our original abstract logical
compilation method on the language of hereditary Harrop formulas. This
language differs from $\mathcal{L}^{s}$ for the presence of conjunction
(formulas of the form $A\,\land\,B$) and truth ($\top$). While our original
treatment could handle them easily (in a clause position, they were compiled
to disjunctions and falsehood respectively), the approach taken in Sections 3
and 4 does not support them directly. The problem is that, as soon as we allow
these connectives, clauses can have multiple heads (or even none). Consider
for example:
$\forall x.\,\forall y.\,q\>x\>y\supset(p_{1}\>x\>y\,\land\,(r\>x\;y\supset
p_{2}\>x))$
This clause has two heads: $p_{1}\>x\>y$ and $p_{2}\>x$. What should it be
compiled to? To ensure immediacy (embodied in the macro-rule
$\mathbf{g1\\_atm^{\prime}}$), our compilation strategy produces a pseudo
clause applied to a residual, thereby exposing the (flattened) head of a
compiled clause as close to the top level as possible. How to achieve this now
that there may be more than one head?
One approach to dealing with this problem is to observe that $\land$
distributes over (the antecedent of) $\supset$ and $\forall$. By doing so to
the above example, we obtain the formula
$(\forall x.\,\forall y.\,q\>x\>y\supset p_{1}\>x\>y)\,\land\,(\forall
x.\,\forall y.\,q\>x\>y\supset r\>x\;y\supset p_{2}\>x)$
Observe that it is a conjunction of $\mathcal{L}^{s}$ clauses. Each of them
can now be compiled as in Section 3 and the results can be combined by means
of a disjunction. This approach generalizes to the full language of hereditary
Harrop formulas. It pushes the conjunctions to the outside, leaving inner
formulas resembling the clauses of $\mathcal{L}^{c}_{0}$ (conjunction and
truth in a goal position are left alone as they are not problematic). Clauses
with no head (e.g., $A\supset\top$) are reduced to $\top$. These preprocessing
steps can be implemented as a source-code transformation or integrated in the
compilation process.
The other abstract logic programming language examined in [Cervesato (1998)]
is the language of linear hereditary Harrop formulas, found at the core of
Lolli [Hodas and Miller (1994)] and LLF [Cervesato and Pfenning (2002)]. The
improved compilation process discussed in this paper extends directly in the
presence of linearity. Because linear hereditary Harrop formulas feature a
form of conjunction and truth, the technical device just outlined is needed to
obtain workable compiled clauses.
## 6 Future Work
The discussion in Section 4 sets the stage for a nearly functional operational
semantics of well-moded programs. Indeed, given an atomic goal with ground
terms in its input positions, proof search will instantiate its output
positions to ground terms, if it succeeds. Being in a logic programming
setting, more than one answer could be returned. Indeed, for well-moded
programs, the clauses for a predicate implement a partial, non-deterministic
function. This observation informed the choice of the notation for the
synthetic operators we exposed:
$\mathsf{call}\>p\>\underline{\check{t}}=:\underline{\hat{t}}$ and
$\Lambda_{p}\underline{\check{x}}.\,\exists\underline{y}.\,(R;\mathsf{return}\>\underline{\hat{t}})$.
Now we believe that, in the case of well-moded programs, a more detailed
operational semantics that exposes variable manipulations using logical
variables and explicit substitutions (and restricts the execution order) can
bring this functional interpretation to the surface. This would provide a
logical justification for the natural impulse to give well-moded programs a
semantics that is typical of functional programming languages, where atomic
predicates carry just input terms and from which the terms in output position
emerge by a process of reduction.
In future work, we intend to carry out this program by giving such a detailed
operational semantics to $\mathcal{L}^{s}$ as well as well-moding rules. The
goal will then be to perform logical transformations, akin to what we did in
this paper, that expose this functional semantics for well-moded programs. It
would also allow us to prove formally that the operator $=:$ of Section 4 can
indeed be implemented as matching rather than general unification.
## Acknowledgments
This work was supported by the Qatar National Research Fund under grant NPRP
09-1107-1-168. We are grateful to Frank Pfenning, Carsten Schürmann, Robert J.
Simmons and Jorge Sacchini for the many fruitful discussions, as well as to
the anonymous reviewers.
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* Sarnat (2010) Sarnat, J. 2010\. Syntactic finitism in the metatheory of programming languages. Ph.D. thesis, Department of Computer Science, Yale University.
* Stirling (2009) Stirling, C. 2009\. Decidability of higher-order matching. Logical Methods in Computer Science 5, 3\.
* Warren (1983) Warren, D. H. D. 1983\. An abstract Prolog instruction set. Technical Note 309, SRI International, Menlo Park, CA. Oct.
* Watkins et al. (2003) Watkins, K., Cervesato, I., Pfenning, F., and Walker, D. 2003\. A Concurrent Logical Framework I: Judgments and Properties. Technical Report CMU-CS-02-101, Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA. March 2002, revised May.
|
arxiv-papers
| 2012-10-05T06:26:26 |
2024-09-04T02:49:36.001494
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Iliano Cervesato",
"submitter": "Iliano Cervesato",
"url": "https://arxiv.org/abs/1210.1653"
}
|
1210.1675
|
# Thermodynamics of Ising Spins on the Star Lattice
Zewei Chen Nvsen Ma Dao-Xin Yao Email: yaodaox@mail.sysu.edu.cn State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics and
Engineering, Sun Yat-sen University, Guangzhou 510275, China
###### Abstract
There is a new class of two-dimensional magnetic materials polymeric iron
(III) acetate fabricated recently in which Fe ions form a star lattice. We
study the thermodynamics of Ising spins on the star lattice with exact
analytic method and Monte Carlo simulations. Mapping the star lattice to the
honeycomb lattice, we obtain the partition function for the system with
asymmetric interactions. The free energy, internal energy, specific heat,
entropy and susceptibility are presented, which can be used to determine the
sign of the interactions in the real materials. Moreover, we find the rich
phase diagrams of the system as a function of interactions, temperature and
external magnetic field. For frustrated interactions without external field,
the ground state is disordered (spin liquid) with residual entropy
$1.522\ldots$ per unit cell. When a weak field is applied, the system enters a
ferrimagnetic phase with residual entropy $ln4$ per unit cell.
###### pacs:
75.10.Hk, 75.30.Kz, 75.40.-s, 64.60.-i
## I Introduction
Spin systems with geometrical frustration have both fundamental and practical
importance. Theoretically, lots of interesting phenomena have been found in
the geometrically frustrated systems, like the antiferromagnetic triangular
lattice, kagome lattice. The systems can remain disordered even at absolute
zero temperature because of the competitive magnetic interactions. For
example, an antiferromagnetic triangular lattice has a residual entropy
$s_{0}=0.3281\ldots$ per unit cell Wannier (1950). The frustration effect has
important application in achieving a lower temperatures through the adiabatic
demagnetization compared with other methods. When the temperature, external
magnetic field, and other factors are considered, the geometrically frustrated
systems can show very rich phase diagrams. Diep (2004) A typical case is that
the magnetocaloric effect can be enhanced near the phase transition points
when a finite external field is applied. Zhitomirsky and Tsunetsugu (2004);
Isakov _et al._ (2004); Aoki _et al._ (2004) Of considerable interest has
been searching for geometrically frustrated systems. Some new frustrated
materials have been fabricated and studied recently, such as
$Ho_{2}Ti_{2}O_{7}$, $Dy_{2}Ti_{2}O_{7}$, and $Cu_{9}X_{2}(cpa)_{6}\cdot
xH_{2}O(X=F,Cl,Br)$. Rosenkranz _et al._ (2000); Harris _et al._ (1997);
Maruti and Ter Haar (1994); Mekata _et al._ (1998); Loh _et al._ (2008,
2008); Yao _et al._ (2008)
In 2007, a new class of geometrically frustrated magnetic materials polymeric
iron (III) acetate Zheng _et al._ (2007) was fabricated, in which Fe ions
form a two-dimensional lattice referred as star lattice. Experiment has found
that the materials exhibit spin frustration and have two kinds of magnetic
interactions: intratrimer $J_{T}$ and intertrimer $J_{D}$ shown in Fig. 1. The
Fe ion has a large spin, which is $S=5/2$. The system may have the
paramagnetic ground state because of the geometrical and quantum fluctuations.
The Ising model on the star lattice with uniform ferromagnetic couplings has
been solved. Barry and Khatun (1995) The critical temperature $K_{c}=0.81201$
was given with $K_{c}=\beta_{c}J$. The Bose-Hubbard model on the star lattice
has also been studied using the quantum Monte Carlo method. Isakov _et al._
(2009) Recently, the edge states and topological orders were found in the spin
liquid phases of star lattice. Huang _et al._ (2012) Even though the $S=1/2$
quantum Heisenberg model is considered to be appropriate to help studying the
new material because of its quantum fluctuations in the ground state, Richter
_et al._ (2004) the Ising model on the star lattice is still very important
especially for the non-uniform case. In real materials, the Fe ion has $S=5/2$
which is close to the classical limit and the magnetic system shows two types
of interactions. Zheng _et al._ (2007) Therefore, it is important to study
Ising spins on the star lattice with the asymmetric interactions, especially
for the frustrated case.
In this paper, we aim at the thermodynamics of Ising spins on the star lattice
with asymmetric interactions using the exact analytic methods and Monte Carlo
simulations. We present the phase diagram as a function of interactions,
temperature and external magnetic field. There is a clear difference between
the $J_{D}>0$ case and the $J_{D}<0$ case. Our study provides useful
information for determining the sign of $J_{D}$.
This paper is organized as follows. The model is described in Sec. II. In Sec.
III, we map the star lattice to honeycomb lattice and get the exact results.
Sec. IV presents the phase diagrams as functions of interactions and external
magnetic field. In Sec. V, the Monte Carlo results for the heat capacity and
susceptibility are given. Finally, in Sec. VI we summarize the results.
Figure 1: (Color online). The star lattice. The dashed frame represents a unit
cell of the star lattice. There are six spins per unit cell.
## II Model
The structure of star lattice and its unit cell are illustrated in Fig. 1. We
study Ising spins on the star lattice with two kinds of nearest-neighbor
interactions, the intratrimer coupling $J_{T}$ and intertrimer coupling
$J_{D}$. The Hamiltonian is
$H=-J_{T}\sum_{<ij>}\sigma_{i}\sigma_{j}-J_{D}\sum_{<i^{\prime}j^{\prime}>}\sigma_{i^{\prime}}\sigma_{j^{\prime}}-h\sum_{i}\sigma_{i},$
(1)
where $<ij>$ runs over all the nearest neighbor spin pairs, $J_{T}$ is the
intra-triangular interaction and $J_{D}$ is the inter-triangular interaction,
$h$ is the external magnetic field, and $\sigma_{i}=\pm 1$. The unit cell of
the star lattice contains six spins shown in Fig. 1. If we use $N_{T}$ and
$N_{D}$ to denote the total numbers of $J_{T}$\- and $J_{D}$-bonds, we have
$N_{T}:N_{D}=2:1$. The analytic result of partition function is obtained for
$h=0$.
For simplicity, we use $|J_{T}|$ as the units of energy in the following. The
corresponding phase diagrams are actually in a three-dimensional parameter
space, $\frac{J_{D}}{|J_{T}|}$,$\frac{T}{|J_{T}|}$ and $\frac{h}{|J_{T}|}$.
## III Exact solution in zero field
In this section, we study the exact analytic results of Ising model on the
star lattice in zero magnetic field ($h=0$). Using a sequence of $\Delta-Y$
transformation and series reductionsLoh _et al._ (2008), we can transform the
Ising model on the star lattice into one on the honeycomb lattice whose
partition function has been exactly solved using the Pfaffian method Fisher
(1966); Kasteleyn (1963). Besides the exact analytical results, we expand the
partition function in series for some special cases.
### III.1 Effective coupling on the equivalent honeycomb lattice
The results of $\Delta-Y$ transformation and series reduction are given in
Ref. Loh _et al._ , 2008. Using the variables $t_{i}=tanh\beta J_{i}$ and
$x_{i}=e^{-2\beta J_{i}}$, the relations among the exchange couplings of Fig.
2 can be written as
$\displaystyle t_{1}=$ $\displaystyle\frac{1}{\sqrt{t_{T}+t_{T}^{-1}-1}}$ (2)
$\displaystyle t_{2}=t_{1}t_{D}$ (3) $\displaystyle t_{h}=t_{1}t_{2}.$ (4)
We write $t_{h}$ in terms of $t_{T}$ and $t_{D}$ directly
$t_{h}=\frac{t_{T}t_{D}}{t_{T}^{2}-t_{T}+1}$ (5)
For convenience, we can rewrite it in terms of $x_{i}$
$x_{h}=\frac{x_{D}+(2+x_{D})x_{T}^{2}}{1+(1+2x_{D})x_{T}^{2}}$ (6)
Figure 2: (Color online). Transformation of star lattice to honeycomb lattice.
(a) Depicts a section of the star lattice with two couplings, $J_{D},J_{T}$.
By applying the $\Delta-Y$ transformation and then we obtain (b). After that
take the two bonds $J_{D}$, $J_{1}$ in series and we obtain (c) where
generates a new coupling $J_{2}$. Finally, take $J_{1}$ and $J_{2}$ in series
and the honeycomb lattice (d) is obtained.
### III.2 Phase boundary
It is known that the critical temperature of the honeycomb lattice Ising model
is given by $x_{h}^{c}=2-\sqrt{3}$. Baxter (1989) Having mapped star lattice
to honeycomb lattice, we can substitute this into the equivalent coupling in
Eq. (6). Thus, an implicit equation for the critical temperature
$\frac{1}{\beta_{c}}$ of star lattice Ising model can be obtained as,
$\frac{e^{-2\beta_{c}J_{D}}+(2+e^{-2\beta_{c}J_{D}})e^{-4\beta_{c}J_{T}}}{1+(1+2e^{-2\beta_{c}J_{D}})e^{-4\beta_{c}J_{T}}}=2-\sqrt{3}$
(7)
This result is plotted in Fig. 3. When $J_{D}$ is ferromagnetic ($J_{D}>0$)
and strong enough, the critical temperature saturates at a finite value, i.e.
$T_{c}/|J_{T}|\approx 4/ln[3/(2\sqrt{3}-3)]\approx 2.14332$. The critical
temperature drops to zero as $J_{D}$ approaches zero. When
$J_{T}\approx|J_{D}|$, the curve is approximately linear with
$T_{c}/|J_{T}|\approx 1.23151$.
Furthermore, when $x_{D}=x_{T}=x$, Eq.( 6) reduces to the result of star
lattice with equivalent couplings. Barry and Khatun (1995) We get
$x_{c}=0.19710$, or equivalently, $K_{c}=0.81201$.
Since all the factors obtained here are analytical, the singularity in the
partition function remains when we transform the star lattice into the
honeycomb lattice. The phase transition is the same as the honeycomb lattice
where a continuous second-order transition happens.
If $J_{D}$ is antiferromagnetic ($J_{D}<0$), we get the negative critical
temperature, which implies no phase transition existing in this case. It can
be used as a criterion for experimentalists to determine if the couplings in a
real material is ferromagnetic or antiferromagnetic. If one finds a phase
transition in the real material, we can conclude that the couplings $J_{D}$
and $J_{T}$ should be both ferromagnetic. In there is no long range order
found, it means that at least one kind of nearest neighbor couplings is
antiferromagnetic in the material.
### III.3 Partition function
Figure 3: (Color online). Phase diagram of the star lattice Ising model in the
$(J_{D},T)$ plane, for $J_{T}=1$ and $h=0$. The thick curve is the exact
solution. It illustrates that when $J_{D}>0$, the phase is ordered. The
ordered phase is ferromagnetic. On the contrary, when $J_{D}<0$ the phase is
immediately disordered (paramagnetic). When $J_{T}=-1$, the phase diagram is
below $T$-axis (not shown in the figure), which implies that there is no phase
transition in this phase.
Since we have utilized the $\Delta-Y$ transformation and series reductions to
map the star lattice to a honeycomb lattice, the partition function per unit
cell, $z_{s}$, of the star lattice Ising model is equivalent to that of the
honeycomb lattice $z_{H}$ multiplied by some coefficients which result from
the transformation. These coefficients are as follows,
$\displaystyle z_{1}=\frac{1}{1+x_{1}^{3}}\sqrt{\frac{x_{1}^{3}}{x_{T}^{3}}}$
(8) $\displaystyle z_{2}$
$\displaystyle=(1+x_{1}x_{D})\sqrt{\frac{x_{2}}{x_{1}x_{D}}}$ (9)
$\displaystyle z_{3}$
$\displaystyle=(1+x_{1}x_{2})\sqrt{\frac{x_{h}}{x_{1}x_{2}}}$ (10)
Therefore, the total partition function of the star lattice is
$z_{s}=z_{1}^{2}z_{2}^{3}z_{3}^{3}z_{H}$ (11)
where $z_{H}$ is calculated using the Pfaffian method.Kasteleyn (1963) We
rewrite it here,
$z_{H}(x_{h})=\frac{\sqrt{2}(1-{x_{h}}^{2})}{x_{h}}\exp\\{\tfrac{1}{2}\Omega\left[w(x_{h})\right]\\}$
(12)
where
$\Omega(w)=\int_{0}^{2\pi}\frac{dp}{2\pi}\int_{0}^{2\pi}\frac{dq}{2\pi}\ln(w-\cos
p-\cos q-\cos(p+q))$ (13)
and
$w(x_{h})=\frac{1-2{x_{h}}+6{x_{h}}^{2}-2{x_{h}}^{3}+{x_{h}}^{4}}{2x_{h}(1-x_{h})^{2}}.$
(14)
We can rewrite $\Omega(w)$ and get a more accurate numerical evaluation
according to the singularities of the integrand.
$\Omega(w)=\frac{2}{\pi}\int_{0}^{\pi/2}dp\ln\left[\cos
p+\text{arccosh}\frac{w-\cos 2p}{2\cos p}\right]$ (15)
The partition function of the star lattice Ising model is therefore
$z_{s}(x_{T},x_{D})=\Psi(x_{T},x_{D})\exp\left[\tfrac{1}{2}\Omega(w(x_{h}(x_{T},x_{D})))\right]$
(16)
where
$\displaystyle\Psi(x_{T}$
$\displaystyle,x_{D})=x_{T}^{-3}x_{D}^{-\frac{3}{2}}(1-x_{T}^{2})(1-x_{D}^{2})$
$\displaystyle\times\sqrt{2(1+x_{T}^{2}+2x_{D}x_{T}^{2})(x_{D}+2x_{T}^{2}+x_{D}x_{T}^{2})}$
(17)
Figure 4: (Color online). Thermodynamic functions for one unit cell vs
temperature $T$ for the unfrustrated case $J_{D}=0.5$ and $J_{T}=1$. The
specific heat (red line) diverges as $T\approx 0.74|J_{T}|$ revealing that
there is a phase transition from the ferromagnetic phase to the parametric
phase. Energy (blue line) is shown as $-u(T)$. Entropy (orange line)
approaches zero when $T\rightarrow 0$ and saturates at $6ln2$ as
$T\rightarrow\infty$.
Figure 5: (Color online). Thermodynamic functions for one unit cell vs
temperature $T$ for frustrated coupling $J_{D}=-0.5$ and $J_{T}=-1$. The
specific heat (red line) is no longer diverging. Entropy (orange line) remains
a none-zero value at $T=0$ and $6ln2$ at high temperature.
The total partition function is given by $Z_{s}=z_{s}^{N}$, where $N$ is the
spin number of unit cell. Since the partition function is obtained, the
internal energy, specific heat, entropy and free energy can be calculated from
it.
### III.4 Energy
Taking derivation of the partition function, the energy per unit cell of the
star lattice Ising model can be obtained.
$\displaystyle u$ $\displaystyle=-\frac{d\ln
z}{d\beta}=-\frac{dx_{T}}{d\beta}\frac{\partial\ln z}{\partial
x_{T}}-\frac{dx_{D}}{d\beta}\frac{\partial\ln z}{\partial x_{D}}$
$\displaystyle=\sum_{i=D,T}J_{i}x_{i}\left[2\frac{\partial\ln\Psi}{\partial
x_{i}}+\frac{\partial x_{h}}{\partial
x_{i}}\frac{dw}{dx_{h}}\frac{d\Omega}{dw}\right],$ (18)
where $\frac{d\Omega}{dw}$ is expressed in terms of the complete elliptic
integral of the first kind, K, Horiguchi _et al._ (1992)
$\displaystyle\frac{d\Omega}{dw}$
$\displaystyle=-\tfrac{2}{\pi(-w-1)^{3/4}(-w+3)^{1/4}}$ $\displaystyle\times$
$\displaystyle
K\left(\tfrac{1}{2}+\tfrac{w^{2}-3}{2(w+1)(-w-1)^{1/2}(-w+3)^{1/2}}\right).$
(19)
The plots of energy in units of $|J_{D}|$, $\frac{u}{|J_{D}|}$ is illustrated
in Figs 4 and 5 for the unfrustrated case and frustrated case respectively.
### III.5 Specific heat
By further derivation, $c=\frac{du}{dT}$, the heat capacity per unit cell can
be obtained. The details are shown in Ref. Loh _et al._ , 2008. Here we show
the plots of $c$ in Figs 4 and 5.
In the unfrustrated case, the specific heat $c$ shows a sharp peak at
$T\approx 0.74|J_{T}|$ where a phase transition happens. The phase transition
point is consistent with the result of Eq. (6). In addition, there is a broad
hump at higher temperature because of the flopping of spins. Moreover, this
hump changes with $R=|\frac{J_{D}}{J_{T}}|$. It is obvious when $R<1$ and
becomes indistinct when $R=1$. However, it arises again when $R\geq 6$ In the
unfrustrated case, the sharp peak vanishes which implies no phase transition,
consistent with the conclusion drawn from the phase diagram.
### III.6 Zero-temperature limit: residual entropy
The plots of entropy are shown in Figs. 4 and 5. Nonetheless, we can expand
the partition function in series to gain more information about the residual
entropy in the low temperature limit.
In the case of $J_{D}>0$, the partition function can be expanded as
$\displaystyle lnz$
$\displaystyle=-\frac{3}{2}lnx_{D}-3lnx_{T}+\frac{3}{2}x_{D}^{2}+...$ (20)
$\displaystyle u$
$\displaystyle=-3|J_{D}|-6|J_{T}|+6|J_{D}|e^{-4\beta|J_{D}|}+...$ (21)
The residual entropy is therefore $0$ when $T\rightarrow 0$.
Figure 6: (Color online). Spin configurations of the degenerate states of
phase II. The two spins are coupled by $J_{T}$. Each triangular has exactly
two spins pointing up.
However, when $J_{D}<0$, the model becomes frustrated. In this way, when
$T\rightarrow 0$, $\beta\rightarrow\infty$, which means
$x_{D},x_{T}\rightarrow\infty$, $w\rightarrow\infty$. Therefore, $ln\Omega(w)$
becomes $\sim ln(w)$. Expanding $ln(z)$, we get
$\displaystyle lnz$
$\displaystyle=\frac{1}{2}ln21+\frac{3}{2}lnx_{D}+lnx_{T}+...$ (22)
$\displaystyle u$ $\displaystyle=(3|J_{D}+2|J_{T}|)+...$ (23)
These results contribute to the residual entropy by
$\displaystyle s_{0}$ $\displaystyle=\lim_{\beta
J_{T}\rightarrow-\infty}\lim_{\beta J_{D}\rightarrow-\infty}\left(\ln Z+\beta
u\right)=\frac{1}{2}ln21.$ (24)
Thus, the frustration of the system leads to a $\frac{1}{2}ln21\approx 1.522$
residual entropy per unit cell when $T\rightarrow 0$. One can confirm that,
this value is consistent with the entropy at $T=0$ in Fig. 5. The residual
entropy per site is approximately $0.254$, smaller than the triangle lattice,
TKL, and kagome latticeWannier (1950); Loh _et al._ (2008); Kanô and Naya
(1953). Therefore, the star lattice is less frustrated compared to them.
## IV Phase diagrams at zero temperature
In this section, we present the phase diagrams at zero temperature along with
some thermodynamic properties such as energy, magnetization and entropy. By
calculating the ground state energy of the star lattice, we derive the full
phase diagram for the system. Since the phase diagram at zero field is already
shown in Fig. 3, we focus on the none-zero field case in this section. The
corresponding results are summarized in Figs. 10 and 11 according to the sign
of $J_{T}$.
Figure 7: (Color online). Spin configurations of the degenerate states of
phase III. The two spins are coupled by $J_{T}$. Each triangle has exactly two
spins pointing up.
### IV.1 Zero Field (Phase V and VI)
The phase diagram for zero field as a function of couplings is showed in Fig.
3. When $J_{D}>0$, the phase is ordered and ferromagnetic. When $J_{D}<0$, the
phase is frustrated with a residual entropy $s_{0}=\frac{1}{2}ln21$. When
$J_{T}=-1$, the phase is located in the negative section of $T$-axis , which
reveals that there is no phase transition in this phase. The disordered and
ordered phases are labeled by V and VI in Figs. 10 and 11 respectively.
In phase V, the system is fully frustrated. We find $18$ degenerate ground
states for each unit cell. However, as shown in Eq. (24), the residual entropy
is not $ln18$ but $\frac{1}{2}ln21$. This is similar as the triangular lattice
whose residual entropy can not be obtained by counting the number of ground
states in a unit cell. Wannier, 1950
Figure 8: (Color online). Spin configurations of the degenerate states of
phase IV. All the configurations have only one spin points down.
Figure 9: (Color online). Spin configurations of the degenerate states of
phase VII. This is a interesting phase because spins in the same triangle
point the same direction. It can map to a honeycomb lattice with equivalent
antiferromagtic coupling with high spins.
### IV.2 Saturated ferromagnetic phase (Phase I)
When the external field is strong enough, i.e.
$h>Max\\{2|J_{T}|,|J_{D}|+2|J_{T}|\\}$, the phase is ferromagnetic where all
spins are lined up. It is obvious that this state has energy
$u=-J_{D}-6J_{T}-6h$, magnetization $m=6$ and entropy $s=0$ per unit cell.
### IV.3 Phase II
When the field is weaker, e.g. $0<h<2|J_{T}|$ and $J_{T}<0,J_{D}>0$, it is a
phase with $u=-J_{D}+2J_{T}-2h,m=2,s=ln5$. The spin configurations of the
degenerate ground states of this phase are shown in Fig. 6. The two spins
connected by $J_{D}$ become aligned due to the positive $J_{D}$ and the weak
field $h$.
Figure 10: (Color online). Phase diagram of the star lattice Ising model in
the $(J_{D},h)$ plane for $J_{T}>0$ and $T=0$. The phase diagram is symmetric
under a sign change of $h$.
Figure 11: (Color online). Phase diagram of the star lattice Ising model in
the $(J_{D},h)$ plane for $J_{T}<0$ and $T=0$.
### IV.4 Phase III
If $0<h<2$, the system is in a frustrated phase. We find four degenerate
ground states in this phase contributing to the residual entropy $s=ln4$. The
other properties are given by $u=J_{D}+2J_{T}-2h$, and $m=2$. The spin
configurations are shown in Fig. 7. In this case, the two spins connected by
$J_{T}$ become antiparallel since $J_{D}$ is antiferromagnetic.
### IV.5 Phase IV
In the case of $2<h<|J_{D}|+2|J_{T}|$ and $J_{T}>0,J_{D}<0$, phase III evolves
into phase IV, which has $m=4$ and $s=ln2$. Only one spin points down in this
phase and it should be one of the two connected by $J_{D}$. The spin
configurations are shown in Fig. 8.
### IV.6 Phase VII
Phase VII is a new phase when $J_{T}$ becomes positive in none-zero field. In
this phase, $h<\frac{1}{3}|J_{D}|$ and $J_{D}<0,J_{T}>0$, the spins on the
same triangle are parallel, however, antiparallel to the neighboring triangles
for the positive $J_{T}$ and negative $J_{D}$. If we treat the three spins on
the same triangle as a higher spin located in the center of the triangle, it
becomes an antiferromagnetic phase in a honeycomb lattice. This state gives
$u=J_{D}-6J_{T},s=ln2$ and m=0.
### IV.7 Phase diagram
According to the discussion above, we now combine all the results together to
obtain a full phase diagram. Fig. 10 show the phase diagram when $J_{T}$ is
antiferromagnetic and thus $J_{T}<0$ and Fig. 11, on the contrary, shows the
case when $J_{T}$ is ferromagnetic. The phase diagram is symmetric under the
sign change of $h$.
## V Monte Carlo Simulations
In this section we show the Monte Carlo (MC) simulation results of the star
lattice Ising model with different combinations of parameters, which helps to
corroborate our analytic predictions. Meanwhile, they allow us to calculate
the magnetization and susceptibility at finite temperature.
Figure 12: (Color online). Temperature dependence of heat capacity per site
from exact solution (red line) and Monte Carlo simulations with $L=8$, $16$
and $32$ for $J_{D}=5J_{T}=1$. The critical temperature $T_{c}\approx
2.1$$|J_{T}|$, consistent with Fig. 3.
Figure 13: (Color online). Temperature dependence of susceptibility from the
Monte Carlo simulations with $L=8$, $16$ and $32$ for $J_{D}=5J_{T}=1$.
Figure 14: (Color online). Temperature dependence of susceptibility from the
Monte Carlo simulations with $L=8$, $16$ and $32$ for $J_{D}=5J_{T}=-1$. There
is no apparent peak found, which means there is no continuous phase transition
for $J_{T}<0$. The exact solution in Fig. 3 gives the same result in this
case.
We choose system size $L=8$, $16$, $32$, where $L$ is the length of the unit
cell for the star lattice, which means that the total number of spins $N$ is
$N=6L^{2}$, as there are six spins in each unit cell. The periodic boundary
condition is used for the simulations.
The specific heat $c$ and magnetic susceptibility $\chi$ during the MC
simulations can be calculated using the fluctuation-dissipation theorem
$\displaystyle c$
$\displaystyle=\frac{\left<H^{2}\right>-\left<H\right>^{2}}{NT^{2}},$ (25)
$\displaystyle\chi$
$\displaystyle=\frac{\left<M^{2}\right>-\left<M\right>^{2}}{NT},$ (26)
where $\left<H\right>$ and $\left<M\right>$ are respectively the Monte Carlo
averages of the total energy (i.e., the Hamiltonian) and magnetization.
Fig. 12 shows the temperature dependence of heat capacity per site at $h=0$
for a typical unfrustrated case $J_{D}/J_{T}=5$, $J_{T}>0$. The MC results are
consistent with the exact analytic results.
We also calculate the susceptibility from the MC simulations, which is shown
in Fig. 13. As $L$ increases, the peaks become sharper and sharper, indicating
a phase transition.
We also study two different combinations of interactions for the unfrustrated
case, which is $J_{T}>0$, in Figs. 13 and 14. The temperature dependence of
susceptibility is found to be sensitive to the sign of $J_{D}$. When
$J_{D}>0$, the susceptibility has a sharp peak at the critical point between
the ferromagnetic phase and paramagnetic phase. However, when $J_{D}<0$, there
is no such peak, which implies no phase transition in this case, just as what
we get in exact solutions. The shape of the susceptibility peak depends on the
size of the system when $J_{D}>0$, whereas for $J_{D}<0$, the size of the
system has no influence on the susceptibility.
## VI CONCLUSIONS
In summary, we have studied the Ising model on the star lattice with two
different exchange couplings $J_{T}$ and $J_{D}$ using both analytical method
and Monte Carlo simulations. We have presented its thermodynamic properties
including internal energy, free energy, specific heat, entropy and
susceptibility in the zero field. The phase transition temperature for
$J_{T}=J_{D}$ is exactly same as the one found in Ref. Barry and Khatun
(1995). There is no phase transition found if one of the couplings is
antiferromagnetic. Moreover, we have obtained the rich phase diagrams in terms
of $J_{T},J_{D}$ and $h$ at zero temperature. Monte Carlo simulation is used
to confirm the exact results and calculate the susceptibility.
In the fully frustrated case, the residual entropy of the system can be
expressed as a closed form ($s_{0}=\frac{1}{2}ln21$ as showed in Eq. (24)
which is consistent with the triangular and Kagome lattices. The system is
less frustrated compared to the other triangulated lattices.
Our study provides a benchmark calculation for the thermodynamics of Ising
spins on the star lattice, which can help experimentalists to investigate the
real materials.
###### Acknowledgements.
We thank Xiao-Ming Chen and Ming-Liang Tong for helpful discussions. This work
is supported by the Fundamental Research Funds for the Central Universities of
China (11lgjc12 and 10lgzd09), NSFC-11074310 and 11275279, MOST of China 973
program (2012CB821400), Specialized Research Fund for the Doctoral Program of
Higher Education (20110171110026), Undergraduate Training Program at SYSU and
NCET-11-0547.
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|
arxiv-papers
| 2012-10-05T08:37:59 |
2024-09-04T02:49:36.015489
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zewei Chen, Nvsen Ma, Dao-Xin Yao",
"submitter": "Nvsen Ma",
"url": "https://arxiv.org/abs/1210.1675"
}
|
1210.1711
|
# Search for Super Earths by Timing of Transits with CoRoT
Juan Cabrera Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstrasse 2,
12489 Berlin, Germany
LUTH, Observatoire de Paris, CNRS, Université Paris Diderot ; 5 Place Jules
Janssen, 92190 Meudon,France
###### Abstract
We explore the possibility of detecting Super Earths via transit timing
variations with the satellite CoRoT.
## 1 Introduction
The satellite CoRoT (Baglin et al., 2006) was launched on 27th December 2006
with a double scientific purpose: the analysis of stellar seismology and the
detection of extrasolar planets by the method of transits. So far, 7 transits
have been published (see Jean Schneider’s Extrasolar Planets
Encyclopaedia111http://exoplanet.eu), but more will come in the near future.
Table 1 gathers the data from these six planets and one brown dwarf. CoRoT’s
photometric precision is below $8\cdot 10^{-4}$ in 2h at $R=15$ (Aigrain et
al., 2009) and candidates are found with transit depths of $0.034\%$ (Leger et
al., 2009). Neptune size planets seem to be common (Gould et al., 2006) and,
there is something even more interesting: they don’t come alone (Mayor et al.,
2009). Super Earths, which should also be numerous, are in the range of
detectability of CoRoT(Leger et al., 2009; Queloz et al., 2009). In the near
future the number of candidates will increase and we will find ourselves with
a collection of planets whose diversity we can only start to imagine.
Section 2 is a short introduction to the CoRoT mission. Section 3 gives a
short overview of different sources of transit timing variations which could
allow the detection of Super Earths with the satellite CoRoT.
## 2 CoRoT
CoRoT is an afocal telescope with a 27 cm diameter pupil, equipped with 4 CCDs
($2048\times 2048$ pixels each); the pixel scale is $2.32^{\prime\prime}$ and
the field of view is $3.05^{\circ}\times 2.8^{\circ}.$ The selection of
observational targets follows two different strategies: the seismology channel
observes a small number (10) of bright stars ($6<m_{v}<9$) with a cadence of
32s whereas the exoplanet channel observes a large number ($\sim 11\,000$) of
faint stars ($12<m_{v}<16$) every 512s (although a limited number of targets
is measured every 32s). CoRoT is placed in a polar low Earth orbit which
determines the observational scheme. The satellite continuously monitors the
same region of the sky during 150 days; but then it has to turn around to
avoid the Sun entering the field of view: these are the _long runs_.
Immediately before or after the turnaround, the satellite is pointed during
roughly 20 days to perform a _short run_ in a different direction. Every year,
CoRoT observes 2 long runs and 2 short runs providing roughly $40\,000$ light
curves. These light curves are narrowed down in the search for transits and a
list of candidates is built. The most promising candidates are followed up
photometrically and spectroscopically from the ground.
The photometric follow-up attempts to discover if the transit is on target or,
on the contrary, if it is produced by a background binary. The PSF of CoRoT is
quite large: the flux for each star is calculated on-board in masks of size 60
pixels on average. In front of the CCD there is a prism used to produce
chromatic light curves with the aim of distinguishing between stellar
(coloured) activity and (achromatic) transits. However, for faint stars it is
not possible to make this distinction. Large masks raise the probability of
observing background binaries, which are a major source of confusion (Pont et
al., 2005).
Spectroscopic follow-up measures the mass of the transiting object. There is a
degeneracy between the mass and the radius of low mass stellar objects, brown
dwarfs and planets (see Fig. 1); to confirm the nature of a transiting object
it is mandatory to perform radial velocity measurements and calculate the
object’s mass. This can become a bottleneck for the characterization because
the measurement of faint candidates is challenging.
Deleuil and Baglin (2008) is a very interesting short summary of CoRoT and its
achievements. Fully detailed recent information about the technical
characteristics of the mission can be found in Fridlund et al. (2006); Barge
et al. (2008b) and Drummond et al. (2008).
Table 1: 5 transits found by CoRoT. | 1b222Barge et al. (2008a). | 2b333Alonso et al. (2008); Bouchy et al. (2008); Alonso et al. (2009). | 4b444Aigrain et al. (2008); Moutou et al. (2008). | 5b555Rauer et al. (2009). | 6b666Fridlund (2009). | 7b777Leger et al. (2009); Queloz et al. (2009). | 3b888Deleuil et al. (2008).
---|---|---|---|---|---|---|---
radius | 1.49 | 1.47 | 1.19 | 1.39 | 1.15 | 0.15 | 1.01
(Jupiter’s radii) | | | | | | |
mass | 1.03 | 3.31 | 0.72 | 0.47 | 3.3 | 0.015 | 21.66
(Jupiter’s masses) | | | | | | |
period | 1.51 | 1.74 | 9.20 | 4.04 | 8.89 | 0.85 | 4.25
(days) | | | | | | |
Figure 1: Figure from Deleuil et al. (2008) showing the mass–radius diagram
for planets and low mass M stars. CoRoT-3b is highlighted. The theoretical
isochrones at 10 and 1 Gyr are from Baraffe et al. (2003).
## 3 Transit Timing Variations
Kepler’s laws of motion assign periodical orbits to planets. However, there
are numerous sources of perturbations which produce deviations from the
periodicity. Not only are there differences in the observed minus calculated
(O$-$C) epochs of transits, but also in their durations and depths. Some
possible sources are general relativity effects, the quadrupolar moment of the
gravitational potential of the star, tidal interaction or even the proper
motion of the star (Miralda-Escudé, 2002; Jordán and Bakos, 2008; Pál and
Kocsis, 2008; Rafikov, 2009). But all these perturbations act on timescales
much longer than the baseline of CoRoT observations and so, in spite of their
interest, are beyond the scope of this study.
However, there are still several other sources of perturbations acting on
shorter timescales, such us those produced by other planets (Schneider, 2004;
Holman and Murray, 2005; Agol et al., 2005; Nesvorný and Morbidelli, 2008),
Trojan planets (Laughlin and Chambers, 2002; Dvorak et al., 2004; Ford and
Holman, 2007), moons (Doyle and Deeg, 2004; Kipping, 2009a, b), orbital
eccentricity (Kipping, 2008) and the light time effect, the so-called LITE
(widely studied in binary systems, see Irwin 1952; Mayer 1990; Borkovits et
al. 2003). In addition to these, Winn (2009) contains a very interesting list
of the information that can be obtained from transits.
### 3.1 Photometric Precision
From Doyle and Deeg (2004), we can calculate the maximal accuracy of $\delta
t_{0}$ that one can achieve when determining the position of a transit of
length $T_{tr}$ and depth $\Delta L$; the photometric accuracy is $\delta L$
and the number of observations is $N$. This accuracy is:
$\delta t_{0}=\delta_{L}\frac{T_{tr}}{2\Delta L\sqrt{N}}.$
In CoRoT, with a photometric accuracy of $0.1\%$, measuring a transit of depth
$1\%$ at the observing cadence of 32s, we can achieve a timing accuracy on the
order of seconds.
### 3.2 Multiple Systems
On the day this manuscript was submitted, there were 374 extrasolar planets
known, among which most are isolated. But there are already 40 known multiple
planet systems and in the future, as Dr. Udry pointed out in this conference,
probably more and more planets will be found in multiple systems. See also the
work by Dr. Wright in this volume.
We can calculate the perturbations in the time of arrival of the transit
signal of a planet if there is another planet in the system in an interior
orbit. Rigorous calculations are done in Agol et al. (2005), but we can
estimate $\delta t$, the amplitude of this perturbation, with the expression:
$\delta
t={\displaystyle\frac{P_{e}}{2\pi}}{\displaystyle\frac{m_{i}}{m_{*}+m_{i}}}{\displaystyle\frac{a_{i}}{a_{e}}};$
(1)
where $P$ stands for period, $m$ for masses, $a$ for the semi-major axis of
the orbits and the subscripts $i$ and $e$ refer to the inner and outer
(exterior) planet respectively. For a Jupiter outer planet with a period of 20
days around a star of one solar mass, an interior Super Earth of 11
terrestrial masses would produce a perturbation of 3 seconds, which is within
the limits of CoRoT.
Dynamics in multiple planets systems is a complicated matter (for example see,
in this volume, the work by Dr. Michtchenko) and resonances are one of the
most important features because they enhance the amplitude of these
perturbations and could open the door to the discovery of low mass planets
(Holman and Murray, 2005; Haghighipour et al., 2007).
In 2008 alone, at least 8 publications have seen the light on the detection of
this kind of perturbation: Agol et al. (2009); Alonso et al. (2009); Díaz et
al. (2008); Hrudková et al. (2009); Irwin et al. (2008); Miller-Ricci et al.
(2008a, b); Shporer et al. (2009).
### 3.3 Trojan planets
In our Solar System, Trojan satellites are a group of asteroids moving close
to the Lagrange points L4 and L5 in 1:1 mean-motion resonance with Jupiter’s
orbit. Many efforts have been done in the search for these kind of objects in
extrasolar systems (see for example Moldovan and Matthews 2008 and Madhusudhan
and Winn 2009). Bodies in these orbits are stable (Ford and Gaudi, 2006;
Dvorak et al., 2004) and can be found not only photometrically or by radial
velocity, but also by the timing variations that they produce in the transits
of the planet whose orbit they share. We can estimate the amplitude of this
perturbation:
$\delta t=\frac{M_{Trojan}}{M_{planet}}\frac{{\alpha}}{2\pi}P_{planet};$ (2)
for a Trojan object with the mass of the Moon and a transiting planet with the
mass of Jupiter in a 20 day orbit, $\alpha$ being the typical angle involved
in the calculation, with $\alpha\sim 30$ degrees (see the references given
above for justification), the amplitude of the perturbation is about 5
seconds. Needless to say, if the transiting planet is a Super Earth, this
perturbation is far more important. Another speculative hypothesis is the
existence of massive Trojan planets. If the ratio between the mass of the
transiting planet plus the mass of the Trojan over the mass of the star is
below $\sim 1/27,$ the system can be stable; this opens the possibility of
Trojan Super Earths (see Nauenberg 2002; Schwarz et al. 2007 and references
therein).
### 3.4 LITE
LITE was first used by the astronomer Ole Römer, working in Paris Observatory
with Jean-Dominique Cassini, to measure the speed of light (Römer, 1676).
Nowadays it is used to find hidden companions to binary systems, even those of
planetary mass (Deeg et al., 2008; Lee et al., 2008). But we can find the same
effect in multiple planet systems (Schneider, 2005). The reflex motion induced
in our Sun by Jupiter has an amplitude of one solar radius, which light covers
in 2 seconds. If we observe the transits of an inner planet and there is an
outer planet in the system, the amplitude of the LITE perturbation is:
$\delta t=2\frac{m_{e}}{M_{*}}\frac{a_{e}}{c};$ (3)
which is of the order of 0.1 seconds for the time baselines of CoRoT (and in
this case, we must concede that this favors the detection of high mass
companions and not of Super Earths).
### 3.5 Moons
No moon has yet been detected around any extrasolar planet, although their
existence is expected (Sartoretti and Schneider, 1999). However, their
detection is difficult (Brown et al., 2001). In general, it is not an easy
task to estimate the magnitude of the perturbation because it depends on the
orbit of the satellite around the planet; and for planets within the specific
period range detectable by CoRoT, we don’t yet have any clue as to how much
this estimate may be. Nevertheless, reasonable assumptions in the general case
give perturbations under 1 s, which is below CoRoT limits.
However, we point out the possibility of finding binary planets (Cabrera and
Schneider, 2007). Binarity is common among stars, from bright massive objects
down to brown dwarfs; and we can also find binary objects from the size of
trans-neptunian objects down to asteroids. Binarity should be possible among
planets and those systems will produce peculiar transit signals.
## 4 Conclusions
The photometric precision achieved by CoRoT allows the detection of Super
Earth planets in transit; but here we have shown that also non-transiting
Super Earths could be detected in multiple systems by the perturbations they
might produce in transiting planets. We have shown several possible scenarios
and discussed their limitations.
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|
arxiv-papers
| 2012-10-05T11:19:47 |
2024-09-04T02:49:36.024542
|
{
"license": "Public Domain",
"authors": "J. Cabrera",
"submitter": "Juan Cabrera",
"url": "https://arxiv.org/abs/1210.1711"
}
|
1210.1714
|
# Formats over Time: Exploring UK Web History
Andrew N. Jackson
The British Library Boston Spa, Wetherby West Yorkshire, LS23 7BQ, UK
Andrew.Jackson@bl.uk
###### Abstract
Is software obsolescence a significant risk? To explore this issue, we
analysed a corpus of over 2.5 billion resources corresponding to the UK Web
domain, as crawled between 1996 and 2010. Using the DROID and Apache Tika
identification tools, we examined each resource and captured the results as
extended MIME types, embedding version, software and hardware identifiers
alongside the format information. The combined results form a detailed
temporal format profile of the corpus, which we have made available as open
data. We present the results of our initial analysis of this dataset. We look
at image, HTML and PDF resources in some detail, showing how the usage of
different formats, versions and software implementations has changed over
time. Furthermore, we show that software obsolescence is rare on the web and
uncover evidence indicating that network effects act to stabilise formats
against obsolescence.
###### category:
H.3.3 Information Storage and Retrieval Information Search and Retrieval
###### keywords:
Information filtering, Selection process
###### category:
H.m Information Systems Miscellaneous
## 1 Introduction
In order to ensure that our digital resources remain accessible over time, we
need to fully understand the software and hardware dependencies required for
playback and re-use. The relationship between bitstreams and the software that
makes them accessible is usually expressed in terms of data ‘format’ - instead
of explicitly linking individual resources to individual pieces of software,
we attach identifiers like file extensions, MIME types and PRONOM IDs to each
and use that to maintain the link. These identifiers can also be attached to
formal format specifications, if such documentation is available.
Successful digital preservation therefore requires us to fully understand the
relationship between data, formats, software and documentation, and how these
things change over time. Critically, we must learn how formats become
obsolete, so that we might understand the warning signs, choices and costs
involved. This issue, and the arguments around the threat of obsolescence, can
be traced back to 1997, when Rothenburg asserted that “Digital Information
Lasts Forever—Or Five Years, Whichever Comes First.” [1]. Fifteen years later,
Rothenberg maintains that this aphorism is still apt [2]. If true, this
implies that all formats should be considered brittle and transient, and that
frequent preservation actions will be required in order to to keep our data
usable. In contrast, Rosenthal maintains that this is simply not the case,
writing in 2010 that “when challenged, proponents of [format migration
strategies] have failed to identify even one format in wide use when
Rothenberg [made that assertion] that has gone obsolete in the intervening
decade and a half.” [3]. Rosenthal argues that the network effects of data
sharing act to inhibit obsolescence and ensure forward migration options will
arise. Similarly, Rothenburg remains skeptical of the common belief that
different types of content are normalising on HTML5 and so reducing the number
of formats we need to address [2]. If these assertions are true, then format
migration or emulation strategies become largely unnecessary, leaving us to
concentrate on storing the content and simply making use the available
rendering software.
The fact that the very existence of software obsolescence remains hotly
disputed therefore undermines our ability to plan for the future. To find a
way forward, we must examine the available evidence and try to test these
competing hypotheses. In this paper, we begin this process by running
identification tools over a suitable corpus, so that we can use the resulting
format profile to explore what happens when formats are born, and when they
fade away. Working in partnership with JISC and the Internet Archive (IA), we
have been able to secure a copy of the IA web archives relating to the UK
domain, and host it on our computer cluster. The collection is composed of
over 2.5 billion resources, crawled between 1996 and 2010, and thus gives us a
sufficiently long timeline over which some reasonable conclusions about web
formats might be drawn.
Determining the format of each resource is not easy, as the MIME type supplied
by the originating server is often malformed [4]. Instead, we apply two format
identification tools to the content of each resource \- DROID and Apache Tika.
Both use internal file signature (or ‘magic numbers’) to identify the likely
format of each bitstream, but differ in coverage, complexity and granularity.
In particular, DROID tuned to determine different versions of formats, while
Apache Tika returns only the general format type, but augments it with more
detailed information gleaned from parsing the bitstream. Thus, by combining
both sets of results, we can come to a more complete understanding of the
corpus. Furthermore, by comparing the results from the different
identification tools, we can also uncover inconsistencies, problematic formats
and weak signatures, and so help drive the refinement of both tools.
## 2 Method
The test corpus is called the JISC UK Web Domain Dataset (1996-2010), and
contains over 2.5 billion resources harvested between 1996 and 2010 (with a
few hundred resources dated from 1994), either hosted on UK domains, or
directly referenced from resources hosted under ‘.uk’. This adds up to 35TB of
compressed content held in 470,466 arc.gz and warc.gz files, now held on the a
50-node HDFS filesystem. As the content is hosted on this distributed
filesystem, we are able to run a range of tools over the whole dataset in a
reasonable time using Hadoop’s Map-Reduce framework.
Due to it’s prominence among the preservation community and the fine-grained
identification of individual versions of formats, DROID was chosen as one of
the tools. To complement this, we also chose to use the popular Apache Tika
identification tool, which has been shown to have much broader format coverage
[5]. Unfortunately, both tools required some modification in order to be used
in this context. DROID was particularly problematic, and we were unable to
completely extract the container-based identification system in a form that
made it re-usable as a Map-Reduce task. However, the binary file format
identification engine could be reused, and the vast majority of the formats
that DROID can identify are based on using that code (and the DROID signature
file it depends upon - we used signature file version 59). Herein, we refer to
this as the ’DROID-B’ tool. Both tools were run directly on the bitstreams,
rather than being passed the URLs or responses in question, and so the
identification was based upon the resource content rather than the name or any
other metadata. For this first experimental scan, we decided to limit the
identification process to the top-level resource associated with each URL and
crawl time - archive or container formats were not unpacked.
In order to compare the results from DROID-B and Apache Tika with the MIME
type supplied by the server, the identification results are normalised in the
form of extended MIME Types. That is, where we know the version of a format as
well as the overall MIME Type, we add that information to the identifier using
a standard type parameter, e.g. “image/png; version=1.0”, corresponding to
PUID fmt/11. In this way, extended MIME types can act as a bridge between the
world of PRONOM identifiers and the standard identification system used on the
web. Broad agreement between tools can be captured by stripping off the
parameters, but their presence lets more detailed information be collected and
compared in simple standard form. A number of formats also embed information
about the particular software or hardware that was using in their creation -
PDF files have a ‘creator’ and a ‘producer’ field, and many image formats have
similar EXIF tags. As we are also interested in the relationship between
software and formats, we have attempted to extract this data and embed it in
the extended MIME type as software and hardware parameters. The full
identification process also extracted the year each resource was crawled, and
combines this with the three different MIME types to form a single ‘key’.
These keys were then collected and the total number of resources calculated
for each. Overall, the analysis was remarkably quick, requiring just over 24
hours of continuous cluster time.
## 3 Results
### 3.1 The Format Profile Dataset
The primary output of this work is the format profile dataset itself111To
download the dataset, see http://dx.doi.org/10.5259/ukwa.ds.2/fmt/1.. Each
line of this dataset captures a particular combination of MIME types (server,
Apache Tika and DROID-B), for a particular year, and indicates how many
resources matched that combination. For example, this line:
image/png image/png image/png; version=1.0 2004 102
means that in this dataset there were 102 resources, crawled in 2004, that the
server, Tika and DROID-B all agreed have the format ‘image/png’, with the
latter also determining the format version to be ‘1.0’. Due disagreements over
MIME types and the number different hardware and software identifiers the
overall profile is rather large, containing over 530,000 distinct combinations
of types and year. Below, we document some initial findings drawn from the
data. However, there is much more to be gleaned from this rich dataset, and we
have made it available under an open licence (CC0) in the hope that others
will explore and re-use it.
Figure 3.1: Identification failure rates for Apache Tika and DROID-B.
### 3.2 Comparing Identification Methods
#### 3.2.1 Coverage & Depth
The identification failure rates for both tools are shown in Figure 3.1, as a
percentage of the total number of resources from each year. Overall, Apache
Tika has significantly lower failure rate than DROID-B - 1% versus around 10%.
There also seems to be a significant downward trend in the DROID-B curve,
which would indicate that DROID copes less well with older formats. However,
initial exploration indicate that this is almost entirely due to the
prevalence of pre-2.0 HTML, which was often poorly formed.
#### 3.2.2 Inconsistencies
By comparing the simple MIME types (no parameters) we were able to compare the
results from both tools, revealing 174 conflicting MIME type combinations. For
example, some 2,957,878 resources that Apache Tika identified as ‘image/jpeg’
we identified as ‘image/x-pict’ by DROID. The PRONOM signature for this format
is rather weak (consisting of a single byte value at a given offset) and can
therefore produce a large number of false positives when run at scale
222Indeed, it appears that this signature has been removed from the latest
version of the DROID binary signature file (version 60, published during
preparation).. Another notable class of weak signatures correspond to text-
based formats like CSS, JavaScript, and older or malformed HTML. Apache Tika
appears to perform slightly better here - for example, the HTML signature is
much more forgiving than the DROID-B signature.
More subtle inconsistencies arose for the Microsoft Office binary formats and
for PDF. In the former case, a full implementation of DROID would probably be
able to resolve many of the discrepancies. The picture for PDF is more
complex. The results were mostly consistent, but DROID-B failed to recognise
1,340,462 resources that Apache Tika identified as PDF. This appears to be
because the corresponding PRONOM signature requires the correct end-of-file
marker (‘%%EOF’) to be present, whereas many functional documents can be
mildly malformed, e.g. ending with ‘%%EO’ instead. Also, the results for
PDF/A-1a and PDF/A-1b were not entirely consistent, with Tika failing to
identify many documents that DROID-B matched, but matching a small number of
PDF/A-1b documents that DROID missed. A detailed examination of the signatures
and software will be required to resolve these issues.
### 3.3 Format Trends
Figure 3.2: Number of resources of each format versus its lifespan. Formats
identified using Apache Tika.
As mentioned in the introduction, one of the core questions we need to
understand is whether formats last a few years and then die off, or whether
(on the web at least) network effects take over and help ensure formats
survive. We start to examine this question by first determining the lifespan
of each format - i.e. the number of years that have elapsed between a format’s
first and last appearance in the archive. This lifespan is plotted against the
number of resources that were found to have that format, such that young and
rare formats appear in the bottom-left corner, whereas older and popular
formats appear in the top-right, as shown in figure 3.2. Due to the extreme
variation in usage between formats, the results are plotted on a logarithmic
scale.
If popularity has no effect on lifespan, we would expect to see a simple
linear trend - i.e. a format that has existed for twice as long as another
would be found in twice as many documents. Due to the logarithmic vertical
axis of figure 3.2, would be shown as a sharp initial increase followed by an
apparent plateau. However, in the presence of network effects we would expect
a much stronger relationship, and indeed this is what we find - a format that
has been around longer is exponentially more common that younger formats (an
exponential fit appears as a straight line in figure 3.2). A large number of
formats have persistent for a long time (47 formats have been around for 15
years), and that since 1997, roughly six new formats have appeared each year
while fewer have been lost (roughly 2 per year). While this confirms the
presence of the network effects Rosenthal proposed, proving that these formats
are more resilient against obsolescence will require a deeper understanding of
obsolescence itself.
Figure 3.3: Selected popular image formats over time. Formats identified using
Apache Tika.
As a first step in that direction, we examine how format usage changes over
time. Figure 3.3 shows the variation in usage of some of the most common image
formats. Unsurprisingly, JPEG has remained consistently popular. In contrast,
the PNG and ICO formats have become more popular over time, and the GIF, TIFF
and XBM formats have decreased in popularity, with the drop in usage of the
XBM format being particularly striking.
Figure 3.4: HTML versions over time. Formats identified using DROID-B. Figure
3.5: PDF versions over time. Formats identified using DROID-B and Apache Tika.
### 3.4 Versions & Software
Figures 3.4 and 3.5 show how the popularity of various versions of HTML and
PDF has changed over time. In general, each new version grows and dominates
the picture for a few years, before very slowly sinking into obscurity. Thus,
while there were just two active versions of HTML in 1996 (2.0 and 3.2), all
six were still active in 2010. Similarly, there were three active versions of
PDF in 1996 (1.0-1.2) and eleven different versions in 2010 (1.0-1.7, 1.7
Extension Level 3, A-1a and A-1b, with 1.2-1.6 dominant). In general, it
appears that format versions, like formats, are quick to arise but slow to
fade away.
Figure 3.6: PDF software identifiers over time. Formats and software
identified using Apache Tika.
Finally, figure 3.6 shows the popularity of different software implementations
over time and the dominance of the Adobe implementations (although later years
have seen an explosion in the number of distinct creator applications, with
over 2100 different implementations of around 600 distinct software packages).
Similarly, the JPEG data revealed over 1900 distinct software identifiers and
over 2100 distinct hardware identifiers. We speculate that the number of
distinct implementations can be taken as an indicator for the maturity,
stability and degree of standardisation of a particular format, although more
thorough analysis across more formats would be required to confirm this.
## 4 Conclusions
We have made a rich dataset available, profiling the format, version, software
and hardware data from large web archive spanning almost one and a half
decades. Our initial analysis supports Rosenthal’s position; that most formats
last much longer than five years, that network effects to appear to stabilise
formats, and that new formats appear at a modest, manageable rate. However, we
have also found a number of formats and versions that are fading from use, and
these should be studied closely in order to understand the process of
obsolescence. Furthermore, we must note that every corpus contains its own
biases, such as crawl size limits or scope parameters333Even the crawl time
itself can be quite misleading, as a newly discovered resource may have been
created or published some years before. Therefore, we recommend that similar
analyses be performed on a wider range of different corpora in order to
attempt to confirm these trends.
We used two different tools (DROID-B and Apache Tika) that perform the
essentially the same task (format identification), and ran them across the
same large and varied corpus. In effect, each can be considered a different
‘opinion’ on the format, and by uncovering the inconsistencies and resolving
them, we can improve the signatures and tools in a very concrete and
measurable way, and more rapidly approach something like a ‘ground truth’
corpus for format identification.
Future work will examine whether the underlying biases of the corpus can be
addressed, whether we can reliably identify resources within container
formats, and whether the raw resource-level data can be made available. This
last point would allow many more format properties to be exposed and make it
easier to resolve inconsistent results by linking back to the actual
resources.
## 5 Acknowledgments
This work was partially supported by JISC (under grant DIINN06) and by the
SCAPE Project. The SCAPE project is co-funded by the European Union under FP7
ICT-2009.4.1 (Grant Agreement number 270137).
## 6 References
* [1]
Rothenberg, Jeff; (1997) “Digital Information Lasts Forever—Or Five Years,
Whichever Comes First.” RAND Video V-079
* [2]
Rothenberg, Jeff; (2012) “Digital Preservation in Perspective: How far have we
come, and what’s next?” Future Perfect 2012, New Zealand (Archived by WebCite®
at
http://www.webcitation.org/68OuQxEHj)
* [3]
Rosenthal, David S.H.; (2010) "Format obsolescence: assessing the threat and
the defenses", Library Hi Tech, Vol. 28 Iss: 2, pp.195 - 210
* [4]
Clausen, Lars R.; (2004) “Handling file formats” (Archived by WebCite® at
http://www.webcitation.org/68PyiaA9w)
* [5]
Radtisch, Markus; May, Peter; Askov Blekinge, Asger; Møldrup-Dalum, Per;
(2012) “SCAPE Deliverable D9.1: Characterisation technology, Release 1 &
release report.” (Archived by WebCite® at
http://www.webcitation.org/68OttmVnn)
|
arxiv-papers
| 2012-10-05T11:34:33 |
2024-09-04T02:49:36.032921
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrew N. Jackson",
"submitter": "Andrew N. Jackson",
"url": "https://arxiv.org/abs/1210.1714"
}
|
1210.1727
|
# Potential-driven eddy current in rippled graphene nanoribbons
Hua-Tong Yang yanght653@nenu.edu.cn Key Laboratory for UV-Emitting Materials
and Technology of Ministry of Education, Department of Physics, Northeast
Normal University, Changchun 130024, China
###### Abstract
It is well known that an eddy current will be induced in a conductor subject
to a varying magnetic field. Here we propose another mechanism of generating
nano-scale eddy current in rippled graphene nanoribbons(GNRs), which is only
driven by an electric potential. In particular, it is found that under
appropriate gate voltages, a local deformation may induce some unexpected
global eddy currents, which form vortices in both rippled and entire flat
areas of the GNR. We will explain that these vortices in flat areas is a
manifestation of the nonlocality of quantum interference.
###### pacs:
73.22.Pr, 72.80.Vp, 73.22.Dj, 73.25.+i
## .1 Introduction
Graphene, as the first experimentally available strictly two-dimensional
crystalNovoselov-sci04 ; Novoselov-PNAS05 , has offered a lot of new
possibilities for both fundamental research and new technologies due to its
exceptional mechanical and electronic propertiesNovoselov-N05 ; ZhangY-05 ;
Neto-09 . These single-atom-thick carbon membranes are very soft and flexible.
Experiments have observed that graphene sheets are rippling even in free-
standing conditionMeyer ; Parga08 ; Bao09 . This feature is expected to have
significant influences on their electronic properties and to result in some
new observable effectsJuan-07 ; Guinea-09 ; Levy ; Juan-11 . Recently, Bao et
al. reported a method to control and create nearly periodic ripples of
sinusoidal form in graphene sheetsBao09 , which provides a possibility to
design and manipulate electronic states in graphene sheets by controlling
their local structure. According to the continuous model, a linear
approximation of the tight-binding Hamiltonian in vicinity of its Dirac
points, the electronic states of graphene can be described by a massless Dirac
equation. In this model the action of a ripple can be represented by an
effective vector fieldSuzuura-02 ; Manes-07 ; Vozmediano_10 , which
essentially is the shift of the Dirac points due to the deformation, and a
velocity tensor representing anisotropy of the energy bandYang . This
continuous model is adequate if the deformation is very smooth. However, this
condition cannot be always satisfied in realistic graphene sheets, since some
corrugations in graphene sheets are not so smooth.
In this work, the local density of states(LDOS) and current flow in rippled
GNRs are investigated by using the tight-binding model and non-equilibrium
Green’s function(NEFG) methodKadanoff ; Keldysh ; Chou . It is found that the
current distributions near each step edge of the conductance staircases are
unstable, they will become non-potential or eddy flows if the GNRs are
rippled, although their driven force is potential. In particular, there also
occur some global eddy currents, which are still vortical in the flat regions
of the GNRs. The vortices in flat areas is rather exotic from the classical
view point, because there is no local deflection mechanism in these areas. We
will explain that this phenomenon is originated from a non-local quantum
interference effect. This eddy current will give rise to a very inhomogeneous
magnetic field in nanometer scale. It not only has crucial influences on the
performance of graphene-based electronic devices and circuits, but may also
have important potential applications.
## .2 Model and Method
The electronic properties of graphene can be described by the nearest tight-
binding Hamiltonian
$\displaystyle\hat{H}=\sum_{<\mathbf{r},\mathbf{r^{\prime}}>}t(\mathbf{r}-\mathbf{r^{\prime}})c^{{\dagger}}(\mathbf{r})c(\mathbf{r^{\prime}})+\textrm{H.c.}$
(1)
where $\mathbf{r},\mathbf{r^{\prime}}$ denote two nearest lattice points,
$c^{{\dagger}},c$ are electron’s creation and annihilation operators,
$t(\mathbf{r}-\mathbf{r^{\prime}})$ is a distance-dependent hopping amplitude
modeling the influence of deformation and can be fitted by
$\displaystyle t(\mathbf{r}-\mathbf{r^{\prime}})\simeq
t_{0}e^{\alpha(1-|\mathbf{r}-\mathbf{r^{\prime}}|/l)},$ (2)
where $t_{0}=-2.75eV$, $\alpha\simeq 3.37$, $l\simeq 1.42{\AA}$Pereira ;
Pellegrino . For the tight-binding model, we define the bond current
$j(\mathbf{r},\mathbf{r^{\prime}})$ from $\mathbf{r}$ to $\mathbf{r^{\prime}}$
by the conservation equation
$\sum_{\mathbf{r^{\prime}}}j(\mathbf{r},\mathbf{r^{\prime}})=-\frac{\partial\langle\hat{n}(\mathbf{r}t)\rangle}{\partial
t}$, where
$\hat{n}(\mathbf{r}t)=2c^{{\dagger}}(\mathbf{r}t)c(\mathbf{r^{\prime}}t)$ is
the electronic number operator, the factor $2$ comes form the spin degree of
freedom. From the equation of motion we have
$\displaystyle
j(\mathbf{r},\mathbf{r^{\prime}},t)=\frac{4}{\hbar}\textrm{Re}[t(\mathbf{r}-\mathbf{r^{\prime}})G^{<}(\mathbf{r^{\prime}}t,\mathbf{r}t)],$
(3)
where $G^{<}(\mathbf{r^{\prime}}t^{\prime},\mathbf{r}t)=i\langle
c^{{\dagger}}(\mathbf{r}t)c(\mathbf{r^{\prime}}t^{\prime})\rangle$ is the
lesser Green’s function. In steady states, the current can be written as
$j(\mathbf{r},\mathbf{r^{\prime}})=\int
j(\mathbf{r},\mathbf{r^{\prime}},E)dE,$ where
$\displaystyle
j(\mathbf{r},\mathbf{r^{\prime}},E)=\frac{4}{h}\textrm{Re}[t(\mathbf{r}-\mathbf{r^{\prime}})G^{<}(\mathbf{r^{\prime}},\mathbf{r},E)]$
(4)
is the current per unit energyAreshkin-07 .
For the sake of simplicity, we only consider zigzag and armchair GNRs with a
finite rippled region. The ripple is a static height fluctuation given by
$\displaystyle
z(\mathbf{r})=\left\\{\begin{array}[]{ll}h\sin(\mathbf{k}\cdot\mathbf{r}+\phi_{0}),&0\leq\mathbf{k}\cdot\mathbf{r}\leq
2m\pi\\\
h\sin\phi_{0},&\mathbf{k}\cdot\mathbf{r}<0~{}\textrm{or}~{}\mathbf{k}\cdot\mathbf{r}>2m\pi,\end{array}\right.$
(7)
with $\phi_{0}$ an arbitrary constant and $m$ an integer, the atomic in-plane
displacement is ignored. A typical configuration of the rippled GNRs
considered in this paper is shown in Fig.1.
Figure 1: An $N=20$ zigzag GNR with a sinusoidal ripple given by Eq.(7), here
$h\simeq 2a$ with $a\simeq 2.46\AA$ the length of the basis vectors, the
ripple wavelength $\lambda=30a$, the included angle $\theta_{\mathbf{k}}$
between the GNR axis and $\mathbf{k}$ is $30^{\circ}$, $\phi_{0}=-\pi/2$,
$m=1$.
In order to calculate the LDOS and current of GNRs by using the NEFG
methodKadanoff ; Keldysh ; Chou , the GNRs have to be considered as consisting
of three connected parts: a rippled region as a sample and two semi-infinite
ideal regions as leadsLopez-Sancho ; Lake . The left and right leads can be
assumed to be in equilibrium states with different chemical potentials
$\mu_{L}$ and $\mu_{R}$, respectively, where $\mu_{L}-\mu_{R}$ is very small.
Thus the current can be written as
$\displaystyle
j(\mathbf{r},\mathbf{r^{\prime}})=\frac{4}{h}\int_{\mu_{R}}^{\mu_{L}}\textrm{Re}[t(\mathbf{r}-\mathbf{r^{\prime}})G^{<}(\mathbf{r^{\prime}},\mathbf{r},E)]dE,$
(8)
where
$\displaystyle G^{<}=G^{r}\Sigma^{<}_{L}G^{a},$ (9a) $\displaystyle
G^{r,a}=(E^{\pm}I-H_{c}-\Sigma^{r,a})^{-1},$ (9b)
with $E^{\pm}\equiv E\pm i0^{+}$ and $H_{c}$ is the Hamiltonian matrix of the
isolated sample. $\Sigma^{r,a}=\Sigma^{r,a}_{L}+\Sigma^{r,a}_{R}$ is the self-
energy arisen from the couplings between the sample and two leads.
$\Sigma^{<}_{L,R}=f_{L,R}(\Sigma^{a}_{R,L}-\Sigma^{r}_{R,L}),$ where the Fermi
functions $f_{L,R}$ satisfy $f_{L,R}(E)\simeq\theta(\mu_{L,R}-E)$ at low
temperature. $\Sigma^{r,a}_{R,L}$ can be obtained by using the iterative
method developed by Lopez-Sancho et.al.Lopez-Sancho . To reduce the
calculation and memory requirement, we orderly denote the layers of the sample
from the left to the right by $1,2,\cdots,l$, thus the necessary sub-matrices
$G^{<}_{n,n-1}$ and $G^{<}_{n,n}$ for calculating current can be obtained by
following back-and-forth recurrence procedure:
$\displaystyle g^{r,a}_{n,n}=(E^{\pm}I-H_{n,n}-\Sigma^{r,a}_{R,n})^{-1},$
(10a) $\displaystyle\Sigma^{r}_{R,n-1}=H_{n-1,n}g^{r}_{n,n}H_{n,n-1},$ (10b)
with $n=l,\cdots,2$ and $\Sigma^{r}_{R,l}=\Sigma^{r}_{R}.$ For $\mu_{R}\leq
E\leq\mu_{L}$,
$\displaystyle G^{<}_{n,n-1}=g^{r}_{n,n}H_{n,n-1}G^{<}_{n-1,n-1},$ (11a)
$\displaystyle G^{<}_{n,n}=G^{<}_{n,n-1}H_{n-1,n}g^{a}_{n,n}$ (11b)
with $n=2,\cdots,l$ and $G^{<}_{1,1}=G^{r}_{1,1}\Sigma^{<}_{L}G^{a}_{1,1}$,
where
$\displaystyle
G^{r,a}_{1,1}=[E^{\pm}I-H_{1,1}-\Sigma^{r,a}_{R,1}-\Sigma^{r,a}_{L,1})]^{-1}.$
(12)
Similarly, the LDOS
$\displaystyle\rho(\mathbf{r},E)=\frac{1}{2\pi}A(\mathbf{r},\mathbf{r}),$ (13)
where $A=i(G^{r}-G^{a})$, and the conductance
$\displaystyle
T(E)=\frac{2e^{2}}{h}\textrm{Tr}[\Gamma_{R}(A-G^{r}\Gamma_{R}G^{a})],$ (14)
with $\Gamma_{R,L}=i(\Sigma^{r}_{R,L}-\Sigma^{a}_{R,L})$, can also be obtained
by the same method.
In order to see the ripple’s influence on the current as clear as possible,
the inherent deflection due to the lattice background has to be averaged out.
To this end we define a cell-average current as an average vector of six bond
currents around a honeycomb cell
$\displaystyle\mathbf{j}(\mathbf{r}_{c})=\frac{1}{6}\sum_{i<j}j(\mathbf{r}_{i},\mathbf{r}_{j})\frac{\mathbf{r}_{j}-\mathbf{r}_{i}}{|\mathbf{r}_{j}-\mathbf{r}_{i}|},$
(15)
where $\mathbf{r}_{i,j}$ denote the vertices of the cell, $\mathbf{r}_{c}$ is
the cell center.
## .3 Results and Discussion
First we compare the LDOSs and conductances for a fixed GNR with different
ripples. Fig.2 shows the LDOSs and conductances for a zigzag and an armchair
GNR in the presence of different ripples given by Eq.(7) with different $h$’s
and orientations (represented by slope angle $\theta_{\mathbf{k}}$).
Figure 2: (a) LDOS and (b) conductance of rippled $N=20$ zigzag GNRs as shown
in Fig.1 for different $h$’s, where $\lambda=30a$,
$\theta_{\mathbf{k}}=30^{\circ}$. (c) LDOS and (d) conductance of identical
GNRs as Fig.1 for different $\theta_{\mathbf{k}}$’s, where $h=1.5a$,
$\lambda=30a$. (e) LDOS and (f) conductance of $N=20$ armchair GNRs for
different $h$’s, here $\lambda=40a$, $\theta_{\mathbf{k}}=20^{\circ}$. (g)
LDOS and (h) conductance of identical GNRs as (e,f) for different
$\theta_{\mathbf{k}}$’s, where $h=1.5a$, $\phi_{0}=-\pi/2$.
The most remarkable change is that the conductance near every step edge of the
conductance staircase is remarkably decreased. Meanwhile, each corresponding
van Hove peak in LDOSs is also broaden and successively split into two sub-
peaks when $h$ reaches critical values, except for zigzag GNRs with
$\theta_{\mathbf{k}}=0^{\circ}$(which will be further discussed in the last
paragraph). The critical degrees of deformation can be represented by the
associated maximum relative bond elongation due to the ripple. This bond
elongation can be roughly estimated from the ratio between the ripple’s height
$h$ and wavelength $\lambda$ if we consider only the atomic height
fluctuation. For zigzag GNRs with $\theta_{\mathbf{k}}=30^{\circ}$, this
critical ratio is about $1/20$, the corresponding maximum relative bond
elongation is about $5\%$.
To get an intuitive understanding of the microscopic behavior of electrons
that gives rise to these changes, now we analyze the spatial distributions of
the LDOS $\rho(\mathbf{r},E)$ and current $\mathbf{j}(\mathbf{r}_{c},E)$ at
the corresponding energies. Fig.3 shows two examples of this kind of LDOS and
current distributions in a rippled $N=20$ zigzag GNR at $0.56eV$ and $0.61eV$,
respectively corresponding to two sub-peaks split off from the first van Hove
peak in conducting sub-band. We can see from the LDOS distributions
(Fig.3(a,c)) that the lower sub-peak is mainly localized in the ripple area;
while the upper one (belonging to the bottom of the first conducting sub-band)
is extended, its LDOS does not decay outside the ripple region. More
importantly, their current distributions both occur remarkable vortices
(Fig.3(b,d)). Some current lines even form closed loops, showing that these
currents are not potential flow. Similarly, the second van Hove peak also
splits into two sub-peaks and their current distributions also have this
vortical feature. The valence bands also occur identical phenomena. When the
bias is reversed, these eddy currents will also be exactly reversed, thus an
alternative bias can drive a varying eddy current in this kind of rippled
GNRs.
Figure 3: (a) LDOS and (b) current distribution at $0.56eV$(the lower sub-
peak split off from 1st van Hove peak) in an $N=20$ zigzag GNR as shown in
Fig.1. (c) LDOS and (d) current at $0.61eV$(the upper sub-peak of the 1st van
Hove peak) in the same GNR. The brightness in (a,c) represents the LDOS, the
contours depict the ripple. The brightness in (b,d) represents the height of
the ripple.
At first sight, these vortices seems can be ascribed to Landau-like quasi-
bound states caused by the ripple-induced pseudo-magnetic field, since
according to the continuous model a geometrical deformation will induce an
effective vector potential
$\displaystyle\mathbf{A}\simeq\frac{1}{2}(\sqrt{3}(t_{3}-t_{2}),t_{3}+t_{2}-2t_{1})$
(16)
with $t_{1,2,3}$ three nearest hopping amplitudesVozmediano_10 . Actually,
some conductance fluctuation phenomena similar to the results shown in Fig.(2)
have been observed in the measurement of differential conductance of graphene
sheet with nanobubbles, and the researchers interpreted all this conductance
fluctuation as the contribution of the Landau levels arising from the pseudo-
magnetic fieldLevy . It is true that most of the above results can be
explained by this pseudo-magnetic field. For example, if $\mathbf{k}$ is
exactly along some special directions, e.g.,
$\theta_{\mathbf{k}}=0^{\circ},30^{\circ},60^{\circ}$, we can deduce some
general properties of $\mathbf{A}$ by qualitative analysis, which can be
compared with the numerical results. In these special cases the $\mathbf{A}$
will has the form $\mathbf{A}_{0}\cos(\mathbf{k}\cdot\mathbf{r}+\phi_{0})$ and
will be parallel or perpendicular to $\mathbf{k}$Yang . More specifically, for
rippled zigzag GNRs of $\theta_{\mathbf{k}}=0^{\circ},60^{\circ}$ or armchair
GNRs of $\theta_{\mathbf{k}}=30^{\circ}$, $\mathbf{A}$ is nearly parallel to
$\mathbf{k}$, so the pseudo-magnetic field
$\nabla\times\mathbf{A}\simeq-\mathbf{k}\times\mathbf{A}_{0}\sin(\mathbf{k}\cdot\mathbf{r}+\phi_{0})$
is very small, thus the ripple influence will be much weaker than other ripple
orientations; while for armchair GNRs of $\theta_{\mathbf{k}}=0^{\circ}$ or
zigzag GNRs of $\theta_{\mathbf{k}}=30^{\circ}$, $\mathbf{A}$ is perpendicular
to $\mathbf{k}$, so the pseudo-magnetic field is strong and will strongly
disturb the electronic states. For these special cases, the above explanation
is qualitatively in accord with the result in Fig.2(there are still some more
subtle problems, which will be discussed in last paragraph).
However, these current distributions also exhibit a remarkable feature
revealing that this is not the whole story. We notice that if the energy
slightly higher than the bottom of a conducting sub-band (or lower than the
top of a valance sub-band), the vortices of the eddy current will not entirely
lie within the ripple region, but also occur in flat regions far from the
ripple, where the pseudo-magnetic field has vanished, as can be seen from
Fig.3(d). This feature reveals that these vortices cannot be interpreted as
Landau states produced by the pseudo-magnetic field, because all ripple-
induced scattering mechanisms, both the pseudo-magnetic field and the velocity
variation, only act locally within the ripple area, hence the electronic waves
should freely propagate in flat areas.
Figure 4: (a) Eddy current in the same GNR as Fig.1 at $1eV$, the upper sub-
peak of the second van Hove peak, and (b) $1.37eV$, the upper sub-peak of the
third van Hove peak. (c) Eddy current at $0.6eV$ in a similar GNR as
Fig.3(c,d) but with a hyperbolic-surface ripple given by Eq.(19). (d) LDOS and
(e) current distribution at $0.69eV$ in an $N=20$ armchair GNR with a ripple
given by Eq.(7), where $h=1.5a$, $\lambda=40a$,
$\theta_{\mathbf{k}}=30^{\circ}$, $\phi_{0}=-\pi/2$. (f) LDOS and (g) current
distribution at $0.631eV$ in a GNR as (d,e) except
$\theta_{\mathbf{k}}=0^{\circ}$.
Fig.4(a,b) give other two examples of this global eddying currents in the same
zigzag GNR as Fig.3, their energies are slightly above the second and third
van Hove peaks of the conducting band, respectively. We can see that both of
them have pronounced vortices in the flat areas far beyond the ripple, where
the pseudo-magnetic field has certainly vanished. Generally, if the ripple is
slope relative to the GNR axis, the current distributions within the energy
range slightly above the bottom of a conducting sub-band or below the top of a
valence sub-band will occur remarkable vortices in entire flat areas. These
vortices appeared in flat areas are rather exotic because there is no local
responsible deflection mechanism.
In order to explain the origin of these exotic vortices, we have to notice two
basic properties of the electronic states of GNRs under perturbation. The
first is that the electronic states will become superposition of partial waves
with approximate energies. The second is that the velocity direction (forward
or backward relative to GNR’s axis) of the states near the bottom of a
conducting sub-band (or the top of a valence sub-band) is unstable, because
its velocity $\partial E(\mathbf{k})/\partial k_{x}$ ($k_{x}$ is the momentum
component along the GNR axis) is very small, so its sign can be easily changed
by a small variation of $E(\mathbf{k})$, as illustrated in Fig.5(a,b).
Figure 5: (a) Conducting band of graphene near a Fermi point, the red dashed
line represents a sub-band of a GNR. The velocity of states near the minimum
of the sub-band (red dashed line) can be easily reversed under a small
variation of the energy band, as shown in (b). (b) Contour map of energy bands
for a perfect (blue-purple) and a deformed (yellow) graphene. For the deformed
one, only two bonds are elongated respectively by $5.3\%$ and $2.8\%$. The
red(yellow) arrow represents the velocities of a state in perfect(deformed)
graphene. (c) Current distribution of three plane waves.
Therefore, if there is a nonuniform deformation, the perturbed state
corresponding to an originally forward state in these energy ranges may
contribute a backward flow in the deformed regions. In particular, this local
backward flow demands the perturbed state to include backward partial waves in
flat areas in order to satisfy continuity condition at the interfaces between
deformed and flat areas, though there exists no local responsible scattering
mechanism. These partial waves of different directions are always spatially
superimposed due to the edge reflection and will produce interference.
Consequently, the current density no longer equals to the sum of the currents
of every partial waves,
$\mathbf{j}(\mathbf{r})\neq\sum\langle\mathbf{k}|\hat{\mathbf{j}}(\mathbf{r})|\mathbf{k}\rangle$,
but must include interference terms
$\langle\mathbf{k}|\hat{\mathbf{j}}(\mathbf{r})|\mathbf{k}^{\prime}\rangle$,
similar to the usual interference for probability density. These kind of
current interference patterns will be winding or eddying flows in areas
without any local deflection mechanism. As a simplest example, Fig.5(c) shows
the current interference pattern of three plane waves
$\psi(\mathbf{r})=e^{i\mathbf{k}_{1}\cdot\mathbf{r}}+e^{i\mathbf{k}_{2}\cdot\mathbf{r}}+e^{i\mathbf{k}_{3}\cdot\mathbf{r}}$,
where $\mathbf{k}_{1,2,3}$ have equal length while their included angles are
$120^{\circ}$. We can see that its current distribution forms a very similar
eddying pattern. Obviously, the vortex scale of these eddy currents arising
from the interference is in proportional to the wavelengths of the partial
waves. For the eddy currents in rippled GNRs, this character can also be
verified by comparing Fig.3(a,d) and Fig.4(a,b). We find that the vortices
become smaller and smaller with the increasing of energy owing to the linear
dispersion relation $E\propto k$. Conversely, the vortex scale will be very
large in wider GNRs because the energies of each corresponding step of the
conductance staircases will be smaller, it is only limited by the electronic
interference length.
According to this explanation, these global eddy currents would be a rather
ubiquitous effect in rippled GNRs, although their patters depend on specific
ripple configurations. Fig.4(c) is another eddy current in an identical zigzag
GNR in the same energy range as Fig.3(b) but with different ripples, which is
a hyperbolic surface
$\displaystyle
z(\mathbf{r})=\left\\{\begin{array}[]{lc}h[1-f(\mathbf{r})],&f(\mathbf{r})\leq
1\\\ 0,&f(\mathbf{r})>1,\end{array}\right.$ (19)
where$f(\mathbf{r})=\sqrt{\left(\frac{\mathbf{\hat{e}}_{1}\cdot\mathbf{r}}{a_{0}}\right)^{2}+\left(\frac{\mathbf{\hat{e}}_{2}\cdot\mathbf{r}}{b_{0}}\right)^{2}}$,
with $h=4a$, $a_{0}=15a$, $b_{0}=25a$,
$\mathbf{\hat{e}}_{1,2}=(\sqrt{2}/2,\mp\sqrt{2}/2)$(its maximum bond
elongation is about $3\%$). By comparing Fig.3(d) and Fig.4(c), we can see
that their current distributions in flat regions are very similar although
their ripple are very different. Similar to this structural insensitivity of
the global eddying character, it is conceivable from the above explanation
that this vortical character will also not very sensitive to the energy
variation. Actually, the representation of these eddying states in the LDOS
curve is not a sharp peak like quasi-bound states, but a broad and smooth one
forming a piece of continuous spectrum (Fig.2).
In addition, there are few special cases worth to be particularly pointed out.
The first case is zigzag GNRs with $\theta_{\mathbf{k}}=0^{\circ}$. In this
case the current lines remain to be straight lines and no vortex occurs,
because in this case the $\mathbf{A}$ is parallel to $\mathbf{k}$ according
Eq.(16) or Yang , so the $\nabla\times\mathbf{A}$ can be ignored; moreover,
the incident current is along a symmetric axis of the energy band, so the
refraction also does not change its direction. The second is zigzag GNRs with
$\theta_{\mathbf{k}}=\pm 60^{\circ}$ or armchairs GNRs with
$\theta_{\mathbf{k}}=\pm 30^{\circ}$. Similar to the first case, here the
$\nabla\times\mathbf{A}$ are also very small, however, there exist apparent
vortices, as shown in Fig.4(e), because in these cases the anisotropic
deformation of the energy contours(see Fig.5(b)) will result in similar
backward flow. The third one is armchair GNRs of
$\theta_{\mathbf{k}}=0^{\circ}$, although the ripple induce a strong pseudo-
magnetic field and results in apparent quasi-bound states (Fig.2(f) and
Fig.4(f)), but their currents do not occur any vortex(Fig.4(g)), because the
direction of the incident current is the symmetric axis of two pseudo-magnetic
fields for two inequivalent Dirac points (mutually symmetric) as well as the
principle axis of the velocity tensor, so the action of the pseudo-magnetic
fields of two Dirac points will be mutually canceled out and the refraction
and dispersion due to the velocity anisotropy also does not change the current
direction.
In summary, the current flows near every step edge of the conductance
staircases of a GNR are rather unstable. They will become eddy currents if
there occurs a slope ripple. These eddy currents can be divided into two
classes. The first one are carried by Landau-like states caused by the pseudo-
magnetic field, these states have slightly lower energies and their current
distributions form vortices only within the ripple region. In contrast, the
second one are carried by some special scattering states, which have slightly
higher energies and include backward partial waves in flat areas.
Consequently, they will form some global eddy currents due to the interference
of these partial waves. This global eddy current is a manifestation of the
non-locality of quantum interference effect.
###### Acknowledgements.
This work was supported by ”the Fundamental Research Funds for the Central
Universities” and NSFC(Grant Nos. 10974027, 50832001).
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|
arxiv-papers
| 2012-10-04T15:37:37 |
2024-09-04T02:49:36.041880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua-Tong Yang",
"submitter": "Hua-Tong Yang",
"url": "https://arxiv.org/abs/1210.1727"
}
|
1210.1878
|
# A PCG Implementation of an Elliptic Kernel in an Ocean Global Circulation
Model Based on GPU Libraries
S. Cuomoa∗ P. De Michelea and R. Farinab and M. Chinnicic
aUniversity of Naples “Federico II” - Dept. of Mathematics and Applications
“R. Caccioppoli” - Via Cinthia, 80126, Naples, Italy; bCMCC, Research Center,
Via Aldo Moro 44 (Bologna), Italy cENEA-UTICT, Casaccia Research Center, S.
Maria di Galeria (Roma), Italy
∗Corresponding author. Email: salvatore.cuomo@unina.it
(released October 2012)
###### Abstract
In this paper an inverse preconditioner for the numerical solution of an
elliptic Laplace problem of a global circulation ocean model is presented. The
inverse preconditiong technique is adopted in order to efficiently compute the
numerical solution of the elliptic kernel by using the Conjugate Gradient (CG)
method. We show how the performance and the rate of convergence of the solver
are linked to the discretized grid resolution and to the Laplace coefficients
of the oceanic model. Finally, we describe an easy-to-implement version of the
solver on the Graphical Processing Units (GPUs) by means of scientific
computing libraries and we discuss its performance.
65Y5; 65Y10; 65F08; 65F35; 37N10.
###### keywords:
Ocean Modelling; Preconditioning Technique; GPU Programming; Scientific
Computing.
††articletype: Research Paper
## 1 Introduction
The ocean is now well known to play a dominant role in the climate system
because it can initiate and amplify climate change on many different time
scales. Hence, the simulation of ocean model is became a relevant but highly
complex task and it involves an intricate interaction of theoretical insight,
data handling and numerical modelling. Over the past several years, ocean
numerical models have become quite realistic as a result of improved methods,
faster computers, and global data sets. Models now treat basin-scale to global
domains while retaining the fine spatial scales that are important for
modelling the transport of heat, salt, and other properties over vast
distances. Currently, there are many models and methods employed in the
rapidly advancing field of numerical ocean circulation modelling as Nemo,
Hops, MOM , POP et al (see [2] for a nice review). However, several of these
numerical models are not yet optimized by using scientific computing libraries
and ”ad hoc” preconditioning techniques. In all these frameworks the numerical
kernel is represented by the discretization of the Navier-Stokes equations [4]
on a three dimensional grid and by the computation of the evolution time of
each variable for each grid point. The high resolution computational grid
requires efficient preconditiong techniques for improving the accuracy in the
computed solution and parallelization strategies for answering to the huge
amount of computational demand.
In this paper we propose a new solver based on preconditioned conjugate
gradient (PCG) method with an approximate inverse preconditioner AINV [7] for
the numerical solution of the elliptic sea-surface equation in NEMO-OPA ocean
model [3], a state of the art modelling framework in the oceanographic
research. The PCG is a widely used iterative method for solving linear systems
with symmetric, positive definite matrix and it has proven its efficiency and
robustness in a lot of applications. The preconditioning is often a bottleneck
in solving the linear systems efficiently and it is well established that a
suitable preconditioner increases the performance of an application
dramatically.
The elliptic sea-surface equation is originally solved in NEMO-OPA by means of
the PCG with diagonal preconditioner, and in our work we prove that it is
inefficient and inaccurate. We build a new inverse preconditioner and we
implement the PCG on a Graphic Processor Unit (GPU) by means of the linear
algebra Scientific Computing libraries. The GPUs are massively parallel
architectures that efficiently work with the linear algebra operations and
give impressive performance improvements. They require a deep understanding of
the underlying computing architecture and the programming with these devices
involves a massive re-thinking of existing CPU based applications. A challenge
is how to optimally use the GPU hardware adopting adequate programming
techniques, models, languages and tools. In this paper, we present an easy-to-
implement version of the elliptic solver with the scientific computing
libraries on Compute Unified Device Architecture (CUDA) [15]. We implement a
code by using CUDA based supported libraries CUBLAS [16] and CUSPARSE [17] for
the sparse linear algebra and the graph computations. The library GPU based
approach allows a short code development times and an easy to use GPUs
implementations that can fruitfully speed up the expensive numerical kernel of
an oceanographic simulator. The paper is set out as follows. In section 2 we
briefly review the mathematical model: elliptic equations that are at the
heart of the model. In section 3, the preconditioned conjugate gradient method
used to invert the elliptic equations are described. In section 4, we outline
a implementation strategy for solving the elliptic solver by using standard
libraries and in section 5 we discuss the mapping of our algorithm onto a
massively parallel machine. Finally, the conclusions are drawn.
## 2 The Mathematical Model
Building and running ocean models able to simulate the world of global
circulation with great realism require a variety of scientific skills. In
modelling the general ocean circulation it is necessary to solve problems of
elliptic nature. These problems are difficult to solve, with the following
issues causing the most trouble in practice[2]:
1. 1.
In simulations with complicated geometry (e.g., multiple islands), topography,
time varying surface forcing, and many space-time scales of variability (i.e.,
the World Ocean), achieving a good first guess for the iterative elliptic
solver is often quite difficult to achieve. This makes it difficult for
elliptic solvers to converge to a solution within a reasonable number of
iterations. For this reason, many climate modellers limit the number of
elliptic solver iterations used, even if the solver has not converged. This
approach is very unsatisfying.
2. 2.
Many elliptic solvers with their associated non-local and time dependent
boundary conditions (be they Neumann or Dirichlet) do not project well onto
parallel distributed computers, which acts to hinder their scaling properties
[23, 24, 25].
In the OPA-NEMO numerical code the primitive equations are discretized within
sea-surface hypothesis [1] and the model is charecterized by the three-
dimensional distribution of currents, potential temperature, salinity,
preassure and density [4]. The numerical method OPA-NEMO is grounded on
discretizing of the primitive equation - by the use of finite differences on a
three dimensional grid - and computing the time evolution to each variable
”ocean” at each grid point for the entire globe [6]. A sketch of the OPA-NEMO
computational model, see Figure 1, shows the complex dynamic processes that
mimic the ocean circulation model, composed by steps that are many time
simulated.
Figure 1: NEMO-OPA model.
The kernel algorithm (highlighted with red color, equation 1) solves the sea-
surface hight equation $\eta$ The elliptic kernel is discretized with a semi-
discrete equations, as following:
$\displaystyle{\eta^{n+1}=\eta^{n-1}}-2\Delta
tD^{n}\qquad\qquad\qquad\quad\qquad\ \ \quad$ (1) $\displaystyle 2\Delta
tgT_{c}\Delta_{h}D^{n+1}=D^{n+1}-D^{n-1}+2\Delta tg\Delta_{h}\eta^{n}\quad\ \
$ (2)
$\displaystyle\Delta_{h}=\nabla\big{[}(H+\eta^{n})\nabla\big{]}.\qquad\qquad\quad\qquad\
\ \qquad\quad$ (3)
where $\eta^{n},\ n\in\mathbb{N}$ is the sea-surface height at the $n-$th step
of the model, which describes the shape of the air-sea interface, $D^{n}$ is
the centered difference approximation of the first time derivative of $\eta$,
$\Delta t$ is the time stepping, $g$ is the gravity constant, $T_{c}$ is a
physical parameter, $\Delta_{h}$ is the horizontal Laplacian operator and $H$
is the depth of the ocean bottom [3]. Whereas the domain of the ocean models
is the Earth sphere (or part of it) the model uses the geographical
coordinates system $(\lambda,\phi,r)$ in which a position is defined by the
latitude $\phi$, the longitude $\lambda$ and the distance from the center of
the earth $r=a+z(k)$ where a is the Earth’s radius and z the altitude above a
reference sea level. The local deformation of the curvilinear geographical
coordinate system is given by $e_{1}$,$e_{2}$ and $e_{3}$:
$\small\begin{array}[]{c}e_{1}=rcos\phi,\quad e_{2}=r,\quad
e_{3}=1.\end{array}$ (4)
The Laplacian Operator in spherical coordinates $\Delta_{h}D^{n+1}$ in (2)
becomes:
$\Delta_{h}D^{n+1}=\frac{1}{e_{1}e_{2}}\Bigg{[}\frac{\partial}{\partial
i}\bigg{(}\alpha(\phi)\frac{\partial D^{n+1}}{\partial
i}\bigg{)}+\frac{\partial}{\partial j}\bigg{(}\beta(\phi)\frac{\partial
D^{n+1}}{\partial i}\bigg{)}\Bigg{]}$ (5)
where:
$\displaystyle\alpha(\phi)=(H+\eta^{n}){{e_{2}}/{e_{1}}}$ (6)
$\displaystyle\beta(\phi)=(H+\eta^{n}){e_{1}}/{e_{2}}$ (7)
For the functions $\alpha(\phi)$ in (6) and $\beta(\phi)$ in (7), we have the
following relations:
$\small\lim_{\phi\longrightarrow\pm\frac{\pi}{2}}\alpha(\phi)=+\infty\quad\wedge\
\lim_{\phi\longrightarrow\pm\frac{\pi}{2}}\beta(\phi)=0$ (8)
From (8), if we choose $M,\epsilon\in\mathbb{R}$ with $M>>\epsilon$ then
exists an interval $\big{[}\frac{\pi}{2}-\delta,\frac{\pi}{2}\big{]}$ or
$\big{[}-\frac{\pi}{2},\ -\frac{\pi}{2}+~{}\delta\big{]}$, such that the
following inequality holds:
$\small\alpha(\phi)>M>>\epsilon>\beta(\phi)$ (9)
In physical terms, in the proximity of the geographical poles,
$(\lambda,\pm\phi/2,r)$, there are several orders of magnitude between the
functions $\alpha(\phi)$ and $\beta(\phi)$
The result (9), will significantly influence the rate of convergence in the
iterative solver.
## 3 Inverse Preconditioned Techniques in the Elliptic Solver of the Ocean
Model
Let us now consider the elliptic NEMO model [3] defined by the following
coefficients:
$\small\begin{array}[]{c}C_{i,j}^{NS}=2\Delta
t^{2}{H(i,j)e_{1}(i,j)}/{e_{2}(i,j)},\quad C_{i,j}^{EW}=2\Delta
t^{2}{H(i,j)e_{2}(i,j)}/{e_{1}(i,j)}\\\\[1.42262pt]
b_{i,j}=\delta_{i}(e_{2}M_{u})-\delta_{j}(e_{1}M_{v})\end{array}$ (10)
where $\delta_{i}$ and $\delta_{j}$ are the discrete derivative operators
along the axes $\mathbf{i}$ and $\mathbf{j}$. The discretization of the
equation (2) by means of a five-point finite difference method gives:
$\small\begin{array}[]{c}C_{i,j}^{NS}D_{i-1,j}+C_{i,j}^{EW}D_{i,j-1}-\big{(}C_{i+1,j}^{NS}+C_{i,j+1}^{EW}+C_{i,j}^{NS}+C_{i,j}^{EW}\big{)}D_{i,j}+\\\\[5.69054pt]
\qquad\qquad+C_{i,j+1}^{EW}D_{i,j+1}+C_{i+1,j}^{NS}D_{i+1,j}=b_{i,j}.\end{array}$
(11)
where the equation (11) is a symmetric system of linear equations. All the
elements of the sparse matrix $\mathbf{A}$ vanish except those of five
diagonals. With the natural ordering of the grid points (i.e. from west to
east and from south to north), the structure of $\mathbf{A}$ is a block-
tridiagonal with tridiagonal or diagonal blocks. The matrix $\mathbf{A}$ is a
positive-definite symmetric matrix with $n=jpi\times jpj$ size, where $jpi$
and $jpj$ are respectively the horizontal dimensions of the grid
discretization of the domain.
The Conjugate Gradient Method is a very efficient iterative method for solving
the linear system (11) and it provides the exact solution in a number of
iterations equal to the size of the matrix. The convergence rate is faster as
the matrix is closer to the identity one. By spectral point of view a
convergence relation between the solution of the linear system and its
approximation $x_{m}$ is given by:
$\|\mathbf{x}-\mathbf{x}_{m}\|_{A}<2\bigg{(}\frac{\sqrt{\mu_{2}(A)}-1}{\sqrt{\mu_{2}(A)}+1}\bigg{)}^{m-1}\|\mathbf{x}-\mathbf{x}_{0}\|_{A}$
(12)
with $\mu_{2}(A)=\lambda_{max}/\lambda_{min}$, where $\lambda_{max}$ and
$\lambda_{min}$ are respectively the greatest and the lowest eigenvalue of
$\mathbf{A}$, and $\|\cdot\|_{A}$ is the A-norm. The preconditioning framework
consists to introduce amatrix $\mathbf{M}$, that is an approximation of
$\mathbf{A}$ easier to invert, and to solve the equivalent linear system:
$\small\mathbf{M^{-1}A}\mathbf{x}=\mathbf{M^{-1}b}$ (13)
The ocean global model NEMO-OPA uses the diagonal preconditioner, where
$\mathbf{M}$ is chosen to the diagonal of $\mathbf{A}$. Let us introduce the
following cardinal coefficients:
$\displaystyle\alpha_{i,j}^{E}={{C_{i,j+1}}^{EW}}/{d_{i,j}}\qquad\alpha_{i,j}^{W}={{C_{i,j}}^{EW}}/{d_{i,j}}\qquad$
(14)
$\displaystyle\beta_{i,j}^{S}={{C_{i,j}}^{NS}}/{d_{i,j}}\qquad\beta_{i,j}^{N}={{C_{i+1,j}}^{NS}}/{d_{i,j}}$
(15)
where
$d_{i,j}=\big{(}C_{i+1,j}^{NS}+C_{i,j+1}^{EW}+C_{i,j}^{NS}+C_{i,j}^{EW}\big{)}$
represents the diagonal of the matrix $\mathbf{A}$. The (11), using the
diagonal preconditioner, can be written as:
$\small\begin{array}[]{c}-\beta_{i,j}^{S}D_{i-1,j}-\alpha_{i,j}^{W}D_{i,j-1}+D_{i,j}-\alpha_{i,j}^{E}D_{i,j+1}+\hfill-\beta_{i,j}^{N}D_{i+1,j}=\bar{b}_{i,j}.\end{array}$
(16)
with $\bar{b}_{i,j}=-b_{i,j}/d_{i,j}$.
Starting from the observations (8) and (9) we proof that the diagonal
preconditioner does not work very well in some critical physical situations
involving curvilinear spherical coordinates.
###### Proposition 3.1.
In the geographical coordinate, if $\phi\rightarrow+{\frac{\pi}{2}}^{-}$,
$\Delta\lambda\rightarrow 0$, $\Delta\phi\rightarrow 0$ then the conditioning
number $\mu(\mathbf{M^{-1}A)}$ goes to $+\infty$.
Proof. In the geographical coordinate, i.e. when
$(i,j)\rightarrow(\lambda,\phi)$ and $(e_{1},e_{2})\rightarrow(r\cos\phi,r)$,
for $\phi\rightarrow+{\frac{\pi}{2}}^{-}$, $\Delta\lambda\rightarrow 0$ and
$\Delta\phi\rightarrow 0$, the functions $\alpha^{W}$ and $\alpha^{E}$ in (14)
go to -1/2 while $\beta^{N}$ and $\beta^{S}$ in (15) go to 0. Hence the limit
of matrix $\mathbf{M^{-1}A}$ is given by:
$\tiny\mathbf{A}^{\prime}=\left[\begin{array}[]{ccccc}1&-1/2&0&\ldots&0\\\
-1/2&1&-1/2&\ddots&0\\\ 0&\ddots&\ddots&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&-1/2\\\ 0&\ddots&\ddots&-1/2&1\\\
\end{array}\right].$ (17)
The eigenvalues of the matrix in (17) are:
$\lambda_{k}=1+\cos\bigg{(}\frac{k\pi}{n+1}\bigg{)}\quad k=1,...,n$ (18)
and then the condition number
$\mu_{2}(M^{-1}{A})=\lambda_{max}/\lambda_{min}\approx n^{2}/2$ (by using the
series expansion of $\cos x=1-x^{2}/2+o(x^{2})$). Moreover for
$\Delta\lambda\rightarrow 0$ and $\Delta\phi\rightarrow 0$ the size $n$ of the
matrix $\mathbf{A}$ goes to $+\infty$ and hence we obtain the thesis.
$\small\blacksquare$
By the proposition (3.1), for $n$ large and $\phi\rightarrow\pm\pi/2$, it is
preferable to adopt more suitable preconditioning techniques or a stategy
based on the local change of the coordinates at poles. In this paper we
propose an alternative approximate sparse inverse preconditioning AINV
techniques [7] for the linear system (11). AINV technique is a critical step
since the inverse of a sparse matrix is usually dense. The problem is how to
build a preconditoner that preserves the sparse structure. We introduce a
factored sparse approximate inverse FSAI preconditioner
$\mathbf{P}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z^{t}}}$ [8, 11], computed by
means conjugate-orthogonalization procedure. Specifically, we propose an “ad
hoc” method for computing an incomplete factorization of the inverse of the
matrix $\mathbf{T}\subset\mathbf{A}$, obtained by $\mathbf{A}$ taking only the
elements $a_{i,j}$ such that $|j-i|\leq 1$. The factorized sparse approximate
inverse of $\mathbf{T}$ is used as explicit preconditioner for (11). In the
following we give some remarks for the sparsity pattern selection S of our
inverse preconditioner $\mathbf{P}$.
###### Proposition 3.2.
If $\mathbf{T}$ is a tridiagonal, symmetric and diagonally dominant matrix,
with diagonal elements all positive $t_{k,k}>0,\ k=1,...,n$, then the
Cholesky’s factor $\mathbf{U}$ of the matrix $\mathbf{T}$ is again diagonally
dominant.
Proof. Since $\mathbf{T}$ is a tridiagonal matrix then $\mathbf{U}$ is a
bidiagonal matrix. Using the inductive method we proof that $\mathbf{U}$ is
diagonally dominant matrix. For $k=1$ is trivially, indeed by hypothesis we
know that
$|a_{1,1}|>|a_{1,2}|\Longleftrightarrow|u_{1,1}^{2}|>|u_{1,1}u_{1,2}|$, then
we obtain $|u_{1,1}|>|u_{1,2}|$. Moreover placed the thesis true for $k-1$
i.e. $|{{u_{k-1,k-1}}}|>|{{u_{k-1,k}}}|$ then by the following inequalities:
$\displaystyle|a_{k,k}|>|a_{k,k-1}|+|a_{k,k+1}|\Longleftrightarrow\qquad\quad\qquad\quad\qquad\quad\qquad\quad$
$\displaystyle|u_{k-1,k}^{2}+u_{k,k}^{2}|>|u_{k-1,k}u_{k-1,k-1}|+|u_{k,k}u_{k,k+1}|>u_{k-1,k}^{2}+|u_{k,k}u_{k,k+1}|.\qquad\quad$
(19)
subtracting the inequality (19) for $u_{k-1,k}^{2}$, then the thesis holds
also for k. $\small\blacksquare$
This result allows to prove the following proposition:
###### Proposition 3.3.
The inverse matrix $\mathbf{Z}$ of a bidiagonal and diagonally dominant matrix
$\mathbf{U}$ has column vectors $\mathbf{z}_{k},k=1,...n$ such that starting
from diagonal element $z_{k,k}$, they contain a finite sequence
$\\{z_{k-i,k}\\}_{i=0,...,k-1}$ strictly decreasing.
Proof. Applying a backward substitution procedure for solving the system of
equations $\mathbf{U}\mathbf{z}_{k}=\mathbf{e}_{k}$, we get:
$z_{k-i,k}=\left\\{\begin{array}[]{l}{1}/{u_{k,k}}\quad{if\ \
i=0}\\\\[14.22636pt] ({-1})^{i}/{u_{k,k}}\cdot\
{\displaystyle\prod_{r=1}^{i}}\big{(}{u_{k-r,k-r+1}}/{u_{k-r,k-r}}\big{)}\\\\[14.22636pt]
\qquad\qquad\qquad\qquad\qquad{if\ \ 0<i\leq k-1}.\end{array}\right.$ (20)
By means of the preposition (3.2) we obtain that $z_{k-i,k}>z_{k-i-1,k}$ with
$\small{0<i\leq k-1}$ and hence the thesis is proved. $\small\blacksquare$
The previous propositions (3.2) and (3.3) enable to select a sparsity pattern
S by the following scheme:
1. 1.
Consider the symmetric, diagonally dominant and triangular matrix
$\mathbf{T}$, obtained by $\mathbf{A}$ taking only the elements $a_{i,j}$ such
that $|j-i|\leq 1$
2. 2.
$\mathbf{T}=\mathbf{U}^{T}\mathbf{U}$ is diagonally dominant matrix.
Consequently its Cholesky factor $\mathbf{U}$ is diagonally dominant
(proposition (3.2) ).
3. 3.
$\mathbf{U}$ is a bidiagonal and diagonally dominant matrix.
$\mathbf{Z}=\mathbf{U}^{-1}$ has columns vector $\mathbf{z}_{k},\ k=1,...,n$
such that $z_{k-i,k}>z_{k-i-1,k}$ with $\small{0<i\leq k-1}$. (proposition
(3.3))
4. 4.
Fixed an upper bandwidth $q$, the entries $z_{i,j}$ with $j>i+q$ of
$\mathbf{Z}$ are considered negligible.
5. 5.
The preconditioner $\mathbf{P}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z^{t}}}$ is
built as:
$\tilde{z}_{i,j}=\left\\{\begin{array}[]{l}z_{i,j}\quad{if\ \ j\leq
i+q}\\\\[14.22636pt] 0\quad{if\ \ j>i+q}\end{array}\right.$ (21)
6. 6.
The sparse factor $\tilde{\mathbf{Z}}$ is computed by $T$-orthogonalization
procedure posing the sparsity pattern S=$\\{(i,j)\ /j>i+q\\}$
$\mathbf{T}$ is a diagonally dominant matrix then the incomplete inverse
factorization of $\mathbf{T}$ exists for any choice of the sparsity pattern S
on $\mathbf{Z}$ [8].
From computationally point of view, the $T$-orthogonalization procedure with
the sparsity pattern S is based on matrix-vector operations with computational
cost of $5(q+1)$ floating point operations. Moreover, for each column vector
$\tilde{\mathbf{z}}_{k}$ of $\tilde{\mathbf{Z}}$ we work only on its $q+1$
components $\tilde{z}_{k}[k-q],\tilde{z}_{k}[k-q+1],...,\tilde{z}_{k}[k]$ with
consequently global complexity of $5q(q+1)O(n)$.
## 4 Practical Considerations
In this section we give some practical details on the elliptic solver
implementation with FSAI preconditioner, on a generic GPU architecture. The
matrices $\mathbf{A}$, $\tilde{\mathbf{Z}}$ and $\tilde{\mathbf{Z}^{T}}$ are
stored with the special storage format Compressed Sparse Row (CSR). The FSAI
is performed in serial on the CPU and its building requires a negligible time
on total execution of the elliptic solver. We show the implementation of the
Algorithm 1 outlines on the GPUs [12, 13, 20].
Algorithm 1 FSAI-PCG solver
1: $k=0$; $\quad\mathbf{x}_{0}=D_{i,j}^{0}=2D_{i,j}^{t-1}$, the initial guess;
2: $\mathbf{r}_{0}=\mathbf{b}-\mathbf{A}\mathbf{x}_{0}$;
3:
${\mathbf{s}_{0}}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z^{t}}}{\mathbf{r}_{0}}$,
with ${\mathbf{P}}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z}^{t}}$ the FSAI
preconditioner;
4: $\mathbf{d}_{0}=\mathbf{s}_{0}$;
5: while $\big{(}{\|\mathbf{r}_{k}\|}/{\|\mathbf{b}\|}>\epsilon\ .and.\ k\leq
n\big{)}$ do
6: $\mathbf{q}_{k}=\mathbf{A}\mathbf{d}_{k}$;
$\quad\alpha_{k}={(\mathbf{s}_{k},\mathbf{r}_{k})}/({\mathbf{d}_{k},\mathbf{q}_{k}})$;
$\quad\mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha_{k}\mathbf{d}_{k}$;
7: ${\mathbf{r}_{k+1}}={\mathbf{r}_{k}}-\alpha_{k}{\mathbf{q}_{k}}$;
$\quad{\mathbf{s}_{k+1}}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z^{t}}}{\mathbf{r}_{k+1}}$;
$\quad\beta_{k}={({\mathbf{s}}_{k+1},\mathbf{r}_{k+1})}/{({\mathbf{s}}_{k},\mathbf{r}_{k})}$;
8: $\mathbf{d}_{k+1}={\mathbf{r}}_{k+1}+\beta_{k}\mathbf{d}_{k}$; $\quad
k=k+1$;
9: end while
In details, our solver is implemented by means of the CUDA language with the
auxiliary linear algebra libraries CUBLAS, for the “dot product ” (xDOT),
“combined scalar multiplication plus vector addition” (xAXPY), “euclidean
norm” (xNRM2) and “vector by a constant scaling” (xSCAL) operations, and
CUSPARSE for the sparse matrix-vector operations in the PCG solver. The linear
algebra scientific libraries are extremely helpful to easily implement a
software on the GPU architecture. A “by-hand” implementation (see Figure 2 and
3) of the solver without the library features in reported as a tedious GPU
programming example. In this type of coding, the manually configuration of the
grid of thread blocks is necessary. For example if we use the TESLA S2050 the
variables warpSize and maxGridSize (respectively at lines 2 and 3) have to be
assigned. In details,warpSize indicates the number of threads (32) in a warp,
which is a sub-division use in the hardware implementation to coalesce memory
access and instruction dispatch; maxGridSize is the maximum number of
simultaneous blocks (65535). Furthermore, warpCount (at line 4) represents the
number of warps and it depends on the dimension n of the problem. In the end,
variables threadCountPerBlock and blockCount (respectively at lines 5 and 6)
are the parameter used for setting the grid and block configuration (lines 7
and 8). For example, if n = 10000 is the size of a vector, then
threadCountPerBlock = 32 and blockCount = 313.
Figure 2: Grid and block configuration.
The figure 3 shows the matrix-vector product, with the matrix stored in CSR
format. We observe that this implementation requires a large amount of kernel
functions, invoked by the “host” (CPU) and executed on the “device” (GPU).
Figure 3: Naive GPU implementation of a function for the matrix-vector product
of a matrix in CSR format.
In the following, we will show how to implement the Algorithm 1 outlines by
using library features. In order to use the CUBLAS library it is necessary
inizialize it by means of the following instructions:
cublasStatus stat; cublasInit();
For the use of the CUSPARSE library two steps are necessary. The first one
consists, as follow, in the library initialization:
cusparseHandle_t handle=0; cusparseCreate(&handle);
moreover, it is recalled the creation and setup of a matrix descriptor:
cusparseMatDescr_t descra=0; cusparseCreateMatDescr(&descra);
cusparseSetMatType(descra,CUSPARSE_MATRIX_TYPE_GENERAL);
cusparseSetMatIndexBase(descra,CUSPARSE_INDEX_BASE_ZERO);
The library avoids to configure the grid of the thread blocks and it allows to
write codes in a very fast way. For example, at line 6 of the Algorithm 1 the
computation of $\mathbf{q}_{k}=\mathbf{Ad}_{k}$ is required, and this
operation can be made simply by calling the CUSPARSE routine cusparseScsrmv(),
that performs the operation $\mathbf{q}=a\mathbf{A∗d}+b\mathbf{q}$ as follows:
cusparseScsrmv(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, n, n, a, descra, A,
start, j, d, b, q);
In our context, A, j and start represent the symmetric positive-definite
matrix $\mathbf{A}$, stored in the CSR format. More precisely, the vector A
denotes the non-zero elements of the matrix $\mathbf{A}$, j is the vector that
stores the column indexes of the non-zero elements, the vector start denotes,
for each row of the matrix, the address of the first non-zero element and n
represents the row and columns number of the square matrix $\mathbf{A}$. The
constants $a$ and $b$ are assigned to $1.0$ and $0.0$ respectively. Moreover
it happens that at line 6 of the Algorithm 1, the computation of
${(\mathbf{s}_{k},\mathbf{r}_{k})}$ is performed by means the CUBLAS routine
for the dot product:
alfa_num = cublasSdot(n, s, INCREMENT_S, r, INCREMENT_R); The constants
INCREMENT$\\_$S and INCREMENT$\\_$R are both assigned to $1$. Last operation
of line 6 in Algorithm 1, is the
$\mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha_{k}\mathbf{d}_{k}$ for updating the
solution and it is implemented by calling the CUBLAS routine for the saxpy
operation:
cublasSaxpy(n, alfa, d, INCREMENT_D, x, INCREMENT_X); The constants
INCREMENT$\\_$D and INCREMENT$\\_$X are both assigned to $1$. In addition, the
computation of
${\mathbf{s_{k+1}}}=\tilde{\mathbf{Z}}\tilde{\mathbf{Z^{t}}}{\mathbf{r_{k+1}}}$
at line 7 is the preconditioning step of the linear system (11) and it is
computed by means of two matrix-vector operations performed as:
cusparseScsrmv(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, n, n, a, descra, Z_t,
start_Z_t, j_Z_t, r, b, zt); cusparseScsrmv(handle,
CUSPARSE_OPERATION_NON_TRANSPOSE, n, n, a, descra, Z_t, start_Z, j_Z, zt, b,
z); In details, in the first call of cusparseScsrmv(),
${\mathbf{zt}}=\tilde{\mathbf{Z^{t}}}{\mathbf{r_{k+1}}}$ and
${\mathbf{s_{k+1}}}=\tilde{\mathbf{Z}}{\mathbf{zt}}$ are computed. We have
outlined just few of computational operations because the other will be
performed in the same way. The parameters handle,
CUSPARSE_OPERATION_NON_TRANSPOSE and descra are discussed in the NVIDIA report
[17] in more detailed way. In summary, we highlight how the use of the
standard library, designed for the GPU architecture, allow to optimize the
computational oceanographic simulation model.
## 5 Numerical Experiments
In this section we focus on the important numerical issues of our elliptic
solver implemented with GPU architecture in single precision. The solver is
tested on three grid size resolutions of the NEMO-OPA ocean model (Table 1).
Matrix Name | Size | Matrix non-zeros elements
---|---|---
ORCA-2 | $180\times 149$ | $133800$
ORCA-05 | $751\times 510$ | $1837528$
ORCA-025 | $1442\times 1021$ | $7359366$
Table 1: NEMO-OPA grid resolutions.
In the Table 2, we compare the performance in terms of PCG iterations of the
proposed inverse bandwidth preconditioner $\mathbf{P}$ respect to
$\mathbf{P}^{-1}$, that is the diagonal NEMO-OPA preconditioner. We fix an
accuracy of $\epsilon=10^{-6}$ on the relative residue
$r=||\mathbf{Ax}-\mathbf{b}||/||\mathbf{b}||$ on the linear system solution.
The experiments are carried out in the case of well-conditioned $\mathbf{A}$
matrix, corresponding to the geographical case of $\phi\approx 0$ and in the
case of ill-conditioned $\mathbf{A}$ with $\phi\approx\pi/2$. We can observe
as in the worst case with $n$ large and $\mathbf{A}$ ill-conditioned the PCG
with $\mathbf{P}^{-1}$ has a very slow convergence with a huge number of
iterations to reach the fixed accuracy. The experiment, reported in the Table
2, highlights the poor performance of the $\mathbf{P}^{-1}$ for solving the
Laplace elliptic problem (11) within OPA-NEMO.
$\mathbf{A}$ Dimension | $\mathbf{P}$ $(\phi\approx 0)$ | $\mathbf{P}^{-1}$ $(\phi\approx 0)$ | $\mathbf{P}$ $(\phi\approx\pi/2)$ | $\mathbf{P}^{-1}$ $(\phi\approx\pi/2)$
---|---|---|---|---
ORCA-2 | 271 | 460 | 8725 | 26820
ORCA-05 | 1128 | 1593 | 22447 | 86280
ORCA-025 | 2458 | 3066 | 28513 | 139742
Table 2: Comparison between $\mathbf{P}$ and $\mathbf{P}^{-1}$ in terms of
Number of Iterations of the PCG in the case $\mathbf{A}$ is well-conditioned
($\phi\approx 0$) and $\mathbf{A}$ ill-conditioned ($\phi\approx\pi/2$),
varying the problem dimensions.
In the following, we show the performance in terms of PCG iterations of
$\mathbf{P}$ respect to the AINV Bridson Class preconditioners, that believe
to CUSP library. To be more specific, let us consider the $\mathbf{P}_{B1}$
and $\mathbf{P}_{B2}$ Bridson’s preconditioners, obtained by means of the
$A$-orthogonalization method. The first is given by posing a (fixed) drop
tolerance and by ignoring the elements below the fixed tolerance [9] and in
the second one is predetermined the number of non-zeros elements on each its
row. [10].
Figure 4: Comparison between $\mathbf{P}$, $\mathbf{P}_{B1}$ and
$\mathbf{P}_{B2}$ in terms of Number of Iterations of the PCG ($y-$axis) when
$\mathbf{A}$ is well-conditioned, varying the problem dimensions ($x-$axis)
Figure 5: Comparison between $\mathbf{P}$, $\mathbf{P}_{B1}$ and
$\mathbf{P}_{B2}$ in terms of Number of Iterations of the PCG ($y-$axis) when
$\mathbf{A}$ is ill-conditioned, varying the problem dimensions ($x-$axis)
The required accuracy on the solution is fixed to $\epsilon=10^{-6}$ on the
relative residue. In Figure 5 we report the PCG iterations of $\mathbf{P}$,
$\mathbf{P}_{B1}$ and $\mathbf{P}_{B2}$ in the case of the matrix $\mathbf{A}$
well-conditioned $(\phi\approx 0)$. In Figure 5 we present the case of the
ill-conditioned $(\phi\approx\pi/2)$ matrix $\mathbf{A}$. The numerical
results show as the number of iterations of the solver $\mathbf{P}$ is
comparable to $\mathbf{P}_{B1}$ and $\mathbf{P}_{B2}$ when the dimensions of
the problem are small or middle. Furthermore, it is strongly indicated to use
$\mathbf{P}$ with a huge problem dimension. We test $\mathbf{P}$,
$\mathbf{P^{-1}}$, $\mathbf{P}_{B1}$ and $\mathbf{P}_{B2}$ on the sparse
matrix NOS6 of the Market Matrix database [22] by setting the required
accuracy on the computed solution to $\epsilon=10^{-6}$ and the band q of the
preconditioner to 4. This sparse matrix is obtained in the Lanczos algorithm
with partial re-orthogonalization Finite difference approximation to Poisson’s
equation in an L-shaped region, mixed boundary conditions.
Figure 6: PCG iterations. P is the proposed preconditioner, P_d is
$\mathbf{P}^{-1}$, P_B1 P_B2 are $\mathbf{P}_{B1}$ and $\mathbf{P}_{B2}$
The figure 6 shows as $\mathbf{P}$ achieves the best performance in terms of
iterations.
Finally, we test the elliptic solver implementation on GPU architecture. The
numerical experiments are carried out on an “NVIDIA TESLA S2050” card, based
on the “FERMI GPU”. The “TESLA S2050” consists of 4 GPGPUs, each of which with
3GB of RAM memory and 448 processing cores working at 1.55 GHz. All runs are
given on 1 GPU device. We have adopted CUDA release 4.0, provided by NVIDIA as
a GPGPU environment and the numerical code is implemented by using the single
precision arithmetic.
As described in the previous sections by using scientific computing library it
is not necessary manually setting up the block and grid configuration on the
memory device. The number of blocks required to store the elliptic solver
input data (in CSR format) do not have to exceed the maximum sizes of each
dimension of a GPU grid device. Schematic results of GPU memory utilization
for ocean model resolutions are presented in the Table 3. Observe that in our
numerical experiments we do not fill the memory of the TESLA GPU and the
simulations run also on cheaper or older boards, as for example the Quadro
4700FX. Generally, it is possible to grow the grid dimensions of the ocean
model according to the memory capacity of the available GPU.
Matrix Name | Non-zeros Elem. | Mem. Occ.
---|---|---
ORCA-2 | $133800$ | $4$MB
ORCA-05 | $1837528$ | $37$MB
ORCA-025 | $7359366$ | $135$MB
Table 3: Matrix memory occupancy. Mem Occ. is the full memory allocated memory
on the GPU.
The elliptic solver requires a large amount of Sparse-Matrix Vector
(cusparseCsrmv) multiplications, vector reductions and other vector operation
to be performed. CPU version is implemented in ANSI C executed in serial on a
2.4GHz “Intel Xeon E5620” CPU, with 12MB of cache memory. Serial and GPU
versions are in single precision. We test the performance of the solver in
terms of Floating Point Operations (FLOPS). The performance of the numerical
experiments (reported in the Figure 8) are given in the case of $A$ ill-
conditioned matrix ($\phi\approx\pi/2$). We count an average of the iterations
of solver and the complexity of all linear algebra operations involved in both
serial and parallel implementations. The ”GPU solver“ (blue) and ”CPU solver“
(orange) curves represent the GFLOPS of the solver, respectively, for the CPU
and GPU versions.
Figure 7: Sparse-Matrix Vector multiplications speed-up.
Figure 8: CPU and GPU comparison of the solver in terms of GFLOPS.
The main recalled computational kernels in the solver are the Sparse-Matrix
Vector. From the Figure 8 we highlight the improvement in terms of GFLOPs
speed-up by replacing gemv() with cusparseCsrmv() function. These results
prove that, increasing the model grid resolution, it is possible to exploit
the computational power of the GPUs. In details, the GPU solver implementation
in the ORCA-025 configuration has a peak performance of 87 GFLOPS respect to
1,43 GFLOPS of the CPU version.
## 6 Conclusions
The ocean modelling is a challenging application where expensive computational
kernels are fundamental tools to investigate the physics of the ocean and the
climate change. In a lot of applications, the elliptic Laplace equations are
used in the complex mathematical models; they represents critical
computational points since the convergence of the numerical solvers to a
solution, within a reasonable number of iterations, it is not always
guaranteed. In our case, this happens to the preconditioning technique of the
OPA-NEMO ocean model, for which we prove to be inefficient and inaccurate. In
this paper, we have proposed a new inverse preconditioner based on the FSAI
method that shows better results respect to the OPA-NEMO diagonal one and to
others of the Bridson class. Moreover, an important contribute is given by an
innovative approach for parallelizing the elliptic solver on the Graphical
Processing Units (GPU) by means of the scientific computing libraries. The
library based implementation of the computing codes allows to optimize oceanic
framework reducing the simulation times and to develop computational solvers
easy-to-implement.
## 7 Acknowledgments
The computing resources and the related technical support used for this work
have been provided by CRESCO/ENEAGRID “High Performance Computing
infrastructure” and its staff with particular acknowledgments to the
researcher Marta Chinnici; CRESCO/ENEAGRID is funded by ENEA, the “Italian
National Agency for New Technologies, Energy and Sustainable Economic
Development” and by national and European research programs. See
www.cresco.enea.it for more information.
## References
* [1] Roullet, G. and G. Madec,, _Salt conservation, free surface, and varying levels : a new formulation for ocean general circulation models_ , J. Geophys. Res, 105, 2000.
* [2] Gries M., Boning C., Bryan F.O., et al. _Developments in ocean climate modeling_ , Ocean modeling Volume 2.
* [3] Madec G, _NEMO-OPA ocean engine_ , Institute Pierre-Simon Laplace (IPSL), France, 2012.
* [4] Higdon R._Numerical modeling of ocean circulation_ , Acta Numerica, Cambridge University Press.
* [5] Epicoco I., S. Mocavero, E. Scoccimarro and G. Aloisio. _ORCA025: Performance Analysis on Scalar Architecture_ , CMCC Research Paper N. 50.
* [6] Arakawa A., F.Mesinger,_Numerical Methods used in Atmospheric Models_ , VOL 1 Garp Publication Series NO 17, France (1976).
* [7] Benzi M., _Preconditioning techniques for large linear systems: A survey_ , J. Comput. Phys., 182, 2002.
* [8] Benzi M., C. D. Meyer, and M. Tuma, _A sparse approximate inverse preconditioner for the conjugate gradient method._ , SIAM J. Sci. Comput. 17, 1995.
* [9] Bridson R. and W.-P. Tang, _Refining an approximate inverse_ ”, J. Comput. Appl. Math. 22, 2000.
* [10] Lin C.J. and J. J. More, _Incomplete Cholesky factorizations with limited memory_ , SIAM J.Sci. Comput., 21, 1999.
* [11] Benzi M. _An explicit preconditioner for the conjugate gradient method_ Proceedings of the Cornelius Lanczos International Centenary Conference, J. D. Brown et al., eds., Society of Industrial and Applied Mathematics, Philadelphia.
* [12] Bell N. and M. Garland,_Efficient sparse matrix-vector multiplication on CUDA_ , NVIDIA Technical Report NVR-2008-004, NVIDIA Corporation, 2008.
* [13] Farina R., S. Cuomo and P. De Michele,_A CUBLAS-CUDA Implementation of PCG Method of an Ocean Circulation Model_ , AIP Conf. Proc. September 14, Volume 1389, 2011.
* [14] Dongarra J. et al.,_Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition_ , SIAM, Philadelphia, PA, 1994.
* [15] NVIDIA, _NVIDIA CUDA programming guide_ http://developer.download.nvidia.com, Nvidia Technical Report, 2012.
* [16] NVIDIA,_CUBLAS Library_ , http://developer.download.nvidia.com, Nvidia Technical Report, 2007.
* [17] NVIDIA,_CUSPARSE Library_ , http://developer.download.nvidia.com, Nvidia Technical Report, 2012.
* [18] _Cusp Library_ , http://code.google.com/p/cusp-library/.
* [19] Farina R., S. Cuomo and P. De Michele _An Inverse Preconditioner for a Free Surface Ocean Circulation Model_ , submitted to AIP Conf. Proc. July 10, 2012.
* [20] Farina R., S. Cuomo, P. De Michele and M. Chinnici. _Inverse Preconditioning Techniques on a GPUs Architecture in Global Ocean Models_ , Euro Siam proceedings 2011.
* [21] Duff I.S., A.M. Erisman, C.W. Gear, and J.K. Reid, _Sparsity structure and Gaussian elimination, SIGNUM Newsl. 23, 2_ (1988).
* [22] Market Matrix web site, _http://math.nist.gov/MatrixMarket/data/Harwell-Boeing/lanpro/nos6.html_ (2012).
* [23] Dukowicz, J.K., Smith, R.D., Malone, R.C., _A reformulation and implementation of the Bryan Cox Semtner ocean model on the connection machine,_ Journal of Atmospheric and Oceanic Technology 10, 195-208.
* [24] Webb, D.J., 1996. _An ocean model code for array processor computers_. Computers and Geophysics 22.
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|
arxiv-papers
| 2012-10-05T22:08:28 |
2024-09-04T02:49:36.053798
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Salvatore Cuomo, Pasquale De Michele, Raffaele Farina and Marta\n Chinnici",
"submitter": "Salvatore Cuomo",
"url": "https://arxiv.org/abs/1210.1878"
}
|
1210.1912
|
yen@math.fju.edu.tw (C.C. Yen)
# Self-gravitational force calculation of infinitesimally thin gaseous disks
C.C. Yen 1,2 R.E. Taam 2,5 Ken H.C. Yeh 2,4 and K.C. Jea 1,3
11affiliationmark: Department of Mathematics, Fu Jen Catholic University, New
Taipei City, Taiwan.
22affiliationmark: Institute of Astronomy & Astrophysics, Academia Sinica,
Taipei, Taiwan.
33affiliationmark: Graduate Institute of Applied Science and Engineering, Fu
Jen Catholic University, New Taipei City, Taiwan.
44affiliationmark: Department of Physics and Astronomy, University of British
Columbia, Vancouver, Canada.
55affiliationmark: Department of Physics and Astronomy, Northwestern
University, 2145 Sheridan Road, Evanston, IL 60208
###### Abstract
A thin gaseous disk has often been investigated in the context of various
phenomena in galaxies, which point to the existence of starburst rings and
dense circumnuclear molecular disks. The effect of self-gravity of the gas in
the $2D$ disk can be important in confronting observations and numerical
simulations in detail. For use in such applications, a new method for the
calculation of the gravitational force of a $2D$ disk is presented. Instead of
solving the complete potential function problem, we calculate the force in
infinite planes in Cartesian and polar coordinates by a reproducing kernel
method. Under the limitation of a $2D$ disk, we specifically represent the
force as a double summation of a convolution of the surface density and a
fundamental kernel and employ a fast Fourier transform technique. In this
method, the entire computational complexity can be reduced from $O(N^{2}\times
N^{2})$ to $O(N^{2}(\log_{2}N)^{2})$, where $N$ is the number of zones in one
dimension. This approach does not require softening. The proposed method is
similar to a spectral method, but without the necessity of imposing a periodic
boundary condition. We further show this approach is of near second order
accuracy for a smooth surface density in a Cartesian coordinate system.
###### keywords:
Self-gravitating force, infinitesimally thin disk, fast Fourier transform,
Poisson equation, reproducing kernel
52B10, 65D18, 68U05, 68U07
## 1 Introduction
The potential $\Phi$ of a given distribution of density $\rho$ in
${\mathbb{R}}^{3}$ satisfies the Poisson equation,
$\displaystyle\Delta\Phi(\boldsymbol{x})=4\pi
G\rho(\boldsymbol{x})=f(\boldsymbol{x}),\quad\boldsymbol{x}\in{\mathbb{R}}^{3},$
(1)
where $G$ is the gravitational constant and $\boldsymbol{x}=(x,y,z)$ is the
position. Without loss of generality, we may assume that the gravitational
constant $G=1$. Provided that the density profile has a continuous second
derivative with respect to the spatial coordinates, the potential is smooth.
In this situation, the numerical approach for solving the potential via (1) by
the finite difference method is adopted. Artificial boundary conditions are
imposed in the numerical approach for solving (1) because the boundary
condition is
$\displaystyle\lim_{|\boldsymbol{x}|\to\infty}\Phi(\boldsymbol{x})=0.$ (2)
The Poisson equation is intrinsically 3-dimensional, and the calculation of
the potential can be computationally prohibitive. A possible solution to
reduce the computation time is to apply the multigrid method [11, 6], but the
computational complexity is $O(N^{3})$, where $N$ is the number of zones in
one dimension.
The solution of (1) can be represented in terms of the fundamental solution,
$\displaystyle\frac{1}{4\pi}{\cal K}(\boldsymbol{x})$, where
$\displaystyle{\cal K}(\boldsymbol{x})=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}},$
as
$\displaystyle\Phi(x,y,z)=-\int\\!\\!\\!\int\\!\\!\\!\int{\cal
K}(\bar{x}-x,\bar{y}-y,\bar{z}-z)f(\bar{x},\bar{y},\bar{z})d\bar{x}d\bar{y}d\bar{z}.$
(3)
The above formula is preferable to (1) when the density is not smooth. The
potential can be solved via the integral equation in (3). Spectral methods are
a common method of choice and a review article has recently been written by
Shen and Wang [8], describing work on the analysis and application of these
methods in unbounded domains. The difficulties encountered in the numerical
approach for solving (1) or (3) are related to the extent of the domain
${\mathbb{R}}^{3}$ and the density which can be singular.
In this paper, we consider the density represented by
$\displaystyle\rho(\boldsymbol{x})=\sigma(x,y)\delta(z),$ (4)
where $\sigma(x,y)$ is so-called surface density equal to
$\displaystyle\sigma(x,y)=\int\rho(\boldsymbol{x})dz.$ (5)
We restrict our attention to calculating the forces directly for the surface
density of compact supports.
For an infinitesimally thin gaseous disk, the multigrid method, which is
intrinsically suited for 3D problems, cannot be reduced for the two
dimensional problem we consider in this paper. The spectral method using
Fourier basis functions on a two dimensional space artificially imposes the
assumption of periodic boundary conditions. This is not realistic for the long
range gravitational force calculations. A direct method without the periodic
assumption requires a softening parameter technqiue, but the accuracy is
reduced simultaneously. A method is proposed which is of linear complexity,
without artificial boundary conditions, and near second order accuracy.
This paper is organized as follows. The framework and assumption are presented
in Section 2. Sections 3 and 4 describe the numerical methods for Cartesian
and polar coordinates, respectively. Section 5 demonstrates the order of
accuracy of the proposed methods as verified by a family of finite disks
(e.g., $D_{2}$ disk; [7]) and a disk of a pair of spirals. A comparison with
several existing methods is also presented in that section. Finally, the
discussion and conclusion are given in section 6.
## 2 Framework and assumption
The evolution of a thin disk is of fundamental interest in astrophysics and
the effect of the self-gravity of gas therein may be important in modeling
observed phenomena in detail. This paper presents a numerical method for
solving the self-gravitating forces in Cartesian and polar coordinates, which
can be used in modeling infinitesimally thin disks in galaxies and
protostellar systems [10].
The self-gravitating force can be determined by taking derivatives of the
potential function which satisfies the Poisson equation in (1). However, the
calculation of the potential (1) is on an unbounded domain and the solution in
a finite region requires the imposition of artificial boundary conditions. The
solution of Poisson’s equation with variable coefficients and Dirichlet
boundary conditions on a two dimensional irregular domain is one of second
order[2].
Let us confine our attention to the density in an infinitesimally thin disk as
defined in (4) and (5). Here, we focus on the self-gravitating force
computation. The approach presented in this paper is to directly calculate the
self-gravitating force by expressing the potential function as a type of a
convolution of the surface density and the fundamental kernel and taking the
derivative of the potential function. This approach is similar to the spectral
method, but less restrictive. Trigonometric bases functions and the artificial
periodic boundary conditions are used for the spectral method, but are not
required in the proposed approach here.
A uniform grid discretization in Cartesian coordinates and a linear
approximation of the surface density on each cell are used to reduce the
computational time and increase the accuracy of the numerical solution,
respectively. Similarly, for polar coordinates, a logarithmic grid
discretization is used instead of a uniform grid discretization. Based on the
discretization and approximation, the self-gravitating force is written as a
convolution form of double summations. It is known that the calculation of
convolution form can be accelerated by the use of a fast Fourier transform
(FFT), see Appendix B. Employing the FFT, the computational complexity is
reduced from $O(N^{4})$ to $O((N\log_{2}N)^{2})$, where $N$ is the number of
zones in one direction. The linear approximation also leads to an order of
convergence that is near second order $O(h^{2})$, where the size of a zone
$h=O(1/N)$.
## 3 Self-gravitating force calculation in Cartesian coordinates
In this section, we describe the method in detail. The potential function
$\Phi$ of (1) can be expressed as
$\displaystyle\Phi(x,y,z)=-\int\\!\\!\\!\int\\!\\!\\!\int{\cal
K}(\bar{x}-x,\bar{y}-y,\bar{z}-z)\rho(\bar{x},\bar{y},\bar{z})d\bar{x}d\bar{y}d\bar{z},$
where $\displaystyle{\cal K}(x,y,z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}.$ By
(4), the forces on the disk in the $x$-direction and the $y$-direction become
$\displaystyle\frac{\partial}{\partial
x}\Phi(x,y,0)=\int\\!\\!\\!\int\frac{\partial}{\partial x}{\cal
K}(\bar{x}-x,\bar{y}-y,0)\sigma(\bar{x},\bar{y})d\bar{x}d\bar{y}$ (6)
and
$\displaystyle\frac{\partial}{\partial
y}\Phi(x,y,0)=\int\\!\\!\\!\int\frac{\partial}{\partial y}{\cal
K}(\bar{x}-x,\bar{y}-y,0)\sigma(\bar{x},\bar{y})d\bar{x}d\bar{y}.$ (7)
We calculate (6) and (7) by a numerical approach. Here, we focus on the
derivation of the force calculation in the $x$-direction. The force in the
$y$-direction is obtained in a similar manner (see Appendix A).
Since the support of the surface density is compact, contained in a domain
$D=[-M,M]\times[-M,M]$ for some number $M>0$, we discretize the region
uniformly as follows. Given a positive integer $N$, we define $\Delta x=2M/N$,
$\Delta y=\Delta x$, $x_{i+1/2}=-M+i\Delta x$, $y_{j+1/2}=-M+j\Delta y$, where
$i,j=0,\ldots,N$. We further define the center of the cell
$D_{ij}=[x_{i-1/2},x_{i+1/2}]\times[y_{j-1/2},y_{j+1/2}]$ as
$x_{i}=(x_{i-1/2}+x_{i+1/2})/2$ and $y_{j}=(y_{j-1/2}+y_{j+1/2})/2$, where
$i,j=1,\ldots,N$. Hence, the domain $D$ is discretized into the $N^{2}$ cells.
The forces in the $x$-direction and the $y$-direction at the center of cells
are
$\displaystyle F^{x}_{i,j}=\frac{\partial}{\partial
x}\Phi(x_{i},y_{j},0),\mbox{\quad and
\quad}F^{y}_{i,j}=\frac{\partial}{\partial y}\Phi(x_{i},y_{j},0).$ (8)
The surface density $\sigma$ on $D_{i,j}$ in (6) is linearly approximated by
$\displaystyle\sigma(\bar{x},\bar{y})\approx\sigma_{i,j}+\delta^{x}_{i,j}(\bar{x}-x_{i})+\delta^{y}_{i,j}(\bar{y}-y_{j}),$
(9)
where $\sigma_{i,j}=\sigma(x_{i},y_{j})$ and
$\delta^{x}_{i,j}=\sigma_{x}(x_{i},y_{j})$ and
$\delta^{y}_{i,j}=\sigma_{y}(x_{i},y_{j})$ are constant in the cell $D_{i,j}$.
The error of the discretization is
$O((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2})$. Equation (9) is the Taylor
expansion in two dimensions. If a higher order accuracy is required,
additional terms in the Taylor expansion can be considered.
Let
$\displaystyle{\cal
K}^{x,0}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y},$
(10) $\displaystyle{\cal
K}^{x,x}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})(\bar{x}-x_{i^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y},$
(11)
and
$\displaystyle{\cal
K}^{x,y}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})(\bar{y}-y_{j^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}.$
(12)
If the surface density is approximated by (9) then the force in the
$x$-direction defined by (8) and (6) can also be approximated by
$\displaystyle F^{x}_{i,j}$ $\displaystyle\approx$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{\partial}{\partial
x}{\cal
K}(\bar{x}-x_{i},\bar{y}-y_{j},0)\left(\sigma_{i^{\prime},j^{\prime}}+\delta^{x}_{i^{\prime},j^{\prime}}(\bar{x}-x_{i^{\prime}})+\delta^{y}_{i^{\prime},j^{\prime}}(\bar{y}-y_{j^{\prime}})\right)d\bar{x}d\bar{y}$
$\displaystyle:=$ $\displaystyle F^{x,0}_{i,j}+F^{x,x}_{i,j}+F^{x,y}_{i,j},$
where
$\displaystyle F^{x,0}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}{\cal
K}^{x,0}_{i-i^{\prime},j-j^{\prime}},$ (13) $\displaystyle F^{x,x}_{i,j}$
$\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{x}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})(\bar{x}-x_{i^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{x}_{i^{\prime},j^{\prime}}{\cal
K}^{x,x}_{i-i^{\prime},j-j^{\prime}},$ (14) $\displaystyle F^{x,y}_{i,j}$
$\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{y}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{x}-x_{i})(\bar{y}-y_{j^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{y}_{i^{\prime},j^{\prime}}{\cal
K}^{x,y}_{i-i^{\prime},j-j^{\prime}}.$ (15)
The evaluation of (10), (11) and (12) can be obtained with the help of the
following simple integrals,
$\displaystyle\int\\!\\!\\!\int\frac{x}{(x^{2}+y^{2})^{3/2}}dxdy=-\ln(y+\sqrt{x^{2}+y^{2}})+C,\quad\quad\>\int\\!\\!\\!\int\frac{xy}{(x^{2}+y^{2})^{3/2}}dxdy=-\sqrt{x^{2}+y^{2}}+C,$
$\displaystyle\int\\!\\!\\!\int\frac{x^{2}}{(x^{2}+y^{2})^{3/2}}dxdy=y\ln(x+\sqrt{x^{2}+y^{2}})+C,\quad\int\\!\\!\\!\int\frac{1}{(x^{2}+y^{2})^{3/2}}dxdy=-\frac{\sqrt{x^{2}+y^{2}}}{xy}+C.$
The value ${\cal K}^{x,0}_{i-i^{\prime},j-j^{\prime}}$ is equal to
$\displaystyle{\cal
K}^{x,0}_{i-i^{\prime},j-j^{\prime}}=-\ln\left((\bar{y}-y_{j})+\sqrt{(\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right.,$
(16)
where the notation $\displaystyle g(x)\left|{}^{b}_{a}\right.=g(b)-g(a)$. The
calculation of ${\cal K}^{x,x}_{i-i^{\prime},j-j^{\prime}}$ and ${\cal
K}^{x,y}_{i-i^{\prime},j-j^{\prime}}$ are split into two parts by the identity
$(\bar{x}-x_{i})(\bar{x}-x_{i^{\prime}})=(\bar{x}-x_{i})^{2}+(\bar{x}-x_{i})(x_{i}-x_{i^{\prime}})$,
and
$(\bar{x}-x_{i})(\bar{y}-y_{j^{\prime}})=(\bar{x}-x_{i})(\bar{y}-y_{j})+(\bar{x}-x_{i})(y_{j}-y_{j^{\prime}})$,
respectively. It follows that
$\displaystyle{\cal K}^{x,x}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle=$
$\displaystyle(x_{i}-x_{i^{\prime}}){\cal
K}^{x,0}_{i-i^{\prime},j-j^{\prime}}+\left((\bar{y}-y_{j})\ln(\bar{x}-x_{i}+\sqrt{(\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2})}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right.,$
$\displaystyle{\cal K}^{x,y}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle=$
$\displaystyle(y_{j}-y_{j^{\prime}}){\cal
K}^{x,0}_{i-i^{\prime},j-j^{\prime}}+\left(-\sqrt{(\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right..$
It is worth noting that the form of $F^{x,0}_{i,j}$, $F^{x,x}_{i,j}$, and
$F^{x,y}_{i,j}$ in (13)-(15) are a type of convolution. It is known that the
computational complexity of a convolution of two vectors can be reduced to
$O(N\log_{2}N)$ with the help of FFT (see Appendix B). It implies that the
complexity of this method is $O(N^{2}(\log_{2}N)^{2})$.
## 4 Self-gravitating force calculation in polar coordinates
A similar approach is used to develop the method for polar coordinates in this
section. The singular integral disappears, but the high order of accuracy is
not attained because there is no explicit form for the integral of an elliptic
function. The method in polar coordinates is described in detail below.
The potential function $\Phi$ of (1) under the assumption $G=1$ in cylindrical
coordinate can be expressed as
$\displaystyle\Phi(r,\theta,z)=-\int\\!\\!\\!\int\\!\\!\\!\int{\cal
K}(\bar{r},r,\bar{\theta},\theta,\bar{z}-z)\rho(\bar{r},\bar{\theta},\bar{z})\bar{r}d\bar{r}d\bar{\theta}d\bar{z},$
where $\displaystyle{\cal
K}(\bar{r},r,\bar{\theta},\theta,z)=\frac{1}{\sqrt{{\bar{r}}^{2}-2\bar{r}r\cos(\bar{\theta}-\theta)+r^{2}+z^{2}}}.$
By (4), the forces on the disk in $r$-direction and $\theta$-direction become
$\displaystyle\frac{\partial}{\partial
r}\Phi(r,\theta,0)=\int\\!\\!\\!\int\frac{\partial}{\partial r}{\cal
K}(\bar{r},r,\bar{\theta},\theta,0)\sigma(\bar{r},\bar{\theta})\bar{r}d\bar{r}d\bar{\theta}$
(17)
and
$\displaystyle\frac{1}{r}\frac{\partial}{\partial\theta}\Phi(r,\theta,0)=\frac{1}{r}\int\\!\\!\\!\int\frac{\partial}{\partial\theta}{\cal
K}(\bar{r},r,\bar{\theta},\theta,0)\sigma(\bar{r},\bar{\theta})\bar{r}d\bar{r}d\bar{\theta}.$
(18)
We calculate (17) and (18) by a numerical approach.
Since the support of the surface density is compact, contained in a region
${\cal R}=[0,M]\times[0,2\pi]$ for some number $M>0$, we discretize the radial
region in logarithmic form and the azimuthal region uniformly as follows.
Given a positive integer $N$, we define $\Delta\theta=2\pi/N$,
$0<\beta_{0}<1$, $\beta=\beta_{0}(1-\Delta\theta)$, $r_{i+1/2}=\beta^{N-i}M$,
$\theta_{j+1/2}=j\Delta\theta$, $i,j=0,\ldots,N$,
$r_{i}=\frac{1}{2}(r_{i-1/2}+r_{i+1/2})$ and
$\theta_{j}=\frac{1}{2}(\theta_{j-1/2}+\theta_{j+1/2})$ where
$i,j=1,\ldots,N$. It is worth noting that the point $r_{i}$ should be the
center of the cell to guarantee the discretization of the surface density is
to second order and the effect of $\beta_{0}$ is to refine the mesh. Here, the
proposed method for polar coordinates is of first order because a singular
integration occurs (see below). Furthermore, the region ${\cal R}$ is
discretized into the $N^{2}$ cells, ${\cal
R}_{ij}=[r_{i-1/2},r_{i+1/2}]\times[\theta_{j-1/2},\theta_{j+1/2}]$ for
$i,j=1,\ldots,N$. For $j=1,\ldots,N$, the cells ${\cal R}_{1,j}$ do not cover
the origin and extra cells $\hat{\cal
R}_{j}=[0,r_{1/2}]\times[\theta_{j-1/2},\theta_{j+1/2}]$ should be included.
For simplification of notation, we denote ${\cal R}_{0,j}=\hat{\cal R}_{j}$,
$j=1,\ldots,N$.
The forces in the $r$-direction and the $\theta$-direction at the point
$(r_{i},\theta_{j})$ of the cell ${\cal R}_{ij}$ are
$\displaystyle F^{r}_{i,j}=\frac{\partial}{\partial
r}\Phi(r_{i},\theta_{j},0),\mbox{\quad and
\quad}F^{\theta}_{i,j}=\frac{1}{r_{i}}\frac{\partial}{\partial\theta}\Phi(r_{i},\theta_{j},0).$
(19)
The surface density $\sigma$ on $R_{i,j}$ in (17) is linearly approximated by
$\displaystyle\sigma(\bar{r},\bar{\theta})\approx\sigma_{i,j}+\delta^{r}_{i,j}(\bar{r}-r_{i})+\delta^{\theta}_{i,j}(\bar{\theta}-\theta_{j}),$
(20)
where $\sigma_{i,j}=\sigma(r_{i},\theta_{j})$ and
$\delta^{r}_{i,j}=\sigma_{r}(r_{i},\theta_{j})$ and
$\delta^{\theta}_{i,j}=\sigma_{\theta}(r_{i},\theta_{j})$ are constant in the
cell $R_{i,j}$. The error of the discretization is
$O((\bar{r}-r_{i})^{2}+(\bar{\theta}-\theta_{j})^{2})$. Equation (20) is the
Taylor expansion in two dimensions.
### 4.1 The calculation of radial forces
Let
$\displaystyle{\cal
K}^{r,0}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos({\bar{\theta}-\theta_{j}}))}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta},$
(21) $\displaystyle r_{i}{\cal
K}^{r,r}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j}))(\bar{r}-r_{i^{\prime}})}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta},$
(22)
and
$\displaystyle{\cal
K}^{r,\theta}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j}))(\bar{\theta}-\theta_{j^{\prime}})}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}.$
(23)
The term $r_{i}$ in (22) is for the formulation of a convolution type. By (17)
and (20), we have
$\displaystyle F^{r}_{i,j}$ $\displaystyle\approx$
$\displaystyle\sum^{N}_{i^{\prime}=0}\sum^{N}_{j^{\prime}=1}\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\partial}{\partial
r}{\cal
K}(\bar{r},r_{i},\bar{\theta},\theta_{j},0)\left(\sigma_{i^{\prime},j^{\prime}}+\delta^{r}_{i^{\prime},j^{\prime}}(\bar{r}-r_{i^{\prime}})+\delta^{\theta}_{i^{\prime},j^{\prime}}(\bar{\theta}-\theta_{j^{\prime}})\right)\bar{r}d\bar{r}d\bar{\theta}$
$\displaystyle:=$ $\displaystyle
F^{r,0}_{i,j}+F^{r,r}_{i,j}+F^{r,\theta}_{i,j},$
where
$\displaystyle F^{r,0}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=0}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j}))}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}$
(24) $\displaystyle F^{r,r}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=0}\sum^{N}_{j^{\prime}=1}\delta^{r}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j})(\bar{r}-r_{i^{\prime}})}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}$
(25) $\displaystyle F^{r,\theta}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=0}\sum^{N}_{j^{\prime}=1}\delta^{\theta}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{R_{i^{\prime},j^{\prime}}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j}))(\bar{\theta}-\theta_{j^{\prime}})}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}$
(26)
Equations (24), (25), and (26) can be rewritten as
$\displaystyle F^{r,0}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}{\cal
K}^{r,0}_{i-i^{\prime},j-j^{\prime}}+\sum^{N}_{j^{\prime}=1}\sigma_{0,j^{\prime}}{\bar{\cal
K}}^{r,0}_{i,j-j^{\prime}},$ (27) $\displaystyle F^{r,r}_{i,j}$
$\displaystyle=$ $\displaystyle
r_{i}\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{r}_{i^{\prime},j^{\prime}}{\cal
K}^{r,r}_{i-i,j-j^{\prime}}+r_{i}\sum^{N}_{j^{\prime}=1}\delta^{r}_{0,j^{\prime}}{\bar{\cal
K}}^{r,r}_{i,j-j^{\prime}},$ (28) $\displaystyle F^{r,\theta}_{i,j}$
$\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{\theta}_{i^{\prime},j^{\prime}}{\cal
K}^{r,\theta}_{i-i^{\prime},j-j^{\prime}}+\sum^{N}_{j^{\prime}=1}\delta^{\theta}_{0,j^{\prime}}\bar{\cal
K}^{r,\theta}_{i,j-j^{\prime}}.$ (29)
Let us define
$F(\tilde{r},\theta)=\sqrt{1+{\tilde{r}}^{2}-2{\tilde{r}}\cos(\theta)}$, where
$\tilde{r}$ is a dimensionless radius. The evaluation of (21), (22) and (23)
can be obtained with the help of the following simple integrals,
$\displaystyle\int\frac{\bar{r}(r-\bar{r}\cos(\theta))}{\left({\bar{r}}^{2}+r^{2}-2r\bar{r}\cos(\theta)\right)^{3/2}}d\bar{r}$
$\displaystyle=$
$\displaystyle-\cos(\theta)\ln(-\cos(\theta)+\frac{\bar{r}}{r}+F(\frac{\bar{r}}{r},\theta))+\frac{2\cos(\theta)\frac{\bar{r}}{r}-1}{F(\frac{\bar{r}}{r},\theta)}+C$
$\displaystyle:=$ $\displaystyle H_{1}(\frac{\bar{r}}{r},\theta)+C$
and
$\displaystyle\int\frac{{\bar{r}}^{2}(r-\bar{r}\cos(\theta))}{\left({\bar{r}}^{2}+r^{2}-2r\bar{r}\cos(\theta)\right)^{3/2}}d\bar{r}$
$\displaystyle=$
$\displaystyle-r\left((3\cos^{2}(\theta)-1)\ln(-\cos(\theta)+\frac{\bar{r}}{r}+F(\frac{\bar{r}}{r},\theta))\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{F(\frac{\bar{r}}{r},\theta)}(-6\frac{\bar{r}}{r}\cos^{2}(\theta)+3\cos(\theta)+\frac{{\bar{r}}^{2}}{r^{2}}\cos(\theta)+\frac{\bar{r}}{r})\right)+C$
$\displaystyle:=$ $\displaystyle rH_{2}(\frac{\bar{r}}{r},\theta)+C.$
Following the definition of $r_{i^{\prime}+1/2}$ and $r_{i}$, we have
$\displaystyle\frac{r_{i^{\prime}+1/2}}{r_{i}}=\frac{2\beta^{i-i^{\prime}}}{1+\beta},\mbox{
and }\frac{r_{i^{\prime}}}{r_{i}}=\beta^{i-i^{\prime}}.$
We calculate the value of the integral
$\displaystyle{\cal K}^{r,0}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle=$
$\displaystyle\int^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\\!\\!\\!\int^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\frac{\bar{r}(r_{i}-\bar{r}\cos(\bar{\theta}-\theta_{j}))}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}$
$\displaystyle=$
$\displaystyle\int^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}-\cos(\bar{\theta}-\theta_{j})\ln(-\cos(\bar{\theta}-\theta_{j})+\bar{r}/r_{i}+F(\bar{r}/r_{i},\bar{\theta}-\theta_{j}))$
$\displaystyle+$
$\displaystyle\frac{2\cos(\bar{\theta}-\theta_{j})\bar{r}/r_{i}-1}{F(\bar{r}/r_{i},\bar{\theta}-\theta_{j})}\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\right.d\bar{\theta}$
The last integral in the above equation is an elliptic integral and a
trapzoidal rule has been applied for its evaluation. It is of second order
accuracy for the integration of a smooth function. Unfortunately, the presence
of a singular function in terms of $\ln(1-\cos(\theta))$ reduces the accuracy
of the proposed method for polar coordinate to first order. Finally, the value
${\cal K}^{r,0}_{i-i^{\prime},j-j^{\prime}}$ is approximated as follows and is
used in the numerical calculation,
$\displaystyle{\cal K}^{r,0}_{i-i^{\prime},j-j^{\prime}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{2}\left(H_{1}(r_{i^{\prime}+1/2}/r_{i},\theta_{j^{\prime}+1/2}-\theta_{j})\quad-
H_{1}(r_{i^{\prime}-1/2}/r_{i},\theta_{j^{\prime}+1/2}-\theta_{j})\right.$
$\displaystyle+$ $\displaystyle
H_{1}(r_{i^{\prime}+1/2}/r_{i},\theta_{j^{\prime}-1/2}-\theta_{j})-H_{1}(r_{i^{\prime}-1/2}/r_{i},\theta_{j^{\prime}-1/2}-\theta_{j})\left.\right)(\theta_{j^{\prime}+1/2}-\theta_{j^{\prime}-1/2})$
$\displaystyle:=$ $\displaystyle
H_{1}(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}})\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\right.\left]{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right.,$
where the notation $f(\cdot)]^{b}_{a}=\frac{1}{2}(f(b)+f(a))(b-a)$. Similarly,
$\displaystyle{\cal K}^{r,r}_{i-i^{\prime},j-j^{\prime}}$
$\displaystyle\approx$ $\displaystyle
H_{2}(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}})\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\right.\left]{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right.-\frac{r_{i^{\prime}}}{r_{i}}{\cal
K}^{r,0}_{i-i^{\prime},j-j^{\prime}},$ $\displaystyle{\cal
K}^{r,\theta}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle\approx$
$\displaystyle(\bar{\theta}-\theta_{j})H_{1}(\frac{\bar{r}}{r_{i}},\bar{\theta}-\theta_{j})\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\right.\left]{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right..$
### 4.2 The calculation of azimuthal forces
Next, we introduce the calculation for ${\cal
K}^{\theta,0}_{i-i^{\prime},j-j^{\prime}}$, ${\cal
K}^{\theta,r}_{i-i^{\prime},j-j^{\prime}}$, and ${\cal
K}^{\theta,\theta}_{i-i^{\prime},j-j^{\prime}}$.
In particular, we calculate the value of the integral
$\displaystyle{\cal K}^{\theta,0}_{i-i^{\prime},j-j^{\prime}}$
$\displaystyle=$ $\displaystyle
r_{i}\int^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\\!\\!\\!\int^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\frac{{\bar{r}}^{2}\sin({\bar{\theta}-\theta_{j}})}{\left({\bar{r}}^{2}+r^{2}_{i}-2\bar{r}r_{i}\cos(\bar{\theta}-\theta_{j})\right)^{3/2}}d\bar{r}d\bar{\theta}$
$\displaystyle=$ $\displaystyle-
r_{i}\left(F(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}})+\frac{\bar{r}}{r_{i}}\ln(-\cos(\bar{\theta}-\theta_{j})+\frac{\bar{r}}{r_{i}}+F(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}}))\right)\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\\!\left|{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right.\right..$
Similarly,
$\displaystyle{\cal K}^{\theta,r}_{i-i^{\prime},j-j^{\prime}}$
$\displaystyle=$ $\displaystyle-
r_{i}(-1+\frac{\bar{r}}{2r_{i}}+\frac{3}{2}\cos(\bar{\theta}-\theta_{j}))F(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}})$
$\displaystyle-$ $\displaystyle
r^{2}_{i}(\frac{3}{2}\cos^{2}({\bar{\theta}-\theta_{j}})-\frac{1}{2}-\cos({\bar{\theta}-\theta_{j}}))\ln(-\cos(\bar{\theta}-\theta_{j})+\frac{\bar{r}}{r_{i}}+F(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}}))\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\\!\left|{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right.\right..$
and
$\displaystyle{\cal K}^{\theta,\theta}_{i-i^{\prime},j-j^{\prime}}\approx
r_{i}\frac{{\bar{\theta}}-\theta_{j}}{-1+\cos^{2}({\bar{\theta}-\theta_{j}})}\left(\sin({\bar{\theta}}-\theta_{j})(\frac{\bar{r}}{r_{i}}-2\cos^{2}({\bar{\theta}}-\theta_{j})\frac{\bar{r}}{r_{i}}+\cos({\bar{\theta}}-\theta_{j}))\right.$
$\displaystyle\left.+(\cos^{2}({\bar{\theta}}-\theta_{j})-\sin({\bar{\theta}}-\theta_{j}))\ln(-\cos({\bar{\theta}}-\theta_{j})+\frac{\bar{r}}{r_{i}}+F(\frac{\bar{r}}{r_{i}},{\bar{\theta}-\theta_{j}})\right)\left|{}^{r_{i^{\prime}+1/2}}_{r_{i^{\prime}-1/2}}\right.\left]{}^{\theta_{j^{\prime}+1/2}}_{\theta_{j^{\prime}-1/2}}\right.$
## 5 Order of accuracy and a comparison study
### 5.1 Order of accuracy
We investigate the numerical errors induced by the truncation introduced in
(9), which is
$\displaystyle O\left(((\Delta x)^{2}+(\Delta
y)^{2})\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{|\bar{x}-x_{i}|}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}\right).$
The last integral in the above estimation is $O(|\ln\Delta x|)$ which gives
the numerical error of order $O((\Delta x)^{2}|\ln\Delta x|)=O((\Delta
x)^{2-})$ with $\Delta x=\Delta y$. Three types of norm are used to measure
the errors between the numerical and analytic solutions. The $L^{1}$, $L^{2}$,
and $L^{\infty}$ norms of a function $f$ on a domain $\Omega$ are defined as
$\displaystyle\|f\|_{p}=\left(\int_{\Omega}|f(\boldsymbol{x})|^{p}d\boldsymbol{x}\right)^{1/p},\mbox{
for }p=1,2,\mbox{ and }\|f\|_{\infty}=\mbox{ess
sup}_{\Omega}|f(\boldsymbol{x})|.$
The errors between the analytic and numerical solutions for various
resolutions using different norms $L^{1}$, $L^{2}$, and $L^{\infty}$
demonstrate the convergence in total variation, energy, and pointwise senses,
respectively.
We verify that the proposed method is of second order accuracy by
demonstrating the following examples, a $D_{2}$ disk [7], a non-axisymmetric
disk consisting of two superposed $D_{2}$ disks and a non-axisymmetric disk
describing a pair of spirals.
Example 1. The $D_{2}$ disk has the surface density
$\displaystyle\Sigma_{D_{2}}(R;\alpha)=\left\\{\begin{array}[]{ll}\sigma_{0}(1-R/\alpha^{2})^{3/2}&\mbox{
for }R<\alpha,\\\ 0&\mbox{ for }R>\alpha,\end{array}\right.$ (32)
where $R=\sqrt{x^{2}+y^{2}}$ and $\alpha$ is a given constant. The
corresponding potential on the $z=0$ plane is
$\displaystyle\Phi_{D_{2}}(R,0;\alpha)=\left\\{\begin{array}[]{ll}-\frac{3\pi^{2}\sigma_{0}RG}{64\alpha^{3}}(8\alpha^{4}-8\alpha^{2}R^{2}+3R^{4})&\mbox{
for }R\leq\alpha\\\
-\frac{3\pi\sigma_{0}G}{32\alpha}\left[(8\alpha^{4}-8\alpha^{2}R^{2}+3R^{4})\sin^{-1}(\frac{\alpha}{R})+3\alpha(2\alpha^{2}-R^{2})\sqrt{R^{2}-\alpha^{2}}\right]&\mbox{
for }R\geq\alpha,\end{array}\right.$
and the radial force is found as
$\displaystyle
F_{R,D_{2}}(R,0;\alpha)=\left\\{\begin{array}[]{ll}-\frac{3\pi^{2}\sigma_{0}RG}{16\alpha^{3}}(4\alpha^{2}-3R^{2})&\mbox{
for }R\leq\alpha\\\
-\frac{3\pi\sigma_{0}G}{8\alpha^{3}}\left[R(4\alpha^{2}-3R^{2})\sin^{-1}(\frac{\alpha}{R})-\alpha(2\alpha^{2}-3R^{2})\sqrt{1-\alpha^{2}/R^{2}}\right]&\mbox{
for }R\geq\alpha.\end{array}\right.$
Without loss of generality for studying the order of accuracy, let us set the
computational domain $\Omega=[-1,1]\times[-1,1]$, $\sigma_{0}=G=1$ and
$\alpha=0.25$. We illustrate the contour plots of the surface density,
$x$-directional force, $y$-directional force, radial force, residuals between
analytic and numerical solutions for $x$ , and radial directions for $N=1024$
in Fig. 1. The residuals show that the largest errors occur in regions
surrounding the edge of the disk where the second derivative of the surface
density with respect to radius is infinite.
In Table 1, the column $E^{p}_{x}$ and $E^{p}_{R}$ is the error of the $x$
directional force and $R$ radial direction by $p$-norm, $p=1,2$, and $\infty$,
between the analytic and numerical solutions. The column $O^{p}_{x}$ in Table
1 is the order of accuracy as measured by
$\log_{2}(E^{p}_{x}(2^{k-1})/E^{p}_{x}(2^{k}))$ for $k=6$ to $10$ and
similarly for $O^{p}_{R}$. These errors show that this method is nearly of
second order accuracy. More precisely, we obtain the order of convergence to
be about $1.8$ or $1.9$ as measured by the $L^{1}$ and $L^{2}$ norms for the
simulation of a $D_{2}$ disk. The $L^{\infty}$ norm only demonstrates the
convergence, since the second derivatives of the surface density of the
$D_{2}$ disk are not bounded.
$N$ | $E^{1}_{x}$ | $E^{2}_{x}$ | $E^{\infty}_{x}$ | $E^{1}_{R}$ | $E^{2}_{R}$ | $L^{\infty}_{R}$
---|---|---|---|---|---|---
32 | 1.156E-2 | 1.134E-2 | 2.231E-2 | 1.788E-2 | 1.589E-2 | 2.315E-2
64 | 3.039E-3 | 3.176E-3 | 7.525E-3 | 4.742E-3 | 4.460E-3 | 8.535E-3
128 | 8.476E-4 | 9.312E-4 | 5.264E-3 | 1.319E-3 | 1.309E-3 | 5.906E-3
256 | 2.161E-4 | 2.444E-4 | 1.932E-3 | 3.379E-4 | 3.439E-4 | 1.994E-3
512 | 5.620E-5 | 6.884E-5 | 9.478E-4 | 8.795E-5 | 9.695E-5 | 9.842E-4
1024 | 1.427E-5 | 1.824E-5 | 3.281E-4 | 2.236E-5 | 2.570E-5 | 3.470E-4
$N_{k-1}/N_{k}$ | $O^{1}_{x}$ | $O^{2}_{x}$ | $O^{\infty}_{x}$ | $O^{1}_{R}$ | $O^{2}_{R}$ | $O^{\infty}_{R}$
32/64 | 1.927 | 1.836 | 1.567 | 1.914 | 1.833 | 1.439
64/128 | 1.842 | 1.770 | 0.515 | 1.846 | 1.768 | 0.531
128/256 | 1.971 | 1.929 | 1.446 | 1.964 | 1.928 | 1.566
256/512 | 1.943 | 1.827 | 1.027 | 1.941 | 1.826 | 1.018
512/1024 | 1.977 | 1.916 | 1.530 | 1.975 | 1.915 | 1.504
Table 1: This table demonstrates the errors and order accuracy of the proposed
method for the $D_{2}$ disk for various number of zones $N=2^{k}$ from $k=5$
to $10$. It shows that the order for the $D_{2}$ disk is about $1.8$ or $1.9$
order in $L^{1}$ and $L^{2}$ norm.
Figure 1: The numerical solutions of a $D_{2}$ disk for $N=1024$, the contour
plots are surface density (upper left), the $y$-directional force (upper
right), the $x$-directional force (middle left), the radial force (middle
right), the difference between analytic and numerical solutions in $x$
direction (lower left), and the difference in radial direction (lower right).
The values in the lower contour plots are the absolute difference in the
common logarithmic scale.
We continue to use the $D_{2}$ disk as an example and a unit disk
$D(0,1)=\Omega=[0,1]\times[0,2\pi]$ as the computational domain to investigate
the self-gravitational force in polar coordinates. The value $\beta_{0}=0.99$
is set. We show the contour plots of the surface density, radial force, and
the difference between analytic and numerical solutions for $N=512$ in Fig. 2
and the order of accuracy is only about 1 as given in Table 2. The largest
errors occur in regions not only surrounding the edge of the disk, but also
close to the origin. Although the surface density at the origin is smooth, the
singular elliptic integral introduces significant error there. Hereafter, we
concentrate on the self-gravitational forces in Cartesian coordinates.
$N$ | $E^{1}_{R}$ | $E^{2}_{R}$ | $L^{\infty}_{R}$ | $N_{k-1}/N_{k}$ | $O^{1}_{R}$ | $O^{2}_{R}$ | $O^{\infty}_{R}$
---|---|---|---|---|---|---|---
32 | 1.603E-1 | 1.725E-1 | 2.725E-1 | | | |
64 | 7.618E-2 | 8.289E-2 | 1.361E-1 | 32/64 | 1.073 | 1.057 | 1.002
128 | 3.646E-2 | 4.045E-2 | 6.806E-2 | 64/128 | 1.063 | 1.035 | 1.000
256 | 1.754E-2 | 2.098E-2 | 3.403E-2 | 128/256 | 1.056 | 0.947 | 1.000
512 | 8.762E-3 | 1.049E-2 | 1.701E-2 | 256/512 | 1.001 | 1.000 | 1.000
Table 2: This table demonstrates the errors and order accuracy of the proposed
method for the $D_{2}$ disk for various number of zones $N=2^{k}$ from $k=5$
to $10$ on polar coordinates. It shows that the order for the $D_{2}$ disk is
about 1 in each norm.
Figure 2: The numerical solutions of a $D_{2}$ disk for $N=512$ to investigate
the self-gravitational force calculation in polar coordinate. From left to
right, the contour plots are surface density, the radial directional force,
the difference between analytic and numerical solutions, respectively. The
values in the right contour plot are the absolute difference in the common
logarithmic scale.
Example 2. The disk $D_{2,2}$ of two superposed $D_{2}$ has the surface
density
$\Sigma_{D_{2,2}}=\Sigma_{D_{2}}(R_{1};\alpha)+\Sigma_{D_{2}}(R_{2};\alpha)$,
where $R_{1}=\sqrt{(x-1/4)^{2}+y^{2}}$ and $R_{2}=\sqrt{(x+1/4)^{2}+y^{2}}$.
This example represents a non-symmetric distribution of the surface density of
a disk. The results are shown in Table 3 and Figure 3. This result is
similarly to Example 1. The factors $O^{\infty}_{x}$ of errors in Table 3 are
non-monotonic as the numerical resolution, $N_{k}$, increases. This may be due
to the distribution of the surface density on grid cells, the centers of which
can shift with varying numerical resolution. However, the total variation and
energy shows the convergence and the order of accuracy is about 1.8 and 1.9
respectively.
$N$ | $E^{1}_{x}$ | $E^{2}_{x}$ | $E^{\infty}_{x}$ | $E^{1}_{y}$ | $E^{2}_{y}$ | $E^{\infty}_{y}$ | $E^{1}_{R}$ | $E^{2}_{R}$ | $L^{\infty}_{R}$
---|---|---|---|---|---|---|---|---|---
32 | 1.56E-2 | 1.29E-2 | 2.19E-2 | 2.09E-2 | 1.71E-2 | 3.57E-2 | 2.56E-2 | 1.95E-2 | 3.55E-2
64 | 4.29E-3 | 3.83E-3 | 7.75E-3 | 5.38E-3 | 4.57E-3 | 1.07E-2 | 6.89E-3 | 5.53E-3 | 1.16E-2
128 | 1.23E-3 | 1.18E-3 | 5.41E-3 | 1.50E-3 | 1.35E-3 | 5.72E-3 | 1.96E-3 | 1.64E-3 | 5.81E-3
256 | 3.17E-4 | 3.12E-4 | 1.94E-3 | 3.83E-4 | 3.54E-4 | 1.96E-3 | 5.06E-4 | 4.34E-4 | 2.01E-3
512 | 8.32E-5 | 9.00E-5 | 9.49E-4 | 9.99E-5 | 9.89E-5 | 9.53E-4 | 1.33E-4 | 1.23E-4 | 9.64E-4
1024 | 2.12E-5 | 2.41E-5 | 3.28E-4 | 2.54E-5 | 2.62E-5 | 3.29E-4 | 3.39E-5 | 3.29E-5 | 3.38E-4
$N$ | $O^{1}_{x}$ | $O^{2}_{x}$ | $O^{\infty}_{x}$ | $O^{1}_{y}$ | $O^{2}_{y}$ | $O^{\infty}_{y}$ | $O^{1}_{R}$ | $O^{2}_{R}$ | $O^{\infty}_{R}$
32/64 | 1.86 | 1.75 | 1.50 | 1.96 | 1.87 | 1.74 | 1.89 | 1.82 | 1.62
64/128 | 1.80 | 1.71 | 0.52 | 1.85 | 1.79 | 0.90 | 1.81 | 1.75 | 1.00
128/256 | 1.96 | 1.91 | 1.48 | 1.97 | 1.93 | 1.55 | 1.95 | 1.92 | 1.53
256/512 | 1.93 | 1.79 | 1.03 | 1.94 | 1.84 | 1.04 | 1.93 | 1.81 | 1.06
512/1024 | 1.97 | 1.90 | 1.53 | 1.98 | 1.91 | 1.53 | 1.97 | 1.90 | 1.51
Table 3: This table demonstrates the errors and order of accuracy of the
proposed method for the $D_{2,2}$ disk for various number of zones $N=2^{k}$
from $k=5$ to $10$. It shows that the order for the $D_{2,2}$ disk is about
$1.8$ or $1.9$ order in $L^{1}$ and $L^{2}$ norm.
Figure 3: The numerical solutions of a $D_{2,2}$ disk for $N=1024$, The top
contour plot is the surface density. The contour plots in the second row are
the $x$-directional, $y$-directional, and radial forces, respectively. The
corresponding errors between the numerical and analytic solutions in the third
row. The values in the contour plots in the third row are the absolute errors
in the common logarithmic scale.
Example 3. As another example of a non-axisymmetric potential, we consider a
logarithmic spiral disk. Since an analytic pair for the surface density and
potential are not known, we assume a surface density profile of the form
$\displaystyle\Sigma_{LS}(r,\theta)=e^{-2r^{2}}(2+\cos(2\theta+16r)).$
To investigate the order of accuracy, the solution at the finest mesh size is
regarded as the true solution. For various coarser resolutions, the values at
some specific position are taken to be the average of the four closest to the
position. The results are shown for the method based on Cartesian coordinates
in Table 4 and Figure 4. It can be seen that the order of accuracy is about
$1.5$ for the $L^{1}$ norm and about $1$ for the $L^{2}$ norm. The
$L^{\infty}$ norm is only convergent.
$N$ | $E^{1}_{x}$ | $E^{2}_{x}$ | $E^{\infty}_{x}$ | $E^{1}_{y}$ | $E^{2}_{y}$ | $E^{\infty}_{y}$ | $E^{1}_{R}$ | $E^{2}_{R}$ | $L^{\infty}_{R}$
---|---|---|---|---|---|---|---|---|---
32 | 3.40E-1 | 2.97E-1 | 1.40E-0 | 3.38E-1 | 2.98E-1 | 1.39E-0 | 4.64E-1 | 3.03E-1 | 4.27E-1
64 | 1.21E-1 | 1.70E-1 | 1.83E-0 | 1.23E-1 | 1.72E-1 | 1.83E-0 | 1.36E-1 | 9.29E-2 | 2.07E-1
128 | 4.71E-2 | 9.28E-2 | 1.92E-0 | 4.83E-2 | 9.51E-2 | 1.92E-0 | 3.97E-2 | 3.93E-2 | 2.02E-1
256 | 1.87E-2 | 4.50E-2 | 1.71E-0 | 1.93E-2 | 4.68E-2 | 1.71E-0 | 1.17E-2 | 2.11E-2 | 1.75E-1
512 | 6.05E-3 | 1.67E-2 | 1.16E-0 | 6.30E-3 | 1.78E-2 | 1.16E-0 | 1.00E-3 | 9.53E-3 | 1.54E-1
$N$ | $O^{1}_{x}$ | $O^{2}_{x}$ | $O^{\infty}_{x}$ | $O^{1}_{y}$ | $O^{2}_{y}$ | $O^{\infty}_{y}$ | $O^{1}_{R}$ | $O^{2}_{R}$ | $O^{\infty}_{R}$
32/64 | 1.49 | 0.81 | -0.39 | 1.46 | 0.79 | -0.40 | 1.77 | 1.70 | 1.05
64/128 | 1.36 | 0.87 | -0.07 | 1.34 | 0.85 | -0.07 | 1.77 | 1.24 | 0.03
128/256 | 1.34 | 1.05 | 0.17 | 1.32 | 1.02 | 0.17 | 1.76 | 0.90 | 0.21
256/512 | 1.63 | 1.43 | 0.56 | 1.62 | 1.62 | 0.56 | 1.96 | 1.14 | 0.18
Table 4: This table demonstrates the errors and order of accuracy of the
proposed method for the spiral disk for various number of zones $N=2^{k}$ from
$k=5$ to $9$. It shows that the order for the spiral disk is about 1.5 or 1.0
in the $L^{1}$ and $L^{2}$ norms, respectively.
Figure 4: The numerical solutions of a logarithmic spiral disk for $N=512$ to
investigate the self-gravitational force calculation. The contour plots
illustrate the surface density (upper left), $x$-force (upper right),
$y$-force (low left), and radial force (lower right).
### 5.2 A comparison study
The goal of this paper is to calculate the self-gravitational forces with as
few restrictions as possible. The most straight forward approach is to solve
for the potential via (1) and obtain the self-gravitational forces by taking
its derivatives. If one uses the finite difference or finite element method on
(1), the discretization is
$\displaystyle\frac{-\Phi_{i+1,j,k}+2\Phi_{i,j,k}-\Phi_{i-1,j,k}}{(\Delta
x)^{2}}+\frac{-\Phi_{i,j+1,k}+2\Phi_{i,j,k}-\Phi_{i,j-1,k}}{(\Delta
y)^{2}}+\frac{-\Phi_{i,j,k+1}+2\Phi_{i,j,k}-\Phi_{i,j,k-1}}{(\Delta
z)^{2}}=-f_{i,j,k}$
where $\Phi_{i,j,k}=\Phi(x_{i},y_{j},z_{k})$ and
$f_{i,j,k}=f(x_{i},y_{j},z_{k})$ based on the uniform mesh grids
$(x_{i},y_{j},z_{k})$. Here, $f_{i,j,k}=0$ for $k\not=0$. For such an
approach, artificial boundary conditions should be imposed and a fully
3-dimensional calculation must be undertaken. We point out that the (1) can
not be reduced to the two dimensional numerical partial differential problem,
i.e.,
$\displaystyle\frac{-\Phi_{i+1,j,0}+2\Phi_{i,j,0}-\Phi_{i-1,j,0}}{(\Delta
x)^{2}}+\frac{-\Phi_{i,j+1,0}+2\Phi_{i,j,0}-\Phi_{i,j-1,0}}{(\Delta
y)^{2}}=-f_{i,j,0}.$
Any numerical solution of the partial differential problem will involve
$O(N^{3})$ unknowns. It follows that the linear complexity of such an
approach, viz. multigrid method, is at least $O(N^{3})$. For an
infinitesimally thin gaseous disk problem, this approach does not appear to be
suitable.
Alternatively, one can solve the reduced equation given by
$\displaystyle\Phi(x,y,0)=-G\int\\!\\!\\!\int\frac{\sigma(\bar{x},\bar{y})}{\sqrt{(\bar{x}-x)^{2}+(\bar{y}-y)^{2}}}d\bar{x}d\bar{y}$
or
$\displaystyle\Phi(r,\theta,0)=-G\int\\!\\!\\!\int\frac{\sigma(\bar{r},\bar{\theta})}{\sqrt{{\bar{r}}^{2}+r^{2}-2\bar{r}r\cos(\bar{\theta}-\theta)}}\bar{r}d\bar{r}d\bar{\theta}.$
In this case, one can consider using bases functions on a two dimensional
space as in a spectral method. Unfortunately, this approach requires a
treatment for the boundary conditions. A possible way to deal with this issue
is to impose periodic boundary conditions. However, it is not realistic for a
gravitational force calculation because gravity is a long range force and not
periodic. As an alternative, a method without the periodic assumption has been
proposed for polar coordinates [3]. The approach in [3] transforms the polar
coordinate $(r,\theta)$ into the coordinate $(u,\theta)$ by setting $r=e^{u}$
or $u=\ln(r)$. The potential-density pair in term of the reduced surface
density and reduced potential is given in [3], and it is
$\displaystyle
e^{3u/2}\sigma(e^{u},\theta)=\frac{1}{4\pi^{2}}\sum_{m}\int^{\infty}_{-\infty}A_{m}(\alpha)e^{i(m\theta+\alpha
u)}d\alpha$
and
$\displaystyle
e^{u/2}\Phi(e^{u},\theta)=-\frac{1}{2\pi}G\sum_{m}\int^{\infty}_{-\infty}K(\alpha,m)A_{m}(\alpha)\exp[i(m\theta+\alpha
u)]d\alpha,$ (35)
where $K$ is real and positive and is defined as
$\displaystyle
K(\alpha,m)\equiv\frac{1}{2}\frac{\Gamma[(m+1/2+i\alpha)/2]\Gamma[(m+1/2-i\alpha)/2]}{\Gamma[(m+3/2+i\alpha)/2]\Gamma[(m+3/2-i\alpha)/2]}.$
We regard this method as one of the spectral methods because Fourier series
$e^{-im\theta}$ and Fourier integral $e^{-i\alpha u}$ are used. To apply this
method to the $D_{2}$ disk using the polar coordinates, we transform the
bounded unit disk $D(0,1)=[0,1]\times[0,2\pi]$ to the unbounded domain
$U=(-\infty,0]\times[0,2\pi]$. In this special case, we only need to compute
$m=0$ and truncate
$\displaystyle
A_{0}(\alpha)=\int^{0}_{-\infty}e^{3u/2}\sigma(e^{u})e^{-i\alpha
u}du\approx\int^{0}_{u_{\min}}e^{3u/2}\sigma(e^{u})e^{-i\alpha u}du,$ (36)
where the value $u_{\min}$ is to approximate $-\infty$. The truncation
produces a hole in the unit disk and can introduce significant errors at the
origin. Given a positive integer $N$ and base on the discretization for the
radial region in the previous subsection, to calculate (36) and (35) by the
trapzoidal rule. The variation of the potential with respect to radius is
illustrated in Figure 5. The profile on the left panel shows that the
numerical and analytic solutions for the Kalnajs’ method agree well except
close to the origin for $N=1024$. The small window embedded within the panel
zooms in on the residuals between numerical and analytic solution on the
interval $[0,0.3]$. It is seen that the truncated portion contributes to
significant errors near the origin. In contrast, the application of our
proposed method to the calculation of potentials leads to the results shown in
the right panel of Figure 5. Although the singular integration still remains
due to the unbounded domain, our proposed method on either Cartesian and polar
coordinates is preferable since a hole near the origin is not introduced.
Figure 5: The variation of the potential with respect to radius using Kalnajs’
method (left) and the proposed method (right). The residuals are shown in the
small window in each panel and show that the Kalnajs’ method have significant
errors near the origin, which are eliminated in the proposed method.
Finally, a third approach is to directly calculate the integrals and obtain
the potential. For any given mesh grid, the total amount of complexity is
$O(N^{4})$ based on the number $O(N^{2})$ of mesh zones. If we restrict
ourselves to a uniform grid and use the FFT technique, the complexity can be
reduced from $O(N^{4})$ to $O(N^{2})$. In other words, a fast algorithm of
linear complexity is obtained. It is common to start with
$\displaystyle\Phi(x,y,0)$ $\displaystyle=$
$\displaystyle-G\int\\!\\!\\!\int{\cal
K}(\bar{x}-x,\bar{y}-y,0)\sigma(\bar{x},\bar{y})d\bar{x}d\bar{y}$
$\displaystyle=$
$\displaystyle-G\sum^{N}_{i=1}\sum^{N}_{j=1}\int\\!\\!\\!\int_{D_{i,j}}{\cal
K}(\bar{x}-x,\bar{y}-y,0)\sigma(\bar{x},\bar{y})d\bar{x}d\bar{y}.$
and to introduce a softening parameter $\epsilon$ to approximate
$\displaystyle\int\\!\\!\\!\int_{D_{i,j}}{\cal
K}(\bar{x}-x,\bar{y}-y)\sigma(\bar{x},\bar{y})\approx-\frac{G}{\sqrt{\epsilon^{2}+(x_{i^{\prime}}-x_{i})^{2}+(y_{j^{\prime}}-y_{j})^{2}}}\int\\!\\!\\!\int_{D_{i,j}}\sigma(\bar{x},\bar{y})d\bar{x}d\bar{y}.$
Since the goal is to calculate the forces, the order of accuracy is reduced
when taking the numerical differentiation on the numerical solution of
potentials. For polar coordinates [1], the value of ${\cal K}$ is approximated
by
$\displaystyle{\cal
K}_{i^{\prime}-i,j^{\prime}-j}:=-\frac{G}{\sqrt{2(\cosh(u_{i^{\prime}}-u_{i})-\cos(\theta_{j^{\prime}}-\theta_{j}))}},$
where $u_{i^{\prime}}=\ln(x_{i^{\prime}})$ and $u_{i}=\ln(x_{i})$. Note that
when $(i^{\prime},j^{\prime})=(i,j)$, ${\cal K}$ is undefined. An approach to
avoid the singularity problem can be found in [1]. On the other hand, the
proposed method avoids the singularity problem by directly evaluating the
forces, hence, raising the order of accuracy. For Cartesian coordinates, we
choose the softening parameters as the mesh size $\epsilon=\Delta x$. The
errors for the disks $D_{2}$ and $D_{2,2}$ are shown in Table 5 and Table 6,
respectively. It reveals that the accuracy when using the softening parameter
approach for the $D_{2}$ and $D_{2,2}$ disks is of first order in the $L^{1}$
and $L^{2}$ norms. For the $L^{\infty}$ norm, the order of accuracy for the
$D_{2}$ disk is about $1$. For the $D_{2,2}$ disk, this method loses accuracy.
In comparison with our proposed method for Example 1 and Example 2, our
methods are more accurate and the order of accuracy is verified.
$N$ | $E^{1}_{x}$ | $E^{2}_{x}$ | $L^{\infty}_{x}$ | $N_{k-1}/N_{k}$ | $O^{1}_{x}$ | $O^{2}_{x}$ | $O^{\infty}_{x}$
---|---|---|---|---|---|---|---
32 | 4.283E-1 | 5.116E-1 | 9.981E-1 | | | |
64 | 2.223E-1 | 2.768E-1 | 5.415E-1 | 32/64 | 0.9461 | 0.8862 | 0.9377
128 | 1.133E-1 | 1.442E-1 | 2.827E-1 | 64/128 | 0.9724 | 0.9408 | 0.9377
256 | 5.721E-2 | 7.364E-2 | 1.440E-1 | 128/256 | 0.9858 | 0.9695 | 0.9732
512 | 2.874E-2 | 3.722E-2 | 7.282E-2 | 256/512 | 0.9932 | 0.9844 | 0.9837
1024 | 1.440E-2 | 1.871E-2 | 3.659E-2 | 512/1024 | 0.9970 | 0.9923 | 0.9929
Table 5: This table demonstrates the errors and order accuracy of the softening parameter method for the $D_{2}$ disk for various number of zones $N=2^{k}$ from $k=5$ to $10$ in Cartesian coordinates. It shows that the accuracy for the $D_{2}$ disk is about first order. $N$ | $E^{1}_{x}$ | $E^{2}_{x}$ | $E^{\infty}_{x}$ | $E^{1}_{y}$ | $E^{2}_{y}$ | $E^{\infty}_{y}$ | $E^{1}_{R}$ | $E^{2}_{R}$ | $L^{\infty}_{R}$
---|---|---|---|---|---|---|---|---|---
32 | 5.95E-1 | 5.61E-1 | 1.00E-0 | 9.13E-1 | 8.31E-2 | 1.46E-0 | 1.16E-0 | 9.32E-1 | 1.45E-0
64 | 3.10E-1 | 3.10E-1 | 5.42E-1 | 4.73E-1 | 4.49E-3 | 8.04E-1 | 5.97E-1 | 5.06E-1 | 8.04E-1
128 | 1.59E-1 | 1.69E-1 | 4.17E-1 | 2.41E-1 | 2.36E-3 | 4.22E-1 | 3.03E-1 | 2.69E-1 | 4.21E-1
256 | 8.04E-2 | 9.29E-2 | 4.17E-1 | 1.22E-1 | 1.24E-4 | 3.02E-1 | 1.53E-1 | 1.44E-1 | 4.17E-1
512 | 4.05E-2 | 5.31E-2 | 4.17E-1 | 6.10E-2 | 6.57E-5 | 3.03E-1 | 7.68E-2 | 7.85E-2 | 4.17E-1
1024 | 2.03E-2 | 3.19E-2 | 4.17E-1 | 3.06E-2 | 3.61E-5 | 3.03E-1 | 3.85E-2 | 4.49E-2 | 4.17E-1
$N$ | $O^{1}_{x}$ | $O^{2}_{x}$ | $O^{\infty}_{x}$ | $O^{1}_{y}$ | $O^{2}_{y}$ | $O^{\infty}_{y}$ | $O^{1}_{R}$ | $O^{2}_{R}$ | $O^{\infty}_{R}$
32/64 | 0.94 | 0.86 | 0.88 | 0.95 | 0.89 | 0.86 | 0.95 | 0.88 | 0.85
64/128 | 0.97 | 0.88 | 0.38 | 0.97 | 0.93 | 0.93 | 0.98 | 0.91 | 0.93
128/256 | 0.98 | 0.86 | 0.00 | 0.99 | 0.93 | 0.47 | 0.99 | 0.90 | 0.02
256/512 | 0.99 | 0.80 | 0.00 | 0.99 | 0.91 | 0.00 | 0.99 | 0.87 | 0.00
512/1024 | 0.99 | 0.73 | 0.00 | 1.00 | 0.86 | 0.00 | 0.99 | 0.80 | 0.00
Table 6: This table demonstrates the errors and order accuracy of the
softening parameter approach for the $D_{2,2}$ disk for various number of
zones $N=2^{k}$ from $k=5$ to $10$. It shows that the order for the $D_{2,2}$
disk is about first order in $L^{1}$ and $L^{2}$ norm. For measurement of
$L^{\infty}$ norm, this method may fail in convergence under the pointwise
sense.
We implement the proposed method using MATLAB 7 software under the computer
system, Intel Core 2 Duo CPU 1.8GHz with 2 GB RAM. The CPU time measurement
information of the proposed method is compared with the direct method in Table
7. We list the CPU times in evaluating the kernels ${\cal K}^{\cdot,\cdot}$,
the force calculations of convolutions, and the whole process. The measurement
is evaluated by the mean of 40 simulations. It shows that the CPU times of
both of the proposed method (P.M.) and the direct method (D.M.) are
comparable.
| Kernel ${\cal K}$ | Force | The whole process
---|---|---|---
$N$ | P.M. | D.M. | P.M. | D.M. | P.M. | D.M.
32 | 9.73E-3 | 7.43E-3 | 6.10E-3 | 3.43E-3 | 1.60E-2 | 2.31E-2
64 | 3.80E-2 | 2.39E-2 | 2.08E-2 | 1.26E-2 | 5.87E-2 | 3.74E-2
128 | 1.27E-1 | 9.67E-2 | 1.06E-1 | 6.43E-2 | 2.43E-1 | 1.60E-1
256 | 5.11E-1 | 3.84E-1 | 6.48E-1 | 3.96E-2 | 1.18E+0 | 7.84E-1
512 | 2.18E+0 | 1.57E+0 | 2.75E+0 | 1.61E+0 | 4.83E+0 | 3.29E+0
1024 | 8.59E+0 | 6.29E+0 | 1.13E+1 | 6.49E+0 | 2.01E+1 | 1.43E+1
Table 7: This table demonstrates the CPU time measurement of the proposed
method (P.M.) and direct method (D.M.) with softening parameters. The whole
process consists of the generation of kernels and the forces of calculations.
It shows that the CPU times of both of P.M. and D.M. are comparable.
## 6 Discussion and conclusion
We have presented a near second order method for calculating the self-
gravitating force of an infinitesimally thin disk for Cartesian coordinates.
For polar coordinates, we find that the method is near first order, $\sim
0.89$, only. To quantify the accuracy, we define
$\displaystyle
E_{k}=\left|\int^{\theta_{k}}_{-\theta_{k}}\ln(1-\cos(\theta))d\theta-\frac{1}{2}(\ln(1-\cos(\theta_{k}))+\ln(1-\cos(-\theta_{k}))2(\theta_{k}),\right|$
where $\theta_{k}=1/2^{k}$. Table 8 reveals that the accuracy of the
trapzoidal rule for the integration of the function $\ln(1-\cos(\theta))$ is
nearly of first order. With an improvement of the singular integration of
$\ln(1-\cos(\theta))$, the accuracy can be increased for the proposed method
in polar coordinates.
(Term, $k$ ) | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$E_{k}$ | 2.86 | 1.73 | 1.07 | 0.55 | 0.34 | 0.20 | 0.11 | 0.06 | 0.03
$\log_{2}(E_{k-1}/E_{k})$ | | 0.75 | 0.79 | 0.82 | 0.84 | 0.85 | 0.87 | 0.88 | 0.89
Table 8: This table demonstrates the accuracy of the trapzoidal rule for the
integration of the function $\ln(1-cos(\theta))$ is near of first order $\sim
0.89$.
We note that the fast Fourier transform is only used to reduce the
computational time. For the practical computation, one can extend the range of
the summation in (13). By setting $\sigma_{i^{\prime},j^{\prime}}=0$ whenever
either $i^{\prime}$ or $j^{\prime}$ is in the range $-N+1$ to $0$, the value
of any of the $F^{x,0}_{i,j}$ is unaffected. Furthermore, we can take
$\sigma_{i^{\prime},j^{\prime}}$ to be periodic since the sum (13) does not
involve any values of $i^{\prime}$ and $j^{\prime}$ outside the first period.
We are also free to take ${\cal K}^{x,0}_{i-i^{\prime},j-j^{\prime}}$ periodic
by defining it to be the periodic function that agrees with (43) for
$i-i^{\prime}$ and $j-j^{\prime}$ in the range $[-N+1,N]$ of the Green
function.
An important feature of our approach is that the boundary is not assumed to be
periodic. Our approach is limited to the Cartesian and polar coordinates with
uniform and logarithmic grid discretization, respectively, which allows for
rapid computation. That is, the restriction of a convolution of two vectors
provides the rapid computation, but it is restricted to a grid discretization
that is either uniform or logarithmic. If the discretization is arbitrary,
then the FFT is not suitable.
We point out that our method may be useful for gravity computations on a
nested grid consisting of uniform grids having different grid spacing designed
to resolve a central region with a finer grid. Such an approach would be
complementary to the fast algorithm for solving the Poisson equation on a
nested grid presented by Matsumoto and Hanawa [5].
## Acknowledgments
This paper is dedicated to the memory of Professor C. Yuan, who initiated the
project on the development of the Antares codes. We thank the two anonymous
referees for their valuable suggestions which significantly improved the
presentation of our method in this paper. The author C. C. Yen thanks the
Institute of Astronomy and Astrophysics, Academia Sinica, Taiwan for their
constant support.
## References
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## Appendix A: The calculation of the force in the $y$-direction in Cartesian
coordinate
Let
$\displaystyle{\cal
K}^{y,0}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y},$
(37) $\displaystyle{\cal
K}^{y,x}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})(\bar{x}-x_{i^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y},$
(38)
and
$\displaystyle{\cal
K}^{y,y}_{i-i^{\prime},j-j^{\prime}}=\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})(\bar{y}-y_{j^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}.$
(39)
By (6) and (9), we have
$\displaystyle{F}^{y}_{i,j}$ $\displaystyle\approx$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{\partial}{\partial
y}{\cal
K}(\bar{x}-x_{i},\bar{y}-y_{j},0)\left(\sigma_{i^{\prime},j^{\prime}}+\delta^{x}_{i^{\prime},j^{\prime}}(\bar{x}-x_{i^{\prime}})+\delta^{y}_{i^{\prime},j^{\prime}}(\bar{y}-y_{j^{\prime}})\right)d\bar{x}d\bar{y}$
$\displaystyle:=$ $\displaystyle F^{y,0}_{i,j}+F^{y,x}_{i,j}+F^{y,y}_{i,j},$
where
$\displaystyle F^{y,0}_{i,j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\sigma_{i^{\prime},j^{\prime}}{\cal
K}^{y,0}_{i-i^{\prime},j-j^{\prime}},$ (40) $\displaystyle F^{y,x}_{i,j}$
$\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{x}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})(\bar{x}-x_{i^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{x}_{i^{\prime},j^{\prime}}{\cal
K}^{y,x}_{i-i^{\prime},j-j^{\prime}},$ (41) $\displaystyle F^{y,y}_{i,j}$
$\displaystyle=$
$\displaystyle\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{y}_{i^{\prime},j^{\prime}}\int\\!\\!\\!\int_{D_{i^{\prime},j^{\prime}}}\frac{(\bar{y}-y_{j})(\bar{y}-y_{j^{\prime}})}{\left((\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}\right)^{3/2}}d\bar{x}d\bar{y}=\sum^{N}_{i^{\prime}=1}\sum^{N}_{j^{\prime}=1}\delta^{y}_{i^{\prime},j^{\prime}}{\cal
K}^{y,y}_{i-i^{\prime},j-j^{\prime}}.$ (42)
The evaluation of (37), (38) and (39) can be obtained with the help of the
following simple integrals,
$\displaystyle\int\\!\\!\\!\int\frac{y}{(x^{2}+y^{2})^{3/2}}dxdy=-\ln(x+\sqrt{x^{2}+y^{2}})+C,\quad\quad\>\int\\!\\!\\!\int\frac{xy}{(x^{2}+y^{2})^{3/2}}dxdy=-\sqrt{x^{2}+y^{2}}+C,$
$\displaystyle\int\\!\\!\\!\int\frac{y^{2}}{(x^{2}+y^{2})^{3/2}}dxdy=x\ln(y+\sqrt{x^{2}+y^{2}})+C,\quad\int\\!\\!\\!\int\frac{1}{(x^{2}+y^{2})^{3/2}}dxdy=-\frac{\sqrt{x^{2}+y^{2}}}{xy}+C.$
The value ${\cal K}^{y,0}_{i-i^{\prime},j-j^{\prime}}$ is equal to
$\displaystyle{\cal
K}^{0}_{i-i^{\prime},j-j^{\prime}}=-\ln\left((\bar{x}-x_{i})+\sqrt{(\bar{x}-x_{i})^{2}+(\bar{y}-y_{j})^{2}}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right.$
(43)
The calculation of ${\cal K}^{y,x}_{i-i^{\prime},j-j^{\prime}}$ and ${\cal
K}^{y,y}_{i-i^{\prime},j-j^{\prime}}$ are split into two parts by the identity
$(\bar{y}-y_{j})(\bar{y}-y_{j^{\prime}})=(\bar{y}-y_{j})^{2}+(\bar{y}-y_{j})(y_{j}-y_{j^{\prime}})$,
and
$(\bar{y}-y_{j})(\bar{x}-x_{i^{\prime}})=(\bar{y}-y_{j})(\bar{x}-x_{i})+(\bar{y}-y_{j})(x_{i}-x_{i^{\prime}})$,
respectively. It follows that
$\displaystyle{\cal K}^{y,x}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle=$
$\displaystyle(y_{j}-y_{j^{\prime}}){\cal
K}^{y,0}_{i-i^{\prime},j-j^{\prime}}+\left((\bar{x}-x_{i})\ln(\bar{y}-y_{j}+\sqrt{(\bar{y}-y_{j})^{2}+(\bar{x}-x_{i})^{2})}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right.,$
$\displaystyle{\cal K}^{y,y}_{i-i^{\prime},j-j^{\prime}}$ $\displaystyle=$
$\displaystyle(x_{i}-x_{i^{\prime}}){\cal
K}^{y,0}_{i-i^{\prime},j-j^{\prime}}+\left(-\sqrt{(\bar{y}-y_{j})^{2}+(\bar{x}-x_{i})^{2}}\right)\left|{}^{x_{i^{\prime}+\frac{1}{2}}}_{x_{i^{\prime}-\frac{1}{2}}}\left|{}^{y_{j^{\prime}+\frac{1}{2}}}_{y_{j^{\prime}-\frac{1}{2}}}\right.\right..$
## Appendix B: Calculations of convolution of two vectors by FFT
It is known that the FFT of a vector $u_{n}$, $n=-N,\ldots,N-1$ can be
rewritten as
$\displaystyle\hat{u}_{k}=\sum^{N-1}_{n=-N}u_{n}e^{-j2\pi kn/2N},\mbox{ for
}k=-N,\ldots,N-1,$
and its inverse FFT is given by
$\displaystyle u_{n}=\frac{1}{2N}\sum^{N-1}_{k=-N}\hat{u}_{k}e^{j2\pi
kn/2N},\mbox{ for }n=-N,\ldots,N-1.$
Let us consider two vectors $u_{n}$, $n=0,\ldots,N-1$ and $v_{n}$,
$n=-N+1,\ldots,N-1$ and their inner product
$\displaystyle w_{k}=\sum^{N-1}_{n=0}u_{n}v_{k-n},\mbox{ for }k=0,\ldots,N-1.$
We set $w_{k}=0$, $k=-N,\ldots,0$ and
$\displaystyle\sum^{N-1}_{k=-N}w_{k}e^{-j2\pi mk/2N}$ $\displaystyle=$
$\displaystyle\sum^{N-1}_{k=-N}\sum^{N-1}_{n=-N}u_{n}v_{k-n}e^{-j2\pi mk/2N}$
$\displaystyle=$ $\displaystyle\sum^{N-1}_{n=-N}u_{n}e^{-j2\pi
mn/2N}\sum^{N-1}_{k=-N}v_{k-n}e^{-j2\pi m(k-n)/2N}$ $\displaystyle=$
$\displaystyle\sum^{N-1}_{n=-N}u_{n}e^{-j2\pi
mn/2N}\sum^{N-1}_{k=-N}v_{k}e^{-j2\pi mk/2N}$
This gives us
$\displaystyle\hat{w}_{m}=\hat{u}_{m}\cdot\hat{v}_{m},\mbox{ for
}m=-N,\ldots,N-1.$
Applying the inverse FFT on the above equation, we recover the vector $w_{m}$,
$m=-N,\ldots,N-1$. The vector $w_{m}$, $m=0,\ldots,N-1$ is the desired result.
|
arxiv-papers
| 2012-10-06T04:58:54 |
2024-09-04T02:49:36.065151
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.C. Yen, R.E. Taam, Ken H.C. Yeh, K.C. Chang",
"submitter": "Chien-Chang Yen",
"url": "https://arxiv.org/abs/1210.1912"
}
|
1210.1963
|
# A new proof of subcritical Trudinger-Moser inequalities on the whole
Euclidean space
Yunyan Yang yunyanyang@ruc.edu.cn Xiaobao Zhu zhuxiaobao@ruc.edu.cn
Department of Mathematics, Renmin University of China, Beijing 100872, P. R.
China
###### Abstract
In this note, we give a new proof of subcritical Trudinger-Moser inequality on
$\mathbb{R}^{n}$. All the existing proofs on this inequality are based on the
rearrangement argument with respect to functions in the Sobolev space
$W^{1,n}(\mathbb{R}^{n})$. Our method avoids this technique and thus can be
used in the Riemannian manifold case and in the entire Heisenberg group.
###### keywords:
Trudinger-Moser inequality, Adams inequality
###### MSC:
46E30
††journal: ***
## 1 Introduction
It was proved by Cao [4], Panda [9] and do Ó [5] that
Theorem A Let $\alpha_{n}=n\omega_{n-1}^{\frac{1}{n-1}}$, where $\omega_{n-1}$
is the measure of the unit sphere in $\mathbb{R}^{n}$. Then for any
$\alpha<\alpha_{n}$ there holds
$\displaystyle\sup_{u\in
W^{1,n}(\mathbb{R}^{n}),\,\,\int_{\mathbb{R}^{n}}(|\nabla\
u|^{n}+|u|^{n})dx\leq
1}\int_{\mathbb{R}^{n}}\left(e^{\alpha|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx<\infty.$
(1.1)
This result has various extensions, among which we mention Adachi and Tanaka
[1], Ruf [11], Li-Ruf [7], Adimurthi-Yang [3]. To the authors’ knowledge, all
the existing proofs of such an inequality are based on rearrangement argument
with respect to functions in the Sobolev space $W^{1,n}(\mathbb{R}^{n})$. The
purpose of this short note is to provide a new method to reprove Theorem A.
Namely, we use a technique of the analogy of unity decomposition. More
precisely, for any $u\in W^{1,n}(\mathbb{R}^{n})$, we first take a cut-off
function $\phi_{i}\in C_{0}^{\infty}(B_{R}(x_{i}))$ such that
$0\leq\phi_{i}\leq 1$ on $B_{R}(x_{i})$, $\phi_{i}\equiv 1$ on
$B_{R/2}(x_{i})$. Then, using the usual Trudinger-Moser inequality [8, 10, 13]
for bounded domain, we prove a key estimate
$\int_{\mathbb{R}^{n}}\left(e^{\alpha|\phi_{i}u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}|\phi_{i}u|^{\frac{nk}{n-1}}}{k!}\right)dx\leq\int_{\mathbb{R}^{n}}|\nabla(\phi_{i}u)|^{n}dx$
(1.2)
under the condition that
$\int_{\mathbb{R}^{n}}|\nabla(\phi_{i}u)|^{n}dx\leq 1.$
The power of (1.2) is evident. It permits us to approximate $u$ by
$\sum_{i}\phi_{i}u$, where every $\phi_{i}$ is supported in $B_{R}(x_{i})$,
$\mathbb{R}^{n}=\cup_{i=1}^{\infty}B_{R/2}(x_{i})$, and any fixed
$x\in\mathbb{R}^{n}$ belongs to at most $c(n)$ balls $B_{R}(x_{i})$ for some
universal constant $c(n)$. If we further take $\phi_{i}$ such that
$|\nabla\phi_{i}|\leq 4/R$. Note that for any $\epsilon>0$ there exists a
constant $C(\epsilon)$ such that
$\int_{\mathbb{R}^{n}}|\nabla(\phi_{i}u)|^{n}dx\leq(1+\epsilon)\int_{\mathbb{R}^{n}}|\nabla
u|^{n}dx+\frac{C(\epsilon)}{R^{n}}\int_{\mathbb{R}^{n}}|u|^{n}dx.$
Selecting $\epsilon>0$ sufficiently small and $R>0$ sufficiently large, we get
the desired result.
Similar idea was used by the first named author to deal with similar problems
on complete Riemannian manifolds [14] or the entire Heisenberg group [16].
Note that due to the complicated geometric structure, we have not obtained
Theorem A on manifolds, but a weaker result. Namely
Theorem B Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold.
Suppose that its Ricci curvature has lower bound, namely ${\rm Rc}_{(M,g)}\geq
Kg$ for some constant $K\in\mathbb{R}$, and its injectivity radius is strictly
positive, namely ${\rm inj}_{(M,g)}\geq i_{0}$ for some constant $i_{0}>0$.
Then we have
$(i)$ for any $0\leq\alpha<\alpha_{n}$ there exists positive constants $\tau$
and $\beta$ depending only on $n$, $\alpha$, $K$ and $i_{0}$ such that
$\sup_{u\in W^{1,n}(M),\,\|u\|_{1,\tau}\leq
1}\int_{M}\left(e^{\alpha|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dv_{g}\leq\beta,$
(1.3)
where
$\|u\|_{1,\tau}=\left(\int_{M}|\nabla_{g}u|^{n}dv_{g}\right)^{1/n}+\tau\left(\int_{M}|u|^{n}dv_{g}\right)^{1/n}.$
(1.4)
As a consequence, $W^{1,n}(M)$ is embedded in $L^{q}(M)$ continuously for all
$q\geq n$;
$(ii)$ for any $\alpha>\alpha_{n}$ and any $\tau>0$, the supremum in (1.3) is
infinite;
$(iii)$ for any $u\in W^{1,n}(M)$ and any $\alpha>0$, the integrals in (1.3)
are still finite.
We say more words about this method. For Sobolev inequalities on complete
noncompact Riemannian manifolds, unity decomposition was employed by Hebey et
al. [6]. In the case of Trudinger-Moser inequality, it is not evidently
applicable. We are lucky to find its analogy ([14], Lemma 4.1).
## 2 Preliminary lemmas
We first give a local estimate concerning the Trudinger-Moser functional.
Precisely we have
Lemma 1 For any $x_{0}\in\mathbb{R}^{n}$ and any $u\in
W_{0}^{1,n}(B_{R}(x_{0}))$, $\int_{B_{R}(x_{0})}|\nabla u|^{n}dx\leq 1$, we
have
$\displaystyle\int_{B_{R}(x_{0})}\left(e^{\alpha_{n}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{n}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx\leq
C(n)R^{n}\int_{B_{R}(x_{0})}|\nabla u|^{n}dx,$ (2.1)
where $C(n)$ is a constant depending only on $n$.
Proof. Essentially this is the same as ([14], Lemma 4.1). For reader’s
convenience we give the details here. It is well known [8, 10, 13] that
$\displaystyle\sup_{u\in W_{0}^{1,n}(B_{R}(x_{0})),\int_{B_{R}(x_{0})}|\nabla
u|^{n}dx\leq 1}\int_{B_{R}(x_{0})}e^{\alpha_{n}|u|^{\frac{n}{n-1}}}dx\leq
C(n)R^{n}.$ (2.2)
Letting $\widetilde{u}=\frac{u}{||\nabla u||_{L^{n}({B_{R}(x_{0})})}}$ for any
$u\in W_{0}^{1,n}(B_{R}(x_{0}))\setminus\\{0\\}$, we have
$\displaystyle\int_{B_{R}(x_{0})}\left(e^{\alpha_{n}|\widetilde{u}|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{n}^{k}|\widetilde{u}|^{\frac{nk}{n-1}}}{k!}\right)dx$
$\displaystyle\geq$ $\displaystyle\frac{1}{||\nabla
u||_{L^{n}({B_{R}(x_{0})})}}\int_{B_{R}(x_{0})}\sum_{k=n-1}^{\infty}\frac{\alpha_{n}^{k}|u|^{\frac{nk}{n-1}}}{k!}dx$
(2.3) $\displaystyle=$ $\displaystyle\frac{1}{||\nabla
u||_{L^{n}({B_{R}(x_{0})})}}\int_{B_{R}(x_{0})}\left(e^{\alpha_{n}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{n}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx.\quad$
Combining (2.2) and (2.3), we get the desired result. $\hfill\Box$
Also we need a covering lemma of $\mathbb{R}^{n}$, see for example ([6], Lemma
1.6).
Lemma 2 For any $R>0$, there exists a sequence
$\\{x_{i}\\}_{i=1}^{\infty}\subset\mathbb{R}^{n}$ such that
$(i)$ $\cup_{i=1}^{\infty}B_{{R}/{2}}(x_{i})=\mathbb{R}^{n}$;
$(ii)$ $\forall i\neq j,\,B_{{R}/{4}}(x_{i})\cap
B_{{R}/{4}}(x_{j})=\varnothing$;
$(iii)$ $\forall x\in\mathbb{R}^{n}$, $x$ belongs to at most $N$ balls
$B_{R}(x_{i})$ for some integer $N$.
## 3 Proof of Theorem A
We shall obtain a global inequality (1.1) by gluing local estimates (2.1).
Proof of Theorem A. Let $R>0$ to be determined later. Let $\phi_{i}$ be the
cut-off function satisfies the following conditions: $(i)$ $\phi_{i}\in
C_{0}^{\infty}(B_{R}(x_{i}))$; $(ii)$ $0\leq\phi_{i}\leq 1$ on $B_{R}(x_{i})$
and $\phi_{i}\equiv 1$ on $B_{{R}/{2}}(x_{i})$; $(iii)$
$|\nabla\phi_{i}(x)|\leq{4}/{R}$. For $u\in W^{1,n}(\mathbb{R}^{n})$
satisfying
$\displaystyle\int_{\mathbb{R}^{n}}(|\nabla u|^{n}+|u|^{n})dx\leq 1,$ (3.1)
we have $\phi_{i}u\in W_{0}^{1,n}(B_{R}(x_{i}))$, using Cauchy inequality with
$\epsilon$ term we obtain
$\displaystyle\int_{B_{R}(x_{i})}|\nabla(\phi_{i}u)|^{n}dx\leq$
$\displaystyle(1+\epsilon)\int_{B_{R}(x_{i})}\phi_{i}^{n}|\nabla
u|^{n}dx+C(\epsilon)\int_{B_{R}(x_{i})}|\nabla\phi_{i}|^{n}|u|^{n}dx$
$\displaystyle\leq$ $\displaystyle(1+\epsilon)\int_{B_{R}(x_{i})}|\nabla
u|^{n}dx+\frac{C(\epsilon)}{R^{n}}\int_{B_{R}(x_{i})}|u|^{n}dx$
$\displaystyle\leq$ $\displaystyle(1+\epsilon)\int_{B_{R}(x_{i})}(|\nabla
u|^{n}+|u|^{n})dx,$ (3.2)
where in the last inequality we choose a sufficiently large $R$ to make sure
$\frac{C(\epsilon)}{R^{n}}\leq(1+\epsilon)$. Let
$\alpha_{\epsilon}=\frac{\alpha_{n}}{(1+\epsilon)^{1/(n-1)}}$ and
$\widetilde{\phi_{i}u}=\frac{\phi_{i}u}{(1+\epsilon)^{1/n}}$. Noting that
$\widetilde{\phi_{i}u}\in W_{0}^{1,n}(B_{R}(x_{i}))$, we have by (3) and Lemma
1
$\displaystyle\int_{B_{\frac{R}{2}}(x_{i})}\left(e^{\alpha_{\epsilon}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{\epsilon}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx\leq$
$\displaystyle\int_{B_{R}(x_{i})}\left(e^{\alpha_{\epsilon}|\phi_{i}u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{\epsilon}^{k}|\phi_{i}u|^{\frac{nk}{n-1}}}{k!}\right)dx$
$\displaystyle=$
$\displaystyle\int_{B_{R}(x_{i})}\left(e^{\alpha_{n}|\widetilde{\phi_{i}u}|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{n}^{k}|\widetilde{\phi_{i}u}|^{\frac{nk}{n-1}}}{k!}\right)dx$
$\displaystyle\leq$ $\displaystyle
C(n)R^{n}\int_{B_{R}(x_{i})}|\nabla(\widetilde{\phi_{i}u})|^{n}dx$
$\displaystyle\leq$ $\displaystyle C(n)R^{n}\int_{B_{R}(x_{i})}(|\nabla
u|^{n}+|u|^{n})dx.$ (3.3)
By Lemma 2 and (3), we have
$\displaystyle\int_{\mathbb{R}^{n}}\left(e^{\alpha_{\epsilon}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{\epsilon}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx\leq$
$\displaystyle\int_{\cup_{i=1}^{\infty}B_{\frac{R}{2}}(x_{i})}\left(e^{\alpha_{\epsilon}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{\epsilon}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{\infty}\int_{B_{\frac{R}{2}}(x_{i})}\left(e^{\alpha_{\epsilon}|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha_{\epsilon}^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dx$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{\infty}C(n)R^{n}\int_{B_{R}(x_{i})}(|\nabla
u|^{n}+|u|^{n})dx$ $\displaystyle\leq$ $\displaystyle
C(n)R^{n}N\int_{\mathbb{R}^{n}}(|\nabla u|^{n}+|u|^{n})dx$ $\displaystyle\leq$
$\displaystyle C(n)R^{n}N.$ (3.4)
For any $\alpha<\alpha_{n}$, we can choose $\epsilon>0$ sufficiently small
such that $\alpha<\alpha_{\epsilon}$. This ends the proof of Theorem A.
$\hfill\Box$
## 4 Concluding remarks
Using the same idea to prove Theorem A, we can also prove the subcritical
Adams inequality in $\mathbb{R}^{n}$ [2, 12, 15], which strengthen ([14],
Theorem 2.6). Since the proof is completely analogous to our proof of Theorem
A, we leave it to the reader.
Acknowledgement. This work is supported by the NSFC 11171347.
## References
* [1] S. Adachi, K. Tanaka, Trudinger type inequalities in $\mathbb{R}^{N}$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000) 2051-2057.
* [2] D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math. 128 (1988) 385-398.
* [3] Adimurthi, Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^{N}$ and its applications, Internat. Mathematics Research Notices 13 (2010) 2394-2426.
* [4] D. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^{2}$, Commun. Partial Differential Equations 17 (1992) 407-435.
* [5] J. M. do Ó, $N$-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal. 2 (1997) 301-315.
* [6] E. Hebey, Sobolev spaces on Riemannian maifolds, Lecture notes in mathematics 1635, Springer, 1996.
* [7] Y. Li, B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Ind. Univ. Math. J. 57 (2008) 451-480.
* [8] J. Moser, A sharp form of an inequality by N.Trudinger, Ind. Univ. Math. J. 20 (1971) 1077-1091.
* [9] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^{n}$, Proc. Indian Acad. Sci. (Math. Sci.) 105 (1995) 425-444.
* [10] S. Pohozaev, The Sobolev embedding in the special case $pl=n$, Proceedings of the technical scientific conference on advances of scientific reseach 1964-1965, Mathematics sections, 158-170, Moscov. Energet. Inst., Moscow, 1965.
* [11] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{2}$, J. Funct. Anal. 219 (2005) 340-367.
* [12] B. Ruf, F. Sani, Sharp Adams-type inequalities in $\mathbb{R}^{n}$. Trans. Amer. Math. Soc. (In press).
* [13] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-484.
* [14] Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal. 263 (2012) 1894-1938.
* [15] Y. Yang, Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differ. Equations 252 (2012) 2266-2295.
* [16] Y. Yang, Trudinger-Moser inequalities on the entire Heisenberg group, arXiv:1201.2993.
|
arxiv-papers
| 2012-10-06T14:27:55 |
2024-09-04T02:49:36.080833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yunyan Yang, Xiaobao Zhu",
"submitter": "Yunyan Yang",
"url": "https://arxiv.org/abs/1210.1963"
}
|
1210.2031
|
# Curvature estimates for minimal submanifolds of higher codimension and small
G-rank
J. Jost, Y. L. Xin and Ling Yang Max Planck Institute for Mathematics in the
Sciences, Inselstr. 22, 04103 Leipzig, Germany, and
Department of Mathematics, University of Leipzig, 04081 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. yanglingfd@fudan.edu.cn
###### Abstract.
We obtain new curvature estimates and Bernstein type results for minimal
$n-$submanifolds in $\mathbb{R}^{n+m},\,m\geq 2$ under the condition that the
rank of its Gauss map is at most 2. In particular, this applies to minimal
surfaces in Euclidean spaces of arbitrary codimension.
###### 1991 Mathematics Subject Classification:
58E20,53A10.
The first author is supported by the ERC Advanced Grant FP7-267087. The second
author and the third author are supported partially by NSFC. They are also
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 says that an entire minimal graph in
$\mathbb{R}^{3}$ has to be an affine plane. The mathematics behind this
theorem has proved to be enormously rich. It connects with differential
geometry, partial differential equations, and complex analysis, and it has
been a stimulus for important developments in all these fields (for some
references, see for instance [17]). In particular, the question emerged and
has been intensively investigated to what extent this result can be
generalized in various directions, that is, under which conditions a minimal
submanifold of some Euclidean space (or a sphere) is necessarily affine linear
(or a sub-sphere).
In particular, Bernstein type theorems for higher dimension and codimension
have been studied. Thus, let $M\to\mathbb{R}^{n+m}$ be an $n$ dimensional
submanifold in Euclidean space $\mathbb{R}^{n+m}$. In recent work ([10], [11]
and [12]), we have systematically used geometric properties of Grassmannian
manifolds and the regularity theory of harmonic maps to obtain new Bernstein
type results for higher dimension $n\geq 3$ and codimension $m\geq 2$. The key
point is that the Gauss map of such a minimal submanifold is a harmonic map
with values in a Grassmann manifold. Thus, our approach combines methods from
differential geometry and partial differential equations. This leads us to the
question whether this can also be combined with the complex analysis approach.
The complex analysis approach is, of course, naturally restricted to the case
$n=2$, if, for the sake of the discussion, we ignore such issues as
subvarieties of complex spaces.
Thus, the present paper is concerned with the case $n=2$ and $m\geq 2$. Now,
the target manifold of the Gauss map is $\mathbb{G}_{2,m}$, the complex
quadric, and the Gauss map is holomorphic. A powerful traditional approach to
this problem investigates the value distribution of the Gauss image within the
framework of complex geometry. This was started by Chern and Osserman [2].
From their results, an analogue of Moser’s Bernstein theorem [13] that works
for $n\geq 2$ and $m=1$ also holds for the case $n=2$ and $m\geq 2$. More
precisely, for an entire minimal graph given by
$f:\mathbb{R}^{2}\to\mathbb{R}^{m}$ if
$\Delta_{f}=[\det(\delta_{ij}+f^{\alpha}_{i}f^{\alpha}_{j})]^{\frac{1}{2}}$ is
uniformly bounded, then $f$ is affine linear, and thus represents an affine
plane in $\mathbb{R}^{2+m}$.
In the present paper, we use curvature estimate techniques to improve this
result. This will also enable us to achieve some technical generalization
which we shall now formulate. For a minimal $n-$submanifold $M$ in
$\mathbb{R}^{n+m}$ we consider the rank of the Gauss map, which is called the
$G-rank$ for simplicity. Our condition then simply is $G-rank\leq 2$.
Obviously, this class of submanifolds contains surfaces in $\mathbb{R}^{2+m}$,
as well as cylinders over surfaces in $\mathbb{R}^{3}$. In [3], Dajczer and
Florit gave a parametric description of all Euclidean minimal submanifolds of
$G-rank=2$. In particular, they showed that complete minimal submanifolds with
$G-rank=2$ have dimension $n=3$ at most (without Euclidean factor).
Here, then, are our main results.
###### Theorem 1.1.
Let $M$ be an $n$-dimensional complete minimal submanifold in ${\tenmsb
R}^{n+m}$ with $G-rank\leq 2$ and positive $w$-function (see (2.3). If $M$ has
polynomial volume growth and the function $v=w^{-1}$ has growth
(1.1) $\max_{D_{R}(p_{0})}v=o(R^{\frac{2}{3}})$
for a fixed point $p_{0}$, then $M$ has to be an affine linear subspace.
Then, we have
###### Theorem 1.2.
Let $M=\\{(x,f(x)):x\in{\tenmsb R}^{n}\\}$ be an entire minimal graph given by
a vector-valued function $f:{\tenmsb R}^{n}\rightarrow{\tenmsb R}^{m}$ with
$G-rank\leq 2$. If the slope of $f$ satisfies
(1.2) $\Delta_{f}=\left[\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\right]^{\frac{1}{2}}=o(R^{\frac{2}{3}}),$
where $R^{2}=|x|^{2}+|f(x)|^{2}$, then $f$ has to be an affine linear
function.
###### Theorem 1.3.
Let $f:{\tenmsb R}^{2}\rightarrow{\tenmsb R}^{m}$
$(x^{1},x^{2})\mapsto(f^{1},\cdots,f^{m})$ be an entire solution of the
minimal surface system
(1.3) $\Big{(}1+\Big{|}\frac{\partial f}{\partial
x^{2}}\Big{|}^{2}\Big{)}\frac{\partial^{2}f}{(\partial
x^{1})^{2}}-2\Big{\langle}\frac{\partial f}{\partial x^{1}},\frac{\partial
f}{\partial x^{2}}\Big{\rangle}\frac{\partial^{2}f}{\partial x^{1}\partial
x^{2}}+\Big{(}1+\Big{|}\frac{\partial f}{\partial
x^{1}}\Big{|}^{2}\Big{)}\frac{\partial^{2}f}{(\partial x^{2})^{2}}=0.$
If there exists $\varepsilon>0$,
(1.4) $\Delta_{f}=\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}^{\frac{1}{2}}=O(R^{1-\varepsilon})$
with $R=|x|$, then $f$ has to be affine linear.
This is an improvement of the Chern-Osserman theorem mentioned above which
required $\Delta_{f}$ to be bounded.
The paper is organized as follows. After §2 on basic notation and formulae we
describe the special features of the $G-rank\leq 2$ case in §3. Then in §4 ,
using subharmonic functions obtained from the geometry of the Grassmann
manifolds and a Bochner type formula for the squared norm of the second
fundamental form $|B|^{2}$ we can obtain $L^{p}-$estimates and point-wise
estimates for $|B|^{2}$. Those estimates lead to Bernstein type results. §5 is
devoted to the graphic situation. In the final section we discuss the
sharpness of our estimates. We find that holomorphic curves reach all the
possible equalities in all the geometric inequalities (2.8), (3.14) and
(3.22).
We thank Marcos Dajczer for informing us about [3].
## 2\. Fundamental formulas
Let $M$ be an $n$-dimensional submanifold in an $(n+m)$-dimensional Riemannian
manifold $\bar{M}$ with second fundamental form $B.$ Let $\bar{\nabla}$ denote
the Levi-Civita connection on $\bar{M}$. It naturally induces connections on
the tangent bundle $TM$, the normal bundle $NM$ and various induced bundles
over $M.$ For notational simplicity all of them are denoted by $\nabla$. For
arbitrary $\nu\in\Gamma(NM)$ the shape operator $A^{\nu}:TM\rightarrow TM$
satisfies $\langle B_{X,Y},\nu\rangle=\langle A^{\nu}(X),Y\rangle$ for every
$X,Y\in\Gamma(TM)$. It is self adjoint in the tangent spaces of $M.$ The mean
curvature field $H$ is defined to be the trace of the second fundamental form.
$M$ is called a minimal submanifold whenever $H$ vanishes on $M$ everywhere.
The second fundamental form, the curvature tensor of the submanifold, the
curvature tensor of the normal bundle and that of the ambient manifold are
connected by the Gauss equations, the Codazzi equations and the Ricci
equations (see [17], §1.1).
In this paper we consider a minimal submanifold $M$ of dimension $n$ in the
Euclidean space ${\tenmsb R}^{n+m}$ with codimension $m\geq 2$.
Now, there is an important tool, the Gauss map. The Gauss map
$\gamma:M\rightarrow\mathbb{G}_{n,m}$ is defined by
$\gamma(p)=T_{p}M\in\mathbb{G}_{n,m}$
via the parallel translation in ${\tenmsb R}^{n+m}$ for every $p\in M$, where
$\mathbb{G}_{n,m}$ is the Grassmann manifold consisting of the oriented linear
$n$-subspaces in ${\tenmsb R}^{n+m}$. One can write
$\gamma(p)=e_{1}\wedge\cdots\wedge e_{n}$
by using Plücker coordinates. Here and in the sequel, $\\{e_{i}\\}$ is a local
orthonormal tangent frame field of $M$ and $\\{\nu_{\alpha}\\}$ denotes a
local orthonormal normal frame field of $M$; we use the summation convention
and agree on the following ranges of indices:
$1\leq i,j,k\leq n;\qquad 1\leq\alpha,\beta,\gamma\leq m.$
Let
$h_{\alpha,ij}:=\langle B_{e_{i}e_{j}},\nu_{\alpha}\rangle$
be the coefficients of the second fundamental form $B$ of $M$ in ${\tenmsb
R}^{n+m}.$ Then,
(2.1) $\gamma_{*}e_{i}=h_{\alpha,ij}e_{j\alpha},$
where $e_{j\alpha}$ is obtained by replacing $e_{j}$ by $\nu_{\alpha}$ in
$e_{1}\wedge\cdots\wedge e_{n}$. The energy density of the Gauss map thus is
nothing but the squared norm of the second fundamental form (see [17] §3.1),
$e(\gamma)=\frac{1}{2}\langle\gamma_{*}e_{i},\gamma_{*}e_{i}\rangle=\frac{1}{2}\sum_{\alpha,i,j}h_{\alpha,ij}^{2}=\frac{1}{2}|B|^{2}.$
Given two unit $n$-vectors
$A=a_{1}\wedge\cdots\wedge a_{n},\qquad B=b_{1}\wedge\cdots\wedge b_{n}$
in the Grassmann manifold $\mathbb{G}_{n,m}$, their inner product is defined
by
(2.2) $\langle A,B\rangle=\det\big{(}\langle a_{i},b_{j}\rangle\big{)}$
Fixing a simple unit $n$-vector
$A=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$, we define the
$w$-function on $M$:
(2.3) $w(p):=\langle e_{1}\wedge\cdots\wedge e_{n},A\rangle=\det\big{(}\langle
e_{i},\varepsilon_{j}\rangle\rangle\big{)}.$
Via the Plücker imbedding, the Grassmann manifold $\mathbb{G}_{n,m}$ can be
viewed as a submanifold in a Euclidean space, and the $w$-function can be
regarded as the composition of the Gauss map and a height function on
$\mathbb{G}_{n,m}$ (see [20], [11] and [12]). In particular, if
$M=\big{(}x,f(x)\big{)}$ is a graph in ${\tenmsb R}^{n+m}$ given by a vector-
valued function $f:{\tenmsb R}^{n}\rightarrow{\tenmsb R}^{m}$, then choosing
$A$ to be one representing $(x_{1},\cdots,x_{n})$ coordinate $n$-plane implies
$w>0$ and moreover
(2.4) $v:=w^{-1}$
equals the volume element of $M$ (see [10]).
The Codazzi equations yield the following formulas for the $w$-function:
###### Lemma 2.1.
[6][18] If $M$ is a submanifold in ${\tenmsb R}^{n+m}$, then
(2.5) $\nabla_{e_{i}}w=h_{\alpha,ij}\langle
e_{j\alpha},\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}\rangle.$
Moreover if $M$ has parallel mean curvature, i.e. $\nabla H\equiv 0$, then
(2.6) $\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
(2.7)
$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.
To have the curvature estimates we need the Simons’ version of the Bochner
type formula for the squared norm of the second fundamental form. A
straightforward calculation shows (see [16], (2.6) in [18])
(2.8) $\Delta|B|^{2}=2|\nabla B|^{2}+2\langle\nabla^{2}B,B\rangle\geq 2|\nabla
B|^{2}-3|B|^{4}.$
## 3\. Small G-rank cases
The rank of the Gauss map for a submanifold $M$ in $\mathbb{R}^{n+m}$ is
closely related to rigidity problems. The classical Beez-Killing theorem is a
local rigidity property for hypersurfaces in $\mathbb{R}^{n+1}$ when
$G-rank\geq 3$. For global investigations, we refer to [4]. Here, we study the
case of $G-rank\leq 2$.
Now, for every $p\in M$, we have
$\dim\text{Ker}(\gamma_{*})_{p}=n-\text{rank}(\gamma_{*})_{p}\geq n-2.$
Then for any $p_{0}\in M$, there exists a local smooth distribution
$\mathcal{K}$ of dimension $n-2$ on $U\ni p_{0}$, such that
$\mathcal{K}_{p}\subset\text{Ker}(\gamma_{*})_{p}$ for any $p\in U$.
$\mathcal{K}$ is called the relative nullity distribution by Chern-Kuiper [1].
This is an integrable distribution. Therefore, one can find a local tangent
orthonormal frame field $\\{e_{i}\\}$, such that
$\mathcal{K}_{p}=\text{span}\\{e_{i}(p):i\geq 3\\}$, i.e.
(3.1) $\gamma_{*}e_{i}=0\qquad 3\leq i\leq n$
and it follows from (2.1) that $h_{\alpha,ij}=0$, i.e.
(3.2) $B_{e_{i}e_{j}}=0\qquad\text{whenever }i\geq 3\text{ or }j\geq 3.$
Hence
(3.3) $0=H=\sum_{i=1}^{n}B_{e_{i}e_{i}}=B_{e_{1}e_{1}}+B_{e_{2}e_{2}}.$
At the considered point, let
(3.4) $G(e_{1},e_{2}):=\left(\begin{array}[]{cc}\langle
B_{e_{1}e_{1}},B_{e_{1}e_{1}}\rangle&\langle
B_{e_{1}e_{1}},B_{e_{1}e_{2}}\rangle\\\ \langle
B_{e_{1}e_{2}},B_{e_{1},e_{1}}\rangle&\langle
B_{e_{1}e_{2}},B_{e_{1}e_{2}}\rangle\end{array}\right).$
$G$ then is a semi-positive definite matrix, whose eigenvalues are denoted by
$\mu_{1}^{2}$ and $\mu_{2}^{2}$ ($\mu_{1}\geq\mu_{2}\geq 0$) . Then there
exists an orthogonal matrix
(3.5) $O=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right)$
such that
(3.6) $G=O\left(\begin{array}[]{cc}\mu_{1}^{2}&\\\
&\mu_{2}^{2}\end{array}\right)O^{T}.$
Now we put
(3.7) $f_{1}=\cos\alpha e_{1}-\sin\alpha e_{2}\qquad f_{2}=\sin\alpha
e_{1}+\cos\alpha e_{2}$
with $\alpha$ to be chosen, then
$\displaystyle B_{f_{1}f_{1}}$ $\displaystyle=\cos^{2}\alpha
B_{e_{1}e_{1}}+\sin^{2}\alpha B_{e_{2}e_{2}}-2\cos\alpha\sin\alpha
B_{e_{1}e_{2}}$
$\displaystyle=\cos(2\alpha)B_{e_{1}e_{1}}-\sin(2\alpha)B_{e_{1}e_{2}}$
$\displaystyle B_{f_{1}f_{2}}$ $\displaystyle=\cos\alpha\sin\alpha
B_{e_{1}e_{1}}-\cos\alpha\sin\alpha
B_{e_{2}e_{2}}+(\cos^{2}\alpha-\sin^{2}\alpha)B_{e_{1}e_{2}}$
$\displaystyle=\sin(2\alpha)B_{e_{1}e_{1}}+\cos(2\alpha)B_{e_{1}e_{2}}.$
Thus
(3.8) $G(f_{1},f_{2})=\left(\begin{array}[]{cc}\cos(2\alpha)&-\sin(2\alpha)\\\
\sin(2\alpha)&\cos(2\alpha)\end{array}\right)G(e_{1},e_{2})\left(\begin{array}[]{cc}\cos(2\alpha)&-\sin(2\alpha)\\\
\sin(2\alpha)&\cos(2\alpha)\end{array}\right)^{T}$
Choosing $\alpha=-\frac{\theta}{2}$ and combining with (3.5), (3.6) and (3.8)
gives
$G(f_{1},f_{2})=\left(\begin{array}[]{cc}\mu_{1}^{2}&\\\
&\mu_{2}^{2}\end{array}\right).$
Therefore, by carefully choosing local tangent frames and normal frames, one
can assume that at the considered point
(3.9) $A^{1}=\left(\begin{array}[]{cccc}\mu_{1}&0&&\\\ 0&-\mu_{1}&&\\\ &&&\\\
&&\lx@intercol\hfil\raisebox{2.15277pt}[0.0pt]{\Huge
o}\hfil\lx@intercol&\end{array}\right)\qquad
A^{2}=\left(\begin{array}[]{cccc}0&\mu_{2}&&\\\ \mu_{2}&0&&\\\ &&&\\\
&&\lx@intercol\hfil\raisebox{2.15277pt}[0.0pt]{\Huge
o}\hfil\lx@intercol&\end{array}\right)$
and $A^{\alpha}=0$ for each $\alpha\geq 3$, where
$A^{\alpha}:=A^{\nu_{\alpha}}$ is the shape operator.
In this case, (2.6) can be rewritten as
(3.10) $\displaystyle\Delta w$ $\displaystyle=-|B|^{2}w+2\sum_{i}\sum_{j\neq
k}h_{1,ij}h_{2,ik}\langle e_{j1,k2},A\rangle$
$\displaystyle=-|B|^{2}w+2h_{1,11}h_{2,12}\langle
e_{11,22},A\rangle+2h_{1,22}h_{2,21}\langle e_{21,12},A\rangle$
$\displaystyle=-|B|^{2}w+4\mu_{1}\mu_{2}\langle e_{11,22},A\rangle$
where $A=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$ and the last step
follows from $e_{11,22}=-e_{21,12}=\nu_{1}\wedge\nu_{2}\wedge
e_{3}\wedge\cdots\wedge e_{n}$. By (2.5),
$\displaystyle\nabla_{e_{1}}w$ $\displaystyle=h_{1,11}\langle
e_{11},A\rangle+h_{2,12}\langle e_{22},A\rangle=\mu_{1}\langle
e_{11},A\rangle+\mu_{2}\langle e_{22},A\rangle$ $\displaystyle\nabla_{e_{2}}w$
$\displaystyle=h_{1,22}\langle e_{21},A\rangle+h_{2,21}\langle
e_{12},A\rangle=-\mu_{1}\langle e_{21},A\rangle+\mu_{2}\langle
e_{12},A\rangle$
and $\nabla_{e_{i}}w=0$ for every $i\geq 3$. Hence
(3.11) $\displaystyle|\nabla w|^{2}=$
$\displaystyle\sum_{i}|\nabla_{e_{i}}w|^{2}=\big{(}\mu_{1}\langle
e_{11},A\rangle+\mu_{2}\langle
e_{22},A\rangle\big{)}^{2}+\big{(}-\mu_{1}\langle
e_{21},A\rangle+\mu_{2}\langle e_{12},A\rangle\big{)}^{2}$ $\displaystyle=$
$\displaystyle\big{(}\mu_{1}\langle e_{11},A\rangle-\mu_{2}\langle
e_{22},A\rangle\big{)}^{2}+\big{(}\mu_{1}\langle
e_{21},A\rangle+\mu_{2}\langle e_{12},A\rangle\big{)}^{2}$
$\displaystyle+4\mu_{1}\mu_{2}\big{(}\langle e_{11},A\rangle\langle
e_{22},A\rangle-\langle e_{21},A\rangle\langle e_{12},A\rangle\big{)}$
By Lemma 3.2 of [12],
(3.12) $\langle e_{1}\wedge\cdots\wedge e_{n},A\rangle\langle
e_{11,22},A\rangle-\langle e_{11},A\rangle\langle e_{22},A\rangle+\langle
e_{12},A\rangle\langle e_{21},A\rangle=0.$
In conjunction with (3.10), (3.11) and (3.12), we have
(3.13) $\displaystyle\Delta\log w$ $\displaystyle=w^{-2}(w\Delta w-|\nabla
w|^{2})$ $\displaystyle=-|B|^{2}-w^{-2}\Big{[}\big{(}\mu_{1}\langle
e_{11},A\rangle-\mu_{2}\langle
e_{22},A\rangle\big{)}^{2}+\big{(}\mu_{1}\langle
e_{21},A\rangle+\mu_{2}\langle e_{12},A\rangle\big{)}^{2}\Big{]}$
whenever $w>0$. We thus have the following results from our previous paper
###### Proposition 3.1.
[12] Let $M$ be a minimal submanifold of ${\tenmsb R}^{n+m}$ with $G-rank\leq
2$ and $w>0$. Then,
(3.14) $\Delta\log w\leq-|B|^{2}.$
###### Definition 3.1.
Let $M$ be an $n$-dimensional minimal submanifold in $\mathbb{R}^{n+m}$ ,
($m\geq 2$). A point $p\in M$ is called a G-conformal point, if there exists
an orthonormal basis $\\{e_{1},\cdots,e_{n}\\}$ of $T_{p}M$, such that
$B_{e_{i}e_{j}}=0$ whenever $i\geq 3$ or $j\geq 3$, and
$\langle
B_{e_{1}e_{1}},B_{e_{1}e_{2}}\rangle=0,\qquad|B_{e_{1}e_{1}}|=|B_{e_{1}e_{2}}|.$
Moreover if each point of $M$ is a G-conformal point, we call $M$ a totally
G-conformal minimal submanifold.
The formula (2.8) is for minimal submanifolds in $\mathbb{R}^{n+m}$ with
codimension $m\geq 2$. In the present situation we can derive it directly and
analyze its accuracy. A straightforward calculation shows (see [16] [17])
(3.15) $\nabla^{2}B=-\tilde{\mathcal{B}}-\underline{\mathcal{B}}.$
Here $\nabla^{2}$ denotes the trace-Laplace operator acting on any cross-
section of a vector bundle over $M$,
(3.16) $\tilde{\mathcal{B}}:=B\circ B^{t}\circ B$
with $B^{t}$ denoting the conjugate map of $B$, and
(3.17)
$\underline{\mathcal{B}}:=\sum_{\alpha=1}^{m}\big{(}B_{A^{\alpha}A^{\alpha}(X),Y}+B_{X,A^{\alpha}A^{\alpha}(Y)}-2B_{A^{\alpha}(X),A^{\alpha}(Y)}\big{)}.$
Hence
(3.18) $\displaystyle\langle\tilde{\mathcal{B}},B\rangle$
$\displaystyle=\langle B\circ B^{t}\circ B,B\rangle=\langle B^{t}\circ
B,B^{t}\circ B\rangle$ $\displaystyle=\langle
B_{e_{i}e_{j}},B_{e_{k}e_{l}}\rangle\langle
B_{e_{i}e_{j}},B_{e_{k}e_{l}}\rangle=h_{\alpha,ij}h_{\alpha,kl}h_{\beta,ij}h_{\beta,kl}$
$\displaystyle=h_{\alpha,ij}h_{\beta,ji}h_{\alpha,kl}h_{\beta,lk}=(A^{\alpha}A^{\beta})_{ii}(A^{\alpha}A^{\beta})_{kk}$
$\displaystyle=\sum_{\alpha,\beta}\big{[}\mbox{tr}(A^{\alpha}A^{\beta})\big{]}^{2}=4\mu_{1}^{4}+4\mu_{2}^{4}$
where the last step follows from (3.9), and
(3.19) $\displaystyle\langle\underline{\mathcal{B}},B\rangle=$
$\displaystyle\langle
B_{A^{\alpha}A^{\alpha}(e_{i}),e_{j}}+B_{e_{i},A^{\alpha}A^{\alpha}(e_{j})}-2B_{A^{\alpha}(e_{i}),A^{\alpha}(e_{j})},\nu_{\beta}\rangle\langle
B_{e_{i},e_{j}},\nu_{\beta}\rangle$ $\displaystyle=$ $\displaystyle\langle
A^{\beta}A^{\alpha}A^{\alpha}(e_{i}),e_{j}\rangle\langle
A^{\beta}(e_{j}),e_{i}\rangle+\langle
A^{\beta}A^{\alpha}A^{\alpha}(e_{j}),e_{i}\rangle\langle
A^{\beta}(e_{i}),e_{j}\rangle$ $\displaystyle-2\langle
A^{\beta}A^{\alpha}(e_{i}),A^{\alpha}(e_{j})\rangle\langle
A^{\beta}(e_{j}),e_{i}\rangle$ $\displaystyle=$
$\displaystyle(A^{\beta}A^{\alpha}A^{\alpha})_{ij}(A^{\beta})_{ji}+(A^{\beta}A^{\alpha}A^{\alpha})_{ji}(A^{\beta})_{ij}-2(A^{\alpha}A^{\beta}A^{\alpha})_{ij}(A^{\beta})_{ji}$
$\displaystyle=$ $\displaystyle
2\mbox{tr}(A^{\beta}A^{\alpha}A^{\alpha}A^{\beta}-A^{\alpha}A^{\beta}A^{\alpha}A^{\beta})=2\mbox{tr}\big{(}[A^{\beta},A^{\alpha}]A^{\alpha}A^{\beta}\big{)}$
$\displaystyle=$
$\displaystyle\mbox{tr}\big{(}[A^{\beta},A^{\alpha}]A^{\alpha}A^{\beta}\big{)}+\mbox{tr}\big{(}[A^{\alpha},A^{\beta}]A^{\beta}A^{\alpha}\big{)}=-\sum_{\alpha,\beta}\mbox{tr}\big{(}[A^{\alpha},A^{\beta}]^{2}\big{)}$
$\displaystyle=$
$\displaystyle-2\mbox{tr}\big{(}[A^{1},A^{2}]^{2}\big{)}=16\mu_{1}^{2}\mu_{2}^{2}$
where
(3.20) $[A^{1},A^{2}]=\left(\begin{array}[]{cccc}0&2\mu_{1}\mu_{2}&&\\\
-2\mu_{1}\mu_{2}&0&&\\\ &&&\\\
&&\lx@intercol\hfil\raisebox{2.15277pt}[0.0pt]{\Huge
o}\hfil\lx@intercol&\end{array}\right).$
Substituting (3.18) and (3.19) into (3.15) gives
(3.21) $\displaystyle-\frac{\langle\nabla^{2}B,B\rangle}{|B|^{4}}$
$\displaystyle=\frac{\langle\tilde{\mathcal{B}}+\underline{\mathcal{B}},B\rangle}{|B|^{4}}=\frac{4\mu_{1}^{4}+4\mu_{2}^{4}+16\mu_{1}^{2}\mu_{2}^{2}}{(2\mu_{1}^{2}+2\mu_{2}^{2})^{2}}$
$\displaystyle=1+\frac{2\mu_{1}^{2}\mu_{2}^{2}}{(\mu_{1}^{2}+\mu_{2}^{2})^{2}}\leq\frac{3}{2}$
where the equality holds if and only if $\mu_{1}=\mu_{2}$.
###### Proposition 3.2.
Let $M$ be an $n$-dimensional minimal submanifold in $\mathbb{R}^{n+m}$ with
codimension $m\geq 2$. Then
$\Delta|B|^{2}\geq 2|\nabla B|^{2}-3|B|^{4}.$
In the case of $G-rank\leq 2$ the equality holds at $p\in M$ if and only if
$p$ is a G-conformal point.
In order to make use of the formula (2.8), we also need to estimate $|\nabla
B|^{2}$ in terms of $|\nabla|B||^{2}$. Schoen-Simon-Yau [15] obtained such an
estimate for the hypersurface case. It was generalized to arbitrary
codimension in [20] and refined and generalized in [19]. In particular, if
$G-rank\leq 2$ for $M$, we have a more precise estimate.
###### Proposition 3.3.
If $M$ is an $n$-dimensional minimal submanifold in ${\tenmsb R}^{n+m}$ with
$G-rank\leq 2$, then
(3.22) $|\nabla B|^{2}\geq 2\big{|}\nabla|B|\big{|}^{2}.$
The equality holds at $p\in M$, if and only if there exist an orthonormal
basis $\\{e_{1},\cdots,e_{n}\\}$ of $T_{p}M$ and
$\lambda_{1},\lambda_{2}\in{\tenmsb R}$, such that $B_{e_{i}e_{j}}=0$ whenever
$i\geq 3$ or $j\geq 3$, $\langle B_{e_{1}e_{1}},B_{e_{1}e_{2}}\rangle=0$,
$(\nabla_{e_{k}}B)_{e_{i}e_{j}}=0$ whenever $i\geq 3$, $j\geq 3$ or $k\geq 3$,
and
(3.23) $\displaystyle(\nabla_{e_{1}}B)_{e_{1}e_{1}}$
$\displaystyle=\lambda_{1}B_{e_{1}e_{1}}-\lambda_{2}B_{e_{1}e_{2}},$
$\displaystyle(\nabla_{e_{2}}B)_{e_{1}e_{1}}$
$\displaystyle=\lambda_{2}B_{e_{1}e_{1}}+\lambda_{1}B_{e_{1}e_{2}}.$
In particular, if $n=2$ and $m=1$, $|\nabla
B|^{2}=2\big{|}\nabla|B|\big{|}^{2}$ holds everywhere.
###### Proof.
It is sufficient for us to prove the inequality at the points where
$|B|^{2}\neq 0$.
With the same notation $A^{\alpha},\mu_{1},\mu_{2}$ as in (3.9), the triangle
inequality yields
(3.24)
$\big{|}\nabla|B|^{2}\big{|}=\Big{|}\sum_{\alpha}\nabla|A^{\alpha}|^{2}\Big{|}\leq\sum_{\alpha}\big{|}\nabla|A^{\alpha}|^{2}\big{|}.$
By the Schwarz inequality, we obtain
(3.25) $\displaystyle\big{|}\nabla|B|\big{|}^{2}$
$\displaystyle=\frac{\big{|}\nabla|B|^{2}\big{|}^{2}}{4|B|^{2}}\leq\frac{\Big{(}\sum_{\alpha}\big{|}\nabla|A^{\alpha}|^{2}\big{|}\Big{)}^{2}}{4\sum_{\alpha}|A^{\alpha}|^{2}}=\frac{\Big{(}\sum_{\alpha}\frac{\big{|}\nabla|A^{\alpha}|^{2}\big{|}}{|A^{\alpha}|}\cdot|A^{\alpha}|\Big{)}^{2}}{4\sum_{\alpha}|A^{\alpha}|^{2}}$
$\displaystyle\leq\frac{\sum\Big{(}\frac{{\big{|}\nabla|A^{\alpha}|^{2}\big{|}}^{2}}{|A^{\alpha}|^{2}}\Big{)}\cdot\sum_{\alpha}|A^{\alpha}|^{2}}{4\sum_{\alpha}|A^{\alpha}|^{2}}=\sum_{\alpha}\frac{{\big{|}\nabla|A^{\alpha}|^{2}\big{|}}^{2}}{4|A^{\alpha}|^{2}}$
$\displaystyle=\frac{{\big{|}\nabla|A^{1}|^{2}\big{|}}^{2}}{4|A^{1}|^{2}}+\frac{{\big{|}\nabla|A^{2}|^{2}\big{|}}^{2}}{4|A^{2}|^{2}}.$
Note that here and in the sequel we set
$\frac{{\big{|}\nabla|A^{\alpha}|^{2}\big{|}}^{2}}{4|A^{\alpha}|^{2}}=0$
whenever $|A^{\alpha}|=0$.
Since $|A^{\alpha}|^{2}=\sum_{i,j}h_{\alpha,ij}^{2}$,
(3.26) $\nabla_{e_{k}}|A^{\alpha}|^{2}=2h_{\alpha,ij}h_{\alpha,ijk}$
with
(3.27)
$h_{\alpha,ijk}:=\langle(\nabla_{e_{k}}B)_{e_{i}e_{j}},\nu_{\alpha}\rangle.$
As shown above, the assumption $G-rank\leq 2$ implies the existence of a local
orthonormal tangent frame field $\\{e_{i}\\}$ on an open domain $U$ as shown
before, such that $B_{e_{i}e_{j}}\equiv 0$ whenever $i\geq 3$ or $j\geq 3$.
Hence for arbitrary $i,j\geq 3$,
$0=\nabla_{e_{k}}(B_{e_{i}e_{j}})=(\nabla_{e_{k}}B)_{e_{i}e_{j}}+B_{\nabla_{e_{k}}e_{i},e_{j}}+B_{e_{i},\nabla_{e_{k}}e_{j}}=(\nabla_{e_{k}}B)_{e_{i}e_{j}}$
holds for all $k$, i.e.
(3.28) $h_{\alpha,ijk}=0\qquad\forall i,j\geq 3.$
It immediately follows that
(3.29)
$0=\langle\nabla_{e_{k}}H,\nu_{\alpha}\rangle=\sum_{i}h_{\alpha,iik}=h_{\alpha,11k}+h_{\alpha,22k}.$
In conjunction with (3.9), (3.26) and (3.29), we get
$\displaystyle\big{|}\nabla|A^{1}|^{2}\big{|}^{2}$
$\displaystyle=4\sum_{k}\big{(}\sum_{i,j}h_{1,ij}h_{1,ijk}\big{)}^{2}=4\sum_{k}(h_{1,11}h_{1,11k}+h_{1,22}h_{1,22k})^{2}$
$\displaystyle=16\mu_{1}^{2}\sum_{k}h_{1,11k}^{2}=8|A^{1}|^{2}\sum_{k}h_{1,11k}^{2}$
and moreover
(3.30)
$\frac{\big{|}\nabla|A^{1}|^{2}\big{|}^{2}}{|A^{1}|^{2}}=8\sum_{k}h_{1,11k}^{2}.$
A similar calculation shows
(3.31)
$\frac{\big{|}\nabla|A^{2}|^{2}\big{|}^{2}}{|A^{2}|^{2}}=8\sum_{k}h_{2,12k}^{2}.$
Substituting (3.30) and (3.31) into (3.25) implies
(3.32) $\big{|}\nabla|B|\big{|}^{2}\leq
2\sum_{k}h_{1,11k}^{2}+2\sum_{k}h_{2,12k}^{2}.$
On the other hand,
(3.33) $\displaystyle|\nabla B|^{2}=$
$\displaystyle\sum_{\alpha,i,j,k}h_{\alpha,ijk}^{2}\geq\sum_{i,j,k}h_{1,ijk}^{2}+\sum_{i,j,k}h_{2,ijk}^{2}$
$\displaystyle=$ $\displaystyle\
(h_{1,111}^{2}+h_{1,221}^{2}+h_{1,122}^{2}+h_{1,212}^{2})+(h_{1,112}^{2}+h_{1,121}^{2}+h_{1,211}^{2}+h_{1,222}^{2})$
$\displaystyle+\sum_{k\geq
3}(h_{1,11k}^{2}+h_{1,1k1}^{2}+h_{1,k11}^{2}+h_{1,22k}^{2}+h_{1,2k2}^{2}+h_{1,k22}^{2})$
$\displaystyle+(h_{2,121}^{2}+h_{2,112}^{2}+h_{2,211}^{2}+h_{2,222}^{2})+(h_{2,122}^{2}+h_{2,212}^{2}+h_{2,221}^{2}+h_{2,111}^{2})$
$\displaystyle+\sum_{k\geq
3}(h_{2,12k}^{2}+h_{2,k12}^{2}+h_{2,2k1}^{2}+h_{2,21k}^{2}+h_{2,k21}^{2}+h_{2,1k2}^{2})$
$\displaystyle\geq$ $\displaystyle
4\sum_{k}h_{1,11k}^{2}+4\sum_{k}h_{2,12k}^{2}.$
Here we have used (3.28), (3.29) and $h_{\alpha,ijk}=h_{\alpha,ikj}$, which is
an immediate corollary of the Codazzi equations. Combining this with (3.32)
and (3.33) yields (3.22).
Now we determine the conditions ensuring that equality in (3.22) holds true at
$p\in M$. Obviously $|\nabla B|^{2}=2\big{|}\nabla|B|\big{|}^{2}$ requires all
the equalities in (3.24), (3.25) and (3.33) hold simultaneously.
It is easily seen that equality holds in (3.33) if and only if
$h_{\alpha,ijk}=0$ whenever one of the indices is no less than $3$. Hence by
(3.26),
(3.34) $\displaystyle\nabla|A^{1}|^{2}=$
$\displaystyle(2h_{1,11}h_{1,111}+2h_{1,22}h_{1,221})e_{1}+(2h_{1,11}h_{1,112}+2h_{1,22}h_{1,222})e_{2}$
$\displaystyle=$ $\displaystyle 4\mu_{1}(h_{1,111}e_{1}+h_{1,112}e_{2}),$
$\displaystyle\nabla|A^{2}|^{2}=$
$\displaystyle(2h_{2,12}h_{2,121}+2h_{2,21}h_{2,211})e_{1}+(2h_{2,12}h_{2,122}+2h_{2,21}h_{2,212})e_{2}$
$\displaystyle=$ $\displaystyle 4\mu_{2}(h_{2,112}e_{1}-h_{2,111}e_{2}),$
and
(3.35) $\displaystyle v_{1}:=\frac{\nabla|A^{1}|^{2}}{|A^{1}|}$
$\displaystyle=2\sqrt{2}(h_{1,111}e_{1}+h_{1,112}e_{2}),$ $\displaystyle
v_{2}:=\frac{\nabla|A^{2}|^{2}}{|A^{2}|}$
$\displaystyle=2\sqrt{2}(h_{2,112}e_{1}-h_{2,111}e_{2}).$
(3.24) and (3.25) hold true if and only if the following 2 conditions are
satisfied: (i) $\nabla|A^{1}|^{2}$ and $\nabla|A^{2}|^{2}$ point in the same
direction, (ii) $(|v_{1}|,|v_{2}|)$ and $(\mu_{1},\mu_{2})$ are linearly
depedent. Hence there exist $\lambda_{1},\lambda_{2}\in{\tenmsb R}$, such that
$\displaystyle h_{1,111}e_{1}+h_{1,112}e_{2}$
$\displaystyle=\mu_{1}(\lambda_{1}e_{1}+\lambda_{2}e_{2}),$ $\displaystyle
h_{2,112}e_{1}-h_{2,111}e_{2}$
$\displaystyle=\mu_{2}(\lambda_{1}e_{1}+\lambda_{2}e_{2}).$
This is equivalent to
(3.36) $\displaystyle(\nabla_{e_{1}}B)_{e_{1}e_{1}}$
$\displaystyle=\mu_{1}\lambda_{1}\nu_{1}-\mu_{2}\lambda_{2}\nu_{2}=\lambda_{1}B_{e_{1}e_{1}}-\lambda_{2}B_{e_{1}e_{2}},$
$\displaystyle(\nabla_{e_{2}}B)_{e_{1}e_{1}}$
$\displaystyle=\mu_{1}\lambda_{2}\nu_{1}+\mu_{2}\lambda_{1}\nu_{2}=\lambda_{2}B_{e_{1}e_{1}}+\lambda_{1}B_{e_{1}e_{2}}.$
∎
In conjunction with (2.8) and (3.22), we arrive at
(3.37) $\Delta|B|^{2}\geq 4\big{|}\nabla|B|\big{|}^{2}-3|B|^{4}.$
## 4\. Curvature estimates
We are ready to derive the curvature estimates, in a manner similar to [5].
When $w>0$, we put $v:=w^{-1}$, then (3.14) is equivalent to
(4.1) $\Delta v\geq|B|^{2}v+v^{-1}|\nabla v|^{2}.$
From (4.1) and (3.37), a straightforward calculation shows
$\displaystyle\Delta\big{(}|B|^{2s}v^{q}\big{)}$ $\displaystyle=$
$\displaystyle\Delta\big{(}|B|^{2s}\big{)}v^{q}+|B|^{2s}\Delta
v^{q}+2\langle\nabla|B|^{2s},\nabla v^{q}\rangle$ $\displaystyle\geq$
$\displaystyle
s|B|^{2s-2}\Big{(}4\big{|}\nabla|B|\big{|}^{2}-3|B|^{4}\Big{)}v^{q}+4s(s-1)|B|^{2s-2}\big{|}\nabla|B|\big{|}^{2}v^{q}$
$\displaystyle+q|B|^{2s}v^{q-1}\big{(}|B|^{2}v+v^{-1}|\nabla
v|^{2}\big{)}+q(q-1)|B|^{2s}v^{q-2}|\nabla v|^{2}$
$\displaystyle+4sq|B|^{2s-1}v^{q-1}\langle\nabla|B|,\nabla v\rangle$
$\displaystyle\geq$
$\displaystyle(-3s+q)|B|^{2s+2}v^{q}+4s^{2}|B|^{2s-2}\big{|}\nabla|B|\big{|}^{2}v^{q}$
$\displaystyle+q^{2}|B|^{2s}v^{q-2}|\nabla
v|^{2}+4sq|B|^{2s-1}v^{q-1}\langle\nabla|B|,\nabla v\rangle.$
It follows that
(4.2) $\Delta\big{(}|B|^{2s}v^{q}\big{)}\geq(-3s+q)|B|^{2s+2}v^{q}$
for arbitrary $s,q\geq 1$.
Let $t=2s+1$, then
(4.3)
$\Delta\big{(}|B|^{t-1}v^{q}\big{)}\geq\big{(}q-\frac{3t-3}{2}\big{)}|B|^{t+1}v^{q}$
for arbitrary $t\geq 3$ and $q\geq 1$. Whenever $q>\frac{3t-3}{2}$, putting
$C_{1}(t,q)=\big{(}q-\frac{3t-3}{2}\big{)}^{-1}$ gives
(4.4) $|B|^{2t}v^{2q}\eta^{2t}\leq
C_{1}\Delta\big{(}|B|^{t-1}v^{q}\big{)}|B|^{t-1}v^{q}\eta^{2t}$
with $\eta$ being an arbitrary smooth function in $M$ with compact supporting
set. Integrating both sides of the above inequality over $M$ implies
$\displaystyle\int_{M}|B|^{2t}v^{2q}\eta^{2t}*1$ $\displaystyle\leq$
$\displaystyle
C_{1}\int_{M}\Delta\big{(}|B|^{t-1}v^{q}\big{)}|B|^{t-1}v^{q}\eta^{2t}*1$
$\displaystyle=$ $\displaystyle-
C_{1}\int_{M}\Big{\langle}\nabla\big{(}|B|^{t-1}v^{q}\big{)},\nabla\big{(}|B|^{t-1}v^{q}\eta^{2t}\big{)}\Big{\rangle}*1$
$\displaystyle=$ $\displaystyle-
C_{1}\int_{M}\Big{|}\nabla\big{(}|B|^{t-1}v^{q}\big{)}\Big{|}^{2}\eta^{2t}*1-2tC_{1}\int_{M}|B|^{t-1}v^{q}\eta^{2t-1}\langle\nabla\big{(}|B|^{t-1}v^{q}\big{)},\nabla\eta\rangle*1$
$\displaystyle\leq$ $\displaystyle-
C_{1}\int_{M}\Big{|}\nabla\big{(}|B|^{t-1}v^{q}\big{)}\Big{|}^{2}\eta^{2t}*1+C_{1}\int_{M}\Big{|}\nabla\big{(}|B|^{t-1}v^{q}\big{)}\Big{|}^{2}\eta^{2t}*1$
$\displaystyle+C_{1}t^{2}\int_{M}|B|^{2t-2}v^{2q}\eta^{2t-2}|\nabla\eta|^{2}*1$
$\displaystyle\leq$ $\displaystyle
C_{1}t^{2}\Big{(}\frac{t-1}{t}\varepsilon^{\frac{t}{t-1}}\int_{M}|B|^{2t}v^{2q}\eta^{2t}*1+\frac{1}{t}\varepsilon^{-t}\int_{M}v^{2q}|\nabla\eta|^{2t}*1\Big{)}$
for arbitrary $\varepsilon>0$. Here we have used Stokes’ theorem and Young’s
inequality. Choosing $\varepsilon$ such that
$C_{1}t(t-1)\varepsilon^{\frac{t}{t-1}}=\frac{1}{2}$ gives
(4.5) $\Big{(}\int_{M}|B|^{2t}v^{2q}\eta^{2t}*1\Big{)}^{\frac{1}{t}}\leq
C_{2}(t,q)\Big{(}\int_{M}v^{2q}|\nabla\eta|^{2t}*1\Big{)}^{\frac{1}{t}}$
for arbitrary $t\geq 3$ and $q>\frac{3t-3}{2}$.
###### Theorem 4.1.
Let $M$ be an $n$-dimensional minimal submanifold (not necessarily complete)
in ${\tenmsb R}^{n+m}$ with $G-rank\leq 2$ and positive $w$-function on $M$.
Let $\rho:M\times M\rightarrow{\tenmsb R}$ be a distance function on $M$, such
that $|\nabla\rho(\cdot,p)|\leq 1$ for each $p\in M$. Fix $p_{0}\in M$, and
denote by $B_{R}=B_{R}(p_{0}):=\\{p\in M:\rho(p,p_{0})<R\\}$ the distance ball
centered at $p_{0}$ and of radius $R$. Assume $B_{R_{0}}\subset
B_{R}\subset\subset M$, then for arbitrary $t\geq 3$ and $q>\frac{3t-3}{2}$,
there exists a positive constant $C_{3}$, depending only on $t$ and $q$, such
that
(4.6) $\big{\|}|B|^{2}v^{\frac{2q}{t}}\big{\|}_{L^{t}(B_{R_{0}})}\leq
C_{3}(R-R_{0})^{-2}\big{\|}v^{\frac{2q}{t}}\big{\|}_{L^{t}(B_{R})}.$
with $v:=w^{-1}$.
###### Proof.
We let $\psi$ be a standard bump function on $[0,\infty)$ with
$\text{supp}(\psi)\subset[0,R)$, $\psi\equiv 1$ on $[0,R_{0}]$ and
$|\psi^{\prime}|\leq c_{0}(R-R_{0})^{-1}$. Inserting
$\eta=\psi\circ\rho(\cdot,p_{0})$ in (4.5), we have
(4.7)
$\displaystyle\big{\|}|B|^{2}v^{\frac{2q}{t}}\big{\|}_{L^{t}(B_{R_{0}})}=\Big{(}\int_{B_{R_{0}}}|B|^{2t}v^{2q}*1\Big{)}^{\frac{1}{t}}\leq\Big{(}\int_{M}|B|^{2t}v^{2q}\eta^{2t}*1\Big{)}^{\frac{1}{t}}$
$\displaystyle\leq$ $\displaystyle
C_{2}\Big{(}\int_{M}v^{2q}|\nabla\eta|^{2t}*1\Big{)}^{\frac{1}{t}}=C_{2}\Big{(}\int_{B_{R}}v^{2q}|\psi^{\prime}|^{2t}|\nabla\rho(\cdot,p_{0})|^{2t}*1\Big{)}^{\frac{1}{t}}$
$\displaystyle\leq$ $\displaystyle
C_{3}(R-R_{0})^{-2}\Big{(}\int_{B_{R}}v^{2q}*1\Big{)}^{\frac{1}{t}}=C_{3}(R-R_{0})^{-2}\big{\|}v^{\frac{2q}{t}}\big{\|}_{L^{t}(B_{R})}.$
∎
Furthermore, the mean value inequality for subharmonic functions on minimal
submanifolds in Euclidean space can be applied to deduce a pointwise estimate
for $|B|^{2}$.
###### Theorem 4.2.
Our assumption of $M$ is the same as in Theorem 4.1. Denote by
$D_{R}=D_{R}(p_{0})$ the exterior ball centered at $p_{0}$ and of radius $R$,
then for every $t\geq 3$, there exists a positive constant $C_{4}$ only
depending on $t$, such that
(4.8) $(|B|^{2}v^{3})(p_{0})\leq
C_{4}R^{-2}(\max_{D_{R}}v)^{3}\Big{(}\frac{V(R)}{V(\frac{R}{2})}\Big{)}^{\frac{1}{t}}.$
Here $V(R)=V(p_{0},R):=\text{Vol}(D_{R}(p_{0})).$
###### Proof.
Let $F:M\rightarrow{\tenmsb R}^{n+m}$ be the isomorphic immersion and denote
by $r:M\times M\rightarrow{\tenmsb R}$ the restriction of the Euclidean
distance function. Without loss of generality one can assume $F(p_{0})=0$ for
$p_{0}\in M$, then $r^{2}(\cdot,p_{0})=\langle F,F\rangle.$ This extrinsic
distance function $r$ on $M$ satisfies the assumptions of Theorem 4.1.
Letting $q=\frac{3t}{2}$ in (4.5) yields
(4.9) $\Big{(}\int_{M}|B|^{2t}v^{3t}\eta^{2t}*1\Big{)}^{\frac{1}{t}}\leq
C_{2}\Big{(}\int_{M}v^{3t}|\nabla\eta|^{2t}*1\Big{)}^{\frac{1}{t}}.$
Let $\eta$ be a cut-off function on $M$ with $\text{supp}\ \eta\subset B_{R}$,
$\eta|_{B_{\frac{R}{2}}}\equiv 1$ and $|\nabla\eta|\leq c_{0}R^{-1}$ (the
construction of the auxiliary function is the same as in Theorem 4.1). Then
(4.10) $\Big{(}\int_{M}v^{3t}|\nabla\eta|^{2t}*1\Big{)}^{\frac{1}{t}}\leq
C_{5}(t)R^{-2}(\max_{D_{R}}v)^{3}V(R)^{\frac{1}{t}}.$
By (4.2), $|B|^{2t}v^{3t}$ is a subharmonic function on $M$, and by the mean
value inequality,
(4.11)
$\Big{(}\int_{M}|B|^{2t}v^{3t}\eta^{2t}*1\Big{)}^{\frac{1}{t}}\geq\Big{(}\int_{D_{\frac{R}{2}}}|B|^{2t}v^{3t}*1\Big{)}^{\frac{1}{t}}\geq(|B|^{2}v^{3})(p_{0})V\left(\frac{R}{2}\right)^{\frac{1}{t}}.$
In conjunction with (4.9)-(4.11) we arrive at (4.8).
∎
From the preceding curvature estimates we immediately get the following
Bernstein type theorem.
###### Theorem 4.3.
Let $M$ be an $n$-dimensional complete minimal submanifold in ${\tenmsb
R}^{n+m}$ with $G-rank\leq 2$ and a positive $w$-function. If $M$ has
polynomial volume growth and the function $v=w^{-1}$ has growth
(4.12) $\max_{D_{R}(p_{0})}v=o(R^{\frac{2}{3}})$
for a fixed point $p_{0}$, then $M$ has to be an affine linear subspace.
###### Remark 4.1.
Here, we say that $M$ has polynomial volume growth iff there exists $l\geq 0$
with $V(R)=V(p_{0},R)=O(R^{l})$.
###### Proof.
Let $c_{1}$ be a positive constant such that
(4.13) $V(R)\leq c_{1}R^{l}.$
Now we claim
(4.14) $\liminf_{k\rightarrow\infty}\frac{V(2^{k+1})}{V(2^{k})}\leq 2^{l}.$
Otherwise, there are $\varepsilon>0$ and a positive integer $N$, such that for
any $k\geq N$,
$\frac{V(2^{k+1})}{V(2^{k})}\geq 2^{l}+\varepsilon.$
Thus,
$\frac{V(2^{k})}{(2^{k})^{l}}\geq\frac{V(2^{N})(2^{l}+\varepsilon)^{k-N}}{(2^{N})^{l}(2^{l})^{k-N}}=\frac{V(2^{N})}{(2^{N})^{l}}\Big{(}\frac{2^{l}+\varepsilon}{2^{l}}\Big{)}^{k-N}.$
It follows that
$\lim_{k\rightarrow\infty}\frac{V(2^{k})}{(2^{k})^{l}}=+\infty$
which contradicts (4.13).
(4.14) implies the existence of a sequence $\\{k_{i}:i\in{\tenmsb N}\\}$, such
that $k_{i}<k_{j}$ whenever $i<j$, $\lim_{i\rightarrow\infty}k_{i}=\infty$ and
$\frac{V(2^{k_{i}+1})}{V(2^{k_{i}})}\leq 2^{l}.$
then putting $R=R_{i}:=2^{k_{i}+1}$ and letting $t=3$ in (4.8) give
(4.15) $(|B|^{2}v^{3})(p_{0})\leq
C_{4}2^{\frac{l}{3}}R_{i}^{-2}(\max_{D_{R_{i}}}v)^{3}$
Since $\max_{D_{R}}v=o(R^{\frac{2}{3}})$, letting $i\rightarrow\infty$ yields
$|B|^{2}=0$ at $p_{0}$.
For arbitrary $p\in M$, put $R_{0}:=r(p,p_{0})$, then the triangle inequality
implies $D_{R}(p)\subset D_{R+R_{0}}(p_{0})$ for any $R\geq 0$, hence
$\frac{V(p,R)}{R^{l}}\leq\frac{V(p_{0},R+R_{0})}{R^{l}}\leq\frac{c_{1}(R+R_{0})^{l}}{R^{l}}$
which means $V(p,R)=O(R^{l})$. Similarly one can show
$\max_{D_{R}(p)}v=o(R^{\frac{2}{3}})$ for arbitrary $p$. Thereby one can
proceed as above to arrive at $|B|^{2}=0$ at $p$. Hence $|B|\equiv 0$ on $M$
and $M$ has to be affine linear.
∎
## 5\. Graphical cases
Let $f=(f^{1},\cdots,f^{n}):\Omega\subset{\tenmsb R}^{n}\rightarrow{\tenmsb
R}^{m}$ be a vector-valued function, then the graph
$M=\\{(x,f(x)):x\in\Omega\\}$ is an embedded submanifold in ${\tenmsb
R}^{n+m}$. Let $\\{\varepsilon_{i},\varepsilon_{n+\alpha}\\}$ be the standard
orthonormal basis, and put
$A=\varepsilon_{1}\wedge\cdots\wedge\varepsilon_{n}$, then as shown in [10],
the $w$-function is positive everywhere on $M$ and the volume element of $M$
is
(5.1) $*1=v\ dx^{1}\wedge\cdots\wedge dx^{n},$
where
(5.2) $v=w^{-1}=\left[\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\right]^{\frac{1}{2}}.$
Without loss of generality we can assume $f(0)=0$. Denote $p_{0}=(0,0)$, then
(5.3) $D_{R}=D_{R}(p_{0})=\\{(x,f(x)):|x|^{2}+|f(x)|^{2}\leq R^{2}\\}.$
Denote
(5.4) $\Omega_{R}=\\{x\in\Omega:|x|^{2}+|f(x)|^{2}\leq R^{2}\\},$
then obviously $\Omega_{R}\subset{\tenmsb D}^{n}(R)$ and $D_{R}$ is just the
graph over $\Omega_{R}$, where ${\tenmsb D}^{n}(R)$ is the $n$-dimensional
Euclidean ball of radius $R$. Hence if
(5.5) $\max_{D_{R}}v\leq CR^{l},$
then
(5.6) $\displaystyle V(R)$
$\displaystyle=\int_{D_{R}}*1=\int_{\Omega_{R}}vdx^{1}\wedge\cdots\wedge
dx^{n}$ $\displaystyle\leq\max_{D_{R}}v\cdot\text{Vol}(\Omega_{R})\leq
CR^{l}\text{Vol}({\tenmsb D}^{n}(R))$ $\displaystyle=C\omega_{n}R^{n+l}$
with $\omega_{n}$ being the volume of the $n$-dimensional unit Euclidean ball.
This means that the exterior balls of a graph have polynomial volume growth
whenever the $v$-function has polynomial growth. This fact leads us to the
following result.
###### Theorem 5.1.
Let $M=\\{(x,f(x)):x\in{\tenmsb R}^{n}\\}$ be an entire minimal graph given by
a vector-valued function $f:{\tenmsb R}^{n}\rightarrow{\tenmsb R}^{m}$ with
$G-rank\leq 2$. If the slope of $f$ satisfies
(5.7) $\Delta_{f}=\left[\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}\right]^{\frac{1}{2}}=o(R^{\frac{2}{3}}),$
where $R^{2}=|x|^{2}+|f(x)|^{2}$, then $f$ has to be an affine linear
function.
Now we study $2$-dimensional cases. It is well-known that every oriented
$2$-dimensional Riemannian manifold $M$ admits a local isothermal coordinate
chart around any point. More precisely, each $p\in M$ has a coordinate
neighborhood $(U;u,v)$, such that
$g=\lambda^{2}(du^{2}+dv^{2})$
on $U$ with a positive function $\lambda$. In fact, for minimal entire graphs,
one can find a global isothermal coordinate chart:
###### Lemma 5.1.
([14] §5) Let $M=\\{(x,f(x):x\in{\tenmsb R}^{2}\\}$ be a $2$-dimensional
entire minimal graph in ${\tenmsb R}^{2+m}$, then there exists a nonsigular
linear transformation
(5.8) $\displaystyle u_{1}$ $\displaystyle=x_{1}$ $\displaystyle u_{2}$
$\displaystyle=ax_{1}+bx_{2},\qquad(b>0)$
such that $(u_{1},u_{2})$ are global isothermal parameters for $M$.
Equipped with this tool, we can obtain another Bernstein type theorem for
entire minimal graphs of dimension $2$.
###### Theorem 5.2.
Let $f:{\tenmsb R}^{2}\rightarrow{\tenmsb R}^{m}$
$(x^{1},x^{2})\mapsto(f^{1},\cdots,f^{m})$ be an entire solution of the
minimal surface equations
(5.9) $\Big{(}1+\Big{|}\frac{\partial f}{\partial
x^{2}}\Big{|}^{2}\Big{)}\frac{\partial^{2}f}{(\partial
x^{1})^{2}}-2\Big{\langle}\frac{\partial f}{\partial x^{1}},\frac{\partial
f}{\partial x^{2}}\Big{\rangle}\frac{\partial^{2}f}{\partial x^{1}\partial
x^{2}}+\Big{(}1+\Big{|}\frac{\partial f}{\partial
x^{1}}\Big{|}^{2}\Big{)}\frac{\partial^{2}f}{(\partial x^{2})^{2}}=0.$
If for some $\varepsilon>0$,
(5.10) $\Delta_{f}=\det\Big{(}\delta_{ij}+\sum_{\alpha}\frac{\partial
f^{\alpha}}{\partial x^{i}}\frac{\partial f^{\alpha}}{\partial
x^{j}}\Big{)}^{\frac{1}{2}}=O(R^{1-\varepsilon})$
with $R=|x|$, then $f$ has to be affine linear.
###### Proof.
By Lemma 5.1, one can find a global isothermal coordinate $(u_{1},u_{2})$ for
the entire minimal graph $M:=\\{(x,f(x)):x\in{\tenmsb R}^{2}\\}$, i.e.
(5.11) $\displaystyle g$
$\displaystyle=\lambda^{2}\big{(}(du^{1})^{2}+(du^{2})^{2}\big{)}=\lambda^{2}\big{(}(dx^{1})^{2}+(a\
dx^{1}+b\ dx^{2})^{2}\big{)}$
$\displaystyle=\lambda^{2}\big{(}(1+a^{2})(dx^{1})^{2}+2ab\
dx^{1}dx^{2}+b^{2}(dx^{2})^{2}\big{)}.$
In other words, the metric is given by
(5.12) $(g_{ij})=\lambda^{2}\left(\begin{array}[]{cc}1+a^{2}&ab\\\
ab&b^{2}\end{array}\right).$
Denote the two eigenvalues of $\left(\begin{array}[]{cc}1+a^{2}&ab\\\
ab&b^{2}\end{array}\right)$ by $\lambda_{1}^{2}\geq\lambda_{2}^{2}>0$, then
(5.13) $v=\det(g_{ij})^{\frac{1}{2}}=\lambda^{2}\lambda_{1}\lambda_{2}.$
Since $M$ is a graph, any function $\varphi$ on $M$ can be regarded as a
function on ${\tenmsb R}^{2}$. Denote
(5.14) $\partial_{i}\varphi=\frac{\partial\varphi}{\partial x^{i}},\qquad
D\varphi=(\partial_{1}\varphi,\partial_{2}\varphi)$
and let $\nabla\varphi$ be the gradient vector of $\varphi$ on $M$ with
respect to $g$. Since the largest eigenvalue of $(g^{ij})$ equals the
multiplicative inverse of the smallest eigenvalue of $(g_{ij})$, which is
$\lambda^{-2}\lambda_{2}^{-2}$, we have
$|\nabla\varphi|^{2}=g^{ij}\partial_{i}\varphi\partial_{j}\varphi\leq\lambda^{-2}\lambda_{2}^{-2}|D\varphi|^{2}$
i.e.
(5.15)
$|\nabla\varphi|\leq\lambda^{-1}\lambda_{2}^{-1}|D\varphi|=\Big{(}\frac{\lambda_{1}}{\lambda_{2}}\Big{)}^{\frac{1}{2}}v^{-\frac{1}{2}}|D\varphi|.$
Given $0<R_{0}<R$, let $\psi$ be a standard bump function, such that
$\text{supp}\ \psi\subset[0,R)$, $\psi\equiv 1$ on $[0,R_{0}]$ and
$|\psi^{\prime}|\leq c_{0}(R-R_{0})^{-1}$. Taking $\eta(x,f(x))=\psi(|x|)$ in
(4.5) gives
(5.16) $\displaystyle\Big{(}\int_{{\tenmsb
D}^{2}(R_{0})}|B|^{2t}v^{2q+1}dx^{1}dx^{2}\Big{)}^{\frac{1}{t}}\leq\Big{(}\int_{M}|B|^{2t}v^{2q}\eta^{2t}*1\Big{)}^{\frac{1}{t}}$
$\displaystyle\leq$ $\displaystyle
C_{2}\Big{(}\int_{M}v^{2q}|\nabla\eta|^{2t}*1\Big{)}^{\frac{1}{t}}=C_{2}\Big{(}\int_{M}v^{2q}\Big{(}\frac{\lambda_{1}}{\lambda_{2}}\Big{)}^{t}v^{-t}|D\eta|^{2t}*1\Big{)}^{\frac{1}{t}}$
$\displaystyle\leq$ $\displaystyle C_{6}(R-R_{0})^{-2}\Big{(}\int_{{\tenmsb
D}^{2}(R)}v^{2q-t+1}dx_{1}dx_{2}\Big{)}^{\frac{1}{t}}$ $\displaystyle\leq$
$\displaystyle C_{6}(R-R_{0})^{-2}(\max_{{\tenmsb
D}^{2}(R)}v)^{\frac{2q+1}{t}-1}(\pi R^{2})^{\frac{1}{t}}$ $\displaystyle=$
$\displaystyle
C_{7}\Big{(}1-\frac{R_{0}}{R}\Big{)}^{-2}R^{-2+\frac{2}{t}}(\max_{{\tenmsb
D}^{2}(R)}v)^{\frac{2q+1}{t}-1}$
with $C_{6}$ and $C_{7}$ being positive constants depending only on $t,q,a$
and $b$. Letting $q=\frac{3t-1}{2}$ gives $\frac{2q+1}{t}-1=2$. Thus the
growth condition of $v$ implies
(5.17) $\Big{(}\int_{{\tenmsb
D}^{2}(R_{0})}|B|^{2t}v^{3t}dx^{1}dx^{2}\Big{)}^{\frac{1}{t}}\leq
C_{8}\Big{(}1-\frac{R_{0}}{R}\Big{)}^{-2}R^{\frac{2}{t}-2\varepsilon}.$
Taking $t=\frac{2}{\varepsilon}$ and then letting $R\rightarrow+\infty$ force
$|B|(x,f(x))=0$ whenever $|x|<R_{0}$. Finally by letting
$R_{0}\rightarrow+\infty$ we get the Bernstein type result. ∎
Given a vector-valued function $f:{\tenmsb R}^{2}\rightarrow{\tenmsb R}^{m}$,
denote by
$Df=Df(x):=\Big{(}\frac{\partial f^{\alpha}}{\partial x^{i}}\Big{)}$
the Jacobi matrix of $f$ at $x\in{\tenmsb R}^{2}$. $Df$ can also be seen as a
linear mapping from ${\tenmsb R}^{2}$ to ${\tenmsb R}^{m}$. Obviously
$Df(Df)^{T}$ is a nonnegative definite symmetric matrix, whose engenvalues are
denoted by $\mu_{1}^{2}\geq\mu_{2}^{2}\geq 0$. It is easy to check that
$\mu_{1}$ and $\mu_{2}$ are just the critical values of the function
$v\in{\tenmsb R}^{2}\backslash 0\mapsto\frac{\big{|}(Df)(v)\big{|}}{|v|}$
and for any bounded domain $\mathcal{D}\subset{\tenmsb R}^{2}$,
$\mu_{1}\mu_{2}=\frac{\text{Area}\big{(}Df(\mathcal{D})\big{)}}{\text{Area}(\mathcal{D})}.$
In matrix terminology, $\mu_{1}^{2}\mu_{2}^{2}$ equals the squared sum of all
the $2\times 2$-minors of $Df$, i.e.
(5.18) $\mu_{1}^{2}\mu_{2}^{2}=\sum_{\alpha<\beta}\Big{(}\frac{\partial
f^{\alpha}}{\partial x^{1}}\frac{\partial f^{\beta}}{\partial
x^{2}}-\frac{\partial f^{\alpha}}{\partial x^{2}}\frac{\partial
f^{\beta}}{\partial x^{1}}\Big{)}^{2}.$
When $m=2$, $\mu_{1}\mu_{2}$ then is the absolute value of $J_{f}:=\det(Df)$.
As shown in (5.2), the metric matrix of the graph given by $f$ is
(5.19) $(g_{ij})=I_{2}+Df(Df)^{T}.$
Thus the two eigenvalues of $(g_{ij})$ are $1+\mu_{1}^{2}$ and
$1+\mu_{2}^{2}$, and
(5.20) $v^{2}=\det(g_{ij})=(1+\mu_{1}^{2})(1+\mu_{2}^{2}).$
Now we additionally assume that $f$ is an entire solution of the minimal
surface equations. Then as shown in (5.12), there exists a positive function
$\lambda$ on $M$ and two positive constants $\lambda_{1},\lambda_{2}$,
depending only on $a$ and $b$, such that
(5.21) $1+\mu_{1}^{2}=\lambda^{2}\lambda_{1}^{2}\qquad
1+\mu_{2}^{2}=\lambda^{2}\lambda_{2}^{2}.$
Hence
(5.22) $\displaystyle\mu_{1}^{2}\mu_{2}^{2}$
$\displaystyle=(\lambda^{2}\lambda_{1}^{2}-1)(\lambda^{2}\lambda_{2}^{2}-1)=\lambda_{1}^{2}\lambda_{2}^{2}\lambda^{4}-(\lambda_{1}^{2}+\lambda_{2}^{2})\lambda^{2}+1$
$\displaystyle=v^{2}-\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\lambda_{1}\lambda_{2}}v+1.$
Note that $\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\lambda_{1}\lambda_{2}}$ is
a constant. Once $v$ has polynomial growth, $\mu_{1}\mu_{2}$ also has
polynomial growth of the same order, and vice versa. Therefore one can obtain
an equivalent form of Theorem 5.2 as follows.
###### Theorem 5.3.
Let $f:{\tenmsb R}^{2}\rightarrow{\tenmsb R}^{m}$
$(x^{1},x^{2})\mapsto(f^{1},\cdots,f^{m})$ be an entire solution of the
minimal surface equations. If for some $\varepsilon>0$,
(5.23) $\sum_{\alpha<\beta}\Big{(}\frac{\partial f^{\alpha}}{\partial
x^{1}}\frac{\partial f^{\beta}}{\partial x^{2}}-\frac{\partial
f^{\alpha}}{\partial x^{2}}\frac{\partial f^{\beta}}{\partial
x^{1}}\Big{)}^{2}=O(R^{2(1-\varepsilon)})$
with $R=|x|$, then $f$ has to be affine linear. If $m=2$, the condition (5.23)
is equivalent to
(5.24) $|J_{f}|:=|\det(Df)|=O(R^{1-\varepsilon}).$
Similarly we have a version of Theorem 5.1 for the minimal surface case.
###### Theorem 5.4.
Let $f:{\tenmsb R}^{2}\rightarrow{\tenmsb R}^{m}$
$(x^{1},x^{2})\mapsto(f^{1},\cdots,f^{m})$ be an entire solution of the
minimal surface equations. If
(5.25) $\sum_{\alpha<\beta}\Big{(}\frac{\partial f^{\alpha}}{\partial
x^{1}}\frac{\partial f^{\beta}}{\partial x^{2}}-\frac{\partial
f^{\alpha}}{\partial x^{2}}\frac{\partial f^{\beta}}{\partial
x^{1}}\Big{)}^{2}=o(R^{\frac{4}{3}})$
with $R^{2}=|x|^{2}+|f(x)|^{2}$, then $f$ has to be affine linear. If $m=2$,
the condition (5.25) is equivalent to
(5.26) $|J_{f}|:=|\det(Df)|=o(R^{\frac{2}{3}}).$
###### Remark 5.1.
Obviously, the above result is also a generalization of that of [7].
## 6\. Discussions
We wish to discuss the case of a minimal surface $M$ in ${\tenmsb R}^{2+m}$.
It is natural to ask under which conditions the equality in (2.8), (3.14) or
(3.22) holds.
With the aid of Lemma 5.1, one can get a sufficient condition for equality in
(3.14).
###### Proposition 6.1.
If $M$ is a $2$-dimensional entire minimal graph in ${\tenmsb R}^{2+m}$, then
(6.1) $\Delta\log w=-|B|^{2}.$
###### Proof.
By Lemma 5.1, there exists a nonsingular linear transformation
(6.2) $\left(\begin{array}[]{c}u^{1}\\\
u^{2}\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
a&b\end{array}\right)\left(\begin{array}[]{c}x^{1}\\\ x^{2}\end{array}\right)$
such that $(u^{1},u^{2})$ are global isothermal parameters for $M$, where $a$
and $b>0$ are constants. Hence there is a positive function $\lambda$ on $M$,
such that the metric $g$ on $M$ can be expressed as
(6.3) $g=\lambda^{2}\big{(}(du^{1})^{2}+(du^{2})^{2}\big{)}.$
As shown in (5.13),
$w^{-1}=v=\lambda^{2}\lambda_{1}\lambda_{2}$
with $\lambda_{1}^{2}\geq\lambda_{2}^{2}>0$ being eigenvalues of
$\left(\begin{array}[]{cc}1+a^{2}&ab\\\ ab&b^{2}\end{array}\right)$. Thus
$\log w=-\log(\lambda^{2})-\log(\lambda_{1}\lambda_{2})$
and moreover
(6.4) $\Delta\log w=-\Delta\log(\lambda^{2}).$
The Gauss curvature $K$ of $M$ is given by (see e.g. [9])
(6.5) $K=-\frac{1}{2}\Delta\log(\lambda^{2}).$
On the other hand, let $\\{e_{1},e_{2}\\}$ be an orthonormal basis of
$T_{p}M$, with $p$ an arbitrary point in $M$. Since $M$ is minimal,
$B_{e_{1}e_{1}}+B_{e_{2}e_{2}}=0$ and the Gauss equation yields
(6.6) $K=\mathrm{det}B_{e_{i}e_{j}}=-\frac{1}{2}|B|^{2}.$
Finally combining (6.4), (6.5), (6.6) yields (6.1). ∎
Let $M$ be a Riemann surface and
$F=(F^{1},\cdots,F^{n+m}):M\rightarrow{\tenmsb R}^{2+m}$ be an isomorphic
immersion. Every $p\in M$ has a coordinate neighborhood $(U;u,v)$ such that
$g=\lambda^{2}(du^{2}+dv^{2})$ on $U$. Now we introduce the complex coordinate
$w=u+\sqrt{-1}v.$
It is well-known that $F$ is minimal if and only if all components of $F$ are
harmonic functions on $M$, i.e. $\frac{\partial F}{\partial w}$ is a vector-
valued holomorphic function on $U$; here and in the sequel
(6.7) $\displaystyle\frac{\partial missing}{\partial
w}=\frac{1}{2}\Big{(}\frac{\partial missing}{\partial
u}-\sqrt{-1}\frac{\partial missing}{\partial v}\Big{)},$
$\displaystyle\frac{\partial}{\partial\bar{w}}=\frac{1}{2}\Big{(}\frac{\partial}{\partial
u}+\sqrt{-1}\frac{\partial}{\partial v}\Big{)},$ $\displaystyle
dw=du+\sqrt{-1}dv,$ $\displaystyle d\bar{w}=du-\sqrt{-1}dv.$
While $\frac{\partial F}{\partial w}$ depends on the choice of local
coordinate, the vector-valued holomorphic $1$-form
$\partial F:=\frac{\partial F}{\partial w}dw$
is independent of these local coordinates and can be well-defined on the whole
surface, where $dw=du+\sqrt{-1}dv$. Similarly we can define
$\bar{\partial}F:=\frac{\partial F}{\partial\bar{w}}d\bar{w}.$
With the symmetric bi-linear form
$\langle(a_{1},\cdots,a_{N}),(b_{1},\cdots,b_{N})\rangle=\sum_{i=1}^{N}a_{i}b_{i},$
since $(u,v)$ are isothermal parameters, it is well known and easy to check
that
(6.8) $\Big{\langle}\frac{\partial F}{\partial w},\frac{\partial F}{\partial
w}\Big{\rangle}=0,\qquad\Big{\langle}\frac{\partial F}{\partial
w},\frac{\partial F}{\partial\bar{w}}\Big{\rangle}>0$
which is equivalent to
(6.9) $\langle\partial F,\partial F\rangle=0,\qquad\langle\partial
F,\bar{\partial}F\rangle>0.$
Similarly, one can define
(6.10) $\partial^{2}F:=\frac{\partial^{2}F}{\partial
w^{2}}dw^{2},\qquad\bar{\partial}^{2}F:=\frac{\partial^{2}F}{\partial\bar{w}^{2}}d\bar{w}^{2}.$
Then the minimality of $F$ implies that $\partial^{2}F$ is a vector-valued
holomorphic $2$-form.
###### Proposition 6.2.
For a fixed point $p$ in a minimal surface $M\subset{\tenmsb R}^{2+m}$, the
following statements are equivalent:
(a) $\Delta|B|^{2}=|\nabla B|^{2}-3|B|^{4}$ at $p$;
(b) $p$ is a G-conformal point;
(c) $\big{\langle}B_{\frac{\partial}{\partial w}\frac{\partial}{\partial
w}},B_{\frac{\partial}{\partial w}\frac{\partial}{\partial w}}\big{\rangle}=0$
at $p$, where $w$ is a local complex coordinate near $p$;
(d) $\langle\partial^{2}F,\partial^{2}F\rangle=0$ at $p$.
###### Proof.
The equivalence of (a) and (b) has been proved in Proposition 3.2.
Since $(u,v)$ is an isothermal coordinate, $\frac{\partial}{\partial u}$ and
$\frac{\partial}{\partial v}$ have the same length and are orthogonal to each
other, hence $p$ is an holomorphic-like point if and only if
(6.11) $|B_{uu}|=|B_{uv}|,\qquad\langle B_{uu},B_{uv}\rangle=0.$
Here and in the sequel, $B_{uu}:=B_{\frac{\partial}{\partial
u}\frac{\partial}{\partial u}}$, $B_{uv}:=B_{\frac{\partial}{\partial
u}\frac{\partial}{\partial v}}$ and so on.
By using (6.7) one can get
(6.12) $B_{ww}=\frac{1}{2}B_{uu}-\frac{\sqrt{-1}}{2}B_{uv}.$
It implies
(6.13) $\displaystyle\langle
B_{ww},B_{ww}\rangle=\frac{1}{4}\big{(}|B_{uu}|^{2}-|B_{uv}|^{2}\big{)}-\frac{\sqrt{-1}}{2}\langle
B_{uu},B_{uv}\rangle$
and hence (b) and (c) are equivalent.
Since
$(T_{p}M)\otimes{\tenmsb C}=\text{span}\Big{\\{}\frac{\partial F}{\partial
w},\frac{\partial F}{\partial\bar{w}}\Big{\\}}$
there exist two complex numbers $\mu_{1}$ and $\mu_{2}$, such that
(6.14) $\nabla_{\frac{\partial}{\partial w}}\frac{\partial}{\partial
w}=\Big{(}\frac{\partial^{2}F}{\partial
w^{2}}\Big{)}^{T}=\mu_{1}\frac{\partial F}{\partial w}+\mu_{2}\frac{\partial
F}{\partial\bar{w}}.$
By (6.8),
(6.15) $\displaystyle 0$ $\displaystyle=\frac{1}{2}\frac{\partial}{\partial
w}\Big{\langle}\frac{\partial F}{\partial w},\frac{\partial F}{\partial
w}\Big{\rangle}=\Big{\langle}\frac{\partial^{2}F}{\partial
w^{2}},\frac{\partial F}{\partial w}\Big{\rangle}$
$\displaystyle=\Big{\langle}\mu_{1}\frac{\partial F}{\partial
w}+\mu_{2}\frac{\partial F}{\partial\bar{w}},\frac{\partial F}{\partial
w}\Big{\rangle}=\mu_{2}\Big{\langle}\frac{\partial
F}{\partial\bar{w}},\frac{\partial F}{\partial w}\Big{\rangle}.$
Hence $\mu_{2}=0$ and moreover
(6.16) $\displaystyle\Big{\langle}\frac{\partial^{2}F}{\partial
w^{2}},\frac{\partial^{2}F}{\partial w^{2}}\Big{\rangle}$
$\displaystyle=\Big{\langle}\Big{(}\frac{\partial^{2}F}{\partial
w^{2}}\Big{)}^{N},\Big{(}\frac{\partial^{2}F}{\partial
w^{2}}\Big{)}^{N}\Big{\rangle}+\Big{\langle}\Big{(}\frac{\partial^{2}F}{\partial
w^{2}}\Big{)}^{T},\Big{(}\frac{\partial^{2}F}{\partial
w^{2}}\Big{)}^{T}\Big{\rangle}$ $\displaystyle=\langle
B_{ww},B_{ww}\rangle+\mu_{1}^{2}\Big{\langle}\frac{\partial F}{\partial
w},\frac{\partial F}{\partial w}\Big{\rangle}=\langle B_{ww},B_{ww}\rangle.$
Thus (c) is equivalent to (d).
∎
Define
(6.17)
$\omega:=\langle\partial^{2}F,\partial^{2}F\rangle=\Big{\langle}\frac{\partial^{2}F}{\partial
w^{2}},\frac{\partial^{2}F}{\partial w^{2}}\Big{\rangle}dw^{4}$
then it is easy to check that the definition of $\omega$ is independent of the
choice of coordinate, and
$\frac{\partial}{\partial\bar{w}}\Big{\langle}\frac{\partial^{2}F}{\partial
w^{2}},\frac{\partial^{2}F}{\partial
w^{2}}\Big{\rangle}=2\Big{\langle}\frac{\partial}{\partial
w}\Big{(}\frac{\partial^{2}F}{\partial
w\partial\bar{w}}\Big{)},\frac{\partial^{2}F}{\partial w^{2}}\Big{\rangle}=0$
implies $\omega$ is a homolomorphic $4$-form on $M$. By using Proposition 6.2
we immediately get the following corollary.
###### Corollary 6.1.
Let $M$ be a minimal surface in $\mathbb{R}^{2+m}$, then $M$ is totally
G-conformal if and only if the holomorphic $4$-form
$\omega:=\langle\partial^{2}F,\partial^{2}F\rangle$ vanishes everywhere.
###### Corollary 6.2.
Let $M=\\{(x,f(x)):x\in{\tenmsb R}^{2}\\}$ be an entire minimal graph in
${\tenmsb R}^{4}$. Then $M$ is totally G-conformal if and only if at least one
of the following 3 cases occurs: (i) $f:{\tenmsb R}^{2}\rightarrow{\tenmsb
R}^{2}$ is a holomorphic function; (ii) $f$ is anti-holomorphic; (iii) $f$ is
affine linear.
###### Proof.
Let $(u_{1},u_{2})$ be the global isothermal parameters on $M$ given in (6.2).
Denote $z:=u_{1}+\sqrt{-1}u_{2}$ and
(6.18) $\phi_{i}=\frac{\partial x^{i}}{\partial
z},\qquad\phi_{2+\alpha}=\frac{\partial f^{\alpha}}{\partial z}.$
then $\frac{\partial F}{\partial z}=(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$ and
(6.8) yields $\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}=0$. By
(6.2), $\phi_{1}$ and $\phi_{2}$ are both constants, denote
(6.19) $d:=\phi_{1}^{2}+\phi_{2}^{2},$
then
(6.20) $\phi_{3}^{2}+\phi_{4}^{2}=-(\phi_{1}^{2}+\phi_{2}^{2})=-d.$
If $d=0$, then $\phi_{4}=\pm\sqrt{-1}\phi_{3}$ and hence
(6.21) $\frac{\partial^{2}F}{\partial
z^{2}}=(\phi^{\prime}_{1},\phi^{\prime}_{2},\phi^{\prime}_{3},\phi^{\prime}_{4})=(0,0,\phi^{\prime}_{3},\pm\sqrt{-1}\phi^{\prime}_{3}).$
It follows that
(6.22) $\Big{\langle}\frac{\partial^{2}F}{\partial
z^{2}},\frac{\partial^{2}F}{\partial
z^{2}}\Big{\rangle}=(\phi^{\prime}_{3})^{2}-(\phi^{\prime}_{3})^{2}=0$
and $M$ is totally holomorphic-like. As show in [8], $d=0$ implies $f$ is
holomorphic or anti-holomorphic, and vice versa.
If $d\neq 0$, then
(6.23)
$-d=\phi_{3}^{2}+\phi_{4}^{2}=(\phi_{3}+\sqrt{-1}\phi_{4})(\phi_{3}-\sqrt{-1}\phi_{4})$
implies $\phi_{3}-\sqrt{-1}\phi_{4}$ is an entire function having no zeros,
hence there is an entire function $H(z)$, such that
(6.24) $\phi_{3}-\sqrt{-1}\phi_{4}=e^{H(z)}.$
Substituting it into (6.23) gives
(6.25) $\phi_{3}+\sqrt{-1}\phi_{4}=-de^{-H(z)}.$
In conjunction with the above two equations we have
(6.26)
$\phi_{3}=\frac{1}{2}(e^{H}-de^{-H}),\qquad\phi_{4}=\frac{\sqrt{-1}}{2}(e^{H}+de^{-H}).$
Thus
(6.27) $\displaystyle\Big{\langle}\frac{\partial^{2}F}{\partial
z^{2}},\frac{\partial^{2}F}{\partial z^{2}}\Big{\rangle}$
$\displaystyle=(\phi^{\prime}_{1})^{2}+(\phi^{\prime}_{2})^{2}+(\phi^{\prime}_{3})^{2}+(\phi^{\prime}_{4})^{2}$
$\displaystyle=\frac{1}{4}(e^{H}+de^{-H})^{2}(H^{\prime})^{2}-\frac{1}{4}(e^{H}-de^{-H})^{2}(H^{\prime})^{2}$
$\displaystyle=d(H^{\prime})^{2}$
which is identically zero if and only if $H$ is a constant function. In this
case, $\phi_{i}$ and $\phi_{2+\alpha}$ are all constants on $M$, hence $M$ has
to be an affine plane.
∎
For the sequel, we put
(6.28) $(\nabla B)_{uuv}:=(\nabla_{\frac{\partial}{\partial
v}}B)\Big{(}\frac{\partial}{\partial u},\frac{\partial}{\partial
u}\Big{)},\qquad(\nabla B)_{www}:=(\nabla_{\frac{\partial}{\partial
w}}B)\Big{(}\frac{\partial}{\partial w},\frac{\partial}{\partial w}\Big{)}$
and so on. Then (3.23) says that there are $\xi_{1},\xi_{2}\in{\tenmsb R}$,
such that
(6.29) $\displaystyle(\nabla B)_{uuu}$
$\displaystyle=\xi_{1}B_{uu}-\xi_{2}B_{uv}$ $\displaystyle(\nabla B)_{uuv}$
$\displaystyle=\xi_{2}B_{uu}+\xi_{1}B_{uv}.$
###### Proposition 6.3.
For a fixed point $p$ in a minimal surface $M\subset{\tenmsb R}^{2+m}$, the
following statements are equivalent:
(a) $|\nabla B|^{2}=2\big{|}\nabla|B|\big{|}^{2}$ at $p$;
(b) There is an isothermal coordinate chart $(U;u,v)$ around $p$, such that
$(\nabla B)_{www}=\zeta B_{ww}$ at $p$, with $w=u+\sqrt{-1}v$ and
$\zeta\in{\tenmsb C}$;
(c) For an arbitrary isothermal coordinate chart $(U;u,v)$ around $p$, there
is $\zeta\in{\tenmsb C}$, such that $(\nabla B)_{www}=\zeta B_{ww}$ at $p$,
with $w=u+\sqrt{-1}v$.
###### Proof.
The equivalence of (b) and (c) is obvious, so it is sufficient to prove the
equivalence of (a) and (b).
Similarly to Section 2, one can choose an isothermal coordinate neighborhood
$(U;u,v)$ of $p$, such that
$\langle B_{uu},B_{uv}\rangle=0\qquad\text{at }p.$
Then by Proposition 3.3, (a) is equivalent to (6.29).
By (6.7), one can obtain
(6.30) $(\nabla B)_{www}=\frac{1}{2}(\nabla
B)_{uuu}-\frac{\sqrt{-1}}{2}(\nabla B)_{uuv}$
with the aid of the Codazzi equations. If (6.29) holds, letting
$\zeta:=\xi_{1}-\sqrt{-1}\xi_{2}$ and combining with (6.12) and (6.30) implies
(6.31) $\displaystyle\zeta B_{ww}$
$\displaystyle=\frac{1}{2}(\xi_{1}B_{uu}-\xi_{2}B_{uv})-\frac{\sqrt{-1}}{2}(\xi_{1}B_{uv}+\xi_{2}B_{uu})$
$\displaystyle=\frac{1}{2}(\nabla B)_{uuu}-\frac{\sqrt{-1}}{2}(\nabla
B)_{uuv}=(\nabla B)_{www}.$
Conversely, if $(\nabla B)_{www}=\zeta B_{ww}$, then by letting
$\xi_{1}=\text{Re}\zeta$ and $\xi_{2}=-\text{Im}\zeta$, one can proceed
similarly to above to get (6.29). Therefore (a) and (b) are equivalent.
∎
###### Corollary 6.3.
Let $M$ be a totally G-conformal minimal surface in ${\tenmsb R}^{4}$, then
(6.32) $|\nabla B|^{2}=2\big{|}\nabla|B|\big{|}^{2}$
holds at any $p\in M$ satisfying $|B|^{2}(p)>0$.
###### Proof.
Since $M$ is totally holomorphic-like, $B_{uu}$ and $B_{uv}$ have the same
length and are orthogonal to each other. Since $\dim N_{p}M=2$ and
$|B|^{2}(p)>0$ we conclude that $N_{p}M=\text{span}\\{B_{uu},B_{uv}\\}$ and
moreover
$N_{p}M\otimes{\tenmsb C}=\text{span}\\{B_{ww},B_{\bar{w}\bar{w}}\\}.$
Thus there are $\mu_{3},\mu_{4}\in{\tenmsb C}$, such that
$(\nabla B)_{www}=\mu_{3}B_{ww}+\mu_{4}B_{\bar{w}\bar{w}}.$
Differentiating both sides of $\langle B_{ww},B_{ww}\rangle=0$ yields
(6.33) $\displaystyle 0$ $\displaystyle=\frac{1}{2}\frac{\partial}{\partial
w}\langle B_{ww},B_{ww}\rangle=\langle\nabla_{\frac{\partial}{\partial
w}}(B_{ww}),B_{ww}\rangle$ $\displaystyle=\langle(\nabla
B)_{www}+2B_{\nabla_{\frac{\partial}{\partial w}}\frac{\partial}{\partial
w},\frac{\partial}{\partial w}},B_{ww}\rangle$ $\displaystyle=\langle(\nabla
B)_{www},B_{ww}\rangle+2\mu_{1}\langle B_{ww},B_{ww}\rangle$
$\displaystyle=(\mu_{3}+2\mu_{1})\langle B_{ww},B_{ww}\rangle+\mu_{4}\langle
B_{\bar{w}\bar{w}},B_{ww}\rangle$ $\displaystyle=\mu_{4}|B_{ww}|^{2}$
where we have used (6.14). Hence $\mu_{4}=0$ and then (6.32) follows from
Proposition 6.3.
∎
## References
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* [11] J. Jost, Y. L. Xin and Ling Yang, The Gauss image of entire graphs of higher codimension and Bernstein type theorems. arXiv: 1009.3901v1, Calcu Var. and PDE. DOI10.1007/S00526-012-0533-0.
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|
arxiv-papers
| 2012-10-07T07:03:57 |
2024-09-04T02:49:36.098061
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Jost, Y. L. Xin and Ling Yang",
"submitter": "Yuanlong Xin",
"url": "https://arxiv.org/abs/1210.2031"
}
|
1210.2080
|
Locally conformally Kähler metrics obtained from pseudoconvex shells
Liviu Ornea111Partially supported by CNCS UEFISCDI, project number PN-II-ID-
PCE-2011-3-0118. and Misha Verbitsky222Partially supported by the RFBR grant
10-01-93113-NCNIL-a, Science Foundation of the SU-HSE award No. 10-09-0015 and
AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023
Keywords: Locally conformally Kähler manifold, Kähler potential, pseudoconvex,
Sasakian manifold, Vaisman manifold, Hopf manifold. 2010 Mathematics Subject
Classification: 53C55, 53C25.
Abstract
A locally conformally Kähler (LCK) manifold is a complex manifold $M$
admitting a Kähler covering $\tilde{M}$, such that its monodromy acts on this
covering by homotheties. A compact LCK manifold is called LCK with potential
if its covering admits an authomorphic Kähler potential. It is known that in
this case $\tilde{M}$ is an algebraic cone, that is, the set of all non-zero
vectors in the total space of an anti-ample line bundle over a projective
orbifold. We start with an algebraic cone $C$, and show that the set of Kähler
metrics with potential which could arise from an LCK structure is in bijective
correspondence with the set of pseudoconvex shells, that is, pseudoconvex
hypersurfaces in $C$ meeting each orbit of the associated
${\mathbb{R}}^{>0}$-action exactly once. This is used to produce explicit LCK
and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-
Ornea and Kamishima-Ornea.
###### Contents
1. 1 Introduction
1. 1.1 Constructions of LCK metrics
2. 1.2 LCK manifolds
3. 1.3 Survey of literature
2. 2 Algebraic cones and LCK manifolds with potential
1. 2.1 Algebraic cones
2. 2.2 CR-geometry and Sasakian manifolds
3. 2.3 Pseudoconvex shells in algebraic cones
4. 2.4 Vaisman metrics and pseudoconvex shells
5. 2.5 Examples and erratum
## 1 Introduction
### 1.1 Constructions of LCK metrics
Definition 1.1: 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.
For equivalent definitions and examples, see [DO] or [OV6].
A (linear) Hopf manifold $H$ is a quotient of ${\mathbb{C}}^{n}\backslash 0$
by an action of ${\mathbb{Z}}$ generated by a linear map
$A:\;{\mathbb{C}}^{n}{\>\longrightarrow\>}{\mathbb{C}}^{n}$, with all
eigenvalues satisfying $|\alpha_{i}|<1$. It is easy to see that the Hopf
manifold is diffeomorphic to $S^{1}\times S^{2n-1}$. Since $b_{1}(H)=1$ (an
odd number), $H$ is non-Kähler. In fact, this manifold is probably the
earliest example of a non-Kähler complex manifold known in mathematics.
However, any Hopf manifold is locally conformally Kähler. This observation
originated in works of Izu Vaisman of 1970-ies; Vaisman defined and studied a
strictly smaller class of manifolds, called by him “generalized Hopf”. Now
these manifolds are known as “Vaisman manifolds”, because the name
“generalized Hopf manifold” was already used by Brieskorn and van de Ven (see
[BV]) for some products of homotopy spheres which do not bear Vaisman’s
structure. Besides, now it is known that not all Hopf manifolds belong to this
smaller class.
These constructions are non-elementary. In fact, even the existence of locally
conformally Kähler metrics on many Hopf surfaces is quite non-trivial. A clean
but complicated argument was given in [GO], where such metrics were
constructed in dimension 2; when the map $A$ is diagonal, the constructed
metric is Vaisman. For non-diagonal $A$ the construction was much less
explicit; in fact, the LCK metric on these Hopf surfaces was obtained only by
deformation.
In the present paper we use the formalism of “locally conformally Kähler
metrics with potential”, that we built in previous papers, e.g. [OV4], to
obtain an explicit, computation-free and extremely simple construction of LCK
metrics on manifolds which are obtained as ${\mathbb{Z}}$-quotients of
algebraic varieties. This gives, among other things, the first explicit (i.e.
not by deformations) construction of an LCK metric on a non-diagonal Hopf
manifolds.
### 1.2 LCK manifolds
This section contains the definitions to be used in the paper. Unless
otherwise stated, we only refer to compact, connected manifolds (although the
definitions work also for noncompact manifolds).
Definition 1.2: A complex Hermitian manifold $(M,J,g)$ is locally conformally
Kähler if its fundamental two-form $\omega:=g\circ J$ satisfies
$d\omega=\theta\wedge\omega,\quad d\theta=0;$
here $\theta$ is called the Lee form. This definition is equivalent to the one
given above.
A particular subclass of LCK manifolds is described in the following:
Definition 1.3: A Vaisman manifold is a LCK manifold whose Lee form is
parallel with respect to the Levi Civita connection of $g$.
Compact Vaisman manifolds are equipped with a Riemannian submersion (a
suspension in fact) to the circle with the fibers isometric to a Sasakian
manifold $N$, see [OV1] (and see [BG] for an introduction to Sasakian
geometry). Their universal coverings are Riemannian cones
$N\times{\mathbb{R}}^{>0}$ on which the deck transform group, isomorphic to
${\mathbb{Z}}$, acts by $(x,t)\mapsto(\varphi(x),qt)$, where $\varphi$ is a
Sasakian automorphism of $N$ and $q\in{\mathbb{N}}$. The diagonal Hopf
manifold is a typical example, see §1.3. On the other hand, it is known, [B],
that non-diagonal Hopf surfaces can never be Vaisman (although they are LCK,
see [GO, OV4]).
A still wider subclass is the following:
Definition 1.4: An LCK manifold $M$ which admits a Kähler covering
$(\tilde{M},\tilde{\omega})$ with the Kähler form $\tilde{\omega}$ having a
global, automorphic potential is called LCK manifold with potential. Here, by
an automorphic potential we understand a function
$\psi:\;\tilde{M}{\>\longrightarrow\>}{\mathbb{R}}$ satisfying
$dd^{c}\psi=\tilde{\omega}$, with the monodromy of $\tilde{M}$ mapping $\psi$
to $\operatorname{\text{\sf const}}\cdot\psi$.
All Vaisman manifolds are LCK with potential (given by the squared norm of the
Lee form with respect to the Kähler metric). As for Vaisman manifolds, the
monodromy of LCK with potential manifolds is isomorphic to ${\mathbb{Z}}$. And
hence this subclass is strict, as shown by the example of the LCK Inoue
surfaces and of the LCK Oeljeklaus-Toma manifolds (see [OT]).
In this paper we shall be concerned with the linear Hopf manifolds. These are
quotients of ${\mathbb{C}}^{n}\setminus\\{0\\}$ by the cyclic group generated
by a linear operator with eigenvalues strictly smaller than $1$ in absolute
value. It is known that Hopf manifolds are LCK with potential, see [OV4], and
Vaisman if the operator is diagonal.
### 1.3 Survey of literature
There are several papers where explicit constructions of LCK metric on
diagonal Hopf manifolds appear.
The first one is [Va], where the metric (therein named after W. Boothby)
$\displaystyle\frac{\sum dz_{i}\otimes dz_{i}}{|\sum
z_{i}\overline{z}_{i}|^{2}}$ was considered on
${\mathbb{C}}^{n}\setminus\\{0\\}/\langle z_{i}\mapsto 2z_{i}\rangle$.
More than twenty years took to pass from operators $A=\alpha\cdot I_{n}$,
$\alpha\in\mathbb{C}$, to diagonal operators with complex non-equal
eigenvalues. In [GO], a LCK metric was constructed on diagonal Hopf surfaces
$H_{\alpha,\beta}:={\mathbb{C}}^{2}\setminus\\{0\\}/\langle(u,v)\mapsto(\alpha
u,\beta v)\rangle$. The construction is based on finding a Kähler potential on
${\mathbb{C}}^{2}\setminus\\{0\\}$ in terms of $\alpha,\beta$, but the formula
is only implicit. This procedure was generalized in [B]. The construction of
Vaisman metrics in the present paper can also be considered as a
generalization of [GO] to arbitrary dimensions.
In [KO] a construction was done for LCK metrics on
${\mathbb{C}}^{n}\setminus\\{0\\}/\langle z_{i}\mapsto\alpha_{i}z_{i}\rangle$,
starting from a deformation of the standard Sasakian structure of $S^{2n-1}$
according to the $S^{1}$ action with weights $\alpha_{i}$ (cf. also [GO,
Section 3]). The paper [KO] also contains a very useful criterion to decide
when a conformal class of LCK metrics on a complex manifold contains a Vaisman
representative, in terms of the existence of a holomorphic complex flow which
lifts to a non-trivial flow of homotheties of the Kähler covering.
A different construction on the same manifold, writing explicitly a Kähler
potential on ${\mathbb{C}}^{n}\setminus\\{0\\}$, appeared in [Ve] and since
then it was cited in almost all our subsequent papers. However, as observed by
Matei Toma and Ryushi Goto, that metric is singular.
To correct this error we provide here a general construction of LCK metrics on
Hopf manifolds. Our approach works for LCK manifolds with potential, giving a
complete list of LCK metrics with potential in terms of a pseudoconvex shells
in the covering (2.3).
For the time being, we don’t know whether is it possible or not to write a
formula for a potential for an LCK metric on a Hopf manifold; in the present
paper, as well as in [GO], the potential is written as a solution of a certain
differential equation.
## 2 Algebraic cones and LCK manifolds with potential
### 2.1 Algebraic cones
Definition 2.1: A closed algebraic cone is an affine variety $C$ admitting a
${\mathbb{C}}^{*}$-action $\tau$ with a unique fixed point $x_{0}$ (called the
origin), which satisfies the following.
1.
$C$ is smooth outside of $x_{0}$.
2.
$\tau$ acts on the Zariski tangent space $T_{x_{0}}{\cal C}$ diagonally, with
all eigenvalues $|\alpha_{i}|<1$.
An open algebraic cone is a closed algebraic cone with the origin removed:
$C\setminus\\{x_{0}\\}$.
Definition 2.2: Let $X$ be a projective orbifold, and let $L$ be an ample line
bundle on $X$. Assume that the total space of $L$ is smooth outside of the
zero divisor. The algebraic cone ${\cal C}(X,L)$ of $X,L$ is the total space
of non-zero vectors in $L^{*}$. A cone structure on ${\cal C}(X,L)$ is the
${\mathbb{C}}^{*}$-action arising this way (by fiberwise multiplication).
In [OV3, Section 4], it was shown that any open algebraic cone $C$ is
isomorphic to ${\cal C}(X,L)$, for appropriate $X$ and $L$. This was shown by
the following argument. Given the algebraic cone $C$, one obtains $X$ as the
quotient of $C$ by ${\mathbb{C}}^{*}$, and then the cone $C$ is naturally
identified with the total space of a principal ${\mathbb{C}}^{*}$-bundle
$L_{1}$. Ampleness of this bundle follows, because the corresponding closed
algebraic cone $C_{c}$ is an affine variety, and algebraic functions on
$C_{c}$ are identified with the section of the line bundle $L$ associated with
$L_{1}$.
Definition 2.3: Let $\gamma$ be an automorphism of a closed algebraic cone. It
is called a holomorphic contraction if for any compact subset $K\subset C$,
and any open neighbourhood $U$ of the origin, there exists a number $N$
sufficiently big such that $\gamma^{N}(K)\subset U$.
Definition 2.4: Let $C$ be a closed algebraic cone, and
$\rho:\;{\mathbb{R}}^{>0}{\>\longrightarrow\>}\operatorname{Aut}(C)$ a
${\mathbb{R}}^{>0}$-action. We say that ${\mathbb{R}}^{>0}$ acts by
holomorphic contractions, if $\rho(t)$ is a holomorphic contraction for all
$t<1$.
Example 2.5: Let $C={\mathbb{C}}^{n}={\cal C}({\mathbb{C}}P^{n-1},{\cal
O}(1))$. Then any linear automorphism of $C$ with all eigenvalues
$|\alpha_{i}|<1$ acts on $C$ by holomorphic contractions.
Example 2.6: Let
$\rho:=\tau{\left|{}_{{\phantom{|}\\!\\!}_{{\mathbb{R}}^{>0}}}\right.}$ be the
action of ${\mathbb{R}}^{>0}$ on an algebraic cone provided by the cone
structure, ${\mathbb{R}}^{>0}\subset{\mathbb{C}}^{*}$. Since $\rho$ acts on
the tangent space $T_{c}C$ to the origin with eigenvalues smaller than 1, it
acts on $C$ by holomorphic contractions ([OV4, Theorem 3.3]).
Remark 2.7: As shown in [OV4, Theorem 3.3], the quotient of an algebraic cone
by a contraction is an LCK manifold with potential, and, conversely, any LCK
manifold with potential is obtained by taking the quotient of an open
algebraic cone by a holomorphic contraction. Such a contraction, being a
priori a ${\mathbb{Z}}$–action, can be extended to a
${\mathbb{R}}^{>0}$–action by holomorphic contractions.
Further on, we use the following version of this result.
Theorem 2.8: Let $M$ be a locally conformally Kähler manifold with potential.
Then $\tilde{M}$, as a complex manifold, is isomorphic to an open algebraic
cone $C$, equipped with an action $\rho$ of ${\mathbb{R}}^{>0}$ by holomorphic
contractions, and the quotient $\tilde{M}/\langle\rho(2^{n})\rangle$ is
isomorphic (as a complex manifold) to $M$.
Proof: In [OV5, Theorem 2.1], it is shown that $M$ can be deformed into a
Vaisman manifold. From its proof it is apparent that this deformation
preserves $\tilde{M}$ (in fact, only the ${\mathbb{Z}}$-action is deformed).
Therefore, $\tilde{M}$ is a covering of a Vaisman manifold. Then, it is an
algebraic cone, as follows from [OV2, Proposition 4.6].
### 2.2 CR-geometry and Sasakian manifolds
In this subsection, we introduce the Sasakian manifolds and some related
notions of CR-geometry. We follow [OV3].
Definition 2.9: A CR-structure (Cauchy-Riemann structure) on a manifold $M$ is
a subbundle $H\subset TM\otimes{\mathbb{C}}$ of the complexified tangent
bundle, which is closed under commutator: $[H,H]\subset H$ and satisfies
$H\cap\overline{H}=0$.
A function $f:M\rightarrow{\mathbb{C}}$ is CR-holomorphic if $D_{V}f=0$ for
any vector field $V\in\overline{H}$.
On a CR manifold $(M,H)$, the bundle $H\oplus\overline{H}$ is preserved by
complex conjugation and hence it is obtained as a complexification of a real
subbundle $H_{\mathbb{R}}$. Then $I_{H}:=-\sqrt{-1}\text{Id}_{\overline{H}}$
defines a complex structure on $H_{\mathbb{R}}$ and $H$ is its
$\sqrt{-1}$-eigenspace of its extension to the complexification
$H_{\mathbb{R}}\otimes{\mathbb{C}}$.
If $\text{codim}_{TM}H_{\mathbb{R}}=1$, and the Frobenius tensor
$L:H_{\mathbb{R}}\times H_{\mathbb{R}}\rightarrow TM/H_{\mathbb{R}}$,
$L(X,Y)=[X,Y]\mod H_{\mathbb{R}}$ is nondegenerate, then $(M,H)$ is a CR
contact manifold and $H_{\mathbb{R}}$ is its contact structure (or
distribution). In this context, $L$ is called the Levi form.
As $L$ vanishes on $H$ and $\overline{H}$, $L$ is $(1,1)$ with respect to
$I_{H}$.
Definition 2.10: A contact CR-manifold $(M,H_{\mathbb{R}},I_{H})$ is called
pseudo-convex if the Levi form is positive or negative, depending on the
choice of orientation. If this form is also sign-definite, then
$(M,H_{\mathbb{R}},I_{H})$ is called strictly pseudoconvex.
Definition 2.11: Let $S$ be a CR-manifold. A CR-holomorphic vector field $v\in
TS$ is called transversal if it is transversal to the CR-distribution
$H_{\mathbb{R}}\subset TS$.
Theorem 2.12: [OV3, Theorem 1.2] Let $M$ be a compact pseudoconvex contact
CR-manifold. Then the following conditions are equivalent.
(i)
$M$ admits a Sasakian metric, compatible with the CR-structure.
(ii)
$M$ admits a proper, transversal CR-holomorphic $S^{1}$-action.
(iii)
$M$ admits a nowhere degenerate, transversal CR-holomorphic vector field.
Theorem 2.13: [OV3, Theorem 1.3] Let $M$ be a compact, strictly pseudoconvex
CR-manifold admitting a proper, transversal CR–holomorphic $S^{1}$–action.
Then $M$ admits a unique (up to an automorphism) $S^{1}$-invariant CR-
embedding into an algebraic cone $({\cal C},\tau)$. Moreover, a Sasakian
metric on $M$ can be induced from a Kähler metric $\tilde{\omega}$ on this
cone, which is automorphic in the following sense: for some constant $c>1$,
one has $\tau(t)^{*}\tilde{\omega}=|t|^{c}\tilde{\omega}$.
### 2.3 Pseudoconvex shells in algebraic cones
Definition 2.14: Let $C$ be an algebraic cone, equipped with an action $\rho$
of ${\mathbb{R}}^{>0}$ by holomorphic contractions. A pseudoconvex shell in
$C$ is a strictly pseudoconvex submanifold in $C$, intersecting each orbit of
$\rho$ exactly once.
Remark 2.15: Please note that the action of $\rho$ may bear no relation to the
cone action $\tau:{\mathbb{C}}^{*}{\>\longrightarrow\>}\operatorname{Aut}(C)$.
Theorem 2.16: Let $M=C/\langle\rho(q)\rangle$ where $(C,\tau)$ be an
algebraic cone, equipped with the action $\rho$ of ${\mathbb{R}}^{>0}$ by
holomorphic contractions and $q>1$. Let $\vec{r}$ be the infinitesimal
generator of $\rho$ and let $S$ be a pseudoconvex shell in $C$. Then for each
$\lambda\in{\mathbb{R}}$ there exists a unique function $\varphi_{\lambda}$
such that
$\operatorname{Lie}_{\vec{r}}\varphi_{\lambda}=\lambda\varphi_{\lambda}$ and
$\varphi_{\lambda}{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}=1$. Moreover,
such $\varphi_{\lambda}$ is plurisubharmonic for sufficiently big
$\lambda>\\!\\!>0$. Conversely, any LCK manifold with potential admits a
metric obtained this way.
Proof: For each $\rho$-orbit and each $\rho$-equivariant potential $\varphi$,
one has:
$\rho(t)\cdot\varphi_{\lambda}=e^{t\lambda}\varphi_{\lambda},\qquad
t\in{\mathbb{R}}^{>0}.$
Let $S$ be a pseudoconvex shell in $C$. Then
$S\times{\mathbb{R}}^{>0}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}C$,
as orbits intersect the shell only once. Hence for any $s\in S$ and for any
$t\in{\mathbb{R}}^{>0}$ we have:
$\varphi_{\lambda}(\rho(t)\cdot s)=e^{\lambda t},$ (2.1)
and this equation uniquely defines $\varphi_{\lambda}$. The problem is to
prove that $dd^{c}\varphi_{\lambda}>0$ and this is not automatic for
$\varphi_{\lambda}$, but it holds for some power of it.
Now let $B:=e^{{\mathbb{R}}\vec{r}}\cdot(TS\cap I(TS))\subset TC$ be the
subbundle obtained by translating $TS\cap I(TS)$ with all $e^{t\vec{r}}$.
Then, by construction,
$dd^{c}\varphi_{\lambda}{\left|{}_{{\phantom{|}\\!\\!}_{B}}\right.}$ is the
Levi form of $B$ and hence it is positive definite.
It will now suffice to show that
$dd^{c}\varphi_{2a\lambda}{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}=dd^{c}\varphi_{\lambda}^{2a}{\left|{}_{{\phantom{|}\\!\\!}_{S}}\right.}>0$
for sufficiently big $a$. But
$dd^{c}\varphi_{\lambda}^{2a}=\varphi_{\lambda}^{2a-2}\big{(}2a\varphi_{\lambda}\cdot
dd^{c}\varphi_{\lambda}+2a(2a-1)d\varphi_{\lambda}\wedge
d^{c}\varphi_{\lambda}\big{)}.$
As the shell $S$ is compact, the result is implied by the following elementary
linear algebra lemma:
Lemma 2.17: Let $h_{1}$, $h_{2}$ be pseudo-Hermitian forms on a complex
vector space $V$ and let $W\subset V$ be a codimension 1 subspace. Assume that
$h_{1}{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}$ and
$h_{2}{\left|{}_{{\phantom{|}\\!\\!}_{V/W}}\right.}$ are strictly positive
(that is, positive definite), and
$h_{2}{\left|{}_{{\phantom{|}\\!\\!}_{W}}\right.}=0$. Then there exists
$u_{0}\in{\mathbb{R}}$, depending continuously on $h_{1},h_{2}$, such that
$h_{u}:=h_{1}+uh_{2}$ is positive definite for any $u>u_{0}$.
The direct part of the Theorem now follows by applying 2.3 (whose proof we
postpone) to $V=TM$, $W=B$, $h_{1}=\varphi_{\lambda}dd^{c}\varphi_{\lambda}$,
$h_{2}=d\varphi_{\lambda}\wedge d^{c}\varphi_{\lambda}$.
For the converse, let $\varphi_{\lambda}$ be any automorphic potential, thus
satisfying $(\gamma^{k})^{*}=e^{\lambda}\varphi_{\lambda}$, and let $\vec{r}$
be the holomorphic vector field which is the logarithm of the monodromy
action. Let then $\rho(t)=e^{-t\lambda}\big{(}e^{t}\vec{r}\big{)}^{*}$ the
corresponding endomorphism of ${\cal C}^{\infty}(M)$. Then
$\rho(k+t)(\varphi_{\lambda})=\rho(t)\varphi_{\lambda}$ and hence the orbit of
$\rho$ through $\varphi$ is compact. We then average $\rho(t)\varphi$ on
$\mathbb{R}$ and obtain a $\rho(t)$-invariant Kähler potential
$\varphi_{\lambda 0}$. This $\varphi_{\lambda 0}$ is obtained from a
pseudoconvex shell $S=\varphi_{\lambda 0}^{-1}(1)$ and from $\vec{r}$ as in
the direct part of the Theorem.
Let us now give the proof of 2.3. For simplicity, we work in the real setting,
and we consider $h_{1},h_{2}$ as bilinear symmetric forms. Let $y\in V$ be a
vector such that $h_{2}(y,y)=1$. Then any $x\in V$ can be written as $x=ay+z$,
for some $z\in W$. This translates to:
$h_{u}(x,x)=ua^{2}+a^{2}h_{1}(y,y)+h_{1}(z,z)+2ah_{1}(z,y)$
which we view as a polynomial in $a$. This one is positive definite for all
$a$ if and only if
$(h_{1}(z,y))^{2}-(u+h_{1}(y,y))\cdot h_{1}(z,z)<0.$ (2.2)
Choose $y^{\prime}\in W$ such that $h_{1}(z,y^{\prime})=h_{1}(z,y)$ for all
$z\in W$ and let $u>u_{0}:=h_{1}(y^{\prime},y^{\prime})-h_{1}(y,y)$. Then
(2.2) becomes
$(h_{1}(z,y^{\prime}))^{2}-h_{1}(y^{\prime},y^{\prime})h_{1}(z,z)<0,$
which is satisfied by Cauchy-Buniakovski-Schwarz inequality, as $h_{1}$ is
positive definite on $W$.
Example 2.18: Let $A$ be a linear operator on ${\mathbb{C}}^{n}$ with
eigenvalues of absolute values strictly smaller than $1$. Let
$C={\mathbb{C}}^{n}\setminus\\{0\\}$ and let $\rho(t)=e^{t\log A}$. Take a
sphere $S=S^{2n-1}\subset{\mathbb{C}}^{n}$; it is easy to see that $S$ is a
pseudoconvex shell. Applying 2.3, we obtain an automorphic potential $\varphi$
on ${\mathbb{C}}^{n}\backslash 0$, giving an LCK metric on
$M=({\mathbb{C}}^{n}\backslash 0)/\langle A\rangle$.
Remark 2.19: When $n=2$ and $A$ is diagonal, the same potential was obtained
in [GO].
In particular, we recover the result proven in [GO, KO, OV4] that all Hopf
manifolds $({\mathbb{C}}^{n}\setminus\\{0\\})/\langle A\rangle$ are LCK.
### 2.4 Vaisman metrics and pseudoconvex shells
Remark 2.20: Vaisman manifolds are LCK with potential and hence they have
canonical pseudoconvex shells (levels of the potential).
Definition 2.21: Let $(M,I,g)$ be a Vaisman manifold, let $(C,\rho)$ be the
associated algebraic cone, equipped with the action $\rho$ of
${\mathbb{R}}^{>0}$ by holomorphic contractions, and let $S$ be its
pseudoconvex shell. The Reeb field of $M$ is the CR-holomorphic vector field
$I\theta^{\sharp}\in TS$ obtained from $\rho$ by complex conjugation.
Remark 2.22: For Vaisman manifolds, the Reeb field is always transversal
([OV3]).
Proposition 2.23: Let $v\in TS$ be a CR-holomorphic vector field on $S$, where
$S$ is a pseudoconvex shell in an algebraic cone $C$. Then $v$ can be uniquely
extended to a holomorphic vector field on the whole of $C$. 111This is called
the holomorphic extension of $v$.
Proof: Let ${\cal O}_{S}$, respectively ${\cal O}_{{S^{\circ}}}$ be the ring
of CR-holomorphic functions on $S$, respectively on the interior ${S^{\circ}}$
of the shell $S$. By the solution of the Neumann problem, $L^{2}({\cal
O}_{S})=L^{2}({\cal O}_{{S^{\circ}}})$ and hence, if we restrict to bounded
functions, ${\cal O}_{{S^{\circ}}}={\cal O}_{S}$ as rings. As vector fields on
$S$ and ${S^{\circ}}$ are derivations of the above rings, the result follows.
Theorem 2.24: Let $(M,I,g)$ be an LCK manifold obtained (as in 2.3) from an
algebraic cone $C$ and a pseudoconvex shell $S$, and let
$\gamma:\;{\mathbb{Z}}{\>\longrightarrow\>}\operatorname{Aut}(C)$ be the deck
transform map. Then the Hermitian manifold $(M,I,g)$ is conformally equivalent
to a Vaisman one if and only if $S$ admits a transversal CR-holomorphic vector
field $\xi$, such that its holomorphic extension to $C$ is
$\gamma(1)$–invariant and $\exp(-I\xi)\cdot\gamma(1)$ preserves $S$.
Proof: Suppose $(M,I,g)$ is Vaisman. Then $\xi=I\theta^{\sharp}$ is an
isometry of the LCK metric. The Kähler metric on $C$ is $dd^{c}\varphi$, with
$\varphi$ given by the equation
$e^{t\theta^{\sharp}}(S)=\varphi^{-1}(t),$
where $S$ is the level set of $\varphi$. Then $S$ is Sasakian and
$I\theta^{\sharp}$ is the Reeb field of the underlying contact structure,
hence it is transversal by definition. By construction,
$M=C/\langle\gamma(1)\rangle$, and the extension of $\xi$ to $C$ is the lift
to $C$ and is $\gamma(1)$ invariant by definition. We have checked all the
conditions of 2.4, except the last one: $\exp(-I\xi)\cdot\gamma(1)(S)=S$.
The Lee field $-I\xi=\theta^{\sharp}$ acts by homotheties on the potential
$\varphi$, hence $\operatorname{Lie}_{\exp(-I\xi)}\varphi=c\varphi$ for some
contant $c$. The monodromy map $\gamma(1)$ also acts by homotheties on
$\varphi$: $\operatorname{Lie}_{\rho(1)}\varphi=c^{\prime}\varphi$ for another
constant $c^{\prime}$. What we have to do is to homothetically modify the
initial metric $g$ such that $c$ becomes $1/c^{\prime}$; in this case
$\exp(-I\xi)\cdot\gamma(1)$ will preserve the potential $\varphi$ and hence
will preserve $S$.
Conversely, the vector field $\xi$ is tangent to $S$ and its flow $\exp(t\xi)$
acts by contractions, hence defining a metric on the cone $C$ over $S$. As
$\gamma(1)$ maps a shell $S_{\lambda}$ to $S_{\operatorname{\text{\sf
const}}\cdot\lambda}$, if follows that $\gamma(1)$ acts by homotheties on $C$.
This means that the complex flow generated on $M=C/\langle\gamma(1)\rangle$ by
$\xi$ and $I\xi$ lifts to a flow of non-trivial homotheties on the cone. By
[KO] this ensures the existence of a Vaisman metric in the conformal class of
$g$.
### 2.5 Examples and erratum
As an application, we add an erratum to several papers where we have given a
wrong formula for an LCK-metric on diagonal Hopf manifolds (e.g. [Ve], [O]),
and give a general and almost explicit construction of a Vaisman metric on any
diagonal Hopf manifold. This constructions originates in the ones in [GO, B,
KO], but unifies them and presents them in a much more synthetic and
transparent way.
We apply 2.4. The data to start with are the cone $C$, the shell $S$, the
vector field $\xi$ and the monodromy $\gamma$.
Let $A=\mathrm{diag}(\alpha_{1},\cdots,\alpha_{n})$, with
$0<|\alpha_{1}|\leqslant|\alpha_{2}|\leqslant\cdots\leqslant|\alpha_{n}|<1$.
Denote by $A_{|\cdot|}$ the matrix
$\mathrm{diag}(|\alpha_{1}|,\cdots,|\alpha_{n}|)$ (in the same basis).
Let $C$ be ${\mathbb{C}}^{n}\setminus\\{0\\}$. As a shell $S$, we take the
sphere $S^{2n-1}$, but one can take for $S$ the boundary of any strictly
pseudoconvex body containing 0 and satisfying $A_{|\cdot|}A^{-1}(S)=S$.
The transversal CR-holomorphic vector field $\xi$ is
$\xi=\sqrt{-1}\mathrm{Re}\log
A=\sqrt{-1}\log\mathrm{diag}(|\alpha_{1}|,\ldots,|\alpha_{n}|)$ (2.3)
and the monodromy map $\gamma$ is given by $\gamma(z)=A\cdot z$, $z\in C$.
To apply 2.4 we need to verify that $\xi$ is $\gamma(1)$–invariant and
$\exp(-I\xi)\cdot\gamma(1)$ preserves $S$. An easy computation shows that
$\exp(-I\xi)\cdot\gamma(1)$ acts as
$A_{|\cdot|}A^{-1}=\mathrm{diag}(\frac{\alpha_{1}}{|\alpha_{1}|},\cdots,\frac{\alpha_{n}}{|\alpha_{n}|})$
and thus it preserves the norm of vectors, hence preserving the spheres.
Finally, as the action of $\gamma(1)$ is linear, the $\gamma(1)$–invariance of
$\xi$ amounts to $A\cdot\xi_{z}=\xi_{A\cdot z}$ which is immediate from (2.3).
This gives an explicit construction of Vaisman structure on diagonal Hopf
manifolds. On the other hand, 2.3 provides a rather explicit construction of
LCK metrics on non-diagonal Hopf manifolds, avoiding the argument by
deformations in [OV4].
Acknowledgments: We thank Matei Toma and Ryushi Goto for drawing our attention
to the wrong formula which originated this paper, and to Max Planck Institute
for Mathematics (Bonn), where this paper was finished, for the excellent
research environment.
## References
* [B] F.A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.
* [BG] C.P. Boyer, K. Galicki, Sasakian geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008.
* [BV] E. Brieskorn, A. van de Ven, Some complex structures on products of homotopy spheres, Topology 7 (1968) 389–393.
* [DO] S. Dragomir and L. Ornea, Locally conformal Kähler geometry, Progress in Math. 155, Birkhäuser, Boston, Basel, 1998.
* [GO] P. Gauduchon and L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier (Grenoble) 48 (1998), 1107–1127.
* [KO] Y. Kamishima and L. Ornea, Geometric flow on compact locally conformally Kähler manifolds, Tohoku Math. J. 57 (2005), 201–221.
* [OT] K. Oeljeklaus and M. Toma, Non-Kähler compact complex manifolds associated to number fields, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 161–171.
* [O] L. Ornea, Locally conformally Kaehler manifolds. A selection of results. Lecture Notes of Seminario Interdisciplinare di Matematica, 4 (2005), 121–152.
* [OV1] L. Ornea and M. Verbitsky, Structure theorem for compact Vaisman manifolds, Math. Res. Lett., 10 (2003), 799–805.
* [OV2] L. Ornea and M. Verbitsky, An immersion theorem for compact Vaisman manifolds, Math. Ann. 332 (2005), no. 1, 121-143, see also math.AG/0306077.
* [OV3] L. Ornea and M. Verbitsky, _Sasakian structures on CR-manifolds_ , math.DG/0606136, Geom. Dedicata 125 (2007), 159–173.
* [OV4] L. Ornea and M. Verbitsky, Locally conformally Kähler manifolds with potential, Mathematische Annalen, 248 (1) (2010), 25–33, math.AG/0407231.
* [OV5] L. Ornea and M. Verbitsky, Topology of locally conformally Kahler manifolds with potential, arXiv:0904.3362, Int. Math. Res. Not., 4, 717–726 (2010).
* [OV6] L. Ornea and M. Verbitsky, A report on locally conformally Kähler manifolds, Contemporary Mathematics 542, 135–150, 2011. arXiv:1002.3473.
* [Va] I. Vaisman, On locally conformal almost Kähler manifolds Israel J. Math. 24 (1976), 338–351.
* [Ve] M. Verbitsky, Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds, Proc. Steklov Inst. Math. 246 (2004) 54–78. arXiv:math/0302219.
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,
National Research University HSE,
7 Vavilova Str. Moscow, Russia, 117312
verbit@mccme.ru, verbit@verbit.ru
|
arxiv-papers
| 2012-10-07T17:33:22 |
2024-09-04T02:49:36.111320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liviu Ornea, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1210.2080"
}
|
1210.2141
|
# Sharp error bounds for Jacobi expansions and Gengenbauer-Gauss quadrature of
analytic functions
Xiaodan Zhao${}^{1},$ Li-Lian Wang1 and Ziqing Xie2
###### Abstract.
This paper provides a rigorous and delicate analysis for exponential decay of
Jacobi polynomial expansions of analytic functions associated with the
Bernstein ellipse. Using an argument that can recover the best estimate for
the Chebyshev expansion, we derive various new and sharp bounds of the
expansion coefficients, which are featured with explicit dependence of all
related parameters and valid for degree $n\geq 1$. We demonstrate the
sharpness of the estimates by comparing with existing ones, in particular, the
very recent results in [38, SIAM J. Numer. Anal., 2012]. We also extend this
argument to estimate the Gegenbauer-Gauss quadrature remainder of analytic
functions, which leads to some new tight bounds for quadrature errors.
###### Key words and phrases:
Bernstein ellipse, exponential convergence, analytic functions, Jacobi
polynomials, Gegenbuer-Gauss quadrature, error bounds, sharp estimate
###### 1991 Mathematics Subject Classification:
65N35, 65E05, 65M70, 41A05, 41A10, 41A25
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 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).
## 1\. Introduction
The spectral method employs global orthogonal polynomials or Fourier complex
exponentials as basis functions, so it enjoys high-order accuracy (with only a
few basis functions), if the underlying function is smooth (and periodic in
the Fourier case). The convergence rate $O(n^{-r})$, where $n$ is the number
of basis functions involved in a spectral expansion and $r$ is related to the
Sobolev-regularity of the underlying function, is typically documented in
various monographs on spectral methods [18, 15, 14, 4, 21, 35, 7, 8, 24, 32].
It is also widely appreciated that if the function under consideration is
analytic, the convergence rate is of exponential order $O(q^{n})$ (for
constant $0<q<1$). However, there appears very limited discussions of such
error bounds (mostly mentioned, but not proved) in [14, 35, 7]. Indeed, as
commented by Hale and Trefethen [23], the general idea of such convergence
goes back to Bernstein in early nineties, but such results do not appear in
many textbooks or monographs, and there is not much uniformity in the
constants in the upper bounds.
An important result in Bernstein [5] (1912) (also see [28]) states that $u$ is
analytic on $[-1,1],$ if and only if
$\sup\lim_{N\to\infty}\sqrt[N]{E_{N}(u)}=\frac{1}{\rho},\quad
E_{N}(u)=\inf_{v\in P_{N}}\|v-u\|_{\infty},$
where $P_{N}$ is the polynomial space of degree no more than $N$, and $\rho>1$
is the sum of the semi-axes of the maximum ellipse ${\mathcal{E}}_{\rho}$ with
foci $\pm 1,$ known as the Bernstein ellipse, on and within which $u$ can be
analytically extended to. One immediate implication is that the best
polynomial approximation in the maximum norm enjoys exponential convergence. A
more precise estimate for the Chebyshev expansion can be found in various
approximation theory texts (see e.g., [31, Theorem 3.8] and [29, Theorem
5.16]):
$|\hat{u}_{n}^{C}|\leq\frac{2M}{\rho^{n}},\;\;\forall n\geq
0;\quad\big{\|}u-S_{N}^{C}u\big{\|}_{\infty}\leq\frac{M}{(\rho-1)\rho^{N}},$
(1.1)
where $M=\max_{z\in{\mathcal{E}}_{\rho}}|u(z)|,$ $\\{\hat{u}_{n}^{C}\\}$ are
Chebyshev expansion coefficients of $u,$ and $S_{N}^{C}u$ is the partial sum
involving the first $N+1$ terms. One also refers to [33, 12, 31, 6, 29, 36,
37] and the references therein for verification/description of exponential
convergence of Fourier, Chebyshev or Legendre expansions. We remark that
Gottlieb and Shu et al [20, 19] studied exponential convergence of Gegenbauer
expansions (when the parameter grows linearly with $n$) in the context of
defeating the Gibbs phenomenon.
Here, we particularly highlight that a very recent paper of Xiang [38]
provided a simple approach to obtain the bounds for Jacobi expansion
coefficients of analytic functions on and within the Bernstein ellipse
${\mathcal{E}}_{\rho}:$
$|\hat{u}_{n}^{\alpha,\beta}|\leq\frac{2M}{\rho^{n-1}(\rho-1)}\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}}\;\;{\rm
where}\;\;\hat{u}_{n}^{\alpha,\beta}=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}u(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)dx.$
(1.2)
Here, $\\{J_{n}^{\alpha,\beta}\\}(\alpha,\beta>-1)$ are Jacobi polynomials
mutually orthogonal with the weight function
$\omega^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}$ and with the
normalization factor $\gamma_{n}^{\alpha,\beta}$ (cf. (2.8)). The key step is
to insert the Chebyshev expansion
$u(x)=\sum_{j=0}^{\infty}\hat{u}_{j}^{C}T_{j}(x)$ into the Jacobi expansion
coefficients and rewrite
$\hat{u}_{n}^{\alpha,\beta}=\frac{1}{\gamma_{n}^{\alpha,\beta}}\sum_{j=0}^{\infty}\hat{u}_{j}^{C}\int_{-1}^{1}T_{j}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)dx,$
so the bound for the Chebyshev coefficient in (1.1) could be used.
The first purpose of the paper is to take a different approach to derive sharp
estimates for general Jacobi expansion of analytic functions. The assertion of
sharpness is in the following sense:
* (i)
The bound for general Jacobi case is tighter than (1.2) (see Remark 2.3).
* (ii)
Refined estimates can be obtained for Gegenbauer expansion
($\alpha=\beta>-1$), Chebyshev-type expansion ($\alpha=k-1/2,\beta=l-1/2$ for
non-negative integers $k,l$), and Legendre-type expansion ($\alpha=k,\beta=l$
for non-negative integers $k,l$). The argument can recover the bounds known to
be the sharpest (e.g., the Chebyshev case), and some obtained estimates are
new and significantly improve the existing ones (see e.g., Remark 2.5).
A second purpose of this work is to extend the argument to analyze Gegenbauer-
Gauss quadrature of analytical functions. Recall that the remainder of Gauss-
quadrature with the nodes and weights $\\{x_{j},\omega_{j}\\}_{j=1}^{n},$
takes the form (see e.g., [13]):
$E_{n}[u]=\int_{-1}^{1}u(x)\omega(x)\,dx-\sum_{j=1}^{n}u(x_{j})\omega_{j}=\frac{1}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{q_{n}(z)}{p_{n}(z)}u(z)\,dz,$ (1.3)
where $\\{x_{j}\\}_{j=1}^{n}$ are the zeros of $p_{n}(x),$ orthogonal with
respect to the weight function $\omega(x),$ and
$q_{n}(z)=\frac{1}{2}\int_{-1}^{1}\frac{p_{n}(x)\omega(x)}{z-x}\,dx.$ (1.4)
The estimate of quadrature errors has attracted much attention (see e.g., [11,
10, 3, 17, 13, 16, 25, 26]). Among these results, intensive discussions have
been centered around the Chebyshev case and its family, e.g., Chebyshev of the
second kind, but with very limited results even for Legendre-Gauss quadrature
(see e.g., [9, 27]). In fact, the analysis heavily relies on the availability
of explicit expression of $p_{n}(z)$ on $\mathcal{E}_{\rho}.$ Armed with a
delicate estimate of $q_{n}(z)$ (in the first part of the paper) and the
explicit formula of Gegenbauer polynomial in [39], we are able to derive sharp
bound for the Gegenbauer-Gauss quadrature errors.
We remark that there has been much interest in estimating spectral
differentiation errors of analytic functions. Tadmor [34] first attempted to
estimate the aliasing errors to verify exponential convergence of Fourier and
Chebyshev spectral differentiation with a different assumption on analyticity.
The results for analyticity characterized by the Bernstein ellipse include
Reddy and Weideman [30] for Chebyshev case, and Xie, Wang and Zhao [39] for
Gegenbauer spectral differentiation. It is also interesting to point out that
Zhang [40, 41, 42] studied superconvergence of spectral interpolation and
differentiation. We stress that the analysis apparatuses and arguments in this
pipeline are different from these in this work.
The rest of this paper is organized as follows. In Section 2, we provide sharp
bounds for general Jacobi expansions of analytic functions, followed by some
refined results for Chebyshev-type and Legendre-type expansions. In Section 3,
we extend the argument to analyze Gegenbauer-Gauss quadrature errors. In the
final section, we provide results to show the sharpness of the bounds by
comparing them with existing ones.
## 2\. Sharp bounds for Jacobi expansions
We derive in this section sharp bounds for Jacobi expansions of functions
analytic on and within the Bernstein ellipse ${\mathcal{E}}_{\rho}$.
### 2.1. Preliminaries
It is known (see e.g., [12]) that the Bernstein ellipse is transformed from
the circle
${\mathcal{C}}_{\rho}=\big{\\{}w=\rho e^{{\rm
i}\theta}:\theta\in[0,2\pi]\big{\\}},\quad\rho>1,$ (2.1)
via the conformal mapping: $z=(w+w^{-1})/2,$ namely,
${\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{\\}},$ (2.2)
where ${\mathbb{C}}$ is the set of all complex numbers, and ${\rm
i}=\sqrt{-1}$ is the complex unit. It has the foci at $\pm 1,$ and the major
and minor semi-axes are
$a=\frac{1}{2}\big{(}\rho+\rho^{-1}\big{)},\quad
b=\frac{1}{2}(\rho-\rho^{-1}),$ (2.3)
respectively, so the sum of two semi-axes is $\rho.$ The perimeter of
${\mathcal{E}}_{\rho}$ has the bound
$L({\mathcal{E}}_{\rho})\leq\pi\sqrt{\rho^{2}+\rho^{-2}},$ (2.4)
which overestimates the perimeter by less than $12$ percent (cf. [30]). The
distance from ${\mathcal{E}}_{\rho}$ to the interval $[-1,1]$ is
$d_{\rho}=\frac{1}{2}(\rho+\rho^{-1})-1.$ (2.5)
We see that $d_{\rho}$ increases with respect to $\rho,$ and $d_{\rho}\to
0^{+}$ as $\rho\to 1^{+}$ (so the ellipse reduces to the interval $[-1,1]$).
Thus, by the theory of analytic continuation, we have that for any analytic
function $u$ on $[-1,1],$ there always exists a Bernstein ellipse
${\mathcal{E}}_{\rho}$ with $\rho>1$ such that the continuation of $u$ is
analytic on and within ${\mathcal{E}}_{\rho}.$ Hereafter, we denote by
${\mathcal{A}}_{\rho}:=\big{\\{}u\,:\,u\;\;\text{is analytic on and
within}\;\;{\mathcal{E}}_{\rho}\big{\\}},\quad 1<\rho<\rho_{\rm max},$ (2.6)
where ${\mathcal{E}}_{\rho_{\rm max}}$ labels the largest ellipse within which
$u$ is analytic. In particular, if $\rho_{\rm max}=\infty,$ $u$ is an entire
function.
Throughout this paper, the Jacobi polynomials, denoted by
$J_{n}^{\alpha,\beta}(x)$ (with $\alpha,\beta>-1$ and $x\in I:=(-1,1)$), are
normalized as in Szegö [33], i.e.,
$\int_{-1}^{1}J_{n}^{\alpha,\beta}(x)J_{m}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx=\gamma_{n}^{\alpha,\beta}\delta_{m,n},$
(2.7)
where $\omega^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta},$ $\delta_{m,n}$
is the Kronecker delta, and
$\gamma_{n}^{\alpha,\beta}=\frac{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n!\Gamma(n+\alpha+\beta+1)}.$
(2.8)
In Appendix A, we collect the relevant properties of Jacobi polynomials.
In the analysis, we also use the following property of the Gamma function,
derived from [1, Eq. (6.1.38)]:
$\Gamma(x+1)=\sqrt{2\pi}x^{x+1/2}\exp\Big{(}-x+\frac{\theta}{12x}\Big{)},\quad\forall\,x>0,\;\;0<\theta<1.$
(2.9)
###### Lemma 2.1.
For any constants $a,b,$ we have that for $n\geq 1,$ $n+a>1$ and $n+b>1,$
$\frac{\Gamma(n+a)}{\Gamma(n+b)}\leq\Upsilon_{n}^{a,b}n^{a-b},$ (2.10)
where
$\Upsilon_{n}^{a,b}=\exp\Big{(}\frac{a-b}{2(n+b-1)}+\frac{1}{12(n+a-1)}+\frac{(a-b)^{2}}{n}\Big{)}.$
(2.11)
###### Proof.
Let $\theta_{1},\theta_{2}$ be two constants in $(0,1).$ We find from (2.9)
that
$\begin{split}\frac{\Gamma(n+a)}{\Gamma(n+b)}&=\frac{(n+a-1)^{n+a-1/2}}{(n+b-1)^{n+b-1/2}}\exp\Big{(}-a+b+\frac{\theta_{1}}{12(n+a-1)}-\frac{\theta_{2}}{12(n+b-1)}\Big{)}\\\
&\leq(n+a-1)^{a-b}\Big{(}1+\frac{a-b}{n+b-1}\Big{)}^{n+b-1/2}\exp\Big{(}-a+b+\frac{1}{12(n+a-1)}\Big{)}\\\
&\leq
n^{a-b}\Big{(}1+\frac{a-b}{n}\Big{)}^{a-b}\exp\Big{(}-a+b+\frac{(a-b)(n+b-1/2)}{n+b-1}+\frac{1}{12(n+a-1)}\Big{)}\\\
&\leq
n^{a-b}\exp\Big{(}\frac{a-b}{2(n+b-1)}+\frac{1}{12(n+a-1)}+\frac{(a-b)^{2}}{n}\Big{)}:=\Upsilon_{n}^{a,b}n^{a-b},\end{split}$
where we used the fact that $1+x\leq e^{x},$ for real $x.$ ∎
###### Remark 2.1.
Applying (2.11) to $\gamma_{n}^{\alpha,\beta}$ leads to that for
$\alpha,\beta>-1$, $n\geq 1$ and $n+\alpha+\beta>0$,
$\gamma_{n}^{\alpha,\beta}\leq\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\Upsilon_{n}^{\alpha+1,1}\Upsilon^{\beta+1,\alpha+\beta+1}_{n}.$
(2.12)
Note that for fixed $a$ and $b,$
$\Upsilon_{n}^{a,b}=1+O(n^{-1}),$ (2.13)
as it behaves like $e^{1/n}.$ ∎
### 2.2. Main tools
Our starting point is the following important representation.
###### Lemma 2.2.
Let $\\{\hat{u}_{n}^{\alpha,\beta}\\}$ be the Jacobi polynomial expansion
coefficients given by
$\hat{u}_{n}^{\alpha,\beta}=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}u(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx,\quad\alpha,\beta>-1,\;\;n\geq
0.$ (2.14)
If $u\in{\mathcal{A}}_{\rho}$ with $\rho>1,$ we have the representation:
$\hat{u}_{n}^{\alpha,\beta}=\frac{1}{\pi{\rm
i}}\sum_{j=0}^{\infty}\sigma_{n,j}^{\alpha,\beta}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{w^{n+j+1}}\,dz,\quad
n\geq 0,$ (2.15)
where $z=(w+w^{-1})/2$ with $w=\rho e^{{\rm i}\theta},\;\theta\in[0,2\pi],$
and
$\sigma_{n,j}^{\alpha,\beta}=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}U_{n+j}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx,\quad
n,j\geq 0.$ (2.16)
Here, $U_{n+j}(x)$ is the Chebyshev polynomial of the second kind of degree
$n+j$ (cf. (A.5)).
Actually, the formula (2.15)-(2.16) can be obtained by assembling several
formulas in Szegö [33], and then using the generating function of $U_{k}(x)$
(cf. [1]). For the readers’ reference, we sketch its derivation in Appendix B.
The establishment of sharp bounds heavily relies on estimating
$\sigma_{n,j}^{\alpha,\beta}.$ The following explicit formulas follow from
(2.16) and some properties of Jacobi polynomials listed in Appendix A. We
remark that the formula (2.19) can be found in various books e.g., [12, 29],
while the formula (2.20) is due to Heine (see [11]). We also highlight that
the formula (2.21) for the general Jacobi case seems new.
###### Corollary 2.1.
Let $n\geq 0.$
* (i)
For $\alpha=\beta>-1$ (ultraspherical/Gegenbauer polynomial )111In this paper,
we do not distinguish between ultraspherical and Gegenbauer polynomials.,
$\sigma_{n,j}^{\alpha,\alpha}=0,\quad\text{for odd}\;\;j.$ (2.17)
* (ii)
For $\alpha=\beta=1/2$ (Chebyshev polynomial of the second kind ),
$\sigma_{n,0}^{1/2,1/2}=\frac{\sqrt{\pi}}{2}\frac{(n+1)!}{\Gamma(n+3/2)};\quad\sigma_{n,j}^{1/2,1/2}=0,\quad\text{for}\;\;j\geq
1.$ (2.18)
* (iii)
For $\alpha=\beta=-1/2$ (Chebyshev polynomial ),
$\sigma_{n,j}^{-1/2,-1/2}=\begin{cases}\dfrac{2\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+1/2)},\quad&\text{for
even}\;\;j,\\\ 0,\quad&\text{for odd}\;\;j.\end{cases}$ (2.19)
* (iv)
For $\alpha=\beta=0$ (Legendre polynomial ),
$\sigma_{n,j}^{0,0}=\begin{cases}\dfrac{2n+1}{2}\dfrac{\Gamma(l+1/2)}{\Gamma(l+1)}\dfrac{\Gamma(n+l+1)}{\Gamma(n+l+3/2)},\quad&\text{for
even}\;\;j=2l,\\\ 0,\quad&\text{for odd}\;\;j.\end{cases}$ (2.20)
* (v)
For general $\alpha,\beta>-1$ (Jacobi polynomial ),
$\begin{split}\sigma_{n,j}^{\alpha,\beta}&=\frac{\sqrt{\pi}(2n+\alpha+\beta+1)\Gamma(n+\alpha+\beta+1)}{2\Gamma(n+\alpha+1)}\\\
&\quad\times\sum_{m=0}^{j}\frac{(-1)^{m}\Gamma(2n+j+m+2)\Gamma(n+m+\alpha+1)}{m!(j-m)!\Gamma(n+m+3/2)\Gamma(2n+m+\alpha+\beta+2)}.\end{split}$
(2.21)
###### Proof.
(i). The property (2.17) is a direct consequence of the parity of
ultraspherical polynomials.
(ii). For $\alpha=\beta=1/2,$ we find from (A.5) and the orthogonality
(2.7)-(2.8) that
$\begin{split}\sigma_{n,j}^{1/2,1/2}&=\sqrt{\frac{\pi}{2}}\frac{1}{\sqrt{\gamma_{n+j}^{1/2,1/2}}}\frac{1}{\gamma_{n}^{1/2,1/2}}\int_{-1}^{1}J_{n+j}^{1/2,1/2}(x)J_{n}^{1/2,1/2}(x)(1-x^{2})^{1/2}dx\\\
&=\sqrt{\frac{\pi}{2}}\frac{1}{\sqrt{\gamma_{n+j}^{1/2,1/2}}}\delta_{j,0},\end{split}$
where $\delta_{j,0}$ is the Kronecker delta. Working out the constant leads to
(2.18).
(iii) For $\alpha=\beta=-1/2$, if $j=2l,$ we have
$\begin{split}&\sigma_{n,2l}^{-1/2,-1/2}\overset{(\ref{chbytype2})}{=}\frac{1}{\gamma_{n}^{-1/2,-1/2}}\frac{1}{n+2l+1}\int_{-1}^{1}T_{n+2l+1}^{\prime}(x)J_{n}^{-1/2,-1/2}(x)(1-x^{2})^{-1/2}dx\\\
&\qquad\overset{(\ref{dtn1})}{=}\frac{2}{\gamma_{n}^{-1/2,-1/2}}\int_{-1}^{1}T_{n}(x)J_{n}^{-1/2,-1/2}(x)(1-x^{2})^{-1/2}dx\overset{(\ref{dtn0})}{=}\frac{2\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+1/2)},\end{split}$
which, together with (2.17), implies (2.19).
(iv) For $\alpha=\beta=0,$ we derive from [11, Eq. (14)] that
$\begin{split}\sigma_{n,2l}^{0,0}&=\frac{1}{\gamma_{n}^{0,0}}\int_{-1}^{1}J_{n}^{0,0}(x)U_{n+2l}(x)dx=\frac{2n+1}{2}\int_{0}^{\pi}J_{n}^{0,0}(\cos\theta)\sin\big{(}(n+2l+1)\theta\big{)}d\theta\\\
&=\frac{2n+1}{2}\frac{\Gamma(l+1/2)}{\Gamma(l+1)}\frac{\Gamma(n+l+1)}{\Gamma(n+l+3/2)},\quad
l\geq 0.\end{split}$
This yields (2.20).
(v) The formula (2.21) follows from a combination of (2.8), (A.4) and (A.5). ∎
With the aid of Lemma 2.2, we can derive the following estimate, from which
our sharp bounds are stemmed.
###### Lemma 2.3.
For any $u\in{\mathcal{A}}_{\rho}$ with $\rho>1,$ we have that for
$\alpha,\beta>-1$ and $n\geq 0,$
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}\leq\frac{M}{\rho^{n}}\Big{(}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{1}{\rho}\big{|}\sigma_{n,1}^{\alpha,\beta}\big{|}+\frac{1}{\rho^{2}}\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{|}\frac{1}{\rho^{j}}\Big{)},\end{split}$
(2.22)
where $M=\max_{z\in\mathcal{E}_{\rho}}|u(z)|$ and
$\\{\sigma_{n,j}^{\alpha,\beta}\\}$ are given by (2.16).
###### Proof.
Since $z=(w+w^{-1})/2\in{\mathcal{E}}_{\rho}$ with $w\in{\mathcal{C}}_{\rho}$
(cf. (2.1)-(2.2)), we can rewrite $\hat{u}_{n}^{\alpha,\beta}$ in (2.15) as
$\begin{split}\hat{u}_{n}^{\alpha,\beta}&=\frac{1}{2\pi{\rm
i}}\sum_{j=0}^{\infty}\sigma_{n,j}^{\alpha,\beta}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+j+1}}\Big{(}1-\frac{1}{w^{2}}\Big{)}\,dw\\\
&=\frac{1}{2\pi{\rm
i}}\sum_{j=0}^{\infty}\sigma_{n,j}^{\alpha,\beta}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+j+1}}\,dw-\frac{1}{2\pi{\rm
i}}\sum_{j=0}^{\infty}\sigma_{n,j}^{\alpha,\beta}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+j+3}}\,dw\\\
&=\frac{1}{2\pi{\rm
i}}\sigma_{n,0}^{\alpha,\beta}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+1}}\,dw+\frac{1}{2\pi{\rm
i}}\sigma_{n,1}^{\alpha,\beta}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+2}}\,dw\\\
&\quad+\frac{1}{2\pi{\rm
i}}\sum_{j=0}^{\infty}\big{(}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{)}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+j+3}}\,dw.\end{split}$
(2.23)
Hence, we arrive at
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}&\leq\frac{M}{2\pi}\frac{2\pi\rho}{\rho^{n+1}}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{M}{2\pi}\frac{2\pi\rho}{\rho^{n+2}}\big{|}\sigma_{n,1}^{\alpha,\beta}\big{|}+\frac{M}{2\pi}\frac{2\pi\rho}{\rho^{n+3}}\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{|}\frac{1}{\rho^{j}}\\\
&=\frac{M}{\rho^{n}}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{M}{\rho^{n+1}}\big{|}\sigma_{n,1}^{\alpha,\beta}\big{|}+\frac{M}{\rho^{n+2}}\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{|}\frac{1}{\rho^{j}}.\end{split}$
(2.24)
This ends the proof. ∎
Observe from the proof that we split the contour integral on
$\mathcal{E}_{\rho}$ into two parts on ${\mathcal{C}}_{\rho},$ which actually
allows us to take the advantage of cancelation of
$\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}.$ Indeed, the bound
(2.22) is tight, as we will see shortly that this argument can recover the
best estimate for the Chebyshev case (see [31, Theorem 3.8] and (1.1)), and
improve the bounds in [38] (see (1.2)).
### 2.3. Main results
For clarity of exposition, we first present the result on the general Jacobi
polynomial expansions, followed by the refined results on the Chebyshev-type
expansions ($\alpha=k-1/2,\beta=l-1/2$ with
$k,l\in{\mathbb{N}}:=\\{0,1,2,\cdots\\}$), and Legendre-type expansions
($\alpha=k,\beta=l$ with $k,l\in{\mathbb{N}}$).
#### 2.3.1. General Jacobi expansions ($\alpha,\beta>-1$)
###### Theorem 2.1.
For any $u\in{\mathcal{A}}_{\rho}$ (with $\rho>1{\rm)},\alpha,\beta>-1$ and
$n\geq 0,$ we have
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}&\leq\frac{M}{\rho^{n}}\bigg{[}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{|\sigma_{n,1}^{\alpha,\beta}|}{\rho}+\frac{2}{\rho(\rho-1)}\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}}\;\bigg{]},\end{split}$
(2.25)
where
$\sigma_{n,0}^{\alpha,\beta}=\frac{\sqrt{\pi}}{2}\frac{(2n+1)!\Gamma(n+\alpha+\beta+1)}{\Gamma(n+3/2)\Gamma(2n+\alpha+\beta+1)},\quad\sigma_{n,1}^{\alpha,\beta}=\frac{(\beta-\alpha)(2n+2)}{2n+\alpha+\beta+2}\sigma_{n,0}^{\alpha,\beta},$
(2.26)
and $\gamma_{n}^{\alpha,\beta}$ is defined in (2.8).
In particular, if $\alpha=\beta$, we have
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\alpha}\big{|}&\leq\frac{M}{\rho^{n}}\bigg{[}\big{|}\sigma_{n,0}^{\alpha,\alpha}\big{|}+\frac{2}{\rho^{2}-1}\sqrt{\frac{\gamma_{0}^{\alpha,\alpha}}{\gamma_{n}^{\alpha,\alpha}}}\;\bigg{]}.\end{split}$
(2.27)
###### Proof.
By (2.22),
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}&\leq\frac{M}{\rho^{n}}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{M}{\rho^{n+1}}\big{|}\sigma_{n,1}^{\alpha,\beta}\big{|}+\frac{M}{\rho^{n+2}}\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{|}\frac{1}{\rho^{j}}.\end{split}$
(2.28)
The factors $\sigma_{n,0}^{\alpha,\beta}$ and $\sigma_{n,1}^{\alpha,\beta}$ in
(2.26) are computed from (2.21) directly, so it suffices to estimate the
infinite sum in (2.28). Recall the identity (cf. [29]):
$U_{k}(x)-U_{k-2}(x)=2T_{k}(x),\quad k\geq 2.$ (2.29)
Then we infer from (2.16) that
$\begin{split}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}&=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}\big{(}U_{n+j+2}(x)-U_{n+j}(x)\big{)}J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx\\\
&=\frac{2}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}T_{n+j+2}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx,\quad
n,j\geq 0.\end{split}$ (2.30)
Thus, using the Cauchy-Schwartz inequality, the orthogonality (2.7), and the
fact $|T_{k}(x)|\leq 1,$ leads to
$\big{|}\sigma_{n,j+2}^{\alpha,\beta}-\sigma_{n,j}^{\alpha,\beta}\big{|}\leq\frac{2}{\sqrt{\gamma_{n}^{\alpha,\beta}}}\Big{(}\int_{-1}^{1}T_{n+j+2}^{2}(x)\omega^{\alpha,\beta}(x)\,dx\Big{)}^{1/2}\leq
2\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}}.$ (2.31)
Therefore, the bound (2.25) follows from
$\sum_{j=0}^{\infty}\rho^{-j}=1/(1-\rho^{-1}),$ as $\rho>1.$
For $\alpha=\beta,$ since $|\sigma_{n,2l+1}^{\alpha,\alpha}|=0,$ for all
$l\geq 0$ (cf. Corollary 2.1 (i)), we have
$\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{\alpha,\alpha}-\sigma_{n,j}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{j}}=\sum_{l=0}^{\infty}\big{|}\sigma_{n,2l+2}^{\alpha,\alpha}-\sigma_{n,2l}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{2l}}\leq
2\sqrt{\frac{\gamma_{0}^{\alpha,\alpha}}{\gamma_{n}^{\alpha,\alpha}}}\frac{1}{1-\rho^{-2}}.$
This yields the refined bound in (2.27). ∎
###### Remark 2.2.
Using Lemma 2.1, we can characterize the explicit dependence of the upper
bounds in (2.25) and (2.27) on $n,\alpha,\beta.$ Indeed, for
$\alpha,\beta>-1,n\geq 1$ and $n+\alpha+\beta>0,$
$\begin{split}\sigma_{n,0}^{\alpha,\beta}&\overset{(\ref{sigman0})}{=}\frac{\sqrt{\pi}}{2}\frac{\Gamma(n+\alpha+\beta+1)}{\Gamma(n+3/2)}\frac{(2n+1)!}{\Gamma(2n+\alpha+\beta+1)}\\\
&\overset{(\ref{Gammaratio})}{\leq}\frac{\sqrt{\pi}}{2}\big{(}\Upsilon_{n}^{\alpha+\beta+1,3/2}n^{\alpha+\beta+1-3/2}\big{)}\big{(}\Upsilon_{2n}^{2,\alpha+\beta+1}(2n)^{2-(\alpha+\beta+1)}\big{)}\\\
&=\frac{\sqrt{\pi
n}}{{2^{\alpha+\beta}}}\Upsilon_{n}^{\alpha+\beta+1,3/2}\Upsilon_{2n}^{2,\alpha+\beta+1}\overset{(\ref{consUps})}{=}\frac{\sqrt{\pi
n}}{{2^{\alpha+\beta}}}\big{(}1+O(n^{-1})\big{)},\end{split}$ (2.32)
which implies
$\begin{split}|\sigma_{n,1}^{\alpha,\beta}|&\overset{(\ref{sigman0})}{=}\frac{|\alpha-\beta|(2n+2)}{2n+\alpha+\beta+2}\sigma_{n,0}^{\alpha,\beta}\leq\frac{|\alpha-\beta|(2n+2)}{2n+\alpha+\beta+2}\frac{\sqrt{\pi
n}}{{2^{\alpha+\beta}}}\Upsilon_{n}^{\alpha+\beta+1,3/2}\Upsilon_{2n}^{2,\alpha+\beta+1}\\\
&=|\alpha-\beta|\frac{\sqrt{\pi
n}}{{2^{\alpha+\beta}}}\big{(}1+O(n^{-1})\big{)}.\end{split}$ (2.33)
Similarly, one verifies
$\begin{split}\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}&\overset{(\ref{gammafd})}{=}(2n+\alpha+\beta+1)\frac{\gamma_{0}^{\alpha,\beta}}{2^{\alpha+\beta+1}}\frac{n!\Gamma(n+\alpha+\beta+1)}{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}\\\
&\leq(2n+\alpha+\beta+1)\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}\Upsilon_{n}^{1,\alpha+1}\Upsilon_{n}^{\alpha+\beta+1,\beta+1}\\\
&\overset{(\ref{consUps})}{=}\frac{2\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}n\big{(}1+O(n^{-1})\big{)}.\end{split}$
(2.34)
Consequently, we infer from the estimate (2.25) that for fixed
$\alpha,\beta>-1$ and $n\gg 1,$
$|\hat{u}_{n}^{\alpha,\beta}|\leq
C_{n}M\bigg{(}\frac{\sqrt{\pi}}{2^{\alpha+\beta}}\Big{(}1+\frac{|\alpha-\beta|}{\rho}\Big{)}+\sqrt{\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)}}\frac{2\sqrt{2}}{\rho(\rho-1)}\bigg{)}\frac{\sqrt{n}}{\rho^{n}},$
(2.35)
and likewise, we find from (2.27) that
$|\hat{u}_{n}^{\alpha,\alpha}|\leq
C_{n}M\bigg{(}\frac{\sqrt{\pi}}{2^{2\alpha}}+\frac{\Gamma(\alpha+1)}{\sqrt{\Gamma(2\alpha+2)}}\frac{2\sqrt{2}}{\rho^{2}-1}\bigg{)}\frac{\sqrt{n}}{\rho^{n}},$
(2.36)
where $C_{n}=1+O(n^{-1}).$ ∎
###### Remark 2.3.
It is worthwhile to show that the bound obtained in this way is tighter than
(1.2) obtained in [38]. Indeed, it follows from (A.5), (A.6b) and (2.7) that
for $n\geq 1$ and $j=0,1,$
$\displaystyle\sigma_{n,j}^{\alpha,\beta}$
$\displaystyle=\frac{1}{\gamma_{n}^{\alpha,\beta}}\frac{1}{n+j+1}\int_{-1}^{1}T_{n+j+1}^{\prime}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)dx$
$\displaystyle=\frac{2}{\gamma_{n}^{\alpha,\beta}}\underset{k+n+j+1\;\text{odd}}{\underset{k=0}{\sum^{n+j}}}\frac{1}{c_{k}}\int_{-1}^{1}T_{k}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)dx$
$\displaystyle=\frac{2}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}T_{n+j}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)dx,$
where $c_{0}=2$ and $c_{k}=1$ for $k\geq 1.$ Following (2.30)-(2.31), we have
$|\sigma_{n,j}^{\alpha,\beta}|\leq
2\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}},\quad
n\geq 1,\;\;j=0,1.$
Finally, a straightforward calculation leads to
$\frac{M}{\rho^{n}}\bigg{[}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}+\frac{|\sigma_{n,1}^{\alpha,\beta}|}{\rho}+\frac{2}{\rho(\rho-1)}\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}}\;\bigg{]}\leq\frac{2M}{\rho^{n-1}(\rho-1)}\sqrt{\frac{\gamma_{0}^{\alpha,\beta}}{\gamma_{n}^{\alpha,\beta}}}.$
(2.37)
Moreover, we claim from (2.27) that the strict inequality holds, when
$\alpha=\beta>-1.$ One may refer to Section 4 for numerical evidences. ∎
#### 2.3.2. Chebyshev-type expansions ($\alpha=k-1/2,\beta=l-1/2$ with
$k,l\in{\mathbb{N}}$)
In view of (2.19), it follows from (2.23) that the Chebyshev coefficient takes
the simplest form:
$\hat{u}_{n}^{-1/2,-1/2}=\frac{\sigma_{n,0}^{-1/2,-1/2}}{2\pi{\rm
i}}\oint_{{\mathcal{C}}_{\rho}}\frac{u(z)}{w^{n+1}}\,dw.$ (2.38)
Thus, using (2.19) and (2.22) leads to
$|\hat{u}_{n}^{-1/2,-1/2}|\leq\dfrac{2\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+1/2)}\frac{M}{\rho^{n}}.$
(2.39)
This leads to the estimate for the expansion coefficients, denoted by
$\\{\hat{u}_{n}^{C}\\}$ as before, in terms of $\\{T_{n}(x)\\}:$
$\big{|}\hat{u}_{n}^{C}\big{|}\leq\frac{2M}{\rho^{n}},\quad n\geq 0,$ (2.40)
as documented in e.g., [31].
For the second-kind Chebyshev case, we find from (2.15) the closed-form
formula like (2.38):
$\displaystyle\hat{u}_{n}^{1/2,1/2}=\frac{\sigma_{n,0}^{1/2,1/2}}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{w^{n+1}}\,dz,$ (2.41)
but the contour integration is on ${\mathcal{E}}_{\rho}.$ It follows from
(2.18) and (2.23) that
$\displaystyle|\hat{u}_{n}^{1/2,1/2}|\leq\frac{1}{2\sqrt{\pi}}\frac{(n+1)!}{\Gamma(n+3/2)}\Big{|}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{w^{n+1}}\,dz\Big{|}\leq\frac{\sqrt{\pi}}{2}\frac{(n+1)!}{\Gamma(n+3/2)}\frac{M}{\rho^{n}}{\Big{(}1+\frac{1}{\rho^{2}}\Big{)}}.$
(2.42)
Like (2.40), if we re-scale the expansion in terms of $\\{U_{n}\\},$ i.e.,
$\hat{u}_{n}^{U}=\frac{2}{\pi}\int_{-1}^{1}u(x)U_{n}(x)\sqrt{1-x^{2}}\,dx,$
then we find from (A.5) and (2.42) that
$\displaystyle\big{|}\hat{u}_{n}^{U}\big{|}=\frac{2}{\sqrt{\pi}}\frac{\Gamma(n+3/2)}{\Gamma(n+2)}|\hat{u}_{n}^{1/2,1/2}|\leq\frac{M}{\rho^{n}}{\Big{(}1+\frac{1}{\rho^{2}}\Big{)}}.$
(2.43)
###### Remark 2.4.
It is seen from (2.41) that the second-kind Chebyshev coefficient takes the
simplest form on the contour $\mathcal{E}_{\rho}.$ This motivates us to
estimate the contour integral directly by
$\Big{|}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{w^{n+1}}\,dz\Big{|}\leq\frac{M}{\rho^{n+1}}\oint_{\mathcal{E}_{\rho}}|dz|=\frac{M}{\rho^{n+1}}L({\mathcal{E}}_{\rho}),$
which implies
$\displaystyle\big{|}\hat{u}_{n}^{U}\big{|}\leq\frac{M}{\rho^{n+1}}\frac{L({\mathcal{E}}_{\rho})}{\pi}.$
(2.44)
By (2.4),
$\frac{L({\mathcal{E}}_{\rho})}{\pi\rho}\leq\sqrt{1+\frac{1}{\rho^{4}}}<1+\frac{1}{\rho^{2}}.$
Therefore, the estimate (2.44) is slightly sharper than (2.43). ∎
Some refined results can also be derived for $\alpha=k+1/2,\beta=l+1/2$ with
$k,l\in\mathbb{N}.$ Indeed, we find that
$\big{\\{}\sigma_{n,j}^{k+1/2,l+1/2}\big{\\}}$ can be computed explicitly by
the following formula.
###### Proposition 2.1.
For any $k,l,n,j\in\mathbb{N},$
$\begin{split}\sigma_{n,j}^{k+1/2,l+1/2}=\sqrt{\frac{\pi}{2}}\frac{1}{\gamma_{n}^{k+1/2,l+1/2}}\sum_{m=n}^{n+k+l}d_{m}^{k+1/2,l+1/2}\sqrt{\gamma_{m}^{1/2,1/2}}\delta_{m,n+j},\end{split}$
(2.45)
where $\big{\\{}d_{m}^{k+1/2,l+1/2}\big{\\}}_{m=n}^{n+k+l}$ are given in
(A.3), and $\delta_{m,n+j}$ is the Kronecker delta.
###### Proof.
Using (A.3) (with $\alpha=\beta=1/2$), (2.16) and the properties of Jacobi
polynomials (cf. (2.7) and (A.5)), leads to
$\begin{split}\sigma_{n,j}^{k+1/2,l+1/2}&=\frac{1}{\gamma_{n}^{k+1/2,l+1/2}}\sum_{m=n}^{n+k+l}d_{m}^{k+1/2,l+1/2}\int_{-1}^{1}U_{n+j}(x)J_{m}^{1/2,1/2}(x)(1-x^{2})^{1/2}dx\\\
&=\frac{1}{\gamma_{n}^{k+1/2,l+1/2}}\sqrt{\frac{\pi}{2}}\sum_{m=n}^{n+k+l}d_{m}^{k+1/2,l+1/2}\sqrt{\gamma_{m}^{1/2,1/2}}\delta_{m,n+j},\end{split}$
(2.46)
This completes the proof. ∎
Equipped with (2.45), we can obtain the bound for Chebyshev-type expansion
coefficients by computing $\\{d_{m}^{k+1/2,l+1/2}\\}$ explicitly. To fix the
idea, we just consider the case: $k=1$ and $l=0.$ One finds
$d_{n}^{3/2,1/2}=1,\quad d_{n+1}^{3/2,1/2}=-\frac{2n+2}{2n+3},$
and
$\sigma_{n,0}^{3/2,1/2}=\frac{\sqrt{\pi}}{4}\frac{n!(n+2)}{\Gamma(n+3/2)},\quad\sigma_{n,1}^{3/2,1/2}=-\frac{\sqrt{\pi}}{4}\frac{(n+1)!}{\Gamma(n+3/2)},\quad\sigma_{n,j}^{3/2,1/2}=0,\;\;j\geq
2.$
The estimate (2.14) reduces to
$\begin{split}\hat{u}_{n}^{3/2,1/2}&\leq\frac{M}{\rho^{n}}\bigg{[}\sigma_{n,0}^{3/2,1/2}+\frac{\sigma_{n,1}^{3/2,1/2}}{\rho}+\frac{\sigma_{n,0}^{3/2,1/2}}{\rho^{2}}+\frac{\sigma_{n,1}^{3/2,1/2}}{\rho^{3}}\bigg{]}\\\
&=\frac{M}{\rho^{n}}\Big{(}1+\frac{1}{\rho^{2}}\Big{)}\bigg{[}\sigma_{n,0}^{3/2,1/2}+\frac{\sigma_{n,1}^{3/2,1/2}}{\rho}\bigg{]}.\end{split}$
Thus, we have
$|\hat{u}_{n}^{3/2,1/2}|\leq\frac{\sqrt{\pi}}{4}\frac{(n+1)!}{\Gamma(n+3/2)}\frac{M}{\rho^{n}}\Big{(}1+\frac{1}{\rho^{2}}\Big{)}\Big{(}\frac{n+2}{n+1}+\frac{1}{\rho}\Big{)},$
(2.47)
and by (2.10), we have for $n\geq 0,$
$\frac{(n+1)!}{\Gamma(n+3/2)}\leq\sqrt{n}\exp\Big{(}\frac{8n+7}{12(2n+1)(n+1)}+\frac{1}{4n}\Big{)}.$
(2.48)
Actually, the infinite sum in (2.22) does not appear for the Chebyshev-type
expansions, which allows us to derive very tight bounds. However, for the
Legendre-type expansions, some care has to be taken to handle this sum.
#### 2.3.3. Legendre-type expansions ($\alpha=k,\beta=l$ with
$k,l\in{\mathbb{N}}$)
We first consider the Legendre case. By (2.8) and (2.26),
$\gamma_{n}^{0,0}=\frac{2}{2n+1},\quad\frac{\gamma_{0}^{0,0}}{\gamma_{n}^{0,0}}=2n+1,\quad\sigma_{n,0}^{0,0}=\frac{\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+1/2)},$
so the estimate (2.27) reduces to
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{M}{\rho^{n}}\Big{[}\frac{\sqrt{\pi}\Gamma(n+1)}{\Gamma(n+1/2)}+\frac{2\sqrt{2n+1}}{\rho^{2}-1}\Big{]}.$
(2.49)
In fact, we can improve this estimate, as highlighted in the following
theorem, by using the explicit information of $\sigma_{n,2l}^{0,0}.$
###### Theorem 2.2.
Let $\\{\hat{u}_{n}^{0,0}\\}$ be the Legendre expansion coefficients of any
$u\in{\mathcal{A}}_{\rho}$ with $\rho>1.$ Then for any $n\geq 1,$
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{M\sqrt{\pi
n}}{\rho^{n}}\Big{(}1+\frac{n+2}{2n+3}\frac{1}{\rho^{2}-1}\Big{)}\exp\Big{(}\frac{8n-1}{12n(2n-1)}\Big{)}.$
(2.50)
###### Proof.
A straightforward calculation from (2.20) yields
$\sigma_{n,2l+2}^{0,0}-\sigma_{n,2l}^{0,0}=-\frac{n+2l+2}{2(l+1)(n+l+3/2)}\sigma_{n,2l}^{0,0},\quad
l\geq 0,$ (2.51)
which implies $\\{\sigma_{n,2l}^{0,0}\\}$ is strictly descending with respect
to $l.$ Hence, we have
$\big{|}\sigma_{n,2l+2}^{0,0}-\sigma_{n,2l}^{0,0}\big{|}=\frac{n+2l+2}{2(l+1)(n+l+3/2)}\sigma_{n,2l}^{0,0}\leq\frac{n+2}{2n+3}\sigma_{n,0}^{0,0},$
(2.52)
where we used the fact that ${n+2l+2}/{((l+1)(n+l+3/2))}$ is strictly
descending with respect to $l.$ Then, we obtain the improved bound from
(2.22):
$\begin{split}\big{|}\hat{u}_{n}^{0,0}\big{|}&\leq\frac{M}{\rho^{n}}\sigma_{n,0}^{0,0}\Big{(}1+\frac{n+2}{2n+3}\sum_{l=0}^{\infty}\frac{1}{\rho^{2l+2}}\Big{)}=\frac{\sqrt{\pi}M}{\rho^{n}}\frac{\Gamma(n+1)}{\Gamma(n+1/2)}\Big{(}1+\frac{n+2}{2n+3}\frac{1}{\rho^{2}-1}\Big{)},\end{split}$
(2.53)
and by (2.10),
$\frac{\Gamma(n+1)}{\Gamma(n+1/2)}\leq\sqrt{n}\exp\Big{(}\frac{8n-1}{12n(2n-1)}\Big{)},\quad
n\geq 1.$ (2.54)
This completes the proof. ∎
###### Remark 2.5.
We compare the bound in (2.50) with the existing ones. Davis [12, Page 313]
stated the bound
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{2n+1}{2}\frac{ML({\mathcal{E}}_{\rho})}{\rho^{n}(\rho-1)}\overset{(\ref{Lehp})}{\leq}\frac{2n+1}{2}\frac{\pi\sqrt{\rho^{2}+\rho^{-2}}M}{\rho^{n}(\rho-1)},$
where clearly the algebraic order of $n$ in the numerator is not optimal. The
following asymptotic bound can be obtained from [27, Eq. (32) and Eq. (38)]
and [12, Eq. (12.4.25)]:
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{M\sqrt{\pi
n}}{\rho^{n}}\frac{\sqrt{\rho^{4}+1}}{\rho^{2}-1},\quad n\gg 1,$
while the asymptotic estimate derived from (2.50) is
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{M\sqrt{\pi
n}}{\rho^{n}}\frac{{\rho^{2}-1/2}}{\rho^{2}-1},\quad n\gg 1,$ (2.55)
which is sharper. Another bound for comparison is obtained in the recent paper
[38]:
$\big{|}\hat{u}_{n}^{0,0}\big{|}\leq\frac{2\sqrt{n}M}{\rho^{n}}\Big{(}1+\frac{1}{\rho^{2}-1}\Big{)},\quad
n\geq 1,$ (2.56)
which is also inferior to our estimate (2.50). Some comparisons in numerical
perspective are given in Section 4. ∎
Like the Chebysheve case, we can derive similar refined estimates for
Legendre-type expansions with $\alpha=k,\beta=l$ and $k,l\in{\mathbb{N}}.$ The
counterpart of Proposition 2.1 is stated as follows, which can be obtained by
using (A.3) (with $\alpha=\beta=0$), (2.16) and the properties of Jacobi
polynomials (e.g., (2.7)) as before.
###### Proposition 2.2.
For any $k,l,n,j\in\mathbb{N},$
$\sigma_{n,j}^{k,l}=\frac{1}{\gamma_{n}^{k,l}}\sum\limits_{m=n}^{n+k+l}d_{m}^{k,l}\gamma_{m}^{0,0}\sigma_{m,n+j-m}^{0,0},$
(2.57)
where $\big{\\{}d_{m}^{k,l}\big{\\}}_{m=n}^{n+k+l}$ are the same as in (A.3),
and $\big{\\{}\sigma_{m,n+j-m}^{0,0}\big{\\}}$ are computed by (2.20).
Once again, to fix the idea, we just consider the case: $k=1$ and $l=0.$ One
finds $d_{n}^{1,0}=1,d_{n+1}^{1,0}=-1,$ and
$\begin{split}\sigma_{n,j}^{1,0}&=\frac{1}{\gamma_{n}^{1,0}}\big{(}\gamma_{n}^{0,0}\sigma_{n,j}^{0,0}-\gamma_{n+1}^{0,0}\sigma_{n+1,j-1}^{0,0}\big{)}=\frac{n+1}{2n+1}\sigma_{n,j}^{0,0}-\frac{n+1}{2n+3}\sigma_{n+1,j-1}^{0,0}.\end{split}$
By (2.20),
$\sigma_{n,2l}^{1,0}=\frac{n+1}{2n+1}\sigma_{n,2l}^{0,0},\quad\sigma_{n,2l+1}^{1,0}=-\frac{n+1}{2n+3}\sigma_{n+1,2l}^{0,0},\quad
l\geq 0.$
Therefore, with (2.51) and (2.52), the estimate (2.22) reduces to
$\begin{split}\big{|}\hat{u}_{n}^{1,0}\big{|}&\leq\frac{M}{\rho^{n}}\big{|}\sigma_{n,0}^{1,0}\big{|}+\frac{M}{\rho^{n+1}}\big{|}\sigma_{n,1}^{1,0}\big{|}+\frac{M}{\rho^{n+2}}\sum_{j=0}^{\infty}\big{|}\sigma_{n,j+2}^{1,0}-\sigma_{n,j}^{1,0}\big{|}\frac{1}{\rho^{j}}\\\
&=\frac{M}{\rho^{n}}\frac{n+1}{2n+1}\Big{(}\sigma_{n,0}^{0,0}+\frac{1}{\rho^{2}}\sum_{l=0}^{\infty}\big{|}\sigma_{n,2l+2}^{0,0}-\sigma_{n,2l}^{0,0}\big{|}\frac{1}{\rho^{2l}}\Big{)}\\\
&\quad+\frac{M}{\rho^{n+1}}\frac{n+1}{2n+3}\Big{(}\sigma_{n+1,0}^{0,0}+\frac{1}{\rho^{2}}\sum_{l=0}^{\infty}\big{|}\sigma_{n+1,2l+2}^{0,0}-\sigma_{n+1,2l}^{0,0}\big{|}\frac{1}{\rho^{2l}}\Big{)}\\\
&\leq\sigma_{n,0}^{0,0}\frac{n+1}{2n+1}\frac{M}{\rho^{n}}\Big{(}1+\frac{n+2}{2n+3}\frac{1}{\rho^{2}-1}\Big{)}+\sigma_{n+1,0}^{0,0}\frac{n+1}{2n+3}\frac{M}{\rho^{n+1}}\Big{(}1+\frac{n+3}{2n+5}\frac{1}{\rho^{2}-1}\Big{)}.\end{split}$
Working out the expressions of $\sigma_{n,0}^{0,0}$ and $\sigma_{n+1,0}^{0,0}$
by (2.26), we have
$\begin{split}\big{|}\hat{u}_{n}^{1,0}\big{|}&\leq\frac{M}{\rho^{n}}\frac{\sqrt{\pi}\Gamma(n+2)}{\Gamma(n+3/2)}\Big{\\{}\frac{1}{2}+\frac{n+2}{2(2n+3)}\frac{1}{\rho^{2}-1}+\frac{1}{\rho}\frac{n+1}{2n+3}\Big{(}1+\frac{n+3}{2n+5}\frac{1}{\rho^{2}-1}\Big{)}\Big{\\}}.\end{split}$
(2.58)
Note that the ratio of the Gamma functions can be bounded as in (2.48).
The same process applies to other $k,l\in{\mathbb{N}},$ but the derivation
seems tedious.
### 2.4. Estimates for truncated Jacobi expansions
Given a cut-off number $N\geq 1$ and $N\in{\mathbb{N}},$ we define the partial
sum
$\big{(}\pi_{N}^{\alpha,\beta}u\big{)}(x)=\sum_{n=0}^{N-1}\hat{u}_{n}^{\alpha,\beta}J_{n}^{\alpha,\beta}(x),$
(2.59)
where $\\{\hat{u}_{n}^{\alpha,\beta}\\}$ are the Jacobi expansion coefficients
defined in (2.14). To this end, let $L^{2}_{\omega^{\alpha,\beta}}(I)$ be the
weighted $L^{2}$-space on $I=(-1,1),$ and its norm is denoted by
$\|\cdot\|_{\omega^{\alpha,\beta}},$ where we drop the weight function, if
$\alpha=\beta=0.$
Notice that $\pi_{N}^{\alpha,\beta}u$ is the
$L^{2}_{\omega^{\alpha,\beta}}$-projection of $u$ upon $P_{N-1}$ (denoting the
set of all algebraic polynomials of degree at most $N-1$), that is,
$\pi_{N}^{\alpha,\beta}u$ is the best approximation to $u$ in the norm
$\|\cdot\|_{\omega^{\alpha,\beta}}.$ With the previous bounds for the
expansion coefficients, we can estimate the truncation error
straightforwardly.
###### Theorem 2.3.
For any $u\in{\mathcal{A}}_{\rho}$ with $\rho>1,$ and $\alpha,\beta>-1,$ we
have
$\begin{split}\big{\|}\pi_{N}^{\alpha,\beta}u-u\big{\|}_{\omega^{\alpha,\beta}}&\leq\bigg{[}\sqrt{\frac{\pi}{2^{\alpha+\beta}}}\Big{(}1+\frac{|\alpha-\beta|}{\rho}\Big{)}+\frac{2\sqrt{\gamma_{0}^{\alpha,\beta}}}{\rho(\rho-1)}\;\bigg{]}\frac{C_{N}M}{\rho^{N-1}\sqrt{\rho^{2}-1}},\end{split}$
(2.60)
where $\gamma_{0}^{\alpha,\beta}$ is given in (2.8) and $C_{N}\approx 1.$
###### Proof.
By the orthogonality (cf (2.7)-(2.8)) of Jacobi polynomials, we have
$\begin{split}\big{\|}\pi_{N}^{\alpha,\beta}u-u\big{\|}_{\omega^{\alpha,\beta}}^{2}&=\sum_{n=N}^{\infty}|\hat{u}_{n}^{\alpha,\beta}|^{2}\gamma_{n}^{\alpha,\beta}.\end{split}$
It follows from the estimate of $\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}$ in
Theorem 2.1, and a combination of (2.12)-(2.13) and (2.32)-(2.33) that for
$n\geq N\gg 1$,
$\begin{split}\big{|}\hat{u}_{n}^{\alpha,\beta}\big{|}\sqrt{\gamma_{n}^{\alpha,\beta}}&\leq\frac{M}{\rho^{n}}\bigg{[}\big{|}\sigma_{n,0}^{\alpha,\beta}\big{|}\sqrt{\gamma_{n}^{\alpha,\beta}}+\frac{1}{\rho}\big{|}\sigma_{n,1}^{\alpha,\beta}\big{|}\sqrt{\gamma_{n}^{\alpha,\beta}}+\frac{2}{\rho(\rho-1)}\sqrt{\gamma_{0}^{\alpha,\beta}}\;\bigg{]}\\\
&\leq\frac{C_{n}M}{\rho^{n}}\bigg{[}\sqrt{\frac{\pi}{2^{\alpha+\beta}}}\Big{(}1+\frac{|\alpha-\beta|}{\rho}\Big{)}+\frac{2}{\rho(\rho-1)}\sqrt{\gamma_{0}^{\alpha,\beta}}\;\bigg{]},\end{split}$
where $C_{n}=1+O(n^{-1}).$ Therefore, we have
$\begin{split}\big{\|}\pi_{N}^{\alpha,\beta}u-u\big{\|}_{\omega^{\alpha,\beta}}&\leq
C_{N}M\bigg{[}\sqrt{\frac{\pi}{2^{\alpha+\beta}}}\Big{(}1+\frac{|\alpha-\beta|}{\rho}\Big{)}+\frac{2\sqrt{\gamma_{0}^{\alpha,\beta}}}{\rho(\rho-1)}\;\bigg{]}\Big{(}\sum_{n=N}^{\infty}\frac{1}{\rho^{2n}}\Big{)}^{1/2}\\\
&\leq\bigg{[}\sqrt{\frac{\pi}{2^{\alpha+\beta}}}\Big{(}1+\frac{|\alpha-\beta|}{\rho}\Big{)}+\frac{2\sqrt{\gamma_{0}^{\alpha,\beta}}}{\rho(\rho-1)}\;\bigg{]}\frac{C_{N}M}{\rho^{N-1}\sqrt{\rho^{2}-1}}.\end{split}$
This ends the proof. ∎
###### Remark 2.6.
Note that $\\{\frac{d^{l}}{dx^{l}}J_{n}^{\alpha,\beta}\\}_{n\geq l}$ are
mutually orthogonal with respect to $\omega^{\alpha+l,\beta+l},$ so we can
estimate
$\big{\|}(\pi_{N}^{\alpha,\beta}u-u)^{(l)}\big{\|}_{\omega^{\alpha+l,\beta+l}}$
in a similar fashion. ∎
###### Remark 2.7.
Some refined estimates can be obtained from the refined bounds for special
cases, e.g., $\alpha=\beta$ or $\alpha=\beta=0,-1/2.$ Here, we just state the
result for the Legendre case:
$\begin{split}\big{\|}\pi_{N}^{0,0}u-u\big{\|}\leq\Big{(}1+\frac{1}{2(\rho^{2}-1)}\Big{)}\frac{C_{N}\sqrt{\pi}M}{\rho^{N-1}\sqrt{\rho^{2}-1}},\end{split}$
(2.61)
where $C_{N}\approx 1$ as before. It follows from Theorem 2.2 and the above
process. Note that Xiang [38] derived the following estimate for the Legendre
expansion:
$\begin{split}\big{\|}\pi_{N}^{0,0}u-u\big{\|}\leq\frac{2\sqrt{2}M}{\rho^{N-2}(\rho-1)^{2}}.\end{split}$
(2.62)
The estimate (2.61) seems tighter than this one. ∎
## 3\. Error estimates for Gegenbauer-Gauss quadrature
### 3.1. Preliminaries
The Gegenbauer-Gauss quadrature remainder (1.3)-(1.4) with the nodes being
zeros of the Gegenbauer polynomial $J_{n}^{\alpha,\alpha}(x)$, takes the form
$E_{n}^{GG}[u]=\frac{\gamma_{n}^{\alpha,\alpha}}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{Q_{n}^{\alpha,\alpha}(z)}{J_{n}^{\alpha,\alpha}(z)}u(z)\,dz,\quad\forall\,u\in{\mathcal{A}}_{\rho},$
(3.1)
where $Q_{n}^{\alpha,\alpha}(z)$ is defined as in (B.2), namely,
$Q_{n}^{\alpha,\alpha}(z)=\frac{1}{2\gamma_{n}^{\alpha,\alpha}}\int_{-1}^{1}\frac{J_{n}^{\alpha,\alpha}(x)\omega^{\alpha,\alpha}(x)}{z-x}\,dx\overset{(\ref{Jcoe3a0s})}{=}\sum_{j=0}^{\infty}\frac{\sigma_{n,j}^{\alpha,\alpha}}{w^{n+j+1}}\overset{(\ref{sigmanjodd})}{=}\sum_{l=0}^{\infty}\frac{\sigma_{n,2l}^{\alpha,\alpha}}{w^{n+2l+1}}.$
(3.2)
As already mentioned, the analysis of quadrature errors (even for the
Chebyshev case) has attracted much attention (see e.g., [11, 10, 3, 17, 13,
16, 25, 26]). Just to mention that Chawla and Jain [11, Theorem 5] obtained
the estimate:
$\big{|}E_{n}^{CG}[u]\big{|}\leq\frac{2\pi
M}{\rho^{2n}-1},\quad\forall\,u\in{\mathcal{A}}_{\rho},\;\;\forall\,n\geq 1,$
(3.3)
Hunter [25] derived the general bound
$\big{|}E_{n}^{GG}[u]\big{|}\leq\frac{4\int_{-1}^{1}(1-x^{2})^{\alpha}dx}{\rho^{2n-2}(\rho^{2}-1)},\quad
n\geq 1,$ (3.4)
and some refined results for $\alpha=\pm 1/2$ and $\beta=\pm 1/2$ by expanding
$Q_{n}^{\alpha,\alpha}/J_{n}^{\alpha,\alpha}$ into the Laurent series of $w$
in the disk enclosed by ${\mathcal{C}}_{\rho},$ and manipulating the series.
It is worthwhile to note that Gautschi and Varga [17] estimated the Jacobi-
Gauss quadrature (with $J_{n}^{\alpha,\beta}$ and $Q_{n}^{\alpha,\beta}$ in
place of $J_{n}^{\alpha,\alpha}$ and $Q_{n}^{\alpha,\alpha}$ in (3.1),
respectively) by
$\big{|}E_{n}^{JG}[u]\big{|}\leq{\pi^{-1}\gamma_{n}^{\alpha,\beta}}ML(\mathcal{E}_{\rho})\max_{z\in{\mathcal{E}}_{\rho}}\big{|}Q_{n}^{\alpha,\beta}(z)/J_{n}^{\alpha,\beta}(z)\big{|},$
(3.5)
and attempted to find the exact maximum value on the Bernstein ellipse, which
was feasible for $\alpha=\pm 1/2$ and $\beta=\pm 1/2$ again. Some conjectures
and empirical results were explored in [17] for the general Jacobi case.
Using the explicit expression of Legendre polynomials on the Bernstein ellipse
(see e.g., [12, Lemma 12.4.1]), Kambo [27] obtained the bound for the
Legendre-Gauss quadrature:
$\big{|}E_{n}^{LG}[u]\big{|}\leq{\pi^{-1}\gamma_{n}^{0,0}}ML(\mathcal{E}_{\rho})\frac{\max_{z\in{\mathcal{E}_{\rho}}}|Q_{n}^{0,0}(z)|}{\min_{z\in{\mathcal{E}_{\rho}}}|J_{n}^{0,0}(z)|}\leq\frac{d_{n}M}{\rho^{2n}}\frac{\rho^{2}+1}{\rho^{2}-2},\quad\rho>\sqrt{2},$
(3.6)
where $0<d_{n}\leq\pi.$ While this bound is only valid for $\rho>\sqrt{2},$ it
holds for all $n,$ when compared with the asymptotic estimate (with $n\gg 1$)
for the Legendre-Gauss quadrature in [9].
In what follows, we aim to extend our analysis to estimate $E_{n}^{GG}[u]$ in
(3.1). The essential tools include the explicit formula for the Gegenbauer
polynomial $J_{n}^{\alpha,\alpha}(z)$ on ${\mathcal{E}}_{\rho}$ derived in our
recent paper [39], and the previous argument for estimating
$Q_{n}^{\alpha,\alpha}(z).$ Let us recall the important formula stated in [39,
Lemma 3.1].
###### Lemma 3.1.
Let $z=\frac{1}{2}(w+w^{-1}).$ Then we have
$J_{n}^{\alpha,\alpha}(z)=A_{n}^{\alpha}\sum_{k=0}^{n}g_{k}^{\alpha}g_{n-k}^{\alpha}w^{n-2k},\quad
n\geq 0,\;\;\alpha>-1,\;\;\alpha\not=-1/2,$ (3.7)
where
$g_{0}^{\alpha}=1,\;\;g_{k}^{\alpha}=\frac{\Gamma(k+\alpha+1/2)}{k!\Gamma(\alpha+1/2)},\;\;1\leq
k\leq
n,\;\;\text{and}\;\;A_{n}^{\alpha}=\frac{\Gamma(2\alpha+1)\Gamma(n+\alpha+1)}{\Gamma(\alpha+1)\Gamma(n+2\alpha+1)}.$
(3.8)
###### Remark 3.1.
This formula excludes the Chebyshev case. For $\alpha=-1/2,$ we define
$g_{0}^{-1/2}=g_{n}^{-1/2}=1,\;\;g_{k}^{-1/2}=0,\;\;1\leq k\leq
n-1,\;\;\text{and}\;\;A_{n}^{-1/2}=\frac{\Gamma(n+1/2)}{2\sqrt{\pi}n!},$ (3.9)
since (see e.g., [12])
$T_{n}(z)=\frac{1}{2}(w^{n}+w^{-n})=\frac{1}{2A_{n}^{-1/2}}J^{-1/2,-1/2}_{n}(z).$
(3.10)
Hence, we understand that (3.7) holds for $\alpha=-1/2$ with the constants
given by (3.9). ∎
### 3.2. Main results
We adopt two approaches to estimate the quadrature remainder. The first one is
to expand $Q_{n}^{\alpha,\alpha}/J_{n}^{\alpha,\alpha}$ in Laurent series of
$w\in{\mathcal{C}}_{\rho},$ and then we use an argument as for Theorem 2.3 to
obtain the tight error bound. However, this situation is reminiscent to that
in Gautschi and Varga [17], that is, computable bounds can be derived for
general $\alpha.$ We highlight that the computational part (see (3.11)) is
independent of $\rho$ and $u.$
The second approach is based on an important relation between the quadrature
remainder and Gegenbauer expansion coefficient (see (3.22)).
The main estimate resulted from the first approach is stated as follows.
###### Theorem 3.1.
For any $u\in{\mathcal{A}}_{\rho}$ with $\rho>1,$ we have that for $\alpha>-1$
and $n\geq 1,$
$\begin{split}\big{|}E_{n}^{GG}[u]\big{|}&\leq\gamma_{n}^{\alpha,\alpha}\Big{[}\big{|}\mu_{n,0}^{\alpha,\alpha}\big{|}+\max_{l\geq
0}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{2}-1}\Big{]}\frac{M}{\rho^{2n}},\end{split}$
(3.11)
where $\\{\mu_{n,2l}^{\alpha,\alpha}\\}_{l\geq 0}$ are computed by the
recursive formula:
$\mu_{n,2l}^{\alpha,\alpha}=\frac{1}{g_{n}^{\alpha}}\Big{(}\frac{\sigma_{n,2l}^{\alpha,\alpha}}{A_{n}^{\alpha}}-\sum_{k=1}^{\min\\{n,l\\}}g_{k}^{\alpha}g_{n-k}^{\alpha}\mu_{n,2l-2k}^{\alpha,\alpha}\Big{)},\;\;l\geq
1,\quad\mu_{n,0}^{\alpha,\alpha}=\frac{\sigma_{n,0}^{\alpha,\alpha}}{A_{n}^{\alpha}g_{n}^{\alpha}}.$
(3.12)
###### Proof.
A straightforward calculation from (3.2) (note:
$\sigma_{n,2l+1}^{\alpha,\alpha}=0$ for all $l\geq 0$) and (3.7) leads to
$\displaystyle\frac{Q_{n}^{\alpha,\alpha}(z)}{J_{n}^{\alpha,\alpha}(z)}=\sum_{l=0}^{\infty}\frac{\mu_{n,2l}^{\alpha,\alpha}}{w^{2n+2l+1}}\;\;\text{with}\;\;\sigma_{n,2l}^{\alpha,\alpha}=A_{n}^{\alpha}\sum_{k=0}^{\min\\{n,l\\}}g_{k}^{\alpha}g_{n-k}^{\alpha}\mu_{n,2l-2k}^{\alpha,\alpha},$
(3.13)
so solving out $\mu_{n,2l}^{\alpha,\alpha}$ yields (3.12).
Next, following the same lines as the derivation of (2.23), we infer from
(3.1) and (3.13) that
$\begin{split}\big{|}E_{n}^{GG}[u]\big{|}&\leq\gamma_{n}^{\alpha,\alpha}\frac{M}{2\pi}\Big{|}\sum_{l=0}^{\infty}\mu_{n,2l}^{\alpha,\alpha}\oint_{{\mathcal{C}}_{\rho}}\frac{1}{w^{2n+2l+1}}\Big{(}1-\frac{1}{w^{2}}\Big{)}\,dw\Big{|}\\\
&\leq\gamma_{n}^{\alpha,\alpha}\frac{M}{2\pi}\Big{[}\frac{2\pi\rho}{\rho^{2n+1}}\big{|}\mu_{n,0}^{\alpha,\alpha}\big{|}+\frac{2\pi\rho}{\rho^{2n+3}}\sum_{l=0}^{\infty}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{2l}}\Big{]}\\\
&=\gamma_{n}^{\alpha,\alpha}\frac{M}{\rho^{2n}}\Big{[}\big{|}\mu_{n,0}^{\alpha,\alpha}\big{|}+\frac{1}{\rho^{2}}\sum_{l=0}^{\infty}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{2l}}\Big{]}\\\
&\leq\gamma_{n}^{\alpha,\alpha}\frac{M}{\rho^{2n}}\Big{[}\big{|}\mu_{n,0}^{\alpha,\alpha}\big{|}+\max_{l\geq
0}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|}\sum_{l=0}^{\infty}\frac{1}{\rho^{2l+2}}\Big{]}\\\
&=\gamma_{n}^{\alpha,\alpha}\frac{M}{\rho^{2n}}\Big{[}\big{|}\mu_{n,0}^{\alpha,\alpha}\big{|}+\max_{l\geq
0}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|}\frac{1}{\rho^{2}-1}\Big{]}.\end{split}$
(3.14)
This completes the proof. ∎
###### Remark 3.2.
We find from (3.12) that for $\alpha=-1/2,$
$\mu_{n,0}^{-1/2,-1/2}=\frac{2\pi}{\gamma_{n}^{-1/2,-1/2}},\quad\big{|}\mu_{n,2l+2}^{-1/2,-1/2}-\mu_{n,2l}^{-1/2,-1/2}\big{|}=\frac{2\pi\delta_{\kappa,0}}{\gamma_{n}^{-1/2,-1/2}},\;\;\kappa:=\text{mod}(l+1,n),$
where $\delta_{\kappa,0}$ is the Kronecker delta. Hence, it follows from
(3.14) that
$\big{|}E_{n}^{CG}[u]\big{|}\leq\frac{2\pi
M}{\rho^{2n}}\Big{[}1+\frac{1}{\rho^{2}}\sum_{j=1}^{\infty}\frac{1}{\rho^{2(jn-1)}}\Big{]}=\frac{2\pi
M}{\rho^{2n}-1},\quad n\geq 1,$ (3.15)
which is the same as (3.3) derived in [11] .∎
###### Remark 3.3.
We find from (3.12) that for $\alpha=1/2,$
$\mu_{n,2l}^{1/2,1/2}=\begin{cases}(-1)^{\kappa}\dfrac{\pi}{2}\dfrac{1}{\gamma_{n}^{1/2,1/2}},&\;\;\text{if}\;\;\kappa:=\text{mod}(l,n+1)=0,1,\\\
0,&\;\;\text{otherwise},\end{cases}$ (3.16)
which implies
$\begin{split}&\sum_{l=0}^{\infty}\big{|}\mu_{n,2l+2}^{1/2,1/2}-\mu_{n,2l}^{1/2,1/2}\big{|}\frac{1}{\rho^{2l}}=\big{|}\mu_{n,2}^{1/2,1/2}-\mu_{n,0}^{1/2,1/2}\big{|}+\sum_{j=1}^{\infty}\big{|}\mu_{n,2j(n+1)}^{1/2,1/2}\big{|}\frac{1}{\rho^{2j(n+1)-2}}\\\
&\qquad+\sum_{j=1}^{\infty}\Big{(}\big{|}\mu_{n,2j(n+1)+2}^{1/2,1/2}-\mu_{n,2j(n+1)}^{1/2,1/2}\big{|}\frac{1}{\rho^{2j(n+1)}}+\big{|}\mu_{n,2j(n+1)+2}^{1/2,1/2}\big{|}\frac{1}{\rho^{2j(n+1)+2}}\Big{)}\\\
&\qquad=\dfrac{\pi}{2}\dfrac{1}{\gamma_{n}^{1/2,1/2}}\Big{(}2+(\rho+\rho^{-1})^{2}\sum_{j=1}^{\infty}\frac{1}{\rho^{2j(n+1)}}\Big{)}=\frac{\pi}{2}\frac{1}{\gamma_{n}^{1/2,1/2}}\Big{(}2+\frac{(\rho+\rho^{-1})^{2}}{\rho^{2n+2}-1}\Big{)}.\end{split}$
Hence, it follows from (3.14) that for the Chebyshev-Gauss quadrature of the
second kind,
$\begin{split}\big{|}E_{n}^{GG}[u]\big{|}&\leq\frac{\pi
M}{2\rho^{2n}}\bigg{(}1+\frac{1}{\rho^{2}}\Big{(}2+\frac{(\rho+\rho^{-1})^{2}}{\rho^{2n+2}-1}\Big{)}\bigg{)}=\frac{\pi
M(\rho^{2}+2+\rho^{-2n-4})}{2(\rho^{2n+2}-1)}.\end{split}$ (3.17)
Note that Hunter [25, (4.8)] obtained the following estimate by a delicate
technique:
$\big{|}E_{n}^{GG}[u]\big{|}\leq\frac{\pi
M(\rho^{2}+2+\rho^{-2})}{2(\rho^{2n+2}-1)}.$ (3.18)
We see that (3.17) is sharper. ∎
For general $\alpha>-1,$ the derivation of an explicit bound for
$\Theta_{n}^{\alpha}:=\max_{l\geq
0}\theta_{n,l}^{\alpha},\quad\theta_{n,l}^{\alpha}:=\gamma_{n}^{\alpha,\alpha}\big{|}\mu_{n,2l+2}^{\alpha,\alpha}-\mu_{n,2l}^{\alpha,\alpha}\big{|},\;\;\;n\geq
1,$ (3.19)
seems nontrivial. We have only empirical results bases on computation. Some
indications are listed as follows.
* (i)
Observe from (3.16) that for fixed $n$, $\\{\theta_{n,l}^{1/2}\\}_{l\geq 0}$
are $(n+1)$-periodic (see Figure 3.1 (a)), and the maximum is attained at
$l=j(n+1),j=0,1,\cdots$. We compute ample samples of $n,l$ and $\alpha,$ and
find very similar “periodic” behaviors (see Figure 3.1 (b)-(c) for
$\alpha=0,1$).
* (ii)
Another interesting empirical observation is that for fixed $\alpha,$ the
maximum value $\Theta_{n}^{\alpha}$ converges to a constant value, and it
decreases as $\alpha$ increases (see Figure 3.1 (d)). Note that for the
Legendre case, $\Theta_{n}^{0}\approx 4.$
(a) $\theta_{36,l}^{1/2}$
(b) $\theta_{36,l}^{0}$
(c) $\theta_{36,l}^{1}$
(d) $\Theta_{n}^{\alpha}$ with various $\alpha$
Figure 3.1. (a)-(c): Profiles of $\theta_{n,l}^{\alpha}$ with
$n=36,\alpha=1/2,0,1$ and $0\leq l\leq 250.$ We mark by “$\square$” the
location of the maximum value $\Theta_{n}^{\alpha}$ is attached. (d): The
maximum value $\Theta_{n}^{\alpha}$ with $\alpha=0,3/2,5,10$ and $10\leq n\leq
100,$ where we compute $\\{\theta_{n,l}^{\alpha}\\}$ for $l$ up to $1000.$
Now, we turn to the second approach. The main result is summarized below.
###### Theorem 3.2.
For any $u\in{\mathcal{A}}_{\rho}$ with $\rho>1,$ and for $\alpha>-1$ and
$\alpha\not=-1/2,$ we have
$\big{|}E_{n}^{GG}[u]\big{|}\leq\frac{C_{n}M\sqrt{\pi}}{\rho^{2n}}\bigg{(}\frac{\sqrt{\pi}}{2^{2\alpha}}+\frac{\Gamma(\alpha+1)}{\sqrt{\Gamma(2\alpha+2)}}\frac{2\sqrt{2}}{\rho^{2}-1}\bigg{)}\begin{cases}(1+\rho^{-2})^{\alpha+1/2},\;&\alpha>-1/2,\\\
(1-\rho^{-2})^{\alpha+1/2},\;&\alpha<-1/2,\end{cases}$ (3.20)
and in particular, for the Legendre case,
$\big{|}E_{n}^{LG}[u]\big{|}\leq\frac{C_{n}M\pi\sqrt{1+\rho^{-2}}}{\rho^{2n}}\Big{(}1+\frac{1}{2(\rho^{2}-1)}\Big{)},$
(3.21)
where the constant $C_{n}\approx 1$.
###### Proof.
We carry out the proof by using the important relation, due to (3.1) and
(B.1):
$\begin{split}\big{|}E_{n}^{GG}[u]\big{|}&\leq\frac{\gamma_{n}^{\alpha,\alpha}}{\min_{z\in\mathcal{E}_{\rho}}|J_{n}^{\alpha,\alpha}(z)|}\Big{|}\frac{1}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}Q_{n}^{\alpha,\alpha}(z)u(z)dz\Big{|}\\\
&\overset{(\ref{Jcoe3a})}{=}\frac{\gamma_{n}^{\alpha,\alpha}|\hat{u}_{n}^{\alpha,\alpha}|}{\min_{z\in\mathcal{E}_{\rho}}|J_{n}^{\alpha,\alpha}(z)|}.\end{split}$
(3.22)
Since the numerator has been estimated in Theorem 2.1 (also see (2.36)), it
suffices to deal with the denominator.
By [39, (4.6)], we have
$\begin{split}|J_{n}^{\alpha,\alpha}(z)|&\geq
C_{n}|A_{n}^{\alpha}|\frac{n^{\alpha-1/2}\rho^{n}}{|\Gamma(\alpha+1/2)|}\begin{cases}(1+\rho^{-2})^{-\alpha-1/2},\quad&{\rm
if}\;\;\alpha>-1/2,\\\ (1-\rho^{-2})^{-\alpha-1/2},\quad&{\rm
if}\;\;\alpha<-1/2,\end{cases}\\\ &\geq
C_{n}\frac{2^{2\alpha}\rho^{n}}{\sqrt{\pi
n}}\begin{cases}(1+\rho^{-2})^{-\alpha-1/2},\quad&{\rm if}\;\;\alpha>-1/2,\\\
(1-\rho^{-2})^{-\alpha-1/2},\quad&{\rm
if}\;\;\alpha<-1/2,\end{cases}\end{split}$ (3.23)
where $C_{n}\approx 1.$ Note that in the last step, we dealt with
$|A_{n}^{\alpha}|$ as
$|A_{n}^{\alpha}|\overset{(\ref{ajexp0})}{=}\frac{|\Gamma(2\alpha+1)|}{\Gamma(\alpha+1)}\frac{\Gamma(n+\alpha+1)}{\Gamma(n+2\alpha+1)}=\frac{|\Gamma(\alpha+1/2)|}{2^{-2\alpha}\sqrt{\pi}}\frac{\Gamma(n+\alpha+1)}{\Gamma(n+2\alpha+1)}\geq
C_{n}\frac{|\Gamma(\alpha+1/2)|}{2^{-2\alpha}\sqrt{\pi}n^{\alpha}},$
where we used Lemma 2.1 and the property of Gamma function (see [1]):
$\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\,\Gamma(2z).$
Therefore, a combination of (2.12), (2.36) and (3.22)-(3.23) leads to the
desired result.
Using the refined estimate (2.50), yields (3.21). ∎
## 4\. Numerical results and comparisons
In this section, we present various numerical results to show the tightness of
the bounds derived in this paper, and to compare them with other existing ones
mentioned in the previous part.
In the first example, we purposely choose the Chebyshev and Legendre
expansions with known expansion coefficients:
$u_{1}(x)=\frac{3}{5-4x}=T_{0}(x)+\sum_{n=1}^{\infty}\frac{T_{n}(x)}{2^{n-1}},\quad
u_{2}(x)=\frac{2}{\sqrt{5-4x}}=\sum_{n=0}^{\infty}\frac{L_{n}(x)}{2^{n}},$
(4.1)
which follow from generating functions of Chebyshev and Legendre polynomials
(cf. [33]).
Note that the function $u_{1}$ has a simple pole at $z=5/4,$ so the semi-major
axis (cf. (2.3)) should satisfy
$1<a=(\rho+\rho^{-1})/2<5/4\;\;\Rightarrow\;\;1<\rho<2.$
One also verifies that
$M=\max_{z\in{\mathcal{E}_{\rho}}}|u_{1}(z)|=\frac{3\rho}{(2\rho-1)(2-\rho)}.$
Then the estimate (2.40) reduces to
$\hat{u}_{n}^{C}=\frac{1}{2^{n-1}}\leq\frac{6}{(2\rho-1)(2-\rho)\rho^{n-1}}:=B_{n}^{C}(\rho),\quad
1<\rho<2,\;\;n\geq 1.$
Similarly, for the Legendre expansion of $u_{2},$ the result (2.50) becomes
$\hat{u}_{n}^{0,0}=\frac{1}{2^{n}}\leq\frac{\sqrt{\pi
n}}{\rho^{n}}\Big{(}1+\frac{n+2}{2n+3}\frac{1}{\rho^{2}-1}\Big{)}\exp\Big{(}\frac{8n-1}{12n(2n-1)}\Big{)}\sqrt{\frac{4\rho}{(2\rho-1)(2-\rho)}}:=B_{n}^{L}(\rho),$
for $1<\rho<2$ and $n\geq 1.$
We take $\rho=1.98,$ and plot the exact coefficients $\hat{u}_{n}^{C}$ and
$\hat{u}_{n}^{0,0}$, and the bounds $B_{n}^{C}$ and $B_{n}^{L}$ in Figure 4.1
(a) and (b), respectively. Actually, the bound for the Chebyshev case (see
(1.1)) can be considered as one benchmark for illustrating tightness of the
upper bound. Indeed, the result for the Legendre case stated in Theorem 2.2
seems as sharp as that for the Chebyshev case.
(a) Chebyshev case
(b) Legendre case
Figure 4.1. Expansion coefficients of $u_{1},u_{2}$ in (4.1) against their
error bounds.
(a) $e_{n}(\rho)$
(b) $e_{n}(\rho)$ with $\rho$ near $1$
Figure 4.2. (a): Comparison of error bounds for Legendre expansions in (2.50)
and (2.56). (b): Samples of $e_{n}(\rho)$ for $\rho$ close to $1.$
Next, we compare the bounds for the Legendre expansion coefficients in Theorem
2.2 and (2.56) (obtained by [38]). For clarity, we drop the common part
$M\sqrt{n}/\rho^{n},$ and denote the remaining factors in the upper bounds by
$\begin{split}&b_{n}(\rho)\overset{(\ref{legenest4})}{=}\sqrt{\pi}\Big{(}1+\frac{n+2}{2n+3}\frac{1}{\rho^{2}-1}\Big{)}\exp\Big{(}\frac{8n-1}{12n(2n-1)}\Big{)},\quad\tilde{b}(\rho)\overset{(\ref{xianglegenbnd})}{=}2\Big{(}1+\frac{1}{\rho^{2}-1}\Big{)}.\end{split}$
In Figure 4.2 (a), we plot the difference
$e_{n}(\rho):=\tilde{b}(\rho)-b_{n}(\rho)$ for various $\rho$ and $1\leq n\leq
80.$ We see that $e_{n}(\rho)>0$ and the difference is of magnitude around
$6$, when $\rho$ is close to $1$. Moreover, for fixed $\rho,$ the difference
is roughly a constant for slightly large $n.$ In Figure 4.2 (b), we plot some
sample $e_{n}(\rho)$ for $\rho$ close to 1, we see that our bound is much
sharper.
(a) $\alpha=1,\beta=0$
(b) $\alpha=\beta=2$
Figure 4.3. (a): Comparison of error bounds for Jacobi expansion with
$\alpha=1,\beta=0$ in (1.2) and (2.58). (b): Comparison of error bounds for
Gegenbauer expansion with $\alpha=\beta=2$ in (1.2) and (2.27).
We next make a similar comparison of bounds for Jacobi and Gegenbauer
expansions. For example, for $\alpha=1$ and $\beta=0,$ we extract the factors
in (1.2) and (2.58) by dropping $M\sqrt{n}/\rho^{n}.$ We plot in Figure 4.3
(a) the difference of two remaining parts (i.e., that of (1.2) subtracts that
of (2.58)). Once again, our bound is much tighter. Likewise, we depict in
Figure 4.3 (b) the extracted bounds from (1.2) and (2.27) with
$\alpha=\beta=2.$ The situation is mimic to the Legendre case, where the
bounds obtained in this paper are sharper.
Finally, we turn to the comparison of error bounds for the Gegenbauer-Gauss
quadrature remainder. For $\alpha=1/2$, we extract the factors in (3.17) and
(3.18) by dropping $M/\rho^{2n}$ as before. We plot in Figure 4.4 (a) the
difference of two remaining parts in (3.18) and in (3.17)). Once again, our
bound is much tighter. Likewise, we depict in Figure 4.4 (b) the extracted
bounds from (3.4) and (3.11) with $\alpha=2,$ and observe similar behaviors.
(a) $\alpha=1/2$
(b) $\alpha=2$
Figure 4.4. (a): Comparison of error bounds for the Gegenbauer-Gauss
quadrature with $\alpha=1/2$ in (3.17) and (3.18). (b): Comparison of error
bounds for the Gegenbauer-Gauss quadrature with $\alpha=2$ in (3.4) and
(3.11).
Concluding remarks In this paper, we derived various new and sharp error
bounds for Jacobi polynomial expansions and Gegenbauer-Gauss quadrature of
analytic functions with analyticity characterized by the Bernstein ellipse. We
adopted an argument that could recover the best known bounds, and attempted to
make the dependence of the estimates on the parameters explicitly. Both
analytic estimates and numerical comparisons with available ones demonstrated
the sharpness of the error bounds.
## Appendix A Jacobi polynomials
We collect some properties of Jacobi polynomials used in the paper. For
$\alpha,\beta>-1,$ the Jacobi polynomials (see e.g., [33]), denoted by
$J_{n}^{\alpha,\beta}(x),x\in I:=(-1,1),$ are defined by the Rodrigues’
formula
$(1-x)^{\alpha}(1+x)^{\beta}J_{n}^{\alpha,\beta}(x)=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}}\Big{[}(1-x)^{\alpha+n}(1+x)^{\beta+n}\Big{]},\quad
n\geq 0.$ (A.1)
The Jacobi polynomials satisfy
$\displaystyle(1-x)J_{n}^{\alpha+1,\beta}(x)=\frac{2}{2n+\alpha+\beta+2}\big{(}(n+\alpha+1)J_{n}^{\alpha,\beta}(x)-(n+1)J_{n+1}^{\alpha,\beta}(x)\big{)},$
(A.2a)
$\displaystyle(1+x)J_{n}^{\alpha,\beta+1}(x)=\frac{2}{2n+\alpha+\beta+2}\big{(}(n+\beta+1)J_{n}^{\alpha,\beta}(x)+(n+1)J_{n+1}^{\alpha,\beta}(x)\big{)}.$
(A.2b)
As a direct consequence of (A.2), we have that for any
$k,l\in{\mathbb{N}}=\\{0,1,\cdots\\},$
$(1-x)^{k}(1+x)^{l}J_{n}^{\alpha+k,\beta+l}(x)=\sum_{i=n}^{n+k+l}d_{i}^{\alpha+k,\beta+l}J_{i}^{\alpha,\beta}(x),$
(A.3)
where $\\{d_{i}^{\alpha+k,\beta+l}\\}_{i=n}^{n+k+l}$ is a unique set of
constants (with $d_{n}^{\alpha,\beta}=1$), computed from (A.2) recursively.
Here, we sketch the proof of (A.3). To this end, let $\\{c_{j}\\}$ be a set of
generic constants. Using (A.2a) and (A.2b) repeatedly leads to
$\begin{split}&(1-x)^{k}(1+x)^{l}J_{n}^{\alpha+k,\beta+l}(x)\\\
&\qquad=(1-x)^{k-1}(1+x)^{l}\big{(}c_{1}J_{n}^{\alpha+k-1,\beta+l}(x)+c_{2}J_{n+1}^{\alpha+k-1,\beta+l}(x)\big{)}\\\
&\qquad=\cdots=(1+x)^{l}\sum_{m=n}^{n+k}c_{m}J_{m}^{\alpha,\beta+l}(x)=\cdots=\sum_{m=n}^{n+k+l}c_{m}J_{m}^{\alpha,\beta}(x).\end{split}$
This yields (A.3). We point out that for $\alpha=\beta=0,$
$\\{(1-x)^{k}(1+x)^{l}J_{n}^{k,l}\\}$ (up to a certain constant factor) are
defined as generalized Jacobi polynomials in [22].
The following formula, derived from [2, Lemma 7.1.1] (also see [32, Theorem
3.21]), was used for the derivation of (2.21):
$\begin{split}\hat{c}_{j}^{n}:&=\hat{c}_{j}^{n}(\alpha,\beta,a,b)=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}J_{n+j}^{a,b}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx\\\
&=\frac{\Gamma(n+j+a+1)}{\Gamma(n+j+a+b+1)}\frac{(2n+\alpha+\beta+1)\Gamma(n+\alpha+\beta+1)}{\Gamma(n+\alpha+1)}\\\
&\quad\times\sum_{m=0}^{j}\frac{(-1)^{m}\Gamma(2n+j+m+a+b+1)\Gamma(n+m+\alpha+1)}{m!(j-m)!\Gamma(n+m+a+1)\Gamma(2n+m+\alpha+\beta+2)},\end{split}$
(A.4)
for $a,b,\alpha,\beta>-1$ and $n,j\geq 0.$
Let $T_{n}(x)=\cos(n\,{\rm arccos}(x))$ be the Chebyshev polynomial of the
first kind of degree $n.$ Then the second-kine Chebyshev polynomial, denoted
by $U_{n}(x),$ can be expressed by
$\begin{split}U_{n}(x)=\frac{\sin\big{(}(n+1)\,{\rm
arccos}(x)\big{)}}{\sqrt{1-x^{2}}}=\frac{T_{n+1}^{\prime}(x)}{n+1}=\sqrt{\frac{\pi}{2}}\frac{J_{n}^{1/2,1/2}(x)}{\sqrt{\gamma_{n}^{1/2,1/2}}}.\end{split}$
(A.5)
The Chebyshev polynomials enjoy the following important properties:
$\displaystyle
J^{-1/2,-1/2}_{n}(x)=J^{-1/2,-1/2}_{n}(1)T_{n}(x)=\frac{\Gamma(n+1/2)}{\sqrt{\pi}n!}T_{n}(x),$
(A.6a) $\displaystyle
T^{\prime}_{n}(x)=2n\underset{k+n\;\text{odd}}{\underset{k=0}{\sum^{n-1}}}\frac{1}{c_{k}}T_{k}(x),$
(A.6b)
where $c_{0}=2$ and $c_{k}=1$ for $k\geq 1.$
## Appendix B Proof of Lemma 2.2
We first show that
$\begin{split}\hat{u}_{n}^{\alpha,\beta}&=\frac{1}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}Q_{n}^{\alpha,\beta}(z)u(z)\,dz,\end{split}$
(B.1)
where
$Q_{n}^{\alpha,\beta}(z):=\frac{1}{2\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}\frac{J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)}{z-x}\,dx,$
(B.2)
and $\gamma_{n}^{\alpha,\beta}$ is given by (2.8). Recall the Cauchy’s
integral formula:
$\begin{split}\frac{d^{n}}{dx^{n}}u(x)=\frac{n!}{2\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{(z-x)^{n+1}}\,dz.\end{split}$ (B.3)
Using the Rodrigues’ formula (A.1) and integration by parts leads to
$\begin{split}\hat{u}_{n}^{\alpha,\beta}&\overset{(\ref{xiangres})}{=}\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}u(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx\\\
&\overset{(\ref{Rodrigues})}{=}\frac{1}{\gamma_{n}^{\alpha,\beta}}\frac{(-1)^{n}}{2^{n}n!}\int_{-1}^{1}\omega^{\alpha+n,\beta+n}(x)\frac{d^{n}}{dx^{n}}u(x)\,dx\\\
&\overset{(\ref{Cauc})}{=}\frac{1}{\gamma_{n}^{\alpha,\beta}}\frac{1}{2^{n}n!}\int_{-1}^{1}\Big{(}\frac{n!}{2\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\frac{u(z)}{(z-x)^{n+1}}\,dz\Big{)}\omega^{\alpha+n,\beta+n}(x)\,dx\\\
&=\frac{1}{2^{n}\gamma_{n}^{\alpha,\beta}}\frac{1}{2\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}\Big{(}\int_{-1}^{1}\frac{\omega^{\alpha+n,\beta+n}(x)}{(z-x)^{n+1}}\,dx\Big{)}u(z)\,dz.\end{split}$
(B.4)
We find from integration by parts that
$\displaystyle\int_{-1}^{1}\frac{\omega^{\alpha+n,\beta+n}(x)}{(z-x)^{n+1}}\,dx=\frac{(-1)^{n}}{n!}\int_{-1}^{1}\frac{1}{z-x}\frac{d^{n}}{dx^{n}}\omega^{\alpha+n,\beta+n}(x)\,dx.$
(B.5)
Inserting (B.5) into (B.4), we derive from the Rodrigues’ formula (A.1) that
$\begin{split}\hat{u}_{n}^{\alpha,\beta}&=\frac{1}{2\pi{\rm
i}}\frac{1}{\gamma_{n}^{\alpha,\beta}}\oint_{\mathcal{E}_{\rho}}\Big{(}\int_{-1}^{1}\frac{\omega^{\alpha,\beta}(x)J_{n}^{\alpha,\beta}(x)}{z-x}\,dx\Big{)}u(z)\,dz=\frac{1}{\pi{\rm
i}}\oint_{\mathcal{E}_{\rho}}Q_{n}^{\alpha,\beta}(z)u(z)\,dz,\end{split}$
where $Q_{n}^{\alpha,\beta}(z)$ is given in (B.2).
Since $z=(w+w^{-1})/2,$ we have from the generating function of the Chebyshev
polynomial of the second-kind (cf. [1]) that
$\frac{1}{z-x}=\frac{2}{w}\frac{1}{w^{-2}-2xw^{-1}+1}=\frac{2}{w}\sum_{k=0}^{\infty}\frac{U_{k}(x)}{w^{k}}.$
(B.6)
Inserting it into (B.2), we find from the orthogonality of the Jacobi
polynomials (cf. (2.7)) that
$\begin{split}Q_{n}^{\alpha,\beta}(z)&=\frac{1}{\gamma_{n}^{\alpha,\beta}}\sum_{k=0}^{\infty}\frac{1}{w^{k+1}}\int_{-1}^{1}U_{k}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx\\\
&=\frac{1}{\gamma_{n}^{\alpha,\beta}}\sum_{k=n}^{\infty}\frac{1}{w^{k+1}}\int_{-1}^{1}U_{k}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx\\\
&=\frac{1}{\gamma_{n}^{\alpha,\beta}}\sum_{j=0}^{\infty}\frac{1}{w^{n+j+1}}\int_{-1}^{1}U_{n+j}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx=\sum_{j=0}^{\infty}\frac{\sigma_{n,j}^{\alpha,\beta}}{w^{n+j+1}},\end{split}$
(B.7)
where we defined
$\sigma_{n,j}^{\alpha,\beta}=\frac{1}{\gamma_{n}^{\alpha,\beta}}\int_{-1}^{1}U_{n+j}(x)J_{n}^{\alpha,\beta}(x)\omega^{\alpha,\beta}(x)\,dx,$
Substituting the last identity of (B.7) into (B.1) leads to the desired
formula (2.15).
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|
arxiv-papers
| 2012-10-08T04:02:52 |
2024-09-04T02:49:36.121992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaodan Zhao, Li-Lian Wang and Ziqing Xie",
"submitter": "Li-Lian Wang Dr.",
"url": "https://arxiv.org/abs/1210.2141"
}
|
1210.2155
|
# Linear maps preserving separability of pure states
Jinchuan Hou Department of Mathematics
Taiyuan University of Technology
Taiyuan 030024, P. R. of China jinchuanhou@yahoo.com.cn and Xiaofei Qi
Department of Mathematics, Shanxi University, Taiyuan 030006, P. R. of China;
xiaofeiqisxu@yahoo.com.cn
###### Abstract.
Linear maps preserving pure states of a quantum system of any dimension are
characterized. This is then used to establish a structure theorem for linear
maps that preserve separable pure states in multipartite systems. As an
application, a characterization of separable pure state preserving affine maps
is obtained.
2010 Mathematics Subject Classification. 47B49; 47N50; 47A80
Key words and phrases. Linear maps, affine maps, quantum states, separability,
tensor products
This work is supported by National Natural Science Foundation of China
(11171249, 11101250) and Youth Foundation of Shanxi Province (2012021004).
## 1\. Introduction
A quantum state $\rho$ is a density operator acting on a complex Hilbert space
which is positive semidefinite and has trace 1. Furthermore, $\rho$ is a pure
state if $\rho^{2}=\rho$ (i.e., $\rho$ is a rank-1 projection); $\rho$ is a
mixed state if $\rho^{2}\not=\rho$. Denote by ${\mathcal{S}}(H)$ the set of
all states on a Hilbert space $H$. In quantum information theory we deal, in
general, with multipartite systems. The underlying space $H$ of a multipartite
composite quantum system is a tensor product of underlying spaces $H_{i}$ of
its subsystems, that is, $H=H_{1}\otimes H_{2}\otimes\cdots\otimes H_{n}$. In
the case $n=2$ the system is called a bipartite system. If $H$ and $K$ are
finite dimensional Hilbert spaces, $\rho\in{\mathcal{S}}(H\otimes K)$ is said
to be separable if $\rho$ can be written as
$None$ $\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$. Otherwise, $\rho$
is said to be inseparable or entangled (ref. [1, 6]). For the case that at
least one of $H$ and $K$ is of infinite dimension, by Werner [7], a state
$\rho$ acting on $H\otimes K$ is called separable if it can be approximated in
the trace norm by states of the form (0.1). Otherwise, $\rho$ is called an
entangled state. The full separability of multipartite states can be defined
similarly.
Entanglement is a basic physical resource to realize various quantum
information and quantum communication tasks [3, 4, 6]. So it is important to
determine whether or not a state in a composite system is separable, which is
also a very difficult task in this field. Thus, this makes it interesting to
find linear maps sending states to states, which will simplify a given state
so that it is easier to detect the entanglement in it. Clearly, such linear
maps should leave the separability of states invariant. So, this proposes the
question of studying linear preservers of separable states. This question was
attacked in [2] for the finite dimensional systems. Let ${\bf H}_{N}$ be the
real linear space of all $N\times N$ Hermitian matrices. It was shown in [2]
that, if a surjective linear map $\Phi:{\bf H}_{n_{1}n_{2}}\rightarrow{\bf
H}_{n_{1}n_{2}}$ preserves separable pure states in the bipartite system
${\mathbb{C}}^{n_{1}}\otimes{\mathbb{C}}^{n_{2}}$, then $\Phi$ sends product
states to product states, that is, $\Phi(A_{1}\otimes
A_{2})=\psi_{1}(A_{p_{1}})\otimes\psi_{2}(A_{p_{2}})$, where $(p_{1},p_{2})$
is a permutation of $(1,2)$, $n_{j}=n_{{p_{j}}}$ and
$\psi_{j}:M_{n_{j}}\rightarrow M_{n_{j}}$ is a linear map of the form
$X\mapsto U_{j}XU_{j}^{*}$ or $X\mapsto U_{j}X^{\rm t}U_{j}^{*}$ for a unitary
matrix $U_{j}\in M_{n_{j}}$. Here $X^{\rm t}$ denotes the transposed matrix of
$X$. A similar result holds for finite dimensional multipartite systems.
The purpose of the present paper is to characterize general linear maps that
preserve separable pure states for both finite and infinite dimensional
systems. We remark that the results for finite dimensional systems are
somewhat different from that for infinite dimensional systems because the
linear maps on infinite dimensional spaces may not be continuous.
Let ${\mathcal{T}}(H)$ be the Banach space of trace-class operators on a
complex Hilbert space $H$ endowed with the trace-norm $\|\cdot\|_{\rm Tr}$.
Denote by ${\mathcal{T}}_{\rm sa}(H)$ and ${\mathcal{F}}_{\rm sa}(H)$ the
subspace of self-adjoint operators and finite-rank self-adjoint operators in
${\mathcal{T}}(H)$, respectively. Denote by ${\mathcal{P}ur}(H)$ the set of
pure states (i.e., rank-one projections) on $H$. We first consider in Section
2 the question of characterizing linear maps preserving pure states since this
is basic for the study of our main question. Let $H$ and $K$ be Hilbert spaces
of any dimension. It is shown that a linear map $\Phi:{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ satisfies
$\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ if and only if $\Phi$
either has the form $A\mapsto{\rm Tr}(A)R+\phi(A)$ for all
$A\in{\mathcal{T}}_{\rm sa}(H)$; or $\dim H\leq\dim K$ and $\Phi$ has the form
$A\mapsto UAU^{*}+\phi(A)$ for all $A\in{\mathcal{T}}_{\rm sa}(H)$, where
$\phi:{\mathcal{T}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ is a
linear map vanishing on each finite-rank operator, $R\in{\mathcal{P}ur}(K)$
and $U:H\rightarrow K$ is a linear or conjugate linear isometry (Theorem 2.2).
Particularly, if $\Phi$ preserves pure states in both directions, then $\Phi$
has the second form with $U$ unitary or conjugate unitary. The main result of
this section generalizes a result in [2] for finite dimensional case.
In Section 3 we discuss the question of characterizing linear maps preserving
separable pure states of bipartite systems. Let ${\mathcal{S}}_{\rm
sep}(H\otimes K)$ stand for the convex set of all separable states on
$H\otimes K$. Denote by ${\mathcal{T}}_{\rm sep}(H\otimes K)$ the linear
manifold generated by ${\mathcal{S}}_{\rm sep}(H\otimes K)$ and
${\mathcal{F}}_{\rm sep}(H\otimes K)={\mathcal{T}}_{\rm sep}(H\otimes
K)\cap{\mathcal{F}}_{\rm sa}(H\otimes K)$. It is obvious that
${\mathcal{T}}_{\rm sep}(H\otimes K)\subseteq{\mathcal{T}}_{\rm sa}(H\otimes
K)$. Our main result gives a characterization of linear maps from
${\mathcal{T}}_{\rm sa}(H\otimes K)$ into itself which preserve separable pure
states (not necessarily in both directions). It turns out such maps have one
of nine forms on ${\mathcal{F}}_{\rm sep}(H\otimes K)$ (see Theorem 3.2).
However, for most situations they have a standard form ((6) or (7) in Theorem
3.2). As an application, we get a characterization of affine maps between
convex sets of states which preserve separable pure states in both directions.
Such a map is either of the form $\Phi(\rho)=(U_{1}\otimes
U_{2})\Lambda(\rho)(U_{1}\otimes U_{2})^{*}$ for all
$\rho\in{\mathcal{S}}_{\rm sep}(H\otimes K)$ or of the form
$\Phi(\rho)=(U_{1}\otimes U_{2})\Lambda(\theta(\rho))(U_{1}\otimes U_{2})^{*}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H\otimes K)$, where $U_{1}$ and
$U_{2}$ are unitary operators on $H$ and $K$, respectively, $\Lambda$ is one
of the identity map, the transpose, the partial transpose with respect to any
fixed product orthonormal basis of $H\otimes K$, and
$\theta:{\mathcal{T}}(H\otimes K)\rightarrow{\mathcal{T}}(K\otimes H)$ is the
swap determined by $\theta(A\otimes B)=B\otimes A$.
In Section 4, a brief discussion of the question for multipartite systems is
given. Some results similar to those in bipartite systems in Section 3 are
presented.
## 2\. Linear maps preserving pure states
The main purpose of this section is to characterize linear preservers of pure
states for infinite dimensional systems, which are also needed to characterize
separable pure state preservers in the next section.
The following proposition comes from [2], which can be viewed as a
characterization of linear preservers of pure states for finite dimensional
systems.
Let ${\bf H}_{m}$ be the real linear space of all $m\times m$ Hermitian
matrices and let ${\mathcal{P}}_{m}$ be the set of all rank-1 $m\times m$
projection matrices.
Proposition 2.1. Suppose that $\phi:{\bf H}_{m}\rightarrow{\bf H}_{n}$ is
linear and satisfies $\phi({\mathcal{P}}_{m})\subseteq{\mathcal{P}}_{n}$. Then
one of the following holds:
(i) There is $Q\in{\mathcal{P}}_{n}$ such that $\phi(A)={\rm Tr}(A)Q$ for all
$A\in{\bf H}_{m}$.
(ii) $m\leq n$ and there is a matrix $U\in M_{n\times m}$ with $U^{*}U=I_{m}$
such that $\phi(A)=UAU^{*}$ for all $A\in{\bf H}_{m}$, or $\phi(A)=UA^{\rm
t}U^{*}$ for all $A\in{\bf H}_{m}$.
By using a result due to [5], we can generalize Proposition 2.1 to the
infinite dimensional case. Recall that a linear map $V:H\rightarrow K$ is an
isometry if $\|Vx\|=\|x\|$ for all $x\in H$, or equivalently, $V^{*}V=I_{H}$,
the identity operator on $H$. A conjugate linear isometry is defined
similarly.
The following is the main result of this section.
Theorem 2.2. Let $H$ and $K$ be Hilbert spaces of any dimension. Suppose that
$\Phi:{\mathcal{T}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ is a real
linear map. Then $\Phi$ satisfies
$\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ if and only if there
exists a linear map $\phi:{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ vanishing on each finite-rank
operator and one of the following holds:
(i) There is some $R\in{\mathcal{P}ur}(K)$ such that $\Phi(A)={\rm
Tr}(A)R+\phi(A)$ for all $A\in{\mathcal{T}}_{\rm sa}(H)$.
(ii) $\dim H\leq\dim K$ and there is a linear or conjugate linear isometry
$U:H\rightarrow K$ such that $\Phi(A)=UAU^{*}+\phi(A)$ for all
$A\in{\mathcal{T}}_{\rm sa}(H)$.
Proof. Only the “only if” part should be checked.
Suppose that $\Phi$ preserves pure states. By Proposition 2.1, we may assume
that $\dim H=\infty$. Define a map $\Psi:{\mathcal{T}}_{\rm sa}(H\oplus
K)\rightarrow{\mathcal{T}}_{\rm sa}(H\oplus K)$ given by
$\Psi(S)=\Psi(\begin{bmatrix}A&C\\\
C^{*}&B\end{bmatrix})=\begin{bmatrix}0&0\\\ 0&\Phi(A)\end{bmatrix}$
for all
$S=\begin{bmatrix}A&C\\\ C^{*}&B\end{bmatrix}\in{\mathcal{T}}_{\rm sa}(H\oplus
K)\ {\rm with}\ A\in{\mathcal{T}}_{\rm sa}(H).$
It is obvious that $\Psi$ is linear and $\Psi(A\oplus 0)=0\oplus\Phi(A)$ for
all $A\in{\mathcal{T}}_{\rm sa}(H)$. Moreover, $\Psi$ is rank one decreasing,
that is, ${\rm rank}(\Psi(S))\leq 1$ whenever ${\rm rank}(S)=1$, since
$\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$. It follows that
$\Psi({\mathcal{F}}_{\rm sa}(H\oplus K))\subseteq{\mathcal{F}}_{\rm
sa}(H\oplus K)$, where ${\mathcal{F}}_{\rm sa}(H)$ stands for the set of all
finite-rank self-adjoint operators on $H$. Hence, by [5, Theorem 2.10], we get
that one of the following is true:
(1) $\Psi(S)=f(S)Q$ for all $S\in{\mathcal{F}}_{\rm sa}(H\oplus K)$, where
$f:{\mathcal{F}}_{\rm sa}(H\oplus K)\rightarrow{\mathbb{R}}$ is a linear map
and $Q$ is a rank one projection;
(2) $\Psi(x\otimes x)=\lambda Tx\otimes Tx$ for all $x\in H\oplus K$, where
$\lambda$ is a nonzero real number and $T:H\oplus K\rightarrow H\oplus K$ is a
linear or conjugate linear operator.
If (1) holds, then there exists some $R\in{\mathcal{P}ur}(K)$ such that
$Q=0\oplus R$ and $\Phi(A)=g(A)R$ for all $A\in{\mathcal{F}}_{\rm sa}(H)$,
where $g(A)=f(A\oplus 0)$. Since
$\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$, we have $g(P)=1$ for
all $P\in{\mathcal{P}ur}(H)$. Now, for any $A\in{\mathcal{F}}_{\rm sa}(H)$,
write $A=\sum_{i=1}^{n}\lambda_{i}P_{i}$, which is the spectral decomposition
of $A$. It follows from the linearity of $\Phi$ that
$g(A)=g(\sum_{i=1}^{n}\lambda_{i}P_{i})=\sum_{i=1}^{n}\lambda_{i}g(P_{i})=\sum_{i=1}^{n}\lambda_{i}={\rm
Tr}(A)$, that is, (i) holds for all finite-rank operators. Next let us show
that the statement (i) of Theorem 2.2 holds for all $A\in{\mathcal{T}}_{\rm
sa}(H)$. Note that $\Phi$ is bounded on the normed space ${\mathcal{F}}_{\rm
sa}(H)$ endowed with the trace-norm. In fact, $\|\Phi|_{{\mathcal{F}}_{\rm
sa}(H)}\|\leq\|R\|$. Let $\widehat{\Phi}:{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ be the bounded linear map defined
by $\widehat{\Phi}(A)={\rm Tr}(A)R$ and let $\phi=\Phi-\widehat{\Phi}$. Then,
$\phi(F)=0$ for each $F\in{\mathcal{F}}_{\rm sa}(H)$ and $\Phi(A)={\rm
Tr}(A)R+\phi(A)$ for all $A$, as desired.
Now assume that (2) holds. Note that $\Psi(A\oplus 0)=0\oplus\Phi(A)$. Then
$\Phi$ has the form $\Phi(x\otimes x)=\lambda Ux\otimes Ux$ for all $x\in H$,
where $U$ is the part of $T$ restricted to $H$. Next we prove that $U$ is
bounded. In fact, since $\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$,
we have
$None$ $\lambda\langle Ux,Ux\rangle=\lambda\|Ux\|^{2}=1$
for all unit vectors $x\in H$, which implies $\lambda>0$. Without loss of
generality, we may assume that $\lambda=1$. Since $U$ is linear or conjugate
linear, by Eq.(2.1), we get $\|Ux\|=\|x\|$ for all $x\in H$. It follows that
$U$ is bounded and $U^{*}U=I_{H}$, that is, $U$ is an isometry or conjugate
isometry. Then $\Phi(x\otimes x)=U(x\otimes x)U^{*}$, and consequently,
$\Phi(A)=UAU^{*}$ for all $A\in{\mathcal{F}}_{\rm sa}(H)$. Let
$\phi:{\mathcal{T}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(H)$ be the
linear map defined by $\phi(A)=\Phi(A)-UAU^{*}$ for every $A$. Then
$\Phi(A)=UAU^{*}+\phi(A)$ for all $A\in{\mathcal{T}}_{\rm sa}(H)$, that is,
(ii) of Theorem 2.2 holds.
The proof of the theorem is complete. $\Box$
Remark 2.3. If the linear map $\phi:{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(H)$ is not zero, then both $\Phi$ and
$\phi$ are not continuous because ${\mathcal{F}}_{\rm sa}(H)$ is dense in
${\mathcal{T}}_{\rm sa}(H)$. Such linear maps exist. For example, take any
nonzero linear map $\widehat{\phi}:{\mathcal{T}}_{\rm
sa}(H)/{\mathcal{F}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$, and let
$\phi$ be defined by $\phi(A)=\widehat{{\phi}}(\pi(A))$ for any
$A\in{\mathcal{T}}_{\rm sa}(H)$, where $\pi:{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(H)/{\mathcal{F}}_{\rm sa}(H)$ is the
quotient map.
The following corollary is immediate from Theorem 2.2.
Corollary 2.4. Let $H$ and $K$ be Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{T}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(K)$ is a
bounded real linear map. Then $\Phi$ satisfies
$\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ if and only if one of
the following holds:
(i) There is some $R\in{\mathcal{P}ur}(K)$ such that $\Phi(A)={\rm Tr}(A)R$
for all $A\in{\mathcal{T}}_{\rm sa}(H)$.
(ii) There is a linear or conjugate linear isometry $U:H\rightarrow K$ with
$U^{*}U=I_{H}$ such that $\Phi(A)=UAU^{*}$ for all $A\in{\mathcal{T}}_{\rm
sa}(H)$.
Note that a bijective affine map from ${\mathcal{S}}(H)$ onto
${\mathcal{S}}(K)$ preserves pure states in both directions. So the following
corollary is a generalization of Kadison’s characterization of affine
isomorphisms on ${\mathcal{S}}(H)$, which says that a bijective affine map has
the form $\rho\mapsto U\rho U^{*}$, where $U$ is a unitary or conjugate
unitary operator (See, for instance, [1, Theorem 8.1]).
Corollary 2.5. Let $H$ and $K$ be Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ is an affine map. Then
$\Phi$ satisfies $\Phi({\mathcal{P}ur}(H))\subseteq{\mathcal{P}ur}(K)$ if and
only if one of the following holds:
(i) There is some $R\in{\mathcal{P}ur}(K)$ such that $\Phi(\rho)=R$ for all
$\rho\in{\mathcal{S}}(H)$.
(ii) There is a linear or conjugate linear isometry $U:H\rightarrow K$ such
that $\Phi(\rho)=U\rho U^{*}$ for all $\rho\in{\mathcal{S}}(H)$.
Proof. We need only show that if
$\Phi:{\mathcal{S}}(H)\rightarrow{\mathcal{S}}(K)$ is affine and preserves
pure states, then $\Phi$ has the form (i) or the form (ii) stated in the
corollary.
To do this, note that the affinity of $\Phi$ allow us to extend it to a linear
map (see Ref. [2] for details), still denoted by $\Phi$, from
${\mathcal{T}}_{\rm sa}(H)$ into ${\mathcal{T}}_{\rm sa}(K)$. Every
$A\in{\mathcal{T}}_{\rm sa}(H)$ has a representation $A=A^{+}-A^{-}$ with
$A^{\pm}\geq 0$ and $A^{+}A^{-}=0$. Thus $\|A\|_{\rm Tr}=\|A^{+}\|_{\rm
Tr}+\|A^{-}\|_{\rm Tr}$. As $\|\Phi(\rho)\|_{\rm Tr}=\|\rho\|_{\rm Tr}$ for
all states $\rho$, we see that $\|\Phi(A)\|_{\rm Tr}\leq\|\Phi(A^{+})\|_{\rm
Tr}+\|\Phi(A^{-})\|_{\rm Tr}=\|A^{+}\|_{\rm Tr}+\|A^{-}\|_{\rm Tr}=\|A\|_{\rm
Tr}$. Hence $\Phi$ is bounded and, by Corollary 2.4, $\Phi$ has the desired
form. $\Box$
## 3\. linear maps preserving separable pure states: bipartite systems
Now we are ready to give a characterization of linear maps preserving
separable pure states for bipartite quantum systems.
Write ${\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)=\\{P\otimes
Q:P\in{\mathcal{P}ur}(H),\ Q\in{\mathcal{P}ur}(K)\\}$.
Lemma 3.1. Let $H$ and $K$ be any Hilbert spaces. Then the set of separable
states ${\mathcal{S}}_{\rm sep}(H\otimes K)$ is a convex set, whose extreme
points is ${\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$.
Proof. Obvious. $\Box$
Denote by ${\mathcal{T}}_{\rm sep}(H\otimes K)$ the real linear space
generated by ${\mathcal{S}}_{\rm sep}(H\otimes K)$, the set of all separable
states on $H\otimes K$; ${\mathcal{F}}_{\rm sep}(H\otimes K)$ the subspace of
all finite-rank operators in ${\mathcal{T}}_{\rm sep}(H\otimes K)$. We denote
by ${\rm Tr}_{i}$ the partial trace of the $i$th subsystem, that is, ${\rm
Tr}_{1}(\rho)={\rm Tr}_{H}(\rho)=({\rm Tr}\otimes I_{K})(\rho)$ and ${\rm
Tr}_{2}(\rho)={\rm Tr}_{K}(\rho)=(I_{H}\otimes{\rm Tr})(\rho)$. Clearly, ${\rm
Tr}_{i}$ is linear.
The following is the main result of this paper.
Theorem 3.2. Let $H$ and $K$ be two Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{T}}_{\rm sa}(H\otimes K)\rightarrow{\mathcal{T}}_{\rm
sa}(H\otimes K)$ is a linear map. Then
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))\subseteq{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
if and only if one of the following holds:
(1) There exists $R_{1}\otimes
R_{2}\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$ such that
$\Phi(F)={\rm Tr}(F)R_{1}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(2) There exist $R_{2}\in{\mathcal{P}ur}(K)$ and a linear or conjugate linear
isometry $U_{1}:H\rightarrow H$ such that
$\Phi(F)=U_{1}[{\rm Tr}_{2}(F)]U_{1}^{*}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(3) There exist $R_{1}\in{\mathcal{P}ur}(H)$ and a linear or conjugate linear
isometry $U_{2}:K\rightarrow K$ such that
$\Phi(F)=R_{1}\otimes U_{2}[{\rm Tr}_{1}(F)]U_{2}^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(4) $\dim H\geq\dim K$, there exist $R_{2}\in{\mathcal{P}ur}(K)$ and a linear
or conjugate linear isometry $U_{1}:K\rightarrow H$ such that
$\Phi(F)=U_{1}[{\rm Tr}_{1}(F)]U_{1}^{*}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(5) $\dim H\leq\dim K$, there exist $R_{1}\in{\mathcal{P}ur}(H)$ and a linear
or conjugate linear isometry $U_{2}:H\rightarrow K$ such that
$\Phi(F)=R_{1}\otimes U_{2}[{\rm Tr}_{2}(F)]U_{2}^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(6) There exist linear or conjugate linear isometries $U_{1}:H\rightarrow H$
and $U_{2}:K\rightarrow K$ such that
$\Phi(F)=(U_{1}\otimes U_{2})F(U_{1}\otimes U_{2})^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(7) $\dim H=\dim K$, there exist linear or conjugate linear isometries
$U_{1}:K\rightarrow H$ and $U_{2}:H\rightarrow K$ such that
$\Phi(F)=(U_{1}\otimes U_{2})\theta(F)(U_{1}\otimes U_{2})^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$, where
$\theta:{\mathcal{T}}(H\otimes K)\rightarrow{\mathcal{T}}(K\otimes H)$ is the
swap determined by $\theta(A\otimes B)=B\otimes A$.
(8) $\dim K\leq\dim H$, there exist $R_{2}\in{\mathcal{P}ur}(K)$ and a linear
map $\phi_{1}:{\mathcal{F}}_{\rm sep}(H\otimes K)\rightarrow{\mathcal{F}}_{\rm
sa}(H)$ such that, for each $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$, $\phi_{1}(P\otimes
Q)=U_{P}QU_{P}^{*}=V_{Q}PV_{Q}^{*}$ for some linear or conjugate linear
isometries $U_{P}:K\rightarrow H$, $V_{Q}:H\rightarrow H$, and
$\Phi(F)=\phi_{1}(F)\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
(9) $\dim H\leq\dim K$, there exist $R_{1}\in{\mathcal{P}ur}(H)$ and a linear
map $\phi_{2}:{\mathcal{F}}_{\rm sep}(H\otimes K)\rightarrow{\mathcal{F}}_{\rm
sa}(K)$ such that, for each $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$, $\phi_{2}(P\otimes
Q)=U_{P}QU_{P}^{*}=V_{Q}PV_{Q}^{*}$ for some linear or conjugate linear
isometries $U_{P}:K\rightarrow K$, $V_{Q}:H\rightarrow K$, and
$\Phi(F)=R_{1}\otimes\phi_{2}(F)$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
We remark that, in cases (6)-(9) of Theorem 3.2, it is possible to have one of
the isometries be linear and the other isometry be conjugate-linear.
Proof of Theorem 3.2. It is clear that if any one of (1)-(9) holds, then
$\Phi$ preserves separable pure states. So we only need to check the converse.
Assume that
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))\subseteq{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$.
Define two maps $\phi_{1}:({\mathcal{T}}_{\rm sa}(H),{\mathcal{T}}_{\rm
sa}(K))\rightarrow{\mathcal{T}}_{\rm sa}(H)$ and $\phi_{2}:({\mathcal{T}}_{\rm
sa}(H),{\mathcal{T}}_{\rm sa}(K))\rightarrow{\mathcal{T}}_{\rm sa}(K)$ by
$None$ $\phi_{1}(A,B)={\rm Tr}_{2}(\Phi(A\otimes B))\quad{\rm
and}\quad\phi_{2}(A,B)={\rm Tr}_{1}(\Phi(A\otimes B)).$
Fix a $Q\in{\mathcal{P}ur}(K)$. Then $\phi_{1}(\cdot,Q):{\mathcal{T}}_{\rm
sa}(H)\rightarrow{\mathcal{T}}_{\rm sa}(H)$ and
$\phi_{2}(\cdot,Q):{\mathcal{T}}_{\rm sa}(H)\rightarrow{\mathcal{T}}_{\rm
sa}(K)$ are both linear. By the assumption, we have
$\Phi(P\otimes Q)={\rm Tr}_{2}(\Phi(P\otimes Q))\otimes{\rm
Tr}_{1}(\Phi(P\otimes Q))=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)$
for all $P\in{\mathcal{P}ur}(H)$ and all $Q\in{\mathcal{P}ur}(K)$. It follows
that
$\phi_{1}({\mathcal{P}ur}(H),Q)\subseteq{\mathcal{P}ur}(H)\quad{\rm
and}\quad\phi_{2}({\mathcal{P}ur}(H),Q)\subseteq{\mathcal{P}ur}(K).$
Thus, applying Theorem 2.2 to $\phi_{1}(\cdot,Q)$ and $\phi_{2}(\cdot,Q)$,
respectively, we get that, either
(i) there is a pure state $R_{iQ}$ such that $\phi_{i}(A,Q)={\rm Tr}(A)R_{iQ}$
for all $A\in{\mathcal{F}}_{\rm sa}(H)$, $i=1,2$, or
(ii) there is a linear or conjugate linear operator $U_{iQ}$ with
$U_{iQ}^{*}U_{iQ}=I_{H}$ such that $\phi_{i}(A,Q)=U_{iQ}AU_{iQ}^{*}$ for all
$A\in{\mathcal{F}}_{\rm sa}(H)$, $i=1,2$.
Similarly, for any fixed $P\in{\mathcal{P}ur}(H)$, considering the maps
$\phi_{1}(P,\cdot):{\mathcal{T}}_{\rm sa}(K)\rightarrow{\mathcal{T}}_{\rm
sa}(H)$ and $\phi_{2}(P,\cdot):{\mathcal{T}}_{\rm
sa}(K)\rightarrow{\mathcal{T}}_{\rm sa}(K)$, we have that, either
(i′) there is a pure state $R_{iP}$ such that $\phi_{i}(P,B)={\rm
Tr}(B)R_{iP}$ for all $B\in{\mathcal{F}}_{\rm sa}(K)$, $i=1,2$, or
(ii′) there is a linear or conjugate linear operator $U_{iP}$ with
$U_{iP}^{*}U_{iP}=I_{K}$ such that $\phi_{i}(P,B)=U_{iP}BU_{iP}^{*}$ for all
$B\in{\mathcal{F}}_{\rm sa}(K)$, $i=1,2$.
Observe that, for $i=1,2$, $\phi_{i}(\cdot,Q)$s and $\phi_{i}(P,\cdot)$s are
continuous on ${\mathcal{F}}_{\rm sa}(H)$ for all $Q\in{\mathcal{P}ur}(K)$ and
on ${\mathcal{F}}_{\rm sa}(K)$ for all $P\in{\mathcal{P}ur}(H)$, respectively.
Now, we consider the map $\phi_{1}(\cdot,Q)$.
Claim 1. Either $\phi_{1}(\cdot,Q)$ has the form (i) for all
$Q\in{\mathcal{P}ur}(K)$ or $\phi_{1}(\cdot,Q)$ has the form (ii) for all
$Q\in{\mathcal{P}ur}(K)$.
Fix $A_{0}=e_{1}\otimes e_{1}-e_{2}\otimes e_{2}\in{\mathcal{F}}_{\rm sa}(H)$
and define a function $F:{\mathcal{P}ur}(K)\rightarrow{\mathbb{R}}$ by
$F(Q)=\|\phi_{1}(A_{0},Q)\|_{\rm Tr}$ for all $Q\in{\mathcal{P}ur}(K)$. Note
that $F(Q)=\|{\rm Tr}(A_{0})R_{1Q}\|_{\rm Tr}=0$ if $\phi_{1}$ has the form
(i) and $F(Q)=\|U_{iQ}A_{0}U_{iQ}^{*}\|_{\rm Tr}=\|A_{0}\|_{\rm Tr}={2}$ if
$\phi_{1}$ has the form (ii).
Take any two distinct $Q_{1},Q_{2}\in{\mathcal{P}ur}(K)$. Then there exist two
linearly independent unit vectors $x,y\in K$ such that $Q_{1}=x\otimes x$ and
$Q_{2}=y\otimes y$. For any $t\in[0,1]$, define
$Q(t)=\frac{1}{\|x+t(y-x)\|^{2}}(x+t(y-x))\otimes(x+t(y-x))\in{\mathcal{P}ur}(K).$
Clearly, $Q(0)=Q_{1}$ and $Q(1)=Q_{2}$. Note that, for each $t\in[0,1]$,
$\phi_{1}(\cdot,Q(t))$ has the form (i) or (ii). Let ${\mathcal{L}}={\rm
span}\\{A_{0}\otimes Q(t):t\in[0,1]\\}$. It is clear that ${\mathcal{L}}$ is a
finite dimensional subspace of ${\mathcal{T}}_{\rm sep}(H\otimes K)$ and hence
$\Phi|_{{\mathcal{L}}}$ is continuous. It follows that
$\phi_{1}(\cdot,Q)|_{{\mathcal{L}}}$ is continuous and hence $t\mapsto
F(Q(t))$ is a continuous map. As $F(Q(t))$ can take only two possible distinct
values, it must be a constant. So Claim 1 holds.
Similarly, we have
Claim 1′. Either $\phi_{2}(\cdot,Q)$ has the form (i) for all
$Q\in{\mathcal{P}ur}(K)$ or $\phi_{2}(\cdot,Q)$ has the form (ii) for all
$Q\in{\mathcal{P}ur}(K)$.
Claim 2. One of the following holds:
(a) For all $Q\in{\mathcal{P}ur}(K)$, both $\phi_{1}(\cdot,Q)$ and
$\phi_{2}(\cdot,Q)$ have the form (i).
(b) For all $Q\in{\mathcal{P}ur}(K)$, $\phi_{1}(\cdot,Q)$ has the form (i) and
$\phi_{2}(\cdot,Q)$ has the form (ii).
(c) For all $Q\in{\mathcal{P}ur}(K)$, $\phi_{1}(\cdot,Q)$ has the form (ii)
and $\phi_{2}(\cdot,Q)$ has the form (i).
We need only to check that, for all $Q\in{\mathcal{P}ur}(K)$,
$\phi_{1}(\cdot,Q)$ and $\phi_{2}(\cdot,Q)$ can not have the form (ii)
simultaneously. Suppose there exists some $Q_{0}\in{\mathcal{P}ur}(K)$ such
that both $\phi_{1}(\cdot,Q_{0})$ and $\phi_{2}(\cdot,Q_{0})$ are of the form
(ii). So there exist isometric or conjugate isometric operators
$U_{1Q_{0}}:H\rightarrow H$ and $U_{2Q_{0}}:H\rightarrow K$ such that
$\phi_{i}(A,Q_{0})=U_{iQ_{0}}AU_{iQ_{0}}^{*}$ for all $A\in{\mathcal{F}}_{\rm
sa}(H)$, $i=1,2$. Thus, we must have $\dim H\leq\dim K$ and
$None$ $\Phi(P\otimes
Q_{0})=\phi_{1}(P,Q_{0})\otimes\phi_{2}(P,Q_{0})=U_{1Q_{0}}PU_{1Q_{0}}^{*}\otimes
U_{2Q_{0}}PU_{2Q_{0}}^{*}=U(P\otimes P)U^{*}$
for all $P\in{\mathcal{P}ur}(H)$, where $U=U_{1Q_{0}}\otimes
U_{2Q_{0}}:H\otimes H\rightarrow H\otimes K$. Particularly, take
$P_{1}=e_{1}\otimes e_{1}$, $P_{2}=e_{2}\otimes e_{2}$,
$P_{3}=\frac{1}{2}(e_{1}\otimes e_{1}+e_{1}\otimes e_{2}+e_{2}\otimes
e_{1}+e_{2}\otimes e_{2})$ and $P_{4}=\frac{1}{2}(e_{1}\otimes
e_{1}-e_{1}\otimes e_{2}-e_{2}\otimes e_{1}+e_{2}\otimes e_{2})$. Then
$P_{1}+P_{2}=P_{3}+P_{4}$ and so $P_{1}\otimes Q_{0}+P_{2}\otimes
Q_{0}=P_{3}\otimes Q_{0}+P_{4}\otimes Q_{0}$. However, by a simple
calculation, $P_{1}\otimes P_{1}+P_{2}\otimes P_{2}\not=P_{3}\otimes
P_{3}+P_{4}\otimes P_{4}$. Note that
$\Phi(P_{1}\otimes Q_{0}+P_{2}\otimes Q_{0})=U(P_{1}\otimes P_{1}+P_{2}\otimes
P_{2})U^{*}$
and
$\Phi(P_{3}\otimes Q_{0}+P_{4}\otimes Q_{0})=U(P_{3}\otimes P_{3}+P_{4}\otimes
P_{4})U^{*}.$
It follows that $\Phi(P_{1}\otimes Q_{0}+P_{2}\otimes
Q_{0})\not=\Phi(P_{3}\otimes Q_{0}+P_{4}\otimes Q_{0})$, a contradiction. So
the claim is true.
Similarly, one can check that
Claim 3. One of the following holds:
(a′) For all $P\in{\mathcal{P}ur}(H)$, both $\phi_{1}(P,\cdot)$ and
$\phi_{2}(P,\cdot)$ have the form (i′).
(b′) For all $P\in{\mathcal{P}ur}(H)$, $\phi_{1}(P,\cdot)$ has the form (i′)
and $\phi_{2}(P,\cdot)$ has the form (ii′).
(c′) For all $P\in{\mathcal{P}ur}(H)$, $\phi_{1}(P,\cdot)$ has the form (ii′)
and $\phi_{2}(P,\cdot)$ has the form (i′).
Claim 4. If (a) and (a′) hold, then there exists $R_{1}\otimes
R_{2}\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$ such that
$\Phi(F)={\rm Tr}(F)R_{1}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. Hence $\Phi$ has the form
(1).
Suppose that (a) and (a′) hold, that is, for all $Q\in{\mathcal{P}ur}(K)$, we
have $\phi_{i}(A,Q)={\rm Tr}(A)R_{iQ}$, and, for all $P\in{\mathcal{P}ur}(H)$,
we have $\phi_{i}(P,B)={\rm Tr}(B)R_{iP}$. Fix $P_{0}\in{\mathcal{P}ur}(H)$
and $Q_{0}\in{\mathcal{P}ur}(K)$. Then we get
$\phi_{i}(P,Q)=\phi_{i}(P,Q_{0})=\phi_{i}(P_{0},Q_{0})=R_{i}.$
Therefore, $\Phi(P\otimes Q)=R_{1}\otimes R_{2}$ for all $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$. By the linearity of $\Phi$,
one sees that Claim 4 is true.
Claim 5. If (a) and (b′) hold, then $\Phi$ has the form (3).
In this case, for any $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$, we have
$\Phi(P\otimes Q)=R_{1Q}\otimes R_{2Q}=R_{1P}\otimes U_{2P}QU_{2P}^{*},$
which implies that $R_{1Q}=R_{1P}$ is independent of $P,Q$ and
$U_{2P}QU_{2P}^{*}=R_{2Q}$ is independent of $P$. So there exist
$R_{1}\in{\mathcal{P}ur}(H)$ and a linear or conjugate linear isometry $U_{2}$
such that $\Phi(P\otimes Q)=R_{1}\otimes U_{2}QU_{2}^{*}=R_{1}\otimes
U_{2}[{\rm Tr}_{1}(P\otimes Q)]U_{2}^{*}$ for all separable pure states
$P\otimes Q$. Now by the linearity of $\Phi$, the claim is true.
Similarly, one can show the following Claims 6-8.
Claim 6. If (a) and (c′) hold, then (4) holds, that is, there exist
$R_{2}\in{\mathcal{P}ur}(K)$ and a linear or conjugate linear isometry
$U_{1}:K\rightarrow H$ such that
$\Phi(F)=U_{1}[{\rm Tr}_{1}(F)]U_{1}^{*}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. In this case we must have
$\dim H\geq\dim K$.
Claim 7. If (b) and (a′) hold, then $\Phi$ has the form (5), that is, there
exist $R_{1}\in{\mathcal{P}ur}(H)$ and a linear or conjugate linear isometry
$U_{2}:H\rightarrow K$ such that
$\Phi(F)=R_{1}\otimes U_{2}[{\rm Tr}_{2}(F)]U_{2}^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. In this case $\dim
H\leq\dim K$.
Claim 8. If (c) and (a′) hold, then there exist $R_{2}\in{\mathcal{P}ur}(K)$
and a linear or conjugate linear isometry $U_{1}:H\rightarrow H$ such that
$\Phi(F)=U_{1}[{\rm Tr}_{2}(F)]U_{1}^{*}\otimes R_{2}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. Hence $\Phi$ takes the form
(2).
Claim 9. If (b) and (c′) hold, then $\Phi$ has the form (7).
Now suppose that (b) and (c′) hold. Then
$\phi_{2}(\cdot,Q_{0})=U_{2Q_{0}}(\cdot)U_{2Q_{0}}^{*}\quad{\rm
and}\quad\phi_{1}(P_{0},\cdot)=U_{1P_{0}}(\cdot)U_{1P_{0}}^{*}$
with $U_{2Q_{0}}^{*}U_{2Q_{0}}=I_{H}$ and $U_{1P_{0}}^{*}U_{1P_{0}}=I_{K}$.
Moreover,
$\phi_{1}(P_{0},Q)={\rm Tr}(P_{0})R_{1Q}={\rm Tr}(P)R_{1Q}=\phi_{1}(P,Q)$
and
$\phi_{2}(P,Q_{0})={\rm Tr}(Q_{0})R_{2P}={\rm Tr}(Q)R_{2P}=\phi_{2}(P,Q).$
Thus, we obtain
$\Phi(P\otimes
Q)=\phi_{1}(P,Q)\otimes\phi_{2}(P,Q)=\phi_{1}(P_{0},Q)\otimes\phi_{2}(P,Q_{0})=U_{1P_{0}}QU_{1P_{0}}^{*}\otimes
U_{2Q_{0}}PU_{2Q_{0}}^{*}$
for all $P\in{\mathcal{P}ur}(H)$ and $Q\in{\mathcal{P}ur}(K)$. Let
$U_{1}=U_{1P_{0}}$ and $U_{2}=U_{2Q_{0}}$. Then $\Phi(P\otimes
Q)=(U_{1}\otimes U_{2})(Q\otimes P)(U_{1}\otimes U_{2})^{*}=(U_{1}\otimes
U_{2})\theta(P\otimes Q)(U_{1}\otimes U_{2})^{*}$ for all separable pure
states $P\otimes Q$. Obviously, $\dim H=\dim K$ in this case. It follows from
the linearity of $\Phi$ that the claim is true.
Similarly, we have
Claim 10. If (c) and (b′) hold, then there exist linear or conjugate linear
isometries $U_{1}:H\rightarrow H$ and $U_{2}:K\rightarrow K$ such that
$\Phi(F)=(U_{1}\otimes U_{2})F(U_{1}\otimes U_{2})^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. Hence in this case we have
(6).
Claim 11. If (b) and (b′) hold, then $\Phi$ has the form (9).
Assume (b) and (b′) hold synchronously. Then for any $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$, we have $\Phi(P\otimes
Q)=R_{1Q}\otimes U_{2Q}PU_{2Q}^{*}=R_{1P}\otimes U_{2P}QU_{2P}^{*}$. It
follows that there exists $R_{1}\in{\mathcal{P}ur}(H)$ such that
$R_{1Q}=R_{1P}=R_{1}$ and $U_{2Q}PU_{2Q}^{*}=U_{2P}QU_{2P}^{*}$ for all $P,Q$.
Thus there exists a linear map $\phi_{2}:{\mathcal{F}}_{\rm sep}(H\otimes
K)\rightarrow{\mathcal{F}}_{\rm sa}(K)$ such that, for each $P\otimes
Q\in{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$, $\phi_{2}(P\otimes
Q)=U_{P}QU_{P}^{*}=V_{Q}PV_{Q}^{*}$ for some linear or conjugate linear
isometries $U_{P}:K\rightarrow K$, $V_{Q}:H\rightarrow K$, and
$\Phi(F)=R_{1}\otimes\phi_{2}(F)$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$. In this case $\dim
H\leq\dim K$. So the claim is true.
Similarly,
Claim 12. If (c) and (c′) hold, then $\Phi$ takes the form (8).
The proof of the theorem is complete. $\Box$
The cases (8) and (9) of Theorem 3.2 seem not as natural as the other forms.
We do not know whether or not they may really occur. It raises another
interesting question of characterizing the real linear maps from
${\mathcal{F}}_{\rm sep}(H\otimes K)$ into ${\mathcal{F}}_{\rm sa}(H)$ that
send separable pure states to pure states.
Corollary 3.3. Let $H$ and $K$ be two Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{T}}_{\rm sa}(H\otimes K)\rightarrow{\mathcal{T}}_{\rm
sa}(H\otimes K)$ is a linear map satisfying
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))\subseteq{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
and $\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))$ contains two elements
$P^{\prime}_{i}\otimes Q^{\prime}_{i}$, $i=1,2$, with
$\\{P_{1}^{\prime},P_{2}^{\prime}\\}$ and
$\\{Q_{1}^{\prime},Q_{2}^{\prime}\\}$ linearly independent sets. Then $\Phi$
has the form (6) or (7) in Theorem 3.2.
In the case of finite dimension, we get a generalization of the main result
obtained in [2]; there the condition
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))={\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
is assumed.
Corollary 3.4. Let $H$ and $K$ be two finite-dimensional Hilbert spaces.
Suppose that $\Phi:{\mathcal{T}}_{\rm sa}(H\otimes
K)\rightarrow{\mathcal{T}}_{\rm sa}(H\otimes K)$ is a linear map. Then
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))\subseteq{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
if and only if one of the statements (1)-(9) holds for all
$A\in{\mathcal{T}}_{\rm sep}(H\otimes K)$.
Let $\\{e_{i}\\}_{i=1}^{\dim H}$ and $\\{u_{j}\\}_{j=1}^{\dim K}$ be
orthonormal bases of $H$ and $K$ respectively. With respect to the product
basis $\\{e_{i}\otimes u_{j}\\}_{i,j}$ of $H\otimes K$, the linear map ${\bf
T}\otimes{\rm id}:{\mathcal{T}}(H\otimes K)\rightarrow{\mathcal{T}}(H\otimes
K)$ determined by $A\otimes B\mapsto A^{T}\otimes B$ is called the partial
transpose of the first system. The partial transpose of the second system
${\rm id}\otimes{\bf T}$ is defined similarly. In terms of partial
transposition, one can restate Theorem 3.2 to avoid the term “conjugate
linear”. In fact, if $U$ is a conjugate isometry, then there exists an
isometry $V$ such that $UAU^{*}=VA^{\rm t}V^{*}$ for all $A$.
In the finite dimensional case, for a linear map $\Phi$, surjectivity and
separable pure state preserving is equivalent to preserving separable pure
states in both directions, and in turn, is equivalent to
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))={\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$.
But for the infinite dimensional case, bijectivity and separable pure state
preserving might not imply that $\Phi$ preserves separable pure states in both
directions.
The next result is a characterization of linear maps preserving separable pure
states in both directions. We state it avoiding the term “conjugate linear”.
Corollary 3.5. Let $H$ and $K$ be two Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{T}}_{\rm sa}(H\otimes K)\rightarrow{\mathcal{T}}_{\rm
sa}(H\otimes K)$ is a linear map. Then the following conditions are
equivalent.
(1) $\Phi$ preserves separable pure states in both directions.
(2)
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))={\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$.
(3) Either
(i) There exist unitary operators $U_{1}\in{\mathcal{B}}(H)$ and
$U_{2}\in{\mathcal{B}}(K)$ such that
$\Phi(F)=(U_{1}\otimes U_{2})\Lambda(F)(U_{1}\otimes U_{2})^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$; or
(ii) $\dim H=\dim K$, there exist unitary operators
$U_{1}\in{\mathcal{B}}(K,H)$ and $U_{2}\in{\mathcal{B}}(H,K)$ such that
$\Phi(F)=(U_{1}\otimes U_{2})\Lambda(\theta(F))(U_{1}\otimes U_{2})^{*}$
for all $F\in{\mathcal{F}}_{\rm sep}(H\otimes K)$.
Here $\Lambda$ is one of the identity map, the transpose, a partial transpose
with respect to any fixed product orthonormal basis of $H\otimes K$, and
$\theta:{\mathcal{T}}(H\otimes K)\rightarrow{\mathcal{T}}(K\otimes H)$ is the
swap determined by $\theta(A\otimes B)=B\otimes A$.
Proof. (3)$\Rightarrow$(1)$\Rightarrow$(2) are obvious. For
(2)$\Rightarrow$(3), by Corollary 3.3, we see that
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))={\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
implies that the statement (6) or (7) in Theorem 3.2 holds. Moreover, the
linear or conjugate linear isometries involved are all surjective and hence
unitary or conjugate unitary. $\Box$
Remark 3.6. Note that, by the proof of Theorem 3.2, the assumption of
$\Phi:{\mathcal{T}}_{\rm sa}(H\otimes K)\rightarrow{\mathcal{T}}_{\rm
sa}(H\otimes K)$ may be replaced by the assumption of $\Phi:{\mathcal{T}}_{\rm
sep}(H\otimes K)\rightarrow{\mathcal{T}}_{\rm sep}(H\otimes K)$, and the
results of Theorem 3.2, Corollaries 3.3-3.5 remain true. This remark is useful
in some applications.
As an application of Theorem 3.2, let us consider the separable pure state
preserving maps between states of bipartite systems.
Theorem 3.7. Let $H$ and $K$ be two Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{S}}(H\otimes K)\rightarrow{\mathcal{S}}(H\otimes K)$ is
an affine map. Then the following statements are equivalent.
(1)
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))\subseteq{\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$
.
(2) The conditions (1)-(9) in Theorem 3.2 hold for all
$\rho\in{\mathcal{S}}_{\rm sep}(H\otimes K)$.
Proof. (2)$\Rightarrow$(1) is obvious.
(1)$\Rightarrow$(2). As $\Phi$ is affine, it can be extended to a real linear
map from ${\mathcal{T}}_{\rm sa}(H\otimes K)$ into ${\mathcal{T}}_{\rm
sa}(H\otimes K)$ which still sends separable pure states to separable pure
states. Furthermore, the fact $\Phi({\mathcal{S}}(H\otimes
K))\subseteq{\mathcal{S}}(H\otimes K)$ implies that $\Phi$ is continuous, and
then, applying Theorem 3.2, one sees that (2) holds. $\Box$
Corollary 3.8. Let $H$ and $K$ be two Hilbert spaces of any dimension. Suppose
that $\Phi:{\mathcal{S}}(H\otimes K)\rightarrow{\mathcal{S}}(H\otimes K)$ is
an affine map. Then the following statements are equivalent.
(1) $\Phi$ preserves separable states in both directions.
(2) $\Phi$ preserves separable pure states in both directions.
(3) Either
(i) There exist unitary operators $U_{1}\in{\mathcal{B}}(H)$ and
$U_{2}\in{\mathcal{B}}(K)$ such that
$\Phi(\rho)=(U_{1}\otimes U_{2})\Lambda(\rho)(U_{1}\otimes U_{2})^{*}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H\otimes K)$; or
(ii) $\dim H=\dim K$, there exist unitary operators
$U_{1}\in{\mathcal{B}}(K,H)$ and $U_{2}\in{\mathcal{B}}(H,K)$ such that
$\Phi(\rho)=(U_{1}\otimes U_{2})\Lambda(\theta(\rho))(U_{1}\otimes U_{2})^{*}$
for all $\rho\in{\mathcal{S}}_{\rm sep}(H\otimes K)$.
Here $\Lambda$ is one of the identity map, the transpose, a partial transpose
with respect to any fixed product orthonormal basis of $H\otimes K$, and
$\theta:{\mathcal{T}}(H\otimes K)\rightarrow{\mathcal{T}}(K\otimes H)$ is the
swap determined by $\theta(A\otimes B)=B\otimes A$.
Proof. (3)$\Rightarrow$(1) is obvious.
(1)$\Rightarrow$(2): $\Phi$ preserves separable states in both directions
implies that $\Phi|_{{\mathcal{S}}_{\rm sep}(H\otimes K)}:{\mathcal{S}}_{\rm
sep}(H\otimes K)\rightarrow{\mathcal{S}}_{\rm sep}(H\otimes K)$ is a bijective
affine map. Note that an affine isomorphism between two convex sets preserves
extreme points in both directions. Hence $\Phi$ preserves separable pure
states in both directions and (2) is true.
(2)$\Rightarrow$(3): Assume that (2) holds; then
$\Phi({\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K))={\mathcal{P}ur}(H)\otimes{\mathcal{P}ur}(K)$.
By Theorem 3.7 and Corollary 3.5, we see that either (i) or (ii) holds for
$\Phi$ and all separable states $\rho$, that is, (3) is true. $\Box$
## 4\. linear maps preserving separable pure states: multipartite systems
Results similar to that in Section 3 for bipartite cases are valid for
multipartite cases also, but with more complicated expressions. The techniques
of the proofs are almost identical to those used in the preceding part of the
paper and in [2]. In this section we only list those results which have
relatively simple expressions and which may have more applications. The
meanings of the notations used here are also similar to that in Section 2. For
example, ${\mathcal{S}}_{\rm sep}(H_{1}\otimes H_{2}\otimes\cdots\otimes
H_{n})$ denotes the closed convex hull of all fully separable states in
${\mathcal{S}}(H_{1}\otimes H_{2}\cdots\otimes H_{n})$.
The following result corresponds to Corollary 3.3.
Theorem 4.1. Let $H_{1},H_{2},\ldots,H_{n}$ be complex Hilbert spaces of any
dimensions and let $\Phi:{\mathcal{T}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{T}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})$ be a linear map. Then
$\Phi({\mathcal{P}ur}(H_{1})\otimes\cdots\otimes{\mathcal{P}ur}(H_{n}))\subseteq{\mathcal{P}ur}(H_{1})\otimes\cdots\otimes{\mathcal{P}ur}(H_{n})$
and there are $P_{1}^{\prime}\otimes\cdots\otimes
P_{n}^{\prime}\in\Phi({\mathcal{P}ur}(H_{1})\otimes\cdots\otimes{\mathcal{P}ur}(H_{n}))$
and $Q_{1}^{\prime}\otimes\cdots\otimes
Q_{n}^{\prime}\in\Phi({\mathcal{P}ur}(H_{1})\otimes\cdots\otimes{\mathcal{P}ur}(H_{n}))$
with $\\{P_{i}^{\prime},Q_{i}^{\prime}\\}$ linearly independent for each
$i=1,2,\ldots,n$, if and only if there is a permutation
$\pi:(1,\ldots,n)\mapsto(p_{1},\ldots,p_{n})$ of $(1,\ldots,n)$ and linear or
conjugate linear isometries $U_{j}:H_{p_{j}}\rightarrow H_{j}$,
$j=1,\ldots,n$, such that
$None$ $\Phi(F)=(U_{1}\otimes\cdots\otimes
U_{n})\theta_{\pi}(F)(U_{1}^{*}\otimes\cdots\otimes U_{n}^{*})$
holds for all $F\in{\mathcal{F}}_{\rm sep}(H_{1}\otimes\cdots\otimes H_{n})$.
Here $\theta_{\pi}:{\mathcal{T}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{T}}_{\rm
sa}(H_{p_{1}}\otimes H_{p_{2}}\otimes\cdots\otimes H_{p_{n}})$ is a linear map
determined by $\theta_{\pi}(A_{1}\otimes A_{2}\otimes\cdots\otimes
A_{n})=A_{p_{1}}\otimes A_{p_{2}}\otimes\cdots\otimes A_{p_{n}}$, and each
$U_{j}$ can be independently linear or conjugate-linear.
It is clear that, if $\Phi$ has the form Eq.(4.1), then $\dim
H_{p_{j}}\leq\dim H_{j}$. So, if $\dim H_{p_{j}}>\dim H_{j}$ for some $j$,
then $\Phi$ cannot take the form (4.1) for the permutation
$\pi:(1,\ldots,n)\mapsto(p_{1},\ldots,p_{n})$. Actually, $\dim H_{j}$ is
constant for all indices $j$ in a cycle of the permutation $\pi$.
The following is a special case corresponding to Theorem 3.7.
Theorem 4.2. A linear map $\Phi:{\mathcal{T}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})\rightarrow{\mathcal{T}}_{\rm sa}(H_{1}\otimes
H_{2}\otimes\cdots\otimes H_{n})$ satisfies the conditions of Theorem 4.1 and
the condition $\Phi({\mathcal{S}}(H_{1}\otimes\cdots\otimes
H_{n}))\subseteq{\mathcal{S}}(H_{1}\otimes\cdots\otimes H_{n})$ if and only if
Eq.(4.1) holds for all $\rho\in{\mathcal{T}}_{\rm
sep}(H_{1}\otimes\cdots\otimes H_{n})$.
Particularly,
Theorem 4.3. Let $H_{1},H_{2},\ldots,H_{n}$ be complex Hilbert spaces of any
dimensions and let $\Phi:{\mathcal{S}}(H_{1}\otimes H_{2}\otimes\cdots\otimes
H_{n})\rightarrow{\mathcal{S}}(H_{1}\otimes H_{2}\otimes\cdots\otimes H_{n})$
be an affine map. Then the following statements are equivalent.
(1) $\Phi$ preserves fully separable states in both directions.
(2) $\Phi$ preserves separable pure states in both directions.
(3) There is a permutation $\pi:(1,\ldots,n)\mapsto(p_{1},\ldots,p_{n})$ of
$(1,\ldots,n)$ and unitary or conjugate unitary operators
$U_{j}:H_{p_{j}}\rightarrow H_{j}$, $j=1,\ldots,n$, such that
$None$ $\Phi(\rho)=(U_{1}\otimes\cdots\otimes
U_{n})\theta_{\pi}(\rho)(U_{1}^{*}\otimes\cdots\otimes U_{n}^{*})$
holds for all $\rho\in{\mathcal{S}}_{\rm sep}(H_{1}\otimes\cdots\otimes
H_{n})$. Here, each $U_{j}$ can be independently linear or conjugate-linear.
Obviously, if $\Phi$ is of the form (4.2), then $\dim H_{p_{j}}=\dim H_{j}$.
Theorems 4.1-4.3 may be restated in terms of linear isometric (unitary)
operators and partial transpositions as in Corollaries 3.5 and 3.8, but with
more complicated expressions.
Acknowledgement. The authors wish to give their thanks to the referees. They
read the original manuscript carefully, made up some gaps and gave many
helpful suggestions to improve the paper.
## References
* [1] I. Bengtsson, K. Zyczkowski, Geometry of Quantum States, An introduction to quantum entangument, Cambridge University Press, Cambridge, 2006.
* [2] S. Friedland, C. K. Li, T. Y. Poon, N. S. Sze, The automorphism group of separable states in quantum information theory, J. Math. Phys., 52 (2011), 042203.
* [3] M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A, 223 (1996), 1.
* [4] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Mod. Phys., 81 (2009), 865.
* [5] M. H. Lim, Additive mappings between Hermitian matrix spaces preserving rank not exceeding one, Lin. Alg. Appl., 408 (2005), 259-267.
* [6] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.
* [7] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A, 40 (1989), 4277.
|
arxiv-papers
| 2012-10-08T06:51:21 |
2024-09-04T02:49:36.135995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinchuan Hou and Xiaofei Qi",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1210.2155"
}
|
1210.2223
|
# Observer dependent entanglement
Paul M. Alsing Air Force Research Laboratory, Information Directorate, Rome,
N.Y., USA Ivette Fuentes School of Mathematical Sciences, University of
Nottingham, Nottingham NG7 2RD United Kingdom
###### Abstract
Understanding the observer-dependent nature of quantum entanglement has been a
central question in relativistic quantum information. In this paper we will
review key results on relativistic entanglement in flat and curved spacetime
and discuss recent work which shows that motion and gravity have observable
effects on entanglement between localized systems.
## 1 Introduction
In quantum information non-classical properties such as entanglement are
exploited to improve information tasks. A prototypical example of this is
quantum teleportation where two observers Alice and Bob use two quantum
systems in an entangled state to transmit information about the state of a
third system. Impressively, cutting-edge experiments involving entanglement
based communications are reaching regimes where relativistic effects can no
longer be neglected. Such is the case of protocols which involve distributing
entanglement over hundreds of kilometers [1, 2]. Understanding entanglement in
relativistic settings has been a key question in relativistic quantum
information. Early results show that entanglement is observer-dependent [3, 4,
5, 6]. The entanglement between two field modes is degraded by the Unruh
effect when observers are in uniform acceleration. We also learned that the
spatial degrees of freedom of global fields are entangled, including the
vacuum state [7, 8, 9, 10]. This entanglement can be extracted by point-like
systems and in principal be used for quantum information processing (see for
example [11, 12, 13, 14]). Most of the early studies on relativistic
entanglement in non-inertial frames involved global modes. However, more
recently, researchers in the field have focused their attention on
understanding entanglement between fields or systems which are localized in
space and time. The motivation for this is that entangled localized systems
can be in principle measured, transformed and exploited for quantum
information tasks. Among the most popular systems considered for this purpose
are moving cavities [15, 16, 17, 18, 19, 20, 21], point-like detectors [11,
22, 23, 24, 25] and localized wave-packets [26, 27, 28]. In this paper we will
review global mode entanglement in flat and curved spacetime which constitutes
the first step in the study of entanglement in quantum field theory. We will
then discuss more recent ideas on entanglement which show that motion and
gravity have observable effects on quantum correlations between localized
systems [16, 17, 18, 19, 20, 21]. Interestingly, in these settings it is
possible to generate quantum gates through motion in spacetime [19, 21, 29].
The observer-dependent nature of entanglement is a consequence of the particle
content being different for different observers in quantum field theory [30,
31]. In flat spacetime, all inertial observers agree on particle number and
therefore, on entanglement. Entanglement is well defined in that case since
inertial observers play a special role. However, in the case of curved
spacetime, the entanglement in a given state varies even for inertial
observers (see discussion in [31]).
In special relativity one also finds that quantum correlations are observer-
dependent. The entanglement between two spin particles is invariant only when
the spin and momentum of the particles are considered to be a single
subsystem. If only spin degrees of freedom are considered, different inertial
observers would disagree on the amount of the entanglement between the
particles. Some works show that spin entanglement in transformed into momentum
entanglement under Lorentz transformations while some recent papers argue that
considering spin degrees of freedom alone (by tracing over momentum) lead to
inconsistencies (this will be discussed further in section 6).
The paper is organized as follows: in the section (2) we will introduce
technical tools in quantum field theory and quantum information. We will
review the basics of field quantization focusing on the free bosonic massless
case. We will describe the interaction of the field with point-like systems
better known as Unruh-DeWitt detectors. By imposing boundary conditions we
will describe fields contained within moving mirrors (cavities) and show how
to construct wave-packets that are localized in space and time. A brief
discussion on fields in curved spacetimes will be presented. We will end the
section by reviewing measures of entanglement in the pure and mixed case as
well as introduce the covariant matrix formalisms which allows for relatively
simple entanglement computations in quantum field theory. In section (3) of
this paper we will review the results on free mode entanglement in non-
inertial frames, in an expanding universe and in a black hole spacetime. We
will present ideas on how to extract field entanglement using Unruh-DeWitt
detectors in section (4). In section (5) we will present a more modern view on
the study of entanglement in quantum field theory where the entanglement
between the modes of moving cavities is analyzed and review recent work on how
localized wave-packets can be used to implement quantum information protocols.
For completeness, in section (6) we review the concept of observer dependent
entanglement for the case of zero acceleration. Here we discuss the Wigner
rotation, the change in state under Lorentz transformations and their effect
on entanglement for spin $\frac{1}{2}$ particles and photons. Finally, in
section (7) we will point out open questions, discuss work in progress and
future directions in the understanding of entanglement in quantum field
theory.
## 2 Technical tools
### 2.1 Quantum field theory
The theoretical framework in which questions of relativistic entanglement are
analyzed is quantum field theory in flat and curved spacetime. In the absence
of a consistent quantum theory of gravity, quantum field theory allows the
exploration of some aspects of the overlap of relativity and quantum theory by
considering quantum fields on a classical spacetime. The most important lesson
we have learned from quantum field theory is that fields are fundamental,
while particles are derived notions (if at all possible) [30]. Field
quantization is inequivalent for different observers and therefore, the
particle content of the field may vary for different observers. For example,
the Minkowski vacuum seen by inertial observers in flat spacetime corresponds
to a state populated with a thermal distribution of particles for observers in
uniform acceleration [32]. The temperature, known as the Unruh temperature, is
a function of the observer’s acceleration. As we will see, a consequence of
this is that the entanglement of free field modes in flat spacetime is
observer-dependent [3, 4], and effects quantum information processing tasks
such a teleporation [5, 33, 11]. Another interesting example is that of an
expanding universe [30]. The vacuum state for observers in the asymptotic past
is populated by particles as seen by observers in the future infinity [30].
The expansion of the universe creates particles and these particles are
entangled [31, 34]. It might at first sight seam surprising that the dynamical
Casimir effect is closely related to the Unruh effect [35, 36]. Both effects
are predictions of quantum field theory. The vacuum state of an inertial
cavity defined by inertial observers is inequivalent to the vacuum state of
the cavity undergoing uniform acceleration as seen by observers moving along
with the cavity (Rindler observers)[15, 16, 17, 18, 19, 20, 21]. Therefore, if
a cavity is at rest and the field is in the vacuum state, entangled particles
will be created when the cavity subsequently undergoes non-uniform accelerated
motion [16, 18]. Related to this effect is the dynamical Casimir effect where
the mirrors of the cavity oscillate [36, 37]. Before we discuss in more detail
the entanglement between the modes of a quantum field in these and other
scenarios we will revisit basic concepts in quantum field theory, considering
the simplest case: the massless uncharged bosonic field (which we denote
$\phi$ ) in a flat (1+1)-dimensional spacetime. Throughout our paper we will
work in natural units $c=\hbar=1$ and the signature of the metric $(+,-)$.
#### 2.1.1 Global fields
The massless real bosonic quantum field obeys the Klein-Gordon equation
$\square\phi=0$, where the d’Alambertian operator $\square$ is defined as
$\square\phi:=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi),$
(1)
where $g=det(g_{ab})$ and $\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$.
In flat $(1+1)$-dimensional spacetime the metric is
$g_{\mu\nu}=\eta_{\mu\nu}=\\{+-\\}$ and thus,
$\square\phi=\partial^{2}_{t}-\partial^{2}_{x}$. Minkowski coordinates $(t,x)$
are a convenient choice for inertial observers. The solutions to the equation
are plane waves
$u_{\omega,M}(t,x)=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega(t-\epsilon x)},$ (2)
where the label $M$ stands for Minkowski and $\epsilon$ takes the value $+1$
for modes with positive momentum (right movers) and $-1$ for modes with
negative momentum (left movers). The modes of frequency $\omega>0$ are
orthonormal with respect to the Lorentz invariant inner product
$(\phi,\psi)=-i\int_{\Sigma}(\psi^{*}\partial_{\mu}\phi-(\partial_{\mu}\psi^{*})\phi)d\Sigma^{\mu},$
(3)
where $\Sigma$ is a spacelike hypersurface. These solutions are known as
global field modes.
To quantize the field the notion of a time-like Killing vector field is
required. A Killing vector field $K^{\mu}$ is the tangent field to a flow
induced by a transformation which leaves the metric invariant. This means that
the Lie derivative of the metric tensor defined by
$\mathcal{L}_{K}g_{\mu\nu}=K^{\lambda}\partial_{\lambda}g_{\mu\nu}+g_{\mu\lambda}\partial_{\nu}K^{\lambda}+g_{\nu\lambda}\partial_{\mu}K^{\lambda},$
must vanish. When a spacetime admits such a structure it is possible to find a
special basis for the solutions of $\square\phi=0$ such that
$\displaystyle\mathcal{L}_{K}u_{k,M}$ $\displaystyle=$ $\displaystyle
K^{\mu}\partial_{\mu}u_{k,M}=-i\omega u_{k,M},$
where we have considered the action of a Lie derivative on a function. Vectors
lying within the light cone at each point are called time-like. Therefore, if
$K^{\mu}$ is a timelike Minkowski vector field, the Lie derivative corresponds
to $\partial_{t}$. By the action of the Lie derivative on the solutions of the
Klein-Gordon equation we can identify the parameter $\omega>0$ with a
frequency, and classify the plane waves such that $u_{k,M}$ are positive
frequency solutions and $u_{k,M}^{*}$ are negative frequency solutions. A few
words about the physical significance of the existence of a Killing vector
field are in order. If a spacetime has as Killing vector $K^{\mu}$, one can
always find a coordinate system in which the metric is independent of one of
the coordinates and the quantity $E=p_{\mu}K^{\mu}$ is constant along a
geodesic with tangent vector $p^{\mu}$ [38]. The quantity $E$ can be
considered as the conserved energy of a photon with 4-momentum $p^{\mu}$. For
static observers, i.e. those whose 4-velocity $U^{\mu}=dx^{\mu}/d\tau$ is
proportional to the timelike Killing vector $K^{\mu}$ as
$K^{\mu}=V(x)U^{\mu}$, one defines the “redshift” factor
$V=(K^{\mu}K_{\mu})^{1/2}$ as the norm of the Killing vector (since
$U^{\mu}U_{\mu}=1$). The frequency $\omega$ of the photon measured by a static
observer with 4-velocity $U^{\mu}$ is given by $\omega=p_{\mu}U^{\mu}$, and
hence $\omega=E/V$. A photon emitted by a static observer $1$ will be observed
by a static observer $2$ to have frequency $\omega_{2}=\omega_{1}V_{1}/V_{2}$.
Note that along the orbit of the Killing vector $K^{\mu}$ (not necessarily a
geodesic), $V$ is constant. For a general $1+1$ spacetime with coordinates
$x=(x^{0},x^{1})$, a photon $p^{\mu}=(\omega_{0},\pm k(x))$ of frequency
$\omega_{0}>0$ and wavevector of magnitude
$k(x)=\omega_{0}\,\sqrt{-g_{00}(x)/g_{1}(x)}$ (such that
$g_{\mu\nu}p^{\mu}p^{\nu}=0$) will be measured to have frequency
$\omega_{K}(x)=\omega_{0}\,\sqrt{g_{00}(x)}\,\sqrt{(1\pm\alpha)/(1\mp\alpha)}$
with $x$-dependent Doppler factor
$\alpha=\sqrt{-g_{00}(x)/g_{1}(x)}\,\big{(}K^{1}(x)/K^{0}(x)\big{)}$ by a
static observer along the orbit of the Killing vector
$K=K^{0}(x)\partial_{x^{0}}+K^{1}(x)\partial_{x^{1}}$. In particular, in flat
Minkowski spacetime with metric $g_{\mu\nu}=(+,-)$ in $(t,x)$ coordinates a
photon of frequency $\omega_{0}$ emitted by an inertial Minkowski observer
will be measured to have the frequency
$\omega_{K}(x)=\omega\,\,\sqrt{(1\pm\alpha)/(1\mp\alpha)}$ with Doppler factor
$\alpha=K^{1}(x)/K^{0}(x)$.
If the metric is static ($\partial_{0}g_{\mu\nu}=0$ and $g_{0\nu}=0$) then the
metric components are independent of the time coordinates $t$ and the Klein-
Gordon equation can be separated into space and time components as
$f_{\omega}(t,\vec{x})=e^{-i\omega t}\bar{f}_{\omega}(\vec{x})$ (here $(t,x)$
are general $1+1$ coordinates). The modes $(f_{\omega},f^{*}_{\omega})$ form a
basis of the wave equation from which to define the notion of particles. By
definition, a detector measures the proper time $\tau$ along its trajectory.
If the detectors’s trajectory follows the orbit of the Killing field (i.e. the
static observers defined above) the proper time will be proportional to the
Killing time $t$. Modes that are positive frequency with respect to this
Killing vector serve as a natural basis for describing the Fock space of
particles [38]. Most importantly, under Lorentz transformations, timelike
vectors are transformed into timelike vectors, so that the separation of modes
into positive and negative frequencies remains invariant under boosts. In a
general curved spacetime, the non-existence of a Killing field implies that
the separation of modes into positive and negative frequencies is different
along each point of the detectors’s trajectory, and hence the concept of
“particle” is lost (for further details, see [30] and Chap. 9 of [38]). Note
that the photons of measured frequency $\omega_{K}(x)$ in the previous
paragraph are not pure plane waves along the Killing orbit, and therefore must
be decomposed into the natural positive and negative frequency modes
$(f_{\omega},f^{*}_{\omega})$.
Having identified positive and negative modes, the quantized field satisfying
$\square\hat{\phi}=0$ is then given by the following operator value function
$\hat{\phi}=\int(u_{k,M}a_{k,M}+u_{k,M}^{*}a_{k,M}^{\dagger})dk,$
where the creation and annihilation Minkowski operators $a_{k,M}^{\dagger}$
and $a_{k,M}$ satisfy the commutation relations
$[a^{\dagger}_{k,M},a_{k^{\prime},M}]=\delta_{k,k^{\prime}}$. Note that the
solutions have been treated differently by associating creation and
annihilation operators with negative and positive frequency modes,
respectively. The vacuum state is defined by the equation
$a_{k,M}{|0\rangle}^{\mathcal{M}}=0$ and can be written as
${|0\rangle}^{\mathcal{M}}=\prod_{k}{|0_{k}\rangle}^{\mathcal{M}}$ where
${|0_{k}\rangle}^{\mathcal{M}}$ is the ground state of mode $k$. Particle
states are constructed by the action of creation operators on the vacuum state
$|n_{1},...,n_{k}\rangle^{\mathcal{M}}=(n_{1}!,...,n_{k}!)^{-1/2}\,a_{1,M}^{\dagger
n_{1}}...a_{k,M}^{\dagger n_{k}}|0\rangle^{\mathcal{M}}.$
Only when there exists a time-like Killing vector field it is meaningful to
define particles. Observers flowing along timelike Killing vector fields are
those who can properly describe particle states. This has important
consequences to relativistic quantum information since the notion of particles
(and therefore, subsystems) are indispensable to store information and thus,
to define entanglement. However, in the most general case, curved spacetimes
do not admit time-like Killing vector fields.
Interestingly, in the case where the spacetime admits a global timelike
Killing vector field, the vector field is not necessarily unique. Consider two
time-like Killing vector fields $\partial_{T}$ and $\partial_{\hat{T}}$. It is
then possible to find in each case a basis for the solutions to the Klein-
Gordon equation $\\{{u}_{k},{u}_{k}^{*}\\}$ and
$\\{\bar{u}_{k},\bar{u}_{k}^{*}\\}$ such that classification into positive and
frequency solutions is possible with respect to $\partial_{T}$ and
$\partial_{\hat{T}}$ respectively. The field is equivalently quantized in both
bases, therefore
$\hat{\phi}=\int(u_{k}a_{k}+u_{k}^{*}a_{k}^{\dagger})dk=\int(\bar{u}_{k^{\prime}}\bar{a}_{k^{\prime}}+\bar{u}_{k^{\prime}}^{*}\bar{a}_{k^{\prime}}^{\dagger})dk^{\prime}.$
Using the inner product, one obtains a transformation between the mode
solutions and correspondingly, between the creation and annihilation
operators,
$a_{k}=\sum_{k^{\prime}}(\alpha^{\ast}_{kk^{\prime}}\bar{a}_{k^{\prime}}-\beta^{\ast}_{kk^{\prime}}\bar{a}_{k^{\prime}}^{\dagger}),$
where $\alpha_{kk^{\prime}}=(u_{k},\bar{u}_{k^{\prime}})$ and
$\beta_{kk^{\prime}}=-(u_{k},\bar{u}^{\ast}_{k^{\prime}})$ are called
Bogoliubov coefficients. Since the vacua are given by
${a}_{k}{|0\rangle}={\bar{a}}_{k}{\bar{|0\rangle}}=0$ it is possible to find a
transformation between the states in the two bases. We note that as long as
one of the Bogoliubov coefficients $\beta_{kk^{\prime}}$ is non-zero, and the
un-barred state is the vacuum state, the state in the bared basis is populated
with particles. Therefore, different Killing observers observe a different
particle content in the field, i.e. particles are observer-dependent notions.
In flat spacetime there are two kinds of observers who can meaningfully
describe particles for all times: inertial observers who follow straight lines
and observers in uniform acceleration who’s trajectories are given by
hyperbolas parameterized for example by
$x=\chi\cosh\left(a\eta\right),\qquad t=\chi\sinh\left(a\eta\right),$ (4)
where $a$ is the proper acceleration at the reference worldline $\chi=1/a$
with proper time $\eta$. (The notion of defining particles in a general curved
spacetime is addressed in e.g. [30, 38]. For the other special cases when the
acceleration (i) is asymptotically uniform in the past/future see e.g. [30,
38, 39], or (ii) asymptotically zero in the past but asymptotically uniform in
the future and see e.g. [40]). The transformation suggests that a suitable
choice of coordinates for uniformly accelerated observers are $(\eta,\chi)$
which are known as Rindler coordinates.
Figure 1: Rindler space-time diagram: lines of constant position $\chi$ are
hyperbolae and all curves of constant $\eta$ are straight lines that come from
the origin. An uniformly accelerated observer Rob travels along a hyperbola
constrained to either region $I$ or region $II$.
The transformation in Eq. (4) is defined in the region $|x|\geq t$ known as
the (right) Rindler wedge I. When $\eta\rightarrow\infty$ then
$t/x=\tanh(a\eta)\rightarrow 1\Rightarrow x=t$. Uniformly accelerated
observers asymptotically approach the speed of light and are constrained to
move in wedge I. Since the transformation does not cover all of Minkowski
spacetime, one must define a second region called (left) Rindler wedge II by
considering a coordinate transformation which differs from Eq. (4) by an
overall sign in both coordinates. Rindler regions I and II are causally
disconnected, and the lines $x\pm t=0$ at 45 degrees define the Rindler
horizon, Fig.(1).
The metric in Rindler coordinates takes the form
$ds^{2}=(a^{2}\chi^{2}\,d\eta^{2}-d\chi^{2})$ where $a^{2}\chi^{2}$ acts as an
effective gravitational potential $g_{\eta\eta}(\chi)$ for the Rindler
observer’s local redshift factor. The Klein-Gordon equation in Rindler
coordinates is $(a^{-2}\partial_{\eta}^{2}-\partial_{ln(a\chi)}^{2})\phi=0$
and the solutions [41, 42, 43] are again plane waves, though now with
logarithmic spatial dependence $\ln\chi$, (compare with (2))
$\displaystyle u_{\omega,I}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{i(\epsilon(\omega/a)\ln\chi-\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{x-\epsilon
t}{l_{\Omega}}\right)^{i\epsilon\Omega}\equiv u_{\Omega,I},$ (5a)
$\displaystyle u^{*}_{\omega,I}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{-i(\epsilon(\omega/a)\ln\chi-\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{x-\epsilon
t}{l_{\Omega}}\right)^{-i\epsilon\Omega}\equiv u^{*}_{\Omega,I}.$ (5b)
In the above $\omega>0$, $\epsilon=1$ corresponds to modes propagating to the
right along lines of constant $x-t$, and $\epsilon=-1$ to modes propagating to
the left along lines of constant $x+t$. In the second equality we have
introduced a positive constant $l_{\Omega}$ of dimension length, and defined
the dimensionless positive constant $\Omega=\omega/a$. Some authors [42]
choose to label the Rindler mode by the (positive) frequency $\omega$, while
other authors [41, 43] label the modes by the (positive) dimensionless
quantity $\Omega$. (Note that $-\infty<\epsilon\omega/a=\epsilon\Omega<\infty$
acts as the effective wavevector for the Unruh modes, if one where to push the
analogy with the inertial Minkowski modes (2)). Here we follow derivations
from [43] and throughout this work, it will be understood that a wavevector
subscript $k$ on Minkowski modes ($u_{k,M}$, etc…) takes values in the range
$-\infty$ to $\infty$, while for Unruh modes ($u_{k,I}$, $u_{k,II}$ etc…) it
takes values from $0$ to $\infty$.
The solutions $u_{k,I}$ and $u^{*}_{k,I}$ are identified as positive and
negative frequency solutions, respectively, with respect to the timelike
Killing vector field $\partial_{\eta}$. These solutions have support only in
the right Rindler wedge and therefore are labeled by the subscript $I$. Note
that they do not constitute a complete set of solutions. The transformation
which defines Rindler region $II$ also gives rise to the same spacetime.
However, the future-directed timelike Killing vector field which in this case
is given by $\partial_{(-\eta)}=-\partial_{\eta}$, and the solutions are
$\displaystyle u_{\omega,II}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{i(-\epsilon(\omega/a)\ln(-\chi)+\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{\epsilon
t-x}{l_{\Omega}}\right)^{-i\epsilon\Omega}\equiv u_{\Omega,II},$ (5fa)
$\displaystyle u^{*}_{\omega,II}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{-i(-\epsilon(\omega/a)\ln(-\chi)+\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{\epsilon
t-x}{l_{\Omega}}\right)^{i\epsilon\Omega}\equiv u^{*}_{\Omega,II},$ (5fb)
with support in region $II$. The solutions of region $I$ together with the
solutions in region II form a complete set of orthonormal solutions.
Therefore, we can quantize the field in this basis as well,
$\hat{\phi}=\int(u_{\Omega,{I}}a_{\Omega,I}+u_{\Omega,{II}}a_{\Omega,{II}}+h.c.)\,d\Omega.$
Since region $I$ is causally disconnected from region $II$, the mode operators
in the separated wedges commute
$[a_{\Omega,I},a^{\dagger}_{\Omega^{\prime},II}]=0$, etc. The vacuum state in
the Rindler basis is
$\left|0\right>_{R}=\left|0\right>^{I}\otimes\left|0\right>^{II}$ where
$a_{k,{I}}\left|0\right>^{I}=0$ and $a_{k,{II}}\left|0\right>^{II}=0$. Making
use of the inner product we find the Bogoliubov transformations,
$\displaystyle a_{k,M}$ $\displaystyle=$
$\displaystyle\int(u_{k,M},u_{\Omega,{I}})a_{\Omega,{I}}+(u_{k,M},u_{\Omega,I}^{\ast})a_{\Omega,I}^{\dagger}$
$\displaystyle+$
$\displaystyle(u_{k,M},u_{\Omega,II})a_{\Omega,II}+(u_{\Omega,M},u_{\Omega,II}^{*})a_{\Omega,II}^{\dagger}d\Omega,$
where, for example,
$(u_{k,M},u_{\Omega,I})=-i\,\int(u^{*}_{k,M}\,\partial_{t}u_{\Omega,I}-(\partial_{t}u_{\Omega,I})\,u_{k,M}^{*})dx$.
Upon computing the inner products [41, 43] the above formula can be written as
$\displaystyle a_{\omega,M}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}d\Omega\,[(\alpha_{\omega\Omega}^{R})^{*}(\cosh(r_{\Omega})a_{\Omega,I}-\sinh(r_{\Omega})a^{\dagger}_{\Omega,II})$
(5fg) $\displaystyle+$
$\displaystyle(\alpha_{\omega\Omega}^{L})^{*}(-\sinh(r_{\Omega})a^{\dagger}_{\Omega,I}+\cosh(r_{\Omega})a_{\Omega,II})],$
where
$\displaystyle\alpha_{\omega\Omega}^{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\omega}}\,(\omega
l)^{i\epsilon\Omega},\qquad\alpha_{\omega\Omega}^{L}=\frac{1}{\sqrt{2\pi\omega}}\,(\omega
l)^{-i\epsilon\Omega},$
are the Bogoliubov coefficients for the massless case, and $l$ is an overall
constant of dimension length, independent of $\Omega$ and $\epsilon$.
The Minkowski creation and annihilation operators result in an infinite sum of
Rindler operators. An alternative basis for the inertial observers known as
the Unruh basis can significantly simplify the transformations between
inertial and uniformly accelerated observers. The Unruh modes $a_{\Omega,R}$,
$a_{\Omega,L}$ are appropriately chosen linear combinations of right-moving
and left-moving Rindler modes respectively such that they are analytic across
both regions $I$ and $II$. That is, $u_{\Omega,I}$ and $u^{*}_{\Omega,II}$ are
both proportional to $(x-\epsilon t)^{i\epsilon\Omega}$ when
$(-1)^{i\epsilon\Omega}=(e^{i\pi})^{i\epsilon\Omega}=e^{-\epsilon\pi\Omega}$
is factored out of the latter region $II$ mode. The Unruh modes are given by
the direct Bogoliubov transformation with region $I$ and $II$ Rindler modes
for each value of $\Omega$ as
$\displaystyle a_{\Omega,R}$ $\displaystyle=$
$\displaystyle\cosh(r_{\Omega})a_{\Omega,I}-\sinh(r_{\Omega})a^{\dagger}_{\Omega,II},$
$\displaystyle a^{\dagger}_{\Omega,L}$ $\displaystyle=$
$\displaystyle-\sinh(r_{\Omega})a_{\Omega,I}+\cosh(r_{\Omega})a^{\dagger}_{\Omega,II},$
(5fh)
where $\tanh(r_{\Omega})=e^{-\pi\Omega}$. Here $a_{\Omega,R}$ annihilates a
right ($R$) moving Unruh mode traveling along lines of constant $x-t$ in both
wedges $I$ and $II$, while $a_{\Omega,L}$ annihilates a left ($L$) moving
Unruh mode traveling along lines of constant $x+t$, with
$[a_{\Omega,R},a^{\dagger}_{\Omega,R}]=1$,
$[a_{\Omega,L},a^{\dagger}_{\Omega,L}]=1$ and all cross commutators vanishing.
In terms of mode functions, the Bogoliubov transformation from the Rindler to
the Unruh modes is given by
$\displaystyle u_{\Omega,R}$ $\displaystyle=$
$\displaystyle\cosh({r})u_{\Omega,I}+\sinh({r})u^{\ast}_{\Omega,II},$
$\displaystyle u^{*}_{\Omega,L}$ $\displaystyle=$
$\displaystyle\sinh({r})u_{\Omega,I}+\cosh({r})u^{\ast}_{\Omega,II},$ (5fi)
which are analytic in $x-t$ across both Rindler wedges $I$ and $II$. Note that
the sign of the momentum $k$ in region $II$ is opposite of that in region $I$,
but coupled with utilizing the complex conjugate of the region $II$ Rindler
mode, renders the resulting Unruh modes $u_{\Omega,R}$ and $u^{*}_{\Omega,L}$
right-movers (see [38] Chap. 9.5 for further details).
The most general Unruh annihilation operator of purely positive Minkowski
frequency is a linear combination of the two $R,L$ Unruh creation operators,
$a_{\Omega,U}=q_{L}a_{\Omega,L}+q_{R}a_{\Omega,R},$ (5fj)
where $q_{L}$ and $q_{R}$ are complex numbers with
$|q_{L}|^{2}+|q_{R}|^{2}=1$. The introduction of the Unruh modes allows us to
write the Minkowski annihilation operator (5fg) as a linear combination of
only Unruh annihilation operators
$a_{\omega,M}=\int^{\infty}_{0}d\Omega\,[(\alpha_{\omega\Omega}^{R})^{*}a_{\Omega,R}+(\alpha_{\omega\Omega}^{L})^{*}a_{\Omega,L}].$
(5fk)
Hence, both Minkowski and Unruh annihilation operators annihilate the
Minkowski vacuum, i.e. $a_{\omega,M}|0\rangle_{\mathcal{M}}=0$,
$a_{\Omega,R}|0\rangle_{\mathcal{M}}=0$, and
$a_{\Omega,L}|0\rangle_{\mathcal{M}}=0$, and therefore, the Unruh vacuum and
the Minkowski vacuum coincide.
Using the direct Bogoliubov transformation between the Unruh and Rindler
annihilation operators it is straight forward to show that (see e.g. [32, 38,
41])
$\left|0_{k}\right>^{\mathcal{M}}=\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\left|n_{k}\right>^{I}\left|n_{k}\right>^{II},$
(5fl)
where $\tanh r\equiv e^{-\pi\omega/a}$. The vacuum state in the Rindler basis
corresponds to a two mode squeezed state. Since the accelerated observer is
constrained to move in region $I$ one must trace over the states in (the
causally disconnected) region $II$. The density matrix of the Minkowski vacuum
is given by $\rho_{0}=\left|0_{k}\right>\left<0_{k}\right|^{\mathcal{M}}$ and
therefore, the state in region $I$ corresponds to the following reduced
density matrix
$\displaystyle\rho_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{\cosh^{2}(r)}\sum_{n}\tanh^{2n}(r)\left|n\right>^{I}\left<n\right|^{I},$
$\displaystyle=$
$\displaystyle(1-e^{-2\pi\omega/a})\sum_{n}(e^{-2\pi\omega/a})^{n}\left|n\right>^{I}\left<n\right|^{I},$
which corresponds to a thermal state with temperature $T_{U}=\frac{a}{2\pi
k_{B}}$ (where $k_{B}$ is the Boltzmann constant) proportional to the
observer’s acceleration. The temperature is known as the Unruh temperature.
This is the well known Unruh effect [32]: the vacuum state as seen by inertial
observers is a thermal state for observers in uniform acceleration.
#### 2.1.2 Unruh-DeWitt detectors
In order to give a more physical interpretation to the Unruh effect, Unruh-
DeWitt detectors were introduced [32, 41, 42, 44]. The detectors consist of a
point-like system endowed with an internal structure which can be either a two
level system or a harmonic oscillator [45, 46]. The detector moves in
spacetime following a classical trajectory given by $x(\tau)$ where $\tau$ is
the detector’s proper time. The detector couples to the field locally via it’s
monopole moment. Therefore, the interaction Hamiltonian for a harmonic
oscillator detector is given by
$\hat{H}_{I}(\tau)=\int
dk\lambda(\tau)(d\,e^{i\Omega\tau}+d^{\dagger}\,e^{-i\Omega\tau})\left(\hat{a}_{k}e^{i(kx(\tau)-\omega
t(\tau))}+\hat{a}_{k}^{\dagger}e^{-i(kx(\tau)-t(\tau))}\right),$ (5fm)
where $d$ and $d^{\dagger}$ are annihilation and creation operators for the
internal degrees of freedom of the detector, $\Omega$ the frequency of the
detector and $\omega=|k|$ the frequency of the field modes. The coupling
function $\lambda(\tau)$ between the field and the detector can be chosen such
that the interaction is switched on and off adiabatically. This removes
transient effects.
Standard calculations [30] consider a detector that follows either an inertial
trajectory (say $x(t)=vt$ for a detector with zero acceleration) or a
uniformly accelerated one ($x=(t^{2}+a^{-2})^{1/2}$). At time $t=-\tau_{0}$
the field is in the vacuum state $|0\rangle_{f}$ and the detector in it’s
ground state $|0\rangle_{d}$. The detector is then turned on and the
transition probability of the detector to an exited state at time $\tau_{0}$
is calculated using perturbation theory. The transition rate per unit of
proper time of the detector to a state with $n=1$ excitations at first order
is given by
$\displaystyle\mathcal{P}=\sum_{n,\,\psi}\frac{1}{2\tau_{0}}\int_{-\infty}^{\infty}d\tau|A_{n}|^{2},\quad
A_{n}={}_{d}\\!\langle
1_{n}|{}_{f}\\!\langle\psi|H_{I}(\tau)|0\rangle_{d}|0\rangle_{f},$ (5fn)
where $|\psi\rangle_{f}$ is the final state of the field which is not
observed, and hence, averaged over all possible outcomes. This leads to the
detector response function [30, 41]
$F(\omega)=\lim_{s\downarrow
0}\lim_{\tau_{0}\uparrow\infty}\frac{1}{2\tau_{0}}\int_{-\tau_{0}}^{\tau_{0}}\int_{-\tau^{\prime}_{0}}^{\tau^{\prime}_{0}}\,e^{-i\omega(\tau-\tau^{\prime})-s|\tau|-s|\tau^{\prime}}\,g(\tau,\tau^{\prime}),$
taking $\lambda(\tau)=\exp(-s|\tau|)$, and defining
$g(\tau,\tau^{\prime})=G(x(\tau),x(\tau^{\prime}))$ where
$G(x,x^{\prime})={}_{f}\\!\langle
0|\,\hat{\phi}(x)\,\hat{\phi}(x^{\prime})\,|0\rangle_{f}$ is the positive
frequency Wightman function (WF), and the field $\hat{\phi}(x)$ is given by
the integral over all momentum of the rightmost term in large parentheses in
(5fm). In the above, the response function $F(\omega)$ is a consequence of
Fermi’s golden rule and depends only on the field and not on the structure of
the detector, since the latter has been completely factored out [30]. In
general the WF is inversely proportional to the squared geodesic distance
$\sigma(\Delta x)=g_{\mu\nu}(x-x^{\prime})^{\mu}(x-x^{\prime})^{\nu}$ and
hence is singular as $x\rightarrow x^{\prime}$. This singular behavior is
typically regularized by the “$i\epsilon$” prescription which treats
$F(\omega)$ as a contour integral in the complex
$\Delta\tau=\tau-\tau^{\prime}$ plane. For an inertial trajectory $x=vt$ one
has [30] $g(\Delta\tau)=-[2\pi(\Delta\tau-i\epsilon)]^{-2}$. For $\omega>0$
the contour is closed in the lower half-plane for the integral to be
convergent. However, since the pole in $g(\Delta\tau)$ is in the upper half-
plane, no contributions are obtained resulting in $F(\omega)=0$ as expected.
For the uniformly accelerated trajectory $x(\tau)=(t^{2}(\tau)+a^{-2})^{1/2}$
with $t(\tau)=a^{-1}\sinh(a\tau)$ one obtains
$g(\Delta\tau)=-[(4\pi/a)\sinh(a\Delta\tau/2-i\epsilon a)]^{-2}$. A contour
integration [30] now picks up contributions from the poles along the negative
imaginary axis, leading to a response function proportional to
$\omega/(\exp(2\pi\omega/a)-1)^{-1}$. The appearance of the Planck factor
$(\exp(2\pi\omega/a)-1)^{-1}$ indicates that the equilibrium reached between
the accelerated detector and the field $\hat{\phi}$ in the vacuum state
$|0\rangle_{f}$ is identical to the case when the detector remained
unaccelerated, but was immersed in a bath at (Unruh) temperature
$T_{U}=a/(2\pi k_{B})$. Transition rates for Unruh-DeWitt detectors in curved
spacetime have been considered in [47]. We will see in the next section that
such accelerated detectors can be used to extract entanglement from the field.
Note that since the detector is point-like it couples with the same strength
to every frequency field mode. However this is not a very realistic situation.
A more physical situation corresponds to a detector which has a spatial
profile [41, 48, 49]. Such detectors couple to a distribution of field modes
which depends on the specifics of the profile. The use of detectors with
infinitesimal, though non-zero, spatial extent can be utilized to give a more
physical interpretation of the mathematically formal “$i\epsilon$”
prescription used above in the computation of the detector response function.
In brief, the use of infinitesimal sized detectors smears out the field
$\hat{\phi}(x)$ along the detector trajectory, thus regularizing the singular
behavior of the WF as $x\rightarrow x^{\prime}$. Following Takagi (see §3.2 of
[41]), one can consider the instantaneous non-rotating rest frame of the
detector (the Fermi-Walker (FW) frame, see §13.6 of [50]). The coordinates
transformation $t(\tau,\zeta)=(a^{-1}+\zeta)\sinh(a\tau)$ and
$x(\tau,\zeta)=(a^{-1}+\zeta)\cosh(a\tau)$ with metric
$ds^{2}=(1+a\zeta)^{2}d\tau^{2}-d\zeta^{2}$, describes the observer’s “local
laboratory” where the detector sits at the origin $\zeta=0$ of the local
spatial coordinates. For a “rigid” detector, its monopole field
${\cal{M}}(x(\tau,\zeta))$ (the term in leftmost parentheses in (5fm) can be
written in a factorized form
${\cal{M}}(x(\tau,\zeta))=\lambda(\tau)\,M(\tau)\,f(\zeta)$. The detector-
field interaction is now given by $H_{int}=\int d\tau\int
d\zeta{\cal{M}}(x(\tau,\zeta))\,\hat{\phi}(x(\tau,\zeta))$ which is now of the
form $\lambda(\tau)\,M(\tau)\,\hat{\phi}^{\prime}(\tau)$, where
$\hat{\phi}^{\prime}(\tau)=\int d\zeta f(\zeta)\,\hat{\phi}(x(\tau,\zeta)$ is
the smeared out quantum field. Using the mode expansion
$\hat{\phi}(x(\tau))=\int dk\,a_{k}U_{k}(x(\tau))$ with plane waves
$U_{k}(x(\tau))=e^{ik_{\mu}x^{\mu}(\tau)}$ allows the smeared field to be
written as $\hat{\phi}^{\prime}(\tau)=\int
dk\,a_{k}U_{k}(x(\tau))\tilde{f}_{k}(\tau)+h.c.$ Here
$\tilde{f}_{k}(\tau)=\int d\zeta
e^{ik_{\mu}[x^{\mu}(\tau,\zeta)-x^{\mu}(\tau)]}$ $=\int d\zeta
e^{ik_{\mu}[\partial x^{\mu}(\tau,\zeta)/\partial\zeta^{j}]\,\zeta^{j}}$ are
wavepackets that can be effectively replaced with $\tilde{f}_{k}\approx
e^{-\epsilon\omega_{k}/2}$ (upon substitution of the coordinate
transformations to the FW frame) where $\epsilon$ is a small positive quantity
on the order $1/a$. This latter form of $\tilde{f}_{k}$ is just the
“$i\epsilon$” prescription, and shows the advantage of utilizing smeared
fields/wavepackets to regularize divergent quantities, as well as lending
physical interpretation to mathematical procedures. Work in progress shows
that a uniformly accelerated detector with a Gaussian spatial profile
naturally couples to a peak distribution of Rindler modes [51]. Such a
detector also naturally couples to distributions of Unruh modes which in this
case yields a frequency distribution with two peaks corresponding to left and
right moving Unruh modes. This detector model will help in the understanding,
from physical perspective, the nature of Unruh modes which have been used to
analyze the degradation of entanglement in non-inertial frames.
#### 2.1.3 Moving cavities
As mentioned previously, in order to preform quantum information tasks
observers need to have access to the state and therefore, localizing field
modes in space and time becomes relevant. This can be achieved by confining
quantum fields in cavities which can move in spacetime [15, 16, 17, 18, 19,
20, 21]. To confine a massless field within a cavity of length $L$ appropriate
Dirichlet boundary conditions must be imposed on the solutions of the Klein-
Gordon equation given by (2) at the cavity mirrors placed at $x_{l}$ and
$x_{r}=x_{l}+L$. The normalized field solutions in this case correspond to
sine functions
$\displaystyle U_{n,M}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n\pi}}\sin\left(\omega_{n}[x-x_{l}]\right)e^{-i\omega_{n}t},$
(5fo)
where $n\in N$ labels the energy states which now have a discrete spectrum
given by $\omega_{n}=n\pi/L$. The Killing vector field is $\partial_{t}$ which
classifies $U_{n,M}$ and $U^{\ast}_{n,M}$ as positive and negative frequency
solutions. The normalization of the solutions gives
$[U_{n,M},U_{m,M}]=[U^{\ast}_{n,M},U^{\ast}_{m,M}]=\delta_{nm}$ and the mix
products vanish. The field is therefore,
$\hat{\phi}=\sum_{n}(U_{n,M}a_{n,M}+U_{n,M}^{*}a_{n,M}^{\dagger}),$
where creation and annihilation operators satisfy standard bosonic commutation
relations. We have used capitalized letters for the cavity modes to
distinguish them from global modes.
We can also consider the field confined to a cavity in uniform acceleration by
imposing uniformly accelerating boundary conditions to the modes in Eq. (5a)
at $\chi=\chi_{r}$ and $\chi=\chi_{l}$. Here we considered Rindler coordinates
$(\eta,\chi)$ which are a suitable choice for this case once more. We consider
the cavity to move in region $I$ without loss of generality. The cavity is
constructed such that it has a constant proper length as measured by a co-
moving observer. Therefore, the mirrors move with different proper
acceleration $A_{r}=1/x_{r}$ and $A_{l}=1/x_{l}$. Taking advantage of the
invariance under the boost Killing vector $\partial_{\eta}$ during
acceleration we write
$\displaystyle U_{n,R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n\pi}}\sin\left(\Omega_{n}[\chi-\chi_{l}]\right)e^{-i\omega_{n}\eta},$
where $\Omega_{n}=n\pi/L^{\prime}$ with
$L^{\prime}=\chi_{r}-\chi_{l}=a^{-1}\ln(1+aL)$. For the case $aL/c^{2}\ll 1$
we have $L^{\prime}\approx L$, and the small difference of the acceleration at
the ends of the mirror can be ignored with respect to the acceleration at the
center of the cavity. Since we are only interested in the field within the
mirrors, we dropped the index $I$ and instead included $R$ to denote the modes
of the cavity, which is undergoing uniform acceleration. The quantum field
inside the cavity as seen by a co-moving observer is given by
$\hat{\phi}_{R}(\eta,\xi)=\sum_{n}(U_{n,R}(\eta,\xi)a_{n,R}+U_{n,R}^{*}(\eta,\xi)a_{n,R}^{\dagger})$
where $a_{n,R}^{\dagger}$ and $a_{n,R}$ are once more creation and
annihilation operators with
$[a_{n,R},a^{\dagger}_{n^{\prime},R}]=\delta_{n,n^{\prime}}$. The ground
state, in this case, is defined by $a_{n,R}|0\rangle^{R}=0$, $\forall n$. Note
that the Rindler coordinates completely cover the region inside the cavity and
that the horizon always lies outside the cavity for all values of $a$. We
assume the cavity’s mirrors to be perfectly reflecting therefore, the states
inside the cavity it will remain the same for all times [15]. This is in
agreement with Schutzhold and Unruh’s [52] comment concerning how the non-
transmissive cavity protects the state from Unruh radiation. If one prepares a
state, for example a pure state, as long as the cavity remains either inertial
or in uniform acceleration, the state remains unchanged, i.e. pure.
More interesting is to consider the case in which the cavity is placed in a
spaceship initially at rest which uniformly accelerates for a finite amount of
time and finally moves at constant velocity, as illustrated in figure 2.
Figure 2: Non-uniformly accelerated trajectory from initial constant velocity
to uniformly accelerated motion.
This trajectory is interesting since it is more realistic (the uniformly
accelerated trajectory assumes constant acceleration from past infinity to
future infinity). Further, general travel scenarios can be crafted by
considering sequences of it, and therefore it is called the basic building
block trajectory[16]. The modes inside an cavity initially at rest will be
affected by the non-uniform accelerated motion giving rise to particle
creation and entanglement [16, 17, 18, 19, 20, 21],. It is possible to find
Bogoliubov transformations between the modes of the inertial cavity and the
modes of the cavity after any given travel scenario. The new modes will be
given by
$a_{m,F}=\sum_{n}{\alpha^{\ast}_{mn}a_{n,M}-\beta^{\ast}_{mn}a^{\dagger}_{n,M}},$
(5fp)
where $F$ denotes the modes in the final region, and
$\alpha_{mn}=(U_{m,F},U_{n,M})$ and $\beta_{mn}=-(U_{m,F},U^{\ast}_{n,M})$ are
Bogoliubov coefficients. To study the above case in more detail it is
convenient to work in the covariant matrix formalism which will be introduced
in the section on quantum information. In the simple case of the building
block trajectory the coefficients are given by
$\displaystyle\alpha^{B}_{mn}=\frac{1}{L}\sqrt{\frac{n}{m}}F_{mn}+\frac{1}{\ln(\frac{x_{r}}{x_{l}})}\sqrt{\frac{m}{n}}G_{mn},$
(5fqa)
$\displaystyle\beta^{B}_{mn}=\frac{1}{L}\sqrt{\frac{n}{m}}F_{mn}-\frac{1}{\ln(\frac{x_{r}}{x_{l}})}\sqrt{\frac{m}{n}}G_{mn},$
(5fqb)
where
$\displaystyle F_{mn}:$ $\displaystyle=$
$\displaystyle\int_{L_{0}}^{R_{0}}dx\sin\left(\omega_{n}(x-L_{0})\right)\sin\left(\Omega_{m}\ln(\frac{x}{L_{0}})\right)$
$\displaystyle G_{mn}:$ $\displaystyle=$
$\displaystyle\int_{L_{0}}^{R_{0}}\frac{dx}{x}\sin\left(\omega_{n}(x-L_{0})\right)\sin\left(\Omega_{m}\ln(\frac{x}{L_{0}})\right).$
Here $B$ stands for basic building block. These coefficients are difficult to
handle both numerically and analytically. However, in the case where $h=aL$ is
small, the terms in the integrand can be expanded in a Maclaurin series such
that, to second order, the coefficients take a simple form [16, 17, 18].
#### 2.1.4 Curved spacetime
For a general curved space metric $g_{\mu\nu}(x)$ the d’Alambertian (see
section 2.1.1) gives rises to a more complicated, and very likely non-
separable, Klein-Gordon equation. However, in some special cases solutions can
be found [30]. An example is a $(1+1)$-dim expanding Robertson-Walker
universe. This spacetime does not admit a global Killing vector field and
therefore, it is not possible to define particles globally. This is also true
for most spacetimes. However, the Robertson-Walker universe has a special
property in that it is asymptotically flat in the past and future infinity
regions where timelike Killing vector fields can be defined and employed to
distinguish positive and negative solutions to the Klein-Gordon equation. We
will consider that a scalar field living in this spacetime is in the vacuum
state as seen by observers in the past infinity region, and show that the
state will be populated with particles in the future infinity.
The spacetime of a Robertson-Walker universe in $(1+1)$-dim is given by
$ds^{2}=dt^{2}-a^{2}(t)d\chi^{2}$ where the spatial sections of the spacetime
are expanding (or contracting) uniformly according to the function $a^{2}(t)$.
Considering the infinitesimal coordinate transformation $d\eta=dt/a(t)$ and
defining $a^{2}(t)=c^{2}(\eta)$ we obtain the metric
$ds^{2}=c^{2}(\eta)(d\eta^{2}-d\chi^{2})$. Consider
$c(\eta)=1+\epsilon(1+\tanh(\sigma\eta))$ where $\epsilon$ and $\sigma$ are
constants. This describes a toy model of a universe undergoing a period of
smooth expansion. The parameter $\epsilon$ is known as the expansion volume
and $\sigma$ is the expansion rate. In the limit $\eta\rightarrow-\infty$ the
metric is $ds^{2}=(d\eta^{2}-d\chi^{2})$ and in the limit
$\eta\rightarrow\infty$ then $ds^{2}=(1+2\epsilon)(d\eta^{2}-d\chi^{2})$. The
metric is flat in these regions where the vector field $\partial_{\eta}$ has
Killing properties.
We now consider the massive Klein-Gordon equation $(\square+m^{2})\phi=0$ in
the Robertson-Walker spacetime described above where $m$ is the mass of the
field. The metric tensor $g_{\mu\nu}$ has components
$g_{\mu\nu}=c(\eta)\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$
therefore, $g=det(g_{\mu\nu})=-c^{2}(\eta)$ and the Klein-Gordon equation
takes the form
$((\partial_{\eta}^{2}-\partial_{\chi}^{2})+c(\eta)m^{2})\phi=0.$ (5fqr)
Exploiting spatial translational invariance we separate the solutions into
$u_{k}=\frac{1}{\sqrt{2\pi\omega}}e^{ik\chi}\xi_{k}(\eta),$
so that the equation becomes
$\partial^{2}_{\eta}\xi_{k}(\eta)+(k^{2}+c(\eta)m^{2})\xi_{k}(\eta)=0.$ (5fqs)
This equation can be solved in terms of two hypergeometric functions. We find
two types of solutions [30]
$\displaystyle u^{(1)}_{k}(\eta,\chi)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega_{in}}}e^{ik\chi-i\omega_{+}\eta-\left(\frac{i\omega_{-}}{\sigma}\right)\ln
2\cosh(\sigma\eta)}{{}_{2}}F_{1}(\alpha,\beta,\gamma_{1},\delta_{+}),$ (5fqta)
$\displaystyle u^{(2)}_{k}(\eta,\chi)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega_{out}}}e^{ik\chi-i\omega_{+}\eta-\left(\frac{i\omega_{-}}{\sigma}\right)\ln
2\cosh(\sigma\eta)}{{}_{2}}F_{1}(\alpha,\beta,\gamma_{2},\delta_{-}),$ (5fqtb)
where the constants above are defined as
$\alpha=1+\frac{i\omega_{-}}{\sigma},\quad\beta=\frac{i\omega_{-}}{\sigma},\quad\gamma_{1}=1-\frac{i\omega_{in}}{\sigma},\quad\gamma_{2}=1+\frac{i\omega_{out}}{\sigma}\quad\delta_{\pm}=\frac{1}{2}(1\pm\tanh(\sigma\eta)),$
(5fqtu)
and the frequencies $\omega_{in}=[k^{2}+m^{2}]^{\frac{1}{2}}$,
$\omega_{out}=[k^{2}+m^{2}(1+2\epsilon)]^{\frac{1}{2}}$ and
$\omega_{\pm}=\frac{1}{2}(\omega_{out}\pm\omega_{in})$. We note that in the
limit $\eta\rightarrow-\infty$ the first solution becomes
$u_{k}^{(1)}\rightarrow\frac{1}{\sqrt{4\pi\omega_{in}}}e^{ik\chi-i\omega_{in}\eta},$
while in the case $\eta\rightarrow+\infty$ the second solution is
$u_{k}^{(2)}\rightarrow\frac{1}{\sqrt{4\pi\omega_{out}}}e^{ik\chi-i\omega_{out}\eta}.$
One can see that in these limits the asymptotic solutions $u_{k}^{(1)}$ and
$u_{k}^{(2)}$ to the Klein-Gordon equation are plane waves which can then be
associated with positive mode solutions. The negative mode solutions
correspond to $u_{k}^{1*}$ and $u_{k}^{2*}$. Since $u^{1}_{k}$ is associated
with a plane wave at $\eta\rightarrow-\infty$ (past infinity) we call these
solutions in-waves $u_{k}^{(1)}\equiv u_{k}^{(in)}$. The solutions
$u_{k}^{(2)}\equiv u_{k}^{(out)}$ which are associated with plane waves at
$\eta\rightarrow+\infty$ (future infinity) will be called out-waves.
Using the linear transformation properties of hypergeometric functions one can
write $u_{k}^{(in)}$ in terms of $u_{k}^{(out)}$. This is easier than
calculating the Bogoliubov coefficients via brute force direct integration
using the inner product. One obtains
$u_{k}^{(in)}(\eta,\chi)=\alpha_{k}u_{k}^{(out)}(\eta,\chi)+\beta_{k}u_{k}^{(out*)}(\eta,\chi),$
(5fqtv)
where
$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{\omega_{out}}{\omega_{in}}\right)^{\frac{1}{2}}\frac{\Gamma\left(1-\frac{i\omega_{in}}{\sigma}\right)\Gamma\left(-\frac{i\omega_{out}}{\sigma}\right)}{\Gamma\left(-\frac{i\omega_{+}}{\sigma}\right)\Gamma\left(1-\frac{i\omega_{+}}{\sigma}\right)},$
(5fqtwa) $\displaystyle\beta_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{\omega_{out}}{\omega_{in}}\right)^{\frac{1}{2}}\frac{\Gamma\left(1-\frac{i\omega_{in}}{\sigma}\right)\Gamma\left(\frac{i\omega_{out}}{\sigma}\right)}{\Gamma\left(\frac{i\omega_{-}}{\sigma}\right)\Gamma\left(1+\frac{i\omega_{-}}{\sigma}\right)}.$
(5fqtwb)
Here $\Gamma$ is the Gamma functions with the properties
$\Gamma(1+z)=z\,\Gamma(z)$, $\Gamma(1-ix)=\pi/[\Gamma(ix)\sinh(\pi x)]$ and
$|\Gamma(ix)|^{2}=\pi/[x\sinh(\pi x)]$ for $z$ complex and $x$ real. From the
above expressions we can read off the Bogoliubov coefficients
$\alpha_{kk^{\prime}}=\alpha_{k}\delta_{kk^{\prime}}$ and
$\beta_{kk^{\prime}}=\beta_{k}\delta_{-kk^{\prime}}$. Therefore the
transformation between annihilation operators yields
$a^{in}_{k}=\alpha_{k}^{*}a^{out}_{k}-\beta^{*}_{k}a_{-k}^{out\dagger}.$ We
now consider that the state of the field in the past infinity is the vacuum
state,
${|0\rangle}^{in}=\bigotimes_{k=-\infty}^{\infty}{|0\rangle_{k}}^{in,}$
(5fqtwx)
and use the expression for the in-mode annihilation operator to calculate the
state in the future infinity. Since the transformation between in and out
annihilation operators only mixes modes of frequency $k$ and $-k$, one finds
that the state seen by observers in the future infinity is,
$|0\rangle_{k}^{in}=\sqrt{1-\gamma}\sum_{n}\gamma^{n}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}.$
(5fqtwy)
where,
$\gamma=\frac{\sinh^{2}(\pi\omega_{-}/\sigma)}{\sinh^{2}(\pi\omega_{+}/\sigma)}.$
The vacuum state from the perspective of observers in the remote past has
particles in the remote future. Due to the expansion of the universe there has
been particle creation. The state $|0\rangle^{in}$ is (once again) a two-mode
squeezed state, and the particle creation process just described has a strong
analogy with the quantum optical process of spontaneous parametric down
conversion if one describes only the signal photons emerging from the end of a
nonlinear crystal, while ignoring the idler photons [53, 33]. (Note: here the
laser pump acts as as the “source” driving the expansion of the universe).
Before reviewing work on entanglement in all of the above scenarios, we will
introduce the basic tools to quantify it.
### 2.2 Quantum information
The main aim of quantum information is to learn how to store, process and read
information using quantum systems. In this section we will briefly revise
basic concepts of quantum entanglement for pure and mixed states which are
considered to be key resources in quantum information.
#### 2.2.1 Entanglement
Entanglement is a quantum property which is a consequence of the superposition
principle and the tensor product structure of the Hilbert space. The pure bi-
partite case is well understood. The state of two particles A and B is a
vector in a $(d_{a}\times d_{b})$-dimensional Hilbert space
$\mathcal{H}_{ab}=\mathcal{H}_{a}\otimes\mathcal{H}_{b}$. The space
$\mathcal{H}_{ab}$ is the tensor product of the subspaces $\mathcal{H}_{a}$
and $\mathcal{H}_{b}$ of each particle. An pure state element of the space
$\mathcal{H}_{ab}$ is written as
$|\psi_{ab}\rangle=\sum_{i,j}A_{ij}|i\rangle_{a}\otimes|j\rangle_{b}$.
A state $|\psi_{ab}\rangle\in\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is
separable if $|\psi_{ab}\rangle=|\phi\rangle_{a}\otimes|\varphi\rangle_{b}$.
Separable states can be prepared by local operations and classical
communication. This means that observers manipulate each particle
independently by making measurements, or applying unitary transformations of
the form $|\psi_{ab}\rangle=U_{a}\otimes
U_{b}\,|\phi\rangle_{a}\otimes|\varphi\rangle_{b}$ where $U_{a}$ and $U_{b}$
are unitaries acting on particles A and B, respectively. The observers are
also allowed to exchange classical information. If the state is not separable
then it is entangled. An entangled state cannot be prepared by local
operations and classical communication, observers must make global operations
on the systems. To determine whether or not a general pure state
$|\psi_{ab}\rangle=\sum A_{ij}|i\rangle_{a}|j\rangle_{b}$ is entangled we
consider the following theorem:
> Schmidt decomposition: Let $\mathcal{H}_{a}$ and $\mathcal{H}_{a}$ be
> Hilbert spaces of dimension $d_{a}$ and $d_{b}$, respectively. For any
> vector $|\psi_{ab}\rangle\in\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ there
> exists a sets of orthonormal vectors
> $\\{|j\rangle_{a}\\}_{a=1,\ldots,d_{a}}\\!\subset\\!\mathcal{H}_{a}\mbox{
> and }\\{|l\rangle_{b}\\}_{b=1,\ldots,d_{b}}\\!\subset\\!\mathcal{H}_{b}$
> such that we can write
> $|\psi_{ab}\rangle=\sum_{i=1,\ldots,d}\lambda_{i}|i\rangle_{a}|i\rangle_{b}$.
> The Schmidt coefficients $\lambda_{i}$ are non-negative scalars such that
> $\sum_{i}\lambda^{2}_{i}=1$, and the sum runs over the minimum dimension
> $d=\textrm{min}(d_{a},d_{b})$ of the two Hilbert spaces.
This special basis is called the Schmidt basis [54, 55]. Note that the
correlations between systems A and B are now made explicit. For example, if
$\lambda_{i}=1$ and $\lambda_{j\neq i}=0$ then the state is separable. In the
case that all the $\lambda_{i}$’s are equal and $\lambda_{i}=1/\sqrt{d}$ then
the state is maximally entangled. It is clear that the distribution of the
Schmidt coefficients determine how entangled the state is. Therefore, to
quantify entanglement in the pure bi-partite case we need a monotonous and
continuous function of the $\lambda_{i}$’s such that
1. 1.
$S(\lambda_{i})=0$ for separable states,
2. 2.
$S(\lambda_{i})=\log(d)$ for maximally entangled states.
Considering the density matrix $\rho_{ab}=|\psi_{ab}\rangle\langle\psi_{ab}|$
and it’s reduced density matrix $\rho_{b}=Tr_{a}(\rho_{ab})$, we find that the
von Neumann entropy
$\displaystyle S(\rho_{b})$ $\displaystyle=$ $\displaystyle-
Tr\left(\rho_{b}\log_{2}\rho_{b}\right)=-\sum_{i}|\lambda_{i}|^{2}\log_{2}|\lambda_{i}|^{2},$
quantifies the entanglement between system A and B. We observe from the
Schmidt decomposition that it is equivalent to trace over either system A or B
and therefore, $S(\rho_{a})=S(\rho_{b})$. The von Neumann entropy of a pure
state is $S(\rho_{ab})=0$.
Quantifying entanglement in the mixed case is more involved since now there is
no analog to the Schmidt decomposition. However, it is still possible to
define a separable mixed state. A bipartite mixed state is separable if we can
write its density matrix as
$\rho_{ab}=\sum_{i}p_{i}\rho_{a}^{i}\otimes\rho_{b}^{i}$ where
$\sum_{i}p_{i}=1$.
To determine whether or not a general mixed state is entangled it is
convenient to define the partial transpose of a density matrix. Consider the
general bipartite mixed state
$\rho_{ab}=\sum_{ijkl}C_{ijkl}\,|i\rangle_{a}|j\rangle_{b}\,{}_{b}\langle
k|{}_{a}\langle l|\equiv\sum_{ijkl}C_{ijkl}\,|i\rangle_{a}{}_{a}\langle
l|\otimes|j\rangle_{b}{}_{b}\langle k|.$
The partial transpose $\rho^{PT}_{ab}$ of $\rho_{ab}$ is (transposing on
system $a$)
$\rho_{ab}^{PT}=\sum_{ijkl}C_{ljki}\,|i\rangle_{a}|j\rangle_{b}{}_{b}\langle
k|_{a}\langle l|,$
or equivalently (transposing on system $b$)
$\rho_{ab}^{PT}=\sum_{ijkl}C_{ikjl}|i\rangle_{a}|j\rangle_{b}{}_{b}\langle
k|_{a}\langle l|.$
Since the partial transpose of a separable state has positive eigenvalues it
is possible to construct a bipartite separability criterion: If the
eigenvalues of $\rho_{ab}^{PT}\geqslant 0$ then $\rho_{ab}$ is separable.
However, this criterion is only sufficient for $2\times 2$ and $2\times 3$
systems [56]. For systems of higher dimension the criterion is only necessary,
meaning that there are (bound) entangled states with positive partial
transpose. Such states are known as bound entangled states.
Adding up the negative eigenvalues gives an estimate of how entangled a state
is. Therefore, we will now define the negativity and logarithmic negativity
[57, 58] which are two entanglement monotones. The negativity of a density
matrix $\rho_{ab}$ is defined as the sum of the absolute values of the
negative eigenvalues of the partial transpose $\rho_{ab}^{PT}$,
$N(\rho_{ab}):=\frac{\|{\rho^{PT}_{ab}}\|-1}{2},$
where $\|\cdot\|$ denotes the trace norm
$\|X\|:=Tr\left[\sqrt{X^{\dagger}X}\right]$. The logarithmic negativity of a
density matrix $\rho_{ab}$ is defined as
$E_{N}(\rho_{ab}):=\log_{2}\|\rho_{ab}^{PT}\|.$
#### 2.2.2 Covariant Matrix formalism
The covariant matrix formalism is usually employed in quantum optics [59]. It
is a framework that involves simple mathematical tools which are applicable to
systems consisting of a finite number of harmonic oscillators. While in
general computing entanglement can be very involved, it has been shown that in
the case of Gaussian states (such as coherent, squeezed and thermal states)
the covariant matrix formalism can be employed to produce computable measures
of quantum information [60]. Gaussian states are described by quasi-
probability distributions of Gaussian shape in phase space [61]. This
technique is very useful in relativistic quantum information since Bogoliubov
transformations map Gaussian states back onto Gaussian states [20, 62, 63]. As
long as one considers initially Gaussian states and a finite number of field
modes it is possible to calculate entanglement and other interesting
quantities using the measures developed in this framework. In the standard
Fock space description of quantum field theory quantum states correspond to
density matrices which contain all the information pertaining to the state. In
the covariant matrix formalism the notion of density matrix is replaced by a
real symplectic covariant matrix $\sigma$ defined by
$\sigma_{ij}\,=\,\left\langle\,\mathrm{X}_{i}\mathrm{X}_{j}\,+\,\mathrm{X}_{j}\mathrm{X}_{i}\,\right\rangle\,-\,2\,\left\langle\,\mathrm{X}_{i}\,\right\rangle\left\langle\,\mathrm{X}_{j}\,\right\rangle,$
(5fqtwz)
where the operators $\mathrm{X}_{i}$ are the generalized positions and
momenta, i.e.
$\displaystyle\mathrm{X}_{(2n-1)}$
$\displaystyle=\frac{1}{\sqrt{2}}(a_{n}\,+\,a_{n}^{\dagger})\,,$ (5fqtwaaa)
$\displaystyle\mathrm{X}_{(2n)}$
$\displaystyle=\frac{-i}{\sqrt{2}}(a_{n}\,-\,a_{n}^{\dagger})\,,$ (5fqtwaab)
and the index $n=1,2,3,\ldots$ labels the modes, and
$\langle\,\mathcal{O}\,\rangle$ denotes the expectation value of the operator
$\mathcal{O}$ with respect to the initial Gaussian state. The matrix $\sigma$
together with the vector of first moments $\langle\,\mathrm{X}_{i}\,\rangle$
completely characterizes all Gaussian states. However, the covariance matrix
is sufficient to compute entanglement. Unitary transformations in the Fock
space are replaced in this formalism by symplectic transformations in phase
space. A transformation $S$ is called symplectic, if it leaves the symplectic
form $\Omega$ (a numerical matrix) invariant, i.e., $S\,\Omega\,S^{T}=\Omega$,
where $[X_{i},X_{j}]=2i\Omega_{ij}$ and $(S_{ij})^{T}=(S_{ji})$. The
expression for the symplectic representation of a Bogoliubov transformation in
terms of its general coefficients $\alpha_{mn}$ and $\beta_{mn}$ takes a
simple form [20]. The matrix $S$ is decomposed into $2\times 2$ blocks
${\mathcal{M}}_{mn}$ as
$S=\left(\begin{array}[]{cccc}{\mathcal{M}}_{11}&{\mathcal{M}}_{12}&{\mathcal{M}}_{13}&\ldots\\\
{\mathcal{M}}_{21}&{\mathcal{M}}_{22}&{\mathcal{M}}_{23}&\ldots\\\
{\mathcal{M}}_{31}&{\mathcal{M}}_{32}&{\mathcal{M}}_{33}&\ldots\\\
\vdots&\vdots&\vdots&\ddots\end{array}\right),$ (5fqtwaaab)
where the sub-blocks are given by
$\mathcal{M}_{mn}\,=\,\left(\begin{array}[]{cc}\Re(\alpha_{mn}\,-\,\beta_{mn})&\Im(\alpha_{mn}\,+\,\beta_{mn})\\\\[4.2679pt]
-\Im(\alpha_{mn}\,-\,\beta_{mn})&\Re(\alpha_{mn}\,+\,\beta_{mn})\\\
\end{array}\right)\,.$ (5fqtwaaac)
Here $\Re(z)$ and $\Im(z)$ denote the real part and imaginary part of the
complex number $z$ respectively. The transformed covariance matrix
$\tilde{\sigma}$ is then simply obtained as
$\tilde{\sigma}\,=\,S\,\sigma\,S^{T}\,.$ This transformation ensures that if
$\sigma$ is Gaussian, $\tilde{\sigma}$ remains Gaussian. Fortunately, for
Gaussian states computable measures of entanglement have been developed [60,
64]. For example, the negativity between two modes is given by
$\mathcal{N}=\max\\{0,(1-\widehat{\nu}_{-})/2\widehat{\nu}_{-}\\},$
(5fqtwaaad)
where $\widehat{\nu}_{-}$ is the smallest symplectic eigenvalue of the partial
transpose matrix
$\,\widehat{\sigma}_{kk^{\prime}}=T_{k^{\prime}}{\sigma}_{kk^{\prime}}\,T_{k^{\prime}}\,$,
where $T_{k^{\prime}}=\textrm{diag}\\{1,1,1,-1\\}$ partially transposes mode
$k^{\prime}$, see Ref. [61].
Computing the negativity between two modes $k$ and $k^{\prime}$ involves first
tracing over all other modes. Impressively, the partial trace over any subset
of modes is computed by eliminating all rows and columns corresponding to all
modes other than $k$ and $k^{\prime}$. The smallest symplectic eigenvalue
$\widehat{\nu}_{-}$ is then obtained by diagonalizing the matrix
$i\Omega\,\widehat{\sigma}_{kk^{\prime}}$ by a symplectic operation $D$,
yielding the two eigenvalues $0\leq\widehat{\nu}_{-}\leq\widehat{\nu}_{+}$.
The state ${\sigma}_{kk^{\prime}}$ is entangled if $0\leq\widehat{\nu}_{-}<1$.
## 3 Entanglement of global modes
### 3.1 Flat spacetime entanglement
In the Unruh effect the Minkowski vacuum is a thermal state for uniformly
accelerated observers. Since the Rindler regions $I$ and $II$ are causally
disconnected, uniformly accelerated observers in one Rindler region (wedge)
have no access to information from the other Rindler region. Therefore, the
state of the Rindler observer, adapted to his/her particular wedge by tracing
out over the unaccessible wedge, is mixed. The state appears more mixed for
observers with increasing acceleration. By this we mean that the temperature
associated to the state by a particular observer is higher for observers with
larger proper accelerations. This has an effect on entanglement [3, 4].
Consider a pure entangled Minkowski state of two field modes each of them
labeled by a different Minkowski frequency. We assume that two inertial
observers, Alice and Bob, are able to distinguish these modes and agree that
the state is maximally entangled. As typically considered in the literature,
Alice and a Rindler observer in region $I$, called Rob, also analyze the
state. Since Rindler observers in Region $I$ must trace over the region $II$
part of the state, the correlations of the state will be partially lost. Since
the vacuum itself exhibits greater noise for observers with higher
accelerations one also expects other states to appear more mixed and less
correlated for higher accelerations. Therefore, there is a degradation of
entanglement due to the Unruh effect. Let us point out that general region $I$
states are not necessarily thermal after tracing out over region $II$ states,
however, they are mixed. A thermal state is a particular type of mixed state
in which the fraction of the ensemble in each pure state is given by a
Boltzmann distribution. In the case of Gaussian states such as the vacuum,
thermal states are obtained after tracing over modes. Calculations involving
Minkowski states are very involved because a single Minkowski mode corresponds
to an infinite superposition (continuous) of Rindler modes (5fg) making the
trace operation highly non-trivial. It is therefore, more convenient from a
mathematical point of view to analyze inertial states involving Unruh modes
such as the Bell state,
$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|0_{k}\rangle^{\mathcal{M}}|0_{\Omega}\rangle^{\mathcal{U}}+|1_{k}\rangle^{\mathcal{M}}|1_{\Omega}\rangle^{\mathcal{U}}\right),$
(5fqtwaaae)
where ${\mathcal{M}}$ and $\mathcal{U}$ label Minkowski and Unruh states
respectively with frequencies $k$ and $\Omega=\omega/a$. This state was
introduced in Bruschi, et.al [43]. For more details on the use of Unruh modes
to analyze entanglement and the single mode approximation see the appendix,
section 8. For inertial observers the modes $k$ and $\Omega>0$ (which must be
two distinguishable modes) are maximally correlated since the Bell state is
maximally entangled. It is then interesting to investigate to what degree the
state is entangled when described by observers in uniform acceleration. In the
simplest scenario shown in Fig. (1), Alice is considered to be inertial and a
uniformly accelerated observer Rob is introduced, who analyzes mode $\Omega$.
To study this situation, the states corresponding to Rob must be transformed
into the appropriate basis, in this case, the Rindler basis. We have already
calculated the transformation for the vacuum state in (2.1.1), and with that
in hand, we find the single particle Unruh state
$a_{\Omega,U}^{\dagger}|0_{\Omega}\rangle^{\mathcal{U}}=\left|1_{\Omega}\right>^{\mathcal{U}}=\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\sqrt{n+1}\left|(n+1)_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II},$
(5fqtwaaaf)
where we have chosen $|q_{R}|=1$ in Eq. (5fj). Thus, the maximally entangled
state seen by inertial Alice and accelerated Rob is
$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|0_{k}\right>^{\mathcal{M}}\otimes\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\left|n_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II}$
(5fqtwaaag) $\displaystyle+$
$\displaystyle\frac{1}{\sqrt{2}}\left|1_{k}\right>^{\mathcal{M}}\otimes\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\sqrt{n+1}\left|(n+1)_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II}.$
Since Rob, with trajectory in region $I$, is causally disconnected from region
$II$ we must take the trace over region $II$. The density matrix for the
Alice-Rob subsystem is
$\rho_{AR}=\frac{1}{\cosh^{2}(r)}\sum_{n=0}^{\infty}\tanh^{2n}(r)\rho_{n},$
(5fqtwaaah)
where
$\displaystyle\rho_{n}$ $\displaystyle=$
$\displaystyle\left|0_{k},n_{\Omega}\right>\left<0_{k},n_{\Omega}\right|+\frac{\sqrt{n+1}}{\cosh(r)}\left(\left|0_{k},n_{\Omega}\right>\left<1_{k},(n+1)_{\Omega}\right|+\left|1_{k},(n+1)_{\Omega}\right>\left<0_{k},n_{\Omega}\right|\right)$
$\displaystyle+$
$\displaystyle\frac{n+1}{\cosh^{2}(r)}\left|1_{k},(n+1)_{\Omega}\right>\left<1_{k},(n+1)_{\Omega}\right|,$
involves only ${\mathcal{M}}$ and region $I$ states. The entanglement between
the modes can be quantified by the logarithmic negativity yielding
$\mathcal{N}(\rho_{AR})=\log_{2}((1/2\cosh^{2}(r))+\Sigma)$ where
$\Sigma=\sum_{n}\frac{\tanh^{2n}(r)}{2\cosh^{2}(r)}\sqrt{\left(\frac{n}{\sinh^{2}(r)}+\tanh^{2}(r)\right)^{2}+\frac{4}{\cosh^{2}(r)}}.$
(5fqtwaaai)
One observes that entanglement is degraded for observers in uniform
acceleration and therefore, it is observer dependent. In the flat spacetime
case, one can argue that this is an effect of Rob being non-inertial. Rob must
be in a spaceship to be accelerated and energy must be supplied to accelerate
the system. In the flat spacetime case, inertial observers play a special role
and therefore, a well defined notion of entanglement corresponds to the
entanglement described from the inertial perspective. However, in curved
spacetime different observers describe a different particle content in the
field which results in different degrees of entanglement in the field. In that
case, there is no well-defined notion of entanglement.
Using Unruh rather than Minkowski states considerably simplifies the
mathematical techniques involved in the analysis of entanglement. However, it
introduces interpretational issues [65]. The physical interpretation of Unruh
states requires deeper understanding. Work in progress shows that finite size
accelerated detectors naturally couple to peaked distributions of Unruh modes
[51]. Such detectors will provide further insights on the states considered in
Eq. (5fqtwaaae). Currently, it is only possible to say that the work of
Bruschi, et.al. [43] studies the entanglement of a family of maximally
entangled states parameterized by the dimensionless $\Omega=\omega/a$ in non-
inertial frames. By fixing the physical frequency $\omega$ and changing $a$
one analyzes the entanglement in a family of states, all of which have the
same frequency $\omega$ as seen by observers with different proper
acceleration $a$. An alternative view is that the entanglement corresponds to
a family of states with different physical frequency $\omega$ as seen by the
same observer moving with fixed proper acceleration $a$ [66].
The analysis of entanglement in non-inertial frames has been generalized in
several directions and different quantum fields have been considered. For
example, the work by Adesso, et. al. [62] considers two-mode squeezed states
in the inertial frame (instead of the Bell states analyzed above) allowing
them to employ the covariance matrix formalism which is only applicable for
Gaussian states. The authors studied the case when both observers accelerate
finding that entanglement vanishes at finite acceleration. This effect is also
known in the literature as sudden death of entanglement [67]. The effect is
similar to that which occurs in parametric down conversion when a two-mode
entangled state is feed into an amplification channel (see also [68]). It was
shown that the rate of entanglement degradation depends on the entanglement in
the inertial frame. States with higher degrees of inertial entanglement
degrade more in non-inertial frames. The degradation of entanglement has also
been analyzed for inertial thermal states and non-maximally entangled states
[69].
Other types of correlations relevant to quantum information theory have also
been studied in non-inertial frames. For example, it was found that classical
correlations are conserved in case one observer is non-inertial [3, 4] and
degraded when both observers accelerate [62]. More recently, the quantum
discord [70, 71], a type of purely quantum correlations which are present even
in separable states, was analyzed in non-inertial frames. Datta [72] found
that in a regime where there is no distillable entanglement there is a finite
amount of quantum discord. Interestingly, in the limit of infinite
acceleration the discord does not vanish. Multi-partite entanglement has also
been subject of study for observers in uniform acceleration [4, 62]. The bi-
partite entanglement between modes is degraded in non-inertial frames,
however, three and four-partite correlations are created between modes in
regions $I$ and $II$ [62, 73]. Multipartite entanglement has also been
analyzed when the state contains tri-partite correlations in the inertial
frame [74].
The analysis of entanglement in non-inertial frames has been considered for
different types of fields. For example, Dirac fields [4, 69, 74, 75, 76, 77,
78], the electromagnetic field [79] and charged bosonic fields have been
studied [66]. The results for fermionic fields show that entanglement is
degraded in non-inertial frames however, the entanglement remains finite in
the infinite acceleration limit [4, 69, 77, 80]. Interestingly, for fermions
the entanglement in the infinite acceleration limit does not violate Bell’s
inequalities [78]. Multipartite entanglement also presents interesting
differences. In the bosonic case genuine tripartite entanglement is generated
amongst the modes of Alice, Rob and Anti-Rob (an observer with uniform
acceleration in region $II$) [62]. These correlations increase with
acceleration. However, in the fermionic case tripartite entanglement is always
zero[4]. Recent work [74] considers an inertial tripartite state which
contains genuine tripartite correlations, as measured by the Svetlichny
inequality [81]. These correlations persist for any finite acceleration for
both GGHZ and MS three-qubit states provided the respective control parameters
are appropriately chosen, vanishing only in the infinite acceleration limit.
This indicates that tripartite entanglement and its associated nonlocal
correlations are more robust to relativistic effects than their bipartite
counterparts. It is important to mention that an ambiguity on the definition
of entanglement between fermionic fields has been pointed out in [82, 83].
However, these results are subject of a present debate [84, 85]. (Note that in
[4], signs arising from the transposition of anti-commuting fermionic
operators could always be absorbed into an explicit phase that was introduced,
and hence into a redefinition of the fermionic creation operators).
The main differences in the entanglement degradation for scalar and fermionic
fields arise due to particle statistics [66, 77, 86, 87] and not to
dimensionality of the Hilbert space. In the fermionic case, a redistribution
of entanglement between particle and antiparticle states seams to prevent the
degradation of entanglement in the infinite acceleration limit [77]. Such
redistribution does not occur in the bosonic case [66]. Initially it was
conjectured that the Hilbert space dimensionality played a role [4, 69]. In
the bosonic case the Hilbert space is infinite dimensional while in the case
of Grassmann scalars, the lowest dimensional fermions, the fermionic Hilbert
space is of dimension two. One might suspect that entanglement is degraded
stronger in the bosonic case since for higher accelerations the bosonic
harmonic oscillators become populated with increasing temperature while
Grassmann scalars are always in mixed states which involve only two states.
Considering higher dimensional fermions [88] and bosonic fields with truncated
Hilbert spaces [86] showed that it is statistics rather than dimensionality
which produces the observed differences in the degradation of entanglement in
non-inertial frames. In the case of entanglement degradation for
electromagnetic fields it was found that photon helicity entangled states do
not degrade with acceleration [79].
### 3.2 Entanglement in curved spacetime.
General curved spacetimes do not have global timelike Killing vector fields
and therefore it is not possible to define subsystems. Without the notion of
subsystem it is not possible to study entanglement. In spite of this, special
cases have been analyzed. For instance, black hole spacetimes where the
spacetime is approximately flat at the horizon, and Robertson-Walker universes
which have two asymptotically flat spacetime regions.
#### 3.2.1 Entanglement across black hole horizons
The Schwarzchild spacetime of an eternal black hole describes the geometry of
a spherical non-rotating mass $m$ [30]. Considering only the radial component,
the metric is
$\displaystyle ds^{2}=(1-(2m/R))dT^{2}-(1-(2m/R))^{-1}dR^{2}.$
where the horizon of the black hole is at $R=2m$. Considering the following
coordinate change $R-2m=x^{2}/8m$, such that
$1-(2m/R)=x^{2}/8mR=\frac{(x^{2}/8m)}{(x^{2}/8m+2m)}=\frac{(Ax)^{2}}{(1+(Ax)^{2})}\approx(Ax)^{2}$
when $x\approx 0$ with $A=1/4m$. This means that $dR^{2}=(Ax^{2})dx^{2}$.
Therefore, very close to the horizon $R\approx 2m$, the Schwarzschild
spacetime can be approximated by Rindler spacetime
$\displaystyle ds^{2}=-(Ax)^{2}dT^{2}+dx^{2},$
where the acceleration parameter $a=A^{-1}=4m$.
This means that, very close to the horizon of the black hole, we can consider
Alice as being an inertial observer and falling into the black hole, while Rob
escapes the fall by being a stationary, accelerated observer. If Alice claims
that the state of the field is the vacuum state, then Rob detects a thermal
state of the form
$\left|0_{k}\right>^{\mathcal{\mathcal{M}}}=\frac{1}{\cosh(q)}\sum_{n}\tanh^{n}(q)\left|n_{k}\right>^{in}\left|n_{k}\right>^{out},$
(5fqtwaaaj)
where $\cosh(q)=(1-e^{2m\pi\omega})^{-1/2}$ is a function of the mass of the
black hole $m$. Here $in$ and $out$ denote the modes inside and outside the
black hole. $\left|0_{k}\right>^{\mathcal{\mathcal{M}}}$ is the state detected
by Alice. If we then consider the case where two modes of the field in Alice’s
frame are maximally entangled, Rob will detect less entanglement between the
modes due to the Hawking effect [3, 73, 89, 90]. Using the covariant matrix
formalism an inertial two-mode squeezed state was analyzed [73]. The
entanglement between the modes is degraded for observers outside the black
hole, however genuine four-partite entanglement between the modes inside and
outside the black hole are created. For low black hole masses entanglement
vanishes and classical correlations also degrade. The analysis of entanglement
degradation has also been considered taking into account the distance of the
observer to the event horizon [91] by writing the metric in terms of the
proper time $\tau_{0}$ of a stationary observer placed at $r_{0}$,
$\displaystyle ds^{2}=-(\tilde{A}x)^{2}d\tau_{0}^{2}+dx^{2}$
which yields a modified acceleration
$\tilde{A}=A/\sqrt{1-2m/r_{0}}=(4m\sqrt{1-2m/r_{0}})^{-1}$ giving rise to the
same state as in Eq. (5fqtwaaaj) with a modified squeezing parameter
$\tilde{q}$ given by
$\cosh(\tilde{q})=(1-e^{2m\pi\omega/\sqrt{1-2m/r_{0}}})^{-1/2}$. This study
has been also carried out in the fermionic case [87, 88], and in higher
dimensional black holes [92].
#### 3.2.2 Entanglement in an expanding universe
Our second curved space example is the Robertson-Walker universe [30]. In this
spacetime particles can be defined in two asymptotically flat regions.
Considering the field in the remote past to be in the vacuum state, it is
found that in the distant future the state contains particles. In the interim
region, where the universe is undergoing expansion, no sensible notion of
particles exist. In [31] it was shown that in the future infinity region
entanglement has been created between field modes. Interestingly, it is
possible to learn about the expansion parameters of the Universe from the
entanglement generated. The vacuum state in the past infinity corresponds to
the following two mode states in the future infinity
$|0\rangle^{in}=\sqrt{1-\gamma}\sum_{n}\gamma^{n}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}.$
Since the state is pure, we can employ the von Neumann entropy to quantify the
entanglement generated the field modes $k$ and $-k$. In order to do this we
need to compute the reduced density matrix for one of the modes. The density
matrix for the state is
$\displaystyle\rho_{0}$ $\displaystyle=$ $\displaystyle|0\rangle^{in}\langle
0|^{in}=(1-\gamma)\sum_{n,m}\gamma^{(n+m)}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}\langle
m|_{k}^{out}\langle m|_{-k}^{out}.$ (5fqtwaaak)
The reduced density matrix for mode $k$ is obtained by tracing over mode $-k$
$\displaystyle\rho_{k}$ $\displaystyle=$ $\displaystyle
tr_{-k}[\rho_{0}]=(1-\gamma)\sum_{n,m}\gamma^{(n+m)}\delta_{nm}|n\rangle_{k}^{out}\langle
m|_{k}^{out}$ (5fqtwaaal) $\displaystyle=$
$\displaystyle(1-\gamma)\sum_{n,m}\gamma^{2n}|n\rangle_{k}^{out}\langle
n|_{k}^{out}.$
Since the reduced denstiy matrix is already in diagonal form with eigenvalues
$\lambda_{n}=(1-\gamma)\gamma^{2n}$, it is straight forward to compute the von
Neumann entropy
$S(\rho_{k})=-(1-\gamma)\sum_{n}\gamma^{2n}\log_{2}((1-\gamma)\gamma^{2n})=\log_{2}\left[\frac{\gamma^{\gamma/(\gamma-1)}}{1-\gamma}\right].$
(5fqtwaaam)
Entanglement has been created in the remote future due to the expansion of the
universe. The entanglement depends on the cosmological constants since the
coefficient $\gamma$ depends on the expansion rate $\sigma$, the expansion
volume $\epsilon$ and the frequency of the modes involved through
$\gamma=\frac{\sinh^{2}(\pi\omega_{-}/\sigma)}{\sinh^{2}(\pi\omega_{+}/\sigma)},$
(5fqtwaaan)
with $\omega_{\pm}=1/2(\omega_{out}\pm\omega_{in})$,
$\omega_{in}=[k^{2}+m^{2}]^{\frac{1}{2}}$ and
$\omega_{out}=[k^{2}+m^{2}(1+2\epsilon)]^{\frac{1}{2}}.$ In the case of light
particles, the equations can be inverted and we can show that we can estimate
the expansion parameters from the entanglement. For a calculation in the
covariant matrix formalism see [20]. The analysis of entanglement in an
expanding universe has been extended to Dirac fields [34] and entanglement in
particle creation has also been studied in [93].
## 4 Extracting global mode entanglement using point-like detectors
Although early investigations on relativistic entanglement allowed us to
understand some aspects of the observer dependent nature of entanglement, the
results are purely academic. Global mode entanglement cannot be measured nor
manipulated by local observers. Alice would not be able to take an entangled
state and apply, for example, a CNOT gate on it since the states live in the
whole spacetime. This would require infinite time and energy making impossible
any practical quantum information task. However, it has been shown that global
mode entanglement can be accessed by local observers employing Unruh-Dewitt
detectors [7, 8, 9, 10, 12]. Two Unruh-deWitt detectors at rest can become
entangled by interacting with the Minkowski vacuum even if they are space-like
separated. This shows that the vacuum state is entangled. The vacuum is not
mode-wise entangled however, it contains entanglement in spatial degrees of
freedom. This entanglement is extracted, or transferred (swapped in the
quantum information language) from the field to the detector’s degrees of
freedom. In principle, it should be possible to use Unruh-Dewitt detectors to
study some aspects of the global mode entanglement analysis within a more
practical approach [94].
Since Unruh-Dewitt detectors are local, one can consider them as suitable
systems for the implementation of quantum information tasks. For example,
Alice and Bob could be spacelike separated and entangle thier detectors
through the interaction with the vacuum. This entanglement could be exploited
for quantum information tasks. The entanglement extracted is currently too
small to be useful, but in the future, enhancement schemes may be developed.
At the moment, the main limitation in using Unruh-Dewitt detectors for quantum
information processing is that the mathematical techniques involved already
become un-manageable with a few systems. For example, for two detectors one
must consider the interaction Hamiltonian
$\displaystyle\hat{H}_{I}(\tau)$ $\displaystyle=$ $\displaystyle\int
dk\lambda(\tau_{1})(d_{1}\,e^{i\Omega_{1}\tau_{1}}+d_{1}^{\dagger}\,e^{-i\Omega_{1}\tau_{1}})\left(a_{k}\,e^{i(kx_{1}(\tau_{1})-\omega
t_{1}(\tau_{1}))}+a_{k}^{\dagger}\,e^{-i(kx_{1}(\tau_{1})-t_{1}(\tau_{1}))}\right),$
$\displaystyle+$ $\displaystyle\int
dk\lambda(\tau_{2})(d_{2}\,e^{i\Omega_{2}\tau_{2}}+d_{2}^{\dagger}\,e^{-i\Omega_{2}\tau_{2}})\left(a_{k}\,e^{i(kx_{2}(\tau_{2})-\omega
t_{2}(\tau_{2}))}+a_{k}^{\dagger}\,e^{-i(kx_{2}(\tau_{2})-t_{2}(\tau_{2}))}\right),$
for which entanglement calculations involve $4th$-order perturbation theory
[8, 9, 10]. Current research is focused on finding ways to modify the
detectors in order to make them more mathematically accessible [51].
In spite of the mathematical difficulties involved, interesting results have
been found in relativistic quantum information using point-like detectors. For
example, the entanglement between an Unruh-DeWitt detector and a massless
bosonic field has been considered in [22] showing that the field and detector
remain entangled at late times. Information stored in the point-like system
flows into the field propagating with the radiation into null infinity. The
information lost by the detector is never restored (non-Markovian regime).
Another analysis considers two detectors, one inertial belonging to Alice, and
one in uniform acceleration belonging to Rob, which do not interact directly
but are coupled to the same field [24]. The detectors are initially entangled
and their entanglement vanishes after a finite time. The time at which the
entanglement vanishes is shorter for higher accelerations, as expected. This
setting has been considered recently for teleportation of a coherent state
[11]. When both detectors are at rest and initially entangled, the detectors
also disentangle and the dependence with their separation is studied in [23].
The case of Rob having a detector in non-uniform acceleration has been studied
in [24]. The authors show that the disentangling time slows down if the motion
is non-uniform. Two uniformly accelerated detectors have also been considered,
each on of them in a different Rindler wedge [7, 9]. The authors show that
entanglement is generated through their interaction with the Minkowski vacuum
even if they are always causally disconnected. This shows again that the
vacuum contains quantum correlations. Another interesting analysis is that of
the creation and loss of entanglement between detectors moving in circular
paths [25].
Recently, Olson and Ralph [10] found that by independently quantizing the past
and future spacetime regions of a massless quantum field it is also possible
to extract vacuum entanglement from timelike separated regions. To extract the
entanglement they considered two timelike separated point-like detectors at
rest characterized by a strongly time-dependent energy gap. Time-like
entanglement has been further analyzed in [95] and exploited in a
teleportation-type protocol [13, 14] which can be implemented in a circuit QED
experiment [14].
In curved spacetimes the entanglement between point-like detectors can been
used to distinguish between two universes. Two inertial detectors in flat
spacetime would become entangled if they interacted with a Minkowski thermal
field however, they would not become entangled in an exponentially expanding
de Sitter spacetime [96].
## 5 Accelerated cavities and localized wave-packet for quantum information
Work on the use of Unruh-Dewitt detectors for relativistic quantum information
processing has inspired efforts to find suitable systems to store information
in the framework of quantum field theory, with the hope of developing
mathematical techniques which render the calculations accessible. At the
moment moving cavities and localized wave-packets seem plausible systems for
this purpose.
### 5.1 Entanglement creation and degradation in moving cavities
The idea of using cavities moving in spacetime for relativistic quantum
information processing was proposed in [5] where two observers Alice and Rob
share an entangled state of two cavities as a resource for quantum
teleportation in non-inertial frames. The mathematical techniques to implement
this idea were introduced in [15] for inertial and uniformly accelerated
cavities and in [16] for non-uniform motion. The authors in [15] showed that
it is possible to entangle two cavities, one inertial and the second in
uniform acceleration, by letting a single two level atom interact with the
cavities. The scheme is a generalization of [97] in which the modes of two
cavities at rest positioned next to each other become maximally entangled
after an excited atom moving through them emits an excitation. Since it is not
possible to know in which cavity the atom emitted, the cavities become
entangled. When one of the cavities is in uniform acceleration the cavity
modes become detuned affecting the atom’s emission probabilities. The modes of
an inertial and accelerated cavity are given in (5fo) and (2.1.3),
respectively. The interaction Hamiltonian considered by the authors is of the
Unruh-Dewitt type given in (4) only in the sense that the atom is a two-level
ponit-like system interacting with discrete cavity modes. The authors show
that the ability to maximally entangle the cavities is reduced with
acceleration. However, by appropriately modifying the cavity length it is
possible to alter the modes and compensate for the effect. An advantage of
this scheme is that the quantum information stored in cavities is protected
from the Unruh effect (unlike for information stored in point-like systems) by
the mirrors. As long as the cavity walls are perfectly reflecting the field
inside the cavity is independent from the field outside. While the cavity
undergoes uniform acceleration (again we consider the case $aL/c^{2}\ll 1$ so
that the acceleration is essentially constant across the proper length of the
cavity), the states within the cavity however, remain pure since the Unruh
horizon is not contained in this spacetime region [52]. The quantum
information stored in field modes remains constant at all times as long as the
cavities are inertial or in uniform acceleration. However the situation
changes when the motion is non-uniform.
Consider a single inertial cavity with all modes in the vacuum state. If the
cavity accelerates for a finite time the modes will become populated as shown
is section 2.1.3. Working in the low acceleration regime $h<1$ it is possible
to find analytical expressions for the Bogoliubov coefficients for bosons [16]
and fermions [18] and calculate the entanglement produced between two modes
$k$ and $k^{\prime}$ using the covariant matrix formalism [98]. The
entanglement quantified by the logarithmic negativity is given by
$\mathcal{N}=|\beta^{(1)}_{k,k^{\prime}}|$ which is the first order correction
to the Bogoliubov coefficient $\beta_{k,k^{\prime}}$ [20]. The entanglement
between the modes can be calculated for any trajectory composed of segments of
inertial and uniformly accelerated segments. The entanglement in a basic
building block composed of initial and final inertial segments with a single
period of uniform acceleration is periodic in the acceleration period. A
consequence of this is that if one maximally entangles two modes each in a
different cavity as proposed in [15] and then accelerates for a finite time,
the entanglement between those modes will be degraded since the initial
information will be transferred to other modes. Since the evolution of
entanglement is periodic, Rob can plan his travel such that at the end of it
his cavity will still be maximally entangled with Alice’s.
Interesting conclusions can be drawn from comparing the analysis of
entanglement generation in fermionic and bosonic cavities [98]. In both cases
entanglement is always generated under non-uniform motion however, the
entanglement generated is different due to particle statistics. In the bosonic
case motion can populate modes with the characteristic that any number of
excitations can be produced in a given mode. However, fermionic statistics
constrains the number of particles in each mode giving rise to differences in
the entanglement produced.
Another interesting property of motion generated entanglement is that for any
trajectory segment constructed from periods of inertial and uniformly
accelerated motion, there is an entanglement resonance when the segment is
repeated $N$ times [19]. It can be shown analytically that when the frequency
associated with the segment period is equal to the sum of the frequencies of
the two modes then the entanglement between those two modes grows linearly
with $N$. The presence of the resonance is independent of the segment travel
details however the degree of entanglement does depend on the specific
trajectory. Using the covariant matrix formalisms it is possible to calculate
the entanglement at resonance of any trajectory analytically. Interestingly,
through these resonances it is possible to produce motion generated quantum
gates [19, 21, 29]. Recent results also show that genuine multipartite
entanglement is generated through the periodic motion of the cavity [21].
The general case of a cavity undergoing an arbitrary acceleration and the
possibility of performing actual experiments with mechanically oscillating
optical cavities are under study [29]. In the cases of a linear sinusoidal or
a uniform circular motion a resonance appears at much lower frequencies for
which no new photons are generated. The resonance leads to the generation of
entanglement between existing and previously non-entangled cavity modes [29].
Understanding the effects of gravity on quantum properties and quantum
information is of great interest. The analysis for entanglement in moving
cavities already allows us to draw some interesting conclusions: via the
equivalence principle [50] the results suggest that gravity can create
entanglement in the field modes of a cavity (as long as the cavity is small).
Consider a cavity in the vacuum held at a fixed position in a uniform
gravitational field. The modes of the cavity would become entangled by
changing the cavity’s position and letting the cavity undergo periods of free
fall.
### 5.2 Localized wave packets
An alternative for constructing localized field states is achieved by
superimposing global plane wave modes $U_{k}(x)=e^{ik_{\mu}x^{\mu}}$ localized
by a complete set of shape functions $f_{m}(k)$ via $U_{mk}(x)=\int
dkf_{m}(k)U_{k}(x)$ [41, 99]. A common example involves “box normalization” in
which one quantizes in a box of length $L$ and imposes periodic boundary
conditions, leading to normalized states $f_{m}^{box}=L^{-1/2}e^{i2\pi mx/L}$.
One can also normalize in both momentum and space degrees of freedom by using
functions [41] $f_{ml}(k)=\chi_{m}(k)e^{-i2\pi lk/\epsilon}$ where
$\chi_{m}(k)=\epsilon^{-1/2}$ for $|k-m\epsilon|<\epsilon/2$. Here $m,l$ are
integers and $\epsilon$ is a small positive constant with dimension of inverse
length. The additional degree of freedom represented by the index $l$
expresses the location of the box. In box normalization one takes the limit as
$L\rightarrow\infty$, while in the latter procedure one effectively uses and
infinite number of boxes of length $\sim 2\pi\epsilon$.
### 5.3 Quantum communication with moving wave-packets
As one recently proposed example of a relativistic quantum information
protocol, Downes et al. [26] consider the case of Alice sending coherent
states to Rob’s accelerated detector, who then performs homodyne detection.
Here the authors bypass the intermediate Unruh modes and consider the full
integral Bogoliubov transformation (5fg) from which they form localized modes
in frequency, as in the first equality in (5fqtwaaawaybe) in the appendix
(section 8). In addition, they eschew the Schrodinger state transformation
approach and instead consider the transformation of observables in the
Heisenberg picture. With Alice stationary at an $x$-coordinate position to the
right of focus of Rob’s hyperbola at $x=1/a$ ($\chi=0$), they consider only
left moving $L$ modes that travel along geodesics with $x+t$ constant. Rob’s
homodyne operator is given by
$\hat{O}(\tau)=a_{i,I}^{S}\,a_{i,I}^{L\dagger}\,e^{i\phi}+h.c.$ where the
superscripts $K=S,L$ denote the signal and local oscillator modes (both of
which are sent to Rob by Alice as localized Minkowski frequency modes) and the
subscript $i$ denotes modes (labeled by components of the wavevector), both
parallel (longitudinal) and perpendicular (transverse) to Rob’s acceleration.
Rob integrates the photocurrent from his detector over a long time compared to
the pulse length of Alice’s signal. Thus the average value of the signal
received by Rob is given by the expectation value $X=\langle\int
d\tau\hat{O}_{i}(\tau)\rangle$ with variance $V=\langle(\int
d\tau\hat{O}_{i}(\tau))^{2}\rangle-X^{2}$, with the Minkowski coherent state
$|\alpha,\beta,t\rangle_{i}=D^{S}_{i}(\alpha)D^{S}_{i}(\beta)|0\rangle_{M}$.
Here $D^{K}_{i}(\gamma)=\exp(\gamma a^{K}_{i}-\gamma^{*}a^{K\dagger}_{i})$ is
the Minkowski displacement operator with $\gamma=\alpha,\beta$ the amplitude
of the signal and local oscillator respectively. The authors make reasonable
approximations for matching the transverse spatial profile of the signal and
local oscillator to Rob’s detector (“beam-like” communication between Alice
and Rob), and further assume that the longitudinal source wavevector is
strongly peaked about a central frequency $\omega_{so}$. They obtain the
result that the normalized variance, defined as the variance $V$ divided by
the amplitude of the local oscillator $\beta_{R}^{2}$ as seen by Rob is given
by the following Planck factor
$\bar{V}=V/\beta^{2}_{R}=1/(e^{2\pi\omega_{so}(x+t)}-1)$.
The “novel” dependence of the Planck factor $\bar{V}$ on $x+t$ is correctly
attributed by the authors to the time dependent Doppler shift of the signal
frequency $\omega_{so}$ sent by Alice as measured by Rob [42, 100, 101]. This
should not be confused with the usual Planck factor
$1/(e^{2\pi\omega_{d,R}}-1)$ associated with the spectrum of frequencies
$\omega_{d,R}$ Rob’s detector measures when intercepting a Minkowski plane
wave $e^{i\omega_{so}(x+t)}$ from Alice. The two are closely related as
follows. The signal sent by Alice is centered on the 4-wavevector
$k^{\mu}_{so}=(\omega_{so},-\omega_{so},0,0)$, where we have taken
$k_{x,so}=-\omega_{so}$ for a left moving photon. For Rob’s trajectory
$x^{\mu}=1/a\,(\sinh(a\tau),\cosh(a\tau),0,0)$ his 4-velocity is given by
$u^{\mu}_{R}=(\cosh(a\tau),\sinh(a\tau),0,0)$. By the previous discussion on
Killing vectors in section 2.1.1, the frequency measured by Rob of this photon
is given by $\omega_{R}(\tau)=k^{\mu}_{so}\,u_{\mu,R}=\omega_{so}e^{a\tau}$.
By substituting in $x+t$, the variance can be written as
$\bar{V}=1/(e^{2\pi\omega_{R}(\tau)/a}-1)$, that is, the variance depends on
the time dependent Doppler shifted frequency $\omega_{R}(\tau)$ that Rob
measures when he detects a Minkowski plane wave
$e^{i\omega_{so}(x+t)}=e^{i\omega_{R}(\tau)/a}$ of fixed Minkowski frequency
$\omega_{so}$. This Minkowski plane wave is not detected as a positive
frequency Rindler plane wave $e^{-i\omega_{d,R}\tau}$ by Rob, but rather as a
combination of positive and negative frequency Rindler plane waves $e^{\mp
i\omega_{d,R}\tau}$. The square of the Fourier transform $S$ of
$e^{i\omega_{so}(x+t)}$ as analyzed by Rob
$S=\int_{-\infty}^{\infty}d\tau\,e^{-i\omega_{d,R}\tau}\,e^{i\omega_{R}(\tau)/a}$,
yields the Planck factor $1/(e^{2\pi\omega_{d,R}}-1)$ [100, 101]. In fact, for
finite integration times
$S(\tau)=\int_{0}^{\tau}d\tau\,e^{-i\omega_{d,R}\tau}\,e^{i\omega_{R}(\tau)/a}\equiv\int_{0}^{\tau}d\tau\,e^{i\varphi(\tau)}$
the phase of the integrand has a stationary point [42, 102] at
$d\varphi(\tau)/d\tau=0$ yielding the condition
$\omega_{d,R}=\omega_{so}\,e^{a\tau}$ which is the frequency that appears in
$\bar{V}$. Previously wave packets for relativistic quantum information where
considered in [103]. Recently a model on how to detect localized field modes
has been introduced [27]. This model can also be used to study entanglement
between localized wavepackets in non-inertal frames [28].
## 6 Change of state under Lorentz transformations
While the main body of this work has been concerned with observer dependent
entanglement for one or more observers undergoing constant acceleration, there
is a large body of research on closely related effects for purely inertial
observers (the zero acceleration case). This work is loosely classified under
the heading of the change in quantum states under Lorentz transformations
(LTs), and is intimately tied to the concept of Wigner rotation and the
unitary transformation of massive and massless particles under the Poincare
group (translations, rotations and boosts). In the following, we review this
research, which continues to be very active today, especially in connection to
the transformation of entangled quantum states in curved spacetime.
#### 6.0.1 Preliminaries: Wigner rotation
In flat (Minkowski) spacetime, the positive energy, single particle states of
a massive particle forms a spinor representation of the inhomogeneous Lorentz
(Poincare) group [104, 105] consisting of ten generators: four translations
$P^{\mu}=(P^{0},P^{i})$, three generators $\\{J^{i}\\}$ of angular momentum
that generate spatial rotations, and three generators $\\{K^{i}\\}$ of pure
boosts. The quantum states, denoted by
$|\mbox{{\boldmath$p$}},\lambda\rangle$, are labeled by their spatial
3-momentum $p$ (where $p^{\mu}=(p^{0},\mbox{{\boldmath$p$}})$ with
$p^{0}=E=\sqrt{\mbox{{\boldmath$p$}}^{\hskip 1.75pt2}+m^{2}}\,$), and
$\lambda$ the component of angular momentum along a quantization axis in its
rest frame (typically taken to be along the third ($z$) spatial direction).
Under a Lorentz transformation $\Lambda$ the one-particle state transforms
under the unitary transformation $U(\Lambda)$ as [104, 105]
$U(\Lambda)|\mbox{{\boldmath$p$}},\lambda\rangle=\sum_{\lambda^{\prime}}\,D^{(j)}_{\lambda^{\prime}\lambda}(W(\Lambda,p))\,|\mbox{{\boldmath$p$}}_{\Lambda},\lambda^{\prime}\rangle,$
(5fqtwaaao)
where $j$ is total angular momentum of the particle (equal to the spin of the
particle in its rest frame), the summation is over
$\lambda^{\prime}=(-j,-j+1,\ldots,j)$, and $\mbox{{\boldmath$p$}}_{\Lambda}$
are the spatial components of the Lorentz transformed 4-momentum, i.e.
$\mbox{{\boldmath$p$}}^{\prime}$ where $p^{\prime\mu}=\Lambda^{\mu}_{\hskip
3.5pt\nu}\,p^{\nu}$. Most research has been primarily concerned with massive
spin-$\frac{1}{2}$ particles $(j=\frac{1}{2})$, and massless spin-1 particles
($j=1$, photons). While we will primarily discuss spin $\frac{1}{2}$
particles, many related issues carry over to the case of photons, with
important technical distinctions due to their absence of mass. In (5fqtwaaao),
$D^{(j)}_{\lambda^{\prime}\lambda}(W(\Lambda,p))$ is a $(2j+1)\times(2j+1)$
matrix spinor representation of the rotation group $SU(2)$. The quantity
$W(\Lambda,p)$ is called the Wigner rotation (matrix), and depends on both the
Lorentz transformation $\Lambda$ describing the (constant velocity) inertial
reference frame of the observer and the 4-momentum $p$ of the particle
observed. The important point to note about (5fqtwaaao) is that a single basis
state $|\mbox{{\boldmath$p$}},\lambda\rangle$ is transformed by a unitary
representation of a momentum dependent rotation $W(\Lambda,p)$ into a
superposition of all $\lambda$ (evaluated at the transformed momentum
$\mbox{{\boldmath$p$}}_{\Lambda}$).
The explicit form of the Wigner rotation matrix is given by
$W(\Lambda,p)=L^{-1}(\Lambda p)\,\Lambda\,L(p),$ (5fqtwaaap)
where $L(p)$ is a standard boost taking the standard rest frame 4-momentum
$k\equiv(m,0,0,0)$ to an arbitrary 4-momentum $p$, $\Lambda$ is an arbitrary
LT taking $p\to\Lambda p$ , and $L^{-1}(\Lambda p)$ is an inverse standard
boost taking the final 4-momentum $\Lambda p$ back to the particle’s rest
frame. Because of the form of the standard rest 4-momentum $k$, this final
rest momentum $k^{\prime}$ can at most be a spatial rotation of the initial
standard 4-momentum $k$, i.e. $k^{{}^{\prime}}=W(\Lambda,p)\,k$. The rotation
group $SO(3)$ is then said to form (Wigner’s) little group for massive
particles, i.e. the invariance group of the particle’s rest 4-momentum. The
explicit form of the standard boost is given by [104]
$\displaystyle L^{0}_{\hskip 1.75pt0}(\xi)$ $\displaystyle=$
$\displaystyle\frac{p^{0}}{m}\equiv\cosh\xi,$ $\displaystyle L^{i}_{\hskip
1.75pt0}(\xi)$ $\displaystyle=$
$\displaystyle\frac{p^{i}}{m}\equiv\sinh\xi\,\hat{p}^{i},\quad L^{0}_{\hskip
1.75pti}=-\sinh\xi\,\hat{p}_{i},$ $\displaystyle L^{i}_{\hskip 1.75ptj}(\xi)$
$\displaystyle=$ $\displaystyle\delta^{i}_{\hskip
3.5ptj}-(\cosh\xi-1)\,\hat{p}^{i}\hat{p}_{j},\qquad i,j=(1,2,3),$ (5fqtwaaaq)
where $\gamma=p^{0}/m=E/m$ is the particle’s energy per unit rest mass. Note
that for the flat spacetime metric $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1)$,
$p_{0}=p^{0}$ and $p_{i}=-p^{i}$, and
$\hat{\mbox{{\boldmath$p$}}}=\mbox{{\boldmath$p$}}/|\mbox{{\boldmath$p$}}|=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)=\hat{p}^{i}$
is the direction of the 3-momentum $p$.
In the rest frame of the particle, with (standard) 4-momentum
$k=(m,\mbox{{\boldmath$k$}}=\mbox{{\boldmath$0$}})$ the basis vectors
$|\mbox{{\boldmath$0$}},\lambda\rangle$ are eigenstates of the four momentum
$P^{\mu}$, total angular momentum $\mbox{{\boldmath$J$}}^{2}=j(j+1)$, and
third component of angular momentum $J^{3}$ as (in units $\hbar=c=1$)
$\displaystyle P^{\mu}|\mbox{{\boldmath$0$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle k^{\mu}|\mbox{{\boldmath$0$}},\lambda\rangle,$
$\displaystyle\mbox{{\boldmath$J$}}^{2}|\mbox{{\boldmath$0$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle j(j+1)|\mbox{{\boldmath$0$}},\lambda\rangle,$
$\displaystyle J^{3}|\mbox{{\boldmath$0$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle\lambda|\mbox{{\boldmath$0$}},\lambda\rangle.$ (5fqtwaaar)
The standard boost (6.0.1) can be decomposed [104, 105] as
$L(p)=R(\hat{\mbox{{\boldmath$p$}}})\,B_{3}(|\mbox{{\boldmath$p$}}|)\,R^{-1}(\hat{\mbox{{\boldmath$p$}}})$
where $R(\hat{\mbox{{\boldmath$p$}}})$ is a rotation that takes the
$\hat{\mbox{{\boldmath$z$}}}$-axis to $\hat{\mbox{{\boldmath$p$}}}$, and
$B_{3}(|\mbox{{\boldmath$p$}}|)$ is a pure boost in the
$\hat{\mbox{{\boldmath$z$}}}$ direction such that
$||\mbox{{\boldmath$p$}}|\hat{\mbox{{\boldmath$z$}}},\sigma\rangle=B_{3}(|\mbox{{\boldmath$p$}}|)\,|\mbox{{\boldmath$0$}},\sigma\rangle$
with $|\mbox{{\boldmath$p$}}|=m\,\sinh\xi$. The unitary representation of
$R(\hat{\mbox{{\boldmath$p$}}})$ on the Hilbert space of states is given by
$U(R(\hat{\mbox{{\boldmath$p$}}}))=e^{i\phi J_{3}}\,e^{i\theta J_{2}}$. Since
the rightmost rotation in $L(p)$ has no effect on
$|\mbox{{\boldmath$0$}},\lambda\rangle$ we have [105]
$|\mbox{{\boldmath$p$}},\lambda\rangle=L(p)\,|\mbox{{\boldmath$0$}},\lambda\rangle=R(\hat{\mbox{{\boldmath$p$}}})\,B_{3}(|\mbox{{\boldmath$p$}}|)\,|\mbox{{\boldmath$0$}},\lambda\rangle\equiv
H(p)\,|\mbox{{\boldmath$0$}},\lambda\rangle.$ (5fqtwaaas)
The states $|\mbox{{\boldmath$p$}},\lambda\rangle$ are eigenstates of
$P^{\mu}$, $\mbox{{\boldmath$J$}}^{2}$, and the helicity operator
$\mbox{{\boldmath$J$}}\cdot\hat{\mbox{{\boldmath$P$}}}/|\hat{\mbox{{\boldmath$P$}}}|$
with eigenvalue $\lambda$, the component of the particle’s angular momentum
along the direction $\hat{\mbox{{\boldmath$p$}}}$ of it’s linear momentum
[105],
$\displaystyle P^{\mu}|\mbox{{\boldmath$p$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle p^{\mu}|\mbox{{\boldmath$p$}},\lambda\rangle,$
$\displaystyle\mbox{{\boldmath$J$}}^{2}|\mbox{{\boldmath$p$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle j(j+1)|\mbox{{\boldmath$p$}},\lambda\rangle,$
$\displaystyle\frac{\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}}{|\mbox{{\boldmath$P$}}|}|\mbox{{\boldmath$p$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle\lambda|\mbox{{\boldmath$p$}},\lambda\rangle.$
(5fqtwaaat)
(Note that
$J^{3}|\mbox{{\boldmath$p$}},\lambda\rangle\neq\lambda|\mbox{{\boldmath$p$}},\lambda\rangle$
since $J^{3}$ now contains a contribution from the orbital motion). The Wigner
rotation in (5fqtwaaap) is given by $W(\Lambda,p)=H^{-1}(\Lambda
p)\,\Lambda\,H(p)$. The $|\mbox{{\boldmath$p$}},\lambda\rangle$ are called
helicity states. One can also define spin states denoted by
$|\mbox{{\boldmath$p$}},\sigma\rangle$ by suitable rotation
$|\mbox{{\boldmath$p$}},\sigma\rangle=\sum_{\lambda}{D^{(j)}_{\lambda\sigma}(R^{-1}(\hat{\mbox{{\boldmath$p$}}}))|\mbox{{\boldmath$p$}},\lambda\rangle}$
[106], which are commonly used in the literature. Here
$R^{-1}(\hat{\mbox{{\boldmath$p$}}})$ rotates the 3-momentum direction
$\hat{\mbox{{\boldmath$p$}}}$ to the $\hat{\mbox{{\boldmath$z$}}}$-axis.
The formalism above carries through for massless particles, but with important
physical differences. Since there is no rest frame for a massless particle,
the standard 4-momentum is chosen to be $k\equiv(1,0,0,1)$ with
$\mbox{{\boldmath$k$}}=\hat{\mbox{{\boldmath$z$}}}$, and basis state
$|\mbox{{\boldmath$k$}},\lambda\rangle$ labeled by $J^{3}$ as in (6.0.1)
(which hold with substitution $\mbox{{\boldmath$0$}}\to\mbox{{\boldmath$k$}}$
). The general state
$|\mbox{{\boldmath$p$}},\lambda\rangle=H(p)\,|\mbox{{\boldmath$k$}},\lambda\rangle$
is built up in exactly the same fashion as the massive case discussed in
(5fqtwaaas), where now the “energy” in $B_{3}(|\mbox{{\boldmath$p$}}|)$ is
given by $p^{0}=|\mbox{{\boldmath$p$}}|=e^{\xi}$. The helicity states
$|\mbox{{\boldmath$p$}},\lambda\rangle$ again satisfy (6.0.1) with $\lambda$
now restricted to the two values $\pm j$. The new feature for massless states
is that the helicity $\lambda$ is a Lorentz invariant since only the phase of
the state $|\mbox{{\boldmath$p$}},\lambda\rangle$ changes in a momentum
dependent fashion under a LT $\Lambda$
$U(\Lambda)|\mbox{{\boldmath$p$}},\lambda\rangle=e^{-i\,\lambda\,\theta(\Lambda,\hat{\bf
p})}\,|\mbox{{\boldmath$p$}},\lambda\rangle.$ (5fqtwaaau)
Here, the notation indicates that
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ depends only upon the LT
$\Lambda$ and the direction $\hat{\mbox{{\boldmath$p$}}}$ of the particle’s
momentum, but not upon its frequency $p^{0}$. The Wigner rotation angle
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ can be computed as
$\langle\mbox{{\boldmath$k$}},\lambda|W(\Lambda,p)|\mbox{{\boldmath$k$}},\lambda\rangle$,
where again $W(\Lambda,p)=H^{-1}(\Lambda p)\,\Lambda\,H(p)$ leaves the
standard momentum $k$ invariant $k^{{}^{\prime}}=W(\Lambda,p)\,k$ (Wigner’s
little group is now $ISO(2)$, rotations in plane perpendicular to $k$). For
$j=1$, $\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ describes the amount of
rotation experienced by the plane of polarization of a linearly polarized
photon (for example calculations of
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ see [104, 107, 108]).
Returning to the massive case, the spin states
$|\mbox{{\boldmath$p$}},\sigma\rangle$ (and similarly for the helicity states)
are normalized as
$\langle{\mbox{{\boldmath$p$}}^{{}^{\prime}},\sigma^{{}^{\prime}}}|\mbox{{\boldmath$p$}},\sigma\rangle=(2\pi)^{3}\,(2p^{0})\delta_{\sigma^{\prime}\sigma}\delta^{3}(\mbox{{\boldmath$p$}}^{\prime}-\mbox{{\boldmath$p$}})$.
One-particle states (wavepackets) are given by [105, 109]
$|\Psi\rangle=\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,\psi_{\sigma}(p)\,|\mbox{{\boldmath$p$}},\sigma\rangle,\,\,\psi_{\sigma}(p)=\langle\mbox{{\boldmath$p$}},\sigma|\Psi\rangle,\,\,\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,|\psi_{\sigma}(p)|^{2}=1,$
(5fqtwaaav)
with respect to the Lorentz invariant measure
$d\mu(p)=d^{3}\mbox{{\boldmath$p$}}/(2\pi)^{3}\,(2p^{0})$. Under a LT
$\Lambda$, the same state $|\Psi\rangle$ described in the boosted frame is
given by
$\displaystyle|\Psi^{{}^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,\psi^{{}^{\prime}}_{\sigma}(p)\,|\mbox{{\boldmath$p$}},\sigma\rangle,$
(5fqtwaaawa) $\displaystyle\psi^{{}^{\prime}}_{\sigma}(p)$ $\displaystyle=$
$\displaystyle\sum_{\eta}D^{(1/2)}_{\sigma\eta}(W(\Lambda,\Lambda^{-1}p))\,\psi_{\sigma}(\Lambda^{-1}p)=\langle\mbox{{\boldmath$p$}},\sigma|U(\Lambda)|\Psi\rangle,$
(5fqtwaaawb)
which can be obtained from a straight forward calculation using (5fqtwaaav)
and (5fqtwaaao), the basis state normalization, a change of variables
$p\to\Lambda^{-1}\,p$ and utilizing $d\mu(\Lambda^{-1}p)=d\mu(p)$, along with
some index relabeling.
#### 6.0.2 Early investigations
Quantum information was first considered in a relativistic setting by Czachor
in 1997 who analyzed a relativistic version of the Einstein-Podolsky-Rosen-
Bohm experiment [107, 108, 109, 110, 111, 112, 113, 114]. Czachor’s work
showed that relativistic effects are relevant to the experiment where the
degree of violation of Bell’s inequalities depends on the velocity of the
entangled particles. However, it was not until 2002 that the importance of
investigating quantum information in the presence of spacetime was recognized.
A. Peres & D. Terno pointed out that most concepts in quantum information
theory may require a reassessment [109, 115]. Further interesting results on
entanglement in flat spacetime were obtained by Alsing & Milburn [107], Adami,
Bergou, & Gingrich [116, 117], and Solano & Pachos [118]. Their work showed
that although entanglement is conserved overall under a change of inertial
frame, it may swap between spin and position degrees of freedom. (For a
comprehensive account of this early work up to 2004, with indications of
future work on accelerated observers, see the review by A. Peres and D.R.
Terno [109]).
The insightful and provoking work of Peres and Terno [109, 115] considered
single particle wavepackets as in (5fqtwaaav) with density matrix
$\rho=|\Psi\rangle\langle\Psi|$ having components
$\rho_{\sigma\sigma^{{}^{\prime}}}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$p$}}^{{}^{\prime}})=\psi^{*}_{\sigma}(p)\,\psi_{\sigma^{{}^{\prime}}}(p^{{}^{\prime}})$,
and traced out over the momentum to produce the reduced spin density matrix
$\rho^{red}_{\sigma\sigma^{{}^{\prime}}}=\int^{\infty}_{-\infty}d\mu(p)\,\psi^{*}_{\sigma}(p)\,\psi_{\sigma^{{}^{\prime}}}(p^{{}^{\prime}}).$
(5fqtwaaawax)
They showed that owing to the momentum dependent Wigner rotations that arise
in the full state (5fqtwaaawa), that are induced by the LT $\Lambda$
describing the reference frame of a new observer, the reduced quantity
$\rho^{red}$ has no covariant transformation law, except in the limiting case
of sharp momentum. Only the complete density $\rho$ has a covariant
transformation law. They concluded that the “spin state” of a single particle
is meaningless if one does not specify its complete state, including momentum
variables. Even though it may be possible to formally define spin in any
Lorentz frame, there will be no relationship between the observable
expectation values in different Lorentz frames [115].
Alsing and Milburn [107] and Terashima and Ueda [108] considered an EPR Bell
state with factorable pure momentum eigenstates. In a second rest frame of
another pair of observers $A,B$ traveling perpendicularly to the initial rest
frame, described by the pure boost $\Lambda$, the initial state $|\Psi\rangle$
is described as
$|\Psi^{{}^{\prime}}\rangle=U_{AB}(\Lambda)\,|\Psi\rangle=U_{A}(\Lambda)\otimes
U_{B}(\Lambda)\,|\Psi\rangle$. While the state $|\Psi^{{}^{\prime}}\rangle$
remains factorable between momentum and spin, the initial Bell state is
coherently superposed with other Bell states. The reason is that
$U_{A}(\Lambda)$ and $U_{B}(\Lambda)$ give Wigner rotation angles of opposite
signs for particles traveling in opposite directions. Maximal EPR correlations
are recovered if the detectors in the boosted frame are rotated to the new
rest frame direction of spin.
Gingrich and Adami [116] considered wavepackets for two massive spin $1/2$
particles, with similar form and transformation properties
$(U_{AB}(\Lambda)=U_{A}(\Lambda)\otimes U_{B}(\Lambda))$ as (5fqtwaaav) and
(5fqtwaaawa), respectively
$\displaystyle|\Psi_{AB}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\sigma\tau}\,\int\int
d\mu(p)\,d\mu(q)\,g_{\sigma\tau}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$q$}})\,|\mbox{{\boldmath$p$}},\sigma\rangle\,|\mbox{{\boldmath$q$}},\tau\rangle,$
(5fqtwaaawaya) $\displaystyle
g_{\sigma\tau}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$q$}})$
$\displaystyle\to$
$\displaystyle\sum_{\sigma^{{}^{\prime}}\tau^{{}^{\prime}}}\,D^{(1/2)}_{\sigma\sigma^{{}^{\prime}}}(W(\Lambda,\Lambda^{-1}p))D^{(1/2)}_{\tau\tau^{{}^{\prime}}}(W(\Lambda,\Lambda^{-1}q))\,g_{\sigma^{{}^{\prime}}\tau^{{}^{\prime}}}(\Lambda^{-1}\mbox{{\boldmath$p$}},\Lambda^{-1}\mbox{{\boldmath$q$}}).$
(5fqtwaaawayb)
Since LTs entangle the spin and momentum degrees of freedom within a single
particle, entanglement can be transferred between them. Their work showed that
this is also the case for pairs of particles, and that LTs affects the
entanglement between the spins of different particles, as measured by the
Wootters concurrence [119] on the reduced density matrix formed by tracing out
over the momentum of the complete state
$\rho_{AB}=|\Psi_{AB}\rangle\langle\Psi_{AB}|$. Thus, spin-spin entanglement
is not a Lorentz invariant.
These works generated an intense study of relativistic EPR correlations for
both spin $1/2$ particles(see [109, 120] and references therein) and photons
[107, 109, 117, 121] that continues today. Much of the work for spin $1/2$
particles (which we will primarily discuss henceforth) has focused on the
relevant covariant observable(s) for spin such that the expectation values
obtained from measurements are the same in all inertial frames. Also in
question is the meaning and validity of the reduced density matrix, especially
in the case of tracing out the momentum from the complete quantum state. Terno
et al. [115, 109] argued that in the relativistic setting momentum is a
primary variable that has relativistic transformation laws that depends only
on the LT matrix $\Lambda$ that acts on the spacetime coordinates. Quantities
such as spin and polarization are secondary variables that depend both on
$\Lambda$ and the primary variable momentum. Thus, even though the reduced
density matrix for the secondary variable may be well defined in any
coordinate system, it has no transformation law relating values in different
Lorentz frames. Further, the unambiguous and seemingly natural construction of
a reduced density matrix by means of tracing out over the primary variables is
possible only if the secondary variables are unconstrained. In the absence of
a general description, a case-by-case treatment is required.
Shortly afterward, Czachor [122], Caban and Rembielinski [123] (see also [124]
and references therein) showed that it is possible to define a Lorentz-
covariant reduced spin density matrix for single massive particles. Such an
object [123] contains information about the average polarization of the
particle, as well as information about its average kinematical state. For
sharp momentum, the reduced density matrix does not change under LTs. However,
in the case of an arbitrary momentum distribution (5fqtwaaav) the entropy of
the reduced density matrix is in general not a Lorentz invariant. The
conclusion of these studies is that while one can define a Lorentz-covariant
finite dimensional matrix describing the polarization of a massive particle,
in the relativistic case one cannot completely separate out kinematical
degrees of freedom. These results ultimately stem from the momentum dependent
spin transformations (5fqtwaaao) induced by a LT. Extensions of this line of
research have been carried out for states of two spin $1/2$ particles (see
e.g. [120] and references therein). Note that as pointed out by Peres and
Terno [109] an invariant definition of entanglement of a pair of spin $1/2$
particles typically utilizes the center-of-mass “rest frame” where
$\langle\sum\mbox{{\boldmath$p$}}\rangle=0$. However, due to the problem of
cluster decomposition [104] this definition is not adequate for more than two
constituent particles since subsets of particles may have different rest
frames. This is a difficult and still unresolved problem that has relevance to
investigations of relativistic multipartite entanglement for more than two
particles.
#### 6.0.3 Spin observable issues
It is worth noting that complications in defining an appropriate reduced spin
density matrix arise from the long appreciated non-unique definition of the
“spin” operator for massive particles [125, 126, 127]. Ultimately, this is
traced back to the fact that spin is only defined in the rest frame of the
particle, where it coincides with the total angular momentum (since for
$\mbox{{\boldmath$p$}}=\mbox{{\boldmath$0$}}$ the orbital angular momentum
contribution vanishes). For studies of relativistic EPR correlations, most
works have settled on some variation of the the spin operator defined from the
Paul-Lubanski (PL) vector [127]
$W_{\mu}=-1/2\epsilon_{\mu\nu\rho\sigma}\,J^{\nu\rho}\,P^{\sigma}$ where
$J_{\mu\nu}$ are generators of the proper orthochronous Lorentz group with
$J_{ij}=-J_{ji}=\epsilon_{ijk}\,J_{k}$ and $J_{i0}=-J_{0i}=K_{i}$, the
generators of rotations and boosts, respectively. The two Casimir operators of
the Poincare group are given by $C_{1}=P^{\mu}P_{\mu}=m^{2}$ and
$C_{2}=W^{\mu}W_{\mu}=-m^{2}\,j(j+1)$ that commute with all the generators
$\\{P^{\mu},J^{\mu\nu}\\}$, and label the basis states
$|\mbox{{\boldmath$p$}},\sigma\rangle$ by their mass $m$ and spin $j$. The PL
vector
$W^{\mu}=(\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}},\,P^{0}\mbox{{\boldmath$J$}}+\mbox{{\boldmath$P$}}\times\mbox{{\boldmath$K$}})$
transforms as a 4-vector under LTs and is proportional to the total angular
momentum in the rest frame of the particle
$W^{\mu}_{rest}=(0,m\mbox{{\boldmath$J$}})$. Hence, spin is well defined in
the rest frame of the particle. In an arbitrary frame, the time component of
the PL vector is proportional to the helicity operator
$\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}/|\mbox{{\boldmath$P$}}|$.
For observers in an arbitrary inertial frame the spin
$\mbox{{\boldmath$S$}}\equiv\mbox{{\boldmath$J$}}-\mbox{{\boldmath$L$}}$ is
defined as the difference between the total angular momentum, which is well
defined as a generator of the Poincare group, and the orbital angular momentum
$\mbox{{\boldmath$L$}}=\mbox{{\boldmath$R$}}\times\mbox{{\boldmath$P$}}$.
While the momentum $P$ is a well defined generator of the Poincare group,
there is no well defined notion of a position operator (and hence the concept
of localizability) in relativistic quantum mechanics. Hence $L$ and $S$ are
not uniquely defined. Popular choices for the position operator [126] include
the center-of-mass operator $\mbox{{\boldmath$R$}}_{c.m.}$ and Newton-Wigner
position operator $\mbox{{\boldmath$R$}}_{NW}$ defined by
$\mbox{{\boldmath$R$}}_{c.m.}=-\frac{1}{2}\left[\frac{1}{P^{0}}\,\mbox{{\boldmath$K$}}+\mbox{{\boldmath$K$}}\,\frac{1}{P^{0}}\right],\quad\mbox{{\boldmath$R$}}_{NW}=\mbox{{\boldmath$R$}}_{c.m.}-\displaystyle\frac{\mbox{{\boldmath$P$}}\times\mbox{{\boldmath$K$}}}{mP^{0}(m+P^{0})}.$
(5fqtwaaawayaz)
This leads to the center-of-mass and Newton-Wigner spin operators
$\mbox{{\boldmath$S$}}_{c.m.}=\displaystyle\frac{\mbox{{\boldmath$W$}}}{P^{0}},\quad\mbox{{\boldmath$S$}}_{NW}=\frac{1}{m}\left(\mbox{{\boldmath$W$}}-\displaystyle\frac{W^{0}\mbox{{\boldmath$P$}}}{P^{0}+m}\right).$
(5fqtwaaawayba)
The operator $\mbox{{\boldmath$S$}}_{NW}$ (favored by researchers such as
Terno et al [115, 128] and Caban et al [123, 120]) satisfies the usual $su(2)$
commutation relations and is the only axial-vector operator that is a linear
combination of the PL vector. Using (5fqtwaaawayaz), a spin observable is
defined by $\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{NW}$ as the
projection of spin along the unit spatial direction
$\hat{\mbox{{\boldmath$n$}}}$. The downside of $\mbox{{\boldmath$S$}}_{NW}$ is
that by itself, it is not part of any known 4-vector, nor second rank tensor
with well defined transformation properties under LTs. (Note that for both
operators
$\mbox{{\boldmath$S$}}\cdot\mbox{{\boldmath$P$}}=\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}$
so that we can also write
$W^{0}=\mbox{{\boldmath$S$}}\cdot\mbox{{\boldmath$P$}}$). The center-of-mass
spin operator $\mbox{{\boldmath$S$}}_{c.m.}$ (favored by researchers such as
Czachor [110, 111, 115, 129, 128, 130]) does not generate the $su(2)$ spin
algebra and its eigenvalues are momentum dependent. Further, in contrast to
the operator $\mbox{{\boldmath$S$}}^{2}_{NW}$, the operator
$\mbox{{\boldmath$S$}}^{2}_{c.m.}$ does not reduce to the second Casimir
operator $C_{2}=j(j+1)$. Despite these flaws, the spin observable
$\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.}$, has many
favorable properties and is often used in the literature, as well as a
normalized (nonlinear) version (introduced by Czachor [110])
$\mbox{{\boldmath$S$}}_{CZ}(\hat{\mbox{{\boldmath$n$}}})=\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$W$}}/(m^{2}+(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$p$}})^{2})^{1/2}$.
A comparison of EPR correlations using both the Newton-Wigner and center-of-
mass of spin observables [126] reveals that for fixed measurement directions,
the EPR correlations do not necessarily decrease monotonically with velocity,
and may exhibit local extrema for certain configurations (which also holds
true for spin 1 particles). Furthermore, this effect occurs for both types of
spin observables, and hence appears to be a generic feature of relativistic
correlations, and in some cases strongly depends on which definition of spin
observable is utilized.
Recent work by Friis et al. [130] utilized $\mbox{{\boldmath$S$}}_{c.m.}$ for
the investigation of a parameterized set of pure states of two spin $1/2$
particles, with various configurations of entanglement between the spin and
momentum degrees of freedom. As indicated by earlier works [107, 108, 110,
116] the maximum EPR correlation can be recovered in any inertial frame if
both the state and the spin observable are Lorentz transformed. In an inertial
frame in which the particle has 4-momentum
$p^{\mu}=(p^{0},\mbox{{\boldmath$p$}})$, the authors show the observable
defined by
$\hat{n}(p)\equiv(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})/|\lambda(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})|$,
with $\lambda(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})$
the eigenvector of the operator
$\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.}$, can also be
written as
$\hat{n}_{local}(p)=\hat{\mbox{{\boldmath$n$}}}_{local}(p)\cdot\mbox{{\boldmath$\sigma$}}$.
Here $\sigma$ is the usual 3-vector of Pauli spin matrices, and
$\hat{\mbox{{\boldmath$n$}}}_{local}(p)=(\sqrt{1-\beta^{2}}\hat{\mbox{{\boldmath$n$}}}_{\perp}+\hat{\mbox{{\boldmath$n$}}}_{\parallel})/(1-\beta^{2}(1-\hat{\mbox{{\boldmath$n$}}}^{2}_{\parallel}))^{1/2}$
can be interpreted as the detector orientation
$\hat{\mbox{{\boldmath$n$}}}_{local}$ as seen from the rest frame of the
particle (where $\beta=v/c$ is the velocity of the inertial frame). This later
expression can also be written as
$\hat{n}_{local}(p)=[L^{-1}(p)n]^{i}\,\sigma_{i}/|[L^{-1}(p)n]^{j}|$ where
$|[L^{-1}(p)n]^{j}|$ is the norm of the spatial portion of the Lorentz
transformed orientation vector $L^{-1}(p)\,n$. Here
$n=(n^{0},\hat{\mbox{{\boldmath$n$}}})$ is the 4-vector whose spatial portion
$\hat{\mbox{{\boldmath$n$}}}$ is the orientation of the detector in the
inertial frame with 4-momentum $p$. Thus, $L^{-1}(p)$ transforms the
orientation 4-vector $n$ back to the rest frame $n_{local}$, whose spatial
portion $\hat{\mbox{{\boldmath$n$}}}_{local}$ indicates the local direction of
the detector. Under a general LT $\Lambda$ the orientation 4-vector is given
by $n^{\prime\prime}=\Lambda\,n=\Lambda\,L(p)n_{local}$ so that
$\hat{n}^{\prime\prime}(p)=[L^{-1}(\Lambda
p)n^{\prime\prime}]^{i}\,\sigma_{i}/|[L^{-1}(\Lambda
p)n^{\prime\prime}]^{j}|=[W(\Lambda,p)n_{local}]^{i}\,\sigma_{i}/|[W(\Lambda,p)n_{local}]^{j}|$.
Since the Wigner rotation $W(\Lambda,p)$ does not change the norm of
$\hat{\mbox{{\boldmath$n$}}}_{local}$ we have that the observable
$\hat{n}^{\prime\prime}(p)$ transforms under LTs as
$\hat{n}^{\prime\prime}(p)=\big{(}W(\Lambda,p)\hat{\mbox{{\boldmath$n$}}}_{local}\big{)}\cdot\mbox{{\boldmath$\sigma$}}=U(\Lambda,p)\,\big{(}\hat{\mbox{{\boldmath$n$}}}_{local}\cdot\mbox{{\boldmath$\sigma$}}\big{)}\,U^{\dagger}(\Lambda,p)$.
Thus, the expectation values of observable $\hat{n}^{\prime\prime}(p)$ are the
same as that in the rest frame of the particle $\hat{n}_{local}(p)$. Most
importantly, while the spin observable $\hat{n}^{\prime\prime}(p)$ depends on
the momentum of the particle, the measurement direction
$\hat{\mbox{{\boldmath$n$}}}^{\prime\prime}$ corresponding to this observable
does not [130].
Recently Saldanha and Vedral [124] have raised concerns about the physical
implementations of the above spin observable measurements. They argue that
while the above formulations are mathematically covariant, in the sense that
spin measurements obtain values that are the same in all inertial frames, such
measurements may be physically inconsistent. Consider a covariant description
of the interaction $H_{PL}$ of a measurement apparatus with spin observable
$S$ constructed from the PL vector $W^{\mu}=(W^{0},\mbox{{\boldmath$W$}})$,
where from (5fqtwaaawayba)
$W^{\mu}=(\mbox{{\boldmath$S$}}_{c.m.}\cdot\mbox{{\boldmath$P$}},P^{0}\mbox{{\boldmath$S$}}_{c.m.})$
or
$W^{\mu}=(\mbox{{\boldmath$S$}}_{NW}\cdot\mbox{{\boldmath$P$}},m\mbox{{\boldmath$S$}}_{NW}+(P^{0}-m)(\mbox{{\boldmath$S$}}_{NW}\cdot\mbox{{\boldmath$P$}})\mbox{{\boldmath$P$}}/|\mbox{{\boldmath$P$}}|^{2})$.
The interaction must be a Lorentz scalar of the form
$H_{PL}=W^{\mu}\,G_{\mu}=W^{0}\,G^{0}-\mbox{{\boldmath$W$}}\cdot\mbox{{\boldmath$G$}}$
for some 4-vector $G^{\mu}=(G^{0},\mbox{{\boldmath$G$}})$ in order that the
expectation values of a spin measurement are the same in all inertial
reference frames. The authors state there are no known couplings of this type,
and in particular, not for measurements where spin couples to the
electromagnetic field. They argue, as have other researchers discussed above,
that it is not possible to uniquely measure the spin of a particle independent
from its momentum. They further conclude that any spin-momentum partition by
means of reduced density matrices is “actually completely meaningless,” while
other researchers have argued that such bipartite partitions must be examined
on a case by case basis, where meaningful information can be obtained.
#### 6.0.4 Extension to curved spacetime
The extension of the above work to curved spacetime for both spin $1/2$
particles and photons has been examined by Terashima and Ueda [131], Alsing et
al. [132, 133], Brodutch and Terno [134] and Palmer et al. [135]. In flat
spacetime the inertial frame, described by the the LT $\Lambda$ is global. In
curved spacetime, this is true only locally at each spacetime point
$x=(x^{0},\mbox{{\boldmath$x$}})$, where by the Equivalence Principle [38,
50], spacetime is locally (Lorentzian) flat, and the rules of special
relativity hold. The LT is now parameterized by the spacetime location $x$ as
$\Lambda^{\mu}_{\hskip 3.5pt\nu}(x)$ \- a local Lorentz transformation (LLT)
which describes transformations between the instantaneous states of motion of
different local observers at the same spacetime point $x$, for example freely
falling, stationary (e.g. fixed radial coordinate), circular motion, or any
trajectory with arbitrary acceleration. These LLTs are to be distinguished
from general coordinate or world transformations $\partial
x^{\prime\mu}/\partial x^{\nu}$ which relates vectors, tensors, etc…
describing the same spacetime quantity in different coordinates $x$ and
$x^{\prime}$. The local Lorentz frame (LLF), i.e. the observer’s local
laboratory [136], has a small finite extent in space and time in as much as
the curvature can be considered constant within this region. The LLF can be
described by a tetrad $e^{\hat{a}}_{\hskip 3.5pt\alpha}(x)$ [50, 137, 136], a
set of 4 (orthonormal) local axes $\\{e^{\hat{a}}(x)\\}$ labeled by (hatted
local Lorentz) indices $(\hat{0},\hat{1},\hat{2},\hat{3})$ with spacetime
components (world coordinate indices) $\alpha=(0,1,2,,3)$. The spatial triad
$\\{e^{\hat{i}}(x)\\}$ describes the orientation of the LLF in the surrounding
spacetime, and the temporal axis ${e^{\hat{0}}(x)}$, equal to the observer’s
(world) 4-velocity (and thus tangent to their worldline), describes the
observer’s local proper time. As the observer moves through spacetime their
LLF, described by their tetrad, twists and turns in the surrounding curved
background. The observer describes events in their local laboratory by
projecting world tensors onto this tetrad, e.g. a particle with momentum
$p^{\alpha}(x)$ passing through the observer’s local laboratory is described
locally as $p^{\hat{a}}(x)=e^{\hat{a}}_{\hskip
3.5pt\alpha}(x)\,p^{\alpha}(x)$.
The quantum mechanical single particle states are now represented as
$|p^{\hat{i}}(x),\sigma\rangle$ where $p^{\hat{i}}(x)$ are the spatial
components of the particle’s 4-momentum $p(x)=p^{\hat{a}}(x)\,e_{\hat{a}}(x)$
(also equal to $p^{\mu}(x)\,e_{\mu}(x)$ where $e_{\mu}(x)$ can be taken as the
coordinate basis vectors in the surrounding spacetime
$e_{\mu}(x)=\partial_{x^{\mu}}$). Terashima and Ueda [131] (see also appendix
in Alsing et al. [132]) showed that as the observer with 4-momentum $p(x)$ at
$x$ travels to a new, infinitesimally close spacetime point $x^{\prime}$ in
proper time $d\tau$, the change $\delta p(x)=\delta
p^{\hat{a}}(x)\,e_{\hat{a}}(x)$ relative to the LLF can be written as a LLT
$\delta p^{\hat{a}}(x)=\lambda^{\hat{a}}_{\hskip
3.5pt\hat{b}}(x)\,p^{\hat{b}}(x)\,\delta\tau$, so that
$\Lambda^{\hat{a}}_{\hskip 3.5pt\hat{b}}(x)=\delta^{\hat{a}}_{\hskip
3.5pt\hat{b}}+\lambda^{\hat{a}}_{\hskip 1.75pt\hat{b}}(x)\,d\tau$. One can now
calculate the effect of the Wigner rotation expressed as a local version of
(5fqtwaaao) (with $\mbox{{\boldmath$p$}}\to p^{\hat{i}}(x)$,
$\Lambda\to\Lambda(x)$, etc…) and (5fqtwaaap) with an infinitesimal Wigner
rotation given by $W^{\hat{a}}_{\hskip
3.5pt\hat{b}}(x)=\delta^{\hat{a}}_{\hskip
3.5pt\hat{b}}+\vartheta^{\hat{a}}_{\hskip 3.5pt\hat{b}}(x)\,d\tau$, where the
antisymmetric $\vartheta^{\hat{a}}_{\hskip 3.5pt\hat{b}}(x)$ only has non-zero
components on the spatial indices (i.e. a rotation). Finite Wigner rotations
are obtained by a time-ordered integration along the observer’s worldline with
respect to his proper time. The implications of these results is that the
orientation of the spin of the particle in the observer’s LLF depends on
instantaneous state of his motion as embodied in his tetrad
$e^{\hat{a}}_{\hskip 3.5pt\alpha}(x)$ describing his local laboratory. This is
not unexpected from the results of flat spacetime investigation when one
invokes the Equivalence Principle. However, the entanglement of a pair of spin
$1/2$ particles (initially created at a single spacetime point) when the
parties are widely spacelike separated, depends not only upon the initial
state of entanglement amongst the spin and momentum degrees of freedom (as in
the flat spacetime case), but also upon the full history of the motion through
spacetime. That is, Wigner rotation induced spin-momentum entanglement occurs
at each spacetime point along the trajectories of the constituent particles.
While general relativity advocates an agnostic description of physics using
arbitrary observers, the work of Alsing et al. [132] and Palmer et al. [135]
shows that in the Fermi-Walker frame (the instantaneous, co-moving, non-
rotating rest frame of the observer [50]) the Wigner rotation due to the
effects of the gravitation field is zero. Though it appears contrary to the
spirit of general relativity to single out a particular reference frame as
special, the Fermi-Walker frame may prove advantageous to describe internal
quantum mechanical degrees of freedom. While in the above discussion we have
primarily concentrated on the Wigner rotation for massive particle in curved
spacetime, many interesting results follow from the consideration of photons
in an arbitrary gravitational field [133, 134, 135, 138, 139].
## 7 Open questions and future directions
In prototypical quantum communication protocols Alice and Bob produce
entangled resources (for example two entangled qubits) to be used to transit
information. Alice and Bob take their systems to their labs. In case spacetime
is flat and Alice and Bob move at non-relativistic speeds the entanglement
resources they produced will remain un-changed. However, what happens to
quantum resources if Alice and Bob live in curved spacetime? If they move in
the presence of a gravitational field? If they move at relativistic speeds?
These are some of the most interesting questions in the field of relativistic
quantum information.
Understanding quantum information in relativistic settings might lead to
finding ways to exploit relativity to improve quantum information tasks. A
first step has been taken, finding suitable systems to store and process
information in quantum field theory. The systems presented in this paper seam
to be suitable candidates for this and hopefully ideas presented here will
stimulate other researchers to find new systems and develop mathematical
techniques that will allow us to find answers to the questions posed above.
Quantum protocols such as teleportation are possible thanks to the tensor
product structure of the Hilbert space. Constructing new protocols which are
only possible in a quantum and relativistic world would indeed be exciting.
An interesting question without doubt is if a covariant notion of entanglement
can be constructed. The analysis of entanglement in relativistic settings has
been performed by using the standard non-relativistic definition of
entanglement and applying it to relativistic settings. By doing this we have
found the entanglement is observer-dependent and that inconsistencies my arise
when defining subsystems in special relativity. An open question is if there
exits a more general notion of quantum correlations in relativistic quantum
theory which coincides with the standard notion in the non-relativistic
regime. The answer to this question is intimately tied to issues related to
the physical implementation of local measurements for observers in arbitrary
motion. In this respect, the current research on a locally covariant
formulation of quantum field theory in curved spacetime may prove useful [140,
141, 142].
Finally, we would like to say that we find the time ripe to find experimental
demonstration of results in the field of relativistic quantum information. On
one hand, cutting edge experiments in quantum information are approaching
regimes where relativity starts playing a role and on the other hand, recent
progress in the experimental demonstration of the dynamical Casimir effect
[143, 144] might provide the requisite techniques to show that gravity has
effects on entanglement .
### 7.1 Acknowledgments
IF would like to acknowledge the support by EPSRC [CAF Grant EP/G00496X/2].
PMA would like to acknowledge the support of the Air Force Office of
Scientific Research (AFOSR) for this work. We thank A. Dragan, J. Louko, D.
Bruschi, N. Friis. A. Lee, E. Martin-Martinez, G. Adesso, J. Doukas, T.G.
Downes, T. Ralph, R. Mann, F. P. Schuller, C. Sabin, R. Jauregui, P. Barberis-
Blostein and D. Sudarsky for stimulating conversations. We strived to be
inclusive in our citations to this exciting and burgeoning field. Any
omissions on the part of the authors are purely unintentional. Any opinions,
findings and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of AFRL.
## 8 Appendix
A brief discussion of some of the states used in the literature to investigate
acceleration/observer dependent entanglement is in order. A highly idealized
scenario is to invoke the “single-mode approximation,” (SMA) as considered by
Alsing and Milburn [5], in which Minkowski annihilation operator is taken to
be one of the right or left moving Unruh modes (since they do not couple to
each other) in the integrand of (5fk)
$a_{\omega,M}\sim
a_{\Omega,R}=\cosh(r_{\Omega})a_{\Omega,I}-\sinh(r_{\Omega})a^{\dagger}_{\Omega,II}.$
(5fqtwaaawaybb)
One can then consider the entangled Minkowski state between Alice and Bob
(left and right kets, respectively)
$|\Phi\rangle=\frac{1}{\sqrt{2}}\,\big{(}|0_{\omega}\rangle^{\mathcal{M}}\,|0_{\omega^{\prime}}\rangle^{\mathcal{M}}+|1_{\omega}\rangle^{\mathcal{M}}\,|1_{\omega^{\prime}}\rangle^{\mathcal{M}}\big{)}$
(5fqtwaaawaybc)
when undergoes uniform acceleration (transforming into his Rindler alter-ego
Rob) in wedge $I$. One can then expand Bob’s Minkowski states
$|0_{\omega^{\prime}}\rangle^{\mathcal{M}}$ and
$|1_{\omega^{\prime}}\rangle^{\mathcal{M}}=a^{\dagger}_{\omega^{\prime},M}|0_{\omega^{\prime}}\rangle^{\mathcal{M}}$
in terms of the Minkowski vacuum expressed in terms of the Rindler wedge modes
$I$ and $II$ given above and the adjoint of (5fqtwaaawaybb). This was done for
idealized leaky cavities by Alsing and Milburn[5] and for free space modes by
Fuentes and Mann [3] (see also the original Alsing and Milburn arxiv version
in [5]).
Although the SMA gives degradation of entanglement with increasing
acceleration, it is not rigourously justifiable since the Bogoliubov phase
factors $\alpha_{\omega\Omega}^{R}\sim(\omega
l)^{i\epsilon\Omega}=e^{i\epsilon\Omega\ln(\omega l)}$ are not localized in
frequency (the SMA was invoked to qualitatively represent the central
frequency of some wavepacket). A more refined approximation was considered by
Bruschi et al. [43] by considering the pre-accelerated initial Minkowski state
to be composed of Unruh modes for Bob (compare with (5fqtwaaawaybc))
$|\Phi^{\prime}\rangle=\frac{1}{\sqrt{2}}\,(|0_{\omega}\rangle^{\mathcal{M}}\,|0_{\Omega}\rangle^{\mathcal{U}}+|1_{\omega}\rangle^{\mathcal{M}}\,|1_{\Omega}\rangle^{\mathcal{U}}).$
(5fqtwaaawaybd)
The authors considered essentially a generalized version of the integrand in
(5fk) by taking
$a_{\omega,M}=q_{R}\,a_{\Omega,R}+q_{L}a_{\Omega,L},$
with $|q_{R}|^{2}+|q_{L}|^{2}=1$, which reduces to the SMA (5fqtwaaawaybb) for
$q_{L}=0$. Again, entanglement was degraded as the acceleration is increased.
The authors further considered the more physically realistic case of states
with a finite spread of frequencies, i.e. wavepackets (see [41]), versus the
highly idealized, sharply defined frequency states considered above (the
latter of which capture the essence of the acceleration-dependent
entanglement). Instead of (5fk) one considers
$a^{\dagger}_{M}=\int_{0}^{\infty}d\omega\,f(\omega)\,a^{\dagger}_{\omega,M}=\int^{\infty}_{0}d\Omega\,[g_{R}(\Omega)\,\alpha_{\omega\Omega}^{R}\,a^{\dagger}_{\Omega,R}+g_{L}(\Omega)\,\alpha_{\omega\Omega}^{L}\,a^{\dagger}_{\Omega,L}]=a^{\dagger}_{R}+a^{\dagger}_{L},$
(5fqtwaaawaybe)
where
$g_{R}(\Omega)=\int_{0}^{\infty}d\omega\,f(\omega)\,\alpha_{\omega\Omega}^{R}$
and
$g_{L}(\Omega)=\int_{0}^{\infty}d\omega\,f(\omega)\,\alpha_{\omega\Omega}^{L}$.
By considering shape functions that are Gaussian in $ln(\omega l)$ and
adjusting parameters to ensure that the wavepacket version of the state
(5fqtwaaawaybd) has negligible overlap between Alice’s and Rob’s state during
acceleration, the SMA (5fqtwaaawaybb) is recovered with one of the $q_{R}$,
$q_{L}$ vanishing (see [43] for further details).
The two global initial states $|\Phi\rangle$ (5fqtwaaawaybc) and
$|\Phi^{\prime}\rangle$ (5fqtwaaawaybd) consider different scenarios when the
second party Bob undergoes uniform acceleration (to become the accelerated
Rob). Both states $|\Phi\rangle$ and $|\Phi^{\prime}\rangle$ are maximally
entangled, global Minkowski states. The first state $|\Phi\rangle$ is a
“fixed” Minkowski state, independent of the acceleration $a$ for both Alice
and Bob, and is composed of the usual Minkowski plane wave $u_{\omega,M}\sim
e^{-i\omega(t-\epsilon x)}$ in (2). The second state $|\Phi^{\prime}\rangle$
is composed of Minkowski states for Alice, and Unruh states for Bob whose mode
$u_{\Omega,U}$ is a linear combination of Rindler modes $u_{\Omega,I}$ and
$u^{*}_{\Omega,II}$ that are both proportional to $\Omega^{-1/2}(x-\epsilon
t)^{i\epsilon\Omega}$ in region $I$ and $II$, respectively. On the one hand,
by the transformation between Minkowski and Unruh modes (5fk),
$|\Phi^{\prime}\rangle$ Bob’s portion of the state can be considered as an
involved integration of global Minkowski modes. The entanglement in this state
is then explored between the inertial Alice, and the Rindler observer Rob, who
is specified by a particular value of the acceleration $a$, and whose mode
function in region $I$ is $u_{\Omega,I}$. On the other hand, as discussed in
section 3.1 on flat spacetime entanglement, the state $|\Phi^{\prime}\rangle$
may also be considered as a family of maximally entangled states parameterized
by the dimensionless variable $\Omega=\omega/a$ in non-inertial frames. By
fixing the physical frequency $\omega$ and changing $a$ one analyzes the
entanglement in a family of states which, all have the same frequency $\omega$
as seen by observers with different proper acceleration $a$. Alternatively,
the entanglement corresponds to a family of states with different physical
frequency $\omega$ as seen by the same observer moving with fixed proper
acceleration $a$ [66].
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|
arxiv-papers
| 2012-10-08T10:41:52 |
2024-09-04T02:49:36.150181
|
{
"license": "Public Domain",
"authors": "Paul M. Alsing and Ivette Fuentes",
"submitter": "Ivette Fuentes",
"url": "https://arxiv.org/abs/1210.2223"
}
|
1210.2228
|
# Summation formula for solutions of Riccati-Abel equation
Robert M. Yamaleev
Joint Institute for Nuclear Research, LIT, Dubna, Russia.
Universidad Nacional Autonoma de Mexico, Mexico.
Email: yamaleev@jinr.ru
###### Abstract
The Riccati-Abel equation defined as an equation between first order
derivative and cubic polynomial is explored. In the case of constant
coefficients this equation is reduced into algebraic equation. The method of
derivation of a summation formula for solutions of Riccati-Abel equation
elaborated. Interrelation with general complex algebra of third order is
established.
## 1 Introduction
Consider the first order differential equation
$f(u,x)=\frac{du}{dx}.$ $None$
If we approximate $f(u,x)$, while $x$ is kept constant, we will get
$Q_{0}(x)+Q_{1}(x)u+Q_{2}(x)u^{2}+Q_{3}(x)u^{3}+\cdots=\frac{du}{dx}.$ $None$
When the series in the left-hand side is restricted with second order
polynomial the equation is the Riccati equation [1].
The Riccati equation is one of the widely used equations of mathematical
physics. The ordinary Riccati equations are closely related with second order
linear differential equations. For the solutions of the ordinary Riccati
equations with constant coefficients a summation formula can be derived. These
solutions are presented by trigonometric functions induced by general complex
algebra.
If, in particular, $f(u,x)$ is a cubic polynomial, then the equation is called
Riccati-Abel equation. Abel’s original equation was written in the form
$(y+s)\frac{dy}{dx}+p+qy+ry^{2}=0.$ $None$
This equation is converted into Riccati-Abel equation by transformation
$y+s=1/z$, which yields
$\frac{dz}{dx}=rz+(q-s^{\prime}-2rs)z^{2}+(p-qs+rs^{2})z^{3}.$ $None$
It is seen that the case $Q_{0}(u,\phi)=0$ was actually considered by Abel
[2].
When the series in the left-hand side of equation (1.2) is given by $n$-order
polynomial we deal with the generalized Riccati equations. The solution of the
generalized Riccati equation with constant coefficients can be denominated as
generalized tangent function. The generalized Riccati equations are used, for
example, in various problems of renorm-group theory [3]. The mean field free
energy concept and the perturbation renormalization group theory deal with
differential equations of first order with polynomial non-linearity.
The aim of this paper is to explore solutions of the Riccati-Abel equation
with constant coefficients and to derive some kind of summation formula for
them. Summation (addition) formulae for solutions of linear differential
equations are considered as important features of these functions. Let us
mention, for example, summation formulae for the trigonometric sine-cosine
functions, the Bessel functions, the hypergeometric functions and their
various generalizations. Whereas solutions of the linear differential
equations with constant coefficients admit universal methods of obtaining of
summation formulas (see, for instance, [4], [5]), the solutions of nonlinear
equations require special investigations. In this context let us mention the
addition formulae for Jacobi and Weierstrass elliptic functions [6].
The solutions of the generalized Riccati equations with cubic and higher
polynomials, in general, do not admit any summation formula. Nevertheless, by
careful analysis we found a new summation law according to which in order to
obtain a summation formula for the solutions of the third order Riccati
equation is necessary to use two independent variables. We will show that the
summation formula can be derived also by using interconnection between
solutions of Riccati-Abel equation and the characteristic functions of
generalized complex algebra of third order.
The paper is presented by the following sections. Section 2 committed to
solution of ordinary Riccati equation with constant coefficients. Summation
formula for the solutions are derived and interrelation with solutions of the
linear differential equations is underlined. In Section 3, the Riccati-Abel
equation is integrated, the corresponding algebraic equation for solutions is
derived, a summation formula for solutions is established. In Section 4, the
solutions of Riccati-Abel equation are constructed within generalized complex
algebra of third order.
## 2 Ordinary Riccati equation, summation formula and general complex
algebra
2.1 The ordinary Riccati equation.
Consider the Riccati equation with constant coefficients
$u^{2}-a_{1}u+a_{0}=\frac{du}{d\phi}.$ $None$
If coefficients $a_{0},a_{1}$ are constants then a great simplification
results because it is possible to obtain the complete solution by means of
quadratures. Thus, equation (2.1) admits direct integration
$\int~{}\frac{dx}{x^{2}-a_{1}x+a_{0}}=\int d\phi.$ $None$
Let $x_{1},x_{2}\in C$ be roots of the polynomial equation
$x^{2}-a_{1}x+a_{0}=0.$ $None$
In order to calculate the integral (2.2) the following formula expansion is
used
$\frac{1}{x^{2}-a_{1}x+a_{0}}=\frac{1}{2x_{1}-a_{1}}\frac{1}{x-x_{1}}+\frac{1}{2x_{2}-a_{1}}\frac{1}{x-x_{2}},$
$None$
where,
${2x_{1}-a_{1}}=(x_{1}-x_{2}),~{}~{}{2x_{2}-a_{1}}=(x_{2}-x_{1}).$
Then the integral (2.2) is easily calculated and the result is given by the
logarithmic functions
$\int^{u}_{w}~{}\frac{dx}{x^{2}-a_{1}x+a_{0}}=\frac{1}{m_{12}}(~{}\log\frac{u-x_{1}}{u-x_{2}}-~{}\log\frac{w-x_{1}}{w-x_{2}}~{})=\phi(u)-\phi(w),$
$None$
where $m_{12}=x_{1}-x_{2}.$ Now, let us keep the first logarithm of (2.5)
depending of the initial limit of the integral, that is
$\frac{1}{m_{12}}\log[\frac{u-x_{1}}{u-x_{2}}]=\phi(u).$
By inverting the logarithm function we come to the algebraic equation for
solution of (2.1),
$\exp(m_{12}\phi)=\frac{u-x_{1}}{u-x_{2}}.$ $None$
Let $u(\phi_{0})=0$, then
$\exp(m_{12}\phi_{0})=\frac{x_{1}}{x_{2}}.$ $None$
As soon as the point $\phi=\phi_{0}$ is determined, one may calculate the
function $u(\phi)$ by making use of algebraic equation (2.6). Since
$a_{1}=x_{1}+x_{2}$, from (2.7) it follows that
$a_{1}=m_{12}\coth(m_{12}\phi_{0}/2).$
Consequently, from (2.6) we obtain
$u(\phi,\phi_{0})=\frac{1}{2}m_{12}\coth(m_{12}\phi_{0}/2)-\frac{1}{2}m_{12}\coth(m_{12}\phi/2).$
2.2 Summation (addition) formula for function $u=u(\phi,\phi_{0})$.
Consider the following integral equation
$\int^{u}~{}\frac{dx}{x^{2}-a_{1}x+a_{0}}+\int^{v}~{}\frac{dx}{x^{2}-a_{1}x+a_{0}}=\int^{w}~{}\frac{dx}{x^{2}-a_{1}x+a_{0}}.$
$None$
The quantity $w$ is a function of $u$ and $v$, if the function $w=f(u,v)$ is
an algebraic function then this function can be considered as the summation
formula. Write (2.8) in the following notations $\phi_{u}+\phi_{v}=\phi_{w}$.
Then, $w(\phi_{w})=w(\phi_{u}+\phi_{v})=f(u(\phi_{u}),~{}v(\phi_{v})~{}).$
Calculating the integrals in (2.8) we come to the following algebraic equation
$\frac{1}{2m}\log\frac{u-x_{1}}{u-x_{2}}\frac{v-x_{1}}{v-x_{2}}=\frac{1}{2m}\log\frac{w-x_{1}}{w-x_{2}}.$
$None$
Thus, the function $w(u,v)$ has to satisfy the equation
$\frac{u-x_{1}}{u-x_{2}}\frac{v-x_{1}}{v-x_{2}}=\frac{w-x_{1}}{w-x_{2}}.$
$None$
Multiplying fractions and taking into account the fact that $x_{1},x_{2}$ obey
(2.3), we get
$\frac{uv-x_{1}(u+v)+a_{1}x_{1}-a_{0}}{uv-x_{2}(u+v)+a_{1}x_{2}-a_{0}}=$
$=\frac{\frac{uv-a_{0}}{u+v-a_{1}}-x_{1}}{\frac{uv-
a_{0}}{u+v-a_{1}}-x_{2}}=\frac{w-x_{1}}{w-x_{2}},$ $w=\frac{uv-
a_{0}}{u+v-a_{1}}.$ $None$
This is the summation formula for function $u(\phi;a_{1},a_{0})$.
2.3 Relationship with General complex algebra.
Like the (co)tangent function can be defined as a ratio of cosine and sine
functions, the solution of the Riccati equation $u(\phi;a_{1},a_{0})$ also can
be represented as a ratio of modified cosine and sine functions. Firstly, let
us construct these functions.
Consider general complex algebra generated by the $(2\times 2)$ matrix [7]
$E=\left(\begin{array}[]{cc}0&-a_{0}\\\ 1&a_{1}/2\end{array}\right)$ $None$
obeying the quadratic equation (2.3):
$E^{2}-a_{1}E+a_{0}I=0,$ $None$
with $I$\- unit matrix. Expansion with respect to $E$ of the exponential
function $\exp(E\phi)$ leads to the Euler formula [8]
$\exp(E\phi)=g_{1}(\phi;a_{0},a_{1})E+g_{0}(\phi;a_{0},a_{1}).$ $None$
In terms of the roots $x_{1},x_{2}$ this matrix equation is separated into two
equations
$\exp(x_{2}\phi)=x_{2}~{}g_{1}(\phi;a_{0},a_{1})+g_{0}(\phi;a_{0},a_{1}),~{}~{}\exp(x_{1}\phi)=x_{1}~{}g_{1}(\phi;a_{0},a_{1})+g_{0}(\phi;a_{0},a_{1}),$
$None$
from which an explicit form of $g$-functions can be obtained. Apparently,
$g_{0}$ and $g_{1}$ are modified (generalized) cosine-sine functions with the
following formulas of differentiation
$\frac{d}{d\phi}g_{1}(\phi;a_{0},a_{1})=g_{0}(\phi;a_{0},a_{1})+a_{1}~{}g_{1}(\phi;a_{0},a_{1}),~{}~{}\frac{d}{d\phi}g_{0}(\phi;a_{0},a_{1})=-a_{0}~{}g_{1}(\phi;a_{0},a_{1}).$
$None$
Form a ratio of two equations of (2.15) as follows
$\exp(m_{21}\phi)=\frac{x_{2}~{}g_{1}(\phi;a_{0},a_{1})+g_{0}(\phi;a_{0},a_{1})}{x_{1}~{}g_{1}(\phi;a_{0},a_{1})+g_{0}(\phi;a_{0},a_{1})}.$
$None$
Let $g_{1}(s;a_{0},a_{1})\neq 0$. Then,
$\exp(m_{21}\phi)=\frac{x_{2}+D}{x_{1}+D},$ $None$
where
$D=\frac{g_{0}(\phi;a_{0},a_{1})}{g_{1}(\phi;a_{0},a_{1})}.$
Differential equation for function $D(\phi)$ is obtained by using (2.16):
$D^{2}+a_{1}D+a_{0}=-\frac{dD}{d\phi}.$ $None$
Thus, we have proved that the function
$u(\phi;a_{0},a_{1})=-D=-\frac{g_{0}(\phi;a_{0},a_{1})}{g_{1}(\phi;a_{0},a_{1})}$
$None$
obeys the Riccati equation.
The summation formulae for $g$-functions are well defined [7]. They are
$g_{0}(a+b)=g_{0}(a)g_{0}(b)-a_{0}g_{1}(a)g_{1}(b),~{}~{}$
$g_{1}(a+b)=g_{1}(a)g_{0}(b)+g_{0}(a)g_{1}(b)+a_{1}g_{1}(a)g_{1}(b).$
$\frac{g_{0}(a+b)}{g_{1}(a+b)}=\frac{g_{0}(a)g_{0}(b)-a_{0}g_{1}(a)g_{1}(b)}{g_{1}(a)g_{0}(b)+g_{0}(a)g_{1}(b)+a_{1}g_{1}(a)g_{1}(b)}.$
$None$
By taking into account (2.20) we get
$u(a+b)=-\frac{g_{0}(a+b)}{g_{1}(a+b)}=\frac{u(a)u(b)-a_{0}}{u(a)+u(b)-a_{1}}.$
$None$
which coincides with (2.11).
## 3 Generalized Riccati equation with cubic order polynomial
3.1 The Riccati-Abel equation.
Consider the following non-linear differential equation with constant
coefficients
$u^{3}-a_{2}u^{2}+a_{1}u-a_{0}=\frac{du}{d\phi},$ $None$
which admits direct integration by
$\int^{u}_{w}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=\phi(w)-\phi(u).$
$None$
This integral is calculated by making use of well-known method of partial
fractional decomposition [9]
$\frac{1}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=\frac{1}{(x-x_{3})(x-x_{2})(x-x_{1})}=$
$\frac{(x_{3}-x_{2})}{V}\frac{1}{x-x_{1}}+\frac{(x_{1}-x_{3})}{V}\frac{1}{x-x_{2}}+\frac{(x_{2}-x_{1})}{V}\frac{1}{x-x_{3}},$
$None$
where $V$ is the Vandermonde’s determinant [10]
$V=(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1}),$ $None$
and the distinct constants $x_{1},x_{2},x_{3}\in C$ are roots of the cubic
polynomial
$f(x)=x^{3}-a_{2}x^{2}+a_{1}x-a_{0}=0.$ $None$
By using expansion (3.3) the integral (3.2) is easily calculated
$\int^{u}_{w}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=$
$\frac{(x_{3}-x_{2})}{V}\log\frac{u-x_{1}}{w-x_{1}}+\frac{(x_{1}-x_{3})}{V}\log\frac{u-x_{2}}{w-x_{2}}+\frac{(x_{2}-x_{1})}{V}\log\frac{u-x_{3}}{w-x_{3}}=\phi(u)-\phi(w).$
$None$
Introduce the following notations
$m_{ij}=(x_{i}-x_{j}),~{}i,j=1,2,3,~{}\mbox{with}~{}m_{21}+m_{32}+m_{13}=0.$
$None$
Equation (3.6) re-write as follows
$\int^{u}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=\log{~{}(u-x_{1})^{m_{32}}(u-x_{2})^{m_{13}}(u-x_{3})^{m_{21}}}=V\phi(u)$
$None$
and invert the logarithm, this leads to the following algebraic equation
$[u-x_{1}]^{m_{32}}[u-x_{2}]^{m_{13}}[u-x_{3}]^{m_{21}}=\exp(V\phi).$ $None$
This equation can be written also in the fractional form
$[\frac{u-x_{1}}{u-x_{3}}]^{m_{32}}[\frac{u-x_{2}}{u-x_{3}}]^{m_{13}}~{}=\exp(V\phi).$
$None$
Thus, the problem of solution of differential (3.1) is reduced to the problem
of solution of the algebraic equation (3.10). Notice, if the roots of cubic
equation and function $u$ are defined in the field of real numbers then this
equation is meaningful only for certain domain of definition of $u(\phi)$.
3.2 Semigroup property of fractions of $n$-order monic polynomials on the set
of roots of $n+1$-order polynomial.
In this section let us recall semigroup property of the fractions of $n$-order
polynomials defined on the set of roots of $n+1$-order polynomial. Let
$F(x,n+1)$ be $(n+1)$ order polynomial with $(n+1)$ distinct roots
$x_{i},i=1,\ldots,n+1$. Denote this set of roots by $FX(n+1)$.
Lemma 3.1
Let $P_{a}(x_{i},n)$ be $n$-order polynomial on $x_{i}\in FX(n+1)$. The
product of two $n$-order polynomials
$P_{a}(x_{i},n)*P_{b}(x_{i},n)$
is also $n$-order polynomial $P_{c}(x_{i},n)$.
Proof.
The product $P_{ab}(x_{i},2n):=P_{a}(x_{i},n)*P_{b}(x_{i},n)$ is polynomial of
$2n$-degree with respect to variable $x_{i}$. Since $x_{i}$ obeys $n+1$-order
polynomial equation all degrees of the variable $x_{i}$ from $(n+1)$ up till
$2n$ degree can be expressed via $n$-degree polynomial. In this way the
polynomial $P_{ab}(x_{i},2n)$ is reduced till $n$ degree polynomial with
respect to variable $x_{i}\in FX(n+1)$.
End of proof.
are $n$-order monic polynomials defined on the set of the roots of polynomial
$F(x,n+1)$.
Consider two monic polynomials of $n$-degree
$P_{a}(x_{i},n),~{}P_{b}(x_{k},n)$ with $x_{i}\neq x_{k}\in FX(n+1)$. Form a
rational algebraic fraction
$\frac{P_{a}(x_{i},n)}{P_{a}(x_{k},n)}.$
The following Corollary 3.2 holds true:
The product of two fractions formed by two $n$-order monic polynomials on the
roots of $(n+1)$-order polynomial is a fraction of the same order monic
polynomials on the variables,
$\frac{P_{a}(x_{i},n)}{P_{a}(x_{k},n)}\frac{P_{b}(x_{i},n)}{P_{b}(x_{k},n)}=\frac{P_{c}(x_{i},n)}{P_{c}(x_{k},n)}.$
3.3 Addition formula for $u(\phi)$.
Let $\phi=\phi_{0}$ be a point where $u(\phi_{0})=0$. Then, (3.10) is reduced
to
$[\frac{x_{1}}{x_{3}}]^{m_{32}}[\frac{x_{2}}{x_{3}}]^{m_{13}}=\exp(V\phi_{0}).$
$None$
If we make simultaneous translations of the roots $x_{k},k=1,2,3$ by some
value $u$ in the left-hand side of (3.11), then in the right-hand side of the
equation $V$ does not change, hence $\phi_{0}$ will undergo some translation
by $\phi=\phi_{0}+\delta$. In this way one may construct the solution of
Riccati-Abel equation (3.1) with initial condition $u(\phi_{0})=0.$
Now, let $u,v,w$ be solutions of equation (3.1) calculated for tree variables
$\phi_{u},\phi_{v},\phi_{w}$, which obey the equation
$\phi_{w}=\phi_{u}+\phi_{v}$. Then, in accordance with (3.10) we write
$\exp(V\phi_{u})\exp(V\phi_{v})=\\{~{}[\frac{u-x_{1}}{u-x_{3}}\frac{v-x_{1}}{v-x_{3}}]^{m_{32}}[\frac{u-x_{2}}{u-x_{3}}\frac{v-x_{2}}{v-x_{3}}]^{m_{21}}~{}~{}\\}$
$=\\{~{}(\frac{w-x_{1}}{w-x_{3}})^{m_{32}}[\frac{w-x_{2}}{w-x_{3}}]^{m_{21}}~{}~{}\\}=\exp(V(\phi_{u}+\phi_{v}).$
$None$
The problem is to find some rational function expressing $w$ via the pair
$(u,v)$, i.e., the function $w=w(u,v)$ has to be a rational function.
Evidently, the method used in the previous section for the ordinary Riccati
equation now is not applicable. According to Lemma we are able to transform a
product of ratios of $n$-order polynomials into the ratio of $n$-order
polynomials if these polynomials are defined on roots of $n+1$-order
polynomial. Thus, we have to seek another way of construction of a summation
formula.
The problem we suggest to resolve as follows.
Let us to present the integral (3.8) as a sum of two integrals by
$\int^{w}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=\int^{u}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}+\int^{v}~{}\frac{dx}{x^{3}-a_{2}x^{2}+a_{1}x-a_{0}}=$
$=\phi=V\log(~{}(~{}\frac{u-x_{1}}{u-x_{3}}~{}\frac{v-x_{1}}{v-x_{3}}~{})^{m_{32}}~{}(~{}\frac{u-x_{2}}{u-x_{3}}~{}\frac{v-x_{2}}{v-x_{3}}~{})^{m_{13}}~{}).$
$None$
In this way we arrive to the following algebraic equation
$~{}(~{}\frac{u-x_{1}}{u-x_{3}}~{}\frac{v-x_{1}}{v-x_{3}}~{})^{m_{32}}(~{}\frac{u-x_{2}}{u-x_{3}}~{}\frac{v-x_{2}}{v-x_{3}}~{})^{m_{13}}~{}~{})=\exp(V\phi(u,v))=\exp(V\phi_{u})\exp(V\phi_{v}).$
$None$
Let $u,v$ be solutions of the quadratic equation
$x^{2}+tx+s=0,~{}t=-(u+v),~{}s=uv.$ $None$
Then, equation (3.14) is written as
$[~{}\frac{x_{1}^{2}+tx_{1}+s}{x_{3}^{2}+tx_{3}+s}~{}]^{m_{32}}*(~{}\frac{x_{2}^{2}+tx_{2}+s}{x_{3}^{2}+tx_{3}+s}~{}~{})^{m_{13}}~{}~{})=\exp(V\phi(t,s)).$
$None$
Thus from the pair of functions $(u,v)$ we come to another pair $(t,s)$. This
pair of functions, in fact, admits a summation rule because the problem is
reduced to the task of transformation four-degree polynomial into quadratic
polynomial at the solutions of the cubic equation. Evidently, this task can be
easily performed by simple algebraic operations.
Theorem 3.3.
The following summation formula for solutions of Riccati-Abel equation holds
true
$(t,s)\bigoplus(v,u)=(r,w),$
where
$r=~{}\frac{~{}(a_{0}-2a_{2}a_{1})-a_{1}(v+t)+(tu+sv)}{(~{}3a_{2}^{2}-a_{1})+a_{2}(v+t)+(s+u+tv)}~{}~{}w=\frac{a_{2}a_{0}+(v+t)a_{0}+su}{(~{}3a_{2}^{2}-a_{1})+a_{2}(v+t)+(s+u+tv)}.$
$None$
Proof.
Consider product of two monic polynomials
$(x^{2}+tx+s)(x^{2}+vx+u)=x^{4}+x^{3}(v+t)+x^{2}(s+u+tv)+x(tu+vs)+su,$
where $x$ is one of roots of cubic equation
$x^{3}-a_{2}x^{2}+a_{1}x-a_{0}=0.$ $None$
From the cubic equation (3.18) we are able to express $x^{3}$ and $x^{4}$ as
polynomials of second order as follows
$x^{3}=a_{2}x^{2}-a_{1}~{}x+a_{0},~{}~{}x^{4}=(~{}3a_{2}^{2}-a_{1})x^{2}+(a_{0}-a_{1}a_{2})~{}x+a_{2}a_{0}.$
Then, four-degree polynomial on roots of the cubic polynomial is reduced into
polynomial of second order
$x^{4}+x^{3}(v+t)+x^{2}(s+u+tv)+x(tu+vs)+su=Ax^{2}+Bx+C,$ $None$
where
$A=(~{}3a_{2}^{2}-a_{1})+a_{2}(v+t)+(s+u+tv),~{}B=(a_{0}-2a_{2}a_{1})-a_{1}(v+t)+(tu+sv),~{}C=a_{2}a_{0}+(v+t)a_{0}+su.$
$None$
Since we deal with the ratios of polynomials the coefficients of the quadratic
polynomial in (3.19) and polynomials in denominator and in numerator have the
same leading coefficient, we are able return to the ratio of monic
polynomials. In this way we come to the relations
$r=\frac{B}{A},~{}~{}w=\frac{C}{A}.$ $None$
End of proof.
## 4 Generalized complex algebra of third order and solutions of Riccati-
Abel equation
In this section we will establish a relationship between characteristic
functions of general complex algebra of third order and solutions Riccati-Abel
equation.
The unique generator $E$ of general complex algebra of third-order, $CG_{3}$,
is defined by cubic equation [11]
$E^{3}-a_{2}E^{2}+a_{1}E-a_{0}=0.$ $None$
The companion matrix $E$ of the cubic equation (4.1) is given by $(3\times 3)$
matrix
$E:=\left(\begin{array}[]{ccc}0&0&a_{0}\\\ 1&0&-a_{1}\\\
0&1&a_{2}\end{array}\right).$ $None$
Consider the expansion
$\exp(E\phi_{1}+E^{2}\phi_{2})=g_{0}(\phi_{1},\phi_{2})+E~{}g_{1}(\phi_{1},\phi_{2})+E^{2}~{}g_{2}(\phi_{1},\phi_{2}).$
$None$
This is an analogue of the Euler formula for exponential function, the
function $g_{0}(\phi_{1},\phi_{2})$ is an analogue of cosine function, and
$g_{k}(\phi_{1},\phi_{2}),k=0,1,2$ are extensions of the sine function. It is
seen, the characteristic functions of $GC_{3}$ algebra depend of pair of
”angles”. Correspondingly, for each of them we have formulae of
differentiation.
$\frac{\partial}{\partial\phi_{1}}\left(\begin{array}[]{c}g_{0}\\\ g_{1}\\\
g_{2}\end{array}\right)=\left(\begin{array}[]{ccc}0&0&a_{0}\\\ 1&0&-a_{1}\\\
0&1&a_{1}\end{array}\right)\left(\begin{array}[]{c}g_{0}\\\ g_{1}\\\
g_{2}\end{array}\right),$ $None$
$\frac{\partial}{\partial\phi_{2}}\left(\begin{array}[]{c}g_{0}\\\ g_{1}\\\
g_{2}\end{array}\right)=\left(\begin{array}[]{ccc}0&a_{0}&a_{0}a_{2}\\\
0&-a_{1}&a_{0}-a_{1}a_{2}\\\
1&a_{2}&-a_{1}+a_{2}^{2}\end{array}\right)\left(\begin{array}[]{c}g_{0}\\\
g_{1}\\\ g_{2}\end{array}\right).$ $None$
The semigroup of multiplications of the exponential functions leads to the
following the addition formulae for $g$\- functions[12]
$\left(\begin{array}[]{c}g_{0}\\\ g_{1}\\\
g_{2}\end{array}\right)_{(\psi_{c}=\psi_{a}+\psi_{b})}=\left(\begin{array}[]{ccc}g_{0}&g_{2}a_{0}&g_{1}a_{0}+g_{2}a_{0}a_{2}\\\
g_{1}&g_{0}-g_{2}a_{1}&-g_{1}a_{1}+g_{2}(a_{0}-a_{1}a_{2})\\\
g_{2}&g_{1}+g_{2}a_{2}&g_{0}+g_{1}a_{2}+g_{2}(-a_{1}+a_{2}^{2})\end{array}\right)_{\psi_{a}}\left(\begin{array}[]{c}g_{0}\\\
g_{1}\\\ g_{2}\end{array}\right)_{\psi_{b}},$ $None$
where the sub-indices of the brackets indicate dependence of the $g$-functions
of the pair of variables $\psi_{i}=(\phi_{1i},\phi_{2i}),i=a,b,c$.
Introduce two fractions of $g$-functions by
$tg=\frac{g_{1}}{g_{2}},~{}~{}sg=\frac{g_{0}}{g_{2}}.$ $None$
It is seen, these functions are analogues of tangent-cotangent functions. From
addition formulae for $g$-functions (4.6) the following summation formulae for
the general tangent functions are derived.
$t_{0}=g_{0}/g_{2},~{}~{}r_{0}=f_{0}/r_{2}.$ $None$
$T_{0}=\frac{t_{0}r_{0}+a_{0}(r_{1}+t_{1})+a_{0}a_{2}}{r_{0}+(t_{1}+a_{2})r_{1}+t_{0}+t_{1}a_{2}+(-a_{1}+a_{2}^{2})~{}},$
$None$
$T_{1}=\frac{t_{1}r_{0}+t_{0}r_{1}-a_{1}(r_{1}+t_{1})+(a_{0}-a_{1}a_{2})~{})}{r_{0}+t_{0}+a_{2}(t_{1}+r_{1})+t_{1}r_{1}+(-a_{1}+a_{2}^{2})~{}}.$
$None$
Here the following notations are used
$T_{0}(\psi_{c})=\frac{g_{0}(\psi_{c})}{g_{2}(\psi_{c})},~{}~{}T_{1}(\psi_{c})=\frac{g_{1}(\psi_{c})}{g_{2}(\psi_{c})},$
$t_{0}(\psi_{a})=\frac{g_{0}(\psi_{a})}{g_{2}(\psi_{a})},~{}r_{0}(\psi_{b})=\frac{g_{0}(\psi_{b})}{g_{2}(\psi_{b})},$
$t_{1}(\psi_{a})=\frac{g_{1}(\psi_{a})}{g_{2}(\psi_{a})},~{}r_{1}(\psi_{b})=\frac{g_{1}(\psi_{b})}{g_{2}(\psi_{b})},$
and $\psi_{i}=(\phi_{1i},\phi_{2i}),i=a,b$,
$\psi_{c}=(\phi_{1c}=\phi_{1a}+\phi_{1b},\phi_{2c}=\phi_{2a}+\phi_{2b})$.
Let $x_{1},x_{2},x_{3}\in C$ be eigenvalues of $E$ given by distinct values.
Then, the matrix equation (4.3) is represented by three separated series
$(k=1,2,3)$:
$\exp(x_{k}\phi_{1}+x_{k}^{2}\phi_{2})=g_{0}(\phi_{1},\phi_{2})+x_{k}~{}g_{1}(\phi_{1},\phi_{2})+x_{k}^{2}~{}g_{2}(\phi_{1},\phi_{2}),$
$None$
Form the following ratios $i\neq k$:
$\exp((x_{i}-x_{k})\phi_{1}+(x_{i}^{2}-x_{k}^{2})\phi_{2})=\frac{g_{0}(\phi_{1},\phi_{2})+x_{i}~{}g_{1}(\phi_{1},\phi_{2})+x_{i}^{2}~{}g_{2}(\phi_{1},\phi_{2})}{g_{0}(\phi_{1},\phi_{2})+x_{k}~{}g_{1}(\phi_{1},\phi_{2})+x_{k}^{2}~{}g_{2}(\phi_{1},\phi_{2})~{}}.$
$None$
Consider two of these ratios, namely,
$\exp(m_{13}\phi_{1}+(x_{1}^{2}-x_{3}^{2})\phi_{2})=\frac{g_{2}x^{2}_{1}+g_{1}x_{1}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}},$
$None$
$\exp(m_{23}\phi_{1}+(x_{2}^{2}-x_{3}^{2})\phi_{2})=\frac{g_{2}x^{2}_{2}+g_{1}x_{2}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}}.$
$None$
where $m_{ij}=x_{i}-x_{j}$. The both sides of equation (4.13a) raise to power
$m_{32}$ and the both sides of equation (4.13b) raise to power $m_{13}$ and
multiply left and right sides of the obtained equations, correspondingly. And,
by taking into account that $m_{13}m_{32}+m_{23}m_{13}=0$, we arrive to the
following equation
$\exp(m_{13}m_{32}\phi_{1}+(x_{1}+x_{3})m_{13}m_{32}\phi_{2})~{}\exp(m_{23}m_{13}\phi_{1}+(x_{2}+x_{3})m_{13}\phi_{2})$
$=[\frac{g_{2}x^{2}_{2}+g_{1}x_{2}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}}]^{m_{13}}~{}[\frac{g_{2}x^{2}_{1}+g_{1}x_{1}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}}]^{m_{32}}.$
$None$
The left hand side of this equation is equal to $\exp(V\phi_{2})$, that is,
$\exp(V\phi_{2})=[\frac{g_{2}x^{2}_{2}+g_{1}x_{2}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}}]^{m_{13}}~{}[\frac{g_{2}x^{2}_{1}+g_{1}x_{1}+g_{0}}{g_{2}x^{2}_{3}+g_{1}x_{3}+g_{0}}]^{m_{32}}.$
$None$
Let $g_{2}\neq 0$, then by dividing numerator and denominator by $g_{2}$ we
obtain
$\exp(V\phi_{2})=[\frac{x^{2}_{2}+tg~{}x_{2}+sg}{x^{2}_{3}+tg~{}x_{3}+sg}]^{m_{13}}~{}[\frac{x^{2}_{1}+tg~{}x_{1}+sg}{x^{2}_{3}+tg~{}x_{3}+sg}]^{m_{32}},$
$None$
where
$tg=\frac{g_{1}}{g_{2}},~{}~{}sg=\frac{g_{0}}{g_{2}}.$
Let $u,v$ be roots of the quadratic equation
$g_{0}(\phi_{1},\phi_{2})+y~{}g_{1}(\phi_{1},\phi_{2})+y^{2}~{}g_{2}(\phi_{1},\phi_{2})=0.$
$None$
Then the ratios (4.13a,b) can be re-written as follows
$\exp((x_{k}-x_{l})\phi_{1}+(x_{k}^{2}-x_{l}^{2})\phi_{2})=\frac{(u-x_{k})}{(u-x_{l})}\frac{(v-x_{k})}{(v-x_{l})}.$
$None$
This equation is true for any $k,l=1,2,3,~{}k\neq l.$ This is to say, for any
index we have same $\phi_{1},\phi_{2}$ and same $u,v$. Here $u,v$ depend of
two parameters $\phi_{1},\phi_{2}$.
Inversely, If we have $u=u(\varphi_{u}),~{}v=v(\varphi_{v})$, then we can find
corresponding $g$ by
$\frac{g_{0}}{g_{2}}=uv,~{}~{}\frac{g_{1}}{g_{2}}=u+v.$
From this two equations we find $\phi_{1}$ and $\phi_{2}$. We expect that
$\exp(V(\varphi_{u}+\varphi_{v}))=\exp(V\phi_{2}),$
or,
$\varphi_{u}+\varphi_{v}=\phi_{2}.$
In this way we have established connection between characteristics of general
complex algebra and solutions of Riccati-Abel equation.
The next task is to prove that the ratio $u=-g_{0}/g_{1}|_{g_{2}=0}$, in fact,
satisfies the Riccati-Abel equation.
Now, let us calculate derivatives of $g_{1},g_{0}$ under the following
condition
$g_{2}(\phi_{1},\phi_{2})=0.$ $None$
From this equation it follows that $\phi_{1}$ is a function of $\phi_{2}$,
$\phi_{1}=\phi_{1}(\phi_{2})$. Thus, we have to prove that the function
$u(\phi_{2})=-\frac{g_{0}(\phi_{1}(\phi_{2}),\phi_{2})}{g_{1}(\phi_{1}(\phi_{2}),\phi_{2})},$
$None$
obeys the Riccati-Abel equation.
Differentiating equation (4.19) we obtain
$\frac{dg_{2}}{d\phi_{2}}+\frac{dg_{2}}{d\phi_{1}}\frac{d\phi_{1}}{d\phi_{2}}=0.$
$None$
From this equation taking into account constraint (4.21) we get
$\frac{d\phi_{1}}{d\phi_{2}}=-\frac{1}{g_{1}}(g_{0}+a_{2}g_{1}).$ $None$
Now we have to use the following formulae
$\frac{dg_{0}}{d\phi_{2}}=a_{0}g_{1}+\frac{dg_{0}}{d\phi_{1}}\frac{d\phi_{1}}{d\phi_{2}}=a_{0}g_{1}-a_{0}\frac{g_{2}}{g_{1}}(g_{0}+a_{2}g_{1})|_{g_{2}=0}=a_{0}g_{1}.$
$\frac{dg_{1}}{d\phi_{2}}=-a_{1}g_{1}-\frac{dg_{1}}{d\phi_{1}}\frac{d\phi_{1}}{d\phi_{2}}=-a_{1}g_{1}-\frac{g_{0}}{g_{1}}(g_{0}+a_{2}g_{1}).$
By using these formulae we are able to calculate derivative of the fraction:
$\frac{d}{d\phi_{2}}\frac{g_{0}}{g_{1}}=\frac{g_{0}^{\prime}g_{1}-g_{0}g_{1}^{\prime}}{g_{1}^{2}}=\frac{1}{g_{1}^{3}}(~{}a_{0}g_{1}^{3}+a_{1}g_{1}^{2}g_{0}+g_{0}^{3}+a_{2}g_{1}g_{0}^{2}).$
$None$
Coming back to definition (4.20) transform (4.23) into Riccati-Abel equation:
$\frac{du}{d\phi_{2}}=-a_{0}+a_{1}u^{2}+u^{3}-a_{2}u^{2}.$ $None$
Concluding remarks.
As the ordinary Riccati equation, also the Riccati-Abel equation has a
relationship with linear differential equation. Seeking a summation formula
for solutions of Riccati-Abel equation we established a certain interrelation
between these solutions with multi-trigonometric functions of third order. We
have elaborated some rule according to which in order to build a summation
formula for solutions of Riccati-Abel equations it is necessary to consider
the pair of solutions, which can be achieved by using an auxiliary variable.
This idea can be successfully used for the solutions of generalized Riccati
equations of any order with constant coefficients. By increasing the order of
the non-linearity the number of auxiliary variables also will increase. For
example, from solutions of generalized Riccati equations of fourth order we
have to compose the triplet of solutions with two auxiliary variables, and for
$n$-order generalized Riccati equations it is necessary to compose
$(n-1)$-pulet of solutions with $(n-1)$ auxiliary variables.
## References
* [1] Davis H.T. Introduction to nonlinear differential and integral equations. United States Atomic Energy Commission. U.S.Goverment Printing Office, washington D.C.Reprinted Dover, New York 1960.
* [2] N.H.Abel, Oeuvres completes du Niels Henrik Abel.-Christiana, 1881.
* [3] Yeomans J.M. Statistical mechanics of phase transitions. Clarendon Press, Oxford. (1992).
John Cardy. Scaling and renormalization in statistical physics. Cambridge
lecture notes in physics. Eds. P.Goddard, J.Yeomans. Cambridge university
press 1996. ISBN 052149959 3.
* [4] A.Ungar, Addition theorems inordinary differential equations. Amer.Math.Monthly 94 (1987) 872-875.
* [5] A.Ungar, Addition theorems for solutions to linear homogeneous constant coefficient differential equations. Aequatios Math. 26 (1983), 104-112.
* [6] N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
* [7] R.M.Yamaleev, Geometrical and physical interpretation of evolution governed by general complex algebra Journal of Mathematical Analysis and Applications doi:10.1016/j.jmaa. (2007)09.018; 340(2008) 1046-1057
* [8] Babusci D.,Dattoli G.,Di Palma E.,Sabia E., Complex-type numbers and generalization of the Euler identity. Adv.Appl.Clifford Al. 22 (2012) 271.
* [9] Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, by Swami Sankaracarya (1884-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965;
Stapel, Elizabeth. ”Partial-Fraction Decomposition: General Techniques.”
Purplemath. Available from http://www.purplemath.com/modules/partfrac.htm.
* [10] Vein R., Dale P., ”Determinants and their applications in mathematical physics”, Springer-Verlag, New York, Inc., 1999. ISBN 0-387-98558-1.
* [11] R.M.Yamaleev Multicomplex algebras on polynomials and generalized Hamilton dynamics Journal of Mathematical Analysis and Applications 322 (2006) 815-824.
* [12] R.M.Yamaleev Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics J. Adv. Appl. Clifford Al. 15 No.2 (2005) 123-150.
|
arxiv-papers
| 2012-10-08T10:52:24 |
2024-09-04T02:49:36.170963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robert M.Yamaleev",
"submitter": "Robert Yamaleev Masgutovich",
"url": "https://arxiv.org/abs/1210.2228"
}
|
1210.2390
|
# Växjö Interpretation of Wave Function: 2012
Andrei Khrennikov
International Center for Mathematical Modelling
in Physics and Cognitive Sciences
Linnaeus University, Växjö, S-35195, Sweden
###### Abstract
We discuss the problem of interpretation of the wave function. The latest
version of the Växjö interpretation is presented. It differs essentially from
the original Växjö interpretation (2001). The main distinguishing feature of
the present Växjö interpretation is the combination of realism on the
subquantum level with nonobjectivity of quantum observables (i.e.,
impossibility to assign their values before measurements). Hence, realism is
destroyed by detectors transforming continuous subquantum reality into
discrete events, clicks of detectors. The Växjö interpretation-2012 is
fundamentally contextual in the sense that the value of an observable depends
on measurement context. This is contextuality in Bohr’s sense. It is more
general than Bell’s contextuality based on joint measurements of compatible
observables.
## 1 Introduction
During the Växjö series of conferences on quantum foundations the most
exciting spectacle started each time when the question of interpretations of
the wave function attracted the attention. Finally, I understood that the
number of different interpretations is in the best case equal to the number of
participants. If you meet two people who say that they are advocates of, e.g.,
the Copenhagen interpretation of quantum mechanics (QM), ask them about the
details. You will see immediately that their views on what is the Copenhagen
interpretation can differ very much. The same is true for other
interpretations. If two scientists tell that they are followers of Albert
Einstein’s ensemble interpretation, ask them about the details… At one of the
round tables (after two hours of debates with opinions for and against
completeness of QM) we had decided to vote on this problem. Incompleteness
advocates have won, but only because a few advocates of completeness voted for
incompleteness. The situation is really disappointing: the basic notion of QM
has not yet been properly interpreted (after 100 years of exciting, but not
very productive debates).111“When I speak with somebody and get to know their
interpretation, I understand immediately it is wrong. The main problem is that
I do not know whether my own interpretation is right.” (Theo Nieuwenhuizen)
This is the standard problem of participants of Växjö conferences. I
specifically appreciate the activity of Arkady Plotnitsky, philosopher
studying Bohr’s views, see, e.g., [1]. He teaches us (participants of Växjö
conferences) about interpretations of the wave functions a lot. First of all
we got to know that the Copenhagen interpretation cannot be rigidly coupled
with Bohr’s views. On many occasions Niels Bohr emphasized that QM is not
about physical processes in microworld, but about our measurements [2]:
“Strictly speaking, the mathematical formalism of quantum mechanics and
electrodynamics merely offers rules of calculation for the deduction of
expectations pertaining to observations obtained under well-defined
experimental conditions specified by classical physical concepts”.
The basic postulate of the Copenhagen interpretation of QM – “the wave
function describes completely the state of a quantum system” (i.e., a concrete
system, not an ensemble) – cannot be assigned to Bohr. Then we learned (again
from Plotnitsky) that Bohr’s views have been crucially changed a few times
during his life. Thus, there can be found many different Bohr’s
interpretations of QM. Bohr was definitely the father of the operational
interpretation of QM. As was already pointed out, Bohr emphasized that the
formalism of QM does not provide the intrinsic description of processes in
microworld, it describes only results of measurements. Bohr also can be
considered as one of fathers of the so-called information interpretation of
QM: the QM-formalism describes information about micro systems extracted by
means of macroscopic measurement devices. Heisenberg (and to some extent
Schrödinger) shared this viewpoint. Nowadays the information interpretation of
QM became very popular, see, e.g., [3], [4], [1]. I can mention Anton
Zeilinger [5] and Christopher Fuchs [6]–[8] among the active promoters of this
interpretation; we can also mention Mermin’s paper [9].
In this paper we discuss three main interpretations of QM: the Copenhagen,
ensemble and Växjö interpretations.
## 2 Interpretations of Wave Function
Everywhere below $H$ denotes complex Hilbert space with the scalar product
$\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$ corresponding to the
scalar product.
Postulate IM. (The mathematical description of quantum states). Quantum (pure)
states (wave functions) are represented by normalized vectors $\psi$ (i.e.,
$\|\psi\|^{2}=\langle\psi,\psi\rangle=1$) of a complex Hilbert space $H.$
Every normalized vector $\psi\in H$ may represent a quantum state. If a vector
$\psi$ corresponding to a state is multiplied by any complex number $c,|c|=1,$
the resulting vector will correspond to the same state.222Thus states are
given by elements of the unit sphere of the Hilbert space $H.$
The physical meaning of “a quantum state” is not defined by this postulate. It
must be provided by a separate postulate; see Postulates IE, IC, IV
(respectively, the ensemble, Copenhagen and Växjö interpretations).
Postulate IE. (The ensemble interpretation). A wave function provides a
description of certain statistical properties of an ensemble of similarly
prepared quantum systems.
This interpretation is upheld, for example by Einstein, Popper, Blokhintsev,
Margenau, Ballentine, Klyshko, and recent years by, e.g., De Muynck, De Baere,
Holevo, Santos, Khrennikov, Nieuwenhuizen, Adenier, Groessing, Hofer and many
others
Postulate IC. (The Copenhagen interpretation). A wave function provides a
complete description of an individual quantum system.
This interpretation was supported by a great variety of members, from
Schrödinger’s original attempt to identify the electron with a wave function
solution of his equation to the several versions of the Copenhagen
interpretation; for example, Heisenberg, Bohr, Pauli, Dirac, von Neumann,
Landau, Fock, …, Greenberger, Mermin, Lahti, Peres.
There is an interesting story about the correspondence between Bohr and Fock
on the individual interpretation. This story was told me by one of former
students of Fock who pointed out that one of the strongest supporters of this
interpretation was Vladimir A. Fock, and that even though Bohr himself had
doubts about its consistency, he, Fock, demonstrated to Bohr inconsistency in
the Einsteinian ensemble interpretation. Thus interpretation which is commonly
known as the Copenhagen interpretation might be as well called the “Leningrad
interpretation.”
Instead of Einstein’s terminology ”ensemble interpretation”, L. Ballentine
[11], [10] used the terminology “statistical interpretation.” However,
Ballentine’s terminology is rather misleading, because the term “statistical
interpretation” was also used by J. von Neumann [12] for individual
randomness! For him “statistical interpretation” had the meaning which is
totally different from the Ballentine’s “ensemble-statistical interpretation.”
J. von Neumann wanted to emphasize the difference between deterministic
(Newtonian) classical mechanics in that the state of a system is determined by
values of two observables (position and momentum) and quantum mechanics in
that the state is determined not by values of observables, but by
probabilities. We shall follow Albert Einstein and use the terminology
“ensemble interpretation”.
## 3 Växjö interpretation of quantum mechanics
The Växjö interpretation [13] was born (in 2001) as an alternative to the
Copenhagen interpretation. The basic assumption of the latter that the wave
function describes completely the state of a quantum system is the main source
of all quantum mysteries. (We remind that Schrödinger elaborated
the“Schrödinger cat” experiment to show the absurdness of this assumption.
Nowadays the origin of the Schrödinger cat illustration of the absurdness of
the Copenhagen interpretation is practically forgotten. The presence of such
cat-states is often considered as one of natural features of QM.) Therefore
the easiest way to resolve the majority of interpretation problems of QM is to
assume that the wave function description is not the final description of
micro phenomena. By proceeding in this way one has to develop prequantum
models inducing the QM formalism as an operational formalism ignoring details
of processes in the microworld and describing only results of measurements.
Measurements are operationally described by Hermitian operators or more
generally by POVM. However, there are known many no-go statements which
prohibit any motion beyond QM (under the natural assumption of locality333We
remind that Einstein, Podolsky and Rosen considered nonlocality as the absurd
alternative to incompleteness of QM [14]. This is also practically forgotten
(ignored?). One can often find just the statement that quantum nonlocality was
originally discussed by Einstein, Podolsky and Rosen [14]. In his reply to the
EPR paper [15] Bohr did not mention nonlocality at all. He also was sure that
QM is local.). The first version [13] of the Växjö interpretation was created
as the result of analysis of the “impossibility statements”, e.g., [12], [16],
[17], playing the crucial role in the contemporary debates on quantum
foundations. Surprisingly for myself, I found that all no-go statements
contain some unphysical assumptions which are not valid for real experimental
situations, see [18] for details; cf. with Bell’s and Ballentine’s critical
analysis of assumptions of von Neumann no-go theorem [17], [10], [11].
Suddenly I understood that usage of the operational quantum formalism for
results of measurements does not contradict with the possibility of creating
prequantum classical models. These models can be both realistic and local.
Here realism (objectivity) is defined as a possibility to assign the values of
quantum observables to quantum systems before measurement. This viewpoint to
realism is common in discussions related to Bell’s inequality [17] and in
general to inter-relation of classical and quantum physics.
The Växjö interpretation-2001 was the (local) realistic interpretation [13].
In 2004 I visited Beckman Institute for Advanced Science and Technology
(University of Illinois at Urbana-Champaign). After my talk (May 3, 2004), I
discussed with A. Leggett a role of no-go theorems in QM. In particular, I
wondered why Einstein had never mentioned the von Neumann’s no-to theorem.
(After Einstein’s death, the book of von Neumann [12] was found in Einstein’s
office.) A. Leggett remarked that Einstein was mainly interested in the real
physical situation rather than in formal mathematical statements. I started to
think about the real physical situation described by QM. I understood that
even if some classical prequantum model is formally possible, in spite of all
no-go “theorems”, this does not imply that this model matches with the real
physical situation. Hence, although local realism cannot be rejected as a
consequence of, e.g., violation of Bell’s inequality, one has carefully to
analyze matching of local realism with the real physical situation. Since, as
for Einstein and Bohr, I did not take seriously quantum nonlocality, for me
the questionable point was the possibility of realistic representation of
quantum observables.
By reading Bohr’s works [15], [2] I understood the fundamental role of
measurement context in quantum measurements. Bohr stressed that the result of
measurement is the sum of impacts of the quantum system and the measurement
device. And it is impossible to distill the contribution of the system from
the integral measurement result. Thus quantum observables are not objective.
We could not assign a value of, e.g., position to a quantum system before
measurement. We remark that we discuss measurement of a single quantum
observable and not of a pair of observables. Hence, the essence of this
discussion is not in the impossibility of the joint measurement of some
quantum observables, so called incompatible observables, e.g., position and
momentum (the Bohr’s principle of complementarity [2]). We discuss
contextuality of even a single quantum observable: the properties of
measurement context cannot be separated from intrinsic properties of a quantum
system.444 May be Bohr would not accept that a quantum system has intrinsic
properties at all. (Bohr’s works are very difficult for understanding; try to
read , e.g., [15].) At least it is clear that he was not an idealist. So, he
might agree that, e.g., an electron has some intrinsic properties. However,
for Bohr, these properties were fundamentally inapproachable in measurements.
Unfortunately, in recent discussions related to Bell’s theorem [17] the notion
of contextuality is coupled to the joint measurement of compatible
observables. In this book (following N. Bohr), we always use the notion of
contextuality as the irreducible dependence of the result of measurement (of
even a single observable) on the whole experimental arrangement.
Under influence of Bohr I rejected Einstein’s realism.555I also mention an
intermediate version of the Växjö interpretation [19] in which I tried to
combine realism with contextuality, see also [18]. This interpretation might
be useful in some applications of quantum information outside of physics. Here
two circumstances of totally different nature were very important.
One of my main scientific interests is the application of the mathematical
formalism of QM outside of physics: to study statistical data from cognitive
science, psychology, finances [20]. In these applications observables, “mental
observables”, are nonobjective by their nature; contexts play a crucial role.
And this is well known for psychologists and researchers working in cognitive
science, sociology, economics. This out-physics activity improved my
understanding of the role of contextuality.
Another strong motivation to reject realism of quantum observables666We state
again that in this discussion realism is regarded as objectivity of quantum
observables, i.e., the possibility to assign the value of a quantum observable
to the quantum system before measurement. We do not reject realism in its
general philosophical meaning. In particular, quantum systems have their own
properties, properties of objects. However, these objective properties could
not be approached by the measurement devices in use. The problem is that the
class of measurement devices is not large enough to approach the subquantum
level. Of course, this theoretical discussion does not imply that it would be
possible (at least for our civilization) to elaborate novel measurement
devices to approach subquantum spatial and temporal scales. was related to the
development of a measurement theory for prequantum classical statistical field
theory, PCSFT [21]–[33]. This is the classical field theory which reproduces
quantum averages, including correlations for quantum observables (e.g., spin
or polarization) for entangled systems. Thus, opposite to claims of adherents
of the Copenhagen interpretation (starting with von Neumann [12]), quantum
randomness can be reduced to randomness of classical systems. There is,
however, a crucial proviso. While creation of PCSFT implies QM can be
interpreted as a form of classical statistical mechanics, this classical
statistical theory is not that of particles, but of fields. This means that
the mathematical formalism of QM must be translated into the mathematical
formalism of classical statistical mechanics on the infinite-dimensional phase
space. The infinite dimension of the phase space of this translation is a
price of classicality. From the mathematical viewpoint this price is very
high, because in this case the theories of measure, dynamical systems, and
distributions are essentially more complicated than in the case of the finite-
dimensional phase space found in classical statistical mechanics of
particles.777However, at the model level (similar to quantum information
theory) one can proceed with the finite-dimensional phase space by
approximating physical prequantum fields by vectors with finite number of
coordinates. On the other hand, from the physical and philosophical
viewpoints, considering QM as classical statistical mechanics of fields can
resolve the basic interpretational problems of QM. As was already pointed out,
for example and in particular, quantum correlations of entangled systems can
be reduced to correlations of classical random fields. From this perspective,
quantum entanglement is not mysterious at all, since quantum correlations are
no longer different from the classical ones.
However, usage of PCSFT reduces quantum randomness to classical only on the
level of averages and not individual events, clicks of detectors. PCSFT is the
theory of classical random fields reproducing quantum averages (including
correlations for entangled systems) as averages of quadratic forms of fields.
Since the basic model of prequantum fields is the Gaussian one, these fields
are continuous. Hence, in the same way as in the classical signal theory,
averages are calculated for variables with a continuous range of values. This
model, PCSFT, is really a prequantum model. In accordance with the Bohr’s
viewpoint, we consider QM as an operational formalism describing (predicting)
results of measurements on micro systems. QM cannot describe intrinsic
physical processes in the microworld, but only measurements performed by
macroscopic classical devices. In contrast, PCSFT describes intrinsic
processes in the microworld. However, it does not describe results of
measurements. In principle one might be satisfied with creation of a
prequantum model which reproduces only quantum averages without establishing a
direct connection with theory of measurement on the level of individual
events. This approach would match with views of Schrödinger and the Bild
concept in general. (The Bild concent was formulated by Hertz in his 1894
Prinzipien der Mechanik. Schrödinger’s views to QM were based on this
concept.) However, historical development of QM demonstrated that the Bild
concept was not attractive for the majority of physicists. Since the
experimental verification is considered as the basic counterpart of any
physical theory, Bild-like prequantum theories are considered as metaphysical.
Measurement theory connecting PCSFT with experiment on the level of individual
events was developed in my paper [34]. In this paper I elaborated a scheme of
discrete measurements of classical random signals which reproduces the basic
rule of QM, the Born’s rule. This is the scheme of threshold type detection:
such a detector clicks after it has “eaten” a special portion of energy
$\epsilon$ of a prequantum random signal. We call such measurement theory
threshold detection (TSD) model. TSD is a classical measurement model which
delivers the same predictions as QM. In contrast to QM, which is an
operational formalism888In QM measurement devices are simply black boxes which
are symbolically represented by self-adjoint operators or more generally (and
even more formally) by positive operator valued measures (POVM). , TSD
provides a detailed description of the energy balance in the process of
interaction of a classical random field (fluctuating at a fine time scale)
with a detector, as well as the conditions inducing an individual quantum
event, a click. Thus QM can be interpreted as theory of measurements of
classical random signals with the aid of threshold type detectors. They
produce discretization of continuous classical fields. This discretization is
the essence of quantum phenomena. Hence, “quantumness is created in
detectors”, there is no quantumness without measurement. In particular, this
viewpoint contradicts the views of early Einstein who claimed that the
electromagnetic field is quantized not only in the process of the energy
exchange with material bodies, but even in vacuum. In the PCSFT/TSD model
photons appear only on the level of detection, so they are nothing else than
the clicks of detectors. Surprisingly this viewpoint on the notion of photon
coincides with the views of some top level experimenters working in quantum
optics, e.g., A. Zeilinger, A. Migdal, and S. Polyakov also identify photon
with a click of a detector (their talks at Växjö conferences). Of course, the
majority of experts in quantum optics (including aforementioned) proceed under
the Bohr’s assumption on completeness of QM.
Our threshold detection model is fundamentally based on the assumption of
ergodicity of prequantum random signals, i.e., the time and ensemble averages
coincide. We stress that this ergodicity is related to the subquantum
processes, i.e., to the intrinsic physical processes in the microworld.
Another important assumption is that the prequantum random signals fluctuate
on a very fine time scale (comparing with the time scale of lab-measurements).
The duration of measurement, i.e., the interaction of a random field with a
detector, is huge on the prequantum time scale. However, it is sufficiently
small with respect to the time scale of lab-measurements. (The rigorous
mathematical justification of the derivation of the Born’s rule in the TSD-
framework is rather complicated mathematically and it is based on the theory
of ergodic stochastic processes.)
We remark that ergodic processes are stationary. In the process of interaction
with a detector, a prequantum field (emitted by a source and evolving in the
free space in accordance with the Schrödinger equation with a random initial
condition approaches a steady state, the field becomes stationary. This first
phase is very short. Then the stationary stochastic process transmits its
energy to the detector and this stage is very long (infinitely long in the
pure mathematical framework). We remark that this discussion is about the
prequantum time scale.
Max Born postulated the probabilistic rule coupling the wave function with
quantum detection probabilities: the probability equals to the squared wave
function. Whether it is possible to derive this rule from some natural
physical principles is still the subject of intensive debates. In [34] we
derived the Born’s rule in the PCSFT/TSD framework: threshold type detectors
interacting with random signals of a special class produce a statistics of
clicks which is (approximately) described by the Born’s rule. Of course, this
framework is based on PCSFT-coupling between the covariance operator of a
random field and the density operator of the quantum system corresponding to
this random field, see [21]–[33] for details.999 This derivation of the
fundamental rule of QM which establishes the basic relation of the theoretical
predictions of QM with experiment definitely would not match with the
expectation of those who consider QM as theory which is full of mysteries.
PCSFT/TSD also proposed very natural solution of quantum measurement problem
[12]. Continuous evolution of the prequantum random field induces discrete
jumps corresponding to approaching the detection threshold. Such jumps are
formally described by the von Neumann projection postulate. However, this
solution of the measurement problem is also far from the expectations of
people excited by quantum exotics. Moreover, since, to solve the measurement
problem, we go beyond QM, our solution would not be considered as the solution
of the problem formulated by von Neumann [12]. The latter is expected to be
solved in the conventional quantum framework.
However, the TSD-transition from (continuous) prequantum random fields to
clicks of detectors (discrete events) destroys the PCSFT-objectivity. A
prequantum field emitted by a source does not predetermine the result of
measurement. There are a few different sources of non-objectivity of quantum
observables:
BF. The presence of the irreducibly random background field. PCSFT can
reproduce quantum correlations for entangled systems only by taking into
account the random background. Its presence (everywhere in space) makes
correlations stronger than they could be expected. In particular, by PCSFT
Bell’s inequality is violated only due to the contribution of the random
background. Hence, by emitting the concrete pulse from a source we are not
able to predict its structure at the moment of arrival to a detector;
moreover, the random background is also present in detectors. Therefore we are
not able to predict the output of the detector, even if we were able to
eliminate the random background in space between the source and the detector.
However, this is simply the well known problem of classical stochasticity. As
in classical physics, we may idealize the situation and to assume that the
random background can be eliminated (although in reality this is definitely
impossible). Nevertheless, objectivity cannot be recovered even under this
assumption, see CONT and DC.
CONT. In TSD (in the total accordance with Bohr’s views) the result of a
measurement depends fundamentally on experimental context, on all parameters
of the detector in use (and, in particular, on the threshold in the detector).
For the same classical field, detector can click or not depending on its
settings. On the level of individual events such “properties” as spin and
polarization cannot be assigned to the prequantum wave. They are determined by
detection context. (We remark that a prequantum wave definitely has such
properties, but in continuous and not discrete representation). Thus PCSFT/TSD
is contextual, in general Bohr’s and not restricted Bell’s sense. Suppose now
that we fixed all parameters of a detector. Can we speak about objectivity of
the corresponding quantum observable? (Of course, under the assumption that
the random background was totally eliminated.) Still not, see DC.
DC. In QM coincidence counts of two detectors reacting to a single system
emitted by a source are interpreted as the artifacts induced by noise. In TSD
such coincidence counts are fundamentally irreducible. This is a consequence
of the threshold detection scheme. Hence, real observables are multi-valued.
Even the concrete prequantum wave interacting with the concrete detector and
in the total absence of the random background can produce clicks in two
detectors for, e.g., measurement of spin projection to two axes. It is
impossible to assign to this wave neither the value spin up nor spin down.
The new Växjö interpretation combines Einstein’s and Bohr positions. As well
as Einstein, we do not believe that the quantum state describes completely the
physical state of a quantum system. The quantum state describes only
statistical features. As well as Bohr, we do not consider quantum observables
as objective quantities (intrinsic properties of objects). Quantum observables
are fundamentally contextual.
Postulate IV. (The Växjö interpretation). A wave function describes
correlations in prequantum random fields (which are symbolically represented
as quantum systems).
This paper was written under support of the grant ”Mathematical Modeling of
Complex Hierarchic Systems” of the Faculty of Natural Science and Engineering
of Linnaeus University.
## References
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* [4] H. J. Folse, “Bohr’s conception of the quantum mechanical state of a system and its role in the framework of complementarity”, in _Quantum Theory: Reconsideration of Foundations_ , edited by A. Khrennikov, Växjö Univ. Press, Växjö, 2002, pp. 83-98.
* [5] A. Zeilinger, _Dance of the Photons: From Einstein to Quantum Teleportation,_ Farrar, Straus and Giroux, New-York, 2010.
* [6] C. Fuchs, “The anti-Växjö interpretation of quantum mechanics”, _Quantum Theory: Reconsideration of Foundations_ , edited by A. Khrennikov, Växjö Univ. Press, Växjö, 2002, pp. 99-116.
* [7] C. Fuchs, “Quantum mechanics as quantum information (and only a little more)”, _Quantum Theory: Reconsideration of Foundations_ , edited by A. Khrennikov, Växjö Univ. Press, Växjö, 2002, pp. 463-543.
* [8] C. Fuchs, “Delirium quantum (or, where I will take quantum mechanics if it will let me)”, in _Foundations of Probability and Physics-3_ , edited by G. Adenier, C. Fuchs and A. Khrennikov, AIP Conference Proceedings 1232, American Institute of Physics, New York, 2010, pp. 438-462.
* [9] N. D. Mermin, “Whose knowledge?” _Quantum Theory: Reconsideration of Foundations_ , edited by A. Khrennikov, Växjö Univ. Press, Växjö, 2002, pp. 261-270.
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* [20] A. Khrennikov, _Ubiquitous Quantum Structure: from Psychology to Finances,_ Springer, Berlin-Heidelberg-New York, 2010\.
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|
arxiv-papers
| 2012-10-07T17:43:13 |
2024-09-04T02:49:36.182459
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrei Khrennikov",
"submitter": "Andrei Khrennikov",
"url": "https://arxiv.org/abs/1210.2390"
}
|
1210.2432
|
11institutetext: 1Institut d${}^{{}^{\prime}}$Astrophysique de Paris (IAP),
CNRS, UPMC, 98 bis boulevard Arago, France
2Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531,
Tehran, Iran
3Department of Physics and Astronomy, University of California, Riverside, CA
92521, USA
# The challenge of large and empty voids in the SDSS DR7 redshift survey
Saeed Tavasoli 1122 Kaveh Vasei and Roya Mohayaee 223311
(Received 21 Novembre 2012)
###### Abstract
Context. We present catalogues of voids for the SDSS DR7 redshift survey and
for Millennium I simulation mock data.
Aims. We aim to compare the observations with simulations based on a
$\Lambda$CDM model and a semi-analytic galaxy formation model. We use the void
statistics as a test for these models.
Methods. We assembled a mock catalogue that closely resembles the SDSS DR7
catalogue and carried out a parallel statistical analysis of the observed and
simulated catalogue.
Results. We find that in the observation and the simulation, voids tend to be
equally spherical. The total volume occupied by the voids and their total
number are slightly larger in the simulation than in the observation. We find
that large voids are less abundant in the simulation and the total luminosity
of the galaxies contained in a void with a given radius is higher on average
than observed by SDSS DR7 survey. We expect these discrepancies to be even
more significant in reality than found here since the present value of
$\sigma_{8}$ given by WMAP7 is lower than the value of $0.9$ used in the
Millennium I simulation.
Conclusions. The reason why the simulation fails to produce enough large and
dark voids might be the failure of certain semi-analytic galaxy formation
models to reduce the small-scale power of $\Lambda$CDM and to produce
sufficient power on large scales.
###### Key Words.:
cosmology, galaxies
## 1 Introduction
Redshift surveys have been demonstrating for several decades that galaxies are
distributed on a cosmic web of filaments, walls, and clumps. These structures,
which form on a hierarchy of scales and span a wide redshift range, border
low-luminosity regions that are mostly devoid of observable galaxies. These
“void” regions occupy more than $80\%$ of the volume of the observable
Universe. Since the discovery of voids using Zwicky clusters (Einasto et.al.,
1980) and the discovery of the first giant or supervoid in the Bootes
constellation (Kirshner et al., 1981) numerous works have followed (Zeldovich
et al., 1982; Davis et al., 1982; de Lapparent et al., 1986; da Costa et al.,
1988; Geller & Huchra, 1989; da Costa et al., 1994) and diverse algorithms for
void identification have been developed and applied to larger and more
complete surveys (see Colberg et al. (2008) for a summary and comparison of
different methods).
Figure 1: Darker shaded areas in the two panels show the SDSS DR7 region
(left) and the volume-limited sample (right) that we selected for this work.
The formation and evolution of voids is well-understood in the framework of
gravitational instability (Zeldovich et al., 1982; Shandarin & Zeldovich,
1989). However, when one compares void properties of observations and
simulations based on $\Lambda$CDM, certain problems still remain to be better
understood. By definition, voids are devoid of galaxies or contain only a
negligible number of faint galaxies. The perplexing issue is that we do not
see a large population of low-mass galaxies populating voids (Klypin et.al.
(1999); Moore et.al. (1999)), and furthermore, the void galaxies that we do
see are basically representative of the general population (Peebles, 2001).
Observed voids seem to contain fewer galaxies and in particular dwarf
galaxies, contrary to what is expected from $\Lambda$CDM (Peebles, 2001; Tully
et al., 2008; Tikhonov & Klypin, 2009). Some studies have also shown that
voids in observations are significantly larger than those in simulations
(Ryden & Turner, 1984). Although modifying models of galaxy formation might
solve these problems and various remedies such as proper biasing and halo
occupation distribution have been proposed (Hoyle et al., 2005; Tinker et al.,
2008), different studies suggest that the problem would still persist (Bothun
et al., 1986; Little & Weinberg, 1994; Plionis & Basilakos, 2002; Gottlöber et
al., 2003; Hoyle & Vogeley, 2004; Goldberg et al., 2005; Hoeft et al., 2006).
The problem of empty and large voids could arise because the $\Lambda$CDM has
too much power on small scales which would in turn lead to the problem of
over-abundance of substructures (Tikhonov & Klypin, 2009). Substructures would
occupy the voids, making them less empty, and statistically, they could break
larger voids into smaller ones. On the other hand, one could equally infer
that $\Lambda$CDM lacks power on large scales, perhaps because the value of
$\sigma_{8}$ is too low.
In this work, we study this problem by analysing voids in the SDSS DR7 data
and by carrying out a parallel and comparative analysis on a mock-SDSS DR7
catalogue based on the Millennium I simulation. Our void-finder algorithm is
an improved and generalised version of the original algorithm proposed by
Aikio & Maehoenen (1998). The important feature of this algorithm is that it
does not assume a priori that voids are spherical and hence can be used to
study the shapes of the voids. We apply our void-finder algorithm to the Sloan
Digital Sky Survey SDSS DR7 and build a catalogue of voids. In parallel, we
also apply our algorithm to a mock-SDSS DR7 catalogue, which we construct out
of the Millennium I simulation. The mock catalogue is given the same magnitude
cut-off as SDSS DR7. In a different version, we also set up a mock catalogue
with the same number density as SDSS, but a different magnitude cut-off. This
allows us to compare various properties of observed voids to those predicted
by $\Lambda$CDM and the semi-analytic model of galaxy formation.
In Section 2, we present our sample taken from the SDSS DR7 catalogue. In
Section 3, we present our mock catalogue. In Section 4, we explain our void-
finder algorithm. In Section 5, we find the voids in the simulation and
observation catalogues and discuss the numbers, sizes, and shapes of the
voids. In Section 6, we study the abundance of large voids in the observations
and the mock catalogues. In Section 7, luminosities of voids as a function of
their sizes are presented and compared between the simulation and the
observation. In Section 8, we conclude.
Figure 2: Right panel: initial voids in the observational data of SDSS DR7.
Left panel: final voids after without small and edge voids.
## 2 SDSS DR7: definition of the sample
We have selected the main galaxy sample of the seventh data release of the
Sloan Digital Sky Survey (SDSS DR7) (Abazajian et al., 2009). The galaxy
redshifts were corrected for the motion of the local group and are given in
the CMB rest frame. The k-corrections for the SDSS galaxies were calculated
using the KCORRECT algorithm developed by Blanton et.al. (2003a) and Blanton &
Roweis (2007). The boundaries of our selected region of SDSS are: $135<{\rm
RA}<235$ and $0<{\rm DEC}<40$, which contains 283076 galaxies. The choice of
boundaries clearly is arbitrary. However, the selected region in our study
covers most of SDSS DR7. We used spectroscopic data and applied a void
algorithm to volume-limited samples. Had we selected high-redshift galaxies,
we would have had to consider very bright galaxies (M ¡ -21, -22), which would
be meaningless for voids. All objects in this selected region have a redshift
error smaller than $2.5\times 10^{-4}$ and the errors in their apparent
”Petrosian” magnitudes of the r band, $m_{r}$, are smaller than $0.1$. The
absolute magnitudes of the galaxies were determined in the r band using
cosmological parameters; $H_{0}=100$ and the density parameters
$\Omega_{m}=0.25$ and $\Omega_{\Lambda}=0.75$. Galaxies belonging to voids
were identified by using a volume-limited sample taken from the selected
region. The final subsample contains 68702 galaxies with absolute magnitudes
$M_{r}<-19.9$, which lie in the comoving distance interval 75-325 h-1Mpc,
corresponding to $0.02<z<0.12$.
The selected region of the SDSS DR7 is shown in the left panel of Fig. 1. The
right panel of this figure shows the plot of the absolute r-band magnitude
versus comoving distance. The dark region in this plot illustrates the
selected volume-limited sample we used.
## 3 Mock Millennium I catalogue: definition of the sample
The Millennium I simulation was with a $N=2160^{3}$ particles in a comoving
box of length $L=500h^{-1}{\rm Mpc}$ and mass resolution of $8.6\times
10^{8}h^{-1}M_{\odot}$. The adopted cosmology is a $\Lambda$CDM model with
$\Omega_{m}=0.25$, $\Omega_{b}=0.045,\Omega_{\Lambda}=0.75,h=0.73,n=1$ and
$\sigma_{8}=0.9$. This value of $\sigma_{8}$ is higher than its present value
of $0.8$ given by WMAP7 (Komatsu et.al., 2011), hence yielding more power on
larger scales. The evolution of baryons within these dark matter halos is
predicted by different semi-analytic models. Current semi-analytic models try
to incorporate various complex processes such as gas cooling, reionization,
star formation, supernova feedback, metal evolution, black hole growth, and
active galactic nuclei (AGN) feedback (e.g. Bower et.al. (2006),De Lucia &
Blaizot (2007),Guo et al. (2011)). Although the semi-analytic models are
designed to match the observational data as closely as possible, they can
still fail in certain aspects, for example the low-mass galaxies with stellar-
mass ($<10^{9}M_{\odot}$) are slightly over-predicted. Consequently, to remedy
this problem, supernova feedback, a modified law for star formation, or a
different cosmological model are evoked (see e.g. Guo et al. (2011); Bower et
al. (2012); Wang et al. (2012); Menci et.al. (2012)).
In this work, we used the mock galaxy redshift catalogue of the Blaizot-
ALLSky-PT-1 1 11footnotetext:
http://www.gvo.org/Millennium/Help?page=databases/
mpamocks/blaizot2006_allsky, which was designed to mimic the SDSS and has an
almost identical redshift distribution and a very similar colour distribution.
This mock catalogue was constructed by Blaizot et al. (2005) using the mock
map facility (MoMaF) code and the semi-analytic model presented in De Lucia &
Blaizot (2007). Furthermore, to have a mock catalogue that resembles the SDSS
DR7 galaxy survey as closely as possible, we selected a region in the
simulation that lies in the same redshift range ($0.02<z<0.12$) and has the
same geometry. Our mock volume-limited sample includes 68701 galaxies with
stellar masses larger than $10^{9}M_{\odot}$ and brighter than $M_{r}<-20.16$,
roughly representing the galaxies brighter than $M_{r}<-19.9$ in the SDSS DR7
sample and covering a volume of $1.2\times 10^{7}{\rm(Mpc/h)^{3}}$ in the
volume-limited SDSS DR7. Consequently, the simulation sample has the same
galaxy number density as the SDSS DR7 sample.
Table 1: Characteristics of our volume-limited samples. | ${\,{\rm Observation}}$ | ${\,{\rm Simulation}}$
---|---|---
${\rm Sample\,\,Volume\,\,}({\rm Mpc/h})^{3}$ | $\approx 1.2\times 10^{7}$ | $\approx 1.2\times 10^{7}$
${\rm Number\,\,of\,\,galaxies}$ | 68702 | 68701
${\rm Number\,\,of\,\,field\,\,galaxies}$ | 5873 | 5377
${\rm Number\,\,of\,\,wall\,\,galaxies}$ | 62829 | 63324
${\rm Number\,\,of\,\,void\,\,galaxies(field+faint)}$ | 26859 | 43666
${\rm Mean\,\,galaxy\,\,separation\,\,}({\rm Mpc/h})$ | 6.22 | 6.35
## 4 Void-finder algorithm
Various definitions of voids have been suggested previously (Kirshner et al.,
1981; Kauffmann & Fairall, 1991; Sahni et al., 1994; Benson et al., 2003) and
a number of void-finding algorithms, some which assuming voids to be nearly
spherical, have been developed (see e.g. Hoyle & Vogeley (2002)). We developed
a method that does not assume a priori that voids are spherical, and is based
on the original algorithm of Aikio & Maehoenen (1998). (Hereafter AM
algorithm). The AM algorithm was originally written in 2D. We extended it to
3D and adapted it for application to large datasets. The algorithm does not
constrain the voids to be of any particular shape and hence can be used to
study the shapes of the voids and their deviations from sphericity. We
emphasise that here we consider the Aikio-Maehoenen statistics only as a tool
for the relative measurement of some parameters of voids (eg. sphericity) in
observational and simulated catalogues, and not as tool which would provide
any absolute measurements.
Prior to applying of AM algorithm to our volume-limited galaxy sample, we
classified galaxies as wall or field galaxies. To distinguish between wall and
field galaxies, we introduced the parameter $d$, which is related to the mean
distance of the third-nearest neighbour, $d_{3}$, and the standard deviation
of its value, $\sigma$, by the following expression: ($d=d_{3}+1.5\sigma$)
(Hoyle & Vogeley, 2002). In our volume-limited galaxy sample, all galaxies
with a third-nearest neighbour distance, $d_{3}$, greater than this selection
parameter, $d$, were taken to be field galaxies and removed from the galaxy
sample. The remaining objects were identified as wall galaxies. We remark that
a field galaxy may lie within a void region, hence a void galaxy, whereas wall
galaxies all lie in the cosmic filaments and clusters and by definition are
not to be found in voids.
We found that the selection parameters, $d$, for observation and simulation
data are 5.96 and 6.16 Mpc/h, respectively, which means that $9\%$ of the
galaxies in the observation and $8\%$ in the simulation are identified as
field galaxies. The details of the samples are given in Table 1.
To implement the AM algorithm, the wall galaxies were gridded up in cells of
size 1 Mpc/h. The AM algorithm starts on the Cartesian gridded wall galaxy
sample by defining a distance field (DF). For a given grid in a 3D galaxy
sample the DF was defined as the distance to the nearest particle. Then
according to the value of DF for the closest neighbours of each grid, the
local maximum of the DF subvoid was calculated. To assign each element in the
grid sample to a subvoid, we employed the climbing algorithm (Schmidt et al.,
2001) where for a unit cell bounded by the grid points, i.e. an elementary
cell, the gradient in DF to each of the neighbouring cell is calculated. In
this method, the elementary cell and every other cell along the climbing route
is then assigned to a subvoid. Finally, if the distance between two subvoids
is less than both DFs, they will be joined into a larger void.
Table 2: Statistics of voids in the observation of SDSS DR7 and the mock simulation catalogue | | Observation | | Simulation
---|---|---|---|---
| | Number | Volume (Mpc/h)3 | | Number | Volume (Mpc/h)3
All voids | | 4616 | 12541454 | | 4847 | 12555147
Edge voids | | 1148 | 7844214 $(62.5\%)$ | | 1193 | 7646672 $(61\%)$
Small voids $(r_{eff}<7{\rm Mpc/h})$ | | 3001 | 722062 $(5.8\%)$ | | 3085 | 845753 $(6.7\%)$
Voids in the final sample | | 467 | 3975178 $(31.7\%)$ | | 569 | 4062722 $(32.3\%)$
The void volume was estimated using the number of grid points inside a given
void multiplied by the volume associated with the grid cell. For each void, we
defined its effective radius ($r_{\rm eff}$) as the radius of a sphere whose
volume is equal to that of the void.
The configuration of each void in this algorithm depends on the grid points,
and subsequently we determined the void centre as the centre of mass
identified by the positions of the grid points that enclose an elementary
cell. Following this standard method and giving the same weight to all
elementary cells, the centre of each void can be written as
$\ X_{V}^{j}=1/N\sum_{i=1}^{N}x_{i}^{j},\ $ (1)
where $x_{i}^{j}$ $(j=1,2,3)$ are the locations of elementary cells and $\it
N$ is the number of cells in the void $\it V$. The shape of a voids is then
characterised by the ratio of the total number of grid points, which lie
between its centre and its effective radius, to its volume. This ratio is an
indicator of the deviation of the void shape from sphericity. Ideally, for a
spherical void this ratio is equal to one.
In the next section, we apply this algorithm to the SDSS DR7 and the mock
catalogue to construct catalogues of voids and study their characteristics.
## 5 Voids in the SDSS DR7 redshift survey and in the mock catalogue
Table 3: Sizes and sphericities of voids in the observation and simulation mock catalogues | | Effective radius $({\rm Mpc/h})$ | | Max-length $({\rm Mpc/h})$ | | Surface $({\rm Mpc/h})^{2}$ | | Sphericity
---|---|---|---|---|---|---|---|---
| | Max | Min | Median | | Max | Min | Median | | Max | Min | Median | | Max | Min | Median
Observation | | 30.47 | 7.02 | 9.65 | | 108.6 | 19.9 | 32.3 | | 35414 | 1214 | 2588 | | 0.82 | 0.22 | 0.71
Simulaion | | 28.15 | 7.00 | 9.08 | | 103.1 | 19.1 | 30.1 | | 33018 | 1210 | 2276 | | 0.84 | 0.12 | 0.72
We identified 4616 and 4847 voids of different sizes and shapes in the SDSS
DR7 survey and in the mock catalogue, respectively. We avoided problems due to
boundary effects by selecting voids that lie completely inside the geometrical
boundaries of our catalogues. Therefore, edge voids, those that touch the
survey boundaries, are removed from our void catalogue because of their under-
estimated volumes and distorted shapes (see Fig.2).
The size of each void is characterised by its effective radius, defined in the
previous section. To avoid counting spurious voids, we set a threshold of 7
Mpc/h for the minimum size of effective radii of voids in both samples. This
threshold is higher than mean distance between galaxies in the sample and
helps to eliminate seemingly small voids from the sample. After removing all
spurious voids, we had about 467 and 569 voids in our volume-limited sample of
the SDSS DR7 survey and the mock simulation data, respectively, which occupy
$\sim 32\%$ of the volumes of the samples. In Table 2, we provide the void
statistics. Hereafter all analyses are carried out on voids in the final
sample, obtained after eliminating small and edge voids.
Table 3 compares the statistical properties of voids in the observed and mock
catalogues. It shows that the median of void sphericity in both samples is
nearly $\sim 0.70$, which indicates that voids tend to be mostly spherical.
Fig. 3 also shows that voids tend to become more spherical with increasing
radii. There is a good agreement between the mock catalogue and the SDSS
observation, although the observed voids seem to be marginally more spherical
in general. More and better data are needed to see if the marginal difference
reported here is of any significance.
Figure 3: Left panel: distribution of sphericity is skewed towards larger
sphericities, i.e. voids are mostly spherical. Right panel: plot of the
sphericities versus the equivalent radii of the voids, demonstrating that
voids become more spherical with increasing radii. There is no significant
difference between the observation and the simulation and more data would be
needed to establish any disagreement between the two.
## 6 Abundance of large voids: the SDSS DR7 observation versus the mock
catalogue
We compared the distribution of the void sizes in the observation with the
simulated mock catalogues. Fig. 4 shows that the volume occupied by voids is
larger in the simulation than in the observation. In particular, both the
histograms and the commulative plots show that the largest voids are absent
from the simulation, whereas they are present in the observation.
The problem of large voids could be related to the over-abundance of small
galaxies, which would subsequently divide large voids into smaller ones.
However, this could be resolved by proper biasing in modelling the galaxy
formation and evolution. Hence, the problem of large voids could be due to the
shortcoming of the semi-analytic model of galaxy formation for the mock
catalogue that we used here. A recent study that also compared the SDSS DR7
voids with those taken from a smoothed particle hydrodynamics (SPH) simulation
and a halo-occupation model and hence used a different model of galaxy
evolution, seems to indicate that the distribution of the void sizes agree in
the two samples (Pan et al., 2012). Hence, these void properties could be of
potential importance in distinguishing between different galaxy formation
scenarios.
Figure 4: Top panel: Distribution of the void sizes in the observation and
the simulation: larger voids are more abundant in the observation. Bottom
panel: Cumulative plots of the number of voids against their equivalent radii
shows again that larger voids are more abundant in the observation. The bottom
plots show the volume/radius cumulative curves where both the commulative
volume and normalised volumes are plotted against the effective radii of the
voids. The histograms show that at large radii, there are more voids in the
observation than in the simulation. The lower panels demonstrate that the
number and volume of voids are, in general, higher in the simulation than in
the observation (see Table.2). Because there are only two catalogues, we
cannot perform a proper error analysis and determine the error bars in these
figures. However, we performed a Kolmogorov-Smirnov test that shows that the
probability of the two samples to have similar distributions is only about
0.004 and hence the difference between the two catalogues reported in these
figures is statistically significant.
## 7 Observed SDSS voids are less luminous than those in the mock catalogue
Prior to comparing the luminosities of the voids between simulation and
observation, we checked that there was no bias between the two samples. In
Fig. 5, we plotted the histogram of the absolute magnitudes of field and faint
galaxies that are found in the voids in the two catalogues. The figure shows
that although there are more void galaxies in the mock catalogue than in the
observation, the distributions are the same in both catalogues. Min and max
magnitudes are nearly the same, namely $M\sim-16.5$ in the and M$\sim$ -22 in
the observation and the simulation. This demonstrates that there is no bias
between the two samples.
Figure 5: Number of void galaxies plotted against their absolute magnitudes.
The luminosity range of void galaxies is nearly the same for the simulation
and observation, which demonstrates that there is no bias imposed on the
calculation of the void luminosities. Voids in the simulation contain more
galaxies in almost all magnitude bands and hence are more luminous than those
in the observation.
We comment that the void galaxies could be field galaxies or be field and
faint galaxies. We recall that the field galaxies are in the luminosity ranges
$M\textless-19.9$ in the observation and $M\textless-20.16$ in the simulation,
but faint galaxies are less luminous than these thresholds set in our volume-
limited sample (see Fig.5). We stress again that to obtain the same number
density in both samples, we have to consider different luminosity thresholds
in our two volume-limited samples (M=-19.9 & M=-20.16). The difference of
luminosities is insignificant (about 0.26). Nonetheless, even if we consider
the same luminosity threshold for both samples (e.g. M=-19.9), we derive the
same result again and the galaxy luminosities in the simulation are higher
than in the observation.
We compared the total luminosity of the voids and their luminosity per unit
volume between the observation and the simulation. The comparisons are shown
in Fig. 6. The lower panel of Fig. 6 shows that if we consider faint and field
galaxies, large voids are clearly more luminous in the mock catalogue than in
the observation. However, the top panel of Fig. 6 shows that if we consider
only field galaxies, this discrepancy becomes less prominent. We emphasise
that the lowest magnitude cut-off for both samples is nearly the same when
faint galaxies are considered (see Fig.5). This discrepancy could be a sign of
the over-abundance of small faint galaxies in the simulation. The problem of
empty voids could be related to the lacking large power of $\Lambda$CDM, even
though the value of $\sigma_{8}$ used here is $0.9$, which is higher than its
present value of 0.8 given by WMAP7. Hence, this discrepancy is expected to be
more significant for the WMAP7 value of $\sigma_{8}$.
Figure 6: Top panel: total luminosity and luminosity density of field
galaxies ploted against the effective radii of the voids to which they belong.
Larger voids are less luminous in the observation than in the simulation. This
disagreement becomes more significant when faint objects are also taken into
account, as shown in the two plots of the lower panel. Observed voids are
clearly less luminous than simulated voids. Note that the luminosity cutoffs
are the same for the observation and the simulation when faint galaxies are
taken into account. We expect this discrepancy to be even more significant
than shown here because our Millennium I simulation uses a higher value of
$\sigma_{8}$ than given by WMAP7.
## 8 Conclusion
We have carried out a parallel study of the voids in the SDSS DR7 redshift
survey and in a mock catalogue. The latter was extracted from the Millennium I
simulation and aims at replicating the observational biases and limitation of
the SDSS DR7 catalogue.
We found that the total number and the volume occupied by the voids are larger
in the simulation than in the observation. We found 467 voids in SDSS DR7 and
569 in the mock catalogue. The voids’ pseudo-radii or effective radii (i.e.
radii of an equivalent spherical volume) range from 7 to 31 Mpc/h. The
sphericities of voids also have similar distributions in the observation and
the simulation. The voids also tend to become more spherical with increasing
effective radii. Furthermore, large voids are less abundant in the simulation
and the mean void luminosities, as defined by the sum of the luminosities of
the galaxies they contain, is higher in the simulation. The aboundance problem
of large voids could be related to the over-abundance problem of small haloes
in $\Lambda$CDM ,which would then divide large voids into smaller ones in the
simulation. However, this problem is usually taken care of in models of galaxy
formation by suitable biasing or quenching of galaxy formation on small
scales. The persistence of this problem could demonstrate that the semi-
analytic model of galaxy formation used in the mock catalogue does not
efficiently suppress galaxy formation in small voids. Recent catalogues of
voids in SDSS including also the luminous red galaxies will be analysed in
future works to obtain better statistics and shed more light on this problem
(Sutter et al., 2012).
We also found that voids are in general more luminous in the simulation than
in the observation. This could be related to the lack of power of $\Lambda$CDM
on large scales. The value of $\sigma_{8}$ used in the Millennium I simulation
is $0.9$ compared to the value of 0.8 given by the WMAP7. The problem of empty
voids could then become even more significant if the current value of
$\sigma_{8}$ were used in the simulation. Hence, either the ingredients used
in the semi-analytic model do not correctly reproduce the observations, or on
a more fundamental level, the power spectrum of $\Lambda$CDM has too much
power on small scales and too little on large scales, which cannot be remedied
by realistic models of galaxy formation.
Acknowledgments: We thank Sepehr Arbabi for collaborations and Habib
Khosroshahi, Guilhem Lavaux, Gary Mamon, Joe Silk, and David Weinberg for
discussions. RM thanks IPM and ST and KV thank IAP for their hospitality.
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|
arxiv-papers
| 2012-10-08T22:14:46 |
2024-09-04T02:49:36.192905
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Saeed Tavasoli, Kaveh Vasei, Roya Mohayaee",
"submitter": "Saeed Tavasoli",
"url": "https://arxiv.org/abs/1210.2432"
}
|
1210.2538
|
# KUNS-2420
Bare Higgs mass at Planck scale
Yuta Hamada, Hikaru Kawai, and Kin-ya Oda
Department of Physics, Kyoto University, Kyoto 606-8502, Japan E-mail:
hamada@gauge.scphys.kyoto-u.ac.jpE-mail: hkawai@gauge.scphys.kyoto-u.ac.jpE-
mail: odakin@gauge.scphys.kyoto-u.ac.jp
###### Abstract
We compute one- and two-loop quadratic divergent contributions to the bare
Higgs mass in terms of the bare couplings in the Standard Model. We
approximate the bare couplings, defined at the ultraviolet cutoff scale, by
the $\overline{\text{MS}}$ ones at the same scale, which are evaluated by the
two-loop renormalization group equations for the Higgs mass around 126 GeV in
the Standard Model. We obtain the cutoff scale dependence of the bare Higgs
mass, and examine where it becomes zero. We find that when we take the current
central value for the top quark pole mass, 173 GeV, the bare Higgs mass
vanishes if the cutoff is about $10^{23}$ GeV. With a 1.3 $\sigma$ smaller
mass, 170 GeV, the scale can be of the order of the Planck scale.
## 1 Introduction
The ATLAS [1] and CMS [2] experiments at the Large Hadron Collider (LHC)
observed a particle at the 5$\sigma$ confidence level (C.L.), which is
consistent with the Standard Model (SM) Higgs boson with mass
$\displaystyle m_{H}$ $\displaystyle=\begin{cases}126.0\pm 0.4\pm
0.4\,\text{GeV},&\text{ATLAS~{}\cite[cite]{[\@@bibref{}{:2012gk}{}{}]}},\\\
125.3\pm 0.4\pm
0.5\,\text{GeV},&\text{CMS~{}\cite[cite]{[\@@bibref{}{:2012gu}{}{}]}}.\end{cases}$
(1)
Such a relatively light Higgs boson is compatible with the electroweak
precision data [3]. Furthermore, this value of Higgs mass allows the SM to be
valid up to the Planck scale, within the unitarity, (meta)stability, and
triviality bounds [4, 5, 6]. Up to now, there are no symptoms of breakdown of
the SM as an effective theory below the Planck scale.
On the other hand, if one wants to solve the Higgs mass fine-tuning problem
within a framework of quantum field theory, it would be natural to assume a
new physics at around the TeV scale. The supersymmetry is a possible solution
to cancel the quadratic divergences in the Higgs mass, see e.g. Ref. [7].
However, a Higgs mass around 126 GeV requires some amount of fine-tuning in
the Higgs sector in the minimal supersymmetric Standard Model; see, e.g., Ref.
[8]. Furthermore, no sign of supersymmetry has been observed at LHC so far
[9].
Given the current experimental situation, it is important to examine a
possibility in which the SM is valid towards a very high ultraviolet (UV)
cutoff scale $\Lambda$. In such a case, a fine-tuning of the Higgs mass must
be done, as is the case for the cosmological constant. There are several
approaches to the fine-tuning. One is simply not to regard it as a problem but
to accept the parameters which nature has chosen. Instead, one may resort to
the anthropic principle in which one explains the parameters by the necessity
of the existence of ourselves; see, e.g., Refs. [10, 11]. Or else, the tuning
may be accounted for by quantum gravitational nonperturbative effects such as
those from a multiverse or baby universe; see, e.g., Ref. [12]. There are yet
other discussions that the tuning is achieved within the context of field
theory such as the classical conformal symmetry; see, e.g., Ref. [13].
In this paper, we do not try to solve the naturalness problem. Rather, we
evaluate the value of the bare parameters in order to investigate the Planck
scale physics. They must be useful to connect the low energy physics to the
underlying microscopic description, such as string theory.
In this paper, we compute the bare Higgs mass by taking into account one- and
two-loop corrections in the SM. When we write in terms of the dimensionless
bare couplings, the bare Higgs mass turns out to be a sum of a quadratically
divergent part ($\propto\Lambda^{2}$), which is independent of the physical
Higgs mass, and a logarithmically divergent one ($\propto\log\Lambda$). The
importance of the coefficient of $\Lambda^{2}$ was first pointed out by
Veltman at the one-loop order [14]. Generalizations to higher loops within the
renormalized perturbation theory have been developed and applied in Ref. [15]
in which the authors have reported the behavior
$\sim\Lambda^{2}(\log\Lambda)^{n}$; see also Ref. [16] for a review. In
contrast, we see that such behavior does not appear in the bare perturbation
theory. The reason why we employ the latter framework is that we are
interested in the scale near the cutoff. These points will be discussed in
detail with explicit calculations in Sec. 2.
We will see that the bare mass can be zero if $\Lambda$ is around the Planck
scale, which gives some interesting suggestions on the Planck scale physics.
First, it may imply that the supersymmetry of the underlying microscopic
theory is restored above the Planck scale. In fact, superstring theory has
many phenomenologically viable perturbative vacua in which supersymmetry is
broken at the Planck scale; see, e.g., Ref. [17]. In the last section, we will
discuss that threshold corrections at the string scale may generate a small
nonvanishing bare mass. Second, the vanishing of the bare Higgs mass together
with that of the quartic Higgs coupling indicates almost flat potential near
the Planck scale, which opens a possibility that the slow-roll inflation is
achieved solely by the Higgs potential [18].
This paper is organized as follows. In the next section, we explain our
convention, and calculate the quadratic divergent contributions to the bare
Higgs mass up to the two-loop orders. In Sec. 3, we present a renormalization
group equation (RGE) analysis in the SM and give our results for the Higgs
quartic coupling at high scales. In Sec. 4, we examine how small the bare
Higgs mass can be at the Planck scale and show at what scale the bare Higgs
mass vanishes. We vary $\alpha_{s}$, $m_{H}$, and $m_{t}^{\text{pole}}$ to see
how the results are affected. The last section contains the summary and
discussions.
## 2 Bare Higgs mass
In this section, we compute the quadratic divergence in the bare Higgs mass.
### 2.1 Bare mass in $\phi^{4}$ theory
Let us explain our treatment of the bare mass by taking a simple example of
the $\phi^{4}$ theory with the bare Lagrangian:
$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{B})^{2}-\frac{m_{B}^{2}}{2}\phi_{B}^{2}-\frac{\lambda_{B}}{4!}\phi_{B}^{4}.$
(2)
In the mass independent renormalization scheme,111 See, e.g., the introduction
and the subsequent section of Ref. [19] for a recent review of the discussion
explained in this paragraph. In particular, our Eq. (2) corresponds to Eq.
(2.6) in Ref. [19]. Note that in Ref. [19] “bare mass” refers to $m_{0}$
whereas our terminology is the same as “the common definition”, that is, we
call $\Delta_{\text{sub}}+m_{0}^{2}$ the bare mass in general, though we
consider only the leading term $\Delta_{\text{sub}}$ in actual computation.
the bare mass $m_{B}^{2}$ is separated into the quadratically divergent part
$\Delta_{\text{sub}}$ and the remaining one $m_{0}^{2}$:
$\displaystyle{\color[rgb]{0,0,0}m_{B}^{2}=\Delta_{\text{sub}}+m_{0}^{2}.}$
(3)
Here $\Delta_{\text{sub}}$ is chosen in such a way that the physical mass
becomes zero when $m_{0}^{2}=0$. Then the mass parameter $m_{0}^{2}$ is
introduced to describe the deviation from it and is multiplicatively
renormalized to absorb the logarithmic divergence. We note that in the
dimensional regularization, $\Delta_{\text{sub}}$ happens to be formally zero
and only $m_{0}^{2}$ remains.222 If one insists on the dimensional
regularization, one might check the $D=2$ pole to see the quadratically
divergent bare mass, which is beyond the scope of this paper. What we discuss
in this paper is not $m_{0}^{2}$ but the whole $m_{B}^{2}$. Since $m_{0}^{2}$
is negligibly small compared to $\Delta_{\text{sub}}$, we concentrate on the
quadratically divergent part $\Delta_{\text{sub}}$ in the following.
From the bare Lagrangian (2), we calculate the bare mass $m_{B}^{2}$ order by
order in the loop expansion so that the physical mass is tuned to be zero333
Precisely speaking, $m^{2}_{B,\,\text{0-loop}}$ corresponds to the physical
mass times the wave function renormalization factor and is negligibly small
compared to the UV cutoff scale.
$m_{B}^{2}=m_{B,\,\text{0-loop}}^{2}+m_{B,\,\text{1-loop}}^{2}+m_{B,\,\text{2-loop}}^{2}+\cdots.$
(4)
At each order, we fix the bare mass as
$\displaystyle m_{B,\,\text{0-loop}}^{2}$ $\displaystyle=0,$ (5)
$\displaystyle
m_{B,\,\text{1-loop}}^{2}+i\left.\left(\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{eq_7.pdf}}\right)\right|_{k=0}$
$\displaystyle=0,$ (6) $\displaystyle
m_{B,\,\text{2-loop}}^{2}+i\left.\left(\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{eq_8a.pdf}}+\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{eq_8b.pdf}}+\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{eq_8c.pdf}}\right)\right|_{k=0}$
$\displaystyle=0.$ (7)
The one-loop integral in Eq. (6) is quadratically divergent and is
proportional to
$\displaystyle I_{1}$
$\displaystyle:=\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{1}{p_{E}^{2}},$ (8)
where $p_{E}$ is a Euclidean four momentum. In the two-loop computation (7),
the momentum integrals in the third and fourth terms are, respectively,
$\displaystyle J_{2}$
$\displaystyle:=\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{d^{4}q_{E}}{(2\pi)^{4}}\frac{1}{p_{E}^{4}q_{E}^{2}},$
(9) $\displaystyle I_{2}$
$\displaystyle:=\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{d^{4}q_{E}}{(2\pi)^{4}}\frac{1}{p_{E}^{2}q_{E}^{2}(p_{E}+q_{E})^{2}}.$
(10)
The integral $J_{2}$ is infrared (IR) divergent:
$J_{2}\propto\Lambda^{2}\ln(\Lambda/\mu_{\text{IR}})$, but is canceled by the
second term in Eq. (7) due to the lower order condition (6). Therefore, we are
left with only $I_{2}$, which does not suffer from the infrared divergence.
This situation does not change in higher orders because a mass should not
contain an IR divergence.
### 2.2 Bare mass in SM
For the SM Higgs sector, we start from the bare Lagrangian of the following
form in a fixed cutoff scheme with cutoff $\Lambda$:444 In general the
effective Lagrangian of an underlying microscopic theory at the cutoff scale
contains higher dimensional operators. Their effects can be absorbed by the
redefinition of the renormalizable and super-renormalizable couplings in the
low energy region. Therefore it suffices to take the form of Eq. (11) without
higher dimensional operators in order to reproduce the low energy physics.
However, the differences among the bare theories emerge when the energy scale
gets close to the cutoff $\Lambda$.
$\displaystyle\mathcal{L}$
$\displaystyle=(D_{\mu}\phi_{B})^{\dagger}(D^{\mu}\phi_{B})-m_{B}^{2}\phi_{B}^{\dagger}\phi_{B}-\lambda_{B}(\phi_{B}^{\dagger}\phi_{B})^{2},$
$\displaystyle\phi_{B}$ $\displaystyle=\begin{pmatrix}\phi_{B}^{+}\\\
\phi_{B}^{0}\end{pmatrix}.$ (11)
We set the physical mass to be zero: $m_{\text{$B$,0-loop}}^{2}=0$, as we are
interested in physics at very high scales.555 We are not intending to realize
the Coleman-Weinberg mechanism, but to neglect the physical mass that is
unimportant for our consideration. The Planck scale is
$M_{\text{Pl}}=\frac{1}{\sqrt{G_{N}}}=1.22\times 10^{19}\,\text{GeV}.$ (12)
We take into account the SM couplings $g_{Y}$, $g_{2}$, $g_{3}$, $\lambda$,
$y_{t}$ and neglect the others.
Now let us follow the prescription, shown in the previous subsection, in the
SM. In the following, we work in the symmetric phase
$\left\langle\phi\right\rangle=0$ as we are interested only in the quadratic
divergent terms. In the evaluation of the Feynman diagrams, it is convenient
to take the Landau gauge for all the $SU(3)\times SU(2)\times U(1)$ gauge
fields. In this gauge, a diagram always vanishes if an external Higgs line is
attached with a gauge boson propagator by a three-point vertex:
$\displaystyle\left.\parbox{71.13188pt}{\includegraphics[width=71.13188pt]{eq_12.pdf}}\right|_{k=0}$
$\displaystyle=0.$ (13)
From the one-loop diagrams we get the quadratic divergent integral $I_{1}$
again [14]
$\displaystyle m_{B,\,\text{1-loop}}^{2}$
$\displaystyle=-\bigg{(}6\lambda_{B}+\frac{3}{4}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-6y_{tB}^{2}\bigg{)}\,I_{1}.$
(14)
Figure 1: Nonvanishing two-loop Feynman diagrams. Arrows are omitted. The
dashed, solid, wavy, and dotted lines represent the scalar, fermion, gauge,
and ghost propagators, respectively.
In Fig. 1, we present the two-loop Feynman diagrams that do not vanish in the
symmetric phase $\left\langle\phi\right\rangle=0$ and in the Landau gauge. In
the second row of Fig. 1, the last diagram cancels the divergences coming from
the one-loop self-energy of the internal Higgs propagators, as in Eq. (7).666
In practice, from each diagram containing a self-energy correction, one
subtracts a term that is obtained by setting the external momentum of its
self-energy to zero. We have also applied this subtraction for diagrams
containing a vacuum polarization. For the gauge boson, this subtraction
introduces a bare mass, which becomes zero in a gauge invariant regularization
scheme such as the Pauli-Villars or dimensional regularizations. All the
momentum integrals can be recast into either $I_{2}$ or $J_{2}$.777 Gauge
invariance is formally satisfied in the sense that the Ward-Takahashi identity
holds if we shift momenta freely without worrying about the ultraviolet
divergences. In this paper, we are interested in the quadratic divergences
that are left after these momentum redefinitions. We have explicitly checked
that the coefficients of the infrared divergent integral $J_{2}$ cancel in
each gauge invariant set of diagrams.888 More precisely, we have assumed
existence of a gauge invariant regularization behind, and have subtracted the
quadratic divergences in the one-loop vacuum polarization. The cancellation of
the coefficients of $J_{2}$ is checked under this assumption. We then obtain
the $g^{4}$ terms in $m_{B,\,\text{2-loop}}^{2}$ as in Table 1.
Table 1: $g^{4}$ terms in $m_{B,\,\text{2-loop}}^{2}$ in units of $I_{2}$.
By collecting these terms, the two-loop contribution to the bare Higgs mass at
$\Lambda$ becomes999As mentioned in Ref. [14], while at the one-loop level,
only a restricted set of particles participates; on the two-loop level, all
kinds of particles up to the Planck mass enter in the discussion. We assume
that there appear only SM degrees of freedom up to the UV cutoff scale.
$\displaystyle m_{B,\,\text{2-loop}}^{2}$
$\displaystyle=-\bigg{\\{}9y_{tB}^{4}+y_{tB}^{2}\left(-\frac{7}{12}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-16g_{3B}^{2}\right)$
$\displaystyle\phantom{=-\bigg{\\{}}{\color[rgb]{0,0,0}\mbox{}-\frac{87}{16}g_{YB}^{4}-\frac{63}{16}g_{2B}^{4}-\frac{15}{8}g_{YB}^{2}g_{2B}^{2}}$
$\displaystyle\phantom{=-\bigg{\\{}}+\lambda_{B}\left(-18y_{tB}^{2}+3g_{YB}^{2}+9g_{2B}^{2}\right)-{\color[rgb]{0,0,0}12}\lambda_{B}^{2}\bigg{\\}}\,I_{2}.$
(15)
This is one of our main results. Note that Eqs. (14) and (15) are minus the
radiative corrections to the physical Higgs mass squared; see Eqs. (6) and
(7).
In Sec. 4, we will examine whether the bare mass can vanish at a particular UV
cutoff scale. For that purpose, we need to relate the integrals $I_{1}$ and
$I_{2}$. This relation necessarily depends on the cutoff scheme.101010 One can
rigorously compute both $I_{1}$ and $I_{2}$ in principle if one fixes a cutoff
scheme, such as an embedding in string theory. For our purpose, the simplified
procedure (16) suffices as we just want to check the smallness of the two-loop
contributions. In particular, if the two-loop contribution to the bare mass
$m^{2}_{B,\,\text{2-loop}}$ becomes sizable compared to
$m^{2}_{B,\,\text{1-loop}}$, the result suffers from a large theoretical
uncertainty. We will verify that it is actually small. With this caution in
mind, let us employ the following regularization:
$\int d^{4}k_{E}\frac{1}{k_{E}^{2}}=\int_{\varepsilon}^{\infty}d\alpha\int
d^{4}k_{E}\,e^{-\alpha k_{E}^{2}},$ (16)
which gives
$\displaystyle I_{1}$
$\displaystyle=\frac{1}{\varepsilon}\frac{1}{16\pi^{2}},$ $\displaystyle
I_{2}$
$\displaystyle=\frac{1}{\varepsilon}\frac{1}{(16\pi^{2})^{2}}\ln{\frac{2^{6}}{3^{3}}}\simeq
0.005\,I_{1}.$ (17)
When we employ a naive momentum cutoff by $\Lambda$, we get
$I_{1}=\frac{\Lambda^{2}}{16\pi^{2}},$ (18)
and hence we can regard $1/\varepsilon=\Lambda^{2}$.
### 2.3 Graviton effects
Let us estimate the graviton loop effects on the above obtained result. The
graviton $h_{\mu\nu}$ in the metric
$\displaystyle g_{\mu\nu}$
$\displaystyle=\eta_{\mu\nu}+\frac{\sqrt{32\pi}}{M_{\text{Pl}}}h_{\mu\nu}$
(19)
couples to the Higgs through the energy-momentum tensor:
$\displaystyle T_{\mu\nu}$
$\displaystyle=\frac{2}{\sqrt{-g}}\frac{\delta}{\delta
g^{\mu\nu}}\sqrt{-g}\mathcal{L}$
$\displaystyle=(D_{\mu}\phi)^{\dagger}(D_{\nu}\phi)+(D_{\nu}\phi)^{\dagger}(D_{\mu}\phi)-g_{\mu\nu}\left[(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)-m_{B}^{2}\phi^{\dagger}\phi-\lambda(\phi^{\dagger}\phi)^{2}\right].$
(20)
The most divergent contributions come from two derivative couplings. A one-
loop diagram containing such a graviton coupling vanishes because it
necessarily picks up an external momentum, which is set to zero. Other
contributions are at most logarithmically divergent. At the two-loop level,
diagrams involving an internal graviton line that does not touch a Higgs
external line give a form $\Lambda^{4}/M_{\text{Pl}}^{2}$. If the UV cutoff is
much smaller than the Planck scale, this becomes negligible, and the higher
loops become further insignificant. Indeed in perturbative string theory,
higher loop corrections are proportional to powers of the string coupling
constant $g_{s}$ and become subleading. If the cutoff scale exceeds the Planck
scale, we cannot neglect the graviton contributions.
## 3 SM RGE evolution toward Planck scale
In Sec. 4, we will approximate the dimensionless bare coupling constants in
the SM at the UV cutoff scale $\Lambda$ by the running ones in the modified
minimal subtraction ($\overline{\text{MS}}$) scheme at the same scale
$\Lambda$; see the Appendix for its justification. We note that the
$\overline{\text{MS}}$ couplings will be used solely to approximate the
dimensionless bare couplings at the cutoff scale and that the bare Higgs mass
does not run.
To get the $\overline{\text{MS}}$ running coupling constant, we apply the RGE
at the two-loop order. For $g_{Y}$, $g_{2}$, $g_{3}$, and $y_{t}$, we use the
ones in Ref. [22].111111We replace $g_{1}$ of the GUT normalization to
$g_{Y}=\sqrt{3/5}\,g_{1}$ and rewrite the quartic coupling as
$\lambda_{\text{\cite[cite]{[\@@bibref{}{Arason:1991ic}{}{}]}}}=2\lambda$,
where $\lambda_{\text{\cite[cite]{[\@@bibref{}{Arason:1991ic}{}{}]}}}$ is the
one employed in Ref. [22]. For the quartic coupling, we employ the one given
in Ref. [23].121212 We use the arXiv version 2 of Ref. [23] with the
replacements $g^{\prime}=g_{Y}$, $g=g_{2}$, $h=y_{t}$, and
$\lambda_{\text{\cite[cite]{[\@@bibref{}{Ford:1992pn}{}{}]}}}=6\lambda$, where
$\lambda_{\text{\cite[cite]{[\@@bibref{}{Ford:1992pn}{}{}]}}}$ is the quartic
coupling employed in Ref. [23]. The RGE for $\lambda$ in Ref. [22] becomes
equal to that of Ref. [23], after correcting $-{3\over 2}g_{2}^{4}Y_{4}(S)$ to
$-{3\over 2}g_{2}^{4}Y_{2}(S)$ and changing the part ${229\over 4}+{50\over
9}n_{g}$ to $-{229\over 24}-{50\over 9}n_{g}$ in Eq. (A.17) in Ref. [22]. To
be explicit,
$\displaystyle\frac{dg_{Y}}{dt}$ $\displaystyle={1\over
16\pi^{2}}\frac{41}{6}g_{Y}^{3}+\frac{g_{Y}^{3}}{(16\pi^{2})^{2}}\left({199\over
18}g_{Y}^{2}+{9\over 2}g_{2}^{2}+{44\over 3}g_{3}^{2}-{17\over
6}y_{t}^{2}\right),$ $\displaystyle\frac{dg_{2}}{dt}$ $\displaystyle=-{1\over
16\pi^{2}}\frac{19}{6}g_{2}^{3}+\frac{g_{2}^{3}}{(16\pi^{2})^{2}}\left({3\over
2}g_{Y}^{2}+{35\over 6}g_{2}^{2}+12g_{3}^{2}-{3\over 2}y_{t}^{2}\right),$
$\displaystyle\frac{dg_{3}}{dt}$
$\displaystyle=-\frac{7}{16\pi^{2}}g_{3}^{3}+\frac{g_{3}^{3}}{(16\pi^{2})^{2}}\left({11\over
6}g_{Y}^{2}+{9\over 2}g_{2}^{2}-26g_{3}^{2}-2y_{t}^{2}\right),$
$\displaystyle\frac{dy_{t}}{dt}$
$\displaystyle=\frac{y_{t}}{16\pi^{2}}\bigg{(}\frac{9}{2}y_{t}^{2}-\frac{17}{12}g_{Y}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}\bigg{)}+\frac{y_{t}}{(16\pi^{2})^{2}}\bigg{(}-12y_{t}^{2}+6\lambda^{2}-12\lambda
y_{t}^{2}$
$\displaystyle\quad+\frac{131}{16}g_{Y}^{2}y_{t}^{2}+\frac{225}{16}g_{2}^{2}y_{t}^{2}+36g_{3}^{2}y_{t}^{2}+\frac{1187}{216}g_{Y}^{4}-\frac{23}{4}g_{2}^{4}-108g_{3}^{4}-\frac{3}{4}g_{Y}^{2}g_{2}^{2}+9g_{2}^{2}g_{3}^{2}+\frac{19}{9}g_{3}^{2}g_{Y}^{2}\bigg{)},$
$\displaystyle\frac{d\lambda}{dt}$
$\displaystyle=\frac{1}{16\pi^{2}}\bigg{(}24\lambda^{2}-3g_{Y}^{2}\lambda-9g_{2}^{2}\lambda+\frac{3}{8}g_{Y}^{4}+\frac{3}{4}g_{Y}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}+12\lambda
y_{t}^{2}-6y_{t}^{4}\bigg{)}$
$\displaystyle\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}-312\lambda^{3}+36\lambda^{2}(g_{Y}^{2}+3g_{2}^{2})-\lambda\left(-{629\over
24}g_{Y}^{4}-{39\over 4}g_{Y}^{2}g_{2}^{2}+{73\over 8}g_{2}^{4}\right)$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}+\frac{305}{16}g_{2}^{6}-\frac{289}{48}g_{Y}^{2}g_{2}^{4}-\frac{559}{48}g_{Y}^{4}g_{2}^{2}-\frac{379}{48}g_{Y}^{6}-32g_{3}^{2}y_{t}^{4}-\frac{8}{3}g_{Y}^{2}y_{t}^{4}-\frac{9}{4}g_{2}^{4}y_{t}^{2}$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}+\lambda
y_{t}^{2}\bigg{(}\frac{85}{6}g_{Y}^{2}+\frac{45}{2}g_{2}^{2}+80g_{3}^{2}\bigg{)}+g_{Y}^{2}y_{t}^{2}\bigg{(}-\frac{19}{4}g_{Y}^{2}+\frac{21}{2}g_{2}^{2}\bigg{)}$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}-144\lambda^{2}y_{t}^{2}-3\lambda
y_{t}^{4}+30y_{t}^{6}\bigg{\\}},$ (21)
where $t=\ln\mu$. Though we do not include the bottom and tau Yukawa couplings
in this paper, we have checked that these are negligible within the precision
that we work in.
We put the boundary condition for the RGE (21) according to Ref. [5]. The
$\overline{\text{MS}}$ gauge coupling of $SU(3)$ is given by the three-loop
RGE running from $m_{Z}$ to $m_{t}^{\text{pole}}$ and matching with six flavor
theory as [5]
$g_{s}(m_{t}^{\text{pole}})=1.1645+0.0031\bigg{(}\frac{\alpha_{s}(m_{Z})-0.1184}{0.0007}\bigg{)}-0.00046\bigg{(}\frac{m_{t}^{\text{pole}}}{\,\text{GeV}}-173.15\bigg{)},$
(22)
where $m_{t}^{\text{pole}}$ is the pole mass of the top quark. The
$\overline{\text{MS}}$ quartic coupling at the top pole mass
$m_{t}^{\text{pole}}$ is given by taking into account the QCD and Yukawa two-
loop corrections [5]
$\lambda(m_{t}^{\text{pole}})=0.12577+0.00205\bigg{(}\frac{m_{H}}{\,\text{GeV}}-125\bigg{)}-0.00004\bigg{(}\frac{m_{t}^{\text{pole}}}{\,\text{GeV}}-173.15\bigg{)}\pm
0.00140_{\text{th}},$ (23)
where $m_{H}$ is the observed Higgs mass which we read off from Eq. (1) as
$\displaystyle m_{H}$ $\displaystyle=125.7\pm 0.6\,\text{GeV}.$ (24)
The $\overline{\text{MS}}$ top Yukawa coupling at the scale
$m_{t}^{\text{pole}}$ is given by taking into account the QCD three-loop,
electroweak one-loop, and $O(\alpha\alpha_{s})$ two-loop corrections [5]:
$\displaystyle y_{t}(m_{t}^{\text{pole}})$
$\displaystyle=0.93587+0.00557\bigg{(}\frac{m_{t}^{\text{pole}}}{\,\text{GeV}}-173.15\bigg{)}-0.00003\bigg{(}\frac{m_{H}}{\,\text{GeV}}-125\bigg{)}$
$\displaystyle\quad-0.00041\bigg{(}\frac{\alpha_{s}(m_{Z})-0.1184}{0.0007}\bigg{)}\pm
0.00200_{\text{th}}.$ (25)
In a more recent work [6], it has been pointed out that the error in the top
quark pole mass, consistently derived from the running one, is larger than
that given in Ref. [5], $173.1\pm 0.7\,\text{GeV}$. The value obtained is [6]
$\displaystyle m_{t}^{\text{pole}}$ $\displaystyle=173.3\pm 2.8\,\text{GeV},$
(26)
which we will use in our analysis.
Figure 2: Left: $\overline{\text{MS}}$ running of the quartic coupling
$\lambda$. The band corresponds to the $1\sigma$ deviation
$m_{t}^{\text{pole}}=173.3\pm 2.8\,\text{GeV}$. Right: The scale
$\mu_{\text{min}}$ at which $\lambda(\mu)$ takes its minimum value, as a
function of $m_{t}^{\text{pole}}$. In both panels, low energy inputs are given
by the central values $\alpha_{s}(m_{Z})=0.1184$ and
$m_{H}=125.7\,\text{GeV}$.
We plot the $\overline{\text{MS}}$ running coupling constant $\lambda(\mu)$ in
Fig. 2. As we increase the scale $\mu$, the coupling $\lambda$ first decreases
due to the term $-6y_{t}^{4}$ and remains small above
$\mu=10^{10}\,\text{GeV}$ for a while. At further higher energies, $y_{t}$
becomes smaller and $\lambda$ starts to increase due to the contribution from
$\frac{3}{8}g_{Y}^{4}$ which is not asymptotically free. At the intermediate
scale, $\lambda$ can become negative but it is shown that a metastability
condition can be met even in this case [4, 5, 6].131313 At first sight,
$\lambda_{B}<0$ seems to indicate a runaway potential. In the SM, radiative
corrections from the top quark loop generates a potential barrier. The
metastability argument does not assume an existence of a true stable vacuum at
a very high scale but computes the vacuum decay rate from the area of the
potential barrier from $\phi=0$ to the other zero point. In our case, it is
possible that the runaway potential can be cured for a negative but small
coupling ($\lambda_{B}<0,\left|\lambda_{B}\right|\ll 1$) by the higher
dimensional operators with positive couplings, such as
$\left|\phi\right|^{6}/\Lambda^{2}$, which become important near the cutoff
scale $\Lambda$. See also footnote 4. The value of $\lambda$ at the Planck
scale $M_{\text{Pl}}$ becomes consistent with Eq. (64) in Ref. [5]:
$\displaystyle\lambda(M_{\text{Pl}})$
$\displaystyle=-0.014-0.018\left({m_{t}^{\text{pole}}-173.3\,\text{GeV}\over
2.8\,\text{GeV}}\right)+0.002\left({\alpha_{s}(m_{Z})-0.1184\over
0.0007}\right)$ $\displaystyle\quad+0.002\left({m_{H}-125.7\,\text{GeV}\over
0.6\,\text{GeV}}\right)\pm 0.004_{\text{th}}.$ (27)
As we can see from the left panel in Fig. 2, the value of the quartic coupling
stays around its minimum in $10^{15}\,\text{GeV}\lesssim\mu\lesssim
10^{20}\,\text{GeV}$. Therefore, the minimum value of $\lambda$ is also given
by Eq. (27) within our precision. In the right panel in Fig. 2, we plot
$\mu_{\text{min}}$ at which the $\lambda(\mu)$ takes its minimum value. The
central value $m_{t}^{\text{pole}}=173.3\,\text{GeV}$ gives
$\mu_{\text{min}}=4\times 10^{17}\,\text{GeV}$.
## 4 Bare Higgs mass at Planck scale
Figure 3: Left: The bare Higgs mass $m_{B}^{2}$ in units of
$\Lambda^{2}/16\pi^{2}$ vs the UV cutoff scale $\Lambda$. The blue (narrower)
and pink (wider) bands represent the one and two sigma deviations of
$m_{t}^{\text{pole}}$, respectively. Right: The UV cutoff scale at which the
bare mass $m_{B}^{2}$ becomes zero as a function of $m_{t}^{\text{pole}}$. The
solid (dashed) line corresponds to the scale where $m_{B}^{2}$
($m^{2}_{B,\,\text{1-loop}}$) becomes zero. In both panels, we have taken the
central values $\alpha_{s}(m_{Z})=0.1184$ and $m_{H}=125.7\,\text{GeV}$.
Figure 4: The blue solid (dashed) line corresponds to the one-plus-two-loop
(one-loop) bare mass $m_{B}^{2}$ ($m_{B,\,\text{1-loop}}^{2}$) in units of
${M_{\text{Pl}}^{2}/16\pi^{2}}$ for $\Lambda=M_{\text{Pl}}$. For comparison,
we also plot the quartic coupling $\lambda$ at the Planck scale with the red
dotted line. The central values $\alpha_{s}(m_{Z})=0.1184$ and
$m_{H}=125.7\,\text{GeV}$ are used.
Now we can estimate the bare Higgs mass at the cutoff scale by substituting
the $\overline{\text{MS}}$ couplings derived in the previous section to the
bare ones in the right-hand sides of Eqs. (14) and (15).
In the left panel of Fig. 3, we plot the dependence of the bare Higgs mass-
squared in units of $\Lambda^{2}/16\pi^{2}$ on the UV cutoff scale $\Lambda$:
$\displaystyle{m_{B}^{2}\over\Lambda^{2}/16\pi^{2}}$
$\displaystyle={m^{2}_{B,\,\text{1-loop}}\over
I_{1}}+{m^{2}_{B,\,\text{2-loop}}\over I_{2}}{I_{2}\over I_{1}},$ (28)
where we have taken $I_{2}/I_{1}=0.005$ as in Eq. (17). In the figure, we can
see that the bare mass $m_{B}^{2}$ monotonically decreases when one increases
$\Lambda$.141414 We note again that the bare Higgs mass is defined for each UV
cutoff $\Lambda$ and is not a running quantity.
We obtain the UV cutoff scale at which the bare mass $m_{B}^{2}$ becomes zero:
$\displaystyle\log_{10}{\left.\Lambda\right|_{m_{B}^{2}=0}\over\,\text{GeV}}$
$\displaystyle=23.5+3.3\left({m_{t}^{\text{pole}}-173.3\,\text{GeV}\over
2.8\,\text{GeV}}\right)-0.2\left({m_{H}-125.7\,\text{GeV}\over
0.6\,\text{GeV}}\right)$
$\displaystyle\quad-0.4\left({\alpha_{s}(m_{Z})-0.1184\over 0.0007}\right)\pm
0.4_{\text{th}}.$ (29)
In the right panel of Fig. 3, we plot this quantity as a function of the top
quark pole mass for the central values of $\alpha_{s}(m_{Z})$ and $m_{H}$,
without referring to the linear approximation (29).
We show an approximate formula for the bare Higgs mass when the cutoff is at
the Planck scale, $\Lambda=M_{\text{Pl}}$:
$\displaystyle m_{B}^{2}$
$\displaystyle=\bigg{[}0.22+0.18\left({m_{t}^{\text{pole}}-173.3\,\text{GeV}\over
2.8\,\text{GeV}}\right)-0.02\left({\alpha_{s}(m_{Z})-0.1184\over
0.0007}\right)$
$\displaystyle\phantom{=\bigg{[}}-0.01\left({m_{H}-125.7\,\text{GeV}\over
0.6\,\text{GeV}}\right)\pm
0.02_{\text{th}}\bigg{]}\frac{M_{\text{Pl}}^{2}}{16\pi^{2}}.$ (30)
This is one of our main results. We verify that the two-loop correction (15)
can be safely neglected:
$m^{2}_{B,\,\text{2-loop}}\simeq-0.005\,M_{\text{Pl}}^{2}/16\pi^{2}$ within
the cutoff scheme (17), as advertised before. In Fig. 4, we plot the bare
Higgs mass-squared in units of $M_{\text{Pl}}^{2}/16\pi^{2}$ as a function of
$m_{t}^{\text{pole}}$ for the central values of $\alpha_{s}(m_{Z})$ and
$m_{H}$, without referring to the linear approximation (30). For comparison,
we also plot the quartic coupling $\lambda$ at the Planck scale.
From Fig. 4 we see that the bare Higgs mass becomes zero if
$m_{t}^{\text{pole}}=169.8\,\text{GeV}$, while the quartic coupling
$\lambda(M_{\text{Pl}})$ vanishes if $m_{t}^{\text{pole}}=171.2\,\text{GeV}$,
when we take the central values for $\alpha_{s}(m_{Z})$ and $m_{H}$. See Refs.
[13] for arguments supporting the vanishing parameter at a cutoff scale, see
also Ref. [24]. There is no low energy parameter set within two sigma that
makes both the quartic coupling and the bare mass vanish simultaneously at the
Planck scale. This might suggest an existence of a small threshold effect from
an underlying UV complete theory.
## 5 Summary and discussions
It is important to fix all the parameters, including the bare Higgs mass, at
the UV cutoff scale of the Standard Model in order to explore the Planck scale
physics. We note again that in this paper we are not trying to solve the fine-
tuning problem but to determine all the bare parameters at the cutoff scale.
In addition, we investigate the scale of the vanishing bare mass as a hint of
that of the supersymmetry restoration.
We have presented a procedure where the quadratic divergence of the bare Higgs
mass is computed in terms of the bare couplings at a UV cutoff scale
$\Lambda$. Using it, we have obtained the bare Higgs mass up to the two-loop
order in the SM. This calculation has been made easier by working in the
symmetric phase $\left\langle\phi\right\rangle=0$ and in the Landau gauge. We
have checked that all the IR divergent terms, which are proportional to
$\Lambda^{2}\ln(\Lambda/\mu_{\text{IR}})$, cancel out as expected.
Approximating the bare couplings at $\Lambda$ by the corresponding
$\overline{\text{MS}}$ ones at the same scale, we can examine whether the
quadratic divergence in the bare Higgs mass vanishes or not. To get the
$\overline{\text{MS}}$ couplings at high scales, we employ the two-loop RGE in
the SM. We have found that it is indeed the case if the top quark mass is
$m_{t}^{\text{pole}}=169.8\,\text{GeV}$, which is $1.3\,\sigma$ smaller than
the current central value.151515 The vanishing of the quadratic divergence
does not immediately indicate that the bare Higgs mass is exactly zero. Our
result does not exclude logarithmically divergent corrections such as
$m_{H}^{2}\ln(\Lambda/m_{H})$ or finite ones. If the quadratic divergence
indeed vanishes exactly for some reason, then such corrections become
important. It would be interesting to study them. One might find it
intriguing that this value is close to
$m_{t}^{\text{pole}}=171.2\,\text{GeV}$, which gives a vanishing quartic
coupling at $M_{\text{Pl}}$.
It is a curious fact that the scale of the vanishing bare Higgs mass
$m_{B}^{2}$ and that for the quartic coupling $\lambda$ are quite close to
each other and to the Planck scale. The fact that the Planck scale appears
only from the SM might indicate that the SM is indeed valid up to the Planck
scale and is a direct consequence of an underlying physics there. Also, it may
imply an almost flat potential near the Planck scale, which opens a
possibility that the slow-roll inflation is achieved solely by the Higgs
potential.
If we take all the central values for $m_{t}^{\text{pole}}$,
$\alpha_{s}(m_{Z})$, and $m_{H}$, then the cancellation occurs not at the
Planck scale but at a scale around $\Lambda\sim 10^{23}\,\text{GeV}$. This may
hint a new physics around that scale. In this case however, we need to take
the graviton effects into account, as discussed in Sec. 2.3.
There can be a different interpretation for the small bare Higgs mass
$m_{B}^{2}$ left at the Planck scale. It might appear as a threshold
correction in string theory. In string theory, the tree-level masses of the
particles are quantized by $m_{s}:=(\alpha^{\prime})^{-1/2}$, and therefore
the Higgs mass is zero at the tree-level. The threshold effect from
integrating out the massive stringy excitations is obtained by computing
insertions of two Higgs emission vertices with zero external momenta into the
world sheet. The result would become
$m^{2}_{B}\sim C\frac{g_{s}^{2}}{16\pi^{2}}m_{s}^{2},$ (31)
where $C$ is a model dependent constant. This calculation can be performed for
a concrete model such as the orbifold and fermionic constructions in heterotic
string. This work will be presented in a separate publication.
We comment on the case where the UV completion of the SM appears as a
supersymmetry. When the supersymmetry is softly broken, there cannot be any
quadratic divergence and our study does not apply. In the case of the
split/high-scale supersymmetry [25, 11] it is possible to perform a parallel
analysis to the current one, which will be shown elsewhere.
If we assume the seesaw mechanism, the right-handed neutrinos are introduced
above an intermediate scale $M_{R}$. Our analysis corresponds to the case
where $M_{R}$ is small enough that all the neutrino Dirac-Yukawa couplings are
negligible $y_{D}\lesssim 10^{-1}$. This condition implies $M_{R}\lesssim
10^{12}\,\text{GeV}$ for the neutrino mass $m_{\nu}\sim
y_{D}^{2}v^{2}/M_{R}\sim 0.1\,\text{eV}$. It would be interesting to extend
our analysis to include larger Dirac-Yukawa couplings for $M_{R}\gtrsim
10^{12}\,\text{GeV}$.
### Note added
It has been pointed out [20] that the formula for the two loop bare Higgs mass
(15) in the previous version does not agree with the result in Ref. [21]. They
obtained it from the residue at $d=3$ in the dimensional reduction using the
background Feynman gauge, whereas we have computed Eq. (15) in the Landau
gauge in four dimensions. The quantity under consideration is an on-shell
quantity, namely the two-point function with zero external momentum in the
massless theory. Therefore it is a gauge invariant quantity, and hence two
results should agree with each other. We have re-examined our calculation and
found errors that are suggested in Ref. [20]. Eq. (15) is the corrected
version. The error does not influence the consequence, that is, the two loop
bare mass is still negligible compared to the one loop one.
### Acknowledgements
We greatly appreciate valuable discussions with Satoshi Iso and useful
information from him, without which this study would never have been started.
We thank D. R. T. Jones for the kind correspondence. This work is in part
supported by the Grants-in-Aid for Scientific Research No. 22540277 (HK), No.
23104009, No. 20244028, and No. 23740192 (KO) and for the Global COE program
“The Next Generation of Physics, Spun from Universality and Emergence.”
## Appendix
## Appendix A Cutoff vs $\overline{\text{MS}}$
We have approximated the dimensionless bare coupling constants in the SM by
the running ones in the $\overline{\text{MS}}$ scheme at $\Lambda$. The
resulting error can be evaluated once the cutoff scheme is explicitly
specified.
More concretely, let us first express the $\overline{\text{MS}}$ couplings at
a scale $\mu$ in terms of the bare couplings defined at the cutoff scale
$\Lambda$:
$\displaystyle\lambda^{i}_{\overline{\text{MS}}}(\mu)$
$\displaystyle=\lambda^{i}_{B}+\sum_{jk}c^{ijk}\\!\left(\mu/\Lambda\right)\,\lambda_{B}^{j}\lambda_{B}^{k}+O(\lambda_{B}^{3}),$
(32) $\displaystyle c^{ijk}(x)$ $\displaystyle:=f^{ijk}+b^{ijk}\ln
x+O(x^{2}),$ (33)
where $b^{ijk}$ is the coefficient in the one-loop beta function and $f^{ijk}$
is the finite part from the one-loop diagrams.
$\left\\{\lambda^{i}_{\overline{\text{MS}}}\right\\}_{i=1,\dots,5}$
($\left\\{\lambda^{i}_{B}\right\\}_{i=1,\dots,5}$) stands for the
$\overline{\text{MS}}$ (bare) couplings of the SM:
$\left\\{g_{Y}^{2},g_{2}^{2},g_{3}^{2},y_{t}^{2},\lambda\right\\}$
($\left\\{g_{YB}^{2},g_{2B}^{2},g_{3B}^{2},y_{tB}^{2},\lambda_{B}\right\\}$).
In our case, the two-loop corrections in the RGE at high scales are small
compared to the one-loop order, which indicates that the two-loop terms
$O(\lambda_{B}^{3})$ in Eq. (32) are negligible as we can take $\mu$ that
satisfies both
$\displaystyle\mu$ $\displaystyle\ll\Lambda,$
$\displaystyle\left|{\lambda^{i}_{\overline{\text{MS}}}\over
16\pi^{2}}\ln(\mu/\Lambda)\right|$ $\displaystyle\ll 1,$ (34)
simultaneously. Thus we have
$\displaystyle\lambda^{i}_{\overline{\text{MS}}}(\mu)$
$\displaystyle=\lambda^{i}_{B}+\sum_{jk}\left(f^{ijk}+b^{ijk}\ln{\mu\over\Lambda}\right)\,\lambda_{B}^{j}\lambda_{B}^{k}.$
(35)
On the other hand, from the RGE, we get
$\displaystyle\lambda^{i}_{\overline{\text{MS}}}(\Lambda)$
$\displaystyle=\lambda^{i}_{\overline{\text{MS}}}(\mu)+\sum_{jk}b^{ijk}\lambda_{\overline{\text{MS}}}^{j}(\mu)\,\lambda_{\overline{\text{MS}}}^{k}(\mu)\ln{\Lambda\over\mu}.$
(36)
From Eqs. (35) and (36), we obtain
$\displaystyle\lambda^{i}_{\overline{\text{MS}}}(\Lambda)$
$\displaystyle=\lambda^{i}_{B}+\sum_{jk}f^{ijk}\lambda^{j}_{B}\lambda^{k}_{B},$
(37)
which gives the relation between the bare and the $\overline{\text{MS}}$
couplings at the same scale.
With the above correction, the formula for the bare Higgs mass is modified by
$\displaystyle\Delta m_{B}^{2}$
$\displaystyle=-\sum_{ijk}a^{i}f^{ijk}\lambda^{j}_{\overline{\text{MS}}}(\Lambda)\,\lambda^{k}_{\overline{\text{MS}}}(\Lambda),$
(38)
where $a^{i}$ are the coefficients in the one-loop bare Higgs mass
$m_{B}^{2}=\sum_{i}a^{i}\lambda^{i}_{B}$ in Eq. (14), and are proportional to
$I_{1}$. The scale at which the bare Higgs mass vanishes
$\left.\Lambda\right|_{m_{B}^{2}=0}$ is changed to
$\left.\Lambda\right|_{m_{B}^{2}=0}e^{\delta t}$, where
$\displaystyle\delta t$
$\displaystyle={\sum_{ijk}a^{i}f^{ijk}\lambda^{j}_{\overline{\text{MS}}}(\Lambda)\,\lambda^{k}_{\overline{\text{MS}}}(\Lambda)\over\sum_{ijk}a^{i}b^{ijk}\lambda^{j}_{\overline{\text{MS}}}(\Lambda)\,\lambda^{k}_{\overline{\text{MS}}}(\Lambda)}.$
(39)
Generically $f^{ijk}$ are of the same order as $b^{ijk}$ and hence the
correction due to the replacement of the bare couplings by the
$\overline{\text{MS}}$ ones, $\Delta m_{B}^{2}$, is as small as the two-loop
corrections. Since $\delta t$ is of order unity, the ambiguity for the scale
$\left.\Lambda\right|_{m_{B}^{2}=0}$ would be at most $e^{\delta t}\lesssim
10$.
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|
arxiv-papers
| 2012-10-09T09:31:14 |
2024-09-04T02:49:36.208890
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuta Hamada, Hikaru Kawai, and Kin-ya Oda",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1210.2538"
}
|
1210.2550
|
# Size Dependent Growth in Metabolic Networks
Henry Dorrian1††H.Dorrian@mmu.ac.uk, Kieran
Smallbone2††Kieran.Smallbone@Manchester.ac.uk and Jon
Borresen1††J.Borresen@mmu.ac.uk 1 Dept of Computing and Mathematics,
Manchester Metropolitan University, Manchester UK M1 5GD
2 School of Mathematics, The University of Manchester,
Manchester UK M60 1QD
###### Abstract
Accurately determining and classifying the structure of complex networks is
the focus of much current research. One class of network of particular
interest are metabolic pathways, which have previously been studied from a
graph theoretical viewpoint in a number of ways. Metabolic networks describe
the chemical reactions within cells and are thus of prime importance from a
biological perspective.
Here we analyse metabolic networks from a section of microorganisms, using a
range of metrics and attempt to address anomalies between the observed metrics
and current descriptions of the graphical structure. We propose that the
growth of the network may in some way be regulated by network size and attempt
to reproduce networks with similar metrics to the metabolic pathways using a
generative approach. We provide some hypotheses as to why biological networks
may evolve according to these model criteria.
###### pacs:
89.75.Hc, 89.75.Fb, 84.35.Sn, 05.65.+b
The graphical structure of metabolic pathways has been extensively studied
Lacroix2008 ; jeong2000large ; wagner2001small ; schuster2011topological and
describing the structure should give insight into functionality Stelling .
Many of the salient features of such networks have been investigated,
particularly any scale free nature barabasi2000scale although the scale free
model is currently the subject of some debate Khanin ; Stumpf . Of particular
interest are the high clustering coefficients observed in metabolic pathways
which have previously been explained using concepts of topological hierarchy
ravasz2003hierarchical and modularity Takemoto . Other concepts describing
the structure such as memory Klemm and decomposition into functional modules
Ma have also been proposed.
A more recent model by Schneider et al. schneider2011scale uses network
depletion, where a fully connected network is degraded according to the degree
of the nodes. This gives rise to the high clustering coefficient and lower
degree distribution values found in metabolic pathways, but the model cannot
agree with how these biological networks would have evolved (i.e it is not
plausible that metabolic networks have evolved from being fully connected and
slowly losing connections as, for one thing, not all metabolites can react
with one another).
Previous work by Dorogovtsev and Mendes Dorogovtsev explains the presence of
high degree decay rates in evolving networks with accelerating growth rates,
where it is shown that for networks where the growth is accelerating and where
the decay rate of the degree distribution $\gamma>2$, the probability
distribution for preferential attachment must be non-stationary.
It has also been shown that the probability of a metabolite reacting with k
other metabolites decays as $P(k)\sim k^{-2.2}$ jeong2000large ;
wagner2001small . However the metabolic pathways investigated here demonstrate
$\gamma<2.2$ in all cases. This leads to the biologically plausible idea that
the growth of such networks may be slowing and that the probability of
connection to existing nodes may not be static.
Initially, we conduct a graphical analysis of eight bacterial metabolic
pathways, concentrating on their clustering coefficients and mean path
lengths. Here microorganisms have been chosen, as the metabolic pathways are
observed to be less modular than higher organisms allowing greater
illustration of the concept. Further to this, a growth model, whereby the rate
of growth decays as a function of network size, is used to demonstrate that
size dependent growth may provide a suitable explanation as to many of the
structural features of metabolic networks.
## I Graphical Analysis
Eight microbial metabolic pathways were considered, they were as follows:
Escherichia Coli, Escherichia Coli iAF1260, Escherichia Coli iJR904,
Helicobacter Pylori, Methanosarcina Barkeri, Staphylococcus Aureus,
Mycobacterium Tuberculosis and Saccharomyces Cerevisiae.
Three formulations of E Coli are chosen to ensure the results are independent
of the methodology used to initially determine the network.
The eight metabolic reconstructions are downloaded in SBML format SBMLonline
from the BiGG database schellenberger2010bigg . The models were imported to
Matlab using libSBML libSBML .
We consider each metabolic pathway as an undirected graph with the adjacency
matrix of the graph being the boolean representation of the chemical
interactions. For each model, nodes $i$ and $j$ are defined as adjacent if
metabolite $i$ appears as a reactant and $j$ as a product, or indeed $i$ as a
product and $j$ as a reactant in any reaction. Although this is a highly
simplified representation this has been shown to be a useful tool in analysing
such systems schuster2000general .
Note: There are a variety of network constructions available, both including
and excluding subcellular compartmentalisation, and with/without considering
the role of water and protons. We consider a formulation with subcellular
compartmentalisation, as this most accurately represents the structural nature
of the biochemical processes within the cells and excluding the presence of
water and protons as is customary in such studies as these Lacroix2008 ;
jeong2000large ; wagner2001small ; schuster2011topological . The results
obtained here are applicable to the other available formulations with some
small modification to the size dependent decay constant $C$ described in
Section I.0.4.
### I.0.1 Clustering Coefficients
The clustering coefficient of a network is a measure of transitivity \- how
the nodes in the network tend to cluster together. We consider a network
average global clustering coefficient watts1998collective of the form.
$<c>=\frac{1}{n}\sum_{i=1}^{n}c_{i},$ (1)
where $n$ is the number of nodes in the network,
$c_{i}=\frac{2e_{i}}{k_{i}(k_{i}-1)},$ (2)
where $k_{i}$ is the degree of node $i$ and $e_{n}$ is the number of connected
pairs between the nodes to which node $i$ is connected.
The observed clustering coefficients $<c>$ for the metabolic data show that
the networks are highly clustered and that the clustering coefficient is
independent of network size (see figure 1).
### I.0.2 Mean Minimum Path Lengths
The minimum path length (or geodesic length) is a measure of the smallest
number of nodes between any two nodes, and represents the shortest route along
the network between them. For our metabolic pathways, the average path length
is surprisingly low given both the size of the networks and their average
degree. This demonstrates a very strong small world effect milgram1967small .
Conversely, the highest of the minimum path lengths are somewhat greater than
one would expect, given such a small world effect. For instance, the H. pylori
metabolic network with $562$ nodes has an average path length of $2.8$ and a
maximum path length of $8$. For a non-directed network of this size, such a
high maximum path length suggests structural qualities not in keeping with
other small world networks.
### I.0.3 Scale Free Structure
It has been previously observed that the graphical structures of metabolic
pathways have much in common with many other complex networks, particularly
with respects to their possible ‘scale-free’ nature jeong2000large ;
fell2000small .
In a scale-free network, the probability, $P(k)$, of a node in the system
having $k$ connections follows a power law distribution of the form $P(k)\sim
k^{-\gamma}$ barab1999emergence . This is in contrast to the much studied
random graphs which follow a Poisson or Binomial distribution
erds1961evolution but are similar to the social networks Milgram
milgram1967small described.
As can be observed (see figure 2), the metabolic networks display some scale-
free property, however they cannot be considered truly scale free as there are
too few connections with low degree and too many with high degree for an
accurate fit of the form $P(k)\sim k^{-\gamma}$ to be valid. Approximations
for the value $\gamma$ (above) for the metabolic pathways via least squares
fits, obtain values in all cases of $\gamma<2$ \- although such a logarithmic
fit of the data yields high least square errors.
Networks which obey a power law distribution of the form $P(k)\sim
k^{-\gamma}$ can be artificially generated. For instance Barabási and Albert
barab1999emergence describe a network growing via preferential attachment of
new nodes to existing nodes with a higher degree, the probability that a new
node connects to an existing node is calculated using equation 3.
$P(\textit{New node connects to node i})=\frac{K_{i}}{\sum_{j}(K_{j})}$ (3)
where $K_{i}$ is the degree of node $i$ and the sum is over all pre-existing
nodes $j$.
Artificially generated networks of this type typically have $\gamma\in[2,3]$
\- greater than the observed metabolic data. The clustering coefficients are
also considerably lower than those observed for the metabolic networks
(typically lower than $0.1$ for any randomly generated Barabási– Albert model
(BA model)) and are observed to decrease with increasing network size.
### I.0.4 A Size Dependent Generative Model
Although the BA model (equation 3) does not provide good fits to the metabolic
networks, a generative model using preferential attachment would seem to have
much to offer when considering the growth of metabolic networks. Here
promiscuous metabolites within the network with high degree ie those which are
present as reactants in higher numbers of reactions will be those which are
more likely to form new connections, whereas co-factors which are more
specific in their function will form less connections.
As such, a generative model similar to the BA model appears as a strong
candidate for describing some of the features which develop in the growth of
metabolic networks. However, as a variety of studies have demonstrated
schneider2011scale ; ravasz2003hierarchical ; ravasz2002hierarchical , the
scale-free model alone is not sufficient to fully describe the networks when a
range of graph metrics are considered.
In considering a generative model approach to recreating networks similar to
the metabolic networks of microorganisms, the concept of size dependent growth
was considered. As such we attempt to introduce some concept of limiting
factors on the generative model. Essentially we attempt to model the growth of
the network such that, initially, it is very easy for new connections to form
and as the network grows the probability of new metabolites attaching is
reduced. We have chosen a linear model for simplicity.
Networks are grown according to the following:
$P(\textit{New node connects to node
i})=\frac{K_{i}}{\sum_{j}(K_{j})}\times\frac{C}{n}$ (4)
where $C$ is some constant and $n$ is the number of nodes in the existing
network. Such a model will produce a globally connected network for
$n^{2}+1\leq C$. For $n^{2}+1>C$ the preferential attachment model begins,
however, the probability of attachment is initially high. As the network grows
and $\sum j\sim C$, new nodes attach in a manner identical to the original BA
model and for $\sum j>C$ the network ceases to grow any further. (Note, this
model assumes that nodes are not self connected and restricts the probability
of any new attachment to $P\leq 1$, for $n^{2}+1\leq C$).
The effect of modifying the BA model thusly has two effects: Firstly the
probabilities of attachment are not static and are rescaled as each new node
enters the network, secondly the probability that any new node will attach is
decreasing as the network grows.
Figure 1: Mean minimum path lengths (top) and clustering coefficients(bottom)
for eight microbial metabolic pathways (red circles) and (black line) average
minimum path lengths and clustering coefficients for randomly generated size
dependent networks with $C=N/2$ (equation 4).
The size to which any network will grow in finite time, is essentially
determined by the value $C$. When $n>C$, the probability of new nodes
attaching is very small and these are new nodes are rejected. Simulations were
conducted until the network had grown to a specified size, attempting to
attach new nodes until this was successful.
Repeated simulations for network growth according to model 4 were performed
for varying constant $C$. It was observed that for $C=N/2$, where $N$ is the
size to which the network is grown, the average path length and the clustering
coefficients of the generated networks fitted the metabolic pathway data
better than any single previous structural description of metabolic networks
(see figure 1).
It should be noted that the networks generated are not truly scale-free, being
somewhere a hybrid of both exponential and scale free type distributions which
is not unlike the original metabolic networks.
The overall effect of such a size dependent modification to the original BA
model is to initially produce a highly connected ’hub’ which then grows via
preferential attachment giving rise to high clustering coefficients and a very
strong small world effect resulting in low average path lengths.
It is straightforward to amend the size dependent model 4 such that no
globally connected initial stage occurs by rescaling the size dependent
modification in Equation 4 to $C/n+\sqrt{C}$. This produces a network which
has a more pronounced scale free structure, while retaining the high
clustering coefficients of the original formulation.
### I.0.5 Example - S Aureus
The bacteria Staphylococcus Aureus has a metabolic network of $741$ nodes, a
clustering coefficient of $<c>=0.124$, a maximum geodesic length of $9$ and a
mean geodesic length of $3.34$.
The (mean) average values for $100$ trials of a network of $741$ nodes,
generated according to equation 4 with $C=37.5$ are: clustering coefficient
$<c>=0.125$, maximum path length $7.84$ and mean geodesic length $3.53$.
The distribution of geodesic lengths over the whole of the S Aureus network
and an example size dependent network are shown in figure 2 in addition to the
degree distributions.
Figure 2: Minimum geodesic lengths (top) and degree distribution plots
(bottom) for S Aureus and example size dependent network of $741$ nodes. Note
the geodesic distribution for the size dependent network matches closely the
metabolic network. The degree distribution for the metabolic network is not
entirely scale free due to fewer nodes of degree $1$ and $2$ than would be
expected. The size dependent growth model demonstrates a degree distribution
which is in many respects closer that of the metabolic pathways. 3
Figure 3: Graphical representations of artificially generated Barabási Albert
Scale-free network (741 nodes), S Aureus isB619 (741 nodes) and artificially
generated size dependent network (741 nodes). All figures are generated using
the Mathematica spring embedding algorithm.
Although somewhat circumstantial as evidence, it is often useful to illustrate
the similarities and differences between graphical structures using pictorial
representations. Such comparison between the original BA model, S. Aureus and
the size dependent growth model demonstrate the similarity between the size
dependent model and the metabolic pathways (see figure 3). Here the highly
connected initial growth stage, although omitting the underlying
compartmentalisation of the S. Aureus metabolic network provides a good
approximation of the structure. Additionally the presence of some ’dead end’
metabolites, which appear as nodes connected to only $1$ other node (thus
giving rise to higher maximum path lengths than may otherwise be observed) is
not modelled by this approach.
## II Discussion
This investigation has demonstrated that the metabolic networks of
microorganisms are more accurately modelled with a network growth model in
which network size is a modifying factor, a concept which has surprisingly not
previously been considered. This model, as well as fitting the graphical
measures examined, also demonstrates a one possible mechanism which may be
significant in the network evolution. This suggests that when the metabolic
networks are growing (or evolving) the size of the network causes it to be
increasingly unlikely for a new metabolite to join the network and participate
in the reactions. Due to the size dependent growth networks having a densely
connected cluster, these networks will have an increased resilience to
targeted attacks than that of the BA model, which are known to be devastated
by targeted attacks newman2003structure . This suggests that the structure of
metabolic pathways gives them a greater resilience to targeted attacks than if
they were examples of scale-free networks, modelled by preferential
attachment.
The value of the rate at which the probabilities of attachment decay, $C$, has
in our modelling been chosen as a single value to fit all metabolic networks,
however we envisage that for specific networks a better fit would be available
using particular values.
We have presented a model which is both simple and biologically plausible. Due
to the fact it does not require any specific seed network it allows for a
generic model which can be used to model various metabolic pathways with only
the network size being known. This allows for various graphical measures to be
estimated for any given metabolic network, without them needing to be
individually analysed.
Clearly the concept of size dependent growth may not be confined to evolving
metabolic networks but may be applicable to a variety of networks where growth
rates may be affected by limited resources. One example where this may be
applicable is that of the London Underground, where an initially a highly
connected network has grown and now as the network has become larger and more
complex it has become increasingly difficult for new stations and connections
to be added to the network.
## Acknowledgements
KS is grateful for the financial support of the EU FP7 (KBBE) grant 289434
“BioPreDyn: New Bioinformatics Methods and Tools for Data-Driven Predictive
Dynamic Modelling in Biotechnological Applications”.
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* [24] E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, and A.L. Barabási. Hierarchical organization of modularity in metabolic networks. Science, 297(5586):1551, 2002.
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|
arxiv-papers
| 2012-10-09T10:22:30 |
2024-09-04T02:49:36.219840
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Henry Dorrian, Kieran Smallbone and Jon borresen",
"submitter": "Jon Borresen",
"url": "https://arxiv.org/abs/1210.2550"
}
|
1210.2625
|
# Synchronization in clustered random networks
Thomas Kauê Dal’Maso Peron thomas.peron@usp.br Instituto de Física de São
Carlos, Universidade de São Paulo, Av. Trabalhador São Carlense 400, Caixa
Postal 369, CEP 13560-970, São Carlos, São Paulo, Brazil Francisco A.
Rodrigues francisco@icmc.usp.br Departamento de Matemática Aplicada e
Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade
de São Paulo, Caixa Postal 668,13560-970 São Carlos, São Paulo, Brazil Jürgen
Kurths Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam,
Germany Department of Physics, Humboldt University, 12489 Berlin, Germany
Institute for Complex Systems and Mathematical Biology, University of
Aberdeen, Aberdeen AB24 3UE, United Kingdom
###### Abstract
In this paper we study synchronization of random clustered networks consisting
of Kuramoto oscillators. More specifically, by developing a mean-field
analysis, we find that the presence of cycles of order three does not play an
important role on network synchronization, showing that the synchronization of
random clustered networks can be described by tree-based theories, even for
high values of clustering. In order to support our findings, we provide
numerical simulations considering clustered and non-clustered networks, which
are in good agreement with our theoretical results.
###### pacs:
89.75.Hc,89.75.-k,89.75.Kd
## I Introduction
Synchronization processes have attracted the interest of scientists for
centuries and is in the focus of intense research today Arenas et al. (2008).
This collective phenomena has been observed in biological, chemical, physical,
and social systems Pikovsky et al. (2003); Acebrón et al. (2005). Many works
have verified that the dynamics of synchronization depends on the connectivity
pattern of networks Arenas et al. (2008). For instance, when the natural
frequency distributions are unimodal and even, the critical coupling depends
on the ratio between the first and second statistical moments of the degree
distribution Ichinomiya (2004); Lee (2005); Restrepo et al. (2005). In
addition, for networks in which there is a positive correlation between the
network structure and dynamics, the critical coupling has an inverse
dependence on the network average degre Peron and Rodrigues (2012).
However most of the analytical results of networks consisting of Kuramoto
oscillators have been obtained for uncorrelated networks generated through the
configuration model Ichinomiya (2004); Lee (2005); Restrepo et al. (2005),
which generates networks with arbitrary degree distributions by randomly
connecting the nodes according to a specified degree sequence. One of the main
properties of this model is that in the thermodynamic limit, i.e.,
$N\rightarrow\infty$, the probability of occurrence of cycles of order three
tends to zero. Such probability can be quantified through the clustering
coefficient $C$, defined as
$C=\frac{3\times\left(\mbox{number of triangules in the
network}\right)}{\mbox{number of connected
triples}}=\frac{3N_{\triangle}}{N_{3}}.$ (1)
In addition to the configuration model, the clustering coefficient also
vanishes for Erdos-Renyi (ER) and Barabasi-Albert (BA) networks when
$N\rightarrow\infty$. Therefore, most of the theoretical results concerning
the Kuramoto model are derived for networks that have locally tree-like
structures, i.e., $C\rightarrow 0$. However, most real-world networks have
clustered topologies Costa et al. (2011). Thus, an analysis of the Kuramoto
model on networks with clustering is necessary to model real-world
synchronization with more accuracy.
The objective of the current work is to study the dynamics of synchronization
on clustered random networks and perform a comparison with non-clustered ones.
More specifically, we develop a mean-field theory for the configuration model
proposed independently by Newman Newman (2009) and Miller Miller (2009), which
generates networks with $C>0$ even in the limit of large networks. Such
analysis is compared with the mean-field theory developed for locally tree-
like networks. Our results show that the mean-field theory for networks with
low values of triangles can be applied with certain accuracy on clustered
networks, indicating that the presence of cycles of order three does not
influence the network synchronization. This result is in agreement with
previous works Melnik et al. (2011); Gleeson et al. (2010); Hackett et al.
(2011); Gleeson et al. (2012), which observed that the clustering coefficient
does not play an important role in other dynamical process, such as bond-
percolation, rumor and epidemic spreading, provided that the networks have low
values of the average shortest path length.
In Sec.II we briefly describe the configuration model proposed in Newman
(2009) and Miller (2009). In Sec.III we derive a sufficient condition for
synchronization through mean-field approximation and in Sec.IV we compare
numerical and theoretical results and give our conclusions.
## II Random clustered networks
In the standard configuration model, the network is generated through the
degree sequence $\\{k_{i}\\}$, connecting the “stubs” at random. The process
to generate random clustered networks is quite similar. Let $s_{i}$ and
$t_{i}$ be the number of single edges and the number of triangles attached to
the node $i$, respectively. Given a network the sequence $\\{s_{i},t_{i}\\}$
is possible to connect the “stubs” in order to generate single edges and also
to connect nodes in order to obtain complete triangles. Hence, it is
convenient to define the joint degree distribution $p_{s,t_{\triangle}}$ of
the network, which is the fraction of vertices connected to $s$ single edges
and $t_{\triangle}$ triangles. Therefore, the conventional degree of each node
is given by $k=s+2t_{\triangle}$, since each triangle contributes with two to
the degree. Also, it is possible to relate the joint degree distribution
$p_{st_{\triangle}}$ with the conventional degree distribution $p_{k}$ through
$p_{k}=\sum_{s,t=0}^{\infty}p_{st_{\triangle}}\delta_{k,s+2t_{\triangle}},$
(2)
where $\delta_{i,j}$ is the Kronecker delta.
With the joint degree distribution $p_{s,t_{\triangle}}$ and the degree
distribution $p_{k}$, we can calculate the clustering coefficient for random
networks. The number of triangles in the network is given by
$3N_{\triangle}=N\sum_{st}tp_{st}$ and the number of connected triples
$N_{3}=N\sum_{k}\binom{k}{2}p_{k}$. Thus, using Eq. 1, the clustering
coefficient is
$C=\frac{\sum_{st}tp_{st_{\triangle}}}{\sum_{k}\binom{k}{2}p_{k}}.$ (3)
Note that the of factors $N$ cancel, letting $C>0$ in the limit of large
networks, i.e., $N\rightarrow\infty$.
## III Synchronization on clustered networks
The Kuramoto model consists of a set of $N$ oscillators coupled by the sine of
their phase differences Strogatz (2000); Acebrón et al. (2005). The state of
each oscillator is characterized by its phase $\theta_{i}(t)$ $i=1,\ldots,N$.
Considering a complex network where each node is a Kuramoto oscillator, the
equations of motion are given by
$\frac{d\theta_{i}(t)}{dt}=\omega_{i}+\lambda\sum_{i=1}^{N}A_{ij}\sin\left(\theta_{j}-\theta_{i}\right),\;i=1,\ldots,N,$
(4)
where $\omega_{i}$ is the natural frequency of the node $i$ , $\lambda$ is the
coupling strength and $A_{ij}$ are the elements of the adjacency matrix
$\mathbf{A}$, where $A_{ij}=1$ if the nodes $i$ and $j$ are connected while
$A_{ij}=0$, otherwise. The synchronization can be quantified through the order
parameter
$re^{i\psi(t)}=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_{j}(t)},$ (5)
where $\psi(t)$ is the average phase of the system. The coherence parameter is
bounded as $0\leq r\leq 1$, where $r=1$ represents the fully synchronized
state and $r=0$ is the incoherent solution. In the fully connected graph
($A_{ij}=1\;\forall i,j$ and $i\neq j$), the order parameter $r$ as a function
of $\lambda$ displays a second-order phase transition characterized by the
critical coupling $\lambda_{c}=2/(\pi g(\bar{\omega}))$ Strogatz (2000);
Acebrón et al. (2005), where $g(\omega)$ is the distribution of the natural
frequencies and $\bar{\omega}$ is the average frequency. In random networks,
the critical coupling $\lambda_{c}$ of such phase transition is rescaled by
the ratio $\left\langle k\right\rangle/\left\langle k^{2}\right\rangle$ Arenas
et al. (2008); Ichinomiya (2004); Lee (2005); Restrepo et al. (2005), i.e.,
$\lambda_{c}=\frac{2}{\pi g(\bar{\omega})}\frac{\left\langle
k\right\rangle}{\left\langle k^{2}\right\rangle},$ (6)
where $\left\langle k^{n}\right\rangle$ is the $n$-th moment of the degree
distribution $p_{k}$ of the network.
### III.1 Mean-field theory
The critical coupling strength necessary for the onset of synchronization in
Eq. 6 was firstly obtained by Ichinomiya through a mean-field analysis
Ichinomiya (2004) for the standard configuration model. The mean-field
analysis has the advantage of allowing an analytical treatment. In order to
extend the mean-field treatment to the configuration model for random
clustered networks, we approximate the Eqs. 4 into the following equation
$\frac{d\theta_{i}(t)}{dt}=\omega_{i}+\lambda\sum_{k^{\prime}}k_{i}P(k^{\prime}|k_{i})\sin\left(\theta_{k^{\prime}}-\theta_{i}\right),$
(7)
where $k_{i}$ is the degree of the node $i$ and $P(k^{\prime}|k)$ is the
probability that an edge emitted by a node with degree $k$ is connected to a
node with $k^{\prime}$. For uncorrelated networks $P(k^{\prime}|k)$ is given
by
$P(k^{\prime}|k)=\frac{k^{\prime}p_{k^{\prime}}}{\left\langle
k\right\rangle}.$ (8)
Substituting Eq. 8 in Eq. 4 we have:
$\frac{d\theta_{i}(t)}{dt}=\omega_{i}+\frac{\lambda k_{i}}{\left\langle
k\right\rangle}\sum_{k^{\prime}}k^{\prime}p_{k^{\prime}}\sin\left(\theta_{k^{\prime}}-\theta_{i}\right).$
(9)
Figure 1: Synchronization diagrams for networks with double Poisson joint
distribution $p_{st_{\triangle}}$ (Eq. 23). The dots are obtained calculating
the equations of motion (Eq. 4) until the system reaches the stationary state
for each value of coupling $\lambda$. The order parameter $r$ is then
calculated with Eq. 5. Each point is an average over 10 network realizations.
Solid lines correspond to the theoretical prediction from Eq. 21.
For the configuration model of clustered random networks, $p_{k}$ is defined
by Eq. 2. Therefore, substituting in Eq. 9 and noting that
$k_{i}=s_{i}+2t_{i}$ we get
$\frac{d\theta_{i}(t)}{dt}=\omega_{i}+\frac{\lambda(s_{i}+2t_{i})}{\left\langle
k\right\rangle}\sum_{s^{\prime},t^{\prime}_{\triangle}}(s^{\prime}+2t^{\prime}_{\triangle})p_{s^{\prime}t^{\prime}_{\triangle}}\sin\left(\theta_{k^{\prime}}-\theta_{i}\right),$
(10)
where $\left\langle k\right\rangle=\left\langle s\right\rangle+2\left\langle
t_{\triangle}\right\rangle$.
For an analytic treatment it is convenient to use the continuum limit of Eq.
10. For this purpose, let us define the density of the nodes with phase
$\theta$ at time $t$, for a given $\omega$, with $s$ singles edges and
$t_{\triangle}$ triangles, denoted by $\rho(s,t_{\triangle},\omega;\theta,t)$.
This density is normalized as
$\int_{0}^{2\pi}\rho(s,t_{\triangle},\omega;\theta,t)d\theta=1.$ (11)
Therefore, Eq. 10 in the continuum limit is given by
$\displaystyle\frac{d\theta(t)}{dt}$ $\displaystyle=$
$\displaystyle\omega+\frac{\lambda(s+2t_{\triangle})}{\left\langle
k\right\rangle}\int ds^{\prime}\int dt^{\prime}_{\triangle}\int
d\theta^{\prime}(s^{\prime}+2t^{\prime}_{\triangle})$ (12)
$\displaystyle\times
p_{s^{\prime}t^{\prime}_{\triangle}}\sin\left(\theta^{\prime}-\theta\right).$
The order parameter can also be redefined in order to account the connectivity
pattern of a random network as
$\displaystyle re^{i\psi(t)}$ $\displaystyle=$
$\displaystyle\frac{1}{\left\langle k\right\rangle}\int d\omega\int ds\int
dt_{\triangle}\int d\theta(s+2t_{\triangle})p_{st_{\triangle}}$ (13)
$\displaystyle\times\rho(s,t_{\triangle},\omega;\theta,t)e^{i\theta}.$
Figure 2: Synchronization diagram calculated as in Fig. 1 for networks with
double power-law joint distribution $p_{st_{\triangle}}$, with
$\gamma_{s}=\gamma_{t}=3$ in Eq. 24. Each point is an average over 10 network
realizations. Solid lines correspond to the theoretical prediction from Eq.
21.
Considering Eq. 13, it allows to rewrite Eq. 10 in terms of the order
parameter, resulting in
$\frac{d\theta}{dt}=\omega+\lambda\left(s+2t_{\triangle}\right)\sin(\psi-\theta).$
(14)
The density $\rho(s,t_{\triangle},\omega;\theta,t)$ will obey the following
continuity equation
$\frac{\partial\rho}{\partial
t}+\frac{\partial}{\partial\theta}\\{v_{\theta}\rho(s,t_{\triangle},\omega;\theta,t)\\}=0,$
(15)
which for the stationary states ($\partial\rho/\partial t=0$) has the
solutions
$\rho(s,t_{\triangle},\omega;\theta)=\begin{cases}\delta\left[\phi-\arcsin\left(\frac{\omega}{(s+2t_{\triangle})\lambda
r}\right)\right]&\mbox{ if
}\frac{\left|\omega\right|}{(s+2t_{\triangle})}\leq\lambda r,\\\
\frac{C(s,t_{\triangle},\theta)}{\left|\omega-\lambda
kr\sin\theta\right|}&\mbox{otherwise.}\end{cases}$ (16)
These solutions correspond to those oscillators that are entrained by the mean
field and those non-entrained, respectively. Thus, separating each
contribution to the order parameter we yield
$\displaystyle\left\langle k\right\rangle r$ $\displaystyle=$
$\displaystyle\int ds\int
dt_{\triangle}d\theta\left[\int_{\frac{\left|\omega\right|}{(s+2t_{\triangle})}\leq\lambda
r}+\int_{\frac{\left|\omega\right|}{(s+2t_{\triangle})}>\lambda r}\right]$
(17) $\displaystyle\times$ $\displaystyle
p_{st_{\triangle}}(s+2t_{\triangle})g(\omega)\rho(s,t_{\triangle};\theta)e^{i\theta}.$
The part of the non-entrained oscillator is given by
$\displaystyle\int d\theta
g(\omega)e^{i\theta}\left[\int_{\lambda(s+2t_{\triangle})r}^{\infty}d\omega\frac{1}{(\omega-\lambda(s+2t_{\triangle})r\sin\theta)}\right.$
$\displaystyle+\left.\int_{-\infty}^{-\lambda(s+2t_{\triangle})r}d\omega\frac{1}{(-\omega+\lambda(s+2t_{\triangle})r\sin\theta)}\right]=0,$
(18)
since the integral over $\theta$ is 0. Thus the only contribution for
coherence the parameter $r$ is due to the synchronous oscillators, which is
accounted in Eq. 17:
$\displaystyle\left\langle k\right\rangle r$ $\displaystyle=$
$\displaystyle\int ds\int dt_{\triangle}\int d\omega\int
d\theta(s+2t_{\triangle})g(\omega)$ (19) $\displaystyle\times
p_{st_{\triangle}}\exp\left[i\arcsin\left(\frac{\omega}{(s+2t_{\triangle})\lambda
r}\right)\right].$
From the real part we get:
$\displaystyle\left\langle k\right\rangle r$ $\displaystyle=$
$\displaystyle\int ds\int dt_{\triangle}\int d\omega\int
d\theta(s+2t_{\triangle})g(\omega)$ (20) $\displaystyle\times
p_{st_{\triangle}}\sqrt{1-\left(\frac{\omega}{(s+2t_{\triangle})\lambda
r}\right)^{2}}.$
Considering the following change of variable
$\omega^{\prime}=\omega/(s+2t_{\triangle})\lambda r$ and considering
$g(\omega)=(\sqrt{2\pi})^{-1}e^{-\omega^{2}/2}$, we obtain the following
implicit equation for the coherence parameter $r$
$\lambda=\sqrt{\frac{\pi}{8}}\left\langle
k\right\rangle\left\\{\int\int(s+2t_{\triangle})^{2}p_{st_{\triangle}}e^{-\lambda^{2}(s+2t_{\triangle})^{2}r^{2}/4}\left[I_{0}\left(\frac{\lambda^{2}(s+2t_{\triangle})^{2}r^{2}}{4}\right)+I_{1}\left(\frac{\lambda^{2}(s+2t_{\triangle})^{2}r^{2}}{4}\right)\right]dsdt_{\triangle}\right\\}^{-1},$
(21)
where $I_{0}$ and $I_{1}$ are the modified Bessel functions of first kind.
Thus, tending $r^{+}\rightarrow 0$, we obtain the critical coupling
$\lambda_{c}$ for the onset of synchronization
$\lambda_{c}=\sqrt{\frac{8}{\pi}}\left\langle
k\right\rangle\left\\{\int\int(s+2t_{\triangle})^{2}p_{st_{\triangle}}dsdt_{\triangle}\right\\}^{-1}.$
(22)
The mean field result obtained for the critical coupling $\lambda_{c}$ for
clustered netwokrs, Eq. 22, is similar to the one for non-clustered networks
(Eq. 6), i.e., in both cases the critical coupling is proportional to the
ratio of the moments of the degree distribution. Note that in the absence of
triangles $(t_{\triangle}=0)$ we recover the result
$\lambda_{c}=\sqrt{8/\pi}\left\langle k\right\rangle/\left\langle
k^{2}\right\rangle$, where the degrees are just due to single edges, $k=s$.
Figure 3: Synchronization diagrams calculated as in the other figures for
random networks with degree distribution $p_{k}=e^{-\left\langle
k\right\rangle}\left\langle k\right\rangle^{k}/k!$ and joint degree
distribution generated by Eq. 25. Each point is an average over 10 network
realizations. Solid lines correspond to the theoretical prediction from Eq.
21.
## IV Numerical simulations
In this section we give some numerical simulations of clustered and non-
clustered networks and will compare them with the theoretical result of Eq.
21. All simulations consider networks which are constructed through the
configuration model presented in Sec.II and the distributions of single edges
and triangles independently. However, it is also possible consider correlated
distributions as well.
Figure 4: Synchronization diagrams calculated as in the other figures for
random networks with degree distribution $p_{k}\propto k^{-\gamma}$, with
$\gamma=3$ and joint degree distribution generated by Eq. 25. Each point is an
average over 10 network realizations. Solid lines correspond to the
theoretical prediction from Eq. 21.
Let us study first networks with the following joint distribution of single
edges and triangles
$p_{st_{\triangle}}=e^{-\left\langle s\right\rangle}\frac{\left\langle
s\right\rangle^{s}}{s!}e^{-\left\langle
t_{\triangle}\right\rangle}\frac{\left\langle
t_{\triangle}\right\rangle^{t_{\triangle}}}{t_{\triangle}!}.$ (23)
In order to analyse systematically the dependence of the order parameter on
the presence of triangles in the network, we kept the average degree
$\left\langle k\right\rangle=\left\langle s\right\rangle+2\left\langle
t_{\triangle}\right\rangle$ fixed and varied the $\left\langle s\right\rangle$
and $\left\langle t_{\triangle}\right\rangle$, calculating the order parameter
$r$ as a function of the critical coupling $\lambda$. Fig. 1 shows the
synchronization diagram for networks with double Poisson degree distributions
(Eq. 23) with average degree $\left\langle k\right\rangle=20$. It is
interesting to note that networks with higher values of $\left\langle
t_{\triangle}\right\rangle$ have the same critical coupling for the onset of
synchronization.
We have also considered networks with joint distribution consisting of a
double power-law distribution
$p_{st_{\triangle}}\propto s^{-\gamma_{s}}t_{\triangle}^{-\gamma_{t}},$ (24)
where $\gamma_{s}=\gamma_{t}=\gamma$ for the sake of simplicity. Fig. 2 shows
order parameter $r$ as a function of $\lambda$ considering $\gamma=3$. As we
can see, the same behavior is observed as in Fig. 1, the presence of
clustering in the network does not affect the network synchronization. The
non-zero values of the order parameter $r$ for small values of the coupling
$\lambda$ in Fig. 1 and 2 are due to finite-size effects Ichinomiya (2004);
Lee (2005); Restrepo et al. (2005).
It is also possible to construct the joint distribution $p_{st_{\triangle}}$
from a given degree distribution $p_{k}$ through the relation Hackett et al.
(2011); Gleeson et al. (2010)
$p_{st_{\triangle}}=p_{k}\delta_{k,s+2t_{\triangle}}\left[(1-f)\delta_{t,0}+f\delta_{t,\left\lfloor(s+2t_{\triangle})/2\right\rfloor}\right]$
(25)
where $0\leq f\leq 1$ and $\left\lfloor\cdot\right\rfloor$ is the floor
function. Through Eq. 25 we can construct $p_{st_{\triangle}}$ keeping the
degree distribution $p_{k}$ fixed with $f$ being the fraction of nodes in the
network attached to the maximum possible number of triangles
$t=\left\lfloor(s+2t_{\triangle})/2\right\rfloor$ and $(1-f)$ the fraction of
nodes which are attached to single edges only. Substituting Eq. 25 into Eq. 3
we obtained Hackett et al. (2011)
$C=f\frac{\sum_{k}(p_{2k}+p_{2k+1})}{\sum_{k}\binom{k}{2}p_{k}}.$ (26)
Eq. 26 establishes a linear relationship between $C$ and $f$, i.e., with $f=0$
we construct a network with the minimum value for the cluster coefficient and
$f=1$ a network with the maximum value of $C$ for a given $p_{k}$, allowing to
study the extreme cases of the topology. Fig. 3 and Fig. 4 show the
synchronization diagrams for networks with $p_{k}=e^{-\left\langle
k\right\rangle}\left\langle k\right\rangle^{k}/k!$ and $p_{k}\propto
k^{-\gamma}$, respectively. Again, we observe a good agreement with the
theoretical curve. Therefore, the clustering coefficient has no effect on the
coherence parameter evolution $r(\lambda)$, comparing the curves with $f=0$
and $f=1$. Also, finite-size effects are observed for small values of
$\lambda$.
In summary, we have shown that the presence of cycles of order three does not
play an important role in network synchronization of Kuramoto oscillators. In
fact, the theoretical results for non-clustered networks are highly accurate
on describing the behavior of the order parameter $r$ for clustered networks,
even when the cluster coefficient $C$ has the maximum accessible value for a
given network. The results presented here are in agreement with previous
findings Melnik et al. (2011), where it was found that the presence of
triangles in the network topology does not influence the performance of other
dynamical processes, such as bond percolation, $k$-core size percolations and
epidemic spreading.
F. A. Rodrigues would like to acknowledge CNPq (305940/2010-4) and FAPESP
(2010/19440-2) for the financial support given to this research. T. Peron
would like to acknowledge FAPESP and J. Kurths would like to acknowledge IRTG
for the sponsorship provided.
## References
* Arenas et al. (2008) A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Physics Reports 469, 93 (2008).
* Pikovsky et al. (2003) A. Pikovsky, M. Rosenblum, and J. Kurths, _Synchronization: A universal concept in nonlinear sciences_ , vol. 12 (Cambridge University Press, 2003).
* Acebrón et al. (2005) J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Reviews of Modern Physics 77, 137 (2005).
* Ichinomiya (2004) T. Ichinomiya, Physical Review E 70, 026116 (2004).
* Lee (2005) D.-S. Lee, Phys. Rev. E 72, 026208 (2005).
* Restrepo et al. (2005) J. G. Restrepo, E. Ott, and B. R. Hunt, Physical Review E 71, 036151 (2005).
* Peron and Rodrigues (2012) T. K. D. M. Peron and F. A. Rodrigues, preprint arXiv:1204.4768 (2012).
* Costa et al. (2011) L. d. F. Costa, O. Oliveira Jr, G. Travieso, F. Rodrigues, P. Boas, L. Antiqueira, M. Viana, and L. Rocha, Advances in Physics 60, 329 (2011).
* Newman (2009) M. Newman, Physical Review Letters 103, 58701 (2009).
* Miller (2009) J. Miller, Physical Review E 80, 020901 (2009).
* Melnik et al. (2011) S. Melnik, A. Hackett, M. Porter, P. Mucha, and J. Gleeson, Physical Review E 83, 036112 (2011).
* Gleeson et al. (2010) J. Gleeson, S. Melnik, and A. Hackett, Physical Review E 81, 066114 (2010).
* Hackett et al. (2011) A. Hackett, S. Melnik, and J. Gleeson, Physical Review E 83, 056107 (2011).
* Gleeson et al. (2012) J. P. Gleeson, S. Melnik, J. A. Ward, M. A. Porter, and P. J. Mucha, Phys. Rev. E 85, 026106 (2012).
* Strogatz (2000) S. Strogatz, Physica D: Nonlinear Phenomena 143, 1 (2000).
|
arxiv-papers
| 2012-10-09T14:58:11 |
2024-09-04T02:49:36.229622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Thomas Kau\\^e Dal'Maso Peron, Francisco Aparecido Rodrigues and\n J\\\"urgen Kurths",
"submitter": "Thomas Peron",
"url": "https://arxiv.org/abs/1210.2625"
}
|
1210.2631
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-287 LHCb-PAPER-2012-022
Evidence for the decay
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ and measurement of the relative branching fractions of
$\mathrm{B}^{0}_{\mathrm{s}}$ meson decays to
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{\prime}$
The LHCb collaboration†††Authors are listed on the following pages.
First evidence of the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ decay is found and the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$ decays are studied using a dataset corresponding to an
integrated luminosity of 1.0 $\mathrm{fb}^{-1}$ collected by the LHCb
experiment in proton-proton collisions at a centre-of-mass energy of
$\sqrt{s}=7$ TeV. The branching fractions of these decays are measured
relative to that of the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$ decay:
$\begin{array}[]{lll}\frac{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega)}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}$=$\>0.89\pm
0.19\,(\mathrm{stat})\,^{+0.07}_{-0.13}\,(\mathrm{syst}),\\\ \vskip
3.0pt\cr\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}$=$\>14.0\pm
1.2\,(\mathrm{stat})\,^{+1.1}_{-1.5}\,(\mathrm{syst})\,^{+1.1}_{-1.0}\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right),\\\
\vskip 3.0pt\cr\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}$=$\>12.7\pm
1.1\,(\mathrm{stat})\,^{+0.5}_{-1.3}\,(\mathrm{syst})\,^{+1.0}_{-0.9}\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right),\end{array}$
where the last uncertainty is due to the knowledge of
$f_{\mathrm{d}}/f_{\mathrm{s}}$, the ratio of b-quark hadronization factors
that accounts for the different production rate of $\mathrm{B}^{0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}$ mesons. The ratio of the branching fractions of
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$ decays is measured to be
$\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}=0.90\pm
0.09\,(\mathrm{stat})\,^{+0.06}_{-0.02}\,(\mathrm{syst}).$
Submitted to Nucl. Phys. B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura36,28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S.
Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R. Bernet37, M.-O.
Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T. Bird51, A.
Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50, J.
Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49, M.
Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N.
Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-Montells33, A.
Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36,
D. Craik45, S. Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F.
Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, D. Esperante Pereira34, A. Falabella14,e, C. Färber11, G. Fardell47,
C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T.
Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, R.S. Huston12, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, M. Karbach35, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20,
Y.M. Kim47, O. Kochebina7, I. Komarov29, V. Komarov36, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C.
Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N.
Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
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Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A.
Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M.
Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6,
P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27,
T. Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50,
R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51,
K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36,35, T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26,
F. Teubert35, C. Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M.
Tobin37, S. Tolk39, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
G. Veneziano36, M. Vesterinen35, B. Viaud7, I. Videau7, D. Vieira2, X.
Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D.
Voong43, A. Vorobyev27, V. Vorobyev31, H. Voss10, C. Voß55, R. Waldi55, R.
Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D.
Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D. Wiedner11, L.
Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p, F.F. Wilson46,
J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A. Wotton44, S. Wright44, S.
Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X.
Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
Decays of B mesons into a ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ and a light meson are dominated by color-suppressed tree diagrams
involving
$\bar{\mathrm{b}}\rightarrow\bar{\mathrm{c}}\mathrm{c}\bar{\mathrm{s}}$ and
$\bar{\mathrm{b}}\rightarrow\bar{\mathrm{c}}\mathrm{c}\bar{\mathrm{d}}$
transitions (see Fig. 1). Contributions from other diagrams are expected to be
small [1]. Measurements of the branching fractions of these decays can help to
shed light on hadronic interactions. The decay
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ has not been observed previously. The CLEO collaboration has
set the most restrictive upper limit to date of ${\cal
B}(\mathrm{B}^{0}~{}\rightarrow~{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega)<2.7\times 10^{-4}$ at 90% confidence level [2].
The
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}$ decays were observed by the Belle collaboration [3]
with branching fractions ${\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)=(5.10\pm 0.50\pm 0.25\,^{+1.14}_{-0.79})\times 10^{-4}$ and
${\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})=(3.71\pm 0.61\pm 0.18\,^{+0.83}_{-0.57})\times
10^{-4}$, where the first uncertainty is statistical, the second is systematic
and the third one is due to an uncertainty of the number of produced
$\mathrm{B}^{0}_{\mathrm{s}}{\mathrm{\bar{B}^{0}_{s}}}$ pairs. Since both
final states are $C\\!P$ eigenstates, time-dependent $C\\!P$ violation studies
and access to the $\mathrm{B}^{0}_{\mathrm{s}}-{\mathrm{\bar{B}^{0}_{s}}}$
mixing phase $\upphi_{\mathrm{s}}$ will be possible in the future [4]. The
theoretical prediction for these branching fractions and their ratio relies on
knowledge of the $\upeta-\upeta^{\prime}$ mixing phase $\upphi_{\mathrm{P}}$.
Taking $\upphi_{\mathrm{P}}=(41.4\pm 0.5)^{\circ}$ [5] and ignoring a possible
gluonic component and corrections due to form factors, the ratio becomes
$\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}\times\frac{\mathcal{F}_{s}^{\upeta}}{\mathcal{F}_{s}^{\upeta^{\prime}}}=\frac{1}{\tan^{2}\upphi_{\mathrm{P}}}=1.28\,^{+0.10}_{-0.08}.$
Here $\mathcal{F}_{s}^{\upeta^{(\prime)}}$ is the phase space factor of the
$\mathrm{B}^{0}_{\mathrm{s}}~{}\rightarrow~{}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}~{}\upeta^{(\prime)}$ decay and the uncertainty is due to the inaccuracy
in the knowledge of the mixing phase. As discussed in Ref. [1], a precise
measurement of this ratio tests $SU(3)$ flavour symmetry. In addition, in
combination with other measurements, the fraction of the gluonic component in
the $\upeta^{\prime}$ meson can eventually be estimated [6].
$\mathrm{B}^{0}$$\mathrm{d}$$\mathrm{\bar{b}}$$\mathrm{W^{-}}$$\mathrm{d}$$\mathrm{\bar{c}}$$\mathrm{\bar{d}}$$\mathrm{c}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$$\uprho^{0},\upomega$$\mathrm{B}^{0}_{\mathrm{s}}$$\mathrm{s}$$\mathrm{\bar{b}}$$\mathrm{W^{-}}$$\mathrm{s}$$\mathrm{\bar{c}}$$\mathrm{\bar{s}}$$\mathrm{c}$${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$$\upeta,\upeta^{\prime}$ Figure 1: Examples of the dominant diagrams
for the
$\mathrm{B}^{0}_{\mathrm{(s)}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}$ decays (where
$\mathrm{X}^{0}=\upeta,\,\upeta^{\prime},\,\upomega$ or $\uprho^{0}$).
The analysis presented here is based on a data sample corresponding to an
integrated luminosity of 1.0 fb-1 collected by the LHCb detector in 2011 in pp
collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV. The branching
fractions of these decays are measured relative to ${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})$ and the ratio $\dfrac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}$ is determined.
## 2 LHCb detector
The LHCb detector [7] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of $\mathrm{b}$\- and
$\mathrm{c}$-hadrons. The detector includes a high precision tracking system
consisting of a silicon-strip vertex detector surrounding the pp interaction
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\rm\,Tm}$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream. The combined
tracking system has a momentum resolution $\Delta p/p$ that varies from 0.4%
at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
and electromagnetic and hadron calorimeters. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. This analysis uses events triggered by one or two
muon candidates. In the case of one muon, the hardware level requirement was
for its $p_{\rm T}$ to be larger than 1.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; in case of two muons the restriction
$\sqrt{\mbox{$p_{\rm T}$}_{1}\cdot\mbox{$p_{\rm T}$}_{2}}>1.3$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ was applied. At the software level, the
two muons were required to have an invariant mass in the interval
$2.97<\mathrm{m}_{\upmu^{+}\upmu^{-}}<3.21$ GeV/c2 and to be consistent with
originating from the same vertex. To avoid the possibility that a few events
with high occupancy dominate the trigger processing time, a set of global
event selection requirements based on hit multiplicities was applied.
For the simulation, pp collisions are generated using Pythia 6.4 [8] with a
specific LHCb configuration [9]. Decays of hadronic particles are described by
EvtGen [10] in which final state radiation is generated using Photos [11]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [12, *Agostinelli:2002hh] as described in
Ref. [14]. The digitized output is passed through a full simulation of both
the hardware and software trigger and then reconstructed in the same way as
the data.
## 3 Data sample and common selection requirements
The decays
$\mathrm{B}^{0}_{(\mathrm{s})}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}$ (where $\mathrm{X}^{0}$ = $\upeta$, $\upeta^{\prime}$,
$\upomega$ and $\uppi^{+}\uppi^{-}$) are reconstructed using the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ decay mode. The X0 candidates are
reconstructed in the $\upeta\rightarrow\upgamma{}\upgamma$,
$\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$,
$\upeta^{\prime}\rightarrow\uprho^{0}\upgamma$,
$\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}$ and
$\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$ final states. Pairs of
oppositely charged particles identified as muons, each having $\mbox{$p_{\rm
T}$}>550~{}\mathrm{MeV}/c$ and originating from a common vertex, are combined
to form ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}$ candidates. Well identified muons are
selected by requiring that the difference in logarithms of the global
likelihood of the muon hypothesis, $\Delta\ln\mathcal{L}_{\upmu\mathrm{h}}$,
provided by the particle identification detectors [15], with respect to the
hadron hypothesis is greater than zero. The fit of the common two-prong vertex
is required to satisfy $\chi^{2}<20$. The vertex is deemed to be well
separated from the reconstructed primary vertex of the pp interaction by
requiring the decay length significance to be greater than 3. Finally, the
invariant mass of the dimuon combination is required to be within $\pm
40~{}\mathrm{MeV}/c^{2}$ of the nominal
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mass [16].
To identify charged pions the difference between the logarithmic likelihoods
of the pion and kaon hypotheses provided by RICH detectors,
$\Delta\ln\mathcal{L}_{\uppi\mathrm{K}}$, should be greater than zero. In the
reconstruction of the
$\mathrm{B}^{0}_{\left(\mathrm{s}\right)}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decay this requirement is tightened to be
$\Delta\ln\mathcal{L}_{\uppi\mathrm{K}}>2$ so as to suppress the contamination
from
$\mathrm{B}^{0}_{\left(\mathrm{s}\right)}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi\mathrm{K}$ decays with misidentified kaons. In addition, the pion
tracks are required to have $\mbox{$p_{\rm
T}$}>250{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. A minimal value of
$\Delta\upchi^{2}_{\mathrm{IP}}$, defined as the difference between the
$\upchi^{2}$ of the primary vertex, reconstructed with and without the
considered track, is required to be larger than four.
Photons are selected from neutral clusters in the electromagnetic calorimeter
with minimal transverse energy in excess of $300~{}\mathrm{MeV}$. To suppress
the large combinatorial background from
$\uppi^{0}\rightarrow\upgamma{}\upgamma$ decays, photons that can form part of
a $\uppi^{0}\rightarrow\upgamma{}\upgamma$ candidate with invariant mass
within $\pm 25~{}\mathrm{MeV}/c^{2}$ of the nominal $\uppi^{0}$ mass are not
used for reconstruction of $\upeta\rightarrow\upgamma{}\upgamma$ and
$\upeta^{\prime}\rightarrow\uprho^{0}\upgamma$ candidates.
The $\upeta\rightarrow\upgamma{}\upgamma$
($\uppi^{0}\rightarrow\upgamma{}\upgamma$) candidates are reconstructed as
diphoton combinations with invariant mass within $\pm
70\,(25)~{}\mathrm{MeV}/c^{2}$ around the nominal $\upeta\,(\uppi^{0})$ mass.
To suppress the combinatorial background to the
$\upeta\rightarrow\upgamma{}\upgamma$ decay, the cosine of the decay angle
$\theta^{*}_{\upeta}$, between the photon momentum in the $\upeta$ rest frame
and the direction of the Lorentz boost from the laboratory frame to the
$\upeta$ rest frame, is required to have
$\left|\cos\theta^{*}_{\upeta}\right|<0.8$.
The $\upeta^{\prime}$ candidates are reconstructed as
$\upeta\uppi^{+}\uppi^{-}$ and $\uprho^{0}\upgamma$ combinations with
invariant mass within $\pm 60~{}\mathrm{MeV}/c^{2}$ from the nominal
$\upeta^{\prime}$ mass. For the $\upeta^{\prime}\rightarrow\uprho^{0}\upgamma$
case, the invariant mass of the $\uppi^{+}\uppi^{-}$ combination is required
to be within $\pm 150~{}\mathrm{MeV}/c^{2}$ of the $\uprho^{0}$ mass. For
$\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$
($\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$) candidates the invariant
mass is required to be within $\pm 50~{}\mathrm{MeV}/c^{2}$ of the nominal
$\upeta\,(\upomega)$ mass.
The $\mathrm{B}^{0}_{\left(\mathrm{s}\right)}$ candidates are formed from
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{X}^{0}$ pairs
with $\mbox{$p_{\rm T}$}>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the
$\mathrm{X}^{0}$. To improve the invariant mass resolution a kinematic fit
[17] is applied. In this fit, constraints are applied on the known masses [16]
of intermediate resonances, except the wide $\uprho^{0}$ and $\upomega$
states, and it is also required that the candidate’s momentum vector points to
the associated primary vertex. The $\chi^{2}$ per degree of freedom for this
fit is required to be less than five. Finally, the decay time (c$\uptau$) of
the $\mathrm{B}^{0}_{\left(\mathrm{s}\right)}$ candidates is required to be in
excess of $150\,\upmu\rm m$.
## 4 Evidence for the
$\boldsymbol{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega}$ decay
$\mathrm{m}$${}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega}$$\left[\mathrm{GeV}/c^{2}\right]$LHCb
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$
Candidates/$\left(25~{}\mathrm{MeV}/c^{2}\right)$
Figure 2: Invariant mass distribution for selected
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ candidates. The black dots correspond to the data
distribution, the thick solid blue line is the total fit function, the blue
dashed line shows the background contribution and the orange thin line is the
signal component of the fit function.
The invariant mass distribution of the selected
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upomega$ candidates
is shown in Fig. 2, where a $\mathrm{B}^{0}$ signal is visible. To determine
the signal yield, an unbinned maximum likelihood fit is performed to this
distribution. The signal is modelled by a Gaussian distribution and the
background by an exponential function. The peak position is found to be
$5284\pm 5$ ṀeV/$c^{2}$, which is consistent with the nominal $\mathrm{B}^{0}$
mass [16] and the resolution is in good agreement with the prediction from
simulation. The event yield is determined to be
$\mathcal{Y}_{\mathrm{B}^{0}}=72\pm 15$.
The statistical significance for the observed signal is determined as
$\mathcal{S}=\sqrt{-2\times\ln(\mathcal{L}_{\mathcal{B}}/\mathcal{L}_{\mathcal{S}+\mathcal{B}})},$
where $\mathcal{L}_{\mathcal{S}+\mathcal{B}}$ and $\mathcal{L}_{\mathcal{B}}$
denote the likelihood of the signal plus background hypothesis and the
background hypothesis, respectively. The statistical significance of the
signal is found to be 5.0 standard deviations. Taking into account the
systematic uncertainty related to the fit function, which is discussed in
detail in Section 7.1, the significance is 4.6$\upsigma$; this also takes into
account the freedom in the peak position and width in the nominal fit.
To demonstrate that the signal originates from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ decays, the sPlot technique [18] has been applied. Using the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}\upgamma{}\upgamma$ invariant mass as the
discriminating variable, the distributions for the invariant masses of the
intermediate resonances $\uppi^{0}\rightarrow\upgamma{}\upgamma$ and
$\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$ have been obtained. The
invariant mass window for each corresponding resonance is released and the
mass constraint is removed.
The invariant mass distributions for $\upgamma{}\upgamma$ and
$\uppi^{+}\uppi^{-}{}\uppi^{0}$ from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ candidates are shown in Fig. 3. Clear signals are seen for
both the $\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$ and
$\uppi^{0}\rightarrow\upgamma{}\upgamma$ decays. The $\upgamma{}\upgamma$
distribution is described by a sum of a Gaussian function and a constant. The
$\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$ signal is modelled by a
convolution of a Gaussian and a Breit-Wigner function with a constant
background. The peak positions are in good agreement with the nominal
$\uppi^{0}$ and $\upomega$ masses and the yields determined from the fits are
compatible with the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ yield. The nonresonant contribution in each case is found to
be consistent with zero.
$\uppi^{0}\rightarrow\upgamma{}\upgamma$$\mathrm{m}_{\upgamma{}\upgamma}$$\left[\mathrm{MeV}/c^{2}\right]$LHCb(a)
Candidates/$\left(10~{}\mathrm{MeV}/c^{2}\right)$
$\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$$\mathrm{m}_{\uppi^{+}\uppi^{-}{}\uppi^{0}}$$\left[\mathrm{MeV}/c^{2}\right]$LHCb(b)
Candidates/$\left(30~{}\mathrm{MeV}/c^{2}\right)$
Figure 3: Background-subtracted (a) $\upgamma{}\upgamma$ and (b)
$\uppi^{+}\uppi^{-}{}\uppi^{0}$ invariant mass distributions for
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}\gamma\gamma$ decays. In both distributions the line
is the result of the fit described in the text.
## 5 Decays into
$\boldsymbol{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}}$ final states
The invariant mass spectra for
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\left(\prime\right)}$ candidates are shown in Fig. 4, where
signals are visible. To determine the signal yields, unbinned maximum
likelihood fits are performed. For all modes apart from
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}\upgamma\right)$,
the $\mathrm{B}^{0}_{\mathrm{s}}$ signal is modelled by a single Gaussian
function. In all cases there is a possible corresponding $\mathrm{B}^{0}$
signal, which is included in the fit model as an additional Gaussian
component. The difference of the means of the two Gaussians is fixed to the
known difference between the $\mathrm{B}^{0}_{\mathrm{s}}$ and the
$\mathrm{B}^{0}$ masses [19]. Simulation studies for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}\upgamma\right)$
mode indicate that in this case a double Gaussian resolution model is more
appropriate. The mean values of the two Gaussian functions are required to be
the same, and the ratio of their resolutions and the fraction of the event
yield carried by each of the Gaussian functions are fixed at the values
obtained from simulation.
The combinatorial background is modelled by an exponential function. In
addition, a component is added to describe the contribution from partially
reconstructed $\mathrm{B}$ decays. It is described with the phase space
function for two particles in a three body decay under the hypothesis of
$\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}\mathrm{X}$ decay, where $\mathrm{X}$ can be either a
kaon or a pion, which escapes detection. The phase space function is convolved
with a resolution factor, which is fixed at the value of the signal
resolution.
(a)$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$$(\upeta\rightarrow\upgamma{}\upgamma$)LHCb(b)$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$$(\upeta\rightarrow
3\uppi)$LHCb(c)$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$$(\upeta^{\prime}\rightarrow\uprho^{0}\upgamma)$LHCb(d)$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$$(\upeta^{\prime}\rightarrow\uppi\uppi\upeta)$LHCb$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta}$$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}}$$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}}$$\left[\mathrm{GeV}/c^{2}\right]$$\left[\mathrm{GeV}/c^{2}\right]$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
Candidates/$\left(10~{}\mathrm{MeV}/c^{2}\right)$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
Figure 4: Invariant mass distributions for selected
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\left(\prime\right)}$ candidates: (a)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\upgamma{}\upgamma\right)$, (b)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)$, (c)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}\upgamma\right)$
and (d)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uppi^{+}\uppi^{-}{}\upeta\right)$.
In all distributions the black dots show the data. The thin solid orange lines
show the signal $\mathrm{B}^{0}_{\mathrm{s}}$ contributions and the orange
dot-dashed lines correspond to the $\mathrm{B}^{0}$ contributions. The blue
dashed lines show the combinatorial background contributions and the dotted
blue lines show the partially reconstructed background components. The total
fit functions are drawn as solid blue lines. The results of the fit are
described in the text.
The fit results are summarized in Table 1. In all cases the position of the
signal peak is consistent with the nominal $\mathrm{B}^{0}_{\mathrm{s}}$ mass
[16] and the resolutions agree with the expectations from simulation. The
statistical significances of all the $\mathrm{B}^{0}_{\mathrm{s}}$ decays
exceed 7$\upsigma$.
Table 1: Signal yields, $\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}}$, the fitted $\mathrm{B}^{0}_{\mathrm{s}}$ mass, $\mathrm{m}_{\mathrm{B}^{0}_{\mathrm{s}}}$ and mass resolutions, $\sigma_{\mathrm{B}^{0}_{\mathrm{s}}}$ for the $\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}{}\upeta^{\left(\prime\right)}$ decays. Mode | $\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}}$ | $\mathrm{m}_{\mathrm{B}^{0}_{\mathrm{s}}}$ | $\sigma_{\mathrm{B}^{0}_{\mathrm{s}}}$
---|---|---|---
$\left[\mathrm{MeV}/c^{2}\right]$ | $\left[\mathrm{MeV}/c^{2}\right]$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta(\upeta\rightarrow\upgamma{}\upgamma)$ | $810\pm 65$ | $5367.2\pm 3.5$ | $40.1\pm 3.6$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0})$ | $94\pm 11$ | $5368.4\pm 2.6$ | $20.3\pm 2.3$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{\prime}(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma)$ | $336\pm 30$ | $5367.0\pm 1.1$ | $8.0\>\pm 1.1$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{\prime}(\upeta^{\prime}\rightarrow\uppi^{+}\uppi^{-}{}\upeta)$ | $79\pm 10$ | $5369.0\pm 2.8$ | $20.7\pm 2.3$
To test the resonance structure of the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}$ decays, the sPlot technique is used. For the
$\uppi^{0}$, $\upeta$ and $\upeta^{\prime}$ candidates the background-
subtracted invariant mass distributions are studied. The restrictions on the
invariant mass for the corresponding resonance are released and the mass
constraints (if any) removed. The background-subtracted distributions are then
fitted with the sum of a Gaussian function and a constant component for the
resonant and nonresonant components respectively. In the fit of the dipion
invariant mass for the $\upeta^{\prime}\rightarrow\uppi^{+}\uppi^{-}\upgamma$
decay a modified relativistic Breit-Wigner function is used as the signal
component [20, 21].
Background-subtracted invariant mass distributions of the intermediate
resonance states from the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}$ decays, are shown in Fig. 5. Clear signals are seen. In
all cases the signal yields determined from the fits are in agreement with the
event yield in the $\mathrm{B}^{0}_{\mathrm{s}}$ signal within one standard
deviation (Table 1). The signal positions are consistent with the nominal
masses of the $\upeta^{(\prime)}$ mesons and the nonresonant contribution
appears to be negligible. In each case the invariant mass resolution agrees
with the expectation from simulation studies.
(a) $\upeta\rightarrow\upgamma\upgamma$LHCb
$\mathrm{m}_{\upgamma\upgamma}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(10~{}\mathrm{MeV}/c^{2}\right)$
(b) $\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}$LHCb
$\mathrm{m}_{\uppi^{+}\uppi^{-}{}\uppi^{0}}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(10~{}\mathrm{MeV}/c^{2}\right)$
(c) $\upeta^{\prime}\rightarrow\uprho^{0}\upgamma$LHCb
$\mathrm{m}_{\uprho^{0}\upgamma}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
(d) $\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}$LHCb
$\mathrm{m}_{\uppi^{+}\uppi^{-}}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(45~{}\mathrm{MeV}/c^{2}\right)$
(e) $\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}$LHCb
$\mathrm{m}_{\upeta\uppi^{+}\uppi^{-}}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
(f) $\upeta\rightarrow\upgamma{}\upgamma$LHCb
$\mathrm{m}_{\upgamma{}\upgamma}$$\left[\mathrm{MeV}/c^{2}\right]$
Candidates/$\left(20~{}\mathrm{MeV}/c^{2}\right)$
Figure 5: Background-subtracted invariant mass distributions for (a)
$\upgamma{}\upgamma$ from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta(\upeta\rightarrow\upgamma{}\upgamma)$; (b)
$\uppi^{+}\uppi^{-}{}\uppi^{0}$ from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0})$; (c) and (d)
$\uppi^{+}\uppi^{-}\upgamma$ and $\uppi^{+}\uppi^{-}$ from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}(\upeta^{\prime}\rightarrow\uprho^{0}\upgamma,\,\uprho\rightarrow\uppi^{+}\uppi^{-})$;
(e) and (f) $\upeta\uppi^{+}\uppi^{-}$ and $\upgamma{}\upgamma$ from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}(\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-})$.
The purple line is the result of the fit described in the text.
## 6 The
$\boldsymbol{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}}$ decay
The $\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}\,(\uprho^{0}\rightarrow\uppi^{+}\uppi^{-})$ decay is used as
a normalization channel [22]. Since it contains a
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson and two pions
in the final state, the systematic uncertainty is reduced in the ratio of the
branching fractions, as the corresponding reconstruction and particle
identification uncertainties are expected to cancel.
The invariant mass spectrum for
$\mathrm{B}^{0}_{\left(\mathrm{s}\right)}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ candidates is presented in Fig. 6, where three clear
signals are visible. Two narrow signals correspond to the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decays. The latter decay has been studied in detail
in Refs. [23, 24]. The peak at lower mass corresponds to contamination from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}\left(\mathrm{K}^{*0}\rightarrow\mathrm{K}^{+}\uppi^{-}\right)$
decays with a kaon being misreconstructed as a pion. A contribution from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ decay is considered to be negligible.
$\mathrm{B}^{0}_{\mathrm{(s)}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$$\mathrm{m}_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}}$$\left[\mathrm{GeV}/c^{2}\right]$LHCb
Candidates/$\left(10~{}\mathrm{MeV}/c^{2}\right)$
Figure 6: Invariant mass distribution for selected
$\mathrm{B}^{0}_{\mathrm{(s)}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ candidates. The black dots show the data. The dot-
dashed thin orange line shows the signal $\mathrm{B}^{0}$ contribution and the
orange solid line shows the signal $\mathrm{B}^{0}_{\mathrm{s}}$ contribution,
a reflection from misidentified
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}(\mathrm{K^{*}}\rightarrow\mathrm{K}\uppi)$ is shown by a blue dotted
line. The blue dashed line shows the background contribution. The total fit
function is shown as a solid blue line.
The invariant mass distribution is fitted with a sum of three Gaussian
functions to describe the three signals, and an exponential function to
represent the background. The fit gives a yield of $1143\pm 39$ for
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$.
Previous studies at BaBar [22] show that the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ final state has contributions from decays of
$\uprho^{0}$ and $\mathrm{K}^{0}_{\mathrm{S}}$ mesons, as well as a broad
S-wave component. A further component from the $\mathrm{f}_{2}(1270)$
resonance is also hinted at in the BaBar study. To study the dipion mass
distribution the sPlot technique is used. With the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uppi^{+}\uppi^{-}$
invariant mass as the discriminating variable, the $\uppi^{+}\uppi^{-}$
invariant mass spectrum from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decays is obtained (see Fig. 7). A dominant
$\uprho^{0}$ signal is observed together with a narrow peak around 498 MeV/c2
due to $\mathrm{K}^{0}_{\mathrm{S}}$ decays. There is also a wide enhancement
at a mass close to $1260~{}\mathrm{MeV}/c^{2}$. The position and width of this
structure are consistent with the interpretation as a contribution from the
$\mathrm{f}_{2}(1270)$ state. This will be the subject of a future
publication.
LHCb $\mathrm{m}_{\uppi^{+}\uppi^{-}}$$\left[\mathrm{GeV}/c^{2}\right]$
Candidates $\left(20~{}\mathrm{MeV}/c^{2}\right)$
Figure 7: Background-subtracted $\uppi^{+}\uppi^{-}$ invariant mass
distribution from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uppi^{+}\uppi^{-}$ decays. The black dots show the data. A violet solid
line denotes the total fit function, the solid orange line shows the
$\uprho^{0}$ signal contribution and the blue dashed line shows the
$\mathrm{f}_{2}(1270)$ contribution. The blue dot-dashed line shows the
contribution from the $\mathrm{f}_{0}(500)$. The region $\pm
40~{}\mathrm{MeV/c}^{2}$ around the $\mathrm{K}^{0}_{\mathrm{S}}$ mass is
excluded from the fit.
The distribution is fitted with the sum of several components. A P-wave
modified relativistic Breit-Wigner function [20, 21] multiplied by a phase
space factor describes the $\uprho^{0}$ signal. A D-wave relativistic Breit-
Wigner function is added to describe the enhancement at
$1260~{}\mathrm{MeV}/c^{2}$. The parameters (width and mean value) of this
function are fixed to the known $\mathrm{f}_{2}(1270)$ mass and decay width
[16]. The S-wave contribution expected from the $\mathrm{f}_{0}(500)$
resonance is modelled by a Zou-Bugg [25, 26] function with parameters from
Ref. [27]. The $\uprho^{0}$ parameters (mass and width) are fixed at their
nominal values and the region around the $\mathrm{K}^{0}_{\mathrm{S}}$ peak is
excluded from the fit. The excluded region is $\pm 40~{}\mathrm{MeV/c}^{2}$
which is four times the mass resolution. A small systematic uncertainty is
induced by neglecting the $\uprho^{0}-\upomega$ interference. The value of the
uncertainty is estimated to be 0.5% relative to the $\uprho^{0}$ event yield.
Table 2: Fitted yields of the $\uprho^{0}$ resonance, the relative yields of the $\mathrm{f}_{2}(1270)$ and $\mathrm{f}_{0}(500)$ components and probabilities, $\mathcal{P}$, of the fits to the uncorrected and efficiency-corrected $\uppi^{+}\uppi^{-}$ invariant mass distributions. | Uncorrected fit | Efficiency-corrected fit
---|---|---
$\uprho^{0}$ event yield | $811\pm 38$ | $\,\,\,\,\,\,\,(27.6\pm 1.3)\times 10^{3}$
$\mathrm{f}_{0}\left(500\right)$ fraction | $0.20\pm 0.04$ | $0.19\pm 0.04$
$\mathrm{f}_{2}\left(1270\right)$ fraction | $0.14\pm 0.03$ | $0.16\pm 0.04$
$\mathcal{P}~{}\left[\%\right]$ | $40$ | $46$
The reconstruction and selection efficiency for the dipion system has some
dependence on the dipion invariant mass. A study using simulated data has
shown that with the increase of the $\uppi^{+}\uppi^{-}$ invariant mass in the
range 300 – 1500 MeV/$c^{2}$ the efficiency decreases by approximately 16%. As
the $\uprho^{0}$ meson has a significant width, this dependence needs to be
accounted for in the determination of the $\uprho^{0}$ signal yield. For this,
the efficiency dependence on $\uppi^{+}\uppi^{-}$ invariant mass extracted
from the simulation is described with a linear function. Then each entry in
the invariant mass distribution is given a weight proportional to the inverse
value of the efficiency function and the efficiency-corrected invariant mass
distribution is refitted with the same sum of functions to extract the
efficiency-corrected event yield for
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$. The resulting fit parameters both for the uncorrected and
efficiency-corrected distributions are listed in Table 2.
## 7 Measurements of ratios of branching fractions
Ratios of branching fractions are measured using the formula
$\mathcal{R}^{\mathrm{B,X}^{0}}_{\mathrm{B,Y}^{0}}\equiv\dfrac{{\cal
B}\left(\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}\right)}{{\cal
B}\left(\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{Y}^{0}\right)}=\dfrac{\mathcal{Y}\left(\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}\right)}{\mathcal{Y}\left(\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{Y}^{0}\right)}\times\dfrac{{\cal B}_{\mathrm{Y}^{0}}}{{\cal
B}_{\mathrm{X}^{0}}}\times\dfrac{\varepsilon^{\mathrm{tot}}_{\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{Y}^{0}}}{\varepsilon^{\mathrm{tot}}_{\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}^{0}}},$
where $\mathcal{Y}$ are the measured event yields,
$\varepsilon^{\mathrm{tot}}$ are the total efficiencies, excluding the
branching fractions of light mesons and ${\cal B}_{\mathrm{X}^{0}}$(${\cal
B}_{\mathrm{Y}^{0}}$) is the relevant branching ratio of the light meson
$\mathrm{X}^{0}$($\mathrm{Y}^{0}$) to the final state under consideration
[16]. In cases where decays of different types of B mesons are compared, the
ratio of the branching fractions is multiplied by the ratio of the
corresponding b-quark hadronization fractions $f_{\mathrm{d}}/f_{\mathrm{s}}$
[28].
The total efficiencies consist of three components: the geometrical acceptance
of the detector, the reconstruction and selection efficiency and the trigger
efficiency. For the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$ decay, the event yield $\mathcal{Y}$ implies the value
weighted by the selection and reconstruction efficiency from Table 2. Only the
acceptance and trigger efficiencies are included in
$\varepsilon^{\mathrm{tot}}_{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}}$. All efficiency components have been determined using the
simulation and the values are listed in Table 3.
For channels with photons and neutral pions in the final states, the
reconstruction and selection efficiencies are corrected for the difference in
the photon reconstruction between the data and simulation. This correction
factor has been determined by comparing the relative yields of the
reconstructed
$\mathrm{B}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*+}(\mathrm{K}^{*+}\rightarrow\mathrm{K}^{+}\uppi^{0})$ and
$\mathrm{B}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{+}$ decays. The results of these studies are convolved with
the background subtracted photon momentum spectra to give the correction
factor for each channel. The values of the correction factors
($\eta^{\mathrm{corr}}$) are also listed in Table 3.
Table 3: Branching fractions of the intermediate resonances, total efficiencies (excluding the branching fractions of the intermediate resonances), $\varepsilon^{\mathrm{tot}}$, and the photon and $\uppi^{0}$ efficiency correction factors $\eta^{\mathrm{corr}}$ for various channels. For the $\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uprho^{0}$ decay the total efficiency includes only the detector acceptance and trigger efficiencies, as the reconstruction and selection efficiency for this channel has been discussed in Sect. 6. Mode | ${\cal B}~{}\left[\%\right]$ | $\varepsilon^{\mathrm{tot}}~{}~{}\left[\%\right]$ | $\eta^{\mathrm{corr}}~{}~{}\left[\%\right]$
---|---|---|---
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta\left(\upeta\rightarrow\upgamma{}\upgamma\right)$ | $39.31\pm 0.20$ | $0.236\pm 0.006$ | $98.0\pm 7.5$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)$ | $22.74\pm 0.28$ | $0.059\pm 0.002$ | $94.1\pm 7.5$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma\right)$ | $29.3\;\;\pm 0.6\;\;$ | $0.142\pm 0.004$ | $98.0\pm 3.7$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uppi^{+}\uppi^{-}{}\upeta\right)$ | $18.6\;\;\pm 0.3\;\;$ | $0.068\pm 0.003$ | $96.0\pm 7.5$
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upomega\left(\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)$ | $89.2\;\;\pm 0.7\;\;$ | $0.043\pm 0.002$ | $94.1\pm 7.5$
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uprho^{0}\left(\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}\right)$ | $98.90\pm 0.16$ | $12.6\pm 0.5\;\;$ | –
### 7.1 Systematic uncertainties
Most systematic uncertainties cancel in the branching fraction ratios, in
particular, those related to the muon and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ reconstruction and
identification. For the final states with photons the largest systematic
uncertainty is related to the efficiency of $\uppi^{0}$/$\upgamma$
reconstruction and identification, as described above. The uncertainties of
the applied corrections reflect simulation statistics, and are taken as
systematic uncertainties on the branching fractions ratios.
Another systematic uncertainty is due to the charged particle reconstruction
efficiency which has been studied through a comparison between data and
simulation. For the ratios where this does not cancel exactly, the
corresponding systematic uncertainty is taken to be 1.8% per pion [29].
The systematic uncertainty related to the trigger efficiency has been obtained
by comparison of the trigger efficiency ratios in data and simulation for the
high yield decay mode
$\mathrm{B}^{\pm}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{\pm}$ with similar kinematics and the same trigger
requirements [30]. This uncertainty is taken to be 1.1$\%$.
In the ratios where decays of B mesons of different types are compared
($\mathrm{B}^{0}$ or $\mathrm{B}^{0}_{\mathrm{s}}$), knowledge of the
hadronization fraction ratio $f_{\mathrm{d}}/f_{\mathrm{s}}$ is required. The
measured value of this ratio [28] has an asymmetric uncertainty of
${}^{+7.9}_{-7.5}\%$.
Systematic uncertainties related to the fit model are estimated using a number
of alternative models for the description of the invariant mass distributions.
For the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}$ decays the tested alternatives include a fit without
the $\mathrm{B}^{0}$ component, a fit with the means of the Gaussians fixed to
the nominal B meson masses, a fit with the width of the Gaussians fixed to the
expected mass resolutions from simulation and substitution of the exponential
background hypothesis with first- and second-order polynomials. This
uncertainty is calculated for the ratios of the event yields. For each
alternative fit model the ratio of the event yields is calculated and the
systematic uncertainty is then determined as the maximum deviation of this
ratio from the ratio obtained with the baseline model.
A similar study is performed for the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ channel. As the fit with one Gaussian function is the baseline
model in this case, here the alternative model is a fit with two Gaussian
functions (allowing a possible $\mathrm{B}^{0}_{\mathrm{s}}$ signal).
In the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$ case, an alternative model replaces the Zou-Bugg
$\mathrm{f}_{0}(500)$ term with a Breit-Wigner shape. The mass and width of
the broad $\mathrm{f}_{0}(500)$ state are not well known. The mass measured by
various experiments varies in a range between 400 and 1200
$\mathrm{MeV}/c^{2}$ and the measured width ranges between 600 and 1000
$\mathrm{MeV}/c^{2}$ [16]. Therefore, the $\mathrm{f}_{0}(500)$ parameters are
varied in this range and the $\uprho^{0}$ yield is determined. Again, the
maximum deviation from the baseline model is treated as the systematic
uncertainty of the fit.
The uncertainties related to the knowledge of the branching fractions of
$\upeta$, $\upeta^{\prime}$, $\uppi^{0}$ and $\upomega$ decays are taken from
Ref. [16]. Other systematic uncertainties, such as those related to the
selection criteria are negligible. The systematic uncertainties are summarized
in Tables 4 and 5. The total systematic uncertainties are estimated using a
simulation technique (see Sect. 7.2).
Table 4: Relative systematic uncertainties for ratios of the branching fractions ($\mathcal{R}$) for the $\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{(\prime)}$ channels $\left[\%\right]$. Parameter | $\mathcal{R}^{\upeta\rightarrow\upgamma{}\upgamma}_{\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}}$ | $\mathcal{R}^{\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}}_{\upeta\rightarrow\upgamma{}\upgamma}$ | $\mathcal{R}^{\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}}_{\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}}$ | $\mathcal{R}^{\upeta^{\prime}\rightarrow\uprho^{0}\gamma}_{\upeta\rightarrow\upgamma{}\upgamma}$ | $\mathcal{R}^{\upeta^{\prime}\rightarrow\uprho^{0}\gamma}_{\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}}$ | $\mathcal{R}^{\upeta^{\prime}\rightarrow\uprho^{0}\gamma}_{\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}}$
---|---|---|---|---|---|---
$\upeta_{\mathrm{corr}}$ | – | – | – | $3.8$ | $3.9$ | $3.9$
$\uppi^{\pm}$ reco | $2\times 1.8$ | $2\times 1.8$ | – | $2\times 1.8$ | – | –
Trigger | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1
Fit function | ${}^{+3.7}_{-0.0}$ | ${}^{+9.9}_{-0.0}$ | ${}^{+1.3}_{-5.6}$ | ${}^{+3.4}_{-0.0}$ | $<0.1$ | ${}^{+0.0}_{-2.8}$
${\cal B}\left(\upeta,\upeta^{\prime},\upomega\right)$ | 1.3 | 1.7 | 2.0 | 2.1 | 1.8 | 2.6
Table 5: Systematic uncertainties for ratios of the branching fractions ($\mathcal{R}$) relative to $\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\uprho^{0}$ $\left[\%\right]$. Parameter | $\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta\rightarrow\upgamma{}\upgamma}_{\mathrm{B}^{0},\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}}$ | $\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}}_{\mathrm{B}^{0},\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}}$ | $\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}\rightarrow\uprho^{0}\gamma}_{\mathrm{B}^{0},\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}}$ | $\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}\rightarrow\upeta\uppi^{+}\uppi^{-}}_{\mathrm{B}^{0},\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}}$ | $\mathcal{R}^{\mathrm{B}^{0},\upomega\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}}_{\mathrm{B}^{0},\uprho^{0}\rightarrow\uppi^{+}\uppi^{-}}$
---|---|---|---|---|---
$\upeta_{\mathrm{corr}}$ | $7.6$ | $8.0$ | $3.8$ | $7.8$ | $8.0$
$\uppi^{\pm}$ reco | $2\times 1.8$ | – | – | – | –
Trigger | 1.1 | 1.1 | 1.1 | 1.1 | 1.1
Fit function | ${}^{+5.1}_{-3.7}$ | ${}^{+5.0}_{-4.3}$ | ${}^{+5.0}_{-5.7}$ | ${}^{+5.0}_{-8.7}$ | ${}^{+6.4}_{-8.8}$
${\cal B}\left(\upeta,\upeta^{\prime},\upomega\right)$ | 0.5 | 1.2 | 2.1 | 1.6 | 0.8
### 7.2 Results
The final ratios
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0}_{\mathrm{s}},\upeta}$,
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\left(\prime\right)}}_{\mathrm{B}^{0},\uprho^{0}}$
and $\mathcal{R}^{\mathrm{B}^{0},\upomega}_{\mathrm{B}^{0},\uprho^{0}}$ are
determined using a procedure that combines $\chi^{2}$-minimization with
constraints and simplified simulation. First, the $\chi^{2}$ is minimized
$\chi^{2}=\sum_{i}\chi^{2}_{i},$
where the sum is performed over the six measured event yields for the six
different modes:
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\upgamma{}\upgamma\right)$,
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\pi^{0}\right)$,
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma\right)$,
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\upeta{}\uppi^{+}\uppi^{-}\right)$,
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ and
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$, and
$\chi^{2}_{i}=\frac{\left(x-\mathcal{Y}_{i}\right)^{2}}{\sigma^{2}_{\mathcal{Y}_{i}}}$.
In this procedure the following constraints are imposed
$\displaystyle\dfrac{\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\upgamma{}\upgamma\right)}}{\varepsilon_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\upgamma{}\upgamma\right)}\times{\cal
B}\left(\upeta\rightarrow\upgamma{}\upgamma\right)}$ $\displaystyle=$
$\displaystyle\dfrac{\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)}}{\varepsilon_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)}\times{\cal
B}\left(\upeta\rightarrow\uppi^{+}\uppi^{-}{}\uppi^{0}\right)},$
$\displaystyle\dfrac{\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma\right)}}{\varepsilon_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma\right)}\times{\cal
B}\left(\upeta^{\prime}\rightarrow\uprho^{0}{}\upgamma\right)}$
$\displaystyle=$
$\displaystyle\dfrac{\mathcal{Y}_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\upeta{}\uppi^{+}\uppi^{-}\right)}}{\varepsilon_{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}\left(\upeta^{\prime}\rightarrow\upeta{}\uppi^{+}\uppi^{-}\right)}\times{\cal
B}\left(\upeta^{\prime}\rightarrow\upeta{}\uppi^{+}\uppi^{-}\right)}.$
The ratios
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0}_{\mathrm{s}},\upeta}$,
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\left(\prime\right)}}_{\mathrm{B}^{0},\uprho^{0}}$
and $\mathcal{R}^{\mathrm{B}^{0},\upomega}_{\mathrm{B}^{0},\uprho^{0}}$ are
determined using the event yields obtained from the minimization procedure.
For this determination the efficiencies $\varepsilon_{i}$ have been varied
using a simplified simulation taking into account correlations between the
various components where appropriate. As both the $\chi^{2}$ and the ratios
$\mathcal{R}$ depend only on the ratios of efficiencies, systematic
uncertainties are minimized. The remaining systematic uncertainties have been
taken into account as uncertainties in the efficiency ratios. In total,
$10^{6}$ simulated experiments with different settings of $\varepsilon_{i}$
have been performed. The symmetric 68% intervals have been assigned as the
systematic uncertainty.
The obtained ratios $\mathcal{R}$ are
$\displaystyle\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0}_{\mathrm{s}},\upeta}$
$\displaystyle=$ $\displaystyle 0.90\pm 0.09\,^{+0.06}_{-0.02},$
$\displaystyle\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta}_{\mathrm{B}^{0},\uprho^{0}}$
$\displaystyle=$ $\displaystyle 3.75\pm
0.31\,^{+0.30}_{-0.40}\times\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right),$
$\displaystyle\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0},\uprho^{0}}$
$\displaystyle=$ $\displaystyle 3.38\pm
0.30\,^{+0.14}_{-0.36}\times\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right),$
$\displaystyle\mathcal{R}^{\mathrm{B}^{0},\upomega}_{\mathrm{B}^{0},\uprho^{0}}$
$\displaystyle=$ $\displaystyle 0.89\pm 0.19\,^{+0.07}_{-0.13},$
where the first uncertainty is statistical and the second is systematic.
## 8 Summary
With 1.0 fb-1 of data, collected in 2011 with the LHCb detector, the first
evidence for the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega$ decay has been found, and its branching fraction, normalized
to that of the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$ decay, is measured to be
$\dfrac{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega)}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}=0.89\pm
0.19\,(\mathrm{stat})\,^{+0.07}_{-0.13}\,(\mathrm{syst}).$
Multiplying by the known value of ${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})=(2.7\pm 0.4)\times 10^{-5}$ [22], the absolute value of the
branching fraction is
${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upomega)=(2.41\pm
0.52\,(\mathrm{stat})\,^{+0.19}_{-0.35}\,(\mathrm{syst})\pm 0.36\,({\cal
B}_{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}}))\times 10^{-5}.$
Using the same dataset, the ratio of the branching fractions of
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$ decays has been measured. As each of the decays has
been reconstructed in two final states, the resulting ratio has been
calculated through an averaging procedure to be
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0}_{\mathrm{s}},\upeta}=\dfrac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}=0.90\pm
0.09\,(\mathrm{stat})\,^{+0.06}_{-0.02}\,(\mathrm{syst}).$
This result is consistent with the previous Belle measurement of
$\mathcal{R}^{\mathrm{B}^{0}_{\mathrm{s}},\upeta^{\prime}}_{\mathrm{B}^{0}_{\mathrm{s}},\upeta}~{}=~{}0.73~{}\pm~{}0.14$
[3], but is more precise. Assuming that the contribution from the purely
gluonic component is negligible, this ratio corresponds to a value of the
$\upeta-\upeta^{\prime}$ mixing phase of
$\upphi_{\mathrm{P}}=\left(45.5\,^{+1.8}_{-1.5}\right)^{\circ}$. The branching
fractions of the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime}$ decays have been determined by normalization to the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}$ decay branching fraction, and using the known value of
$f_{\mathrm{s}}/f_{\mathrm{d}}=0.267\,^{+0.021}_{-0.020}$ [28] their ratios
are
$\displaystyle\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}$ $\displaystyle=$ $\displaystyle\>14.0\pm
1.2\,(\mathrm{stat})\,^{+1.1}_{-1.5}\,(\mathrm{syst})\,^{+1.1}_{-1.0}\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right),$
$\displaystyle\frac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})}$ $\displaystyle=$ $\displaystyle\>12.7\pm
1.1\,(\mathrm{stat})\,^{+0.5}_{-1.3}\,(\mathrm{syst})\,^{+1.0}_{-0.9}\left(\frac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right).$
When multiplying by the known value of ${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0})$, the branching fractions are measured as
$\displaystyle{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta)$ $\displaystyle=$ $\displaystyle\left(3.79\pm
0.31\,(\mathrm{stat})\,^{+0.20}_{-0.41}\,(\mathrm{syst})\,^{+0.29}_{-0.27}\left(\tfrac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right)\pm
0.56\,({\cal
B}_{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}})\right)\times 10^{-4},$ $\displaystyle{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{\prime})$ $\displaystyle=$ $\displaystyle\left(3.42\pm
0.30\,(\mathrm{stat})\,^{+0.14}_{-0.35}\,(\mathrm{syst})\,^{+0.26}_{-0.25}\left(\tfrac{f_{\mathrm{d}}}{f_{\mathrm{s}}}\right)\pm
0.51\,({\cal
B}_{\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\uprho^{0}})\right)\times 10^{-4}.$
The branching fractions measured here correspond to the time integrated
quantities, while theory predictions usually refer to the branching fractions
at $t=0$. Special care needs to be taken when the
$\mathrm{B}^{0}_{\mathrm{s}}$ and $\mathrm{B}^{0}$ decays are compared at the
amplitude level, corresponding to the branching ratio at $t=0$ [31]. Since the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upeta^{(\prime)}$
final states are $C\\!P$-eigenstates, the size of this effect can be as large
as 10%, and can be corrected for using input from theory or determined from
effective lifetime measurements [31]. With a larger dataset such measurements,
as well as studies of $\upeta-\upeta^{\prime}$ mixing and measurements of
$C\\!P$ asymmetries in the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upeta^{(\prime)}$ modes will be possible.
## Acknowledgements
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 (The 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.
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|
arxiv-papers
| 2012-10-09T15:04:33 |
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T.\n 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, M. Britsch, T.\n Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A.\n Bursche, J. Buytaert, S. Cadeddu, 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, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, 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, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F.\n Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, C.\n Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V.\n Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.\n C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T.\n Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K.\n Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, D.\n Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\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, M. Kaballo,\n S. Kandybei, M. Karacson, M. Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel,\n T. Ketel, A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, I. Komarov, V.\n Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, 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, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac\n Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M.\n Maino, S. Malde, 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, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, 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. K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K. Sobczak, F. J.\n P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, C.\n Vo{\\ss}, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1210.2631"
}
|
1210.2645
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-284 LHCb-PAPER-2012-020 October 9, 2012
First observation of the decay $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
A discovery of the rare decay $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
presented. This decay is observed for the first time, with 5.2 $\sigma$
significance. The observation is made using $pp$ collision data, corresponding
to an integrated luminosity of 1.0 fb-1, collected with the LHCb detector. The
measured branching fraction is (2.3 $\pm$ 0.6 (stat.) $\pm$ 0.1
(syst.))$\times 10^{-8}$, and the ratio of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is measured to be 0.053 $\pm$ 0.014
(stat.) $\pm$ 0.001 (syst.).
Published in the Journal of High Energy Physics
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, C. Bauer10, Th. Bauer38, A. Bay36, J. Beddow48, I.
Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R.
Bernet37, M.-O. Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N.
Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-Montells33, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D.
Craik45, S. Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, D. Esperante Pereira34, A. Falabella14,e, C. Färber11, G. Fardell47,
C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M.
Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C.
Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M.
Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33,
G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, R.S. Huston12, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U.
Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52,
A. Martín Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z.
Mathe12, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8,
G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C.
Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro33,
W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I.
Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos
Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, J. Rouvinet36, T. Ruf35,
H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48,
B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C.
Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The ratio of Cabibbo-Kobayshi-Maskawa matrix [1] elements
$|V_{\text{td}}|/|V_{\text{ts}}|$ has been measured in $B$ mixing processes,
where it is probed in box diagrams through the ratio of $B^{0}$ and
$B^{0}_{s}$ mixing frequencies [2, 3, 4, 5]. The ratio of these matrix
elements has also been measured using the ratio of branching fractions of
$b\\!\rightarrow s\gamma$ and $b\\!\rightarrow d\gamma$ decays, where
radiative penguin diagrams mediate the transition [6, 7, 8]. These
measurements of $|V_{\text{td}}|/|V_{\text{ts}}|$ are consistent, within the
(dominant) $\sim$10% uncertainty on the determination from radiative decays.
The decays $b\\!\rightarrow s\mu^{+}\mu^{-}$ and $b\\!\rightarrow
d\mu^{+}\mu^{-}$ offer an alternative way of measuring
$|V_{\text{td}}|/|V_{\text{ts}}|$ which is sensitive to different classes of
operators than the radiative decay modes [9]. These
$b\\!\rightarrow(s,d)\mu^{+}\mu^{-}$ transitions are flavour-changing neutral
current processes which are forbidden at tree level in the Standard Model
(SM). In the SM, the branching fractions for $b\\!\rightarrow
d\ell^{+}\ell^{-}$ transitions are suppressed relative to $b\\!\rightarrow
s\ell^{+}\ell^{-}$ processes by the ratio
$|V_{\text{td}}|^{2}/|V_{\text{ts}}|^{2}$. This suppression does not
necessarily apply to models beyond the SM, and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decays111Charge conjugation is
implicit throughout this paper. may be more sensitive to the effect of new
particles than $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays. In the SM,
the ratio of branching fractions for these exclusive modes
$R\equiv{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}~{}/~{}{{\cal
B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})}$ (1)
is given by $R=V^{2}f^{2}$, where $V=|V_{\text{td}}|/|V_{\text{ts}}|$ and $f$
is the ratio of the relevant form factors and Wilson coefficients, integrated
over the relevant phase space. A difference between the measured value of $R$
and $V^{2}f^{2}$ would indicate a deviation from the minimal flavour violation
hypothesis [10, 11], and would rule out certain approximate flavour symmetry
models [12].
No $b\\!\rightarrow d\ell^{+}\ell^{-}$ transitions have previously been
detected, and the observation of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay would therefore be the first
time such a process has been seen. The predicted SM branching fraction for
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is (2.0 $\pm$ 0.2)$\times 10^{-8}$
[13]. The most stringent limit to date is $\cal
B$($B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$) $<6.9\times 10^{-8}$ at 90%
confidence level [14]. The analogous $b\\!\rightarrow s\ell^{+}\ell^{-}$
decay, $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, has been observed with a
branching fraction of (4.36$~{}\pm~{}$0.15$~{}\pm~{}$0.18) $\times~{}10^{-7}$
[15].
This paper describes the search for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay using $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected with the LHCb
detector. The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction is
measured with respect to that of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$, and the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is also determined.
The LHCb detector [16] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$. The experiment is designed for the study of
particles containing $b$ or $c$ quarks. The apparatus includes a high
precision tracking system, consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, and a large-area silicon-strip
detector located upstream of a dipole magnet. The dipole magnet has a bending
power of about $4{\rm\,Tm}$. Three stations of silicon-strip detectors and
straw drift-tubes are placed downstream of the magnet. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from 0.4% at momenta
of 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The tracking system gives an impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with a high transverse
momentum ($p_{\rm T}$). Charged hadrons are identified using two ring-imaging
Cherenkov detectors. Photon, electron and hadron candidates are identified by
a calorimeter system consisting of scintillating-pad and preshower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and either
multi-wire proportional chambers or triple gaseous electron multipliers.
In the present analysis, events are first required to have passed a hardware
trigger which selects high-$p_{\rm T}$ single muons or dimuons. In the first
stage of the subsequent software trigger, a single high impact parameter and
high-$p_{\rm T}$ track is required. In the second stage of the software
trigger, events are reconstructed and then selected for storage based on
either the (partially) reconstructed $B$ candidate or the dimuon candidate
[17, 18].
To produce simulated samples of signal and background decays, $pp$ collisions
are generated using Pythia 6.4 [19] with a specific LHCb configuration [20].
Decays of hadronic particles are described by the EvtGen package [21] in which
final state radiation is generated using Photos [22]. The interaction of the
generated particles with the detector and the detector response are
implemented using the Geant4 toolkit [23, *Agostinelli:2002hh], as described
in Ref. [25].
The small branching fractions of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ signal decays necessitate good control of the backgrounds
and the use of suitably constrained models to fit the invariant-mass
distributions. The decay
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$ (hereafter denoted
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$) is
used to extract both the shape of the signal mass peaks and, in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ case, the invariant mass
distribution of the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
events. These misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ events
form the main residual background after the application of the selection
requirements.
## 2 Event selection
The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates are selected by combining pairs of oppositely
charged muons with a charged pion or kaon. The selection includes requirements
on the impact parameters of the final-state particles and $B$ candidate, the
vertex quality and displacement of the $B$ candidate, particle identification
(PID) requirements on the muons and a requirement that the $B$ candidate
momentum vector points to one of the primary vertices in the event. The rate
of events containing more than one reconstructed candidate is 1 in
$\sim$20,000 for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$. No restriction is therefore placed on the number of candidates
per event.
The pion identification requirements select a sample of pions with an
efficiency of $\sim$70% and a kaon rejection of 99%. The kaon identification
requirements allow the selection of a mutually exclusive sample with similar
efficiencies. The muon identification requirements have an efficiency of
$\sim$80%, with a pion rejection of $\sim$99.5%. The PID requirements have a
momentum dependent efficiency which is measured from data, in bins of
momentum, pseudorapidity and track multiplicity. The efficiency of the hadron
PID requirements is measured from a sample of
$D^{*+}\rightarrow(D^{0}\rightarrow K^{-}\pi^{+})\pi^{+}$ candidates that
allows the hadrons to be unambiguously identified based on their kinematics.
The muon PID efficiencies are measured using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates, using a tag and probe method.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
resonances, where ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},{\psi{(2S)}}\rightarrow\mu^{+}\mu^{-}$, are excluded using a veto on
the dimuon mass. This veto has a total width of 200 (150)
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ($\psi(2S)$) mass [26], and
takes into account the radiative tail of these decays. Candidates where the
dimuon mass is poorly measured have a correlated mismeasurement in the
$h\mu\mu$ mass. The veto therefore includes a component which shifts with
$h\mu\mu$ mass to exclude such candidates. Several other backgrounds are
considered: combinatorial backgrounds, where the particles selected do not
originate from a single decay; peaking backgrounds, where a single decay is
selected but with one or more particles misidentified; and partially
reconstructed backgrounds, where one or more final-state particles from a $B$
decay are not reconstructed. These backgrounds are each described below.
### 2.1 Combinatorial backgrounds
A boosted decision tree (BDT) [27] which employs the AdaBoost algorithm[28] is
used to separate signal candidates from the combinatorial background.
Kinematic and geometric properties of the $B^{+}$ candidate and final state
particles, $B^{+}$ candidate vertex quality and final state particle track
quality are input variables to the BDT.
The BDT is trained on a simulated $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal sample, and a background sample taken from sidebands in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ invariant mass distributions. These invariant masses are
denoted $M_{\pi^{+}\mu^{+}\mu^{-}}$ and $M_{K^{+}\mu^{+}\mu^{-}}$,
respectively. The background sample consists of 20% of the candidates with
$M_{\pi^{+}\mu^{+}\mu^{-}}$ or $M_{K^{+}\mu^{+}\mu^{-}}$ $>$ 5500
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This sample is not used for any of
the subsequent analysis. Signal candidates are required to have a BDT output
which exceeds a set value. This value is determined by simulating an ensemble
of datasets with the expected signal and background yields, and choosing the
cut value which gives the best statistical significance for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal yield. The same method is
used to select the optimal set of PID requirements. The BDT output
distribution for simulated $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ events
and for mass sideband candidates is shown in Fig. 1. A cut on the BDT output
$>$ 0.325 reduces the expected combinatorial background from 652 $\pm$ 11 to 9
$\pm$ 2 candidates in a $\pm$60 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
window around the nominal $B$ mass, while retaining 68% of signal events.
Assuming the SM branching fraction and the single event sensitivity defined in
Sect. 4, 21 $\pm$ 3 $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal events
are expected in the data sample.
Figure 1: BDT output distribution for simulated
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ events (black solid line) and
candidates taken from the mass sidebands in the data (red dotted line). Both
distributions are normalised to unit area. The vertical line indicates the
chosen cut value of 0.325.
### 2.2 Peaking and partially reconstructed backgrounds
Backgrounds from fully reconstructed $B^{+}$ decays with one or more
misidentified particles have a peaking mass structure. After applying the PID
requirements, the fraction of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
candidates misidentified as $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
0.9%, giving a residual background expectation of 6.2 $\pm$ 0.3 candidates.
This expectation is computed by weighting $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates, isolated using a kaon PID requirement,
according to the PID efficiency obtained from the $D^{*+}$ calibration sample.
The only other decay found to give a significant peaking background in the
search for $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
$B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$, where both a $\pi^{+}$ and a
$\pi^{-}$ are misidentified as muons. For $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ decays, the only significant peaking background is
$B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$, which includes the contribution
from $B^{+}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\rightarrow K^{+}\pi^{-})\pi^{+}$.
The expected background levels from $B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$
($B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$) decays are computed to be 0.39
$\pm$ 0.04 (1.56 $\pm$ 0.16) residual background candidates, using simulated
events.
Backgrounds from decays that have one or more final state particles which are
not reconstructed have a mass below the nominal $B$ mass, and do not extend
into the signal window. However, in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ case, these backgrounds overlap
with the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ component
described above, and must therefore be included in the fit. In the
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ case such partially reconstructed
backgrounds are negligible.
### 2.3 Control channels
The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay candidates are isolated by
replacing the pion PID criteria with a requirement to select kaons. In
addition, instead of the dimuon mass vetoes described above, the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates are required to have dimuon mass within $\pm$50
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass (the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass resolution is 14.5
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The remainder of the selection is
the same as that used for $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$. This
minimises the systematic uncertainty on the ratio of branching fractions,
although the selection is considerably tighter than that which would give the
lowest statistical uncertainty on the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ event yield. The
${B^{+}\rightarrow(J\\!/\\!\psi\rightarrow\mu^{+}\mu^{-})\pi^{+}}$ candidates
(denoted ${B^{+}\rightarrow J\\!/\\!\psi\pi^{+}}$), which are discussed below,
are selected using the same BDT, the pion PID criteria, and the above window
on the dimuon invariant mass. There is no significant peaking background for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
For $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays the only significant peaking background is misidentified
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events.
## 3 Signal yield determination
The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yields
are determined from a simultaneous unbinned maximum likelihood fit to four
invariant mass distributions which contain:
1. 1.
Reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates;
2. 2.
Reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates, with the kaon attributed to have the pion mass;
3. 3.
Reconstructed $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates; and
4. 4.
Reconstructed $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates.
The signal probability density functions (PDFs) for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
modes are modelled with the sum of two Gaussian functions. The PDFs for all of
these decay modes share the same mean, widths and fraction of the total PDF
between the two Gaussians. The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ PDF
is adjusted for the difference between the widths of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
distributions, which is observed to be at the percent level in simulation. The
peaking backgrounds described in Sect. 2.2 are taken into account in the fit
by including PDFs with shapes determined from simulation. The combinatorial
backgrounds are modelled with a single exponential PDF, with the exponent
allowed to vary independently for each distribution. The partially
reconstructed candidates are modelled using a PDF consisting of an exponential
distribution cut-off at a threshold mass, with the transition smeared by the
experimental resolution. The shape parameters are again allowed to vary
independently for each distribution. The misidentified
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates are modelled with a Crystal Ball function [29], as it describes the
shape well. In order to describe the relevant background components for
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, the fit is performed in the mass
range 4900 $<$ $M_{\pi^{+}\mu^{+}\mu^{-}}$ $<$ 7000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To avoid fitting the partially
reconstructed background for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, which
is irrelevant for the analysis, the fit is performed in the mass range 5170
$<$ $M_{K^{+}\mu^{+}\mu^{-}}$ $<$ 7000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
### 3.1 Reconstructed
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates
The reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates are shown in the $M_{K^{+}\mu^{+}\mu^{-}}$
distribution in Fig. 2(a). The fitted
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield
is 106,230 $\pm$ 330. This large event yield determines the lineshape for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ signal distributions, and provides the normalisation for
the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction.
### 3.2 Reconstructed
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates with the pion mass hypothesis
The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates reconstructed under the pion mass hypothesis provide the lineshape
for the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates
that are a background to the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal. The equivalent background from
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ in the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ sample is negligible.
Figure 2: Invariant mass distribution for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates under the (a) $K^{+}\mu^{+}\mu^{-}$ and (b) $\pi^{+}\mu^{+}\mu^{-}$
mass hypotheses with the fit projections overlaid. In the legend, “part. reco”
refers to partially reconstructed background. The fit models are described in
the text.
The PID requirements used in the selection have a momentum dependent
efficiency and therefore change the mass distribution of any backgrounds with
candidates that have misidentified particles. In order to correct for this
effect, the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates are reweighted according to the PID efficiencies
derived from data, as described in Sect. 2.2. This adjusts the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
invariant mass distribution to remove the effect of the kaon PID requirement
used to isolate $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$, and to reproduce the effect of the pion PID requirement used to
isolate $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$. In addition, there is a
difference in the lineshapes of the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ invariant mass distributions under
the pion mass hypothesis. This effect arises from the differences between the
two decay modes’ dimuon energy and hadron momentum spectra, and is therefore
corrected by reweighting
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates in terms of these variables. The $M_{\pi^{+}\mu^{+}\mu^{-}}$
distribution after both weighting procedures have been applied is shown in
Fig. 2(b).
### 3.3 Reconstructed $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates
The yield of misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
candidates in the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ invariant mass
distribution is constrained to the expectation given in Sect. 2.2. Performing
the fit without this constraint gives a yield of 5.6 $\pm$ 6.4 misidentified
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates. The yields for the
peaking background components are constrained to the expectations given in
Sect. 2.2. For both the $M_{\pi^{+}\mu^{+}\mu^{-}}$ and
$M_{K^{+}\mu^{+}\mu^{-}}$ distributions, the exponential PDF used to model the
combinatorial background has a step in the normalisation at 5500
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to account for the data used for
training the BDT.
The $M_{\pi^{+}\mu^{+}\mu^{-}}$ and $M_{K^{+}\mu^{+}\mu^{-}}$ distributions
are shown in Figs 3 and 4, respectively. The fit gives a
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal yield of 25.3
${}^{+6.7}_{-6.4}$, and a $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal
yield of 553 ${}^{+24}_{-25}$.
Figure 3: Invariant mass distribution of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates with the fit projection
overlaid (a) in the full mass range and (b) in the region around the $B$ mass.
In the legend, “part. reco.” and “combinatorial” refer to partially
reconstructed and combinatorial backgrounds respectively. The discontinuity at
5500 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is due to the removal of data
used for training the BDT.
Figure 4: Invariant mass distribution of $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates with the fit projection overlaid (a) in the
full mass range and (b) in the region around the $B$ mass. In the legend,
“combinatorial” refers to the combinatorial background.
### 3.4 Cross check of the fit procedure
The fit procedure was cross-checked on
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays, accounting for the background from
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
The resulting fit is shown in Fig. 5. The shape of the combined
${B^{+}\rightarrow J\\!/\\!\psi\pi^{+}}$ and ${B^{+}\rightarrow J\\!/\\!\psi
K^{+}}$ mass distribution is well reproduced. The ${B^{+}\rightarrow
J\\!/\\!\psi K^{+}}$ yield is not constrained in this fit. The fitted yield of
1024 $\pm$ 61 candidates is consistent with the expectation of 958 $\pm$ 31
$\mathrm{(stat.)}$ candidates. This expectation is again computed by weighting
the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates, which are isolated using a kaon PID requirement, according to the
PID efficiency derived from $D^{*+}\rightarrow(D^{0}\rightarrow
K^{-}\pi^{+})\pi^{+}$ events.
Figure 5: Invariant mass distribution of ${B^{+}\rightarrow
J\\!/\\!\psi\pi^{+}}$ candidates with the fit projection overlaid. In the
legend, “part. reco.” and “combinatorial” refer to partially reconstructed and
combinatorial backgrounds respectively. The fit model is described in the
text.
## 4 Determination of branching fractions
The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction is given by
$\displaystyle{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle\frac{{\cal
B}({B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}})}{N_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}}{\frac{\epsilon_{{B^{+}\rightarrow J\\!/\\!\psi
K^{+}}}}{{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}}}N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$
(2) $\displaystyle=$ $\displaystyle\alpha\cdot
N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}\,,$ (3)
where $\cal B$$(X)$, $N_{X}$ and $\epsilon_{X}$ are the branching fraction,
the number of events and the total efficiency, respectively, for decay mode
$X$, and $\alpha$ is the single event sensitivity. The total efficiency
includes reconstruction, trigger and selection efficiencies. The ratio
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ is
determined to be 1.60 $\pm$ 0.01 using simulated events, where the uncertainty
is due to the limited sizes of the simulated samples only. Other sources of
systematic uncertainty are discussed in Sect. 5. The difference in
efficiencies between $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
${B^{+}\rightarrow J\\!/\\!\psi K^{+}}$ events is largely due to the mass
vetoes used to remove the charmonium resonances, and the different PID
requirements. The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$ branching fraction is (6.02 $\pm$
0.20)$\times 10^{-5}$ [26]. Together with the other quantities in Eq. 2, this
gives a single event sensitivity of $\alpha$ = (9.1 $\pm$ 0.1)$\times
10^{-10}$, where the uncertainty is due to the limited sizes of the simulated
samples only.
The ratio of $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ branching fractions is given by
$R=\frac{N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}{N_{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}\frac{\epsilon_{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}\,,$
(4)
where simulated events give ${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$
= 1.15 $\pm$ 0.01.
## 5 Systematic uncertainties
Two sources of systematic uncertainties are considered: those affecting the
determination of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal yields, and those affecting
only the normalisation.
Uncertainties in the shape parameters for the misidentified
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ PDF in the fit are taken into
account by including Gaussian constraints on their values. The most
significant sources of uncertainty in the determination of these shape
parameters arise from the procedure for correcting the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ mass
shape to match that of the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay,
and the correction for the hadron PID requirements. The uncertainty on the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield determined with the fit takes
these shape parameter uncertainties into account, and they are therefore
included in the statistical rather than the systematic uncertainty. These
uncertainties affect the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield at
below the one percent level. None of these effects give rise to any
significant uncertainty for the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
decay.
Uncertainties on the two efficiency ratios
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$
affect the conversion of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield
into a branching fraction, and the measurement of the ratio of branching
fractions $R$. The largest systematic uncertainty on these efficiency ratios
is the choice of form factors used to generate the simulated events. Using an
alternative set of form factors changes the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ efficiency by 3%, and this
difference is taken as a systematic uncertainty. For the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, the alternative form factors are used for both
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, giving a systematic uncertainty of 1.7%. To estimate the
uncertainty arising from the PID efficiency, the ratio of corrected yields
between the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ decay modes is measured, varying the PID requirements. The
largest resulting difference with respect to the nominal value is 1.1%, which
is taken as the systematic uncertainty.
The systematic uncertainty arising from the knowledge of the trigger
efficiency is determined using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates in the data. Taking the events which pass the trigger independently
of the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidate, the fraction of these events which also pass the trigger based on
the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidate provides a determination of the trigger efficiency. The efficiency
determined in this way is compared to that calculated in simulated events
using the same method, and the difference is taken as the systematic
uncertainty. This gives a 1.4% uncertainty on
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$.
For all decays under consideration, there are small differences between the
distributions of some reconstructed quantities in the data and in the
simulated events. These differences are assessed by comparing the
distributions of data and simulated events for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates. The simulation is corrected to match the data where it disagrees,
and the resulting 0.4% difference between the raw and corrected ratio of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ efficiencies is taken as a
systematic uncertainty. The statistical uncertainty from the limited
simulation sample size is 0.7%. When normalising to
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, the
measured $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $J/\psi\rightarrow\mu^{+}\mu^{-}$ branching fractions
contribute an uncertainty of 3.5% to the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction. The systematic
uncertainties are summarised in Table 1.
Table 1: Summary of systematic uncertainties. Source | $\cal B$($B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$) (%) | $\frac{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})}$ (%)
---|---|---
Form factors | 3.0 | 1.7
Trigger efficiency | 1.4 | 1.4
PID performance | 1.1 | 1.1
Data simulation differences | 0.4 | 0.4
Simulation sample size | 0.7 | 0.7
$\cal B$($B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$) | 3.5 | –
Total | 5.0 | 2.6
## 6 Results and conclusion
The statistical significance of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal observed in Fig. 3 is computed from the difference in the minimum log-
likelihood between the signal-plus-background and background-only hypotheses.
Both the statistical and systematic uncertainties on the shape parameters
(which affect the significance) are taken into account. The fitted yield
corresponds to an observation of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay with 5.2 $\sigma$
significance. This is the first observation of a $b\\!\rightarrow
d\ell^{+}\ell^{-}$ transition. Normalising the observed signal to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay,
using the single event sensitivity given in Sect. 4, the branching fraction of
the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay is measured to be
${\cal
B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})=(2.3~{}\pm~{}0.6~{}\mathrm{(stat.)}~{}\pm~{}0.1~{}\mathrm{(syst.)})\times
10^{-8}\,.$
This is compatible with the SM expectation of (2.0 $\pm$ 0.2)$\times 10^{-8}$
[13]. Given the agreement between the present measurement and the SM
prediction, contributions from physics beyond the SM can only modify the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction by a small
amount. A significant improvement in the precision of both the experimental
measurements and the theoretical prediction will therefore be required to
resolve any new physics contributions.
Taking the measured $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ yield and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$,
the ratio of $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ branching fractions is measured to
be
$\frac{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal
B}(B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-})}=0.053~{}\pm~{}0.014~{}\mathrm{(stat.)}~{}\pm~{}0.001~{}\mathrm{(syst.)}\,.$
In order to extract $|V_{\text{td}}|/|V_{\text{ts}}|$ from this ratio of
branching fractions, the SM expectation for the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is calculated using the EvtGen
package [21], which implements the calculation in Ref. [30]. This calculation
has been updated with the expressions for Wilson coefficients and power
corrections from Ref. [31], and formulae for the $q^{2}$ dependence of these
coefficients from Refs. [32, 33]. Using this calculation, and form factors
taken from Ref. [34] (“set II”), the integrated ratio of form factors and
Wilson coefficients is determined to be $f=0.87$. Neglecting theoretical
uncertainties, the measured ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions then gives
$|V_{\text{td}}|/|V_{\text{ts}}|=\frac{1}{f}\sqrt{\frac{{\cal
B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal B}(B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-})}}=0.266~{}\pm~{}0.035~{}\mathrm{(stat.)}~{}\pm~{}0.003~{}\mathrm{(syst.)},$
which is compatible with previous determinations [5, 6, 7, 8]. An additional
uncertainty will arise from the knowledge of the form factors. As an estimate
of the scale of this uncertainty, the “set IV” parameters available in Ref.
[34] change the value of $|V_{\text{td}}|/|V_{\text{ts}}|$ by 5.1%. This
estimate is unlikely to cover a one sigma range on the form factor
uncertainty, and does not take into account additional sources of uncertainty
beyond the form factors. A full theoretical calculation taking into account
such additional uncertainties, which also accurately determines the
uncertainty on the ratio of form factors, would allow a determination of
$|V_{\text{td}}|/|V_{\text{ts}}|$ with comparable precision to that from
radiative penguin decays.
## 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 (The 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.
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|
arxiv-papers
| 2012-10-09T15:40:39 |
2024-09-04T02:49:36.253189
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, V. Balagura, W. Baldini, R. J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J.\n Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M.\n Benayoun, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.\n A. Cowan, D. Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P. N. Y.\n David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H.\n Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, 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, D. Gascon, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S. T. Harnew,\n J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, D. Hill, M.\n Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, R. S.\n 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, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, O. Kochebina, I.\n Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, 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, Y.\n Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J.\n Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave,\n U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, 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. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus,\n F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi,\n A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, M. Smith, K. Sobczak, F. J.\n P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U.\n Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, R.\n Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D.\n Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Greg Ciezarek",
"url": "https://arxiv.org/abs/1210.2645"
}
|
1210.2770
|
# Siquieros accidental painting technique: a fluid mechanics point of view
Sandra Zetina1 and Roberto Zenit2
1. Instituto de Investigaciones Estéticas
2.Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México
Cd. Universitaria, México D.F., 04510
MÉXICO
###### Abstract
This is an entry for the Gallery of Fluid Motion of the 65st Annual Meeting of
the APS-DFD ( fluid dynamics video ). This video shows an analysis of the
‘accidental painting’ technique developed by D.A. Siqueiros, a famous Mexican
muralist. We reproduced the technique that he used: pouring layers of paint of
different colors on top of each other. We found that the layers mix, creating
aesthetically pleasing patterns, as a result of a Rayleigh-Taylor instability.
Due to the pigments used to give paints their color, they can have different
densities. When poured on top of each other, if the top layer is denser than
the lower one, the viscous gravity current undergoes unstable as it spread
radially. We photograph the process and produced slowed-down video to
visualize the process.
## 1 Introduction
In the spring of 1936, the famous Mexican muralist David Alfaro Siqueiros [1]
organized an experimental painting workshop in New York: a group of artists
focused in developing painting techniques through empirical experimentation of
modern and industrial materials and tools. Among the young artists attending
the workshop was Jackson Pollock [2]. They tested different lacquers and a
number of experimental techniques. One of the techniques, named by Siqueiros
as a “controlled accident,” consisted in pouring layers of paint of different
colors on top of each other. After a brief time, the paint from the lower
layer emerged from bottom to top creating a relatively regular pattern of
blobs. This technique led to the creation of explosion-inspired textures and
catastrophic images. We conducted an analysis of this process. We
experimentally reproduced the patterns “discovered” by Siqueiros and analyzed
the behavior of the flow. We found that the flow is driven by the well-known
Rayleigh Taylor instability [3]: different colors paints have different
densities; a heavy layer on top of a light one is an unstable configuration.
The blobs and plumes that result from the instability create the aesthetically
pleasing patterns. We discuss the importance of fluid mechanics in artistic
creation.
## 2 Experimental Conditions
We used the same type of paints that Siqueiros used. Their physical properties
are shown in Table 1. With these combinations, both the Reynolds and Artwood
numbers are small. The paint layers where dripping on top of a horizontal
glass sheet. Volumes of approximately 50 and 25 ml were deposited for the
bottom and bottom layers, respectively. The flow was photographed with a
computer-controlled HD digital camera (FinePix Si Pro, Fujifilm), such that
long term time sequences could be obtained. The time interval between photos
was either 500 or 143 ms.
Fluid | density, kg/m3 | viscosity Pa | Ar | Re
---|---|---|---|---
white paint | 1110 | 2.5 | 5.1$\times 10^{-2}$ | 6.9$\times 10^{-5}$
black paint | 1002 | 11.7 | |
yellow paint | 1080 | 3.6 | 3.4$\times 10^{-2}$ | 1.1$\times 10^{-4}$
transparent lacquer | 1008 | 12.9 | |
Table 1: Properties of the fluids and fluid combinations used.
$Ar=(\rho_{1}-\rho_{2})/(\rho_{1}+\rho_{2})$, is the Artwood number and
$Re=U_{f}H\rho/\mu$, is the Reynolds number, where $U_{f}$ and $H$ and the
velocity of the front and the thickness of the layer, respectively.
To our knowledge an formal analysis of this process has not been studied to
date.
## 3 Videos
Our video contributions can be found at:
* •
Video 1, mpeg4, full resolution
* •
Video 2, mpeg2, low resolution
## References
* [1] P. Stein, _Siqueiros: His Life and Works_. (Intl. Pub., 1994).
* [2] E. G. Landau, _Jackson Pollock_. (Abrams, 2010).
* [3] F. Charru, _Hydrodynamic Instabilities_. (Cambridge University Press, 2011).
|
arxiv-papers
| 2012-10-09T22:14:36 |
2024-09-04T02:49:36.266884
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sandra Zetina and Roberto Zenit",
"submitter": "Roberto Zenit",
"url": "https://arxiv.org/abs/1210.2770"
}
|
1210.2806
|
# Risk-Sensitive Mean Field Games
Hamidou Tembine, Quanyan Zhu, Tamer Başar We are grateful to many seminar and
conference participants such as those in the Workshop on Mean Field Games
(Rome, Italy, May 2011) and IFAC World Congress (Milan, Italy, August-
September 2011) for their valuable comments and suggestions on the preliminary
versions of this work. An earlier version of this work appeared in the
Proceedings of 18th IFAC World Congress (Milan, Italy; August 29 - September
2, 2011). Research of second and third authors was supported in part by the
U.S. Air Force Office of Scientific Research (AFOSR) under the MURI Grant
FA9550-10-1-0573. The first author acknowledges the financial support from the
CNRS mean-field game project “MEAN-MACCS”. H. Tembine is with Ecole Supérieure
d’Electricité (SUPELEC), France. E-mail: tembine@ieee.org Q. Zhu and T. Başar
are with Coordinated Science Laboratory and the Department of Electrical and
Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL,
USA. {zhu31, basar1}@illinois.edu
###### Abstract
In this paper, we study a class of risk-sensitive mean-field stochastic
differential games. We show that under appropriate regularity conditions, the
mean-field value of the stochastic differential game with exponentiated
integral cost functional coincides with the value function described by a
Hamilton-Jacobi-Bellman (HJB) equation with an additional quadratic term. We
provide an explicit solution of the mean-field best response when the
instantaneous cost functions are log-quadratic and the state dynamics are
affine in the control. An equivalent mean-field risk-neutral problem is
formulated and the corresponding mean-field equilibria are characterized in
terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-
Kolmogorov equations, and HJB equations. We provide numerical examples on the
mean field behavior to illustrate both linear and McKean-Vlasov dynamics.
## I Introduction
Most formulations of mean-field (MF) models such as anonymous sequential
population games [19, 7], MF stochastic controls [17, 15, 36], MF
optimization, MF teams [33], MF stochastic games [34, 1, 33, 31], MF
stochastic difference games [14], and MF stochastic differential games [23,
13, 32] have been of risk-neutral type where the cost (or payoff, utility)
functions to be minimized (or to be maximized) are the expected values of
stage-additive loss functions.
Not all behavior, however, can be captured by risk-neutral cost functions. One
way of capturing risk-seeking or risk-averse behavior is by exponentiating
loss functions before expectation (see [2, 18] and the references therein).
The particular risk-sensitive mean-field stochastic differential game that we
consider in this paper involves an exponential term in the stochastic long-
term cost function. This approach was first taken by Jacobson in [18], when
considering the risk-sensitive Linear-Quadratic-Gaussian (LQG) problem with
state feedback. Jacobson demonstrated a link between the exponential cost
criterion and deterministic linear-quadratic differential games. He showed
that the risk-sensitive approach provides a method for varying the robustness
of the controller and noted that in the case of no risk, or risk-neutral case,
the well known LQR solution would result (see, for follow-up work on risk-
sensitive stochastic control problems with noisy state measurements, [35, 6,
27]).
In this paper, we examine the risk-sensitive stochastic differential game in a
regime of large population of players. We first present a mean-field
stochastic differential game model where the players are coupled not only via
their risk-sensitive cost functionals but also via their states. The main
coupling term is the mean-field process, also called the occupancy process or
population profile process. Each player reacts to the mean field or a subset
of the mean field generated by the states of the other players in an area, and
at the same time the mean field evolves according to a controlled Kolmogorov
forward equation.
Our contribution can be summarized as follows. Using a particular structure of
state dynamics, we derive the mean-field limit of the individual state
dynamics leading to a non-linear controlled macroscopic McKean-Vlasov
equation; see [21]. Combining this with a limiting risk-sensitive cost
functional, we arrive at the mean-field response framework, and establish its
compatibility with the density distribution using the controlled Fokker-
Planck-Kolmogorov forward equation. The mean-field equilibria are
characterized by coupled backward-forward equations. In general a backward-
forward system may not have solution (a simple example is provided in section
III-D). An explicit solution of the Hamilton-Jacobi-Bellman (HJB) equation is
provided for the affine-exponentiated-Gaussian mean-field problem. An
equivalent risk-neutral mean-field problem (in terms of value function) is
formulated and the solution of the mean-field game problem is characterized.
Finally, we provide a sufficiency condition for having at most one smooth
solution to the risk-sensitive mean field system in the local sense.
The rest of the paper is organized as follows. In Section II, we present the
model description. We provide an overview of the mean-field convergence result
in Section II-A. In Section III, we present the risk-sensitive mean-field
stochastic differential game formulation and its equivalences. In Section IV,
we analyze a special class of risk-sensitive mean-field games where the state
dynamics are linear and independent of the mean field. In Section V, we
provide a numerical example, and section VI concludes the paper. An appendix
includes proofs of two main results in the main body of the paper. We
summarize some of the notations used in the paper in Table I.
TABLE I: Summary of Notations Symbol | Meaning
---|---
$f$ | drift function (finite dimensional)
$\sigma$ | diffusion function (finite dimensional)
${x}^{n}_{j}(t)$ | state of Player $j$ in a population of size $n$
$\bar{x}_{j}(t)$ | solution of macroscopic McKean-Vlasov equation
${x}_{j}(t)$ | limit of state process ${x}^{n}_{j}(t)$
$U_{j}$ | space of feasible control actions of Player $j$
$\tilde{\gamma}_{j}$ | state feedback strategy of Player $j$
$\bar{\gamma}_{j}$ | individual state-feedback strategy of Player $j$
$\tilde{\Gamma}_{j}$ | set of admissible state feedback strategies of Player $j$
$\bar{\Gamma}_{j}$ | set of admissible individual state-feedback strategies of Player $j$
$u_{j}$ | control action of Player $j$ under a generic control strategy
$c$ | instantaneous cost function
$g$ | terminal cost function
$\delta$ | risk-sensitivity index
$\mathbb{B}_{j}$ | standard Brownian motion process for Player $j$’s dynamics
$\mathbb{E}$ | Expectation operator
$L$ | risk-sensitive cost functional
$\partial_{x}$ | partial derivative with respect to $x$ (gradient)
$\partial^{2}_{xx}$ | second partial derivative (Hessian operator) with the respect to $x$
$x^{\prime}$ | transpose of $x$
$m^{n}_{t}$ | empirical measure of the states of the players
$m_{t}$ | limit of $m_{t}^{n}$ when $n\rightarrow\infty$
$m^{n}$ | limit of $m_{t}^{n}$ when $t\rightarrow\infty$
tr($M$) | trace of a square matrix $M$, i.e., $tr(M):=\sum_{i}M_{ii}.$
$A\succ B$ | $A-B$ is positive definite, where $A$, $B$ are square symmetric matrices of the same dimension.
## II The problem setting
We consider a class of $n-$person stochastic differential games, where Player
$j$’s individual state, $x_{j}^{n}$, evolves according to the Itô stochastic
differential equation (S) as follows:
$\begin{array}[]{ccl}dx_{j}^{n}(t)&=&\frac{1}{n}\displaystyle\sum_{i=1}^{n}f_{ji}(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))dt+\frac{\sqrt{\epsilon}}{n}\displaystyle\sum_{i=1}^{n}\sigma_{ji}(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))d\mathbb{B}_{j}(t),\\\
x_{j}^{n}(0)&=&x_{j,0}\in\mathcal{X}\subseteq\mathbb{R}^{k},\ k\geq
1,j\in\\{1,\ldots,n\\},\end{array}\ (\textrm{S})$
where $x_{j}^{n}(t)$ is the $k$-dimensional state of Player $j$;
$u_{j}^{n}(t)\in{U}_{j},$ is the control of Player $j$ at time $t$ with
${U}_{j}$ being a subset of the $p_{j}$-dimensional Euclidean space
$\mathbb{R}^{p_{j}}$; $\mathbb{B}_{j}(t)$ are mutually independent standard
Brownian motion processes in $\mathbb{R}^{k}$; and $\epsilon$ is a small
positive parameter, which will play a role in the analysis in the later
sections. We will assume in (S) that there is some symmetry in $f_{ji}$ and
$\sigma_{ji}$, in the sense that there exist $f$ and $\sigma$ (conditions on
which will be specified shortly) such that for all $j$ and $i$,
$f_{ji}(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))\equiv
f(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))$
and
$\sigma_{ji}(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))\equiv\sigma(t,x_{j}^{n}(t),u^{n}_{j}(t),x_{i}^{n}(t))\,.$
The system (S) is a controlled McKean-Vlasov dynamics. Historically, the
McKean-Vlasov stochastic differential equation (SDE) is a kind of mean field
forward SDE suggested by Kac in 1956 as a stochastic toy model for the Vlasov
kinetic equation of plasma and the study of which was initiated by McKean in
1966. Since then, many authors have made contributions to McKean-Vlasov type
SDEs and related applications [20, 10].
The uncontrolled version of state dynamics (S) captures many interesting
problems involving interactions between agents. We list below a few examples.
###### Example 1 (Stochastic Kuramoto model).
Consider $n$ oscillators where each of the oscillators is considered to have
its own intrinsic natural frequency $\omega_{j}$, and each is coupled
symmetrically to all other oscillators. For
$f_{ji}(x_{i},u_{i},x_{j})=f(x_{i},u_{i},x_{j})=K\sin(x_{j}-x_{i})+\omega_{j}$
and $\sigma_{ji}$ a constant in (S), the state dynamics without control is
known as (stochastic) Kuramoto oscillator [22] where the goal is convergence
to some common value (consensus) or alignment of the players’ parameters. The
stochastic Kuramoto model is given by
${d\theta_{j}}(t)=\left(\omega_{j}(t)+\dfrac{K}{n}\sum_{i=1}^{n}\sin(\theta_{i}(t)-\theta_{j}(t))\right)dt+Dd\mathbb{B}_{j}(t),$
where $D,K>0.$
###### Example 2 (Stochastic Cucker-Smale dynamics:).
Consider a population, say of birds or fish that move in the three dimensional
space. It has been observed that for some initial conditions, for example on
their positions and velocities, the state of the flock converges to one in
which all birds fly with the same velocity. See, for example, Cucker-Smale
flocking dynamics [9, 8] where each vector $x_{i}=(y_{i},v_{i})$ is composed
of position dynamics and velocity dynamics of the corresponding player. For
$f(x_{i},u_{i},x_{j})=(\epsilon^{2}+\parallel
x_{j}-x_{i}\parallel^{2})^{-\alpha}c(x_{j}-x_{i})$ in (S), where
$\epsilon>0,\alpha>0$ and $c(\cdot)$ is a continuous function, one arrives at
a generic class of consensus algorithms developed for flocking problems.
###### Example 3 (Temperature dynamics for energy-efficient buildings).
Consider a heating system serving a finite number of zones. In each zone, the
goal is to maintain a certain temperature. Denote by $T_{j}$ the temperature
of zone $j,$ and by $T^{ext}$ the ambient temperature. The law of conservation
of energy can be written down as the following equation for zone $j,$
$dT_{j}(t)=\sigma
d\mathbb{B}_{j}(t)+\left[r_{j}(t)+\frac{\gamma}{\beta}(T^{ext}(t)-T_{j}(t))+\sum_{i\neq
j}\alpha_{ij}(t)(T_{i}(t)-T_{j}(t))\right]dt,$
where $r_{j}$ denotes the heat input rate of the heater in zone $j,$
$\gamma,\beta>0,$ $\alpha_{ij}$ is the thermal conductance between zone $i$
and zone $j$ and $\sigma$ is a small variance term. The evolution of the
temperature has a McKean-Vlasov structure of the type in system (S). We can
introduce a control variable into $r_{j}$ such that the heater can be turned
on and off in each zone.
The three examples above can be viewed as special cases of the system (S). The
controlled dynamics in (S) allows one to address several interesting
questions. For example, how to control the flocking dynamics and consensus
algorithms of the first two examples above to a certain target? How to control
the temperature in the third example in order to achieve a specific thermal
comfort while minimizing energy cost? In order to define the controlled
dynamical system in precise terms, we have to specify the nature of
information that players are allowed in the choice of their control at each
point in time. This brings us to the first definition below.
###### Definition 1.
A state-feedback strategy for Player $j$ is a mapping $\tilde{\gamma}_{j}:\
\mathbb{R}_{+}\times(\mathbb{R}^{k})^{n}\longrightarrow{U}_{j}$, whereas an
individual state-feedback strategy for Player $j$ is a mapping
$\bar{\gamma}_{j}:\ \mathbb{R}_{+}\times\mathbb{R}^{k}\longrightarrow{U}_{j}.$
Note that the individual state-feedback strategy involves only the self state
of a player, whereas the state-feedback strategy involves the entire
$nk-$dimensional state vector. The individual strategy spaces in each case
have to be chosen in such a way that the resulting system of stochastic
differential equations (S) admits a unique solution (in the sense specified
shortly) when the players pick their strategies independently; furthermore,
the feasible sets are time invariant and independent of the controls. We
denote by $\bar{\Gamma}_{j}$ the set of such admissible control laws
$\bar{\gamma}_{j}:[0,T]\times\mathbb{R}^{k}\rightarrow{U}_{j}$ for Player $j$;
a similar set, $\tilde{\Gamma}_{j}$, can be defined for state-feedback
strategies $\tilde{\gamma}_{j}$.
We assume the following standard conditions on $f,\sigma,\bar{\gamma}_{j}$ and
the action sets $U_{j}$, for all $j=1,2,\cdots,n$.
* (i)
$f$ is $C^{1}$ in $(t,x,u,m)$, and Lipschitz in $(x,u,m)$.
* (ii)
The entries of the matrix $\sigma$ are $C^{2}$ and $\sigma\sigma^{\prime}$ is
strictly positive;
* (iii)
$f,\partial_{x}f$ are uniformly bounded;
* (iv)
${U}_{j}$ is non-empty, closed and bounded;
* (v)
$\bar{\gamma}_{j}:\ [0,T]\times\mathbb{R}^{k}\longrightarrow{U}_{j}$ is
piecewise continuous in $t$ and Lipschitz in $x.$
Normally, when we have a cost function for Player $j$, which depends also on
the state variables of the other players, either directly, or implicitly
through the coupling of the state dynamics (as in (S)), then any state-
feedback Nash equilibrium solution will generally depend not only on self
states but also on the other states, i.e., it will not be in the set
$\bar{\Gamma}_{j},j=1,\cdots,n$. However, this paper aims to characterize the
solution in the high-population regime (i.e., as $n\rightarrow\infty$) in
which case the dependence on other players’ states will be through the
distribution of the player states. Hence each player will respond (in an
optimal, cost minimizing manner) to the behavior of the mass population and
not to behaviors of individual players. Validity of this property will be
established later in Section III of the paper, but in anticipation of this, we
first introduce the quantity
$m^{n}_{t}=\frac{1}{n}\sum_{j=1}^{n}\delta_{x_{j}^{n}(t)},$ (1)
as an empirical measure of the collection of states of the players, where
$\delta$ is a Dirac measure on the state space. This enables us to introduce
the long-term cost function of Player $j$ (to be minimized by him) in terms of
only the self variables ($x_{j}$ and $u_{j}$) and $m_{t}^{n},t\geq 0$, where
the latter can be viewed as an exogenous process (not directly influenced by
Player $j$). But we first introduce a mean-field representation of the
dynamics (S), which uses $m^{n}_{t}$ and will be used in the description of
the cost.
### II-A Mean-field representation
The system (S) can be written into a measure representation using the formula
$\int\phi(w)\left[\sum_{i=1}^{n}\bar{\omega}_{i}\delta_{x_{i}}\right](dw)=\sum_{i=1}^{n}\bar{\omega}_{i}\phi(x_{i}),$
where $\delta_{z},z\in\mathcal{X}$ is a Dirac measure concentrated at $z$,
$\phi$ is a measurable bounded function defined on the state space and
$\bar{\omega}_{i}\in\mathbb{R}$. Then, the system (S) reduces to the system
$\displaystyle dx_{j}^{n}(t)$ $\displaystyle=$
$\displaystyle\left(\int_{w}f(t,x_{j}^{n}(t),u^{n}_{j}(t),w)\left[\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}^{n}(t)}\right](dw)\right)dt$
$\displaystyle+$
$\displaystyle\sqrt{\epsilon}\left(\int_{w}\sigma(t,x_{j}^{n}(t),u^{n}_{j}(t),w)\left[\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}^{n}(t)}\right](dw)\right)d\mathbb{B}_{j}(t),$
$\displaystyle x_{j}^{n}(0)$ $\displaystyle=$ $\displaystyle
x_{j,0}\in\mathbb{R}^{k},\ k\geq 1,j\in\\{1,\ldots,n\\},$
which, by (1), is equivalent to the following system (SM):
$\begin{array}[]{lll}\displaystyle
dx_{j}^{n}(t)&=&\left(\displaystyle\int_{w}f(t,x_{j}^{n}(t),u^{n}_{j}(t),w)m^{n}_{t}(dw)\right)dt\\\
&&+\sqrt{\epsilon}\left(\displaystyle\int_{w}\sigma(t,x_{j}^{n}(t),u^{n}_{j}(t),w)m^{n}_{t}(dw)\right)d\mathbb{B}_{j}(t),\\\
x_{j}^{n}(0)&=&x_{j,0}\in\mathbb{R}^{k},\ k\geq
1,j\in\\{1,\ldots,n\\}.\end{array}$
The above representation of the system (SM) can be seen as a controlled
interacting particles representation of a macroscopic McKean-Vlasov equation
where $m^{n}_{t}$ represents the discrete density of the population. Next, we
address the mean field convergence of the population profile process $m^{n}.$
To do so, we introduce the key notion of indistinguishability.
###### Definition 2 (Indistinguishability).
We say that a family of processes $(x^{n}_{1},x^{n}_{2},\ldots,x^{n}_{n})$ is
indistinguishable (or exchangeable) if the law of $x^{n}$ is invariant by
permutation over the index set $\\{1,\ldots,n\\}.$
The solution of (S) obtained under fixed control $u(\cdot)$ generates
indistinguishable processes. For any permutation $\pi$ over
$\\{1,2,\ldots,n\\}$, one has
$\mathcal{L}(x^{n}_{j_{1}},\ldots,x^{n}_{j_{n}})=\mathcal{L}(x^{n}_{\pi(j_{1})},\ldots,x^{n}_{\pi(j_{n})}),\
$ where $\mathcal{L}(X)$ denotes the law of the random variable $X.$ For
indistinguishable (exchangeable) processes, the convergence of the empirical
measure has been widely studied (see [29] and the references therein). To
preserve this property for the controlled system we restrict ourselves to
admissible homogeneous controls. Then, the mean field convergence is
equivalent to the existence of a random measure $\mu$ such that the system is
$\mu-$chaotic, i.e.,
$\lim_{n}\int\prod_{l=1}^{L}\phi_{l}({x}^{n}_{j_{l}})\mu^{n}(dx^{n})=\prod_{l=1}^{L}\left(\int\phi_{l}d\mu\right),$
for any fixed natural number $L\geq 2$ and a collection of measurable bounded
functions $\\{\phi_{l}\\}_{1\leq l\leq L}$ defined over the state space
$\mathcal{X}.$ Following the indistinguishability property, one has that the
law of $x^{n}_{j}=(x_{j}^{n}(t),\ t\geq 0)$ is $\mathbb{E}[m^{n}].$ The same
result is obtained by proving the weak convergence of the individual state
dynamics to a macroscopic McKean-Vlasov equation (see later Proposition 5).
Then, when the initial states are i.i.d. and given some homogeneous control
actions $u,$ the solution of the state dynamics generates an indistinguishable
random process and the weak convergence of the population profile process
$m^{n}$ to $\mu$ is equivalent to the $\mu-$chaoticity. For general results on
mean-field convergence of controlled stochastic differential equations, we
refer to [14]. These processes depend implicitly on the strategies used by the
players. Note that an admissible control law $\bar{\gamma}$ may depend on time
$t$, the value of the individual state $x_{j}(t)$ and the mean-field process
$m_{t}$. The weak convergence of the process $m^{n}$ implies the weak
convergence of its marginal $m^{n}_{t}$ and one can characterize the
distribution of $m_{t}$ by the Fokker-Planck-Kolmogorov (FPK) equation:
$\partial_{t}m_{t}+D^{1}_{x}\left(m_{t}\displaystyle\int_{w}f(t,x,u(t),w)m_{t}(dw)\right)$
$\displaystyle=$
$\displaystyle\frac{\epsilon}{2}D^{2}_{xx}\left(m_{t}\left(\int_{w}\sigma^{\prime}(t,x,u(t),w)m_{t}(dw)\right)\cdot\left(\int_{w}\sigma(t,x,u(t),w)m_{t}(dw)\right)\right).$
(2)
Here $f(\cdot)\in\mathbb{R}^{k},$ which we denote by
$(f_{k^{\prime}}(\cdot))_{1\leq k^{\prime}\leq k},$ where $f_{k^{\prime}}$ is
scalar. We let
$\underline{\sigma}[t,x,u(t),m_{t}]:=\int_{w}\sigma(t,x,u(t),w)m_{t}(dw),$
$\Gamma(\cdot):=\underline{\sigma}(\cdot)\underline{\sigma}^{\prime}(\cdot)$
is a square matrix with dimension $k\times k.$ The term $D^{1}_{x}(\cdot)$
denotes
$\sum_{k^{\prime}=1}^{k}\frac{\partial}{\partial
x_{k^{\prime}}}\left(m_{t}\int_{w}f_{k^{\prime}}(t,x,u(t),w)m_{t}(dw)\right),$
and the last term on $D^{2}_{xx}(\cdot)$ is
$\sum_{k^{\prime\prime}=1}^{k}\sum_{k^{\prime}=1}^{k}\frac{\partial^{2}}{\partial
x_{k^{\prime}}\partial
x_{k^{\prime\prime}}}\left(m_{t}\Gamma_{k^{\prime}k^{\prime\prime}}(\cdot)\right).$
In the one-dimensional case, the terms $D^{1},D^{2}$ reduce to the divergence
“div” and the Laplacian operator $\Delta$, respectively.
It is important to note that the existence of a unique rest point
(distribution) in FPK does not automatically imply that the mean-field
converges to the rest point when $t$ goes to infinity. This is because the
rest point may not be stable.
###### Remark 1.
In mathematical physics, convergence to an independent and identically
distributed system is sometimes referred to as chaoticity [28, 29, 11], and
the fact that chaoticity at the initial time implies chaoticity at further
times is called propagation of chaos. However in our setting the chaoticity
property needs to be studied together with the controls of the players. In
general the chaoticity property may not hold. One particular case should be
mentioned, which is when the rest point $m^{*}$ is related to the
$\delta_{m^{*}}-$ chaoticity. If the mean-field dynamics has a unique global
attractor $m^{*}$, then the propagation of chaos property holds for the
measure $\delta_{m^{*}}.$ Beyond this particular case, one may have multiple
rest points but also the double limit, $\lim_{n}\lim_{t}m^{n}_{t}$ may differ
from the one when the order is swapped, $\lim_{t}\lim_{n}m^{n}_{t}$ leading a
non-commutative diagram. Thus, a deep study of the underlying dynamical system
is required if one wants to analyze a performance metric for a stationary
regime. A counterexample of non-commutativity of the double limit is provided
in [30].
### II-B Cost Function
We now introduce the cost functions for the differential game. Risk-sensitive
behaviors can be captured by cost functions which exponentiate loss functions
before the expectation operator. For each $t\in[0,T]$, and $m_{t}^{n},x_{j}$
initialized at a generic feasible pair $\underline{m},\underline{x}$ at $t$,
the risk-sensitive cost function for Player $j$ is given by
$\displaystyle
L(\bar{\gamma}_{j},m^{n}_{[t,T]};t,\underline{x},\underline{m})=\delta\log\mathbb{E}\left(e^{\frac{1}{\delta}[\displaystyle
g(x_{T})+\int_{t}^{T}c(s,x_{j}^{n}(s),u_{j}^{n}(s),m^{n}(s))\ ds]}\ \Bigg{|}\
x_{j}(t)=\underline{x},m^{n}_{t}=\underline{m}\right),$ (3)
where $c(\cdot)$ is the instantaneous cost at time $s$; $g(\cdot)$ is the
terminal cost; $\delta>0$ is the risk-sensitivity index; $m^{n}_{[t,T]}$
denotes the process $\\{m^{n}_{s},t\leq s\leq T\\}$; and
$u_{j}^{n}(s)=\bar{\gamma}_{j}(s,x_{j}^{n}(s),m^{n}(s)),$ with
$\bar{\gamma}_{j}\in\bar{\Gamma}_{j}$. Note that because of the symmetry
assumption across players, the cost function of Player $j$ is not indexed by
$j$, since it is in the same structural form for all players. This is still a
game problem (and not a team problem), however, because each such cost
function depends only on the self variables (indexed by $j$ for Player $j$) as
well as the common population variable $m^{n}$.
We assume the following standard conditions on $c$ and $g$.
* (vi)
$c$ is $C^{1}$ in $(t,x,u,m)$; $g$ is $C^{2}$ in $x$; $c,g$ are non-negative;
* (vii)
$c,\partial_{x}c,g,\partial_{x}g$ are uniformly bounded.
The cost function (3) is called the risk-sensitive cost functional or the
exponentiated integral cost, which measures risk-sensitivity for the long-run
and not at each instant of time (see [18, 35, 6, 2]). We note that the
McKean-Vlasov mean field game considered here differs from the model in [16];
specifically, in this paper, the volatility term in (SM) is a function of
state, control and the mean field, and further, the cost functional is of the
risk-sensitive type.
###### Remark 2 (Connection with mean-variance cost).
Consider the function $c^{\lambda}:\
\lambda\longmapsto\frac{1}{\lambda}\log(\mathbb{E}e^{\lambda C}).$ It is
obvious that the risk-sensitive cost $c^{\lambda}$ takes into consideration
all the moments of the cost $C$, and not only its mean value. Around zero, the
Taylor expansion of $c^{\lambda}$ is given by
$c^{\lambda}\underbrace{\approx}_{\lambda\sim
0}\mathbb{E}(C)+\frac{\lambda}{2}\textrm{var}(C)+o(\lambda),$
where the important terms are the mean cost and the variance of the cost for
small $\lambda.$ Hence risk-sensitive cost entails a weighted sum of the mean
and variance of the cost, to some level of approximation.
With the dynamics (SM) and cost functionals as introduced, we seek an
individual state-feedback non-cooperative Nash equilibrium
$\\{\bar{\gamma}^{*}_{i},i\in\\{1,\cdots,n\\}\\}$, satisfying the set of
inequalities
$L(\bar{\gamma}^{*}_{j},m^{n}_{[0,T]};0,x_{j,0},\underline{m})\leq
L(\bar{\gamma}_{j},\tilde{m}^{n,j}_{[0,T]};0,x_{j,0},\underline{m}),$ (4)
for all $\bar{\gamma}_{j}\in\bar{\Gamma}_{j},j\in\\{1,2,\cdots,n\\}$, where
$m^{n}[0,T]$ is generated by the $\bar{\gamma}_{j}^{*}$’s, and
$\tilde{m}^{n,j}_{[0,T]}$ by $(\bar{\gamma},\bar{\gamma}_{-j}^{*})$,
$\bar{\gamma}_{-j}^{*}=\\{\bar{\gamma}_{i}^{*},i=1,2,\cdots,n,i\neq j\\}$;
$u^{*}_{j}$ and $u_{j}$ are control actions generated by control laws
$\bar{\gamma}^{*}_{j}$ and $\bar{\gamma}_{j}$, respectively, i.e.,
$u^{*}_{j}=\bar{\gamma}^{*}_{j}(t,x_{j})$ and
$u_{j}=\bar{\gamma}_{j}(t,x_{j})$; $m^{n}_{t}=m^{n}_{t}[u^{*}]$ laws are given
by forward FPK equation under the strategy $\bar{\gamma}^{*},$ and
$\tilde{m}^{n,j}_{t}=\tilde{m}^{n,j}_{t}[u_{j},u_{-j}^{*}]$ is the induced
measure under the strategy $(\bar{\gamma}_{j},\bar{\gamma}_{-j}^{*}).$
A more stringent equilibrium solution concept is that of strongly time-
consistent individual state-feedback Nash equilibrium satisfying,
$L(\bar{\gamma}^{*}_{j},m^{n}_{[t,T]};t,x_{j},\underline{m})\leq
L(\bar{\gamma}_{j},\tilde{m}^{n,j}_{[t,T]};t,x_{j},\underline{m}),$ (5)
for all $x_{j}\in\mathcal{X}$, $t\in[0,T)$,
$\bar{\gamma}_{j}\in\bar{\Gamma}_{j},j\in\\{1,2,\cdots,n\\}.$
Note that the two measures $m^{n}_{t}$ and $\tilde{m}^{n,j}_{t}$ differ only
in the component $j$ and have a common term which is
$\frac{1}{n}\sum_{j^{\prime}\neq j}\delta_{x^{n}_{j^{\prime}}(t)}$, which
converges in distribution to some measure with a distribution that is a
solution of the forward PFK partial differential equation.
## III Risk-sensitive best response to mean-field and equilibria
In this section, we present the risk-sensitive mean-field results. We first
provide an overview of the mean-field (feedback) best response for a given
mean-field trajectory $m^{n}=(m^{n}(s),\ s\geq 0).$ A mean-field best-response
strategy of a generic Player $j$ to a given mean field $m_{t}^{n}$ is a
measurable mapping $\bar{\gamma}^{*}_{j}$ satisfying: $\ \forall\
\bar{\gamma}_{j}\in\bar{{\Gamma}}_{j}$, with $x_{j}$ and $m^{n}_{t}$
initialized at $x_{j,0},\underline{m}$, respectively,
$L(\bar{\gamma}_{j}^{*},m^{n}_{[0,T]},0,x_{j,0},\underline{m})\leq
L(\gamma_{j},m^{n}_{[0,T]},0,x_{j,0},\underline{m}).$
where law of $m^{n}_{t}$ is given by the forward FPK equation in the whole
space $\mathcal{X}^{n}$, and is an exogenous process. Let
$v^{n}(t,x_{j},\underline{m})=\inf_{u_{j}}L({u}_{j},m^{n}_{[0,T]},t,x_{j},\underline{m}).$
The next proposition establishes the risk-sensitive Hamilton-Jacobi-Bellman
(HJB) equation of the risk-sensitive cost function satisfied by a smooth
optimal value function of a generic player. The main difference from the
standard HJB equation is the presence of the term
$\frac{\epsilon}{2\delta}\parallel\sigma\partial_{x_{j}}v^{n}\parallel^{2}.$
###### Proposition 1.
Suppose that the trajectory of $m^{n}_{t}$ is given. If $v^{n}$ is twice
continuously differentiable, then $v^{n}$ is solution of the risk-sensitive
HJB equation
$\displaystyle\partial_{t}v^{n}+\inf_{u_{j}}\left\\{f\cdot\partial_{x_{j}}v^{n}+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\partial^{2}_{x_{j}x_{j}}v^{n}_{j})+\frac{\epsilon}{2\delta}\parallel\sigma\partial_{x_{j}}v^{n}\parallel^{2}+c\right\\}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle v^{n}(T,x_{j})$
$\displaystyle=$ $\displaystyle g(x_{j}).$
Moreover, any strategy satisfying
$\bar{\gamma}^{n}_{j}(\cdot)\in\arg\min_{u_{j}}\left\\{f\cdot\partial_{x_{j}}v^{n}+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\partial^{2}_{x_{j}x_{j}}v^{n})+\frac{\epsilon}{2\delta}\parallel\sigma\partial_{x_{j}}v^{n}\parallel^{2}+c\right\\},$
constitutes a best response strategy to the mean-field $m^{n}.$
###### Proof of Proposition 1.
For feasible initial conditions $\underline{x}$ and $\underline{m}$, we define
$\phi^{n}(t,\underline{x},\underline{m}):=\inf_{{u}^{n}_{j}}\mathbb{E}\left(e^{\frac{1}{\delta}[g(x_{T})+\int_{t}^{T}c(s,x^{n}(s),u_{j}(t),m^{n}_{s})\
ds]}\ |\ x_{j}(t)=\underline{x},m^{n}_{t}=\underline{m}\right).$
It is clear that $v^{n}(t,x_{j},\underline{m})=\inf
L=\delta\log\phi^{n}(t,x_{j},\underline{m}).$ Under the regularity assumptions
of Section II, the function $\phi^{n}$ is $C^{1}$ in $t$ and $C^{2}$ in $x.$
Using Itô’s formula,
$d\phi^{n}(t,x_{j})=[\partial_{t}\phi^{n}(t,x_{j})+f\cdot\partial_{x_{j}}\phi^{n}+\frac{\epsilon}{2}\textrm{tr}(\sigma\sigma^{\prime}\partial^{2}_{x_{j}x_{j}}\phi^{n})]dt.$
Using the Ito-Dynkin’s formula (see [26, 6, 27]), the dynamic optimization
yields
$\inf_{\bar{u}_{j}}\\{d\phi^{n}+\frac{1}{\delta}c\phi^{n}dt\\}=0.$
Thus, one obtains
$\displaystyle\partial_{t}\phi^{n}+\inf_{u_{j}}\left\\{f\cdot\partial_{x_{j}}\phi^{n}+\frac{\epsilon}{2}\textrm{tr}(\sigma\sigma^{\prime}\partial^{2}_{xx}\phi^{n})+\frac{1}{\delta}c\phi^{n}\right\\}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\phi^{n}(T,x_{j})$
$\displaystyle=$ $\displaystyle e^{\frac{1}{\delta}g(x_{j})}.$
To establish the connection with the risk-sensitive cost value, we use the
relation $\phi^{n}=e^{\frac{1}{\delta}v^{n}}$. One can compute the partial
derivatives:
$\partial_{t}\phi^{n}=\left(\partial_{t}v^{n}\right)\frac{1}{\delta}\phi^{n},\
\
\partial_{x_{j}}\phi^{n}=\left(\partial_{x_{j}}v^{n}\right)\frac{1}{\delta}\phi^{n},$
and
$\partial^{2}_{x_{j}x_{j}}\phi^{n}=\left(\partial^{2}_{x_{j}x_{j}}v^{n}\right)\frac{1}{\delta}\phi^{n}+\frac{1}{\delta^{2}}\left(\partial_{x_{j}}v^{n}\right)^{\prime}\left(\partial_{x_{j}}v^{n}\right)\phi^{n},$
where the latter immediately yields
$\textrm{tr}(\partial^{2}_{x_{j}x_{j}}\phi^{n}\sigma\sigma^{\prime})=\textrm{tr}(\partial^{2}_{x_{j}x_{j}}v^{n}\sigma\sigma^{\prime})\frac{1}{\delta}\phi^{n}+\frac{1}{\delta^{2}}\parallel\sigma\partial_{x_{j}}v^{n}\parallel^{2}\phi^{n}.$
Combining together and dividing by $\phi^{n}/\delta,$ we arrive at the HJB
equation (1).
∎
###### Remark 3.
Let us introduce the Hamiltonian $H$ as
$H(t,x,\tilde{p},\tilde{M})=\inf_{u}\left\\{\tilde{p}\cdot
f+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\tilde{M})+\frac{\epsilon}{2\delta}\parallel\sigma\tilde{p}\parallel^{2}+c\right\\},$
for a vector $\tilde{p}$ and a matrix $\tilde{M}$ which is the same as the
Hessian of $v^{n}.$
If $\sigma$ does not depend on the control, then the above expression reduces
to
$\inf_{u}\\{\tilde{p}\cdot
f+c\\}+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\tilde{M})+\frac{\epsilon}{2\delta}\parallel\sigma\tilde{p}\parallel^{2},$
and the term to be minimized is
$H^{2}(t,x,\tilde{p},\tilde{M})=\inf_{u}\\{\tilde{p}\cdot f+c\\},$ which is
related to the Legendre-Fenchel transform for linear dynamics, i.e., the case
where $f$ is linear in the control $u.$
In that case,
$\partial_{\tilde{p}}H^{2}(t,x,\tilde{p},\tilde{M})=\alpha u^{*}$
for some non-singular $\alpha$ of proper dimension. This says that the
derivative of the modified Hamiltonian is related to the optimal feedback
control. Now, for non-linear drift $f$ the same technique can be used but the
function $f$ needs to be inverted to obtain a generic closed form expression
the optimal feedback control and is given by
$u^{*}_{j}=\tilde{g}^{-1}(\partial_{\tilde{p}}H^{2}(t,x,\tilde{p},\tilde{M})),$
where $\tilde{g}^{-1}$ is the inverse of the map
$u\longmapsto f(t,x,u,m).$
This generic expression of the optimal control will play an important role in
non-linear McKean-Vlasov mean field games.
The next proposition provides the best-response control to the affine-
quadratic in $u$-exponentiated cost-Gaussian mean-field game, and the
proposition that follows that deals with the case of affine-quadratic in both
$u$ and $x$.
###### Proposition 2.
Suppose $\sigma(t,x)=\sigma(t)$ and
$\displaystyle f(t,x_{j},u_{j},m)$ $\displaystyle=$
$\displaystyle\bar{f}(t,x_{j},m)+B(t,x_{j},m)u_{j},$ $\displaystyle
c(t,x_{j},u_{j},m)$ $\displaystyle=$
$\displaystyle\bar{c}(t,x_{j},m)+\parallel u_{j}\parallel^{2}.$
Then, the best-response control of Player $j$ is
$\bar{\gamma}_{j}^{n,*}=-\frac{1}{2}B\partial_{x_{j}}v^{n}.$
###### Proof.
Following Proposition 1, we know
$\bar{u}^{n,*}_{j}=\bar{\gamma}^{n,*}_{j}(\cdot)\in\arg\min_{u_{j}}\\{c(t,x_{j}(t),u_{j}(t),m_{t})+f(t,x_{j}(t),u_{j},m_{t})\cdot\partial_{x_{j}}v^{n}\\}.$
With the assumptions on $\sigma,f,c,g$, the condition reduces to
$\arg\min_{u_{j}}\left\\{[\bar{f}+Bu_{j}]\partial_{x_{j}}v^{n}+\bar{c}+\parallel
u_{j}\parallel^{2}\right\\}.$
and hence, we obtain
$\bar{\gamma}_{j}^{n,*}=-\frac{1}{2}B\partial_{x_{j}}v^{n}$ by convexity and
coercivity of the mapping
$u_{j}\longmapsto[\bar{f}+Bu_{j}]\partial_{x_{j}}v^{n}+\bar{c}+\parallel
u_{j}\parallel^{2}.$ ∎
###### Proposition 3 (Explicit optimal control and cost, [2]).
Consider the risk-sensitive mean-field stochastic game described in
Proposition 2 with $\bar{f}=A(t)x$, $B$ a constant matrix,
$c=x^{\prime}Q(t)x,\ Q(t)\geq 0,\ g(x)=x^{\prime}Q_{T}x,Q_{T}\geq 0,$ where
the symmetric matrix $Q(\cdot)$ is continuous. Then, the solution to HJB
equation in Proposition 1 (whenever it exists) is given by
$v^{n}(t,x)=x^{\prime}Z(t)x+\epsilon\int_{t}^{T}{tr}(Z(s)\sigma\sigma^{\prime})\
ds.$ where $Z(s)$ is the nonnegative definite solution of the generalized
Riccati differential equation
$\dot{Z}+A^{\prime}Z+ZA+Q-Z\left(BB^{\prime}-\frac{1}{\rho^{2}}\sigma\sigma^{\prime}\right)Z=0,\
Z(T)=Q_{T},$
where $\rho=(\frac{\delta}{2\epsilon})^{1/2}$ and the optimal response
strategy is
$u^{*}_{j}(t)=\bar{\gamma}^{*}_{j}(\cdot)=-B^{\prime}Zx.$ (6)
Using Proposition 3, one has the following result for any given trajectory
$(m^{n}_{t})_{t\geq 0},$ which enters the cost function in a particular way.
###### Proposition 4.
If $c$ is in the form $c=x^{\prime}(Q(t)-\Lambda(t,m^{n}_{t}))x$, where
$\Lambda$ is symmetric and continuous in $(t,m)$, then the generalized Riccati
equation becomes
$\dot{Z}^{*}+A^{\prime}Z^{*}+Z^{*}A+Q-\Lambda(t,m^{n}_{t})-Z^{*}\left(BB^{\prime}-\frac{1}{\rho^{2}}\sigma\sigma^{\prime}\right)Z^{*}=0,Z^{*}(T)=Q_{T},$
and
$v^{n}(t,x)=x^{\prime}Z^{*}x+\epsilon\int_{t}^{T}{tr}(Z^{*}(s)\sigma\sigma^{\prime})\
ds.$
### III-A Macroscopic McKean-Vlasov equation
Since the controls used by the players influence the mean-field limit via the
state dynamics, we need to characterize the evolution of the mean-field limit
as a function of the controls. The law of $m_{t}$ is the solution of the
Fokker-Planck-Kolmogorov equation given by (2) and the individual state
dynamics follows the so-called macroscopic McKean-Vlasov equation
$\displaystyle
d\bar{x}_{j}(t)=\left(\int_{w}f(t,\bar{x}_{j}(t),u_{j}^{*}(t),w)m_{t}(dw)\right)dt+\sqrt{\epsilon}\left(\int_{w}\sigma(t,\bar{x}_{j}(t),u_{j}^{*}(t),w)m_{t}(dw)\right)d\mathbb{B}_{j}(t).$
(7)
In order to obtain an error bound, we introduce the following notion: Given
two measures $\mu$ and $\nu$ the Monge-Kontorovich metric (also called
Wasserstein metric) between $\mu$ and $\nu$ is
$\mathcal{W}_{1}(\mu,\nu)=\inf_{X\sim\mu,Y\sim\nu}\mathbb{E}|X-Y|.$
In other words, let $E(\mu,\nu)$ be the set of probability measures
$\mathbb{P}$ on the product space such that the image of $\mathbb{P}$ under
the projection on the first argument (resp. on the second argument) is $\mu$
(resp. $\nu$). Then,
$\displaystyle\mathcal{W}_{1}(\mu,\nu)=\inf_{\mathbb{P}\in
E(\mu,\nu)}\int\int|z-z^{\prime}|\mathbb{P}(dz,dz^{\prime}).$ (8)
This is known indeed as a distance (it can be checked that the separation, the
triangle inequality and positivity properties are satisfied) and it metricizes
the weak topology.
###### Proposition 5.
Under the conditions (i)-(vii), the following holds: For any $t,$ if the
control law $\gamma^{*}_{j}(\cdot)$ is used, then there exists
$\tilde{y}_{t}>0$ such that
$\mathbb{E}\left(\parallel
x^{n}_{j}(t)-\tilde{x}_{j}(t)\parallel\right)\leq\frac{\tilde{y}_{t}}{\sqrt{n}}.$
Moreover, for any $T<\infty,$ there exists $C_{T}>0$ such that
$\displaystyle\mathcal{W}_{1}\left(\mathcal{L}((x^{n}_{j}(t))_{t\in[0,T]}),\mathcal{L}((\tilde{x}_{j}(t))_{t\in[0,T]})\right)\leq\frac{C_{T}}{\sqrt{n}},$
(9)
where $\mathcal{L}(X_{t})$ denotes the law of the random variable $X_{t}$.
The last inequality says that the error bound is at most of
$O(\frac{1}{\sqrt{n}})$ for any fixed compact interval. The proof of this
assertion follows the following steps: Let $x^{n}_{j}(t)$ and
$\tilde{x}_{j}(t)$ be the solutions of the two SDEs with initial gap less than
$\frac{1}{\sqrt{n}}.$ Then, we take the difference between the two solutions.
In a second step, use triangle inequality of norms and take the expectation.
Gronwall inequality allows one to complete the proof. A detailed proof is
provided in the Appendix.
#### III-A1 Risk-sensitive mean-field cost
Based on the fact that $m^{n}_{t}$ converges weakly to $m_{t}$ under the
admissible controls $(u^{n}_{j}(s),\ s\geq 0)\longrightarrow(u_{j}(s),\ s\geq
0)$ when $n$ goes to infinity, one can show the weak convergence of the risk-
sensitive cost function (3) under the regularity conditions (vi) and (vii) on
functions $c$ and $g$, i.e., as $n\rightarrow\infty$,
$\displaystyle
L(\bar{\gamma}_{j},m^{n}_{[t,T]};t,\underline{x},\underline{m})$
$\displaystyle\rightarrow$ $\displaystyle
L(u_{j},m_{[t,T]},t,\underline{x},\underline{m})$
$\displaystyle=\delta\log\mathbb{E}\left(e^{\frac{1}{\delta}[g(x_{j}(T))+\int_{t}^{T}c(s,x_{j}(s),u_{j}(s),m_{s})\
ds]}\bigg{|}\ x_{j}(t)=\underline{x},m_{t}=\underline{m}\right).$
Based on this limiting cost, we can construct the best response to mean field
in the limit. Given $\\{m_{s}\\}_{s\in[t,T]}$, we minimize
$L(u_{j},m_{[t,T]};t,x,m)$ subject to the state-dynamics constraints.
### III-B Fixed-point problem
We now define the mean field equilibrium problem as the following fixed-point
problem.
###### Definition 3.
The mean field equilibrium problem (P) is one where each player solves the
optimal control problem, i.e.,
${\inf}_{u_{j}}\
\delta\log\mathbb{E}\left(e^{\frac{1}{\delta}[g(x_{j}(T))+\int_{t}^{T}c(s,x_{j}(s),u_{j}(s),m_{s}^{*})\
ds]}\bigg{|}\ x_{j}(t)=\underline{x},m_{t}=\underline{m}\right),$
subject to the dynamics of $x_{j}(t)$ given by the dynamics in Section III-A,
where the mean field $m_{t}$ is replaced by $m^{*}_{t}$ and $\bar{m}_{t}^{*}$
is the mean of the optimal mean field trajectory. The optimal feedback control
$u_{j}^{*}[t,x,m^{*}]$ depends on $m^{*}$, and $m^{*}$ is the mean field
reproduced by all the $u_{j}^{*}$, i.e., $m^{*}_{t}=m[t,u^{*}]$ solution of
the Fokker-Planck-Kolmogorov forward equation (2). The equilibrium is called
an individual feedback mean field equilibrium if every player adopts an
individual state-feedback strategy.
Note that this problem differs from the risk-sensitive mean field stochastic
optimal control problem where the objective is
$\delta\log\mathbb{E}\left(e^{\frac{1}{\delta}[g(x_{j}(T))+\int_{t}^{T}c(s,x_{j}(s),u_{j}(s),m_{s}[u])\
ds]}\bigg{|}\ x_{j}(t)=\underline{x},m_{t}=\underline{m}\right),$
with $m_{s}[u]$ the distribution of the state dynamics $x_{j}(s)$ driven by
the control $u_{j}.$
### III-C Risk-sensitive FPK-McV equations
The regular solutions to problem (P) introduced above are solutions to HJB
backward equation combined with FPK equation and macroscopic McKean-Vlasov
version of the limiting individual dynamics, i.e.,
$\displaystyle d{x}_{j}(t)$ $\displaystyle=$
$\displaystyle\left(\int_{w}f(t,x_{j}(t),u_{j}^{*}(t),w)m_{t}(dw)\right)dt$
$\displaystyle+$
$\displaystyle\sqrt{\epsilon}\left(\int_{w}\sigma(t,x_{j}(t),u_{j}^{*}(t),w)m_{t}(dw)\right)d\mathbb{B}_{j}(t),$
$\displaystyle x_{j}(0)=x_{j,0}=x$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{t}v+\inf_{u_{j}}\left\\{f\cdot\partial_{x}v+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\partial^{2}_{xx}v)+\frac{\epsilon}{2\delta}\parallel\sigma\partial_{x}v\parallel^{2}+c\right\\},$
$\displaystyle x_{j}:=x;\ \ v(T,x)=g(x)$ $\displaystyle\partial_{t}m_{t}$
$\displaystyle=$ $\displaystyle-
D_{x}^{1}\left(m_{t}\int_{w}f(t,x,u^{*},w)m_{t}(dw)\right)$
$\displaystyle+\frac{\epsilon}{2}D^{2}_{xx}\left(m_{t}\left(\int_{w}\sigma^{\prime}(t,x,u^{*},w)m_{t}(dw)\right)\right.\left.\cdot\left(\int_{w}\sigma(t,x,u^{*},w)m_{t}(dw)\right)\right)$
$\displaystyle m_{0}(\cdot)\ \mbox{fixed}.$
Then, the question of existence of a solution to the above system arises. This
is a backward-forward system. Very little is known about the existence of a
solution to such a system. In general, a solution may not exist as the
following example demonstrates.
### III-D Non-existence of solution to backward-forward boundary value
problems
There are many examples of systems of backward-forward equations which do not
admit solutions. As a very simple example from [37], consider the system:
$\dot{v}=m,\ \dot{m}=-v,m(0)=m_{0};v(T)=-m_{T}.$
It is obvious that the coefficients of this pair of backward-forward
differential equations are all uniformly Lipschitz. However, depending on $T$,
this may not be solvable for $m_{0}\neq 0.$ We can easily show that for
$T=k\pi+3\pi/4$ ($k$, a nonnegative integer), the above two-point boundary
value problem does not admit a solution for any $m_{0}\neq 0$ and it admits
infinitely many solutions for $m_{0}=0.$
Following the same ideas, one can show that the system of stochastic
differential equations (SDEs)
$d{v}=mdt+\sigma d\mathbb{B}(t),\ d{m}=-vdt+\nu d\mathbb{B}(t),$
where $\mathbb{B}(t)$ is the standard Brownian motion in $\mathbb{R}$. With
the initial conditions:
$m(0)=m_{0}\neq 0;v(T)=-m_{T},$
and $T=7\pi/4$, the system of SDEs has no solution.
This example shows us that the system needs to be normalized and the boundary
conditions will have to be well posed. In view of this, we will introduce the
notion of reduced mean field system in Section IV to establish the existence
of equilibrium for a specific class of risk-sensitive games.
### III-E Risk-sensitive mean-field equilibria
###### Theorem 1.
Consider a risk-sensitive mean-field stochastic differential game as
formulated above. Assume that $\sigma=\sigma(t)$ and there exists a unique
pair $(u^{*},m^{*})$ such that
(i) The coupled backward-forward PDEs
$\displaystyle\partial_{t}v^{*}+\inf_{u_{j}}\left\\{f^{*}\cdot\partial_{x}v^{*}+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\partial^{2}_{xx}v^{*})\right.\left.+\frac{\epsilon}{2\delta}\parallel\sigma\partial_{x}v\parallel^{2}+c^{*}\right\\}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle v(T,x)=g(x),\ m^{*}_{0}(x)\
{\textrm{fixed}}.$ $\displaystyle\partial_{t}m^{*}_{t}$ $\displaystyle+$
$\displaystyle
D^{1}_{x}\left(m^{*}_{t}\int_{w}f^{*}(t,x,u^{*},w)m^{*}_{t}(dw)\right)$
$\displaystyle=$
$\displaystyle\frac{\epsilon}{2}D^{2}_{xx}\left(m^{*}_{t}\left(\int_{w}\sigma^{\prime}m^{*}_{t}(dw)\right)\left(\int_{w}\sigma
m^{*}_{t}(dw)\right)\right)$
admit a pair a bounded nonnegative solutions $v^{*},m^{*}$; and
(ii) $u^{*}$ minimizes the Hamiltonian, i.e.,
$f(t,x,u,m^{*})\cdot\partial_{x}v^{*}+c(t,x,u,m^{*})$.
Under these conditions, the pair $(u^{*},m^{*})$ is a strongly time-consistent
mean-field equilibrium and $L(t,u^{*},m^{*})=v^{*}.$ In addition, if
$c=x^{\prime}(Q(t)-\Lambda_{t}(m^{n}_{t}))x$ where $\Lambda(t,\cdot)$ is a
measurable symmetric matrix-valued function, then any convergent subsequence
of optimal control laws $\bar{\gamma}^{\alpha(n)}_{j}$ leads to a best
strategy for $m.$
###### Proof.
See the Appendix. ∎
###### Remark 4.
This result can be extended to finitely multiple classes of players (see [25,
3, 23] for discussions). To do so, consider a finite number of classes indexed
by $\theta\in\Theta.$ The individual dynamics are indexed by $\theta,$ i.e.
the function $f$ becomes $f_{\theta}$ and $\sigma$ becomes $\sigma_{\theta}.$
This means that the indistinguishability property is not satisfied anymore.
The law depends on $\theta$ (it is not invariant by permutation of index).
However, the invariance property holds within each class. This allows us to
establish a weak convergence of the individual dynamics of each generic player
for each class, and we obtain $\tilde{x}_{\theta}(t).$ The multi-class mean-
field equilibrium will be defined by a system for each class and the classes
are interdependent via the mean field and the value functions per class.
#### Limiting behavior with respect to $\epsilon$
We scale the parameters $\delta,\epsilon$ and $\rho$ such that
$\delta=2\epsilon\rho^{2}.$ The PDE given in Proposition 1 becomes
$\partial_{t}v+\inf_{u}\left\\{f^{*}\cdot\partial_{x}v+\frac{\epsilon}{2}{tr}(\sigma\sigma^{\prime}\partial^{2}_{xx}v)+\frac{1}{4\rho^{2}}\parallel\sigma\partial_{x}v\parallel^{2}+c^{*}\right\\}=0,\
v(T,x)=g(x).$
When the parameter $\epsilon$ goes to zero, one arrives at a deterministic
PDE. This situation captures the large deviation limit:
$\partial_{t}v+\inf_{u}\left\\{f^{*}\cdot\partial_{x}v+\frac{1}{4\rho^{2}}\parallel\sigma\partial_{x}v\parallel^{2}+c^{*}\right\\}=0,\
v(T,x)=g(x).$
### III-F Equivalent stochastic mean-field problem
In this subsection, we formulate an equivalent $(n+1)-$player game in which
the state dynamics of the $n$ players are given by the system (ESM) as
follows:
$\begin{array}[]{lll}dx_{j}^{n}(t)&=&\left(\displaystyle\int_{w}f(t,x_{j}^{n}(t),u^{n}_{j}(t),w)m^{n}_{t}(dw)+\sigma\zeta(t)\right)dt+\sqrt{\epsilon}\sigma
d\mathbb{B}_{j}(t),\\\ x_{j}^{n}(0)&=&x_{j,0}\in\mathbb{R}^{k},\ k\geq
1,j\in\\{1,\ldots,n\\},\end{array}$
where $\zeta(t)$ is the control parameter of the “fictitious” $(n+1)-$th
player. In parallel to (3), we define the risk-neutral cost function of the
$n$ players as follows:
$\tilde{L}(\bar{\gamma}_{j},\bar{\zeta},x^{n}_{j},m_{[0,T]}^{n};t,\underline{x},\underline{m})=$
$\mathbb{E}\left(g(x^{n}_{j}(T))+\int_{t}^{T}c(s,x_{j}^{n}(s),u_{j}^{n}(s),m^{n}_{s})\
ds-\rho^{2}\int_{t}^{T}\parallel\zeta(s)\parallel^{2}\ ds\
\bigg{|}x_{j}(t)=\underline{x},m^{n}_{t}=\underline{m}\right),$ (10)
where $\bar{\zeta}:[0,T]\times\mathbb{R}^{k}\rightarrow U_{n+1}$ is the
individual feedback control strategy of the fictitious Player $n+1$ that
yields an admissible control action $\zeta(t)$ in a set of feasible actions
$U_{n+1}$.
Every player $j\in\\{1,2,\ldots,n\\}$ minimizes $\tilde{L}$ by taking the
worst over the feedback strategy $\bar{\zeta}$ of player $n+1$ which is
piecewise continuous in $t$ and Lipschitz in $x_{j}.$ We refer to this game
described by (ESM) and (10) as the robust mean-field game. In the following
Proposition, we describe the connection between the mean-field risk-sensitive
game problem described in (SM) and (3) and the robust mean-field game problem
described in (ESM) and (10),
###### Proposition 6.
Under the regularity assumptions (i)-(vii), given a mean field $m^{n}_{t}$,
the value functions of the risk-sensitive game and the robust game problems
are identical, and the mean-field best-response control strategy of the risk-
sensitive stochastic differential game is identical to the one for the
corresponding robust mean-field game.
###### Proof.
Let
$\tilde{v}^{n}=\inf_{{u}_{j}}\sup_{{\zeta}}\tilde{L}({u}_{j},{\zeta},x^{n}_{j},m_{[0,T]}^{n},t,x_{j},\underline{m})$
denote the upper-value function associated with this robust mean-field game.
Then, under the regularity assumptions (i)-(vii), if $\tilde{v}^{n}$ is
$C^{1}$ in $t$ and $C^{2}$ in $x$, it satisfies the Hamilton-Jacobi-Isaacs
(HJI) equation
$\displaystyle\inf_{u}\sup_{\zeta}\left\\{\partial_{t}\tilde{v}^{n}_{j}+(f+\sigma\zeta)\cdot\partial_{x_{j}}\tilde{v}^{n}+c-\rho^{2}\parallel\zeta\parallel^{2}+\frac{\epsilon}{2}{tr}(\partial^{2}_{x_{j}x_{j}}\tilde{v}^{n}\sigma\sigma^{\prime})\right\\}$
$\displaystyle=$ $\displaystyle 0,$ (11) $\displaystyle\tilde{v}^{n}(T,x_{j})$
$\displaystyle=$ $\displaystyle g(x_{j}).$
Note that (11) can be rewritten as $\inf_{u}\sup_{\zeta}H^{3}$, where
$H^{3}:=H+(\sigma\zeta)\cdot\partial_{x_{j}}\tilde{v}^{n}-\rho^{2}\parallel\zeta\parallel^{2}$
is the Hamiltonian associated with this robust game.
Since the dependence on $u$ and $\zeta$ above are separable, the Isaacs
condition (see [4]) holds, i.e.,
$\inf_{u}\sup_{\zeta}H^{3}=\sup_{\zeta}\inf_{u}H^{3}$
and hence the function $\tilde{v}^{n}_{j}$ satisfies the following after
obtaining the best-response strategy for $\zeta$:
$\displaystyle-\partial_{t}\tilde{v}^{n}$ $\displaystyle=$
$\displaystyle\inf_{u}\left\\{f\cdot\partial_{x_{j}}\tilde{v}^{n}+c+\frac{1}{4\rho^{2}}\parallel\sigma^{\prime}\partial_{x_{j}}\tilde{v}^{n}\parallel^{2}+\frac{\epsilon}{2}{\textrm{tr}}(\partial^{2}_{x_{j}x_{j}}\tilde{v}^{n}\sigma\sigma^{\prime})\right\\}.$
(12) $\displaystyle\tilde{v}^{n}(T,x_{j})$ $\displaystyle=$ $\displaystyle
g(x_{j}).$
Note that the two PDEs, (12) and the one given in Proposition 1, are identical
with $\rho^{2}=\frac{\delta}{2\epsilon}$. Moreover, the optimal cost and the
optimal control laws in the two problems are the same. ∎
###### Remark 5.
The FPK forward equation will have to be modified to include the control of
fictitious player in the robust mean field game formulation accordingly by
including the term $\sigma\zeta$ in (ESM). Hence the mean field equilibrium
solutions to the two games are not necessarily identical.
## IV Linear state dynamics
In this section, we analyze a specific class of risk-sensitive games where
state dynamics are linear and do not depend explicitly on the mean field. We
first state a related result from [24, 12] for the risk-neutral case.
###### Theorem 2 ([24]).
Consider the reduced mean field system (rMFG):
$\displaystyle\partial_{x}v+{H}(x,\nabla_{x}v,m_{t}(x))+\frac{\sigma^{2}}{2}\partial^{2}_{xx}v$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\partial_{x}m_{t}+\textrm{div}(m_{t}\partial_{p}{H}(x,\nabla_{x}v,m_{t}(x))-\frac{\sigma^{2}}{2}\partial^{2}_{xx}m_{t}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle m_{0}(\cdot)\
\mbox{{fixed}},v(T,\cdot)\ \mbox{{fixed}},$ $\displaystyle v,m\ \mbox{ {are
1-periodic.}},$ $\displaystyle x\in(0,1)^{d}:=\mathcal{X},$
where $H$ is the Legendre transform (with respect to the control) of the
instantaneous cost function.
Suppose that $(x,p,z)\longmapsto H(x,p,z)$ is twice continuously
differentiable with the respect to $(p,z)$ and for all
$(x,p,z)\in\mathcal{X}\times\mathbb{R}^{p}\times\mathbb{R}_{+}^{*},$
$\left(\begin{array}[]{cc}\partial^{2}_{pp}H(x,p,z)&\frac{1}{2}\partial_{pz}^{2}H(x,p,z)\\\
\frac{1}{2}[\partial_{pz}^{2}H(x,p,z)]^{\prime}&-\frac{1}{z}\partial_{z}H(x,p,z)\end{array}\right)\succ
0$
Then, there exists at most one smooth solution to the (rMFG).
###### Remark 6.
We have a number of observations and notes.
* •
The Hamilitonian function $H$ in the result above requires a special
structure. Instead of a direct dependence on the mean field distribution
$m_{t}$, its dependence on the mean field is through the value of $m_{t}$
evaluated at state $x$.
* •
For global dependence on $m,$ a sufficiency condition for uniqueness can be
found in [23] for the case where the Hamiltonian is separable, i.e.,
$H(x,p,m)=\xi(x,p)+\tilde{f}(x,m)$ with $\tilde{f}$ monotone in $m$ and $\xi$
strictly convex in $p.$
* •
The solution of (rMFG) can be unique even if the above conditions are
violated. Further, the uniqueness condition is independent of the horizon of
the game.
* •
For the linear-quadratic mean field case, it has been shown in [3] that the
normalized system may have a unique i.i.d. solution or infinitely many
solutions depending on the system parameters. See also [5] for recent analysis
on risk-neutral linear-quadratic mean field games.
The next result provides the counterpart of Theorem 2 in the risk-sensitive
case. It provides sufficient conditions for having at most one smooth solution
in the risk-sensitive mean field system by exploiting the presence of the
additive quadratic term (which is strictly convex in $p$).
###### Theorem 3.
Consider the risk-sensitive (reduced) mean field system (RS-rMFG). Let
$\delta>0,$ and $H(x,p,z)$ be twice continuously differentiable in
$(p,z)\in\mathbb{R}^{d}\times\mathbb{R}_{+},$ satisfying the following
conditions:
* •
$H$ is strictly convex in $p,$
* •
$H$ is decreasing in $z,$
* •
$\left(-\frac{\partial_{z}H}{z}\right)\cdot\left(\partial^{2}_{pp}H\right)\succ(\partial^{2}_{pz}H-\frac{\epsilon\sigma^{2}}{2\delta}p/z)^{\prime}\cdot(\partial^{2}_{pz}H-\frac{\epsilon\sigma^{2}}{2\delta}p/z)$.
Then, (RS-rMFG) has at most one smooth solution.
###### Proof.
See the Appendix. ∎
###### Remark 7.
We observe that in contrast to Theorem 2 (risk-neutral case), the sufficiency
condition for having at most one smooth solution in (RS-rMFG) now depends on
the variance term.
## V Numerical Illustration
In this section, we provide two numerical examples to illustrate the risk-
sensitive mean-field game under affine state dynamics and McKean-Vlasov
dynamics.
### V-A Affine state dynamics
Figure 1: The evolution of distribution $m^{*}_{t},0\leq t\leq 5,-19\leq x\leq
21$. Figure 2: Mean value $\mathbb{E}(m^{*}_{t})$ as a function of time,
$0\leq t\leq 5$. Figure 3: Variance of the distribution $m^{*}_{t}$ as a
function of time, $0\leq t\leq 5$. Figure 4: $z(t)$ as a function of time,
$0\leq t\leq T$.
We let Player $j$’s state evolution be described by a decoupled stochastic
differential equation
$dx_{j}^{n}(t)=u_{j}(t)dt+\sqrt{\epsilon}\sigma d\mathbb{B}_{j}(t).$
The risk-sensitive cost functional is given by
$L(\bar{\gamma}_{j},m^{n};t,\underline{x},\underline{m})=\delta\log\mathbb{E}_{\underline{x},\underline{m}}\left\\{\exp\left[\frac{1}{\delta}\left(Q(x^{n}_{j})^{2}\right.\right.\right.\left.\left.\left.+\int_{0}^{T}(q-\mathbb{E}(m^{n}_{t}))(x_{j}^{n})^{2}(t)+\bar{u}_{j}^{2}(t)dt\right)\right]\right\\},$
where $\delta,Q,q$ are positive parameters; hence coupling of the players is
only through the cost. The optimal strategy of Player $j$ has the form of
$\bar{u}_{j}^{*}(t)=-z(t)x,$ (13)
where $z(t)$ is a solution to the Riccati equation
$\dot{z}(t)+q-\mathbb{E}(m^{n})-z^{2}(t)(1-\sigma^{2}/\rho^{2})=0,$
with boundary condition $z(T)=Q$. An explicit solution is given by
$z(t)=-\frac{\sqrt{q-M}}{\sqrt{L}}\text{tan}\left[\sqrt{L}\sqrt{q-M}(t-T)+\right.\left.\text{arctan}\left(\frac{\sqrt{L}Q}{\sqrt{q-M}}\right)\right],0\leq
t\leq T,$
where $L:=1-\sigma^{2}/\rho^{2}$ and $M:=\mathbb{E}(m^{n})$. The FPK-McV
equation reduces to
$\partial_{t}m_{t}^{*}+\partial_{x}(m_{t}^{*}z(t)x(t))=\frac{\epsilon}{2}\sigma^{2}\partial_{xx}^{2}m^{*}_{t}.$
We set the parameters as follows: $q=1.2,Q=0.1$, $\delta=100,000$,
$\sigma=2.0$, $T=5$ and $\epsilon=5.0$. Let $m^{*}_{0}(x)$ be a normal
distribution $\mathcal{N}(1,1)$ and for every $0\leq t\leq T$, $m^{*}_{t}$
vanishes at infinity. In Figure 1, we show the evolution of the distribution
$m^{*}_{t}$ and in Figures 2 and 3, we show the mean and the variance of the
distribution which affects the optimal strategies in (13). The optimal linear
feedback $z(t)$ is illustrated in Figure 4. We can observe that the mean value
$\mathbb{E}(m_{t}^{*})$ monotonically decreases from 1.0 and hence the unit
cost on state is monotonically increasing. As the state cost increases, the
control effort becomes relatively cheaper and therefore we can observe an
increment in the magnitude of $z(t)$. However, when the mean value goes beyond
1.08, we observe that the control effort reduces to avoid undershooting in the
state.
### V-B McKean-Vlasov dynamics
We let the dynamics of an individual player be
$dx_{j}^{n}(t)=\left(\frac{\beta}{n}\sum_{i=1}^{n}x_{i}^{n}(t)+u_{j}^{n}(t)\right)dt+\sqrt{\epsilon}\sigma
d\mathbb{B}_{j}(t),$ (14)
and take the risk-sensitive cost function to be
$L=\delta\log
E\left\\{\exp\left[\frac{1}{\delta}\int_{0}^{T}q(x_{j}^{n}(t))^{2}+(u_{j}^{n}(t))^{2}\right]\right\\}.$
Note that the cost function is independent of other players’ controls or
states. As $n\rightarrow\infty$, under regularity conditions,
$\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{1}{n}x_{i}^{n}(t)=M(t),$
where $M(t)$ is the mean of the population. The feedback optimal control
$\bar{u}_{j}$ in response to the mean field $M(t)$ is characterized by
$\bar{u}_{j}(t)=-z(t)x_{j}(t)-k(t),$
where
$\displaystyle\dot{z}(t)+q-z^{2}(1-\sigma^{2}/\rho^{2})$ $\displaystyle=$
$\displaystyle 0,\ \ z(T)=0,$ $\displaystyle\dot{k}(t)-z(t)k(t)+z(t)M(t)$
$\displaystyle=$ $\displaystyle 0,\ \ k(T)=0,$
and $\rho^{2}=\frac{\delta}{2\epsilon}$ and
$M(t)=\int_{x\in\mathcal{X}}xm(x,t)dx$. The Fokker-Planck-Kolmogorov (FPK)
equation is
$\partial_{t}m(x,t)+\partial_{x}\left(m(x,t)\left(-z(t)x(t)-k(t)\right)+\beta
m(x,t)\int_{w}wm(w,t)dw\right)=\frac{\epsilon}{2}\sigma^{2}\partial^{2}_{xx}m(x,t)$
By solving the ODEs, we find that
$z(t)=-\sqrt{\bar{q}}\textrm{tan}\left(\sqrt{\hat{q}}(t-T)\right),\ \ 0\leq
t\leq T.$
where $\bar{q}=q/(1-\sigma^{2}/\rho^{2})$ and
$\hat{q}=q(1-\sigma^{2}/\rho^{2})$. Let $q=r=1$ and we find the solution
$k(t)=\textrm{cos}(t-T)\left(\int_{1}^{T}m(\tau)\textrm{sec}(T-\tau)\textrm{tan}(T-\tau)d\tau-\int_{t}^{T}m(\tau^{\prime})\textrm{sec}(T-\tau^{\prime})\textrm{tan}(T-\tau^{\prime})d\tau^{\prime}\right).$
Let $\sigma=1,\rho=2,\beta=1$ and we show in Figure 5 the evolution of the
probability density function $m(x,t)$. The mean $M(t)$ and the variance are
shown in Figure 6 and Figure 7, respectively.
Figure 5: Evolution of the probability density function $m(x,t)$ Figure 6: The
mean $M(t)$ under equilibrium solution Figure 7: Variance over time under
equilibrium solution
## VI Concluding remarks
We have studied risk-sensitive mean-field stochastic differential games with
state dynamics given by an Itô stochastic differential equation and the cost
function being the expected value of an exponentiated integral.
Using a particular structure of state dynamics, we have shown that the mean-
field limit of the individual state dynamics leads to a controlled macroscopic
McKean-Vlasov equation. We have formulated a risk-sensitive mean-field
response framework, and established its compatibility with the density
distribution using the controlled Fokker-Planck-Kolmogorov forward equation.
The risk-sensitive mean-field equilibria are characterized by coupled
backward-forward equations. For the general case, the resulting mean field
system is very hard to solve (numerically or analytically) even if the number
of equations have been reduced. We have, however, provided generic explicit
forms in the particular case of the affine-exponentiated-Gaussian mean-field
problem. In addition, we have shown that the risk-sensitive problem can be
transformed into a risk-neutral mean-field game problem with the introduction
of an additional fictitious player. This allows one to study a novel class of
mean field games, robust mean field games, under the Isaacs condition.
An interesting direction that we leave for future research is to extend the
model to accommodate multiple classes of players and a drift function which
may depend on the other players’ controls. Another direction would be to
soften the conditions under which Proposition 5 is valid, such as boundedness
and Lipschitz continuity, and extend the result to games with non-smooth
coefficients. In this context, one could address a mean field central limit
question on the asymptotic behavior of the process
$\sqrt{n}\mathbb{E}\left(\parallel
x^{n}_{j}(t)-\tilde{x}_{j}(t)\parallel\right).$ Yet another extension would be
to the time average risk-sensitive cost functional. Finally, the approach
needs to be compared with other risk-sensitive approaches such as the mean-
variance criterion and extended to the case where the drift is a function of
the state-mean field and the control-mean field.
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###### Proof of Proposition 5.
Under the stated standard assumptions on the drift $f$ and variance $\sigma$,
the forward stochastic differential equation has a unique solution adapted to
the filtration generated by the Brownian motions. We want to show that
$\mathbb{E}\left(\sup_{t\in[0,T]}\parallel
x^{n}_{j}(t)-\tilde{x}_{j}(t)\parallel\right)\leq\frac{C_{T}}{\sqrt{n}},$
where $C_{T}$ is a positive number which only depends on the bounds, $T$ and
the Lipschitz constants of the coefficients of the drifts and the variance
term. First we observe that for a fixed control $u,$ the averaging terms
$\frac{1}{n}\sum_{i=1}^{n}f(t,x_{j},u,x_{i})$ and
$\frac{1}{n}\sum_{i=1}^{n}\sigma(t,x_{j},u,x_{i})$ are measurable, bounded and
Lipschitz with the respect to the state and uniformly with the respect to
time.
Second, we observe that the bound on the Lipschitz constants of the
coefficients do not depend on the population size $n.$
Hence, $\int f(t,x,u,x^{\prime})\ m_{t}(dx^{\prime})$ and
$\int\sigma(t,x,u,x^{\prime})\ m_{t}(dx^{\prime})$ are bounded and Lipschitz
uniformly with the respect to $t.$ Moreover, these coefficients are
deterministic. This means that there is a unique solution to the limiting SDE
and that solution is measurable with the filtration generated by the mutually
independent Brownian motions.
Third, we evaluate the gap between the coefficients in order to obtain an
estimate of the two processes. We start by evaluating the gap
$\mathbb{E}\left(\left\|\frac{1}{n}\sum_{i=1}^{n}f(t,x,u,x_{i})-\int
f(t,x,u,x^{\prime})\ m_{t}(dx^{\prime})\right\|^{2}\right)$
Notice that $f$ returns a $k-$dimensional vector and $x$ belongs to
$\mathbb{R}^{k}$. By reordering the above expression (in $2-$norm), we obtain
$\sum_{l=1}^{k}\mbox{var}\left(\frac{1}{n}\sum_{i=1}^{n}f_{l}(t,x_{j},u,x_{i})\right)\leq\frac{k}{n}(1+\max_{l}b_{l})^{2}\leq\frac{C_{T}}{n},$
where $\mbox{var}(X)$ denotes the variance of $X$ and $b_{l}$ is a bound on
the $l-$th component of the drift term. (This exists because we have assumed
boundedness conditions on the coefficients).
Following a similar reasoning, we obtain the bounds on the second term in
$\sigma$, i.e.,
$\sum_{l,l^{\prime}}\mbox{var}\left(\frac{1}{n}\sum_{i=1}^{n}\sigma_{ll^{\prime}}(t,x_{j},u,x_{i})\right)\leq\frac{k}{n}(1+\max_{ll^{\prime}}c_{ll^{\prime}})^{2}\leq\frac{C_{T}}{n},$
where $c_{ll^{\prime}}$ is a bound on the entries $(l,l^{\prime})-$ of the
matrix $\sigma$.
Now we use the Lispchitz conditions and standard Gronwall estimates to deduce
that the mean of the quadratic gap between the two stochastic processes
(starting from $x$ at time $0$) is in order of $\frac{1}{n}.$
∎
###### Proof of Theorem 1.
Under the stated regularity and boundedness assumptions, there is a solution
to the McKean-Vlasov FPK equation. Suppose that (i) and (ii) are satisfied.
Then, $m_{t}=m^{*}(t,u^{*}(t))$ is the solution of the mean-field limit state
dynamics, i.e., the macroscopic McKean-Vlasov PDE when $m$ is substituted into
the HJB equation. By fixing $f^{*},c^{*},\sigma,$ we obtain a novel HJB
equation for the mean-field stochastic game. Since the new PDE admits a
solution according to (ii), the control $u^{*}(t)=u(t,x)$ minimizing
$\partial_{x}v\cdot f+c,$ is a best response to $m^{*}$ at time $t.$ The
optimal response of the individual player generates a mean-field limit which
in law is a solution of the FPK PDE and the players compute their controls as
a function of this mean-field. Thus, the consistency between the control, the
state and the mean field is guaranteed by assumption (i). It follows that
$(u^{*},m^{*})$ is a solution to the fixed-point problem i.e., a mean-field
equilibrium, and a strongly time-consistent one.
Now, we look at the quadratic instantaneous cost case. In that case, we obtain
the risk-sensitive equations provided in Proposition 3. The fact that any
convergent subsequence of best-response to $m^{n}$ is a best response to
$m^{*}$ and the fact that $u^{*}$ is an $\epsilon^{*}-$best response to the
mean-field limit $m^{*}$ follow from mean-field convergence of order
$O\left(\frac{1}{\sqrt{n}}\right)$ and the continuity of the risk-sensitive
quadratic cost functional. ∎
###### Proof of Theorem 3.
We provide a sufficient condition for the risk-sensitive mean field game to
have at most one smooth solution. Suppose $\delta>0,$ and $\sigma$ is positive
constant. Let $H$ be the Hamiltonian associated with the risk-neutral mean
field system. Then the Hamiltonian for the risk-sensitive mean field system is
$\tilde{H}(x,p,m)=H+(\frac{\epsilon\sigma^{2}}{2\delta})\parallel
p\parallel^{2}.$ Assume that the dependence on $m$ is local, i.e., it is
function of $m(x).$
The generic expression for the optimal control is given by
$u^{*}=\partial_{p}{H}(x,\partial_{x}v,m_{t}(x))$ (note that the generic
feedback control is expressed in terms of $H$, and not of $\tilde{H}$).
Suppose that there exist two smooth solutions $(\hat{v}_{1},\hat{m}_{1}),$
$(\hat{v}_{2},\hat{m}_{2})$ to the (normalized) risk-sensitive mean field
system. Now, consider the function
$t\longmapsto\int_{x\in\mathcal{X}}(\hat{v}_{2}(x)-\hat{v}_{1}(x))(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx.$
Observe that this function is $0$ at time $t=0$ because the measures coincide
initially, and the function is equal to $0$ at time $t=T$ because the final
values coincide. Therefore, the function will be identically $0$ in $[0,T]$ if
we show that it is monotone. This will imply that the integrand is zero, and
hence one of the two terms $(\hat{v}_{2}(x)-\hat{v}_{1}(x))$ or
$(\hat{m}_{2,t}(x)-\hat{m}_{1,t}(x))$ should be $0.$ Then, if the measures are
identical, we use the HJB equation to obtain the result. If the value
functions are identical, we can use the FPK equation to show the uniqueness of
the measure. Thus, it remains to find a sufficient condition for monotonicity,
that is, a sufficient condition under which the quantity
$\int_{x\in\mathcal{X}}(\hat{v}_{2}(x)-\hat{v}_{1}(x))(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
is monotone in time. We compute the following time derivative:
$S(t):=\frac{d}{dt}\left[\int_{x\in\mathcal{X}}(\hat{v}_{2}(x)-\hat{v}_{1}(x))(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx\right].$
We interchange the order of the integral and the differentiation and use time
derivative of a product to arrive at;
$\displaystyle S(t)$ $\displaystyle=$
$\displaystyle\int_{x\in\mathcal{X}}(\partial_{t}\hat{v}_{2}-\partial_{t}\hat{v}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx+$
$\displaystyle\int_{x\in\mathcal{X}}(\hat{v}_{2}-\hat{v}_{1})(\partial_{t}\hat{m}_{2}(x)-\partial_{t}\hat{m}_{1}(x))dx$
Now we expand the first term
$A:=\int_{x\in\mathcal{X}}(\partial_{t}\hat{v}_{2}-\partial_{t}\hat{v}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx.$
Consider the two HJB equations:
$\displaystyle\partial_{t}\hat{v}_{1}+\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1}(x))+\frac{1}{2}\sigma^{2}\partial^{2}_{xx}\hat{v}_{1}=0,$
$\displaystyle\partial_{t}\hat{v}_{2}+\tilde{H}(x,\partial_{x}\hat{v}_{2},\hat{m}_{2}(x))+\frac{1}{2}\sigma^{2}\partial^{2}_{xx}\hat{v}_{2}=0$
To compute $A$, we take the difference between the two HJB equations above and
multiply by $\hat{m}_{2}-\hat{m}_{1},$ which gives
$\partial_{t}\hat{v}_{2}-\partial_{t}\hat{v}_{1}=-\tilde{H}(x,\partial_{x}\hat{v}_{2},\hat{m}_{2})+\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})-\frac{1}{2}\sigma^{2}\partial^{2}_{xx}\hat{v}_{2}+\frac{1}{2}\sigma^{2}\partial^{2}_{xx}\hat{v}_{1}$
Hence,
$\displaystyle{A}$ $\displaystyle:=$
$\displaystyle\int_{x}[\partial_{t}\hat{v}_{2}-\partial_{t}\hat{v}_{1}](\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle=$
$\displaystyle-\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{2},\hat{m}_{2})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle-\int_{x}\frac{1}{2}\sigma^{2}\partial^{2}_{xx}(\hat{v}_{2})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\sigma^{2}\frac{1}{2}\partial^{2}_{xx}(\hat{v}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
Next we expand the second term
$B:=\int_{x\in\mathcal{X}}(\partial_{t}\hat{m}_{2}-\partial_{t}\hat{m}_{1})(\hat{v}_{2}-\hat{v}_{1})dx.$
Note that the Laplacian terms are canceled by integration by parts in the
expression $A+B$. By collecting all the terms in $A+B$, we obtain
$\displaystyle A+B$ $\displaystyle=$
$\displaystyle-\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{2},\hat{m}_{2})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\hat{m}_{2}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{2},\hat{m}_{2})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle-\int_{x}\hat{m}_{1}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
Letting $S(t)=A+B$, we introduce
$\hat{m}_{\lambda}:=(1-\lambda)\hat{m}_{1}+\lambda\hat{m}_{2}=\hat{m}_{1}+\lambda(\hat{m}_{2}-\hat{m}_{1}).$
The measure $\hat{m}_{\lambda}$ starts with $\hat{m}_{1}$ for the parameter
$\lambda=0$ and yields the measure $\hat{m}_{2}$ for $\lambda=1.$ Similarly
define
$\hat{v}_{\lambda}:=(1-\lambda)\hat{v}_{1}+\lambda\hat{v}_{2}.$
Introduce an auxiliary integral parameterized by $\lambda.$
$\displaystyle C_{\lambda}$ $\displaystyle:=$
$\displaystyle-\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\hat{m}_{\lambda}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})(\hat{m}_{\lambda}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\hat{m}_{\lambda}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})](\partial_{x}\hat{v}_{\lambda}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle-\int_{x}\hat{m}_{1}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})](\partial_{x}\hat{v}_{\lambda}-\partial_{x}\hat{v}_{1})dx$
Substituting the terms
$\hat{v}_{\lambda}-\hat{v}_{1}=\lambda(\hat{v}_{2}-\hat{v}_{1})$ and
$\hat{m}_{\lambda}-\hat{m}_{1}=\lambda(\hat{m}_{2}-\hat{m}_{1}),$ we obtain
$\displaystyle\frac{C_{\lambda}}{\lambda}$ $\displaystyle:=$
$\displaystyle-\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\tilde{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\hat{m}_{\lambda}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle-\int_{x}\hat{m}_{1}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{1},\hat{m}_{1})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
Using the continuity of the terms (of the RHS) above and the compactness of
$\mathcal{X},$ we deduce that
$\lim_{\lambda\longrightarrow 0}\frac{C_{\lambda}}{\lambda}=0.$
We next find a condition under which the one-dimensional function
$\lambda\longmapsto\frac{C_{\lambda}}{\lambda}$ is monotone in $\lambda.$ We
need to compute the variations of
$\frac{d}{d\lambda}\left(\frac{C_{\lambda}}{\lambda}\right).$
Suppose that $(x,p,m)\longmapsto\tilde{H}(x,p,m)$ is twice continuously
differentiable with the respect to $(p,m).$ Then,
$\displaystyle\frac{d}{d\lambda}\left(\frac{C_{\lambda}}{\lambda}\right)$
$\displaystyle=$
$\displaystyle-\int_{x}\left[\partial_{p}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})\right](\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle-\int_{x}\left[\partial_{m}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\hat{m}_{2}(x)-\hat{m}_{1}(x))\right](\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle+\int_{x}\partial_{\lambda}\left(\hat{m}_{\lambda}(x)[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})]\right)(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle\frac{d}{d\lambda}\left(\frac{C_{\lambda}}{\lambda}\right)$
$\displaystyle=$
$\displaystyle-\int_{x}\partial_{p}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle-\int_{x}\partial_{m}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\hat{m}_{2}(x)-\hat{m}_{1}(x))^{2}dx$
$\displaystyle+\int_{x}(\hat{m}_{2}-\hat{m}_{1})[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle+\int_{x}\hat{m}_{\lambda}\partial_{\lambda}[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
Computation of the term
$\hat{m}_{\lambda}(x)\partial_{\lambda}\left([\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})]\right)$
yields
$\displaystyle D_{\lambda}$ $\displaystyle=$
$\displaystyle\partial_{\lambda}[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})]$
$\displaystyle=$
$\displaystyle\partial^{2}_{pp}{H}.(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})+\partial^{2}_{mp}{H}.(\hat{m}_{2}-\hat{m}_{1})$
and we obtain
$\displaystyle\frac{d}{d\lambda}\left(\frac{C_{\lambda}}{\lambda}\right)$
$\displaystyle=$
$\displaystyle-\int_{x}\partial_{p}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})(\hat{m}_{2}(x)-\hat{m}_{1}(x))dx$
$\displaystyle-\int_{x}\partial_{m}\tilde{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})(\hat{m}_{2}(x)-\hat{m}_{1}(x))^{2}dx$
$\displaystyle+\int_{x}(\hat{m}_{2}-\hat{m}_{1})[\partial_{p}{H}(x,\partial_{x}\hat{v}_{\lambda},\hat{m}_{\lambda})](\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})dx$
$\displaystyle+\int_{x}\hat{m}_{\lambda}\partial^{2}_{pp}{H}.(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})^{2}+\hat{m}_{\lambda}\partial^{2}_{mp}{H}.(\hat{m}_{2}-\hat{m}_{1})(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1})$
The first and the third lines differ by
$-\int_{x}\left(\frac{\epsilon\sigma^{2}}{\delta}\langle.,\nabla_{x}\hat{v}\rangle\right)\left(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1}\right)\left(\hat{m}_{2}(x)-\hat{m}_{1}(x)\right)dx.$
Hence, we obtain
$\displaystyle\frac{d}{d\lambda}\left(\frac{C_{\lambda}}{\lambda}\right)$
$\displaystyle=$
$\displaystyle\int_{x}m_{\lambda}(\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1},\hat{m}_{2}-\hat{m}_{1})\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)\left(\begin{array}[]{c}\partial_{x}\hat{v}_{2}-\partial_{x}\hat{v}_{1}\\\
\hat{m}_{2}-\hat{m}_{1}\end{array}\right)dx,$ (19)
where
$a_{11}:=\partial_{pp}^{2}{H},$
$a_{21}:=\frac{1}{2}\partial_{mp}^{2}\tilde{H}=\frac{1}{2}\partial_{mp}^{2}H-\frac{\epsilon\sigma^{2}}{2\delta}p/m,$
$a_{21}:=\frac{1}{2}(\partial_{pm}^{2}\tilde{H})^{\prime}-\frac{\epsilon\sigma^{2}}{2\delta}\frac{p}{m}=\frac{1}{2}(\partial_{pm}^{2}H)^{\prime}-\frac{\epsilon\sigma^{2}}{2\delta}\frac{p}{m},$
$a_{22}:=-\frac{\partial_{m}\tilde{H}}{m}.$
Suppose that for all
$(x,p,m)\in\mathcal{X}\times\mathbb{R}^{d}\times\mathbb{R}_{+},$ the matrix
$\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)\succ 0.$
Then, the monotonicity follows, and this completes the proof. ∎
|
arxiv-papers
| 2012-10-10T05:29:07 |
2024-09-04T02:49:36.275618
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hamidou Tembine, Quanyan Zhu, Tamer Basar",
"submitter": "Quanyan Zhu",
"url": "https://arxiv.org/abs/1210.2806"
}
|
1210.2883
|
Predator-prey dynamics on infinite trees Extinction probability and total
progeny of predator-prey dynamics on infinite trees Charles Bordenave111CNRS &
Université Toulouse III, France. charles.bordenave@math.univ-toulouse.frSIR
models ; predator-prey dynamics ; branching processes 60J80 October 10, 2012
January 10, 2014 1210.2883 pvkyj 19 2014 20 v19-2361 We consider the spreading
dynamics of two nested invasion clusters on an infinite tree. This model was
defined as the chase-escape model by Kordzakhia and it admits a limit process,
the birth-and-assassination process, previously introduced by Aldous and
Krebs. On both models, we prove an asymptotic equivalent of the extinction
probability near criticality. In the subcritical regime, we give a tail bound
on the total progeny of the preys before extinction.
## 1 Introduction
The chase-escape process is a stochastic predator-prey dynamics which was
studied by Kordzakhia [15] on a regular tree. In an earlier paper, Aldous and
Krebs [4] had introduced the birth-and-assassination (BA) process. The latter
model can be seen as a natural limit of the chase-escape model. In [8] the two
models were merged into the rumor scotching process. The original motivation
of Aldous and Krebs was then to analyze a scaling limit of a queueing process
with blocking which appeared in database processing, see Tsitsiklis,
Papadimitriou and Humblet [24]. As pointed in [8], the BA process is also the
scaling limit of a rumor spreading model which is motivated by network
epidemics and dynamic data dissemination (see for example, [19], [5], [20]).
We may conveniently define the chase-escape processes as a SIR dynamics (see
for example [19] or [5] for some background on standard SIR dynamics). This
process represents the dynamics of a rumor/epidemic spreading on the vertices
of a graph along its edges. A vertex may be unaware of the rumor/susceptible
(S), aware of the rumor and spreading it as true/infected (I), or aware of the
rumor and trying to scotch it/recovered (R).
We fix a locally finite connected graph $G=(V,E)$. The chase-escape process is
described by a Markov process on $\mathcal{X}=\\{S,I,R\\}^{V}$. If
$\\{u,v\\}\in E$, we write $u\sim v$. For $v\in V$, we also define the
$\mathcal{X}\to\mathcal{X}$ maps $I_{v}$ and $R_{v}$ by : for $x=(x_{u})_{u\in
V}$, $(I_{v}(x))_{u}=(R_{v}(x))_{u}=x_{u}$, if $u\neq v$ and
$(I_{v}(x))_{v}=I$, $(R_{v}(x))_{v}=R$. Let $\lambda\in(0,1)$ be a fixed
infection intensity. We then define the Markov process with transition rates:
$\displaystyle K(x,I_{v}(x))$ $\displaystyle=$
$\displaystyle\lambda\mathbf{1}(x_{v}=S)\sum_{u{\sim}v}\mathbf{1}(x_{u}=I),$
$\displaystyle K(x,R_{v}(x))$ $\displaystyle=$
$\displaystyle\mathbf{1}(x_{v}=I)\sum_{u{\sim}v}\mathbf{1}(x_{u}=R),$
and all other transitions have rate $0$. In words, a susceptible vertex is
infected at rate $\lambda$ by its infected neighbors, and an infected vertex
is recovered at rate $1$ by its recovered neighbors. The absorbing states of
this process are the states without $I$-vertices or with only $I$ vertices. In
this paper, we are interested by the behavior of the process when at time $0$
there is a non-empty finite set of $I$ and $R$-vertices.
In [15], this model was described as a predator-prey dynamics: each vertex may
be empty (S), occupied by a prey (I) or occupied by a predator (R). The preys
spread on unoccupied vertices and predators spread on vertices occupied by
preys. If $G$ is the $\mathbb{Z}^{d}$-lattice and if there is no $R$-vertex,
the process is the original Richardson’s model [21]. With $R$-vertices, this
process is a variant of the two-species Richardson model with prey and
predators, see for example Häggström and Pemantle [12], Kordzakhia and Lalley
[16]. There is a growing cluster of (I)-vertices spreading over (S)-vertices
and a nested growing cluster of (R)-vertices spreading on (I)-vertices.
The chase-escape process differs from the classical SIR dynamics on the
transition from $I$ to $R$: in the classical SIR dynamics, a $I$-vertex is
recovered at rate $1$ independently of its neighborhood.
#### Chase-escape process on a tree
If the graph $G=T=(V,E)$ is a rooted tree, the process is much simpler to
study. We denote by ø the root of $T$. For the range of initial conditions of
interest (non-empty finite set of $I$ and $R$-vertices), there is no real loss
of generality to study the chase-escape process on the tree $T^{\downarrow}$
obtained from $T$ by adding a particular vertex, say $o$, connected to the
root of the tree. At time $0$, vertex $o$ is in state $R$, the root ø is in
state $I$, while all other vertices are in state $S$ (see figure 1). We shall
denote by $X(t)\in\\{S,I,R\\}^{V}$ our Markov process on the tree
$T^{\downarrow}$. Under $\mathbb{P}_{\lambda}$, $X$ is the chase escape
process on $T^{\downarrow}$ with infection rate $\lambda$.
$o$ø
Figure 1: The initial condition : the root is $I$, $o$ is $R$, all other
vertices are $S$.
We say that the Markov process $X$ gets extinct if at some (random) time
$\tau<\infty,$ there is no $I$-particle. Otherwise the process is said to
survive. We define the probability of extinction as
$q_{T}(\lambda)=\mathbb{P}_{\lambda}(X\hbox{ gets extinct}).$
Obviously, if $T$ is finite then $q_{T}(\lambda)=1$ for any $\lambda\geq 0$.
Before stating our results, we first need to introduce some extra terminology.
There is a canonical way to represent the vertex set $V$ as a subset of
$\mathbb{N}^{f}=\cup_{k=0}^{\infty}\mathbb{N}^{k}$ with
$\mathbb{N}^{0}=\text{\o}$ and $\mathbb{N}=\\{1,2\cdots\\}$. If $k\geq 1$ and
$v\in V$ is at distance $k$ from the root, then $v=(i_{1},\cdots,i_{k})\in
V\cap\mathbb{N}^{k}$. The genitor of $v$ is $(i_{1},\cdots,i_{k-1})$: it is
the first vertex on the path from $v$ to the root ø of length $k$. The
offsprings of $v$ are set of vertices who have genitor $v$. They are indexed
by $(i_{1},\cdots,i_{k},1),\cdots,(i_{1},\cdots,i_{k},n_{v})$, where $n_{v}$
is the number of offsprings of $v$. The ancestors of $v$ is the set of
vertices $(i_{1},\cdots,i_{\ell})$, $0\leq\ell\leq k-1$ with the convention
$i_{0}=\o$. Similarly, the _$n$ -th generation offsprings_ of $v$ are the
vertices in $V\cap\mathbb{N}^{k+n}$ of the form $(v,i_{k+1},\cdots,i_{k+n})$.
Recall that the _upper growth rate_ $d\in[1,\infty]$ of a rooted infinite tree
$T$ is defined as
$d=\limsup_{k\to\infty}|V_{k}|^{1/k},$
where $V_{k}=V\cap\mathbb{N}^{k}$ is the set of vertices at distance $k$ from
the root $\o$ and $|\cdot|$ denotes the cardinal of a finite set. The lower
growth rate is defined similarly with a $\liminf$. When the $\liminf$ and the
$\limsup$ coincide, this defines the _growth rate_ of the tree.
For example, for integer $d\geq 1$, we define the _$d$ -ary tree_ as the tree
where all vertices have exactly $d$ offsprings222It would be more proper to
call this tree the complete infinite $d$-ary tree.. Obviously, the $d$-ary
tree has growth rate $d$. More generally, consider a realization $T$ of a
Galton-Watson tree with mean number of offsprings $d\in(1,\infty)$. Then, the
Seneta-Heyde Theorem [23, 14] implies that, conditioned on $T$ infinite, the
growth rate of $T$ is a.s. equal to $d$. For background on random trees and
branching processes, we refer to [6, 22].
For integer $n\geq 1$, we define $T^{*n}$ as the rooted tree on $V$ obtained
from $T$ by putting an edge between all vertices and their $n$-th generation
offsprings. For real $d>1$, we say that $T$ is a _lower $d$-ary_ if for any
$1<\delta<d$, there exist an integer $n\geq 1$ and $v\in V$ such that the
subtree of the descendants of $v$ in $T^{*n}$ contains a
$\lceil{\delta}^{n}\rceil$-ary tree. Note that if $T$ is lower $d$-ary then
its lower growth rate is at least $d$. Also, if $T$ is the realization of a
Galton-Watson tree with mean number of offsprings $d\in(1,\infty)$ then,
conditioned on $T$ infinite, $T$ is a.s. lower $d$-ary (for a proof see Lemma
5.9 in appendix).
The first result is an extension of [15, Theorem 1] where it is proved for
$d$-ary trees. It describes the phase transition of the event of survival.
###### Theorem 1.1.
Let $d>1$ and
$\lambda_{1}=2d-1-2\sqrt{d(d-1)}.$
If $0<\lambda<\lambda_{1}$ and the upper growth rate of $T$ is at most $d$,
then $q_{T}(\lambda)=1$. If $\lambda>\lambda_{1}$ and $T$ is lower $d$-ary,
then $0<q_{T}(\lambda)<1$.
Note that in the classical SIR dynamics, it is easy to check that the critical
value of $\lambda$ is $\lambda=1/(d-1)$. Also, for any $d>1$, $\lambda_{1}<1$
and,
$\lambda_{1}\sim_{d\uparrow\infty}\frac{1}{4d}.$ (1)
The proof of Theorem 1.1 will follow a strategy parallel to [15, 4]. We employ
techniques akin to the study the infection process in the Richardson model.
They will be based on large deviation estimates on the probability that a
single vertex is $I$ at time $t$.
To our knowledge, there is no known closed form expression for the extinction
probability $q_{T}(\lambda)$. Our next result determines an asymptotic
equivalent for the probability of survival for $\lambda$ close to
$\lambda_{1}$. Our method does not seem to work on the sole assumption that
$T$ has growth rate $d>1$ and is lower $d$-ary. We shall assume that $T$ is a
realization of a Galton-Watson tree with offspring distribution $P$ and
$d=\sum_{k=1}^{\infty}kP(k)>1.$
We consider the annealed probability of extinction:
$q(\lambda)=\mathbb{E}^{\prime}[q_{T}(\lambda)]=\mathbb{P}^{\prime}_{\lambda}(X\hbox{
gets extinct}),$
where the expectation $\mathbb{E}^{\prime}(\cdot)$ is with respect to the
randomness of the tree and
$\mathbb{P}^{\prime}_{\lambda}(\cdot)=\mathbb{E}^{\prime}\left(\mathbb{P}_{\lambda}(\cdot)\right)$
is the probability measure with respect to the joint randomness of $T$ and
$X$. Note that in the specific case $d$ integer and $P(d)=1$, $T$ is the
$d$-ary tree and the measures $\mathbb{P}^{\prime}_{\lambda}$ and
$\mathbb{P}_{\lambda}$ coincide.
###### Theorem 1.2.
Assume further that the offspring distribution has finite second moment. There
exist constants $c_{0},c_{1}>0$ such that for all $\lambda_{1}<\lambda<1$,
$c_{0}\omega^{3}e^{-\frac{(1-\lambda_{1})\pi}{2(d(d-1))^{1/4}}\omega^{-1}}\leq
1-q(\lambda)\leq
c_{1}e^{-\frac{(1-\lambda_{1})\pi}{2(d(d-1))^{1/4}}\omega^{-1}},$
with
$\omega=\sqrt{\lambda-\lambda_{1}}.$
Note that the behavior depicted in Theorem 1.2 contrasts with the classical
SIR dynamics, where $1-q(\lambda)$ is of order $(\lambda(d-1)-1)_{+}$. This
result should however be compared to similar results in the Brunet-Derrida
model of branching random walk killed below a linear barrier, see Gantert, Hu
and Shi [11] and also Bérard and Gouéré [7]. As in this last reference, our
approach is purely analytic. We will first check that $q(\lambda)$ is related
to a second order non-linear differential equation. Then, we will rely on
comparisons with linear differential equations. A similar technique was
already used by Brunet and Derrida [9], and notably also in Mueller, Mytnik
and Quastel [18, section 2].
A possible parallel with the Brunet-Derrida model of branching random walk
killed below a linear barrier is the following. Consider a branching random
walk on $\mathbb{Z}$ started from a single particle at site $0$ where the
particles may only move by one step on the right. If we are only concerned by
the extinction, we can think of this process as some branching process without
walks where a particle at site $k$ gives birth to particles at site $k+1$. We
can in turn represent this process by a growing random tree where the set of
vertices at depth $k$ is the set of particles at site $k$. Hence (I)-vertices
play the role of the particles, the branching mechanism is the spreading of
the (I)-vertices over the (S)-vertices and the set of (R)-vertices is a
randomly growing barrier which absorbs the particles/(I)-vertices. Kortchemski
[17] has recently built an explicit coupling of a branching random walk with
the chase-escape process on a tree.
In the case $0<\lambda<\lambda_{1}$, the process $X$ stops a.s. evolving after
some finite $\tau$. We define $Z$ as the total progeny of the root, i.e. the
total number of recovered vertices (excluding the vertex $o$ of
$T^{\downarrow}$) at time $\tau$. It is the number of vertices which will have
been infected before the process reaches its absorbing state. We define the
annealed parameter:
$\gamma(\lambda)=\sup\left\\{u\geq
0:\mathbb{E}^{\prime}_{\lambda}[Z^{u}]<\infty\right\\}.$
The scalar $\gamma(\lambda)$ can be though as a power-tail exponent of the
variable $Z$ under the annealed measure $\mathbb{P}^{\prime}_{\lambda}$. In
particular, for any $0<\gamma<\gamma(\lambda)$, from Markov Inequality, there
exists a constant $c>0$ such that for all $t\geq 1$,
$\mathbb{P}^{\prime}_{\lambda}(Z\geq t)\leq ct^{-\gamma}.$ Conversely, if
there exist $c,\gamma>0$ such that for all $t\geq 1$,
$\mathbb{P}^{\prime}_{\lambda}(Z\geq t)\leq ct^{-\gamma}$, then
$\gamma(\lambda)\geq\gamma$. We define
$\gamma_{P}=\sup\left\\{u\geq
1:\sum_{k=1}^{\infty}k^{u}P(k)<\infty\right\\}\geq 1.$
###### Theorem 1.3.
* (i)
For any $0<\lambda<\lambda_{1}$,
$\gamma(\lambda)=\min\left(\frac{\lambda^{2}-2d\lambda+1-(1-\lambda)\sqrt{\lambda^{2}-2\lambda(2d-1)+1}}{2\lambda(d-1)},\gamma_{P}\right).$
* (ii)
Let $1\leq u<\gamma_{P}$, $A_{u}=u^{2}(d-1)+2ud+(d-1)$, and
$\lambda_{u}=\frac{A_{u}-\sqrt{A_{u}^{2}-4u^{2}}}{2u}.$
If $\lambda<\lambda_{u}$ then $\mathbb{E}^{\prime}_{\lambda}[Z^{u}]$ is
finite. If $\lambda>\lambda_{u}$, $\mathbb{E}^{\prime}_{\lambda}[Z^{u}]$ is
infinite.
It is straightforward to check that $(i)$ is equivalent to $(ii)$. Also, for
$u=1$, $\lambda_{u}$ coincides with $\lambda_{1}$ defined in Theorem 1.1. It
follows that $\gamma(\lambda)\geq 1$ for all $0<\lambda<\lambda_{1}$. Theorem
1.3 contrasts with classical SIR dynamics. For example, if $T$ is the $d$-ary
tree, for all $\lambda<1/(d-1)$ there exists a constant $c>0$ such that
$\mathbb{E}^{\prime}_{\lambda}\exp(cS)<\infty$ where $S$ is the total progeny
in the classical SIR dynamics. Here, the heavy-tail phenomenon is an
interesting feature of the chase-escape process. Intuition suggest that large
values of $Z$ come from a (I)-vertex which is not recovered before an
exceptionally long time. Indeed, in the chase escape process, a (I)-vertex
which is not recovered by time $t$ will typically have a progeny which is
exponentially large in $t$ (this is not the case in the classical sub-critical
SIR dynamics, the progeny of such vertex will typically be of order $1$) . A
similar phenomenon appears also in the Brunet-Derrida model, see Addario-Berry
and Broutin [1], Aïdékon [2] and Aïdékon Hu and Zindy [3]. Note finally that
$\gamma(\lambda)\sim_{\lambda\downarrow
0}\min{{\left(\frac{1}{(d-1)\lambda},\gamma_{P}\right)}}\quad\hbox{ and
}\quad\gamma(\lambda)\sim_{\lambda\uparrow\lambda_{1}}1.$
By recursion, we will also compute the moments of $Z$. The computation of the
first moment gives
###### Theorem 1.4.
If $0<\lambda\leq\lambda_{1}$ and $\Delta=\lambda^{2}-2\lambda(2d-1)+1$, then
$\mathbb{E}^{\prime}_{\lambda}[Z]=\frac{2d}{(d-1)(1+\lambda+\sqrt{\Delta})}-\frac{1}{d-1}.$
Theorem 1.4 implies a surprising right discontinuity of the function
$\lambda\mapsto\mathbb{E}^{\prime}_{\lambda}Z$ at the critical intensity
$\lambda=\lambda_{1}$:
$\mathbb{E}^{\prime}_{\lambda_{1}}Z=2d/((d-1)(1+\lambda_{1}))-1/(d-1)<\infty$.
Again, this discontinuity contrasts with what happens in a standard Galton-
Watson process near criticality, where for $0<\lambda<1/(d-1)$,
$\mathbb{E}^{\prime}_{\lambda}Z$ is of order $(1-(d-1)\lambda)^{-1}$. From
Theorem 1.4, we may fill the gap in Theorem 1.1 in the specific case of a
realization of a Galton-Watson tree.
###### Corollary 1.5.
Let $T$ be a Galton-Watson tree with mean number of offsprings $d$. Then a.s.
$q_{T}(\lambda_{1})=1$.
The method of proofs of Theorems 1.3-1.4 will be parallel to arguments in [8]
on the birth-and-assassination process.
#### The birth-and-assassination process
We now turn to the BA process. It is a scaling limit in $d\to\infty$ of the
chase-escape process on the $d$-ary tree when $\lambda$ is rescaled in
$\lambda/d$.
Informally, the process can be described as follows. We start from a root
vertex that produces offsprings according to a Poisson process of rate
$\lambda.$ Each offspring in turn produces children according to independent
Poisson processes and so on. The children of the root are said to belong to
the first generation and their children to the second generation and so forth.
Independently, the root vertex is _at risk_ at time $0$ and dies after a
random time $D_{\text{\o}}$ that is exponentially distributed with mean $1$.
Its offsprings become at risk after time $D_{\text{\o}}$ and the process
continues in the next generations. We now make precise the above description.
As above, $\mathbb{N}^{f}=\cup_{k=0}^{\infty}\mathbb{N}^{k}$ denotes the set
of finite $k-$tuples of positive integers (with $N^{0}=\text{\o}$). Elements
from this set are used to index the offspring in the BA process. Let
$\\{\Xi_{v}\\},v\in\mathbb{N}^{f}$, be a family of independent Poisson
processes with common arrival rate $\lambda$; these will be used to define the
offsprings. Let $\\{D_{v}\\},v\in\mathbb{N}^{f}$, be a family of independent,
identically distributed (iid) exponential random variables with mean $1;$ we
use them to assign the lifetime for the appropriate offspring. The families
$\\{\Xi_{v}\\}$ and $\\{D_{v}\\}$ are independent. The process starts at time
$0$ with only the root, indexed by ø. This produces offspring at the arrival
times determined by $\Xi_{\text{\o}}$ that enter the system with indices
$(1)$, $(2)$, $\cdots$ according to their birth order. Each new vertex $v,$
immediately begins producing offspring determined by the arrival times of
$\Xi_{v}.$ The offspring of $v$ are indexed $(v,1)$, $(v,2)$, $\cdots$ also
according to birth order. The root is at risk at time $0$. It continues to
produce offspring until time $T_{\text{\o}}=D_{\text{\o}}$, when it dies. Let
$k>0$ and let $v=(n_{1},\cdots,n_{k-1},n_{k})$,
$v^{\prime}=(n_{1},...,n_{k-1})$ denote a vertex and its genitor. When a
particle $v^{\prime}$ dies (at time $T_{v^{\prime}}$), the particle $v$ then
becomes at risk; it in turn continues to produce offspring until time
$T_{v}=T_{v^{\prime}}+D_{v}$, when it dies (see figure 2).
The BA process can be equivalently described as a Markov process $X(t)$ on
$\\{S,I,R\\}^{\mathbb{N}^{f}}$, where a particle/vertex in state $S$ is not
yet born, a particle in state $I$ is alive and a particle in state $R$ is
dead. A particle is at risk if it is in state $I$ and its genitor is in state
$R$. We use the same notation as above : under $\mathbb{P}_{\lambda}$, the
process $X(t)$ has infection rate $\lambda>0$, $q(\lambda)$ is the probability
of extinction and so on.
Figure 2: Illustration of the birth-and-assassination process, living
particles are in red, dead particles in blue, particles at risk are encircled.
The following result from [4] describes the phase transition on the
probability of survival as a function of $\lambda$.
###### Theorem 1.6.
Consider the BA process with rate $\lambda>0$. If $\lambda\in[0,1/4],$ then
$q(\lambda)=1$, while if $\lambda>\frac{1}{4},$ $0<q(\lambda)<1.$
The critical case $\lambda=1/4$ was established in [8]. Note also that the
threshold $\lambda=1/4$ is consistent with (1).
Our final result is the analog of Theorem 1.2.
###### Theorem 1.7.
Consider the BA process and assume that $\lambda>1/4$. There exist constants
$c_{0},c_{1}>0$ such that for all $1/4<\lambda<1$,
$c_{0}\omega^{3}e^{-\frac{\pi}{2}\omega^{-1}}\leq 1-q(\lambda)\leq
c_{1}\omega^{-1}e^{-\frac{\pi}{2}\omega^{-1}},$
with
$\omega=\sqrt{\lambda-\frac{1}{4}}.$
Note that the analog of Theorems 1.3-1.4 was already performed in [8]. The
remainder of the paper is organized as follows. In section 2, we start with
the study of the BA process and prove Theorem 1.7. Proofs on the BA process
are simpler and this section is independent of the rest of the paper. We then
come back to the chase escape process: in section 3, we prove Theorem 1.1, in
section 4, we prove Theorem 1.2. Finally, in section 5, we prove Theorems
1.3-1.4.
## 2 Proof of Theorem 1.7
### 2.1 Differential equation for the survival probability
We first determine the differential equation associated to the probability of
extinction for the BA process. Define $Q_{\lambda}(t)$ to be the extinction
probability given that the root dies at time $t>0$ so that
$q(\lambda)=\int_{0}^{\infty}Q_{\lambda}(t)e^{-t}dt$ (2)
and $Q_{\lambda}(0)=1$. Let $\\{\xi_{i}\\}_{i\geq 1}$ be the arrival times of
$\Xi_{\text{\o}}$ with $0\leq\xi_{1}\leq\xi_{2}\leq\cdots$. For integer $i$
with $1\leq\xi_{i}\leq D_{\text{\o}}$, we define $\mathcal{B}_{i}$ as the
subprocess on particles $i\mathbb{N}^{f}$ with ancestors $i$. For the process
$\mathcal{B}$ to get extinct, all the processes $\mathcal{B}_{i}$ must get
extinct. Conditioned on $\Xi_{\text{\o}}$, and on the root to die at time
$D_{\text{\o}}=t$, the evolutions of the $(\mathcal{B}_{i})$ then become
independent. Moreover, on this conditioning, $\mathcal{B}_{i}$ is a birth-and-
assassination process conditioned on their root to be at risk at time
$t-\xi_{i}$. Hence, we get
$Q_{\lambda}(t)=\mathbb{E}_{\lambda}{{\left[\prod_{\xi_{i}\leq
t}Q_{\lambda}(t-\xi_{i}+D_{i})\right]}}=\mathbb{E}_{\lambda}{{\left[\prod_{\xi_{i}\leq
t}Q_{\lambda}(\xi_{i}+D_{i})\right]}},$
where $\\{\xi_{i}\\}_{i\geq 1}$ is a Poisson point process of intensity
$\lambda$ and $(D_{i}),i\geq 1,$ independent exponential variables with
parameter $1$. Using Lévy-Khinchin formula, we deduce
$\displaystyle Q_{\lambda}(t)$ $\displaystyle=$
$\displaystyle\exp{{\left(\lambda\int_{0}^{t}(\mathbb{E}Q_{\lambda}(x+D_{1})-1)dx\right)}}$
$\displaystyle=$
$\displaystyle\exp{{\left(\lambda\int_{0}^{t}\int_{0}^{\infty}(Q_{\lambda}(x+s)-1)e^{-s}dsdx\right)}}.$
So finally, for any $t\geq 0$,
$\displaystyle Q_{\lambda}(t)$ $\displaystyle=$
$\displaystyle\exp{{\left(-\lambda
t+\lambda\int_{0}^{t}e^{x}\int_{x}^{\infty}Q_{\lambda}(s)e^{-s}dsdx\right)}}.$
We perform the change of variable
$x(t)=-\ln Q_{\lambda}(t).$ (3)
We find that for any $t\geq 0$,
$\displaystyle
x(t)=\int_{0}^{t}e^{x}\int_{x}^{\infty}\varphi(x(s))e^{-s}dsdx,$ (4)
where
$\varphi(y)=\lambda(1-e^{-y}).$
Differentiating (4) once gives
$x^{\prime}(t)=e^{t}\int_{t}^{\infty}\varphi(x(s))e^{-s}ds,$ (5)
Now, multiplying the above expression by $e^{-t}$ and differentiating once
again, we find that $x(t)$ satisfies the differential equation
$x^{\prime\prime}-x^{\prime}+\varphi(x)=0.$ (6)
This non-linear ordinary differential equation is not a priori easy to solve.
However, in the neighborhood of $\lambda=1/4$ it is possible to obtain an
asymptotic expansion as explained below. The idea will be to linearize the ODE
near $(x(0),x^{\prime}(0))=(0,0)$ and look at the long time behavior of the
solutions of the linearized ODE. The critical value $\lambda=1/4$ appears to
be the threshold for oscillating solutions of the linearized ODE. From a
priori knowledge on the long time behavior of the solution of (6) of interest
(studied in §2.2), we will obtain an asymptotic equivalent for
$(x(0),x^{\prime}(0))$ as $\lambda\downarrow 1/4$ (in §2.4).
### 2.2 A fixed point equation
Let $\mathcal{H}$ be the set of measurable functions
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $f(0)=0$ and for any $a>0$,
$\lim_{s\to\infty}e^{-as}f(s)=0.$
We define the map $A:\mathcal{H}\to\mathcal{H}$ defined by
$A(y)(t)=\int_{0}^{t}e^{x}\int_{x}^{\infty}\varphi(y(s))e^{-s}dsdx.$ (7)
Using $\|\varphi\|_{\infty}=\lambda<\infty$, it is straightforward to check
that $A(y)$ is indeed an element of $\mathcal{H}$ ($A(y)(t)$ it is bounded by
$\|\varphi\|_{\infty}t$). Note also that $y\equiv 0$ is a solution of the
fixed point equation
$y=A(y).$
Consider the function $x$ defined by (3). Using (4) we find that
$x\in\mathcal{H}$ and satisfies also the fixed point $x=A(x)$. If
$\lambda>1/4$, we know from Theorem 1.6 that $x\not\equiv 0$.
In the sequel, we are going to study any non-trivial fixed point of $A$. To
this end, let $x\in\mathcal{H}$ such that $x=A(x)$ and $x\not\equiv 0$. By
induction, it follows easily that $t\mapsto x(t)$ is twice continuously
differentiable. In particular, since $x(s)\geq 0$, $x^{\prime}(t)\geq 0$ and
the function $x:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is non-decreasing. Moreover,
by assumption there exists $t_{0}>0$ such that $x(t_{0})>0$. Since $x$ is non-
decreasing, we deduce that $x(t)>0$ for all $t>t_{0}$. Then, using again (5),
we find that for all $t\geq 0$,
$0<x^{\prime}(t)<\lambda.$ (8)
From the argument leading to (6), $x$ satisfies (6) and we are looking for a
specific non-negative solution of (6) which satisfies $x(0)=0$. To
characterize completely this solution, it would be enough to compute
$x^{\prime}(0)$ (which is necessary positive since $x(0)=x^{\prime}(0)=0$
corresponds to the trivial solution $x\equiv 0$). We first give some basic
properties of the phase portrait, see figure 3. We define
$X(t)=(x(t),x^{\prime}(t))$ so that
$X^{\prime}=F(X)$ (9)
with $F((x_{1},x_{2}))=(x_{2},x_{2}-\varphi(x_{1}))$. We also introduce the
set
$\Delta=\\{(x_{1},x_{2})\in\mathbb{R}_{+}^{2}:\varphi(x_{1})<x_{2}<\lambda\\}.$
$0$$\lambda$$x$$x^{\prime}$
Figure 3: Illustration of the phase portrait. In blue, the curve $L$, in red
the curve $\Phi$ of Lemma 2.1.
###### Lemma 2.1.
Let $x\in\mathcal{H}$ such that $x=A(x)$ and $x\not\equiv 0$. Then
$x^{\prime}(0)>0$, $x$ satisfies (6) and for all $t\geq 0$,
$X(t)\in\,\Delta.$ (10)
Moreover
$\lim_{t\to\infty}x^{\prime}(t)=\lambda.$
###### Proof 2.2.
We have already checked that $x$ satisfies (6) and $x^{\prime}(0)>0$. Let us
now prove that (10) holds. Define the trajectory
$\Phi=\\{X(t)\in\mathbb{R}_{+}^{2}:t\geq 0\\}$. Since for all $t\geq 0$,
$X(t)^{\prime}_{1}=F(X(t))_{1}>0$, $\Phi$ is the graph of a differentiable
function $f:[0,S)\to\mathbb{R}_{+}$ with $f(0)=x^{\prime}(0)>0$:
$\Phi=\\{(s,f(s)):s\in[0,S)\\},$
with $S=\lim_{t\to\infty}x(t)\in(0,\infty]$, see figure 3. Moreover
$f^{\prime}(s)=\frac{F((s,f(s)))_{2}}{F((s,f(s)))_{1}}=\frac{f(s)-\varphi(s)}{f(s)}.$
(11)
The graph of the function $\varphi$ is the curve
$L=\\{(s,\varphi(s)):s\in\mathbb{R}_{+}\\}$ and the set
$\Delta^{\prime}=\\{(x_{1},x_{2})\in[0,\infty)^{2}:x_{2}<\varphi(x_{1})\\}$
is the set of points below $L$. Assume that (10) does not hold. Then by (8)
and the intermediate value Theorem, the curves $L$ and $\Phi$ intersect. Then
the exists $s_{0}>0$ such that
$f(s_{0})=\varphi(s_{0}).$
From (11), $f^{\prime}(s_{0})=0$ while $\varphi^{\prime}(s_{0})>0$. It follows
that $(s,f(s))\in\Delta^{\prime}$ for all $s\in(s_{0},s_{1})$ for some
$s_{1}>s_{0}$. Since $f^{\prime}(s)<0$ if $(s,f(s))\in\Delta^{\prime}$ while
$\varphi^{\prime}(s)>0$, the curves $L$ and $\Phi$ cannot intersect again. We
get that all $s>s_{0}$, $(s,f(s))\in\Delta^{\prime}$.
On the other hand, since $f^{\prime}(s)<0$, for all $s>s_{1}$,
$f(s)<f(s_{1})<\varphi(s_{1})$. If $x(t_{1})=s_{1}$ and
$\delta=\varphi(s_{1})-f(s_{1})>0$, we deduce that for all $t>t_{1}$ ,
$x^{\prime\prime}(t)=x^{\prime}(t)-\varphi(x(t))<-\delta$. Integrating, this
implies that $\lim_{t\to\infty}x^{\prime}(t)=-\infty$ which contradicts (8).
We have proved so far that for all $t\geq 0$, $X(t)\in\Delta$. This implies
that $x^{\prime}(t)$ is increasing. In particular
$\lim_{t\to\infty}x(t)=\infty$ and $S=\infty$. Since
$\lim_{s\to\infty}\varphi(s)=\lambda$, by (5), we readily deduce that
$x^{\prime}(t)$ converges to $\lambda$ as $t\to\infty$.
### 2.3 Comparison of second order differential equations
It is possible to compare the trajectories of solutions of second order ODE by
using the phase diagram. Let $\mathcal{D}$ be the set of increasing Lipschitz-
continuous functions $\psi$ on $\mathbb{R}_{+}$ such that $\psi(0)=0$. For two
functions $\psi_{1},\psi_{2}$ in $\mathcal{D}$, we write
$\psi_{1}\leq\psi_{2}$ if for all $t\geq 0$, $\psi_{1}(t)\leq\psi_{2}(t)$.
###### Lemma 2.3.
Let $x\in\mathcal{H}$ such that $x=A(x)$ and $x\not\equiv 0$. Let
$\psi\in\mathcal{D}$ and $y$ be a solution of
$y^{\prime\prime}-y^{\prime}+\psi(y)=0$ with $y(0)=0$, $y^{\prime}(0)>0$. We
define the exit times
$T=\inf\\{t\geq 0:(y(t),y^{\prime}(t))\notin\mathbb{R}^{2}_{+}\\},$
$T_{+}=\inf\\{t\geq 0:y^{\prime}(t)\geq\lambda\\}\quad\hbox{ and }\quad
T_{-}=\inf\\{t\geq 0:\varphi(y(t))\leq y^{\prime}(t)\\}.$
1. (i)
If $T_{+}<T$, $T_{+}<\infty$ and $\varphi\leq\psi$ then $y^{\prime}(0)\geq
x^{\prime}(0)$.
2. (ii)
If $T_{-}<T$, $T_{-}<\infty$ and $\psi\geq\varphi$ then $x^{\prime}(0)\geq
y^{\prime}(0)$.
###### Proof 2.4.
Let us start with the hypothesis of (i). The proof is by contradiction : we
also assume that $y^{\prime}(0)<x^{\prime}(0)$. We set
$Y(t)=(y(t),y^{\prime}(t))$ and $G(y_{1},y_{2})=(y_{2},y_{2}-\psi(y_{1}))$.
Define the trajectories $\Phi=\\{X(t)\in\mathbb{R}_{+}^{2}:t\geq 0\\}$, and
for $\tau>0$, $\Psi(\tau)=\\{Y(t)\in\mathbb{R}_{+}^{2}:0\leq t\leq\tau\\}$. By
Lemma 2.1, $\Phi$ is the graph of an increasing function
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $f(0)=x^{\prime}(0)>0$ and
$\Phi=\\{(s,f(s)):s\in\mathbb{R}_{+}\\}.$
Similarly, if $t\in[0,T)$, $y^{\prime}(t)>0$. Thus, there exists a
differentiable function $g:[0,y(T)]\to\mathbb{R}_{+}$ such that
$\Psi(T)=\\{(s,g(s)):s\in[0,y(T)]\\},$
with
$g^{\prime}(s)=1-\frac{\psi(s)}{g(s)}.$
Now, the assumption $0<y^{\prime}(0)<x^{\prime}(0)$ reads $0<g(0)<f(0)$. Since
$T_{+}<T$, for $s\in[0,T_{+}]$, $g(s)>0$ and there is a time $s_{0}>0$ such
that $g(s_{0})\geq\lambda$. In particular, by (8), $f(s_{0})<g(s_{0})$. Hence,
by the intermediate value Theorem, there exists a first time $0<s_{1}<s_{0}$
such that the curves intersect: $g(s_{1})=f(s_{1})$ and $g(s)<f(s)$ on
$[0,s_{1})$. However, it follows from (11) and $\varphi\leq\psi$ that for
$s\in[0,s_{1})$,
$g^{\prime}(s)=1-\frac{\psi(s)}{g(s)}\leq
1-\frac{\varphi(s)}{g(s)}<1-\frac{\varphi(s)}{f(s)}=f^{\prime}(s).$
Hence, integrating over $[0,s_{1}]$ the above inequality gives
$g(s_{1})-g(0)=\int_{0}^{s_{1}}g^{\prime}(s)ds<\int_{0}^{s_{1}}f^{\prime}(s)ds=f(s_{1})-f(0).$
However, by construction, $f(s_{1})=g(s_{1})$. Thus, the above inequality
contradicts $g(0)<f(0)$ and we have proved (i). The proof of (ii) is identical
and is omitted.
### 2.4 Proof of Theorem 1.7
We first linearize (6) with $\varphi(s)=\lambda(1-e^{-s})$ in the neighborhood
of $\lambda=1/4$.
#### Step one : Linearization from above.
We have $\varphi^{\prime}(0)=\lambda$, and from the concavity of $\varphi$,
$\varphi(s)\leq\lambda s.$ (12)
We take $\lambda>1/4$ and consider the linearized differential equation
$y^{\prime\prime}-y^{\prime}+\lambda y=0.$ (13)
The solutions of this differential equation are
$y(t)=a\sin(\omega t)e^{\frac{t}{2}}+b\cos(\omega t)e^{\frac{t}{2}},$
with
$\omega=\sqrt{\lambda-\frac{1}{4}}.$
We use this ODE to upper bound $x^{\prime}(0)$ if $x=A(x)$. Recall that $A$
depends implicitly on $\lambda$.
###### Lemma 2.5.
For any $\lambda>1/4$, let $x\in\mathcal{H}$ such that $x=A(x)$ and
$x\not\equiv 0$. We have
$x^{\prime}(0)\leq\frac{e^{2}}{4}e^{-\frac{\pi}{2\omega}}(1+O(\omega^{2})).$
###### Proof 2.6.
Let $a>0$ and consider the function
$y(t)=a\sin(\omega t)e^{\frac{t}{2}}.$ (14)
We have $y(0)=0$, $y^{\prime}(0)=a\omega$,
$y^{\prime}(t)=ae^{\frac{t}{2}}(\omega\cos(\omega t)+\frac{1}{2}\sin(\omega
t)),$ (15)
and
$y^{\prime\prime}(t)=ae^{\frac{t}{2}}{{\left(\omega\cos(\omega
t)+{{\left(\frac{1}{4}-\omega^{2}\right)}}\sin(\omega t)\right)}}.$
Define
$\tau=\frac{\pi}{\omega}-\frac{1}{\omega}\arctan{{\left(\frac{\omega}{\frac{1}{4}-\omega^{2}}\right)}}=\frac{\pi}{\omega}-4+O(\omega^{2}).$
(16)
On the interval $[0,\tau]$, $y^{\prime\prime}(t)\geq 0$ and
$y^{\prime\prime}(\tau)=0$. Thus the function $y^{\prime}(t)$ is increasing on
$[0,\tau]$. Moreover, since $\cos(\omega\tau)=-1+O(\omega^{2})$ and
$\sin(\omega\tau)=4\omega+O(\omega^{3})$, we get from (15) that
$y^{\prime}(\tau)=e^{-2}ae^{\frac{\pi}{2\omega}}(\omega+O(\omega^{3})).$
Using (16), we have $\exp(\tau/2)=\exp(\pi/(2\omega)-2)(1+O(\omega^{2}))$.
Hence, we may choose $a$ in (14) such that
$y^{\prime}(\tau)=\lambda=\frac{1}{4}+\omega^{2}$ with
$a=\frac{e^{2}}{4}\frac{e^{-\frac{\pi}{2\omega}}}{\omega}(1+O(\omega^{2})).$
From what precedes, on the interval $[0,\tau]$, $y(t)>0$ and
$y^{\prime}(t)>0$. From (12), we may thus use Lemma 2.3$(i)$ with $T_{+}=\tau$
and $\psi(s)=\lambda s$. We get $x^{\prime}(0)\leq y^{\prime}(0)=a\omega$.
#### Step two : linearization from below.
For $0<\kappa<1$, we define
$\ell=\frac{1}{4}+\kappa^{2}\omega^{2}<\lambda,$
and the function in $\mathcal{D}$
$\psi(s)=\min\left(\ell s,\varphi(s)\right).$
In particular
$\varphi\geq\psi.$ (17)
We shall now consider the linear differential equation
$y^{\prime\prime}-y^{\prime}+\ell y=0,$ (18)
The solutions of (18) are
$y(t)=a\sin(\omega\kappa t)e^{\frac{t}{2}}+b\cos(\omega\kappa
t)e^{\frac{t}{2}}.$
A careful choice of $a,\kappa$ will lead to the following lower bound.
###### Lemma 2.7.
For any $\lambda>1/4$, let $x\in\mathcal{H}$ such that $x=A(x)$ and
$x\not\equiv 0$. We have
$x^{\prime}(0)\geq\frac{8e}{\pi}\omega^{3}e^{-\frac{\pi}{2\omega}}(1+O(\omega^{2})).$
###### Proof 2.8.
For $a>0$, we look at the solution
$y(t)=a\sin(\omega\kappa t)e^{\frac{t}{2}}.$ (19)
We have $y(0)=0$, $y^{\prime}(0)=a\kappa\omega$.
$y^{\prime}(t)=ae^{\frac{t}{2}}(\omega\kappa\cos(\omega\kappa
t)+\frac{1}{2}\sin(\omega\kappa t)).$
We repeat the argument of Lemma 2.5. On the interval $[0,\tau]$,
$y^{\prime\prime}(t)\geq 0$ and $y^{\prime\prime}(\tau)=0$, where
$\tau=\frac{\pi}{\omega\kappa}-\frac{1}{\omega\kappa}\arctan{{\left(\frac{\omega\kappa}{\frac{1}{4}-\omega^{2}\kappa^{2}}\right)}}=\frac{\pi}{\omega\kappa}-4+O(\omega^{2}),$
(20)
and the $O(\cdot)$ is uniform over all $\kappa>1/2$. The function
$y^{\prime}(t)$ is increasing on $[0,\tau]$ and
$y^{\prime}(\tau)=ae^{-2}e^{\frac{\pi}{2\omega}}\omega\kappa(1+O(\omega^{2})).$
Now, we have $\ell s\leq\varphi(s)$ for all $s\in[0,\sigma]$ with
$\ell\sigma=\lambda(1-e^{-\sigma}).$
It gives
$\sigma=2{{\left(1-\frac{\ell}{\lambda}\right)}}+O{{\left(1-\frac{\ell}{\lambda}\right)}}^{2}=8(1-\kappa^{2})\omega^{2}+O((1-\kappa^{2})\omega^{4}).$
However from (18), for $t=\tau$, since $y^{\prime\prime}(\tau)=0$, we have
$\frac{y^{\prime}(\tau)}{y(\tau)}=\ell=\frac{1}{4}+\kappa^{2}\omega^{2}.$
From (20), we have $\sin(\omega\kappa\tau)=4\omega\kappa+O(\omega^{3})$ and
$\exp(\tau/2)=\exp(\pi/(2\omega\kappa)-2)(1+O(\omega^{2}))$. In (19), we may
thus choose $a$ such that $y(\tau)=\sigma$ by setting
$a=\sigma\frac{e^{2}}{4}\frac{e^{-\frac{\pi}{2\omega\kappa}}}{\omega\kappa}(1+O(\omega^{2}))=2e^{2}e^{-\frac{\pi}{2\omega\kappa}}\frac{(1-\kappa^{2})\omega}{\kappa}(1+O(\omega^{2})).$
Now, in the domain $0\leq y\leq\sigma$, $\psi(y)=\ell\sigma$ and the non-
linear differential equation $y^{\prime\prime}-y^{\prime}+\psi(y)$ obviously
coincides with (18). Thus, using (17) and Lemma 2.3$(ii)$ with $T_{-}=\tau$,
it leads to
$x^{\prime}(0)\geq
y^{\prime}(0)=a\kappa\omega=2e^{2}e^{-\frac{\pi}{2\omega\kappa}}(1-\kappa^{2})\omega^{2}(1+O(\omega^{2})).$
Taking finally $\kappa=1-2\omega/\pi$ gives the statement.
#### Step three : End of proof.
We now complete the proof of Theorem 1.7. We should consider the function
$x(t)$ defined by (3). We have seen that $x=A(x)$ and $x\not\equiv 0$ if
$\lambda>1/4$. We start with the left hand side inequality. From (10) in Lemma
2.1, $x^{\prime}(t)$ is increasing and we have
$x(t)\geq x^{\prime}(0)t.$
It follows from (2) that
$q(\lambda)=\int_{0}^{\infty}e^{-x(t)}e^{-t}dt\leq\int_{0}^{\infty}e^{-x^{\prime}(0)t}e^{-t}dt=\frac{1}{1+x^{\prime}(0)}.$
It remains to use Lemma 2.7 and we obtain the left hand side of Theorem 1.7.
We now turn the right hand side inequality. For
$X=(x_{1},x_{2})\in\mathbb{R}^{2}$, define $G(X)=(x_{2},x_{2})$. From the
definition of $F$ in (9), we have, component-wise, for any
$X\in\mathbb{R}^{2}$,
$F(X)\leq G(X).$
Note also that $G$ is monotone : if component-wise $X\leq Y$ then $G(X)\leq
G(Y)$. A vector-valued extension of Gronwall’s inequality implies that if
$X(0)=Y(0)$, $X^{\prime}=F(X)$ and $Y^{\prime}=G(Y)$ then, component-wise,
$X(t)\leq Y(t),$
(see e.g. [13, Exercise 4.6]). Looking at the solution of
$y^{\prime\prime}-y^{\prime}=0$ such that $y(0)=0$ and
$y^{\prime}(0)=x^{\prime}(0)$, we get that
$x(t)\leq x^{\prime}(0)(e^{t}-1).$
We deduce from (2) that, for any $T>0$,
$\displaystyle q(\lambda)=\int_{0}^{\infty}e^{-x(t)}e^{-t}dt$
$\displaystyle\geq\int_{0}^{T}e^{-x^{\prime}(0)(e^{t}-1)}e^{-t}dt$
$\displaystyle\geq\int_{0}^{T}(1-x^{\prime}(0)(e^{t}-1))e^{-t}dt$
$\displaystyle\geq 1-e^{-T}-x^{\prime}(0)T.$
Now, we notice that in order to prove Theorem 1.7, by Lemma 2.5, we may choose
$\lambda$ close enough to $1/4$ so that $x^{\prime}(0)<1$. We finally take
$T=-\ln(x^{\prime}(0))$ and apply Lemma 2.5. This concludes the proof of
Theorem 1.7.
## 3 Proof of Theorem 1.1
We define the set recovered and infected vertices as $R(t)=\\{v\in
V:X_{v}(t)=R\\}$ and $I(t)=\\{v\in V:X_{v}(t)=I\\}$. The set $R(t)$ being non-
decreasing, we may define $R(\infty)=\cup_{t>0}R(t)$ and
$Z=|R(\infty)|\in\mathbb{N}\cup\\{\infty\\}$. Note that also a.s.
$R(\infty)=\\{v\in V:\exists t>0,X_{v}(t)=I\\}$, in words, $R(\infty)$ is the
set of vertices which have been infected at some time.
Throughout this section, the chase-escape process is constructed thanks to
i.i.d. $\mathrm{Exp}(\lambda)$ variables $(\xi_{v})_{v\in V}$ and independent
i.i.d. $\mathrm{Exp}(1)$ variables $(D_{v})_{v\in V}$. The variable $\xi_{v}$
(resp. $D_{v}$) is the time by which $v\in V$ will be infected (resp.
recovered) once its ancestor is infected (resp. recovered).
### 3.1 Subcritical regime
We fix $0<\lambda<\lambda_{1}$. In this paragraph we prove that
$q_{T}(\lambda)=1$ if $T$ has upper growth rate at most $d$. It is sufficient
to prove that $\mathbb{E}_{\lambda}Z$. To this end, we will upper bound the
probability that $v\in R(\infty)$ for any $v\in V$. Let $V_{k}$ be the set of
vertices of $V$ which are at distance $k$ from the root $\o$. Let $v\in V_{n}$
and $v_{0},\cdots,v_{n}$ be the ancestor line of $v$: $v_{0}=\o$ and
$v_{n}=v$. The vertex $v$ will have been infected if and only if for all
$1\leq m\leq n$, $v_{m-1}$ has infected $v_{m}$ before being recovered. We
thus find
$\mathbb{P}_{\lambda}(v\in R(\infty))=\mathbb{P}_{\lambda}{{\left(\forall
1\leq m\leq
n,\sum_{i=1}^{m}\xi_{v_{i}}<\sum_{i=1}^{m}D_{v_{i-1}}\right)}}\leq\mathbb{P}_{\lambda}{{\left(\sum_{i=1}^{n}\xi_{v_{i}}<\sum_{i=1}^{n}D_{v_{i-1}}\right)}}.$
The Chernov bound gives for any $0<\theta<1$,
$\displaystyle\mathbb{P}_{\lambda}{{\left(\sum_{i=1}^{n}\xi_{v_{i}}<\sum_{i=1}^{n}D_{v_{i-1}}\right)}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\lambda}\exp{{\left\\{\theta{{\left(\sum_{i=1}^{n}D_{v_{i-1}}-\sum_{i=1}^{n}\xi_{v_{i}}\right)}}\right\\}}}$
$\displaystyle=$
$\displaystyle{{\left(\frac{1}{1-\theta}\right)}}^{n}{{\left(\frac{\lambda}{\lambda+\theta}\right)}}^{n},$
where we have used the independence of all variables at the last line. Now,
the above expression is minimized for $\theta=(1-\lambda)/2>0$ (since
$\lambda<\lambda_{1}<1$). We find
$\mathbb{P}_{\lambda}(v\in
R(\infty))\leq{{\left(\frac{4\lambda}{(\lambda+1)^{2}}\right)}}^{n}.$
Also, from the growth-rate assumption, there exists a sequence
$\varepsilon_{n}\to 0$,
$|V_{n}|\leq(d+\varepsilon_{n})^{n}.$
It follows that
$\mathbb{E}_{\lambda}Z=\sum_{v\in V}\mathbb{P}_{\lambda}(v\in
R(\infty))\leq\sum_{n=0}^{\infty}{{\left(\frac{4(d+\varepsilon_{n})\lambda}{(\lambda+1)^{2}}\right)}}^{n}.$
It is now straightforward to check that
$\frac{4d\lambda}{(\lambda+1)^{2}}<1,$
if $\lambda<\lambda_{1}$. This concludes the first part of the proof.
### 3.2 Supercritical regime
We now fix $\lambda>\lambda_{1}$. We should prove that $q_{T}(\lambda)<1$
under the assumption that $T$ is lower $d$-ary. We are going to construct a
random subtree of $T$ whose vertices are elements of $R(\infty)$ and which is
a supercritical Galton-Watson tree.
First observe that we can couple two chase-escape processes with intensities
$\lambda>\lambda^{\prime}$ on the same probability space in such a way that
they share the same variables $(D_{v})_{v\in V}$ and for all $v\in V$,
$\xi_{v}^{(\lambda)}\leq\xi_{v}^{(\lambda^{\prime})}$ (for example, we take
$\xi_{v}^{(\lambda^{\prime})}=(\lambda/\lambda^{\prime})\xi_{v}^{(\lambda)}$).
The event of non-extinction is easily seen to be non-increasing in the
variables $(\xi_{v})_{v\in V}$ for the partial order on $\mathbb{R}_{+}^{V}$
of component-wise comparison. It follows that the function $\lambda\mapsto
q_{T}(\lambda)$ is non-increasing. We may thus assume without generality that
$\lambda_{1}<\lambda<1$. For $\delta>0,$ we define the function $g_{\delta}$
by, for all $x>0$,
$\displaystyle g_{\delta}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{x}-\log\left(\frac{1}{x}\right)+\frac{\lambda}{x}-\log\left(\frac{\lambda}{x}\right)-2-\log(\delta)$
$\displaystyle=$
$\displaystyle\frac{1+\lambda}{x}+\log{{\left(\frac{x^{2}}{\lambda\delta}\right)}}-2.$
Taking derivative, the minimum of $g_{\delta}$ is reached at
$c=(1+\lambda)/2$. We deduce easily the following property of the function
$g_{d}$.
###### Lemma 3.1.
If $\lambda_{1}<\lambda<1$, $\min_{x>0}g_{d}(x)<0$.
By Lemma 3.1, using continuity, we deduce that there exist $c>0$ and
$1<\delta<d$ such that
$g_{\delta}(c)<0.$
In the remainder of the proof, we fix such pair $(c,\delta)$.
#### Construction of a nested branching process.
We fix an integer $m\geq 1$ that we will be completely specified later on. We
assume that $m$ is large enough such that $T^{*m}$ contains a
$\lceil\delta^{m}\rceil$-ary subtree. We denote by $T^{\prime}$ this subtree
and by $\rho\in V$ its root. For integer $k\geq 0$, we define $V^{\prime}_{k}$
as the set of vertices of generation $k$ in $T^{\prime}$. Note that by
assumption
$|V^{\prime}_{k}|=\lceil\delta^{m}\rceil^{k}.$
We may assume that the generation of $\rho$ in $T$ is larger than $m$. We
denote $a(\rho)\in V$ the $m$-th ancestor of $\rho$ in $T$. For $z\in
V^{\prime}_{k}$ and $k\geq 1$, we denote by $a(z)\in V^{\prime}_{k-1}$ its
ancestor in $T^{\prime}$. For example, if $z\in V^{\prime}_{1}$, $a(z)=\rho$.
We now start a branching process as follows. We set $\rho$ to be the root of
the process, $\mathcal{S}_{0}=\\{\rho\\}$. For integer $k\geq 1$, we define
recursively the offsprings of the $k$-th generation as the set
$\mathcal{S}_{k}$ of vertices $z\in V^{\prime}_{k}$ satisfying the following
three conditions :
1. 1.
the vertex $a(z)\in V^{\prime}_{k-1}$ belongs to $\mathcal{S}_{k-1}$;
2. 2.
$\sum_{i=1}^{m}\xi_{v_{i}}\leq\frac{m}{c}$ where $(v_{0},v_{1},\cdots,v_{m})$
is the set of the vertices on the path from $a(z)$ to $z$, $v_{0}=a(z)$,
$v_{m}=z$;
3. 3.
$\sum_{i=1}^{m}D_{v_{i-1}}\geq\frac{m}{c}$ with $(v_{0},v_{1},\cdots,v_{m})$
as above.
Thus for $z\in V^{\prime}_{k},$ such that its ancestor
$a(z)\in\mathcal{S}_{k-1}$, we have that
$\displaystyle\mathbb{P}_{\lambda}{{\left(z\in{\mathcal{S}}_{k}|\mathcal{S}_{k-1}\right)}}=\mathbb{P}_{\lambda}\left(\sum_{i=1}^{m}\xi_{v_{i}}\leq\frac{m}{c}\right)\mathbb{P}_{\lambda}\left(\sum_{i=1}^{m}D_{v_{i-1}}\geq\frac{m}{c}\right).$
(21)
Notice that by construction, the number of offsprings of $z\neq z^{\prime}$ in
$\mathcal{S}_{k-1}$ are identically distributed and are independent. It
follows that the process forms a Galton-Watson branching process. In the next
paragraph, we will check that this branching process is supercritical, i.e. we
will prove that
$M=\sum_{v\in
V^{\prime}_{1}}\mathbb{P}_{\lambda}{{\left(z\in{\mathcal{S}}_{1}\right)}}>1.$
(22)
It implies that with positive probability, the branching process does not die
out (see Athreya and Ney [6, chapter 1]).
Before proving (22), let us first check that it implies Theorem 1.1. Assume
that at some time $t>0$, the vertex $\rho$ becomes infected and that $a(\rho)$
is still infected. Assume further that $\sum_{i=1}^{m}D_{v_{i-1}}\geq m/c$,
where $(v_{0},v_{1},\cdots,v_{m})$ is the set of the vertices on the path from
$a(\rho)$ to $\rho$. Note that the existence of such finite time $t>0$ and
such sequence $(D_{v_{i}})_{0\leq i\leq m-1}$ has positive probability. Let us
denote by $E$ such event. We set $t_{0}=t$ and, for integer $k\geq 1$,
$t_{k}=t_{k-1}+\frac{m}{c}.$
By construction, if $E$ holds and $z\in\mathcal{S}_{k}$ then, at time $t_{k}$,
$z$ and $a(z)$ are both infected. Hence on the events of $E$ and of non-
extinction of the nested branching process, the chase-escape process does not
get extinct. It thus remains to prove that (22) holds.
#### The nested branching process is supercritical.
We need a standard large deviation estimate. We define
$J(x)=x-\log{x}-1.$
The next lemma is an immediate consequence of Cramer’s Theorem for exponential
variables (see [10, §2.2.1]).
###### Lemma 3.2.
Let $(\zeta_{i})_{i\geq 1},$ be i.i.d. $\mathrm{Exp}(\lambda)$ variables. For
any $a>1/\lambda,$ we have that
$\liminf_{m\to\infty}\frac{1}{m}\log\mathbb{P}\left(\sum_{i=1}^{m}\zeta_{i}\geq
am\right)\geq-J(\lambda a),$
while, for any $a<1/\lambda,$
$\liminf_{m\to\infty}\frac{1}{m}\log\mathbb{P}\left(\sum_{i=1}^{m}\zeta_{i}\leq
am\right)\geq-J(\lambda a).$
Note that the bounds of Lemma 3.2 hold for all $a>0$ (even if they are sharp
only for the above ranges). We may now estimate the terms in (21). We have
from Lemma 3.2 that
$\displaystyle\mathbb{P}_{\lambda}\left(\sum_{i=1}^{m}\xi_{v_{i}}\leq\frac{m}{c}\right)$
$\displaystyle\geq$
$\displaystyle\exp{{\left\\{-mJ\left(\frac{\lambda}{c}\right)+o(m)\right\\}}}$
and
$\displaystyle\mathbb{P}_{\lambda}\left(\sum_{i=1}^{m}D_{v_{i-1}}\geq\frac{m}{c}\right)$
$\displaystyle\geq$
$\displaystyle\exp{{\left\\{-mJ\left(\frac{1}{c}\right)+o(m)\right\\}}}.$
Thus we obtain a lower bound on the mean number of offspring in the first
generation to be
$\displaystyle M$ $\displaystyle=$ $\displaystyle\sum_{z\in
V^{\prime}_{1}}\mathbb{P}_{\lambda}(z\in{\mathcal{S}}_{1})$
$\displaystyle\geq$
$\displaystyle\lceil\delta^{m}\rceil\exp{{\left\\{-m{{\left(J\left(\frac{1}{c}\right)+J\left(\frac{\lambda}{c}\right)+o(m)\right)}}\right\\}}}$
$\displaystyle\geq$ $\displaystyle\exp\left(-mg_{\delta}(c)+o(m)\right),$
where $g_{\delta}(.)$ is as defined in (3.2). If $m$ was chosen large enough,
we have that $M>1$ and hence that the branching process is supercritical.
Therefore with positive probability, this branching process does not die out.
This proves the theorem.
## 4 Proof of Theorem 1.2
The proof of Theorem 1.2 parallels the proof of Theorem 1.7. Even if the
strategy is the same, we will meet some extra difficulties in the study of the
phase diagram (notably in the forthcoming Lemma 4.5).
### 4.1 Differential equation for the survival probability
We first determine a differential equation associated to the probability of
extinction. Under $\mathbb{P}^{\prime}_{\lambda}$, define $Q_{\lambda}(t)$ to
be the extinction probability given that the root ø is recovered at time
$t\geq 0$ so that
$q(\lambda)=\int_{0}^{\infty}Q_{\lambda}(t)e^{-t}dt$ (23)
and $Q_{\lambda}(0)=1$.
Now, in $T$, the offsprings of the root are $\\{1,\cdots,N\\}$, where $N$ has
distribution $P$. The root infects each of its offspring after an independent
exponential variable with intensity $\lambda$. Let $\\{\xi_{i}\\}_{1\leq i\leq
N}$ be the infection times. Note that in $T$, the subtrees generated by each
of the offsprings of the root are iid copies of $T$. Hence, if for integer $i$
with $1\leq\xi_{i}\leq D_{\text{\o}}$, we define $X^{i}$ as the subprocess on
vertices $(i\mathbb{N}^{f})\cap V$ with ancestors $i$. Conditioned on
$D_{\text{\o}}=t$, on $N$ and $(\xi_{i})_{1\leq i\leq N}$, the processes
$(X^{i})$ are independent chase-escape processes conditioned on the fact that
root becomes at risk at time $t-\xi_{i}$ (where we say that a $I$-vertex is at
risk if its genitor is in state $R$).
For the process $X$ to get extinct, all the processes $X^{i}$ must get
extinct. So finally, we get
$\displaystyle Q_{\lambda}(t)$ $\displaystyle=$
$\displaystyle\mathbb{E}^{\prime}_{\lambda}{{\left[\prod_{1\leq i\leq
N}{{\left(\mathbf{1}(\xi_{i}>t)+\mathbf{1}(\xi_{i}\leq
t)Q_{\lambda}(t-\xi_{i}+D_{i})\right)}}\right]}}$
where $(D_{i}),i\geq 1,$ are independent exponential variables with parameter
$1$. Consider the generating function of $P$
$\psi(x)=\mathbb{E}^{\prime}_{\lambda}{{\left[x^{N}\right]}}=\sum_{k=0}^{\infty}x^{k}P(k).$
Recall that $\psi$ is strictly increasing and convex on $[0,1]$ and
$\psi^{\prime}(1)=\mathbb{E}N=d$. We find, for any $t\geq 0$,
$\displaystyle Q_{\lambda}(t)$ $\displaystyle=$
$\displaystyle\psi{{\left(e^{-\lambda t}+\lambda\int_{0}^{t}e^{-\lambda
x}\int_{0}^{\infty}Q_{\lambda}(t-x+s)e^{-s}dsdx\right)}}$ $\displaystyle=$
$\displaystyle\psi{{\left(e^{-\lambda t}+\lambda e^{-\lambda
t}\int_{0}^{t}e^{\lambda
x}\int_{0}^{\infty}Q_{\lambda}(x+s)e^{-s}dsdx\right)}}$ $\displaystyle=$
$\displaystyle\psi{{\left(e^{-\lambda t}+\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}Q_{\lambda}(s)e^{-s}dsdx\right)}}.$
Performing the change of variable
$x(t)=\psi^{-1}(Q_{\lambda}(t))\in[0,1],$ (24)
leads to
$x(t)=e^{-\lambda t}+\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}\psi(x(s))e^{-s}dsdx.$ (25)
We multiply the above expression by $e^{\lambda t}$ and differentiate once, it
gives
$e^{\lambda t}(\lambda x(t)+x^{\prime}(t))=\lambda
e^{(\lambda+1)t}\int_{t}^{\infty}\psi(x(s))e^{-s}ds,$ (26)
Now, multiplying the above expression by $e^{-(\lambda+1)t}$ and
differentiating once again, we find that $x(t)$ satisfies the differential
equation
$x^{\prime\prime}-(1-\lambda)x^{\prime}+\varphi(x)=0$ (27)
with
$\varphi(x)=\lambda\psi(x)-\lambda x.$
### 4.2 A fixed point equation
We define $\rho\in[0,1)$ as the extinction probability in the Galton-Watson
tree:
$\rho=\psi(\rho).$
We note that $\varphi$ is convex, $\varphi(1)=\varphi(\rho)=0$, $\varphi$ is
negative on $(\rho,1)$ and it is increasing in a neighborhood of $1$,
$\varphi^{\prime}(1)=\lambda(d-1)>0$. The fact that $\varphi$ is not monotone
is the main difference with the proof of Theorem 1.7 .
Let $\mathcal{H}$ be the set of non-increasing functions
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $f(0)=1$,
$\lim_{t\to\infty}f(t)=\rho$. The next lemma is an easy consequence of the
monotony of the process.
###### Lemma 4.1.
For any $\lambda>\lambda_{1}$, the function $x(\cdot)$ defined by (24) is in
$\mathcal{H}$.
###### Proof 4.2.
As in the previous section, we may construct the chase escape process
conditioned on the root is recovered at time $t$ thanks to i.i.d.
$\mathrm{Exp}(\lambda)$ variables $(\xi_{v})_{v\in V}$ and independent i.i.d.
$\mathrm{Exp}(1)$ variables $(D_{v})_{v\in V\neq\text{\o}}$ and set
$D_{\text{\o}}=t$. The variable $\xi_{v}$ (resp. $D_{v}$) is the time by which
$v\in V$ will be infected (resp. recovered) once its ancestor is infected
(resp. recovered). The event of extinction is then non-increasing in $t$. It
follows that the map $t\mapsto Q_{\lambda}(t)$ is non-increasing. From (24),
it follows that $x(t)$ is also non-increasing. We may thus define
$a=\lim_{t\to\infty}x(t)$. Using the continuity of $\psi$ leads to
$\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}\psi(x(s))e^{-s}dsdx=\lambda
e^{-\lambda t}\int_{0}^{t}e^{\lambda x}\psi(a)(1+o(1))dx,$
This last integral being divergent as $t\to\infty$, we deduce that
$\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}\psi(x(s))e^{-s}dsdx=\psi(a)+o(1).$
From (25), we get that $a=\psi(a)$ which implies that $a\in\\{\rho,1\\}$. Note
however that Theorem 1.1 and Lemma 5.9 imply that $q(\lambda)<1$ for all
$\lambda>\lambda_{1}$. Then (23) and the monotony of $t\mapsto Q_{\lambda}(t)$
give that for all $t\geq t_{0}$ large enough, $Q_{\lambda}(t)<1$. From (24) it
implies in turn that for all $t\geq t_{0}$, $x(t)<1$. So finally $a\leq
x(t_{0})<1$ and $a=\rho$.
From now on in this section, we fix a small $u>0$ and we assume that
$\lambda_{1}<\lambda<1-u.$ (28)
We define the map $A:\mathcal{H}\to L^{\infty}(\mathbb{R}_{+},\mathbb{R}_{+})$
defined by
$A(y)(t)=e^{-\lambda t}+\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}\psi(y(s))e^{-s}dsdx.$ (29)
Since $\max_{x\in[0,1]}|\psi(x)|=1$ it is indeed straightforward to check that
$A(y)$ is in $L^{\infty}(\mathbb{R}_{+},\mathbb{R}_{+})$: $A(y)(t)$ is bounded
by $1$. Note also that $y\equiv 1$ is a solution of the fixed point equation
$y=A(y).$
By (25), we find that the function $x$ defined by (24) satisfies also the
fixed point $x=A(x)$. In the sequel, we are going to analyze the non trivial
fixed points of $A$.
Let $x\in\mathcal{H}$ such that $x=A(x)$. Then $x\not\equiv 1$. By induction,
it follows easily that $t\mapsto x(t)$ is twice differentiable. In particular,
from the argument following (25), $x$ satisfies (27) and we are looking for a
specific non-negative solution of (27) with $x(0)=1$. To characterize
completely this solution, it would be enough to compute $x^{\prime}(0)$ (which
is necessarily negative since $x(0)=1$, $x^{\prime}(0)=0$ corresponds to the
trivial solution $x\equiv 1$). We will perform this in the next subsection in
an asymptotic regime. We start with some properties obtained from the phase
diagram of the ODE (27).
###### Lemma 4.3.
Let $x\in\mathcal{H}$ such that $x=A(x)$. Then,
1. (i)
for all $t>0$, $\rho<x(t)<1$;
2. (ii)
for all $t\geq 0$, $-1<x^{\prime}(t)<0.$
###### Proof 4.4.
Let us prove $(i)$. We first observe that since $x(t)$ is non-increasing,
$x(0)=1$ and $x^{\prime}(0)<0$, we have that for all $t>0$, $x(t)<1$. Also, if
$x(t)=\rho$ for some $t>0$, then $x(s)=\rho$ for all $s\geq t$ (since $x$ is
non-increasing and has limit $\rho$). However $y\equiv\rho$ being a distinct
solution of (27), $x$ and $y$ cannot coincide on an interval. We thus have for
all $t>0$, $\rho<x(t)<1$.
We now prove $(ii)$. Assume that there is a time $t>0$ such that
$x^{\prime}(t)=0$. Then, from (27) and $\rho<x(t)<1$, we deduce that
$x^{\prime\prime}(t)>0$. In particular, $x^{\prime}(s)>0$ for all
$s\in(t,t+\delta)$ for some $\delta>0$. This contradicts that $x(\cdot)$ is
non-increasing. Also, from (26), for any $t\geq 0$, $\lambda
x(t)+x^{\prime}(t)>0$. Since $x(t)\leq 1$, we deduce that for all $t\geq 0$,
$-\lambda<x^{\prime}(t)<0$.
We define $X(t)=(x(t),x^{\prime}(t))$ and
$F(x_{1},x_{2})=(x_{2},(1-\lambda)x_{2}-\varphi(x_{1}))$
so that
$X^{\prime}=F(X).$ (30)
We define the trajectory $\Phi=\\{X(t):t\geq 0\\}$. Recall that
$\rho=\lim_{t\to\infty}x(t)$. Also, since for all $t\geq 0$,
$X(t)^{\prime}_{1}=F(X(t))_{1}<0$, $\Phi$ is the graph of a differentiable
function $f:(\rho,1]\to(-1,0)$ with $f(1)=x^{\prime}(0)<0$,
$\Phi=\\{(s,f(s)):s\in(\rho,1]\\}.$
Moreover
$f^{\prime}(s)=\frac{F((s,f(s)))_{2}}{F((s,f(s)))_{1}}=1-\lambda-\frac{\varphi(s)}{f(s)}.$
(31)
$\rho$$-1$$0$$x^{\prime}$$x$$-\eta$$1$$\sigma$$\beta$$\alpha$
Figure 4: Illustration of the phase portrait. In blue, the curve $\Gamma$, in
red the curve $\Phi$.
We notice that on the curve
$\Gamma=\\{(x_{1},x_{2})\in[\rho,1]\times[-1,0]:(1-\lambda)x_{2}=\varphi(x_{1})\\}$
the second coordinate of $F$ vanishes (see figure 4). The next lemma shows
that our function $(x(t),x^{\prime}(t))$ cannot cross $\Gamma$ near its origin
$(x(0),x^{\prime}(0))$.
###### Lemma 4.5.
There exists $\delta>0$ depending only on $\psi$ and $u$ defined in (28) such
that the following holds. Let $x\in\mathcal{H}$ such that $x=A(x)$. If
$(x(t),x^{\prime}(t))\in\Gamma$ for some $t>0$, then
$x^{\prime}(t)\leq-\delta$ and $x(t)\leq 1-\delta$.
###### Proof 4.6.
Define $\sigma$ as the largest $s$ such that $(s,f(s))\in\Gamma$ (see figure
4). We have $\sigma<1$. We should prove that $\sigma\leq 1-\delta$ and
$f(\sigma)\leq-\delta$. Recall that $f(1)=x^{\prime}(0)<0$. Thus, on
$(\sigma,1]$, $(s,f(s))$ is below $\Gamma$ and it follows that $f$ is
increasing. By construction,
$f(\sigma)=\varphi(\sigma)/(1-\lambda)=\lambda(\psi(\sigma)-\sigma)/(1-\lambda)$
and $f^{\prime}(\sigma)=0$. We deduce that
$f^{\prime\prime}(\sigma)=-\frac{\varphi^{\prime}(\sigma)}{f(\sigma)}+\frac{f^{\prime}(\sigma)\varphi(\sigma)}{f^{2}(\sigma)}=-\frac{\varphi^{\prime}(\sigma)}{f(\sigma)}\geq
0,$ (32)
where the last inequality comes from $f$ is increasing on $[\sigma,1]$ and
$f^{\prime}(\sigma)=0$.
We define $\alpha\in(\rho,1)$ as the point where the function
$\kappa(x)=x-\psi(x)$ reaches its maximum. Since $\varphi^{\prime}(s)<0$ on
$[\rho,\alpha)$, from (32) we find that $\sigma\in[\alpha,1)$. We set
$f(\sigma)=-\eta$. We will prove that $\eta\geq\lambda\delta_{0}/(1-\lambda)$
for some $\delta_{0}>0$ depending only on $u$ and $\psi$. This will conclude
the proof of the lemma. Indeed, by construction
$(1-\lambda)\eta=-\varphi(\sigma)=\lambda(\sigma-\psi(\sigma))$. The function
$\kappa(x)=x-\psi(x)$ has a continuous decreasing inverse in $[\alpha,1]$ with
$\kappa^{-1}(0)=1$. Hence, $\sigma=\kappa^{-1}((1-\lambda)\eta/\lambda)$ and,
if $\eta\geq\lambda\delta_{0}/(1-\lambda)$, we deduce that the statement of
the lemma holds with
$\delta=\min(\lambda_{1}\delta_{0}/(1-\lambda_{1}),1-\kappa^{-1}(\delta_{0}))$.
To this end, we fix any $\beta\in(\alpha,1)$, we set $b=\kappa(\beta)>0$, and
$\delta_{0}=\min\left(\frac{b}{2},(1-u)\sqrt{\frac{b(\beta-\alpha)}{u}}\right).$
(33)
We assume that $\eta<\lambda\delta_{0}/(1-\lambda)$ and look for a
contradiction. We first notice that $\delta_{0}<b$ implies that
$\sigma=\kappa^{-1}((1-\lambda)\eta/\lambda)>\kappa^{-1}(b)=\beta$.
Consider the solution $Y(t)=(y(t),y^{\prime}(t))$ of the ODE (30) with initial
condition $Y(0)=(\beta,-\eta)$. The trajectory of $Y(t)$ is denoted by
$\tilde{\Phi}=\\{Y(t):t\geq 0\\}$. We define the set
$\Gamma_{+}=\\{(x_{1},x_{2})\in[\alpha,1)\times[-\eta,0):(1-\lambda)x_{2}\geq\varphi(x_{1})\\}$.
On $\Gamma_{+}$, $F(x)_{1}<0$ and $F(x)_{2}\geq 0$. It follows that the
trajectories $\Phi$ and $\tilde{\Phi}$ exit $\Gamma_{+}$ either on its left
side $\\{(\alpha,x_{2}),x_{2}\in[-\eta,0]\\}$ or its upper side
$\\{(x_{1},0),x_{1}\in[\alpha,1)\\}$. However, Lemma 4.3$(ii)$ implies that
$\Phi$ exits $\Gamma_{+}$ on the left side. Since $\Phi$ and $\tilde{\Phi}$
cannot intersect and $\tilde{\Phi}$ is on the left side of $\Phi$ in
$\Gamma_{+}$ (since $\beta<\sigma$), we deduce that necessarily,
$\tilde{\Phi}$ also exits $\Gamma_{+}$ on the left side. We now check that,
with our choice of $\delta_{0}$ in (33), it contradicts
$\eta<\lambda\delta_{0}/(1-\lambda)$.
Define $\tau>0$ as the exit time of $Y(t)$ from $\Gamma_{+}$. If $0\leq
t\leq\tau$, using that $\varphi$ is increasing on $[\alpha,\beta]$, we find
$y^{\prime\prime}(t)\geq-(1-\lambda)\eta-\varphi(\beta)\geq-\lambda\delta_{0}+\lambda
b\geq\lambda b/2,$
since $\delta_{0}\leq b/2$. We deduce that for all $t\in[0,\tau]$,
$y^{\prime}(t)\geq z^{\prime}(t)$ and $y(t)\geq z(t)$ with $z(t)=(\lambda
b/4)t^{2}-\eta t+\beta$. We set $t_{e}=2\eta/(\lambda b)$. Since
$z^{\prime}(t_{e})=0$, we have $\tau\leq t_{e}$. Also $z$ being decreasing on
$[0,t_{e}]$, we have $y(\tau)\geq z(t_{e})=-\eta^{2}/(\lambda b)+\beta$.
Thanks to (33), $\eta^{2}<\lambda^{2}\delta_{0}^{2}/(1-\lambda)^{2}<\lambda
b(\beta-\alpha)$ and we deduce that $y(\tau)\geq-\eta^{2}/(\lambda
b)+\beta>\alpha$. In particular, $\tilde{\Phi}$ exits $\Gamma_{+}$ on the
upper side. It leads to a contradiction. We have thus proved that
$\eta\geq\lambda\delta_{0}/(1-\lambda)$.
### 4.3 Comparison of second order differential equations
For two functions $\varphi_{1},\varphi_{2}$ on $[0,1]$, we write
$\varphi_{1}\leq\varphi_{2}$ if for all $t\in[0,1]$,
$\varphi_{1}(t)\leq\varphi_{2}(t)$. The next lemma is proved as Lemma 2.3, we
omit its proof.
###### Lemma 4.7.
Let $\delta>0$ be as in Lemma 4.5. Let $x\in\mathcal{H}$ such that $x=Ax$. Let
$\tilde{\varphi}$ be a Lipshitz-continuous function and $y$ be solution of
$y^{\prime\prime}-(1-\lambda)y^{\prime}+\tilde{\varphi}(y)=0$ with $y(0)=1$,
$y^{\prime}(0)<0$. We define the exit times
$T=\inf\\{t\geq 0:(y(t),y^{\prime}(t))\notin[0,1]\times[-1,0]\\},$
$T_{-}=\inf\\{t\geq 0:y^{\prime}(t)\leq-1\\}\quad\hbox{ and }\quad
T_{+}=\inf\\{t\geq 0:(1-\lambda)y^{\prime}(t)=\varphi(y(t)),y(t)\geq
1-\delta\\}.$
1. (i)
If $T_{+}<T<\infty$ and $\tilde{\varphi}\geq\varphi$ then $y^{\prime}(0)\geq
x^{\prime}(0)$.
2. (ii)
If $T_{-}=T<\infty$ and $\tilde{\varphi}\leq\varphi$ then $x^{\prime}(0)\geq
y^{\prime}(0)$.
### 4.4 Proof of Theorem 1.2
The proof of Theorem 1.2 follows now closely the proof of Theorem 1.7. We
first linearize (27) in the neighborhood of $\lambda_{1}$.
#### Step one : linearization from below.
We have $\varphi(1)=0$, $\varphi^{\prime}(1)=\lambda(d-1)>0$, and from the
convexity of $\varphi$,
$\varphi(s)\geq\lambda(d-1)(s-1).$ (34)
We take $\lambda_{1}<\lambda<1$ and consider the linearized differential
equation
$y^{\prime\prime}-(1-\lambda)y^{\prime}+\lambda(d-1)(y-1)=0.$ (35)
The solutions of this differential equation are
$y(t)=1+a\sin(\omega t)e^{\frac{(1-\lambda)t}{2}}+b\cos(\omega
t)e^{\frac{(1-\lambda)t}{2}},$
where
$\omega=\frac{1}{2}\sqrt{-\lambda^{2}+2(2d-1)\lambda-1}=c(\lambda)\sqrt{\lambda-\lambda_{1}},$
and
$c(\lambda)=\frac{1}{2}\sqrt{(2d-1)+2\sqrt{d(d-1)}-\lambda}=\left(d(d-1)\right)^{1/4}+O(|\lambda-\lambda_{1}|).$
We use this ODE to bound from below $x^{\prime}(0)$ if $A(x)=x$.
###### Lemma 4.8.
The exists a constant $c_{0}>0$ such that for any $\lambda_{1}<\lambda<1$, if
$x\in\mathcal{H}$ satisfies $A(x)=x$ then
$x^{\prime}(0)\geq-c_{0}e^{-\frac{\pi(1-\lambda)}{2\omega}}(1+O(\omega^{2})).$
###### Proof 4.9.
We can assume without loss of generality that $\lambda$ satisfies (28). Let
$a<0$, $b=(1-\lambda)/2$, and consider the function
$y(t)=1+a\sin(\omega t)e^{bt}.$
We have $y(0)=1$, $y^{\prime}(0)=a\omega$,
$y^{\prime}(t)=ae^{bt}(\omega\cos(\omega t)+b\sin(\omega t)),$
and
$y^{\prime\prime}(t)=ae^{bt}{{\left(2b\omega\cos(\omega
t)+{{\left(b^{2}-\omega^{2}\right)}}\sin(\omega t)\right)}}.$
Define
$\tau=\frac{\pi}{\omega}-\frac{1}{\omega}\arctan{{\left(\frac{2b\omega}{b^{2}-\omega^{2}}\right)}}=\frac{\pi}{\omega}-\frac{2}{b}+O(\omega^{2}).$
On the interval $(0,\tau)$, $y^{\prime\prime}(t)<0$ and
$y^{\prime\prime}(\tau)=0$. Thus the function $y^{\prime}(t)$ is decreasing on
$[0,\tau]$ and
$y^{\prime}(\tau)=e^{-\frac{2}{b}}ae^{\frac{\pi
b}{\omega}}(\omega+O(\omega^{3})).$
Hence, we may choose $a$ such that $y^{\prime}(\tau)=-1$ with
$a=-\omega^{-1}e^{\frac{2}{b}}e^{-\frac{\pi b}{\omega}}(1+O(\omega^{2})).$
It remains to use (34) with Lemma 4.7$(ii)$ and $\tau=T_{-}$.
#### Step two : linearization from above.
For $0<\eta<\min(1,c(\lambda)/(d-1))$, we define
$\ell=(1-\eta)\lambda+\eta\lambda_{1}<\lambda,$
and the Lipschitz-continuous function
$\tilde{\varphi}(s)=\max\left(\varphi(s),\ell(d-1)(s-1)\right).$
In particular
$\varphi\leq\tilde{\varphi}.$ (36)
We define the linear differential equation
$y^{\prime\prime}-(1-\lambda)y^{\prime}+\ell(d-1)(y-1)=0.$ (37)
The solutions of (37) are
$y(t)=1+a\sin(\omega^{\prime}t)e^{\frac{(1-\lambda)t}{2}}+b\cos(\omega^{\prime}t)e^{\frac{(1-\lambda)t}{2}},$
with
$\omega^{\prime}=\frac{1}{2}\sqrt{-\lambda^{2}+2(2d-1)\lambda-1-4\eta(d-1)(\lambda-\lambda_{1})}=\omega\sqrt{1-\frac{\eta(d-1)}{c(\lambda)}}.$
In the sequel, $o(1)$ denotes a function which goes to $0$ as $\omega$ goes to
$0$.
###### Lemma 4.10.
If $P$ has finite second moment, then there exists a constant $c_{1}>0$ such
that for all $\lambda_{1}<\lambda<1$, if $x\in\mathcal{H}$ satisfies $A(x)=x$
then
$x^{\prime}(0)\leq-
c_{1}\omega^{3}e^{-\frac{\pi(1-\lambda)}{2\omega}}(1+o(1)).$
###### Proof 4.11.
We can assume without loss of generality that $\lambda$ satisfies (28). We set
$b=\frac{1-\lambda}{2}\quad\hbox{ and
}\quad\kappa=\sqrt{1-\frac{\eta(d-1)}{c(\lambda)}}.$
We parametrize in terms of $\kappa$, so that
$\ell=\lambda-(1-\kappa^{2})\omega^{2}\quad\hbox{ and
}\quad\omega^{\prime}=\kappa\omega.$ (38)
For $a<0$, we look at the solution
$y(t)=1+a\sin(\omega\kappa t)e^{bt}.$
We have $y(0)=1$, $y^{\prime}(0)=a\kappa\omega$.
$y^{\prime}(t)=ae^{bt}(\omega\kappa\cos(\omega\kappa t)+b\sin(\omega\kappa
t)).$
We repeat the argument of Lemma 4.8. On the interval $[0,\tau]$,
$y^{\prime\prime}(t)\geq 0$ and $y^{\prime\prime}(\tau)=0$, where
$\tau=\frac{\pi}{\omega\kappa}-\frac{1}{\omega\kappa}\arctan{{\left(\frac{2b\omega\kappa}{b^{2}-\omega^{2}\kappa^{2}}\right)}}=\frac{\pi}{\omega\kappa}-\frac{2}{b}+O(\omega^{2}),$
and the $O(\cdot)$ is uniform over all $\kappa>1/2$. The function
$y^{\prime}(t)$ is increasing on $[0,\tau]$ and
$y^{\prime}(\tau)=e^{-2}ae^{\frac{\pi
b}{\omega\kappa}}(\omega\kappa+O(\omega^{2})).$
Now, we have $\varphi(s)<\ell(d-1)(s-1)$ for all $s\in[1-\sigma,1]$ with
$-\ell(d-1)\sigma=\varphi(1-\sigma)=\lambda(\psi(1-\sigma)-1+\sigma).$
If $P$ has finite second moment then, from Abel’s Theorem,
$\psi^{\prime\prime}$ is continuous on $[0,1]$. Also from Jensen’s inequality,
$\psi^{\prime\prime}(1)\geq d(d-1)>0$. We expand $\psi$ in a neighborhood of
$1$, as $\omega\to 0$, it leads to,
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{2(d-1)}{\psi^{\prime\prime}(1)\lambda}{{\left(\lambda-\ell\right)}}(1+o(1))$
$\displaystyle=$
$\displaystyle\frac{2(d-1)}{\psi^{\prime\prime}(1)\lambda}(1-\kappa^{2})\omega^{2}(1+o(1)),$
where $o(1)$ is uniform over all $0<\kappa<1$. In particular, for all $\omega$
small enough, $\sigma<\delta$ with $\delta$ as in Lemma 4.5. Also, from (37),
for $t=\tau$, since $y^{\prime\prime}(\tau)=0$, we have
$\frac{y(\tau)-1}{y^{\prime}(\tau)}=\frac{1-\lambda}{\ell(d-1)}=\frac{2b}{\ell(d-1)}.$
We may choose $a$ such that $y(\tau)=1-\sigma$ by setting
$a=-\sigma e^{2}\frac{\ell(d-1)}{2b}\frac{e^{-\frac{\pi
b}{\omega\kappa}}}{\omega\kappa}(1+O(\omega^{2}))=-e^{2}\frac{\ell(d-1)^{2}}{\lambda\psi^{\prime\prime}(1)b}e^{-\frac{\pi
b}{\omega\kappa}}\frac{(1-\kappa^{2})\omega}{\kappa}(1+o(1)).$
By construction, with this choice of $a$, we have
$(1-\lambda)y^{\prime}(\tau)=\varphi(y(\tau))$. Now, in the domain
$1-\sigma\leq y\leq 1$ the non-linear differential equation
$y^{\prime\prime}-(1-\lambda)y^{\prime}+\tilde{\varphi}(y)$ coincides with
(37). Thus, using (36) and Lemma 4.7$(i)$ with $\tau=T_{+}$, we find that
$x^{\prime}(0)\leq
y^{\prime}(0)=-e^{2}\frac{\ell(d-1)^{2}}{\lambda\psi^{\prime\prime}(1)b}e^{-\frac{\pi
b}{\omega\kappa}}(1-\kappa^{2})\omega^{2}(1+o(1)).$
We finally take $\kappa=1-\omega/(\pi b)$ and use (38). It proves the lemma.
#### Step three : end of proof.
We may now complete the proof of Theorem 1.2. We start with the left hand side
inequality. We first note that, by Lemma 4.5, $x^{\prime}(t)$ is decreasing on
the interval $[0,t_{0}]$ where $t_{0}$ is the time where
$(x(t_{0}),x^{\prime}(t_{0}))\in\Gamma=\\{(x_{1},x_{2})\in[\rho,1]\times[-1,0]:(1-\lambda)x_{2}=\varphi(x_{1})\\}$.
Moreover by Lemma 4.5, we find $x(t_{0})\leq 1-\delta$. However, by Lemma
(4.3)$(ii)$, we have
$x(t)\geq 1-t.$
Hence $t_{0}\geq\delta$. Then, by construction, on the interval $[0,t_{0}]$,
$x(t)\leq 1+x^{\prime}(0)t=1-|x^{\prime}(0)|t.$
Since $\psi(x)\leq x$ on $[\rho,1]$, it follows from (23) that the survival
probability may be lower bounded as
$\displaystyle 1-q(\lambda)=\int_{0}^{\infty}(1-\psi(x(t)))e^{-t}dt$
$\displaystyle\geq$ $\displaystyle\int_{0}^{\infty}(1-x(t))e^{-t}dt$
$\displaystyle\geq$ $\displaystyle\int_{0}^{t_{0}}|x^{\prime}(0)|te^{-t}dt$
$\displaystyle\geq$ $\displaystyle|x^{\prime}(0)|\int_{0}^{\delta}te^{-t}dt.$
It remains to use Lemma 4.10 and we obtain the left hand side of Theorem 1.2.
We turn to the right hand side inequality. For
$X=(x_{1},x_{2})\in[\rho,1]\times(-\infty,0)$, define
$G(X)=(x_{2},(1-\lambda)x_{2})$. From the definition of $F$ in (30), we have,
component-wise, for any $X\in[\rho,1]\times(-\infty,0)$,
$F(X)\geq G(X).$
Note also that $G$ is monotone : if component-wise $X\geq Y$ then $G(X)\geq
G(Y)$. It follows that if $X(0)=Y(0)$, $X^{\prime}=F(X)$ and $Y^{\prime}=G(Y)$
then component-wise
$X(t)\geq Y(t),$
(see e.g. [13, Exercise 4.6]). Looking at the solution of
$y^{\prime\prime}-(1-\lambda)y^{\prime}=0$ such that $y(0)=1$ and
$y^{\prime}(0)=x^{\prime}(0)$, we get that
$x(t)\geq 1+x^{\prime}(0)(e^{(1-\lambda)t}-1).$
We deduce from (23)-(24) and the convexity of $\psi$ that,
$\displaystyle q(\lambda)$ $\displaystyle=\int_{0}^{\infty}\psi(x(t))e^{-t}dt$
$\displaystyle\geq\int_{0}^{\infty}\psi(1+x^{\prime}(0)(e^{(1-\lambda)t}-1))e^{-t}dt$
$\displaystyle\geq\int_{0}^{\infty}{{\left(1+dx^{\prime}(0)(e^{(1-\lambda)t}-1)\right)}}e^{-t}dt$
$\displaystyle\geq 1+dx^{\prime}(0)/\lambda.$
We finally apply Lemma 4.8 and this concludes the proof of Theorem 1.2.
## 5 Proofs of Theorems 1.3-1.4
### 5.1 Proof of Theorem 1.4
We are first going to find a recursive distributional equation (RDE)
associated to the total progeny of the chase-escape process on a Galton-Watson
tree.
As already pointed, we can build the chase escape process on the tree
$T^{\downarrow}$ thanks to i.i.d. $\mathrm{Exp}(\lambda)$ variables
$(\xi_{v})_{v\in V}$ and independent i.i.d. $\mathrm{Exp}(1)$ variables
$(D_{v})_{v\in V}$. The variable $\xi_{v}$ (resp. $D_{v}$) is the time by
which $v\in V$ will be infected (resp. recovered) once its ancestor is
infected (resp. recovered). For $t\geq 0$, we define $Y(t)$ as the total
number of recovered vertices when the process reach its absorbing state
(without counting $o$) when we replace $D_{\text{\o}}$ by $t$. The variable
$Y(t)$ is the conditional variable $Z$ conditioned on the root is recovered at
time $t$. By definition, if $D$ is an independent exponential variable with
mean $1$, then
$Z\stackrel{{\scriptstyle d}}{{=}}Y(D),$
where the symbol $\stackrel{{\scriptstyle d}}{{=}}$ stands for distributional
equality.
In $T$, we denote the offsprings of the root by $\\{1,\cdots,N\\}$. The random
variable $N$ has distribution $P$. The root infects each of its offspring
after an independent exponential variable with intensity $\lambda$. Note that
in $T$, the subtrees generated by each of the offsprings of the root are iid
copies of $T$. Hence, the recursive structure of the tree $T$ leads to the
following equality in distribution
$\displaystyle Y(t)$ $\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle 1+\sum_{i=1}^{N}1{1}(\xi_{i}\leq t)Y_{i}(t-\xi_{i}+D_{i}).$
(39)
where $(\xi_{i})_{i\in\mathbb{N}}$ are iid exponential variables with
intensity $\lambda$, $(Y_{i})_{1\leq i\leq N}$ and $(D_{i})_{1\leq i\leq N}$
are independent copies of $Y$ and $D$ respectively. Note that since all
variables are non-negative, there is no issue with the case $Y(t)=+\infty$ in
the above RDE.
The RDE (39) is the cornerstone of the argument. In the remainder of this
subsection, we will use it to derive a linear second order ODE for the first
moment of $Y(t)$. In the following subsection §5.2, we will extend this exact
computation to any integer moment. Finally, using convexity inequalities, we
will push further the method and obtain sharp lower and upper bounds for any
moment of $Y(t)$ (in §5.3 and §5.4).
We start with a lemma
###### Lemma 5.1.
Let $t>0$ and $u\geq 1$, if $\mathbb{E}^{\prime}_{\lambda}[Z^{u}]<\infty$ then
$\mathbb{E}^{\prime}_{\lambda}[Y(t)^{u}]<\infty$.
###### Proof 5.2.
Since $Z\stackrel{{\scriptstyle d}}{{=}}Y(D)$, from Fubini’s Theorem,
$\mathbb{E}^{\prime}_{\lambda}[Z^{u}]=\int_{0}^{\infty}\mathbb{E}^{\prime}_{\lambda}[Y(t)^{u}]e^{-t}dt$.
Therefore $\mathbb{E}^{\prime}_{\lambda}[Y(t)^{u}]<\infty$ for almost all
$t\geq 0$. Note however that since $t\mapsto Y(t)$ is monotone for the
stochastic domination, it implies that
$\mathbb{E}^{\prime}_{\lambda}[Y(t)^{u}]<\infty$ for all $t\geq 0$.
Now, assume that $\mathbb{E}^{\prime}_{\lambda}Z<\infty$. We may then take
expectation in (39):
$\displaystyle\mathbb{E}^{\prime}_{\lambda}Y(t)$ $\displaystyle=$
$\displaystyle
1+d\int_{0}^{t}\int_{0}^{\infty}\mathbb{E}^{\prime}_{\lambda}[Y(t-x+s)]e^{-s}ds\lambda
e^{-\lambda x}dx.$
Let $f_{1}(t)=\mathbb{E}^{\prime}_{\lambda}Y(t)$, it satisfies the integral
equation, for all $t\geq 0$,
$f_{1}(t)=1+\lambda de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{1}(s)e^{-s}dsdx.$ (40)
Multiplying by $e^{\lambda t}$ and taking derivative, we get:
$(f^{\prime}_{1}(t)+\lambda f_{1}(t))e^{\lambda t}=\lambda e^{\lambda
t}+\lambda de^{(\lambda+1)t}\int_{t}^{\infty}f_{1}(s)e^{-s}ds.$
Then, multiplying by $e^{-(\lambda+1)t}$, taking the derivative a second time
and then re-multiplying by $e^{t}$, we obtain:
$f_{1}^{\prime\prime}(t)-(1-\lambda)f_{1}^{\prime}(t)-\lambda
f_{1}=-\lambda-\lambda df_{1}(t).$ So, finally, $f_{1}$ solves a linear
ordinary differential equation of the second order
$\displaystyle x^{\prime\prime}-(1-\lambda)x^{\prime}+\lambda(d-1)x=-\lambda,$
with initial condition $f_{1}(0)=1$. We get that
$f_{1}(t)=x(t)-\frac{1}{d-1},$
where $x(t)$ solves the ordinary differential equation
$x^{\prime\prime}-(1-\lambda)x^{\prime}+\lambda(d-1)x=0.$ (41)
with $x(0)=d/(d-1)$. The discriminant of the polynomial
$X^{2}-(1-\lambda)X+\lambda(d-1)=0$ is
$\Delta=\lambda^{2}-2\lambda(2d-1)+1.$
If $0<\lambda<\lambda_{1}$, the discriminant is positive. The roots of the
polynomial are
$\alpha=\frac{1-\lambda-\sqrt{\Delta}}{2}\quad\hbox{ and
}\quad\beta=\frac{1-\lambda+\sqrt{\Delta}}{2}.$ (42)
The solutions of (41) are
$x(t)=\frac{d}{d-1}\left((1-a)e^{\alpha t}+ae^{\beta t}\right)$ (43)
for some constant $a$.
Similarly, if $\lambda=\lambda_{1}$, then $\alpha=\beta=1-d+\sqrt{d(d-1)}$ and
the solutions of (41) are
$x(t)=\frac{d}{d-1}(at+1)e^{\alpha t}.$ (44)
For $0<\lambda\leq\lambda_{1}$, we check easily that the functions $x(\cdot)$
with $a\geq 0$ are the nonnegative solutions of the integral equation (40).
It remains to prove that if $0<\lambda\leq\lambda_{1}$ then
$\mathbb{E}Z<\infty$ and
$f_{1}(t)=\frac{de^{\alpha t}-1}{d-1}.$
Indeed, we would get
$\mathbb{E}Z=\int_{0}^{\infty}\mathbb{E}Y(t)e^{-t}dt=\frac{d}{d-1}\frac{1}{1-\alpha}-\frac{1}{d-1}$
as stated in Theorem 1.4. To this end, define $T_{n}$ as the tree $T$ stopped
at generation $n$. As above, we denote by $Y^{(n)}(t)$ the total number of
recovered particles in $T_{n}$ when the root is recovered at time $t$
($D_{\text{\o}}$ is replaced by $t$). As $n\to\infty$, $Y_{n}(t)$ is non-
decreasing and converges to $Y(t)$. We have $Y^{(0)}(t)=1$ and for all $n\geq
0$, as in RDE (39),
$\displaystyle Y^{(n+1)}(t)$ $\displaystyle\stackrel{{\scriptstyle d}}{{=}}$
$\displaystyle 1+\sum_{i=1}^{N}1{1}(\xi_{i}\leq
t)Y^{(n)}_{i}(t-\xi_{i}+D_{i}),$
where $Y^{(n)}_{i},$ and $D_{i}$ are independent copies of $Y^{(n)}$ and $D$
respectively. Since $\mathbb{E}N<\infty$, the expectation of the number of
vertices in $T_{n}$ is finite. In particular, for all $n\geq 0$,
$g_{n}(t)=\mathbb{E}Y^{(n)}(t)<\infty$ is bounded uniformly in $t$. Also,
taking expectation in (5.1), we have for all $t\geq 0$,
$g_{n+1}(t)=1+\lambda de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}g_{n}(s)e^{-s}dsdx=\Phi(g_{n})(t),$
(45)
where $\Phi$ is the mapping
$\Phi:g\mapsto 1+\lambda de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}g(s)e^{-s}dsdx.$
It is easy to check that $\Phi$ is indeed a mapping from $\mathcal{H}_{1}$ to
$\mathcal{H}_{1}$, where $\mathcal{H}_{1}$ is the set of non-decreasing
functions $g:[0,\infty)\to[1,\infty)$ such that $\sup_{t\geq 0}g(t)e^{-\alpha
t}<\infty$. Now, from what precedes the function
$h(t)=\frac{de^{\alpha t}-1}{d-1}.$
is a fixed point of $\Phi$.
Denote by $\leq$ the partial order on $\mathcal{H}_{1}$ of point-wise
domination: $g\leq f$ if for all $t\geq 0$, $g(t)\leq f(t)$. The mapping
$\Phi$ is non-decreasing on $\mathcal{H}_{1}$ for this partial order. We
notice that $g_{0}\leq h$ and $g_{0}\leq g_{1}$. Composing by $\Phi$, we
obtain: $g_{1}=\Phi(g_{0})\leq\Phi(h)=h$ and $g_{1}\leq g_{2}$. By recursion,
it follows for any $n\geq 1$, $g_{n}\leq h$ and $(g_{n})_{n\geq 0}$ is non-
decreasing. By monotone convergence, for any $t\geq 0$, the limit
$g(t)=\lim_{n\to\infty}g_{n}(t)$ exists and is bounded by $h(t)$. Also, since
$g_{n}\leq h$, by dominated convergence, for any $t\geq 0$,
$\lim_{n\to\infty}\Phi(g_{n})(t)=\Phi(g)(t)$. Therefore $g$ solves the
integral equation (40) and is equal to $x-1/(d-1)$ where $x$ is given by (43)
(or (44) if $\lambda=\lambda_{1}$) for some $a\geq 0$. However, from what
precedes, we get $x(t)\leq h(t)+1/(d-1)$ and the only possibility is $a=0$ and
$g(t)=h(t)$.
Finally, since $Y^{(n)}(t)$ is non-decreasing and converges to $Y(t)$, by
monotone convergence we have that
$f_{1}(t)=\mathbb{E}Y(t)=\lim_{n\to\infty}\mathbb{E}Y^{(n)}(t)=g(t)$. This
concludes the proof of Theorem 1.4.
### 5.2 Proof of Theorem 1.3 for integer moments
For $0<\lambda<\lambda_{1}$, we define
$\overline{\gamma}=\frac{\lambda^{2}-2d\lambda+1-(1-\lambda)\sqrt{\Delta}}{2\lambda(d-1)}=\frac{\beta}{\alpha},$
(46)
where $\alpha$ and $\beta$ are given by (42). The key property of
${\overline{\gamma}}(\lambda)$ is that
$(1-\lambda)u\alpha-\lambda(d-1)-u^{2}\alpha^{2}>0$ if and only if
$1<u<{\overline{\gamma}}(\lambda)$. We also note that if $u>1$,
$u<{\overline{\gamma}}$ is equivalent to $\lambda\in(0,\lambda_{u})$. We first
state an important lemma. Let $1<u<{\overline{\gamma}}$, we define
$\mathcal{H}_{u}$, the set of measurable functions $h:[0,\infty)\to[0,\infty)$
such that $\sup_{t\geq 0}h(t)e^{-u\alpha t}<\infty$. Let $L>0$, we define the
mapping from $\mathcal{H}_{u}$ to $\mathcal{H}_{u}$,
$\Psi:h\mapsto Le^{u\alpha t}+\lambda de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}h(s)e^{-s}dsdx.$
In order to check that $\Psi$ is indeed a mapping from $\mathcal{H}_{u}$ to
$\mathcal{H}_{u}$, we use the fact that if
$1<u<\overline{\gamma}=\beta/\alpha$ then $u\alpha<\beta<1$.
###### Lemma 5.3.
Let $1<u<\overline{\gamma}$ and $f\in\mathcal{H}_{u}$ such that
$f\leq\Psi(f)$. Then for all $t\geq 0$,
$f(t)\leq\frac{L(u\alpha+\lambda)(1-u\alpha)}{(1-\lambda)u\alpha-\lambda(d-1)-u^{2}\alpha^{2}}\,e^{u\alpha
t}.$
###### Proof 5.4.
We set $g_{0}=f$ and for $k\geq 1$, we define $g_{k}=\Psi(g_{k-1})$. First,
since $1<u<\overline{\gamma}$ then $(u\alpha+\lambda)(1-u\alpha)>\lambda d$.
We use the formula for all $u\geq 0$ such that $u\alpha<1$:
$\lambda e^{-\lambda t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}e^{\alpha
us}e^{-s}dsdx=\frac{\lambda(e^{\alpha ut}-e^{-\lambda
t})}{(u\alpha+\lambda)(1-u\alpha)}.$ (47)
We deduce easily that if $g_{0}(t)\leq C_{0}e^{u\alpha t}$ then
$g_{1}(t)=\Psi(g_{0})(t)\leq Le^{u\alpha t}+\frac{L\lambda
d}{(u\alpha+\lambda)(1-u\alpha)}(e^{u\alpha t}-e^{-\lambda t})\leq
C_{1}e^{u\alpha t},$
with $C_{1}=L+\frac{C_{0}\lambda d}{(u\alpha+\lambda)(1-u\alpha)}$. By
recursion, we obtain that $\limsup_{k}g_{k}(t)\leq Ce^{u\alpha t}$, with
$C=L(u\alpha+\lambda)(1-u\alpha)/((1-\lambda)u\alpha-\lambda(d-1)-u^{2}\alpha^{2})<\infty$.
We may now conclude the proof. Notice that $\Psi$ is monotone: if for all
$t\geq 0$, $h_{1}(t)\geq h_{2}(t)$ then for all $t\geq 0$,
$\Psi(h_{1})(t)\geq\Psi(h_{2})(t)$. Hence, by recursion, from the assumption
$f\leq\Psi(f)=g_{1}$, we deduce that for all integer $k\geq 1$, $f\leq g_{k}$.
It remains to take the limit in $k$.
Now, let $p$ be an integer, and define
$f_{p}(t)=\mathbb{E}^{\prime}_{\lambda}[Y(t)^{p}]$. The main result of this
subsection is the following lemma.
###### Lemma 5.5.
Let $1\leq p<\gamma_{P}$, if $\lambda\in(0,\lambda_{p})$, then $f_{p}$ is
finite and there exists a constant $C_{p}$ such that for all $t>0$
$f_{p}(t)\leq C_{p}e^{p\alpha t}.$
###### Proof 5.6.
In §5.1, we have computed $f_{p}$ for $p=1$ and found $f_{1}(t)=(de^{\alpha
t}-1)/(d-1)$. Let $p\geq 2$ and assume now that the statement of Lemma 5.5
holds for $q=1,\cdots,p-1$. Let $\kappa>0$,
$Y^{(\kappa)}(t)=\min(Y(t),\kappa)$ and let $\leq_{st}$ denote the stochastic
domination (beware that $Y^{(\kappa)}(t)$ is different from $Y^{(n)}(t)$
defined in §5.1). We use that if $y_{i}\geq 0$,
$\min(\sum_{i}y_{i},\kappa)\leq\sum_{i}\min(y_{i},\kappa)$. Hence, from RDE
(39), we have
$Y^{(\kappa)}(t)\leq_{st}1+\sum_{i=1}^{N}1{1}(\xi_{i}\leq
t)Y^{(\kappa)}_{i}(t-\xi_{i}+D_{i}).$ (48)
Recall the multinomial formula
$\left(\sum_{i=1}^{n}y_{i}\right)^{p}=\sum_{p_{1},\cdots,p_{n}}{{n}\choose{p_{1}\cdots
p_{n}}}y_{1}^{p_{1}}\cdots y_{n}^{p_{n}}.$
where the summation is taken over $n$-tuples of integers that sum up to $p$.
Taking power $p$ in the above stochastic inequality and expanding brutally, we
thus get
$Y^{(\kappa)}(t)^{p}\leq_{st}\sum_{p_{1},\cdots,p_{N+1}}{{N+1}\choose{p_{1}\cdots
p_{N+1}}}\prod_{i=1}^{N}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq
1}1{1}(\xi_{i}\leq t)Y^{(\kappa)}_{i}(t-\xi_{i}+D_{i})^{p_{i}}\right),$
where the summation is taken over $N+1$-tuples of integers that sum up to $p$.
Now we define
$f_{p}^{(\kappa)}(t)=\mathbb{E}^{\prime}_{\lambda}\left[Y^{(\kappa)}(t)^{p}\right]=\mathbb{E}^{\prime}_{\lambda}\left[\min(Y(t),\kappa)^{p}\right].$
Taking expectation and using independence leads to
$\displaystyle f^{(\kappa)}_{p}(t)$ $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{\infty}P(n)\sum_{p_{1},\cdots,p_{n+1}}{{n+1}\choose{p_{1}\cdots
p_{n+1}}}\prod_{i=1}^{n}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq
1}\mathbb{E}^{\prime}_{\lambda}\left[1{1}(\xi\leq
t)Y^{(\kappa)}(t-\xi+D)^{p_{i}}\right]\right)$ (49) $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{\infty}P(n)\sum_{p_{1},\cdots,p_{n+1}}{{n+1}\choose{p_{1}\cdots
p_{n+1}}}$ $\displaystyle\hskip
65.0pt\times\prod_{i=1}^{n}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq 1}\lambda
e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f^{(\kappa)}_{p_{i}}(s)e^{-s}dsdx\right).$
Consider a $n+1$-tuple that sums up to $p$ such that for all $i=1,\cdots,n$,
$p_{i}\leq p-1$, $\sum_{i=1}^{n}p_{i}=q\leq p$ and
$\sum_{i=1}^{n}1{1}_{p_{i}\geq 1}=m\leq p$. From the recursive hypothesis and
(47), with $L=\max_{1\leq k\leq p-1}C_{k}$, we get
$\displaystyle\prod_{i=1}^{n}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq 1}\lambda
e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f^{(\kappa)}_{p_{i}}(s)e^{-s}dsdx\right)$
$\displaystyle\leq$ $\displaystyle\prod_{i:p_{i}\geq 1}\frac{C_{p_{i}}\lambda
e^{\alpha p_{i}t}}{(\lambda+p_{i}\alpha)(1-p_{i}\alpha)}$ $\displaystyle\leq$
$\displaystyle L^{m}e^{\alpha qt}e^{-\sum_{i=1}^{n}\ln(1-p_{i}\alpha)}.$
Now recall that $|\ln(1-y)+y|\leq\frac{y^{2}}{2(1-y)}$ for $y\in(0,1)$. Since
$\sum_{i=1}^{n}p_{i}^{2}\leq q^{2}$, we get
$\displaystyle\prod_{i=1}^{n}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq 1}\lambda
e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{p_{i}}(s)e^{-s}dsdx\right)$
$\displaystyle\leq$ $\displaystyle L^{m}e^{\alpha qt}e^{\alpha
q+\frac{\alpha^{2}q^{2}}{2(1-p\alpha)}}$ $\displaystyle\leq$ $\displaystyle
L^{\prime}e^{\alpha pt}.$
Then, grouping together all such $n+1$-tuples, from (49) we deduce
$\displaystyle f^{(\kappa)}_{p}(t)$ $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{\infty}P(n)\left((n+1)^{p}L^{\prime}e^{\alpha
pt}+n\lambda e^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f^{(\kappa)}_{p}(s)e^{-s}dsdx\right)$
(50) $\displaystyle\leq$ $\displaystyle L^{\prime\prime}e^{\alpha pt}+\lambda
de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f^{(\kappa)}_{p}(s)e^{-s}dsdx,$
where we have used the hypothesis $p<\gamma_{P}$. We may then apply Lemma 5.3:
there exists a constant $C_{p}$ such that
$f_{1}^{(\kappa)}(t)\leq C_{p}e^{\alpha pt}.$
The monotone convergence Theorem implies that
$f_{p}(t)=\lim_{\kappa\to\infty}f_{p}^{\kappa)}(t)$ exists and is bounded by
$C_{p}e^{\alpha pt}$. The recursion is complete.
### 5.3 Proof of Theorem 1.3 : lower bound on $\gamma(\lambda)$
To prove Theorem 1.3, we shall prove two statements
$\hbox{If $\mathbb{E}^{\prime}_{\lambda}[Z^{u}]<\infty$ then
}u\leq\overline{\gamma},$ (51)
$\hbox{If $1<u<\min(\overline{\gamma},\gamma_{P})$ then
}\mathbb{E}^{\prime}_{\lambda}[Z^{u}]<\infty.$ (52)
In this paragraph, we prove (52). The argument is a refinement of the argument
in §5.2. Let $\kappa>0$ and let
$f_{u}^{(\kappa)}(t)=\mathbb{E}^{\prime}_{\lambda}[\min(Y(t),\kappa)^{u}]$, we
have the following lemma.
###### Lemma 5.7.
If $1<u<\min(\overline{\gamma},\gamma_{P})$, there exists a constant $L_{u}>0$
such that for all $t\geq 0$ and $\kappa>0$,
$f_{u}^{(\kappa)}(t)\leq L_{u}e^{u\alpha t}+\lambda de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f^{(\kappa)}_{u}(s)e^{-s}dsdx.$
###### Proof 5.8.
The lemma is already proved if $u$ is an integer in (50). The general case
extends of the same argument. We write $u=p+v$ with $v\in(0,1)$ and integer
$p\geq 1$. We use the inequality, for all $y_{i}\geq 0$, $1\leq i\leq n$,
$\left(\sum_{i=1}^{n}y_{i}\right)^{u}\leq\sum_{i=1}^{n}\sum_{p_{1},\cdots,p_{n}}{{n}\choose{p_{1}\cdots
p_{n}}}y_{i}^{p_{i}+v}\prod_{1\leq j\leq n,j\neq i}y_{j}^{p_{j}},$ (53)
where the summation is taken over $n$-tuples of integers that sum up to $p$
(which follows from the inequality $(\sum y_{i})^{v}\leq\sum y_{i}^{v}$ and
the multinomial formula). Then from (48), we get the stochastic domination
$\displaystyle Y^{(\kappa)}(t)^{u}$ $\displaystyle\leq_{st}$
$\displaystyle\sum_{i=1}^{N}\sum_{p_{1},\cdots,p_{N+1}}{{N+1}\choose{p_{1}\cdots
p_{N+1}}}\left(1{1}_{p_{i}=0}+1{1}_{p_{i}\geq 1}1{1}(\xi_{i}\leq
t)Y^{(\kappa)}_{i}(t-\xi_{i}+D_{i})^{p_{i}+v}\right)$ (54)
$\displaystyle\hskip 50.0pt\times\prod_{1\leq j\leq N,j\neq
i}\left(1{1}_{p_{j}=0}+1{1}_{p_{j}\geq 1}1{1}(\xi_{j}\leq
t)Y^{(\kappa)}_{j}(t-\xi_{j}+D_{i})^{p_{j}}\right),$
where the summation is taken over $N+1$-tuples of integers that sum up to $p$.
From Lemma 5.5, for all $1\leq q\leq p$, $f_{q}(t)\leq C_{q}e^{q\alpha t}$.
Note also, by Jensen inequality, that for all $1\leq q\leq p-1$,
$f_{q+v}(t)\leq f_{p}(t)^{\frac{q+v}{p}}\leq C_{p}e^{(q+v)\alpha t}$. The same
argument (with $p$ replaced by $u$) which led to (50) in the proof of Lemma
5.5 leads to the result.
Statement (52) is a consequence of Lemma 5.3 and Lemma 5.7. Indeed, by Lemma
5.3, for all $t\geq 0$, $f_{u}^{(\kappa)}(t)\leq C_{u}e^{u\alpha t}$ for some
positive constant $C_{u}$ independent of $\kappa$. From the monotone
convergence Theorem, we deduce that, for all $t\geq 0$, $f_{u}(t)\leq
C_{u}e^{u\alpha t}$. However from $Z\stackrel{{\scriptstyle d}}{{=}}Y(D)$, we
find
$\mathbb{E}^{\prime}_{\lambda}Z^{u}=\int_{0}^{\infty}f_{u}(t)e^{-t}dt\leq\int_{0}^{\infty}C_{u}e^{u\alpha
t}e^{-t}dt.$
Then, statement (52) follows from $u\alpha<1$.
### 5.4 Proof of Theorem 1.3 : upper bound on $\gamma(\lambda)$
In this paragraph, we prove statement (51). This will conclude the proof of
Theorem 1.3. Let $u>1$, we assume that
$\mathbb{E}^{\prime}_{\lambda}[Z^{u}]<\infty$ we need to show that
$\lambda<\lambda_{u}$. Without loss of generality we can assume that
$\lambda<\lambda_{1}$. From Lemma 5.1 and (39), we get
$f_{u}(t)=\mathbb{E}^{\prime}_{\lambda}[Y(t)^{u}]=\mathbb{E}^{\prime}_{\lambda}\left(1+\sum_{i=1}^{N}1{1}(\xi_{i}\leq
t)Y_{i}(t-\xi_{i}+D_{i})\right)^{u}.$
Taking expectation and using the inequality $(x+y)^{u}\geq x^{u}+y^{u}$, for
all positive $x$ and $y$, we get:
$\displaystyle f_{u}(t)$ $\displaystyle\geq$ $\displaystyle 1+\lambda
de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx.$ (55)
From Jensen’s Inequality, $f_{u}(t)\geq f_{1}(t)^{u}\geq e^{u\alpha t}$. Note
that the integral $\int_{x}^{\infty}e^{\alpha us}e^{-s}ds$ is finite if and
only if $u<\alpha^{-1}$. Suppose now that $\overline{\gamma}<u<\alpha^{-1}$.
We use the fact that if $u>\overline{\gamma}$ then
$u^{2}\alpha^{2}-(1-\lambda)u\alpha+\lambda(d-1)>0$. It implies that there
exists $0<\epsilon<\lambda$ such that
$u^{2}\alpha^{2}-(1-\lambda)u\alpha+\lambda(d-1)>\epsilon d.$ (56)
We define $\tilde{\lambda}=\lambda-\epsilon$,
$\tilde{\Delta}(\epsilon)=(1-\lambda)^{2}-4(\tilde{\lambda}d-\lambda)$. Note
that $\tilde{\Delta}(0)=\Delta$. Since $\lambda<\lambda_{1}$, for $\epsilon$
small enough, $\tilde{\Delta}$ is non-negative, we may then consider the real
roots of $X^{2}-(1-\lambda)X+\tilde{\lambda}d-\lambda$:
$\tilde{\alpha}(\epsilon)=\frac{1-\lambda-\sqrt{\tilde{\Delta}}}{2}\quad\hbox{
and }\quad\tilde{\beta}(\epsilon)=\frac{1-\lambda+\sqrt{\tilde{\Delta}}}{2}.$
Again, for $\epsilon=0$, $\tilde{\alpha}(0)=\alpha$ and
$\tilde{\beta}(0)=\beta$. Hence, since $u>{\overline{\gamma}}=\beta/\alpha$,
by continuity, for $\epsilon$ small enough,
$u\alpha>\tilde{\beta}.$ (57)
We compute a lower bound from (55) as follows:
$\displaystyle f_{u}(t)$ $\displaystyle\geq$ $\displaystyle 1+\epsilon
de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx+\tilde{\lambda}de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx$
$\displaystyle\geq$ $\displaystyle 1+\epsilon de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}e^{u\alpha
s}e^{-s}dsdx+\tilde{\lambda}de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx$
$\displaystyle\geq$ $\displaystyle 1+L(e^{u\alpha t}-e^{-\lambda
t})+\tilde{\lambda}de^{-\lambda
t}\int_{0}^{t}e^{(\lambda+1)x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx,$
with
$L=\frac{\epsilon d}{(\lambda+u\alpha)(1-u\alpha)}>0.$
We consider the mapping $\Psi:h\mapsto 1+L(e^{u\alpha t}-e^{-\lambda
t})+\tilde{\lambda}\int_{0}^{t}e^{x}\int_{x}^{\infty}f_{u}(s)e^{-s}dsdx$.
$\Psi$ is monotone: if for all $t\geq 0$, $h_{1}(t)\geq h_{2}(t)$ then for all
$t\geq 0$, $\Psi(h_{1})(t)\geq\Psi(h_{2})(t)$. Since, for all $t\geq 0$,
$f_{u}(t)\geq\Psi(f_{u})(t)\geq 1$, we deduce by iteration that there exists a
function $h$ such that $h=\Psi(h)\geq 1$. As in §5.1, solving $h=\Psi(h)$ is
simple, taking twice the derivative, we get,
$h^{\prime\prime}-(1-\lambda)h^{\prime}+(\tilde{\lambda}d-\lambda)h=-\lambda-L(\lambda+u\alpha)(1-u\alpha)e^{u\alpha
t}.$
Therefore,
$h=ae^{\tilde{\alpha}t}+be^{\tilde{\beta}t}-\epsilon(u^{2}\alpha^{2}-(1-\lambda)u\alpha+\lambda(d-1))^{-1}e^{u\alpha
t}$ for some constant $a$ and $b$. From (57) the leading term as $t$ goes to
infinity is equal to
$-\epsilon(u^{2}\alpha^{2}-(1-\lambda)u\alpha+\lambda(d-1))^{-1}e^{u\alpha
t}$. However from (56),
$-\epsilon(u^{2}\alpha^{2}-(1-\lambda)u\alpha+\lambda(d-1))^{-1}<0$ and it
contradicts the assumption that $h(t)\geq 1$ for all $t\geq 0$. Therefore we
have proved that $u\leq\overline{\gamma}$.
## Appendix
In this appendix, for the sake of completeness we include the proof of the
following lemma on Galton-Watson trees.
###### Lemma 5.9.
Let $T$ be a Galton-Watson tree with mean number of offsprings $d>1$.
Conditioned on $T$ is infinite, $T$ is a.s. lower $d$-ary.
###### Proof 5.10.
Let $1<\delta<d$ and $Z_{n}=|V_{n}|$ be the number of offsprings of generation
$n$. From Seneta-Heyde Theorem (see [22, Chapter 5]), conditioned on $T$ is
infinite, a.s.
$\lim_{n\to\infty}\frac{1}{n}\log Z_{n}=d.$
Let $p>0$ be the probability that $T$ is infinite. It implies that for any
$\varepsilon>0$, for all $n$ large enough, we have
$\mathbb{P}(Z_{n}\geq\delta^{n})\geq p-\varepsilon$.
Now, consider a new Galton-Watson tree $T^{\prime}$ starting from the root
$\o$ where each vertex produces independently $m=\lfloor\delta^{n}\rfloor$
offsprings with probability $p-\varepsilon$ and $0$ offspring otherwise. From
what precedes, we may couple $T$ and $T^{\prime}$ such that $T^{\prime}$ is a
subtree of $T^{*n}$.
We are now going to prove that $T^{\prime}$ contains a large regular tree with
probability at least $p-2\varepsilon$. To this end, we set
$q=q(\varepsilon)=1-p+\varepsilon.$
Note that we may have chosen $n=n(\varepsilon)$ large enough so that
$q+(1-q)e^{-m\varepsilon^{2}/2}\leq q+\varepsilon.$ (58)
We consider the following pruning algorithm on $T^{\prime}$. At step $0$, we
start with all vertices of $T^{\prime}$. At step $1$, we remove all vertices
which have less than $(1-q-2\varepsilon)m$ offsprings. We now iterate: at step
$k\geq 1$, we remove all vertices which have less than $(1-q-2\varepsilon)m$
offsprings left by step $k-1$.
Denote by $\rho_{k}$ the probability that the root of $T^{\prime}$ is removed
by step $k$. We have $\rho_{0}=0$, $\rho_{1}=q$ and $(\rho_{k})_{k\geq 0}$ is
an non-decreasing sequence. We are going to check by recursion that for all
$k\geq 1$,
$\rho_{k}\leq q+\varepsilon.$ (59)
Indeed, let $k\geq 1$ and assume that $\rho_{k-1}\leq q+\varepsilon$. Note
that if the root is removed by step $k$, then either it has $0$ offspring or
more than $(q+2\varepsilon)m$ of its offsprings were removed by step $k-1$.
From the recursive structure of the Galton-Watson tree, the probability that
an offspring was removed by step $k-1$ is $\rho_{k-1}$ and these events for
each offspring are independent. It follows that
$\displaystyle\rho_{k}\leq
q+(1-q)\mathbb{P}{{\left(\sum_{i=1}^{m}X_{i}\geq(q+2\varepsilon)m\right)}},$
where $(X_{i})_{1\leq i\leq m}$ are i.i.d. $\\{0,1\\}$-Bernoulli variables
with mean $\rho_{k-1}$. By recursion hypothesis, $\rho_{k-1}\leq
q+\varepsilon$. Hence, Hoeffding’s inequality leads to
$\mathbb{P}{{\left(\sum_{i=1}^{m}X_{i}\geq(q+2\varepsilon)m\right)}}\leq\mathbb{P}{{\left(\sum_{i=1}^{m}(X_{i}-\mathbb{E}X_{i})\geq\varepsilon
m\right)}}\leq e^{-m\varepsilon^{2}/2}.$
From (58), we deduce that $\rho_{k}\leq q+\varepsilon$. This proves (59).
We have thus proven that with probability at least
$1-(q+\varepsilon)=p-2\varepsilon$, the root of $T$ is never removed by the
pruning algorithm. However, on the latter event, by construction $T^{\prime}$
contains a $\lfloor(1-q-2\varepsilon)m\rfloor$-ary tree rooted at $\o$ (note
that $1-q-2\varepsilon=p-3\varepsilon$).
We may now conclude the proof. We apply the above argument to some
$\delta^{\prime}\in(\delta,d)$. This proves that for any $0<\varepsilon<1$,
there exists an integer $n_{\varepsilon}$ such that with probability at least
$(1-\varepsilon)p$, $T^{*n_{\varepsilon}}$ contains a
$\lceil\delta^{n_{\varepsilon}}\rceil$-ary tree. Note that the latter event is
contained in the event that $T$ is infinite. It follows that the conditional
probability that $T^{*n_{\varepsilon}}$ contains a
$\lceil\delta^{n_{\varepsilon}}\rceil$-ary tree, given $T$ infinite, is at
least $1-\varepsilon$.
We finally consider the sequence $\varepsilon_{k}=1/k^{2}$. From Borel-
Cantelli lemma, conditioned on $T$ infinite, a.s. there exists $k$ such that
$T^{*n_{\varepsilon_{k}}}$ contains a
$\lceil\delta^{n_{\varepsilon_{k}}}\rceil$-ary tree.
## References
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This work has benefited from the active participation of Ghurumuruhan Ganesan,
notably on Section 3. The author thanks also Elie Aïdékon and Jeremy Quastel
for enlightening discussions on the analogy between the Brunet-Derrida model
and the chase-escape process.
|
arxiv-papers
| 2012-10-10T12:25:07 |
2024-09-04T02:49:36.296638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Charles Bordenave",
"submitter": "Managing Editor",
"url": "https://arxiv.org/abs/1210.2883"
}
|
1210.2923
|
#
${{~{}~{}~{}~{}~{}~{}~{}}^{{}^{{}^{``Celestial~{}Mechanics~{}and~{}Dynamical~{}Astronomy"\,,~{}Vol.~{}114\,,~{}pp.~{}387\,-\,414\,~{}(2012).}}}}$
Bodily tides near the $\,1:1\,$ spin-orbit resonance.
Correction to Goldreich’s dynamical model
James G. Williams
Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA
91109 USA
e-mail: james.g.williams @ jpl.nasa.gov
and
Michael Efroimsky
US Naval Observatory, Washington DC 20392 USA
e-mail: michael.efroimsky @ usno.navy.mil
###### Abstract
Spin-orbit coupling is often described in an approach known as _“the MacDonald
torque”_ , which has long become the textbook standard due to its apparent
simplicity. Within this method, a concise expression for the additional tidal
potential, derived by MacDonald (1964; Rev. Geophys. 2, 467 - 541), is
combined with a convenient assumption that the quality factor $\,Q\,$ is
frequency-independent (or, equivalently, that the geometric lag angle is
constant in time). This makes the treatment unphysical because MacDonald’s
derivation of the said formula was, very implicitly, based on keeping the time
lag frequency-independent, which is equivalent to setting $\,Q\,$ to scale as
the inverse tidal frequency. This contradiction requires the entire MacDonald
treatment of both non-resonant and resonant rotation to be rewritten.
The non-resonant case was reconsidered by Efroimsky & Williams (2009;
Cel.Mech.$\,\&$ Dyn.Astr. 104, 257 - 289), in application to spin modes
distant from the major commensurabilities. In the current paper, we continue
this work by introducing the necessary alterations into the MacDonald-torque-
based model of falling into a 1-to-1 resonance. (The original version of this
model was offered by Goldreich 1966; AJ 71, 1 - 7.)
Although the MacDonald torque, both in its original formulation and in its
corrected version, is incompatible with realistic rheologies of minerals and
mantles, it remains a useful toy model, which enables one to obtain, in some
situations, qualitatively meaningful results without resorting to the more
rigorous (and complicated) theory of Darwin and Kaula.
We first address this simplified model in application to an oblate primary
body, with tides raised on it by an orbiting zero-inclination secondary. (Here
the role of the tidally-perturbed primary can be played by a satellite, the
perturbing secondary being its host planet. A planet may as well be the
perturbed primary, its host star acting as the tide-raising secondary.) We
then extend the model to a triaxial primary body experiencing both a tidal and
a permanent-figure torque exerted by an orbiting secondary. We consider the
effect of the triaxiality on both circulating and librating rotation near the
synchronous state. Circulating rotation may evolve toward the libration region
or toward a spin faster than synchronous (the so-called pseudosynchronous
spin). Which behaviour depends on the orbit eccentricity, the triaxial figure
of the primary, and the mass ratio of the secondary and primary bodies. The
spin evolution will always stall for the oblate case. For libration with a
small amplitude, expressions are derived for the libration frequency, damping
rate, and average orientation.
Importantly, the stability of pseudosynchronous spin hinges upon the
dissipation model employed. Makarov and Efroimsky (2013; arXiv:1209.1616) have
found that a more realistic tidal dissipation model than the corrected
MacDonald torque makes pseudosynchronous spin unstable. Besides, for a
sufficiently large triaxiality, pseudosynchronism is impossible, no matter
what dissipation model is used.
## 1 Motivation
Bodily tides in a near-spherical homogeneous primary perturbed by a point-like
secondary are described by the theory developed mainly by Darwin (1879, 1880)
and Kaula (1966). Sometimes this theory is referred to as the Darwin torque.
Based on a Fourier-like expansions of the perturbing potential and of the
tidally-induced potential of the disturbed primary, their theory permits an
arbitrary rheology of the primary.
Sometimes a much simpler empirical model, offered by MacDonald (1964) and
often named as the MacDonald torque, is used in the literature for obtaining
very approximate but still qualitatively reasonable description of tidal
evolution. This model can be derived from the Darwin-Kaula theory, under
several simplifying assumptions. Among these is the assumption that the tidal
quality factor $\,Q\,$ of the perturbed primary should scale as the inverse of
the tidal frequency. Being incompatible with the rheologies of actual
minerals, this key assumption prohibits the use of the MacDonald model in
long-term orbital calculations.
Despite the unrealistic rheology instilled into the MacDonald model, the model
remains a lab which permits one to gain qualitative understanding of tidal
evolution (rotational and orbital), without resorting to the lengthy
calculations required in the accurate Darwin-Kaula approach. It should be
noted however that employment of the MacDonald model needs some care.
Historically, the orbital calculations performed with aid of this model by one
of its creators, MacDonald (1964), contained an inherent contradiction.
MacDonald began his paper with deriving a concise expression for the
additional tidal potential, and then combined this expression with a
convenient assumption that the geometric lag angle is constant (or,
equivalently, that the tidal quality factor is a frequency independent
constant). This made the treatment unphysical because MacDonald’s derivation
of the said formula was, very implicitly, based on keeping the time lag
frequency-independent, which is equivalent to setting $\,Q\,$ to scale as the
inverse tidal frequency. This contradiction made his theory inconsistent. The
said oversight has then been repeated many a time in the literature (Kaula
1968, eqn. 4.5.37; Murray & Dermott 1999, eqn. 5.14). 111 Due to an error in
our translation from German, we mis-assumed in our previous papers Efroimsky &
Williams (2009) and Efroimsky (2012a) that Gerstenkorn (1955) had based his
development on a constant-$Q$ model. Therefore we stated that his theory
contained the same genuine inconsistency as the theory by MacDonald (1964).
Accurate translation of the work by Gerstenkorn (1955) has shown that his
method was based on a constant-time-lag model. Therefore we retract our
statement about Gerstenkorn’s approach sharing the inconsistency of
MacDonald’s theory. We also thank Hauke Hussmann and Peter Noerdlinger for
their kind help in translating excerpts from Gerstenkorn’s work. In
particular, the MacDonald method was used by Goldreich (1966) in his theory of
dynamical evolution near the 1:1 spin-orbit resonance. We reconsider this
theory, employing the corrected version of the MacDonald torque, i.e., setting
the quality factor to scale as inverse frequency.
## 2 Linear bodily tide in a near-spherical primary
Consider a near-spherical primary of radius $\,R\,$ and a secondary of mass
$M^{*}_{sec}$ located at
${\mbox{{\boldmath$\vec{r}$}}}^{\;*}=(r^{*},\,\phi^{*},\,\lambda^{*})$, where
$r^{*}\geq R$. The tidal potential created by the secondary alters the
primary’s shape and, as a result, its potential. For linear tides, the
amendment to the primary’s exterior potential is known (e.g., Efroimsky &
Williams 2009, Efroimsky 2012a,b) to read as
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\;=\;\,-\,{G\;M^{*}_{sec}}\sum_{{\it{l}}=2}^{\infty}k_{\it
l}\;\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r^{\textstyle{{}^{\it{l}+1}}}{r^{\;*}}^{\textstyle{{}^{\it{l}+1}}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it
l}+m)!}(2-\delta_{0m})P_{{\it{l}}m}(\sin\phi)P_{{\it{l}}m}(\sin\phi^{*})\;\cos
m(\lambda-\lambda^{*})\,~{},~{}\quad$ (1)
$\delta_{ij}\,$ being the Kronecker delta, $\,G=6.7\times
10^{-11}\,\mbox{m}^{3}\,\mbox{kg}^{-1}\mbox{s}^{-2}\,$ being Newton’s gravity
constant, and $\,\gamma\,$ being the angle between the vectors
$\,\mbox{{\boldmath$\vec{r}$}}^{\;*}\,$ and $\vec{r}$ pointing from the
primary’s centre. As agreed above,
$\,{\mbox{{\boldmath$\vec{r}$}}}^{\;*}\,=\,(r^{*}\,,\,\phi^{*}\,,\,\lambda^{*})$
denotes the position of the perturber, while
$\,\mbox{{\boldmath$\vec{r}$}}\,=\,(r\,,\,\phi\,,\,\lambda)\,$ is an exterior
point, at a radius $\,r\,\geq\,R\,$, where the tidal potential amendment
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ is measured. The longitudes
$\lambda,\,\lambda^{*}$ are reckoned from a fixed meridian on the primary,
while the latitudes $\phi,\,\phi^{*}$ are reckoned from its equator. Indices
$\,l\,$ and $\,m\,$ are traditionally referred to as the degree and order,
accordingly. The associated Legendre functions $\,P_{lm}(x)\,$ (termed
associated Legendre polynomials when their argument is sine or cosine of some
angle) are introduced as in Kaula (1966, 1968), and may be called unnormalised
associated Legendre functions, to distinguish them from their normalised
counterparts.222 The Legendre polynomials may be defined through the Rodriguez
formula $~{}P_{l}(x)\,=\,\frac{\textstyle 1}{\textstyle
2^{l}\,l!}~{}\frac{\textstyle d^{\,l}}{\textstyle
dx^{\,l}}\left(\,x^{2}\,-\,1\,\right)^{l}~{}.~{}~{}$ In most literature, the
associated Legendre functions are introduced, for a nonnegative $\,m\,$, as
$\displaystyle
P_{lm}(x)~{}=~{}\left(\,1\,-\,x^{2}\,\right)^{m/2}~{}\frac{d^{m}~{}}{dx^{m}}\,P_{l}(x)~{}\quad~{}\quad~{}\mbox{and}~{}\quad\quad
P_{l}^{m}(x)~{}=~{}(-1)^{m}\,\left(\,1\,-\,x^{2}\,\right)^{m/2}~{}\frac{d^{m}~{}}{dx^{m}}\,P_{l}(x)~{}~{}~{},$
so that $~{}P_{lm}(x)~{}=~{}(-1)^{m}\,P_{l}^{m}(x)~{}.~{}$ The above
definition agrees with the one offered by Abramowitz & Stegun (1972, p. 332).
A different convention is accepted in those books (e.g., Arfken & Weber 1995,
p. 623) where $\,P_{l}^{m}(x)\,$ lacks the $\,(-1)^{m}\,$ factor and thus
coincides with $\,P_{lm}(x)\,$. The Love numbers $\,k_{\it l}~{}$ can be
calculated from the geophysical properties of the primary.
A different formula for the tidal-response-generated change in the potential
was suggested by Kaula (1961, 1964). Kaula devised a method of switching
variables, from the spherical coordinates to the orbital elements
$\,(\,a^{*},\,e^{*},\,{\it i}^{*},\,\Omega^{*},\,\omega^{*},\,{\cal
M}^{*}\,)\,$ and $\,(\,a,\,e,\,{\it i},\,\Omega,\,\omega,\,{\cal M}\,)\,$ of
the secondaries located at $\,\mbox{{\boldmath$\vec{r}$}}^{\;*}\,$ and
$\vec{r}$ . The goal was to explore how a tide-raising secondary at
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}\,$ acts on a secondary at $\vec{r}$
through the medium of the tides it creates on their mutual primary.
The development enabled Kaula to process (1) into a series, which was a
disguised form of a Fourier expansion of the tide. Interestingly, Kaula
himself never referred to that expansion as a Fourier series, nor did he ever
write down explicitly the expressions for the Fourier modes. At the same time,
the way in which Kaula introduced the phase lags indicates that he was aware
of how the modes were expressed via the perturber’s orbital elements and the
primary’s rotation rate. The original works by Kaula (1961, 1964) were written
in a terse manner, with many technicalities omitted. A comprehensive
elucidation of his approach can be found in the Efroimsky & Makarov (2013).
Referring the reader to that paper for details, here we cite only the
resulting formula for the secular part of $\,U\,$, in the special case when
the tide-raising secondary itself experiences perturbation from the tide it
creates on the primary (so
$\,\mbox{{\boldmath$\vec{r}$}}^{~{}*}=\mbox{{\boldmath$\vec{r}$}}\,$):
$\displaystyle
U^{(sec)}(\mbox{{\boldmath$\vec{r}$}})\,=~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad$
$\displaystyle\,-~{}\frac{G\,M_{sec}}{a}\,\sum_{l=2}^{\infty}\,\left(\,\frac{R}{a}\,\right)^{\textstyle{{}^{2l+1}}}\sum_{m=0}^{\it
l}\frac{({\it l}-m)!}{({\it l}+m)!}\left(2-\delta_{0m}\right)\sum_{p=0}^{\it
l}F^{2}_{{\it l}mp}({\it i})\sum_{q=-\infty}^{\infty}G^{2}_{{\it
l}pq}(e)\;k_{l}(\omega_{lmpq})\;\cos\epsilon_{l}(\omega_{lmpq})~{}_{\textstyle{{}_{\textstyle~{}~{},}}}~{}~{}~{}$
(2)
where $\,l,\,m,\,p,\,q\,$ are integers, $\,F_{lmp}({\it i})\,$ are the
inclination functions (Gooding & Wagner 2008), $\,G_{lpq}(e)\,$ are the
eccentricity polynomials coinciding with the Hansen coefficients
$\,X^{\textstyle{{}^{(-l-1),\,(l-2p)}}}_{\textstyle{{}_{(l-2p+q)}}}(e)\,$,
while the superscript “sec” means: secular.
The dynamical Love numbers $\,k_{l}\,$ and the phase lags $\,\epsilon_{l}\,$
are functions of the Fourier modes
$\displaystyle\omega_{lmpq}\;\equiv\;({\it l}-2p)\;\dot{\omega}\,+\,({\it
l}-2p+q)\;\dot{\cal{M}}\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})\,~{},~{}~{}~{}$
(3)
$\theta\,$ being the primary’s sidereal angle, and $\,\dot{\theta}\,$ being
its angular velocity. While the tidal modes (3) can be of either sign, the
physical forcing frequencies
$\displaystyle\chi_{lmpq}\,\equiv\,|\,\omega_{lmpq}\,|\,=\,|\,({\it
l}-2p)\,\dot{\omega}\,+\,({\it
l}-2p+q)\,\dot{\cal{M}}\,+\,m\,(\dot{\Omega}\,-\,\dot{\theta})~{}|$ (4)
at which the stress and strain oscillate in the primary are positive definite.
A partial sum of series (2), with $\,|{\it{l}}|,\,|q|,\,|j|\,\leq\,2\,$, was
offered earlier by Darwin (1879). An explanation of Darwin’s method in modern
notation can be found in Ferraz-Mello, Rodríguez & Hussmann (2008).333 In
Ferraz-Mello, Rodríguez & Hussmann (2008), the meaning of notations $\vec{r}$
and $\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,$ is opposite to ours.
The power of the Darwin-Kaula approach lies in its compatibility with any
rheology, i.e., with an arbitrary form of the mode-dependence of the products
$\,k_{l}~{}\cos\epsilon_{l}\,$. It can be demonstrated that, for a homogeneous
near-spherical primary, the functional form of the mode dependence of the
product $\,k_{l}\,\cos\epsilon_{lmpq}\,$ is defined by index $\,l\,$ solely:
$\,k_{l}(\omega_{lmpq})\,\cos\epsilon_{l}(\omega_{lmpq})\,$, the other three
indices being attributed to the tidal mode. One can assume that this product
depends not on the tidal mode $\,\omega_{lmpq}\,$ but on the positive definite
physical frequency $\,\chi_{lmpq}\,$. This however will require some care in
derivation of the tidal torque from the above expressions for the potential –
see Efroimsky (2012a,b) for details.
We had to write down the secular part of the Kaula expansion of tide, because
we shall use it as a benchmark wherewith to compare the empirical expression
by MacDonald (1964), which we shall derive below in an accurate manner.
Lacking the ability to accommodate an arbitrary rheology, the MacDonald
approach (or the MacDonald torque) produces, after a necessary correction, a
simple method which can sometimes be employed for getting qualitative
understanding of the picture.
## 3 Tidal torque
Consider a secondary body of mass $\,M_{sec}\,$, located relative to its
primary at $\,\mbox{{\boldmath$\vec{r}$}}\,=\,(r,\,\lambda,\,\phi)\,$, where
$\,\phi\,$ is the latitude, and $\,\lambda\,$ denotes the longitude reckoned
from a meridian fixed on the primary. Let $\,U\,$ stand for the tidal-response
amendment to the primary’s potential. This amendment can be generated either
by this secondary itself or by some other secondary of mass $\,M_{sec}^{*}\,$
located at
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,=\,(r^{*},\,\lambda^{*},\,\phi^{*})\,$.
In either case, the primary’s tidal response to the gravity of the secondary
will render a tidal force and torque acting on the secondary of mass
$\,M_{sec}\,$. The torque’s component perpendicular to the equator of the
primary will be given by:
$\displaystyle T_{z}\;=\;-\;M_{sec}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\lambda}\;\;\;.$ (5)
We would reiterate that (5) is a component of the torque wherewith the primary
acts on the secondary of mass $\,M_{sec}\,$.
Then its negative will be the appropriate (i.e., orthogonal to the primary’s
equator) component of the torque wherewith the secondary acts on the primary:
$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;T_{z}\;=\;\,M_{sec}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\lambda}~{}~{}~{}.~{}~{}~{}$ (6)
Derivation of formulae (5 \- 6) is presented in Appendix A. These expressions
are convenient when the tidal-response potential amendment $\,U\,$ is
expressed through the spherical coordinates $~{}r\,,\,\lambda\,,\,\phi~{}$ and
$~{}r^{\,*}\,,\,\lambda^{\,*}\,,\,\phi^{\,*}~{}$, as in formula (1).
Whenever the tidal response is expressed as a function of the orbital elements
of the secondary and the sidereal angle $\,\theta\,$ of the primary, it is
practical to write down that the perpendicular-to-equator component of the
torque acting on the primary as
$\displaystyle{\cal{T}}_{z}(\mbox{{\boldmath$\vec{r}$}})\;=\;-\;M_{sec}\;\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\theta}\;\;\;,$ (7)
$\theta\,$ standing for the primary’s sidereal angle (Efroimsky 2012a,b).
We prefer to employ the terms _primary_ and _secondary_ rather than _planet_
and _satellite_. This choice of terms is dictated by our intention to apply
the below-developed machinery to research of tidal dissipation and spin
evolution of a satellite. In this setting, the satellite is effectively
playing the role of a tidally-distorted primary, its host planet acting as a
tide-generating secondary. Whenever the method is applied to exploring the
problem of planet despinning, the planet is understood as a primary, the host
star being a tide-raising secondary.
## 4 MacDonald (1964)
When it comes to taking dissipation into account, expression (1) turns out to
be far more restrictive than (2), in that (1) becomes applicable to a very
specific rheological model. This happens because a straightforward444 A more
accurate treatment, which cannot be employed within the MacDonald model but is
implementable within the Darwin-Kaula approach, is to expand the tide-raising
potential $\,W\,$ and the tidal-response potential change $\,U\,$ into Fourier
modes $\,\omega_{lmpq}\,$, and then to introduce the Love numbers
$\,k_{lmpq}\,=\,k_{l}(\omega_{lmpq})\,$, phase lags
$\,\epsilon_{lmpq}\,=\,\epsilon_{l}(\omega_{lmpq})\,$, and time lags $\,\Delta
t_{lmpq}\,=\,\Delta t_{l}(\omega_{lmpq})\,$. This formalism (explained in
detail in Efroimsky 2012a,b) enables one to express the so-introduced Love
numbers through the rheological properties of the primary’s mantle, and
thereby to model adequately the frequency-dependence of these Love numbers. An
intermediate, purely empirical, option would be to introduce “Love numbers”
$\,k_{lm}\,$, as if they were functions of both the degree $\,l\,$ and order
$\,m\,$. This idea is implemented in the IERS Conventions on the Earth
rotation (Petit & Luzum 2011). In the LLR (Lunar Laser Ranging) integration
software, tides in the Earth are parameterised by $\,k_{lm}\,$ and $\,\Delta
t_{lm}\,$ with $\,l\,=\,2\,$ and $\,m\,=\,0\,,\,1\,,\,2~{}~{}$ (Standish and
Williams 2012). option of instilling the delay into (1) is to replace in this
expression the perturber’s coordinates
$\,r^{*}(t)\,,\;\phi^{*}(t)\,,\;\lambda^{*}(t)\,$ with their delayed values,
$\,r^{*}(t-\Delta t)\,,~{}\phi^{*}(t-\Delta t)\,,~{}\lambda^{*}(t-\Delta
t)\,$. For example, instead of $\cos m(\lambda-\lambda^{*})$ we should employ
$\displaystyle\cos\left(~{}m~{}\lambda~{}-~{}m~{}\lambda^{{}^{*^{\,\textstyle{{}^{(delayed)}}}}}~{}\right)~{}=~{}\cos\left(\;m\;\lambda\;-\;(\,m\lambda^{{}^{*}}\,-\,m\stackrel{{\scriptstyle\centerdot}}{{\lambda}}{{}^{{}^{*}}}\Delta
t~{})~{}\right)\;\;\;,$ (8)
insertion whereof into (1) yields:
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})=\,-\,\sum_{{\it{l}}=2}^{\infty}\frac{\;k_{\it
l}\;{G\,M^{*}_{sec}}\;\,R^{\textstyle{{}^{2\it{l}+1}}}}{~{}r(t)^{\textstyle{{}^{\it{l}+1}}}\;\;\;{r^{\;*}}(t-\Delta
t)^{\textstyle{{}^{\it{l}+1}}}\;}\sum_{m=0}^{\it l}\frac{({\it l}-m)!}{({\it
l}+m)!}\left(2\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\delta_{0m}\right)P_{{\it{l}}m}(\sin\phi(t))P_{{\it{l}}m}(\sin\phi^{*}(t-\Delta
t))\;\cos m(\lambda-\lambda^{*}+\dot{\lambda}^{*}\,\Delta t)~{}.~{}~{}$ (9)
Expressing the longitude via the true anomaly $\,\nu\,$,
$\displaystyle\lambda\;=\;-\;\theta\;+\;\Omega\;+\;\omega\;+\;\nu\;+\;O({\it
i}^{2})\;=\;-\;\theta\;+\;\Omega\;+\;\omega\;+\;{\cal{M}}\;+\;2\;e\;\sin{\cal{M}}\;+\;O(e^{2})\;+\;O({\it
i}^{2})\;\;\;,\;\;\;\;$ (10)
and neglecting the apsidal and nodal precessions, we obtain:
$\displaystyle\cos\left(\;(\,m\;\lambda\;-\;m\lambda^{{}^{*}}\,)\;+\,m\dot{\lambda}^{*}\,\Delta
t\;\right)\;=\;\cos\left(\;(\,m\;\lambda\;-\;m\lambda^{{}^{*}}\,)\;+\,m\;(\dot{\nu}^{*}\,-\;\dot{\theta}^{*})\;\Delta
t\;+O~{}({i^{*}}^{2})\;\right)\quad\quad$ (11)
or, in terms of the mean anomaly:
$\displaystyle\cos\left((m\lambda-m\lambda^{{}^{*}})+m\stackrel{{\scriptstyle\centerdot}}{{\lambda}}{{}^{*}}\,\Delta
t\right)=\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}\quad~{}$
$\displaystyle\cos\left((m\lambda-m\lambda^{{}^{*}})+m(n^{*}-\dot{\theta}^{*})\Delta
t+2me^{*}n^{*}\Delta t\,\cos{\cal
M}^{*}+~{}O({e^{*}}^{2})~{}+O~{}({i^{*}}^{2})~{}+~{}O(i^{2})\,\right)~{}~{}~{}.~{}\quad$
(12)
This enables us to write down the potential as
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})=\,-\,G\,M_{sec}^{*}\,\sum_{{\it{l}}=2}^{\infty}k_{l}~{}\frac{R^{\textstyle{{}^{2\it{l}+1}}}}{r(t)^{\textstyle{{}^{\it{l}+1}}}{r^{{}^{*}}}(t-\Delta
t)^{\textstyle{{}^{\it{l}+1}}}}\,\sum_{m=0}^{\it l}\frac{({\it l}-m)!}{({\it
l}+m)!}\;\left(\,2\;-\;\delta_{0m}\right)P_{{\it{l}}m}(\sin\phi(t)\,)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle P_{{\it{l}}m}\left(\sin\phi^{*}\left(t-\Delta
t\right)\,\right)\;\cos\left(\,m\,(\lambda-\lambda^{{}^{*}})+\,m\,(\dot{\nu}^{*}-\dot{\theta}^{*})\,\Delta
t\,+O({\it i}^{2})+O({{\it i}^{*}}^{2})\,\right)~{}\,,~{}~{}$ (13)
where $~{}P_{{\it{l}}m}(\sin\phi(t)\,)P_{{\it{l}}m}(\sin\phi^{*}(t-\Delta
t)\,)~{}$ may be replaced, in the order of $~{}O({\it i}^{2})+O({{\it
i}^{*}}^{2})+O({\it i}{\it i}^{*})\;$, with
$\;P_{{\it{l}}m}(0)P_{{\it{l}}m}(0)\;\,$. A further simplification can be
achieved by taking into account the $\;{\it l}\,=\,2\;$ contribution only.
Omitting the $\,\lambda$-independent term with $\,m=0\,$ (in the expression
for the torque, $\,m\,$ will become a multiplier after the differentiation of
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$ with respect to $\,\lambda\,$), and
omitting the $\,m=1\,$ term (as $\;P_{21}(0)\,=\,0\;$), we arrive at the
expression
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})=\;-\;\frac{3}{4}\;\frac{G\;M_{sec}^{*}\;k_{2}\;R^{\textstyle{{}^{5}}}}{r(t)^{\textstyle{{}^{3}}}{r^{{}^{*}}}(t-\Delta
t)^{\textstyle{{}^{3}}}}\;\left[1\;+\;O({\it i}^{2})+O({{\it
i}^{*}}^{2})+O({\it i}{\it
i}^{*})\right]\;\cos\left(\;(\,2\;\lambda\;-\;2\lambda^{{}^{*}}\,)\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\;2\;(\dot{\nu}^{*}\,-\;\dot{\theta}^{*})\;\Delta
t~{}+~{}O({\it i}^{2})~{}+~{}O({{\it
i}^{*}}^{2})\;\right)~{}~{}~{}.~{}~{}~{}~{}$ (14)
In the special case when the tide-raising satellite is the same body as the
tidally-perturbed one (i.e., when $\,r(t)=r^{*}(t)\,$,
$\,M_{sec}=M_{sec}^{*}\,$, and $\,\lambda=\lambda^{*}\,$), this expression
happens to coincide, in the leading order of $\,{\it i}\,$, and $\,{\it
i}^{*}\,$, with the potential employed by MacDonald (1964). Thus we have
reproduced his empirical approach, by starting with the rigorous formula (1),
and by performing the following sequence of approximations:
* •
First, when accommodating dissipation, we set all the time delays to be equal,
for all the physical frequencies
$\,\chi_{\textstyle{{}_{\it{l}mpq}}}\equiv\,|\,\omega_{\textstyle{{}_{\it{l}mpq}}}\,|\,$
involved in the tide:
$\displaystyle\Delta t_{\textstyle{{}_{\it{l}mpq}}}\;\equiv\;\Delta
t(\chi_{\textstyle{{}_{\it{l}mpq}}})\;=\;\Delta t\;\;\;.$ (15)
This point is explained in great detail in Efroimsky & Makarov (2013).
* •
Second, we assume the smallness of the inclinations and lag, through the
neglect of the relative errors $\,O({\it i}^{2})~{}$, $~{}O({{\it
i}^{*}}^{2})~{}$, and $~{}O({\it i}{\it i}^{*})$ .
* •
Third, we truncate the series by leaving only the $\,{\it{l}}\,=\,m\,=\,2\,$
term.
These three steps take us from (1) to the approximation
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\approx\;-\;\frac{3}{4}\;\frac{G\;M_{sec}^{*}\;k_{2}\;R^{\textstyle{{}^{5}}}}{r(t)^{\textstyle{{}^{3}}}{r^{{}^{*}}}(t-\Delta
t)^{\textstyle{{}^{3}}}}\;\cos\left(\,2\;\lambda\;-\;2\lambda^{{}^{*}}~{}+~{}\epsilon~{}\right)\,~{},$
(16a) With the tide-raising secondary set to coincide with the one perturbed
by the tides on the primary (so $\,r(t)=r^{*}(t)\,$,
$\,M_{sec}=M_{sec}^{*}\,$, and $\,\lambda=\lambda^{*}\,$), the above
expression assumes the form of $\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\approx\;-\;\frac{3}{4}\;G\;M_{sec}\;k_{2}\;\frac{\,R^{\textstyle{{}^{5}}}\,}{\,r(t)^{3}~{}r(t-\Delta
t)\,}\;\,\cos\epsilon\,~{}.$ (16b) In the denominator, $\,r(t-\Delta t)\,$ can
be replaced, 555 From $\,r=a(1-e^{2})/(1+e\,\cos\nu)\,$ and
$\,\partial\nu/\partial M=(1+e\,\cos\nu)^{2}/(1-e^{2})^{3/2}\,$ it is
straightforward that $\displaystyle\Delta r\equiv r(t)-r(t-\Delta
t)=-\frac{a\,e\,(1\,-\,e^{2})}{(1\,+\,e~{}\cos\nu)^{2}}~{}\sin\nu~{}\Delta\nu\,+\,O\left(e\,(\Delta\nu)^{2}\,\right)\,=\,-\,\frac{a\,e\;\sin\nu}{(1-e^{2})^{1/2}}\,n\,\Delta
t\,+\,O\left(e\,(n\;\Delta t)^{2}\,\right)~{}~{}~{}.$ Thus our replacement of
$\,r^{*}(t-\Delta t)\,$ with $\,r^{*}(t)\,=\,r(t)\,$ entails a relative error
of order $\,O(en\Delta t)\,$. In expressions (9), (13), (14), and (16), the
absolute error will be of the same order, for $\,\lambda\neq\lambda^{*}\,$.
However for $\,\lambda=\lambda^{*}\,$ the absolute error will become
$\,O(enQ^{-1}\Delta t)\,$, since $\,\sin\epsilon\,$ is of the same order as
the inverse quality factor $\,Q\,$. In our estimates of errors, it is
irrelevant whether we define $Q$ as that appropriate to the principal tidal
mode or via formula (19) below; so we simply use the generic notation $Q$. As
we explained in Efroimsky & Williams (2009), after averaging over one
revolution of a nonresonant secondary about the primary, the absolute error
reduces to $\,O\left(e^{2}n^{2}Q^{-2}(\Delta t)^{2}\right)~{}$. In a resonant
case, though, we cannot enjoy this reduction, because in this case the
averaging procedure looks different as the torque changes its sign over a
period of the moon’s revolution. For this reason, the absolute error remains
$\,O(enQ^{-1}\Delta t)\,$. in the order of $\,O(e^{*}n^{*}\Delta t)\,$, with
$\,r^{*}(t)\,$. With this simplification implemented, we end up with
$\displaystyle
U(\mbox{{\boldmath$\vec{r}$}})\approx\;-\;\frac{3}{4}\;G\;M_{sec}\;k_{2}\;\frac{\,R^{\textstyle{{}^{5}}}\,}{\,r^{6}\,}\;\,\cos\epsilon\,~{}.$
(16c)
Expressions (14) and (16) contain the longitudinal lag
$\displaystyle\epsilon\;\equiv\;m\,\stackrel{{\scriptstyle\centerdot}}{{\lambda}}{{}^{{}^{*}}}\Delta
t\;=\;2\,(\dot{\nu}^{*}\,-\,\dot{\theta})\,\Delta t\;+\;O({\it i}^{2})\,~{},$
(17a) $\,\nu^{*}\,$ and $\,\theta\,$ being the true anomaly of the perturber
and the sidereal angle of the primary. In the special case (16b \- 16c), when
the tide-generating secondary and the secondary disturbed by the tide on the
primary are one and the same body, the asterisks may be dropped:
$\displaystyle\epsilon\;\equiv\;m\,\stackrel{{\scriptstyle\centerdot}}{{\lambda}}\Delta
t\;=\;2\,(\dot{\nu}\,-\,\dot{\theta})\,\Delta t\;+\;O({\it i}^{2})\,~{}.$
(17b)
The spin rate $\,\dot{\theta}\,$ is a slow variable, in that it may be assumed
constant over one orbiting cycle. The true anomaly is a fast variable. For a
nonvanishing eccentricity, $\,\dot{\nu}\,$ too is a fast variable, and so is
the lag $\,\epsilon\,$. This means that we should take into account these two
quantities’ variations over an orbital period.
Expression (16b) coincides, up to an irrelevant constant,666 The additional
tidal potential $\,U\,$, as given by equation (21) in the work by MacDonald
(1964), fails to vanish in the limit of zero geometric lag $\,\delta\,$. This
minor irregularity, though, does not influence MacDonald’s calculation of the
tidal torque. with the leading term of the appropriate formula from MacDonald
(1964, eqn 21). To appreciate this fact, notice that, up to $\,O({\it
i}^{2})\,$, the absolute value of the longitudinal lag $\,\epsilon\,$ is the
double of the geometrical angle $\,\delta=|(\dot{\theta}-\dot{\nu})\,\Delta
t|\,$ subtended at the primary’s centre between the directions to the
secondary and to the bulge,777 Be mindful that the double of the geometrical
angle is not equal to the absolute value of
$\,\epsilon_{\textstyle{{}_{2200}}}\,=\,2(n\,-\,\dot{\theta})\,\Delta
t_{\textstyle{{}_{2200}}}\,$. provided $\,\Delta t\,$ is postulated to be the
same for all tidal modes.
The analogy between the MacDonald theory and that of Darwin and Kaula can be
traced also by starting from the series (2). Consider the case of the tidally
perturbed secondary coinciding with the tide-raising one, so
$\,\lambda\,=\,\lambda^{*}\,$, and all the orbital variables are identical to
their counterparts with an asterisk. It will then be easy to notice that,
formally (just formally), expression (14) mimics the principal term of the
series (2), provided in this term the multiplier $\,G^{2}_{200}\,$ is replaced
with unity, and the principal phase lag
$\,\epsilon_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\,\equiv\,2\,(n-\dot{\theta})\,\Delta
t_{\textstyle{{}_{\textstyle{{}_{2200}}}}}\,$ is replaced with the
longitudinal lag (17). This way, within the MacDonald formalism, the
longitudinal lag (17) is playing the role of an _instantaneous_ phase lag
associated with double the _instantaneous_ synodic frequency
$\displaystyle\chi\;=\;2\;|\,\dot{\nu}\;-\;\dot{\theta}\,|\;\;\;,$ (18)
which is, up to $\,O({\it i}^{2})\,$, the double of the angular velocity
wherewith the point located under the secondary (with the same latitude and
longitude) is moving over the surface of the primary.
To extend further the analogy between the MacDonald and Darwin-Kaula models,
one can _define_ an auxiliary quantity
$\displaystyle``Q"\,=\,\frac{1}{\,\sin|\epsilon|\,}$ (19)
and derive from (17) and (19) that, in the leading order of $\,\epsilon\,$ and
$\,{\it i}\,$, this quantity satisfies
$\displaystyle``Q"\,=\,\frac{1}{\chi\;\Delta t}\;\;\;.$ (20a) A popular
fallacy would then be to interpret (20a) as a rheological scaling law
$\,Q\,\sim\,\chi^{\alpha}\,$ with $\,\alpha=-1\,$. That this interpretation is
generally incorrect follows from the fact that the quantity $\,``Q"\,$,
_defined_ through (19), is not obliged to coincide with the quality factor.888
Within the Darwin-Kaula theory, for each tidal mode $\,\omega_{{\it l}mpq}\,$,
we introduce the phase lag as $~{}\epsilon_{{\it
l}mpq}\,\equiv\,\omega_{lmpq}\,\Delta t_{{\it l}mpq}~{}$. Then we introduce
the appropriate quality factor $\;Q_{{\it l}mpq}\,\equiv\,Q(\chi_{{\it
l}mpq})\,=\,Q(\,|\omega_{{\it l}mpq}|\,)\;$ via the expression for the one-
cycle energy loss: $\displaystyle\Delta
E_{cycle}(\chi_{\textstyle{{}_{lmpq}}})\;=\;-\;\frac{2\;\pi\;E_{peak}(\chi_{\textstyle{{}_{lmpq}}})}{Q(\chi_{\textstyle{{}_{lmpq}}})}\;\;\;.$
Finally, using the fact that
$\,\chi_{\textstyle{{}_{lmpq}}}\equiv|\omega_{\textstyle{{}_{lmpq}}}|\,$ is
the frequency of a _sinusoidal_ load, we prove that the afore introduced
$\,\epsilon_{\textstyle{{}_{lmpq}}}\,$ and $\,Q_{\textstyle{{}_{lmpq}}}\,$ are
interconnected as $\,1/Q=\sin\epsilon\,$, if $\,E_{peak}\,$ denotes the
maximal energy, or in a more complex way, if $\,E_{peak}\,$ stands for the
maximal work (Efroimsky 2012a,b). Within the MacDonald method, original or
corrected as (15), it is not _a priori_ clear if the overall peak work (or the
overall peak energy stored) and the overall energy loss over a cycle are
interconnected via the auxiliary quantity $\,``Q"\,$ in exactly the same
manner as $\,E_{peak}(\chi_{\textstyle{{}_{lmpq}}})\,$ and $\,\Delta
E_{cycle}(\chi_{\textstyle{{}_{lmpq}}})\,$ are interconnected by
$\,Q(\chi_{\textstyle{{}_{lmpq}}})\,$ in the above expression. (Recall that
the total cycle of the tidal load is, generally, nonsinusoidal.) Whenever we
can prove that $\displaystyle\Delta
E_{cycle}^{(overall)}\;=\;-\;\frac{2\;\pi\;E_{peak}^{(overall)}}{``Q"}\;\;\;,$
our $\,``Q"\,$ can be spared of the quotation marks and can be called
_instantaneous_ quality factor, while (20a) can be treated as a reasonable
approximation to scaling law $\,Q\sim\chi^{\alpha}\,$ with $\,\alpha=-1\,$.
However, this should be justified in each particular case, not taken for
granted.
To sidestep these difficulties, it would be safer to write the constant-time-
lag rheological law, for small lags, simply as
$\displaystyle|\,\epsilon\,|\;=\;\chi\;\Delta t~{}~{}~{}.$ (20b)
Historically, MacDonald (1964) arrived at his model via empirical reasoning.
He certainly realised that the model was applicable to low inclinations only.
At the same time, this author failed to notice that the model also implied the
frequency-independence of the time lag. In fact, this frequency-independence,
(15), is necessary to derive the MacDonald model (16) from the generic
expression (1) for the tidal amendment to the primary’s potential. This way,
equality (15), is a priori instilled into the model. In other words, the
MacDonald model of tides includes in itself the rheological scaling law (20b)
with a constant $\,\Delta t\,$. Unaware of this circumstance, MacDonald (1964)
set the angular lag to be a frequency-independent constant, an assertion
equivalent to the time lag scaling as inverse tidal frequency. However, as we
just saw above, a consistent derivation of the MacDonald tidal model requires
that the time lag be set frequency-independent.999 In application to a non-
resonant setting, a constant-$\Delta t$ approach was taken yet by Darwin
(1879). Later, MacDonald (1964) abandoned this method in favour of a constant-
geometric lag calculation. Soon afterwards, though, Singer (1968) advocated
for reinstallment of the constant-$\Delta t$ method. Doing so, he was
motivated by an apparent paradox in MacDonald’s treatment. As explained by
Efroimsky & Williams (2009), the paradox is nonexistent. Nonetheless, the work
by Singer (1968) was fruitful at the time, as it renewed the interest in the
constant-$\Delta t$ treatment. The full might of the model was revealed by
Mignard (1979, 1980, 1981) who used it to develop closed expressions for the
tidal force and torque. Thus, to be consistent, the MacDonald method must be
corrected by applying (15) or, equivalently, (20b). In greater detail, the
necessity of this amendment is considered in Efroimsky & Makarov (2013).
Even after the model is combined with rheology (15), predictions of such a
theory are of limited use. The problem is that this rheology is radically
different from the actual behaviour of solids. As a result, calculations
relying on (15) render implausibly long times of tidal despinning – see, for
example the discussion and references in Castillo-Rogez et al. (2011). This
tells us that in realistic settings the MacDonald-style approach (14) based on
(15) is inferior, compared to the Darwin and Kaula method (2), which may, in
principle, be applied to any rheology. Despite this, the MacDonald torque
remains a convenient toy model, capable of furnishing results which are
qualitatively acceptable over not too long timescales (Hut 1981, Dobrovolskis
2007).
## 5 The MacDonald torque and the ensuing dynamical model by Goldreich. The
case of an oblate body
In this section, we shall trace how the MacDonald theory of bodily tides
yields the model of near-resonance spin dynamics by Goldreich (1966). Then we
shall recall an oversight in the MacDonald theory, and shall demonstrate that
correction of that oversight brings a minor alteration into Goldreich’s model
of spin evolution.
The goal of this section is limited to consideration of the tidal torque
solely. So we assume that the body is oblate, and the triaxiality-caused
torque does not show up. Appropriate for spinning gaseous objects, this
treatment indicates that the tidal torque stays finite for synchronous
rotation and vanishes at an angular velocity slightly faster than synchronous.
In Section 6 below we shall address the more general setting appropriate to
telluric bodies, with triaxiality included. While some authors (e.g., Heller
et al. 2011) ignore the triaxiality-caused torque in their treatment of solid
planets, it turns out that inclusion of the permanent-figure torque renders
important physical consequences and changes the picture completely, as will be
seen in Section 6.
### 5.1 The MacDonald torque
We shall restrict ourselves to the case of the tidally perturbed secondary
coinciding with the tide-raising one, so
$\,{{M}}_{sec}\,=\,{{M}}_{sec}^{*}\,$, and all the orbital variables are
identical to their counterparts with an asterisk.
Differentiating (14) with respect to $\,\lambda\,$, and then setting
$\,\lambda\,=\,\lambda^{*}\,$, we obtain the following expression for the
polar component of the torque, for low $\,{\it i}\;$:
$\displaystyle{\cal{T}}_{z}$ $\displaystyle=$
$\displaystyle\frac{3}{2}~{}{G\,M_{sec}^{2}}\;k_{2}~{}\frac{R^{\textstyle{{}^{5}}}}{~{}r^{\textstyle{{}^{3}}}(t)\;\;\;{r^{\;*}}^{\textstyle{{}^{3}}}(t-\Delta
t)\;}\,\;\sin\left(\,2\,(\dot{\nu}\,-\,\dot{\theta})\,\Delta t\,\right)+O({\it
i}^{2}/Q)$ $\displaystyle=$
$\displaystyle\frac{3}{2}~{}{G\,M_{sec}^{2}}\;k_{2}~{}\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}~{}\,\sin\left(\,2\,(\dot{\nu}\,-\,\dot{\theta})\,\Delta
t\,\right)+O({\it i}^{2}/Q)+O(en\Delta t/Q)\;\;,~{}~{}$
where the error $\,O(en\Delta t/Q)\,$ emerges when we identify the lagging
distance $\,r^{*}(t-\Delta t)\,$ with $\,r^{*}(t)\,=\,r(t)\,$, as explained in
footnote 5 in the preceding section. Referring the Reader to Efroimsky &
Williams (2009) for this and other technicalities, we would mention that
(LABEL:17) is equivalent to the Darwin torque only under the condition that
the rheological model (15 \- 20a) is accepted. For potentials, employment of
model (15 \- 20a) enables one to wrap up the infinite series (2) into the
elegant finite form (9). For torques (truncated to $\,{\it{l}}=2\,$ only),
similar wrapping of the appropriate series is available within the said model.
In the preceding section, we explained the geometric meaning of the
longitudinal lag (17): its absolute value is the double of the geometric angle
separating the directions to the bulge and the secondary as seen from the
primary’s centre.
If we _define_ a quantity $\,``Q"\,$ via (20a), the MacDonald torque will
look:
$\displaystyle{\cal{T}}_{z}$ $\displaystyle=$
$\displaystyle\frac{3}{2}~{}G~{}M_{sec}^{\,2}\;k_{2}\frac{R^{\textstyle{{}^{5}}}}{~{}r^{\textstyle{{}^{3}}}(t)\;\;\;{r^{\;*}}^{\textstyle{{}^{3}}}(t-\Delta
t)\;}\;\,\sin\epsilon+O({\it
i}^{2}/Q)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=$
$\displaystyle\frac{3}{2}~{}{G~{}M_{sec}^{\,2}}\;k_{2}~{}\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}~{}\frac{1}{``Q"}\,~{}\mbox{sgn}(\dot{\nu}-\dot{\theta})+O({\it
i}^{2}/Q)+O(en\Delta t/Q)~{}~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$
For a nonzero eccentricity, the quantity $\,``Q"\,$ should not be interpreted
as an instantaneous quality factor, because it is not guaranteed to
interconnect the peak work or peak energy and the one-cycle energy loss in a
manner appropriate to a quality factor – see footnote 8 in the previous
section. Therefore a more reasonable and practical way of writing the
MacDonald torque (LABEL:17) would be through using (20b) or (17):
$\displaystyle{\cal{T}}_{z}~{}=~{}\frac{3}{2}~{}{GM_{sec}^{2}}~{}k_{2}\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}~{}\Delta
t~{}2~{}(\dot{\nu}-\dot{\theta})~{}+~{}O({\it i}^{2}/Q)~{}+~{}O(en\Delta
t/Q)~{}~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}$ (23)
As we saw above, the MacDonald model is self-consistent only for $\,\Delta
t\,$ being a frequency-independent constant. However the factor
$~{}\dot{\nu}-\dot{\theta}~{}$ showing up in (23) varies over a cycle, for
which reason the torque needs averaging. This averaging is carried out in
Appendix B. In the vicinity of the $\,1:1\,$ resonance (for $\,\dot{\theta}\,$
close to $\,n\,$), the sign of $~{}\dot{\nu}-\dot{\theta}~{}$ changes twice
over a cycle, which makes the averaging procedure nontrivial. This situation
is to be addressed in subsections 5.2 and 5.3.
### 5.2 Goldreich (1966): treatment based on the MacDonald torque
In this subsection, we shall briefly recall a presently conventional method
pioneered almost half a century ago by Goldreich (1966). Based on the
MacDonald tide theory, this method has inherited both its simplicity and its
flaws.
The 1:1 resonance takes place when the spin rate $\,\dot{\theta}\,$ of the
primary (the satellite) is equal to the mean motion $\,n\,$ wherewith the
secondary body (the planet) is apparently orbiting the primary. The formula
(LABEL:19) for the MacDonald torque contains not the difference
$\,\dot{\theta}\,-\,n\,$ but the difference $\,\dot{\theta}\,-\,\dot{\nu}\,$,
for which reason the expression under the integral may twice change its sign
in the course of one revolution.
With aid of the formula
$\displaystyle\nu\;=\;{\cal{M}}\;+\;2\;e\;\sin{\cal{M}}\;+\;\frac{5}{4}\;e^{2}\;\sin
2{\cal{M}}\;+\;O(e^{3})\;\;\;$ (24)
and under the assumption that $\dot{e}\ll n$, we shape the difference of our
concern into the form of
$\displaystyle\dot{\theta}\;-\;\dot{\nu}\;=\,\left(\dot{\theta}-n\right)\;-\;2\;n\;e\;\left(\cos{\cal{M}}\;+\;\frac{5}{4}\;e\;\cos
2{\cal{M}}\right)\;+\;O(e^{3})$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\right.=\;\dot{\eta}\;-\;2\;n\;e\;\left(\,\cos{\cal{M}}\;+\;\frac{5}{4}\;e\;\cos
2{\cal{M}}\,\right)\;+\;O(e^{3})\;\;\;$ (25)
or, equivalently,
$\displaystyle\frac{\dot{\theta}\;-\;\dot{\nu}}{2\;n\;e}\;=\;-\;\left[\;\cos{\cal{M}}\;+\;\frac{5}{4}\;e\;\cos
2{\cal{M}}\;-\;\frac{\dot{\eta}}{2\;n\;e}\;+\;O(e^{2})\,\right]\;\;\;.$ (26)
Here
$\displaystyle{\eta}\;\equiv\;\theta\;-\;{\cal{M}}\;-\;\omega\;-\;\Omega\;\;\;$
(27)
is a slowly changing quantity, whose time-derivative
$\;\dot{\eta}\;\equiv\;\dot{\theta}\;-\;n\;$ becomes nil when the system goes
through the resonance.101010 Goldreich (1966) defined this quantity simply as
${\eta}\;\equiv\;\theta\;-\;{\cal{M}}\;,$ because he reckoned $\,\theta\,$
from from a fixed perihelion direction. We however reference our $\,\theta\,$
from a direction fixed in space. (It is the same direction wherefrom the node
is reckoned.) This convention originates from our definition of $\,\theta\,$
as the sidereal angle – it is in this capacity that $\,\theta\,$ was
introduced back in equations (3 \- 4). As we are not considering the nodal or
apsidal precession, our subsequent formulae containing $\,\dot{\theta}\,$ will
be equivalent to those ensuing from Goldreich’s definition of $\,\theta\,$. To
impart the words “slowly changing” with a definite meaning, we assert that
$\;\dot{\eta}/n\;$ is of order $\,e^{2}\,$ – a claim to be justified _a
posteriori_.
Expression (25) changes its sign at the points where
$\displaystyle\cos{\cal{M}}\;=\;-\;\frac{5}{4}\;e\;\cos
2{\cal{M}}\;+\;\frac{\dot{\eta}}{2\;n\;e}\;+\;O(e^{2})\;\;\;.$ (28)
As all the terms on the right-hand side of (28) are of order $~{}e~{}$ at
most, so must be the term on the left-hand side. Hence condition (28) is
obeyed in the two points whose mean anomaly (and therefore also true anomaly)
is close to $\,\pm\,\pi/2\,$:
$\displaystyle\nu\;=\;\pm\;\left(\,\frac{\pi}{2}\;-\;\delta\,\right)\;\;\;,$
(29)
$\delta\,$ being of order $\,e\,$. From (29) and (24) we obtain:
$\displaystyle\sin\delta\;=\;\cos\nu\;=\;\cos\left(\,{\cal{M}}\;+\;2\;e\;\sin{\cal{M}}\;+\;O(e^{2})\,\right)\;=\;\cos{\cal{M}}\;-\;2\;e\;\sin^{2}{\cal{M}}\;+\;O(e^{2})\;\;\;,\;\;\;$
(30)
which, in combination with (28), entails:
$\displaystyle\sin\delta\,=\,-\,\frac{5}{4}\,e\;\cos
2{\cal{M}}\,+\,\frac{\dot{\eta}}{2\,n\,e}\,-\,2\,e\;\sin^{2}{\cal{M}}\,+O(e^{2})\,=\,\frac{\dot{\eta}}{2\,n\,e}\,-\,e\,\left(\,\frac{5}{4}\,-\,\frac{1}{2}\;\sin^{2}{\cal{M}}\,\right)\,+O(e^{2})\;\;\;.\;\;\;$
(31)
Insertion of (29) into (24) also yields
$\,\;\pm\,\sin{\cal{M}}\,=\,\cos\delta\,+\,O(e)\,$. Recalling that
$\,\delta\,$ is of order $\,e\,$, we obtain:
$\,\;\sin^{2}{\cal{M}}\,=\,1\,-\,\sin^{2}\delta\,+\,O(e)\,=\,1\,+\,O(e)\,$.
This enables us to rewrite (31) as
$\displaystyle\delta\;=\;\frac{\dot{\eta}}{2\,n\,e}\;-\;\frac{3}{4}\;e\;+\;O(e^{2})\;\;\;.$
(32)
Before finding the rate $\,\dot{\eta}\,\equiv\,\dot{\theta}\,-\,n~{}$ at which
the resonance is traversed, let us enquire if perhaps it could be simply put
nil, the satellite being permanently kept in the resonance. The answer is
negative, because for a vanishing $\,\dot{\theta}\,-\,n\,$ the difference
$\,\dot{\theta}\,-\,\dot{\nu}\,$ emerging in (25) becomes a varying quantity
of an alternating sign, and so becomes the torque. On general grounds, one
should not expect that the average torque becomes nil for a vanishing
$\,\dot{\eta}\,$, though it may vanish for some finite value of
$\,\dot{\eta}\,$.
To find this value of $\,\dot{\eta}\,$, many authors (Goldreich 1966, eqn. 15;
Kaula 1968, eqn 4.5.29; Murray & Dermott 1999, eqn. 5.11) simply integrated
the MacDonald torque (LABEL:19), assuming the quality factor $\,Q\,$ constant,
and thus keeping it outside the integral:
$\displaystyle\langle{\cal{T}}_{z}\rangle^{{{}^{(Goldreich)}}}\;$
$\displaystyle=$
$\displaystyle\;-~{}\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\,k_{2}\,R}{4\;\pi\;a^{2}\;Q}\;\,\frac{1}{\,\left(1\;-\;e^{2}\right)^{1/2}\,}\;\int_{0}^{2\pi}\;\frac{R^{\textstyle{{}^{4}}}}{r^{4}}\;\,{\mbox{sgn}(\dot{\theta}\,-\,\,\dot{\nu})}\,\;{d\nu}\;$
$\displaystyle=$
$\displaystyle\;-~{}\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\,k_{2}\,R}{4\;\pi\;a^{2}\;Q}\;\,\frac{2}{\,\left(1\;-\;e^{2}\right)^{1/2}\,}\;\left[\;\int_{0}^{\pi/2-\delta}\;\frac{R^{\textstyle{{}^{4}}}}{r^{4}}\;\,\,\;{d\nu}\;-\;\int_{\pi/2-\delta\;}^{\pi}\;\frac{R^{\textstyle{{}^{4}}}}{r^{4}}\;\,\,\;{d\nu}\;\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=~{}\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\,k_{2}\,R^{5}}{2\;\pi\;a^{6}\;Q}\,\left[\int_{0}^{\pi/2-\delta}-\int_{\pi/2-\delta}^{\pi}\right]\,(1+4e\;\cos\nu)\,{d\nu}+O(e^{2})~{}$
(33)
$\displaystyle=~{}\frac{3~{}G~{}M_{sec}^{\textstyle{{}^{\,2}}}~{}k_{2}~{}R^{5}}{\pi\;a^{6}\;Q}\,~{}(4e\;\cos\delta~{}-~{}\delta)~{}+~{}O(e^{2})~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(34)
In a trapping situation, the average torque vanishes. So (34) entails:
$\displaystyle\delta\;=\;4\;e\;\cos\delta\;=\;4\;e\;+\;O(e^{2})\;\;\;.$ (35)
By combining the latter with (32), the afore quoted authors arrived at
$\displaystyle\dot{\eta}_{\textstyle{{}_{stall}}}\;=\;\frac{19}{2}\;n\;e^{2}\;\;\;,$
(36)
an expression repeated later in papers and textbooks (e.g., eqn. 4.5.37 in
Kaula 1968, or eqn. 5.14 in Murray & Dermott 1999). This result however needs
to be corrected, because the MacDonald approach is incompatible with the
frequency-independence of $\,Q\,$ assumed in (34).
Be mindful that (36), as well as its corrected version (44) to be derived
below, indicate that $\,\dot{\eta}\,$ is of order $\,e^{2}\,$. This justifies
our assertion made in the paragraph after formula (27).
### 5.3 The corrected MacDonald model
To impart the MacDonald treatment consistently, one has to calculate the
averaged torque, with the frequency-dependence of $\,Q\,$ taken into
consideration. As we saw in subsection 4, the MacDonald approach fixes this
dependence in a manner that can, with some reservations, be approximated with
$\displaystyle Q\;=\;\frac{1}{\chi\;\Delta t}$ (37)
or, in more general notations,
$\displaystyle Q\;=\;{\cal
E}^{\alpha}\;\chi^{\alpha}~{}~{}~{},~{}~{}~{}\mbox{with}~{}~{}~{}\alpha\;=\;-\;1~{}~{}~{},$
(38)
where the double instantaneous synodic frequency is given by (18).
The form (38) of the rheological law is more convenient, as it leaves us an
opportunity to consider values of $\,\alpha\,$ different from $\;-1\,$. For
any value of $\,\alpha\,$, the constant $\,{\cal{E}}\,$ is an integral
rheological parameter, which has the dimension of time, and whose physical
meaning is discussed in Efroimsky & Lainey (2007). It can be demonstrated that
in the special case of $\,\alpha=-1\,$ the parameter $\,{\cal{E}}\,$ coincides
with the time lag $\,\Delta t\,$. For actual terrestrial bodies, $\,\alpha\,$
is different from $~{}-\,1\,$, and the integral rheological parameter
$\,{\cal{E}}\,$ is related to the time lag in a more complicated manner
(Ibid.).
As demonstrated in Appendix B, insertion of (38) and (18) into (LABEL:19), for
$\,\alpha=-1\,$, or equivalently, direct employment of (23), entails the
following expression for the orbit-averaged torque acting on a secondary:
$\displaystyle\langle\;{\cal{T}}_{z}\;\,\rangle\;=\;-\;{\cal
Z}\;\left[\;\dot{\theta}\;\,{\cal A}(e)\;-\;n\;{\cal N}(e)\;\right]+O({\it
i}^{2}/Q)+O(Q^{-3})+O(en\Delta t/Q)\;\;\;,$ (39)
where
$\displaystyle{\cal
A}(e)\;=\;\left(\,1\;+\;3\;e^{2}\;+\;\frac{3}{8}\;e^{4}\,\right)\;\left(\,1\,-\,e^{2}\,\right)^{-9/2}~{}=~{}1~{}+~{}\frac{15}{2}~{}e^{2}~{}+~{}\frac{105}{4}~{}e^{4}~{}+~{}O(e^{6})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(40)
and
$\displaystyle{\cal
N}(e)\;=\;\left(\,1\;+\;\frac{15}{2}\;e^{2}\;+\;\frac{45}{8}\;e^{4}\;+\;\frac{5}{16}\;e^{6}\,\right)\;\left(\,1\,-\,e^{2}\,\right)^{-6}~{}=~{}1~{}+~{}\frac{27}{2}~{}e^{2}~{}+~{}\frac{573}{8}~{}e^{4}~{}+~{}O(e^{6})~{}~{}~{},\quad$
(41)
while the factor $\,{\cal{Z}}\,$ is given by 111111 Recall that, for
$\,\alpha=-1\,$, the rheological parameter $\,{\cal{E}}\,$ is simply the time
lag $\,\Delta t$.
$\displaystyle{\cal
Z}\,=\,\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\cal
E}}{R}\;\frac{R^{\textstyle{{}^{6}}}}{a^{6}}\,=\,\frac{3\,n^{2}\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\Delta
t}}{(M_{prim}\,+\,M_{sec})}\;\frac{R^{\textstyle{{}^{5}}}}{a^{3}}\,=\,\frac{3\,n\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}}{Q\;(M_{prim}\,+\,M_{sec})}\;\frac{R^{\textstyle{{}^{5}}}}{a^{3}}\;\,\frac{n}{\chi}\;\;\;,~{}~{}~{}~{}$
(42)
$M_{prim}\,$ and $\,M_{sec}\,$ being the masses of the primary and the
secondary.
Be mindful that the right-hand side of (42) contains a multiplier
$~{}\frac{\textstyle n}{\textstyle\chi}\,=\,\frac{\textstyle n}{\textstyle
2\,|\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}-\stackrel{{\scriptstyle\bf\centerdot}}{{\nu\,}}|}\,~{}$
which is missing in the despinning formula employed by Correia & Laskar (2004,
2009). This happened because in Ibid. the quality factor was introduced as
$\,1/(n\,\Delta t)\,$ and not as $\,1/(\chi\,\Delta t)\,$ – see the line after
formula (9) in Correia & Laskar (2009). In reality, the quality factor $\,Q\,$
must, of course, be a function of the forcing frequency $\,\chi\,$ (which
happens to coincide with the mean motion $\,n\,$ in the 3:2 and 1:2 resonances
but differs from $\,n\,$ outside these).
The quality factor $\,Q\,$ being (within this model) inversely proportional to
$\,\chi\,$, the presence of the $\,\frac{\textstyle n}{\textstyle\chi}\,$
factor in (42) makes the overall factor $\,{\cal{Z}}\,$ a frequency-
independent constant.
Rewriting (39) as
$\displaystyle\langle\,{\cal{T}}_{z}\,\rangle$ $\displaystyle=$
$\displaystyle-\,{\cal
Z}\left[\dot{\theta}\,\left(1+\,\frac{15}{2}\;e^{2}+\,\frac{105}{4}\;e^{4}\right)-n\left(1+\,\frac{27}{2}\;e^{2}+\,\frac{573}{8}\;e^{4}\right)\right]+O(e^{6})+O({\it
i}^{2}/Q)+O(en\Delta t/Q)$ (43) $\displaystyle=$ $\displaystyle-\,{\cal
Z}\left[\dot{\eta}\,\left(1+\,\frac{15}{2}\;e^{2}+\,\frac{105}{4}\;e^{4}\right)-6ne^{2}\left(1+\,\frac{121}{16}\;e^{2}\right)\right]+O(e^{6})+O({\it
i}^{2}/Q)+O(en\Delta t/Q)\;\;,\quad\quad\quad$
we see that it vanishes when the rate of change of
$\,{\eta}\,=\,{\theta}\,-\,{\cal{M}}\,-\,\omega\,-\,\Omega\;$ accepts the
value
$\displaystyle\dot{\eta}_{\textstyle{{}_{stall}}}\,=\;{6}\;n\;e^{2}\,+\;\frac{3}{8}~{}n~{}e^{4}\,+~{}O(e^{6})~{}+~{}O({\it
i}^{2}/Q)~{}+~{}O(en\Delta t/Q)\;\;\;.\;$ (44)
For $\,\dot{\eta}\,$ larger or smaller than
$\,\dot{\eta}_{\textstyle{{}_{stall}}}\,$, the average torque (43) is nonzero
and impels $\,\dot{\eta}\,$ to evolve towards the stall value (44).
On the right-hand side of (44), the leading-order term contains a numerical
factor of $\,6\,$, as different from the factor $\,19/2\,$ showing up in (36).
The necessity to change $\,19/2\,$ to $\,6\,$ in the $\,e^{2}\,$ term was
pointed out by Rodríguez, Ferraz-Mello & Hussmann (2008, eqn. 2.4), who had
arrived at this conclusion through some different considerations (which, too,
were based on the frequency-dependence (38) ). The same result can be obtained
within the Darwin-Kaula approach, provided the scaling law (38) is employed
(Efroimsky 2012a,b). 121212 Implicitly, this result is present also in Correia
et al. (2011, eqn 20), Laskar & Correia (2003, eqn 9), and in Hut (1981). The
earliest implicit occurrence of this result was in Goldreich & Peale (1966,
eqn 24).
Although correction of the oversight in the MacDonald torque renders a number
different from the one furnished by Goldreich’s development (6 instead of
19/2), qualitatively the principal conclusion by Goldreich (1966) remains
unchanged: when an oblate body’s spin is evolving toward the resonance,
vanishing of the average tidal torque entails spin slightly faster than
resonant, a so-called pseudosynchronous rotation. It however should be
strongly emphasised that the possibility of pseudosynchronous rotation hinges
upon the dissipation model employed. Makarov and Efroimsky (2013) have found
that a more realistic tidal dissipation model than the corrected MacDonald
torque makes pseudosynchronous rotation impossible.
## 6 Evolution of rotation near the 1:1 resonance.
The case of a triaxial body
In distinction from a gaseous or liquid body, a solid body would be expected
to have a permanent figure in addition to the tidal distortion discussed so
far. Goldreich (1966) demonstrated that this permanent figure plays an
important role in determining if a primary, whose rotation is being slowed
down or sped up by the tidal torque caused by a secondary, can be captured
into the synchronous rotation state. This section follows Goldreich’s
derivation, but substitutes the MacDonald tidal torque with its corrected
version, and also uses a more general mass expression.
### 6.1 Rotating primary subject to a triaxiality-caused torque.
The equation of motion and the first integral.
Consider a triaxial primary body, which has its principal moments of inertia
ordered as $\,A\,<\,B\,<\,C\,$, and which is rotating about the maximal-
inertia axis associated with moment $\,C\,$. About the primary, a secondary
body describes a near-equatorial orbit (so its inclination on the primary’s
equator, $\,i\,$, may be neglected). The secondary exerts on the primary two
torques. One being tidal, the other is triaxiality-caused, i.e., generated by
the existence of the permanent figure of the primary. Its component acting on
the primary about the maximal-inertia axis is
$\displaystyle{\cal{T}}_{{}_{triax}}\;=\;\frac{3}{2}\;(B~{}-~{}A)\;\frac{G~{}M_{sec}}{r^{3}}\;\,\sin
2\lambda~{}~{}~{},$ (45)
the longitude $\,\lambda\,$ being furnished by formula (10). In that formula,
the sidereal angle $\,\theta\,$ is reckoned from a reference direction in
space to the principal axis associated with moment $\,A\,$.
The acceleration of the sidereal angle then obeys
$\displaystyle
C\;\ddot{\theta}-\frac{3}{2}\;(B~{}-~{}A)\;\frac{G~{}M_{sec}}{r^{3}}\,~{}\sin
2\lambda~{}=~{}0~{}~{}~{}.$ (46)
In neglect of the nodal and apsidal precession, as well as of
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{{\cal{M}}_{\textstyle{{}_{0}}}}}\,$,
definition (27) yields: 131313 The caveat about three neglected items implies
that our $\,n\,$ is the _osculating_ mean motion
$\,n(t)\,\equiv\,\sqrt{\mu/a(t)^{3}}\,$, and that we extend this definition to
perturbed settings. The so-defined mean anomaly evolves in time as
$\,{\cal{M}}\,=\,{\cal{M}}_{0}(t)+\int_{t_{o}}n(t)\,dt\,$, whence
$\,\dot{\cal{M}}=\dot{\cal{M}}_{0}+n(t)\,$. The said caveat becomes redundant
when $\,n\,$ is defined as the _apparent_ mean motion, i.e., either as the
mean-anomaly rate $\,d{\cal M}/dt\,$ or as the mean-longitude rate
$\,dL/dt\,=\,d\Omega/dt\,+\,d\omega/dt\,+\,d{\cal{M}}/dt\,$ (Williams et al.
2001). While the first-order perturbations of $\,a(t)\,$ and of the osculating
mean motion $\,\sqrt{\mu/a(t)^{3}}\,$ include no secular terms, such terms are
often contained in the epoch terms $\,\dot{\Omega}\,$, $\,\dot{\omega}\,$, and
$\,\dot{\cal{M}}_{0}\,$. This produces the difference between the apparent
mean motion defined as $\,dL/dt\,$ (or as $\,d{\cal M}/dt\,$) and the
osculating mean motion $\,\sqrt{\mu/a(t)^{3}}\,$.
$\displaystyle\dot{\eta}\;=\;\dot{\theta}\;-\;n~{}~{}~{}.$ (47)
Ignoring changes in the mean motion,141414 Our neglect of evolution of the
mean motion is acceptable, because orbital acceleration $\,\dot{n}\,$ is
normally much smaller than the spin acceleration $\,\ddot{\theta}\,$. Indeed,
for torques arising from the primary,
$\,{\textstyle\dot{n}}/{\textstyle\ddot{\theta}}\,$ is of the same order as
the ratio of $\,C\,$ to the orbital moment of inertia. we write:
$\displaystyle\ddot{\theta}=\ddot{\eta}~{}~{}~{},$ (48)
so the first term in (46) becomes simply $\,C\ddot{\eta}\,$.
To process the second term in (46), we would compare (10) with (27):
$\displaystyle\lambda~{}=~{}-~{}\theta~{}+~{}\Omega~{}+~{}\omega~{}+~{}\nu~{}+~{}O(i^{2})$
$\displaystyle=$
$\displaystyle(\,-\,\theta~{}+~{}\Omega~{}+~{}\omega~{}+~{}{\cal{M}})~{}+~{}(\nu\,-\,{\cal{M}})~{}+~{}O(i^{2})$
(49) $\displaystyle=$
$\displaystyle-~{}\eta~{}+~{}(\nu\,-\,{\cal{M}})~{}+~{}O(i^{2})$
In neglect of the inclination, the following approximation is acceptable in
the vicinity of the 1:1 resonance:151515 Pioneered by Goldreich & Peale
(1966), the approximation is explained in more detail by Murray & Dermott
(1999) whose treatment omits terms of the order of $\,e^{3}\,$ and higher (see
formulae 5.59 - 5.60 in Ibid.). To justify the omission, recall that before
averaging the expansion of $~{}r^{-3}~{}{\sin 2\,\lambda}~{}$ includes a
series of terms with different arguments of the form $\,\sin(2\eta+q{\cal
M})\,$, where $\,q\,$ is an integer. Since we are interested in dynamics in
the vicinity of the 1:1 resonance, where $\,\eta\,$ is a slow variable, then
only the $\,\sin(2\eta)\,$ term remains after the average. The averaged-out
terms are of the order of $\,e\,$ and higher powers including the $\,e^{3}\,$
terms.
$\displaystyle\langle\,r^{-3}~{}{\sin
2\,\lambda}\,\rangle~{}=~{}-~{}G_{200}(e)\,a^{-3}\,{\sin 2\eta}~{}~{}~{},$
(50)
where $\,\langle...\rangle\,$ signifies orbital averaging, while the
eccentricity function can be approximated with
$\displaystyle
G_{200}(e)\,=~{}1~{}-~{}\frac{5}{2}~{}e^{2}\,+~{}O(e^{4})~{}~{}~{}.$ (51)
This way, omitting $\,O(i^{2})\,$ in (49) and substituting $\,\sin 2\lambda\,$
with its average in (46), we transform the second term in (46) to:
$\displaystyle+~{}\frac{3}{2}~{}(B\,-\,A)~{}\frac{G\;M_{sec}}{a^{3}}~{}G_{200}(e)~{}\sin
2\eta~{}~{}~{}.$ (52)
While Goldreich (1966) assumed that the mass of the secondary is much larger
than the mass of the primary, we do not impose this restriction. Combining
Kepler’s third law, $~{}G(M_{sec}+M_{prim})/a^{3}=n^{2}~{}$, with formulae
(46), (48), and (52), we finally arrive at
$\displaystyle
C\ddot{\eta}~{}+~{}\frac{3}{2}~{}(B\,-\,A)~{}\frac{M_{sec}}{M_{sec}\,+\,M_{prim}}~{}n^{2}~{}G_{200}(e)~{}\sin
2\eta~{}=~{}0~{}~{}~{},$ (53)
an equation describing the evolution of the rotation angle $\,\eta\,$. As
pointed out by Goldreich (1966), this equation is equivalent to the one
describing a simple pendulum. Indeed, in terms of $\,\beta=2\eta\,$, equation
(53) becomes $~{}\ddot{\beta}+\chi^{2}_{\textstyle{{}_{lib-
max}}}\,\sin\beta=0~{}$, with a constant positive
$\,\chi^{2}_{\textstyle{{}_{lib-max}}}\,$ and with $\beta$ playing the role of
the pendulum angle. 161616 In subsection 6.5 below, we write down the
expression for the libration frequency $\,\chi_{\textstyle{{}_{lib-max}}}\,$
and also explain the reason why we equip it with such a subscript – see
formulae (68) and (69).
It follows directly from (53) that the spin acceleration $\,\ddot{\eta}\,$
vanishes if $\,\eta\,$ assumes the values of $\,0\,$ or $\,\pi\,$. As can be
seen from the pendulum analogy, the initial conditions
$\,(\eta\,,~{}\dot{\eta}\,)_{\textstyle{{}_{\,t=t_{\textstyle{{}_{0}}}}}}\,=\,(0\,,~{}0)\,$,
as well as the conditions
$\,(\eta\,,~{}\dot{\eta}\,)_{\textstyle{{}_{\,t=t_{\textstyle{{}_{0}}}}}}\,=\,(\pi\,,~{}0)\,$,
correspond to the situation where the pendulum comes to a stall in the lower
point (so $\,\dot{\beta}=0\,$ when $\beta=0$ or $2\pi$). Under such initial
conditions, $\,\eta\,$ stays $\,0\,$ or $\pi$ all the time, which implies a
uniform synchronous rotation of the secondary about the primary. 171717
According to (53), the spin acceleration $\,\ddot{\eta}\,$ vanishes also for
$\,\eta=\pm\pi/2\,$, which is the upper point of the pendulum.
Multiplication of equation (53) by $\,\dot{\eta}\,$, with subsequent
integration over time $\,t\,$, gives the first integral of motion,
$\displaystyle\frac{1}{2}~{}C~{}\dot{\eta}^{2}~{}-~{}\frac{3}{4}~{}(B\,-\,A)~{}\frac{M_{sec}}{M_{sec}\,+\,M_{prim}}~{}n^{2}~{}G_{200}(e)~{}\cos
2\eta~{}=~{}E~{}~{}~{},$ (54)
whose value depends on the initial conditions.
The period of the variable $\,\eta\,$ is given by a quadrupled integral over a
quarter-cycle of $\,\eta~{}$:
$\displaystyle
P~{}=~{}4\,\int_{\eta=0}^{\eta=\eta_{\textstyle{{}_{max}}}}\;\frac{d\eta}{\dot{\eta}}~{}~{}~{}.$
(55)
For circulation, use $\,\eta_{\textstyle{{}_{max}}}=\pi/2\,$. In the case of
libration about $\,\eta=0\,$, the value of $\,\eta_{\textstyle{{}_{max}}}\,$
is less than $\,\pi/2\,$ and can be expressed via $\,E\,$ by setting
$\,\dot{\eta}=0\,$ in equation (54). To get an explicit expression of $\,P\,$,
one should first express $\,\dot{\eta}\,$ via $\,E\,$ using (54) and taking
the positive root, and then should plug the so-obtained expression for
$\,\dot{\eta}\,$ into (55). Also be mindful that libration about $\,\pi\,$ can
be converted to the same integral by adding or subtracting $\,\pi\,$ from the
variable of integration.
In what follows, averaging over the period $\,P\,$ will be denoted by
$\,\langle...\rangle_{\textstyle{{}_{P}}}~{}$, the subscript serving to
distinguish the operation from averaging over the orbit employed in Section 5
and in formula (50).
### 6.2 Parallels with pendulum. The auxiliary quantity $\,W$
Goldreich (1966) noted the similarity of the differential equations (53 \- 54)
to the classical pendulum problem. If the initial conditions place the body
outside of the 1:1 spin-orbit resonance, then $\,\eta\,$ circulates with a
forced oscillation in rotation that depends on the mean value
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ of the $\,\dot{\eta}\,$ frequency.
If however the initial conditions place the body’s spin within a sufficiently
close proximity of the 1:1 resonance (the “resonance region”), then $\,\eta\,$
will librate about $\,0\,$ or $\,\pi\,$. This will be a free physical
libration with an amplitude smaller than $\,\pi/2\,$. Inside the resonance
region, the initial conditions establish the free-libration amplitude and
phase. However the restricted nature of the motion will keep the mean value of
$\,\eta\,$ constant: it will be either $\,0\,$ or $\,\pi\,$. For the same
reason, $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ will remain zero in the
libration regime. In contrast to this, outside of the resonance region the
mean rate of circulation $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ will
possess a value determined by the initial conditions on $\,\eta\,$ and
$\,\dot{\eta}\,$.
Employing (54) to express $\,\dot{\eta}\,$ via $\,E\,$, and plugging this
expression into (55), Goldreich (1966) demonstrated that the period $\,P\,$
can be expressed via $\,E\,$, inside or outside the resonance region, by a
complete elliptic integral of the first kind. This is natural, as the
expression of $\,\dot{\eta}\,$ through $\,E\,$ involves a square root.
Just as in the pendulum case, there exists a critical $\,E\,$ for which the
period diverges. This is the boundary $\,E\,=\,E_{b}\,$ separating circulation
from libration. To locate the boundary, one should set simultaneously
$\,\eta=\pi/2\,$ and $\,\dot{\eta}=0~{}$ in formula (54):
$\displaystyle
E_{b}~{}=~{}\frac{3}{4}~{}(B\,-\,A)~{}\frac{M_{sec}}{M_{sec}\,+\,M_{prim}}~{}G_{200}(e)~{}n^{2}~{}~{}~{}.$
(56)
Mathematically, the emergence of a logarithmic singularity in $\,P\,$ at
$\,E\,=\,E_{b}\,$ can be observed from formulae (9 - 10) in Ibid. Physically,
this situation resembles the slowing-down of a pendulum at the
circulation/libration border. In our problem, though, this division
corresponds to $\,\eta=\pm\pi/2\,$.
By setting simultaneously $\,\eta=0\,$ and $\,\dot{\eta}=0\,$ in (54), we
obtain the minimal value that the integral $\,E\,$ can assume:
$\displaystyle\mbox{min}\,E~{}=~{}-~{}E_{b}~{}~{}~{}.$ (57)
The values $\,E\,>\,E_{b}\,$ correspond to circulation in either direction,
with oscillations of the rotation rate. The values falling within the interval
$\,E_{b}\,>\,E\,>\,-\,E_{b}\,$ give libration. The minimum value
$\,E\,=\,-\,E_{b}\,$ corresponds to synchronous rotation without libration.
The quantity $\,\eta\,$ being slowly varying (compared to the mean motion
$\,n\,$), the period $\,P\,$, with which $\,\eta\,$ is changing, naturally
turns out to be much longer than the orbital period, both outside and inside
the 1:1 resonance:
$\displaystyle P\,\gg\,\frac{2\,\pi}{n}~{}~{}~{}.$ (58)
This can be appreciated from the evident equalities
$\displaystyle
P~{}=~{}\frac{2~{}\pi}{\,|\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,|\,}\quad\quad\quad\mbox{outside~{}of~{}the~{}1:1~{}resonance}~{},$
(59a) $\displaystyle
P~{}=~{}\frac{2~{}\pi}{\chi_{\textstyle{{}_{lib}}}}\quad~{}\quad~{}\quad~{}\mbox{inside~{}the~{}1:1~{}resonance}~{}.\quad~{}$
(59b)
In (59b), $\,\chi_{\textstyle{{}_{lib}}}\,$ denotes the libration frequency
which too is much smaller than the mean motion $\,n\,$, as we shall see
shortly.
A useful quantity defined by Goldreich (1966) was
$\displaystyle
W~{}\equiv~{}P~{}\langle\dot{\eta}^{2}\rangle_{{{}_{P}}}~{}\equiv~{}\int_{0}^{P}\dot{\eta}^{2}~{}dt~{}=~{}4~{}\int_{\eta=0}^{\eta=\eta_{\textstyle{{}_{max}}}}\;\dot{\eta}~{}d\eta~{}~{}~{}.$
(60)
Similarly to (55), $\,\eta_{\textstyle{{}_{max}}}=\pi/2\,$ for circulation,
and $\,\eta_{\textstyle{{}_{max}}}<\pi/2\,$ for libration. Just as $\,P\,$, so
$\,W\,$ can be expressed through $\,E\,$ by complete elliptic integrals. For
circulating $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$, Goldreich described
$\,P\,$ and $\,W\,$ as applying to one oscillation, but they describe one
circulation of $\,\eta\,$ with two oscillations. For librations, $\,P\,$ and
$\,W\,$ describe one libration cycle.
One more quantity of use in this problem will be the mean square variation of
$\,\dot{\eta}\,$ about $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$, given by
$\,\langle\dot{\eta}^{2}\rangle_{{{}_{P}}}-\langle\dot{\eta}\rangle_{{{}_{P}}}^{2}\,$.
Be mindful that the notation $\,\langle...\rangle_{{{}_{P}}}\,$ is employed to
indicate averaging over a circulation cycle as well as that over a libration
cycle.
When the values of the mean motion $\,n\,$, the mass factor $\,{\textstyle
M_{sec}}/({\textstyle M_{sec}\,+\,M_{prim}})\,$, and the ratio $\,(B-A)/C\,$
are given, and the initial conditions on $\,\eta\,$ and $\,\dot{\eta}\,$ are
set, equation (54) furnishes the value of $\,E/C\,$. The comparison of this
value with $\,E_{b}/C\,$ distinguishes circulation from libration. With the
values of $\,(B-A)/C\,$ and $\,E/C\,$ known, the complete elliptic integrals
can be evaluated and the values of the quantities $\,P\,$, $\,W\,$,
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$, and
$\,\langle\dot{\eta}^{2}\rangle_{{{}_{P}}}\,$ can be found. It should also be
possible to reverse this procedure. For example, the knowledge of the
circulating value of $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$, along with the
knowledge of $\,n\,$, the mass factor $\,{\textstyle M_{sec}}/({\textstyle
M_{sec}\,+\,M_{prim}})\,$, and the ratio $\,(B-A)/C\,$, should allow
$\,E/C\,$, $\,W\,$, and $\,\langle\dot{\eta}^{2}\rangle_{{{}_{P}}}\,$ to be
computed.
### 6.3 The tidal torque and the libration bias
To account for tidal dissipation, Goldreich (1966) added the averaged (over an
orbital period) tidal torque (34) to the right-hand side of equation (53). We
however shall employ the corrected average torque (43) instead of (34). This
will result in
$\displaystyle
C\ddot{\eta}+\,\frac{3}{2}(B-A)\,\frac{M_{sec}}{M_{sec}+M_{prim}}~{}n^{2}\,G_{200}(e)~{}\sin
2\eta~{}=~{}\quad\quad\quad\quad~{}~{}~{}~{}\quad~{}~{}~{}~{}~{}\quad\quad\quad\quad~{}~{}~{}~{}~{}\quad\quad\quad\quad~{}~{}~{}~{}$
$\displaystyle~{}-~{}{\cal{Z}}\left[\dot{\eta}\left(1+\,\frac{\textstyle
15}{\textstyle
2}~{}e^{2}\right)\,-\,6\,n\,e^{2}\left(1+\,\frac{121}{16}\;e^{2}\right)\right]~{}+~{}O(e^{6})~{}+~{}O({\it
i}^{2}/Q)~{}+~{}O(en\Delta t/Q)\,~{},~{}~{}~{}~{}\quad~{}~{}$ (61)
where the coefficient $\,{\cal{Z}}\,$ depends via (42) on the orbital
variables, the tidal parameters, and the masses.
As we already mentioned at the beginning of Section 5, ignoring the permanent-
figure term would lead one to the conclusions that the tidal torque is finite
for synchronous rotation and that it vanishes for a spin rate slightly higher
than synchronous. However inclusion of the permanent-figure term into the
picture alters the results radically. The constant term on the right-hand side
of equation (61) causes the mean value of $\,\eta\,$ to be slightly larger
than $\,0\,$ or $\,\pi\,$. For small librations, this bias is
$\displaystyle\eta_{\textstyle{{}_{bias}}}$ $\displaystyle=$
$\displaystyle\frac{2\;{\cal{Z}}\;e^{2}}{(B-A)\,n}~{}\frac{M_{sec}+M_{prim}}{M_{sec}}\left(1+\,\frac{121}{16}\,e^{2}\right)\,\frac{~{}~{}1}{G_{200}(e)}\,+\,O(e^{6})\,+\,O({\it
i}^{2}/Q)\,+\,O(en\Delta t/Q)~{}\quad~{}\,~{}\quad~{}\,\quad$ (62a)
$\displaystyle=$
$\displaystyle\frac{2~{}{\cal{Z}}~{}e^{2}}{(B-A)\,n}~{}\frac{M_{sec}+M_{prim}}{M_{sec}}\left(1+\,\frac{161}{16}\,e^{2}\right)\,+\,O(e^{6})\,+\,O({\it
i}^{2}/Q)\,+\,O(en\Delta t/Q)\,~{},~{}\,\quad~{}\quad~{}\,\quad$ (62b)
with $\,{\cal{Z}}\,$ calculated through (42). The value of the time lag
$\,\Delta t\,$ entering the expression (42) for $\,{\cal{Z}}\,$ should be set
appropriate to the orbital frequency (mean motion).
As evident from (62), the bias comes into being due to the eccentric shape of
the orbit. Bodies with permanent figures can achieve synchronous rotation
because the bias in $\,\eta\,$ gives birth to a permanent-figure torque that
balances the constant dissipative torque.
At this point, an important caveat will be in order. As we explained in
Section 4, the MacDonald model of tides becomes self-consistent only when the
time delay $\,\Delta t\,$ emerging in (42) is set frequency-independent. This
circumstance limits the precision of the MacDonald torque and of the Goldreich
dynamical model based thereon, whenever the model gets employed to determine
the timescales of spin evolution (Efroimsky & Lainey 2007). Nevertheless, for
very slow evolution the model may be employed for obtaining qualitative
estimates. In this case, $\,\Delta t\,$ should be treated as a parameter that
itself depends upon the forcing frequency in the material. For rotation
outside the $\,1:1\,$ spin-orbit lock, it would be a tolerable approximation
to use $\,\Delta t\,$ appropriate to the principal tidal frequency
$\,\chi_{\textstyle{{}_{2200}}}\,$ or to the double instantaneous synodic
frequency (18). However inside the $\,1:1\,$ resonance, $\,\Delta t\,$ would
correspond to the libration frequency $\,\chi_{\textstyle{{}_{lib}}}\,$ which
may be very different from the usual tidal frequencies for nonsynchronous
rotation. This circumstance may change the value of $\,\Delta t\,$ and
therefore of $\,{\cal{Z}}\,$ noticeably.
### 6.4 Tidal dissipation and the point of stall
Be mindful that on the right-hand side of equation (61) we have the tidal
torque averaged over the orbit (see Appendix B for details). Similarly, the
second term on the left-hand side of (61) is the permanent-figure torque
averaged over the orbit – see expression (52).181818 For the first term,
$\,C\ddot{\eta}\,$, the caveat about orbital averaging is unimportant, because
$\,\dot{\eta}\,$ bears no dependence upon the mean anomaly. Therefore equation
(61) renders us the behaviour of $\,\eta\,$ over times longer than the orbital
period. This is acceptable, because the orbital period is much shorter than
the timescales of our interest. Specifically, it is much shorter than $\,P\,$,
see equation (58).
Without dissipation, $\,E\,$ is conserved during oscillations of $\,\eta\,$,
while in the presence of weak dissipation $\,E\,$ changes slowly. Indeed,
multiplying both sides of (61) by $\,\dot{\eta}\,$ and making use of (54), we
arrive at
$\displaystyle\frac{dE}{dt}\,=\,-\,{\cal{Z}}\,\left[\,\dot{\eta}^{2}\,\left(\,1\,+\,\frac{15}{2}\,e^{2}\,\right)\,-\,6\,n\,\dot{\eta}\,e^{2}\,\left(\,1\,+\,\frac{121}{16}\,e^{2}\,\right)\,\right]\,+\,O(e^{6})\,+\,O({\it
i}^{2}/Q)\,+\,O(en\Delta t/Q)~{}~{}.~{}~{}~{}$ (63)
Thus we see that a dissipative tidal torque influences the spin, while causing
changes also in the first integral $\,E\,$ (which is not identical to the
actual kinetic energy). Considering equation (54) and noting that the cosine
term is periodic, we see that weak tidal dissipation will cause a change in
$\,\dot{\eta}^{2}\,$ over time scales long compared to $\,P\,$.
For nonsynchronous rotation, weak dissipation causes both a slow secular
change and a small oscillation. The latter arises from the oscillating part of
$\,\dot{\eta}\,$ as $\,\eta\,$ circulates or librates.
Goldreich (1966) also averages over the period $\,P\,$. For libration, the so-
averaged rate of change of $\,E\,$ is
$\displaystyle\langle\,{dE}/{dt}\,\rangle_{{{}_{P}}}~{}=~{}-~{}\frac{{\cal{Z}}~{}W}{P}~{}\left(\,1~{}+~{}\frac{15}{2}~{}e^{2}\,\right)\,+\,O(e^{6})\,+\,O({\it
i}^{2}/Q)\,+\,O(en\Delta t/Q)~{}~{}~{},$ (64)
where the positive definite quantity $\,W\,$ is introduced via (60). The
negative $~{}\langle\,{dE}/{dt}\,\rangle_{{{}_{P}}}~{}$ for libration means
that the maximum $\,\dot{\eta}^{2}\,$ at $\,\cos 2\eta=1\,$ in equation (54)
is decreasing and the free libration damps with time. Libration evolves toward
synchronous rotation with $\,\dot{\eta}=0\,$ and $\,\eta\,$ equal to $\,0\,$
or to $\,\pi\,$.
The $\,\dot{\eta}\,$ term causes the damping of the libration amplitude. When
the amplitude is sufficiently small, its decrease obeys the exponential law
$\,\exp(-D_{\textstyle{{}_{L}}}\,t)\,$ with
$\displaystyle D_{\textstyle{{}_{L}}}~{}=~{}\frac{\cal
Z}{2\,C}~{}\left(\,1~{}+~{}\frac{15}{2}~{}e^{2}\,\right)~{}~{}~{}.$ (65)
As we emphasised in the paragraph after equation (61), inside the $\,1:1\,$
resonance the time delay $\,\Delta t\,$ showing up in the expression for
$\,{\cal Z}\,$ should be appropriate to the libration frequency
$\,\chi_{\textstyle{{}_{lib}}}\,$ which may differ greatly from the usual
tidal frequencies for nonsynchronous spin. This choice of $\,\Delta t\,$ will
influence the value of $\,{\cal{Z}}\,$.
For circulation, averaging of equation (63) over one orbital period leads to
$\displaystyle\langle\,{dE}/{dt}\,\rangle_{{{}_{P}}}~{}=$ $\displaystyle-$
$\displaystyle\frac{\cal
Z}{P}~{}\left[\,W~{}\left(1~{}+~{}\frac{15}{2}~{}e^{2}\right)~{}-~{}12~{}\pi~{}n~{}e^{2}~{}\left(\,1\;+\;\frac{121}{16}\;e^{2}\,\right)~{}\mbox{sgn}\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,\right]~{}~{}~{},$
(66) $\displaystyle+\,O(e^{6})\,+\,O({\it i}^{2}/Q)\,+\,O(en\Delta t/Q)$
where we have used definition (59a) for $\,P\,$ and definition (60) for
$\,W\,$.
While $\,W=P\langle\dot{\eta}^{2}\rangle_{{{}_{P}}}\,$ is positive definite,
the second term on the right-hand side of equation (66) can, for circulation,
have either sign. In the case when its sign is positive, the expression (66)
for $\,\langle\,{dE}/{dt}\,\rangle_{{{}_{P}}}\,$ will vanish for $\,W\,$ equal
to
$\displaystyle
W_{\textstyle{{}_{stall}}}~{}=~{}12~{}\pi~{}n~{}{e^{2}}\left(1\;+\;\frac{\textstyle
e^{2}}{\textstyle 16}\,\right)\,+\,O(e^{6})\,+\,O({\it
i}^{2}/Q)\,+\,O(en\Delta t/Q)~{}~{}~{}.$ (67)
No matter whether $\,W\,$ begins its evolution with an initial value larger or
smaller than $\,W_{\textstyle{{}_{stall}}}\,$, it will never cross
$\,W_{\textstyle{{}_{stall}}}~{}$.
Another important value of $\,W\,$ is the one corresponding to the boundary
between libration and circulation, $\,W_{b}\,$. Below we shall obtain its
value and shall explain that for $\,W_{b}<W_{\textstyle{{}_{stall}}}\,$ there
exists a positive value of $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ at which
the evolution of a circulating $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$
should stall.
### 6.5 The free-libration frequency
Exploring librations, we start out with the small-amplitude case. Replacing
$\,\sin 2\eta\,$ with $\,2\eta\,$ in equation (53), we write down the
frequency for small librations of $\,\eta~{}$:
$\displaystyle\chi_{\textstyle{{}_{lib-
max}}}~{}=~{}\frac{2\,\pi}{P_{min}}~{}=~{}\left[\,3~{}\frac{B-A}{C}~{}\,\frac{M_{sec}}{\,M_{sec}\,+\,M_{prim}\,}~{}G_{200}(e)\,\right]^{1/2}n~{}~{}~{}.$
(68)
In many realistic situations, the condition $\,\frac{\textstyle
B-A}{\textstyle C}~{}\frac{\textstyle M_{sec}}{\textstyle
M_{sec}\,+\,M_{prim}}\,\ll\,1~{}$ is fulfilled,191919 This condition is
satisfied safely if the primary is a planet (like Mercury) or a large
satellite (like our Moon). Its fulfilment is not guaranteed, though, for small
satellites (like Phobos or Hyperion). which ensures that
$\,\chi_{\textstyle{{}_{lib-max}}}\ll n\,$. Larger amplitudes increase the
period $\,P\,$ and render a smaller frequency
$\,\chi_{\textstyle{{}_{lib}}}\,$, so expression (68) gives the smallest
librating period and the largest frequency
$\displaystyle\chi_{\textstyle{{}_{lib-
max}}}\,=\,\mbox{max}\,\chi_{\textstyle{{}_{lib}}}~{}~{}~{}.$ (69)
The linear $\,\dot{\eta}\,$ term in (61) will alter the frequency, but for
slow tide-caused evolution the correction will be small, so (68) still will
serve well as an approximation for the maximal frequency of libration. On all
these grounds, we now accept that for both small-amplitude and large-amplitude
librations the inequality
$\displaystyle\chi_{\textstyle{{}_{lib}}}~{}\ll~{}n$ (70)
holds. This justifies, a posteriori, the assertion made after (59b).
In the literature on the physical libration of the Moon, the expression for
the free-libration frequency is ubiquitous, though often without the mass
factor. Versions of the expressions for $\,\eta_{\textstyle{{}_{bias}}}\,$ and
the free-libration damping rate appeared in Williams et al. (2001). That
paper, though, did not rely on the corrected MacDonald model (constant time
delay), but simply used separate $\,Qs\,$ for libration and orbital
frequencies.
### 6.6 The boundary between circulation and libration
The boundary between circulation and the resonance zone corresponds to a value
$\,W\,=\,W_{b}\,$. To find it, we combine (54) with (56) and arrive at
$\displaystyle\frac{1}{2}\;C\;\dot{\eta}^{2}\,=\;E_{b}\,\left(\,1\,+~{}\cos
2\eta\,\right)~{}~{}~{},$ (71a) which is the same as
$\displaystyle\dot{\eta}^{2}\,=\;4\;\frac{E_{b}}{C}\,\cos^{2}\eta~{}~{}~{}.$
(71b)
Insertion of the resulting expression for $\,\dot{\eta}\,$ into (60) entails:
$\displaystyle
W_{b}~{}=~{}8\,\sqrt{\frac{E_{b}}{C}}\,\int_{0}^{\pi/2}\cos\eta~{}d\eta~{}=~{}8~{}\sqrt{\frac{E_{b}}{C}}~{}=~{}4~{}n~{}\left[\,3~{}\frac{B-A}{C}~{}\frac{M_{sec}}{\,M_{sec}\,+\,M_{prim}\,}~{}G_{200}(e)\,\right]^{1/2}~{}~{},~{}~{}~{}$
(72)
comparison whereof to (68) yields another expression for the boundary value:
$\displaystyle W_{b}=4~{}\chi_{\textstyle{{}_{lib-max}}}~{}~{}~{}.$
While $\,W\,$ is continuous across the boundary, $\,P\,$ has a logarithmic
singularity, as was mentioned in the paragraph after equation (56). As $\,P\,$
diverges, the evolution rate of $\,\langle\,{dE}/{dt}\,\rangle_{{{}_{P}}}\,$,
given by (66) , vanishes at the boundary. Nonetheless a small perturbation
allows the boundary to be crossed – for more on this, see the two paragraphs
after formula (22) in Goldreich (1966).
### 6.7 Three regimes
If $\,W_{\textstyle{{}_{stall}}}\,<\,W_{b}\,$, then there is no stall point in
the region of circulation ($\,W_{b}\,<\,W\,$). So $\,W\,$ evolves towards
$\,W_{b}\,$, while $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ of either sign
evolves towards zero at the libration/circulation boundary. Figure 1a
illustrates the evolution of $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$.
Goldreich (1966) comments that the boundary will be crossed, the free
libration will damp, and the rotation will evolve toward the synchronous
state. The synchronous state has a zero $\,\dot{\eta}\,$, with $\,\eta\,$
biased slightly off 202020 The value $\,\pi\,$ shows up because of the factor
2 accompanying $\,\eta\,$ when this variable enters $\,\sin 2\eta\,$ in the
equation for the torque and $\,\cos 2\eta\,$ in the expression $\,E\,$. The
tiny bias off $\,\eta=0\,$ or off $\,\eta=\pi\,$, given by (62), will emerge
due to dissipation. of either $\,0\,$ or $\,\pi\,$. The figure does not show
the libration region.212121 A figure with $\,W\,$ rather than
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ would show the libration region,
along with the circulating region. On such a figure, though, positive and
negative values of $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ would overlap, so
that the one-sided nature of the stall would not be evident.
Figure 1: Three possible scenarios of evolution of
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$. Arrows on the top diagramme
illustrate the evolution of circulating
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ toward the libration boundary for
$\,W_{\textstyle{{}_{stall}}}\,<\,W_{b}\,$. Arrows on the left of the bottom
diagramme depict the evolution of negative
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ toward the libration boundary.
Arrows in the midst and on the right of the bottom diagramme show the
evolution of positive $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ toward the
stall point for $\,W_{b}\,<\,W_{\textstyle{{}_{stall}}}\,$.
If $\,W_{b}\,<\,W_{\textstyle{{}_{stall}}}\,$, then a stall point exists in
the circulating region ($\,W_{b}\,<\,W\,$). All in all, Figure 1b summarises
the following three cases:
* A.
For an initially negative $\,{\langle\dot{\eta}\rangle}_{{{}_{P}}}\,$,
circulation is taking place. We have $\,W_{b}<W\,$, but the stall point is
never located on the negative $\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ side of
zero. The rate $\,\langle\,dE/dt\,\rangle_{{{}_{P}}}\,$ is negative, as can be
seen from (63). We then have a slow decrease of the three quantities:222222 Be
mindful that $\,E\,$ remains a constant over a cycle of $\,P\,$, except for a
tiny amount of dissipation during a cycle. $\,E\,\rightarrow\,E_{b}~{}$,
$~{}W\,\rightarrow\,W_{b}\,$, and
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,\rightarrow\,0\,$. This decrease takes
the system to the circulation/libration boundary. Goldreich (1966) commented
that it is ambiguous whether $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ would
cross zero and proceed to increase or whether the boundary would be crossed
passing into the libration region followed by damping of the libration and by
evolution toward the synchronous state.
Be mindful that, while the first integral $\,E\,$ is decreasing in the
circulation case, the actual kinetic energy of rotation is increasing. Indeed,
when the negative $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ is evolving
towards zero, the spin rate $\,\dot{\theta}\,$ is growing, and so is the
rotational energy $\,C\dot{\theta}^{2}/2\,$.
* B.
For an initially positive $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ lying
between the circulation/libration boundary and the stall point, i.e., for
$\,W_{b}\,<\,W\,<\,W_{\textstyle{{}_{stall}}}\,$, the rate
$\,\langle\,dE/dt\,\rangle_{{{}_{P}}}\,$ is positive, and $\,E\,$ increases.
The quantities $\,W\,$, $\,\langle\dot{\eta}^{\,2}\rangle_{{{}_{P}}}\,$ and
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ will evolve to larger values until
the evolution stalls as $\,W\,$ approaches $\,W_{\textstyle{{}_{stall}}}\,$
from below.
As the positive $\,\dot{\eta}\,$ is increasing, so are the spin rate
$\,\dot{\theta}\,$ and the rotational energy.232323 In the first integral
(54), the kinetic-energy-like part is given by
$\,C\dot{\eta}^{2}/2=(C/2)(\dot{\theta}-n)^{2}=(C/2)(\dot{\theta}^{2}-2n\dot{\theta}+n^{2})~{}$.
When the spin accelerates, the additional kinetic energy is borrowed from the
orbital motion. Because of the dissipation, the overall spin + orbit energy
must nevertheless decrease. (The orbital energy includes a kinetic part and
the $\,-GM/r\,$ potential term.)
* C.
For an initially positive $\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ beyond the
stall point we have: $\,W_{b}\,<\,W_{\textstyle{{}_{stall}}}\,<\,W\,$. Then
$\,\langle\,dE/dt\,\rangle_{{{}_{P}}}\,$ is negative and $\,E\,$ is slowly
decreasing.242424 Just as in item A, here $\,E\,$ stays virtually unchanged
over $\,P\,$. Likewise, $\,W\,$,
$\,\langle\dot{\eta}^{\,2}\rangle_{{{}_{P}}}\,$ and
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ evolve to lower values until the
evolution stalls as $\,W\,$ approaches $\,W_{\textstyle{{}_{stall}}}\,$ from
above.
The decrease of the positive $\,\dot{\eta}\,$ renders a decrease in the
rotation rate $\,\dot{\theta}\,$ and thus leads to damping of the kinetic
energy of rotation.
Calculation of the frequency at the stall point
$\,\langle\dot{\eta}_{\textstyle{{}_{stall}}}\,\rangle_{{}_{P}}\,$ would
require two steps: first, reversing the elliptic integral computation,
starting with $\,W_{stall}\,$; and second, deriving $\,E/C\,$, $\,P_{stall}\,$
and $\,\langle\dot{\eta}_{\textstyle{{}_{stall}}}\,\rangle_{{{}_{P}}}~{}$.
At the stall point, formulae (59a) and (60) acquire the form of
$\displaystyle
P_{\textstyle{{}_{stall}}}\,=~{}\frac{2~{}\pi~{}}{~{}\langle\dot{\eta}_{\textstyle{{}_{stall}}}\rangle_{{{}_{P}}}}\quad~{}\quad\mbox{and}\quad~{}\quad
W_{\textstyle{{}_{stall}}}\,=~{}P_{\textstyle{{}_{stall}}}~{}\langle\dot{\eta}^{\,2}_{\textstyle{{}_{stall}}}\,\rangle_{{{}_{P}}}~{}~{}~{},$
(73)
correspondingly. In combination with (67), this entails:
$\displaystyle\frac{\langle\dot{\eta}_{\textstyle{{}_{stall}}}^{\,2}\rangle_{{{}_{P}}}}{\langle\dot{\eta}_{\textstyle{{}_{stall}}}\rangle_{{{}_{P}}}}~{}=~{}{6\,n\,e^{2}}\left(1\;+\;\frac{\textstyle
e^{2}}{\textstyle 16}\,\right)~{}~{}~{},$ (74)
whence we once again note that the frequency
$\,\langle\dot{\eta}_{\textstyle{{}_{stall}}}\,\rangle_{{{}_{P}}}\,$ must be
positive. The comparison of $\,W_{\textstyle{{}_{stall}}}\,$ with $\,W_{b}\,$
in the inequalities mentioned in the above items A, B, C is then equivalent to
comparing
$\,\langle\dot{\eta}^{2}_{\textstyle{{}_{stall}}}\,\rangle_{{{}_{P}}}/\langle\dot{\eta}_{\textstyle{{}_{stall}}}\rangle_{{{}_{P}}}\,$
with $\,(2/\pi)\,\chi_{\textstyle{{}_{lib-max}}}\,$ or to comparing252525The
latter may also be expressed as comparison of $\,3\pi
e^{2}\left(1+\frac{\textstyle 21}{\textstyle 16}\,e^{2}\right)\,$ with
$\,\left[3\,\frac{\textstyle B-A}{\textstyle C}~{}\frac{\textstyle
M_{Earth}}{\textstyle M_{Earth}+M_{Moon}}\right]^{1/2}\,$.
$\,3\,\pi\,e^{2}\,(1+e^{2}/16)\,$ with $\,\left[3~{}\frac{\textstyle
B-A}{\textstyle C}~{}\frac{\textstyle M_{Earth}}{\textstyle
M_{Earth}+M_{Moon}}~{}G_{200}(e)~{}\right]^{1/2}\,$.
A comment on equation (74) would be in order. Since oscillations of $\,\eta\,$
result from the existence of the permanent triaxiality, evolution of
$\,\eta\,$ becomes smooth in the $\,A\,=\,B\,$ limit. Averaging becomes
unnecessary, so
$~{}\langle\dot{\eta}_{\textstyle{{}_{stall}}}^{\,2}\rangle_{{{}_{P}}}~{}$
becomes simply $~{}\dot{\eta}_{\textstyle{{}_{stall}}}^{\,2}~{}$, while
$\,\langle\dot{\eta}_{\textstyle{{}_{stall}}}\rangle_{{{}_{P}}}\,$ becomes
$\,\dot{\eta}_{\textstyle{{}_{stall}}}\,$. This way, in the oblate-body case
considered back in Section 5, equation (74) acquires the form of
$~{}\dot{\eta}_{\textstyle{{}_{stall}}}=\,{6\,n\,e^{2}}\left(1\;+\;{\textstyle
e^{2}}/{\textstyle 16}\,\right)~{}$, which agrees with (44).
### 6.8 Application to the Moon
When Goldreich (1966) applied his inequality expressions to the Moon, he found
that the stall point would have interrupted evolution of rotation from faster
spin to synchronous rotation. Here we have repeated his study, though with the
corrected average torque (43) instead of (34). For $~{}\frac{\textstyle
B-A}{\textstyle C}\,=\,2.278\,\times\,10^{-4}\,$ (Williams & Boggs 2012) ,
$~{}e=0.0549\,$, and $~{}\frac{\textstyle M_{Earth}}{\textstyle
M_{Earth}+M_{Moon}}=0.98785~{}$, the libration period turns out to be 38 times
the orbital period, while $\,W_{stall}\,$ is $\,10\,\%\,$ larger than
$\,W_{b}\,$. This renders a value of $\,W_{stall}/W_{b}\,$ much smaller than
the one found by Goldreich (1966), and the difference is mainly due to our use
of the corrected average torque (43). Despite the so-different value of
$\,W_{stall}/W_{b}\,$, despinning of the Moon would still be interrupted by a
stall. In principle, a tidal spin-up scenario remains an option too, though
this option does not look probable.
Goldreich (1966) noted that the lunar orbital eccentricity is changing. To
make the stall point disappear, i.e., to ensure that $\,W_{stall}<W_{b}\,$.
the eccentricity of the Moon would need to be less than $\,95\,\%\,$ of its
present value. While we lack data on the ancient evolution of the lunar
eccentricity, the modern eccentricity rate of about $2\times 10^{-11}\,$yr-1
has been reliably determined through analysis of the Lunar Laser Ranging data
(Williams et al. 2001, Williams & Boggs 2009). For the measured eccentricity
rate, the eccentricity would be small enough to prevent a stall prior to
$\,1.4\times 10^{8}\,$ yr ago. The Moon has clearly been a satellite of the
Earth for billions of years, and the capture into the synchronous spin state
should have occurred very early in its history.
For the Moon as it exists today, the free libration in longitude has a
$\,2.9\,$ yr period. The damping time is four orders of magnitude longer
(Williams et al. 2001), so evolution of this libration is slow. Despite the
damping time being short compared to the lunar age, the Moon has a small free
libration amplitude of $\,1.3"\,$ (Rambaux and Williams 2011). There has been
geologically recent stimulation, probably due to resonance crossing (Eckhardt
1993).
The ambiguity of evolution of the negative
$\,\langle\dot{\eta}\rangle_{{{}_{P}}}\,$ past the circulation/libration
boundary deserves a comment. For Mercury, Makarov (2012) finds that a more
realistic tidal dissipation model than the corrected MacDonald torque strongly
changes computations of the evolution of planetary spin rate near the
$\,3:2\,$ and higher spin-orbit resonances.
## 7 Conclusions
In the article thus far, we have provided a detailed explanation of how the
empirical MacDonald model can be derived from a more accurate and
comprehensive Darwin-Kaula theory of bodily tides. We have demonstrated that
the derivation hinges on a key assertion that the quality factor $\,Q\,$ of
the primary should be inversely proportional to the tidal frequency. This
crucial circumstance was missed by MacDonald (1964), who made his theory
inherently self-contradictory by setting the quality factor to be a frequency-
independent constant.
We have corrected this oversight in the MacDonald approach, and have developed
an appropriate correction to Goldreich’s model of spin dynamics and evolution
near the 1:1 spin-orbit resonance. Although we got different numbers,
qualitatively the main conclusion by Goldreich (1966) stays unaltered: when an
oblate body’s spin is evolving toward the resonance, vanishing of the average
tidal torque still implies a pseudosynchronous rotation (rotation slightly
faster than resonant), while synchronicity requires a small compensating
torque. For a triaxial body, the picture gets more complex due to the
emergence of a triaxiality-caused torque. (While the oblate case is
appropriate for spinning gaseous or liquid planets and moons, the triaxial
case applies to rocky objects.)
Goldreich (1966) linked the possible trapping of a body in the synchronous
state during its tidal evolution of rotation to its triaxiality. In light of
the correction of expression (36) to (44), that limiting condition between the
triaxiality and $\,e^{2}\,$ must change. After capture into the synchronous
state, the tidal torque is compensated by a triaxiality torque by aligning the
principal axis associated with the smallest moment of inertia slightly off of
the mean direction to the external body. A constant is thereby introduced into
the physical libration in longitude.
Setting the tidal quality factor to scale as inverse frequency is incompatible
with the actual dissipative properties of realistic mantles and crusts.
Nevertheless, after the afore-explained correction is implemented, both the
corrected MacDonald description of tides and the dynamical theory based
thereon remain valuable toy models capable of providing a good qualitative
handle on tidal dynamics over not too long timescales. Specifically, this
approach renders a simple qualitative description of the interplay between the
tidal torque and the triaxiality-caused torque exerted on a body near the 1:1
spin-orbit resonance.
It should also be remembered that the stability of pseudosynchronous rotation
hinges upon the dissipation model employed. Makarov and Efroimsky (2013) have
found that a more realistic tidal dissipation model than the corrected
MacDonald torque makes pseudosynchronous rotation unstable. Finally,
pseudosynchronism becomes impossible when the triaxiality is too large (see
subsection 6.7, specifically footnote 24).
Acknowledgements
We are indebted to Sylvio Ferraz-Mello for numerous fruitful discussions on
the theory of bodily tides and for referring us to the paper by Rodríguez,
Ferraz-Mello & Hussmann (2008). We also express our deep thanks to Anthony
Dobrovolskis whose review of our manuscript was very comprehensive and
extremely helpful.
A portion of the research described in this paper was carried out at the Jet
Propulsion Laboratory of the California Institute of Technology, under a
contract with the National Aeronautics and Space Administration. Government
sponsorship acknowledged.
Appendices.
## Appendix A The tidal-torque vector
and its components in spherical coordinates
A secondary with spherical coordinates
$\,\mbox{{\boldmath$\vec{r}$}}^{\,*}\,=\,(r^{\,*},\,\lambda^{*},\,\phi^{*})\,$
and mass $\,M_{sec}^{\,*}\,$ raises a tidal bulge on the primary. The
gravitational attraction between the tidal bulge and a secondary at
$\,\mbox{{\boldmath$\vec{r}$}}\,=\,(r\,,\,\lambda\,,\,\phi)\,$ with mass
$\,M_{sec}\,$ causes equal but opposite torques on the primary and the
secondary. For the external tidal potential
$\,U(\mbox{{\boldmath$\vec{r}$}})\,$, the torque components depend on the
partial derivatives of the potential $\,U\,$ along great circle arcs.
To calculate these expressions, let us recall some basics. The torque
$\,\vec{\bf T}\,$ wherewith the primary is acting on the secondary is given by
the cross-product
$\displaystyle\vec{\bf T}~{}=~{}\mbox{{\boldmath$\vec{r}$}}\times{\vec{\cal
F}}~{}~{}~{},$ (75)
$\vec{\cal F}\,$ being the tidal force exerted by the primary on the
secondary. This force is given by
$\displaystyle{\vec{\cal F}}~{}=~{}-~{}M_{sec}~{}\frac{\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\partial\mbox{{\boldmath$\vec{r}$}}}~{}=~{}-~{}M_{sec}~{}\left(~{}\frac{\partial
U}{\partial
r}~{}\hat{\bf{e}}_{\textstyle{{}_{r}}}\,+~{}\frac{1}{r}~{}\frac{1}{\cos\phi}~{}\frac{\partial
U}{\partial\lambda}~{}\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,-~{}\frac{1}{r}~{}\frac{\partial
U}{\partial\phi}~{}\hat{\bf{e}}_{\textstyle{{}_{\phi}}}~{}\right)~{}~{}~{},$
(76)
$\hat{\bf{e}}_{\textstyle{{}_{r}}}\,,\,\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,,\,\hat{\bf{e}}_{\textstyle{{}_{\phi}}}\,$
being the unit vectors of a spherical coordinate system associated with the
primary’s equator and corotating with it.262626 Were we using the polar angle
$\,\varphi=\pi/2-\phi\,$ instead of the latitude $\,\phi\,$, the right-handed
triple of unit vectors would be:
$\,\hat{\bf{e}}_{\textstyle{{}_{r}}}\,,\,\hat{\bf{e}}_{\textstyle{{}_{\varphi}}}\,,\,\hat{\bf{e}}_{\lambda}\,$
(“radial – south – east”), while the gradient would read as
$~{}\frac{\textstyle\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\textstyle\partial\mbox{{\boldmath$\vec{r}$}}}~{}=~{}\frac{\textstyle\partial
U}{\textstyle\partial
r}~{}\hat{\bf{e}}_{\textstyle{{}_{r}}}\,+~{}\frac{\textstyle 1}{\textstyle
r}~{}\frac{\textstyle\partial
U}{\textstyle\partial\varphi}~{}\hat{\bf{e}}_{\textstyle{{}_{\varphi}}}\,+~{}\frac{\textstyle
1}{\textstyle r}~{}\frac{\textstyle
1}{\textstyle\sin\varphi}~{}\frac{\textstyle\partial
U}{\textstyle\partial\lambda}~{}\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}~{}$.
As we are employing the latitude, the right-handed triple changes to
$\,\hat{\bf{e}}_{\textstyle{{}_{r}}}\,,\,\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,,\,\hat{\bf{e}}_{\textstyle{{}_{\phi}}}\,$
(“radial – east – north”), and the gradient becomes
$~{}\frac{\textstyle\partial
U(\mbox{{\boldmath$\vec{r}$}})}{\textstyle\partial\mbox{{\boldmath$\vec{r}$}}}~{}=~{}\frac{\textstyle\partial
U}{\textstyle\partial
r}~{}\hat{\bf{e}}_{\textstyle{{}_{r}}}\,+~{}\frac{\textstyle 1}{\textstyle
r}~{}\frac{\textstyle 1}{\textstyle\cos\phi}~{}\frac{\textstyle\partial
U}{\textstyle\partial\lambda}~{}\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,-~{}\frac{\textstyle
1}{\textstyle r}~{}\frac{\textstyle\partial
U}{\textstyle\partial\phi}~{}\hat{\bf{e}}_{\textstyle{{}_{\phi}}}~{}$.
Insertion of (76) into (75) results in
$\displaystyle\vec{\bf
T}~{}=~{}-~{}M_{sec}~{}\left(~{}0~{}\hat{\bf{e}}_{\textstyle{{}_{r}}}\,+~{}\frac{\partial
U}{\partial\phi}~{}\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,+~{}\frac{1}{\cos\phi}~{}\frac{\partial
U}{\partial\lambda}~{}\hat{\bf{e}}_{\textstyle{{}_{\phi}}}~{}\right)~{}~{},$
(77)
The torque $\vec{\tau}$ wherewith the secondary is acting on the primary will
be the negative of (77):
$\displaystyle\mbox{{\boldmath$\vec{\tau}$}}~{}=~{}M_{sec}~{}\left(~{}0~{}\hat{\bf{e}}_{\textstyle{{}_{r}}}\,+~{}\frac{\partial
U}{\partial\phi}~{}\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}\,+~{}\frac{1}{\cos\phi}~{}\frac{\partial
U}{\partial\lambda}~{}\hat{\bf{e}}_{\textstyle{{}_{\phi}}}~{}\right)$ (78)
Its east component, the one aimed along
$\hat{\bf{e}}_{\textstyle{{}_{\lambda}}}$, is parallel to the primary’s
equatorial plane. The north component, aimed along
$\hat{\bf{e}}_{\textstyle{{}_{\phi}}}$, is tangent to the meridian. In our
paper however we employ the projection of the torque vector onto the primary’s
spin axis, i.e., the component $\,\tau_{\textstyle{{}_{z}}}\,$ orthogonal to
the equator plane. This component is $\,\cos\phi\,$ times the north component:
$\displaystyle{\cal{T}}_{z}~{}=~{}M_{sec}~{}\frac{\partial
U}{\partial\lambda}~{}~{}~{}.$ (79)
This torque component, equation (6), slows down the rotation rate of the
primary. The decelerating torque acting on the secondary has an opposite sign
and is rendered by (5).
The east component and the projection of the north component onto the equator
plane act to change the orientation of the primary’s spin axis, a subject we
shall not pursue in this paper.
## Appendix B Calculation of the average torque within the corrected
MacDonald model
To calculate the orbital average of the tidal torque acting on a librating
secondary obeying the corrected MacDonald tidal model, substitute (38) and
(18) into (LABEL:19), and then choose $\,\alpha=-1\,$ (and recall that, for
$\,\alpha=-1\,$, the integral rheological parameter $\,{\cal{E}}\,$ is simply
the time lag: $\,{\cal{E}}\,=\,\Delta t\,$). An equivalent option would be to
employ (23) directly. This will lead us to the following expression for the
torque:
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}\right.{\cal{T}}_{z}~{}=~{}-~{}\frac{3}{2}~{}{GM_{sec}^{2}}\;k_{2}\,\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}\;{\cal{E}}\;\chi\;\,\mbox{sgn}(\dot{\theta}-\dot{\nu})+O({\it
i}^{2}/Q)+O(en\Delta
t/Q)+O(Q^{-3})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.\right.=~{}-~{}\frac{3}{2}~{}{GM_{sec}^{2}}\;k_{2}\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}\;2\;{\cal{E}}\;\,|\,\dot{\theta}-\dot{\nu}\,|\;\,\mbox{sgn}(\dot{\theta}-\dot{\nu})+O({\it
i}^{2}/Q)+O(en\Delta t/Q)+O(Q^{-3})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=\;-\;{3\;G\;M_{sec}^{2}}\;k_{2}\frac{R^{\textstyle{{}^{5}}}}{r^{\textstyle{{}^{6}}}}\;{\cal{E}}\;(\dot{\theta}-\dot{\nu})+O({\it
i}^{2}/Q)+O(en\Delta
t/Q)+O(Q^{-3})~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(80)
and for its average over one orbiting cycle:
$\displaystyle\langle\,{\cal{T}}_{z}\,\rangle=\,-\;\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\cal
E}}{R}\;\;\langle\;\,(\dot{\theta}\,-\,\dot{\nu})\,\;\frac{R^{\textstyle{{}^{6}}}}{r^{6}}\;\;\rangle~{}~{}+O({\it
i}^{2}/Q)+O(Q^{-3})+O(en\Delta
t/Q)\;=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}\,\right.-\,\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;k_{2}\,{\cal
E}}{R}\;\dot{\theta}\;\;\langle\;\,\frac{R^{\textstyle{{}^{6}}}}{r^{6}}\;\;\rangle\,\;+\;\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\cal
E}}{R}\;\langle\;\,\dot{\nu}\;\frac{R^{\textstyle{{}^{6}}}}{r^{6}}\;\;\rangle+O({\it
i}^{2}/Q)+O(Q^{-3})+O(en\Delta t/Q)~{}~{}~{}~{}~{}~{}\,$ (81a)
$\displaystyle=$ $\displaystyle-$
$\displaystyle\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;\,k_{2}\;{\cal
E}}{R}\;\dot{\theta}\;\;\frac{R^{6}}{a^{6}}\left(1\,-\,e^{2}\right)^{-9/2}~{}\,\frac{1}{2\,\pi}\;\int_{0}^{2\pi}\left(1+e\;\cos\nu\right)^{4}\,d\nu\;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(81b) $\displaystyle+$
$\displaystyle\frac{3\,G\,M_{sec}^{\textstyle{{}^{\,2}}}\;k_{2}\,{\cal
E}}{R}\,n\,\frac{R^{\textstyle{{}^{6}}}}{a^{6}}\left(1-e^{2}\right)^{-6}\frac{1}{2\pi}\int_{0}^{2\pi}\left(1+e\,\cos\nu\right)^{6}d{\nu}+O({\it
i}^{2}/Q)+O(Q^{-3})+O(en\Delta t/Q)~{}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}\,$
Evaluation of the integrals is straightforward and entails (39 \- 42).
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|
arxiv-papers
| 2012-10-10T13:58:41 |
2024-09-04T02:49:36.319475
|
{
"license": "Public Domain",
"authors": "James G. Williams and Michael Efroimsky",
"submitter": "Michael Efroimsky",
"url": "https://arxiv.org/abs/1210.2923"
}
|
1210.3025
|
# From interpretation of the three classical mechanics actions to the wave
function in quantum mechanics
Michel Gondran University Paris Dauphine, 75 016 Paris.
michel.gondran@polytechnique.org Alexandre Gondran Ecole Nationale de
l’Aviation Civile, 31000 Toulouse, France
###### Abstract
First, we show that there exists in classical mechanics three actions
corresponding to different boundary conditions: two well-known actions, the
Euler-Lagrange classical action $S_{cl}(\textbf{x},t;\textbf{x}_{0})$, which
links the initial position $\textbf{x}_{0}$ and its position x at time t, the
Hamilton-Jacobi action $S(\textbf{x},t)$, which links a family of particles of
initial action $S_{0}(\textbf{x})$ to their various positions x at time t, and
a new action, the deterministic action
$S(\textbf{x},t;\textbf{x}_{0},\textbf{x}_{0})$, which links a particle in
initial position $\textbf{x}_{0}$ and initial velocity $\textbf{v}_{0}$ to its
position x at time t. Mathematically, the Euler-Lagrange action can be
considered as the elementary solution to the Hamilton-Jacobi equation in a new
branch of non-linear mathematics, the Minplus analysis. We study, in the semi-
classical approximation, the convergence of the quantum density and the
quantum action, solutions to the Madelung equations, when the Planck constant
h tends to 0. We find two different solutions which depend on the initial
density. In the first case, where the initial quantum density is a classical
density $\rho_{0}(\mathbf{x})$, the quantum density and the quantum action
converge to a classical action and a classical density which satisfy the
statistical Hamilton-Jacobi equations. These are the equations of a set of
classical particles whose initial positions are known only by the density
$\rho_{0}(\mathbf{x})$. In the second case where initial density converges to
a Dirac density, the density converges to the Dirac function and the quantum
action converges to a deterministic action. Therefore we introduce into
classical mechanics non-discerned particles, which satisfy the statistical
Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned
particles, which satisfy the deterministic Hamilton-Jacobi equations. When the
semi-classical approximation is not valid, we conclude that the Schrödinger
equation cannot give a deterministic interpretation and the statistical Born
interpretation is the only valid one. Finally, we propose an interpretation of
the Schrödinger wave function that depends on the initial conditions
(preparation). This double interpretation seems to be the interpretation of
Louis de Broglie’s "double solution" idea.
## I Introduction
The aim of this paper is to show how the interpretation of the wave function
in quantum mechanics can be deduced from the interpretation of the action in
classical mechanics and from the study of the convergence QM-CM when the
Planck constant tends to 0\. First, in section 2, we show that there exist in
classical mechanics three actions corresponding to different boundary
conditions: two well-known actions, the Euler-Lagrange action and the
Hamilton-Jacobi action, and a new action, the deterministic action. We
introduce these three actions and present the fundamental relation between the
Hamilton-Jacobi and Euler-Lagrange actions. Second, in section 3, we present a
new branch of non-linear mathematics, the Minplus analysis that we have
developed following Maslov. In this new analysis, the Hamilton-Jacobi equation
can be considered as linear. Third, in section 4, we present in the
semiclassical case approximation, the QM-CM convergence when the Planck
constant tends to 0. It is necessary to introduce two cases: the statistical
semi-classical case and the deterministic semi-classical case. Fourth, in
section 5, we discuss the case where the semi-classical approximation is not
valid. Finally, we propose a realistic interpretation of quantum mechanics,
which is a synthesis of the three interpretations of the founding fathers of
quantum mechanics at the Solvay congress in 1927: the de Broglie
interpretation, the Schrödinger interpretation and the Copenhagen
interpretation.
## II The three classical mechanics actions
Let us consider a system evolving from the position $\textbf{x}_{0}$ at
initial time $t_{0}=0$ to the position x at time t where the variable of
control u(s) is the velocity:
$\frac{d\textbf{x}\left(s\right)}{ds}=\mathbf{u}(s)~{}~{}~{}~{}for~{}~{}~{}~{}s\in\left[0,t\right]$
(1) $\textbf{x}(0)=\mathbf{x}_{0},~{}~{}~{}~{}~{}~{}\textbf{x}(t)=\mathbf{x}.$
(2)
If $L(\textbf{x},\dot{\textbf{x}},t)$ is the Lagrangian of the system, when
the two positions $\textbf{x}_{0}$ and x are given, _the Euler-Lagrange
action_ $S_{cl}(\mathbf{x},t;\textbf{x}_{0})$ is the function defined by:
$S_{cl}(\mathbf{x},t;\textbf{x}_{0})=\min_{\mathbf{u}\left(s\right),0\leq
s\leq t}\left\\{\int_{0}^{t}L(\textbf{x}(s),\mathbf{u}(s),s)ds\right\\},$ (3)
where the minimum (or more generally an extremum) is taken on the controls
$\mathbf{u}(s)$, $s\in$ $\left[0,t\right]$, with the state $\textbf{x}(s)$
given by the equations (1)(2). The solution
$(\widetilde{\textbf{u}}(s),\widetilde{\textbf{x}}(s))$ of (3) satisfies the
Euler-Lagrange equations on the interval $[0,t]$:
$\frac{d}{ds}\frac{\partial
L}{\partial\dot{\textbf{x}}}(\textbf{x}(s),\dot{\textbf{x}}(s),s)-\frac{\partial
L}{\partial\textbf{x}}(\textbf{x}(s),\dot{\textbf{x}}(s),s)=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(0\leq
s\leq t)$ (4)
$\textbf{x}(0)=\mathbf{x}_{0},~{}~{}~{}~{}~{}~{}\textbf{x}(t)=\mathbf{x}.$ (5)
If
$L(\mathbf{x},\dot{\textbf{x}},t)=\frac{1}{2}m\dot{\textbf{x}}^{2}+\textbf{K}.\textbf{x}$,
then the Euler-Lagrange action is
$S_{cl}(\mathbf{x},t;\textbf{x}_{0})=m\frac{(\textbf{x}-\textbf{x}_{0})^{2}}{2t}+\frac{K.(\textbf{x}+\textbf{x}_{0})}{2}t-\frac{K^{2}}{24m}t^{3}$
and the initial velocity is given by
$\textbf{v}_{0}=\dot{\textbf{x}}(o)=-\frac{1}{m}\frac{\partial
S_{cl}}{\partial\textbf{x}_{0}}(\mathbf{x},t;\textbf{x}_{0})=\frac{\textbf{x}-\textbf{x}_{0}}{t}-\frac{Kt}{2m}.$
Let us now consider that an initial action $S_{0}(\textbf{x})$ is given, then
_the Hamilton-Jacobi action_ $S(\mathbf{x},t)$ is the function defined by:
$S(\mathbf{x},t)=\min_{\textbf{x}_{0};\mathbf{u}\left(s\right),0\leq s\leq
t}\left\\{S_{0}\left(\mathbf{x}_{0}\right)+\int_{0}^{t}L(\textbf{x}(s),\mathbf{u}(s),s)ds\right\\}$
(6)
where the minimum is taken on all initial positions $\textbf{x}_{0}$, on the
controls $\mathbf{u}(s)$, $s\in$ $\left[0,t\right]$, with the state
$\textbf{x}(s)$ given by the equations (1)(2). Because the term
$S_{0}(\textbf{x}_{0})$ has no effect in equation (6) for the minimization on
the control $\textbf{u}(s)$, we deduce the important relation between the
Hamilton-Jacobi action and Euler-Lagrange action:
$S(\mathbf{x},t)=\min_{\textbf{x}_{0}}(S_{0}\left(\mathbf{x}_{0}\right)+S_{cl}(\textbf{x},t;\textbf{x}_{0})).$
(7)
This equation is similar to the Hopf-Lax or Lax-Oleinik formula Evans .
If
$L(\mathbf{x},\dot{\textbf{x}},t)=\frac{1}{2}m\dot{\textbf{x}}^{2}+\textbf{K}.\textbf{x}$
with the initial action $S_{0}(\textbf{x})=m\textbf{v}_{0}\cdot\textbf{x}$,
then the Hamilton-Jacobi action is equal to
$S\left(\mathbf{x},t\right)=m\textbf{v}_{0}\cdot\textbf{x}-\frac{1}{2}m\textbf{v}_{0}^{2}t+\textbf{K}.\textbf{x}t-\frac{1}{2}\textbf{K}.\textbf{v}_{0}t^{2}-\frac{\textbf{K}^{2}t^{3}}{6m}.$
For a non-relativistic particle with the Lagrangian
$L(\mathbf{x},\mathbf{\dot{x}},t)=\frac{1}{2}m\mathbf{\dot{x}}^{2}-V(\textbf{x},t)$,
we obtain the well-known result: _The velocity of a non-relativistic classical
particle in a potential field is given for each point_
$\left(\mathbf{x,}t\right)$ _by_ :
$\mathbf{v}\left(\mathbf{x,}t\right)=\frac{\mathbf{\nabla}S\left(\mathbf{x,}t\right)}{m}$
(8)
where $S\left(\mathbf{x,}t\right)$ is the Hamilton-Jacobi action, a solution
to the Hamilton-Jacobi equations:
$\frac{\partial S(\textbf{x},t)}{\partial t}+\frac{1}{2m}(\nabla
S(\textbf{x},t))^{2}+V(\textbf{x},t)=0$ (9)
$S(\textbf{x},0)=S_{0}(\textbf{x}).$ (10)
The Hamilton-Jacobi action corresponds to a velocity field
$\mathbf{v}\left(\mathbf{x,}t\right)=\frac{\mathbf{\nabla}S\left(\mathbf{x,}t\right)}{m}$.
The Hamilton-Jacobi action $S(\mathbf{x},t)$ does not solve only a given
problem with a single initial condition
$\left(\mathbf{x}_{0},\frac{\mathbf{\nabla}S_{0}\left(\mathbf{x}_{0}\right)}{m}\right)$,
but a set of problems with an infinity of initial conditions
$\left(\mathbf{y},\frac{\mathbf{\nabla}S_{0}\left(\mathbf{y}\right)}{m}\right)$.
It is the problem solved by Nature with the principle of least action.
In the absence of an initial velocity field as in the Hamilton-Jacobi action,
the Euler-Lagrange action answers a problem posed by the observer, and not by
Nature: "If we see that a particle in $\textbf{x}_{0}$ at the initial time
arrives in x at time t, what was its initial velocity $\textbf{v}_{0}$?"
Let us now consider that we know the initial conditions ($\textbf{x}_{0}$,
$\textbf{v}_{0}$) and the Lagrangian of the system. If $\xi(t)$ is the
classical trajectory in the field $V(\textbf{x},t)$ of the particle with the
initial conditions $\left(\mathbf{x}_{0}\mathbf{,v}_{0}\right)$, then we
define _the deterministic action_
$S(\mathbf{x},t;\textbf{x}_{0},\textbf{v}_{0})$ by the equation:
$S(\mathbf{x},t;\textbf{x}_{0},\textbf{v}_{0})=m\frac{d\xi(t)}{dt}\cdot\textbf{x}+g(t)$
(11)
where
$g(t)=-\int^{t}_{0}{\frac{1}{2}m(\frac{d\xi(s)}{ds})^{2}+V(\xi(s))+m\frac{d^{2}\xi(s)}{ds^{2}}\cdot\xi(s)}ds$.
###### THEOREME 1
The deterministic action is a solution to the deterministic Hamilton-Jacobi
equations:
$\frac{\partial S(\textbf{x},t;\textbf{x}_{0},\textbf{v}_{0})}{\partial
t}|_{\textbf{x}=\xi(t)}+\frac{1}{2m}(\nabla
S(\textbf{x},t;\textbf{x}_{0},\textbf{v}_{0}))^{2}|_{\textbf{x}=\xi(t)}+V(\textbf{x})|_{\textbf{x}=\xi(t)}=0$
(12) $\frac{d\xi(t)}{dt}=\frac{\nabla
S(\textbf{x},t;\textbf{x}_{0},\textbf{v}_{0})}{m}|_{\textbf{x}=\xi(t)}$ (13)
$S(\textbf{x},0;\textbf{x}_{0},\textbf{v}_{0})=m\textbf{v}_{0}\textbf{x}~{}~{}~{}and~{}~{}~{}\xi(0)=\textbf{x}_{0}.$
(14)
The deterministic action satisfies the Hamilton-Jacobi equations only along
the classical trajectory $\xi(t)$. It is the action introduced by Rybakov
Rybakov for a soliton. We will interpret these equations in section 4 when we
will study the QM-CM convergence.
## III Interpretation of the Euler-Lagrange in Minplus analysis
There exists a new branch of mathematics, the Minplus analysis, which studies
nonlinear problems through a linear approach, cf. Maslov Maslov ; Maslov2 and
Gondran Gondran_1996 ; GondranMinoux . The idea is to substitute the usual
scalar product $\int_{X}f(x)g(x)dx$ with the Minplus scalar product:
$(f,g)=\inf_{x\in X}\left\\{f(x)+g(x)\right\\}$ (15)
In the scalar product we replace the field of the real number $(R,+,\times)$
with the algebraic structure Minplus $(R\cup\\{+\,\infty\\},\min,+)$, i.e. the
set of real numbers (with the element infinity $\\{+\infty\\}$) endowed with
the operation Min (minimum of two reals), which remplaces the usual addition,
and with the operation + (sum of two reals), which remplaces the usual
multiplication. The element $\\{+\,\infty\\}$ corresponds to the neutral
element for the operation Min, Min$(\\{+\infty\\},a)=a$ $\forall a\in R$. This
approach bears a close similarity to _the theory of distributions for the
nonlinear case_ ; here, the operator is "linear" and continuous with respect
to the Minplus structure, though _nonlinear_ with respect to the classical
structure $\left(R,+,\times\right)$. In this Minplus structure, the Hamilton-
Jacobi equation is linear, because if $S_{1}(\textbf{x},t)$ and
$S_{2}(\textbf{x},t)$ are solutions to (9), then
$\min\\{\lambda+S_{1}(\textbf{x},t),\mu+S_{2}(\textbf{x},t)\\}$ is also a
solution to the Hamilton-Jacobi equation (9).
The analog to the Dirac distribution $\delta(\textbf{x})$ in Minplus analysis
is the nonlinear distribution
$\delta_{\min}(\textbf{x})=\\{0~{}if~{}\textbf{x}=\textbf{0},+\infty~{}if~{}not\\}$.
With this nonlinear Dirac distribution, we can define elementary solutions as
in classical distribution theory. In particular, we have:
_The classical Euler-Lagrange action $S_{cl}(\textbf{x},t;\textbf{x}_{0})$ is
the elementary solution to the Hamilton-Jacobi equations (9)(10) in the
Minplus analysis with the initial condition_
$S(\textbf{x},0)=\delta_{\min}(\textbf{x}-\textbf{x}_{0})=\\{0~{}~{}if~{}~{}\textbf{x}=\textbf{x}_{0},~{}+\infty~{}~{}if~{}not\\}.$
(16)
The Hamilton-Jacobi action $S(\textbf{x},t)$ is then given by the Minplus
integral:
$S(\textbf{x},t)=\inf_{\textbf{x}_{0}}\\{S_{0}(\textbf{x}_{0})+S_{cl}(\textbf{x},t;\textbf{x}_{0})\\}$
(17)
in analogy with the solution to the heat transfer equation given by the
classical integral:
$S(x,t)=\int S_{0}(x_{0})\frac{1}{2\sqrt{\pi
t}}e^{-\frac{\left(x-x_{0}\right)^{2}}{4t}}dx_{0}.$ (18)
In this Minplus analysis, the Legendre-Fenchel transform is the analog to the
Fourier transform. This transform is known to have many applications in
physics: it sets the correspondence between the Lagrangian and the Hamiltonian
of a physical system; it sets the correspondence between microscopic and
macroscopic models; it is also at the basis of multifractal analysis relevant
to modeling turbulence in fluid mechanics GondranMinoux .
## IV The two limits of the Schrödinger equation in the semi-classical
approximation
Let us consider the wave function solution to the Schrödinger equation
$\Psi(\textbf{x},t)$:
$i\hbar\frac{\partial\Psi}{\partial
t}=\mathcal{-}\frac{\hbar^{2}}{2m}\triangle\Psi+V(\mathbf{x},t)\Psi$ (19)
$\Psi(\mathbf{x},0)=\Psi_{0}(\mathbf{x}).$ (20)
With the variable change
$\Psi(\mathbf{x},t)=\sqrt{\rho^{\hbar}(\mathbf{x},t)}\exp(i\frac{S^{\hbar}(\textbf{x},t)}{\hbar})$,
the Schrödinger equation can be decomposed into Madelung equations
Madelung_1926 (1926):
$\frac{\partial S^{\hbar}(\mathbf{x},t)}{\partial t}+\frac{1}{2m}(\nabla
S^{\hbar}(\mathbf{x},t))^{2}+V(\mathbf{x},t)-\frac{\hbar^{2}}{2m}\frac{\triangle\sqrt{\rho^{\hbar}(\mathbf{x},t)}}{\sqrt{\rho^{\hbar}(\mathbf{x},t)}}=0$
(21) $\frac{\partial\rho^{\hbar}(\mathbf{x},t)}{\partial
t}+div(\rho^{\hbar}(\mathbf{x},t)\frac{\nabla S^{\hbar}(\mathbf{x},t)}{m})=0$
(22)
with initial conditions
$\rho^{\hbar}(\mathbf{x},0)=\rho^{\hbar}_{0}(\mathbf{x})\qquad and\qquad
S^{\hbar}(\mathbf{x},0)=S^{\hbar}_{0}(\mathbf{x}).$ (23)
We consider two cases depending on the preparation of the particles
Gondran2011 ; Gondran2012 .
###### Définition 1
\- The statistical semi-classical case where
\- the initial probability density $\rho^{\hbar}_{0}(\mathbf{x})$ and the
initial action $S^{\hbar}_{0}(\mathbf{x})$ are regular functions
$\rho_{0}(\mathbf{x})$ and $S_{0}(\mathbf{x})$ not depending on $\hbar$.
\- the interaction with the potential field $V(\textbf{x},t)$ can be described
classically.
It is the case of a set of non-interacting particles all prepared in the same
way: a free particle beam in a linear potential, an electronic or $C_{60}$
beam in the Young’s slits diffraction, or an atomic beam in the Stern and
Gerlach experiment.
###### Définition 2
\- The determinist semi-classical case where
\- the initial probability density $\rho^{\hbar}_{0}(\mathbf{x})$ converges,
when $\hbar\to 0$, to a Dirac distribution and the initial action
$S^{\hbar}_{0}(\mathbf{x})$ is a regular function $S_{0}(\mathbf{x})$ not
depending on $\hbar$.
\- the interaction with the potential field $V(\textbf{x},t)$ can be described
classically.
This situation occurs when the wave packet corresponds to a quasi-classical
coherent state, introduced in 1926 by Schrödinger Schrodinger_26 . The field
quantum theory and the second quantification are built on these coherent
states Glauber_65 . The existence for the hydrogen atom of a localized wave
packet whose motion is on the classical trajectory (an old dream of
Schrödinger’s) was predicted in 1994 by Bialynicki-Birula, Kalinski, Eberly,
Buchleitner et Delande Bialynicki_1994 ; Delande_1995 ; Delande_2002 , and
discovered recently by Maeda and Gallagher Gallagher on Rydberg atoms.
###### THEOREME 2
Gondran2011 ; Gondran2012 For particles in the statistical semi-classical
case, the probability density $\rho^{\hbar}(\textbf{x},t)$ and the action
$S^{\hbar}(\textbf{x},t)$, solutions to the Madelung equations (21)(22)(23),
converge, when $\hbar\to 0$, to the classical density $\rho(\textbf{x},t)$ and
the classical action $S(\textbf{x},t)$, solutions to the statistical Hamilton-
Jacobi equations:
$\frac{\partial S\left(\textbf{x},t\right)}{\partial t}+\frac{1}{2m}(\nabla
S(\textbf{x},t))^{2}+V(\textbf{x},t)=0$ (24)
$\frac{\partial\mathcal{\rho}\left(\textbf{x},t\right)}{\partial
t}+div\left(\rho\left(\textbf{x},t\right)\frac{\nabla
S\left(\textbf{x},t\right)}{m}\right)=0\textit{ \ \ \ \ \ \ \
}\forall\left(\textbf{x},t\right)$ (25)
$\rho(\mathbf{x},0)=\rho_{0}(\mathbf{x})\qquad and\qquad
S(\textbf{x},0)=S_{0}(\textbf{x}).$ (26)
We give some indications on the demonstration of this theorem and we propose
its interpretation. Let us consider the case where the wave function
$\Psi(\textbf{x},t)$ at time t is written as a function of the initial wave
function $\Psi_{0}(\textbf{x})$ by the Feynman paths integral formula
Feynman_1965 (p. 58):
$\Psi(\textbf{x},t)=\int
F(t,\hbar)\exp(\frac{i}{\hbar}S_{cl}(\textbf{x},t;\textbf{x}_{0})\Psi_{0}(\textbf{x}_{0})d\textbf{x}_{0}$
where $F(t,\hbar)$ is an independent function of x and of $\textbf{x}_{0}$ and
where $S_{cl}(\textbf{x},t;\textbf{x}_{0})$ is the classical action. In the
statistical semi-classical case, the wave function is written
$\Psi(\textbf{x},t)=F(t,\hbar)\int\sqrt{\rho_{0}(\mathbf{x}_{0})}\exp(\frac{i}{\hbar}(S_{0}(\textbf{x}_{0})+S_{cl}(\textbf{x},t;\textbf{x}_{0}))d\textbf{x}_{0}$.
The theorem of the stationary phase shows that, if $\hbar$ tends towards 0, we
have
$\Psi(\textbf{x},t)\sim\exp(\frac{i}{\hbar}min_{\textbf{x}_{0}}(S_{0}(\textbf{x}_{0})+S_{cl}(\textbf{x},t;\textbf{x}_{0}))$,
that is to say that the quantum action $S^{h}(\textbf{x},t)$ converges to the
function
$S(\textbf{x},t)=min_{\textbf{x}_{0}}(S_{0}(\textbf{x}_{0})+S_{cl}(\textbf{x},t;\textbf{x}_{0}))$
(27)
which is the solution to the Hamilton-Jacobi equation (9) with the initial
condition (10). Moreover, as the quantum density $\rho^{h}(\textbf{x},t)$
satisfies the continuity equation (22), we deduce, since $S^{h}(\textbf{x},t)$
tends towards $S(\textbf{x},t)$, that $\rho^{h}(x,t)$ converges to the
classical density $\rho(\textbf{x},t)$, which satisfies the continuity
equation (25). We obtain both announced convergences.
The statistical Hamilton-Jacobi equations correspond to a set of independent
classical particles, in a potential field $V(\mathbf{x},t)$, and for which we
only know at the initial time the probability density
$\rho_{0}\left(\mathbf{x}\right)$ and the velocity $\mathbf{v(x)}=\frac{\nabla
S_{0}(\mathbf{x},t)}{m}$.
###### Définition 3
\- N identical particles, prepared in the same way, with the same initial
density $\rho_{0}\left(\textbf{x}\right)$, the same initial action
$S_{0}(\textbf{x})$, and evolving in the same potential $V(\textbf{x},t)$ are
called non-discerned.
We refer to these particles as non-discerned and not as indistinguishable
because, if their initial positions are known, their trajectories will also be
known. Nevertheless, when one counts them, they will have the same properties
as the indistinguishable ones. Thus, if the initial density
$\rho_{0}\left(\textbf{x}\right)$ is given, and one randomly chooses $N$
particles, the N! permutations are strictly equivalent and do not correspond
to the same configuration as for indistinguishable particles. This
indistinguishability of classical particles provides a very simple and natural
explanation to the Gibbs paradox.
In the statistical semi-classical case, the uncertainity about the position of
a quantum particle corresponds to an uncertainity about the position of a
classical particle, whose initial density alone has been defined. _In
classical mechanics, this uncertainity is removed by giving the initial
position of the particle. It would be illogical not to do the same in quantum
mechanics._ We assume that for the statistical semi-classical case, a quantum
particle is not well described by its wave function. One therefore needs to
add its initial position and it follows that we introduce the so-called de
Broglie-Bohm trajectories deBroglie_1927 ; Bohm_52 with the velocity
$\textbf{v}^{\hbar}(\textbf{x},t)=\frac{1}{m}\nabla S^{\hbar}(\textbf{x},t)$.
The convergence study of the determinist semi-classical case is mathematically
very difficult. We only study the example of a coherent state where an
explicit calculation is possible.
For the two dimensional harmonic oscillator,
$V(\textbf{x})=\frac{1}{2}m\omega^{2}\textbf{x}^{2}$, coherent states are
built CohenTannoudji_1977 from the initial wave function
$\Psi_{0}(\textbf{x})$ which corresponds to the density and initial action
$\rho^{\hbar}_{0}(\mathbf{x})=(2\pi\sigma_{\hbar}^{2})^{-1}e^{-\frac{(\textbf{x}-\textbf{x}_{0})^{2}}{2\sigma_{\hbar}^{2}}}$
and
$S_{0}(\mathbf{x})=S^{\hbar}_{0}(\mathbf{x})=m\textbf{v}_{0}\cdot\textbf{x}$
with $\sigma_{\hbar}=\sqrt{\frac{\hbar}{2m\omega}}$. Here, $\textbf{v}_{0}$
and $\textbf{x}_{0}$ are still constant vectors and independent from $\hbar$,
but $\sigma_{\hbar}$ will tend to $0$ as $\hbar$. With initial conditions, the
density $\rho^{\hbar}(\textbf{x},t)$ and the action $S^{\hbar}(\textbf{x},t)$,
solutions to the Madelung equations (21)(22)(23), are equal to
CohenTannoudji_1977 :
$\rho^{\hbar}(\textbf{x},t)=\left(2\pi\sigma_{\hbar}^{2}\right)^{-1}e^{-\frac{(\textbf{x}-\xi(t))^{2}}{2\sigma_{\hbar}^{2}}}$
and
$S^{\hbar}(\textbf{x},t)=+m\frac{d\xi(t)}{dt}\cdot\textbf{x}+g(t)-\hbar\omega
t$, where $\xi(t)$ is the trajectory of a classical particle evolving in the
potential $V(\textbf{x})=\frac{1}{2}m\omega^{2}\textbf{x}^{2}$, with
$\textbf{x}_{0}$ and $\textbf{v}_{0}$ as initial position and velocity and
$g(t)=\int_{0}^{t}(-\frac{1}{2}m(\frac{d\xi(s)}{ds})^{2}+\frac{1}{2}m\omega^{2}\xi(s)^{2})ds$.
###### THEOREME 3
Gondran2011 ; Gondran2012 \- When $\hbar\to 0$, the density
$\rho^{\hbar}(\textbf{x},t)$ and the action $S^{\hbar}(\textbf{x},t)$ converge
to
$\rho(\textbf{x},t)=\delta(\textbf{x}-\xi(t))~{}~{}and~{}~{}S(\textbf{x},t)=m\frac{d\xi(t)}{dt}\cdot\textbf{x}+g(t)$
(28)
where $S(\textbf{x},t)$ and the trajectory $\xi(t)$ are solutions to the
determinist Hamilton-Jacobi equations:
$\frac{\partial S\left(\textbf{x},t\right)}{\partial
t}|_{\textbf{x}=\xi(t)}+\frac{1}{2m}(\nabla
S(\textbf{x},t))^{2}|_{\textbf{x}=\xi(t)}+V(\textbf{x})|_{\textbf{x}=\xi(t)}=0$
(29) $\frac{d\xi(t)}{dt}=\frac{\nabla S(\xi(t),t)}{m}$ (30)
$S(\textbf{x},0)=m\textbf{v}_{0}\cdot\textbf{x}~{}~{}~{}and~{}~{}~{}\xi(0)=\textbf{x}_{0}.$
(31)
Therefore, the kinematic of the wave packet converges to the single harmonic
oscillator described by $\xi(t)$. Because this classical particle is
completely defined by its initial conditions $\textbf{x}_{0}$ and
$\textbf{v}_{0}$, it can be considered as _a discerned particle_. It is then
possible to consider, unlike in the statistical semi-classical case, that the
wave function can be viewed as a single quantum particle. The determinist
semi-classical case is in line with the Copenhagen interpretation of the wave
function, which contains all the information on the particle. A natural
interpretation is proposed by Schrödinger Schrodinger_26 in 1926 for the
coherent states of the harmonic oscillator: the quantum particle is a
spatially extended particle, represented by a wave packet whose center follows
the classical trajectory.
## V The non semi-classical case
The Broglie-Bohm and Schrödinger interpretations correspond to the semi-
classical approximation. They correspond to the two interpretations proposed
in 1927 at the Solvay congress by de Broglie and Schrödinger. The principle of
an interpretation that depends on the particle preparation conditions is not
really new. It had already been figured out by Einstein and de Broglie. For
Louis de Broglie, its real interpretation was the double solution theory
introduced in 1927 in which the pilot-wave is just a low-level product Broglie
:
"I introduced as a ’double solution theory’ the idea that it was necessary to
distinguish two different solutions but both linked to the wave equation, one
that I called wave $u$ which was a real physical wave but not normalizable
having a local anomaly defining the particle and represented by a singularity,
the other one as the Schrödinger $\Psi$ wave, which is normalizable without
singularities and being a probability representation."
We consider as interesting L. de Broglie’s idea of the existence of a
statistical wave, $\Psi$ and of a soliton wave $u$; however, it is not a
double solution that appears here but a double interpretation of the wave
function according to the initial conditions.
Einstein’s point of view is well summed up in one of his final papers (1953),
"Elementary reflections concerning the foundation of quantum mechanics" in
homage to Max Born Einstein :
"The fact that the Schrödinger equation associated with the Born
interpretation does not lead to a description of the "real states" of an
individual system, naturally incites one to find a theory that is not subject
to this limitation. Up to now, the two attempts have in common that they
conserve the Schrödinger equation and abandon the Born interpretation. _The
first one, which marks de Broglie’s comeback, was continued by Bohm…. The
second one, which aimed to get a "real description" of an individual system
and which might be based on the Schrödinger equation is very late and is from
Schrödinger himself. The general idea is briefly the following : the function
$\psi$ represents in itself the reality and it is not necessary to add it to
Born’s statistical interpretation._[…] From previous considerations, it
results that the only acceptable interpretation of the Schrödinger equation is
the statistical interpretation given by Born. Nevertheless, this
interpretation doesn’t give the "real description" of an individual system, it
just gives statistical statements of entire systems."
Thus, it is because de Broglie and Schrödinger maintain the Schrödinger
equation that Einstein, who considers it as fundamentaly statistical, rejected
each of their interpretations. Einstein thought that it was not possible to
obtain an individual deterministic behavior from the Schrödinger equation. It
is the same for Heisenberg who developped matrix mechanics and the second
quantization from the example of transitions in a hydrogen atom.
But there exist situations in which the semi-classical approximation is not
valid. It is in particular the case of state transitions in a hydrogen atom.
Indeed, since Delmelt’experiment Delmelt_1986 in 1986, the physical reality
of individual quantum jumps has been fully validated. The semi-classical
approximation, where the interaction with the potential field can be described
classically, is no longer possible and it is necessary to quantify the
electromagnetic field since the exchanges occur photon by photon. In this
situation, the Schrödinger equation cannot give a deterministic interpretation
and the statistical Born interpretation seems to be the only valid one. It was
the third interpretation proposed in 1927 at the Solvay congress, the
interpretation that was recognized as the right one in spite of Einstein’s, de
Broglie’s and Schrödinger’s criticisms.
This doesn’t mean that it is necessary to abandon determinism and realism in
quantum mechanics, but rather that the Schrödinger wave function doesn’t
allow, in this case, to obtain an individual behavior of a particle. An
individual interpretation needs to use the creation and annihilation operators
of the quantum Field Theory, but this interpretation still remains
statistical.
We hypothesize that it is possible to construct a deterministic quantum field
theory that extends to the non semi-classical interpretation of the double
semi-classical interpretation. First, as shown by de Muynck Muynck , we can
construct a theory with discerned (labeled) creation and annihilation
operators in addition to the usual non-discerned creation and annihilation
operators. But, to satisfy the determinism, it is necessary to search, at
lower scale, the mechanisms that allow the emergence of the creation operator.
## VI Conclusion
The study of the convergence of the Madelung equations when $h\rightarrow 0$,
gives the following results:
\- In the statistical semi-classical case, the quantum particles converge to
classical non-discerned ones satisfying the statistical Hamilton-Jacobi
equations, and the Broglie-Bohm pilot-wave interpretation is relevant.
\- In the determinist semi-classical case, the quantum particles converge to
classical discerned ones satisfying the determinist Hamilton-Jacobi equations.
And we can make a realistic and deterministic assumption such as the
Schrödinger interpretation.
This double interpretation seems to be the interpretation of Louis de
Broglie’s "double solution" idea.
\- In the case where the semi-classical approximation is no longer valid, as
in the transition states in the hydrogen atom, Louis de Broglie’s "double
solution" is not directly applicable. But, we hypothesize that it is possible
to construct a deterministic quantum field theory that extends this double
interpretation to the non semi-classical case.
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|
arxiv-papers
| 2012-10-10T15:37:56 |
2024-09-04T02:49:36.340126
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michel Gondran and Alexandre Gondran",
"submitter": "Michel Gondran",
"url": "https://arxiv.org/abs/1210.3025"
}
|
1210.3028
|
††institutetext: Department of Physics
University of California, San Diego
La Jolla, CA 92093
# Spontaneous $R$-symmetry breaking from the renormalization group flow
Antonio Amariti and David Stone amariti@ucsd.edu dcstone@physics.ucsd.edu
###### Abstract
We propose a mechanism of $R$-symmetry breaking in four-dimensional DSB models
based on the RG properties of the coupling constants. By constraining the UV
sector, we generate new hierarchies amongst the couplings that allow a
spontaneously broken $R$-symmetry in models with pure chiral fields of
$R$-charges $R=0$ and $R=2$ only. The result is obtained by a combination of
one- and two-loop effects, both at the origin of field space and in the region
dominated by leading log potentials.
††preprint: UCSD-PTH-12-14
## 1 Introduction
In the last decade many different mechanisms of supersymmetry breaking have
been investigated. Dynamical supersymmetry breaking (DSB) is an attractive
possibility because it can evade constraints imposed by the supertrace formula
STr$(\mathcal{M}^{2})$. Unfortunately, DSB models often lead to non-calculable
strongly coupled sectors, in which the knowledge of the spectrum requires the
use of non-perturbative techniques that are not always available. A new
scenario for DSB was proposed in Intriligator:2006dd . There a weakly coupled
IR supersymmetry breaking sector was obtained from supersymmetric duality. A
mass deformed $\mathcal{N}=1$ asymptotically free supersymmetric field theory
flows in the IR to a weakly coupled dual theory with parametrically long-lived
metastable minima that break the supersymmetry. At the lowest orders in the
perturbative expansion the dynamics are dominated by a model of pure chiral
fields, like the O’Raifeartaigh model. It is therefore important to know the
exact and general properties of O’Raifeartaigh-like models for the study of
DSB. To provide a phenomenologically viable scenario, we must also break the
$R$-symmetry that generically accompanies these models to give the gaugino a
non-zero Majorana mass.
In this work we focus on $R$-symmetric O’Raifeartaigh-like models whose field
content has $R$-charge $R=0$ or $R=2$ only. This property is typical of
generalizations of the ISS mechanism but these models suffer from broad
constraints that limit the possibility of spontaneous $R$-symmetry breaking,
which is what we seek to achieve in this work. In particular, the generic
(pseudo)moduli fields that accompany the supersymmetry breaking superpotential
that have $R=2$ receive positive corrections from the one-loop Coleman-
Weinberg potential, eliminating the possibility of $R$-symmetry breaking via a
non-zero modulus field vev, or they remain flat. There is no general proof for
the behavior of the pseudomoduli at higher loops, leaving open the possibility
of spontaneous $R$-symmetry breaking via higher loop corrections to
pseudomoduli that are one-loop flat. In fact, most examples that have one-loop
flat directions receive negative two-loop corrections that destabilize the
origin Giveon:2008wp ; Amariti:2008uz . Unfortunately, in all of these
examples the tachyonic behavior near the origin is never stabilized at a non-
zero pseudomodulus vev, and the potentials run away to a supersymmetric vacuum
or infinite field value.
In principle, a model could be constructed that stabilizes these potentials
with tachyonic behavior at the origin by going far out in field space and
using the quantum effective potential methods developed by Intriligator:2008fe
. It then becomes necessary to introduce one-loop corrections to the
pseudomodulus; however, these corrections, at least at the origin, must be
subdominant to the tachyonic two-loop effect. We give a rough sketch that
shows that having both a tachyonic origin and a stabilizing (i.e. positive)
slope in the far field potential cannot be accomplished with a single
superpotential coupling if one-loop effects are subdominant at the origin, in
the perturbative regime, they will continue to be subdominant to higher-loop
order effects far in field space.
This suggests that the myriad obstructions already evident might be evaded by
using more than one superpotential coupling. A mechanism could be introduced
to invert the behavior of the couplings in the two regions of field space and
induce the desired behavior of the effective potential. More concretely, we
invert the natural hierarchy of the perturbative expansion so that at the
origin of field space two-loop effects are dominant, but, far in field space,
the one-loop effects become more important. We achieve this through a new
coupling associated with massive degrees of freedom that are integrated out at
small field values but that contribute far from the origin. This is
reminiscent of the interplay between the gauge and interaction couplings in
Witten:1981kv , where the coupling hierarchy is inverted in the field space
because of asymptotic freedom.
In section 2 we elaborate on the obstructions to spontaneous $R$-symmetry
breaking at one and two loops in models with charges $R=0$ and $R=2$ only.
Then in section 3 we propose our mechanism, explicitly check its validity in a
toy model, and provide a UV completion. In section 4 we conclude and discuss
some open questions.
## 2 $R$-symmetry breaking with $R=0$ and $R=2$: Obstructions
The one-loop correction to the mass of the O’Raifeartaigh field is non-
negative in models of pure chiral fields with charges $R=0$ and $R=2$
Shih:2007av . This result holds when more than one pseudomodulus is present
Curtin:2012yu ; however, the fate of these pseudomoduli at higher-loop order
is generically unconstrained and $R$-symmetry breaking is left as a
possibility. Unlike the one-loop Coleman-Weinberg effective potential, which
can be calculated in terms of the mass matrices only, at two-loop order the
effective potential must be explicitly calculated by including the Yukawa and
quartic couplings111If the supersymmetry breaking scale $F$ is smaller than
the messenger scale $M$, $F\ll M^{2}$, there are simpler results for the two-
loop effective potential. Nibbelink:2005wc .
Explicit examples show that at two-loop order there are no non-negativity
constraints on the pseudomoduli masses as in the one-loop case. For example,
in the model studied in Giveon:2008wp the superpotential is222The model
studied in Giveon:2008wp is slightly different, but the quantum corrections
are computed in a similar manner and the final result is the same.
$W=fX+hX\phi_{1}^{2}+h\mu\phi_{1}\phi_{2}+hY\phi_{1}\phi_{4}+hZ\phi_{4}^{2}+hm\phi_{4}\phi_{5}$
(1)
where $\sqrt{f}$, $\mu$, and $m$ are mass scales in the theory, $h$ is the
superpotential coupling, the $\phi_{i}$ fields are tree-level stable at the
origin, the $X$ and $Y$ are pseudomoduli stabilized at one-loop, and the $Z$
field is still a pseudomodulus at one loop that acquires a negative mass at
two loops.
A different possibility has been studied in Amariti:2008uz , by starting from
the superpotential
$W=fX+hX\phi_{1}^{2}+h\mu\phi_{1}\phi_{2}+hY\phi_{1}\phi_{5}+hZ\phi_{4}^{2}+hm\phi_{4}\phi_{5}.$
(2)
In this case the one-loop pseudoflat direction $Z$ has a positive two-loop
mass that is stabilized around the $R$-symmetric vacuum $\langle Z\rangle=0$.
Clearly, while in (2) the $R$-symmetry is not spontaneously broken, the
possibility to break the $R$-symmetry exists in (1). The vacuum structure for
this model must be determined by calculating the behavior of the $Z$ potential
away from the origin. This can be explored by applying the analysis of
Intriligator:2008fe . There, one reconstructs the effective potential for a
pseudoflat direction far from the origin but below the cutoff scale by
computing the discontinuity in the anomalous dimension of the massive
messengers in the theory. The pseudomodulus is treated as a background field
with non-zero vev. By applying this idea the leading log potential is obtained
order-by-order in perturbation theory- schematically, with loop order $n$, one
has
$V_{eff}(\Phi)\simeq
const.+\sum_{n}(-1)^{(n+1)}\frac{2}{n!}|f|^{2}\Delta{\Omega_{X}^{(n)}}\log^{n}\frac{|\Phi|}{m_{0}}$
(3)
and the sign of the coefficient $(-1)^{(n+1)}\Delta{\Omega_{X}^{(n)}}$
determines the sign of the potential of the pseudomodulus at large $\Phi$. The
discontinuity in the anomalous dimension is captured in
$\Delta{\Omega_{X}^{(n)}}=\left.\frac{d^{n-1}\gamma_{X}}{dt^{n-1}}\right|^{t_{\Phi}^{+}}_{t_{\Phi}^{-}}$.
As explained in Intriligator:2008fe , each derivative of $\gamma_{X}$ gives a
loop factor. This formula is only valid in the region
$\sqrt{F_{\Phi}}\ll\langle\Phi\rangle\ll\Lambda$, where $F_{\Phi}$ is the
scale set by the supersymmetry-breaking $F$-terms of $\Phi$.
In the case of (1) the leading log potential for $Z$ is negative and the
potential flows towards a supersymmetric minimum (or a runaway)333The
supersymmetric vacuum structure is usually associated with the UV completion
of the model.. So there are no $R$-symmetry breaking vacua in (1), even though
the potential is destabilized at the origin.
One can still try to break the $R$-symmetry with the addition of a tree-level
term $W\supset f_{2}Z$ to the superpotential. Indeed, this term generates a
one-loop contribution to the mass of $Z$ (which is automatically positive) and
there is a tension between the one- and two-loop contributions, potentially
giving a non-supersymmetric vacuum at $\langle Z\rangle\neq 0$. One can then
distinguish the two cases $f_{2}\simeq f$ and $f_{2}\ll f$444The case
$f_{2}\gg f$ is irrelevant because it reverses the role of $Z$ and $X$ in the
$f_{2}\ll f$ case. If $f_{2}\simeq f$ the positive one-loop correction
dominates at the origin and the negative two-loop effect dominates at large
vev, so the potential has a local maximum at the origin. On the contrary, if
$f_{2}\ll f$ the negative two-loop potential dominates everywhere, since the
one-loop effects at the origin are suppressed by $f_{2}/f\ll 1$. In both cases
there are no $R$-symmetry breaking minima.
It would appear that this outcome is generic in the models presented. This is
argued as follows: To achieve spontaneous $R$-symmetry breaking in these
O’Raifeartaigh-like models, we require that the $Z$ potential be (a) tachyonic
at the origin and (b) increasing (i.e. with positive slope) somewhere further
out in field space. To satisfy (a), we must have two-loop effects that are
dominant at the origin, since one-loop effects will never afford this
behavior. Since the two-loop effects are suppressed by a factor of $h^{2}$
compared to the one-loop effects (but aided by a factor of $F_{X}/F_{\Phi}\gg
1$), this puts a lower bound on the value of $h$555Hereafter we assume $h$ is
a real coupling by absorbing the imaginary part in the phases of the fields..
As we move farther out in field space to the regime where (3) is applicable we
begin to lose perturbativity as higher loops become increasingly important.
However, in a model of only chiral fields with one coupling, if the two-loop
contribution dominates the one-loop contributions at the origin it will
dominate the one-loop contribution everywhere in field space, since no new
field content is introduced. To satisfy (b), one could argue that the three-
loop behavior might accomplish what the one-loop contribution sought to do,
but then our “leading” log arguments are foregone as we begin to consider all
loop contributions. More quantitatively, the requirement from (b) that the
(positive) one-loop leading log dominate the two-loop leading log out in field
space puts an upper bound on the value of $h$, which will eliminate any
parameter space in $h$ left from the previous lower bound.
We now search for a loophole in this argument based upon the RG properties of
the model in the perturbative large field region with multiple couplings. In
the next section we provide a way to invert the hierarchy amongst the one- and
two-loop effects when the potential is dominated by the leading log.
## 3 $R$-symmetry breaking from the renormalization group flow
We have seen that in O’Raifeartaigh-like models with only $R=0$ and $R=2$
fields $R$-symmetry breaking is quite constrained. One-loop quantum
corrections will leave pseudomoduli flat or stabilize them at the origin,
while two-loop corrections can be either positive or negative. At the quantum
level, this means that there can exist tension between a positive one-loop and
a negative two-loop correction666We ignore higher loop corrections.. In the
models previously studied this leads to runaway behavior, but here we will
attempt to circumvent their fate with a loophole based upon the $RG$
properties of superpotentials and their moduli spaces.
### 3.1 Generalities
Consider a model with chiral fields, a canonical Kähler potential and a
superpotential $W$ with all fields assigned $R$-charges $R=0$ or $R=2$ such
that the $R$-symmetry is preserved. Let the superpotential be of the form
$W=W_{1}(X,\Phi_{i},\phi_{i})+W_{2}(\Phi_{i},\varphi_{i})$ (4)
where we identify the tree-level flat direction with $X$ and $\Phi_{i}$ and
the other fields are the $\phi_{i}$ and $\varphi_{i}$. The $W_{1}$ sector has
the usual O’Raifeartaigh field $X$ in addition to other pseudomoduli
$\Phi_{i}$ and massive messengers $\phi_{i}$. The second sector, $W_{2}$,
contains some of the (pseudo)moduli $\Phi_{i}$ with non-zero $F$-terms such
that $F_{\Phi}\ll F_{X}$ and some massive fields $\varphi_{i}$. We assume the
masses of the $\varphi_{i}$ are much larger than those of the $\phi_{i}$ from
the first sector, $m_{\varphi_{i}}\gg m_{\phi_{i}}$.
In this limit the $W_{2}$ sector decouples around the origin of the
(pseudo)moduli space and the non-supersymmetric vacuum structure is encrypted
in $W_{1}$777There is still a non-zero $F$-term associated to $\Phi_{i}$ in
$W_{2}$ but it is subleading in the limit $F_{\Phi}\ll F_{X}$.. We consider
$W_{1}$ such that one of the $\Phi_{i}$ has a vanishing one-loop mass
correction but a non-zero, negative two-loop correction. The effective
potential for this field is negative around the origin and it remains negative
in the region $|F_{X}|\ll|\langle\Phi_{i}\rangle|\ll\Lambda$, where $\Lambda$
is the strong coupling scale determined by the UV completion of the model
Intriligator:2008fe . This model does not generically break the $R$-symmetry
spontaneously, at least not without the $W_{2}$ sector. The contributions from
$W_{2}$ become important at a scale $|\langle\Phi_{i}\rangle|\simeq
m_{\varphi_{i}}$, where the presence of a non-zero $F$-term for $\Phi_{i}$
gives a positive leading log correction to the effective potential. The
potential in this region is
$V(\Phi_{i})\simeq V^{(1)}(h_{2},F_{\Phi_{i}})+V^{(2)}(h_{1},F_{X})$ (5)
where $h_{i}$ is the coupling in the $W_{i}$ sector. The $R$-symmetry can be
broken if $h_{2}\gg h_{1}\eta(F_{X},F_{\Phi_{i}})$ where
$\eta(F_{X},F_{\Phi_{i}})$ is a model-dependent function. Figure 1 gives a
schematic picture of the effective potential for the field $\Phi$.
Figure 1: A schematic picture of the effective potential for the field $\Phi$.
Near the origin in $\langle\Phi\rangle$ there is a positive one-loop
correction to the tree-level flat potential for $\langle\Phi\rangle$. This
contribution is suppressed by $\sim\frac{F_{\Phi}}{F_{X}}$ in comparison to a
negative two-loop correction that dominates the one-loop contribution. Both
are computed perturbatively. As we move away from the origin and lose
computational control, we approach the far-field region, where
$\langle\Phi\rangle\sim m_{\varphi_{i}}$ but $\langle\Phi\rangle\ll\Lambda$,
where $\Lambda$ is the cutoff scale. Here the potential is computed using the
leading log expansion and the one-loop leading $\log\langle\Phi\rangle$
dominates the two-loop leading $\log^{2}\langle\Phi\rangle$ by a careful
choice of parameters in the model. As $\langle\Phi\rangle\sim\Lambda$, we lose
all perturbative control over the behavior of the potential.
### 3.2 A toy model
Here we propose a toy model that spontaneously breaks the $R$-symmetry in an
O’Raifeartaigh-like model with fields that have $R$-charges $0$ and $2$ only.
We follow the strategy explained above. The superpotential $W_{1}$ is
$W_{1}=f_{X}X+h_{X}X\phi_{1}^{2}+m_{1}\phi_{1}\phi_{2}+Y\phi_{1}\phi_{4}+h_{1}Z\phi_{4}^{2}+m_{2}\phi_{4}\phi_{5}$
(6)
while $W_{2}$ is
$W_{2}=f_{Z}Z+h_{2}Z\xi_{4}^{2}+m_{3}\xi_{4}\xi_{5}$ (7)
We impose a hierarchy amongst the scales
$\sqrt{f}_{Z}\ll\sqrt{f}_{X}\ll m_{1},m_{2}\ll m_{3}\ll\Lambda.$ (8)
Around the origin, $\xi_{4}$ and $\xi_{5}$ are integrated out at zero vev and
the vacuum structure is well described by $W_{1}$. The fields $\phi_{i}$
acquire a tree-level mass at zero vev while the fields $X$ and $Y$ are tree-
level flat directions, stabilized at the origin by one-loop corrections. The
field $Z$ is flat at tree level and its quantum mass is dominated by the two-
loop effect if
$\epsilon\equiv\frac{f_{Z}^{2}}{f_{X}^{2}h_{X}^{2}}\ll 1.$ (9)
At larger $\langle Z\rangle$ the effects of $m_{3}$ are no longer suppressed.
In the region
$m_{3}\ll\langle Z\rangle\ll\Lambda$ (10)
the leading log potential is
$V_{eff}=f_{Z}^{2}(h_{1}^{2}+h_{2}^{2})\log
Z-f_{X}^{2}h_{X}^{2}h_{1}^{2}\log^{2}Z.$ (11)
There can still be an $R$-symmetry breaking minimum if the inequality
$h_{2}>h_{1}\sqrt{\frac{2\log Z}{\epsilon}-1}$ (12)
is satisfied (note this is compatible with (8)). The $R$-symmetry is broken at
the quantum level by the vev of $Z$, with $R(Z)=2$. The presence of two
couplings in this simple example follows the construction outlined in section
3.1 and accomplishes spontaneous $R$-symmetry breaking.
### 3.3 A UV completion
In this section we discuss a supersymmetric gauge theory with supersymmetry-
breaking metastable vacua that also break its $R$-symmetry. In the IR the
model reduces to the class introduced above, where $W_{1}$ and $W_{2}$ provide
a generalization of the toy model.
In this model we tune the masses of the fields in the UV sector, while the
tuning on the couplings is dynamical. This provides a more natural explanation
of the necessary hierarchies amongst the couplings required by our
construction. The field content is (see Figure 2 for a quiver representation
of the model)
Field | $SU(N_{F_{1}})$ | $SU(N_{c})$ | $SU(N_{F_{2}})$ | $SU(M)$
---|---|---|---|---
$Q_{1}\oplus\tilde{Q}_{1}$ | $N_{F_{1}}+\widetilde{N}_{F_{1}}$ | $\widetilde{N}_{c}\oplus N_{c}$ | $1\oplus 1$ | $1\oplus 1$
$Q_{2}\oplus Q_{2}$ | $1\oplus 1$ | $N_{c}\oplus\widetilde{N}_{c}$ | $\widetilde{N}_{F_{2}}+{N_{F_{2}}}$ | $1\oplus 1$
$q_{3}^{(i)}\oplus\tilde{q}_{3}^{(i)}$ with $i=1,2$ | $1\oplus 1$ | $1\oplus 1$ | ${N_{F_{2}}}+\widetilde{N}_{F_{2}}$ | $\widetilde{M}\oplus M$
with superpotential
$W=m_{1}Q_{1}\tilde{Q}_{1}+m_{2}Q_{2}\tilde{Q}_{2}+m_{3}q_{3}^{(1)}\tilde{q}_{3}^{(2)}+m_{3}q_{3}^{(2)}\tilde{q}_{3}^{(1)}+\frac{1}{\Lambda_{0}}Q_{2}\tilde{Q}_{2}q_{3}^{(1)}\tilde{q}_{3}^{(1)}$
(13)
Figure 2: A quiver representing the electric theory. The green boxes are
flavor nodes, the red the gauge node. We do not fix the nature of the blue
node: it can be either a flavor symmetry or a weakly gauged global symmetry.
The groups $SU(N_{F_{1}})$ and $SU(N_{F_{2}})$ are flavor symmetries while
$SU(N_{c})$ is the gauge symmetry. At this level we do not specify the
dynamics of $SU(M)$; Figure 2 indicates the possibilities for this $SU(M)$ in
the context of a quiver diagram.
We consider this $SU(N_{c})$ gauge symmetry in the free magnetic range,
$N_{c}+1<N_{F_{1}}+N_{F_{2}}<\frac{3}{2}N_{c}$ (14)
so that the model is described in the IR by the Seiberg dual with field
content (see Figure 3 for the quiver representation)
Field | $SU(N_{F_{1}})$ | $SU(\widetilde{N}_{c})$ | $SU(N_{F_{2}})$ | $SU(M)$
---|---|---|---|---
$q_{1}\oplus\tilde{q}_{1}$ | $N_{F_{1}}+\widetilde{N}_{F_{1}}$ | $\widetilde{N}_{c}\oplus N_{c}$ | $1\oplus 1$ | $1\oplus 1$
$q_{2}\oplus q_{2}$ | $1\oplus 1$ | $N_{c}\oplus\widetilde{N}_{c}$ | $\widetilde{N}_{F_{2}}+{N_{F_{2}}}$ | $1\oplus 1$
$M_{11}$ | $N_{F_{1}}\times\widetilde{N}_{F_{1}}$ | $1$ | $1$ | $1$
$M_{12}\oplus M_{21}$ | $\widetilde{N}_{F_{1}}+N_{F_{1}}$ | $1$ | $N_{F_{2}}+\widetilde{N}_{F_{2}}$ | $1$
$M_{22}$ | $1$ | $1$ | $N_{F_{2}}\times\widetilde{N}_{F_{2}}$ | $1$
$q_{3}^{(i)}\oplus\tilde{q}_{3}^{(i)}$ with $i=1,2$ | $1\oplus 1$ | $1\oplus 1$ | ${N_{F_{2}}}+\widetilde{N}_{F_{2}}$ | $\widetilde{M}\oplus M$
where $\widetilde{N}_{c}=N_{F_{1}}+N_{F_{2}}-N_{c}$ and the superpotential is
$\displaystyle W$ $\displaystyle=$ $\displaystyle
h\mu_{1}^{2}M_{11}+h\mu_{2}^{2}M_{22}+h\left(\begin{array}[]{cc}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{array}\right)\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)\left(\begin{array}[]{cc}\tilde{q}_{1}&\tilde{q}_{2}\end{array}\right)$
(20) $\displaystyle+$
$\displaystyle\frac{\Lambda_{g}}{\Lambda_{0}}M_{22}q_{3}^{(1)}\tilde{q}_{3}^{(1)}+m_{3}\left(q_{3}^{(1)}\tilde{q}_{3}^{(2)}+q_{3}^{(2)}\tilde{q}_{3}^{(1)}\right).$
(21)
If we fix the hierarchy among the electric masses as
$m_{3}\gg m_{1}\gg m_{2}$ (22)
there is a classical vacuum solution that breaks supersymmetry which can be
written as
$\displaystyle\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)=\left(\begin{array}[]{c}\mu_{1}{\mathbf{1}}_{\tilde{N}_{c}\times\tilde{N}_{c}}\\\
{\mathbf{0}}_{(N_{F_{1}}-\tilde{N}_{c})\times\tilde{N}_{c}}\vspace{-.3cm}\\\
\\\ \hline\cr\vspace{-.4cm}\hfil\\\
{\mathbf{0}}_{N_{F_{2}}\times\tilde{N}_{c}}\end{array}\right)\,,\,\left(\begin{array}[]{cc}\tilde{q}_{1}&\tilde{q}_{2}\end{array}\right)=\left(\begin{array}[]{cccc}\mu_{1}{\mathbf{1}}_{\tilde{N}_{c}\times\tilde{N}_{c}}&{\mathbf{0}}_{\tilde{N}_{c}\times(N_{F_{1}}-\tilde{N}_{c})}&\vline&{\mathbf{0}}_{\tilde{N}_{c}\times
N_{F_{2}}}\end{array}\right)$ (32)
Figure 3: A quiver representing the magnetic theory. The green boxes are
flavor nodes, the red one is the gauge node, while the blue one can be both.
with the rest of the fields at zero expectation value. We can expand about
this vacuum and choose a convenient parametrization of the field fluctuations:
$\displaystyle\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)=\left(\begin{array}[]{c}\mu_{1}+\sigma_{1}\\\
\phi_{1}\vspace{-.3cm}\\\ \\\ \hline\cr\vspace{-.4cm}\hfil\\\
\phi_{5}\end{array}\right)\,,\,\left(\begin{array}[]{cc}\tilde{q}_{1}&\tilde{q}_{2}\end{array}\right)=\left(\begin{array}[]{cccc}\mu_{1}+\tilde{\sigma}_{1}&\phi_{2}&\vline&\phi_{4}\end{array}\right)$
(42) $\displaystyle M_{11}=\left(\begin{array}[]{cc}\Sigma_{11}&\phi_{6}\\\
\phi_{7}&X\end{array}\right)\,,\,M_{12}=\left(\begin{array}[]{c}\phi_{8}\\\
\tilde{Y}\end{array}\right)\,,\,M_{21}=\left(\begin{array}[]{cc}\phi_{9}&Y\end{array}\right)\,,\,M_{22}=Z$
(48) $\displaystyle\left(\begin{array}[]{c}q_{3}^{(1)}\\\
q_{3}^{(2)}\end{array}\right)=\left(\begin{array}[]{c}\xi_{4}\\\
\xi_{6}\end{array}\right)\,,\,\left(\begin{array}[]{cc}\tilde{q}_{3}^{(1)}&\tilde{q}_{3}^{(2)}\end{array}\right)=\left(\begin{array}[]{cc}\xi_{5}&\xi_{7}\end{array}\right)$
(55)
This yields the IR superpotential
$\displaystyle W$ $\displaystyle=$ $\displaystyle{\rm
Tr}[\,h\mu_{1}^{2}X+hX\phi_{1}\phi_{2}+h\mu_{1}(\phi_{1}\phi_{6}+\phi_{2}\phi_{7}+\phi_{4}\phi_{8}+\phi_{5}\phi_{9})$
(56) $\displaystyle+$ $\displaystyle
h\mu_{2}^{2}Z+hZ\phi_{4}\phi_{5}+h_{2}Z\xi_{4}\xi_{5}+m_{3}(\xi_{4}\xi_{6}+\xi_{5}\xi_{7})$
$\displaystyle+$ $\displaystyle
h\phi_{1}Y\phi_{4}+h\phi_{2}\tilde{Y}\phi_{5}]$
plus terms that are supersymmetric at two loops, which is the order to which
we study supersymmetry breaking effects in this work. Here we have defined
$h_{2}\equiv\Lambda_{g}/\Lambda_{0}$.
From (22) we have
$m_{3}^{2}\gg h\mu_{1}^{2}\gg h\mu_{2}^{2}$ (57)
and we can integrate out the $q_{3}^{(i)}$ and $\tilde{q}_{3}^{(i)}$ (with
$\langle q_{3}^{(i)}\rangle=\langle\tilde{q}_{3}^{(i)}\rangle=0$). Deep in the
IR we have the usual $W_{1}$ model, with $m_{Z}^{2}<0$ from two-loop quantum
effects. There is also a one-loop contribution that is suppressed
($\mu_{1}\gg\mu_{2}$). In particular, the one- and two-loop $Z$ masses are
$\displaystyle m_{Z}^{(1)\,2}$
$\displaystyle=\epsilon^{4}\frac{h^{2}\mu_{1}^{2}}{24\pi^{2}}N_{F_{2}}\tilde{N}_{c}\left\\{h^{2}+h_{2}^{2}\left(\frac{\mu_{1}}{m_{3}}\right)^{2}\right\\}+\mathcal{O}\left(\epsilon^{6}\right)$
$\displaystyle m_{Z}^{(2)\,2}$
$\displaystyle=\frac{2h^{4}\mu_{1}^{2}}{(16\pi^{2})^{2}}N_{F_{2}}\tilde{N}_{c}\left\\{\log
4-1-\frac{\pi^{2}}{6}+\epsilon^{4}g(\Lambda)\right\\}+\mathcal{O}\left(\epsilon^{6}\right)$
(58)
where $g(\Lambda)$ is a complicated but well-behaved function that depends
on888The dependence on the cutoff in $m_{Z}^{(2)\,2}$ is introduced through
the $Z$-self corrections, and vanishes in the limit that $\mu_{2}\rightarrow
0$. $\log^{2}\Lambda$ and we have defined $\epsilon\equiv\mu_{2}/\mu_{1}$.
The trace in (56) gives a factor of $N_{F_{2}}\tilde{N}_{c}$.
These masses indicate how the potential for $Z$ behaves at the origin.
Clearly, to have a $R$-symmetry breaking minimum, we must have
$m_{Z}^{2}=m_{Z}^{(1)\,2}+m_{Z}^{(2)\,2}<0$. However, there is another
constraint on the parameters in $m_{Z}^{2}$ that comes from the behavior of
the potential for $Z$ in the far field region. Here $\mu_{1}<<\langle
Z\rangle<<\Lambda$; the one- and two-loop contributions to the potential in
this region are in tension with one another, since they are introduced with
opposite signs (cf. (3)), and we must include the effects of the $m_{3}$ mass
terms. Then, to two-loop order,
$\displaystyle V_{{\rm eff}}(Z)$ $\displaystyle=$ $\displaystyle V^{(1)}_{{\rm
eff}}(Z)+V^{(2)}_{{\rm eff}}(Z)$ (59) $\displaystyle=$
$\displaystyle\frac{2\mu_{2}^{4}}{16\pi^{2}}\left(h^{2}+h_{2}^{2}\right)\log\frac{\langle
Z\rangle}{\mu_{2}}$
$\displaystyle\,-\frac{1}{(16\pi^{2})^{2}}\left(4\mu_{1}^{4}h^{4}\log^{2}\frac{\langle
Z\rangle}{\mu_{1}}+2\mu_{2}^{4}\left(h^{2}+h_{2}^{2}\right)^{2}\log^{2}\frac{\langle
Z\rangle}{\mu_{2}}\right)$
up to an unimportant constant. To have a stable $R$-symmetry breaking minimum
in the pseudomodulus $Z$, we require that the slope of the potential far in
field space be positive, so that an intermediate minimum is guaranteed (cf.
Figure 1). This further constricts the allowed values of $\epsilon$, $h$, and
$h_{2}$; however, the allowed parameter space is substantial, depending on the
ratio between $h$ and $h_{2}$, which we define as
$\rho\equiv\frac{h^{2}}{h_{2}^{2}}$. Figure 4 illustrates the allowed values
of $\epsilon$ and $h_{2}$ as a function of the ratio $\rho$.
Figure 4: The parameter space that satisfies the requirements (a) that the
mass of the $Z$ pseudomodulus is tachyonic at the origin and (b) that the
slope of the far field potential be positive. This space is parametrized by
the ratio of scales $\epsilon\equiv\frac{\mu_{2}}{\mu_{1}}$ and the $m_{3}$
sector coupling $h_{2}$ as a function of the ratio
$\rho=\frac{h^{2}}{h_{2}^{2}}$. Note that small values of $\rho$ are
preferred, but not too small. The behavior of the allowed regions is smooth
everywhere.
This model is a _UV completion_ of the former toy model in the sense that it
provides a gauge theory that underlies the model of the chiral fields. This
completion has two sources of tuning, the first being the mass hierarchy that
is necessary to enforce the decoupling of the $W_{2}$ messenger sector in (4).
There is also tuning in the value of $\rho$. According to Figure 4, values
where $\rho\ll 1$999But not too small, as the two-loop effects in (6) would
vanish completely as $\rho\rightarrow 0$! Figure 4 shows that very small
values of $\rho$ are disfavored. are preferred to maximize the available
parameter space. For the lowest value depicted, $\rho=0.0001$, this
corresponds to $h\sim\frac{1}{100}h_{2}$, but there is still appreciable
available parameter space for $h\sim\frac{1}{10}h_{2}$ $(\rho=0.01)$. We also
know that this ratio is related to the scales in (13) and (20),
$\rho=h^{2}\left(\frac{\Lambda_{0}}{\Lambda_{g}}\right)^{2}$. Indeed,
dynamically it is more natural to have
$\frac{1}{h}\frac{\Lambda_{g}}{\Lambda_{0}}=\frac{1}{\sqrt{\rho}}\ll 1$ than
the case preferred here, where $\frac{\Lambda_{g}}{\Lambda_{0}}\gg h$ or
$\rho\ll 1$. For example, if the quartic term in (13) arises from a massive
field that is integrated out at $\Lambda_{0}$, then $\Lambda_{0}$ is roughly
its mass and is generically larger than $\Lambda_{g}$, the duality scale.
The tuning in $\rho$ can be accommodated by assuming that the $h_{2}$ sector
is a generic strongly coupled sector. After integrating out the massive field
associated to, $\Lambda_{0}$ the RG flow reduces the effective
$\Lambda_{0}/\Lambda_{g}$ such that
$\frac{\Lambda_{g}}{\Lambda_{0}(\Lambda_{g})}\gg h$ at the scale
$\Lambda_{g}$, where the flow changes. These ideas are illustrated in Figure
5.
Figure 5: A complete arrangement of the scales introduced into (56) so as to
accomplish spontaneous $R$-symmetry breaking. The ratio
$\rho\sim\Lambda_{0}(\Lambda_{g})/\Lambda_{g}$ is arranged to be smaller than
one via running in a strongly coupled sector between the scales $\Lambda_{0}$
down to $\Lambda_{g}$. The tuning in $\rho$ is actually quite mild- for
appreciable parameter space that allows a spontaneously broken $R$-symmetry,
$\rho$ can be as large as $\frac{1}{100}$ (corresponding to
$h\sim\frac{1}{10}h_{2}$\- see Figure 4).
## 4 Conclusions
We have shown that there exist $R$-symmetric O’Raifeartaigh-like models with
fields having only $R$-charge 0 and 2 that spontaneously break their
$R$-symmetry. The model we examined had two couplings in the superpotential
that exhibited distinct behaviors under their renormalization group flow; in
particular, one of the couplings ($h_{2}$ in (56)) had to be tuned to within
$\sim 10$% by the RG evolution to achieve a spontaneously broken $R$-symmetry.
There are also two scales in the model that were arranged in a hierarchy, with
tuning of order $\sim 10-20$%. The $R$-symmetry is broken by the non-zero vev
of a $R$-charged pseudomodulus in the model that has a potential dictated by
one- and two-loop quantum corrections to its tree-level flat potential. The
parameter space that allows a non-trivial minimum of the potential is
substantial but prefers the tuning in the scales and couplings already
mentioned.
Many extensions of our work are possible. One may look at a brane engineering
of the UV model (or some generalizations), as done in Ooguri:2006bg ;
Franco:2006ht ; Bena:2006rg for the ISS model. It would be interesting to
check if the brane action can capture the physics of the non-supersymmetric
state that we discovered in this field theory.
Because there is tuning in its marginal couplings, a better understanding of
our UV completion is also necessary. A possible explanation of this tuning can
come from the strong dynamics of the UV sector- for example, one can suppose
that the UV dynamics are governed by an approximate CFT that generates a
hierarchy amongst the couplings from their anomalous dimensions, as in
Nelson:2000sn .
One can ponder the possibility of a general result (like in Shih:2007av ) for
the sign of the two-loop masses, possibly associated to some (global) charge
assignment. In the case of $R=0$ and $R=2$ there is no sign constraint on the
mass at two loops, but extra conditions might provide such a constraint (at
least at the origin).
We conclude by discussing the embedding of the model in a phenomenological
scenario. One can imagine gauging some of the global symmetries and gauge
mediating the supersymmetry breaking effects to a SSM sector. This requires
the existence of an explicit $R$-symmetry breaking sector to prevent massless
axions Bagger:1994hh . It would be important to generate the explicit
$R$-symmetry breaking term in the UV theory and study the possible constraints
of such a term on the other couplings.
## Acknowledgments
We are grateful to B. Grinstein, K. Intriligator and A. Mariotti for
discussions and comments. This work was supported by DOE grant DOE-
FG03-97ER40546.
## References
* (1) K. A. Intriligator, N. Seiberg, and D. Shih, Dynamical SUSY breaking in meta-stable vacua, JHEP 0604 (2006) 021, [hep-th/0602239].
* (2) A. Giveon, A. Katz, and Z. Komargodski, On SQCD with massive and massless flavors, JHEP 0806 (2008) 003, [arXiv:0804.1805].
* (3) A. Amariti and A. Mariotti, Two Loop R-Symmetry Breaking, JHEP 0907 (2009) 071, [arXiv:0812.3633].
* (4) K. Intriligator, D. Shih, and M. Sudano, Surveying Pseudomoduli: The Good, the Bad and the Incalculable, JHEP 0903 (2009) 106, [arXiv:0809.3981].
* (5) E. Witten, Mass Hierarchies in Supersymmetric Theories, Phys.Lett. B105 (1981) 267.
* (6) D. Shih, Spontaneous R-symmetry breaking in O’Raifeartaigh models, JHEP 0802 (2008) 091, [hep-th/0703196].
* (7) D. Curtin, Z. Komargodski, D. Shih, and Y. Tsai, Spontaneous R-symmetry Breaking with Multiple Pseudomoduli, Phys.Rev. D85 (2012) 125031, [arXiv:1202.5331].
* (8) S. Nibbelink Groot and T. S. Nyawelo, Two Loop effective Kahler potential of (non-)renormalizable supersymmetric models, JHEP 0601 (2006) 034, [hep-th/0511004].
* (9) H. Ooguri and Y. Ookouchi, Meta-Stable Supersymmetry Breaking Vacua on Intersecting Branes, Phys.Lett. B641 (2006) 323–328, [hep-th/0607183].
* (10) S. Franco, I. Garcia-Etxebarria, and A. M. Uranga, Non-supersymmetric meta-stable vacua from brane configurations, JHEP 0701 (2007) 085, [hep-th/0607218].
* (11) I. Bena, E. Gorbatov, S. Hellerman, N. Seiberg, and D. Shih, A Note on (Meta)stable Brane Configurations in MQCD, JHEP 0611 (2006) 088, [hep-th/0608157].
* (12) A. E. Nelson and M. J. Strassler, Suppressing flavor anarchy, JHEP 0009 (2000) 030, [hep-ph/0006251].
* (13) J. Bagger, E. Poppitz, and L. Randall, The R axion from dynamical supersymmetry breaking, Nucl.Phys. B426 (1994) 3–18, [hep-ph/9405345].
|
arxiv-papers
| 2012-10-10T20:00:00 |
2024-09-04T02:49:36.349262
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antonio Amariti and David Stone",
"submitter": "David Stone",
"url": "https://arxiv.org/abs/1210.3028"
}
|
1210.3171
|
Data Interpolation
An Efficient Sampling Alternative for Big Data Aggregation
by
Hadassa Daltrophe, Shlomi Dolev and Zvi Lotker
Technical Report #13-01
September 2012
###### Abstract
Given a large set of measurement sensor data, in order to identify a simple
function that captures the essence of the data gathered by the sensors, we
suggest representing the data by (spatial) functions, in particular by
polynomials. Given a (sampled) set of values, we interpolate the datapoints to
define a polynomial that would represent the data. The interpolation is
challenging, since in practice the data can be noisy and even Byzantine, where
the Byzantine data represents an adversarial value that is not limited to
being close to the correct measured data. We present two solutions, one that
extends the Welch-Berlekamp technique in the case of multidimensional data,
and copes with discrete noise and Byzantine data, and the other based on Arora
and Khot techniques, extending them in the case of multidimensional noisy and
Byzantine data.
## 1 Introduction
Consider the task of representing information in an error-tolerant way, such
that it can be introduced even if it contains noise or even if the data is
partially corrupted and destroyed. Polynomials are a common venue for such
approximation, where the goal is to find a polynomial $p$ of degree at most
$d$ that would represent the entire data correctly.
Our motivation comes from sensor data aggregation, and the need to extend the
distributed aggregation to distributed interpolation, use sampling to cope
with huge data and anticipate the value of missing data. For example, a sensor
network may interact with the physical environment, while each node in the
network is may sense the surrounding environment (e.g., temperature, humidity
etc). The environmental measured values should be transmitted to a remote
repository or remote server. Note that the environmental values usually
contain noise, and there can be malicious inputs, i.e., part of the data may
be corrupted.
In contrast to distributed data aggregation where the resulting computation is
a function such as COUNT, SUM and AVERAGE (e.g. [16, 9, 13]), in distributed
data interpolation, our goal is to represent every value of the data by a
single (abstracting) function. Our computational model consists of sampling
the sensor network data and estimating the missing information using
polynomial manipulations.
The management of big data systems also gives motivation for the distributed
interpolation method. The abstraction of big data becomes one of the most
important tasks in the presence of the enormous amount of data produced these
days. Communicating and analyzing the entire data does not scale, even when
data aggregation techniques are used. This study suggests a method to
represent the distributing big data by a simple abstract function (such as
polynomial) which will lead to effective use of that data.
We suggest interpolating the big data in the scope of distributed systems by
using local data centers. Each data center samples the data around it and
computes a polynomial that reflects the local data. The local polynomials are
merged to a global one by interpolation in a hierarchical manner. In the
process of calculating the local polynomials noise and Byzantine data samples
are eliminated.
### Basic Definitions.
* •
For multivariate polynomial
$p(\textbf{{x}})\in\mathbb{R}[\textbf{{x}}]=\mathbb{R}[x_{1},...,x_{k}]$ let
$\left\|p\right\|_{\infty}=sup\left\\{\left|p(x_{1},...,x_{k})\right|:x_{1},...,x_{k}\in\mathbb{R}\right\\}$.
* •
A monomial in a collection of variable $x_{i},...,x_{n}$ is a product
$x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{n}^{\alpha_{n}}$
where $\alpha_{i}$ are non-negative integers.
* •
The total degree of a multivariate polynomial $p$ is the maximum degree of any
term in $p$, where the degree of particular term is the sum of the variable
exponents.
* •
A polynomial $q$ is a $\delta$-approximation to $p$ if
$\left\|p-q\right\|_{\infty}\leq\delta$.
### Polynomial Fitting to Noisy and Byzantine Data.
Formally, in this paper, we learn the following problem:
###### Definition 1.1 (Polynomial Fitting to Noisy and Byzantine Data
Problem).
Given a sample $S$ of $k$ dimension datapoints
$\left\\{\left(x_{1_{i}},...,x_{k_{i}}\right)\right\\}^{N}_{i=1}$ and a
function $f$ defined on those points $f(x_{1_{i}},...,x_{k_{i}})=y_{i}$, a
noise parameter $\delta>0$ and a threshold $\rho>0$, we have to find a
polynomial $p$ of total degree $d$ satisfying
$\displaystyle p(x_{1},...,x_{k})\in\left[y-\delta,y+\delta\right]\text{ for
at least $\rho$ fraction of $S$}$ (1)
Generally, we propose the use of polynomials to represent large amounts of
sensor data. The process works by sampling the data and then using this sample
to construct a polynomial whose distance (according to the $\ell_{\infty}$
metric) from the polynomial constructed using the whole data set is small. The
main challenges to this approach are $(i)$ the presence of noise (identified
by the $\delta$ parameter), and $(ii)$ arbitrarily corrupted data (Byzantine
data, denoted by $\rho$) that can cause inaccurate sampling and, thus, lead to
badly constructed polynomials.
Given that the function $f$ is continuous, by the Weierstrass approximation
Theorem [4] we know that for any given $\epsilon>0$, there exists a polynomial
$p^{\prime}$ such that
$\left\|f-p^{\prime}\right\|_{\infty}<\epsilon$ (2)
This can tell us that our desired polynomial $p$ exists (i.e., $p^{\prime}=p$
and $\epsilon=\delta$, satisfying eq.1), and we can relate the data as arising
from polynomial function (i.e., the unknown function $f$ is $d$ degree
polynomial we need to reconstruct), and this is the underlying model assumed
in the paper.
One obvious candidate to construct approximating polynomial is interpolation
at equidistant points. However, the sequence of interpolation polynomials does
not converge uniformly to $f$ for all $f\in C[0,1]$ due to Runge’s phenomenon
[7]. Chebyshev interpolation (i.e., interpolate $f$ using the points defined
by the Chebyshev polynomial) minimizes Runge’s oscillation, but it is not
suffice the polynomial fitting problem presented above (Definition 1.1) due to
the randomly distributed data we have assumed.
Taylor polynomials are also not appropriate; for even setting aside questions
of convergence, they are applicable only to functions that are infinitely
differentiable, and not to all continuous functions.
Another classical polynomial sequence is suggested by S. Bernstein [3] as
constructive proof of the Weierstrass Theorem. Bernstein polynomial:
$B^{f}_{n}(x)=\displaystyle\sum^{n}_{i=0}f\left(\frac{i}{n}\right){n\choose
i}x^{i}(1-x)^{n-i}$ converges uniformly to any continuous function $f$ which
is bounded on $[0,1]$. The formal Berenstein polynomial samples the function
$f$ in an equidistant fashion. To handle a random sample data, we can use
Vitale [21] results which consider that the datapoints $S=x_{1},...,x_{N}$ are
i.i.d observations drawn from an unknown density function $f$. The Bernstein
polynomial estimate of $f$ defined as
$\tilde{B}^{f}_{n}(x)=\frac{n+1}{N}\displaystyle\sum^{n}_{i=0}\mu_{in}^{N}{n\choose
i}x^{i}(1-x)^{n-i}$ where $\mu_{in}^{N}$ is the number of points ($x_{i}$’s)
appear in the interval $[\frac{i}{n+1},\frac{i+1}{n+1}]$. Vitale [21] showed
that $\left\|\tilde{B}^{f}_{n}(x)-f\right\|_{\infty}\leq\epsilon$ for every
given $\epsilon>0$.
Tenbusch [20] extended Vitale’s idea to multidimensional densities, where
there is need to note that those works hold only when the datapoints are i.i.d
observations. Another reason not to use the Bernstien polynomial is the slow
convergence rate (Voronovskaya’s Theorem states that for functions that are
twice differentiable, the rate of convergence is about $1/n$, see Davis [7]).
Considering other classical curve-fitting and approximation theories [17],
most research has used the $\ell_{2}$ norm of noise, such as the method of
least square errors. These attitudes not suffice the adversarial noise we have
assumed here. To our knowledge, only [2] referred the $\ell_{\infty}$ noise
that fits our considered problem and we further relate [2] study.
The polynomial fitting problem as stated in Definition 1.1 can also be studied
by Error-Correcting Code Theory. From that point of view, extensive literature
exists dealing with the noise-free case (i.e., $\delta=0$ and $\rho<1$). In
the next section, we present an algorithm that handles a combination of
discrete noise and Byzantine data based on the Welch-Berlekamp [22] error-
elimination method. Moreover, the fundamental Welch-Berlekamp algorithm treats
only the one-dimension case, where we suggest a means to deal with corrupted-
noisy data appearing at one and multi-dimensional inputs.
Related to unrestricted noise, we refer to the polynomial-fitting problem as
defined by Arora and Khot [2]. Based on their results in Section 3, we
introduce the polynomial fit generalization, where we provide a polynomial-
time algorithm dealing with multi-variate data.
Summarizing, this work provides the following contributions:
* •
We describe an algorithm that constructs a polynomial using the Welch-
Berlekamp (WB) method as a subroutine. The algorithm is tolerated to discrete-
noise and Byzantine data.
* •
We identify how the previous method can be generalized to handle multi-
dimensional data. Moreover, we present a multivariate analogue of the WB
method, under conditions which will be specified.
* •
Using linear programing minimization and the Markov-Bernstein Theorem, we
generalized Arora and Khot algorithm to reconstruct an unknown multi-
dimensional polynomial. Furthermore, we detail the way to eliminate the
Byzantine appearance when such inputs exist.
Those three points stated in the three algorithms presented in the paper. The
first Algorithm handles one-dimensional Byzantine data that contains discrete-
noise. Algorithm 2 generalized the WB idea to deal with multivariate malicious
data. Finally, Algorithm 3 summarized our approach to cope with unrestricted
noise appeared in the (partially corrupted) data.
## 2 Discrete Finite Noise
In this section, we will study a simple aspect of the polynomial fitting
problem posed in Definition 1.1, where the data function is a polynomial, and
we relaxed the noise constraint to be finite and discrete, i.e., the noise
$\delta$ is defined on a finite field $\mathbb{F}_{q}$ containing $q$
elements.
Welch and Berlekamp related the problem of polynomial reconstruction in their
decoding algorithm for Reed Solomon codes [22]. The main idea of their
algorithm is to describe the received (partially corrupted) data as a ratio of
polynomials. Their solution holds for noise-free cases and a limited fraction
of the corrupted data ($\delta=0,\rho>1/2$). Almost $30$ years later, Sudan’s
list decoding algorithm [19] relaxed the Byzantine constraint ($\delta=0,\rho$
can be less than $1/2$) by using bivariate polynomial interpolation. Those
concepts do not hold up well in the noisy case since they use the roots of the
polynomial and the divisibility of one polynomial by other methods that are
problematic for noisy data (as shown in [2], Section $1.2$). Here, we will use
the WB algorithm [22] as a “black box” to obtain an algorithm that handles the
discrete-noise notation of the polynomial-fitting problem.
Given a data set $\left\\{\left(x_{i},y_{i}\right)\right\\}^{N}_{i=1}$ that is
within a distance of $t=\rho N$ from some polynomial $p(x)$ of degree $<d$,
the WB approach to eliminate the irrelevant data is to use the roots of an
object called the error-locating polynomial, $e$. In other words, we want
$e(x_{i})=0$ whenever $p(x_{i})\neq y_{i}$. This is done by defining another
polynomial $q(x)\equiv p(x)e(x)$. To resolve these polynomials we need to
solve the linear system, $q(x_{i})=y_{i}e(x_{i})\text{ for all $i$.}$
Welch and Berlekamp show that $e(x)|q(x)$ and $p(x)$ can be found by the ratio
$p(x)=q(x)/e(x)$ at $O(N^{\omega})$ running time (where $\omega$ is the matrix
multiplication complexity). In Algorithm 1, we are use the WB method as a
subroutine to manage the noisy-corrupted data.
Algorithm 1 Reconstruct the polynomial $p(x)$ representing the true data
0: $S,S^{\prime}\subseteq S,\rho,d,\delta,\Delta={v_{1},...,v_{|S^{\prime}|}}$
$i\leftarrow 0$
repeat
$i\leftarrow i+1$
$S^{\prime}_{i}\leftarrow S^{\prime}+v_{i}$
$p_{i}(x)\leftarrow WB(S^{\prime}_{i},d,\rho)$
until $p_{i}(x_{j})\in[y_{j}-\delta,y_{j}+\delta]$ for at least $\rho$
fraction of $j$’s; $(x_{j},y_{j})\in S-S^{\prime}_{i}$
return $p(x)\leftarrow p_{i}(x)$
Given any sample $S$ such that $\rho$ fraction of $S$ is not corrupted, we
will choose a subset $S^{\prime}\subseteq S$ in a size related to the desired
degree $d$ and $\rho$ (the WB algorithm requires $2t+d$ points, where $t=\rho
N$ is the number of the corrupted points). At every step $i$, we will add
$S^{\prime}$ different values of noise as defined by the set $\Delta$ which
contain all the vectors of length $|S^{\prime}|$ assigned the elements of
$\mathbb{F}_{q}$ in lexicographic order, i.e.,
$\Delta=\left\\{(a_{1},...,a_{|S^{\prime}|}):a_{i}\in\mathbb{F}_{q}\right\\}$.
Now, we can reconstruct the polynomial $p_{i}$ using the WB algorithm. The
resulting polynomial $p_{i}$ is tested by the original dataset $S$, where the
criteria is that $p_{i}$ is within $\delta$ from all nodes but the Byzantine
nodes (according to the maximal number of Byzantine as defined by $\rho$).
Since we assume a discrete finite noise ($\delta\in\mathbb{F}_{q}$), for each
datapoint at the subset $S^{\prime}$ (of size $O(d+\rho|S|)$), there is a
possibly of $q$ values (where $q$ is a constant). Thus, in the worst case,
when we run the WB polynomial algorithm for every possible value, it will cost
$poly(d+N)$ time.
Note that if the desired polynomial’s degree $d$ is not given, we can search
for the minimal degree of a polynomial that fits the $\delta$ and number of
Byzantine node restrictions in a binary search fashion.
### Multidimensional Data.
To generalize the former algorithm to handle multidimensional data, there is
need to formalize the WB algorithm to deal with multivariate polynomials. This
is a challenging task due to the infinite roots those polynomials may have
(and as previously mentioned, the WB method is strictly based on the
polynomials’ roots).
A suggested method to handle $3$-dimensional data is to assume that the values
of datapoints in one direction (e.g., x-direction) are distinct. This can be
achieved by assuming the inputs
$S=(x_{1},y_{1},f(x_{1},y_{1}))...,(x_{N},y_{N},f(x_{N},y_{N}))$ are i.i.d
observations. Moreover, we allow the malicious authority to change the
observation input but not its distribution (i.e., to determine
$z_{i}=f(x_{i},y_{i})$ value only). This assumption forces the data to have
different $x_{i}$’s values, which help us to define the error locating
polynomial $e$ in the x-direction only (or symmetrically over the y-axis).
The 3-dimensional polynomial reconstruction is described in Algorithm 2.
Algorithm 2 Reconstruct the polynomial $p(x,y)$ representing the true data
* •
Input: $0<t=\rho N$ which is the Byzantine appearance bound, the total degree
$d>1$ of the goal polynomial and $N$ triples ${(x_{i},y_{i},z_{i})}_{i=1}^{N}$
with distinct $x_{i}$’s .
* •
Output: Polynomial $p(x,y)$ of total degree at most $d$ or fail.
* •
Step 1: Compute a non-zero univariate polynomial $e(x)$ of degree exactly $t$
and a bivariate polynomial $q(x,y)$ of total degree $d+t$ such that:
$\displaystyle z_{i}e(x_{i})=q(x_{i},y_{i})$ $\displaystyle 1\leq i\leq N$ (3)
If such polynomials do not exist, output fail.
* •
Step 2: If $e$ does not divide $q$, output fail, else compute
$p(x,y)=\displaystyle\frac{q(x,y)}{e(x)}$. If
$\Delta(z_{i},p(x_{i},y_{i})_{i})>t$, output fail. else output $p(x,y)$.
###### Theorem 2.1.
Let $p$ be an unknown $d$ total degree polynomial with two variables. Given a
threshold $\rho>0$ and a sample $S$ of $N={d+t+m\choose d+m}+t$ ($t=\rho N$)
random points ${(x_{i},y_{i},z_{i})}_{i=1}^{N}$ such that
$z_{i}=p(x_{i},y_{i})\text{ for at least $\rho$ fraction of $S$.}$
The algorithm above reconstructs $p$ at $O(N^{\omega})$ running time (where
$\omega$ is the matrix multiplication complexity).
###### Proof.
The proof of the Theorem above follows from the subsequent claims.
###### Claim 2.2 (Correctness).
There exist a pair of polynomials $e(x)$ and $q(x,y)$ that satisfy Step 1 such
that $q(x,y)=p(x,y)e(x)$.
###### Proof.
Taking the error locator polynomial $e(x)$ and $q(x,y)=p(x,y)e(x)$, where
$deg(q)\leq deg(p)+deg(e)\leq t+d$. By definition, $e(x)$ is a degree $t$
polynomial with the following property:
$e(x)=0\text{ iff }z_{i}\neq p(x,y)$
We now argue that $e(x)$ and $q(x,y)$ satisfy eq. 3. Note that if
$e(x_{i})=0$, then $q(x_{i},y_{i})=z_{i}e(x_{i})=0$. When $e(x_{i})\neq 0$, we
know $p(x_{i},y_{i})=z_{i}$ and so we still have
$p(x_{i},y_{i})e(x_{i})=z_{i}e(x_{i})$, as desired. ∎
###### Claim 2.3 (Uniqueness).
If any two distinct solutions $(q_{1}(x,y),e_{1}(x))\neq(q_{2}(x,y),e_{2}(x))$
satisfy Step 1, then they will satisfy
$\displaystyle\frac{q_{1}(x,y)}{e_{1}(x)}=\frac{q_{2}(x,y)}{e_{2}(x)}$.
###### Proof.
It suffices us to prove that $q_{1}(x,y)e_{2}(x)=(q_{2}(x,y)e_{1}(x)$.
Multiply this with $z_{i}$ and substitute $x,y$ with $x_{i},y_{i}$,
respectively,
$q_{1}(x_{i},y_{i})e_{2}(x_{i})z_{i}=q_{2}(x_{i},y_{i})e_{1}(x_{i})z_{i}$
We know, $\forall i\in[N]$ $q_{1}(x_{i},y_{i})=e_{1}(x_{i})z_{i}$ and
$q_{2}(x_{i},y_{i})=e_{2}(x_{i})z_{i}$ If $z_{i}=0$, then we are done.
Otherwise, if $z_{i}\neq 0$, then
$q_{1}(x_{i},y_{i})=0,q_{(}x_{i},y_{i})=0\Rightarrow
q_{1}(x,y)e_{2}(x)=(q_{2}(x,y)e_{1}(x)$ as desired. ∎
###### Claim 2.4 (Time complexity).
Given $N=t+{d+t+2\choose d+t}$ data samples, we can reconstruct $p(x,y)$ using
$O(N^{\omega})$ running time.
###### Proof.
Generally, for $m$ variate polynomial with degree $d$, there are ${d+m\choose
d}$ terms [18]; thus, it is a necessary condition that we have
$t+{d+t+2\choose d+t}$ distinct points for $q$ and $e$ to be uniquely defined.
We have $N$ linear equation in at most $N$ variables, which we can solve e.g.,
by Gaussian elimination in time $O(N^{\omega})$ (where $\omega$ is the matrix
multiplication complexity).
Finally, Step 2 can be implemented in time $O(NlogN)$ by long division [1].
Note that the general problem of deciding whether one multivariate polynomial
divides another is related to computational algebraic geometry (specifically,
this can be done using the Gröbner base). However, since the divider is a
univariate polynomial, we can mimic long division, where we consider $x$ to be
the “variable” and $y$ to just be some “number.” ∎
∎
Example 1. Suppose the unknown polynomial is $p(x,y)=x+y$. Given the
parameters: $d=1$ (degree of $p$), $m=2$ (number of variable at $p$) and $t=1$
(number of corrupted inputs) and the set of $t+{d+t+2\choose d+t}=7$ points:
(1,2,2),(2,2,4),(6,1,7),(4,3,7),(8,2,0),(9,1,10),(3,7,10)
that lie on $z=p(x,y)$. Following the algorithm, we define:
$deg(e)=1,deg(q)=2$ and
$q_{i}=\alpha_{1}x_{i}^{2}+\alpha_{2}x_{i}y_{i}+\alpha_{3}y_{i}^{2}+\alpha_{4}x_{i}+\alpha_{5}y_{i}+\alpha_{6}=z_{i}(x_{i}+\alpha_{7})$
for coefficients $\alpha_{1},...,\alpha_{6},\beta$ and $1\leq i\leq 12$. Note
that we force $e(x)$ not to be the zero polynomial by define it to be monic
(i.e., the leading coefficient equals to 1). Thus, we derive the linear
system:
$\displaystyle\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}=2\beta+2$
$\displaystyle
4\alpha_{1}+4\alpha_{2}+4\alpha_{3}+2\alpha_{4}+2\alpha_{5}+\alpha_{6}=4\beta+8$
$\displaystyle
36\alpha_{1}+6\alpha_{2}+\alpha_{3}+6\alpha_{4}+\alpha_{5}+\alpha_{6}=7\beta+42$
$\displaystyle
16\alpha_{1}+12\alpha_{2}+9\alpha_{3}+4\alpha_{4}+3\alpha_{5}+\alpha_{6}=7\beta+28$
$\displaystyle
64\alpha_{1}+16\alpha_{2}+4\alpha_{3}+8\alpha_{4}+2\alpha_{5}+\alpha_{6}=0$
$\displaystyle
81\alpha_{1}+9\alpha_{2}+\alpha_{3}+9\alpha_{4}+\alpha_{5}+\alpha_{6}=10\beta+90$
$\displaystyle
9\alpha_{1}+21\alpha_{2}+49\alpha_{3}+3\alpha_{4}+7\alpha_{5}+\alpha_{6}=10\beta+30$
Solving the system gives: $q(x,y)=x^{2}+xy-8x-8y$ and $e(x)=x-8$. Dividing
those polynomials, we get the expected solution: $q(x,y)/e(x)=p(x)=x+y$.
###### Corollary 2.5 (Multivariate Polynomial Reconstruction).
Let $p$ be an unknown $d$ total degree polynomial with $m$ variable. Given a
threshold $\rho>0$, a noise parameter $\delta$ and a sample $S$ of $N$ random
points ${(x_{1_{i}},...,x_{m_{i}},y_{i})}_{i=1}^{N}$ such that
$y_{i}\in[p(x_{1_{i}},...,x_{m_{i}})-\delta,p(x_{1_{i}},...,x_{m_{i}})+\delta]\text{
for at least $\rho$ fraction of $S$}$
$p$ can be reconstructed using $N={d+m+\rho m\choose d+m}+\rho m$ datapoints.
###### Proof.
Following Theorem 2.1, we can rewrite its proof for the multidimensional
generalization. An interesting question is if there is any advantage to define
the error-location polynomial $e$ to be multi-variate (instead of univariate
as we previously presented). One “advantage” may be decreasing the required
number of datapoints or improvement of the complexity. The size of the input
data is strictly defined by the given bound on the corrupted data ($t=\rho N$)
and the goal polynomial degree ($d$). Thus, there is no sense if the unknown
coefficient comes from the highly degree polynomial or from the highly
dimension polynomial, i.e., both options require the same size of sample, as
illustrated in the Appendix.
Related to the complexity change, when the error-locating polynomial is
multivariate, Step 2 of Algorithm 2 is more challenging since it contains
multivariate polynomial division. A related reference is [10] which is the
most efficient implementation for the computation of Gröbner bases relies on
linear algebra. Using Gröbner bases we can implement the division at close to
$O(NlogN)$ time, as done in Algorithm 2.
To finish the proof, there is the need to explain how to deal with the noise.
Since we assume only discrete noise, we can dismiss it using the method
illustrated at Algorithm 1. We consistently insert a vector of possible noise
and try to reconstruct the polynomial using Algorithm 2. ∎
## 3 Random Sample with Unrestricted Noise
Motivated by applications in vision, Arora and Khot [2] studied the univariate
polynomial fitting to noisy data using $O(d^{2})$ datapoints, where $d$ is the
polynomial degree. In this part, we generalized their results to
$k$-dimensional data.
Since our motivation comes from sensor planar aggregation, we will focus on
bivariate polynomial reconstruction, where the multivariate proof is
symmetric. We assume by rescaling the data that each
$x_{i},y_{i},f(x_{i},y_{i})\in[-1,1]$. Allowing small noise at every point and
large noise occasionally then there may be too many polynomials agreeing with
the given data. Thus, given the noise parameter $\delta$, our goal is to find
a polynomial $p$ that is a $\delta$-approximation of $f$, i.e., $p$ is
$\delta$-close in $\ell_{\infty}$ norm to the unknown polynomial.
Let $I$ be a set of $d^{5}$ equally spaced points that cover the interval
$[-1,1]$. Given the random sample $S\subset
I,|S|=\frac{d^{2}}{\delta}log(\frac{d}{\delta})$, we approach the
reconstruction problem by defining a linear programming system with the
fitting polynomial as its solution. To incorporate the constraint that the
unknown polynomial must take values of $[-1,1]$, we move to Chebyshev’s
representation of the polynomial. Thus, each of its coefficients is at most
$\sqrt{2}$ (see eq. 5). We represent Chebyshev’s polynomial by
$T_{i}(\cdot),T_{j}(\cdot)$, and the variables $c_{ij}$ at the system are the
Chebyshev coefficients. We construct the LP:
$\displaystyle\text{minimize }\delta$ $\displaystyle s.t.$ $\displaystyle
f(x_{k},y_{k})-\delta\leq\sum_{i}^{n}{\sum_{j}^{m}{c_{ij}T_{i}(x_{k})T_{j}(y_{k})}}\leq
f(x_{k},y_{k})+\delta,$ $\displaystyle k=1,...,|S|$ (4)
$\displaystyle|c_{ij}|\leq\sqrt{2},$ $\displaystyle{i=1,...,n;j=1,...,m}$ (5)
$\displaystyle|\sum_{i}^{n}{\sum_{j}^{m}{c_{ij}T_{i}(x)T_{j}(y)}}|\leq 1,$
$\displaystyle\forall_{x,y}\in I$ (6)
The following Theorem presents our main result for solving the polynomial
fitting problem:
###### Theorem 3.1.
Let $f$ be an unknown $d$ total degree polynomial with two variables, such
that $f(x,y)\in[-1,1]$ when $x,y\in[-1,1]$. Given a noise parameter
$\delta>0$, a threshold $\rho>0$, a constant $c>0$ (dependent on the dimension
of the data) and a sample $S$ of
$O(\frac{d^{2}}{\delta}log(\frac{d}{\delta}))$ random points
$x_{i},y_{i},z_{i}\in[-1,1]$ such that
$z_{i}\in[f(x_{i},y_{i})-\delta,f(x_{i},y_{i})+\delta]$ for at least $\rho$
fraction of $S$. With probability at least $\frac{1}{2}$ (over the choice of
$S$), any feasible solution $p$ to the above LP is $c\delta$-approximation of
$f$.
###### Proof.
For our proof, we need Bernstein-Markov inequality which we state below.
###### Theorem 3.2.
(Bernstein-Markov [8]) For a polynomial $P_{d}$ of total degree $d$, a
direction $\xi$ and a bounded convex set $A\subset R^{k}$
$\displaystyle\left\|\displaystyle\frac{\partial}{\partial\xi}P_{d}\right\|_{\infty}\leq
c_{\text{\tiny{A}}}d^{2}\left\|P_{d}\right\|_{\infty}$ (7)
where $c_{\text{\tiny{A}}}$ is independent of $d$ (and dependent on the
geometric structure of A).
Let $p=p(x,y)$ be the $d=n+m$ total degree polynomial obtained from taking any
solution to the above LP. We know $p$ exists, i.e., the LP is feasible because
the coefficient of $f$ satisfies it.
Note that although the LP constraint that
$f-\delta\leq p\leq f+\delta$
it is NOT immediate that
$\left\|f-p\right\|_{\infty}\leq\delta$
(i.e., $p$ is the $\delta$-approximation of $f$) since eq. 4 stands for the
discrete sample of $S$, where here we need to prove the $\delta$-approximation
in the continuous state.
###### Claim 3.3.
$\left\|p\right\|_{\infty}\leq 1+O(\frac{n^{3}m+m^{3}n}{|I|})$.
Since $\left\|T_{i}(x)\right\|_{\infty}=\left\|T_{j}(y)\right\|_{\infty}=1$,
from Bernstein-Markov (Theorem 3.2), we get $|T^{\prime}_{i}(x)|=O(i^{2})$.
Thus:
$\displaystyle\left|p^{\prime}_{x}\right|\leq\sum_{i}^{n}{\sum_{j}^{m}|{a_{ij}|\Big{(}T_{i}(x)T_{j}(y)}}\Big{)}^{\prime}_{x}$
$\displaystyle\leq\sum_{i}^{n}{\sum_{j}^{m}\sqrt{2}O(i^{2})}\leq O(n^{3}m)$
From symmetric consideration, we get:
$\displaystyle\left|p^{\prime}_{y}\right|\leq O(nm^{3})$
By construction, $p$ takes all values in $[-1,1]$ for all points in $I$, and
the distance (in $x$ direction or $y$ direction) between successive points of
$I$ is $2/|I|$ ($I$ is equidistant). The claim follows from the fact that the
derivative $p^{\prime}$ by definition gives the rate of change in $p$:
$\left\|p\right\|_{\infty}$ is between two successive points of $I$ that their
values $=1$ where the possible change at the interval of length $2/|I|$ is
$p^{\prime}=O(n^{3}m+m^{3}n)$.
###### Claim 3.4.
$\left\|p^{\prime}_{x}\right\|_{\infty},\left\|p^{\prime}_{y}\right\|_{\infty}\leq
O((n+m)^{2})$.
This follows from Bernstein-Markov (Theorem 3.2) and the estimate
$\left\|p\right\|_{\infty}=1+O(1)$.
Let $\epsilon$ denote the largest distance between two successive points out
of $(x_{1},y_{1}),...,(x_{|S|},y_{|S|})$. Every interval of size $\epsilon$
contains at least one of the datapoints (forming $\epsilon$-net). With high
probability, $\epsilon=O(log|S|/|S|)=O(\frac{\delta}{(n+m)^{2}})$. Now, $p$
and $f$ are functions satisfying
$\left\|p^{\prime}_{x}\right\|_{\infty},\left\|f^{\prime}_{x}\right\|_{\infty},\left\|p^{\prime}_{y}\right\|_{\infty},\left\|f^{\prime}_{y}\right\|_{\infty}\leq
O((n+m)^{2})$; hence,
$\left\|(p-f)^{\prime}_{x}\right\|_{\infty},\left\|(p-f)^{\prime}_{y}\right\|_{\infty}\leq
O((n+m)^{2})$. Due to the LP constraint, $p,f$ differs by at most $\delta$ on
the points in the $\epsilon$-net, so we get
$\displaystyle\left\|(p-f)\right\|_{\infty}\leq
2\delta+O(\epsilon(n+m)^{2})=c\delta$ (10)
which is the finished proof of Theorem 3.1; ∎
Remarks:
* •
If we know that the derivative is bounded by $\Delta$ (i.e.,
$f^{\prime}_{x},f^{\prime}_{y}\leq\Delta$), the above proof that gives us
$\frac{\Delta}{\delta}{log\frac{\Delta}{\delta}}$ points is sufficient.
* •
The Bernstein-Markov Theorem 3.2 also holds for multivariate trigonometric
polynomials (see [8]), thus, we can generalize the above proof also for this
class of function. This generalization is important in the scope of wireless
sensor networks since the use of trigonometric function is the appropriate way
to represent the sensor data behavior (e.g., temperature).
* •
The presented method holds only when we assume equidistance or random sampling
(as opposed to Section 2 that handles any given sample). Otherwise, when the
dataset is dense, since we allow $\delta$ perturbation of the data, it can
cause a sharp slope in the resulting function although the original data is
close to the constant at the sampling interval.
###### Corollary 3.5.
Given the set $S$ of $O(\frac{d^{2}}{\delta}log(\frac{d}{\delta}))$
$k$-dimensional random datapoints and a constant $c(S,k)$ dependent only on
the geometry and the dimension of the data, we can reconstruct the unknown
polynomial within $c(S,k)\delta$ error in $\ell_{\infty}$ norm with high
probability over the choice of the sample.
###### Proof.
The two-dimensional proof holds for the general dimension, where the
approximation accuracy dependent on the constant $c(S,k)$ comes from Theorem
3.2. This constant is independent of the polynomial degree, but dependent on
the set of the data points (see [8]). Note that $c(S,k)$ increases
exponentially when increasing the dimension. ∎
### Byzantine Elimination.
Arora and Khot [2] do not deal with Byzantine inputs; however, the method they
presented in Section $6$ can be rewritten to eliminate corrupted data such
that the input datapoints will contain only true (but noisy) values.
Assume that $\rho$ fraction of the data is uncorrupted. For any point
$x_{i},y_{i}\in[-1,1]$, consider a small square-interval
$\Lambda=[x_{i}-\frac{\delta}{d^{3}},x_{i}+\frac{\delta}{d^{3}}]\times[y_{i}-\frac{\delta}{d^{3}},y_{i}+\frac{\delta}{d^{3}}]$
(where $d$ is the total degree of the polynomial we need to find). For a
sample of $d^{4}\frac{log(1/\delta)}{\delta}$ points, with high probability
$\Omega(log(d))$ of the samples lie in this square. We are given that $\rho$
fraction of these sample points gives an approximate value of
$f(x_{i},y_{i})$, i.e., the correct value lies in the interval
$[f(x_{i},y_{i})-\delta,f(x_{i},y_{i})+\delta]$ and the rest of the sample is
corrupted and, thus, is NOT in
$[f(x_{i},y_{i})-\delta,f(x_{i},y_{i})+\delta]$. As shown in Claim 3.4, the
derivatives are bounded by $O(d^{2})$; thus, the value of the polynomial is
essentially constant over $\Lambda$. Hence, at least $\rho$ fraction of the
values seen in this square will lie in
$[f(x_{i},y_{i})-\delta,f(x_{i},y_{i})+\delta]$ and the rest is irrelevant
corrupted data. Thus, at every point $(x_{i},y_{i})$, we can reconstruct
$f(x_{i},y_{i})$. The sample is large enough so that we can reconstruct the
values of the polynomial at say, $d^{2}/\delta$ equally spaced points. Now,
applying the techniques presented in Section 3 enables us to recover the
polynomial.
### Reconstructing the Multivariate Polynomial.
To conclude this section, we summarize the presented results in Algorithm 3:
Algorithm 3 Reconstruct the polynomial $p(x,y)$ representing the true data
0: $S,\rho,d,\delta$
$S^{\prime}\leftarrow\emptyset$
$i\leftarrow 1$
repeat
$\Lambda=[x_{i}-\frac{\delta}{d^{3}},x_{i}+\frac{\delta}{d^{3}}]\times[y_{i}-\frac{\delta}{d^{3}},y_{i}+\frac{\delta}{d^{3}}]$
$c\leftarrow\frac{z_{1}+...+z_{k}}{k},z_{j}:(x_{j},y_{i})\in\Lambda$
$S^{\prime}\leftarrow\left\\{(x_{j},y_{j},z_{j})|(x_{j},y_{j})\in\Lambda\wedge
z_{j}\approx c\right\\}$
$i\leftarrow i+1$
until $|S^{\prime}|>\frac{d^{2}}{\delta}$
$p(x,y)\leftarrow$LP minimization (Equations 4-6) on the set $S^{\prime}$
return $p(x,y)$
The algorithm requires the dataset $S$, the true-data fraction $\rho$, the
total degree of the expected polynomial $d$ and the noise parameter $\delta$.
In the first phase, we eliminate the Byzantine occurrence, as described in the
former subsection. Assuming the given data lie in $[-1,1]\times[-1,1]$ (or
translate to that interval), for the points in $S$, we are looking at the
$\frac{\delta}{d^{3}}$-close interval and choose all the points that have
constant value at this interval (this is done by the average operation). We
repeat this process until we collect enough true-datapoints, i.e., at least
$\frac{d^{2}}{\delta}$ points. This set (sign as $S^{\prime}$ in the
algorithm) is the input for the linear-programming equations which finally
give us the expected polynomial as proof at Theorem 3.1.
## 4 Conclusions
We have presented the concept of data interpolation in the scope of sensor
data aggregation and representation, as well as the new big data challenge,
where abstraction of the data is essential in order to understand the
semantics and usefulness of the data. Interestingly, we found that classical
techniques used in numeric analysis and function approximation, such as the
Welsh-Berlekamp efficient removal of corrupted data, Arora Khot and the like,
relate to the data interpolation problem. Since the sensor aggregation task is
usually a collection of inputs from spatial sensors, for the first time we
have extended existing classical techniques for the case of three or even more
function dimensions, finding polynomials that approximate the data in the
presence of noise and limited portion of completely corrupted data.
We believe that the mathematical techniques we have presented have
applications beyond the scope of sensor data collection or big data, in
addition to being an interesting problem that lies between the fields of
error-correcting and the classical theory of approximation and curve fitting.
Throughout the research we have distinguished two different measures for the
polynomial fitting to the Byzantine noisy data problem: the first being the
Welsh-Berlekamp generalization for discrete-noise multidimensional data and
the second being the linear-programming evaluation for multivariate
polynomials.
Approached by the error-correcting code methods, we have suggested a way to
represent a noisy-malicious input with a multivariate polynomial. This method
assumes that the noise is discrete. When the noise is unrestricted, based on
Bernstein-Markov Theorem and Arora & Khot algorithm, we have suggested a
method to reconstruct algebraic or trigonometric polynomial that traverses
$\rho$ fraction of the the noisy multidimensional data.
We suggest to use polynomial to represent the abstract data since polynomial
is dense in the function space on bounded domains (i.e., they can approximate
other functions arbitrarily well) and have a simple and compact representation
as oppose to spline e.g., [11] or others image processing methods.
Directions for further investigation might include the use of interval
computation for representing the noisy data with interval polynomials.
## References
* [1] A. V. Aho, J. E. Hopcroft and J. D. Ullman, “The Design and Analysis of Computer Algorithms”, Addison-Wesley Publishing Company, 8, 1974.
* [2] S. Arora, and S. Khot,“Fitting algebraic curves to noisy data”, STOC, pp. 162-169, 2002.
* [3] S. Bernstein, “Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities”, Communications of the Kharkov Mathematical Society, 2(13) pp. 1-2, 1912-1913.
* [4] E. Bishop “A generalization of the Stone Weierstrass theorem”, Pacific Journal of Mathematics 11 (3) pp. 777 783, 1961.
* [5] C. W. Clenshow, and J. G. Hayes,“Curve and Surface Fitting”, J. Inst. Maths Applics, 1, pp. 164-183, 1965.
* [6] C. de Boor, A. Ron, “Computational aspects of polynomial interpolation in several variables”, Math. Comp., 58 pp. 705-727, 1992.
* [7] P. J. Davis, “Interpolation and approximation”, Dover, 1975.
* [8] Z. Ditzian ,“Multivariate Bernstein and Markov inequalities”, Journal of Approximation Theory, 70(3) pp. 273-283, 1992.
* [9] E. Fasolo, M. Rossi, J. Widmer, M. Zorzi, “In-network aggregation techniques for wireless sensor networks: a survey”, IEEE Wireless Commun., 14 (2) pp. 70 87, 2007.
* [10] J. C. Faug re, “A new efficient algorithm for computing Gröbner base”, Journal of Pure and Applied Algebra, 139(1 3) pp. 61 88, 1999.
* [11] N. Guenther, “Approximation by spline functions”, Springer, Berlin, 1989.
* [12] M. Hilbert and P. Lepez, “The World’s Technological Capacity to Store, Communicate, and Compute Information”, Science, 332(6025) pp. 60-65 2011.
* [13] P. Jesus, C. Baquero and P. S. Almeida, “A Survey of Distributed Data Aggregation Algorithms”, CoRR, abs/1110.0725, 2011.
* [14] E. H. Kingsley, “Bernstein polynomials for functions of two variables of class $C^{(k)}$”, Proceedings of the American Mathematical Society, pp. 64-71, 1951.
* [15] C. Lynch, “How do your data grow?”, Nature, 455 pp. 28-29, 2008.
* [16] R. Rajagopalan and P.K. Varshney, “Data aggregation techniques in sensor networks: a survey”, IEEE Commun. Surveys Tutorials, 8 (4), 2006\.
* [17] T. J. Rivlin, “An introduction to the approximation of function”, Blaisdell publishing company , 1969.
* [18] K. Saniee, “A Simple Expression for Multivariate Lagrange Interpolation”, SIAM Undergraduate Research Online, 1(1) ,2008.
* [19] M. Sudan, “Decoding of Reed Solomon codes beyond the error-correction bound”, Journal of Complexity, 13(1), pp.180-193 1997.
* [20] A. Tenbusch, “Two-dimensional Bernstein polynomial density estimators”, Metrika, 41(1), pp.233-253 1994.
* [21] R. A. Vitale, “A Bernstein polynomial approach to density estimation”, Statistical Inference and Related Topics, 2, pp.87-100 1975.
* [22] L. R. Welch, and E. R. Berlekamp, “Error correction for algebraic block codes”, US Patent 4 633 470, 1986.
## Appendix A Appendix
Given that the goal unknown polynomial $p$ has $m=2$ variable, $deg(p)=1$ and
that the data contain $t=2$ Byzantine appearance, we can define the error-
correcting polynomial $e$ to be univariate polynomial, $deg(e)=2$ and get the
linear equation:
$\alpha_{1}x^{3}+\alpha_{2}x^{2}y+\alpha_{3}xy^{2}+\alpha_{4}y^{3}+\alpha_{5}x^{2}+\alpha_{6}xy+\alpha_{7}y^{2}+\alpha_{8}x+\alpha_{9}y+\alpha_{10}=z(x^{2}+\beta_{1}x+\beta_{2})$
(11)
when substitute the given data
(1,2,2),(-2,6,0;),(2,2,4),(6,1,7),(4,3,7),(9,1,10),(3,7,10),(5,7,12),(7,4,11),(10,3,13),(11,2,13),(12,4,16)
at (eq.11) we get:
$\displaystyle q_{1}(x,y)=xy-2y-2x+x^{2}y+x^{2}+x^{3}$ $\displaystyle
e_{1}(x)=x^{2}+x-2$
Or, by defining $e$ to be bivariate polynomial,$deg(e)=1$:
$\alpha_{1}x^{3}+\alpha_{2}x^{2}y+\alpha_{3}xy^{2}+\alpha_{4}y^{3}+\alpha_{5}x^{2}+\alpha_{6}xy+\alpha_{7}y^{2}+\alpha_{8}x+\alpha_{9}y+\alpha_{10}=z(x+\beta_{1}y+\beta_{2})$
(12)
which its solution is:
$\displaystyle q_{2}(x,y)=x^{2}+7xy/4-5x/2+3y^{2}/4-5y/2$ $\displaystyle
e_{2}(x,y)=x+3y/4-5/2$
At both cases the polynomial devision result is equals and gives the expected
solution:
$q_{1}(x,y)/e_{1}(x,y)=q_{2}(x,y)/e_{2}(x,y)=x+y=p(x).$ (13)
|
arxiv-papers
| 2012-10-11T10:25:44 |
2024-09-04T02:49:36.361699
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hadassa Daltrophe, Shlomi Dolev and Zvi Lotker",
"submitter": "Hadassa Daltrophe",
"url": "https://arxiv.org/abs/1210.3171"
}
|
1210.3193
|
# On moments of a polytope
Nick Gravin School of Physical and Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, 637371 Singapore
ngravin@pmail.ntu.edu.sg , Dmitrii V. Pasechnik School of Physical and
Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link,
637371 Singapore dima@ntu.edu.sg , Boris Shapiro Department of Mathematics,
Stockholm University, SE-106 91 Stockholm, Sweden shapiro@math.su.se and
Michael Shapiro Department of Mathematics, Michigan State University, East
Lansing, MI 48824-1027, USA mshapiro@math.msu.edu To the memory of Mikael
Passare
###### Abstract.
We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a
compact polytope $\mathcal{P}\subset\mathbb{R}^{d}$ is a rational function.
Its denominator is the product of linear forms dual to the vertices of
$\mathcal{P}$ raised to the power equal to the degree of the density function.
Using this, we solve the inverse moment problem for the set of, not
necessarily convex, polytopes having a given set $S$ of vertices. Under a weak
non-degeneracy assumption we also show that the uniform measure supported on
any such polytope is a linear combination of uniform measures supported on
simplices with vertices in $S$.
###### Key words and phrases:
moments of a polytope, generating function
###### 2010 Mathematics Subject Classification:
Primary 44A60; Secondary 31B20
## 1\. Introduction
The initial motivation for the present paper came from proposed in [GLPR]
efficient algorithm recovering an arbitrary convex polytope from axial moments
of a polynomial measure supported on it. This algorithm is based on the
formulas for the axial moments of polytopes found over 20 years ago
independently by M. Brion, J. Lawrence, A. Khovanskii-A. Pukhlikov, and A.
Barvinok [Bri, La, MR1190788, Bar2], see [MR2455889, BR] for accessible
explanation. In [GLPR] the authors made an essential, although implicit, use
of a univariate rational generating function for appropriately normalized
axial moments. Here a multivariate, and explicit, analog of the latter
function is developed. It turns out it provides a very convenient encoding of
non-convex polytopes, which is of independent interest. E.g. it leads to a
natural definition of _vertices_ of such non-convex polytopes, which have
similar properties to vertices of convex polytopes. It also allows to find the
_exact solutions_ of a class of inverse moment problems on non-convex
polytopes.
###### Notation.
In what follows we shall always assume that $\mathbb{R}^{d}$ is endowed with a
fixed coordinate system $(x_{1},...,x_{d})$, orthonormal with respect to the
standard scalar product $\langle\cdot,\cdot\rangle$. Let $\mu$ be a finite
complex-valued Borel measure in $\mathbb{R}^{d}$. (For standard measure-
theoretic notions we follow [MR924157].) Given a multiindex
$I=(i_{1},\dots,i_{d})$, let $\mathbf{x}^{I}$ be the shorthand of the monomial
$x_{1}^{i_{1}}\dots x_{d}^{i_{d}}$ and $|I|$ the shorthand for
$i_{1}+\dots+i_{d}$. For any multiindex $I$, define the moment $m_{I}(\mu)$ of
$\mu$ as
$m_{I}(\mu):=\int_{\mathbb{R}^{d}}x_{1}^{i_{1}}x_{2}^{i_{2}}...x_{d}^{i_{d}}d\mu(x_{1},x_{2},\ldots,x_{d})=\int_{\mathbb{R}^{d}}{\mathbf{x}}^{I}d\mu(\mathbf{x}).$
(1.1)
Define the normalized moment generating function
$F_{\mu}(\mathbf{u})=F_{\mu}(u_{1},\dots,u_{d})$ of $\mu$ by
$F_{\mu}(\mathbf{u}):=\sum_{I:=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d)!}{i_{1}!\cdots
i_{d}!}m_{I}(\mu)\mathbf{u}^{I},\quad\text{where
$\mathbf{u}^{I}=u_{1}^{i_{1}}\dots u_{d}^{i_{d}}$.}$ (1.2)
Note that $F_{\mu}(\mathbf{u})$ admits the integral representation
$F_{\mu}(\mathbf{u})=d!\int_{\mathbb{R}^{d}}\frac{d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+1}},$
(1.3)
which is a special case of a _Fantappiè transformation_. For details on the
latter, see e.g. [APS]*Chapter 3. A proof of (1.3) will be given at the end of
Section 2; see also Remark 10.
Given any complex-valued finite measure $\mu$ and any degree $\delta$
homogeneous $d$-variate polynomial $\rho$, it is convenient to define the
(re)normalized moment generating function $F_{\mu}^{\rho}(\mathbf{u})$ for the
measure $\rho\mu$, where by definition,
$\int_{\mathbb{R}^{d}}fd(\rho\mu)=\int_{\mathbb{R}^{d}}f\rho d\mu$, in such a
way that it can be obtained from $F_{\mu}(u)$ by application of the
differential operator $\rho\left(\frac{\partial}{\partial\mathbf{u}}\right)$.
Namely, set
$F_{\mu}^{\rho}(\mathbf{u}):=\sum_{I:=(i_{1},...,i_{d})\geq
0}\frac{(|I|+d+\delta)!}{i_{1}!\cdots i_{d}!}m_{I}(\rho\mu)\mathbf{u}^{I}.$
(1.4)
Note that $F_{\mu}^{\rho}(\mathbf{u})\neq F_{\rho\mu}(\mathbf{u})$ for non-
constant $\rho$. However, they are also connected, by an explicit differential
operator as follows.
###### Theorem 1.
For any complex-valued finite measure $\mu$ and any homogeneous polynomial
$\rho$ of degree $\delta$,
$\displaystyle F_{\mu}^{\rho}(\mathbf{u})$
$\displaystyle=\prod_{\ell=d}^{d+\delta-1}\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ F_{\rho\mu}(\mathbf{u})$ (1.5)
$\displaystyle=\rho\left(\frac{\partial}{\partial\mathbf{u}}\right)\circ
F_{\mu}(\mathbf{u})$ (1.6)
$\displaystyle=(d+\delta)!\int_{\mathbb{R}^{d}}\frac{\rho(\mathbf{x})d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+\delta+1}}.$
(1.7)
Here and in what follows $\circ$ denotes the application of a differential
operator to a function. The proof of the latter result is basically an
exercise in manipulating formal power series, and we do not claim its novelty.
For the sake of completeness, we include a proof in Section 2.
### Results on convex polytopes.
A finite set $S\subset\mathbb{R}^{d}$ is called spanning if it is not
contained in any (affine) hyperplane in $\mathbb{R}^{d}$. (Obviously,
$\text{card}(S)\geq d+1$.) As usual, by a (compact, convex) _polytope_
$\mathcal{P}\subset\mathbb{R}^{d}$ we mean the convex hull of a finite
spanning set in $\mathbb{R}^{d}$. The set of vertices of a convex polytope
$\mathcal{P}$ is the inclusion-minimal finite set with convex hull
$\mathcal{P}$. A $d$-simplex in $\mathbb{R}^{d}$ is the convex hull of a
spanning $(d+1)$-tuple of points. By an open polytope (resp. simplex) we mean
the set of interior points of a compact polytope (resp. simplex).
Given a convex polytope $\mathcal{P}$ let
$\mathcal{V}=(\mathbf{v}_{1},...,\mathbf{v}_{N})$ denote the set of its
vertices. Assume that $\mathcal{P}$ is simple, i.e. each
$\mathbf{v}\in\mathcal{V}$ has exactly $d$ incident edges
$\mathbf{v}\mathbf{v}_{e_{1}}$, …, $\mathbf{v}\mathbf{v}_{e_{d}}$. Set
$w_{k}(\mathbf{v}):=\mathbf{v}_{e_{k}}-\mathbf{v}$, for $1\leq k\leq d$. The
non-negative real span $K_{\mathbf{v}}$ of $w_{1}(\mathbf{v})$,…,
$w_{d}(\mathbf{v})$ is called the tangent cone of $\mathcal{P}$ at
$\mathbf{v}$. For each $K_{\mathbf{v}}$ define $|\det
K_{\mathbf{v}}|=|\det(w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v}))|$ to be the
volume of the parallelepiped formed by
$w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v})$.
Given a bounded domain $\Omega\subset\mathbb{R}^{d}$, we call the measure
$\mu_{\Omega}=\chi_{\Omega}dx_{1}dx_{2}\ldots dx_{d},$
where $\chi_{\Omega}$ is the characteristic function of $\Omega$, the standard
measure of $\Omega$.
For a simple convex polytope $\mathcal{P}$ we have the following explicit
representation of $F_{\mu_{\mathcal{P}}}(\mathbf{u}).$
###### Theorem 2.
For an arbitrary simple convex polytope $\mathcal{P}$,
$\displaystyle F_{\mathcal{P}}(\mathbf{u}):=F_{\mu_{\mathcal{P}}}(\mathbf{u})$
$\displaystyle=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{u}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod\limits_{j=1}^{d}\langle
w_{j}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v},\mathbf{u}\rangle}$
(1.8) $\displaystyle=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{|\det
K_{\mathbf{v}}|}{\prod\limits_{j=1}^{d}\langle
w_{j}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v},\mathbf{u}\rangle}.$
(1.9)
###### Remark 1.
Instead of the explicit choice of $w_{k}(\mathbf{v})$ for
$\mathbf{v}\in\mathcal{V}$ made above, we can take any fixed set of non-zero
vectors $w_{1}(\mathbf{v}),\ldots,w_{d}(\mathbf{v})$, spanning the tangent
cone of $\mathbf{v}$ in $\mathcal{P}$. This does not affect the validity of
(1.8) and (1.9).
Theorem 2 implies
###### Corollary 3.
Let $\Delta=\mathrm{conv}(\mathcal{V})\subset\mathbb{R}^{d}$ be an arbitrary
$d$-simplex. Then
$F_{\Delta}(\mathbf{u})=\frac{d!\mathrm{Vol}(\Delta)}{\prod\limits_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)}.$
(1.10)
###### Remark 2.
As we discovered after we proved the above results, statements similar to
Corollary 3 in the complex setting can be found in [APS]*Section 3.5 and in
particular [APS]*Corollary 3.5.6.
A variation of (1.10) also appears in [BBDL], in the context of designing an
efficient procedure for integration of polynomials over simplices.
Notice that an arbitrary convex polytope $\mathcal{P}$ admits a triangulation
which only uses the existing vertices of $\mathcal{P}$, see e.g. [BR]*Theorem
3.1. Applying Corollary 3 and Theorem 1 to the sum of measures corresponding
to such a triangulation we get the following.
###### Corollary 4.
The normalized moment generating function $F_{\mathcal{P}}^{\rho}(\mathbf{u})$
of any convex polytope $\mathcal{P}$ with respect to any homogeneous
polynomial density function $\rho$ of degree $\delta$ is a rational function
with denominator dividing
$\prod_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)^{\delta}.$
###### Example 1.
Let $\Delta$ be a triangle in $\mathbb{R}^{2}$ with vertices
$v_{1}=(1,1),\;v_{2}=(2,5)$ and $v_{3}=(3,2)$. Its normalized moment
generating function equals
$F_{\Delta}(u_{1},u_{2})=\frac{7}{(1-u_{1}-u_{2})(1-2u_{1}-5u_{2})(1-3u_{1}-2u_{2})}.$
Its Taylor expansion about the origin up to the terms of degree $7$ is given
by
$7+42u_{1}+56u_{2}+175u_{1}^{2}+455u_{1}u_{2}+329u_{2}^{2}+630u_{1}^{3}+2387u_{1}^{2}u_{2}+3367u_{1}u_{2}^{2}+1750u_{2}^{3}+2107u_{1}^{4}+10318u_{1}^{3}u_{2}\\\
+21217u_{1}^{2}u_{2}^{2}+21546u_{1}u_{2}^{3}+8967u_{2}^{4}+6762u_{1}^{5}+40082u_{1}^{4}u_{2}+106526u_{1}^{3}u_{2}^{2}+157976u_{1}^{2}u_{2}^{3}+128772u_{1}u_{2}^{4}\\\
+45276u_{2}^{5}+21175u_{1}^{6}+145845u_{1}^{5}u_{2}+468895u_{1}^{4}u_{2}^{2}+900123u_{1}^{3}u_{2}^{3}+10744451u_{1}^{2}u_{2}^{4}+741993u_{1}u_{2}^{5}+227269u_{2}^{6},$
which implies that
$m_{00}=\frac{7}{2},m_{10}=7,m_{01}=\frac{28}{3},m_{20}=\frac{175}{12},m_{11}=\frac{455}{24},m_{02}=\frac{329}{12},m_{30}=\frac{63}{2},m_{21}=\frac{2387}{60},\\\
m_{12}=\frac{3591}{20},m_{03}=\frac{175}{2},m_{40}=\frac{2107}{30},m_{31}=\frac{5159}{60},m_{22}=\frac{21217}{180},m_{13}=\frac{3591}{20},m_{04}=\frac{2989}{10},\\\
m_{50}={161},m_{41}=\frac{2863}{15},m_{32}=\frac{7609}{30},m_{23}=\frac{5642}{15},m_{14}=\frac{3066}{5},m_{05}=1078,m_{60}=\frac{3025}{8},\\\
m_{51}=\frac{6945}{16},m_{42}=\frac{13397}{24},m_{33}=\frac{128589}{160},m_{24}=\frac{153493}{120},m_{15}=\frac{35333}{16},m_{06}=\frac{32467}{8}.$
### Results on non-convex polytopes.
Our second group of results addresses the problem of distinguishing different
polytopes with the same underlying set of vertices from information on their
moments. The problem of restoring the vertices of a polygon or a polytope with
a constant mass density from information on its moments was addressed earlier
in e.g. [Bro1] [BroSt] [GMP] [GGMPV] [GMV00] [GLPR]. However, the latter do
not provide the recovery of the vertices in the generality required in the
present paper. Below we concentrate on the case of constant density and known
vertices, and plan to return to the general inverse problem for polytopes with
unknown polynomial density and unknown location of their vertices in the
future.
First we need to define what we mean by a polytope. It turned out that there
is no general consensus about this notion. Instead there exist several
competing definitions having their own advantages in different situations. We
shall study the following class of polytopal objects.
###### Definition 1.
A subset $\mathcal{P}\subset\mathbb{R}^{d}$ coinciding with a finite union of
arbitrary convex $d$-dimensional polytopes is called a generalized polytope.
###### Definition 2.
The number of components of a generalized polytope $\mathcal{P}$ is the number
of connected components of the set $\mathcal{P}^{o}\subset\mathcal{P}$ of
interior points of $\mathcal{P}$. The closure of each connected component of
$\mathcal{P}^{o}$ is called a component of $\mathcal{P}$. A generalized
polytope with one component is called indecomposable.
###### Remark 3.
We say that a simplicial complex in $\mathbb{R}^{d}$ is pure if all its
maximal simplices have dimension $d$. Clearly any generalized polytope in
$\mathbb{R}^{d}$ can be represented as the topological space of an appropriate
pure simplicial complex.
###### Remark 4.
Often one considers a more restricted class of objects, namely polytopes. A
polytope $\mathcal{P}\subset\mathbb{R}^{d}$ is a generalized polytope
homeomorphic to a $d$-dimensional manifold with boundary.
We need to introduce the notion of a vertex of a generalized polytope.
###### Definition 3.
Given a generalized polytope $\mathcal{P}\subset\mathbb{R}^{d}$ we call a
finite collection of open disjoint $d$-dimensional simplices in
$\mathbb{R}^{d}$ a dissection of $\mathcal{P}$ if the closure of their union
coincides with $\mathcal{P}$.
A wealth of material on dissections of polytopes can be found in [Pak08], see
also [MR2743368].
###### Definition 4.
Given a generalized polytope $\mathcal{P}\subset\mathbb{R}^{d}$ we call a
point $\mathbf{v}$ a vertex of $\mathcal{P}$, if $\mathbf{v}$ is a vertex of
(the closure of) some open simplex in every dissection of $\mathcal{P}$.
###### Definition 5.
Given a point $p\in\mathcal{P}$ of a generalized polytope $\mathcal{P}$ we
denote by the tangent cone $T_{p}(\mathcal{P})$ of $\mathcal{P}$ at $p$ the
set obtained as follows. For a sufficiently small $\epsilon>0$ set
$\mathcal{P}_{p}(\epsilon)=\mathcal{P}\cap B_{p}(\epsilon)$ where
$B_{p}(\epsilon)$ is the $\epsilon$-ball centered at $p$. Define
$T_{p}(\mathcal{P})$ as the set obtained by taking a ray though $p$ and every
point of $\mathcal{P}_{p}(\epsilon)$. In other words, $T_{p}(\mathcal{P})$ is
the cone with the apex at $p$ and the base $B_{p}(\epsilon)$. (Obviously,
$T_{p}(\mathcal{P})$ is independent of $\epsilon$ for a sufficiently small
$\epsilon>0$.)
###### Lemma 5.
A point $\mathbf{v}$ is a vertex of $\mathcal{P}$ if and only if
$T_{\mathbf{v}}(\mathcal{P})$ does not admit a decomposition in the disjoint
union of convex polygonal subcones, such that each subcone in the
decomposition has a translation-invariant direction. In particular, if the
tangent cone to $\mathcal{P}$ at $\mathbf{v}$ has a connected component with
no translation-invariant direction then $\mathbf{v}$ is a vertex.
We denote by $\mathrm{conv}(S)$ the convex hull of an arbitrary set
$S\subset\mathbb{R}^{d}$. The above lemma implies that any vertex of
$\mathrm{conv}(\mathcal{P})$ is a vertex of $\mathcal{P}$.
The following result extends Corollary 4 to the case of generalized polytopes.
###### Proposition 6.
For any generalized polytope $\mathcal{P}$ with the set of vertices
$\mathcal{V}(\mathcal{P})$, the denominator of its normalized moment
generating function $F^{\rho}_{\mathcal{P}}(\mathbf{u})$ with respect to a
homogeneous polynomial density function $\rho$ of degree $\delta$ divides
$\Phi_{\mathcal{P}}(\mathbf{u}):=\prod_{\mathbf{v}\in\mathcal{V}(\mathcal{P})}(1-\langle\mathbf{v},\mathbf{u}\rangle)^{\delta}.$
###### Remark 5.
There exist generalized polytopes which do not admit dissections with only
existing vertices. The simplest example of this kind is the Schönhardt
polyhedron, see Figure 1 and [Sch]. Absence of a dissection $\mathcal{T}$
which uses only its $6$ vertices can be established by observing that none of
the edges $AC$, $A^{\prime}B$, and $B^{\prime}C^{\prime}$ can appear in a
simplex of $\mathcal{T}$, yet any simplex on these $6$ vertices must contain
one of them. Therefore, Proposition 6 is not an immediate consequence of
Corollary 3.
$0$$C^{\prime}$$A$$C$$B$$B^{\prime}$$A^{\prime}$$C$$C^{\prime}$$A^{\prime}$$B$$B^{\prime}$$A$
Figure 1. Schönhardt polyhedron obtained from an octahedron (on the left) by
removing tetrahedra $[ABB^{\prime}C]$, $[AA^{\prime}B^{\prime}C^{\prime}]$,
and $[A^{\prime}BCC^{\prime}]$.
###### Remark 6.
For “generic” generalized polytopes $\mathcal{P}$ the denominator
$\Omega(\mathbf{u})$ of $F_{\mathcal{P}}(\mathbf{u})$ equals
$\Phi_{\mathcal{P}}(\mathbf{u})$, but for certain special polytopes the
denominator $\Omega(\mathbf{u})$ may be its proper divisor, as can be seen
from the following example. Let
$A=\\{0,a_{1},a_{2},a_{3}\\}\subset\mathbb{R}^{3}$ be a spanning set, and
$v\in\mathbb{R}^{3}$. Let $\mathcal{P}_{\pm}:=\mathrm{conv}(v\pm A)$ and
$\mathcal{P}:=\mathcal{P}_{+}\cup\mathcal{P}_{-}$. Then
$1-\langle\mathbf{u},v\rangle$ does not appear in $\Omega(\mathbf{u})$, as
$F_{\mathcal{P}}(\mathbf{u})=F_{\mathcal{P}_{+}}(\mathbf{u})+F_{\mathcal{P}_{-}}(\mathbf{u})=K\frac{\sum\limits_{1\leq
i<j\leq
3}\langle\mathbf{u},a_{i}\rangle\langle\mathbf{u},a_{j}\rangle+(1-\langle\mathbf{u},v\rangle)^{2}}{\prod\limits_{1\leq
i\leq
3}((1-\langle\mathbf{u},v\rangle)^{2}-\langle\mathbf{u},a_{i}\rangle^{2})},$
where $K\neq 0$ is a real constant.
Now we introduce several finite-dimensional linear spaces related to a given
finite spanning set $S\subset\mathbb{R}^{d}$. Let $\mathcal{P}(S)$ be the set
of all generalized polytopes $\mathcal{P}$ whose sets
$\mathcal{V}(\mathcal{P})$ of vertices are contained in $S$. For
$\mathcal{P}\in\mathcal{P}(S)$ we denote by $\mu_{\mathcal{P}}$ its standard
measure. (Obviously, $\mu_{\mathcal{P}}$ is supported on
$\mathcal{P}\subseteq\mathrm{conv}(S)$.)
Denote by $\mathfrak{M}(S)$ the linear space of all signed measures, i.e. the
linear span of all standard measures $\mu_{\mathcal{P}}$ for
$\mathcal{P}\in\mathcal{P}(S)$. Let
$\mathfrak{M}^{\Delta}(S)\subseteq\mathfrak{M}(S)$ be its subspace spanned by
$\mu_{\Delta}$, for $\Delta\in\mathcal{P}(S)$ a $d$-dimensional simplex. (The
space $\mathfrak{M}^{\Delta}(S)$ has earlier appeared in [AlGelZel], [Al1],
[Al2] in a somewhat different context.) We shall refer to elements of
$\mathfrak{M}(S)$ as to polytopal measures with the vertex set $S$. The next
conjecture is central to our study.
###### Conjecture 7.
For an arbitrary spanning set $S$ and any generalized polytope $\mathcal{P}$
with set of vertices contained in $S$, its standard measure
$\mu_{\mathcal{P}}$ belongs to $\mathfrak{M}^{\Delta}(S)$. In other words,
$\mathfrak{M}(S)=\mathfrak{M}^{\Delta}(S)$.
By Remark 5, the above conjecture is non-trivial. While we do not have a proof
of Conjecture 7 in its full generality, we have succeeded in proving it for a
rather large class of spanning sets. Roughly speaking, the latter should be
close to “generic”. Specifically, given a finite spanning set
$S\subset\mathbb{R}^{d},$ we say that $S$ is weakly non-degenerate if any
$(d+2)$-tuple of points from $S$ is spanning. If $S$ satisfies the stronger
condition that each $(d+1)$-subset of $S$ is spanning then we call the latter
$S$ strongly non-degenerate.
###### Theorem 8.
Conjecture 7 holds for any weakly non-degenerate finite set $S$.
###### Remark 7.
Theorem 8 would imply Conjecture 7 if one could prove that the standard
measure of an arbitrary generalized polytope $\mathcal{P}$ can be obtained as
the limit of the standard measures of a $1$-parameter family of generalized
polytopes $\mathcal{P}(t)$ with $\mathcal{P}(0)=\mathcal{P}$ such that for
$t\neq 0$ the vertices of $\mathcal{P}(t)$ are weakly non-degenerate, and each
vertex of $\mathcal{P}(t)$ tending to a vertex of $\mathcal{P}$ as $t\to 0$.
Unfortunately we are unable to prove the existence of such deformations in
general.
The key idea in the proof of Theorem 8 is to study the corresponding spaces of
Fantappiè transformations of signed measures in $\mathfrak{M}(S)$. In
particular, we are able to compute the corresponding dimensions111 Note that
presently we are not aware of a formula or a recipe for calculating the
dimension of $\mathfrak{M}^{\Delta}(S)$ without the assumption of the theorem.
(The manuscript [Al1] contains an algorithm constructing a basis of this
space.) . In more detail, let $\mathfrak{F}(S)$ (resp.
$\mathfrak{F}^{\Delta}(S)$) be the linear space of Fantappiè transformations
of signed measures in $\mathfrak{M}(S)$ (resp. $\mathfrak{M}^{\Delta}(S)$). In
other words, $\mathfrak{F}(S)$ (resp. $\mathfrak{F}^{\Delta}(S)$) is the space
of normalized moment generating functions of signed measures in
$\mathfrak{M}(S)$ (resp. $\mathfrak{M}^{\Delta}(S)$).
Since each compactly supported measure is uniquely determined by its complete
set of moments, the map
$F_{\mu}:\mathfrak{M}(S)\to\mathfrak{F}(S),$ (1.11)
induced by the Fantappiè transformation is a linear isomorphism, cf.
[APS]*Section 3.5.
Finally, given a spanning set
$S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d},$ denote by
$\mathfrak{Rat}(S)$ the linear space of all rational functions with the
denominator $\Phi_{S}(\mathbf{u})$ as in (1.12),
$\Phi_{S}(\mathbf{u})=\prod_{i=1}^{N}(1-\langle\mathbf{v}_{i},\mathbf{u}\rangle),$
(1.12)
and with the numerator an arbitrary real (inhomogeneous) polynomial of degree
at most $N-d-1$. Here the numerator and the denominator might have common
factors.
###### Proposition 9.
$\mathfrak{F}^{\Delta}(S)$ coincides with $\mathfrak{Rat}(S)$ if and only if
$S$ is strongly non-degenerate.
###### Corollary 10.
If $S$ is strongly non-degenerate then
$\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$.
Corollary 10 implies that for strongly non-degenerate $S$ the dimension of all
these linear spaces equals $\binom{N-1}{d}$. Note that Corollary 10 settles
Theorem 8 for the strongly non-degenerate $S$.
Our final goal is to explicitly solve the following inverse moment problem.
###### Problem 1.
Given a strongly non-degenerate spanning set $S\subset\mathbb{R}^{d}$,
$|S|=N$, find the unique polytopal measure in $\mathfrak{M}(S)$ with a given
set of all moments up to order $N-d-1$.
We start with the following simple observation.
###### Lemma 11.
Given an arbitrary spanning set $S\subset\mathbb{R}^{d}$, $|S|=N$, and an
arbitrary polynomial $T(\mathbf{u})$ of degree at most $N-d-1$ there exists a
unique rational function $R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$
with Taylor polynomial of degree $N-d-1$ at the origin equal to
$T(\mathbf{u})$. Namely,
$P(\mathbf{u})=\left[T(\mathbf{u})\Phi_{S}(\mathbf{u})\right]_{N-d-1}$, where
$\left[\cdot\right]_{N-d-1}$ stands for the truncated polynomial with all
monomials up to degree $N-d-1$.
For $S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d}$
strongly non-degenerate, we give an explicit inversion formula determining the
densities of an unknown polytopal measure having a given set of moments up to
order $N-d-1$ on each simplex in a natural basis of
$\mathfrak{M}^{\Delta}(S)$. In view of Lemma 11 we can assume that we are
already given an arbitrary rational function
$R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where $\deg
P(\mathbf{u})\leq N-d-1$, and we want to determine the densities of the
required signed measure from $\mathfrak{M}(S)$ in terms of numerator
$P(\mathbf{u})$.
From now on we shall choose the basis of $\mathfrak{M}^{\Delta}(S)$ consisting
of the standard measures of all simplices containing the last vertex
$\mathbf{v}_{N}$, see Lemma 15 below. Let
$\mathfrak{L}=\\{l_{1},l_{2},....,l_{N-1}\\}$ be the $(N-1)$-tuple of linear
forms corresponding to vertices
$\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{N-1}$, where
$l_{i}(\mathbf{u})=1-\langle\mathbf{v}_{i},\mathbf{u}\rangle$. Consider the
linear span $V_{\mathfrak{L}}$ of all possible products of the form
$l_{j_{1}}\cdot l_{j_{2}}\cdot...\cdot l_{j_{N-d-1}},\;1\leq
j_{1}<j_{2}<...<j_{N-d-1}$. There are $\binom{N-1}{d}$ such products, and each
of them is a polynomial of degree at most $N-d-1$. On the other hand, the
dimension of the space $Pol(N-d-1,d)$ of all (inhomogeneous) polynomials of
degree at most $N-d-1$ in $d$ variables equals $\binom{N-1}{d}$, as well.
Define the square matrix $Mat_{S}$ of size $\binom{N-1}{d}$ with entries being
coefficients of the above products of linear forms with respect to the
standard monomial basis in $Pol(N-d-1,d)$. We assume that $Mat_{S}$ acts on
the space $V_{\mathfrak{L}}$ of column vectors.
###### Theorem 12.
For an arbitrary strongly non-degenerate spanning set
$S\subset\mathbb{R}^{d},$ $|S|=N$, the matrix $Mat_{S}$ is invertible.
Moreover, for a rational function
$R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where $P(\mathbf{u})$ is
an arbitrary polynomial of degree $N-d-1$, there exists a unique measure
$\mu_{R}\in\mathfrak{M}(S)$ with Fantappiè transform $R(\mathbf{u})$. Namely,
$\mu_{R}=Mat_{S}^{-1}(P(\mathbf{u})).$ (1.13)
###### Remark 8.
A detailed explanation of the meaning of (1.13) can be found in the proof of
Theorem 12, see also Example 2 below. An explicit formula for the matrix
$Mat_{S}^{-1}$ is given in Lemma 18.
Recall that a spanning set $S$ is weakly non-degenerate if any $(d+2)$-tuple
of its points is spanning. With minor changes, the above solution of the
inverse moment problem can be adapted to this more general case. In order not
to overload the introduction we refer the readers interested in this situation
to Section 4. The case of an arbitrary spanning set $S$, however, remains
unsolved and offers several interesting challenges in matroid theory. We hope
to return to it in the future.
It will be convenient to work with scaled volumes of simplices, which we call
weights.
###### Definition 6.
Given a signed measure $\mu$ in $\mathbb{R}^{d}$ and a $d$-dimensional simplex
$\Delta\subset\mathbb{R}^{d}$ we define the weight $w_{\Delta}$ of $\Delta$ by
the formula:
$w_{\Delta}=d!\int_{\Delta}d\mu.$ (1.14)
In other words, the density $d_{\Delta}$ of the measure in question which
should be placed at $\Delta$ equals
$d_{\Delta}=\frac{w_{\Delta}}{d!\mathrm{Vol}(\Delta)}.$
We finish the introduction by explicitly solving the above inverse problem for
a concrete $5$-tuple of points in $\mathbb{R}^{2}$.
###### Example 2.
Set
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=(1,0),\mathbf{v}_{2}=(2,1),\mathbf{v}_{3}=(1,2),\mathbf{v}_{4}=(0,1),\mathbf{v}_{5}=(0,0)$.
The corresponding set $\mathfrak{L}=\\{l_{1},l_{2},l_{3},l_{4}\\}$ of linear
forms is given by
$l_{1}=1-u_{1},\,l_{2}=1-2u_{1}-u_{2},\,l_{3}=1-u_{1}-2u_{2},\,l_{4}=1-u_{2}$.
Additionally, $l_{5}=1$. We are considering the basis of
$\mathfrak{M}^{\Delta}(S)$ consisting of (the standard measures of) 6
triangles containing $\mathbf{v}_{5}$. Therefore we need 6 quadratic forms
obtained as pairwise products $l_{i}l_{j},\,1\leq i<j\leq 4$. We get
$\begin{cases}l_{1}l_{2}=1-3u_{1}-u_{2}+2u_{1}^{2}+u_{1}u_{2}\\\
l_{1}l_{3}=1-2u_{1}-2u_{2}+u_{1}^{2}+2u_{1}u_{2}\\\
l_{1}l_{4}=1-u_{1}-u_{2}+u_{1}u_{2}\\\
l_{2}l_{3}=1-3u_{1}-3u_{2}+2u_{1}^{2}+5u_{1}u_{2}+2u_{2}\\\
l_{2}l_{4}=1-2u_{1}-2u_{2}+2u_{1}u_{2}+u_{2}^{2}\\\
l_{3}l_{4}=1-u_{1}-3u_{2}+u_{1}u_{2}+2u_{2}^{2}.\\\ \end{cases}$
Notice that $l_{1}l_{2}$ corresponds to triangle $\Delta_{345}$, $l_{1}l_{3}$
to $\Delta_{245}$, $l_{1}l_{3}$ to $\Delta_{245}$, $l_{1}l_{4}$ to
$\Delta_{234}$, $l_{2}l_{3}$ to $\Delta_{145}$, $l_{2}l_{4}$ to
$\Delta_{135}$, and $l_{3}l_{4}$ to $\Delta_{125}$. Ordering monomials
spanning the space $Pol(2,2)$ as
$(1,u_{1},u_{2},u_{1}^{2},u_{1}u_{2},u_{2}^{2})$ we get the $6\times 6$-matrix
$Mat_{S}$ and its inverse $Mat_{S}^{-1}$ as follows
$\blockarray{rrrrrrr}\block{cc\BAmulticolumn{4}{c}c}&Mat_{S}=\\\
&l_{1}l_{2}l_{1}l_{3}l_{1}l_{4}l_{2}l_{3}l_{2}l_{4}l_{3}l_{4}\\\
\block{r(rrrrrr)}1111111\\\ u_{1}-3-2-1-3-2-1\\\ u_{2}-1-2-1-3-2-3\\\
u_{1}^{2}210200\\\ u_{1}u_{2}121521\\\ u_{2}^{2}000212\\\
\quad\blockarray{rrrrrrr}\block{cc\BAmulticolumn{4}{c}c}&4Mat_{S}^{-1}=\\\
&1u_{1}u_{2}u_{1}^{2}u_{1}u_{2}u_{2}^{2}\\\
\block{r(rrrrrr)}l_{1}l_{2}1-1111-1\\\ l_{1}l_{3}-40-400-4\\\
l_{1}l_{4}933111\\\ l_{2}l_{3}111111\\\ l_{2}l_{4}-4-40-400\\\
l_{3}l_{4}11-11-11\\\ $
(For TeXnical reasons we give $4Mat_{S}^{-1}$ above.) Thus, given an arbitrary
rational function $R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is a polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we get
$\begin{cases}w_{345}=\frac{1}{4}(a_{00}-a_{10}+a_{01}+a_{20}-a_{11}+a_{02})\\\
w_{245}=-a_{00}-a_{01}-a_{02}\\\
w_{235}=\frac{1}{4}(9a_{00}+3a_{01}+3a_{10}+a_{20}+a_{11}+a_{02})\\\
w_{145}=\frac{1}{4}(a_{00}+a_{01}+a_{10}+a_{20}+a_{11}+a_{02})\\\
w_{135}=-a_{00}-a_{10}-a_{20}\\\
w_{125}=\frac{1}{4}(a_{00}+a_{10}-a_{01}+a_{20}-a_{11}+a_{02}),\\\
\end{cases}$
where $w_{ijk}$ is the weight of the signed measure to be placed on
$\Delta_{ijk}$, see (1.14).
To illustrate all steps of solution of our inverse moment problem assume that
we are looking for a polygonal measure with the vertex set $S$ and (ad hoc
chosen) moments $m_{00}=1,m_{10}=2,m_{01}=3,m_{20}=4,m_{11}=5,m_{02}=6$. Then
its normalized moment generating function $F_{\mu}(\mathbf{u})$ satisfies the
relation
$F_{\mu}(\mathbf{u})=1\frac{2!}{0!0!}+2\frac{3!}{1!0!}u_{1}+3\frac{3!}{0!1!}u_{2}+4\frac{4!}{2!0!}u_{1}^{2}+5\frac{4!}{1!1!}u_{1}u_{2}+6\frac{4!}{0!2!}u_{2}^{2}+\dots=\frac{P(u_{1},u_{2})}{l_{1}l_{2}l_{3}l_{4}l_{5}},$
where $P(u_{1},u_{2})$ is a (non-homogeneous) polynomial of at most second
degree. Thus, truncating the product of the left-hand side and
${l_{1}l_{2}l_{3}l_{4}l_{5}}$ up to the second degree, we obtain
$P(u_{1},u_{2})=2+4u_{1}+10u_{2}+10u_{1}^{2}+24u_{1}u_{2}+10u_{2}^{2},$
i.e. $a_{00}=2,a_{10}=4,a_{01}=10,a_{20}=10,a_{11}=24,a_{02}=10$. Thus
$w_{345}=1,w_{245}=-22,w_{235}=26,w_{145}=15,w_{135}=-16,w_{125}=-2$. The
areas of the corresponding triangles are equal to:
$Area{(\Delta_{345})}=\frac{1}{2};Area({\Delta_{245}})=1;Area{(\Delta_{235})}=\frac{3}{2};Area{(\Delta_{145})}=\frac{1}{2};Area{(\Delta_{135})}=1;Area{(\Delta_{125})}=\frac{1}{2}.$
This implies that the densities of the measure of the corresponding triangles
are equal to
$d_{345}=1,d_{245}=-11,d_{235}=\frac{26}{3},d_{145}=15,d_{135}=-8,d_{125}=-2$.
To obtain the final densities in the convex hull $\mathrm{conv}(S)$ of $S$ one
has to decompose $\mathrm{conv}(S)$ into domains obtained by removing from
$\mathrm{conv}(S)$ the set of all hyperplanes spanned by vertices in $S$. For
each such domain we should add up the densities of all basic simplices
containing this domain. The resulting measure is shown in Fig. 2.
$\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{1}=(1,0)$$\mathbf{v}_{2}=(2,1)$$\mathbf{v}_{3}=(1,2)$$\mathbf{v}_{4}=(0,1)$$5$$-10$$-2$$\frac{26}{3}$$\frac{31}{3}$$-\frac{7}{3}$$\frac{2}{3}$$1$$\frac{14}{3}$$5$$-10$
Figure 2. Final measure in Example 2.
###### Remark 9.
Domains into which the convex hull $\mathrm{conv}(S)$ is cut by the
hyperplanes spanned by $S$ were introduced in [AlGelZel] where they were
called chambers. The incidence matrix of the simplices spanned by $S$ and
those chambers was studied in some detail in [Al1],[Al2]. This matrix allows
to formalize the last step of construction of the above polygonal measure,
where information on the densities of the simplices is transformed into
information on the densities of the chambers. But, in general, already the
number of chambers is a complicated invariant of the set $S$. It seems that
the general problem of constructing the set of chambers and the corresponding
incidence matrix in terms of a given $S$ is quite non-trivial.
###### Acknowledgement.
The second author is grateful to the Mathematics Department of Stockholm
University for the hospitality in June 2011 when this project was initiated.
The third author wants to acknowledge the hospitality of the School of
Physical and Mathematical Sciences, Nanyang Technological University in April
2012 when this project was completed. We want to thank Sinai Robins for
numerous discussions of the topic. We acknowledge extremely helpful answers
and comments on our questions on mathoverflow.net, in particular ones by David
Eppstein, Dirk Lorenz, Igor Pak, David Speyer, and Gjergji Zaimi. Finally, the
third author wants to thank late Mikael Passare (who unfortunately left us so
early) for discussions of the properties of Fantappiè transformation and for
pointing out reference [APS] in September 2011.
## 2\. Proving results on convex polytopes
Following Brion-Lawrence-Khovanskii-Pukhlikov-Barvinok, see [Bar2, BR, GLPR,
La], we define for each vector $\mathbf{z}\in\mathbb{R}^{d}$ the $j$-th
_axial_ moment $\mu_{j}(\mathbf{z})$ of a simple convex polytope $\mathcal{P}$
with respect to $\mathbf{z}$ as
$\mu_{j}(\mathbf{z})=\int_{\mathcal{P}}\langle\mathbf{x},\mathbf{z}\rangle^{j}d\mathbf{x}.$
We will use the following important statement, cf. e.g. [BR]*Theorem 10.5.
###### Theorem 13.
The moment $\mu_{j}(\mathbf{z})$ satisfies
$\mu_{j}(\mathbf{z})=\frac{(-1)^{d}j!}{(j+d)!}\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j+d}D_{\mathbf{v}}(\mathbf{z}),$
(2.1)
where $D_{\mathbf{v}}(\mathbf{z}):=\frac{|\det
K_{\mathbf{v}}|}{\prod_{j=1}^{d}\langle w_{j}(\mathbf{v}),\mathbf{z}\rangle}$,
and $\mathbf{z}$ is an arbitrary vector for which the products
$\prod_{j=1}^{d}\langle w_{j}(\mathbf{v}),\mathbf{z}\rangle$,
$\mathbf{v}\in\mathcal{V}$, do not vanish. Moreover, the following identities
hold:
$\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j}D_{\mathbf{v}}(\mathbf{z})=0,\quad\text{for
$0\leq j\leq d-1$.}$ (2.2)
###### Proof of Theorem 2.
To prove (1.8), consider the generating function
$\Psi_{\mathbf{z}}(u)=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\mu_{j}(\mathbf{z})u^{j},$
where $u\in\mathbb{R}$. Formula (2.1) implies that $\Psi_{\mathbf{z}}(u)$ is
rational. Indeed,
$\Psi_{\mathbf{z}}(u)=\sum_{j=0}^{\infty}(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\langle\mathbf{v},\mathbf{z}\rangle^{j+d}\frac{|\det
K_{\mathbf{v}}|u^{j}}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{z}\rangle}=\\\
=(-1)^{d}\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{z}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{z}\rangle}\sum_{j=0}^{\infty}\langle\mathbf{v},\mathbf{z}\rangle^{j}u^{j}=\sum_{\mathbf{v}\in\mathcal{V}}\frac{\langle\mathbf{v},\mathbf{u}\rangle^{d}|\det
K_{\mathbf{v}}|}{\prod_{k=1}^{d}\langle
w_{k}(\mathbf{v}),\mathbf{u}\rangle}\cdot\frac{(-1)^{d}}{1-\langle\mathbf{v},\mathbf{u}\rangle},$
where $\mathbf{u}=u\mathbf{z}$. On the other hand, using the multinomial
coefficients $\binom{|J|}{J}=\frac{|J|!}{j_{1}!\dots j_{d}!}$ of multiindices
$J=(j_{1},\dots,j_{d})\vdash|J|$, one gets
$\int_{\mathcal{P}}\langle\mathbf{x},\mathbf{z}\rangle^{j}d\mathbf{x}=\int_{\mathcal{P}}\left(\sum_{i=1}^{d}x_{i}z_{i}\right)^{j}d\mathbf{x}=\sum_{J\vdash
j}\binom{j}{J}\mathbf{z}^{J}\int_{\mathcal{P}}\mathbf{x}^{J}d\mathbf{x}=\sum_{J\vdash
j}\binom{j}{J}\mathbf{z}^{J}m_{{}_{J}}(\mathcal{P}),$
where $m_{{}_{J}}(\mathcal{P})=m_{{}_{J}}(\mu_{\mathcal{P}})$. Therefore,
$F_{\mathcal{P}}(u\mathbf{z}):=F_{\mathcal{P}}(uz_{1},...,uz_{d}):=\sum_{j=0}^{\infty}\sum_{(j_{1},...,j_{d})\vdash
j}\frac{(j+d)!}{j_{1}!\dots
j_{d}!}m_{j_{1},\dotsc,j_{d}}(uz_{1})^{j_{1}}\cdots(uz_{d})^{j_{d}}\\\
=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\left[\sum_{J:=(j_{1},...,j_{d})\vdash
j}\binom{j}{J}m_{{}_{J}}(\mathcal{P})\mathbf{z}^{J}\right]u^{j}=\sum_{j=0}^{\infty}\frac{(j+d)!}{j!}\mu_{j}(\mathbf{z})u^{j}=\Psi_{\mathbf{z}}(u),$
and (1.8) follows.
In view of relations (2.2) the right-hand side of (1.8) can be rewritten as
(1.9). Indeed, writing
$(1-\langle\mathbf{v},\mathbf{u}\rangle)^{-1}=\sum_{j=0}^{\infty}\langle\mathbf{v},\mathbf{u}\rangle^{j}$
and expanding (1.8) with respect to $j$th powers of
$\langle\mathbf{v},\mathbf{u}\rangle$, we see that (2.2) implies that for
$j<d$ the sum of all terms $\langle\mathbf{v},\mathbf{u}\rangle^{j}$ vanishes.
∎
###### Proof of Corollary 3.
Let $\mathcal{V}=(\mathbf{v}_{0},\mathbf{v}_{1},\ldots,\mathbf{v}_{d})$. Then
for each $j\neq i$ we have
$w_{j}(\mathbf{v}_{i})=\mathbf{v}_{j}-\mathbf{v}_{i}$. Hence $|\det
K_{\mathbf{v}_{i}}|$ does not depend upon $i$ and equals
$d!\mathrm{Vol}(\Delta)$. The right-hand side of (1.8) becomes
$(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\langle\mathbf{v}_{i},\mathbf{u}\rangle^{d}}{\prod_{j=1}^{d}\langle
w_{j}(\mathbf{v}_{i}),\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v}_{i},\mathbf{u}\rangle}=\\\
=(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\langle\mathbf{v}_{i},\mathbf{u}\rangle^{d}}{\prod_{j\neq
i}\langle\mathbf{v}_{j}-\mathbf{v}_{i},\mathbf{u}\rangle}\cdot\frac{1}{1-\langle\mathbf{v}_{i},\mathbf{u}\rangle}=\\\
=(-1)^{d}d!\mathrm{Vol}(\Delta)\sum_{i=0}^{d}\frac{\zeta_{i}^{d}}{\prod_{j\neq
i}(\zeta_{j}-\zeta_{i})}\cdot\frac{1}{1-\zeta_{i}},$
where $\zeta_{i}=\langle\mathbf{v}_{i},\mathbf{u}\rangle$. Computing the
common denominator of the latter, we obtain
$F_{\Delta}(\mathbf{u})=\frac{(-1)^{d}d!\mathrm{Vol}(\Delta)}{\prod_{i=0}^{d}(1-\zeta_{i})}\frac{\sum\limits_{i=0}^{d}\left[\prod\limits_{k>l,k\neq
i\neq l}(\zeta_{k}-\zeta_{l})\prod\limits_{j\neq
i}(1-\zeta_{j})\right](-1)^{i}\zeta_{i}^{d}}{\prod_{s>t}(\zeta_{s}-\zeta_{t})}.$
It is convenient to introduce one more linear form $\zeta_{d+1}:=1$, so that
the last expression reads as
$F_{\Delta}(\mathbf{u})=(-1)^{d}d!\mathrm{Vol}(\Delta)\frac{\sum\limits_{i=0}^{d}\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq
l\end{subarray}}(\zeta_{k}-\zeta_{l})\right](-1)^{i}\zeta_{i}^{d}}{\prod\limits_{d+1\geq
s>t\geq 0}(\zeta_{s}-\zeta_{t})}.$ (2.3)
To complete the proof we notice that
$0=\det\begin{pmatrix}1&1&\ldots&1&1\\\
\zeta_{0}&\zeta_{1}&\ldots&\zeta_{d}&1\\\
\zeta_{0}^{2}&\zeta_{1}^{2}&\ldots&\zeta_{d}^{2}&1\\\ \ldots\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}&1\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}&1\\\
\end{pmatrix}=1\cdot\det\begin{pmatrix}1&1&\ldots&1\\\
\zeta_{0}&\zeta_{1}&\ldots&\zeta_{d}\\\
\zeta_{0}^{2}&\zeta_{1}^{2}&\ldots&\zeta_{d}^{2}\\\ \ldots\\\
\zeta_{0}^{d}&\zeta_{1}^{d}&\ldots&\zeta_{d}^{d}\\\ \end{pmatrix}+\\\
+(-1)^{d+1}\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\det\begin{pmatrix}1&\ldots&1&1&\ldots&1\\\
\zeta_{0}&\ldots&\zeta_{i-1}&\zeta_{i+1}&\ldots&\zeta_{d+1}\\\
\zeta_{0}^{2}&\ldots&\zeta_{i-1}^{2}&\zeta_{i+1}^{2}&\ldots&\zeta_{d+1}^{2}\\\
\ldots\\\
\zeta_{0}^{d}&\ldots&\zeta_{i-1}^{d}&\zeta_{i+1}^{d}&\ldots&\zeta_{d+1}^{d}\\\
\end{pmatrix}=\\\ =\prod_{d\geq k>l\geq
0}(\zeta_{k}-\zeta_{l})+(-1)^{d+1}\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq l\end{subarray}}(\zeta_{k}-\zeta_{l})\right].$
Indeed, the first matrix has two identical rows and thus vanishing
determinant, which we expand with respect to the last row. The last equality
is the standard formula for the Vandermonde determinant. Thus we have
$\sum_{i=0}^{d}\zeta_{i}^{d}(-1)^{i}\cdot\left[\prod\limits_{\begin{subarray}{c}d+1\geq
k>l\geq 0,\\\ k\neq i\neq
l\end{subarray}}(\zeta_{k}-\zeta_{l})\right]=(-1)^{d}\prod_{d\geq k>l\geq
0}(\zeta_{k}-\zeta_{l}).$
Now we plug this formula into (2.3) and get
$F_{\Delta}(\mathbf{u})=d!\mathrm{Vol}(\Delta)\frac{1}{\prod\limits_{t=0}^{d}(1-\zeta_{t})}=\frac{d!\mathrm{Vol}(\Delta)}{\prod\limits_{\mathbf{v}\in\mathcal{V}}(1-\langle\mathbf{v},\mathbf{u}\rangle)}.\qed$
###### Lemma 14.
Let $\mathbf{u}=(u_{1},\dots,u_{d})$ and $\mathbf{x}=(x_{1},\dots,x_{d})$ be
formal variables, and $\ell\in\mathbb{R}$ . Then
$\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}.$
(2.4)
###### Proof.
Note that $u_{k}\frac{\partial}{\partial
u_{k}}\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=x_{k}u_{k}\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}$.
Thus
$\left(\sum_{k}u_{k}\frac{\partial}{\partial
u_{k}}+\ell\right)\circ(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}\langle\mathbf{x},\mathbf{u}\rangle+\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell}\\\
=\ell(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-\ell-1}.\qed$
###### Proof of (1.3).
For a $d$-variate polynomial $g(\mathbf{z})$ we denote by
$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)$ the differential
operator $g\left(u_{1}\frac{\partial}{\partial
u_{1}},\dots,u_{d}\frac{\partial}{\partial u_{d}}\right)$. We use the identity
$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\sum_{I}a_{{}_{I}}\mathbf{x}^{I}\mathbf{u}^{I}=\sum_{I\geq
0}g(I)a_{{}_{I}}\mathbf{x}^{I}\mathbf{u}^{I},$ (2.5)
which holds for any formal $d$-variate power series
$\sum_{I}a_{I}\mathbf{x}^{I}\mathbf{u}^{I}$ and any $d$-variate polynomial
$g(\mathbf{z})$. (It can be easily verified for monomial $g(\mathbf{z})$ and
then extended by linearity.) Setting
$h(\mathbf{z}):=\prod_{\ell=1}^{d}\left(\sum_{k=1}^{d}z_{k}+\ell\right),$
notice that $h(I)=(|I|+1)(|I|+2)\cdots(|I|+d)$. Now using (2.5) together with
the obvious identity:
$(1-\langle\mathbf{x},\mathbf{u}\rangle)^{-1}=\sum_{I\geq
0}\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I},$
one obtains
$F_{\mu}(\mathbf{u}):=\sum_{I\geq
0}\binom{|I|+d}{I}m_{I}(\mu)\mathbf{u}^{I}=\sum_{I\geq
0}h(I)\binom{|I|}{I}m_{I}(\mu)\mathbf{u}^{I}\\\
=\int_{\mathbb{R}^{d}}\sum_{I\geq
0}h(I)\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I}d\mu(\mathbf{x})=\int\limits_{\mathbb{R}^{d}}h\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\sum_{I\geq
0}\binom{|I|}{I}\mathbf{x}^{I}\mathbf{u}^{I}d\mu(\mathbf{x})\\\
=\int\limits_{\mathbb{R}^{d}}h\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)\circ\frac{d\mu(\mathbf{x})}{1-\langle\mathbf{x},\mathbf{u}\rangle}=\int\limits_{\mathbb{R}^{d}}\frac{d!\
d\mu(\mathbf{x})}{(1-\langle\mathbf{x},\mathbf{u}\rangle)^{d+1}},$
where in the final derivation we repeatedly made use of (2.4), for
$1\leq\ell\leq d.$ ∎
###### Remark 10.
Another point of view on (1.3) is that it is the result of the application of
the differential operator
$g\left(\mathbf{u}\frac{\partial}{\partial\mathbf{u}}\right)$ to the integral
transformation
$\int_{\mathbb{R}^{d}}\frac{d\mu(\mathbf{x})}{1-\langle\mathbf{x},\mathbf{u}\rangle}$
of the measure $\mu$ (also known as the Fantappiè transform of $\mu$); see
e.g. [MR2274973].
In [PS12] a similar idea was applied to the harmonic polygonal measures in the
plane.
###### Proof of Theorem 1.
Assume first that $\rho(x_{1},...,x_{d})={\bf x}^{K}=x_{1}^{k_{1}}\cdots
x_{d}^{k_{d}}$ is a monomial and consider
$\rho\left(\frac{\partial}{\partial{\mathbf{u}}}\right)\circ
F_{\mu}({\mathbf{u}})=\frac{\partial^{|{K}|}}{\partial
u_{1}^{k_{1}}\cdots\partial u_{d}^{k_{d}}}\circ\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d)!}{i_{1}!\cdots i_{d}!}m_{I}(\mu)\mathbf{u}^{I}.$
One gets
$\frac{\partial^{|K|}}{\partial{\mathbf{u}}^{K}}\circ
F_{\mu}({\mathbf{u}})=\sum_{I=(i_{1},...,i_{d})\geq
0}\left(\frac{(|I|+|{K}|+d)!}{\prod\limits_{j=1}^{d}(i_{j}+k_{j})!}\prod_{j=1}^{d}\frac{(i_{j}+k_{j})!}{i_{j}!}\right)m_{I+K}(\mu)\mathbf{u}^{I}\\\
=\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d+|{K}|)!}{\prod_{j=1}^{d}i_{j}!}m_{I}(\mathbf{x}^{K}\mu)\mathbf{u}^{I}.$
Observe that the normalizing coefficients of $m_{I}(\mathbf{x}^{K}\mu)$ in the
latter expression depend only on $I$ and $|K|$ but not on particular entries
of $K$. Therefore for an arbitrary homogeneous $\rho$ of degree $\delta$ one
gets by additivity
$\rho\left(\frac{\partial}{\partial{\mathbf{u}}}\right)\circ
F_{\mu}({\mathbf{u}})=\sum_{I=(i_{1},\dots,i_{d})\geq
0}\frac{(|I|+d+\delta)!}{\prod\limits_{j=1}^{d}i_{j}!}m_{I}(\rho\mu).$
This shows (1.6). Repeated application of (2.4), for $d+1\leq\ell\leq
d+\delta$, to the integral representation (1.3), respectively, to the
representation (1.2), of $F_{\rho\mu}(\mathbf{u})$ implies (1.7),
respectively, (1.5). ∎
## 3\. Inverse moment problem for strongly non-degenerate $S$
###### Proof of Lemma 5.
We prove first that the tangent cone at any non-vertex allows a decomposition
into convex polytopal cones each having a translation-invariant direction.
Let $\mathbf{v}$ be a point in $\mathcal{P}$ which is not a vertex. Then there
is a dissection $\mathcal{T}$ of $\mathcal{P}$ such that $\mathbf{v}$ is not a
vertex of any simplex of $\mathcal{T}$. Let $U$ be the set of simplices
$S_{u}$ of $\mathcal{T}$ with closures containing $\mathbf{v}$. Take the
dissection of the tangent cone $T_{v}(\mathcal{P})$ into the tangent cones to
simplices from $U$, $T_{\mathbf{v}}(\mathcal{P})=\cup_{u\in
U}T_{\mathbf{v}}(S_{u})$. Clearly, every subcone $T_{v}(S_{u})$ contains a
translation-invariant direction (any direction parallel to the minimal face
containing $\mathbf{v}$).
Vice versa, to prove the converse implication, let us take a dissection of the
tangent cone $T_{\mathbf{v}}(\mathcal{P})$ into a disjoint union of convex
polytopal cones $Q_{1},\dots,Q_{k}$. By definition of the tangent cone and
since $\mathcal{P}$ can be represented as a finite union of simplices we
obtain that any sufficiently small neighborhood of $\mathbf{v}$ in the tangent
cone $T_{\mathbf{v}}(\mathcal{P})$ is a neighborhood of $\mathbf{v}$ in the
entire $\mathcal{P}$. Consider the parallelepiped $\text{Box}_{\varepsilon}$
centered at $\mathbf{v}$ that is $\varepsilon$ ball centered at $\mathbf{v}$,
in the $L_{1}$-norm. Note that each convex polytopal set
$Q_{i}\cap\text{Box}_{\varepsilon}$ can be decomposed into a union of
simplices that do not contain $\mathbf{v}$ as a vertex.
Further notice that the set $\mathcal{P}\setminus\text{Box}_{\varepsilon}$ can
be represented as a finite disjoint union of simplices, since
$\text{Box}_{\varepsilon}$ is the intersection of a finite number of half-
spaces and $\mathcal{P}$ is a disjoint union of simplices. Clearly, every
simplex in this union should not have $\mathbf{v}$ as a vertex. Now combining
the dissections of each $Q_{i}\cap\text{Box}_{\varepsilon}$ and
$\mathcal{P}\setminus\text{Box}_{\varepsilon}$ we obtain the required
dissection of $\mathcal{P}$. ∎
###### Proof of Proposition 6.
We begin by considering the case $\rho\equiv 1$. Let $\mathcal{T}$ be a
dissection of $\mathcal{P}$ with vertices $\mathcal{V}(\mathcal{T})$.
Corollary 3 implies that $F_{\mathcal{P}}(\mathbf{u})$ has a denominator
dividing
$g_{\mathcal{T}}(\mathbf{u})=\prod_{\mathbf{v}\in\mathcal{V}(\mathcal{T})}(1-\langle\mathbf{v},\mathbf{u}\rangle)$.
Take
$\mathbf{v}_{1}\in\mathcal{V}(\mathcal{T})\setminus\mathcal{V}(\mathcal{P})$.
Then there exists another dissection $\mathcal{T}^{\prime}$ such that
$\mathbf{v}_{1}\not\in\mathcal{V}(\mathcal{T}^{\prime})$. Expressing
$F_{\mathcal{P}}(\mathbf{u})$ as ratios of polynomials, we have
$F_{\mathcal{P}}(\mathbf{u})=\frac{f_{\mathcal{T}}(\mathbf{u})}{h_{\mathcal{T}}(\mathbf{u})(1-\langle\mathbf{v}_{1},\mathbf{u}\rangle)}=\frac{f_{\mathcal{T}^{\prime}}(\mathbf{u})}{g_{\mathcal{T}^{\prime}}(\mathbf{u})},\quad\text{where
}g_{\mathcal{T}}(\mathbf{u})={h_{\mathcal{T}}(\mathbf{u})(1-\langle\mathbf{v}_{1},\mathbf{u}\rangle)}.$
Here $g_{\mathcal{T}^{\prime}}$ is not divisible by
$1-\langle\mathbf{v}_{1},\mathbf{u}\rangle$, by the choice of
$\mathcal{T}^{\prime}$. Thus $f_{\mathcal{T}}$ is divisible by
$1-\langle\mathbf{v}_{1},\mathbf{u}\rangle$, and can be canceled out in the
expression for $F_{\mathcal{P}}(\mathbf{u})$.
The case of arbitrary homogeneous $\rho$ follows immediately by applying
Theorem 1 to the already covered case $\rho\equiv 1$. ∎
###### Proof of Proposition 9.
First we show that for an arbitrary finite spanning set
$S\subset\mathbb{R}^{d}$ the space $\mathfrak{M}^{\Delta}(S)$ has a basis of
$d$-dimensional simplices containing a fixed vertex $\mathbf{v}\in S$. In
particular, the set of all $d$-dimensional simplices containing $\mathbf{v}$
spans $\mathfrak{M}^{\Delta}(S)$ but is not necessarily a basis. Consequently,
their Fantappiè transformations spans $\mathfrak{F}^{\Delta}(S)$. The
following result is formulated as Theorem 4.2 of [Al1] and in a different form
in [AlGelZel]. (We omit the proof of this statement here.)
Given two points $p$ and $q$ and a set $M$ in $\mathbb{R}^{d}$, we say that
$q$ is visible from $p$ with respect to $M$ if the straight interval $pq$ is
disjoint from $M$.
###### Lemma 15.
Given a $d$-dimensional simplex $\sigma\subset\mathbb{R}^{d}$ denote by
$\mathcal{V}(\sigma)$ the set of vertices of $\sigma$. Let $\sigma^{0}$ be the
interior of $\sigma$. Let $p$ be any point in $\mathbb{R}^{d}$ and let $Q^{+}$
(resp. $Q^{-}$) be the set of all $(d-1)$-dimensional faces of $\sigma$ which
are visible (resp. not visible) from $p$ with respect to $\sigma^{0}$. Then
the standard measures of all $d$-dimensional simplices with vertices in
$\mathcal{V}(\sigma)\cup\\{p\\}$ satisfy
$\mu_{\sigma}=\sum_{\sigma_{i}\in Q^{+}}\mu_{\sigma_{i},p}-\sum_{\sigma_{i}\in
Q^{-}}\mu_{\sigma_{i},p},$
where $\mu_{\sigma_{i},p}$ is the standard measure of the $d$-dimensional
simplex spanned by the vertices of $\sigma_{i}$ and the point $p$.
###### Remark 11.
If $\sigma_{i,p}$ is a degenerate simplex, i.e. $p$ lies in the hyperplane
spanned by $\sigma_{i}$, we simply exclude the corresponding term
$\mu_{\sigma_{i,p}}$ from the above formula.
To prove Proposition 9, we need to show that $\mathfrak{F}^{\Delta}(S)$
coincides with $\mathfrak{Rat}(S)$ if and only if $S$ is strongly non-
degenerate. Indeed, $\mathfrak{F}^{\Delta}(S)\subseteq\mathfrak{Rat}(S)$ for
an arbitrary spanning $S$, by Proposition 6. The Fantappiè transform
$F_{\mu}:\mathfrak{M}^{\Delta}(S)\to\mathfrak{F}^{\Delta}(S)$ is a linear
isomorphism which implies that
$\dim\mathfrak{M}^{\Delta}(S)=\dim\mathfrak{F}^{\Delta}(S)$. By Lemma 15 the
space $\mathfrak{M}^{\Delta}(S)$ is spanned by the standard measures
$\mu_{\Delta}$ of the set $\mathcal{B}_{i}$ of all $d$-dimensional simplices
containing the fixed vertex $\mathbf{v}_{i}$. Let us fix the vertex
$\mathbf{v}_{{}_{N}}$ and consider the set $\mathcal{B}_{{}_{N}}$. For $S$
strongly non-degenerate the cardinality of $\mathcal{B}_{{}_{N}}$ equals
$\binom{N-1}{d}$.
Now we show that $\mathfrak{F}^{\Delta}(S)=\mathfrak{Rat}(S)$, where
$\mathfrak{Rat}(S)$ has the dimension $\binom{N-1}{d}$, as it is isomorphic to
the space $Pol(N-d-1,d)$ of all $d$-variate polynomials of degree at most
$N-d-1$. This would immediately imply that the standard measures of simplices
in $\mathcal{B}_{{}_{N}}$ are linearly independent.
###### Lemma 16.
If $S$ is strongly non-degenerate, then
$\mathfrak{F}^{\Delta}(S)=\mathfrak{Rat}(S)$.
###### Proof.
We recall that $\mathfrak{F}^{\Delta}(S)$ comprises all linear combinations of
the rational functions
$\frac{d!\mathrm{Vol}(\mathbf{v}_{i_{1}},\dots,\mathbf{v}_{i_{d}},\mathbf{v}_{{}_{N}})}{(1-\langle\mathbf{v}_{i_{1}},\mathbf{u}\rangle)\cdot\dots\cdot(1-\langle\mathbf{v}_{i_{d}},\mathbf{u}\rangle)\cdot(1-\langle\mathbf{v}_{{}_{N}},\mathbf{u}\rangle)}.$
For each term $1-\langle\mathbf{v}_{i},\mathbf{u}\rangle$ we consider a
(homogeneous) linear form
$l_{i}(u_{0},\mathbf{u})=u_{0}-\langle\mathbf{v}_{i},\mathbf{u}\rangle$ in
$d+1$ variables $u_{0},\dots,u_{d}$. Set $n=N-1$ where $n\geq d+1$. For the
$n$-tuple $\mathfrak{L}=\\{l_{1},l_{2},....,l_{n}\\}$ of linear
$(d+1)$-variate forms, let $V_{\mathfrak{L}}$ be the linear span of all
possible products of the form $l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d}},\;1\leq
i_{1}<i_{2}<...<i_{n-d}\leq n$. Observe that $V_{\mathfrak{L}}$ is the space
of all numerators that one can obtain in $\mathfrak{F}^{\Delta}(S)$. We need
to show that $V_{\mathfrak{L}}$ contains $HPoly(n-d,d+1)$, the space of all
$d+1$-variate homogeneous polynomials of degree $n-d$. Recall that any
$d+1$-tuple of linear forms $l_{i_{1}},\dots,l_{i_{d+1}}$ is linearly
independent due to the strong degeneracy assumption. Thus we can express each
single variable $u_{0},\dots,u_{d}$ as a linear combination of these forms.
Since $V_{\mathfrak{L}}$ contains all products $l_{i_{1}}l_{i_{2}}\dots
l_{i_{n-d-1}}l_{j}$, where
$j\in\\{1,\dots,n\\}\setminus\\{i_{1},\dots,i_{n-d-1}\\}$, we conclude that
$V_{\mathfrak{L}}$ contains all homogeneous polynomials of the form
$l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d-1}}u_{k},\quad\text{ for $0\leq k\leq d$.}$
From that we deduce that $V_{\mathfrak{L}}$ contains all homogeneous
polynomials of the form $l_{i_{1}}l_{i_{2}}\dots l_{i_{n-d-2}}u_{k}u_{j}$,
where $j,k\in[n]\setminus\\{i_{1},\dots,i_{n-d-1}\\}$. Continuing along the
same lines we derive by induction that $V_{\mathfrak{L}}$ contains
$HPoly(n-d,d+1)$. ∎
For an arbitrary spanning $S$ the cardinality of $\mathcal{B}_{{}_{N}}$ is at
most $\binom{N-1}{d}=\dim\mathfrak{Rat}(S)$. Furthermore, if $S$ is not
strongly non-degenerate the cardinality of $\mathcal{B}_{{}_{N}}$ is strictly
smaller than $\binom{N-1}{d}$, as there will be linear dependencies among the
standard measures on the simplices in $B_{{}_{N}}$. Therefore
$\dim\mathfrak{F}^{\Delta}(S)<\dim\mathfrak{Rat}(S)$. ∎
We define the square matrix $Mat_{\mathfrak{L}}$ of size $\binom{n}{d}$ with
entries being coefficients of the above products of linear forms w.r.t. the
standard monomial basis in $HPol(n-d,d+1)$.
###### Lemma 17.
The determinant of $Mat_{\mathfrak{L}}$ is proportional to the product of the
determinants of all $(d+1)$-tuples
$(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}}),\;i_{1}<i_{2}<\ldots<i_{d+1}$. (By
the determinant of a $(d+1)$-tuple of vectors in $\mathbb{R}^{d+1}$ with a
fixed basis we mean the determinant of the matrix formed by the coordinates of
these vectors in a chosen basis.)
###### Proof.
Indeed, $\det(Mat_{\mathfrak{L}})$ is a form of degree $(d+1)\binom{n}{d}$ in
the coefficients of the linear forms $l_{1},\ldots,l_{n}$. Thus the product
$\prod_{i_{1},\ldots,{i_{d+1}}}\det(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}})$
has the same degree as $\det(Mat_{\mathfrak{L}})$. Therefore it suffices to
show that $\det(Mat_{\mathfrak{L}})$ vanishes as soon as some of
$\det(l_{i_{1}},l_{i_{2}},\ldots,l_{i_{d+1}})$ vanishes. (Observe that all
polynomials $\det(Mat_{\mathfrak{L}})$ are coprime.) Without loss of
generality assume that $l_{1}$ is a linear combination of
$l_{2},\ldots,l_{d+1}$. But then the column of $Mat_{\mathfrak{L}}$
corresponding to the $(n-d)$-tuple $(1,d+2,d+3,\ldots,n)$ will be a linear
combination of those corresponding to $(2,d+2,d+3,\ldots,n),$
…$(d+1,d+2,d+3,\ldots,n)$. ∎
###### Proof of Corollary 10.
As we mentioned above, $\mathfrak{M}(S)$ is isomorphic to $\mathfrak{F}(S)$
and, analogously, $\mathfrak{M}^{\Delta}(S)$ is isomorphic to
$\mathfrak{F}^{\Delta}(S)$. Thus, if we prove the equality
$\mathfrak{F}(S)=\mathfrak{F}^{\Delta}(S)$ then we get
$\mathfrak{M}(S)=\mathfrak{M}^{\Delta}(S)$. By Lemma 16, the space
$\mathfrak{F}^{\Delta}(S)$ coincides with the linear space of all rational
functions with the numerator an arbitrary polynomial of degree at most $N-d-1$
and the denominator $\Phi_{S}(\mathbf{u})$ equal to the product of all linear
forms dual to all vertices in $S$. By Proposition 6 an arbitrary function in
$\mathfrak{F}(S)$ is a rational function with denominator of desired form and
numerator of degree at most $N-d-1$, for obvious reasons—take an arbitrary
dissection and sum over its simplices. Since all such functions are already in
$\mathfrak{F}^{\Delta}(S)$ we are done. ∎
###### Proof of Theorem 12.
Given a strongly non-degenerate set
$S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N-1},\mathbf{v}_{N}\\}$ and the
Fantappiè transform $R(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where
$\Phi_{S}(\mathbf{u})=\prod_{j=1}^{N}l_{j}(\mathbf{u})$, we want to solve the
inverse moment problem. (It is easy to obtain $P(\mathbf{u})$ from information
on the moments of order at most $N-d-1$ using Lemma 11.)
To solve the latter inverse problem using Corollary 3, we need to find an
appropriate set of weights
$\mathbf{w}=\\{w_{i_{1},\ldots,i_{d}}\\},\;i_{1}<i_{1}<\ldots<i_{d}$, where
$w_{i_{1},\ldots,i_{d}}$ is the weight (recall Definition 1.14) of the
$d$-dimensional simplex
$\mathrm{conv}(\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\ldots,\mathbf{v}_{i_{d}},\mathbf{v}_{N})$
so that
$\sum_{i_{1}<i_{2}<\ldots<i_{d}}\frac{w_{i_{1},\ldots,i_{d}}}{l_{i_{1}}(\mathbf{u})l_{i_{2}}(\mathbf{u})\dots
l_{i_{d}}(\mathbf{u})l_{{}_{N}}(\mathbf{u})}=\frac{P(\mathbf{u})}{l_{1}(\mathbf{u})l_{2}(\mathbf{u})\dots
l_{{}_{N}}(\mathbf{u})}.$
Clearing the denominators, we get the equation
$\sum_{i_{1}<i_{2}<\ldots<i_{d}}{w_{i_{1},\ldots,i_{d}}}{l_{j_{1}}(\mathbf{u})l_{j_{2}}(\mathbf{u})\dots
l_{j_{N-d-1}}(\mathbf{u})l_{N}(\mathbf{u})}={P(\mathbf{u})},$
where $\\{j_{1},\ldots
j_{N-d-1}\\}=\\{1,2,\ldots,N-1\\}\setminus\\{i_{1},\ldots,i_{d}\\}$. The
latter equation is obviously equivalent to the system of linear equations
$Mat_{S}\cdot\mathbf{w}=(p_{I_{1}},\dots,p_{I_{t}}),\quad\text{where\
}P(\mathbf{u})=\sum_{I}p_{I}\mathbf{u}^{I},$
and $\mathbf{w}=\\{w_{i_{1},\ldots,i_{d}}\\}$ is the vector consisting of the
weights of all simplices containing $\mathbf{v}_{N}$. ∎
Theorem 12 solves the inverse moment problem for strongly non-degenerate
spanning set $S$. We can make this solution more explicit by giving a closed
formula for the inverse matrix $Mat_{S}^{-1}$. To do this we introduce an
extra variable $u_{0}\in\mathbb{R}$ and identify the space $Pol(N-d-1,d)$ with
the space $HPol(N-d-1,d+1)$ of homogeneous forms of degree $N-d-1$ in $d+1$
variables $(u_{0},u_{1},\dots,u_{d})$. We homogenize each linear form
$l_{i}(\mathbf{u})$ in $\mathfrak{L}$ as
$l_{i}(\mathbf{u},u_{0})=u_{0}-\langle\mathbf{v}_{i},\mathbf{u}\rangle$. (The
matrix $Mat_{S}$ remains unchanged.)
We also need the following $(d+1)\times(N-1)$ matrix $\mathbb{L}$
$\mathbb{L}=\bordermatrix{~{}&l_{1}&l_{2}&\dots&l_{N-1}\cr
u_{0}&1&1&\dots&1\cr
u_{1}&-\langle\mathbf{v}_{1},e_{1}\rangle&-\langle\mathbf{v}_{2},e_{1}\rangle&\dots&-\langle\mathbf{v}_{N-1},e_{1}\rangle\cr~{}\vdots&\vdots&\vdots&&\vdots\cr
u_{d}&-\langle\mathbf{v}_{1},e_{d}\rangle&-\langle\mathbf{v}_{2},e_{d}\rangle&\dots&-\langle\mathbf{v}_{N-1},e_{d}\rangle\cr}$
(3.1)
associated with $Mat_{S}$. For all possible subsets
$\mathbf{i}[d]=\\{i_{1},\dots,i_{d}\\},$ $i_{1}<i_{2}<\dots<i_{d}$ of $d$
distinct columns of $\mathbb{L}$ consider the linear in $\mathbf{u}$ function
$\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})$ given by:
$\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})=\text{det}\begin{bmatrix}u_{0}&1&\dots&1\\\
u_{1}&-\langle\mathbf{v}_{i_{1}},e_{1}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{1}\rangle\\\
\vdots&&\dots&\\\
u_{d}&-\langle\mathbf{v}_{i_{1}},e_{d}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{d}\rangle\end{bmatrix}.$
(3.2)
Denote by $\mathbb{L}_{\mathbf{i}[d],j}$ the coefficient of $u_{j}$ in the
linear form $\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})$. For each $1\leq j\leq
N-1$ define $\mathbb{L}(j,\mathbf{i}[d])$ as
$\mathbb{L}(j,\mathbf{i}[d])=\text{det}\begin{bmatrix}1&1&\dots&1\\\
-\langle\mathbf{v}_{j},e_{1}\rangle&-\langle\mathbf{v}_{i_{1}},e_{1}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{1}\rangle\\\
\vdots&&\dots&\\\
-\langle\mathbf{v}_{j},e_{d}\rangle&-\langle\mathbf{v}_{i_{1}},e_{d}\rangle&\dots&-\langle\mathbf{v}_{i_{d}},e_{d}\rangle\end{bmatrix}.$
(3.3)
Note that if $j\notin\mathbf{i}[d]$ then $\mathbb{L}(j,\mathbf{i}[d])\neq 0$,
as by the assumption of strong non-degeneracy of $S$ the corresponding $d+1$
linear forms are linearly independent. On the other hand, if
$j\in\mathbf{i}[d]$ then we have $\mathbb{L}(j,\mathbf{i}[d])=0.$
The matrix $Mat_{S}^{-1}$ has the following explicit description.
###### Lemma 18.
For each $(N-d-1)$-tuple of forms
$\\{l_{j_{1}},l_{j_{2}},\dots,l_{j_{N-d-1}}\\}$ set
$\mathbf{i}[d]=\\{i_{1},\dots,i_{d}\\}=[N-1]\setminus\\{j_{1},\dots,j_{N-d-1}\\},$
where $i_{1}<i_{2}\dots<i_{d}.$ Then,
$Mat_{S}^{-1}=\bordermatrix{~{}&\dots&u_{0}^{n_{0}}u_{1}^{n_{1}}\dots
u_{d}^{n_{d}}&\dots\cr\quad\vdots&&\quad\vdots&\cr l_{j_{1}}\dots
l_{j_{N-d-1}}&\dots&\frac{\prod\limits_{j=0}^{d}\mathbb{L}_{\mathbf{i}[d],j}^{n_{j}}}{\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])}&\dots\cr\quad\vdots&&\quad\vdots&\cr}.$
(3.4)
###### Proof of Lemma 18.
In order to show that $Mat_{S}^{-1}$ defined by (3.4) is indeed the inverse of
$Mat_{S}$ we need to verify that $Mat_{S}^{-1}\cdot Mat_{S}$ is the identity
operator on $V_{\mathfrak{L}}$.
Let $\mathbf{e}^{\prime}$ be the standard basis vector of $V_{\mathfrak{L}}$
corresponding to the product of linear forms $l_{j^{\prime}_{1}}\dots
l_{j^{\prime}_{N-d-1}}$. Then $Mat_{S}\cdot\mathbf{e}^{\prime}$ is the vector
consisting of the monomial coefficients of the homogeneous form
$l_{j^{\prime}_{1}}\dots l_{j^{\prime}_{N-d-1}}$ in the variables
$u_{0},\dots,u_{d}$. Let $\mathbf{e}^{T}$ be the row vector of $Mat_{S}^{-1}$
corresponding to the product $l_{j_{1}}\dots l_{j_{N-d-1}}$. We note that in
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}^{\prime}\right)$ one can
factor out the common denominator
$\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])$ of all fractions
in $\mathbf{e}^{T}$; the remaining factor is of the form
$\sum\limits_{\begin{subarray}{c}I=(n_{0},\dots,n_{d})\\\
|I|=N-d-1\end{subarray}}Mat_{S}[\mathbf{u}^{I},\mathbf{e}^{\prime}]\prod\limits_{j=0}^{d}\mathbb{L}_{\mathbf{i}[d],j}^{n_{j}}=l_{j^{\prime}_{1}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u}))\dots
l_{j^{\prime}_{N-d-1}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u})).$ (3.5)
Note that
$l_{j^{\prime}_{k}}(\mathbb{L}_{\mathbf{i}[d]}(\mathbf{u}))=\mathbb{L}(j^{\prime}_{k},\mathbf{i}[d])$
for $1\leq k\leq N-d-1$, i.e. this holds for all terms in the product on the
right-hand side of (3.5). Hence, if $\mathbf{e}\neq\mathbf{e}^{\prime}$ then
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}^{\prime}\right)=0$, as among
$j^{\prime}_{1},\dots,j^{\prime}_{N-d-1}$ one can find
$j^{\prime}\in\mathbf{i}[d]$ with $\mathbb{L}(j^{\prime},\mathbf{i}[d])=0$. On
the other hand, if $\mathbf{e}$ and $\mathbf{e}^{\prime}$ coincide, then the
right-hand side of (3.5) is equal to
$\prod\limits_{k=1}^{N-d-1}\mathbb{L}(j_{k},\mathbf{i}[d])$. Dividing by the
common denominator of the fractions in $\mathbf{e}$, we obtain
$\mathbf{e}^{T}\cdot\left(Mat_{S}\cdot\mathbf{e}\right)=1.$ ∎
## 4\. Inverse moment problem for weakly non-degenerate $S$
Given an arbitrary spanning set
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{N}\\}$, consider the
linear space $\Theta(S)\subseteq Pol(N-d-1,d)$ spanned by all products
$l_{j_{1}}l_{j_{2}}\ldots l_{j_{N-d-1}},\;j_{1}<j_{2}<\ldots<j_{N-d-1}$. The
next statement explains why we can extend our solution of the inverse moment
problem from the case of strongly non-degenerate $S$ to the case of weakly
non-degenerate $S$.
###### Lemma 19.
$\Theta(S)=Pol(N-d-1,d)$ if and only if $S$ is weakly non-degenerate, i.e.
each $(d+2)$-tuple of points of $S$ is spanning.
###### Proof.
We have $N$ (non-homogeneous) linear forms $l_{1}\dots,l_{N}$ in variables
$\mathbf{u}=(u_{1},\dots,u_{d})$ and the linear space $V_{\mathfrak{L}}$
spanned by all possible products of $(N-d-1)$-tuples of distinct forms. We
need to investigate whether $V_{\mathfrak{L}}$ coincides with $Pol(N-d-1,d)$.
Homogenizing, we consider the same question for the linear homogeneous forms
and the homogeneous polynomials of degree $N-d-1$ in variables
$(u_{0},u_{1},\dots,u_{d})$.
First assume that there are $d+2$ linear forms $l_{1},\dots,l_{d+2}$ which are
not spanning. Then one can find a non-zero vector
$\mathbf{z}_{0}\in\mathbb{R}^{d+1}$, such that $l_{1}(z)=\dots=l_{d+2}(z)=0$.
Note that each product of $N-d-1$ different forms chosen from
$l_{1},\dots,l_{N}$ contains at least one form among
$\\{l_{1},\dots,l_{d+2}\\}$. Therefore any linear combination of products of
$N-d-1$ forms vanishes at $\mathbf{z}_{0}$. Thus $V_{\mathfrak{L}}$ cannot
coincide with $HPol(N-d-1,d+1)$.
Conversely, assume that every $(d+2)$-tuple of distinct forms among
$l_{1},\dots,l_{N}$ is spanning. First, we notice that $HPol(N-d-1,d+1)$ can
be spanned by the all possible products of $N-d-1$ linear forms (not
necessarily pairwise distinct). Indeed, since first $d+2$ forms span the dual
space of $\mathbb{R}^{d+1}$, we can express each variable $x_{i}$ as a linear
combination of these forms. Therefore every monomial of degree $N-d-1$ can be
expressed as a linear combination of products of $N-d-1$ forms.
Now we show that each product of $N-d-1$, not necessarily distinct, forms can
be expressed as a linear combination of the products of distinct ones. Assume
the contrary and consider monomials $l_{1}^{i_{1}}\dots l_{N}^{i_{N}}$ of
degree $N-d-1$ which cannot be expressed as a linear combination of products
with all distinct forms. Among those monomials we take a monomial
$\mathbf{m}=l_{1}^{k_{1}}\dots l_{N}^{k_{N}}$ having the maximal number of
distinct forms in the product. Since $\mathbf{m}$ is not a product of all
distinct forms, it should contain a form $l_{i}$ in some power $k_{i}\geq 2$.
Given that $k_{i}\geq 2$ and the degree of $\mathbf{m}$ is $N-d-1$, one can
find $d+2$ distinct forms $l_{i_{1}},\dots,l_{i_{d+2}}$ that do not appear in
$\mathbf{m}$. Since any $d+2$ of our forms span the dual space of
$\mathbb{R}^{d+1}$, we can express $l_{i}$ as a linear combination of
$l_{i_{1}},\dots,l_{i_{d+2}}$. Now rewrite $\mathbf{m}$ as
$\left(\alpha_{1}\cdot l_{i_{1}}+\dots+\alpha_{d+2}\cdot
l_{i_{d+2}}\right)l_{1}^{k_{1}}\dots l_{i}^{k_{i}-1}\dots l_{N}^{k_{N}}$,
where $\alpha_{1}\cdot l_{i_{1}}+\dots+\alpha_{d+2}\cdot l_{i_{d+2}}=l_{i}.$
Thus we get an expression of $\mathbf{m}$ as a linear combination of monomials
$\alpha_{j}\cdot l_{i_{j}}l_{1}^{k_{1}}\dots l_{i}^{k_{i}-1}\dots
l_{N}^{k_{N}}$, where each such monomial has more distinct forms than
$\mathbf{m}$. Each of such monomials can be expressed as a linear combination
of products of all distinct forms, since $\mathbf{m}$ was chosen as a monomial
with the maximal possible number of distinct forms, which cannot be expressed
in such a way. This is a contradiction. Therefore $\mathbf{m}$ can also be
expressed as a linear combination of products of all distinct forms. ∎
Below we consider the inverse problem for a weakly non-degenerate $S$, using
notation from (3.1) and (3.3). Here we no longer have a natural basis of all
simplices sharing a common vertex $\mathbf{v}_{N}$. Because of that we need to
consider all $N$ points and include one more linear form $l_{N}$ into the
corresponding matrix $\mathbb{L}$. Slightly abusing our notation, we denote by
$\mathbb{L}$ the same matrix as before, although it contains one more (last)
column corresponding to $\mathbf{v}_{N}$. Similarly to notation (3.3), for a
given set $J$ of $d+1$ linear forms we denote by $\mathbb{L}(J)$ the
determinant of the corresponding $(d+1)\times(d+1)$-minor of $\mathbb{L}$.
We introduce the extended $\binom{N-1}{d}\times\binom{N}{d+1}$-matrix
$\widetilde{Mat}_{S}$ with columns consisting of the coefficients of the
homogeneous polynomial $l_{i_{1}}(\mathbf{u})\dots l_{i_{N-d-1}}(\mathbf{u})$
with respect to the monomial basis in the variables
$(u_{0},u_{1},\dots,u_{d}).$ By Lemma 19, $\widetilde{Mat}_{S}$ has full rank,
since it determines a surjective linear map onto $HPol(N-d-1,d+1)$. Thus
$\widetilde{Mat}_{S}$ has a maximal minor with a non-vanishing determinant.
Formula (4.1) holds for the determinant of any maximal minor of
$\widetilde{Mat}_{S}$.
###### Lemma 20.
Let $\mathfrak{S}$ be any set of $\binom{N-1}{d}$ columns of
$\widetilde{Mat}_{S}$. We label each column $T\in\mathfrak{S}$ by the
corresponding subset of the linear forms $l_{1},\dots,l_{N}$ of cardinality
$N-d-1$. Then the determinant of the maximal minor
$\widetilde{Mat}_{S}(\mathfrak{S})$ formed by the columns of $\mathfrak{S}$ is
given by:
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]=k(\mathfrak{S})\cdot\prod_{\begin{subarray}{c}J\in{N\brack
d+1}:\\\ \forall T\in\mathfrak{S}~{}T\cap
J\neq\emptyset\end{subarray}}\mathbb{L}(J),$ (4.1)
where $k(\mathfrak{S})$ is a constant (possibly equal to zero) depending only
on the combinatorial structure of the $(N-d-1)$-tuples in the set
$\mathfrak{S}$.
###### Proof.
Fix the set $\mathfrak{S}$ as above. In what follows we treat both sides of
(4.1) as complex-valued polynomials in $N\cdot(d+1)$ variables, these
variables being the entries of matrix $\mathbb{L}$.
We first show that every determinant $\mathbb{L}(J)$ divides
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$. Indeed, let
$J=\\{j_{1},\dots,j_{d+1}\\}$ be a set of $(d+1)$ forms which has a nonempty
intersection with any $(N-d-1)$-tuple of forms in $\mathfrak{S}$. Let
$\mathbf{z}=(z_{1},\dots,z_{N\cdot(d+1)})$ be a zero of the polynomial
$\mathbb{L}(J)$, which means that forms $l_{j_{1}},\dots,l_{j_{d+1}}$
comprised of the corresponding coordinates of $\mathbf{z}$ are linearly
dependent. Therefore, there is a non-zero vector
$\mathbf{u}_{0}\in\mathbb{R}^{d+1}$, such that
$l_{j_{1}}(\mathbf{u}_{0})=\dots=l_{j_{d+1}}(\mathbf{u}_{0})=0$. Consider the
row vector $(\mathbf{u}_{0}^{I})$ consisting of $\binom{N-1}{d}$ homogeneous
monomials of degree $N-d-1$ evaluated at $\mathbf{u}_{0}.$ We notice that
$(\mathbf{u}_{0}^{I})$ is in the kernel of
$\widetilde{Mat}_{S}(\mathfrak{S})$, as the product of $(\mathbf{u}_{0}^{I})$
with each column vector $T\in\mathfrak{S}$ of
$\widetilde{Mat}_{S}(\mathfrak{S})$ is equal to $\prod_{j\in
T}l_{j}(\mathbf{u}_{0})$; and every set $T\in\mathfrak{S}$ contains at least
one of the forms $l_{j_{1}},\dots,l_{j_{d+1}}$ in such a product. Thus
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$ also vanishes at such
$\mathbf{z}$.
We recall a well-known fact (cf. e.g. [Boch]*Theorem 61.1) that
$\mathbb{L}(J)=\det\bordermatrix{~{}&l_{j_{1}}&l_{j_{2}}&\dots&l_{j_{d+1}}\cr
u_{0}&z_{j_{1}}&z_{j_{2}}&\cdots&z_{j_{d+1}}\cr
u_{1}&z_{j_{1}+N}&z_{j_{2}+N}&\cdots&z_{j_{d+1}+N}\cr\vdots&\vdots&\vdots&\ddots&\vdots\cr
u_{d}&z_{j_{1}+dN}&z_{j_{2}+dN}&\cdots&z_{j_{d+1}+dN}\cr},$
is an irreducible complex-valued polynomial in variables
$z_{j_{1}},z_{j_{2}}\dots,z_{j_{d+1}+Nd}$. Now if every zero of an irreducible
polynomial $p(z_{1},\dots,z_{N(d+1)})$ annihilates another polynomial
$q(z_{1},\dots,z_{N(d+1)})$ then $p$ divides $q$. We conclude that
$\mathbb{L}(J)$ divides $\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$.
Using the fact that each $\mathbb{L}(J)$ is an irreducible polynomial and all
$\mathbb{L}(J)$’s are pairwise distinct (i.e. have distinct sets of projective
zeros) we conclude that the product of $\mathbb{L}(J)$’s in the right-hand
side of (4.1) divides $\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right].$
Finally, the product of $\mathbb{L}(J)$’s has the degree
$(d+1)\left|\left.\left\\{J\in{N\brack d+1}\right|\forall
T\in\mathfrak{S}\quad T\cap J\neq\emptyset\right\\}\right|.$ (4.2)
We observe that for each $T\in\mathfrak{S}$ the complementary set of $d+1$
forms cannot be taken as a feasible $J$. We notice further that these
complements are the only exceptions for the choice of $J$. Therefore as a
feasible $J$ we can pick any of $\binom{N}{d+1}$ $(d+1)$-tuples except those
$\binom{N-1}{d}$ complements of a $T\in\mathfrak{S}$. Therefore, (4.2) equals
$(d+1)\left(\binom{N}{d+1}-\binom{N-1}{d}\right)=(d+1)\binom{N-1}{d+1}=(N-d-1)\binom{N-1}{d}.$
The latter expression coincides with the degree of the polynomial
$\det\left[\widetilde{Mat}_{S}(\mathfrak{S})\right]$ (assuming that it is not
a zero), as $\widetilde{Mat}_{S}(\mathfrak{S})$ has $\binom{N-1}{d}$ columns
and each entry is a homogeneous polynomial of degree $N-d-1$.
Hence $\widetilde{Mat}_{S}(\mathfrak{S})$ coincides with the product of
$\mathbb{L}(J)$’s up to a constant factor which might vanish. This constant
does not depend on the entries of matrix $\mathbb{L}$ and hence it is
completely determined by the set $\mathfrak{S}$, regardless of the location of
points of $S$ in $\mathbb{R}^{d}$. ∎
Lemma 19 allows us to solve the inverse moment problem for a given weakly non-
degenerate $S=\\{\mathbf{v}_{1},\dots,\mathbf{v}_{N}\\}$ in a certain linear
space $\widetilde{\mathfrak{M}}(S)\supseteq\mathfrak{M}^{\Delta}(S)$ of
measures supported on $\mathrm{conv}(S)$. Namely,
$\widetilde{\mathfrak{M}}(S)$ is spanned by measures
$\mu\in\widetilde{\mathfrak{M}}(S)$ whose normalized moment generating
functions $F_{\mu}(\mathbf{u})$ belong to $\mathfrak{Rat}(S)$, i.e.
$F_{\mu}(\mathbf{u})=P(\mathbf{u})/\Phi_{S}(\mathbf{u})$, where
$\Phi_{S}(\mathbf{u})=\prod_{j=1}^{N}l_{j}(\mathbf{u})$ and $P(\mathbf{u})$ is
a polynomial of degree at most $N-d-1$. Indeed, by Lemma 19 any
$R(\mathbf{u})\in\mathfrak{Rat}(S)$ can be represented in the form
$R(\mathbf{u})=\sum_{i_{1}<i_{2}<\dots<i_{d+1}\leq N}\frac{K_{i_{1}i_{2}\dots
i_{d+1}}}{l_{i_{1}}l_{i_{2}}\dots l_{i_{d+1}}},$ (4.3)
with some real constants $K_{i_{1}i_{2}\dots i_{d+1}}$. If
$\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$ span
$\mathbb{R}^{d}$ then the term $\frac{K_{i_{1}i_{2}\dots
i_{d+1}}}{l_{i_{1}}l_{i_{2}}\dots l_{i_{d+1}}}$ can be interpreted as the
normalized moment generating function of an appropriately scaled standard
measure of the $d$-dimensional simplex spanned by these vertices.
If $\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$ only
span a hyperplane $H$ in $\mathbb{R}^{d}$ then (4.3) corresponds to a singular
(w.r.t. to the Lebesgue measure on $\mathbb{R}^{d}$) measure $\mu_{\delta}$
supported on
$\delta=\mathrm{conv}(\mathbf{v}_{i_{1}},\dots,\mathbf{v}_{i_{d+1}})$. One way
to define it as the weak limit of a sequence of (absolutely continuous with
respect to the Lebesgue measure on $\mathbb{R}^{d}$) measures—the
appropriately scaled standard measures $\mu_{\delta_{t}}$ of family of
$d$-dimensional simlices $\delta_{t}$ which degenerate into $\delta$ when
$t=0$. There is no loss in generality in assuming $K_{i_{1}i_{2}\dots
i_{d+1}}=1$, i.e. to deal with probability measures.
###### Proposition 21.
Let $\mathbf{W}=\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d},\mathbf{w}_{d+1}\\}$ be
a $(d+1)$-tuple of points in $\mathbb{R}^{d}$ such that $\mathbf{W}$ spans a
hyperplane $H\subset\mathbb{R}^{d}$. Denote by
$l_{\mathbf{w}_{1}}=1-\langle\mathbf{w}_{1},\mathbf{u}\rangle,\dots,l_{\mathbf{w}_{d+1}}=1-\langle\mathbf{w}_{d+1},\mathbf{u}\rangle$
the associated linear forms. There exists a unique measure $\mu_{\mathbf{W}}$
supported on $\delta=\mathrm{conv}({\mathbf{W}})$ with the normalized moment
generating function $F_{\mu_{\mathbf{W}}}(\mathbf{u})$ given by
$F_{\mu_{\mathbf{W}}}(\mathbf{u})=\frac{1}{l_{\mathbf{w}_{1}}l_{\mathbf{w}_{2}}\dots
l_{\mathbf{w}_{d+1}}}.$ (4.4)
###### Proof.
Without loss of generality assume that
$\mathbf{W}=\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d},\mathbf{w}_{d+1}\\}$ is
ordered in such a way that $\\{\mathbf{w}_{1},\dots,\mathbf{w}_{d}\\}$ span
$H$. Then, $\delta_{t}$ is defined as
$\delta_{t}=\mathrm{conv}(\delta,\mathbf{w}_{i_{d+1}}+t\mathbf{z})$, with
$\mathbf{z}$ a unit normal to $H$, and $\mu_{\delta_{t}}$ as the uniform
density probability measure supported on $\delta_{t}$. Then $\lim\limits_{t\to
0}\mu_{\delta_{t}}=\mu_{\delta}$, where $\lim$ is understood in sense of _weak
convergence of distributions (measures)_ , i.e. that $\lim\limits_{t\to 0}\int
fd\mu_{\delta_{t}}=\int fd\mu_{\delta}$ for any bounded, continuous real
function on $\mathbb{R}^{d}$, cf. e.g. [MR1700749]. Then, this measure has
compact support, and thus is determined by its moments, cf. e.g.
[MR2244695]*Proposition 3.2. ∎
###### Remark 12.
One can prove that the integration of a smooth compactly supported function
$\phi$ with respect to the limiting measure $\mu_{\mathbf{W}}$ is given by the
integration of $\phi$ over $\delta$ with a continuous piecewise linear weight
function uniquely determined by $\delta$. Similar limits appear frequently in
the theory of splines. Since we only need the existence of $\mu_{\mathbf{W}}$
we do not pursue this topic here.
Our solution of the inverse moment problem for the linear space
$\widetilde{\mathfrak{M}}(S)$ closely follows the pattern presented in Example
2. In other words, given a weakly non-degenerate $S$ and the set of moments up
to order $N-d-1$ we
1. (i)
produce the rational function $R(\mathbf{u})\in\mathfrak{Rat}(S)$ with Taylor
coefficients coinciding with the normalized moments;
2. (ii)
represent $R(\mathbf{u})$ in the form (4.3);
3. (iii)
for each term as in (4.4) determine the underlying measure supported on the
(probably degenerate) convex hull of the vertices
$\mathbf{v}_{i_{1}},\mathbf{v}_{i_{2}},\dots,\mathbf{v}_{i_{d+1}}$.
We can now prove our central result claiming that
$\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$ for a weakly non-degenerate $S$.
###### Proof of Theorem 8.
Theorem 8 is already settled in Corollary 10 for the case of strongly non-
degenerate $S$. It remains to consider the case of weakly non-degenerate $S$.
The denominator of the moment generating function
$F_{\mathcal{P}}(\mathbf{u})$ for an arbitrary generalized polytope
$\mathcal{P}$ with the vertex set $S$ is of the form $\Pi_{i=1}^{N}l_{i}$ by
Proposition 6, and its numerator belongs to $Pol(N-d-1,d)$. As $S$ is weakly
non-degenerate, $F_{\mathcal{P}}(\mathbf{u})$ can be written as a linear
combination of the fractions as in (4.3), where $(i_{1},i_{2},\ldots i_{d+1})$
runs over the set of $(d+1)$-tuples of indices. If a $(d+1)$-tuple
$l_{i_{1}},l_{i_{2}},\ldots l_{i_{d+1}}$ is spanning then
$\frac{K}{l_{i_{1}}l_{i_{2}}\ldots l_{i_{d+1}}}$ is the moment generating
function of the measure supported on the simplex $\Delta$, determined by its
denominator, with the uniform density $K/d!\text{Vol}(\Delta)$. By Proposition
21, if a $(d+1)$-tuple $l_{i_{1}},l_{i_{2}},\ldots l_{i_{d+1}}$ is not
spanning then $\frac{K}{l_{i_{1}}l_{i_{2}}\ldots l_{i_{d+1}}}$ is the moment
generating function of a singular measure supported on a degenerate simplex.
As $\mathcal{P}$ is a generalized polytope, its standard measure has no
singular components. Therefore, no degenerate simplices can appear in its
decomposition. ∎
###### Remark 13.
The latter proof demonstrates that if one starts from the set of moments of
the standard measure $\mu$ of a polytope with the vertex set $S$ then we never
obtain degenerate simplices while solving the inverse moment problem. This is
why $\mathfrak{M}^{\Delta}(S)=\mathfrak{M}(S)$. However, an explicit
description of $\mathfrak{F}^{\Delta}(S)$ for a general weakly non-degenerate
$S$ is missing at present. For concrete Examples 3 and 4 we give these
descriptions below.
Our final result computes $\dim\mathfrak{M}^{\Delta}(S)$ and describes a
procedure to construct a basis for $\mathfrak{M}^{\Delta}(S)$.
###### Proposition 22.
Let $S=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\\}\subset\mathbb{R}^{d}$ be an
arbitrary weakly non-degenerate spanning set. Then
1. (i)
$\dim\mathfrak{M}^{\Delta}(S)=\binom{N-1}{d}-\sharp_{deg}$ where
$\sharp_{deg}$ is the number of degenerate simplices, i.e. the number of non-
spanning $(d+1)$-tuples of points of $S$.
2. (ii)
If $\delta$ is a degenerate $d$-dimensional simplex with vertices in
$S\setminus\\{\mathbf{v}_{i}\\}$ then there is exactly one linear dependence
among the standard measures of all $d$-dimensional simplices on
$\mathbf{v}_{i}$ and $d$ vertices of $\delta$.
3. (iii)
The standard measure of any $d$-dimensional simplex on $\mathbf{v}_{i}$ is
contained in at most one dependence as in (ii).
4. (iv)
For any vertex $\mathbf{v}_{i}$ one can construct a (in general, non-unique)
basis $\mathcal{B}_{i}$ of $\mathfrak{M}^{\Delta}(S)$ consisting of standard
measures of $d$-dimensional simplices on $\mathbf{v}_{i}$, as follows.
1. (a)
Start from the set $\mathcal{B}_{i}$ of the $d$-dimensional simplices on
$\mathbf{v}_{i}$.
2. (b)
For each degenerate simplex $\delta$ not containing $\mathbf{v}_{i}$, remove
from $\mathcal{B}_{i}$ the standard measure of an arbitrary simplex on
$\mathbf{v}_{i}$ from the corresponding to $\delta$ linear dependence, cf.
(i).
Thus we obtain $\binom{N-1}{d}-\sharp_{deg}$ standard measures of
$d$-dimensional simplices, forming a basis of $\mathfrak{M}^{\Delta}(S)$.
###### Proof.
To prove (i), notice that
$\dim\widetilde{\mathfrak{M}}(S)=\dim\mathfrak{Rat}(S)=\binom{N-1}{d}$. As
well,
$\widetilde{\mathfrak{M}}(S)=\mathfrak{M}(S)\oplus\mathfrak{M}_{deg}(S)$,
where $\mathfrak{M}_{deg}(S)$ is the linear span of the measures
$\mu_{\delta}^{(1)}$ with $\delta$ running over the set of all degenerate
simplices spanned by $(d+1)$-tuples of dependent vertices in $S$, cf.
Proposition 21. Observe that these measures $\mu_{\delta}^{(1)}$ are linearly
independent, as each degenerate simplex defines a singular measure supported
in a proper hyperplane, and these hyperplanes differ for different degenerate
simplices. We are done with (i).
Let $\Sigma_{0}$ be a dependent $d+1$-subset of $S$, and
$\delta=\mathrm{conv}(\Sigma_{0})$ be as in (ii). Then $\delta$ spans a
hyperplane $H_{0}$. As each $d$-dimensional simplex on
$\sigma_{0}:=\mathbf{v}_{i}$ and $d$ vertices from $\Sigma_{0}$ is uniquely
defined by the latter, it suffices to analyze dependencies between the
standard measures of $d-1$-simplices with vertices in $\Sigma_{0}$.
We can view $\Sigma_{0}$ as a weakly non-degenerate subset in
$\mathbb{R}^{d-1}\cong H_{0}$. By (i), we have
$\dim\mathfrak{M}^{\Delta}(\Sigma_{0})=\binom{d}{d-1}-\sharp_{deg}(\Sigma_{0})$.
If $\Sigma_{0}$ is strongly non-degenerate as a subset of
$H_{0}\cong\mathbb{R}^{d-1}$, i.e. $\sharp_{deg}(\Sigma_{0})=0$, then
$\dim\mathfrak{M}^{\Delta}(\Sigma_{0})=d$, i.e. there is exactly one linear
dependence between the standard measures of $d-1$-simplices with vertices in
$\Sigma_{0}$, and we are done. Otherwise,
$\Sigma_{0}=\\{\sigma_{1}\\}\cup\Sigma_{1}$, with $\Sigma_{1}$ spanning a
hyperplane $H_{1}$ in $H_{0}$. Moreover, this can only happen if $d\geq 3$.
Now, we can repeat the whole argument with $\sigma_{1}$ in place of
$\sigma_{0}$, $\Sigma_{1}$ in place of $\Sigma_{0}$, and $H_{1}$ in place of
$H_{0}$. Again, we either have $\Sigma_{1}$ strongly degenerate, and we are
done, or we repeat this argument, etc., until we hit a strongly non-degenerate
$\Sigma_{k}$, which is bound to happen, as the dimension goes down each
iteration. This completes the proof of (ii).
Then, (iii) stems from the fact that the vertices of $d$-dimensional simplex
on $\mathbf{v}_{i}$ distinct from $\mathbf{v}_{i}$ span a hyperplane, and the
only possibility for a degenerate simplex $\delta$ as in (ii) is to lie in
this hyperplane.
Finally, to prove (iv), observe that the set $\mathcal{B}_{i}^{\prime}$ of the
standard measures of $d$-dimensional simplices containing a given vertex
$\mathbf{v}_{i}$ always spans $\mathfrak{M}^{\Delta}(S)$, see Lemma 15. Now
for each degenerate $d$-simplex $\delta$ we prune $\mathcal{B}_{i}^{\prime}$
by removing the standard measure of a simplex in the linear dependence
corresponding to $\delta$. In view of (ii) and (iii) this process is well-
defined and unambiguous. In the end we obtain $\binom{N-1}{d}-\sharp_{deg}$
standard measures of $d$-dimensional simplices. In view of (i) they form a
basis of $\mathfrak{M}^{\Delta}(S)$, as claimed. ∎
###### Remark 14.
The above discussions show that the columns of $\widetilde{Mat}_{S}$
corresponding to degenerate simplices must necessarily be included in any non-
vanishing maximal minor $\widetilde{Mat}_{S}(\mathfrak{S})$.
We conclude our discussion of the weakly non-degenerate case with two
examples.
$\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{2}=(2,0)$$\mathbf{v}_{1}=(1,1)$$\mathbf{v}_{3}=(2,2)$$\mathbf{v}_{4}=(0,2)$$\mathbf{v}_{1}=\mathbf{v}_{5}=(0,0)$$\mathbf{v}_{2}=(2,0)$$\mathbf{v}_{3}=(1,1)$$\mathbf{v}_{4}=(0,2)$
Figure 3. Vertices for Examples 3 and 4.
###### Example 3.
Let
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=(1,1),\mathbf{v}_{2}=(2,0),\mathbf{v}_{3}=(2,2),\mathbf{v}_{4}=(0,2),\mathbf{v}_{5}=(0,0)$.
Then
$l_{1}=1-u_{1}-u_{2},l_{2}=1-2u_{1},l_{3}=1-2u_{1}-2u_{2},l_{4}=1-2u_{2},l_{5}=1$.
Calculating the products $l_{i}l_{j},\,i<j$ and taking their coefficients in
the standard monomial basis of $Pol(2,2)$ we obtain the following $6\times
10$-matrix $\widetilde{Mat}_{S}$
$\widetilde{Mat}_{S}=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{3}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1&1&1&1&1\cr u_{1}&-3&-3&-1&-1&-4&-2&-2&-2&-2&0&\cr
u_{2}&-1&-3&-3&-1&-2&-2&0&-4&-2&-2\cr u_{1}^{2}&2&2&0&0&4&0&0&0&0&0\cr
u_{1}u_{2}&2&4&2&0&4&4&0&4&0&0\cr u_{2}^{2}&0&2&2&0&0&0&0&4&0&0}.$
Its rank equals $6$ and one of non-vanishing maximal minors consists of the
columns with numbers $\mathfrak{S}=\\{5,6,7,8,9,10\\}$. (Recall that any non-
vanishing maximal minor must include columns 6 and 9 corresponding to
degenerate triples $(\mathbf{v}_{1},\mathbf{v}_{3},\mathbf{v}_{5})$ and
$(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{4})$ resp.) The corresponding
submatrix $\widetilde{Mat}_{S}(\mathfrak{S})$ equals
$\widetilde{Mat}_{S}(\mathfrak{S})=\bordermatrix{~{}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1\cr u_{1}&-4&-2&-2&-2&-2&0\cr u_{2}&-2&-2&0&-4&-2&-2&\cr
u_{1}^{2}&4&0&0&0&0&0\cr u_{1}u_{2}&4&4&0&4&0&0\cr u_{2}^{2}&0&0&0&4&0&0}.$
Further,
$4\widetilde{Mat}_{S}^{-1}(\mathfrak{S})=\bordermatrix{~{}&1&u_{1}&u_{2}&u_{1}^{2}&u_{1}u_{2}&u_{2}^{2}\cr
l_{2}l_{3}&0&0&4&0&-4&4\cr l_{2}l_{4}&0&0&0&0&-2&2\cr
l_{2}l_{5}&0&0&2&0&-2&0&\cr l_{3}l_{4}&1&-1&0&0&-1&1\cr
l_{3}l_{5}&0&1&0&0&-1&0&\cr l_{4}l_{5}&0&-1&1&1&-1&0}.$
Thus, given an arbitrary rational function
$R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is any polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we obtain
$\begin{cases}w_{145}=-a_{10}-a_{20}-a_{11}\\\
w_{135}=-\frac{1}{2}(a_{11}-a_{02})\\\ w_{134}=\frac{1}{2}(a_{01}-a_{11})\\\
w_{125}=\frac{1}{4}(a_{00}-a_{10}-a_{11}+a_{02})\\\
w_{124}=\frac{1}{4}(a_{10}-a_{11})\\\
w_{123}=-\frac{1}{4}(a_{10}-a_{01}-a_{20}+a_{11}).\\\ \end{cases}$
Triangles $\Delta_{135}$ and $\Delta_{124}$ are degenerate which implies that
if the original measure we are recovering is polygonal then
$w_{135}=w_{124}=0$. Therefore the linear space of numerators $P(u_{1},u_{2})$
for the space $\mathfrak{F}^{\Delta}(S)$ in this example is given by the
relation
$a_{01}=a_{11}=a_{02}.$
Our last example is more degenerate than the previous one, although still
weakly non-degenerate. In fact, in this example $S$ is a multiset since
$\mathbf{v}_{1}=\mathbf{v}_{5}$. It shows that our technique can be
generalized to a certain class of multisets as well.
###### Example 4.
Let
$S=\\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\\}$
where
$\mathbf{v}_{1}=\mathbf{v}_{5}=(0,0),\mathbf{v}_{2}=(2,0),\mathbf{v}_{3}=(1,1),\mathbf{v}_{4}=(0,2)$.
Then $l_{1}=l_{5}=1,l_{2}=1-2u_{1},l_{3}=1-u_{1}-u_{2},l_{4}=1-2u_{2}$.
Calculating all products $l_{i}l_{j},\,i<j$ and taking their coefficients in
the standard monomial basis of $Pol(2,2)$, we obtain the following $6\times
10$-matrix $\widetilde{Mat}_{S}$:
$\widetilde{Mat}_{S}=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{3}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{2}l_{5}&l_{3}l_{4}&l_{3}l_{5}&l_{4}l_{5}\cr
1&1&1&1&1&1&1&1&1&1&1\cr u_{1}&-2&-1&0&0&-3&-2&0&-1&-1&0\cr
u_{2}&0&-1&-2&0&-1&-2&-2&-3&-1&-2\cr u_{1}^{2}&0&0&0&0&2&0&0&0&0&0\cr
u_{1}u_{2}&0&0&0&0&2&4&0&2&0&0\cr u_{2}^{2}&0&0&0&0&0&0&0&2&0&0}.$
Its rank equals $6$ and a non-vanishing maximal minor consists of the columns
with numbers $\mathfrak{S}=\\{1,3,4,5,6,8\\}$. The corresponding submatrix
$\widetilde{Mat}_{S}(\mathfrak{S})$ equals
$\widetilde{Mat}_{S}(\mathfrak{S})=\bordermatrix{~{}&l_{1}l_{2}&l_{1}l_{4}&l_{1}l_{5}&l_{2}l_{3}&l_{2}l_{4}&l_{3}l_{4}\cr
1&1&1&1&1&1&1\cr u_{1}&-2&0&0&-3&-2&-1\cr u_{2}&0&-2&0&-1&-2&-3&\cr
u_{1}^{2}&0&0&0&2&0&0\cr u_{1}u_{2}&0&0&0&2&4&2\cr u_{2}^{2}&0&0&0&0&0&2}.$
Further,
$4\widetilde{Mat}_{S}^{-1}(\mathfrak{S})=\bordermatrix{~{}&1&u_{1}&u_{2}&u_{1}^{2}&u_{1}u_{2}&u_{2}^{2}\cr
l_{1}l_{2}&0&-2&0&-2&-1&0\cr l_{1}l_{4}&0&0&-2&0&-1&-2\cr
l_{1}l_{5}&4&2&2&1&1&1&\cr l_{2}l_{3}&0&0&0&2&0&0\cr
l_{2}l_{4}&0&0&0&-1&1&-1&\cr l_{3}l_{4}&0&0&0&0&0&2}.$
Thus, given an arbitrary rational function
$R(u_{1},u_{2})=P(u_{1},u_{2})/\Phi_{S}(u_{1},u_{2})$ where
$P(u_{1},u_{2})=a_{00}+a_{1,0}u_{1}+a_{0,1}u_{2}+a_{2,0}u_{1}^{2}+a_{11}u_{1}u_{2}+a_{02}u_{2}^{2}$
is any polynomial of degree at most $2$ and
$\Phi_{S}(u_{1},u_{2})=l_{1}l_{2}l_{3}l_{4}l_{5}$, we obtain
$\begin{cases}w_{345}=-\frac{1}{4}(2a_{10}+2a_{20}+a_{11})\\\
w_{235}=-\frac{1}{4}((2a_{01}+a_{11}+2a_{02})\\\
w_{234}=4a_{00}+2a_{01}+2a_{10}+a_{20}+a_{11}+a_{02}\\\ w_{145}=2a_{20}\\\
w_{135}=-a_{10}+a_{11}-a_{20}\\\ w_{125}=2a_{02}.\\\ \end{cases}$
Notice that triangles $\Delta_{125},\Delta_{135},\Delta_{145},\Delta_{234}$
are degenerate. If we know that the original measure we are recovering is
polygonal then one should get $w_{125}=w_{135}=w_{145}=w_{234}=0$. Therefore,
the linear space of numerators $P(u_{1},u_{2})$ for the space
$\mathfrak{F}^{\Delta}(S)$ in this example is given by the system of
equations:
$\begin{cases}a_{20}=0\\\ a_{10}=a_{11}\\\ a_{02}=0\\\
4a_{00}+2a_{01}+3a_{10}=0.\\\ \end{cases}$
## 5\. Remarks and open problems
###### Remark 15.
A weaker form of Corollary 4 (i.e. the rationality of
$F_{\mathcal{P}}^{\rho}(\mathbf{u})$, but without the claim on the particular
shape of the denominator) can be derived directly from (1.3) by using Stokes
formula, along the lines of [Bar2]*Lemma 1.
###### Problem 2.
Find an appropriate version of Theorem 2, applicable to non-simple and/or non-
convex polytopes.
###### Remark 16.
Choose an arbitrary basis $\\{\Delta_{j}\\}$ of $\mathfrak{M}^{\Delta}(S)$
consisting of the standard measures of simplices. The set $\\{\Delta_{j}\\}$
spans an integer lattice in $\mathfrak{M}^{\Delta}(S)$. (One can easily see
that this lattice is invariantly defined independently of the choice of a
basis of standard measures of simplices.) Denote by
$\mathfrak{M}_{\mathbb{Z}}^{\Delta}(S)$ the space $\mathfrak{M}^{\Delta}(S)$
with the latter lattice. We can prove the following.
###### Proposition 23.
If Conjecture 7 holds then any generalized polytope
$\mathcal{P}\in\mathcal{P}(S)$ with standard measure $\mu_{\mathcal{P}}$ in
$\mathfrak{M}^{\Delta}(S)$ corresponds to a rational point in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$.
###### Proof.
One can easily show that any $\mathcal{P}$ as above can be represented as the
union of the closures of connected components of $\mathbb{R}^{d}\setminus
H(S)$, where $H(S)$ is the hyperplane arrangement consisting of all
hyperplanes spanned by $d$-tuples of points in $S$. (The converse is obviously
not true.) Let $\tilde{S}\supseteq S$ be the extended set of vertices obtained
by adding to $S$ all vertices of the hyperplane arrangement $H(S)$. Since each
connected component in $\mathbb{R}^{d}\setminus H(S)$ is convex, it can be
triangulated on $\tilde{S}$. Consider the space
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$. Obviously,
$\mu_{\mathcal{P}}$ is an integer point in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$. Also,
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$ is contained in
$\mathfrak{M}^{\Delta}_{\mathbb{Z}}(\tilde{S})$ as a sublattice. Thus, if
$\mu_{\mathcal{P}}$ belongs to $\mathfrak{M}^{\Delta}_{\mathbb{Z}}(S)$ it is a
rational point there. ∎
###### Remark 17.
If either $\mathcal{P}$ itself or its complement in $\mathrm{conv}(S)$ can be
triangulated by simplices with vertices in $S$ then $\mu_{\mathcal{P}}$ is an
integer point in $\mathfrak{M}_{\mathbb{Z}}(S)$.
Let us pose the following tantalizing question.
###### Problem 3.
Does there exist a generalized polytope $\mathcal{P}$ with the set of vertices
$S(\mathcal{P})$ and with non-integer coordinates in
$\mathfrak{M}_{\mathbb{Z}}(S(\mathcal{P}))$?
###### Problem 4.
One can also define a rational convex cone
$\mathfrak{Pos}(S)\subset\mathfrak{M}_{\mathbb{Z}}(S)$ by taking non-negative
linear combinations of all $\mu_{\mathcal{P}}$, where $\mathcal{P}$ runs over
the set of all generalized polytopes in $\mathcal{P}(S)$.
###### Conjecture 24.
The rational cone $\mathfrak{Pos}(S)$ is uniquely determined by the oriented
matroid associated to $S$.
We conclude this section with the following question.
###### Problem 5.
Is it possible to describe the extremal rays of $\mathfrak{Pos}(S)$?
One can easily show that a simplex from $\mathcal{P}(S)$ spans an extremal ray
of $\mathfrak{Pos}(S)$ if and only if it does not contain any points of $S$
distinct from its vertices. Problem 5 is apparently closely related to the
problem of classification of combinatorial types of point arrangements, see
e.g. [FuMiMo] and references therein.
## References
|
arxiv-papers
| 2012-10-11T12:01:05 |
2024-09-04T02:49:36.373478
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Nick Gravin, Dmitrii V. Pasechnik, Boris Shapiro, Michael Shapiro",
"submitter": "Dmitrii V. Pasechnik",
"url": "https://arxiv.org/abs/1210.3193"
}
|
1210.3234
|
# Risks of Friendships on Social Networks
Cuneyt Gurcan Akcora, Barbara Carminati, Elena Ferrari
DISTA, Università degli Studi dell’Insubria
Via Mazzini 5, Varese, Italy
{cuneyt.akcora, elena.ferrari, barbara.carminati}@uninsubria.it
###### Abstract
In this paper, we explore the risks of friends in social networks caused by
their friendship patterns, by using real life social network data and starting
from a previously defined risk model. Particularly, we observe that risks of
friendships can be mined by analyzing users’ attitude towards friends of
friends. This allows us to give new insights into friendship and risk dynamics
on social networks.
500mm(.15-6cm) To Appear in the 2012 IEEE International Conference on Data
Mining (ICDM).
## I Introduction
Users register on social networks to keep in touch with friends, as well as to
meet with new people. Research works have shown that a big majority of people
that we meet online and add as friends are not random social network users;
these people are introduced into our social graph by friends [2]. Although
friends can enrich the social graph of users, they can also be a source of
privacy risk, because a new relationship always implies the release of some
personal information to the new friend as well as to friends of the new
friend, which are strangers for the user. This problem is aggravated by the
fact that users can reference resources of other users in their social graph;
and make it very difficult to control the resources published by a user. This
uncontrolled information flow highlights the fact that creating a new
relationship might expose users to some privacy risks.
We cannot assume that friends will make the right choices about friendships,
because friends may have a different view on people they want to be friends
with. Considering this, privacy of a social network user should be protected
by building a model that observes friendship choices of friends, and assigns a
risk label to friends accordingly. Such a model requires knowing a user’s
perception on the risks of friends of friends. We made a first effort in this
direction in [3] by proposing a risk model to learn risk labels of strangers
by considering several dimensions. To validate the model, we developed a
browser extension showing for each stranger (i) his/her profile features, (ii)
his/her privacy settings, and (iii) mutual friends. Based on this information,
the user is asked to give a risk label $l\in\\{1,2,3\\}$ to the stranger.
These risk labels correspond to _not risky, risky_ and _very risky_
classification of a stranger. Through the extension, 47 users (32 male, 15
female users) have labeled 4013 strangers. However, we did not consider risk
of friends.
This new work starts with considering two factors in assigned risk labels.
First, strangers can be risky only because of their profile features. Second,
a friend himself can increase or decrease the risk of a stranger. Increases
and decreases will be termed as negative and positive friend impacts,
respectively. In any case, if a risky stranger is introduced into the user’s
social graph it is because of his/her friendship with a friend. However,
determining the friend impacts can help us to determine which privacy actions
should be taken to avoid data disclosure. We aim at learning how risk labels
are assigned to strangers depending only on their profile features, and how
much a friend can impact (i.e., increase or decrease) these labels. If
strangers are risky just because of their profile features, privacy settings
can be restricted to avoid only these strangers. On the other hand, if a
friend increases the risk labels of strangers, all of his/her strangers should
be avoided.
We begin our discussion with reviewing the related work in Section II. In
Section III we explain the building blocks of our model and Section IV shows
how we use our dataset efficiently. In Section V we discuss the role profile
features in risk labels, and in Section VI we show how impacts of friends are
modeled. Section VII explains finding risk labels of friends from friend
impacts, and in Section VIII we give the experimental results.
## II Related Work
Friends’ role in user interactions has been studied in sociology [19], but
observing it on a wide scale has not been possible until online social
networks attracted millions of users and provided researchers with social
network data. For online social networks, Ellison et al. [7] defined friends
as social capital in terms of an individual’s ability to stay connected with
members of a previously inhabited community. Differing from this work, we
study how friends can help users to interact with new people on social
networks. Although these interactions can increase users’ contributions to the
network [21] and help the social network evolve by creation of new friendships
[23], they can also impact the privacy of users by disclosing profile data.
Squicciarini et al. [20] have addressed concerns of data disclosure by
defining access rules that are tailored by 1) the users’ privacy preferences,
2) sensitivity of profile data and 3) the objective risk of disclosing this
data to other users. Similarly, Terzi et al. [14] has considered the
sensitivity of data to compute a privacy score for users. Although these works
regulate profile data disclosure during user interactions, they do not study
the role of friends who connect users on the social network graph and
facilitate interactions. Indeed, research works (see [5] for a review) have
been limited to finding the best privacy settings by observing the interaction
intensity of user-friend pairs [4] or by asking the user to choose privacy
settings [8]. Without explicit user involvement, Leskovec et al. [12] have
shown that the attitude of a user toward another can be estimated from
evidence provided by their relationships with other members of the social
network. Similar works try to find friendship levels of two social network
users (see [1] for a survey). Although these work can explain relations
between social network users, they cannot show how existence of mutual friends
can change these relations.
Privacy risks that are associated by friends’ actions in information
disclosure has been studied in [22], but the authors work with direct actions
(e.g., re-sharing user’s photos) of friends, rather than their friendship
patterns. Recent privacy research focused on creating global models of risk or
privacy rather than finding the best privacy settings, so that ideal privacy
settings can be mined automatically and presented to the user more easily. In
[3], Akcora et al. prepared a risk model for social network users in order to
regulate personal data disclosure. Similarly, Terzi et al. [14] has modeled
privacy by considering how sensitive personal data is disclosed in
interactions. Although users assign global privacy or risk scores to other
social network users, friend roles in information disclosure are ignored in
these work.
An advantage of global models is that once they are learned, privacy settings
can be transfered and applied to other users. In such a shared privacy work,
Bonneau et al. [6] use suites of privacy settings which are specified by
friends or trusted experts. However, the authors do not use a global
risk/privacy model, and users should know which suites to use without knowing
the risk of social network users surrounding him/her.
## III Overall Approach
We will start this section by explaining the terminology that will be used in
the paper. In what follows, on a social graph $\mathcal{G}_{u}$, 1 hop
distance nodes from $u$ are called friends of $u$, and 2 hop distance nodes
are called strangers of $u$, i.e., strangers of user $u$ are friends of
friends of $u$. We will denote all strangers of user $u$ with $S_{u}$, and
risk label of each stranger $s\in S_{u}$ that was labeled by $u$ will be
denoted as $l_{us}\in\\{1,2,3\\}$.
A social network $\mathcal{G}=(N,E,Profiles)$ is a collection of $N$ nodes and
$E\subseteq N\times N$ undirected edges. $Profiles$ is a set of profiles, one
for each node $n\in\\{1,...,\left|N\right|\\}$. A social graph
$\mathcal{G}_{u}=(V,R,F)$ is constructed from the social network $\mathcal{G}$
for each user $u\in N$, such that, the node set $V=\\{\forall n\in
N|distance(n,u)\leq 2\\}$. Nodes in $\mathcal{G}_{u}$ consist of friends and
strangers of $u$. Similarly, edge set R consists of all edges in $\mathcal{G}$
among nodes in $V$. Each node $v\in V$ in a social graph will be associated
with a feature vector $f_{v}\in F$. Cells of $f_{v}$ correspond to profile
feature values from the associated user profile in $Profiles$.
The goal of our model is to assign risk labels to friends according to the
risk labels of their friends (i.e., strangers). As we stated before, risk
labels of strangers depend on stranger features as well as mutual friends [3].
We do not assume that all friends can change users’ risk perception in the
same way. Some friends can make strangers look less risky and facilitate
interactions with them (i.e., friends decrease the risk of strangers). On the
other hand, some friends can make strangers more risky (i.e., friends increase
the risk of strangers). For example, if users do not want to interact with
some friends, they might avoid friends of these friends as well. We will use
positive and negative impacts to refer to decreases and increases in stranger
risk labels, respectively. To understand whether friends have negative or
positive impacts, our model must be able to know what risk label the stranger
would receive from the user if there were no mutual friends. This corresponds
to the case where the user given label depends only on stranger features. We
will term this projected label as the baseline label, and show it with
$b_{us}$. For instance, assume that if there are no mutual friends, a user $u$
considers all male users as very risky, and avoids interacting with them. In
this case, the baseline label for a male stranger $s$ is very risky, i.e.,
$b_{us}=very~{}risky$. However, if the same male stranger $s$ has a mutual
friend with user $u$, we assume that the user given label $l_{us}$ might not
be equal to the baseline label $b_{us}$ (i.e.,$l_{us}\neq b_{us}$), because
the mutual friend might increase or decrease the risk perception of the user.
This difference between the baseline and user given labels will be used to
find out friend impacts.
Finding baseline labels and friend impacts requires different approaches. In
baseline estimates, we use logistic regression on stranger features, and for
the friend impacts we use multiple linear regression [17]. Both of these
regression techniques require many user given labels to compute baseline
labels and friend impacts with high confidence. However, users are reluctant
to label many strangers, therefore we have to exploit few labels to achieve
better results. To this end, we transform our risk dataset, and use the
resulting dataset in regression analyses. In the next sections, this
transformation and regression steps will be described in detail. Overall, we
divide our work into four phases as follows:
1. 1.
Transformation: Exploit the risk label dataset in such a way that regression
analyses for baseline labels and friend impacts can find results with high
confidence. With this step, we increase the number of labels that can be used
to estimate baseline labels and friend impacts.
2. 2.
Baseline Estimation: Find baseline labels of strangers by logistic regression
analysis of their features.
3. 3.
Learning Friend Impacts: Create a multiple linear regression model to find
friends that can change users’ opinion about strangers and result in a
different stranger label than the one found by baseline estimation.
4. 4.
Assigning Risk Labels to Friends: Analyze the sign of friend impacts, and
assign higher risk labels to friends who have negative impacts.
## IV Transforming Data
By transforming the data, we aim at using the available data efficiently to
find friend impacts with higher confidence. To this end, we first transform
profile features of friends and strangers to use k-means and hierarchical
clustering algorithms [9] on the resulting profile data. This section will
discuss the transformation, and briefly explain the clustering algorithms.
Our model has to work with few stranger labels, because users are reluctant to
label many strangers. This limitation is also shared in Recommender Systems
(RS) [16] where the goal is to predict ratings for items with minimum number
of past ratings. In neighborhood based RS [11], ratings of other similar users
are exploited to predict ratings for a specific user. Traditionally, the
definition of similarity depend on the characteristics of data (e.g., ordinal
or categorical data), and it has to be chosen carefully. We use profile data
of friends and strangers in defining similar friends and strangers,
respectively. Friend impacts of a user $u$ is learned from impacts of similar
friends from all other users. To this end, we transform profile data of
friends and strangers in such a way that friends and strangers of different
users are clustered into global friend and stranger clusters. Next sections
will describe the aims and methods of friend and stranger clustering in
detail.
### IV-A Clustering Friends
Clustering friends aim at learning friend impacts for a cluster of friends.
This is because we might not have enough stranger labels to learn impacts of
individual friends with high confidence. To overcome this data disadvantage,
impact of a friend $f$ can be used to find the impacts of other friends who
belong to the same cluster. For example, a user from Milano can have a friend
from Milano, whereas a user from Berlin can have a friend from Berlin.
Although these two friends have different hometown values (Milano and Berlin),
we can assume that both friends can be clustered together because their
hometown feature values are similar to user values. This hometown example
demonstrates a clustering based on a single friend profile feature and it
results in only two clusters: friends who are from Milano/Berlin and friends
who are from somewhere else. However, in real life social networks, friends
have many values for a feature, some of which can be more similar to the
user’s value than others. For example, Italian friends of a user from Milano
can be from Italian cities other than Milano, and these friends should not be
considered as dissimilar as friends from Berlin. By considering these, we
transform categorical friend values to numerical values in such a way that
similarities between friend and user values become more accurate.
Our transformation uses the homophily [15] assumption which states that people
create friendships with other people who are similar to them along profile
features such as gender, education etc. In other words, we assume that all
friends of a user $u$ can be used to judge the similarity of a social network
user to $u$. For example, considering the case where the user $u$ is from
Milano, a social network user from Rome is similar to the user if the user has
many friends from Rome. Moreover, we assume that different users will have
similar clusters of friends, e.g., friends from user’s hometown, alma mater
etc. and friend impact values will be correlated with their corresponding
clusters, e.g., friends from hometowns will have similar impact values. More
precisely, the transformation of friends’ data maps a categorical feature
value of a friend, such as hometown:Milano, to a numerical value which is
equal to the frequency of the feature value among profiles of all friends of a
user. For example, if a friend $f$ has profile feature value hometown:Milano,
and there are 15 out of 100 friends with similar hometown:Milano values,
hometown feature of $f$ will be represented with $15/100=0.15$. After applying
this numerical transformation to all friends of all users, we compute a Social
Frequency Matrix for Friends (SFMF) where each row represents numerical
transformation of feature vector of a user’s friend.
###### Definition 1 (Social Freq. Matrix for friends)
The Social Frequency Matrix associated with a social network $\mathcal{G}$ is
defined as $|N|\times|F|\times n$, where N is the set of users in
$\mathcal{G}$, $F\subset N$ is the set of user in $\mathcal{G}$ that are
friends of at least one user $u\in N$, and n is the number of features of user
profiles. Each element value of the matrix is given by:
$SFMF[u,f,v]=\frac{Sup({\vec{f}_{v}})}{\left|F_{u}\right|}$
where $F_{u}\subset F$ is the set of friends of $u$,
$Sup(\vec{f}_{v})=\left|\\{g\in F_{u}|\vec{g}_{v}=\vec{f}_{v}\\}\right|$ and
$f\in F_{u}$, whereas $\vec{g}_{v}$ and $\vec{f}_{v}$ show the value of
profile feature $v$ for users $g$ and $f$, respectively.
Having transformed friend data into numerical form, we can now use a
clustering algorithm to create clusters of friends. After applying a
clustering algorithm to the Social Frequency Matrix for friends, output friend
clusters will be denoted by $FC$.
### IV-B Clustering Strangers
By clustering friends, we can learn impacts of friends from different
clusters, but this raises another question: do friends have impact on all
strangers of users? Our assumption is that correlation between stranger and
friend profile features can reduce or increase friend impact. For example, if
a student user $u$ labels friends of a classmate friend $f$, we might expect
friends of $f$ who are professors to have higher risk labels than student
friends of $f$, because $u$ might not want his/her professors to see his/her
activities and photos. Here the work feature of strangers changes friend
impact of $f$ by increasing the risk label of professor friends of $f$. To see
how friend and stranger features change friend impacts, we transform
strangers’ profile data to numerical data and cluster the resulting matrix
just like we clustered friends. This clustered stranger representation helps
us detect clusters of strangers for whom certain clusters of friends can
change risk perception of users the most. Formally, we prepare a social
frequency matrix as follows:
###### Definition 2 (Social Freq. Matrix for strangers)
The Social Frequency Matrix for Strangers associated with a social network
$\mathcal{G}$ is defined as $|N|\times|S|\times n$, where N is the set of
users in $\mathcal{G}$, $S\subset N$ is the set of user in $\mathcal{G}$ that
are strangers of at least one user $u\in N$, and n is the number of features
of user profiles. Each element value of the matrix is given by:
$SFMS[u,s,v]=\frac{Sup({\vec{s}_{v}})}{\left|F_{u}\right|}$
where $Sup(\vec{s}_{v})=\left|\\{g\in F_{u}|\vec{g}_{v}=\vec{s}_{v}\\}\right|$
and $S\in N$, whereas $\vec{s}_{v}$ shows the value of feature v for stranger
$s$.
Note that we still use friend profiles in the denominator to transform
stranger data. This is because we cannot see all strangers of a friend due to
API limitations of popular social networks. To overcome this problem, we use
friend profiles because we expect them to be similar to profiles of their own
friends (strangers). We again use the Social Frequency Matrix for strangers to
create clusters of strangers. We will denote stranger these stranger clusters
by $SC$.
### IV-C Clustering Algorithms
In our experiments, we used the k-means and hierarchical algorithms [9] to
produce clusters of friends and strangers. This section will briefly explain
these algorithms. In what follows, we will use data points and
strangers/friends interchangeably to mean elements in a cluster.
The k-means clustering algorithm takes the number of final clusters as input
and clusters the data by successively choosing cluster seeds and refining the
distance within cluster data points. The required input for the number of
final clusters is usually unknown beforehand and this makes k-means unfeasible
in some scenarios. However, in our model it gives us the flexibility to
experiment with different sizes of clusters. k-means is also a fast clustering
algorithm which suits our model for the cases where all friends of all users
can reach a few thousands. In our experiments, we used different $k$ values to
find optimal performance. In hierarchical clustering111We used the
agglomerative form where a new stranger is added to clusters by considering
the complete distance. Height of the tree was 3., a tree structure is formed
by joining clusters and the tree is cut horizontally at some level to produce
a number of clusters.
In friend and stranger clustering, choosing the number of final clusters or
the horizontal level requires some trade-offs. The advantage of using many
clusters is that data points in each cluster are more similar to each other
(i.e., friends or strangers in a cluster are more similar in profile feature
values). On the other hand, too many clusters decreases the average number of
data points in a cluster, and our model may not be trained on these clusters
with high confidence, i.e., there may not be enough data points in a cluster
to prove anything. Using too few clusters also has a disadvantage. Final
clusters may contain too many data points that are not very similar to each
other. This decreases the quality of inferences because what we infer from
some data points might not be valid for others in the same cluster. Despite
this, if data points are naturally homogeneous, the similarity among data
points in a big cluster can be high. As a result, a big cluster may offer more
data to prove our inferences with more confidence.
After transforming our data and creating friend and stranger clusters, we will
now explain baseline label estimation for stranger clusters.
## V Baseline Estimation
Baseline estimation analyzes how feature values on stranger profiles bring
users to assign specific risk labels to strangers. The baseline estimation
process results in baseline labels for each stranger $s\in S$. These labels
are found by using statistical regression methods on already given user labels
and stranger profile features. In this section we will discuss this process.
Figure 1: Features and risk labels
Baseline estimation corresponds to the case where a user would assign a risk
label to a stranger without knowing which one of his/her friends are also
friends with the stranger. Figure 1 shows an example of baseline estimation.
In the figure, each stranger $s\in S_{u}$ is a node surrounded by a ring
representing his/her feature vector $f_{s}$. Each cell in the feature vector
corresponds to a feature value of the stranger (e.g., hometown:Milano).
Different colors for the same cell position represent different values for the
same feature on different stranger profiles. In the example shown in Figure 1
strangers $S_{2}$, $S_{4}$ and $S_{5}$ are labeled with 2 (i.e., the risky
label). These three strangers share the same feature vector as shown with the
same colored cells. Based on these observations, if any stranger has the same
feature vector with $S_{2}$, $S_{4}$ and $S_{5}$, the stranger will be given
label 2. The evidence to support this statement comes from the three strangers
($S_{2}$, $S_{4}$ and $S_{5}$), and the number of such strangers determine the
confidence of the system in assigning baseline labels.
Although in Figure 1 stranger features are shown to be the only parameter in
defining stranger labels, in our dataset labels of strangers have been
collected from users by explicitly showing at least one mutual friend in
addition to the stranger feature values. Because of this, stranger labels that
are learned from users can be different from baseline labels; they can be
higher (more risky) or lower (less risky) depending on the friend impact.
Considering this, in baseline estimation we use the labels of strangers who
have the least number of mutual friends with users. These are the subset of
labels which were given to strangers who have only one friend in common with
users, i.e., for user $u$ and stranger $s$, $\left|F_{u}\cap F_{s}\right|=1$.
In what follows, we will use first group dataset to refer to these strangers.
In our approach, we use logistic regression to learn the baseline labels from
available data. This allows us to work with categorical response variables
(i.e., one of the tree risk labels). Stranger features are used as explanatory
(independent) variables and risk labels as the response (dependent) variable
which is determined by values of explanatory variables (i.e., feature values).
Although the response variable has categorical values, it can be considered
ordinal because risk labels can be ordered as not risky (label 1), risky
(label 2) and very risky (label 3).
Ordinary Logistic Regression is used to model cases with binary response
values, such as 1 (a specific event happens) or 0 (that specific event does
not happen), whereas multinomial logistic regression is used when there are
more than two response values. As multinomial logistic regression a basic
variant of logistic regression, we will first start with the definition of
logistic regression. For this purpose, assume that our three risk labels are
reduced to two (risky, not risky).
Suppose that $\pi$ represents the probability of a particular outcome, such as
a stranger being labeled with risky, given his/her profile features as a set
of explanatory variables $x_{1},...,x_{n}$:
$P(l=risky)=\pi=\frac{e^{(\alpha+\sum{\beta_{k}X_{k}})}}{1+e^{(\alpha+\sum{\beta_{k}X_{k}})}}$
where $0\leq\pi\leq 1$, $X_{k}$ is a feature value, $\alpha$ is an intercept
and $\beta$s are feature coefficients, i.e., weights for feature values. The
logit transformation $log[\frac{\pi}{1-\pi}]$ is used to linearize the
regression model:
$log[\frac{\pi}{1-\pi}]=\alpha+\sum{\beta_{k}X_{k}}$
By transforming the probability ($\pi$) of the response variable to an odd-
ratio ($log[\frac{\pi}{1-\pi}]$), we can now use a linear model. Given the
already known stranger features and labels, we use Maximum Likelihood
Estimation [18] to learn the intercept value and the coefficients of all
features.
Although standard binary logistic regression and multinomial logistic
regression use the same definition, they differ in one aspect: multinomial
regression chooses a reference category and works with not one but $N-1$ log
odds where $N$ is the number of response categories. In our model, $N=3$,
because the response has three labels (1, 2, 3). In both binary and
multinomial logistic regression, intercept and coefficient values are found by
using numerical methods to solve the linearized equation(s). With the found
values, we can write the odd ratio as an equation. For example, in equation
$log[\frac{\pi}{1-\pi}]=0.7+1.2\times X_{1}+0.3\times X_{2}$, the intercept
value is (0.7) and feature coefficients ($\beta_{1}=1.2$ and $\beta_{2}=0.3$).
We can then plug in a new set of values (e.g., $X_{1}=0.5$) for features, and
get the probabilities of response value being one of three labels. For
example, for a specific stranger, the model can tell us that risk label
probabilities of the stranger is distributed as %0.9 very risky, %0.09 risky
and %0.01 not risky. As we can compute baseline label in real values, a
stranger $s\in S$ is assigned a baseline label by weight averaging the
probabilities of risk labels.
## VI Friend Impact
So far, we have discussed clustering and baseline label estimation. In this
section we will first discuss how these two aspects of our model are combined
to compute friend impacts. After finding friend impacts, we will discuss how
risk labels can be assigned to friends by considering the sign of impact
values. In computing friend impacts, we use multiple linear regression [17],
which learns friend impacts by comparing baseline and user given labels to
strangers. To this end, we define an estimated label parameter to use in
linear regression as follows:
###### Definition 3 (Estimated label)
For a stranger $s$ and a user $u$, an estimated label is defined as:
$\hat{l}_{us}=b_{us}+\sum\limits_{FC_{i}\in FC}{FI(FC_{i},SC_{j})}\times
Past(u,s)$
where $\hat{l}_{us}$ and $b_{us}$ are estimated and baseline labels for a
stranger $s$, and $s$ belongs to the stranger cluster $SC_{j}\in SC$. Friend
clusters $FC$ are found by applying a algorithm to the mutual friends of user
$u$ and stranger $s$. $Past(u,s)$ denotes an intermediary value based on
stranger labels given by user $u$, whereas $FI(FC_{i},SC_{j})$ represents
impact of a friend $f$ from a friend cluster $FC_{i}$ on the label of stranger
$s$ from a stranger cluster $SC_{j}$.
In the rest of this section, we will define the $Past(.,.)$ and $FI(.,.)$
parameters, and explain how they are used to compute friend impacts.
### VI-A The Past Labeling Parameter
We start by discussing the past parameter $Past(.,.)$ which returns a value
from past labellings of strangers by user $u$.
The past parameter is traditionally used in recommender systems to adjust
baseline estimate [16]. The need for this parameter arises from the fact that
baseline estimation is computed from labels of all strangers who have only one
mutual friend with user $u$ (i.e., first group dataset), and it tends to be a
rough average. To overcome this, a subset of strangers, who are very _similar_
to $s$ and who have been labeled in the past by $u$, are observed and the
baseline label is increased or decreased to make it more similar to the user
given labels of these strangers.
In defining the past parameter, we consider two factors: how many similar
strangers should be considered in this adjustment and what is an accurate
metric for finding similarity of two strangers? For the first question, we use
the computed stranger clusters. For a stranger $s$, similar strangers from the
first group dataset are those (i) that are labeled by the same user $u$, and
(ii) that belong to the same stranger cluster with $s$. Although we use
stranger clusters to choose similar strangers, the similarity of strangers in
a cluster can be low or high depending on the clustering process. With too few
clusters and too many clusters, similarity of strangers in a cluster can be
low and high respectively. We adjust the baseline labels by considering labels
given to most similar users. To this end, we use the profile similarity
measure by Akcora et al. [2]. This measure assigns a similarity value of 1 to
strangers with identical profiles, and for non-identical profiles the
similarity value is higher for strangers whose profile feature values are more
common in profile features of $u$’s friends. Formally, we define the past
labeling as follows:
###### Definition 4 (Past Labeling Parameter)
For a given user $u$ and stranger $s$, the past labeling parameter is defined
as:
$Past(u,s)=\frac{1}{\left|SC_{i}\right|}\sum\limits_{x\in
C_{i}}{PS(s,x)\times(l_{ux}-b_{ux})}$
where $PS()$ denotes the profile similarity between two strangers, $l_{ux}$ is
the user given label of stranger x, and $b_{ux}$ is the baseline label of x.
Strangers $s$ and $x$ belong to the same stranger cluster $C_{i}$.
### VI-B The Friend Impact Parameter
The second parameter from Definition 3, $FI(f,s)$, is used to show impacts of
mutual friends on the risk label given to $s$ by $u$. In modeling friend
impacts, we wanted to see how friends from different clusters changed the
baseline label. By using this approach, we explain impacts of friend clusters
in terms of friend features that shape friend clusters. If there is at least
one mutual friend from a friend cluster $FC_{i}$, we say that friend cluster
$FC_{i}$ may have impacted the label given to the stranger $s$. For the cases
where a stranger $s$ has two or more mutual friends from a friend cluster
$FC_{i}$, we experimented with both options for $FI(f,s)$. Next, we will
explain these options.
(a) Multiple impacts for a friend cluster.
(b) Single impact for a friend cluster.
Figure 2: Friend impact definitions by considering the number of friends from
the same cluster. In the single impact definition, two friends do not increase
the friend impact.
#### VI-B1 Multiple Impact for the Friend Cluster
In our first approach, we assume that a bigger number of mutual friends from
friend cluster $FC_{i}\in FC$ will impact user labeling. Assume that from a
friend cluster $FC_{i}\in FC$, we are given a set of mutual friends
$MF_{i}=\\{\forall f|f\in FC_{i},f\in\\{F_{u}\cap F_{s}\\}\\}$ of user $u$ and
stranger $s$. We define the impact of friend cluster $FC_{i}$ on the label of
stranger $s\in SC_{j}$ as follows:
$FI_{2}(FC_{i},SC_{j})=\left|MF_{i}\right|\times I_{FC_{i},SC_{j}}$
where $I_{FC_{i},SC_{j}}$ is the impact of a cluster $FC_{i}|f\in\\{FC_{i}\cap
MF_{i}\\}$ on the label of stranger $s\in SC_{j}$. Note that this impact
($I_{FC_{i},SC_{j}}$) is the unknown value that our system will learn.
#### VI-B2 Single Impact for the Friend Cluster
In the second approach, we assume that a bigger number of friends from the
same cluster does not make a difference in user labeling; at least one friend
from the cluster is required, but more friends do not bring additional impact.
This approach is shown in Figure 2(b), where friends are shown with their
cluster ids, and two friends from friend cluster $FC_{2}$ bring a single
impact. Assume that from a friend cluster $FC_{i}\in FC$, we are given a set
of mutual friends of user $u$ and stranger $s$. We give the impact of friend
cluster $FC_{i}$ on the label of stranger $s$ as follows:
$FI_{1}(FC_{i},SC_{j})=I_{FC_{i},SC_{j}}$
where $I_{FC_{i},SC_{j}}$ is the impact of a friend cluster $FC_{i}$ on label
of stranger $s\in SC_{j}$.
These different friend impact approaches change the model by including
different numbers of friend impacts. The unknown impact variable
$I_{FC_{*},SC_{*}}$ is learned by the least squares method [10]. The least
squares method provides an approximate solution when there are more equations
than unknown variables. In our model, each stranger’s label provides an
equation to compute impacts of $k_{1}$ friend clusters on $k_{2}$ stranger
clusters ($k_{1}$ and $k_{2}$ are the final numbers of friend and stranger
clusters in the k-means algorithm). In Example VI.1, we will explain these
points and give equations of one stranger for single and multiple impact
definitions.
###### Example VI.1
Given a stranger $s_{1}\in SC_{1}$ who is labeled by $u$, assume that the user
given label $l_{us_{1}}=2.3$, while the baseline label is $b_{us_{1}}=2.7$.
Again assume that $Past(u,s)=-0.2$. Equations for the stranger $s$ with single
and multiple friend impact definitions are respectively given as follows:
$2.3=2.7+(I_{FC_{2},SC_{1}}+I_{FC_{1},SC_{1}})\times-0.2$ $2.3=2.7+(2\times
I_{FC_{2},SC_{1}}+I_{FC_{1},SC_{1}})\times-0.2$
After choosing one of these definitions of friend impact, we input one
equation for each stranger $s$ to the least squares method to compute impact
values of friend clusters on stranger clusters. In the experimental results,
we will discuss the definition that yielded the best results.
## VII Friend Risk Labels
Learning impact values allows us to see the percentage of positive and
negative impacts for each friend cluster. Negative impact values for a friend
cluster shows that the friend cluster increases the risk label of strangers.
Depending on a user’s choice, friend clusters which have negative impacts less
than $x\%$ of the time can be considered not risky. Similarly, a threshold
$y\%$ can be chosen to determine very risky friend clusters. In our
experiments, we heuristically chose $x=20$ and $y=50$. With these threshold
values for risk labels, we formally define the risk label of a friend $f$ as
follows:
###### Definition 5 (Friend Risk Label)
Assume that the percentage of positive and negative impact values for a
cluster $FC_{i}\in FC$ are denoted with $Im_{i}^{+}$ and $Im_{i}^{-}$
respectively, where $Im_{i}^{+}+Im_{i}^{-}=1$. We assign a risk label to a
friend $f$ who is a member of the friend cluster $FC_{i}$ (i.e., $f\in
FC_{i}$) according to the negative impact percentage of the friend cluster
$FC_{i}$ as follows:
$l(u,f)=\left\\{\begin{array}[]{l l}\text{not risky}&\quad\text{if}\quad
Im_{i}^{-}<0.2\\\ \text{risky}&\quad\text{if}\quad 0.2\leq Im_{i}^{-}<0.5\\\
\text{very risky}&\quad\text{if}\quad Im_{i}^{-}\geq 0.5\\\
\end{array}\right\\}$
Next we will give the experimental results of our model performance.
## VIII Experimental Results
In this section we will validate our model assumptions, and then continue to
give detailed analysis of performance under different parameter/setting
scenarios.
### VIII-A Validating Model Assumptions
Before finding friend impacts, we validated our model assumption (i.e., mutual
friends have an impact on the risk label of a stranger) by using logistic
regression on the whole dataset (4013 stranger labels and profiles). For this,
we included the number of mutual friends as a parameter, and computed the
significance222Significance is measured by p-values. The p-value is the
probability of having a result at least as extreme as the one that was
actually observed in the sample. Traditionally, a $p-value$ of less than 0.05
is considered significant. of model parameters. In overall regression, photo
visibility, wall visibility, education and work parameters were excluded from
the model because they were found to be non-significant. For significant
parameters, $Pr(>\left|t\right|)$ values are shown in Table I.
In the regression, there are two friend related parameters: the number of
mutual friends and the friendlist visibility. Differing from the number of
mutual friends, friendlist visibility is a categorical variable which takes 0
when the stranger hides his/her friendlist from the user and 1 otherwise. From
Table I333 Notes: Reference category for the equation is label 2. Standard
errors in parentheses. Significance codes: ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’
0.1 ’ ’ 1 , we see that seeing a stranger’s friendlist increases the
probability of the stranger getting label 1, whereas it is not an important
parameter for label 3. Our main focus in regression analysis was to verify
that the number of mutual friends parameter is significant. We found that an
increasing number of mutual friends indeed helps a stranger get label 1, and
decreases the probability of getting label 3. This result tells us that
friends have an impact on user decisions and our assumption about the
existence of friend impacts holds true. After validating our model assumption,
we continue to the baseline label estimations.
TABLE I: Regression results for all data points. p-value=2.22e-16. Total N=4013. | Label 1 | Label 3
---|---|---
Intercept | -1.2668850*** | -0.7626810***
| (0.146) | (0.138)
Mutual friends | 0.0379547*** | -0.0467834***
| (0.008) | (0.012)
Gender | -0.3696749** | 0.3480055**
| (0.118) | (0.113)
Friendlist visibility | 0.6203365*** | -0.0642952
| (0.125) | (0.118)
Locale | 0.6167273*** | 0.7070663***
| (0.180) | (0.172)
Location | 0.1347104 | 0.2708697*
| (0.128) | (0.125)
N | 1116 | 1161
### VIII-B Training for Baseline
Baseline calculation predicts labels for strangers without friend impacts. For
this purpose we take strangers who have one mutual friend with users
($|MF|=1$) into a new dataset (first group dataset), and train a logistic
regression model. Logistic regression on the first group dataset finds how
stranger features bring users to label strangers. Table II shows model
parameters and their corresponding $p$-values.
TABLE II: Regression results for the first group data points. p-value = 5.6701e-11. Total N=1520. | Label 1 | Label 3
---|---|---
Intercept | -2.5400*** | -0.8661***
| (0.6305) | (0.2791)
Gender | -1.1026** | 0.6985*
| (0.4108) | (0.3350)
Friendlist visibility | 0.4705* | 0.5214
| (0.2075) | (0.1706)
Wall | 0.4173. | -0.1595
| (0.2463) | (0.2262)
Photo | 1.9425** | 0.1361
| (0.6093) | (0.2339)
Locale | 0.1446 | 0.5846*
| (0.2881) | (0.2277)
N | 278 | 588
In Table II, we see that when users label the first group strangers, photo and
wall visibility are significant parameters. If these items are visible on
stranger profiles, the probability of strangers getting label 1 increases. In
the whole dataset (see Table I), these two parameters were found to be
insignificant. Another interesting result is that locale444Locale is the web
interface language of the user on the social networking site (e.g., IT for
Italian and RU for Russian). is significant for label 3 whereas it is
non/significant for label 1. A high locale value means that the stranger is
similar to existing friends of users, but this high similarity is shown to
increase the probability of strangers being labeled as very risky, i.e.,
receiving label 3.
After computing a baseline label for all strangers, we use the difference
between user given and baseline labels (${l}_{us}-b_{us}$) to model the friend
impact. These differences (deviations from the baseline label) are shown in
Figure 3. In the figure, we see that user given labels are lower than the
computed baseline label, which shows that in overall friends have positive
impacts (i.e., thanks to mutual friends, users assign lower risk labels to
strangers.). Overall, we found that there is not a linear relation between the
number of mutual friends and the deviation values. This non-linearity changes
how we define the impacts of friend clusters. In Section VI we gave two
definitions for friend impacts (see Figure 2) to account for deviations from
the baseline label.
Figure 3: Deviation of user given labels from baseline labels. Values in the
x-axis are the number of mutual friends between a stranger and user.
In multiple friend impacts we assumed that more mutual friends from a friend
cluster bring additional impacts. On the other hand, in single friend impact
one friend was enough to have the impact of a friend cluster. This finding
implies that more friends of the same cluster do not provide any benefits to
strangers on Facebook and mutual friends from different clusters are more
suitable to change the user’s risk perception about a stranger. We believe
that this can be generalized to other undirected social networks.
In the rest of the experiments, we will give the results computed by using the
single friend impact definition. We will now explain the model performance
under different clustering settings.
### VIII-C Clustering
For clustering 12659 friends, and 4013 strangers we experimented with k-means
and hierarchical clustering algorithms. In our experiments with different
numbers of final clusters, the k-means algorithm yielded the best results for
friend clustering, whereas hierarchical clustering was better for stranger
clustering. Due to space limitations, we will omit hierarchical clustering
results for friends and k-means results for strangers.
Figure 4: Coefficient of determination ($R^{2}$) values for 2 and 9 friend
clusters
Friend Clustering: In Figures 4 and 5, we show the adjusted coefficient of
determination555The adjusted coefficient of determination is the proportion of
variability in a data set that can be explained by the statistical model. This
value shows how well future outcomes can be predicted by the model. $R^{2}$
can take 0 as minimum, and 1 as maximum. ($R^{2}$) of our multiple regression
model with different $k$ values for friend clustering. The x-axis gives the
number of stranger clusters for which at least one friend cluster has an
impact. In Figure 4 we see the performance for maximum and minimum number of
friend clusters. For $k=2$, friend clusters are very roughly clustered, and
each cluster is not homogeneous enough (i.e., contains different types of
friends) to mine friend impacts666We use F-ratio probability to test the
significance of parameters, i.e., a low probability (we use .05 as cutoff) for
the F-ratio suggests that at least some of the friend cluster impacts are
significant.. As a result, we can observe friend impacts on very few clusters.
For $k=9$, friend clusters are more homogeneous, but in this case our multiple
regression model does not have many data points (strangers) to learn the
impacts of friend clusters.
Figure 5: Coefficient of determination ($R^{2}$) values for 5, 6 and 7 friend
clusters
Figure 5 shows the results for $k=5,6,7$ values. For two $k$ values, 5 and 6,
we have the best results. Our model hence suggests that friends of social
network users can be put into 5 or 6 clusters when considering how much they
can affect user decisions on stranger labeling.
Stranger Clustering: In Figure 6 we show how the $R^{2}$ values change for the
biggest and smallest numbers of stranger clusters. With 8 stranger clusters,
our model can detect friend cluster impacts on 5 out of 8 stranger clusters
only, whereas for 158 clusters the number is 15 out of 158. For 158 stranger
clusters, $R^{2}$ values are generally low because strangers are distributed
into too many clusters, and each stranger cluster does not have many data
points (strangers) to learn from. Although finding impacts on 5 out of 8
stranger clusters seems like a good performance, low $R^{2}$ values (lower
than 0.5) show that the model can explain less than 50% of the variation in
data. In Figure 7 we see that more stranger clusters can improve the model
performance and this leads to $R^{2}$ values close to 1. For 26 stranger
clusters, $R^{2}$ values are better, and we can find friend impacts in 16 out
of 26 stranger clusters.
Cross Validation: A major point in statistical modeling is the response to
out of sample validation; a statistical model can be over-fitted to the
training data, and it can perform poorly when applied to new testing data.
After clustering and prior to learning friend cluster impacts, we prepare a
test set for validating our model. We remove 10% of strangers from stranger
clusters and set those aside as the test strangers ($T$). Once friend impacts
are found for stranger clusters, we plug in the set of test strangers, and
calculate the root mean square value (RMSE) of their labels. RMSE for a
stranger $s$ and user $u$ is defined by using the predicted label
$\hat{L}_{us}$ and user given label $L_{us}$ as
$RMSE=\sqrt{\frac{\sum\limits_{s\in
T}{(L_{us}-\hat{L}_{us})}}{\left|T\right|}}$.
Figure 6: Coefficient of determination ($R^{2}$) values of friend impacts for
158 and 8 stranger clusters.
Cross validation results for different numbers of stranger clusters is
detailed in Table III by using 6 friend clusters. The first row of the table
shows the number of stranger clusters, whereas the second row shows the
average $R^{2}$ values in these clusters. In the third row, we show the median
size of stranger clusters; with increasing numbers of clusters, the number of
strangers in each cluster decreases. In the case of 158, the average number of
strangers in a cluster is reduced to 7, and this results in a poor performance
because the model cannot have enough data to learn friend impacts on stranger
clusters. The average number of validation points are shown in the fourth row.
An increasing number of stranger clusters results in fewer validation points
because some clusters will have less than 10 strangers themselves. In the
fifth row, the root mean square values (RMSE) are shown for these validation
points. In 26 stranger cluster our model yields the best $R^{2}$ and $RMSE$
pair results.
These experimental results suggest that the optimal number of stranger
clusters (26) is bigger than the optimal number of friend clusters ($k=5,6$).
We explain this by the fact that although users can choose friends of specific
characteristics, they cannot do so with strangers. As a result, strangers are
more diverse than friends, and they need to be clustered differently from
friends.
Figure 7: Coefficient of determination ($R^{2}$) values of friend impacts for 26 and 49 stranger clusters TABLE III: Performance values for different numbers of stranger clusters. Cluster count | 8 | 26 | 49 | 82 | 158
---|---|---|---|---|---
$R^{2}$ | 0.51 | 0.64 | 0.48 | 0.54 | 0.45
Median Size | 62 | 25 | 16 | 12 | 7
Validation points | 179 | 99 | 69 | 48 | 27
RMSE | 0.35 | 0.45 | 0.62 | 0.97 | 0.94
### VIII-D Friend Impacts and Risk Labels
In this section we will give computed friend cluster impacts, and show how
friends are assigned risk labels.
(a) 5 friend clusters
(b) 6 friend clusters
(c) 7 friend clusters
Figure 8: Percentage of positive and negative impact values for friend
clusters.
The rationale behind clustering was to observe different friend cluster
impacts on different stranger clusters. Although a friend cluster can have an
overall positive impact (i.e., reduces the risk label of most strangers),
friend clusters might have different signs and multitudes of impact values on
stranger clusters. In Figure 8 we show how different friend clusters can have
positive and negative impact values for different $k$ values (number of friend
clusters). Note that clusters are not identical across these figures, i.e.,
cluster 1 can have different members in each figure. This is because with
different number of final clusters, the clustering algorithms produce
potentially different clusters of data points. As seen in Figure 8(a), when we
increase the number of friend clusters from $k=5$ to $k=6$, positive and
negative impact frequencies change for each cluster because either friend
clusters became more homogeneous or some clusters did not have enough data
points to learn from. Figure 8(b) shows two friend clusters with overall
negative impacts (friend clusters 1 and 6). Figure 8(c) shows the positive and
negative impact frequencies for $k=7$, where frequencies are more emphasized
for negative and positive impacts of a cluster. Note that the number of
overall negative clusters is reduced from 2 to 1 here. Similar to a transition
from 5 to 6 clusters, friends of two negative clusters might be put into the
same cluster (cluster 1) or there were no longer enough strangers for some
friend clusters to learn a negative impact.
The existence of both positive and negative impact values for each friend
cluster confirms our intuition that impacts of friend clusters vary depending
on a stranger cluster. A friend is assigned a higher risk label when a friend
cluster has a big percentage of negative impact values. In Section VII, we
gave definitions of friend risk labels according to two threshold values
(x=20, y=50) of negative impact percentages. By using k=6 friend clusters,
from Figure 8(b) we see that friends from friend clusters 1 and 6 are labeled
as very risky because the negative impact percentages for the clusters are
$>0.6$. In the figure, we also see that none of the clusters have $<0.2$
negative impacts, hence no friends cluster is said to be not risky (label 1).
We tested the accuracy of our risk definition for friends by observing 261
deleted friendships of users. As a performance measure, we assumed that the
deleted friends should come from friends who are labeled as very risky, i.e.,
friends who belong to the 1st and 6th clusters. We have found 117 of the 261
deleted friends were found to belong to the 1st and 6th friend clusters.
Although we chose to use specific values for very risky and not risky label
thresholds (x=20, y=50) in assigning risk labels to strangers, our model can
ask social network users to define these threshold values on their own. With
this approach, our risk model for friends can be personalized by users and
applied to privacy settings on social networks.
## IX Conclusion and Future Work
In this work, we looked into risks of friendships and analyzed how the risk
labels of friends of friends can be used to compute risk labels of friends. We
found that the number of mutual friends is not very important to change the
risk perception of a user towards a friend of friend. On the other hand,
having different types of mutual friends (i.e., friends from different friend
clusters) with a friend of friend plays a bigger role in users’ risk
perception. Our results showed that in terms of risk, friends can be grouped
into 6-7 clusters, whereas the number of groups for strangers can reach 26 or
more. These results show that even though user numbers reach millions, friends
for each user have similar roles. We have validated risk labels of friends on
deleted Facebook friendships, and showed that risks of friendships can indeed
be learned by considering users’ risk perception towards friends of friends.
In the future, we want to create sets of global privacy settings by using our
risk model, so that privacy settings can be automatically applied to different
social network users.
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|
arxiv-papers
| 2012-10-11T13:38:55 |
2024-09-04T02:49:36.389623
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cuneyt Gurcan Akcora, Barbara Carminati, Elena Ferrari",
"submitter": "Cuneyt Gurcan Akcora",
"url": "https://arxiv.org/abs/1210.3234"
}
|
1210.3306
|
# Gravitational anomalies and entropy
Bibhas Ranjan Majhi
IUCAA, Post Bag 4, Ganeshkhind,
Pune University Campus, Pune 411 007, India
E-mail: bibhas@iucaa.ernet.in
###### Abstract
A derivation of entropy from the expressions for two dimensional gravitation
anomalies is given. Starting from the near horizon anomalous energy-momentum
tensors corresponding to particular anomalies, the Virasoro algebra with
central extension is obtained. The central charge is identified by comparing
with the standard form of the algebra. Then the conserved charge in the ground
state is computed. Finally, using the Cardy formula the entropy is obtained.
Here both the vector and chiral theories are discussed.
## 1 Introduction
Since the discovery of Hawking effect [1], there are several methods in the
literature to understand this. The important feature is that all of them yield
the universal result: temperature and entropy of a black hole are proportional
to the surface gravity and the horizon area, respectively [1, 2]. Still it
remains as a long standing and important question whether the black hole
thermodynamics has a statistical description in terms of microscopic states.
Till now there exists several ways towards this attempt: loop quantum gravity
[3], string theory [4], the conformal field theory [5, 6] technique, etc. But
none of them is complete.
After the discovery of the Hawking effect, people started thinking that it may
play a major role to shed some light towards the quantum nature of gravity.
Therefore it is necessary to have deeper understanding of this effect. Among
the existing approaches, anomaly method attracted a lot. The idea is that: it
is not possible to simultaneously preserve conformal and diffeomorphism
symmetries at the quantum level. In vector theory, any one of them must be
sacrificed. As the latter symmetry is usually retained, there is, in general,
a violation of the conformal invariance which is manifested by a non-vanishing
trace of the energy-momentum tensor and leads to trace anomaly [7]. Whereas in
the chiral theory both the trace and diffeomorphism anomalies exist [8, 9,
10]. Long ago, Christansen and Fulling [11] first determined the Hawking flux
in the vector theory for a two dimensional black hole.
Recently, the method has been extended to the higher dimensional black holes
($D\geq 4$). This is mainly based on the fact that near the event horizon the
effective theory is two dimensional and the background is effectively the two
dimensional ($t-r$) black hole metric [12, 13, 14]. Now since, the ingoing
modes are absorbed classically by the black hole, the theory must be chiral
near the horizon and at the quantum level both the trace and diffeomorphism
anomalies will appear. In this case, there exists two types of anomaly
expressions: consistent and covariant expressions. They are related by a local
counter term [8, 9, 10]. Robinson and Wilczek [12] showed that the Hawking
flux can be obtained from the consistent anomaly expression with the use of
the covariant boundary condition. In this case, both the Ward identities were
required. Later on, an elegant and conceptually clearer derivation was given
in [15] which was based on the covariant anomaly expression 111For more
details and several important issues, see [16, 17, 18, 19].. Interestingly,
only one Ward identity was required.
Although, the anomaly approach has been applied for several metrics to find
the emission flux, nevertheless, a connection between these anomalies and the
entropy has so far been missing in the literature. Such a investigation is
necessary to have complete understanding of the role played by the anomalies
in gravitational physics. In this paper, we establish this connection
precisely in the case of an arbitrary dimensional static black hole.
We briefly give our methodology to compute the entropy using the near horizon
gravitational anomalies. The method, we shall follow here, is the conformal
field theory (CFT) technique, particularly the work of Solodukhin [21]. It is
well known that the relevant conformal theory is the Liouville theory [20]
living on the two dimensional boundary. In this theory, the conserved charges
corresponding to near horizon diffeomorphism symmetry generators obey the
standard Virasoro algebra with central extension and it leads to central
charge [21] 222For a Hamiltonian or Lagrangian approach and its related issues
to find the Virasoro algebra, see [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41].. Then evaluating the ground state conserved
charge one can find the entropy using Cardy formula [22]. Here, we shall first
cast the anomalous energy-momentum tensors corresponding to respective
theories (vector and chiral theories) to the identical form of that of
Liouville theory, since this is a relevant theory for the conformal field
theory [20]. This will be done by using the transformations of metric and the
auxiliary scalar field. Using this transformed tensor, the conserved charges
$L_{n}$, corresponding to the near horizon diffeomorphism symmetry generators
[23, 24], will be defined. Computation of the commutator among $L_{n}$ will
lead to the standard Virasoro algebra with the central extension. Then one can
immediately identify the central charge. The other important quantity is the
conserved charge corresponding to the zero mode, i.e. $L_{0}$. This will be
obtained by straight forward evaluation. We shall then apply the Cardy formula
[22] to obtain the expression for entropy. Generally two copies of Virasoro
algebra, one for $L_{n}$ (holomorphic) and other for ${\bar{L}}_{n}$ (anti-
holomorphic), contribute to the entropy. However, in our case since the theory
is near horizon where only ingoing modes contribute, and so only one copy,
namely $L_{n}$, is sufficient to calculate the entropy.
The organization of the paper is as follows. In the next section, the near
horizon effective two dimensional metric will be introduced. Here it will be
written in the null coordinates which will be relevant for our subsequent
analysis. In section 3, the energy-momentum tensor in two dimensional vector
theory will be casted to that of Liouville theory. Defining the conserved
charge, we will obtain the Virasoro algebra. Then by the Cardy formula the
entropy will be obtained. Next section will be devoted for the same in the
case of chiral theory. Final section will contain the conclusions.
## 2 Metric and null coordinates
The physics of a black hole is mainly dominated by the event horizon
properties. Near this horizon, the theory is effectively determined by the two
dimensional ($t-r$) metric [12, 13, 14]:
$\displaystyle ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}$ (1)
whose event horizon is defined by the relation $F(r=r_{H})=0$ and the surface
gravity is given by $\kappa=\frac{F^{\prime}(r_{H})}{2}$. For our later
analysis, it will be appropriate to use the null tortoise coordinates, defined
as,
$\displaystyle u=t-r^{*},\,\,\ v=t+r^{*};\,\,\,\ dr^{*}=\frac{dr}{F(r)}~{}.$
(2)
In these coordinates metric (1) takes to the following form
$\displaystyle ds^{2}=\frac{F(r)}{2}(dudv+dvdu)~{}.$ (3)
The justification of choosing the effective $2D$ metric (1) for our purpose of
finding entropy by two dimensional gravitational anomalies in the context of
Virasoro algebra is as follows. Consider first the following four dimensional
metric:
$\displaystyle
ds^{2}_{4}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})~{}.$ (4)
The horizon (apparent) is defined as the curve $H$ such that [42],
$\displaystyle g^{ab}\nabla_{a}r\nabla_{b}r|_{H}=0~{},$ (5)
where $g_{ab}$ is the ($t-r$) sector of the full metric (4). This clearly says
$F(r=r_{H})=0$, as stated earlier. Now the above condition is invariant under
conformal transformation $g_{ab}\rightarrow\Omega^{2}g_{ab}$ and so the two
dimensional metric is determined by this condition only upto a conformal
factor. Interestingly, very near to the horizon, the diffeomorphisms which
preserve the condition (5), form the infinite dimensional group of conformal
transformations in two dimensions. The corresponding generators satisfy
Virasoro algebra. Also, the fields become massless. For details, see section 2
of [21]. Therefore, the physics, near the horizon, is conformal and governed
by an effective two dimensional field theory. Thus we see that the horizon
defining condition (5) is essentially a condition on the radial function $r$
of the full metric (4) and the two dimensional metric is determined upto a
conformal factor. So the horizon dynamics is encoded in the radial function
while the $2$D metric is chosen from the same conformal class. Therefore, here
we have chosen the two dimensional metric as (1) and it is feasible to study
conformal field theory on this background.
## 3 Vector theory, Virasoro algebra and entropy
In this section, the entropy will be evaluated from the anomalous stress
tensor for the vector theory. The idea is that near the event horizon, the
theory is effectively dominated by the two dimensional metric of the form (3).
For details, see [12, 13, 14]. Now the energy-momentum tensor for the scalar
field under this effective two dimensional metric, at the quantum level, leads
to non-vanishing trace since usually the diffeomorphism symmetry is retained.
Such theory is called as vector theory.
In this case, the trace anomaly is given by [43]
$\displaystyle\tilde{T}{{}^{a}_{a}}=\frac{\tilde{R}}{24\pi}~{},$ (6)
and the energy-momentum tensor satisfies the conservation equation
$\tilde{\nabla}^{a}\tilde{T}_{ab}=0$. The anomaly induced effective action
$S_{P}$, obtained from the functional integration of
$\tilde{T^{a}_{a}}=\frac{2}{\sqrt{-\tilde{g}}}\tilde{g}^{ab}\frac{\delta
S_{P}}{\delta\tilde{g}^{ab}}$, is given by the Polyakov action [44],
$\displaystyle S_{P}=-\frac{1}{96\pi}\int
d^{2}x\sqrt{-{\tilde{g}}}{\tilde{R}}\frac{1}{\tilde{\Box}}{\tilde{R}}~{}.$ (7)
This action is non-local and it can be written in local form by introducing an
auxiliary field $\tilde{\Phi}$:
$\displaystyle S_{P}=-\frac{1}{96\pi}\int
d^{2}x{\sqrt{-\tilde{g}}}(-\tilde{\Phi}\tilde{\Box}\tilde{\Phi}+2\tilde{\Phi}\tilde{R})~{}.$
(8)
The two dimensional anomalous energy-momentum tensor is given by,
$\displaystyle{\tilde{T}}_{ab}=\frac{2}{\sqrt{-\tilde{g}}}\frac{\delta
S_{P}}{\delta\tilde{g}^{ab}}=\frac{1}{48\pi}\Big{[}\tilde{\nabla}_{a}\tilde{\Phi}\tilde{\nabla}_{b}\tilde{\Phi}-2\tilde{\nabla}_{a}\tilde{\nabla}_{b}\tilde{\Phi}+\tilde{g}_{ab}\Big{(}2\tilde{R}-\frac{1}{2}\tilde{\nabla}_{c}\tilde{\Phi}\tilde{\nabla}^{c}\tilde{\Phi}\Big{)}\Big{]}$
(9)
with the auxiliary scalar $\tilde{\Phi}$ satisfies the equation of motion,
$\displaystyle\tilde{\Box}\tilde{\Phi}=\tilde{R}~{}.$ (10)
Here $\tilde{R}$ is the two dimensional Ricci scalar. Christensen and Fulling
[11] evaluated the Hawking flux by solving the conservation equation and the
trace anomaly expression (6) for the Schwarzschild black hole. Here, in the
following, we will find the entropy from the expression for anomalous energy-
momentum tensor (9).
Now it is known that the Liouville theory is conformally invariant and
possesses two sets of Virasoro generators $L_{n}$ and $\bar{L}_{n}$ [20].
Therefore, it is a relevant CFT and so it is convenient to recast the energy-
momentum tensor (9) into that of the Liouville theory so that it fits into the
Virasoro scheme. The same technique was also adopted in [21], where the four
dimensional Einstein-Hilbert action was casted to Liouville action by
dimensional reduction technique and then imposing a set of particular
transformations. Motivated by this fact, here also we will do the same. This
is done by the following set of transformations,
$\displaystyle\tilde{g}_{ab}=e^{\Big{(}\frac{2}{qr_{H}}-\frac{qr_{H}}{16}\Big{)}\Phi}g_{ab};\,\,\,\
\tilde{\Phi}=\frac{qr_{H}}{8}\Phi~{}.$ (11)
Under these, (9) and (10) transform to
$\displaystyle
T_{ab}=\frac{1}{48\pi}\Big{[}\frac{1}{2}\nabla_{a}\Phi\nabla_{b}\Phi-\frac{qr_{H}}{4}\nabla_{a}\nabla_{b}\Phi+g_{ab}\Big{(}2R-\frac{1}{4}\nabla_{c}\Phi\nabla^{c}\Phi-\frac{4}{qr_{H}}\nabla_{c}\nabla^{c}\Phi$
$\displaystyle+\frac{qr_{H}}{8}\nabla_{c}\nabla^{c}\Phi\Big{)}\Big{]}$ (12)
$\displaystyle\Box\Phi=\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}R$
(13)
where $r_{H}$ is the radius of the event horizon and $q$ is a constant.
Substituting (13) in (12), the energy-momentum tensor reduces to the following
form
$\displaystyle
T_{ab}=\frac{1}{48\pi}\Big{[}\frac{1}{2}\nabla_{a}\Phi\nabla_{b}\Phi-\frac{qr_{H}}{4}\nabla_{a}\nabla_{b}\Phi+g_{ab}\Big{(}\frac{qr_{H}}{4}\nabla_{c}\nabla^{c}\Phi-\frac{1}{4}\nabla_{c}\Phi\nabla^{c}\Phi\Big{)}\Big{]}~{}.$
(14)
This is similar to the energy-momentum tensor of Liouville theory as obtained
by Solodukhin [21] from the four dimensional Einstein-Hilbert action. Here, we
obtained this from the two dimensional trace anomaly expression by applying
the transformations (11).
Before proceeding further, note that because of non-vanishing of trace
$\displaystyle T=\frac{1}{48\pi}\frac{qr_{H}}{4}\Box\Phi~{},$ (15)
corresponding to the stress tensor (14), the theory is not conformal. However,
within the small vicinity of the horizon, we will show below that the theory
becomes conformal. In the ($t,r^{*}$) coordinate, equation of motion for
$\Phi$ (13) under the background (1) reads,
$\displaystyle-\partial^{2}_{t}\Phi+\partial^{2}_{r^{*}}\Phi=F\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}R~{}.$
(16)
Now for the metric (1), $R$ is non-zero and since $F\rightarrow 0$ near the
horizon, right hand side of the above is negligible. Therefore, (16) reduces
to
$\displaystyle\partial^{2}_{t}\Phi-\partial^{2}_{r^{*}}\Phi=0~{},$ (17)
which signifies that due to existence of the horizon, the free scalar field is
propagating in flat spacetime near the horizon. Hence the trace (15) vanishes
and it implies that the Poisson algebra of the components of trace tensor (14)
closes and they form the Virasoro algebra very near to the horizon. Therefore,
the theory of the scalar field $\Phi$, whose tress tensor is given by (14), is
conformal when one considers it in the small vicinity of the horizon.
Let us now proceed to obtain the Virasoro algebra among the charges
corresponding to the near horizon diffeomorphism symmetry generators. Since
near the horizon the ingoing modes contribute, only the $T_{vv}$ component of
(14) is important. Under the metric (3), $T_{vv}$ component comes out to be
$\displaystyle
T_{vv}=\frac{1}{48\pi}\Big{[}\frac{1}{2}\partial_{v}\Phi\partial_{v}\Phi-\frac{qr_{H}}{4}\Big{(}\partial_{v}^{2}\Phi-\frac{\partial_{v}F(r)}{F(r)}\partial_{v}\Phi\Big{)}\Big{]}~{}.$
(18)
Again, since we are interested only near the horizon where
$\frac{\partial_{v}F(r)}{F(r)}\rightarrow\frac{F^{\prime}(r_{H})}{2}=\kappa$,
the above reduces to the following form,
$\displaystyle
T_{vv}=\frac{1}{48\pi}\Big{[}\frac{1}{2}\partial_{v}\Phi\partial_{v}\Phi-\frac{qr_{H}}{4}\partial_{v}^{2}\Phi+\frac{qr_{H}\kappa}{4}\partial_{v}\Phi\Big{]}~{}.$
(19)
Here, we shall determine the entropy by counting the horizon microstates of
(1) via Cardy formula [22],
$\displaystyle S=2\pi\sqrt{\frac{CL_{0}}{6}}$ (20)
corresponding to the quantum Virasoro algebra
$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{C}{12}n(n^{2}-1)\delta_{n+m,0}~{}.$
(21)
In the above, $[,]$ is the commutator while $C$ and $L_{n}$ are the central
charge and the conserved charge for $n$th mode respectively. For the present
case we define $L_{n}$ as,
$\displaystyle L_{n}=\int dv\xi_{n}(v)T_{vv}(v)$ (22)
where $\xi_{n}(v)$ is the near horizon diffeomophism symmetry generator,
obeying one sub-algebra isomorphic to Diff $S^{1}$,
$\displaystyle[\xi_{n},\xi_{m}]=(n-m)\xi_{n+m}~{}.$ (23)
Following [24], the explicit form of these generators for the metric (1)
satisfying (23) can be obtained. These are given by,
$\displaystyle\xi_{n}=\frac{1}{\kappa}e^{in\kappa v}$ (24)
where $n$ is an integer, can be positive and negative as well. The coordinate
$v$ is periodic within the range $0\leq v\leq\frac{2\pi}{\kappa}$ and hence we
choose this as the limits of the integration (22).
Now, in order to obtain the commutator of the conserved charges (22), let us
first calculate the Poisson bracket $\\{L_{n},L_{m}\\}$. This will contain a
Poisson bracket $\\{T_{vv}(v),T_{vv}(\bar{v})\\}$. To evaluate the bracket, we
need the following basic Poisson bracket relation
$\displaystyle\\{\Phi(r^{*},t),\partial_{t}\Phi(\bar{r^{*}},t)\\}=48\pi\delta(r^{*}-\bar{r^{*}})$
(25)
and so consequently, we have,
$\displaystyle\\{(\partial_{t}+\partial_{r^{*}})\Phi(r^{*},t),(\partial_{t}+\partial_{\bar{r^{*}}})\partial_{t}\Phi(\bar{r^{*}},t)\\}=96\pi\partial_{r^{*}}\delta(r^{*}-\bar{r^{*}})~{}.$
(26)
The bracket (25) is actually the bracket among the field and its conjugate
momentum. The normalization in the right hand side has been fixed by the
normalization in the corresponding action. In radial null coordinates, above
bracket reduces to the following form,
$\displaystyle\\{\partial_{v}\Phi(v),\partial_{\bar{v}}\Phi(\bar{v})\\}=48\pi\partial_{v}\delta(v-\bar{v})~{}.$
(27)
For the present case $T_{vv}(v)$ is given by (19). So using (27) we obtain
$\displaystyle\\{T_{vv}(v),T_{vv}(\bar{v})\\}$ $\displaystyle=$
$\displaystyle\frac{1}{48\pi}\Big{[}\partial_{v}\Phi(v)\partial_{\bar{v}}\Phi(\bar{v})\partial_{v}\delta(v-\bar{v})-\frac{qr_{H}}{4}\partial_{v}\Phi(v)\partial_{v}\partial_{\bar{v}}\delta(v-\bar{v})$
(28) $\displaystyle+$
$\displaystyle\frac{qr_{H}\kappa}{4}\partial_{v}\Phi(v)\partial_{v}\delta(v-\bar{v})-\frac{qr_{H}}{4}\partial_{\bar{v}}\Phi(\bar{v})\partial_{v}^{2}\delta(v-\bar{v})$
$\displaystyle+$
$\displaystyle\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\partial_{\bar{v}}\partial_{v}^{2}\delta(v-\bar{v})-\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\kappa\partial_{v}^{2}\delta(v-\bar{v})$
$\displaystyle+$
$\displaystyle\frac{qr_{H}\kappa}{4}\partial_{\bar{v}}\Phi(\bar{v})\partial_{v}\delta(v-\bar{v})-\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\kappa\partial_{v}\partial_{\bar{v}}\delta(v-\bar{v})$
$\displaystyle+$
$\displaystyle\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\kappa^{2}\partial_{v}\delta(v-\bar{v})\Big{]}~{}.$
Hence, considering that the auxiliary field $\Phi$ and its derivatives with
respect to $v$ are periodic within the interval $0\leq v\leq 2\pi/\kappa$, we
find,
$\displaystyle\\{L_{n},L_{m}\\}$ $\displaystyle=$ $\displaystyle\int
dv\Big{(}\xi_{n}\partial_{v}\xi_{m}-\xi_{m}\partial_{v}\xi_{n}\Big{)}T_{vv}-\frac{1}{48\pi}\Big{(}\frac{qr_{H}\kappa}{4}\Big{)}^{2}\int
dv\xi_{m}\partial_{v}\xi_{n}$ (29) $\displaystyle-$
$\displaystyle\frac{1}{48\pi}\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\int
dv\partial_{v}\xi_{m}\partial_{v}^{2}\xi_{n}~{}.$
Finally, substituting the expressions for $\xi_{n}$ from (24) in the above and
performing the integration we obtain,
$\displaystyle
i\\{L_{n},L_{m}\\}=(n-m)L_{n+m}+\frac{1}{24}\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}(n^{3}+n)\delta_{n+m,0}~{}.$
(30)
Application of the usual prescription for getting the commutator relation from
the Poisson bracket yields, in the unit $\hbar=1$,
$\displaystyle[L_{n},L_{m}]=i\\{L_{n},L_{m}\\}=(n-m)L_{n+m}+\frac{1}{24}\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}(n^{3}+n)\delta_{n+m,0}~{}.$
(31)
In order to put this similar to the standard form (21), let us redefine
$L_{n}$ as,
$\displaystyle L_{n}\rightarrow
L_{n}-\frac{1}{24}\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}\delta_{n,0}~{}.$ (32)
This redefinition leads to the following algebra:
$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{1}{24}\Big{(}\frac{qr_{H}}{4}\Big{)}^{2}n(n^{2}-1)\delta_{n+m,0}~{}.$
(33)
Comparing this with (21) we obtain the central charge for the black hole case,
$\displaystyle C=\frac{q^{2}r_{H}^{2}}{32}~{}.$ (34)
Next step is to find the value of $L_{0}$. By the definition for $L_{n}$ (22)
we have,
$\displaystyle L_{0}=\frac{1}{48\pi\kappa}\int
dv\Big{[}\frac{1}{2}\partial_{v}\Phi\partial_{v}\Phi-\frac{qr_{H}}{4}\partial_{v}^{2}\Phi+\frac{qr_{H}\kappa}{4}\partial_{v}\Phi\Big{]}~{}.$
(35)
Here, since $\Phi$ and its derivative terms with respective to coordinate $v$
are periodic within the interval $0\leq v\leq 2\pi/\kappa$, the following
intregal vanishes:
$\displaystyle\int_{0}^{2\pi/\kappa}dv~{}\partial_{v}^{2}\Phi=\partial_{v}\Phi|_{0}^{2\pi/\kappa}=0,$
$\displaystyle\int_{0}^{2\pi/\kappa}dv~{}\partial_{v}\Phi=\Phi|_{0}^{2\pi/\kappa}=0~{}.$
(36)
To evaluate the first integral in (35) we will use the following trick. Let us
first find the solution of $\Phi$ from (13). For simplicity we consider the
Schwarzschild black hole. Also, our region of interest is near the event
horizon and therefore we need only ($t-r$)-sector of the metric. In this case,
the solution is given by [45],
$\displaystyle\Phi=\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}\Big{[}at-\ln\Big{(}1-\frac{2M}{r}\Big{)}+A\Big{\\{}r+2M\ln\Big{(}r-2M\Big{)}\Big{\\}}+B\Big{]}~{},$
(37)
where $a$, $A$ and $B$ are arbitrary constants. Now it has been shown that the
relevant vacuum state to obtain the Hawking flux, in the context of
gravitational anomalies (both in vector case [45] and in chiral case [17]), is
the Unruh vacuum. In this vacuum, the solution for $\Phi$ near the horizon
comes out to be [45],
$\displaystyle\Phi\approx-\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}\frac{v}{4M}+\textrm{constant}~{}.$
(38)
Using this value we obtain,
$\displaystyle\int_{0}^{2\pi/\kappa}dv~{}(\partial_{v}\Phi)^{2}$
$\displaystyle=$
$\displaystyle-\int_{0}^{2\pi/\kappa}dv~{}\Phi\partial^{2}\Phi\approx\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}\frac{1}{4M}\int_{0}^{2\pi/\kappa}dv~{}v\partial^{2}_{v}\Phi$
(39) $\displaystyle=$
$\displaystyle-\Big{(}\frac{qr_{H}}{16}+\frac{2}{qr_{H}}\Big{)}^{-1}\frac{1}{4M}\int_{0}^{2\pi/\kappa}dv~{}\partial_{v}\Phi=0~{},$
and so $L_{0}$ vanishes. Therefore, the redefined $L_{n}$ (32) yields,
$\displaystyle L_{0}=\frac{q^{2}r_{H}^{2}}{384}$ (40)
Now, substituting (34) and (40) in the Cardy formula (20), we obtain the
entropy,
$\displaystyle S=\frac{q^{2}l_{p}^{2}}{96\sqrt{2}}S_{BH}$ (41)
where $S_{BH}=\frac{\pi r_{H}^{2}}{l_{p}^{2}}$ is the Bekenstein-Hawking
entropy and $l_{p}$ is the Planck length. Note that the conformal entropy $S$
is equal to the Bekenstein-Hawking entropy if we choose
$q^{2}=\frac{96\sqrt{2}}{l_{p}^{2}}$.
Now several comments are in order. (i) To see that there is no dimensional
ambiguity in the transformations (11), remember, in two dimensions $\Phi$ is
dimensionless if we choose Lorentz-Heaviside unit. Hence $qr_{H}$ must be
dimensionless and this is actually the case since the dimension of $q$, as we
just observed, is inverse of length. (ii) Here we obtained the entropy from
the anomalies which are the result of the quantization of the matter fields
near the horizon. So the entropy (41) is not the entropy of the black hole,
rather it can be interpreted as that of the matter. But interestingly, the
matter entropy is proportional to the horizon area. Similar has been observed
earlier in [46] for a freely falling box of ideal gas in a black hole. (iii)
Both the central charge $C$ and the zero mode eigenvalue $L_{0}$ depend on
$q$. The dependence of $C$ on $q$ is similar to [21] while that of $L_{0}$ on
$q$ is different. This is because the we applied a different method to find
$L_{0}$. Solodukhin determined it from a more general zero mode solution of
field equation near the horizon and imposing specific boundary conditions.
While we found it by just redefining $L_{n}$ and hence in our case $L_{0}$ is
proportional to $q^{2}$. Therefore the final expression for entropy contains
the parameter $q$. This ambiguity is because a direct evaluation leads to
vanishing $L_{0}$ for the classical configuration. Depending on the
regularization of the ground state one can obtain a non-zero value and
different regularization leads to different value. Of course, one might be
interested to see if there exists any method in which $L_{0}$ will come out to
be inversely proportional to $q^{2}$ and in that case the expression for
entropy will not contain any undetermined coefficient.
## 4 Chiral theory, Virasoro algebra and entropy
In chiral theory, if one of the symmetries (conformal or diffeomorphism)
breaks down then naturally other must breaks down. Hence both trace and
diffeomorphism anomalies will appear in this theory. The analogous energy-
momentum tensor is given by [47, 16, 17],
$\displaystyle\tilde{T}_{ab}=\frac{1}{48\pi}\Big{[}\frac{1}{4}\tilde{D}_{a}\tilde{G}\tilde{D}_{b}\tilde{G}-\frac{1}{2}\tilde{D}_{a}\tilde{D}_{b}\tilde{G}+\frac{1}{2}\tilde{g}_{ab}\tilde{R}\Big{]}$
(42)
where
$\tilde{D}_{a}=\tilde{\nabla}_{a}\pm\tilde{\bar{\epsilon}}_{ab}\tilde{\nabla}^{b}$
is the chiral derivative and the auxiliary field $\tilde{G}$ satisfies the
equation of motion,
$\displaystyle\tilde{\Box}\tilde{G}=\tilde{R}~{}.$ (43)
Here $+$ ($-$) corresponds to the ingoing (outgoing) mode. Since, again the
near horizon theory will be considered where only ingoing mode exists, we will
concentrate on the plus sign. This stress tensor corresponds to both the trace
and diffeomorphism anomalies which are in covariant form. The expressions are:
$\displaystyle\tilde{T}^{a}_{a}=\frac{\tilde{R}}{48\pi};\,\,\,\
\tilde{\nabla}^{a}\tilde{T}_{ab}=-\frac{1}{96\pi}\tilde{\bar{\epsilon}}_{bc}\tilde{\nabla}^{c}\tilde{R}~{}.$
(44)
Now, as earlier, to get a form of energy-momentum tensor similar to that of
Liouville theory the transformations (11) will be applied on (42) and (43).
Then proceeding in the identical way we obtain the following transformed
energy-momentum tensor,
$\displaystyle
T_{ab}=\frac{1}{48\pi}\Big{[}\frac{1}{8}D_{a}GD_{b}G-\frac{qr_{H}}{16}D_{a}D_{b}G+\frac{1}{4}g_{ab}\Big{(}\frac{qr_{H}}{4}\Box
G-\frac{1}{4}\nabla_{c}G\nabla^{c}G+\frac{1}{2}(\frac{qr_{H}}{8})^{2}\nabla_{c}G\nabla^{c}G\Big{)}\Big{]}~{}.$
(45)
Near the horizon, since only ingoing modes will be considered, under the
background metric (3) we must have,
$\displaystyle D_{v}=2\nabla_{v};\,\,\,\ D_{u}=0~{}.$ (46)
Therefore, $T_{vv}$ component is given by (18). Hence following the identical
steps we find the entropy as (41).
## 5 Conclusions
The effective theory near the event horizon of black hole corresponds to
($1+1$)-dimensional metric and is conformal. Such situation gives rise to the
two dimensional gravitational anomaly at the quantum level. Depending on the
nature of the theory (vector or chiral), there exits trace [7] or both trace
and diffeomorphism anomalies [8, 9, 10]. It has been shown that the Hawking
flux can be derived from these anomaly expressions [11, 12, 13, 15]. So it may
be possible that the anomalies can play a crucial role to illuminate the
quantum nature of a black hole. Hence, one has to see if they have a wide
applicability in the different aspects of the gravity.
In this paper, we made an attempt to the thermodynamics of the black hole
system. Here we found the entropy from the anomalous energy-momentum tensor
corresponding to the particular anomaly/anomalies. We discussed both vector as
well as chiral theories. We adopted the conformal field theory technique
proposed earlier in [5, 6, 21]. First, the expressions for energy-momentum
tensors were casted to that of Liouville theory, since this a relevant
conformal theory. Then the conserved charges corresponding to the near horizon
ingoing tensor and diffeomorphism symmetry generators were defined. An
explicit calculation of commutator between these charges gave rise to the
Virasoro algebra which had the central extension. Finally, identifying the
central charge and calculating the charge of the zero mode, we derived the
entropy by the Cardy formula.
It was noted that the entropy came out to be proportional to the well known
Bekenstein-Hawking expression. The proportionality constant contained a
undetermined parameter “$q$”, originally appeared in the transformations (11).
To obtain the exact expression one needed to choose a particular value of it.
The similar situation has been encountered earlier in [38, 40]. The
independent derivation of such choice may be interesting and will make this
method more appealing. Unfortunately, it is not known to the author.
Despite this limitation, our paper addressed a glaring omission in the
literature. A way to find the entropy from the gravitational anomalies has
been given. This may explore the importance of anomalies in studying the
quantum nature of gravity.
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|
arxiv-papers
| 2012-10-11T17:50:23 |
2024-09-04T02:49:36.407207
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/1210.3306"
}
|
1210.3312
|
11institutetext: Laboratoire Informatique d’Avignon,
BP 91228 84911, Avignon, Cedex 09, France
11email: juan-manuel.torres@univ-avignon.fr 22institutetext: École
Polytechnique de Montréal,
CP. 6128 succursale Centre-ville, Montréal, Québec, Canada
# Artex is AnotheR TEXt summarizer
Juan-Manuel Torres-Moreno, 1122
###### Abstract
This paper describes Artex, another algorithm for Automatic Text
Summarization. In order to rank sentences, a simple inner product is
calculated between each sentence, a document vector (text topic) and a lexical
vector (vocabulary used by a sentence). Summaries are then generated by
assembling the highest ranked sentences. No ruled-based linguistic post-
processing is necessary in order to obtain summaries. Tests over several
datasets (coming from Document Understanding Conferences (DUC), Text Analysis
Conference (TAC), evaluation campaigns, etc.) in French, English and Spanish
have shown that Artex summarizer achieves interesting results.
Keywords: Automatic Text Summarization, Space Vector Model, Text extraction,
Ultra-stemming
## 0.1 Introduction
Automatic Text Summarization (ATS) is the process to automatically generate a
compressed version of a source document [15]. Query-oriented summaries focus
on a user’s request, and extract the information related to the specified
topic given explicitly in the form of a query [2]. Generic mono-document
summarization tries to cover as much as possible the information content.
Multi-document summarization is a oriented task to create a summary from a
heterogeneous set of documents on a focused topic. Over the past years,
extensive experiments on query-oriented multi-document summarization have been
carried out. Extractive Summarization produces summaries choosing a subset of
representative sentences from original documents. Sentences are ordered and
then assembled according to their relevance to generate the final summary
[10].
This article introduces a new method of summarization based in sentences
extraction on Vector Space Model (VSM). We score each sentence by calculating
their inner product with a pseudo-sentence vector and a pseudo-word vector.
Results show that Artex not only preserves the content of the summaries
generated using this new representation, but often, surprisingly the
performance can be improved. Artex could be an interesting and simple
algorithm using the extractive summarization paradigm. Our tests on trilingual
corpora (English, Spanish and French) evaluated by the Fresa algorithm
(without human references) confirm the good performance of Artex.
In this paper, related work is given in Section 0.2. Section 0.3 presents the
new algorithm of Automatic Text Summarization. Experiments are presented in
Section 0.4, followed by Results in Section 0.5 and Conclusions in Section
0.6.
## 0.2 Related works
Research in Automatic Text Summarization was introduced by H.P. Luhn in 1958
[9]. In the strategy proposed by Luhn, the sentences are scored for their
component word values as determined by tf*idf-like weights. Scored sentences
are then ranked and selected from the top until some summary length threshold
is reached. Finally, the summary is generated by assembling the selected
sentences in the original source order. Although fairly simple, this
extractive methodology is still used in current approaches. Later on, [3]
extended this work by adding simple heuristic features such as the position of
sentences in the text or some key phrases indicate the importance of the
sentences. As the range of possible features for source characterization
widened, choosing appropriate features, feature weights and feature
combinations have become a central issue.
A natural way to tackle this problem is to consider sentence extraction as a
classification task. To this end, several machine learning approaches that
uses document-summary pairs have been proposed [6, 12], An hybrid method
mixing statistical and linguistics algorithms is presented in [1]. [10] and
[15] propose a good state-of-art of Automatic Text Summarization tasks and
algorithms.
### 0.2.1 Document Pre-processing
The first step to represent documents in a suitable space is the pre-
processing. As we use extractive summarization, documents have to be chunked
into cohesive textual segments that will be assembled to produce the summary.
Pre-processing is very important because the selection of segments is based on
words or bigrams of words. The choice was made to split documents into full
sentences, in this way obtaining textual segments that are likely to be
grammatically corrects. Afterwards, sentences pass through several basic
normalization steps in order to reduce computational complexity.
The process is composed by the following steps:
1. 1.
Sentence splitting. Simple rule-based method is used for sentence splitting.
Documents are chunked at the period, exclamation and question mark.
2. 2.
Sentence filtering. Words lowercased and cleared up from sloppy punctuation.
Words with less than 2 occurrences ($f<2$) are eliminated (Hapax legomenon
presents once in a document). Words that do not carry meaning such as
functional or very common words are removed. Small stop-lists (depending of
language) are used in this step.
3. 3.
Word normalization. Remaining words are replaced by their canonical form using
lemmatization, stemming, ultra-stemming or none of them (raw text). Four
methods of normalization were applied after filtering:
* •
Lemmatization by simples dictionaries of morphological families. These
dictionaries have 1.32M, 208K and 316K words-entries in Spanish, English and
French, respectively.
* •
Porter’s Stemming, available at Snowball (web site
http://snowball.tartarus.org/texts/stemmersoverview.html) for English,
Spanish, French among other languages.
* •
Ultra-stemming. This normalization seems be very efficient and it produces a
compact matrix representation [16]. Ultra-stemming consider only the $n$ first
letters of each word. For example, in the case of ultra-stemming (first
letter, Fix1), inflected verbs like “sing”, “song”, “sings”, “singing”… or
proper names “smith”, “snowboard”, “sex”,… are replaced by the letter “s”.
4. 4.
Text Vectorization. Documents are vectorized in a matrix $S_{[P\times N]}$ of
$P$ sentences and $N$ columns. Each element $s_{i,j}$ represents the
occurrences of an object $j$ (a letter in the case of ultra-stemming, a word
in the case of lemmatization or a stem for stemming), $j=1,2,...,N$ in the
sentence $i$, $i=1,2,...,P$.
## 0.3 AnotheR TEXt summarizer (Artex)
Artex111In French, Artex est un Autre Résumeur TEXtuel. is a simple extractive
algorithm for Automatic Text Summarization. The main idea is the next one:
First, we represent the text in a suitable space model (VSM). Then, we
construct an average document vector that represents the average (the “global
topic”) of all sentences vectors. At the same time, we obtain the “lexical
weight” for each sentence, i.e. the number of words in the sentence. After
that, it is calculated the angle between the average document and each
sentence; narrow angles indicate that the sentences near of the “global topic”
should be important and therefore extracted. See on the figure 1 the VSM of
words: $P$ vector sentences and the average “global topic” are represented in
a $N$ dimensional space of words.
Figure 1: The “global topic” in a Vector Space Model of $N$ words.
Next, a score for each sentence is calculated using their proximity with the
“global topic” and their “lexical weight”. In the figure 2, the “lexical
weight” is represented in a VSM of $P$ sentences.
Finally, the summary is generated concatenating the sentences with the highest
scores following their order in the original document.
Figure 2: The “lexical weight” in a Vector Space Model of $P$ sentences.
### 0.3.1 Algorithm
Formally, Artex algorithm computes the score of each sentence by calculating
the inner product between a sentence vector, an average pseudo-sentence vector
(the “global topic”) and an average pseudo-word vector (the “lexical weight”).
Once a pre-processing (word normalization and filtering of stop words) is
completed, it is created a matrix $S_{[P\times N]}$, using the Vector Space
Model, that contains $N$ words (or letters) and $P$ sentences.
Let $s_{i}=(s_{1},s_{2},...,s_{N})$ be a vector of the sentence $i$,
$i=1,2,...,P$. We defined ${\vec{a}}$ the average pseudo-word vector, as the
average number of occurrences of $N$ words used in the sentence $i$:
(1) $a_{i}=\frac{1}{N}\sum_{j}s_{i,j}$
and ${\vec{b}}$ the average pseudo-sentence vector as the average number of
occurrences of each word $j$ used trough the $P$ sentences:
(2) $b_{j}=\frac{1}{P}\sum_{i}s_{i,j}$
The score or weight of each sentence $s_{i}$ is calculated as follows:
(3) $\textrm{score}(s_{i})=\left(\vec{s}\times\vec{b}\right)\
\times\vec{a}=\frac{1}{NP}\left(\sum_{j}s_{i,j}\times b_{j}\right)\times
a_{i}\,;\,i=1,2,...,P\,;j=1,1,...,N$
The score($\bullet$) computed by equation 3 must be normalized between the
interval [0,1]. The calculation of $\vec{s}\times\vec{b}$ indicates the
proximity between the sentence $\vec{s}$ and the average pseudo-sentence
$\vec{b}$. The product ($\vec{s}\times\vec{b})\times\vec{a}$ weigh this
proximity using the average pseudo-word $a_{i}$.
If a sentence $s_{i}$ is near of $\vec{b}$ and their corresponding element
$a_{i}$ has a high value, $s_{i}$ will have, therefore, a high score.
Moreover, a sentence $i$ far of main topic (i.e. $\vec{s}_{i}\times\vec{b}$ is
near 0) or a less informative sentence $i$ (i.e. $a_{i}$ are near 0) will have
a low score.
In computational terms, it is not really necessary to divide the scalar
product by the constant $\frac{1}{NP}$, because the angle
$\alpha=\arccos{\vec{b}}.{\vec{s}}/|{\vec{b}}||{\vec{s}}|$ between ${\vec{b}}$
and $\vec{s}$ is the same if we use
${\vec{b}}={\vec{b}}^{\prime}=\sum_{i}s_{i,j}$. The element $a_{i}$ is only a
scale factor that does not modify $\alpha$.
In fact, if the matrix $S_{[P\times N]}$ is approximated to a binary
matrix222This is a reasonable approximation in this context, because
$S_{[P\times N]}$ is a sparsed matrix with many term occurrences equal to one
or zero. $S^{\prime}_{[P\times N]}$, where each element
$s^{\prime}_{i,j}=\\{0,1\\}$ has a probability of $p=\frac{1}{2}$, we can
normalize vectors $\vec{a}$, $\vec{b}$ and matrix $S$, as follows:
(4) $\displaystyle|\vec{a}|$ $\displaystyle=$
$\displaystyle\sum_{i}^{P}\sqrt{{s^{\prime}_{i,j}}^{2}}=\sum_{i}^{P}\sqrt{(\\{0,1\\}^{P})^{2}}=N\sqrt{P}$
(5) $\displaystyle|\vec{b}|$ $\displaystyle=$
$\displaystyle\sum_{j}^{N}\sqrt{{s^{\prime}_{i,j}}^{2}}=\sum_{j}^{N}\sqrt{(\\{0,1\\}^{N})^{2}}=\sqrt{N}P$
(6) $\displaystyle|\vec{s}_{i}|$ $\displaystyle=$
$\displaystyle\sum_{j}^{N}\sqrt{{s^{\prime}_{i,j}}^{2}}=\sum_{j}^{N}\sqrt{\\{0,1\\}^{2}}=N$
Vectors then will be represented in hyper-spheres of $N$ or $P$ dimensions,
and the normalized score’ in this space would be:
(7) $\displaystyle\textrm{score'}(s_{i})$ $\displaystyle=$
$\displaystyle\left(\frac{\vec{s}}{|\vec{s}|}\times\frac{\vec{b}}{|\vec{b}|}\right)\
\times\frac{\vec{a}}{|\vec{a}|}=\frac{1}{N\sqrt{N}PN\sqrt{P}}\left(\sum_{j}s_{i,j}\times
b_{j}\right)\times a_{i}\,$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N^{5}P^{3}}}\left(\sum_{j}s_{i,j}\times
b_{j}\right)\times a_{i}\,;\,i=1,2,...,P\,;j=1,2,...,N$
However, the term $1/\sqrt{N^{5}P^{3}}$ is a constant value (i.e. a simple
scale factor), and then the $\textrm{score}(\bullet)$ calculated using the
equation 3) and the $\textrm{score'}(\bullet)$ using the equation 7, are both
equivalent.
## 0.4 Experiments
Artex algorithm described in the previous section has been implemented and
evaluated in corpora in several languages.
We have conducted our experimentation with the following languages,
summarization tasks, summarizers and data sets: 1) Generic multi-document-
summarization in English with the corpus DUC’04; 2) Generic single-document
summarization in Spanish with the corpus Medicina Clínica and 3) Generic
single document summarization in French with the corpus Pistes.
We have applied the summarization algorithms and finally, the results have
been evaluated using Fresa while processing times for each summarizer have
been measured and compared.
The following subsections present formally the details of the summarizers,
corpora and evaluations studied in different experiments.
### 0.4.1 Other Summarizers
To compare the performances, two other summarization systems were used in our
experiments: Cortex and Enertex. To be in the same conditions, these two
systems have used exactly the same textual representation based on Vector
Space Model, described in Section 0.2.1.
* •
Cortex is a single-document summarization system using several metrics and an
optimal decision algorithm [4, 14, 15, 18].
* •
Enertex is a summarization system based in Textual Energy concept [5]: text is
represented as a spin system where spins $\uparrow$ represents words that
their occurrences are $f>1$ (spins $\downarrow$ if the word is not present).
### 0.4.2 Summarization Corpora Description
To study the impact of our summarizer, we used corpora in three languages:
English, Spanish and French. The corpora are heterogeneous, and different
tasks are representatives of Automatic Text Summarization: generic multi-
document summary and mono-document guided by a subject.
* •
Corpus in English. Piloted by NIST in Document Understanding
Conference333http://duc.nist.gov (DUC) the Task 2 of DUC’04444http://www-
nlpir.nist.gov/projects/duc/guidelines/2004.html, aims to produce a short
summary of a cluster of related documents. We studied generic multi-document-
summarization in English using data from DUC’04. This corpus with 300K words
(17 780 types) is compound of 50 clusters, 10 documents each.
* •
Corpus in Spanish. Generic single-document summarization using a corpus from
the scientific journal Medicina
Clínica555http://www.elsevier.es/revistas/ctl_servlet?_f=7032&revistaid=2,
which is composed of 50 medical articles in Spanish, each one with its
corresponding author abstract. This corpus contains 125K words (9 657 types).
* •
Corpus in French. We have studied generic single-document summarization using
the Canadian French Sociological Articles corpus, generated from the journal
Perspectives interdisciplinaires sur le travail et la santé
(Pistes)666http://www.pistes.uqam.ca/. It contains 50 sociological articles in
French, each one with its corresponding author abstract. This corpus contains
near 400K words (18 887 types).
### 0.4.3 Summaries Content Evaluation
DUC conferences have introduced the ROUGE content evaluation [7], wich
measures the overlap of $n$-grams between a candidate summary and reference
summaries written by humans. However, to write the human summaries necessaries
for ROUGE is a very expensive task.
Recently metrics without references have been defined and experimented at DUC
and Text Analysis Conferences (TAC)777www.nist.gov/tac workshops.
Fresa content evaluation [13, 17] is similar to ROUGE evaluation, but human
reference summaries are not necessary. Fresa calculates the divergence of
probabilities between the candidate summary and the document source. Among
these metrics, Kullback-Leibler (KL) and Jensen-Shannon (JS) divergences have
been widely used by [8, 17] to evaluate the informativeness of summaries.
In this article, we use Fresa, based in KL divergence with Dirichlet
smoothing, like in the 2010 and 2011 INEX edition [11], to evaluate the
informative content of summaries by comparing their $n$-gram distributions
with those from source documents.
Fresa only considered absolute log-diff between the terms occurrences of the
source and the summary. Let $T$ be the set of terms in the source. For every
$t\in T$, we denote by $C_{t}^{T}$ its occurrences in the source and
$C_{t}^{S}$ its occurrences in the summary.
The Fresa package computed the divergence between the document source and the
summaries as follows:
(8) ${\mathcal{D}}(T||S)=\sum_{t\in
T}\left|\log\left(\frac{C_{t}^{T}}{|T|}+1\right)-\log\left(\frac{C_{t}^{S}}{|S|}+1\right)\right|$
To evaluate the information content (the “quality”) of the generated
summaries, after removing stop-words, several automatic measures were
computed: Fresa1 (Unigrams of single stems), Fresa2 (Bigrams of pairs of
consecutive stems), FresaSU4 (Bigrams with 2-gaps also made of pairs of
consecutive stems) and finally, $\langle\textsc{Fresa}\rangle$, i.e. the
average of all Fresa values.
The Fresa values (scores) are normalized between 0 and 1. High Fresa values
mean less divergence regarding the source document summary, reflecting a
greater amount of information content. All summaries produced by the systems
were evaluated automatically using Fresa package.
## 0.5 Results
In this section we present the results for each corpus with different
summarizers and the several normalization strategies used. Based on these
results, firstly, we have verified that ultra-stemming improves the
performance of summarizers. Secondly, we show that Artex is a system that has
a similar performances –in terms of information content and processing times–
to other state-of-art summarizers.
### 0.5.1 Content evaluation
* •
English corpus. Figure 3 shows the performance of the three summarizers using
Fix1, stemming and lemmatization. Results show that ultra-stemming improves
the score of the three automatic summarizer systems. Artex and Cortex expose a
similar performances in information content.
Figure 3: Histogram plot of content evaluation for corpus DUC’04 Task 2, with
$\langle$Fresa$\rangle$ measures, for each summarizer and each normalization.
* •
Spanish corpus. Spanish is a language with a greater variability than English.
Results in figure 4 shown that Artex summarizer outperforms Cortex and Enertex
if stemming or lemmatization are used as normalization.
Figure 4: Histogram plot of content evaluation for Spanish corpus Medicina
Clínica with $\langle$Fresa$\rangle$ scores for each summarizer.
* •
French corpus. French is a language with a large variability too. Figure 5
shows the score $\langle$Fresa$\rangle$ on the French corpus Pistes. Results
show a similar behavior: Ultra-stemming improves the score of the three
automatic summarization systems used. In particular, the efficacy of Artex is
less sensible to word normalization than others summarizers.
Figure 5: Histogram plot of content evaluation for French corpus Pistes with
$\langle$Fresa$\rangle$ scores for each summarizer.
### 0.5.2 Processing Times Evaluation
Table 1 shows processing times for each corpus, following the normalization
method for Cortex, Artex and Enertex summarizers888All times are measured in a
7.8 GB of RAM computer, Core i7-2640M CPU @ 2.80GHz $\times$ 4 processor,
running under 32 bits GNU/Linux (Ubuntu Version 12.04).. Processing times of
ultra-stemming Fix1 are shorter compared to all others methods. By example,
Cortex is a very fast summarizer with $O(\log\rho^{2})$ (where $\rho=P\times
N$), and processing times for stemming and Fix1 are close. In other hand,
Enertex summarizer has a complexity of $O(\rho^{2})$, then it needs more time
to process the same corpus. Performances of Artex algorithm remain close to
Cortex.
| Summarizer Average Time
---|---
| (all corpora)
Normalization | Cortex | Artex | Enertex
Lemmatization | 1.60’ | 2.50’ | 10.42’
Stemming | 0.54’ | 1.29’ | 9.47’
fix1 | 0.32’ | 0.40’ | 4.25’
Table 1: Statistics of processing times (in minutes) of three summarizers over
three corpora.
## 0.6 Conclusions
In this article we have introduced and tested a simple method for Automatic
Text Summarization. Artex is a fast and very simple algorithm based in VSM
model and the extractive paradigm. The method uses a matrix representation to
calculate a normalized score for each sentence, using the inner product of
pseudo-(sentences|words) vectors. The algorithm retains the salient
information of each sentence of document. An important aspect of our approach
is that it does not requires linguistic knowledge or resources which makes it
a simple and efficient summarizer method to tackle the issue of Automatic Text
Summarization.
Summaries generated by Artex system are pertinents. The results obtained on
corpora in English, Spanish and French show that Artex can achieve good
results for content quality. Tests with other corpora (DUC and TAC evaluation
campaigns, INEX, etc.) in mono-and multi-document guided by a subject, using
content evaluation with (ROUGE evaluations) or without reference summaries
still in progress.
## References
* [1] Iria da Cunha, Silvia Fernández, Patricia Velázquez-Morales, Jorge Vivaldi, Eric SanJuan, and Juan Manuel Torres-Moreno. A new hybrid summarizer based on vector space model, statistical physics and linguistics. In Proceedings of the 6th Mexican International Conference on Advances in Artificial Intelligence (MICAI’07), pages 872–882, Aguascalientes, Mexico, 2007. Springer-Verlag.
* [2] Harold Daumé III. Practical structured learning techniques for natural language processing. PhD thesis, Los Angeles, CA, 2006.
* [3] H. P. Edmundson. New Methods in Automatic Extraction. Journal of the Association for Computing Machinery, 16(2):264–285, 1969.
* [4] B. Favre, F. Béchet, P. Bellot, F. Boudin, M. El-Bèze, L. Gillard, G. Lapalme, and J-M. Torres-Moreno. The LIA-Thales summarization system at DUC-2006. In Proceedings of the Document Understanding Conference (DUC’06), Brooklyn, New York, United States, 2006. http://duc.nist.gov.
* [5] Silvia Fernández, Eric SanJuan, and Juan-Manuel Torres-Moreno. Textual Energy of Associative Memories: performants applications of Enertex algorithm in text summarization and topic segmentation. In Proceedings of the Mexican International Conference on Artificial Intelligence (MICAI’07), pages 861–871, Aguascalientes, Mexico, 2007\. Springer-Verlag.
* [6] J. Kupiec, J. Pedersen, and F. Chen. A trainable document summarizer. In Proceedings of the 18th Conference ACM Special Interest Group on Information Retrieval (SIGIR’95), pages 68–73, Seattle, WA, United States, 1995. ACM Press, New York.
* [7] Chin-Yew Lin. ROUGE: A Package for Automatic Evaluation of Summaries. In Marie-Francine Moens and Stan Szpakowicz, editors, Proceedings of the Workshop Text Summarization Branches Out (ACL’04), pages 74–81, Barcelone, Spain, july 2004. ACL.
* [8] Annie Louis and Ani Nenkova. Automatic Summary Evaluation without Human Models. In First Text Analysis Conference (TAC’08), Gaithersburg, MD, United States, 17-19 November 2008.
* [9] H.P. Luhn. The Automatic Creation of Literature Abstracts. IBM Journal of Research and Development, 2(2):159–165, 1958.
* [10] I. Mani and M. Mayburi. Advances in Automatic Text Summarization. MIT Press, Cambridge, 1999.
* [11] Eric SanJuan, Patrice Bellot, Véronique Moriceau, and Xavier Tannier. Overview of the INEX 2010 Question Answering Track (QA@INEX). In Shlomo Geva, Jaap Kamps, Ralf Schenkel, and Andrew Trotman, editors, Comparative Evaluation of Focused Retrieval, volume 6932 of Lecture Notes in Computer Science, pages 269–281. Springer Berlin / Heidelberg, 2011.
* [12] Simone Teufel and Marc Moens. Sentence extraction as a classification task. In I. Mani and M. Maybury, editors, Proceedings of the ACL/EACL’97 Workshop on Intelligent Scalable Text Summarization, Madrid, Spain, 11 July 1997.
* [13] J.-M. Torres-Moreno, Horacio Saggion, I. da Cunha, P. Velazquez-Morales, and E. SanJuan. Evaluation automatique de résumés avec et sans réferences. In Proceedings de la conference Traitement Automatique des Langagues Naturelles (TALN’10), Montréal, QC, Canada, 19-23 July 2010. ATALA.
* [14] J.-M. Torres-Moreno, P.-L. St-Onge, M. Gagnon, M. El-Bèze, and P. Bellot. Automatic Summarization System coupled with a Question-Answering System (QAAS). CoRR, abs/0905.2990, 2009.
* [15] Juan-Manuel Torres-Moreno. Résumé automatique de documents: une approche statistique. Hermès-Lavoisier, Paris, 2011.
* [16] Juan-Manuel Torres-Moreno. Beyond Stemming and Lemmatization: Ultra-stemming to Improve Automatic Text Summarization. CoRR, arXiv:1209.3126 [cs.IR], 2012.
* [17] Juan-Manuel Torres-Moreno, Horacio Saggion, Iria da Cunha, and Eric SanJuan. Summary Evaluation With and Without References. Polibits: Research journal on Computer science and computer engineering with applications, 42:13–19, 2010.
* [18] Juan-Manuel Torres-Moreno, Patricia Velázquez-Morales, and Jean-Guy Meunier. Cortex : un algorithme pour la condensation automatique des textes. In Proceedings of the Conference de l’Association pour la Recherche Cognitive, volume 2, pages 365–366, Lyon, France, 2001.
|
arxiv-papers
| 2012-10-11T18:21:01 |
2024-09-04T02:49:36.416459
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan-Manuel Torres-Moreno",
"submitter": "Juan Manuel Torres Moreno",
"url": "https://arxiv.org/abs/1210.3312"
}
|
1210.3343
|
# Supergranules as Probes of the Sun’s Meridional Circulation
David H. Hathaway NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
david.hathaway@nasa.gov
###### Abstract
Recent analysis revealed that supergranules (convection cells seen at the
Sun’s surface) are advected by the zonal flows at depths equal to the widths
of the cells themselves. Here we probe the structure of the meridional
circulation by cross-correlating maps of the Doppler velocity signal using a
series of successively longer time lags between maps. We find that the
poleward meridional flow decreases in amplitude with time lag and reverses
direction to become an equatorward return flow at time lags $>24$ hours. These
cross-correlation results are dominated by larger and deeper cells at longer
time lags. (The smaller cells have shorter lifetimes and do not contribute to
the correlated signal at longer time lags.) We determine the characteristic
cell size associated with each time lag by comparing the equatorial zonal
flows measured at different time lags with the zonal flows associated with
different cell sizes from a Fourier analysis. This association gives a
characteristic cell size of $\sim 50$ Mm at a 24 hour time lag. This indicates
that the poleward meridional flow returns equatorward at depths $>50$ Mm –
just below the base of the surface shear layer. A substantial and highly
significant equatorward flow (4.6 $\pm$ 0.4 m s-1) is found at a time lag of
28 hours corresponding to a depth of $\sim 70$ Mm. This represents one of the
first positive detections of the Sun’s meridional return flow and illustrates
the power of using supergranules to probe the Sun’s internal dynamics.
convection, Sun: rotation
## 1 INTRODUCTION
Supergranules are cellular flows observed in the Sun’s photosphere as a
surface filling pattern of largely horizontal flows (Rieutord & Rincon, 2010).
Typical supergranules have diameters of $\sim 30$ Mm and lifetimes of $\sim 1$
day – much larger and longer lived than granules (diameters of $\sim$ 1 Mm and
lifetimes of $\sim$ 5 minutes).
Since their discovery by Hart (1954) and their initial characterization by
Leighton et al. (1962), supergranules have been recognized as the principal
organizer of the Sun’s chromospheric/magnetic network and as key to the
transport of magnetic flux in the photosphere (Leighton, 1964; DeVore et al.,
1984).
The rotation rate of the supergranules has been measured by a number of
investigators with interesting results. Duvall (1980) cross-correlated the
Doppler velocity pattern from equatorial spectral scans obtained over several
days and found that the pattern rotates about 3% faster than the photospheric
plasma. Faster rates are found for the 24-hr time lags from day-to-day than
for the 8-hr time lags from the beginning to end of an observing day. Beck &
Schou (2000) measured the rotation of the Doppler velocity pattern using a
Fourier technique and found that the larger cells rotate more rapidly than the
smaller cells but with apparent rotation rates that exceed the peak internal
rotation rate. This “superrotation” was attributed to line-of-sight projection
effects on the Doppler velocity pattern by Hathaway et al. (2006). In fact,
Schou (2003) had earlier found rotation consistent with the internal plasma
flow profile when this line-of-sight projection was removed.
These observations are consistent with the original conclusion by Duvall
(1980) that larger cells dominate the longer time lags and that these larger
cells are more deeply anchored in a surface shear layer in which the rotation
rate increases with depth. While it should be noted that some characteristics
of the motions of the supergranules have been attributed to wave-like
properties (Gizon et al., 2003; Schou, 2003), it is clear that the cellular
structures are advected largely by the zonal flows in the surface shear layer.
The existence of this shear layer was suggested by Foukal & Jokipii (1975) as
a consequence of the conservation of angular momentum by convective elements
moving inward and outward in the near surface layers. Helioseismology
(Thompson et al., 1996; Schou et al., 1998) now indicates that the shear layer
extends to a depth of $\sim 50$ Mm.
Recently Hathaway (2012) found a one-to-one correspondence between the
rotation rate of supergranules with increasing size and the rotation rate in
this surface shear layer with increasing depth. Results, using a Fourier
technique like that of Beck & Schou (2000) with the line-of-sight projection
effects removed using the method of Schou (2003), indicate that supergranules
are advected by the flows in the shear layer at depths equal to their widths.
We expect the advection of the supergranules to be dominated by flows near
their bases since the plasma density increases rapidly with depth. However,
this does imply that the cells must extend even deeper than this anchoring
level.
Note that while cells that are as deep as they are wide are typical in
numerical simulations of solar convection (Stein et al., 2011), results from
local helioseismology have suggested flattened supergranules – but with
conflicting results. Duvall (1998) found that the outflows in the
supergranules reversed at a depth of 5 Mm and then disappeared by 8 Mm depth.
Zhao & Kosovichev (2003) found reversal at a depth of 8 Mm and disappearance
at 15 Mm. Švanda et al. (2009) found disappearance at an even greater depth –
25 Mm.
Here we measure the meridional motion of the pattern of supergranules by
analyzing the same data used in Beck & Schou (2000), Schou (2003), and
Hathaway (2012).
The Sun’s meridional flow is extremely weak when compared to the other
photospheric flows. Both the strength and the direction of the meridional flow
were uncertain until data and data analysis techniques were sufficiently
improved (Komm et al., 1993; Hathaway et al., 1996; Giles et al., 1997;
Ulrich, 2010). The meridional flow is now known to be poleward in each
hemisphere with a peak velocity of about 10-20 m s-1 while the zonal flows
(differential rotation) are an order of magnitude stronger and the convective
flows (granules and supergranules) are yet another order of magnitude
stronger.
The Sun’s meridional flow plays a critical role in the transport of magnetic
flux (DeVore et al., 1984; Wang et al., 2009) and in some dynamo models for
the Sun’s magnetic cycle (Dikpati & Charbonneau, 1999; Nandy & Choudhuri,
2002). Although the nonaxisymmetric convective motions transport magnetic
elements at much higher speeds to the cell boundaries, those motions are in
random directions and give diffusion away from flux concentrations. The net
poleward transport has a larger contribution from the slow, but direct,
meridional flow. This poleward transport is directly responsible for the
reversal of the Sun’s polar magnetic fields every solar cycle and for
producing the polar fields that determine the strength of the following solar
cycle (Schatten et al., 1978; Svalgaard et al., 2005).
Fourier techniques (which do give explicit information on the cell sizes) are,
unfortunately, not available for measuring the meridional motion of the
supergranule Doppler velocity pattern. Here we use cross-correlations with
time lags that vary from 1 hour to 28 hours. The displacement of the peak in
the cross-correlation gives both zonal and meridional velocities which vary
systematically with time lag. We determine the characteristic cell size for
each time lag by comparing the zonal velocities with those obtained with the
Fourier technique (Hathaway, 2012). This then gives the meridional flow at a
series of anchoring depths corresponding to the series of time lags.
## 2 DATA PREPARATION
The data consist of $1024^{2}$ pixel images of the line-of-sight velocity
determined from the Doppler shift of a spectral line due to the trace element
nickel in the solar atmosphere by the Michelson Doppler Imager (Scherrer et
al., 1995, MDI) on the ESA/NASA Solar and Heliospheric Observatory (SOHO).
The images are acquired at a 1 minute cadence and cover the full visible disk
of the Sun. We average the data over 31 minutes with a Gaussian weighting
function which filters out variations on time scales less than about 16
minutes, and sample that data at 15 minute intervals. We then map these
temporally filtered images onto a $1024^{2}$ grid in heliographic latitude
from pole to pole and in longitude $\pm 90\arcdeg$ from the central meridian
(Figure 1). This mapping accounts for the position angle of the Sun’s rotation
axis relative to the imaging CCD and the tilt angle of the Sun’s rotation axis
toward or away from the spacecraft. Both of these angles include modifications
in line with the most recent determinations of the orientation of the Sun’s
rotation axis (Beck & Giles, 2005; Hathaway & Rightmire, 2010). We analyze
data obtained during two 60 day periods of continuous coverage: one in 1996
from May 24 to July 22; the other in 1997 from April 14 to June 17.
Each filtered image is analyzed using the methods described by Hathaway (1992)
to determine and remove the stationary signals due to differential rotation,
convective blue shift, and the meridional flow – as well as the time varying
signal due to the motion of the spacecraft. Following this, an average image
from all of the images in each 60 day period was subtracted from each image to
remove any instrumental artifacts. This leaves behind the non-axisymmetric and
time-varying signal due to the convective flows.
Figure 1: Heliographic map detail of the line-of-sight (Doppler) velocity
from SOHO/MDI. The map detail extends $90\arcdeg$ in longitude from the
central meridian on the left and about $35\arcdeg$ in latitude from the
equator (the thick horizontal line). The mottled pattern is the Doppler signal
(blue for blue shifts and red for red shifts) due to the supergranules.
## 3 CROSS-CORRELATION ANALYSIS
We determine the zonal and meridional velocity of the supergranule pattern by
cross-correlating strips of data 11 pixels high in latitude by 601 pixels long
in longitude. The strips are centered on each of the 860 latitude positions
from $75\arcdeg$ south to $75\arcdeg$ north of the equator with overlap from
adjacent strips. These data strips are cross-correlated with corresponding
strips from images obtained at time lags of 1, 2, 4, 8, 16, 24, and 28 hours.
This process is done both forward and backward in time. A data strip centered
on the central meridian is correlated with strips offset to the left at later
times and then data centered on the central meridian at later times is
correlated with strips offset to the right in the initial data map. This
minimizes any systematic errors.
The shift of the Doppler pattern is determined to the nearest pixel by finding
the displacement of the strip in the later Doppler map that gives the maximum
correlation. This pixel-wise shift is then refined by fitting parabolas
through the correlation amplitudes at the displacement that gives the maximum
correlation and those at plus and minus one pixel in each direction. The
maxima of these parabolas then give the Doppler pattern shift to a fraction of
a pixel. The shifts in longitude give the apparent zonal velocity while the
shifts in latitude give the meridional velocity. The 60 days of data in each
data set give some 1400 independent measurements of these velocities at each
of the 860 latitude positions for each time lag.
The meridional flow measurements for time lags from 1 to 24 hours are shown in
Figure 2. Here the data are smoothed in latitude with a tapered Gaussian
having a FWHM of $10\arcdeg$. (A tapered Gaussian is a Gaussian with a
quadratic function subtracted so that the smoothing filter and its first
derivative go to zero at the endpoints.) The flow is poleward in each
hemisphere from the equator up to latitudes of at least $60\arcdeg$. The speed
of the flow decreases as the time lag increases and the poleward flow
virtually vanishes when measured with the 24 hour time lags. The slow down for
time lags from 2 to 16 hours was previously noted by Hathaway et al. (2010).
The lack of a poleward meridional flow for 24 hour time lags was previously
noted (and referred to as “anomalous”) by Gizon et al. (2003).
Figure 2: Meridional flow profiles for time lags from 1 to 24 hours. The
measurements from the 1996 data are shown with the red line while the
measurements from the 1997 data are shown with the blue line. The dotted lines
indicate $2\sigma$ error limits for each year. The meridional flow is poleward
up to latitudes of at least $60\arcdeg$. The amplitude of this poleward
meridional flow decreases as the time lag increases and disappears at 24 hour
time lags.
Measurements with 28 hour time lags are shown in Figure 3. Again, the data are
smoothed in latitude with a tapered Gaussian having a FWHM of $10\arcdeg$.
With this longer time lag the measured meridional flow is equatorward at
virtually all latitudes with a peak of $\sim 4$ m s-1. The $2\sigma$
variations on the measurements at each latitude are $\sim 2$ m s-1 making
nearly all of the individual latitude measurements statistically significant.
Fitting the measurements to a 4th-order polynomial in $\sin\theta$, shown by
the thick black line in Figure 3, gives a peak velocity of 4.6 $\pm$ 0.4 m s-1
with $2\sigma$ errors. As such this represents the first positive measurement
and significant ($>10\sigma$) detection of the Sun’s equatorward return flow.
Figure 3: Meridional flow profiles for time lags of 28 hours. The
measurements from the 1996 data are shown with the red line while the
measurements from the 1997 data are shown with the blue line. The dotted lines
indicate $2\sigma$ error limits for each year. The 4th-order fit to the
combined data is shown with the thick, black line.
## 4 DEPTH DETERMINATIONS
Duvall (1980) suggested that at different time lags the cells that survive are
advected by flows with different speed (and direction) depending upon the
depth to which those cells extend. The different time lags thus probe the
depth dependence of the meridional flow. Hathaway (2012) found that
supergranules of a given cell size (given by their wavenumber in a Fourier
analysis) are advected by the zonal flows at depths equal to their widths.
This was determined by comparing their zonal velocities from the Fourier
analysis with zonal velocities in the surface shear layer determined with
global helioseismology by Schou et al. (1998).
Small cells have short lifetimes. Granules with typical diameters of $\sim 1$
Mm have lifetimes of $\sim 5$ minutes (Title et al., 1989). Supergranules with
typical diameters of $\sim 30$ Mm have lifetimes of $\sim 1$ day (Simon &
Leighton, 1964; Wang & Zirin, 1989; Hirzberger et al., 2008). Cells of
intermediate (mesogranular) size with diameters 5-10 Mm have intermediate
lifetimes of $\sim 2$ hours (November et al., 1981). As the time lag used in
the cross-correlation increases, the cells with lifetimes shorter than the
time lag present an uncorrelated random pattern that does not influence the
determination of the pattern motion. Longer time lags are associated with
larger cells which extend deeper into the Sun. This assertion is supported by
observations of the monotonic decline in the cross-correlations for granules
(Title et al., 1989), the monotonic decline in the cross-correlations for the
supergranule Doppler pattern (Hathaway et al., 2010),and the increase in
lifetimes for larger supergranule cells determined from local helioseismology
(Hirzberger et al., 2008).
We can determine the characteristic cell size at each time lag by matching the
zonal velocities from the cross-correlations with those from the Fourier
method. This Fourier method is described in detail by Beck & Schou (2000) and
Hathaway (2012). Lines of mapped Doppler data at equatorial latitudes are
Fourier transformed in longitude to give spectral coefficients. Spectral
coefficients from 6 10-day series of Doppler maps obtained at 15-minute
cadence are then Fourier transformed in time. The location of the peak power
gives the zonal (longitudinal) velocity for each spatial wavelength or cell
size.
The line-of-sight projection for the Doppler signal influences the measured
zonal flow (but, as shown by Hathaway et al. (2010), not the meridional flow).
When this projection is divided-out (as can only be done at the equator) the
zonal flow as a function of wavelength from the Fourier analysis matches the
zonal flow as a function of depth from global helioseismology. Figure 4 shows
zonal velocities relative to the Carrington Rotation frame of reference for
supergranules of varying wavelength along with the zonal velocity as a
function of depth from global helioseismology (Schou et al., 1998).
Figure 4: Equatorial zonal velocities as functions of cell wavelength and
depth within the Sun. Zonal velocities as functions of wavelength from the
1996 MDI data in which the line-of-sight projection is divided out are shown
with the small black circles. Zonal velocities as functions of depth from
global helioseismology (Schou et al., 1998) are shown with the large red
circles. The match between these measurements at virtually all wavelengths and
depths indicates that supergranules are advected by the flows at depths equal
to their wavelengths.
The zonal velocities for the raw data (without the line-of-sight projection
removed) are shown in Figure 5. Comparing these zonal velocities with those
obtained at the equator (also with the raw data) by the cross-correlation
analysis gives a correspondence between time lag and characteristic wavelength
of the cells that dominate at that time lag. This correspondence is given in
Table 1.
Figure 5: Equatorial zonal velocities as functions of cell wavelength and depth within the Sun. Zonal velocities as functions of wavelength from the 1996 MDI data are shown with the small black circles. Zonal velocities as functions of depth from helioseismology (Schou et al., 1998) are shown with the large red circles. The zonal velocities at the equator from the different time lags are indicated along the vertical axis with lines indicating where they match the zonal velocities from the Fourier analysis. Time Lag | Velocity | Wavelength
---|---|---
1 hour | $12\pm 1$ m s-1 | $7\pm 1$ Mm
2 hours | $24\pm 1$ m s-1 | $14\pm 1$ Mm
4 hours | $40\pm 1$ m s-1 | $22\pm 1$ Mm
8 hours | $49\pm 2$ m s-1 | $26\pm 2$ Mm
16 hours | $59\pm 2$ m s-1 | $37\pm 2$ Mm
24 hours | $71\pm 3$ m s-1 | $50\pm 5$ Mm
28 hours | $80\pm 3$ m s-1 | $70\pm 5$ Mm
Table 1: Correspondence between time lag, equatorial zonal velocity, and cell
wavelength from Figure 5.
## 5 CONCLUSIONS
We conclude that the Sun’s poleward meridional flow is confined to it’s
surface shear layer (from the photosphere to a depth of $\sim 50$ Mm). We draw
this conclusion from the fact that the poleward flow weakens and then
disappears as the cross-correlation time lag increases from 1 to 24 hours
while the measured rotation rates indicate that the cross-correlated patterns
are dominated by cells with sizes, and therefor depths, that increase from
$\sim 7$ Mm to $\sim 50$ Mm. A positive detection of the equatorward return
flow is found with time lags of 28 hours corresponding to depths of $\sim 70$
Mm.
The fact that the return flow signal was significantly detected at a longer
time lag in a deeper layer also indicates that the lack of a poleward flow at
the 24 hour time lag is due to an actual lack of poleward flow rather than an
insensitivity to the signal.
Estimating the poleward mass flux at $30\arcdeg$ latitude within the surface
shear layer using the measured flow speeds and the run of plasma density with
depth, indicates that a balancing equatorward mass flux at a flow speed of
$\sim 4$ m s-1 should only extend another 20-30 Mm below the base of this
surface shear layer.
This shallow return flow is somewhat surprising given earlier results from
helioseismology. Braun & Fan (1998) used frequency shifts between northward
and southward propagating waves and concluded that the meridional flow was
poleward to a depth of at least 100 Mm. Giles (2000) used time-distance
helioseismology with a mass flux constraint and concluded that the return flow
was below a depth of about 140 Mm. However, Duvall & Hanasoge (2009) have
found that these measurements are plagued by systematic errors leading Gough &
Hindman (2010) to state that, with helioseismology, the meridional flow below
a depth of 30 Mm remains uncertain.
Recently Zhao et al. (2012A) have characterized these systematic signals,
removed them from the data, and retrieved a meridional flow profile in the
near surface layers that agrees well with measurements obtained from magnetic
feature motions (Hathaway & Rightmire, 2010) and direct Doppler measurements
(Ulrich, 2010). Zhao et al. (2012B) have now reported that they find evidence
for the meridional return flow at depths much shallower than the tachocline –
supporting the conclusions reached here.
A shallow return flow is also supported by Sivaraman et al. (2010) who
measured the meridional motions of sunspot umbrae. They compared the rotation
rates of sunspot umbrae with rotation rates derived from helioseismology to
find anchoring depths for umbrae of different sizes (similar to what was done
here with supergranules) and found that small sunspots with umbral areas
$<5\mu$Hem (a $\mu$Hem is $10^{-6}$ the area of a solar hemisphere) are
anchored at depths of $\sim 90$ Mm and are moving equatorward at speeds of
$\sim 7$ m s-1. Slightly deeper anchoring depths and slower equatorward
velocities were found for larger sunspots.
## 6 FURTHER IMPLICATIONS
The shallow depth of the meridional return flow has implications for our
understanding of both solar convection zone dynamics and solar dynamo theory.
The axisymmetric meridional flow is maintained by a balance between the
combined effects of three driving forces (latitudinal pressure gradients,
Reynolds stresses resulting from correlations between radial and meridional
motions in the cellular flows, and the Coriolis force acting on the
axisymmetric zonal flow) and the viscous diffusion due to the cellular flows.
Simulations of the effects of the Sun’s rotation on supergranules in plane-
parallel geometry (Hathaway, 1982) indicate that the Reynolds stresses
produced by the supergranules themselves can maintain both the zonal shear
(slower rotation at the surface and faster rotation below) and a meridional
circulation with poleward flow near the surface and equatorward flow below.
The meridional Reynolds stresses (correlated poleward and upward motions in
the cellular flows) are reinforced by the Coriolis force on the axisymmetric
zonal flow in these simulations. (Latitudinal pressure gradients cannot be
maintained or included due to the periodic boundaries in both horizontal
directions.)
The meridional circulations produced by larger convection cells in rotating
spherical shells extending over several density scale heights (but thus far
not including the surface shear layer) are usually highly structured in
latitude and radius and are highly variable in time (Miesch et al., 2000).
More recent spherical shell simulations (Miesch et al., 2008) impose a
latitudinal entropy (temperature) gradient at the base of the convection zone
to give rotation profiles and meridional circulations more in line with
observations but still produce highly time dependent meridional flows. The
source of this structure and variability can be attributed to the small number
($\sim 100$) of convection cells that populate the simulated global volume. A
meridional circulation driven in the surface shear layer by ($\sim 10000$)
supergranules (the convection cells that populate the surface shear layer) is
far more likely to be less structured and variable – as is observed.
The shallow return flow we find here has implications for flux transport
dynamos (Dikpati & Charbonneau, 1999; Nandy & Choudhuri, 2002) – it violates a
key assumption. These dynamo models assume that the return flow begins at a
depth of 100 Mm and reaches a maximum equatorward speed of just 1-2 m s-1 at
the base of the convection zone. This is a critical assumption in that the
equatorward drift of the sunspot zones is directly produced in these dynamo
models by this slow equatorward flow at the base of the convection zone.
The results presented here support the idea of using supergranules to probe
the dynamics of the solar convection zone. This method can complement the
probing done with acoustic waves by the methods of helioseismology and provide
valuable information on the dynamics of solar and stellar convection zones.
The method used here, finding the peak in the cross-correlated Doppler
velocity pattern for data pairs with different time lags, is limited to time
lags less than $\sim 32$ hours. At longer time lags the correlation amplitudes
from individual data pairs are similar to the noise level from the cross-
correlations. However, averaging the cross-correlations over as few as 24
hourly measurements could extend the results to longer time lags and deeper
layers. Local Correlation Tracking using smaller patches might also yield
measurements of the large scale nonaxisymmetric flows – Giant Cells.
Finally, it should be noted that, while these results indicate an equatorward
return flow at a fairly shallow depth, they do not preclude the possible
existence of additional cells of poleward and equatorward flow at even greater
depths.
The author would like to thank NASA for its support of this research through
grants from the Heliophysics Causes and Consequences of the Minimum of Solar
Cycle 23/24 Program and the Living With a Star Program to NASA Marshall Space
Flight Center. He is indebted to Ron Moore and Lisa (Rightmire) Upton and an
anomymous referee whose comments greatly improved the manuscript and to John
Beck who produced the temporally filtered data from the original MDI Doppler
data. He would also like to thank the American taxpayers who support
scientific research in general and this research in particular. SOHO, is a
project of international cooperation between ESA and NASA.
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* Beck & Schou (2000) Beck, J. G., & Schou, J. 2000, Sol. Phys. 193, 333
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|
arxiv-papers
| 2012-10-11T19:45:09 |
2024-09-04T02:49:36.425551
|
{
"license": "Public Domain",
"authors": "David H. Hathaway",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1210.3343"
}
|
1210.3429
|
# Global Well-posedness of the Parabolic-parabolic Keller- Segel Model in
$L^{1}(\mathbb{R}^{2})\times\\!{L}^{\\!\infty}(\mathbb{R}^{2})$ and
$H^{1}_{b}(\mathbb{R}^{2})\times\\!{H}^{1}(\mathbb{R}^{2})$
Chao Deng, Congming Li
###### Abstract
In this paper, we study global well-posedness of the two-dimensional Keller-
Segel model in Lebesgue space and Sobolev space. Recall that in the paper
“Existence and uniqueness theorem on mild solutions to the Keller-Segel system
in the scaling invariant space, _J. Differential Equations_ , 252 (2012),
1213–1228”, Kozono, Sugiyama & Wachi studied global well-posedness of
$n$($\geq 3$) dimensional Keller-Segel system and posted a question about the
even local in time existence for the Keller-Segel system with
$L^{1}(\mathbb{R}^{2})\times{L}^{\infty}(\mathbb{R}^{2})$ initial data. Here
we give an affirmative answer to this question: in fact, we show the global in
time existence and uniqueness for
$L^{1}(\mathbb{R}^{2})\times{L}^{\\!\infty}(\mathbb{R}^{2})$ initial data.
Furthermore, we prove that for any
$H^{1}_{b}(\mathbb{R}^{2})\times{H}^{1}(\mathbb{R}^{2})$ initial data with
$H^{1}_{b}(\mathbb{R}^{2}):=H^{1}(\mathbb{R}^{2})\cap{L}^{\infty}(\mathbb{R}^{2})$,
there also exists a unique global mild solution to the parabolic-parabolic
Keller-Segel model. The estimates of
${\sup_{t>0}}t^{1-\frac{n}{p}}\|u\|_{L^{p}}$ for $(n,p)=(2,\infty)$ and the
introduced special half norm, i.e.
$\sup_{t>0}t^{\frac{1}{2}}(1\\!+t)^{\\!-\frac{1}{2}}\|\nabla{v}\|_{L^{\infty}}$,
are crucial in our proof.
Keywords: Keller-Segel model; Fourier transformation; well-posedness; decay
property; parabolic-parabolic system.
Mathematics Subject Classification: 92C17; 35K55.
## 1 Introduction
In this article, we study the following two-dimensional (2D) Keller-Segel
model:
$\displaystyle{{u}_{t}}-\Delta{u}+\nabla\cdot({u}\nabla{v})=0$ in
$\displaystyle\hskip 6.544pt(0,\infty)\times\mathbb{R}^{2},$ (1.1)
$\displaystyle{{v}_{t}}-\Delta{v}+{v}-{u}=0$ in $\displaystyle\hskip
6.544pt(0,\infty)\times\mathbb{R}^{2},$ (1.2)
$\displaystyle(u,v)|_{t=0}=(u_{0},v_{0})$ in $\displaystyle\hskip
6.544pt\mathbb{R}^{2},$ (1.3)
where $(t,x)\in(0,\infty)\times\mathbb{R}^{2}$, $u=u(t,x)$ and $v=v(t,x)$ are
the scalar valued density of amoebae and the scalar valued concentration of
chemical attractant, respectively, while $(u_{0},v_{0})$ is the given initial
data. For the derivation of the equation, we refer to Childress and Percus [3]
and Keller and Segel [14].
Noticing that (1.1)–(1.2) is “almost” scale invariant since
${{u}_{t}}-\Delta{u}+\nabla\cdot({u}\nabla{v})=0$ and
${{v}_{t}}-\Delta{v}-{u}=0$ are invariant under the following transformations
$(u(t,x),v(t,x))\\!\rightarrow\\!(\lambda^{2}u(\lambda^{2}t,\lambda
x),v(\lambda^{2}t,\lambda{x}))\quad\quad\text{for }\lambda>0.$
The idea of using a functional setting invariant by scaling is now classical
and originates several works, see for instance, global existence of mild
solutions to system (1.1)–(1.3) for initial
$(u_{0},v_{0})\\!\in\\!H^{\frac{n}{r}-2,r}(\mathbb{R}^{n})\times{H}^{\frac{n}{r},r}(\mathbb{R}^{n})$
with $\max\\{1,\frac{n}{4}\\}<r<\frac{n}{2}$ in [17], for initial
$(u_{0},v_{0})\in\\!L^{\\!{n}/{2}}_{w}(\mathbb{R}^{n})\times$BMO$(\mathbb{R}^{n})$
with $n\geq 3$ in [18], and for initial
$(u_{0},v_{0})\in\\!L^{\\!\frac{n}{2}}(\mathbb{R}^{n})\times\dot{H}^{2\alpha,\frac{n}{2\alpha}}(\mathbb{R}^{n})$
with $n\geq 3$ and $\frac{n}{2(n+2)}<\alpha\leq\frac{1}{2}$ in [19]. It is
also known that apart from existence and uniqueness of mild solutions in scale
invariant spaces, there are papers on asymptotic behaviors (see e.g. [12],
[32]) and stationary solutions (see e.g. [9], [24]). We also refer readers to,
for instance [11] and references cited therein, to see results on the
quasilinear degenerate Keller-Segel system.
The first goal of this paper is to answer Kozono, Sugiyama and Wachi’s
question in [19] of figuring out whether there exists a solution to system
(1.1)–(1.3) even locally in time for
$(u_{0},v_{0})\in{L}^{1}(\mathbb{R}^{2})\times{L}^{\infty}(\mathbb{R}^{2})$.
In fact, we prove that there does exist a unique global mild solution to
system (1.1)–(1.3) with
$(u_{0},v_{0})\in{L^{1}(\mathbb{R}^{2})\times{L}^{\infty}}(\mathbb{R}^{2})$ by
estimating $t\|u(t,\cdot)\|_{L^{\infty}}$, $\|u(t,\cdot)\|_{L^{1}}$ and
${t}^{\frac{1}{2}}\|\nabla{v}(t,\cdot)\|_{L^{\infty}}$ in the
$L^{p}$-framework, see for instance [13]. Moreover, by exploring the special
structure of system (1.1)–(1.2), Deng and Li [2] established global existence
of mild solution for initial data
$(u_{0},v_{0})\in{L}^{q}(\mathbb{R}^{2})\times{L}^{\infty}(\mathbb{R}^{2})$
with $1<q<\infty$, where global existence of mild solution for initial data
$(u_{0},v_{0})\in{L^{\infty}(\mathbb{R}^{2})}\times{L}^{\infty}(\mathbb{R}^{2})$
was left as an open question.
The second goal of this paper is to study global well-posedness of system
(1.1)–(1.3) with $H^{1}_{b}(\mathbb{R}^{2})\times H^{1}(\mathbb{R}^{2})$
initial data. Up to now, there are several results on local and global
existence of system (1.1)–(1.3) for
$(u_{0},v_{0})\in{H}^{\nu}(\mathbb{R}^{2})\times{H}^{\nu}(\mathbb{R}^{2})$
with $\nu>\\!1$ (cf. Nagai, Senba and Yoshida [26], and Yagi [33]) and result
on global existence of system (1.1)–(1.3) with initial data $(u_{0},v_{0})\in
H^{-1}(\mathbb{R}^{2})\times{H}^{1}(\mathbb{R}^{2})$ (cf. [2]). Recalling that
$H^{\nu}(\mathbb{R}^{2})$ $\\!\hookrightarrow\\!{H}^{1}_{b}(\mathbb{R}^{2})$
and $H^{1}(\mathbb{R}^{2})$ can not be embedded into
${L}^{\infty}(\mathbb{R}^{2})$, hence global existence of mild solution to
system (1.1)–(1.3) with
$(u_{0},v_{0})\in{H}^{1}_{b}(\mathbb{R}^{2})\times{H}^{1}(\mathbb{R}^{2})$
improves the previous results. The proof is based on a combination of the
$L^{2}$-Fourier multiplier theory, the smoothing properties of heat kernel and
the new half norm of $v$, i.e.
$\sup_{t>0}{t^{\frac{1}{2}}}{(1+t)^{-\frac{1}{2}}}\|\nabla v\|_{L^{\infty}}$
which balances the need for $t$ near zero and $t$ near infinity. With this
unusual half norm, different form the usual scaling invariant ones, enables us
to overcome the main difficulty and to close the iteration scheme. At last,
global well-posedness of system (1.1)–(1.3) with initial data
$(u_{0},v_{0})\in H^{1}(\mathbb{R}^{2}){\times}H^{1}(\mathbb{R}^{2})$ is left
as another open question.
Next we recall some results concerning the parabolic-elliptic/parabolic-
hyperbolic Keller-Segel systems. Concerning the parabolic-elliptic Keller-
Segel model
$\displaystyle{{u}_{t}}=\Delta{u}-\nabla\cdot({u}\nabla{v}),\quad\quad$
$\displaystyle\Delta{v}={v}-{u}.$
It was conjectured by Childress and Percus [4] that in a two-dimensional
domain $\Omega$ there exists a critical number $c^{\ast}$ such that if
$\int_{\Omega}u_{0}(x)dx<c^{\ast}$ then the solution exists globally in time,
and if $\int_{\Omega}u_{0}(x)dx>c^{\ast}$ then blowup happens. For different
versions of the Keller-Segel model, the conjecture has been essentially
proved; for a complete review of this topic, we refer the reader to the paper
[10] and the references therein, also see e.g. Diaz, Nagai, and Rakotoson [7],
Blanchet, Dolbeault and Perthame [1]. As for the hyperbolic-hyperbolic Keller-
Segel model
$\displaystyle\partial_{t}u=\Delta
u+\nabla\cdot(u\nabla{w}),\quad\quad\partial_{t}{w}=u,$
it was used in [31] for one dimensional case and was extended to
multidimensional cases in [22], and has been studied in [21, 27] and a
comprehensive qualitative and numerical analysis was provided there. We refer
readers to references [5, 6, 8, 15, 16, 23, 25, 28, 29, 34] for more
discussions in this direction.
Throughout this paper, both $\mathcal{F}f$ and $\widehat{f}$ stand for Fourier
transform of $f$ with respect to space variable and $\mathcal{F}^{-\\!1}$
stands the inverse Fourier transform. Let $C$ and $c$ be positive constants
that may vary from line to line. $A\\!\lesssim\\!B$ stands for
$A\\!\leq\\!{CB}$ and $A\sim{B}$ stands for $A\lesssim{B}\lesssim{A}$. For any
$(p,q)\in[1,\infty]^{2}$, we denote $L^{p}(0,\infty)$,
$L^{q}(\mathbb{R}^{2})$, $H^{s}(\mathbb{R}^{2})$ and
$L^{p}(0,\infty;L^{q}(\mathbb{R}^{2}))$ by $L^{p}_{t}$, $L^{q}$, $H^{s}$ and
$L^{p}_{t}L^{q}$, respectively.
###### Theorem 1.1.
For any initial data $(u_{0},v_{0})\\!\in\\!{L}^{1}(\mathbb{R}^{2})\times
L^{\infty}(\mathbb{R}^{2})$ with ${\sup_{t>0}}\|e^{t\Delta}u_{0}\|_{L^{1}}$
and ${\sup_{t>0}}t^{\frac{1}{2}}\|\nabla{e}^{t\Delta}v_{0}\|_{L^{\infty}}$
being small, there exist a unique global mild solution $(u,v)$ to system
(1.1)–(1.3) and positive constant $c$ such that
$\displaystyle(u,v)\in{C}([0,\infty);L^{1}(\mathbb{R}^{2}))\times{C}_{w}([0,\infty);L^{\infty}(\mathbb{R}^{2}))$
with $C_{w}([0,\infty);X)$ being the set of weakly-star continuous functions
on $[0,\infty)$ valued in Banach space $X$, and
$\displaystyle\displaystyle{\sup_{t>0}}\,(\|u\|_{L^{1}}$
$\displaystyle+t\|u\|_{L^{\infty}}+\frac{1}{4c}t^{\frac{1}{2}}\|{\nabla{v}}\|_{L^{\infty}})$
$\displaystyle\leq{2}\,\displaystyle{\sup_{t>0}}\,(\|e^{t\Delta}u_{0}\|_{L^{1}}+t\|e^{t\Delta}u_{0}\|_{L^{\infty}}+\frac{1}{4c}t^{\frac{1}{2}}\|e^{t\Delta}{\nabla{v}_{0}}\|_{L^{\infty}}),$
(1.4)
which yields that $\|u\|_{L^{\infty}}\leq{o}(t^{-1})$ and $\|\nabla
v\|_{L^{\infty}}\leq o(t^{-\frac{1}{2}})$ as $t\rightarrow\infty$.
Remark: ($i$) Applying Lemma 2.5 to Proposition 3.1, we observe that (1.1)
holds if
$\displaystyle{\sup_{t>0}}\,(\|e^{t\Delta}u_{0}\|_{L^{1}}+t\|e^{t\Delta}u_{0}\|_{L^{\infty}}+\frac{1}{4c}t^{\frac{1}{2}}\|e^{t\Delta}{\nabla{v}_{0}}\|_{L^{\infty}})\leq\frac{3}{32c^{2}}.$
(1.5)
Applying $\|\frac{1}{4\pi{t}}e^{-\frac{|\cdot|^{2}}{4t}}\|_{L^{p}}\leq
t^{-1+\frac{1}{p}}$ and Proposition 2.4 to the left hand side of (1.5), it
suffices to assume that
$2\|u_{0}\|_{L^{1}}+\frac{1}{4c}\|\nabla{v}_{0}\|_{\dot{B}^{-1}_{\infty,\infty}}\\!\leq\\!\frac{3}{32c^{2}}$,
where
$\displaystyle\|v_{0}\|_{\dot{B}^{0}_{\infty,\infty}}\sim\|\nabla{v}_{0}\|_{\dot{B}^{-1}_{\infty,\infty}}=\sup_{t>0}t^{\frac{1}{2}}\|e^{t\Delta}\nabla
v_{0}\|_{L^{\infty}}\leq\|v_{0}\|_{L^{\infty}}$
since Riesz transforms $\frac{\nabla}{\sqrt{-\Delta}}$ are bounded in
homogeneous Besov spaces, $\sqrt{-\Delta}$ maps $\dot{B}^{0}_{\infty,\infty}$
isomorphically onto $\dot{B}^{-1}_{\infty,\,\infty}$ and
$\frac{1}{\sqrt{-\Delta}}$ maps $\dot{B}^{-1}_{\infty,\infty}$ isomorphically
onto $\dot{B}^{0}_{\infty,\,\infty}$ (see [30], Theorem 1, p.242). Therefore,
only $\dot{B}^{0}_{\\!\infty,\infty}$ smallness of $v_{0}$ and $L^{1}$
smallnessof $u_{0}$ are needed. Local existence of mild solution follows
directly by changing time interval $[0,\infty)$ into $[0,T]$. However, if
$v_{0}(x_{1},x_{2})=1_{[0,1]}(x_{1})$ and
$(t,x_{1},x_{2})\in(0,\frac{1}{64})\times({{t}^{\frac{1}{2}}}\\!,\,2{{t}^{\frac{1}{2}}})\times(-\infty,\infty)$,
then there holds
$\displaystyle t^{\frac{1}{2}}|\partial_{1}e^{t\Delta}v_{0}(x_{1},x_{2})|$
$\displaystyle=\\!\\!\int_{-\infty}^{\infty}\\!\int_{-\infty}^{\infty}\frac{1}{4\pi{t}}\frac{|x_{1}\\!-\\!y_{1}|}{\sqrt{4t}}e^{-\frac{(x_{1}-y_{2})^{2}+(x_{2}-y_{2})^{2}}{4t}}1_{[0,1]}(y_{1})dy_{1}dy_{2}$
$\displaystyle=\\!\frac{1}{\sqrt{\pi}}\\!\int_{y_{1}\in[0,1]}\\!\frac{|x_{1}\\!-y_{1}|}{\sqrt{4t}}e^{-\frac{(x_{1}-y_{1})^{2}}{4t}}d\frac{y_{1}-x_{1}}{\sqrt{4{t}}}$
$\displaystyle=\\!\frac{1}{\sqrt{\pi}}\\!\int_{\frac{-x_{1}}{\sqrt{4t}}}^{\frac{1-x_{1}}{\sqrt{4t}}}\\!ze^{-z^{2}}dz=\\!\frac{1}{\sqrt{\pi}}\\!\int_{\frac{x_{1}}{\sqrt{4t}}}^{\frac{1-x_{1}}{\sqrt{4t}}}\\!ze^{-z^{2}}dz$
$\displaystyle\geq\\!\frac{1}{\sqrt{\pi}}\\!\int_{1}^{3}\\!ze^{-z^{2}}dz=c_{0}.$
For such $v_{0}$, if we set $\widetilde{v}_{0}=v_{0}/c_{0}$, then we have
${\lim_{T\rightarrow 0^{+}}\sup_{0<t<T}}t^{\frac{1}{2}}\|\nabla
e^{t\Delta}\widetilde{v}_{0}\|_{L^{\infty}}\geq 1.$
Hence it seems difficult to prove local (global) existence of mild solution
for arbitrary large $L^{1}(\mathbb{R}^{2})\times{L}^{\infty}(\mathbb{R}^{2})$
initial data.
($ii$) Proof of Theorem 1.1 also applies for $u_{t}-\Delta
u+\nabla\cdot({u\nabla{v}})=0$ and $v_{t}-\Delta v+u=0$ with initial data
$(u_{0},v_{0})\in L^{1}(\mathbb{R}^{2})\times{L^{\infty}}(\mathbb{R}^{2})$.
Here and hereafter, we set $\sigma(t)\\!=t^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}$.
Then we state the following result.
###### Theorem 1.2.
For any initial data $(u_{0},v_{0})\in{H^{1}_{b}}(\mathbb{R}^{2})\times
H^{1}(\mathbb{R}^{2})$, there exist positive constants $\varepsilon_{0}$ and
$c$ so that if
$\|u_{0}\|_{L^{\infty}}+\|u_{0}\|_{H^{1}}+\frac{1}{4c}\|v_{0}\|_{H^{1}}\leq\varepsilon_{0}$,
then system (1.1)–(1.3) has a unique global solution $(u,v)$ satisfying
$\displaystyle(u\pm{v},\,u\pm{\sigma}\hskip
0.28436pt\nabla{v},\,\nabla{u}\pm\nabla
v)\in{C}([0,\infty);H^{1}(\mathbb{R}^{2}))\times{L}^{\\!\infty}_{t}\\!L^{\\!\infty}\times{L}^{2}_{t}H^{1}.$
Moreover,
$\|u\pm\frac{v}{4c}\|_{L^{\infty}_{t}H^{1}}+\|u\pm\frac{{\sigma}}{4c}\nabla{v}\|_{L^{\infty}_{t}L^{\infty}}+\|\nabla{u}\|_{L^{2}_{t}L^{2}}+\frac{1}{4c}\|\nabla{v}\|_{L^{2}_{t}H^{1}}\leq{2}\hskip
0.56917pt\varepsilon_{0}$.
Plan of the paper: In Sect. 2 we introduce several preliminary lemmas, while
in Sect. 3 we prove Theorems 1.1 and 1.2.
## 2 Preliminaries
In this section, we list several known lemma and prove some key lemmas which
will be used in proving the well-posedness of the parabolic-parabolic
chemotaxis. The first lemma given below is concerned with initial data
belonging to $H^{s}(\mathbb{R}^{2})$. For simplicity, here and hereafter, we
omit the space domain in various function spaces, for instance
$H^{1}(\mathbb{R}^{2})$ is denoted by $H^{1}$, if there is no confusion.
###### Lemma 2.1.
Let $n=2$, $\Lambda=\sqrt{-\Delta}$, $(s,\hskip 1.13791pt\delta,\hskip
1.13791ptr,\hskip
1.13791pt\rho)\in(-\infty,\infty)\times[0,\infty)\times[1,\infty]\times[2,\infty]$
and $v\in{H}^{s}$. If $m(t,\xi)\in{L}^{r}_{t}L^{\\!\infty}_{\xi}$ and
$m(t,D)v=\\!\mathcal{F}^{-1}m(t,\xi)\widehat{v}(\xi)$, then we get
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}H^{s}}\lesssim\|m\|_{{L}^{r}_{t}L^{\infty}_{\xi}}\|v\|_{H^{s}};$
(2.1)
Else if
${m}_{\delta}(t,\xi):=m(t,\xi)|\xi|^{\delta}\in\\!{L^{\\!\infty}_{\xi}{L}^{\rho}_{t}}$
and
$m_{\delta}(t,D)v\\!=\\!\mathcal{F}^{-1}m_{\delta}(t,\xi)\widehat{v}(\xi)$,
then we get
$\displaystyle\|m_{\delta}(t,D)v\|_{L^{\rho}_{t}{H}^{s}}\lesssim\|m_{\delta}\|_{L^{\infty}_{\xi}{L}^{\rho}_{t}}\|v\|_{H^{s}}.$
(2.2)
###### Proof.
Proof of (2.1) follows from classical Fourier multiplier theory and, for
readers convinience, we give the detail proof as follows:
$\displaystyle\|m(t,D)v\|_{L^{r}_{t}H^{s}}$
$\displaystyle\lesssim\|m(t,\xi)(1+|\cdot\\!|^{2})^{\frac{s}{2}}\widehat{v}(\cdot)\|_{L^{r}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|m\|_{{L}^{r}_{t}L^{\infty}_{\xi}}\|(1+|\cdot\\!|^{2})^{\frac{s}{2}}\widehat{v}(\cdot)\|_{L^{2}_{\xi}}\;\;$
$\displaystyle\lesssim\|m\|_{{L}^{r}_{t}L^{\infty}_{\xi}}\|v\|_{H^{s}},$ (2.3)
where we have used Plancherel equality twice.
In order to prove (2.2), by making use of Plancherel equality, Minkowski’s
inequality, Hölder’s inequality and Plancherel equality again, we get
$\displaystyle\|m_{\delta}(t,D){v}\|_{L^{\rho}_{t}{H}^{s}}$
$\displaystyle\lesssim\|m_{\delta}\;(1+|\cdot\\!|^{2})^{\frac{s}{2}}\widehat{v}(\cdot)\|_{L^{\rho}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|m_{\delta}\;(1+|\cdot\\!|^{2})^{\frac{s}{2}}\widehat{v}(\cdot)\|_{L^{2}_{\xi}{L^{\rho}_{t}}}$
$\displaystyle\lesssim\|m_{\delta}\|_{L^{\infty}_{\xi}{L}^{\rho}_{t}}\|(1+|\cdot\\!|^{2})^{\frac{s}{2}}\widehat{v}(\cdot)\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|m_{\delta}\|_{L^{\infty}_{\xi}{L}^{\rho}_{t}}\|v\|_{H^{s}}.$
(2.4)
Hence, we finish the proof. ∎
The skill used in the above Lemma will be used repeatedly in the following
parts. The next Lemma is devoted to estimate the bilinear term which is known
as the maximal $L^{p}_{t}L^{q}$ regularity result for the heat kernel (cf.
[20], Theorem 7.3, p. 64).
###### Lemma 2.2.
$({\rm Maximal}$ $L^{p}_{t}L^{q}$ ${\rm regularity\;for\;heat\;kernel})$ The
operator $T$ defined by
$\displaystyle
g(t,x)\mapsto{Tg(t,x)}=\\!\int_{0}^{t}\\!e^{(t-\tau)\Delta}\Delta{g}(\tau,x)d\tau$
(2.5)
is bounded from $L^{p}_{t}L^{q}$ to $L^{p}_{t}L^{q}$ with $1\\!<p<\\!\infty$
and $1\\!<q\\!<\infty$.
The next Lemma is also dedicated to estimating the bilinear term.
###### Lemma 2.3.
For any $(s,\hskip 0.56917ptc,\hskip 0.56917ptc_{1},\hskip 0.56917ptp,\hskip
0.56917ptr,\hskip
0.56917ptp_{1})\in\mathbb{R}\times(0,\infty)^{2}\times[2,\infty]\times[1,2]\times[1,\infty]$,
${p}_{1}\geq{r}$ and $0\leq\theta\\!<2(1+\\!\frac{1}{p_{1}}-\\!\frac{1}{r})$,
if $m(t,\xi)=\frac{c_{1}}{e^{\,ct|\xi|^{2}}}$ and
$\mu(t,\xi)=\frac{{c}_{1}}{e^{\,ct+ct|\xi|^{2}}}$, then there exists constant
$C_{\theta,p_{1},r}$ depending on $\theta$, $p_{1}$ and $r$ such that have
$\displaystyle\|\\!\int_{0}^{t}\\!\\!m(t\\!-\\!\tau,D)\Lambda^{{2+\frac{2}{p}-\frac{2}{r}}}F(\tau,x)\,d\tau\|_{L^{p}_{t}H^{s}}\lesssim\|F\|_{L^{r}_{t}H^{s}},$
(2.6)
$\displaystyle\|\\!\int_{0}^{t}\\!\mu(t\\!-\\!\tau,D)\Lambda^{\theta}F(\tau,x)\,d\tau\|_{L^{p_{1}}_{t}\dot{H}^{s}}\lesssim{C}_{\theta,p_{1},r}\|F\|_{L^{r}_{t}\dot{H}^{s}}.$
(2.7)
###### Proof.
In order to prove (2.6), setting $\langle\xi\rangle=\sqrt{1+|\xi|^{2}}$ and by
using Plancherel equality, the Minkowski inequality, the Young inequality and
the Minkowski inequality as well as Plancherel equality, we have
$\displaystyle\|\\!\int_{0}^{t}\\!m(t\\!-\\!\tau,D)$
$\displaystyle\Lambda^{2+\frac{2}{p}-\frac{2}{r}}F(\tau,x)d\tau\|_{L^{p}_{t}{H}^{s}}\\!=\\!\|\\!\int_{0}^{t}\\!m(t\\!-\\!\tau,\xi)|\xi|^{2+\frac{2}{p}-\frac{2}{r}}\langle\xi\rangle^{s}\widehat{F}(\tau,\xi)d\tau\|_{L^{p}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|\int_{0}^{t}m(t-\tau,\xi)|\xi|^{2+\frac{2}{p}-\frac{2}{r}}\langle\xi\rangle^{s}\widehat{F}(\tau,\xi)d\tau\|_{L^{2}_{\xi}{L}^{p}_{t}}$
$\displaystyle\lesssim\Big{\|}\|m(\cdot,\xi)|\xi|^{2+\frac{2}{p}-\frac{2}{r}}\|_{L^{\frac{p\hskip
1.13791ptr}{p\hskip
1.13791ptr+r-p}}_{t}}\|\langle\xi\rangle^{s}\widehat{F}(\cdot,\xi)\|_{L^{r}_{t}}\Big{\|}_{L^{2}_{\xi}}$
$\displaystyle\lesssim\sup_{\xi\in\mathbb{R}^{2}}\|m(\cdot,\xi)|\xi|^{2+\frac{2}{p}-\frac{2}{r}}\|_{L^{\frac{p\hskip
1.13791ptr}{p\hskip
1.13791ptr+r-p}}_{t}}\|\langle\xi\rangle^{s}\widehat{F}(\cdot,\xi)\|_{L^{2}_{\xi}L^{r}_{t}}$
$\displaystyle\lesssim\|\langle\xi\rangle^{s}\widehat{F}(\cdot,\xi)\|_{L^{r}_{t}L^{2}_{\xi}}$
$\displaystyle\lesssim\|F\|_{L^{r}_{t}H^{s}}.$
It remains to prove (2.7). Using the $L^{1}$ integrability of
$e^{-ct}t^{-\frac{\theta}{2+\frac{2}{p_{1}}-\frac{2}{r}}}$, we get
$\displaystyle\|\\!\int_{0}^{t}\\!\mu(t\\!-\\!\tau,D)\Lambda^{\theta}$
$\displaystyle
F(\tau,x)d\tau\|_{L^{p_{1}}_{t}\dot{H}^{s}}=\|\int_{0}^{t}e^{-c(t-\tau)}(t\\!-\\!\tau)^{-\frac{\theta}{2}}\|{F}\|_{\dot{H}^{s}}d\tau\|_{L^{p_{1}}_{t}}$
$\displaystyle\lesssim\Big{(}\int_{0}^{t}e^{-c{\frac{p_{1}\hskip
1.13791ptr}{p_{1}\hskip 1.13791ptr+r-p_{1}}}t}t^{-\frac{\theta{p_{1}\hskip
1.13791ptr}}{2(p_{1}\hskip
1.13791ptr+r-p)}}dt\Big{)}^{1+\frac{1}{p_{1}}-\frac{1}{r}}\|F\|_{L^{r}_{t}\dot{H}^{s}}$
$\displaystyle\lesssim{C}_{\theta,p_{1},r}\|F\|_{L^{r}_{t}\dot{H}^{s}}.$
Therefore, we finish the whole proof. ∎
Let us state the equivalent definition of Besov space
$\dot{B}^{s}_{p,q}:=\dot{B}^{s}_{p,q}(\mathbb{R}^{2})$ using heat semigroup
method (for a proof see, for instance [30] p.192 or [20] Theorem 5.4, p.45).
###### Proposition 2.4.
Let $(s,p,q)\\!\in\\!(-\infty,0)\times\\![1,\infty]^{2}\\!.\\!$ The
homogeneous Beosv space $\dot{B}^{s}_{\\!p,q}$ is defined as the set of
tempered distribution $f$ such that
$\displaystyle\|f\|_{\dot{B}^{s}_{p,q}}=\Big{(}\int_{0}^{\infty}(t^{\frac{-s}{2}}\|e^{t\Delta}f\|_{L^{p}})^{q}\frac{dt}{t}\Big{)}^{\frac{1}{q}}\quad\;\text{
if }\;1\leq q<\infty,$
$\displaystyle\|f\|_{\dot{B}^{s}_{p,\infty}}\\!=\sup_{t>0}t^{\frac{-s}{2}}\|e^{t\Delta}f\|_{L^{p}}\quad\hskip
46.09332pt\text{ if }\;q=\infty.\hskip 19.91684pt$
The last lemma of this section is a slightly generalized version about the
well-known Picard contraction principle (see for instance [20], Theorem 13.2,
p.124) which is used to prove the main results concerning well-posedness of
(1.1)–(1.3) with $(u_{0},v_{0})$ either belonging to
$L^{1}(\mathbb{R}^{2})\times L^{\infty}(\mathbb{R}^{2})$ or belonging to
$H^{1}_{b}(\mathbb{R}^{2})\times H^{1}(\mathbb{R}^{2})$.
###### Lemma 2.5.
$(\mathrm{The\;Picard\;contraction\;principle})$ Let
$({X}\times{Y},\;\|\cdot\\!\|_{X}\\!+\\!\|\cdot\\!\|_{Y})$ be an abstract
Banach product space, $L:{X}\rightarrow{Y}$ and $B:{X}\times{Y}\rightarrow{X}$
are a linear operator and a bilinear operator, respectively, such that for any
$(u,v)\in{X}\times{Y}$, there exist positive constant $c$ and if
$\displaystyle\|L(u)\|_{Y}\leq
c\|u\|_{X},\;\;\|B(u,v)\|_{X}\leq{c}\|u\|_{X}\|v\|_{Y},$ (2.8)
then for any $(e^{t\Delta}u_{0},e^{t(\Delta-1)}v_{0})\in{X\times{Y}}$ with
$\|(e^{t\Delta}u_{0},\frac{1}{4c}e^{t(\Delta-1)}v_{0})\|_{X\times{Y}}<\frac{3}{32c^{2}}$,
the following system
$(u,v)=(e^{t\Delta}u_{0},e^{t(\Delta-1)}v_{0})+(B(u,v),\ L(u))$ (2.9)
has a solution $(u,v)$ in ${X}\times{Y}$. In particular, the solution is such
that
$\|(u,\frac{v}{4c})\|_{{X}\times{Y}}\leq{2}\|(e^{t\Delta}u_{0},\frac{1}{4c}e^{t(\Delta-1)}v_{0})\|_{{X\times{Y}}}$
and it is the only one such that
$\|(u,\frac{v}{4c})\|_{{X}\times{Y}}<\frac{3}{16c^{2}}.$
###### Proof.
The proof is standard now. However, for reader’s convenience, we give a brief
proof. We first define a mapping ${\Phi}:X\times{Y}\rightarrow X\times Y$ such
that
$\displaystyle\Phi(u,v)=(e^{t\Delta}u_{0},e^{t(\Delta-1)}v_{0})+(B(u,v),\
L(u)).$ (2.10)
Applying simple transformations, i.e. $w=\frac{v}{4c}$ and
$w_{0}=\frac{1}{4c}v_{0}$ to (2.10), we get
$\displaystyle\Phi(u,w)=(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})+(4cB(u,w),\frac{1}{4c}L(u)).$
(2.11)
By applying (2.8) to (2.11), we have
$\displaystyle\|\Phi(u,w)\|_{X\times{Y}}$
$\displaystyle\leq\|(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})\|_{X\times
Y}+4c^{2}\|u\|_{X}\|w\|_{Y}+\frac{1}{4}\|u\|_{Y}$
$\displaystyle\leq{A_{0}}+c^{2}\|(u,w)\|_{X\times{Y}}^{2}+\frac{1}{4}\|(u,w)\|_{X\times{Y}},$
(2.12)
where $A_{0}:=\|(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})\|_{X\times Y}$. Let
$\overline{B(0,2A_{0})}\subset{X\times Y}$ be a closed ball centered at origin
with radius $2A_{0}$. From (2), we observe that $\Phi$ is well defined in
$\overline{B(0,2A_{0})}$ and maps $\overline{B(0,2A_{0})}$ into itself.
Moreover, for any $(u_{1},w_{1})$, $(u_{2},w_{2})\in\overline{B(0,2A_{0})}$,
by making use of (2.8), we get
$\displaystyle\|\Phi(u_{1},w_{1}\\!)-\\!\Phi(u_{2},w_{2})\|_{X\times
Y}\\!=\\!\|(4cB(u_{1},w_{1}\\!)\\!-\\!4cB(u_{2},w_{2}),\,\frac{L(u_{1}\\!)\\!-\\!L(u_{2})}{4c})\|_{X\times{Y}}$
$\displaystyle\leq
4c^{2}\max\\{\|u_{2}\|_{X},\|w_{1}\|_{Y}\\}\|(u_{1}\\!-u_{2},w_{1}\\!-w_{2})\|_{X\times{Y}}+\frac{1}{4}\|(u_{1}\\!-u_{2},w_{1}\\!-w_{2})\|_{X\times{Y}}$
$\displaystyle\leq
8c^{2}A_{0}\|(u_{1}-u_{2},w_{1}-w_{2})\|_{X\times{Y}}\\!+\\!\frac{1}{4}\|(u_{1}-u_{2},w_{1}-w_{2})\|_{X\times{Y}}$
$\displaystyle\leq(8c^{2}A_{0}+\frac{1}{4})\|(u_{1}-u_{2},w_{1}-w_{2})\|_{X\times{Y}},$
(2.13)
where $8c^{2}A_{0}+\frac{1}{4}<1$ since $A_{0}<\frac{3}{32c^{2}}$. From (2),
we observe that $\Phi:(u,w)\mapsto\Phi(u,w)$ in (2.11) is contractive. Thus
there exists a unique solution $(u,w)$ to (2.11), which shows that (2.10) also
has a unique solution $(u,v)$ to (2.10) provides that
$\|(e^{t\Delta}u_{0},\frac{1}{4c}e^{t(\Delta-1)}v_{0})\|_{X\times
Y}<{3}/{32c^{2}}$. ∎
## 3 Proof of Theorem 1.1
As usual, we apply the heat semigroup $e^{t\Delta}$ with heat kernel
$\frac{1}{4\pi t}e^{-\frac{|x|^{2}}{4t}}$ to invert system (1.1)–(1.3) into
the following integral equations via the Duhamel principle:
$\displaystyle\hskip-29.90369pt\left\\{\begin{aligned}
{u}&=e^{t\Delta}{u}_{0}-\int_{0}^{t}e^{(t-\tau)\Delta}\nabla\cdot({u}\nabla{v})d\tau:=e^{t\Delta}{u}_{0}-B(u,v),\\\
v&=e^{t(\Delta-1)}v_{0}+\int_{0}^{t}e^{(t-\tau)(\Delta-1)}{u}d\tau:=e^{t(\Delta-1)}v_{0}+L(u).\end{aligned}$
Let $c$ be the largest positive constant that appears in the linear and
bilinear estimates and depends only on dimension. By denote $\frac{v}{4c}$ by
$w$, we get the following system
$\displaystyle\left\\{\begin{aligned}
{u}&=e^{t\Delta}{u}_{0}-4c\int_{0}^{t}e^{(t-\tau)\Delta}\nabla\cdot({u}\nabla{w})d\tau:=e^{t\Delta}{u}_{0}-4cB(u,w),\\\
w&=e^{t(\Delta-1)}w_{0}+\frac{1}{4c}\int_{0}^{t}e^{(t-\tau)(\Delta-1)}{u}d\tau:=e^{t(\Delta-1)}w_{0}+\frac{1}{4c}L(u),\end{aligned}$
where we regard equations (3) as a fixed point system and let mapping $\Phi$
be
$\displaystyle\Phi:(u,w)\mapsto\Big{(}e^{t\Delta}u_{0},\,e^{t(\Delta-1)}{w_{0}}\Big{)}+\Big{(}-4cB(u,w),\,\frac{1}{4c}L(u)\Big{)}.\quad\quad$
(3.1)
We call solution $(u,4cw)$ to (3) mild solution of (1.1)–(1.3) if $(u,w)$
solves (3).
### 3.1 Proof of Theorem 1.1
In this subsection, we prove global well-posedness of system (3) with initial
data $(u_{0},w_{0})\in L^{1}(\mathbb{R}^{2})\times L^{\infty}(\mathbb{R}^{2})$
by making use of the Kato’s $L^{p}$-framework. At first, we set
$\displaystyle\begin{aligned}
&X=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times(0,\infty));\
\sup_{t>0}\|u(\cdot,t)\|_{L^{1}}+\sup_{t>0}\;t\|u(\cdot,t)\|_{L^{\infty}}<\infty\;\\},\\\
&Y=\\{w\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times(0,\infty));\
\sup_{t>0}t^{\frac{1}{2}}\|\nabla{w}(\cdot,t)\|_{L^{\infty}}\\!<\\!\infty\\}.\end{aligned}$
(3.2)
Then we prove that for suitably small initial data $(u_{0},w_{0})$ the mapping
$\Phi$ is contractive and maps a closed ball of $X\times{Y}$ into itself.
###### Proposition 3.1.
For any initial data $({u}_{0},w_{0})\in{L}^{1}\times{L}^{\infty}$, there
exists positive constant $c$ such that
$\displaystyle\left\\{\begin{aligned}
&\|(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})\|_{X\times{Y}}\leq c\hskip
0.56917pt\|(u_{0},w_{0})\|_{L^{1}\times\dot{B}^{0}_{\infty,\infty}}\\!\\!\leq
c\hskip 0.56917pt\|(u_{0},w_{0})\|_{L^{1}\times L^{\infty}},\\\
&\|4cB(u,w)\|_{X}\leq 4c^{2}\|u\|_{X}\|w\|_{Y}\;\;\;\text{ and
}\;\;\;\|\frac{1}{4c}L(u))\|_{{Y}}\leq\frac{1}{4}\|u\|_{X}.\end{aligned}\right.$
(3.3)
###### Proof.
We divide the whole proof into two parts concerning with $e^{t\Delta}u_{0}$,
$e^{-t}e^{t\Delta}w_{0}$ and $B(u,w),$ $L(u)$, respectively.
Part I. Estimates for $\|e^{t\Delta}u_{0}\|_{X}$ and
$\|e^{-t}e^{t\Delta}w_{0}\|_{Y}$. Recall that the heat kernel is
$\frac{1}{4\pi{t}}e^{-\frac{|x|^{2}}{4t}}$. Then for any $t>0$ and $1\leq
p\leq\infty$, there hold
$\displaystyle\|\frac{1}{4\pi{t}}e^{-\frac{\,\,\,|\cdot|^{2}}{4t}}\|_{L^{p}}\leq{t}^{-1+\frac{1}{p}}\quad\text{and
}\ \|\frac{1}{4\pi{t}}\nabla e^{-\frac{\,\,|\cdot|^{2}}{4t}}\|_{L^{p}}\leq
t^{-\frac{3}{2}+\frac{1}{p}}.$ (3.4)
Applying Young’s inequality and (3.4) to
$e^{t\Delta}u_{0}(x)\\!=\\!\int_{\\!\mathbb{R}^{2}}\\!\frac{1}{4\pi
t}e^{-\frac{|y|^{2}}{4t}}u_{0}(x\\!-\\!y)dy$, we get
$\displaystyle\|e^{t\Delta}u_{0}\|_{X}$
$\displaystyle=\sup_{t>0}\|e^{t\Delta}u_{0}\|_{L^{1}}+\sup_{t>0}\,t\hskip
0.28436pt\|e^{t\Delta}u_{0}\|_{L^{\\!\infty}}$
$\displaystyle\leq\sup_{t>0}\|\frac{1}{4\pi
t}e^{-\frac{|\cdot|^{2}}{4t}}\|_{L^{1}}\|u_{0}\|_{L^{1}}+\sup_{t>0}t\|\frac{1}{4\pi
t}e^{-\frac{|\cdot|^{2}}{4t}}\|_{L^{\infty}}\|u_{0}\|_{L^{1}}$
$\displaystyle\leq 2\|u_{0}\|_{L^{1}}$ (3.5)
or from Proposition 2.4 and embedding theorem
$L^{1}\hookrightarrow\dot{B}^{-2}_{\infty,\infty}$, we have
$\sup_{t>0}\,t\hskip
0.28436pt\|e^{t\Delta}u_{0}\|_{L^{\\!\infty}}=\|u_{0}\|_{\dot{B}^{-2}_{\infty,\infty}}\leq{c_{1}}\|u_{0}\|_{L^{1}}.$
We emphasize here that for any
$(s,\alpha,p,q)\in\mathbb{R}^{2}\times[1,\infty]^{2}$,
$(-\Delta)^{\frac{\alpha}{2}}$ maps $\dot{B}^{s}_{p,q}$ isomorphically onto
$\dot{B}^{s-\alpha}_{p,\,q}$ (cf. [30], Theorem 1, p.242), which is a direct
consequence of the well known Bernstein’s inequalities. Thus following similar
arguments of $e^{t\Delta}$ and using (3.4) as well as Proposition 2.4, we get
$\displaystyle\|e^{-t}e^{t\Delta}w_{0}\|_{Y}$
$\displaystyle=\sup_{t>0}\,t^{\frac{1}{2}}\hskip
0.28436pt\|e^{-t}e^{t\Delta}\nabla
w_{0}\|_{L^{\\!\infty}}\leq\sup_{t>0}\,t^{\frac{1}{2}}\hskip
0.28436pt\|e^{t\Delta}\nabla{w}_{0}\|_{L^{\\!\infty}}$ $\displaystyle=\|\nabla
w_{0}\|_{\dot{B}^{-1}_{\infty,\infty}}\sim\|w_{0}\|_{\dot{B}^{0}_{\infty,\infty}}$
$\displaystyle\leq c_{1}\|w_{0}\|_{L^{\infty}},$ (3.6)
where the fourth and the fifth inequalities follow from Theorem 1 of [30]
p.242 and boundedness of Riesz transforms $\frac{\nabla}{\sqrt{-\Delta}}$ as
well as $L^{\infty}\\!\hookrightarrow\\!\dot{B}^{0}_{\infty,\infty}$.
Part II. Estimates for $\|B(u,w)\|_{X}$ and $L(u)\|_{Y}$. As for
$\|B(u,w)\|_{X}$, we have
$\displaystyle\|B(u,w)\|_{X}$
$\displaystyle=\sup_{t>0}\|\\!\\!\int_{0}^{t}\\!\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{1}}+\sup_{t>0}\,t\|\\!\\!\int_{0}^{t}\\!\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}}$
$\displaystyle\leq\sup_{t>0}\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}\|u\nabla{w}\|_{L^{1}}d\tau+\sup_{t>0}t\\!\\!\int_{0}^{\frac{t}{2}}(t-\tau)^{-\frac{3}{2}}\|u\nabla{w}\|_{L^{1}}d\tau$
$\displaystyle\,\;\;+\sup_{t>0}t\\!\\!\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{1}{2}}\|u\nabla{w}\|_{L^{\infty}}d\tau$
$\displaystyle\leq\sup_{t>0}\Big{(}\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-\frac{1}{2}}d\tau+c_{2}\frac{1}{\\!\\!\sqrt{t}}\\!\int_{0}^{\frac{t}{2}}\tau^{-\frac{1}{2}}d\tau\Big{)}\sup_{\tau>0}\tau^{\frac{1}{2}}\|(u\nabla{w})(\tau)\|_{L^{1}}$
$\displaystyle\,\;\;+\sup_{t>0}t\\!\\!\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-\frac{3}{2}}d\tau\sup_{\tau>0}\tau^{\frac{3}{2}}\|(u\nabla{w})(\tau)\|_{L^{\infty}}$
$\displaystyle\leq
c_{2}\sup_{\tau>0}\,(\|u(\tau)\|_{L^{1}}+\tau\|u(\tau)\|_{L^{\infty}})\sup_{\tau>0}\tau^{\frac{1}{2}}\|\nabla{w}(\tau)\|_{L^{\infty}}$
$\displaystyle\leq{c_{2}}\|u\|_{X}\|w\|_{Y}.$ (3.7)
As for $\|L(u)\|_{Y}$, from definition of $\|\cdot\|_{Y}$, we need to estimate
$\displaystyle\sup_{t>0}{t}^{\frac{1}{2}}\|\nabla{L}(u)\|_{L^{\infty}}=\sup_{t>0}t^{\frac{1}{2}}\|\int_{0}^{t}e^{-(t-\tau)}e^{(t-\tau)\Delta}\nabla{u}d\tau\|_{L^{\infty}}$
$\displaystyle\leq\sup_{t>0}\Big{(}t^{\frac{1}{2}}\\!\\!\int_{0}^{\frac{t}{2}}(t-\tau)^{-\frac{3}{2}}\|{u}(\tau)\|_{L^{1}}d\tau+t^{\frac{1}{2}}\\!\\!\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-1}\tau\|{u}(\tau)\|_{L^{\infty}}d\tau\Big{)}$
$\displaystyle\leq{c_{3}}\sup_{\tau>0}\hskip
0.56917pt(\|u(\tau)\|_{L^{1}}+\tau\|u(\tau)\|_{L^{\infty}})$
$\displaystyle\leq{c}_{3}\|u\|_{X}.$ (3.8)
Setting $c\\!=\max\\{c_{1},c_{2},c_{3}\\}$, combining (3.2)–(3.8), multiplying
$B(u,w)$ by $4c$ and multiplying $L(u)$ by $\frac{1}{4c}$, we prove (3.3). ∎
Proof of Theorem 1.1: At first, applying Proposition 3.1, following similar
arguments as in the proof of Lemma 2.5, we prove that there exists a unique
solution $(u,w)\in\overline{B(0,2A_{10})}\subset{X}\times{Y}$ to system (3) if
$A_{10}:=\|(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})\|_{X\times{Y}}\\!<\\!\\!{3}/{32c^{2}}$.
Moreover, this solution also satisfies $\Phi(u,w)=(u,w)$. From (3.1)–(3.6), it
suffices to assume that
$\|(u_{0},w_{0})\|_{L^{1}\times\dot{B}^{0}_{\infty,\infty}}\\!\\!<\\!{3}/{32c^{3}}$
since $A_{10}\leq
c\|(u_{0},w_{0})\|_{L^{1}\times\dot{B}^{0}_{\infty,\infty}}\\!\\!\\!<\\!{3}/{32c^{2}}$.
Next we show that $w\in{C}_{w}([0,\infty);L^{\infty}(\mathbb{R}^{2}))$. From
(3) and (3.4), we have
$\displaystyle\sup_{t>0}\|w\|_{L^{\infty}}$
$\displaystyle=\sup_{t>0}\|e^{t(\Delta-1)}w_{0}+\frac{1}{4c}L(u)\|_{L^{\infty}}\leq\|w_{0}\|_{L^{\infty}}+\frac{1}{4}\|u\|_{X},$
where in (3.8),
$c_{3}=\sup_{t>0}(t^{\frac{1}{2}}\\!\int_{0}^{\frac{t}{2}}(t-\tau)^{-\frac{3}{2}}d\tau+t^{\frac{1}{2}}\\!\int_{t/2}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-1}d\tau)$
and similarly
$\displaystyle\|\frac{1}{4c}L(u)\|_{L^{\infty}}$
$\displaystyle=\frac{1}{4c}\|\int_{0}^{t}e^{(t-\tau)(\Delta-1)}ud\tau\|_{L^{\infty}}\leq\frac{1}{4c}\|\int_{0}^{t}e^{(t-\tau)\Delta}ud\tau\|_{L^{\infty}}$
$\displaystyle\leq\frac{1}{4c}\int_{0}^{\frac{t}{2}}\\!(t-\\!\tau)^{-\frac{2}{2}(\frac{1}{1}-\frac{1}{\infty})}\|u(\tau)\|_{L^{\infty}}d\tau+\frac{1}{4c}\int_{\frac{t}{2}}^{t}\tau^{-1}\tau\|u(\tau)\|_{L^{\infty}}d\tau$
$\displaystyle\leq\frac{c_{3}}{4c}\|u\|_{X}\leq\frac{1}{4}\|u\|_{X}$
since
$\int_{0}^{\frac{t}{2}}(t-\tau)^{-1}d\tau<t^{\frac{1}{2}}\\!\int_{0}^{\frac{t}{2}}(t-\tau)^{-\frac{3}{2}}d\tau$,
$\int_{\frac{t}{2}}^{t}\tau^{-1}d\tau<t^{\frac{1}{2}}\\!\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-1}d\tau$
and $c\geq{c}_{3}$. Moreover, following a dense argument in
$L^{1}(\mathbb{R}^{2})$ we can prove the time continuity of $u$. Since
Schwartz function space is not dense in $L^{\infty}(\mathbb{R}^{2})$, we can
only obtain the weakly star time continuity of solution $w$.
Finally, performing transformation: $(u,v)=(u,4cw)$, we get the unique
solution $(u,v)$ of (1.1)–(1.3).
### 3.2 Proof of Theorem 1.2
In this subsection, we prove global well-posedness of system (3) with initial
data $(u_{0},w_{0})\in H^{1}_{b}(\mathbb{R}^{2})\times H^{1}(\mathbb{R}^{2})$
by making use of the Kato’s framework, see [13] for instance. At first, we
recall that $\sigma(t)=t^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}$ and then we set
$\displaystyle\begin{aligned}
&X=\\!\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\\!\times\\!(0,\infty));\
\sup_{t>0}\|u(\cdot,t)\|_{H^{1}}+\|\nabla{u}\|_{L^{2}_{t}H^{1}}\\!+\|u\|_{L^{\infty}_{t}L^{\infty}}<\infty\;\\},\\\
&Y=\\!\\{w\in\\!\mathcal{S}^{\prime}(\mathbb{R}^{2}\\!\times\\!(0,\infty));\
\sup_{t>0}\|w(\cdot,t)\|_{H^{1}}\\!+\\!\|\nabla{w}\|_{L^{2}_{t}H^{1}}\\!+\|{\sigma\nabla}{w}\|_{L^{\\!\infty}_{t}\\!L^{\\!\infty}}\\!<\\!\infty\\}.\end{aligned}$
(3.9)
The following Proposition will play a central role in proving Theorem 1.2.
###### Proposition 3.2.
For any initial data $({u}_{0},w_{0})\in H^{1}_{b}(\mathbb{R}^{2})\times
H^{1}(\mathbb{R}^{2})$, there exists positive constant $c$ such that
$\displaystyle\|4cB(u,w)\|_{X}\leq{4c^{2}}\|u\|_{X}\|w\|_{Y}\leq{c^{2}}\|(u,w)\|_{X\times{Y}}^{2},\quad\|\frac{1}{4c}L(u)\|_{Y}\leq\frac{1}{4}\|u\|_{X}$
(3.10)
and $\|(e^{t\Delta}u_{0},e^{t(\Delta-1)}w_{0})\|_{X\times
Y}\leq{c}\|(u_{0},v_{0})\|_{H^{1}_{b}\times{H}^{1}}$.
###### Proof.
We divide the whole proof into two parts concerning with
$(e^{t\Delta}u_{0},\,e^{-t}e^{t\Delta}w_{0})$ and
$(-4cB(u,w),\,\frac{1}{4c}L(u))$, respectively.
Part I. Estimates for $\|e^{t\Delta}u_{0}\|_{X}$ and
$\|e^{-t}e^{t\Delta}w_{0}\|_{Y}$. As for $\|e^{t\Delta}u_{0}\|_{X}$, noticing
that $e^{-ct|\xi|^{2}}\in L^{\infty}_{t}L^{\infty}_{\xi}$ and
$e^{-ct|\xi|^{2}}\xi\in L^{\infty}_{\xi}L^{2}_{t}$, then by applying Lemma 2.1
and Young’s inequality, we have
$\displaystyle\|e^{t\Delta}u_{0}\|_{X}$
$\displaystyle=\|e^{t\Delta}u_{0}\|_{L^{\infty}_{t}H^{1}}+\|e^{t\Delta}\nabla{u_{0}}\|_{L^{2}_{t}H^{1}}+\|e^{t\Delta}u_{0}\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq\|u_{0}\|_{H^{1}}+\|u_{0}\|_{H^{1}}+\|u_{0}\|_{L^{\infty}}$
$\displaystyle\leq c\|u_{0}\|_{H^{1}_{b}}.$ (3.11)
Similarly, we have
$\displaystyle\|e^{-t}e^{t\Delta}w_{0}\|_{L^{\infty}_{t}H^{1}}+\|e^{-t}e^{t\Delta}\nabla{w}_{0}\|_{L^{2}_{t}H^{1}}\leq{c}\|w_{0}\|_{H^{1}}.$
(3.12)
Recall that $\sigma(t)=t^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}$. Then we get
$\displaystyle\|{\sigma\nabla}e^{-t}e^{t\Delta}w_{0}\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle=\sup_{t>0}t^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}e^{-t}\|e^{t\Delta}\nabla{w}_{0}\|_{L^{\infty}}$
$\displaystyle\leq\sup_{t>0}t^{\frac{1}{2}}\|e^{t\Delta}\nabla{w}_{0}\|_{L^{\infty}}\\!=\\!\|\nabla{w}_{0}\|_{\dot{B}^{-1}_{\infty,\infty}}$
$\displaystyle\leq{c}\\!\;\|\nabla w_{0}\|_{L^{2}}\leq
c\\!\;\|w_{0}\|_{H^{1}},$ (3.13)
where we have used Proposition 2.4, embedding theorems of Besov spaces (cf.
[30]).
Part II. Estimates for $\|B(u,w)\|_{X}$ and $\|L(u)\|_{Y}$. As for
$\|B(u,v)\|_{X}$, we get
$\displaystyle\|B(u,w)\|_{X}$
$\displaystyle=\|B(u,w)\|_{L^{\infty}_{t}H^{1}}+\|\nabla
B(u,w)\|_{L^{2}_{t}H^{1}}+\|B(u,w)\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq\|\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}_{t}L^{2}}+\|\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}_{t}L^{2}}$
$\displaystyle\;\;+\|\\!\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\nabla\cdot(u\nabla
w)d\tau\|_{L^{2}_{t}L^{2}}\\!+\|\\!\int_{0}^{t}\\!e^{(t-\tau)\Delta}\Delta\\!\nabla\cdot(u\nabla
w)d\tau\|_{L^{2}_{t}L^{2}}$
$\displaystyle\;\;+\|\\!\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle:=I_{1}+I_{2}+I_{3}+I_{4}+I_{5},$
where by applying Lemma 2.3, we have
$\displaystyle I_{1}\leq c\,\|u\nabla
w\|_{L^{2}_{t}L^{2}}\leq{c}\,\|u\|_{L^{\infty}_{t}L^{\infty}}\|\nabla{w}\|_{L^{2}_{t}L^{2}},$
(3.14) $\displaystyle I_{2}\leq c\,\|u\nabla
w\|_{L^{\infty}_{t}L^{2}}\leq{c}\,\|u\|_{L^{\infty}_{t}L^{\infty}}\|\nabla
w\|_{L^{\infty}_{t}L^{2}}$ (3.15)
and by applying Lemma 2.2, Hölder’s inequality and interpolation theorem, we
have
$\displaystyle
I_{3}\leq{c}\,\|u\nabla{w}\|_{L^{2}_{t}L^{2}}\leq{c}\,\|u\|_{L^{\infty}_{t}L^{\infty}}\|\nabla{w}\|_{L^{2}_{t}L^{2}},.$
(3.16)
Similarly, we have
$\displaystyle
I_{4}\leq{c}\|\nabla\cdot({u\nabla{w}})\|_{L^{2}_{t}L^{2}}\leq{c}\hskip
1.13791pt(\|\nabla{u}\cdot\nabla{w}\|_{L^{2}_{t}L^{2}}+\|u\Delta{w}\|_{L^{2}_{t}L^{2}})$
$\displaystyle\;\;\;\leq{c}\hskip
0.56917pt(\|\nabla{u}\|_{L^{4}_{t}L^{4}}\|\nabla{w}\|_{L^{4}_{t}L^{4}}+\|u\|_{L^{\infty}_{t}L^{\infty}}\|\nabla{w}\|_{L^{2}_{t}\dot{H}^{1}})$
$\displaystyle\;\;\;\leq{c}\hskip
0.56917pt(\|{u}\|_{L^{\infty}_{t}H^{1}}^{\frac{1}{2}}\|\nabla{u}\|_{L^{2}_{t}{H}^{1}}^{\frac{1}{2}}\|{w}\|_{L^{\infty}_{t}H^{1}}^{\frac{1}{2}}\|\nabla{w}\|_{L^{2}_{t}{H}^{1}}^{\frac{1}{2}}+\|u\|_{L^{\infty}_{t}\\!L^{\\!\infty}}\|\nabla{w}\|_{L^{2}_{t}{H}^{1}}),$
(3.17)
where $\|f\|_{L^{4}_{t}L^{4}}^{2}\leq c\hskip
0.28436pt\|f\|_{L^{4}_{t}\dot{H}^{\frac{1}{2}}}^{2}\leq{c}\hskip
0.28436pt\|f\|_{L^{\infty}_{t}L^{2}}\|\nabla{f}\|_{L^{2}_{t}{H}^{1}}$. As for
$I_{5}$, by splitting the time interval, we obtain that if $t>2$, then
$1<t-1<\tau<t$,
$1<\frac{1}{{\sigma(\tau)}}=\frac{\sqrt{1+\tau}}{\sqrt{\tau}}<2$ and
$\displaystyle I_{5}$
$\displaystyle=\|\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq\|\\!\int_{0}^{t-1}\\!e^{(t-\tau)\Delta}\nabla\\!\cdot(u\nabla
w)d\tau\|_{L^{\\!\infty}_{t}L^{\\!\infty}}+\|\\!\int_{t-1}^{t}\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\\!\infty}_{t}L^{\\!\infty}}$ $\displaystyle\leq
c\int_{0}^{t-1}\\!\\!(t-\tau)^{-\frac{3}{2}}\|u\nabla{w}\|_{L^{1}}d\tau+c\int_{t-1}^{t}(t-\tau)^{-\frac{1}{2}}\|u\nabla
w\|_{L^{\infty}}d\tau$ $\displaystyle\leq c\,(\|u\nabla
w\|_{L^{\infty}_{t}L^{1}}+\|u\nabla w\|_{L^{\infty}_{t}L^{\infty}})$
$\displaystyle\leq
c\,(\|u\|_{L^{\infty}_{t}L^{2}}\|\nabla{w}\|_{L^{\infty}_{t}L^{2}}+\|u\|_{L^{\infty}_{t}L^{\infty}}\|{\sigma\nabla}{w}\|_{L^{\infty}_{t}L^{\infty}});$
(3.18)
Else if $0<t\leq 2$ and $0<\tau<t$, then we get
$\tau^{\frac{1}{2}}/2<{\sigma(\tau)}<\tau^{\frac{1}{2}}$ and
$\displaystyle I_{5}$
$\displaystyle=\|\int_{0}^{t}\\!e^{(t-\tau)\Delta}\nabla\cdot(u\nabla
w)d\tau\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq\|\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}\frac{1}{{\sigma(\tau)}}{\sigma(\tau)}\|(u\nabla
w)(\tau)\|_{L^{\infty}}d\tau$ $\displaystyle\leq
c\\!\int_{0}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-\frac{1}{2}}\|u\|_{L^{\infty}_{t}L^{\infty}}\|{\sigma\nabla}{w}\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq
c\,\|u\|_{L^{\infty}_{t}L^{\infty}}\|{\sigma\nabla}{w}\|_{L^{\infty}_{t}L^{\infty}}.$
(3.19)
In order to estimate $\|L(u)\|_{Y}$, we have
$\displaystyle\|L(u)\|_{Y}$
$\displaystyle=\|L(u)\|_{L^{\infty}_{t}H^{1}}+\|\nabla{L}(u)\|_{L^{2}_{t}H^{1}}+\|{\sigma\nabla}{w}\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle:$ $\displaystyle=I_{6}+I_{7}+I_{8},$ (3.20)
where by applying Lemma 2.3 (2.7) to $I_{5}$, we have
$\displaystyle I_{6}$
$\displaystyle=\|L(u)\|_{L^{\infty}_{t}H^{1}}=\|\int_{0}^{t}e^{-t+\tau}e^{(t-\tau)\Delta}u\|_{L^{\infty}_{t}H^{1}}$
$\displaystyle\leq\sup_{t>0}\int_{0}^{t}e^{-t+\tau}\|u\|_{H^{1}}d\tau$
$\displaystyle\leq c\|u\|_{L^{\infty}_{t}H^{1}}$ (3.21)
and by applying (2.7) to $\nabla{L(u)}=L(\nabla u)$ and
$\Delta{L}(u)=\nabla{L}(\nabla u)$ with $\theta=0$ and $\theta=1$, we have
$\displaystyle I_{7}$
$\displaystyle=\|\nabla{L}(u)\|_{L^{2}_{t}H^{1}}\leq\|\nabla{L}(u)\|_{L^{2}_{t}L^{2}}+\|\nabla\nabla{L}(u)\|_{L^{2}_{t}L^{2}}$
$\displaystyle\leq\|{L}(\nabla u)\|_{L^{2}_{t}L^{2}}+\|\nabla{L}(\nabla
u)\|_{L^{2}_{t}L^{2}}$ $\displaystyle\leq c\,\|\nabla u\|_{L^{2}_{t}L^{2}}.$
(3.22)
It remains to estimate
$I_{8}=\|{\sigma\nabla}{w}\|_{L^{\infty}_{t}L^{\infty}}$. Recalling the
definition of $w$, we get
$\displaystyle I_{8}$
$\displaystyle=\sup_{t>0}t^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}\|\int_{0}^{t}e^{-t+\tau}e^{(t-\tau)\Delta}\nabla{u}d\tau\|_{L^{\infty}}$
$\displaystyle\leq
c\sup_{t>0}\int_{0}^{t}e^{-t+\tau}(t-\tau)^{-\frac{1}{2}}d\tau\|u\|_{L^{\infty}_{t}L^{\infty}}$
$\displaystyle\leq c\,\|u\|_{L^{\infty}_{t}L^{\infty}}.$ (3.23)
Combining (3.11)–(3.23), we prove (3.10) and hence finish the proof. ∎
Proof of Theorem 1.1: Applying Proposition 3.2, following similar arguments as
in Lemma 2.5 we can prove Theorem 1.2 and hence we omit the details.
Acknowledgmens: Chao Deng is supported by PAPD of Jiangsu Higher Education
Institutions and Jiangsu Normal University under Grant No. 9212112101; He is
also supported by the China Natural Science Foundation under Grant No.
11171357 and No. 11271166; He would like to express his gratitude to the
Applied Mathematics Department of Colorado University at Boulder for its
hospitality. Congming Li is partially supported by the NSF grants DMS-0908097.
## References
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Department of Mathematics, Jiangsu Normal University Email:
deng315@yahoo.com.cn and Department of Applied Mathematics, Colorado
University at Boulder. Congming Li, Department of Applied Mathematics,
Colorado University at Boulder Email: cli@colorado.edu and Department of
Mathematics and MOE-LSC, Shanghai Jiao Tong University.
|
arxiv-papers
| 2012-10-12T04:36:48 |
2024-09-04T02:49:36.445118
|
{
"license": "Public Domain",
"authors": "Chao Deng and Congming Li",
"submitter": "Chao Deng",
"url": "https://arxiv.org/abs/1210.3429"
}
|
1210.3459
|
SF2A 2012
# Know (better) your neighbour:
New Hi structures in Messier 33 unveiled by a multiple peak analysis of high-
resolution 21-cm data
Laurent Chemin LAB, CNRS UMR 5804, Université de Bordeaux, F-33270, Floirac,
France, contact: chemin@obs.u-bordeaux1.fr Claude Carignan Dept. of
Astronomy, University of Cape Town, Rondebosch 7700, South Africa Tyler
Foster Dept. of Physics and Astronomy, Brandon University, Brandon, MB R7A
6A9, Canada Zacharie Sie Kam Dépt. de physique, Université de Montréal,
Montréal, QC H3C 3J7, Canada
###### Abstract
In our quest to constrain the dynamical and structural properties of Local
Group spirals from high-quality interferometric data, we have performed a
neutral hydrogen survey in the direction of Messier 33. Here we present a few
preliminary results from the survey and show the benefits of fitting the Hi
spectra by multiple peaks on constraining the structure of the Messier 33
disk. In particular we report on the discovery of new inner spiral-like and
outer annular structures overlaying with the well-known main Hi disk of
Messier 33. Possible origins of the additional outer annular structure are
presented.
###### keywords:
galaxies: individual (M33, NGC 598) – galaxies: ISM – galaxies: kinematics and
dynamics – galaxies: structure – Local Group – Techniques: imaging
spectroscopy
## 1 Context
The dynamical and structural properties of Hi disks of nearby spirals mainly
result from the analysis of the 0th and 1st moments of Hi spectra obtained
from single-dish and interferometric observations. Curiously more thorough
analyses of Hi spectra making profit from current high spectroscopic precision
and sensitive cm-data remain rare.
In 2006 we have started a Hi survey of the most massive spiral disks from the
Local Group (except the Milky Way) to revisit their structure, kinematics and
dynamics. Aperture synthesis at DRAO combined with short spacing data have
been used to perform 21-cm observations of the Andromeda galaxy (Messier 31)
at spectral resolution $\lesssim 5$ km s-1, angular resolution $\sim 300$ pc
($D\sim 800$ kpc) and sensitivity down to $\sim 2\times 10^{19}$ cm-2 (Chemin
et al. 2009).
Since many spectra are far from being dominated by one single Hi component we
have shown that the moment analysis of datacubes was not appropriate (see Fig.
1 and §3.2 of Chemin et al. 2009, and left panel of Fig. 1 below). This is the
reason why we developed a ‘search and fit’ algorithm of multiple (gaussian)
components. Applied to new Messier 31 data this algorithm has allowed the
detection of sometimes up to five Hi significant components per profile, which
had never been reported beforehand for nearby Hi spirals. So many multiple
peaks likely result from the combination of extreme projection effects of the
warped Messier 31 disk with internal and external dynamical perturbations
(spiral density wave, lagging halo, expanding gas shells, accretion of gas
from the intergalactic medium or from nearby minor companions, etc). The
discovery of outer Hi spurs and spiral arm was also reported, as well as the
characterization of the disk warp in terms of twist and tilt angles and the
measurement of the most extended rotation curve for Messier 31.
We note that this kind of hyperspectral decomposition within multiple gaussian
peaks is not new and has been used several times (e.g. Sicotte & Carignan
1997; Oh et al. 2008). It is nonetheless not generalized in Hi studies. From a
dynamical point-of-view, the multiple peak analysis has led to (marginally)
different rotation velocities and inclinations than those derived with another
recent and high-quality Hi datacube of Messier 31 from the 0th- and 1st-moment
analysis (Braun et al. 2009; Corbelli et al. 2010). Again, such differences
have already been reported (see e.g. Figs. 11 and 13 of Oh et al. 2008).
## 2 Yet another new Hi survey of Messier 33
In pursuit of our project we present here very preliminary results for Messier
33, a late-type spiral whose Hi disk is known to be warped (Corbelli &
Schneider 1997). The 21-cm interferometric data were still obtained at DRAO
(combined with the Arecibo data of Putman et al. 2009) but at a larger
spectral resolution (3.3 km s-1) than for Messier 31 observations. Of course
it is very likely to detect multiple components with highly resolved spectra.
However this does not guarantee the success of detecting realistic ones
because noise becomes important at high resolution. Furthermore the number of
components that can be fitted per spectra depends on the resolution. With more
and more peaks found in an individual spectrum (as for Messier 31), it becomes
less and less straightforward to interpret the data and identify for instance
the component that is the most representative of the disk circular rotation to
those that are caused by all abovementioned perturbing effects. The Hi
datacube of Messier 33 has thus been filtered to lower resolution to simplify
the hyperspectral decomposition.
## 3 Preliminary results: evidence for new Hi structures in Messier 33
Other recent Hi surveys of Messier 33 have been performed at VLA and Arecibo
(Thilker et al. 2002; Putman et al. 2009). The VLA data of Thilker et al.
(2002) have allowed to determine for the first time the inner structure of the
Hi disk with unprecedented details (resolution of 20 pc). The Arecibo data of
Putman et al. (2009) were more appropriate to study the nearby environment of
Messier 33 at a resolution of about 1 kpc. In particular they have shown the
Hi disk of Messier 33 is surrounded by arc-like structures and clumps. A hint
of such perturbations had been presented in another (earlier) Arecibo view of
Messier 33 (Corbelli, Schneider, & Salpeter 1989). Our DRAO survey has thus an
intermediate angular resolution to them.
Working with a datacube of effective spectral resolution of 10 km s-1 our
‘search and fit’ algorithm of multiple peaks identifies sometimes up to 3
significant Hi components in the datacube. An example of two distinct
components is shown in Fig 1 (left-hand panel). Here the components are
separated by $\sim$ 45 km s-1. The total integrated Hi emission of Messier 33
is shown in Fig 1 (central panel). The external arc-lile structure and the SW
clump are clearly detected, even within our $\sim 300$-pc resolution data, as
well as the ‘main’ inner disk. Multiple components are not observed over the
whole field-of-view, as seen in Figure 1 (right panel), but are preferentially
distributed along a ‘secondary’ spiral-like structure in the inner disk and an
annular structure in the outer regions ($r\sim 80^{\prime}$ or 19 kpc). It is
obvious that none of these new structures would have been identified with a
moment analysis of the datacube.
A preliminary tilted-ring model has been fitted to the velocity field of the
‘main’ Hi component shown in the left-hand panel of Fig. 2. A significant
twist of the orientation of the major kinematical axis is evidenced, as well
as a tilt of the Hi disk (Fig. 3). This result thus confirms the warped nature
of the Hi disk of Messier 33. The kinematics of the external arc-like
structure does not differ so much from that of the inner disk, implying that
this perturbation is bound to the disk. We have not yet fitted the warp
parameters for it, as shown by constant inclination and position angles at
those locations ($r>100^{\prime}$, Fig. 3).
Figure 1: Illustration of a Hi profile with two distinct components (left).
Total integrated Hi emission map of Messier 33 (middle). Hi integrated
emission map of the ‘secondary’ fitted Hi component in Messier 33 (right). A
logarithmic stretch is used for them. The ellipse in the bottom-left corner of
the midle panel represents the size of the synthesized beam.
The kinematics of the outer annular structure is shown in the middle panel of
Fig. 2 and its residual field when subtracted from the velocity field of the
‘main’ Hi component in the right-hand panel of Fig. 2. Differences of radial
velocities sometimes reach 40-50 km s-1 in absolute values. At this stage of
our analysis it is too early to firmly identify which of the multiple
components is the real tracer of the ‘main’ disk kinematics to that of the
inner ‘secondary’ spiral-like structure on one hand, and to that of the outer
Hi annulus on another hand. Indeed the disk kinematics is strongly perturbed
in those regions (warp, connection with the external arc-like structure, etc).
It is also too early to constrain the exact origins of the inner ‘secondary’
spiral-like pattern and the outer annular structure. Origins for this later
could be:
Figure 2: Velocity fields of the ‘main’ Hi component of Messier 33 (left) and
of the secondary component (middle). The residual field from their mutual
subtraction is shown in the right-hand panel.
* •
The annular structure has external origins to Messier 33. Gas accretion on the
outer disk parts from e.g. the external arc-like strucure or the intergalactic
medium could be ongoing. Messier 33 has an obvious perturbed environment, and
past tidal interactions with other galaxies may not be excluded. Numerical
simulations would be needed to test those assumptions.
* •
The annular structure is a genuine ring, with internal origins. For instance
it could have been developed by gas accumulation at the outer Lindblad
resonance. In this case an obvious perturbing density wave could be grand-
design spiral structure of Messier 33. This hypothesis could be tested by
measuring the pattern speed of the spiral density wave with a modified version
of the Tremaine-Weinberg method, and by determining the locations of various
Lindblad resonances.
* •
The annular structure is not a real ring but is only caused by a fortuitous
projection effect of a peculiar warping of Messier 33 (and maybe also a disk
flaring) at the periphery of the Hi disk. One would need here gas orbits that
have orientation angles significantly different from the constant one
displayed in Fig. 3 from $r\sim 85^{\prime}$ to generate a distinct structure
in superimposition to the outer disk.
Noteworthy is the fact that insights for asymmetric Hi profiles along a ring-
like structure as caused by the warped gas orbits has been reported in
Corbelli & Schneider (1997). The location of that ring-like structure found by
Corbelli & Schneider (1997) corresponds with that of the external arc-like
structure, but not to that of the outer Hi annulus we evidence here.
Furthermore the Hi annulus does not share the same orientation parameters than
the external arc-like structure (Fig. 1). Two different ring-like structures
thus seem to coexist in the outer regions of Messier 33.
Figure 3: Preliminary results of the tilted-ring model fitted to the ‘main’ Hi
velocity field of Messier 33 from central panel of Fig. 2. Blue squares are
for the disk inclination and green triangles for the position angle of the
major kinematial axis.
## 4 Conclusions
Provisional results from a new Hi survey in the direction of Messier 33
performed with aperture synthesis observations at the Dominion Radio
Astrophysical Observatory have been presented. Evidence for new Hi structures
in Messier 33 have been found from a multiple Hi peak analysis of the
datacube. Among them is the detection of an annular-like structure in the
outer regions of the Hi disk. That annulus does not correspond to the already
known arc-like structure around Messier 33. Complete details of the observing
campaign, the data reduction and the hyperspectral decomposition will be
presented soonly (Chemin et al. 2013). Our main objectives are to revisit the
structure and dynamics of Messier 33, derive an accurate and extended rotation
curve for it, and model its mass distribution. With the results already
obtained for the Andromeda galaxy, this new dataset should help to better
constrain the evolution of massive spirals in the Local Group.
###### Acknowledgements.
We are very grateful to Mary Putman and Nigel Douglas for having provided us
with their single dish data.
## References
* Braun et al. (2009) Braun R., Thilker D. A., Walterbos R. A. M., & Corbelli E, ApJ, 695, 937
* Chemin et al. (2009) Chemin L., Carignan C., & Foster T., 2009, ApJ, 705, 1395
* Chemin et al. (2013) Chemin L., Carignan C., et al., 2013, ApJ, in preparation
* Corbelli, Schneider, & Salpeter (1989) Corbelli E., Schneider S. E., & Salpeter E. E., 1989, AJ, 97, 390
* Corbelli & Schneider (1997) Corbelli E., & Schneider S. E., 1997, ApJ, 479, 244
* Corbelli et al. (2010) Corbelli E., Lorenzoni S., Walterbos R., Braun R., & Thilker D., 2010, A&A, 511, 89
* Oh et al. (2008) Oh S.-H., de Blok W. J. G., Walter F., Brinks E., & Kennicutt R. C. Jr., 2008, AJ, 136, 2761
* Putman et al. (2009) Putman M. E., et al., 2009, 703, 1486
* Sicotte & Carignan (1997) Sicotte V., & Carignan C., 1997, AJ, 113, 609
* Thilker et al. (2002) Thilker D. A., Braun R., & Walterbos R. A. M., 2002, ASPC, 276, 370
|
arxiv-papers
| 2012-10-12T09:23:00 |
2024-09-04T02:49:36.455760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Laurent Chemin (Univ. de Bordeaux, CNRS), Claude Carignan (Univ. of\n Cape Town), Tyler Foster (Brandon Univ.), Zacharie Sie Kam (Univ. de\n Montreal)",
"submitter": "Laurent Chemin",
"url": "https://arxiv.org/abs/1210.3459"
}
|
1210.3487
|
# Topological Magnon Insulator in Insulating Ferromagnet
Lifa Zhang phyzlf@gmail.com Department of Physics and Centre for
Computational Science and Engineering, National University of Singapore,
Singapore 117542, Republic of Singapore Jie Ren renjie@lanl.gov Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Jian-Sheng Wang Department of Physics and Centre for Computational Science
and Engineering, National University of Singapore, Singapore 117542, Republic
of Singapore Baowen Li Department of Physics and Centre for Computational
Science and Engineering, National University of Singapore, Singapore 117542,
Republic of Singapore NUS Graduate School for Integrative Sciences and
Engineering, Singapore 117456, Republic of Singapore NUS-Tongji Center for
Phononics and Thermal Energy Science and Department of Physics, Tongji
University, 200092 Shanghai, PR China
(Mar 27, 2013)
###### Abstract
In the ferromagnetic insulator with the Dzyaloshinskii-Moriya interaction, we
theoretically predict and numerically verify a topological magnon insulator,
where the charge-free magnon is topologically protected for transporting along
the edge/surface while it is insulating in the bulk. The chiral edge states
form a connected loop as a $4\pi$\- or $8\pi$-period Möbius strip in the spin-
wave vector space, showing the nontrivial topology of magnonic bands. Using
the nonequilibrium Green’s function method, we explicitly demonstrate that the
one-way chiral edge transport is indeed topologically protected from defects
or disorders. Moreover, we show that the topological edge state mainly
localizes around edges and leaks into the bulk with oscillatory decay.
Although the chiral edge magnons and energy current prefer to travel along one
edge from the hot region to the cold one, the anomalous transports are
identified in the opposite edge, which reversely flow from the cold region to
the hot one. Our findings could be validated within wide energy ranges in
various magnonic crystals, such as Lu2V2O7.
###### pacs:
85.75.-d, 75.30.Ds, 75.47.-m, 75.70.Ak
## I Introduction
Topological insulator, as a novel state of quantum matter, is characterized by
an insulating bulk band gap and conducting gapless edge/surface states
protected by symmetries hasan10 ; qi11 . It has been theoretically predicted
and experimentally observed in a variety of systems and becomes a hot spot
because of its theoretical importance in condensed matter physics and wide
potential applications in dissipationless spin-based electronics (spintronics)
spin1 . However, due to the fact that the spin transport in topological
insulators is carried by electrons, dissipations can not be really avoided.
Magnon Hall effect, as a consequence of the Dzyaloshinskii-Moriya (DM)
interaction dzya58 ; moriya60 that plays a role of vector potential similar
to the Lorentz force, has been predicted and observed in magnetic insulators
katsura10 ; onose10 ; matsumoto11 . Compared with spin current, where the
dissipation from Joule heating is still inevitable due to the electronic
carriers, the magnon Hall effect is more promising in device applications
because of the long-range coherence of charge-free spin wave hirsch99 ;
murakami03 ; sinova04 . Magnons are collective excitations of localized spins
in a crystal lattice and can be viewed as quantized quasiparticles of spin
waves. Recently, magnon excitation serga04 ; demidov09 , localization
jorzick02 and interference podbielski06 have been experimentally realized.
The technical advancements offer the perspective of various magnonic devices,
and a new discipline – magnonics – has emerged and is growing exponentially
kruglyak06 ; neusser09 ; kruglyak10 ; serga10 ; lenk11 . The charge-free
property of magnon makes it promising to achieve dissipationless transport and
control in insulating magnets.
Figure 1: (color online). (a) Pyrochlore crystallographic structure of the
sublattice of magnetic atoms V of Lu2V2O7. (b) Two tetrahedrons in the
pyrochlore lattice, where DM vectors on bonds 1-3, 1-2, and 2-4 are indicated
by orange arrows. (c) Schematic magnetic flux due to DM interaction in the
[111] plane of the pyrochlore lattice, i.e., a kagome lattice. The coupling of
two sites along the arrows is $(J+iD)S$ while the opposite direction
corresponds to $(J-iD)S$. (d) The quasi-one-dimensional kagome-lattice strip.
The area enclosed by the dotted line can be regarded as a center which is
connected to two semi-infinite leads in equilibrium at temperatures $T_{L}$
(left) and $T_{R}$ (right). The two big arrows schematically depict the
magnitudes and directions of energy flows along the lattice edges when
$T_{L}>T_{R}$. The width of the strip example is $W=5$, which is defined as
the number of atoms in the left column of each unit cell.
Therefore, it will be of great general interest for both theorists and
experimentalists that we find in this work a new intriguing quantum state that
magnon while insulated in the bulk, can nondissipatively transport along
edges/surfaces in the absence of backscattering from defects and disorders due
to the nontrivial topology of magnon’s band structures. We name this novel
state topological magnon insulator (TMI) and believe that due to the robust
dissipationless magnon transport, the TMI in insulating magnets could provide
widely potential applications in nondissipative magnonics and micro-
spintronics.
## II Spin-wave Hamiltonian
The magnon Hall effect was experimentally observed in insulating ferromagnet
Lu2V2O7 onose10 with a pyrochlore lattice, in which the magnetic atom
vanadium has a corner-sharing-tetrahedra sublattice, that is, a stacking of
alternating kagome and triangular lattices along the [111] direction, as shown
in Fig. 1 (a). To study magnon transport in the ferromagnetic insulator, the
Hamiltonian can be written as dzya58 ; moriya60 ; bose94 :
$\mathcal{H}=\sum\limits_{\left\langle{mn}\right\rangle}{[-J\vec{S}_{m}\cdot\vec{S}_{n}+\vec{D}_{mn}\cdot(\vec{S}_{m}\times\vec{S}_{n})]}-g\mu_{B}\vec{H}_{0}\cdot\sum\limits_{n}{\vec{S}_{n}},$
(1)
where $\vec{S}_{n}$ is the spin angular momentum at site $n$; $-J$ denotes the
nearest-neighbor coupling; $\vec{D}_{mn}$ is the DM interaction between site
$m$ and $n$; the last term comes from the Zeeman effect under an external
field $\vec{H}_{0}$.
As shown in Fig. 1 (b), in a single tetrahedron, the DM vector is
perpendicular to the corresponding bond and parallel to the surface of the
surrounding cube onose10 ; elhajal05 ; kotov05 . Since the component of
$\vec{D}_{mn}$ perpendicular to $\vec{z}=\vec{H}_{0}/H_{0}$ does not
contribute to the Hamiltonian (1) up to quadratic order of the deviation of
$\vec{S}$ onose10 , we only retain $D_{mn}=\vec{D}_{mn}\cdot\vec{z}$. When
applying a magnetic field along $\vec{z}=[111]$ direction, all the projections
of the DM interaction between inter-layer sites $m$ and $n$ are zero
($D_{13}=D_{23}=D_{43}=0$); and all the ones for inner-layer sites are nonzero
and equal ($D_{12}=D_{24}=D_{41}=D$). Therefore, with the magnetic field along
$[111]$ direction, the kagome sublattice structure will play a key role for
the presence of TMI effect in Lu2V2O7. In addition, the two-dimensional kagome
lattice sheet could be obtained through doping one-quarter of the sites [e.g.,
site 3 in Figure 1 (b)] of a pyrochlore lattice by nonmagnetic atoms shores05
; olariu08 ; colman08 .
In the following we first discuss a general kagome lattice with DM
interaction; later we will incorporate actual parameters of a thin film of
Lu2V2O7 with a kagome layer of vanadium sublattice. Using the relation of
$S^{x}={1\over{2}}(S^{+}+S^{-})$ and $S^{y}={1\over{2i}}(S^{+}-S^{-})$, we can
rewrite the Hamiltonian (1) on a kagome lattice as
$\displaystyle\mathcal{H}=$ $\displaystyle-$
$\displaystyle\sum_{\langle{mn}\rangle}{\left({{J+iD}\over
2}S_{m}^{-}S_{n}^{+}+{{J-iD}\over 2}S_{m}^{+}S_{n}^{-}\right)}$ (2)
$\displaystyle-$
$\displaystyle\sum_{\langle{mn}\rangle}{JS_{m}^{z}S_{n}^{z}}-h\sum\limits_{n}{S_{n}^{z}},$
where $h=g\mu_{B}H_{0}$. Now, applying the standard Holstein-Primakoff
transformation holstein40 ; zhang08 , one can straightforwardly obtain the
quadratic spin-wave Hamiltonian:
$\mathcal{H}=\sum\limits_{mn}{b_{m}^{+}H_{mn}b_{n}}+E_{0},$ (3)
where $b^{+}$ ($b$) denotes the operator raising (lowering) the spin component
along $\vec{z}$ direction. $E_{0}=-JS^{2}\sum_{n}{M_{n}/2}-NhS$ is the ground-
state energy with $N$ the total number of sites and $M_{n}$ the number of
nearest neighbors of the site $n$. $H_{mn}=H_{nm}^{*}=(J\pm iD)S$ and
$H_{nn}=JSM_{n}+hS$. Figure 1 (c) illustrates the direction of the DM
interaction vector, that is, the coupling between two sites along the
direction of that arrow corresponds to $(J+iD)S$, while the coupling of the
opposite direction corresponds to $(J-iD)S$. Due to the different types of
loops in a unit cell of the kagome lattice, the DM interaction avoids
cancellation thus induces the Hall effect katsura10 . As a consequence, the
preserved DM interaction acts as a vector potential for the propagation of
magnons similar to the magnetic field for the propagation of electrons, which
is crucial for the manifestation of TMI effect. The magnetic field decides the
direction of spins at the ground state, and the induced Zeeman effect term
just shifts the dispersion relation. We set magnetic field $H_{0}=0^{+}$ in
the part of theoretical model calculations, and will input finite $H_{0}$ in
the part for real-material calculations. Except in Sec. VII, dimensionless
units and $S=1/2$, $J=1$ are used without loss of generality.
Figure 2: (a), (b) and (c) The Berry curvature of the three bands at zero DM
interaction; (d), (e) and (f) The Berry curvature of the three bands at
nonzero DM interaction (D=0.2). For all the insets (a)-(f), the horizontal and
vertical axes correspond to wave vector $k_{x}$ and $k_{y}$, respectively; the
unit is $2\pi/a$, where $a$ is the lattice spacing. (g) The Chern numbers of
the three energy bands for the two-dimensional periodic kagome lattice. The
dotted, solid, and dashed lines correspond to Chern numbers of the first, the
second, and the third bands, respectively.
## III Chern numbers of bulk states
The Eq. (3) resembles the tight-binding model, and each unit cell has three
sites. For a two-dimensional periodic kagome spin lattice, we can perform the
Fourier transformation as
$b_{\vec{R}_{l}+\vec{r}_{m}}=\frac{1}{N_{u}}\sum\limits_{\vec{k}}{e^{-i\vec{k}(\vec{R}_{l}+\vec{r}_{m})}}b_{m}(\vec{k}).$
(4)
Here, $N_{u}$ is the number of unit cells. $\vec{R}_{l}+\vec{r}_{m}$ is the
position of the $m$-th site in the $l$-th unit cell. Thus the spin-wave
Hamiltonian can be written in the momentum space.
Figure 3: The dispersion relations of the periodic kagome strip lattices with
different width sizes. The insets (a), (b), (c), and (d) correspond to $W=2$,
$W=5$, $W=20$, and $W=80$, respectively.
Following the standard method to calculate the Berry phase berry84 ; xiao10 ;
zhang11 , we can obtain the Berry curvature of the $n$-th band as:
$B_{k_{x}k_{y}}^{n}=i\sum\limits_{n^{\prime}\neq
n}{\frac{{\varphi_{n}^{{\dagger}}\frac{{\partial H_{\rm{SW}}}}{{\partial
k_{x}}}\varphi_{n^{\prime}}\varphi_{n^{\prime}}^{{\dagger}}\frac{{\partial
H_{\rm{SW}}}}{{\partial k_{y}}}\varphi_{n}-(k_{x}\leftrightarrow
k_{y})}}{{(\varepsilon_{n}-\varepsilon_{n^{\prime}})^{2}}}}.$ (5)
Here $\varepsilon_{n}$ and $\varphi_{n}$ are the eigenvalue and eigenvector of
the spin-wave Hamiltonian. The associated topological Chern number is obtained
through integrating the Berry curvature over the first Brillouin zone as
$C^{n}=\frac{1}{{2\pi}}\int_{BZ}{dk_{x}dk_{y}B_{k_{x}k_{y}}^{n}}.$ (6)
If the DM interaction is zero, the Berry curvatures of the three bands are
shown in Fig. 2(a), (b) and (c): the maximum points have the opposite values;
the sum of the Berry curvatures are zero, that is, the Chern numbers are zero
at zero DM interaction as shown in Fig. 2(g). Therefore, the magnon Hall
effect and topological magnon insulator effect is absent. If the DM
interaction is nonzero, the Berry curvatures change dramatically and can not
cancel each other. As shown in Fig. 2(d), in the whole momentum space, the
Berry curvature of the first band is positive, which corresponds to the Chern
number with the value of 1 as shown in Fig. 2(g). And the Berry curvatures of
the third band shown in Fig. 2(f) also can not cancel each other and the Chern
number is $-1$. The Berry curvature of the second band also changes, but the
Chern number keeps zero. The topological magnon insulator is only possible
when the DM interaction is nonzero.
## IV Finite size effect for the dispersion relation of quasi-1D kagome
lattice
According to the spin wave Hamiltonian Eq. (3), we can calculate the
dispersion relation ($\varepsilon$ vs $k_{x}$) of the quasi-1D periodic
lattice, as shown in the Fig. 1 (d). In this figure, the left most column has
5 sites, thus we denote the width as $W=5$, in each unit cell of the quasi-1D
kagome lattice there are $6W-1$ sites. As shown in Fig. 3, with increasing
strip width, more modes appear in the energy bands; the edge states will be
gradually fixed and independent of size hatsugai93a . If the width $W\geq 20$,
we find that the states in the bulk gap, that is, the edge states tend to be
fixed, and in the bulk energy bands there are more and more branches. From
$W=20$ to $W=80$, the edge states almost have no changes.
Figure 4: The energy differences vs the width of the periodic kagome strip at
the anti-crossing points. The solid, dashed, and dotted lines correspond to
energy differences at anti-crossing points A, B, and C, respectively in Fig. 3
(d).
In the bulk energy gaps, we find the edge states have the trend to touch each
other at the points $A,B,C$ as shown in Fig. 3 (d). The energy difference of
the corresponding edge states at the points $A,B,C$ is shown in Fig. 4. As the
width increases, the energy difference decreases exponentially. If the system
width is finite, the states in two edges have nearly equal energy and momentum
near the anticrossing points $A,B,C$; thus they can couple together to open an
energy gap which decays exponentially with width increasing zhou08 . When the
strip width increases to infinity, the edges are separated too far to interact
with each other; thus they could have degeneracy in the dispersion relation.
As shown in Fig. 4, the gap at point $A$ decays faster than that at $B$ and
$C$, thus we can find the crossing at $A$ in the upper gap earlier than that
in the lower gap. And after $W\geq 20$, the energy differences at all the
three points $A,B,C$ are very small, therefore it is very reasonable that we
use $W=80$ in all the following numerical calculations to study the chiral
edge state transport in the quasi-1D lattice with large-enough width.
## V Topological magnon edge modes
Figure 5: (color online). The dispersion relation of chiral magnonic edge
modes with non-zero DM interactions. (a) and (b) are dispersion relations in
range of $k_{x}\in[0,8\pi/a]$ for $D/J=0.1$ and 0.4, respectively. (c) A
conventional cylinder strip with two boundaries, of which each has a period of
$2\pi$. (d) A Möbius strip which only has one boundary of $4\pi$ period. (e) A
looped Möbius strip which only has one boundary of $8\pi$ period.
Because of the DM interaction, two edge states within both energy gaps are
twisted so that for each state
$\varepsilon(\pi/a-k_{x})\neq\varepsilon(\pi/a+k_{x})$ and they cross at
$k_{x}=\pi/a$ in the first Brillouin zone, where $a$ is the lattice spacing.
As shown in Fig. 5(a) for the case of $D/J=0.1$, in the upper bulk band gap,
the two edge modes form a continuous state with a period of $4\pi$, which can
not be disturbed to open a gap by weak disorders so that edge modes are
topologically protected. However, when DM interaction is zero, the two edge
states are easy to be perturbed to separate and open a gap, because they do
not cross each other although they degenerate. In the lower bulk band gap, we
find that four edge states will contribute to magnon transport within the
energy gap. When $D/J\approx 0.4$ or larger, in both energy gaps there are
four edge states [see Fig. 5(b)]. In a period of $8\pi$, any two of the four
edge states have degeneracies and cross each other at different points in the
momentum space. All the four edge states form a continuous state with a period
of $8\pi$ to transport magnons along two edges of the strip.
We can understand the topology of the edge states as follows. With zero DM
interaction, the edge states are similar to the two boundaries of the
conventional cylindric strip as shown in Fig. 5 (c), both of which have a
period of $2\pi$ in Brillouin zone and transport separately along two edges.
Due to the nonzero DM interaction, two edge states are twisted so that they
cross each other and go into the other energy band after $2\pi$ in momentum
space, thus form a closed loop with a period of $4\pi$. These edge states are
similar to the one-sided Möbius strip with only one boundary, as shown in Fig.
5 (d), where a line drawn starting from a point at the boundary will meet back
at the “other side” after a circle of $2\pi$, then go back to the original
point after a whole period of $4\pi$. The two edge states form one closed loop
winding the bulk energy gap between the two bands, which are thus
topologically protected from distortions. At larger DM interaction, four edge
states contribute to the transport in the bulk gap, cross each other, and
connect to form a closed $8\pi$-period loop which can be interpreted as the
only one boundary of a looped Möbius strip as shown in Fig. 5 (e). This looped
Möbius strip also has only one boundary winding around the strip surface,
thereby having the same topology as that of the conventional Möbius strip
shown in Fig. 5 (d).
The topological chiral edge state is related to the band topology
characterized by Chern numbers of the bulk states hatsugai93a ; qi06 ;
scarola07 ; yao09 ; zhang11 , as shown in Fig. 2(b). Since there are three
sites in each unit cell, the two-dimensional infinite kagome lattice with
Hamiltonian (3) has three bands. When the DM interaction is absent, all the
Chern numbers of three bands are zero so that there is no TMI effect.
Accordingly, the winding numbers of edge states are zero as well, thus they
are not topologically protected. With nonzero DM interaction, the Chern
numbers of the lowest and highest energy bands become $\pm 1$ that indicate
the nontrivial topology; the one of the middle energy band is still zero.
According to the relation between the Chern number and the winding number
hatsugai93a , the winding numbers of edge states in both bulk gaps have the
same value of 1 or $-1$, which is consistent with the only one closed loop
winding the bulk gap regardless of the period of $4\pi$, $8\pi$ or others.
Figure 6: (color online). The local energy current and density of state for
edge magnon transport at equilibrium. (a) The uniform kagome lattice. (b) The
lattice with a defect at the upmost site of the left fifth column. The red
arrows, the blue dots, and the small black dots correspond to the local energy
current, the local density of magnon, and atom sites, respectively. The color
of the arrows and dots indicate the magnitude of the local current and density
of states, respectively . Parameters are $\varepsilon=1.5$, $T_{L}=T_{R}=1.0$,
$D/J=0.1$, and $a=1$.
## VI Topological magnon transport: the NEGF method
To intuitively illustrate the topological magnon transport carried by chiral
edge states, we choose some unit cells of the kagome lattice strip as a center
region and set the rest as two semi-infinite leads in equilibrium at
temperatures $T_{L}$ and $T_{R}$, respectively [see Fig. 1 (d)]. We then apply
the nonequilibrium Green’s function method haug96 to calculate the local
density of magnons and the local energy current density of magnons. For the
nonequilibrium magnon transport in such system, the Hamiltonian can be written
as follows
$\mathcal{H}=\sum\limits{\mathcal{H}_{\alpha}}+\Bigl{(}\sum\limits_{lm}({b_{l}^{L+}H_{lm}^{LC}b_{m}^{C}+b_{m}^{C+}H_{ml}^{CR}b_{l}^{R}})+{\rm
h.c.}\Bigr{)},$ (7)
where
$\mathcal{H}_{\alpha}=\sum\limits_{lm}{b_{l}^{\alpha+}H_{lm}^{\alpha}b_{m}^{\alpha}},\alpha=L,C,R$,
here ‘$L,C,R$’ denote the left lead, the center part and the right lead,
respectively. The Hamiltonian matrix of the full system is
$H=\left({\begin{array}[]{*{20}c}{H_{L}}&{H_{LC}}&0\\\
{H_{CL}}&{H_{C}}&{H_{CR}}\\\ 0&{H_{RC}}&{H_{R}}\\\ \end{array}}\right).$ (8)
The retarded Green’s function is defined as
$G^{r}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle[b(t),b^{+}(t^{\prime})]\rangle,$
(9)
where we set $\hbar=1$ for notational simplicity. In nonequilibrium steady
states, the Green’s function is time-translationally invariant so that it
depends only on the difference in time. Thus, the Fourier transform of
$G^{r}(t-t^{\prime})=G^{r}(t,t^{\prime})$ is obtained as
$G^{r}[\varepsilon]=\int_{-\infty}^{+\infty}G^{r}(t)e^{i\varepsilon t}dt.$
(10)
Without interaction, the free Green’s functions for three parts in equilibrium
can be written as:
$\begin{array}[]{l}((\varepsilon+i\eta)-H_{\alpha})g_{\alpha}^{r}[\varepsilon]=I,\quad\alpha=L,C,R,\\\
g_{\alpha}^{a}[\varepsilon]=g_{\alpha}^{r}[\varepsilon]^{\dagger}.\end{array}$
(11)
And there is an additional equation relating $g^{r}$ and $g^{<}$:
$g_{\alpha}^{<}[\varepsilon]=f_{\alpha}(\varepsilon)[g_{\alpha}^{a}[\varepsilon]-g_{\alpha}^{r}[\varepsilon]],$
(12)
where $f_{\alpha}(\varepsilon)=\langle
b^{+}b\rangle=[e^{\varepsilon/T_{\alpha}}-1]^{-1}$ is the Bose-Einstein
distribution function at the ${\alpha}$ part with temperature $T_{a}$; we have
set $k_{B}=1$.
For the quadratic Hamiltonian, the magnon transport is ballistic. The lesser
Green’s function can be solved as
$G^{<}[\varepsilon]=G^{r}[\varepsilon]\Sigma^{<}[\varepsilon]G^{a}[\varepsilon]$
(13)
where $G^{a}=(G^{r})^{\dagger}$ and the self energy
$\Sigma^{r,<}[\varepsilon]=H_{CL}g_{L}^{r,<}[\varepsilon]H_{LC}+H_{CR}g_{R}^{r,<}[\varepsilon]H_{RC}.$
(14)
The retarded Green’s function has the same form as for the electron case
$G^{r}[\varepsilon]=\Bigl{[}\varepsilon+i\eta-
H_{C}-\Sigma^{r}[\varepsilon]]^{-1}.$ (15)
The local density of magnon is given by zhang07
$\rho_{n}={{i\hbar G_{nn}^{<}(\varepsilon)}\over{\pi a}}.$ (16)
The local energy current is given by jing09
$\mathfrak{j}_{mn}(\varepsilon)={\varepsilon\over{2\pi}}{\mathop{\rm
Re}\nolimits}[G_{mn}^{<}[\varepsilon]H_{nm}-G_{nm}^{<}[\varepsilon]H_{mn}].$
(17)
At the interface between the left lead and the center part, it reads
$\mathfrak{j}_{mn}(\varepsilon)={\varepsilon\over{2\pi}}{\mathop{\rm
Re}\nolimits}[G_{mn}^{CL,<}[\varepsilon]H_{nm}^{LC}-G_{nm}^{LC,<}[\varepsilon]H_{mn}^{CL}].$
(18)
Taking the trace to sum over all the local current in the interface, and
integrating over all the energy, we then get the Landauer-like formula as
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\sum\limits{j_{mn}}}d\varepsilon$ (19)
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\varepsilon{\varepsilon\over{2\pi}}{\rm{Tr}}\bigl{\\{}{\mathop{\rm
Re}\nolimits}\bigl{(}G^{CL,<}[\varepsilon]H^{LC}-G^{LC,<}[\varepsilon]H^{CL}\bigr{)}\bigr{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{{2\pi}}\int_{0}^{\infty}{\varepsilon\;[f_{L}(\varepsilon)-f_{R}(\varepsilon)]T[\varepsilon]}d\varepsilon$
where the transmission is
$T[\varepsilon]={\rm{Tr}}\bigl{\\{}G^{r}[\varepsilon]\Gamma_{L}[\varepsilon]G^{a}[\varepsilon]\Gamma_{R}[\varepsilon]\bigr{\\}}.$
(20)
with the $\Gamma_{\alpha}$ functions given by
$\Gamma_{\alpha}=i(\Sigma_{\alpha}^{r}-\Sigma_{\alpha}^{a}).$
Figure 7: (a) Local energy current vs the coordinate along the $y$ direction.
The solid and dashed lines correspond to the local currents in two different
columns in one unit cell. (b) Local density of states of magnons vs the
coordinate along the $y$ direction. The solid, dashed, and dotted lines
correspond to the local density of states in three different columns in one
unit cell. The energy of magnon is $\varepsilon=1.5$ in the lower bulk band
gap.
Based on the formula Eq. (17), we can calculate the equilibrium or
nonequilibrium magnon transport in the lattice. Fig. 6 shows the edge state
magnon transport in the bulk gap at a fixed magnon energy $\varepsilon=1.5$ in
the thermal equilibrium. The forward (left-to-right) thermal current carried
by magnons travels along one edge, and the backward (right-to-left) current
with the same magnitude transports along the other one, as shown in Fig. 6
(a). Near both edges the local magnon currents form many chiral vortices due
to the nonzero DM interaction. Moreover, both the current and the magnon
density of states are symmetrically localized at two edges. We plot the local
current and the local density of states for the edge mode with
$\varepsilon=1.5$ in Fig. 7. We find both the local current and the local
density of states decay exponentially from the edge to the center with some
oscillations. The oscillations come from the vortex of the energy current in
the kagome lattice. Thus the magnon with energy $\varepsilon=1.5$ in the lower
bulk band gap indeed localizes at the two edges of the quasi-1D lattice. The
magnon with other energies in the bulk gaps also has the similar picture.
When a defect is present at one edge, the current take a detour around it and
transports ahead without backscattering, as illustrated in Fig. 6 (b).
Although the defect dramatically affects the local density of magnons and
destroys the local current vortex, the global currents along two edges keep
intact compared to those of the uniform lattice, and their summation keeps
zero since the net transport vanishes at equilibrium.
Figure 8: (color online). The local energy current and density of state for
edge magnon transport at nonequilibrium. The red arrows, the blue dots, and
the small black dots correspond to the local energy current, the local density
of magnon, and atom sites, respectively. The color of the arrows and dots
indicate the magnitude of the local current and density of states,
respectively. The uniform kagome lattice are with (a) $T_{L}=1.2,T_{R}=0.8$,
and (b) $T_{L}=0.8,T_{R}=1.2$. The lattice with a defect at the upmost site of
the left fifth column are with (c) $T_{L}=1.2,T_{R}=0.8$, and (d)
$T_{L}=0.8,T_{R}=1.2$. Other parameters are $\varepsilon=1.5$, $D/J=0.1$,
$a=1$, $W=80$.
As shown in Fig. 8 (a) and (b), when two leads are held at different
temperatures, the magnons and energy current prefer to flow along one edge
from the hot lead to the cold one. The transport around the other edge however
shows an interesting anomalous behavior that the magnons and energy reversely
flow from the cold lead to the hot one. Nevertheless, we note that this does
not violate the second law of thermodynamics because the forward (hot-to-cold)
energy current transported along one edge is larger than the backward one
(cold-to-hot) along the other edge so that the total transport is still from
the hot part to the cold one. If we only swap two temperatures
($T_{L}\leftrightarrow T_{R}$), the transport will prefer the other edge but
with the directions of local edge currents unchanged. If we merely reverse the
DM interaction ($D\rightarrow-D$), the transport will change to prefer the
other edge with the local currents reversed but with the total average current
unchanged. If we swap both the temperatures and the DM interaction, the local
currents will just reverse their directions but keep the same magnitudes,
which is a consequence of the time-reversal invariance.
It is worthy to notice that for the chiral edge state, although both the
current and the magnon density of states mainly localize around two edges,
they leak into the bulk with oscillatory decay. The oscillatory motion results
from the quantum interference due to the edge boundaries, which is similar to
the properties of the localized edge phonon modes jiang09 and electron
transport in graphene qiao10 . This phenomenon indicates that even for the
topological chiral edge state, the transport within the bulk of a topological
insulator can not be really avoided. A topological insulator is not a perfect
“insulator”, not only referring to the edges/surfaces, but also for the bulk.
When a defect is present around one edge, the one-way edge current in the TMI
is able to take a detour around it and transport ahead without backscattering,
see Fig. 8 (c) and (d). Although the defect dramatically affects the local
density of magnons and the vortex pattern of local currents, the global
currents along two edges keep intact compared to those of the uniform lattice.
This means that the chiral edge state in the bulk gap is indeed topologically
protected from the lattice defect or weak disorders.
## VII Thin film of Lu2V2O7
Figure 9: Two tetrahedrons in the pyrochlore lattice of the atom vanadium of
the ferromagnet Lu2V2O7. On a single tetrahedron, all the DM vectors on bonds
1-2, 2-4, 4-1, 1-3, 2-3, and 4-3 are shown by the arrows.
In the insulating ferromagnet Lu2V2O7, the orbitals of the $d$ electrons are
ordered to point to the center of mass of the vanadium tetrahedron and a
virtual hopping process stabilizes the ferromagnetic order of the vanadium
spin in this orbital-ordered state onose10 ; ichikawa05 . The vanadium
sublattice in Lu2V2O7 has a pyrochlore structure composed of corner-sharing
tetrahedra, that is, a stacking of alternating kagome and triangular lattices
along the [111] direction. Considering the strong constraint of the crystal
symmetry and using Moriya’s rules moriya60 , possible DM (Dzyaloshinskii-
Moriya) interactions on a single tetrahedron can be determined as onose10
$\vec{D}_{12}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{y}-\hat{e}_{z}),\,\vec{D}_{24}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}-\hat{e}_{y}),\,\vec{D}_{41}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}-\hat{e}_{z}),\,\vec{D}_{13}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{x}+\hat{e}_{y}),\,\vec{D}_{23}={D_{0}\over{\sqrt{2}}}(+\hat{e}_{x}-\hat{e}_{z}),\,\vec{D}_{43}={D_{0}\over{\sqrt{2}}}(-\hat{e}_{y}+\hat{e}_{z})$.
Here $D_{0}$ denotes the strength of the DM interaction; the number 1,2,3, and
4 denote the site in a single tetrahedron. If we apply a magnetic field
$\vec{H}_{0}$, then all the spin angular momentum in the direction along
$\vec{l}=\vec{H}_{0}/H_{0}$, with $H_{0}$ the magnitude of $\vec{H}_{0}$. We
know the component of the DM vector perpendicular to $\vec{l}$ does not
contribute to the spin-wave Hamiltonian, thus we only retain the projections
of the DM interaction along $\vec{l}$ direction, i.e.,
$D_{mn}^{l}=\vec{D}_{mn}\cdot\vec{l}$. If we apply a magnetic field along
$\vec{l}=[111]$ direction, then $D_{13}^{l}=D_{23}^{l}=D_{43}^{l}=0$ and
$D_{12}^{l}=D_{24}^{l}=D_{41}^{l}=\frac{-\sqrt{2}}{\sqrt{3}}D_{0}$ explanD .
Therefore, if the magnetic field is applied along $[111]$ direction, the
magnon Hall effect and topological magnon insulator effect only come from the
noncancellation of different types of DM interaction loops in the unit cell of
the kagome lattice. The effective DM interactions between inter-layer sites
have no contributions. According to the experimental observation in Ref.
onose10 , the DM interaction is obtained $D_{0}/J=0.32$, thus we use
$D={-\sqrt{2}\over{\sqrt{3}}}D_{0}={-\sqrt{2}\over{\sqrt{3}}}\cdot
0.32J=-0.26J$ for the DM interaction of the thin film of Lu2V2O7 with kagome
layer. Since in Ref. onose10 , we have $JS=8D_{s}/a_{0}^{2}$ with
$a_{0}=9.94{\rm\AA}$ the spacing between unit cells of the pyrochlore
structure and $D_{s}=21{\rm meV}{\rm\AA}^{2}$ the spin stiffness constant,
then we get the coupling $J=3.4$ meV and $a=\frac{\sqrt{2}}{2}a_{0}=7.03$ Å.
Based on these parameters, we calculate the dispersion relations for the
quasi-1D kagome lattice, as shown in Fig. 10 (b).
Figure 10: (color online). (a) The current density vs energy of magnon for
uniform and edge-defect kagome lattices with the parameters of Lu2V2O7. The
solid and dotted lines correspond to the energy current in the bulk band gaps
for uniform and edge-defect lattices, respectively. (b) The dispersion
relation of the kagome lattice with the parameters of Lu2V2O7. $J=3.4$ meV,
$D/J=-0.26$, $H_{0}=1$ T, $T_{L}=21$ K, and $T_{R}=19$ K.
As shown in Fig. 10 (a), the energy current of magnon is not affected by
defect or disorder in the range of $[4.45,5.98]\cup[8.79,10.31]$ meV. These
energy intervals coincide with the bulk gaps in the magnon spectrum where the
topological magnonic edge states can be identified [see Fig. 10 (b)]. Although
certain distortions of edge states will occur as results of the defect or
disorder, the total energy current carried by edge magnons does not change in
the whole bulk energy gap. This indicates that the defect or disorder does not
open a gap in the magnon spectrum so that the topology of the chiral magnon
edge state is robust. According to the energy ranges, the topological magnon
states have frequencies within $[1.08,1.45]\cup[2.13,2.49]$ THz.
Applying different external magnetic fields will not change the main
properties of magnons, but shift the corresponding dispersion relations, so
that the frequency of topological edge magnons can be tuned flexibly with a
wide range. Also, when the inter-layer exchange couplings are considered, they
only play the role of effective on-site potentials, which just shift the whole
bands and leave the main band structural properties unchanged. In addition,
the two-dimensional kagome lattice sheet could be obtained by doping with
nonmagnetic atoms as we mentioned before so that the inter-layer exchange
couplings are ignorable. Therefore, we expect that one can observe the TMI for
the thin film of Lu2V2O7 in a wide energy range of magnons. Our findings about
the TMI could also be applied for other magnetic crystals, including even
antiferromagnetic materials where the existence of magnons is possible.
## VIII Possible Experiments
To realize magnonic devices as well as the predicted TMI, the excitation and
detection of magnons is the major challenge. Recent years have witnessed a
fast development in experimental techniques such as ferromagnetic resonance
podbielski06 , pulse-inductive microwave magnetometer silva99 , time-resolved
scanning Kerr microscopy barman03 , optical pump-probe techniques kimel07 , as
well as Brillouin light scattering (BLS) perzlmaier05 which takes a special
role since it allows the direct measure of dispersions and band structures. We
could observe the topological edge modes by using these techniques to measure
the magnon dispersion relation and could also verify the TMI by detecting the
magnon transport in the bulk band gap where magnons could be selectively
excited by non-thermal optical pulses kimel04 ; bigot09 ; zhangy12 ; kamp11 ;
nish12 or induced by external spin-polarized current kiselev03 , so that we
can avoid the thermal transport from bulk states but extract the one only from
edge channels. We hope our theoretical predictions about TMI could open a new
window into the application of nondissipative magnon transport, especially for
the novel magnonic device design, which could also shed light on the
information technology based on magnonics, and micro-spintronics.
_Note added:_ Recently, we have learned of a submission note thanks to its
authors, studying the similar topological chiral magnonic edge mode.
Differently, their results are based on a linearized Landau-Lifshitz equation
and account for the dipolar interaction instead of the DM one studied here.
###### Acknowledgements.
L. Z. and B. L. are supported by the grant R-144-000-285-646 from Ministry of
Education of Republic of Singapore. J. R. acknowledges the support of the U.S.
Department of Energy through the LANL/LDRD Program. J.-S. W. acknowledges
support from a URC research grant R-144-000-257-112 of NUS.
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|
arxiv-papers
| 2012-10-12T12:11:25 |
2024-09-04T02:49:36.463631
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lifa Zhang, Jie Ren, Jian-Sheng Wang, and Baowen Li",
"submitter": "Lifa Zhang",
"url": "https://arxiv.org/abs/1210.3487"
}
|
1210.3527
|
# One lump or two?††thanks: This preprint first appeared on Nature Precedings
on 1 December 2009 [doi:10.1038/npre.2009.4033.1].
Kieran Smallbone
_Manchester Centre for Integrative Systems Biology_
_131 Princess Street, Manchester M1 7DN, UK_
kieran.smallbone@manchester.ac.uk
###### Abstract
We investigate methods for modelling metabolism within populations of cells.
Typically one represents the interaction of a cloned population of cells with
their environment as though it were one large cell. The question is as to
whether any dynamics are lost by this assumption, and as to whether it might
be more appropriate to instead model each cell individually. We show that it
is sufficient to model at an intermediate level of granularity, representing
the population as two interacting lumps of tissue.
## 1 Introduction
The emerging field of systems biology seeks to reconcile subcellular-level
components (such as enzymatic reactions) with cellular- and organism-level
behaviour (such as metabolism). Non-linear processes dominate these
interactions; experience from other areas of science has taught us that
mathematical models, continuously revised by new information, must be used to
describe and interpret complex biological phenomena [1, 2].
As systems biology grows, so we see a proliferation of mathematical models of
cell metabolism and signalling – see the many examples at the model
repositories BioModels.net [3] and CellML.org [4]. Given the inherent
difficulties in in performing single cell experiments, one property held in
common by many of these models is the assumption of “lumped dynamics”. To
explain this term, consider a typical scenario in which a million S.
cerevisiae are grown in a chemostat. Experiments are performed to measure
average metabolite concentrations over the population of yeast cells. A
mathematical model of metabolism is then built in which the cell has these
average characteristics, but a volume equivalent to a million cells (see Fig.
1).
Given the identical metabolic characteristics of each clonal cell, it would
seem natural to approximate the system by lumping the population as a single
mass. Intuition would suggest that dynamics are unchanged but, as we shall see
below, this linear, verbal reasoning approach is incorrect. However, we show
that it is not necessary to consider each individual cell – which would lead
to a million times as many ODEs – rather correct dynamics can be captured by
considering two interacting lumps of cells.
(a)
(b)
Figure 1: Modelling at different scales. See Eqs. (1)–(2) for a mathematical
representation, where $y$ denotes extracellular and $x$ intracellular
concentrations. Typically, (a) one models the cell population as one bulked
compartment; at the other end of the granularity scale, (b) one could consider
each of the $n$ cells individually, which would lead to approximately $n$
times as many ODEs.
## 2 A theorem
We frame the problem mathematically. Let $x_{i}$ denote a set of metabolite
concentrations within cell $i$, and $y$ a set of external concentrations (see
Fig. 1). Assuming each cell has identical characteristics, we may write
$\displaystyle x_{i}^{\prime}$ $\displaystyle=$ $\displaystyle
f(x_{i},y)\qquad\qquad i=1,\ldots,n$ (1) $\displaystyle y^{\prime}$
$\displaystyle=$ $\displaystyle g(y)-\frac{1}{n}\sum_{i}h(x_{i},y)$ (2)
Here $f$ denotes intracellular reactions, $h$ transport into cells and $g$ the
rate of metabolite supply.
Linearise about a steady-state $x_{i}=x^{\ast}$, $y=y^{\ast}$ to give
stability matrix
$A_{n}=\left(\begin{array}[]{c c c c | c}f_{x}&0&\cdots&0&f_{y}\\\ 0&\ddots&&&\vdots\\\ \vdots&&\ddots&&\vdots\\\ 0&&&f_{x}&f_{y}\\\ \hline\cr-\frac{1}{n}h_{x}&\cdots&\cdots&-\frac{1}{n}h_{x}&g_{y}-h_{y}\end{array}\right)$ (3)
We propose that
$\lambda(A_{1})\subseteq\lambda(A_{2})=\lambda(A_{3})=\ldots$ (4)
where $\lambda$ denotes the spectrum. That is, the system bulked into two
compartments has the same eigenvalues as the system with three compartments,
but more than the system with one compartment.
To show $\lambda(A_{n})\subseteq\lambda(A_{n+1})$, let
$v_{n}=(x_{1},\ldots,x_{n}|y)^{\prime}$ and suppose $A_{n}v_{n}=\lambda
v_{n}$. Taking
$v_{n+1}=\left(\left.x_{1},\ldots,x_{n},\frac{1}{n}\sum
x_{i}\right|y\right)^{\prime}$ (5)
we find $A_{n+1}v_{n+1}=\lambda v_{n+1}$.
Now suppose $u_{n+1}=(x_{1},\ldots,x_{n+1}|y)^{\prime}$ and suppose
$A_{n+1}u_{n+1}=\lambda u_{n+1}$. Taking
$u_{n}=\left(\left.\frac{nx_{1}+x_{n+1}}{n+1},\ldots,\frac{nx_{n}+x_{n+1}}{n+1}\right|y\right)^{\prime}$
(6)
we find $A_{n}u_{n}=\lambda u_{n}$.
Finally, we must consider the possibility that $u_{n}=0$, i.e.
$x_{i}=-x_{n+1}/n$ $\forall i$. If $n\geq 2$, this may be overcome by first
creating a new eigenvector
$u_{n+1}^{\prime}=(x_{n+1},x_{2},\ldots,x_{n},x_{1}|y)$ by swapping two
elements, then constructing $u_{n}$ as above. Thus we may conclude
$\lambda(A_{n})\supseteq\lambda(A_{n+1})$ for $n\geq 2$ as required.
The practical implication of the above theorem is that, the dynamic behaviour
(or at least the linear dynamic behaviour) of a full system of cells may be
captured by bulking the cells into two compartments. If cells are instead
bulked as one, some behaviour will be lost.
Moving back to specifics, we may construct the two sets of eigenvectors
associated with the system. If $u_{1}=(x|y)^{\prime}$ is an eigenvector of
$A_{1}$, then $u_{n}=(x,\ldots,x|y)^{\prime}$ is the corresponding eigenvector
of $A_{n}$. If $v=x$ is an eigenvector of $f_{x}$, then
$v_{n}=(x,0,\ldots,0,-x,0,\ldots,0|0)$ are the corresponding eigenvectors of
$A_{n}$.
## 3 An example
From a stability perspective, the system
$x^{\prime}=f(x,y^{\ast})$ (7)
may be naturally unstable at $x=x^{\ast}$, but this instability may be masked
in the model through tight control in $y$ – leading to the eigenvalues of
$A_{1}$ all having negative real part. However, if the cells are not bulked as
one, but rather as two (or more) compartments, the feedback exposes the
realities of the system as $A_{n}$ now inherits positive real part eigenvalues
from $f_{x}$.
For example, the Brusselator is a model proposed in 1968 for an autocatalytic,
oscillating chemical reaction [5]. In dimensionless form, dynamics may be
written as
$\displaystyle u^{\prime}$ $\displaystyle=$ $\displaystyle 1-(b+1)u+au^{2}v$
(8) $\displaystyle v^{\prime}$ $\displaystyle=$ $\displaystyle bu-au^{2}v$ (9)
Its steady-state is given by $(u,v)=(1,b/a)$ and if $b>a+1$ there exists a
globally-stable limit-cycle (see Fig. 2).
Figure 2: (From Eqs. (8)–(9)). Stable limit cycle of the Brusselator.
Parameter values used are $a=1$, $b=3$, $u(0)=1.01$ and $v(0)=b/a$.
This model may be transformed by setting $x=(u,v)$ and letting $y=b$ now be a
variable representing the externally-supplied nutrient (similar results may be
obtained by setting $y=a$).
$\displaystyle u_{i}^{\prime}$ $\displaystyle=$ $\displaystyle
1-(b+1)u_{i}+au_{i}^{2}v_{i}$ (10) $\displaystyle v_{i}^{\prime}$
$\displaystyle=$ $\displaystyle bu_{i}-au_{i}^{2}v_{i}$ (11) $\displaystyle
b^{\prime}$ $\displaystyle=$ $\displaystyle
g-\frac{1}{n}\sum_{i}\left(h_{1}u_{i}+h_{2}v_{i}+h_{3}b\right)$ (12)
For certain parameter values, control on $b$ will seem to stabilise the system
(n=1). (see Fig. 3 (a)). However, when the bulked cells are split, the
underlying oscillations return (b). Similar dynamics are observed when
comparing $n=2$ and $n=3$ (c).
(a)
(b)
(c)
Figure 3: (From Eqs. (10)–(12)). (a) $n=1$: steady state stabilisation.
Parameter values used are as in Fig. 2, with $g=2$, $h_{1}=-4$, $h_{2}=0$,
$h_{3}=2$ and $b(0)=3$. (b) $n=2$: stable limit cycle obtained by dividing
populations. Parameter values used are as before, with initial conditions
$u_{1}(0)=1.01$, $u_{2}(0)=0.99$. (c) $n=3$: initial conditions
$u_{1}(0)=1.01$, $u_{2}(0)=1$ and $u_{3}(0)=0.99$.
## 4 Discussion
Returning to Fig. 1, we see the two scales of granularity typically used in
metabolic modelling. Typically one represents a population of cells as a
single compartment, rather than considering the dynamics of n individual
cells. The reasons for this are not clear. It may be that it is assumed that a
population of clonal cells would behave in the same way as this.
Alternatively, it may be assumed that in order to capture the interactive
dynamics, around n times as many differential equations would be required.
As we have shown, both mathematically and via the example of the Brusselator,
neither of these assumptions are true. Rather, to answer the titular question,
two lumps are required. It is hoped that by using this methodology as
standard, new dynamics may be exposed that were previously hidden by the
standard assumptions.
### Acknowledgements
I acknowledge the support of the BBSRC/EPSRC Grant BB/ C008219/1 “The
Manchester Centre for Integrative Systems Biology (MCISB)”. Thanks to Dave
Broomhead for fruitful discussions.
## References
* [1] Lazebnik Y: Can a biologist fix a radio? – Or, what I learned while studying apoptosis. Cancer Cell 2002, 2:179–82.
* [2] Wiechert W: Modeling and simulation: tools for metabolic engineering. J Biotechnol 2002, 94:37 63.
* [3] Le Novère N, Bornstein B, Broicher A, Courtot M, Donizelli M, Dharuri H, Li L, Sauro H, Schilstra M, Shapiro B, Snoep JL, Hucka M: BioModels Database: a free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucleic Acids Res 2006, 34:D689 91.
* [4] Lloyd CM, Halstead MDB, Nielsen PF: CellML: its future, present and past. Prog Biophys Mol Biol 2004, 85:433 50.
* [5] Prigogin I, Lefever R: Symmetry breaking instabilities in dissipative systems II. J Chem Phys 1968, 48:1695 1700.
|
arxiv-papers
| 2012-10-12T14:23:06 |
2024-09-04T02:49:36.474300
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kieran Smallbone",
"submitter": "Kieran Smallbone",
"url": "https://arxiv.org/abs/1210.3527"
}
|
1210.3538
|
# Droplets bouncing over a vibrating fluid layer
Pablo Cabrera-Garcia1 and Roberto Zenit2
1 Facultad de Ciencias
2 Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México
Cd. Universitaria, México D.F., 04510
MÉXICO
###### Abstract
This is an entry for the Gallery of Fluid Motion of the 65st Annual Meeting of
the APS-DFD ( fluid dynamics video ). This video shows the motion of levitated
liquid droplets. The levitation is produced by the vertical vibration of a
liquid container. We made visualizations of the motion of many droplets to
study the formation of clusters and their stability.
## 1 Introduction
If a liquid drop is deposited over a liquid surface, the drop will, first,
rebound, then arrest and eventually coalesce. Couder et al. [1] reported a
technique to retard indefinitely the coalescence phase. By making the liquid
container, the drop can be made to ‘sit’ on top of the surface for a long time
period. We built a similar experiment to study how several droplets cluster.
## 2 Experimental Conditions
A short glass container was mounted on top of a commercial loudspeaker. The
loudspeaker was fed with an amplified signal from a function generator. The
frequency and amplitude of the signal were chosen such that large surface
instabilities were not observed (Faraday waves). The liquid used was tap
water. A small amount of liquid soap was used; we found that in this manner
the drops where more stable. The process was filmed with a high speed camera.
## 3 Videos
Our video contributions can be found at:
* •
Video 1, mpeg4, full resolution
* •
Video 2, mpeg2, low resolution
## References
* [1] Y. Couder, E. Fort, C.-H. Gautier, and A. Boudaoud, From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath, Phys. Rev. Lett. 94, 177801 (2005).
|
arxiv-papers
| 2012-10-12T14:45:43 |
2024-09-04T02:49:36.480840
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pablo Cabrera-Garcia and Roberto Zenit",
"submitter": "Roberto Zenit",
"url": "https://arxiv.org/abs/1210.3538"
}
|
1210.3594
|
# $R(s)$ and Z decay in order $\alpha_{s}^{4}$: complete results
P. A. Baikov,a ,b J. H. Kühnb and J. Rittingerb
a Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Moscow 119991, Russia
bInstitut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology
(KIT), Wolfgang-Gaede-Straße 1, 726128 Karlsruhe, Germany E-mail:
konstantin.chetyrkin@kit.edu
###### Abstract:
We report on our calculation of the order $\alpha_{s}^{4}$ axial singlet
contributions for the decay rates of the $Z$-boson as well as the vector
singlet contribution to the cross section for electron-positron annihilation
into hadrons. Together with recently finished ${\cal O}(\alpha_{s}^{4})$
calculations of the non-signlet corrections [1, 2], the new results directly
lead us to the first complete ${\cal O}(\alpha_{s}^{4})$ predictions for the
total hadronic decay rate of the Z-boson and the ratio
$R(s)={\sigma(e^{+}e^{-}\to{\rm
hadrons})\over\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})}\,$.
## 1 Introduction
Inclusive quark production through a decay of a heavy virtual photon, Z boson
or $\tau$ is a process of importance for QCD as the theory of strong
interactions. Perturbative QCD (pQCD) provides theoretically clean prediction
for the process (see, e.g. [3, 4]).
Combined with the precise determination of the $Z$-boson decay rate into
hadrons at LEP [5] this has led to one of the most precise determinations of
the strong coupling constant $\alpha_{s}(M_{Z})$. An alternative and also very
precise determination of $\alpha_{s}(M_{Z})$ as derived from
$\alpha_{s}(M_{\tau})$ has been recently obtained from the ${\cal
O}(\alpha_{s}^{4})$ prediction [1] for the ratio
$R_{\tau}=\frac{\Gamma(\tau\rightarrow{\rm hadrons})}{\Gamma(\tau\rightarrow
l+\bar{\nu}_{l}+\nu_{\tau})}$ and the experimental determinations of
$R_{\tau}$ by ALEPH, CLEO and OPAL collaborations (see, e.g. [4]).
Note that while the ${\cal O}(\alpha_{s}^{4})$ predictions for $R_{\tau}$ are
complete this is not the case for $R(s)$ and the Z-decay rate. The missing
pieces are related to so-called singlet diagrams (see Fig. 1 below). Note that
while the top quarks can not be produced in Z-decays due to kinematical
reasons, the (axial) singlet diagrams containing internal quark loops are not
power suppressed (unlike similar loops for the vector singlet (and non-
singlet) diagrams. This remarkable phenomenon first shows up at order
$\alpha_{s}^{2}$ and was first established and fully investigated in works [6,
7]. The full account of singlet diagrams at order $\alpha_{s}^{3}$ was
performed in papers [8, 9, 10] (vector case) and in [11, 12, 13, 10] (axial
case).
In the present work we present the results of the calculations of the order
$\alpha_{s}^{4}$ axial singlet contributions for the decay rates of the
$Z$-boson as well as the vector singlet contribution to the cross section for
electron-positron annihilation into hadrons. Note that we will not dwell on
any phenomenological applications of our calculations as they have been
recently discussed in some detail in [14].
## 2 Preliminaries
The interaction of the Z boson to quarks is described (in the lowest order
approximation in the weak coupling constant) by adding to the QCD Lagrangian
an extra term of the form
$M_{Z}\left(\frac{G_{F}}{2\sqrt{2}}\right)^{1/2}Z^{\alpha}J^{0}_{\alpha}$,
with
$J^{0}_{\alpha}=\sum_{i}\overline{\psi}_{i}\gamma_{\alpha}(g^{V}_{i}-g^{A}_{i}\gamma_{5})\psi_{i}\
$ being the neutral quark current. As a result, the hadronic decay rate of the
Z boson ($\Gamma^{h}_{Z}$) including all strong interaction corrections may be
viewed as an incoherent sum of vector ($\Gamma^{V}_{Z}$) and axial
($\Gamma^{A}_{Z}$) contributions. By the optical theorem both contributions
can be conveniently related to the correlators of vector and axial vector
quark currents. The general definition for the latter reads:
$\displaystyle\Pi_{\mu\nu;i,j}^{V/A}(q)$ $\displaystyle=i\int e^{iqx}\langle
0|~{}T~{}j_{\mu,i}^{V/A}(x)j_{\nu,j}^{V/A}(0)~{}|0\rangle~{}\mathrm{d}x$
$\displaystyle=g^{\mu\nu}q^{2}\Pi^{V/A}_{1;i,j}(-q^{2})+q^{\mu}q^{\nu}\Pi^{V/A}_{2;i,j}(-q^{2})$
(1)
with $j_{\mu,i}^{V}=\overline{\psi}_{i}~{}\gamma_{\mu}~{}\psi_{i}=V^{i}_{\mu}$
and
$j_{\mu,i}^{A}=\overline{\psi}_{i}~{}\gamma_{\mu}~{}\gamma_{5}\psi_{i}=A^{i}_{\mu}$.
The corresponding absorptive parts are defined as follows:
$\displaystyle R^{V/A}_{i,j}(s)=12\pi\Im\
\Pi^{V/A}_{1;i,j}(-s-i\,\varepsilon){}.$ (2)
The $Z$ decay rate
$\Gamma(Z\rightarrow\textnormal{hadrons})=\Gamma_{0}\bigl{(}\,R^{V}(M_{Z}^{2})+R^{A}(M_{Z}^{2})\bigr{)}$,
where $\Gamma_{0}=G_{F}M_{Z}^{3}/(24\pi\sqrt{2})$ and $R^{V/A}$ can be
expressed in terms of $R^{V/A}_{i,j}$ defined in eq. (2), namely
$\displaystyle
R^{V}=\sum_{i,j}g_{i}^{V}\,g_{j}^{V}\,R^{V}_{i,j}~{},\qquad\qquad
R^{A}=\sum_{i,j}g_{i}^{A}\,g_{j}^{A}\,R^{A}_{i,j}~{}.$ (3)
Similarly, the inclusive cross-section reaction of the reaction $e^{+}e^{-}$
annihilation into hadrons through the photon is described by the current
correlation function
$\Pi_{\mu\nu}(q)=\int{\rm d}xe^{iqx}\langle 0|T[\;\;j_{\mu}^{\rm
em}(x)j_{\nu}^{\rm
em}(0)\;]|0\rangle=\displaystyle(-g_{\mu\nu}q^{2}+q_{\mu}q_{\nu})\Pi^{EM}(-q^{2}){}\,,$
(4)
with the hadronic EM current
$j^{\rm em}_{\mu}=\sum_{{f}}q_{{f}}\overline{\psi}_{{f}}\gamma_{\mu}\psi_{f}\
\ \ \mbox{and}\ \ \ R(s)=12\pi\,\Im\Pi^{EM}(-s-i\,\varepsilon){},$
with $q_{f}$ being the EM charge of the quark $f$. As a result, we arrive to
the following representation for the ratio R(s) valid in massless
approximation (precise definitions of $R^{NS}$ and $R^{V,S}$ will be given
below)
$R(s)=\sum_{i,j}q_{i}\,q_{j}\,\,R^{V}_{i,j}(s)=\Bigl{(}\sum_{i}q_{i}^{2}\Bigr{)}\,R^{NS}(s)+\Bigl{(}\sum_{i}q_{i}\Bigr{)}^{2}\,R^{VS}(s){}.$
(5)
(0,20)(15,20)32 (55,20)(70,20)32 (35,20)(20,0,180) (35,20)(20,180,360)
(25,37)(25,3)34 (45,37)(45,3)-34 (5,25)[rb]$Z$(65,25)[lb]$Z$ (a)
(0,20)(15,20)32 (55,20)(70,20)32 (35,20)(20,0,180) (35,20)(20,180,360)
(35,40)(35,28)-31 (35,0)(35,12)-31 (35,20)(8,0,360)
(5,25)[rb]$Z$(65,25)[lb]$Z$(45,20)[lc]$t$ (b)
(0,20)(15,20)32 (65,20)(80,20)32 (15,20)(30,35) (30,35)(30,05) (30,05)(15,20)
(50,35)(65,20) (50,05)(50,35) (65,20)(50,05) (30,20)(50,20)32 (30,35)(50,35)32
(30,5)(50,5)-32 (5,25)[rb]$Z$(75,25)[lb]$Z$ (c)
(0,20)(15,20)32 (85,20)(100,20)32 (15,20)(30,35) (30,35)(30,05) (30,05)(15,20)
(70,35)(85,20) (70,05)(70,35) (85,20)(70,05) (30,20)(70,20)34 (30,5)(70,5)-34
(30,35)(42,35)31 (58,35)(70,35)31 (50,35)(8,90,450)
(5,25)[rb]$Z$(95,25)[lb]$Z$(58,42)[lb]$t$ (d)
Figure 1: Examples of non-singlet diagrams (a), (b), where the two $Z$
vertices are connected by a fermion line, and of singlet diagrams (c),(d),
where the diagram can be split by only cutting gluon lines. The imaginary part
of the non-singlet diagrams gives $R^{V/A,NS}$, while the imaginary part of
the singlet diagrams is denoted by $R^{V/A,S}$.
As the Z-boson is much heavier than all known quarks but the top one, it is
natural111 Mass corrections to both vector and axial vector correlator due to
other massive quarks are dominated by the bottom quark and can be classified
by orders in $m_{b}^{2}/M_{Z}^{2}$ and $\alpha_{s}$. Up to ${\cal
O}(\alpha_{s}^{2}m_{b}^{2}/M_{Z}^{2})$ and ${\cal
O}(\alpha_{s}^{2}m_{b}^{4}/M_{Z}^{4})$ they can be found in [3], as well terms
of order $\alpha_{s}^{2}m_{b}^{2}/M_{Z}^{2}$ (const + $\log\
m_{b}^{2}/M_{Z}^{2}$) and $\alpha_{s}^{2}m_{b}^{2}/M_{t}^{2}$ (const + $\log\
m_{b}^{2}/M_{Z}^{2}$) that arise from axial vector singlet contributions.
Terms of order $\alpha_{s}^{3}m_{b}^{4}/M_{Z}^{4}$ and
$\alpha_{s}^{4}m_{b}^{2}/M_{Z}^{2}$ can be found in [15] and [16]
respectively. Corrections of order $\alpha_{s}^{2}m_{Z}^{2}/m_{t}^{2}$ and
$\alpha_{s}^{3}m_{Z}^{2}/m_{t}^{2}$ from singlet and non-singlet terms are
known from [7, 6, 17] and [10] respectively. to neglect all power suppressed
light quark mass corrections when dealing with ${\cal O}(\alpha_{s}^{4})$
contribution to $\Gamma^{h}_{Z}$. It is customary to split $R_{i,j}^{V/A}$
into two contributions as described in Fig. 1
$\displaystyle
R_{i,j}^{V/A}(M_{Z}^{2})=\delta_{ij}^{\ell}\,R^{NS}(M_{Z}^{2})+R^{V/A,S}_{ij}(M_{Z}^{2})~{}{},$
(6)
with the delta function $\delta_{ij}^{\ell}\equiv\delta_{ij}$ if both flavours
$i$ and $j$ are light and $\delta^{\ell}_{ij}=0$ if either $i$ or/and $j$
refer to the top quark. In the non-singlet diagrams there is no top quark
present in the fermion loop connecting the two external currents, because
these diagrams have no physical cut and therefore have no imaginary part
contributing to $R^{NS}(s=M_{Z}^{2})$. This, together with the assumed
masslessness of all quarks but top leads to the factorized form of the non-
singlet term in eq. (6). Note that the internal top quark loops like in
diagram (b) of Fig. 1 still contribute to $R^{NS}(s=M_{Z}^{2})$ if the strong
coupling is defined for 6 flavours. However, it is well known that such
contributions could be naturally described (up to power suppressed terms) by
transition from the full $n_{f}=6$ QCD to the effective massless $n_{f}=5$ one
(see, e.g. [3] and references therein). In fact, the same is true for the
vector singlet term ($\theta^{h}_{ij}$ below is defined as 1 if either i
or/and j refer to the top quark and 0 in all other cases)
$R^{V,S}_{ij}(M_{Z})\equiv(1-\theta^{h}_{ij})\,R^{V,S}(M_{Z})+{\cal
O}(M_{Z}^{2}/M^{2}_{t}){}.$
The corresponding massless calculations of $R^{NS}$ in order $\alpha_{s}^{4}$
have been recently finished [1, 2]. In what follows we concentrate on the
singlet terms $R^{V,S}$ and $R^{A,S}$.
## 3 $\gamma_{5}$-treatment
As is well-known the treatment of $\gamma_{5}$ within dimensional
regularization is a non-trivial problem by itself (for an excellent review see
[18]). Following works [11, 12] in all our calculations we employ, in fact,
two different definitions of $\gamma_{5}$. First, for all non-singlet diagrams
completely anticommuting naive $\gamma_{5}$ have been used. Second, for
singlet diagrams we employ essentially the ’t Hooft-Veltman definition [19]
$A^{i}_{\alpha}=\overline{\psi}_{i}\gamma_{\alpha}\gamma_{5}\psi_{i}\equiv\frac{\xi^{A}_{5}(a_{s})}{6}\,{\mathrm{i}}\,\epsilon_{\alpha\beta\nu\rho}\overline{\psi_{i}}\gamma_{\beta}\gamma_{\nu}\gamma_{\rho}\psi_{i}.{},$
(7)
where the current
$\overline{\psi_{i}}\gamma_{\beta}\gamma_{\nu}\gamma_{\rho}\psi_{i}$ is
assumed to be minimally renormalized.
The finite normalization factor $\xi_{5}^{A}=1-4{a_{s}}/{3}+O(a_{s}^{2})$ on
the rhs of (7) is necessary [20, 21] for the current (7) to obey the usual
(non-anomalous) Ward identities which in turn are crucial in renormalizing the
Standard Model.
In principle, one could (and even have to!) use one and the same definition
(7) also for non-singlet diagrams. This would result to much more complicated
calculations due to significantly longer traces encountered. Fortunately, it
is not necessary because the factor $\xi_{5}^{A}$ is chosen in such a way to
restore the anti-commutativity of the $\gamma_{5}$ (for a detailed discussion,
see [21]).
## 4 Vector $\mathcal{O}(\alpha^{4})$ singlet term $R^{V,S}$
From purely technical point of view the calculation of the massless five-loop
diagrams contributing to $\Pi^{V,S}_{ij}$ is not much different from those
contributing to $\Pi^{V,NS}$. Using the same methods as described in [1, 2] we
have obtained (below $a_{s}={\alpha_{s}(\mu)}/{\pi}$ and $\mu$ is the
renormalization scale in the $\overline{\mathrm{MS}}$ scheme)
$\displaystyle R^{V,S}(s)=$ $\displaystyle
a_{s}^{3}\,\Bigl{(}\frac{55}{72}-\frac{5}{3}\zeta_{3}\Bigr{)}$ (8)
$\displaystyle{+}$ $\displaystyle
a_{s}^{4}\,\Biggl{(}n_{l}\,\left[-\frac{745}{432}+\frac{65}{24}\zeta_{3}+\frac{5}{6}\,\zeta_{3}^{2}-\frac{25}{12}\zeta_{5}-\frac{55}{144}\ln\frac{\mu^{2}}{s}+\frac{5}{6}\zeta_{3}\ln\frac{\mu^{2}}{s}\right]$
$\displaystyle{+}$
$\displaystyle\frac{5795}{192}-\frac{8245}{144}\zeta_{3}-\frac{55}{4}\,\zeta_{3}^{2}+\frac{2825}{72}\zeta_{5}+\frac{605}{96}\ln\frac{\mu^{2}}{s}-\frac{55}{4}\zeta_{3}\ln\frac{\mu^{2}}{s}\Biggr{)}{}.$
## 5 Axial vector $\mathcal{O}(\alpha^{4})$ singlet term $R^{A,S}_{i,j}$
Due to the obvious property222Obvious, thanks to the existence of the unitary
$SU(n_{l})$ symmetry in the flavour subspace of the first $1\dots n_{l}=5$
massless quarks. $R^{A,S}_{i,j}=R^{A,S}_{i^{\prime},j^{\prime}}$ if all 4
indexes refer to the massless quarks and the fact that
$g_{A}^{u}+g_{A}^{d}=g_{A}^{c}+g_{A}^{s}=0$, we can write the axial singlet
part of the Z decay rate as follows:333Note that separate terms on the rhs of
(9) are not scale-invariant, while their sum is [11, 12].
$R^{A,S}=R^{A,S}_{tt}-2\,R^{A,S}_{tb}+R^{A,S}_{bb}{}.$ (9)
All diagrams contributing to the first two terms of (9) contain at least one
top quark loop. The third term receives contributions by both the completely
massless diagrams and those with top quark loop (the latter start from order
$\alpha_{s}^{3}$, an example is given by Fig. 1 (d)).
As $M_{Z}\ll 2\,M_{t}$, one can use the effective theory methods to compute
top-mass-dependent diagrams as a series in the ratio
$\frac{M_{Z}^{2}}{4M_{t}^{2}}$. The procedure was elaborated long ago and
successfully employed (see works [22, 11, 12]) to get all ingredients of eq.
(9) at order $\alpha_{s}^{3}$ at leading order in $1/M_{t}$ expansion (still
keeping all power non-suppressed terms, including those which depends on
$\ln(\mu^{2}/M_{t}^{2})$). From purely technical point of view the evaluation
at order $\alpha_{s}^{4}$ involves absorptive parts of five-loop diagrams with
massless propagators and, in addition, absorptive parts of four-loop diagrams
combined with one-loop massive tadpoles, etc. down to one-loop massless
diagrams together with four-loop massive tadpoles. The latter have been
computed with the help of the Laporta’s algorithm [23] implemented in Crusher
[24]. The massive tadpoles with number of loops less or equal three have been
independently recalculated with the help of the FORM program MATAD [25]. Our
results for $R^{A,S}_{tt}$, $R^{A,S}_{tb}$, $R^{A,S}_{bb}$ and $R^{A,S}$ read
${R^{A,S}_{tt}}=(a_{s}^{5})^{4}\left[\frac{15}{64}-\frac{15}{8}\ell_{\mu
t}+\frac{15}{4}\ell^{2}_{\mu t}\right]{},$ (10) $\displaystyle
R^{A,S}_{tb}=\,(a_{s}^{5})^{2}\left[\frac{3}{8}-\frac{3}{2}\ell_{\mu
t}\right]{+}\,(a_{s}^{5})^{3}\left[-\frac{3869}{288}+\frac{55}{8}\zeta_{3}-\frac{45}{8}\ell_{\mu
t}-\frac{25}{8}\ell^{2}_{\mu t}\right]$ (11) $\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{4}\left[-\frac{370478273}{14515200}-\zeta_{2}+\frac{1309601}{16800}\zeta_{3}-\frac{4225817}{34560}\zeta_{4}-\frac{10453}{288}\zeta_{5}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}-2\zeta_{2}\,\ln(2)-\frac{89}{48}\zeta_{4}\,\ln(2)-\frac{5861}{1080}\zeta_{2}\,(\ln(2))^{2}+\frac{2}{9}\zeta_{2}\,(\ln(2))^{3}+\frac{5861}{6480}\,(\ln(2))^{4}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}-\frac{1}{45}\,(\ln(2))^{5}+\frac{5861}{270}\,a_{4}+\frac{8}{3}\,a_{5}-\frac{37}{32}\ell_{\mu
Z}\,-\frac{47015}{576}\ell_{\mu t}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}+\frac{709}{8}\zeta_{3}\ell_{\mu
t}+\frac{37}{8}\ell_{\mu Z}\,\ell_{\mu t}-\frac{363}{16}\ell^{2}_{\mu
t}-\frac{193}{32}\ell^{3}_{\mu t}\right]{},$ $\displaystyle
R^{A,S}_{bb}=(a_{s}^{5})^{2}\left[-\frac{17}{2}-3\ell_{\mu Z}\,\right]$ (12)
$\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{3}\left[-\frac{4673}{48}+\frac{23}{2}\zeta_{2}+\frac{67}{4}\zeta_{3}-\frac{373}{8}\ell_{\mu
Z}\,-\frac{23}{4}\ell^{2}_{\mu Z}\,\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{3}}-\frac{1}{12}\ell_{\mu
t}-\frac{1}{2}\ell^{2}_{\mu t}\right]$ $\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{4}\left[-\frac{79017683}{82944}+\frac{8747}{32}\zeta_{2}+\frac{54179}{128}\zeta_{3}+\frac{1481}{128}\zeta_{4}-\frac{6455}{96}\zeta_{5}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}-\zeta_{2}\,(\ln(2))^{2}+\frac{1}{6}\,(\ln(2))^{4}+4\,a_{4}-\frac{174767}{288}\ell_{\mu
Z}\,+\frac{529}{8}\zeta_{2}\ell_{\mu Z}\,\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}+\frac{1519}{8}\zeta_{3}\ell_{\mu
Z}\,-\frac{8747}{64}\ell^{2}_{\mu Z}\,-\frac{529}{48}\ell^{3}_{\mu
Z}\,-\frac{1975}{288}\ell_{\mu t}+\frac{37}{8}\zeta_{3}\ell_{\mu t}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}-\frac{247}{48}\ell^{2}_{\mu
t}-\frac{25}{24}\ell^{3}_{\mu t}\right]{},$ $\displaystyle
R^{A,S}=\,(a_{s}^{5})^{2}\left[-\frac{37}{4}-3\ell_{\mu Z}\,+3\ell_{\mu
t}\right]$ (13) $\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{3}\left[-\frac{5075}{72}+\frac{23}{2}+\zeta_{2}+3\zeta_{3}-\frac{373}{8}\ell_{\mu
Z}\,-\frac{23}{4}\ell^{2}_{\mu Z}\,\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{3}}+\frac{67}{6}\ell_{\mu
t}+\frac{23}{4}\ell^{2}_{\mu t}\right]$ $\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{4}\left[-\frac{13083735979}{14515200}+\frac{8811}{32}\zeta_{2}+\frac{17967167}{67200}\zeta_{3}+\frac{553219}{2160}\zeta_{4}+\frac{1541}{288}\zeta_{5}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}+4\zeta_{2}\,\ln(2)+\frac{89}{24}\zeta_{4}\,\ln(2)+\frac{5321}{540}\zeta_{2}\,(\ln(2))^{2}-\frac{4}{9}\zeta_{2}\,(\ln(2))^{3}-\frac{5321}{3240}\,(\ln(2))^{4}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}+\frac{2}{45}\,(\ln(2))^{5}-\frac{5321}{135}\,a_{4}-\frac{16}{3}\,a_{5}-\frac{174101}{288}\ell_{\mu
Z}\,+\frac{529}{8}\zeta_{2}\ell_{\mu Z}\,\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}+\frac{1519}{8}\zeta_{3}\ell_{\mu
Z}\,-\frac{8747}{64}\ell^{2}_{\mu Z}\,-\frac{529}{48}\ell^{3}_{\mu
Z}\,+\frac{11125}{72}\ell_{\mu t}-\frac{1381}{8}\zeta_{3}\ell_{\mu t}\right.$
$\displaystyle\left.\phantom{+\,(a_{s}^{5})^{4}}-\frac{37}{4}\ell_{\mu
Z}\,\ell_{\mu t}+\frac{2111}{48}\ell^{2}_{\mu t}+\frac{529}{48}\ell^{3}_{\mu
t}\right]{}.$
Here $a^{5}_{s}={\alpha_{s}(\mu)}/{\pi}$ in the effective (topless) $n_{f}=5$
QCD, $\ell_{\mu Z}=\ln\frac{\mu^{2}}{M_{Z}^{2}}$, $\ell_{\mu
t}=\ln\frac{\mu^{2}}{M_{t}^{2}}$, and $M_{t}$ is the pole top quark mass. In
addition, $\zeta_{n}=\zeta(n)$ is Riemann’s zeta function and $a_{n}={\rm
Li}_{n}(1/2)=\sum_{i=1}^{\infty}1/(2^{i}i^{n})$.
Finally, setting $\mu=M_{Z}$, we arrive at the following numerical form of
(13)
$\displaystyle R^{A,S}=\,(a_{s}^{5})^{2}\left[-9.25+3.{\,\rm
ln}\frac{M_{Z}^{2}}{M_{t}}\right]$ (14) $\displaystyle{+}$
$\displaystyle\,(a_{s}^{5})^{3}\left[-47.9632+11.1667{\,\rm
ln}\frac{M_{Z}^{2}}{M_{t}}+5.75{\,\rm ln}^{2}\frac{M_{Z}^{2}}{M_{t}}\right]$
$\displaystyle{+}$ $\displaystyle\,(a_{s}^{5})^{4}\left[147.093-52.9912{\,\rm
ln}\frac{M_{Z}^{2}}{M_{t}}+43.9792{\,\rm
ln}^{2}\frac{M_{Z}^{2}}{M_{t}}+11.0208{\,\rm
ln}^{3}\frac{M_{Z}^{2}}{M_{t}}\right]{}.$
## 6 Conclusion
All our calculations have been performed on a SGI ALTIX 24-node IB-
interconnected cluster of 8-cores Xeon computers and on the HP XC4000
supercomputer of the federal state Baden-Württemberg using parallel MPI-based
[26] as well as thread-based [27] versions of FORM [28]. For evaluation of
color factors we have used the FORM program COLOR [29]. The diagrams have been
generated with QGRAF [30]. This work was supported by the Deutsche
Forschungsgemeinschaft in the Sonderforschungsbereich/Transregio SFB/TR-9
“Computational Particle Physics” and by RFBR grants 11-02-01196 and
10-02-00525.
We thank P. Marquard for his friendly help with the package Crusher.
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* [11] K.G. Chetyrkin and J.H. Kühn, Phys. Lett. B308 (1993) 127.
* [12] K.G. Chetyrkin and O.V. Tarasov, Phys. Lett. B327 (1994) 114, hep-ph/9312323.
* [13] S.A. Larin, T. van Ritbergen and J.A.M. Vermaseren, Phys. Lett. B320 (1994) 159, hep-ph/9310378.
* [14] P.A. Baikov et al., (2012), 1201.5804.
* [15] K.G. Chetyrkin, R.V. Harlander and J.H. Kühn, Nucl. Phys. B586 (2000) 56, hep-ph/0005139.
* [16] P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Nucl. Phys. Proc. Suppl. 135 (2004) 243.
* [17] K.G. Chetyrkin, Phys. Lett. B307 (1993) 169.
* [18] F. Jegerlehner, Eur. Phys. J. C18 (2001) 673, hep-th/0005255.
* [19] G. ’t Hooft and M.J.G. Veltman, Nucl. Phys. B44 (1972) 189.
* [20] T.L. Trueman, Phys. Lett. B88 (1979) 331.
* [21] S.A. Larin, Phys. Lett. B303 (1993) 113, hep-ph/9302240.
* [22] K.G. Chetyrkin and J.H. Kühn, Z. Phys. C60 (1993) 497.
* [23] S. Laporta, Int. J. Mod. Phys. A15 (2000) 5087, hep-ph/0102033.
* [24] P. Marquard and D. Seidel, Crusher, an automated integral reduction setup, unpublished.
* [25] M. Steinhauser, Comput. Phys. Commun. 134 (2001) 335, hep-ph/0009029.
* [26] M. Tentyukov et al., (2004), cs/0407066.
* [27] M. Tentyukov and J.A.M. Vermaseren, Comput. Phys. Commun. 181 (2010) 1419, hep-ph/0702279.
* [28] J.A.M. Vermaseren, (2000), math-ph/0010025.
* [29] T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Int. J. Mod. Phys. A14 (1999) 41, hep-ph/9802376.
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|
arxiv-papers
| 2012-10-12T18:25:46 |
2024-09-04T02:49:36.488157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. A. Baikov, K. G. Chetyrkin, J. H. K\\\"uhn and J. Rittinger",
"submitter": "Konstantin Chetyrkin G.",
"url": "https://arxiv.org/abs/1210.3594"
}
|
1210.3690
|
# The effects of strong magnetic fields on the neutron star structure: lowest
order constrained variational calculations
Gholam Hossein Bordbar1,2111Corresponding author. E-mail:
bordbar@physics.susc.ac.ir and Zeinab Rezaei 1 Department of Physics, Shiraz
University, Shiraz 71454, Iran
and
Center for Excellence in Astronomy and Astrophysics (CEAA-RIAAM) - Maragha,
P.O. Box 55134-441, Maragha 55177-36698, Iran
###### Abstract
We investigate the effects of strong magnetic fields upon the gross properties
of neutron and protoneutron stars. In our calculations, the neutron star
matter was approximated by the pure neutron matter. Using the lowest order
constrained variational approach at zero and finite temperatures, and
employing $AV_{18}$ potential, we present the effects of strong magnetic
fields on the gravitational mass, radius, and gravitational redshift of the
neutron and protoneutron stars. It is found that the equation of state of
neutron star becomes stiffer with increase of the magnetic field and
temperature. This leads to larger values of the maximum mass and radius for
the neutron stars.
## I Introduction
Compression of magnetic flux inherited from the progenitor star could form the
strong magnetic field in the interior of a neutron star (Reisenegger
Reisenegger (2007)). Using this point of view, Woltjer has predicted a
magnetic field strength of order $10^{15}\ G$ for neutron stars (Woltjer
Woltjer (1964)). In the core of high density inhomogeneous gravitationally
bound neutron stars, the magnetic field strength can be as large as $10^{20}\
G$ (Ferrer Ferrer (2010)). In addition, considering the formation of a quark
core in the high density interior of a neutron star, the maximum field reaches
up to about $10^{20}\ G$ (Ferrer Ferrer (2010); Tatsumi Tatsumi (2000)).
According to the scalar virial theorem which is based on the Newtonian
gravity, the magnetic field strength is allowed up to $10^{18}\ G$ in the
interior of a magnetar (Lai & Shapiro Lai1 (1991)). On the other hand, general
relativity predicts the allowed maximum value of the neutron star magnetic
field to be about $10^{18}\ -10^{20}\ G$ (Shapiro & Teukolsky Shapiro (1983)).
By comparing with the observational data, Yuan et al. obtained a magnetic
field strength of order $10^{19}\ G$ for the neutron stars (Yuan & Zhang Yuan
(1998)).
Strong magnetic field could have an important influence on the structure of a
neutron star. Some authors have studied the effects of strong magnetic fields
on the properties of neutron stars. Bocquet et al. extended the numerical code
for computing the perfect fluid rotating neutron stars in general relativity
to include the electromagnetic fields and studied the rapidly rotating neutron
stars endowed with magnetic fields (Bocquet et al. Bocquet (1995)). The
results show that for a magnetic field $B\sim 10^{18}\ G$, the maximum mass
increases by $13$ to $29\%$ (depending upon the equation of state) with
respect to the maximum mass of non-magnetized stars. Within a relativistic
Hartree approach in a simple linear $\sigma-\omega-\rho$ model, Chakrabarty et
al. studied the gross properties of cold symmetric nuclear matter and nuclear
matter in beta equilibrium under the influence of strong magnetic fields
(Chakrabarty et al. Chakrabarty (1997)). They showed that for magnetic fields
$B_{m}=0$, $4.4\times 10^{17}$ and $10^{20}\ G$, the maximum masses are
$M_{max}=3.10M_{\odot}$, $2.99M_{\odot}$ and $2.91M_{\odot}$, with radii
$R_{M_{max}}=15.02$, $14.95$, $12.25\ km$, respectively. Based on two
nonlinear $\sigma-\omega$ models of nuclear matter, Yuan et al. considered the
properties of neutron stars under the influence of strong magnetic fields
(Yuan & Zhang Yuan1999 (1999)). They found that the equation of state became
softer with increase of the magnetic field. The results show that for the $ZM$
model, the maximum masses are $M_{max}=1.70M_{\odot}$ and $1.62M_{\odot}$ for
$B=0$, $10^{20}\ G$, with corresponding radii $R_{M_{max}}=9.82$ and $8.70\
km$. Furthermore, for the $BB$ model, the maximum masses are
$M_{max}=2.26M_{\odot}$ and $2.07M_{\odot}$ for $B=0$, $10^{20}$, with radii
$R_{M_{max}}=12.07$, $10.09\ km$. Cardall et al. studied static neutron stars
with magnetic fields and a simple class of electric current distributions
consistent with the stationarity requirement (Cardall et al. Cardall (2001)).
It has been demonstrated that the maximum mass of static neutron stars with
magnetic fields determined by a constant current function is noticeably larger
than that attainable with uniform rotation and no magnetic field. Within a
relativistic field theory, Mao et al. considered a neutron-star matter
consisting of neutrons, protons and electrons interacting through the exchange
of $\sigma$, $\omega$ and $\rho$ mesons in the presence of a magnetic field
which decreases from the center to the surface of a neutron star (Mao et al.
Mao (2003)). It has been found that the equation of state becomes stiffer by
increasing the magnetic field that led to an increase of $40\%$ on the neutron
star maximum mass.
In our previous studies, we have investigated the properties of neutron stars
and protoneutron stars in the absence of magnetic field (Bordbar et al. 2006a
; Bordbar & Hayati 2006b ; Bordbar et al. Bordbar61 (2009); Yazdizadeh &
Bordbar 2011a ). Recently, we have calculated the properties of spin polarized
neutron matter in the presence of strong magnetic fields at zero (Bordbar et
al. 2011b ) and finite temperatures (Bordbar & Rezaei Bordbar2011 (2012))
using LOCV technique employing $AV_{18}$ potential. In the present work, the
neutron star matter is approximated by the pure neutron matter to investigate
the effects of strong magnetic fields on the gross properties of neuron stars
and protoneutron stars using the equations of state of neutron matter in the
presence of strong magnetic fields (Bordbar et al. 2011b ; Bordbar & Rezaei
Bordbar2011 (2012)).
## II Neutron star structure in the presence of strong magnetic fields
In the present study, we calculate the neutron star and protoneutron star
structure using the equations of state of cold and hot neutron matter in the
presence of strong magnetic fields (Bordbar et al. 2011b ; Bordbar & Rezaei
Bordbar2011 (2012)). In our study, we employ $AV_{18}$ nuclear potential
(Wiringa et al. Wiringa (1995)) and use the lowest order constrained
variational method to calculate the equation of state. For more details, we
refer the reader to (Bordbar et al. 2011b ; Bordbar & Rezaei Bordbar2011
(2012)). Our results for the equation of state of neutron matter in the
presence of strong magnetic fields are given in Figs. 1-3. Figs. 1(b) and 2(b)
indicate that for the cold and hot neutron matter, at each density, the
pressure increases with increase of the magnetic field. This stiffening of the
equation of state is due to the inclusion of neutron anomalous magnetic
moments. In other words, in the presence of high magnetic fields, the fraction
of polarized neutrons at the equilibrium state increases and therefore the
degeneracy pressure grows. This is in agreement with the results obtained in
Refs. (Broderick et al. Broderick (2000); Yue & Shen Yue (2006)). We have
found that at low densities, the influence of magnetic field on the pressure
is negligible. Fig. 3(b) shows that at each density, the pressure grows by
increasing the temperature. Consequently, for hot neutron matter, the equation
of state is stiffer compared with the cold one. Fig. 3(a) also shows that the
effect of finite temperature on the equation of state is more significant at
high densities.
The equilibrium configurations could be obtained by solving the general
relativistic equations of hydrostatic equilibrium, Tolman-Oppenheimer-Volkoff
(TOV) (Shapiro & Teukolsky Shapiro (1983)),
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\varepsilon(r),$ $\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{Gm(r)\varepsilon(r)}{r^{2}}\left(1+\frac{P(r)}{\varepsilon(r)c^{2}}\right)\left(1+\frac{4\pi
r^{3}P(r)}{m(r)c^{2}}\right)\left(1-\frac{2Gm(r)}{c^{2}r}\right)^{-1},$ (1)
where $\varepsilon(r)$ is the energy density, $G$ is the gravitational
constant, and
$m(r)=\int_{0}^{r}4\pi r^{\prime 2}\varepsilon(r^{\prime})dr^{\prime}$ (2)
gives the gravitational mass inside a radius $r$. By selecting a central
energy density $\varepsilon_{c}$, under the boundary conditions $P(0)=P_{c}$,
$m(0)=0$, we integrate the TOV equations outwards to a radius $r=R$, at which
$P$ vanishes. This yields the radius $R$ and mass $M=m(R)$ (Shapiro &
Teukolsky Shapiro (1983)). Gravitational redshift, the criterion for the star
compactness, is given by
$\displaystyle Z=[1-2(\frac{GM}{c^{2}R})]^{-1/2}-1,$ (3)
where R is the radius of the neutron star. In our calculations of neutron star
structure, for densities greater than $0.05\ fm^{-3}$, we use the equations of
state presented in Figs. 1-3. However, at lower densities, because the
magnetic field and finite temperature have insignificant effects on EoS, we
employ the equation of state calculated by Baym et al. (Baym et al. Baym
(1971)) for all magnetic fields and temperatures.
The effects of magnetic fields on the gravitational masses of neutron stars
and protoneutron stars at a temperature about $15\ MeV$ with different central
densities are presented in Fig. 4. Obviously, at very low central densities,
the gravitational masses are independent of the equation of state; but at
higher densities, the gravitational mass increases by increasing both magnetic
field and temperature. The limiting value of neutron star mass (maximum mass)
also reaches the larger amount when the magnetic field and temperature rise.
For a cold neutron star at $B=10^{19}\ G$, the maximum mass is about $1.17\%$
larger than the cold non magnetized one. Considering two stars (a protoneutron
star at $T=15\ MeV$ and a cold neutron star) in the presence of a magnetic
field $B=10^{19}\ G$, the protoneutron star maximum mass is about $1.16\%$
greater than the cold neutron star. Besides, for a protoneutron star at $T=15\
MeV$ and $B=10^{19}\ G$, the maximum mass increases about $2.36\%$ compared to
a cold non magnetized one. These results are due to the stiffening of the
equation of state (Figs. 1-3).
Fig. 5 presents the gravitational mass versus radius (M-R relation) for
different magnetic fields at zero and finite temperatures. For all magnetic
fields and temperatures, the neutron star mass decreases by increasing the
radius. It is clear from Fig. 5 that for a given radius, the gravitational
mass increases whenever the equation of state becomes stiffer. We have found
that the effect of the equation of state on the M-R relation is more
significant for the neutron stars with smaller radius.
Fig. 6 shows the gravitational redshift versus the gravitational mass of the
neutron star for different magnetic fields at zero and finite temperatures.
Clearly, the stiffness of the equation of state reduces the gravitational
redshift. Fig. 6 also indicates that the maximum redshift (redshift
corresponding to the maximum mass) decreases with the increase of maximum
mass. For a cold and a hot neutron star at $T=15\ MeV$ with $B=10^{19}\ G$,
the values of maximum redshift are $z_{s}^{max}=0.49$, and $z_{s}^{max}=0.47$,
respectively. In addition, we have found that in the case of a cold neutron
star, for magnetic fields $B=0$, $5\times 10^{18}$, and $10^{19}\ G$, the
values of $z_{s}^{max}$ are $0.56$, $0.53$, and $0.49$. Therefore, the maximum
surface redshift of our calculations, i.e. $34.18\%$ (for a cold neutron star
at $B=10^{19}G$), is lower than the upper bound on the surface redshift for
subluminal equation of states, i.e. $z^{CL}_{s}=0.8509$ (Haensel et al.
Haensel (1999)).
Tables 1 and 2 show a summary of our results for the maximum mass and the
corresponding radius predicted for different neutron stars. We have found that
the effects of magnetic fields with magnitude $B\leq 10^{18}\ G$ are almost
negligible. Obviously, for cold neutron stars as well as protoneutron stars,
the maximum mass and the corresponding radius increase by increasing the
magnetic field. Tables 1 and 2 show that at any magnetic field, the maximum
mass and the corresponding radius of the protoneuton star are larger than the
cold neutron star. Therefore, we conclude that the stiffer equation of state
leads to a neutron star with a larger maximum mass and radius. According to
our results, for a cold neutron star, the maximum mass can vary between
$1.69M_{\odot}$ and $1.71M_{\odot}$, depending on the interior magnetic field,
but for a protoneutron star with $T=15\ MeV$, this variation is between
$1.70M_{\odot}$ and $1.73M_{\odot}$. Therefore, the effect of magnetic field
on the protoneutron star maximum mass is more important than the cold neutron
star. Our results for the neutron star maximum mass are higher than the
observational results from X-ray binaries presented in Table 3. Moreover, the
study of the statistics of $61$ measured masses of neutron stars in binary
pulsar systems gives a mass average of $M=1.46\pm 0.3M_{\odot}$ (Zhang et al.
Zhang (2011)). Their results indicate that the mass average of the more
rapidly rotating millisecond pulsars (MSPs) is $M=1.57\pm 0.35M_{\odot}$. In
the present work the values of the protoneutron star radius at higher magnetic
fields are near the values obtained using M R relationships (Zhang et al.
Zhang2 (2007)) which shows the neutron star radius varies in the range of
$10-20\ km$. We have also found that the effect of magnetic field on the
radius of the protoneutron star is less important than the cold neutron star.
## III Summary and Conclusion
Different properties of the neutron star and protoneutron star structure have
been investigated using the equation of state of neutron matter in the
presence of strong magnetic fields. In our calculations, we have employed the
lowest order constrained variational method and applied $AV_{18}$ potential to
find the equation of state at zero and finite temperature in the presence of
strong magnetic fields. Our results show that the stiffer equation of state at
higher magnetic fields and larger values of temperatures lead to the higher
values for the maximum mass and radius. For the maximum value of the magnetic
field considered in this study, i.e. $10^{19}\ G$, the maximum masses of a
cold neutron star and protoneutron star at $T=15\ MeV$ are $1.71M_{\odot}$ and
$1.73M_{\odot}$, respectively. The corresponding radii are also $9.16$ and
$9.22\ km$. Our results indicate that the effects of magnetic field on the
maximum mass of the protoneutron stars are more important than cold neutron
stars, while the effects of magnetic fields are more visible on the radius of
cold neutron stars. It has been shown that the effects of the equation of
state on the M-R relation are more important for neutron stars with smaller
radii. Our calculations also demonstrate that the maximum value of the
gravitational surface redshift decreases by increasing the neutron star
maximum mass.
## Acknowledgements
This work has been supported financially by Center for Excellence in Astronomy
and Astrophysics (CEAA-RIAAM). We wish also to thank the Shiraz University
Research Council.
## References
* (1) Baym, G., Pethick, C., & Sutherland, P. 1971, Astrophys. J. 170, 299.
* (2) Bocquet, M., Bonazzola, S., Gourgoulhon, E., & Novak, J. 1995, Astron. Astrophys. 301, 757\.
* (3) Bordbar, G.H., Bigdeli, M., & Yazdizadeh, T. 2006, Int. J. Mod. Phys. A 21, 5991.
* (4) Bordbar, G.H., & Hayati, M. 2006, Int. J. Mod. Phys. A 21, 1555\.
* (5) Bordbar, G.H., Zebarjad, S. M., & Zahedinia, R. 2009, Int. J. Theor. Phys. 48, 61.
* (6) Bordbar, G.H., Rezaei, Z., & Montakhab, A. 2011, Phys. Rev. C 83, 044310.
* (7) Bordbar, G.H., & Rezaei, Z. 2012, submitted for publication.
* (8) Broderick, A., Prakash, M., & Lattimer, J.M. 2000, Astrophys. J. 537, 351.
* (9) Cardall, C.Y., Prakash, M., & Lattimer, J.M. 2001, Astrophys. J. 554, 322.
* (10) Chakrabarty, S., Bandyopadhyay, D., & Pal, S. 1997, Phys. Rev. Lett. 78, 2898.
* (11) Ferrer, E.J., de la Incera, V., Keith, J.P., Portillo, I., & Springsteen, P.L. 2010, Phys. Rev. C 82, 065802.
* (12) Haensel, P., Lasota, J.P., & Zdunik, J.L. 1999, Astron. Astrophys. 344, 151.
* (13) Lai, D., & Shapiro, S. L. 1991, Astrophys. J. 383, 745.
* (14) Mao, G., Iwamoto, A., & Li, Z. 2003, Chin. J. Astron. Astrophys. 3, 359.
* (15) Reisenegger, A. 2007, Astron. Nachr. 328, 1173.
* (16) Shapiro, S., & Teukolsky, S. 1983, _Black Holes, White Dwarfs and Neutron Stars_ , (Wiley,New York).
* (17) Steeghs D., & Jonker P. G. 2007, ApJ, 669L, 85S.
* (18) Tatsumi, T. 2000, Phys. Lett. B 489, 280.
* (19) van der Meer A., Kaper L., van Kerkwijk M. H., & van den Heuvel E. P. J. 2005, AIP, 623.
* (20) Wiringa, R. B., Stoks, V. G. J., & Schiavilla, R. 1995, Phys. Rev. C 51, 38.
* (21) Woltjer, L. 1964, Astrophys. J. 140, 1309.
* (22) Yazdizadeh, T., & Bordbar, G. H. 2011, Res. Astron. Asrtophys. 11, 471.
* (23) Yuan, Y. F., & Zhang, J. L. 1998, Astron. Astrophys. 335, 969.
* (24) Yuan, Y. F., & Zhang, J. L. 1999, Astrophys. J. 525, 950.
* (25) Yue, P., & Shen, H. 2006, Phys. Rev. C 74, 045807.
* (26) Zhang C.M., Wang J., & Zhao Y.H., et al. 2011, A&A, 527, 83.
* (27) Zhang C.M., Yin H. X., & Kojima Y., et al. 2007, Mon. Not. R. Astron. Soc. 374, 232.
Table 1: Maximum gravitational mass, $M_{max}$, and the corresponding radius, $R_{M_{max}}$, obtained for different values of magnetic field, $B$, at $T=0\ MeV$. $B(G)$ | $M_{max}(M_{\odot})$ | $R_{M_{max}}(km)$
---|---|---
$0$ | 1.69 | 8.59
$5\times 10^{18}$ | 1.70 | 8.73
$10^{19}$ | 1.71 | 9.16
Table 2: Same as Table 1 but at $T=15MeV$. $B(G)$ | $M_{max}(M_{\odot})$ | $R_{M_{max}}(km)$
---|---|---
$0$ | 1.70 | 8.70
$5\times 10^{18}$ | 1.71 | 8.83
$10^{19}$ | 1.73 | 9.22
Table 3: Measured masses of neutron stars in X-ray binaries. System | $M(M_{\odot})$ | References
---|---|---
SMC X-1 | $1.05\pm 0.09$ | (van der Meer et al. Meer (2005))
Cen X-3 | $1.24\pm 0.24$ | (van der Meer et al. Meer (2005))
LMC X-4 | $1.31\pm 0.14$ | (van der Meer et al. Meer (2005))
V395 CAR/2S 0921C630 | $1.44\pm 0.10$ | (Steeghs et al. Steeghs (2007))
Figure 1: (a) Pressure, $P$, versus energy density, $\varepsilon$, for the
cases $B=0\ G$ (solid curve), $B=5\times 10^{18}\ G$ (dashed curve) and
$B=10^{19}\ G$ (dashdot curve) at a fixed value of the temperature, $T=0\
MeV$. (b) Same as in the top panel but for a different range of density.
Figure 2: (a) Pressure, $P$, versus energy density, $\varepsilon$, for the
cases $B=0\ G$ (solid curve), $B=5\times 10^{18}\ G$ (dashed curve) and
$B=10^{19}\ G$ (dashdot curve) at a fixed value of the temperature, $T=15\
MeV$. (b) Same as in the top panel but for a different range of density.
Figure 3: (a) Pressure, $P$, versus energy density, $\varepsilon$, for the
cases $T=0\ MeV$ (solid curve) and $T=15\ MeV$ (dashed curve) at a fixed value
of the magnetic field, $B=5\times 10^{18}\ G$. (b) Same as in the top panel
but for a different range of density.
Figure 4: (a) Gravitational mass of neutron star (in units of the solar mass,
$M_{\odot}$) versus central energy density, $\varepsilon_{c}$, at $T=0\ MeV$.
All curves correspond to those of Fig. 1. (b) Same as (a) but at $T=15\ MeV$.
All curves correspond to those of Fig. 2. (c) Gravitational mass of neutron
star (in units of the solar mass, $M_{\odot}$) versus central energy density,
$\varepsilon_{c}$, at $B=5\times 10^{18}\ G$. All curves correspond to those
of Fig. 3.
Figure 5: (a) Mass-radius relation at $T=0\ MeV$. All curves correspond to
those of Fig. 1. (b) Same as (a) but at $T=15\ MeV$. All curves correspond to
those of Fig. 2. (c) Mass-radius relation at $B=5\times 10^{18}\ G$. All
curves correspond to those of Fig. 3.
Figure 6: Gravitational redshift, $Z_{s}$, vs. total mass for neutron stars
at $T=0\ MeV$. All curves correspond to those of Fig. 1. (b) Same as (a) but
at $T=15\ MeV$. All curves correspond to those of Fig. 2. (c) Gravitational
redshift, $Z_{s}$, vs. total mass for neutron stars at $B=5\times 10^{18}\ G$.
All curves correspond to those of Fig. 3.
|
arxiv-papers
| 2012-10-13T09:43:08 |
2024-09-04T02:49:36.505593
|
{
"license": "Public Domain",
"authors": "Gholam Hossein Bordbar and Zeinab Rezaei",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1210.3690"
}
|
1210.3804
|
# Quantum singularities in conformally static spacetimes
Deborah A. Konkowski1 and Thomas M. Helliwell2 1 Mathematics Department, U.S.
Naval Academy, Annapolis, Maryland 21402 USA 2 Physics Department, Harvey
Mudd College, Claremont, California 91711 USA dak@usna.edu, helliwell@hmc.edu
###### Abstract
After a brief review of the standard definition and analysis of classical
singularities in general relativistic spacetimes, and of quantum singularities
in static spacetimes with timelike classical singularities, an extension of
quantum singularities to conformally static spacetimes is summarized and
applied to two test cases. The timelike classical singularities in a
Friedmann-Robertson-Walker (FRW) universe with a cosmic string, and in Roberts
spacetime, are shown to be quantum mechanically singular when tested by either
minimally coupled or conformally coupled scalar waves. In the Roberts case,
however, non-minimally coupled scalar waves with a coupling constant $\xi\geq
2$ do not detect the classical singularity.
## 1 Introduction
We study quantum wave packet propagation in conformally static spacetimes with
timelike classical singularities. If the wave propagation turns out to be well
defined, the spacetimes are said to be quantum mechanically non-singular.
The order of the paper is as follows: First, classical and quantum
singularities are defined with the latter restricted (as usual) to static
spacetimes with timelike singularities. Next, the definition of quantum
singularity is extended to conformally static spacetimes with a timelike
singularity (spacelike singularities, if present, are not tested). In
particular, two spacetimes are tested with generally coupled scalar waves: a
Friedmann-Robertson-Walker (FRW) spacetime with a cosmic string and the
Roberts spacetime. Finally, conclusions are given, together with ideas for
further research.
## 2 Classical singularities
A spacetime $(M,g)$ is taken to be a paracompact, $C^{\infty}$, connected,
Hausdorff manifold $M$ with a Lorentzian metric $g$ [3]. So what is a
classical _singularity_? A spacetime is by definition smooth, so ‘singular’
points are not part of the spacetime; they must be cut out of the spacetime
manifold. This leaves a ‘hole’, with incomplete curves, a seeming boundary to
spacetime. How do we complete spacetime, and how do we define a boundary
$\partial M$ to spacetime? There have been a number of attempts, none of them
entirely satisfactory. Note that Cauchy completeness works only in Riemannian
metrics, not Lorentzian. Boundary definitions have included the
a(abstract)-boundary of Scott and Szekeres [17], the b(bundle)-boundary of
Schmidt [16], the c(causal)- boundary of Geroch, Kronheimer, and Penrose [5]
and the g(geodesic)-boundary of Geroch [6]. In this discussion we will use
Geroch’s 1968 description of a classical singularity. He states that “a
singularity is indicated by incomplete geodesics or incomplete curves of
bounded acceleration in a maximal spacetime.” This is closest to the
definition of classical singularity used in the famous singularity theorems of
Hawking and Penrose, which predict that singularities are ubiquitous in exact
solutions of Einstein’s equations (see, e.g., [7])).
Ellis and Schmidt have classified singular points into three types according
to their strength [3]: quasi-regular (mild, topological singularities), non-
scalar curvature (diverging tidal forces on curves ending at the singularity;
finite tidal forces on some nearby curves) and scalar curvature (diverging
scalars – usually one considers only $C^{0}$ scalar polynomial invariants).
Conical singularities, as in idealized cosmic strings, are a good example of
quasiregular singularities. The other two types of singularities are stronger,
curvature singularities. Nonscalar curvature singularities include those in
whimper cosmologies and certain plane-wave spacetimes, whereas scalar
curvature singularities are the best-known, occurring at the centers of black
holes or the beginning of big bang cosmologies.
### 2.1 Quantum Singularities
What happens if instead of classical particle paths (e.g., null and timelike
geodesics) one uses quantum mechanical particles (quantum wave packets) to
identify singularities? Following pioneering work by Wald [19], Horowitz and
Marolf answered this question for static spacetimes with timelike classical
singularities. In their 1995 paper they posit that a spacetime is quantum
mechanically (QM) _non_ singular if the evolution of a test scalar wave
packet, representing a quantum particle, is uniquely determined by the initial
wave packet, the manifold and the metric, without having to place boundary
conditions at the classical singularity. Technically, a static spacetime is
QM-singular if the spatial portion of the Klein-Gordon operator is not
essentially self-adjoint on $C_{o}^{\infty}(\Sigma)$ in the space of square
integrable functions $L^{2}(\Sigma)$, where $\Sigma$ is a spatial
hypersurface.
The term “essentially self adjoint” arises in functional analysis [14]. An
operator $A$ is called self-adjoint if (i) $A=A^{*}$ and (ii) $Dom(A)$ =
$Dom(A^{*})$, where $A^{*}$ is the adjoint of $A$ and $Dom$ is short for
domain. An operator is _essentially_ self-adjoint if (i) is met and (ii) can
be met by expanding the domain of the operator $A$ or its adjoint $A^{*}$ so
that it is true.
There are two basic tests for essential self-adjointness [14]. The first uses
the von Neumann criterion of deficiency indices [18]; one studies solutions of
$A\Psi=\pm i\Psi$, where $A$ is the spatial portion of the Klein-Gordon
operator, and finds the number of solutions for each sign that are self-
adjoint. The second technique uses the so-called Weyl limit point - limit
circle criterion [20], which relates essential self-adjointness of the
Hamiltonian operator to the behavior of the “potential” in an effective one-
dimensional Schrödinger equation, which in turn determines the behavior of the
scalar wave packet. Relevant theorems that simplify the analysis can be found
in Reed and Simon [14].
Many authors have used the definition of quantum singularity to study the
singularity structure of spacetimes. For a summary, see, for example, the
review article by Pitelli and Letelier [13] or the conference proceeding by
the authors [12] and the references therein. Also, there is the alternative
concept of ‘wave regularity’ introduced by Ishibashi and Hosoya [10], which is
relevant to the discussion. It uses a non-standard Hilbert space, $H^{1}$, the
first Sobolev space.
## 3 Conformally Static Spacetimes
A spacetime $g_{\mu\nu}(x^{\alpha})$ that is conformally static is related to
a static spacetime $\bar{g}_{\mu\nu}(x^{a})$ by a conformal transformation
$C(\eta)$ of the metric. Here $C(\eta)$ is the conformal factor, where $\eta$
is the conformal time, related to the time $t$ by $dt=Cd\eta$. Simply put,
$g_{\mu\nu}(x^{\alpha})=C^{2}(\eta)\bar{g}_{\mu\nu}(x^{a})$. Here Greek
letters $\alpha,\beta,...$ label spacetime indices and have the range over 0,
1, 2, 3, and Latin letters $a,b,c,...$ label spatial indicies that range over
1, 2, 3.
The Lagrangian density for a generally coupled scalar field is [1],
$\mathcal{L}=1/2(-g)^{1/2}[g^{\mu\nu}\Phi,_{\mu}\Phi,_{\nu}-(M^{2}+\xi
R)\Phi^{2}],$ (1)
where $M$ is the mass if the scalar particle, $R$ is the scalar curvature, and
$\xi$ is the coupling (in particular, $\xi=0$ for minimal coupling and
$\xi=1/6$ for conformal coupling). Varying the action $S=\int\mathcal{L}\
d^{4}x$ gives the Klein-Gordon field equation,
$|g|^{-1/2}\left(|g|^{1/2}g^{\mu\nu}\Phi,_{\nu}\right),_{\mu}-\xi
R\Phi=M^{2}\Phi.$ (2)
.
In the massless case with conformal coupling, the field equation above is
conformally invariant under a conformal transformation of the metric and
field; in this case the inner product respecting the stress tensor for the
field is also conformally invariant. This led Ishibashi and Hosoya to state
[10], in the case of wave regularity, that “the calculation is as simple as
that in the static case when singularities in conformally static space-times
are probed with conformally coupled scalar fields.”
Here we study the quantum particle propagation in spacetimes with massive
scalar particles described by the Klein-Gordon equation and the limit point -
limit circle criterion of Weyl [20] [14]. In particular, after separating
variables we study the radial equation in a one-dimensional Schrödinger form
with a ‘potential’ and determine the number of solutions that are square
integrable. If we obtain a unique solution, without placing boundary
conditions at the location of the classical singularity, we can say that the
solution to the full Klein-Gordon equation is quantum mechanically (QM)
nonsingular. The results depend on the spacetime metric parameters and wave
equation modes.
After separating variables we take the spatial portion to be an operator
equation on a Hilbert space $L^{2}(\Sigma)$ with inner product (see, e.g.,
[11]),
$(\chi,\zeta)=\int d^{3}x|\bar{g}_{3}/g_{00}|^{1/2}\chi(x^{a})\zeta(x^{b}),$
(3)
where $\bar{g}_{3}$ is the determinant of the spatial portion of the static
metric, $\chi$ and $\zeta$ are spatial mode solutions and $a,b$ range over 1,
2, 3. Then we consider the radial portion alone, change variables and write
the radial equation in one-dimensional Schrodinger form, $Hu(x)=Eu(x)$, where
the operator $H=-d^{2}/dx^{2}+V(x)$ and $E$ is a constant, with the
singularity at $x=0$. The inner product here is simply $\int dx|u(x)|^{2}$ and
the Hilbert space is $L^{2}(0,\infty)$. At this point one can simply apply the
limit point - limit circle criterion as easily as in the static case in order
to determine the quantum singularity structure.
### 3.1 FRW with a Cosmic String
A simple metric modeling a Friedmann-Robertson-Walker cosmology with a cosmic
string [2] is given by
$ds^{2}=a^{2}(t)(-dt^{2}+dr^{2}+\beta^{2}r^{2}d\phi^{2}+dz^{2})$ (4)
where $\beta=1-4\mu$ and $\mu$ is the mass per unit length of the cosmic
string. This metric is conformally static (actually conformally flat).
Classically it has a scalar curvature singularity when $a(t)$ is zero and a
quasiregular singularity when $\beta^{2}\neq 1$. Here we will consider the
timelike quasiregular singularity alone. The Klein-Gordon equation with
general coupling can be separated into mode solutions
$\Phi=T(t)H(r)e^{im\phi}e^{ikz}$ (5)
where
$\ddot{T}+2\left(\frac{\dot{a}}{a}\right)\dot{T}+(M^{2}a^{2}+\xi Ra^{2}-q)T=0$
(6)
and
$H^{\prime\prime}+\frac{1}{r}H^{\prime}+(-k^{2}-q-\frac{m^{2}}{\beta^{2}r^{2}})H=0.$
(7)
The $T$-equation alone contains $M$ and $R$. Rewriting the dependent and
independent variables as $r=x$ and $H=xu(x)$, we get the correct inner product
form and a one-dimensional Schrödinger equation,
$u^{\prime\prime}+(E-V(x))u=0$ (8)
where $E=-k^{2}-q$ and
$V(x)=\frac{m^{2}-\beta^{2}/4}{\beta^{2}x^{2}}.$ (9)
.
Near zero one can show that the potential $V(x)$ is limit point if
$m^{2}/\beta^{2}\geq 1$. Therefore any modes with sufficiently large $m$ are
limit point, but $m=0$ is limit circle; thus generically this conformally
static space-time is quantum mechanically singular.
### 3.2 Roberts Spacetime
The Roberts metric [15] is
$ds^{2}=e^{2t}(-dt^{2}+dr^{2}+G^{2}(r)d\Omega^{2})$ (10)
where $G^{2}(r)=(1/4)[1+p-(1-p)e^{-2r}](e^{2r}-1)$. The spacetime is
conformally static, spherically symmetric, and self-similar (see, e.g., [10]).
It has a timelike classical scalar curvature singularity at $r=0$ for $0<p<1$.
The Klein-Gordon equation can be solved by separation of variables with mode
solutions given by $\Phi=T(t)H(r)Y_{lm}(\theta,\phi)$. The radial operator can
be put in one-dimensional Schrödinger form and the limit point - limit circle
criterion applied. Details are given in [8]. One finds that the spacetime is
quantum mechanically singular if $\xi<2$ and quantum mechanically non-singular
if $\xi\geq 2$. Therefore, the classical timelike singularity remains singular
when probed by minimally coupled ($\xi=0$) waves or by conformally coupled
($\xi=1/6$) waves.
## 4 Conclusions
After a brief review of the standard definition and analysis of classical
singularities in general relativistic spacetimes, and of quantum singularities
in static spacetimes with timelike classical singularities, an extension of
quantum singularites to conformally static spacetimes was summarized and
applied to two test cases. The timelike classical singularities in a FRW
universe with a cosmic string and in Roberts spacetime were shown to be
quantum mechanically singular when tested by either minimally coupled or
conformally coupled scalar waves. In the Roberts case, however, non-minimally
coupled scalar waves with a coupling constant $\xi\geq 2$ did not detect the
classical singularity.
Further analysis of the singularity structure of conformally static spacetimes
is underway [8]. A class of spherically symmetric conformally static
spacetimes is being analyzed; this class includes the spacetimes of HMN[9] and
Fonarev[4], as well as the Roberts spacetime.
## Acknowledgements
One of us (DAK) thanks B. Yaptinchay for useful discussions.
## References
## References
* [1] Birrell, M.D. and Davies, P.C.W., Quantum fields in curved space, (Cambridge University Press, Cambridge, 1982).
* [2] Davies, P.C.W. and Sahni, V., “Quantum gravitational effects near cosmic strings”, Class. Quantum Grav., 5, 1–17, (1988).
* [3] Ellis, G.F.R. and Schmidt, B.G., “Singular space-times”, Gen. Rel. Grav., 8, 915–988, (1977).
* [4] Fonarev, O.A., “Exact Einstein scalar field solutions for formation of black holes in a cosmological setting”, Class. Quantum Grav., 12, 1739–1752, (1995). [arXiv:gr-qc/9409020].
* [5] Geroch, R.P., Kronheimer E.H. and Penrose, R., “Ideal points in spacetime”, Proc. Roy. Soc. Lond. A, 327, 545–567, (1972).
* [6] Geroch, R.P., “Local characterization of singularities in general relativity”, J. Math. Phys., 9, 450–465, (1968).
* [7] Hawking, S.W. and Ellis, G.F.R., The Large-Scale Structure of Space-time, (Cambridge University Press, Cambridge, 1973).
* [8] Helliwell, T.M. and Konkowski, D.A., “Quantum singularity of spherically symmetric conformally static space-times”, in preparation, (2012).
* [9] Husain, V., Martinez E.A. and Nuñez, D., “Exact solution for scalar field collapse”, Phys. Rev. D, 50, 3783–3786, (1994). [arXiv:gr-qc/9402021].
* [10] Ishibashi, A. and Hosoya, A., “Who’s afraid of naked singularities? Probing timelike singularities with finite energy waves”, Phys. Rev. D, 60, 104028, (1999). [arXiv:gr-qc/9907009].
* [11] Kandrup, H.E., “Statistical mechanics of the gravitational field in a conformally static setting”, J. Math. Phys., 25, 3286–3296, (1984).
* [12] Konkowski, D.A. and Helliwell, T.M., “Quantum singularities in static and conformally static space-times”, Int. J. Mod. Phys. A, 26, 3878–3888, (2011). [arXiv:1112.5488].
* [13] Pitelli, J.P.M. and Letelier, P.S., “Quantum singularities in static spacetimes”, Int. J. Mod. Phys. D, 20, 729–743, (2011). [arXiv:1010.3052].
* [14] Reed, M. and Simon, B., Fourier Analysis and Self-Adjointness, (Academic Press, New York, 1972).
* [15] Roberts, M.D., “Scalar field counterexamples to cosmic censorship hypothesis”, Gen. Rel. Grav., 21, 907–939, (1989).
* [16] Schmidt, B.G., “A new definition of singular points in general relativity”, Gen. Rel. Grav., 1, 269–280, (1971).
* [17] Scott, S.M. and Szekeres, P, “The abstract boundary – a new approach to singularities of manifolds”, J. Geom. Phys., 13, 223–253, (1994). [arXiv:gr-qc/9405063].
* [18] von Neumann, J., “Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren”, Math. Ann., 102, 49–131, (1929).
* [19] Wald, R.M., “Dynamics in non-globally hyperbolic, static spacetimes”, J. Math Phys., 21, 2802–2806, (1980).
* [20] Weyl, H., “Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen”, Math. Ann., 68, 220–269, (1910).
|
arxiv-papers
| 2012-10-14T15:34:19 |
2024-09-04T02:49:36.515340
|
{
"license": "Public Domain",
"authors": "Deborah A. Konkowski and Thomas M. Helliwell",
"submitter": "Deborah A. Konkowski",
"url": "https://arxiv.org/abs/1210.3804"
}
|
1210.3861
|
# Fast, large amplitude vibrations of compliant cylindrical shells carrying a
fluid
Pawel Zimoch, Eliott Tixier, Julia Hsu, Amos Winter and Anette Hosoi
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
###### Abstract
In this fluid dynamics video, we demonstrate the first three circumferential
modes of fast, large amplitude vibrations of compliant cylindrical shells
carrying a fluid.
## 1 Details
We use shells made of vinylpolysiloxane (a silicone-based elastomer) with
elastic moduli of $0.2$, $0.6$ and $1.0$ MPa, as well as shells made of latex.
The shells range from $5$–$10$ mm in diameter, $5$–$50$ mm in length and
approximately $0.1$–$0.2$ mm in thickness. We clamp one end of these shells
onto a rigid nozzle and pass air through them at flow rates ranging from
$0.2$–$2.5$ liters per second. When the flow rate of air in the shells exceeds
a certain critical value, dependent on the dimensions and material properties
of each shell, the shell becomes unstable and begins to vibrate. The mode of
vibration corresponds to one of the circumferential normal modes of vibration
of cantilevered cylindrical shells. Which mode is observed depends on the
dimensions and material properties of the shell. We observed the first three
modes.
In the first mode, commonly known as the “garden hose mode,” the shells
oscillate side-to-side with the frequency of approximately $15$ Hz.
In the second mode, the surface of the shell bends inwards, obstructing the
fluid flow and causing a large jump in the pressure drop across the nozzle. In
this mode, the shell can vibrate with frequencies from $200$–$700$ Hz,
depending on the volumetric flow rate of air. We observed that the frequency
of oscillation is directly proportional to air flow rate. Additionally, when
the shell vibrates in the second mode with average frequencies ranging from
approximately $350$–$550$ Hz, the vibration is unstable and the oscillation
frequency varies widely between periods. The second mode is the most robust
and can be observed in the largest range of parameters in our shells.
In the third mode of vibration, the circumference of the free end of the shell
is divided into three “flaps” oscillating inwards and outwards. In this mode,
the shells vibrate with frequency of approximately $600$–$1000$ Hz.
The images were captured with a Phantom v5.2 high speed color camera. The
images of the second and third mode were captured using a stroboscopic
technique and the final video is a concatenation of frames taken from
different oscillation periods. Each frame is slightly offset in phase,
yielding a slow-motion effect.
## 2 Acknowledgments
We would like to express our gratitude to Felice Frankel for her help with
cinematography, and to Prof. John Bush who kindly lent us the high speed
camera. We would also like to thank Prof. Pedro Reis and Dr. Arnaud Lazarus
for sharing with us their data on elastic properties of vinylpolysiloxane
elastomers.
## References
* [1] Paidoussis, M. P. and Denise, J.P. Flutter of thin cylindrical shells conveying fluid. _Journal of Sound and Vibration_. 1, 9-26, 1972.
|
arxiv-papers
| 2012-10-15T00:19:14 |
2024-09-04T02:49:36.528730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawel Zimoch, Eliott Tixier, Julia Hsu, Amos Winter, Anette Hosoi",
"submitter": "Pawel Zimoch",
"url": "https://arxiv.org/abs/1210.3861"
}
|
1210.3908
|
# The Mean: Axiomatics, Generalizations, Applications
John E. Gray label=e1]John.E.Gray@navy.mil [ Naval Surface Warfare Center
Dahlgren
Dahlgren Division
Sensor Technology Branch Q31
18444 Frontage Road Suite 327
Dahlgren, VA 22448-5161
Naval Surface Warfare Center Dahlgren Andrew
Vogtlabel=e2]vogta@georgetown.edu [ Department of Mathematics and Statistics
Georgetown University
Washington DC 20057-1233
Georgetown University
###### Abstract
We present an axiomatic approach to the mean and discuss generalizations of
the mean, including one due to Kolmogorov based on the Weak Law of Large
Numbers. We offer examples and counterexamples, describe conventional and
unconventional uses of the mean in statistical mechanics, and resolve an
anomaly in quantum theory concerning apparent simultaneous coexistence of
means and variances of observables. These issues all arise from the familiar
definition of the mean.
60A05,
26E60, 62P35, 82B03, 94A17, 93A10, 81P10,
axiomatics for the mean,
median,
entropy,
Jaynes’ Maximum Entropy Principle,
weak mean,
###### keywords:
[class=AMS]
###### keywords:
and
## 1 Introduction
The most important number summarizing a data set is generally thought to be
the mean. Some have questioned its utility, comparing it unfavorably with the
median, the mode, the midrange. Capitalists and communists used to argue over
whether mean income or median income was the truer measure of citizen well-
being. For another example, see Kosko [9]. The mean is not robust against
outliers: it can be strongly influenced by a single observation. This is both
a strength and a weakness. Kosko objected that not only does a Cauchy random
variable not have a well-defined mean but the average of independent
identically distributed Cauchy random variables is itself a Cauchy variable
with the same distribution and thus averaging does not reduce variability at
all.
Investigators often pursue the quest for a single number or a small set of
numbers that capture the essence of a data set, make multiple data sets
comparable, and provide order to the world of data sets. As data sets get
larger and larger, thanks to the digital explosion, scrutiny of measures that
compress data becomes more important. Candidates, in addition to those
mentioned above, include entropy and various generalized means, but no one has
arrived at measures clearly superior to the mean and its associated measure,
the root mean squared deviation or standard deviation.
In work with sample data the mean is easy to understand, in contrast with
other notions from probability theory - such as independence, conditional
probability, and even probability itself. Some have argued (e.g., de Finetti
[2], Pollard [12], and Whittle [14]) that the mean is the fundamental notion
in probability theory and should occupy the central place in all treatments of
probability.
In this note we review some properties of the mean, consider some
generalizations for cases when the ordinary mean does not exist, and
investigate the significance of the mean in state space theory and quantum
mechanics.
We begin by axiomatizing the notion of sample mean. Along with familiar axioms
for symmetry, homogenity, and translation invariance, we introduce a
_condensation_ axiom that describes the result of replacing arbitrary values
by their sample mean. We then use the Strong Law of Large Numbers to arrive at
the familiar mathematical notion of mean, $E(X)$.
Thereafter we consider generalizations of the mean. These are not needed for
bounded or semi-bounded random variables, but really only for variables that
have heavy-tailed distributions on both right and left, with tails of similar
size. We consider what happens when a random variable is restricted to an
interval $[c-M,c+M]$ and $M$ is allowed to tend to infinity. We state a
theorem (Theorem 3.1) describing the different kinds of behavior possible and
provide examples of each. One generalization, which is due to Kolmogorov, is
what we have chosen to call the _weak mean_ , $E_{w}(X)$, and corresponds
precisely to validity of the Weak Law of Large Numbers. Yet another
generalization, the _doubly weak mean_ , $E_{ww}(X)$, applies to the Cauchy
distribution. We also discuss multipliers that can be applied to a variable
$X$ to finitize the mean in the spirit of Feynman and note the dangers of such
finitizations. Nonetheless, we recognize that attempts to scrutinize the
notion of mean in connection with the Cauchy distribution and other long-
tailed distributions are timely.
Turning to applications, we point out that the mean is a natural tool in state
space theory for the transition from deterministic models to statistical
models. We discuss entropy and observe that although it is regarded as a mean
it is very different from means arising from ordinary observables. We recall
Jaynes’ Maximum Entropy Principle, which seeks to maximize entropy subject to
given values of conventional means.
Lastly, we discuss the role the mean plays in quantum theory, and provide a
precise answer to the question of when the mean and variance exist for a
particular quantum state and a particular quantum observable.
The conclusion, implicit in this discussion, is that the mean is the paramount
measure, of great and wide utility, instructive even when it falls short.
There is little prospect of it losing its longtime preeminence.
## 2 Axiomatics for the Sample Mean and the Strong Law of Large Numbers
Prior to introducing probability measures, let us consider potential axioms
for the mean of a finite set. In this setting, with
$\mathcal{R}=(-\infty,\infty)$, the mean can be thought of as a family of
functions $\\{f_{n}\\}$ for $n\geq 1$ with
$f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$. Its properties include the
following:
M-1) (Homogeneity) $f_{n}(\lambda x_{1},\ldots,\lambda x_{n})=\lambda
f_{n}(x_{1},\ldots,x_{n})$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all $\lambda\in\mathcal{R}$;
M-2) (Symmetry)
$f_{n}(x_{1},\ldots,x_{n})=f_{n}(x_{\sigma(1)},\ldots,x_{\sigma(n)})$ for all
permutations $\sigma$ of the set $\\{1,2,...,n\\}$;
M-3) (Translation Invariance)
$f_{n}(x_{1}+c,\ldots,x_{n}+c)=f_{n}(x_{1},\ldots,x_{n})+c$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all c $\in\mathcal{R}$.
Other properties are the following:
(Positive Homogeneity) $f_{n}(\lambda x_{1},\ldots,\lambda x_{n})=\lambda
f_{n}(x_{1},\ldots,x_{n})$ for all
$(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$ and all $\lambda>0$;
(Nonnegativity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{1}\leq y_{1},\ldots,x_{n}\leq y_{n}$, then $f_{n}(x_{1},\ldots,x_{n})\leq
f_{n}(y_{1},\ldots,y_{n})$;
(Positivity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{1}\leq y_{1},\ldots,x_{n}\leq y_{n}$, and $x_{i}<y_{i}$ for some i, then
$f_{n}(x_{1},\ldots,x_{n})<f_{n}(y_{1},\ldots,y_{n})$.
(Strict Positivity) If for some $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$
$x_{i}<y_{i}$ for all i = 1, …, n, then
$f_{n}(x_{1},\ldots,x_{n})<f_{n}(y_{1},\ldots,y_{n})$.
(Additivity)
$f_{n}(x_{1}+y_{1},\ldots,x_{n}+y_{n})=f_{n}(x_{1},\ldots,x_{n})+f_{n}(y_{1},\ldots,y_{n})$
for all $(x_{1},\ldots,x_{n})\mbox{ and
}(y_{1},\ldots,y_{n})\in\mathcal{R}^{n}$;
The above axioms seem reasonable except for additivity. The measure should be
independent of units, thus homogeneous, and independent of the choice of zero
point, and a function of the set rather than the ordered set. In addition, to
capture characteristics of the data, ordering properties - nonnegativity and
perhaps positivity - are not unreasonable. However, additivity asserts a
relationship between the ordering of two data sets that survives reordering of
one set, and this seems much too restrictive.
Consider a rival measure to the mean, namely, the median. The median of a
finite data set $\\{x_{1},\ldots,x_{n}\\}$ is defined as the midmost of the
numbers when they are arranged in increasing order if $n$ is odd, and half the
sum of the two midmost numbers in such an arrangement if $n$ is even.
The median satisfies homogeneity, symmetry, translation invariance, and
nonnegativity. Furthermore, any fixed convex combination of the mean and the
median other than the median itself satisfies homogeneity, symmetry,
translation invariance, nonnegativity, positivity, and strict positivity.
Indeed, not only the median, but the maximum and the minimum of
$\\{x_{1},\ldots,x_{n}\\}$ (and other rank functions and convex combinations)
satisfy positive homogeneity, symmetry, translation invariance, and
nonnegativity.
###### Proposition 2.1
Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, and translation invariance.
1) If n = 1, then $f_{1}(x)=x$ for all $x\in\mathcal{R}$.
2) If n = 2, then $f_{2}(x_{1},x_{2})=\frac{x_{1}+x_{2}}{2}$ for all
$(x_{1},x_{2})\in\mathcal{R}^{2}$.
Proof: Homogeneity implies that $f_{1}(0)=f_{2}(0,0)=0$. Translation
invariance then indicates that $f_{1}(x)=f_{1}(0+x)=f_{1}(0)+x=x$. When n = 2,
$\displaystyle f_{2}(a,b)$ $\displaystyle=$ $\displaystyle
f_{2}(-\frac{b-a}{2}+\frac{a+b}{2},\frac{b-a}{2}+\frac{a+b}{2})$
$\displaystyle=$ $\displaystyle
f_{2}(-\frac{b-a}{2},\frac{b-a}{2})+\frac{a+b}{2}$ $\displaystyle=$
$\displaystyle(\frac{b-a}{2})f_{2}(-1,1)+\frac{a+b}{2}.$
However, by homogeneity and symmetry $f_{2}(-1,1)=-f_{2}(1,-1)$
$=-f_{2}(-1,1)$, and $f_{2}(-1,1)=0$.
When n = 1 or 2, the median and the mean coincide. However, it is obvious that
they do not coincide in general when n is 3 or larger. Without the requirement
of additivity it is natural to inquire whether there is another suitable
property that will distinguish between the median and the mean. One property
that we consider and reject is that $f_{n}$ shall have continuous partial
derivatives.
###### Proposition 2.2
Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, and translation invariance that has partial derivatives
at each point with the partial derivatives continuous at
$(0,\ldots,0)\in\mathcal{R}^{n}$. Then
$\displaystyle f_{n}(x_{1},\ldots,x_{n})=\frac{x_{1}+\ldots+x_{n}}{n}$
for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.
Proof: If we differentiate the equation $f_{n}(\lambda x_{1},\ldots,\lambda
x_{n})=\lambda f_{n}(x_{1},\ldots,x_{n})$ with respect to $x_{i}$, we obtain:
$\displaystyle\lambda\frac{\partial f_{n}}{\partial x_{i}}(\lambda
x_{1},\ldots,\lambda x_{n})$ $\displaystyle=$
$\displaystyle\lambda\frac{\partial f_{n}}{\partial
x_{i}}(x_{1},\ldots,x_{n}).$
Cancelling $\lambda$ from each side and taking a limit as $\lambda$ approaches
0, we obtain:
$\displaystyle\frac{\partial f_{n}}{\partial x_{i}}(0,\ldots,0)$
$\displaystyle=$ $\displaystyle\frac{\partial f_{n}}{\partial
x_{i}}(x_{1},\ldots,x_{n}).$
Thus all partial derivatives are constant. Since $f_{n}(0,\ldots,0)=0$ by
homogeneity, $f_{n}$ has the form:
$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
a_{1}x_{1}+\ldots+a_{n}x_{n}.$
Symmetry now dictates that $a_{1}=\ldots=a_{n}$ and the fact that
$f_{n}(1,\ldots,1)=f_{n}(0,\ldots,0)+1=0+1=1$ accordingly implies that each
$a_{i}=\frac{1}{n}$.
The continuous differentiability assumption seems to be aimed primarily at
elimination of the median. So we reject it. Instead we offer as an axiom a
different property characteristic of the mean.
M-4) (Condensation) For $n>m$,
$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(f_{m}(x_{1},\ldots,x_{m}),\ldots,f_{m}(x_{1},\ldots,x_{m}),x_{m+1},\dots,x_{n})$
for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.
This property asserts that if a subset of data is replaced by its “mean”, the
grand “mean” is not changed. This is the first property that proposes a
definite relationship between means of sets of different sizes. In view of the
symmetry axiom (M-3), the statement does not really restrict the order of the
subset, and as we shall see shortly the statement is only really needed in
special cases.
###### Proposition 2.3
Let $f_{n}:\mathcal{R}^{n}\rightarrow\mathcal{R}$ be a function satisfying
homogeneity, symmetry, translation invariance, and condensation, that is, M-1,
M-2, M-3, and M-4. Then
$\displaystyle f_{n}(x_{1},\ldots,x_{n})=\frac{x_{1}+\ldots+x_{n}}{n}$
for all $(x_{1},\ldots,x_{n})\in\mathcal{R}^{n}$.
Proof: In view of Proposition 2.1 we need only perform an inductive step
showing that the mean formula holds for $n\geq 3$ when it holds for $n-1$.
Consider
$\displaystyle f_{n}(x_{1},\ldots,x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(\frac{1}{n-1}(x_{1}+\ldots x_{n-1}),\ldots,\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}),x_{n})$ $\displaystyle=$ $\displaystyle
f_{n}(0,\ldots,0,x_{n}-\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}))+\frac{1}{n-1}(x_{1}+\ldots x_{n-1})$ $\displaystyle=$
$\displaystyle(x_{n}-\frac{1}{n-1}(x_{1}+\ldots
x_{n-1}))f_{n}(0,\ldots,0,1)+\frac{1}{n-1}(x_{1}+\ldots x_{n-1})$
$\displaystyle=$ $\displaystyle a_{1}x_{1}+...+a_{n}x_{n}.$
This shows that $f_{n}$ is a linear function of $x_{1},\ldots,x_{n}$. It now
follows from Proposition 2.2 that it is the mean.
The proof of Proposition 2.3 requires that M-4 holds in the case when $m=n-1$.
In fact we can get by with the assumption that M-4 holds when $m=2$. It is
easy to see that in this case M-4 also holds for $m=2^{k}$. Now set
$n=2^{k}+j$, where $0\leq j<2^{k}$. If $j=0$,
$f_{n}(x_{1},...,x_{n})=f_{n}(c,...,c)=cf_{n}(1,..,1)$ where
$c=(x_{1}+...+x_{n})/n$. If $j>0$, then
$f_{n}(x_{1},...,x_{n})=f_{n}(c,...,c,x_{m+1},...x_{n})$ where $m=2^{k}$ and
$c=(x_{1}+...+x_{m})/m$. We now replace $x_{1},...,x_{m}$ by
$x_{1}^{\prime},...,x_{m-j}^{\prime},0,...,0$ so that
$c=(x_{1}+...+x_{m})/m=(x_{1}^{\prime}+...+x_{m-j}^{\prime}+0+...+0)/m$. Using
the symmetry and homogeneity axioms, we obtain:
$f_{n}(x_{1},...,x_{n})=f_{n}((mc+x_{m+1}+...+x_{n})/m,...,(mc+x_{m+1}+...+x_{n})/m,0,...0)=((mc+x_{m+1}+...+x_{n})/m)f_{n}(1,...,1,0,...,0)=(x_{1}+...+x_{n})/m)f_{n}(1.,,,.1,0,...,0)$.
Thus we have established that $f_{n}$ is linear in $x_{1},...,x_{n}$. By
Proposition 2.3 $f_{n}$ is the ordinary mean.
A further note on axiomatics is that the translation invariance axiom can be
replaced by $f_{n}(1,...,1)=1$ if we also assume that
$f_{2}(x_{1},x_{2})=(x_{1}+x_{2})/2$.
To pass from the sample mean of a finite set to the usual general notion of
mean, we introduce a real-valued random variable $X$. We suppose that
associated with $X$ is a Borel probability measure $P_{X}$ taking each Borel
subset A of the real numbers to:
$\displaystyle P_{X}(A)=\mbox{ the probability that X belongs to the set A}.$
The mean of X, denoted by E(X) or $\mu_{X}$, is defined when $x$ is integrable
with respect to $P_{X}$ to be:
$\displaystyle E(X)=\int_{\mathcal{R}}xP_{X}(dx).$
One direction of the remarkable Strong Law of Large Numbers (see Pollard [12,
p. 78 and pp. 37-8]) states that if $\\{X_{n}\\}$ is a sequence of independent
random variables with common distribution $P_{X}$ and there exists a constant
$m$ such that
$\displaystyle\frac{X_{1}+\ldots+X_{n}}{n}\mbox{ converges almost surely to
}m$
as $n\rightarrow\infty$, then each $X_{n}$ has mean $m$. Here “almost surely”
means outside a set of measure zero in the countably infinite product space
induced by the measure $P_{X}$ (see [12, pp. 99-102]). More briefly, if sample
means of independent copies of $X$ settle down to something, then that
something is $E(X)$. This can be regarded as the motivation for the transition
from the sample mean to the mathematical mean $E(X)$. The general notion of
mean is derived from the finitary notion considered earlier.
The other direction of the Strong Law of Large Numbers asserts that if $E(X)$
exists, then the sample mean of $n$ identical independent copies of $X$
converges almost surely to $E(X)$ as $n$ tend to infinity. For a proof of both
directions of the Strong Law, see [12, pp. 95-102, p. 105]. For an alternate
proof due to N. Etemadi, see [12, pp. 106-7].
The transition here from finite samples to infinite populations distinguishes
the deductive method from the inductive method. While true science deals
comfortably with induction based on finite samples, the deductive method of
the Greeks (and Isaac Newton) relies on axioms whose relationship to reality
is only approximate and always contingent.
Indeed, in using the Strong Law of Large Number we are admittedly introducing
the full panoply of probability theory. It is possible, as noted in the
Introduction, to represent all of probability theory using the mean as the
primitive notion. Thus $P_{X}(A)$ can be defined as $E(\chi_{A}(X))$, the mean
of $\chi_{A}(X)$, where $\chi_{A}(X)$ is the random variable that equals 1
when $X$ is in $A$ and $0$ when $X$ is not in $A$. However, since in what
follows we plan to use probability theory in its conventional form (i.e.,
according to the axioms of Kolmogorov [6]), we see no reason to restate
measure-theoretic facts in terms of the mean as primitive. Indeed, a reason
not to do so is that the concept of independence, which is also fundamental in
probability, is awkward when expressed exclusively in terms of means.
## 3 Extending the Mean
When $x$ is not integrable with respect to $P_{X}$, the notion $E(X)$ above is
inapplicable and we must rely on other notions of mean. Richard Feynman was
famous for his integration tricks, and some of these are recorded in the book
of Mathews and Walker [11], based on lectures Feynman gave at Cornell.
Feynman’s tricks partly motivated our investigation.
Perhaps the most obvious generalization is the following:
$\displaystyle\mbox{Let
}L(c)=\lim_{M\rightarrow\infty}\int_{[c-M,c+M]}\,x\,P_{X}(dx)$
for a real number $c$.
By the Lebesgue Dominated Convergence Theorem this notion coincides with the
ordinary mean when $x$ is integrable with respect to $P_{X}$. Kolmogorov [6,
p. 40], in his great foundational work, noted this option in the case when
$c=0$ and observed that it does not require integrability of $|x|$. Indeed if
$X$ is a random variable obeying the Cauchy distribution
$f(x)=1/\pi(1+x^{2})$, then $X$ satisfies $L(c)\equiv 0$ for any choice of
$c$.
We mention two related notions of mean:
1. L-1)
$\lim_{M\rightarrow\infty}\int_{[a-M,b+M]}\,xP_{X}(dx)\mbox{, and}$
2. L-2)
$\lim_{\min{\\{M,K\\}}\rightarrow\infty}\int_{[a-M,b+K]}\,xP_{X}(dx),$
where $a\leq b$.
It is easily seen that L-1 coincides with $L((a+b)/2)$ since
$[a-M,b+M]=[\frac{a+b}{2}-(M+\frac{b-a}{2}),\frac{a+b}{2}+(M+\frac{b-a}{2})].$
As for L-2, we have the following result.
###### Proposition 3.1
Let X be a random variable with probability measure $P_{X}$. Then for some $a$
and $b$ with $a\leq b$
$\lim_{\min{\\{K,M\\}}\rightarrow\infty}\int_{[a-M,b+K]}\,xP_{X}(dx)$
exists if and only if $x$ is integrable with respect to $P_{X}$.
Proof: If $x$ is integrable on $\mathcal{R}$, the limit exists and equals the
mean of $x$ by Lebesgue’s Dominated Convergence Theorem. Conversely, if the
limit exists, then
$0\leq\int_{(b+K,b+K^{\prime}]}x\,P_{X}(dx)<\epsilon$
for $K<K^{\prime}$, both sufficiently large, and any given $\epsilon$.
Likewise
$-\epsilon<\int_{[a-M^{\prime},a-M)}x\,P_{X}(dx)\leq 0$
for $M<M^{\prime}$, both sufficiently large. Fatou’s Lemma or Levi’s Theorem
[4, p. 172] thus implies that
$0\leq\int_{(b+k,\infty)}x\,P_{X}(dx)\leq\epsilon,\hskip
14.45377pt-\epsilon\leq\int_{(-\infty,a-M)}x\,P_{X}(dx)\leq 0$
and thus $x$ is integrable on $[0,\infty)$ as well as $(\infty,0]$ and so is
integrable on $\mathcal{R}=(-\infty,\infty)$.
So, when L-2 exists, it coincides with E(X).
We now return to the study of $L(c)$. We shall allow $-\infty\leq
L(c)\leq\infty$. This gives us a bit more flexibility in characterizing what
can happen.
###### Lemma 3.1
Let X be a random variable with probability measure $P_{X}$, and let $c_{1}$
and $c_{2}$ be real numbers with $c_{1}<c_{2}$. Then there are three
possibilities:
i) If $L(c_{1})$ exists in $[-\infty,\infty]$, then
$L(c_{1})\leq\liminf_{M\rightarrow\infty}\int_{[c_{2}-M,c_{2}+M]}\,x\,P_{X}(dx);$
ii) If $L(c_{2})$ exists in $[-\infty,\infty]$, then
$L(c_{2})\geq\limsup_{M\rightarrow\infty}\int_{[c_{1}-M,c_{1}+M]}\,x\,P_{X}(dx);$
iii) If $L(c_{1})$ and $L(c_{2})$ both exist in $[-\infty,\infty]$ , then
$L(c_{1})=L(c_{2})$.
Proof: Suppose $c_{1}<c_{2}$. Then
$\displaystyle\int_{[c_{2}-M,c_{2}+M]}\,xP_{X}(dx)=\int_{[c_{1}-M,c_{1}+M]}\,xP_{X}(dx)$
$\displaystyle+\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)-\int_{[c_{1}-M,c_{2}-M)}\,xP_{X}(dx).$
The second and third terms on the right are both non-negative and accordingly
i) and ii) follow. In the case of iii), note that i) and ii) imply that if
both $L(c_{1})$ and $L(c_{2})$ exist, then $L(c_{1})\leq L(c_{2})$.
If $L(c_{2})-L(c_{1})>0$, then there is a positive constant $K$ (for example,
any positive number $<L(c_{2})-L(c_{1})$) such that for $M$ sufficiently
large:
$\displaystyle
K<\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)-\int_{[c_{1}-M,c_{2}-M)}\,xP_{X}(dx)$
$\displaystyle\leq$
$\displaystyle(c_{2}+M)P_{X}((c_{1}+M,c_{2}+M])+(M-c_{1})P_{X}([c_{1}-M,c_{2}-M))$
$\displaystyle\leq$
$\displaystyle(M+d)(P_{X}((c_{1}+M,c_{2}+M]\cup[c_{1}-M,c_{2}-M))$
where $d=\max{\\{|c_{2}|,|c_{1}|\\}}$. Thus
$\frac{K}{M+d}<P_{X}((c_{1}+M,c_{2}+M]\cup[c_{1}-M,c_{2}-M)).$
Now replace $M$ by $M_{j}=M+j(c_{2}-c_{1})$ for each integer $j\geq 0$ to get:
$\frac{K}{M_{j}+d}<P_{X}((c_{1}+M_{j},c_{2}+M_{j}]\cup[c_{1}-M_{j},c_{2}-M_{j})).$
Summing over these inequalities and noting that $c_{2}+M_{j}=c_{1}+M_{j+1}$
and $c_{1}-M_{j}=c_{2}-M_{j+1}$, we obtain:
$\infty=\sum_{j=0}^{\infty}\frac{K}{M+d+j(c_{2}-c_{1})}\leq
P_{X}((c_{1}+M,\infty)\cup(-\infty,c_{2}-M))\leq 1$
for a contradiction. Thus, this case is eliminated. So $L(c_{1})=L(c_{2})$.
###### Theorem 3.1
Let X be a random variable with probability measure $P_{X}$. Then exactly one
of the following possibilities holds:
i) $L(c)$ does not exist in $[-\infty,\infty]$ for any real number $c$;
ii) $L(c)$ exists in $(-\infty,\infty)$ for exactly one real number $c$;
iii) $L(c)$ exists in $[-\infty,\infty]$ for all real numbers $c$ and is
independent of $c$;
iv) there is a number $c_{0}$ such that $L(c)=\infty$ for $c>c_{0}$ and $L(c)$
does not exist for $c<c_{0}$; or
v) there is a number $c_{0}$ such that $L(c)=-\infty$ for $c<c_{0}$ and $L(c)$
does not exist for $c>c_{0}$.
Proof: By Lemma 3.1 it suffices to show what happens when $L(c_{2})=L(c_{1})$
is finite. In this case the last two terms in the equation at the beginning of
the proof of Lemma 3.1 each tend to $0$ as $M$ tends to infinity. By a change
of variable, we obtain for the positive number $c=c_{2}-c_{1}$.
$\lim_{M\rightarrow\infty}\int_{(M,c+M]}xP_{X}(dx)=\lim_{M\rightarrow\infty}\int_{[-M-c,-M)}xP_{X}(dx)=0;$
Assume $0<d<c$ and $M\geq 0$. Then
$0\leq\int_{(M,d+M]}xP_{X}(dx)\leq\int_{(M,c+M]}xP_{X}(dx)$
and
$\int_{[-M-c,-M)}xP_{X}(dx)\leq\int_{[-M-d,-M)}xP_{X}(dx)\leq 0$
Thus if ii) holds for $c$, it holds for $d$. On the other hand, if ii) holds
for $c$ it also holds for $nc$ where $n$ is any fixed positive integer since
$\int_{(M,nc+M]}xP_{X}(dx)=\sum_{j=1}^{n}\int_{((j-1)c+M,jc+M]}xP_{X}(dx)$
and
$\int_{[-M-nc,-M)}xP_{X}(dx)=\sum_{j=1}^{n}\int_{[-M-(n+1-j)c,-M-(n-j)c)}xP_{X}(dx),$
and if the $M^{\prime}s$ are chosen far enough out so that the individual
integrals are closer to zero than $\epsilon/n$, the sum integral is within
$\epsilon$ of $0$. Finally since any positive real number $d$ is smaller than
$nc$ for some positive integer $n$, all cases are covered. Accordingly, ii)
implies iii).
The argument for i) implies ii) can now be used to show that for any two real
numbers $c_{1}$ and $c_{2}$, if either $L(c_{1})$ or $L(c_{2})$ exists in
$[-\infty,\infty]$, then the other exists and equals it since the
approximating integrals differ by two integrals on intervals of length
$|c_{1}-c_{2}|$ that tend to zero as $M$ tends to infinity. Thus iii) implies
iv).
We give some examples to illustrate that each of the possibilities enumerated
in Theorem 3.1 can occur.
Consider a random variable $X$ whose probability measure is of the form
$P_{X}(A)=\sum_{n=1}^{\infty}(\frac{1}{2^{2n}}\delta_{2^{2n}}(A)+\frac{1}{2^{2n-1}}\delta_{-2^{2n-1}}(A))$
where $\delta_{z}$ is the (Dirac) probability measure whose value is $1$ on
any Borel subset $A$ of $\mathcal{R}$ that contains the real number $z$ and
whose value is zero otherwise. The sum of the nonzero values is one, so this
obviously defines a probability measure. However the integral of $x$ over the
interval $[c-M,c+M]$ is the difference between the size of the first set and
the size of the second set below:
$\displaystyle\,\mbox{ The size of the set }\\{n\mbox{ : }1\leq
n,2^{2n}\leq(c+M)\\}$ $\displaystyle=$
$\displaystyle\lfloor{\frac{\log{(c+M)}}{2\log{2}}}\rfloor$
$\displaystyle\mbox{ The size of the set }\\{n\mbox{ : }1\leq
n,2^{2n-1}\leq(M-c)\\}$ $\displaystyle=$
$\displaystyle\lfloor{\frac{\log{(M-c)}}{2\log{2}}+\frac{1}{2}}\rfloor$
where $\lfloor\,\,\rfloor$ is the floor function. For fixed $c$ and
sufficiently large $M$ the difference of the above quantities can assume the
values $0$ and $-1$ and the integral does not settle down to either one. This
is an instance of Theorem 3.1, part i).
A random variable $X$ can also be defined with probability measure of the form
$P_{X}(A)=\sum_{n=1}^{\infty}\frac{1}{2^{n+1}}(\delta_{2^{n}}(A)+\delta_{(-2^{n})}(A)).$
So $P_{X}$ is concentrated at the points $\pm 2^{n}$ and assigns probability
$1/(2^{n+1})$ to such points. For this measure, $L(0)$ equals $0$ by symmetry.
However, $L(c)$ does not exist for other choices of $c$. If $c$ is positive,
the integral of $x$ over the closed interval $[c-M,c+M]$ reduces to its
integral over the open interval $(M-c,M+c]$ and this integral oscillates
between $0$ and $\frac{1}{2}$ for large $M$ depending on whether $2^{n}$ is in
the interval $(M-c,M+c]$ or not. Similar behavior occurs when $c<0$. This
example is an instance of Theorem 3.1, part ii).
Now consider a random variable $X$ having a probability density (with respect
to Lebesgue measure on the real line) of the form
$f(x)=\left\\{\begin{array}[]{ll}\frac{1}{1+Cx^{a}}\mbox{ if $x\geq 0$}&\\\
\frac{1}{1+D|x|^{b}}\mbox{ if $x<0$}&\end{array}\right.$
where $a$ and $b$ are numbers in $(1,2)$ and $C$ and $D$ are suitable positive
constants that guarantee that the density integrates to $1$. Notice that this
random variable satisfies iii) of Proposition 3.1. It is easy to see that
$L(c)\equiv\infty$ for all $c$ or $-\infty$ for all $c$ according as $b>a$ or
$a>b$. The example illustrates part iii) of Theorem 3.1 (as does the Cauchy
distribution with $L(c)\equiv 0$) .
The probability measure
$P_{X}(A)=\sum_{n=1}^{\infty}(\frac{2^{n-1}}{3^{n+1}}\delta_{3^{n}}(A)+\frac{2^{n-2}}{3^{n+1}}\delta_{-3^{n}}(A))$
illustrates part iv) of Theorem 3.1. If $c\geq 0$, the integral of $x$ over
$[c-M,c+M]$ is given by:
$\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
c+M\\}}\frac{2^{n-1}}{3}-\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M-c\\}}\frac{2^{n-2}}{3},$
and this expression has the value $(2^{n_{0}-1}-2^{-1})/3$ or
$(2^{n_{0}-1}+2^{n_{0}-2}-2^{-1})/3$ where $n_{0}\approx(\log{c+M})/\log{3}$
for large $M$. Since $M$ and $n_{0}$ tend to infinity together, it follows
that $L(c)\equiv\infty$ for $c\geq 0$.
On the other hand, if $c=-d$ where $d>0$, the integral of $x$ over
$[c-M,c+M]=[-M-d,M-d]$ is given by:
$\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M-d\\}}\frac{2^{n-1}}{3}-\sum_{\\{n\mbox{ : }1\leq n,3^{n}\leq
M+d\\}}\frac{2^{n-2}}{3},$
and this reduces to $(2^{n_{0}-1}-2^{-1})/3$ or to $(-1)/6$ for large $M$
where $n_{0}\approx(\log{M-d})/\log{3}$ depending on whether a positive
integer lies in the interval $(\log{M-d}/\log{3},\log{M+d}/\log{3}]$ or not.
Thus, $L(c)$ does not exist for $c<0$.
A final example (also for part iv) of Theorem 3.1 is the case where the
probability measure is given by:
$P_{X}(A)=K\sum_{n=1}^{\infty}(\frac{2^{n}}{3^{n}+(1/n)}\delta_{3^{n}+(1/n)}(A)+\frac{2^{n-1}}{3^{n}}\delta_{-3^{n}}(A)).$
Here $K$ is a suitably chosen positive normalizer, which is easily seen to be
smaller than $1/3$. For $c>0$, the integral of $x$ over $[c-M,c+M]$ is
$K\sum_{\\{n\mbox{ : }1\leq n,(3^{n}+1/n)\leq M+c\\}}2^{n}-K\sum_{\\{n\mbox{ :
}1\leq n,3^{n}\leq M-c\\}}2^{n-1},$
and this reduces to $K(2^{n_{0}}-1)$ for $M$ sufficiently large where $n_{0}$
is the largest integer such that $3^{n_{0}}\leq M-c$. Since $n_{0}$ and $M$
tend to infinity together, $L(c)\equiv\infty$ for all $c>0$.
When $c=0$, the integral of $x$ over $[-M,M]$ reduces to $K(2^{n_{0}}-1)$ or
to $-K$ where $n_{0}$ is the largest integer such that
$3^{n_{0}}+(1/n_{0})\leq M$ and the first or second reduction occurs according
as $M<3^{n_{0}+1}$ or not. Thus $L(0)$ does not exist. By Theorem 3.1 $L(c)$
does not exist for $c<0$.
Other cases arising in Theorem 3.1, such as part v), are obtained by modifying
the examples above, e.g., replacing $X$ by $-X$ or by $X+a$.
## 4 Weak Means and Multipliers
One of the implications of Theorem 3.1 is that if $L(c)$ exists for more than
one choice of $c$ and is finite in some case then it is finite for all $c$ and
is independent of $c$. The case of the Cauchy distribution shows that this can
happen without the ordinary mean existing. Accordingly for a random variable
$X$, we define the doubly weak mean of $X$, denoted by $E_{ww}(X)$, to be the
common value of $L(c)$ for all $c$ when this common value exists and is in
$(-\infty,\infty)$.
We also introduce an intermediate notion due to Kolmogorov between the
ordinary mean and the doubly weak that motivates our terminology. The weak
mean of $X$, denoted by $E_{w}(X)$, is defined as follows: $E_{w}(X)$ is the
quantity $L(0)$ provided the latter exists in $(-\infty,\infty)$ and provided
$\lim_{n\rightarrow\infty}nP_{X}(|X|>n)=0$.
The following proposition is due to Kolmogorov. It indicates that existence of
the weak mean coincides precisely with the existence of a number for which the
Weak Law of Large Numbers holds.
###### Proposition 4.1 (Kolmogorov, 1928)
Let $X$ be a random variable. Suppose that $\\{X_{1},\cdots,X_{n},\cdots\\}$
are independent identically distributed copies of $X$ with $P_{n}$ the n-fold
product distribution. Then there is a real number $m$ such that for each
$\epsilon>0$
$\lim_{n\rightarrow\infty}{P_{n}(|\frac{X_{1}+...+X_{n}}{n}-m|>\epsilon)}=0$
if and only if $X$ has weak mean $E_{w}(X)=m$.
Proof: See [6, p. 65], [7], and [8, Theorems XII and XIII].
###### Corollary 4.1
Let $X$ be a random variable.
i) If $X$ has a mean, then $X$ has a weak mean and $E(X)=E_{w}(X)$;
and
ii) if $X$ has a weak mean, then $X$ has a doubly weak mean and
$E_{w}(X)=E_{ww}(X)$.
Proof: In case $X$ has a mean, then the identity function $x\mapsto x$ is
integrable with respect to the probability measure $P_{X}$ on the real line.
In particular the tail integrals
$\int_{[n,\infty)}xP_{X}(dx)\mbox{ and }\int_{(-\infty,-n]}xP_{X}(dx)$
tend to zero as $n$ tends to infinity. Since the absolute values of these
integrals are larger respectively than $nP_{X}(X\geq n)$ and
$nP_{X}(X\leq-n)$, it follows that $\lim_{n\rightarrow\infty}nP_{X}(|X|>n)=0$.
Likewise by Lebesgue’s Dominated Convergence Theorem, $L(0)=E(X)$, Suppose $X$
has a weak mean. Then if $c_{1}<c_{2}$ and $\epsilon>0$ and a sufficiently
large $M$ are given,
$\displaystyle 0\leq\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)$ $\displaystyle\leq$
$\displaystyle(c_{2}+M)P_{X}((c_{1}+M,c_{2}+M])$ $\displaystyle\leq$
$\displaystyle(c_{2}-c_{1}+c_{1}+M)P_{X}(|X|>c_{1}+M)$ $\displaystyle\leq$
$\displaystyle(c_{2}-c_{1}+c_{1}+M)P_{X}(|X|\geq n)$ $\displaystyle\leq$
$\displaystyle(\frac{c_{2}-c_{1}}{n}+\frac{c_{1}+M}{n})\epsilon$
where $n=\lfloor{c_{1}+M}\rfloor$. For $M$ sufficiently large, the right side
is as close to $\epsilon$ as we like. Thus
$\lim_{M\rightarrow\infty}\int_{(c_{1}+M,c_{2}+M]}\,xP_{X}(dx)=0.$
Similarly,
$\lim_{M\rightarrow\infty}\int_{[c_{1}-M,c_{2}-m)}\,xP_{X}(dx)=0.$
Accordingly from the first equation in the proof of Lemma 3.1, it follows that
when one of $L(c_{2})$ or $L(c_{1})$ exists and is finite, the other exists
and is equal to it. Since $L(0)=m$, it follows that $L(c)$ exists for all $c$,
$L(c)\equiv m$ and $m$ is the doubly weak mean of $X$.
Kolmogorov in [6, p. 66] gives an example where the Weak Law holds but the
Strong Law does not. Cauchy random variables have $L(c)$ existing for all $c$,
independent of $c$, but violate the Weak Law by not decaying rapidly enough at
infinity. Thus the mean, weak mean, and doubly weak mean are strictly distinct
notions.
We make one more observation on generalizations of the mean, based on using
multipliers to attempt to finitize the mean. These multipliers are a type of
“mollifier.” Usually mollifiers are used to aid approximation of the delta
function and to smooth functions, but another use is to regularize behavior at
$\pm\infty$. The idea is to introduce a function $\phi_{\lambda}(x)$ that
depends on a parameter $\lambda$ so that $x\mapsto\phi_{\lambda}(x)x$ is
integrable with respect to $P_{X}$ for $\lambda\neq\lambda_{0}$ and
$\phi_{\lambda}(x)\rightarrow 1$ for a. e. x as
$\lambda\rightarrow\lambda_{0}$. In the case of L(c), the multiplier can be
taken to be
$\phi_{\lambda}(x)=\chi_{[c-1/\lambda,c+1/\lambda]}(x)$
where $\chi_{A}$ is the characteristic function of the set $A$ and
$\lambda=1/M$.
Multipliers, and indeed other straight-forward generalizations of the mean
including the weak and doubly weak mean, are useful only when the following
equations hold:
$\displaystyle\int_{[0,\infty)}xP_{X}(dx)$ $\displaystyle=$
$\displaystyle\infty$ $\displaystyle\int_{(-\infty,0]}xP_{X}(dx)$
$\displaystyle=$ $\displaystyle-\infty.$
If neither of these equations holds, x is integrable and the mean is well-
defined. If only the first equation holds, the mean is $+\infty$, and if only
the second equation holds, the mean is $-\infty$. If both equations hold, then
there is some room for maneuver. $L(c)$ cannot exist finitely unless the
infinities on each end are of the same order. If for example $P_{X}$ is given
by a density function $f$ with respect to Lebesgue measure such that $f(x)$
decays as $1/x^{2}$ as $x\rightarrow\infty$ and decays as $1/|x|^{3/2}$ as
$x\rightarrow-\infty$, then $L(c)\equiv-\infty$ for all $c$.
Multipliers offer possibilities for extending the notion of the mean. They can
be of use in such activities as renormalization where the aim is to
reinterpret integrals to make them finite. In our case we set:
$E_{mult}(X)=\lim_{\lambda\rightarrow\lambda_{0}}{E(\phi_{\lambda}(X)X)}$
provided this limit exists. This method is used to “evaluate” the integrals of
$\sin{bx}$ and $\sin{x}/x$ on $[0,\infty)$ in [11, p. 60 and p. 91].)
However, there are dangers that the following example illustrates.
Define a function $\phi_{\lambda,c}$ for $\lambda>0$ and $c$ in $\mathcal{R}$
by:
$\phi_{\lambda,c}(x)=\left\\{\begin{array}[]{ll}e^{-\lambda x}&\mbox{if
$x>0$}\\\ e^{\lambda x}(1+\pi c\lambda x)&\mbox{if $x<0$.}\end{array}\right.$
Here $c$ is an arbitrary constant. Evidently $\phi_{\lambda,c}$ is a well-
behaved function, integrable and dying off at $\pm\infty$. Also
$\\{\phi_{\lambda,c}\\}$ converges pointwise to the constant function one as
$\lambda$ tend to $0^{+}$ with fixed $c$.
Suppose we use this family of functions as a multiplier to determine a mean
for a variable obeying the Cauchy distribution. Let $m(\lambda,c)$ be defined
by:
$m(\lambda,c)=\int_{-\infty}^{\infty}\phi_{\lambda,c}(x)\frac{x}{\pi(1+x^{2})}\,dx=\int_{-\infty}^{0}\frac{c\lambda
e^{\lambda x}x^{2}}{1+x^{2}}\,dx=cm(\lambda,1).$
Now
$\displaystyle 1=e^{\lambda x}|_{-\infty}^{0}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\lambda e^{\lambda x}\,dx$
$\displaystyle\geq\int_{-\infty}^{0}\frac{\lambda e^{\lambda
x}x^{2}}{1+x^{2}}\,dx$ $\displaystyle=$ $\displaystyle
m(\lambda,1)=\int_{0}^{\infty}\frac{\lambda e^{-\lambda x}x^{2}}{1+x^{2}}\,dx$
$\displaystyle\geq\int_{K}^{\infty}\frac{\lambda e^{-\lambda
x}x^{2}}{1+x^{2}}\,dx$ $\displaystyle\geq$
$\displaystyle\frac{K^{2}e^{-\lambda K}}{1+K^{2}}$
for any positive real number $K$. Thus
$1\geq\limsup_{\lambda\rightarrow
0+}m(\lambda,1)\geq\liminf_{\lambda\rightarrow
0+}m(\lambda,1)\geq\frac{K^{2}}{1+K^{2}}.$
Letting $K$ tend to infinity, we find that $\lim_{\lambda\rightarrow
0+}m(\lambda,1)=1$.
Hence the multiplier-induced mean of the standard Cauchy distribution is:
$\displaystyle E_{mult}(X)=\lim_{\lambda\rightarrow
0+}\int_{-\infty}^{\infty}\phi_{\lambda,c}(x)\frac{x}{\pi(1+x^{2})}\,dx$
$\displaystyle=$ $\displaystyle\lim_{\lambda\rightarrow 0+}m(\lambda,c)$
$\displaystyle=\lim_{\lambda\rightarrow 0+}cm(\lambda,1)$ $\displaystyle=$
$\displaystyle c\lim_{\lambda\rightarrow 0+}m(\lambda,1)=c.$
However, $c$ was arbitrary depending on the choice of the multiplier!
Although some may consider the Cauchy distribution anomalous, we remind the
reader that its legitimacy and importance stem in part from the fact that it
is the quotient of two independent standard normal random variables. It has
application in physics under the name of the Lorentz distribution. Indeed
long-tailed and counter-intuitive distributions are increasingly important in
recent times (see Gumble [3] or Taleb [13]) in financial mathematics, the
study of natural and man-made disasters, and computer network analysis.
Extending the notion of mean to such distributions, and investigating the
limits of the notion of mean in such settings, are among the ways of moving
beyond the normal regime.
## 5 State Space Theory
State Space Theory or System Theory is widely used to provide a mathematical
description of physical systems including those of classical mechanics as well
as other systems such as biological and social systems. The state of the
system at any time is taken to be an element of a set $S$ called state space.
The evolution of the state is given by a function $T_{t}:S\rightarrow S$
taking the state $s$ at time $0$ to the state $T_{t}(s)$ at time t. A (real-
valued) _observable_ is any function $f:S\rightarrow R$ which assigns to each
state $s$ a number $f(s)$ (see Mackey[10]). All observables may be determined
from the state, and indeed the state can be viewed as a maximal independent
set of observables that characterize the system at a given time. The dynamic
evolution of the state is deterministic and time may be taken to be either
discrete or continuous. Evolution of an observable $f$ can be expressed by
$t\mapsto f\circ T_{t}(s)$, i.e., the value of the observable at time t is
obtained by applying the observable function to the state at time t.
A familiar example of the state space approach is Hamiltonian mechanics. The
state space in this case is phase space, and a state is a 2n-tuple
$(q,p)=(q_{1},...,q_{n},p_{1},...,p_{n})$ consisting of position coordinates
$q_{i}$ and momentum coordinates $p_{i}$. The evolution is
$T_{t}(q,p)=(q(t),p(t))$, where the latter is the solution to Hamilton’s
equations with initial data $(q(0),p(0))=(q,p)$:
$\displaystyle\frac{dq_{i}}{dt}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial q_{i}}$
for $i=1,2,...,n$. Here $H(q,p)$ is the Hamiltonian function of the system,
which is assumed to be a continuously differentiable function on state space
representing the total energy of the system. The function $H$ is an example of
an observable, as are the position and momentum coordinates, angular momenta
$q_{i}p_{j}-q_{j}p_{i}$, et cetera. A differentiable observable $f$ evolves
according to the equation:
$\frac{df}{dt}=\sum_{i=1}^{n}(\frac{\partial f}{\partial q_{i}}\frac{\partial
H}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial H}{\partial
q_{i}}),$
the right-hand side being the definition of the Poisson bracket $[f,H]$, under
which operation $C^{\infty}$ observables form a Lie algebra.
Given a deterministic state space it is natural to pass to a statistical
setting as follows. We replace the old states $s$ by new states that are
(Borel) probability measures $P$ on the state space $S$. The old observables
$f$ on the original state space are replaced by new observables that are the
means of the old observables with respect to the probability measure $P$. Thus
for any original state space observable $f$, the map
$P\mapsto E_{P}(f)$
defines an observable on the set of probability measures. If $f$ is bounded,
this observable is defined for all probability measures. If not, it is defined
for those measures with respect to which $f$ is integrable.
If the only observables allowed were obtained in this manner, this would
appear to be a severe limitation. However, the variance of f and all moments
of f can themselves be regarded as means of original observables. Indeed, even
the probability distribution for $f$ can be regarded as a mean. This is
because on a Borel subset $A$ of the reals, the probability that $f$ takes a
value in $A$ is given by $E_{P}(\chi_{A}\circ f)$, where $\chi_{A}$ is the
characteristic function of the set $A$.
The evolution of the probabilistic state can be induced by an underlying
deterministic evolution. The probability measure at time $t$, $P_{t}$, is
given by $P_{t}(A)=P(T_{-t}(A))$ where $A$ is any (Borel) subset of $S$. This
permits us to talk about the evolution of observables since the mapping
$t\mapsto E_{P_{t}}(f)$ describes such an evolution. In the Hamiltonian
formalism phase space has a natural 2n-dimensional Lebesgue measure $\lambda$
called Liouville measure with infinitesimal volume element
$dq_{1}...dq_{n}dp_{1}...dp_{n}$, and $\lambda(T_{t}(A))=\lambda(A)$ for all
Borel subsets $A$ of $S$ and all times t. Dynamics in phase space can be
thought of as a fluid flow that permits change of shape but no change in
volume. The probability state $P$ can often be taken to be the integral of a
probability density function $\rho(q,p)$ with respect to $\lambda$. At the
other extreme $P$ can be taken to be a delta function
$\delta(q-q_{0})\delta(p-p_{0})$, which reduces to the deterministic theory
with state $s=(q_{0},p_{0})$. The probabilistic setting also permits us to
abandon the deterministic evolution $\\{T_{t}\\}$ and work with a stochastic
evolution exclusively, e.g., one of Markov type.
A use of means in state space theory that we have not touched on here relates
to ergodic theory, in which time averages of observables over trajectories are
compared with averages over state space regions using a suitable normalized
volume measure.
The essential point is that means provide the transition from classical
observables for deterministic systems to statistical observables for
stochastic systems.
## 6 Entropy
A subtlety occurs in statistical mechanics that is not present in ordinary
probability theory. An observable is commonly defined as a real-valued
function of the state, and in statistical mechanics the state is a probability
measure $P$ on state space. Thus any real-valued function of $P$ can be taken
to be an observable, e. g., $P\mapsto P(B)$ is an observable where $B$ is any
fixed Borel set in the state space $S$. This observable is an expected value
since $P(B)=E_{P}(\chi_{B})$. However, not all observables arise as expected
values of original observables. The most familiar example of such an
observable is the entropy function, which can be interpreted as an expected
value (mean) but is not a conventional mean.
To avoid certain difficulties associated with the continuous case we will
confine our attention to the case where the underlying state space is a finite
set. Let $S$ be a finite state consisting of n states. A classical observable
is a function $f:S\rightarrow(\infty,\infty)$. A discrete classical evolution
might be a function $T:S\rightarrow S$ such that if $i$ is the state at a
given time then $T(i)$ is the state one time unit later. (A continuum of times
presents a problem for deterministic evolution in a finite state space,
although that problem does not arise in the probabilistic setting.)
When we pass to a statistical notion of state, we arrive at a probability
vector $p=(p_{1},...,p_{n})$ where $p_{i}$ is the probability that the system
is in the deterministic state $i$. We can now form expected values of
classical observables $f$, i.e.,
$E(f)=\sum_{i}p_{i}f(i)$
as noted before. We can also form such expressions as the entropy:
$H(p)=\sum_{i}p_{i}\log(\frac{1}{p_{i}}).$
Superficially the entropy appears to be another mean value, the mean value of
the “uncertainty” $\log(1/p_{i})$, also called the “surprise value” (The log
here is usually taken with base $2$.) Thus the entropy of a probability state
is the mean uncertainty of the state. This is not the mean of a classical
observable since the function $i\mapsto\log(1/p_{i})$ is not a classical
observable. Classical observables should exist and be measurable prior to
assignment of probabilities, but it makes no sense to consider the uncertainty
function until probabilities have been introduced. The dependency of the
uncertainty function on $i$ is not intrinsic and is only determined through
the postulated probability state $p_{i}$.
It happens that entropy has another relationship to means of considerable
importance, namely through the _Maximum Entropy Principle (MEP)_ , also known
as _Jaynes’ Principle_. In the absence of an evolutionary law $T$ and an
initial assignment, we are faced with the problem of determining the
probability state $p$, i. e., an assignment of probabilities to the
deterministic states $i$. The MEP [5, p. 370] asserts that:
> The probability state $p$ maximizing entropy subject to the given values
> $\alpha_{1},...\alpha_{k}$ for the means of known classical observables
> $g_{1},...,g_{k}$ provides predictions ”most strongly indicated by our
> present information.”
Using the calculus of variations, we can in general determine a unique
distribution among those that satisfy the constraints
$\sum_{i}p_{i}g_{j}(i)=\alpha_{j}$
for $j=1,...,k$ and maximizing $H(p)$, namely, the one with the probability
assignment
$p_{i}=C\exp\\{-(\sum_{j}\beta_{j}g_{j}(i))\\}$
for $i=1,...,n$ where $\beta_{1},\dots,\beta_{k}$ are constants determined
from the $\alpha_{i}\mbox{'s}$, and $C$ is a positive normalizing constant
chosen so that the sum of the $p_{i}$’s is $1$.
The interpretation of this result takes two forms (at least). Suppose the
states are those of an individual particle in a gas of $N$ particles. Then the
quantities $\alpha_{1},...\alpha_{k}$ represent measured values of the total
value of $g_{1},...,g_{k}$ over the entire gas divided by $N$. The
probabilities $p_{i}$, derived from the MEP, are the probabilities that a
particle picked at random from among the $N$ particles is in the i-th state.
They may also be regarded as the fraction of particles that are in the i-th
state. We may not care about individual particles but we do care about these
fractions, which can be taken to define the macroscopic state of the gas
(volume, pressure, temperature, and the like). This is the _ensemble_
viewpoint of Gibbs. Yet another perspective is to regard what we usually
observe as a small perturbation about values induced by means.
## 7 Quantum Issues
The mean plays a pivotal role in quantum theory, even if this role has not
been examined closely in most treatments of quantum theory. In quantum
mechanics the state of a physical system is described by a wave function
$\psi$ that is an element of a Hilbert space $\mathcal{H}$. (Strictly speaking
$\psi$ is not a function but an equivalence class of functions, and in
addition each state is associated with a ray in Hilbert space.) Each physical
observable that takes on real-number values (e.g., a position coordinate, a
momentum coordinate, the energy, a spin component) is associate with a self-
adjoint operator $A$ in $\mathcal{H}$. For simplicity each observable is
denoted by the same symbol “A” as the associated operator. Any self-adjoint
operator A has in turn an associated projection-valued measure $P_{A}$ (see,
for example, [10]) that assigns to each Borel set $S$ in $\mathcal{R}$ an
orthogonal projection $P_{A}(S)$ in the Hilbert space:
$S\longmapsto P_{A}(S)$
in such a way that $A$ is an integral combination of these orthogonal
projections, represented symbolically by:
$A=\int_{\mathcal{R}}x\,P_{A}(dx),$
or by:
$A(\psi)=\int_{\mathcal{R}}x\,P_{A}(dx)(\psi),$
where $\psi$ is in the domain of $A$. If a measurement is made, the
probability that the value of A is in the set $S$ when the system state is
$\psi$ defined to be:
$\left\langle P_{A}(S)(\psi),\psi\right\rangle=||P_{A}(S)(\psi)||^{2}$
where $<\,,\,>$ is the inner product on $\mathcal{H}$, linear in the first
variable and conjugate-linear in the second variable, and $||\,||$ is the norm
on $\mathcal{H}$.
Quantum Mechanics is thus a statistical theory based on a family of
probability measures defined by:
$S\longmapsto||P_{A}(S)(\psi)||^{2}.$
These are the Borel probability measures associated with observables $A$ when
the system state is $\psi$. One consequence of this is that the set of
possible values of $A$ is the spectrum of the operator $A$, and another is
that the mean of $A$, when the state is $\psi$, is given by:
$\left\langle A(\psi),\psi\right\rangle=\int_{\mathcal{R}}x\,\left\langle
P_{A}(dx)(\psi),\psi\right\rangle=\int_{\mathcal{R}}x||P_{A}(dx)(\psi)||^{2}.$
In particular the quantity $\left\langle
P_{A}(S)(\psi),\psi\right\rangle=||P_{A}(S)(\psi)||^{2}$ can be interpreted as
the mean of the observable $P_{A}(S)$ when the state is $\psi$. The observable
$P_{A}(S)$ is an orthogonal projection, taking the value $1$ when the value of
$A$ is in is $S$ and the value $0$ when the value of $A$ is not in $S$. Thus
$||P_{A}(S)(\psi)||^{2}$ also represents the probability that $A$ is in $S$
when the state is $\psi$. This is a reminder that all probabilities are means.
The mean, $\left\langle A(\psi),\psi\right\rangle$, is the integral over the
real line of the real variable $x$ with respect to the Borel probability
measure $||P_{A}(\,)(\psi)||^{2}$. Thus the mean exists, it appears, if and
only if $x$ is integrable with respect to this measure, thus if and only if
$\psi$ is in the domain of $A$. Self-adjoint operators have domains that are
dense in $\mathcal{H}$ but many of the most prominent ones (e.g., those
associated with position and momentum and often energy) do not have domain
equal to $\mathcal{H}$. Hence there will be states for which the means of some
observable may not be well-defined. Whether these states are realizable in
practice is uncertain, but there is no good theoretical reason why they should
be ignored. (Our discussion focuses on mathematical definition and
characterization. The spectrum of a self-adjoint operator is identified with
the possible values of a measured quantity. If the spectrum is discrete, a
measurement may be able to distinguish one value from another; if the spectrum
is continuous, measurement will only be able to determine an interval that
contains the value, not the exact value. Repeated measurements when the system
is in the same state thus only arrive at a rough approximation of the
distribution and a rough estimate of the mean for a state.)
A curiosity in quantum mechanics, not ordinarily seen in other applications of
probability, is the following. Suppose $\mu=\left\langle
A(\psi),\psi\right\rangle$ is the mean of some observable $A$ when the state
is $\psi$. Then the variance of the observable in this state is naturally
given by:
$\int_{\mathcal{R}}(x-\mu)^{2}\,\left\langle
P_{A}(dx)(\psi),\psi\right\rangle\,=\,\left\langle(A-\mu
I)^{2}(\psi),\psi\right\rangle=||(A-\mu I)(\psi)||^{2}.$
So the variance exists if and only if $\psi$ is in the domain of $A$. The
condition for the mean to exist is the same as the condition for the variance
to exist. In quantum mechanics we are led to think that the only distributions
for which the mean is finite are ones in which the variance is also finite.
However, a closer look at this situation reveals some discrepancies.
The chief discrepancy is the following. Suppose that the original observable
$A$ can be written in the form
$A=C-D$
where $C$ and $D$ are non-negative self-adjoint operators. Non-negative self-
adjoint operators can be written as squares of self-adjoint operators, so that
$C=E^{2}$ and $D=F^{2}$ with $E$ and $F$ self-adjoint. Then
$\displaystyle\left\langle
A(\psi),\psi\right\rangle=\left\langle(C-D)(\psi),\psi\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle(E^{2}-F^{2})(\psi),\psi\right\rangle$
$\displaystyle=\left\langle E^{2}(\psi),\psi\right\rangle-\left\langle
F^{2}(\psi),\psi\right\rangle$ $\displaystyle=$
$\displaystyle||E(\psi)||^{2}-||F(\psi)||^{2}.$
Thus, the mean of $A$ exists if and only if $\psi$ is in the intersection of
the domain of $E$ and the domain of $F$. It is easy, incidentally, to
construct examples of elements of $\mathcal{H}$ that are in the domain of a
self-adjoint operator $E$ but are not in the domain of its square $E^{2}$. In
addition, as it happens, it is possible to offer explicit candidates for the
operators $E$ and $F$ given $A$. Set
$E=\int_{(0,\infty)}\sqrt{x}\,P_{A}(dx)\mbox{ and
}F=\int_{(-\infty,0)}\sqrt{-x}\,P_{A}(dx).$
We conclude that the mean of $A$ exists when $\psi$ is in $\mbox{dom
}E\cap\mbox{dom }F$ and the variance of $A$ exists when $\psi$ is in
$\mbox{dom }A$. If $\psi$ is not in $\mbox{dom }E$ but is in $\mbox{dom }F$,
then it is reasonable to say that the mean of $A$ is $\infty$. Likewise if
$\psi$ is in $\mbox{dom }E$ but not in $\mbox{dom }F$, the mean of $A$ is
$-\infty$. If $\psi$ is in neither $\mbox{dom }E$ nor $\mbox{dom }F$, then the
ordinary mean does not exist. In the spirit of our discussion of $L(c)$
earlier, it is possible to truncate the integrals for $E$ and $F$ in the last
display, replacing $\infty$ by $M$ and $-\infty$ by $-M$ and investigate the
existence of an appropriate combined limit as $M$ tend to infinity.
Similar considerations can be applied when the “pure” state $\psi$ is replaced
by a density matrix representing a statistical ensemble of pure states, or in
rigged Hilbert spaces where the existence of states varies according to the
properties of the observable, or to cases arising by the use of positive-
operator-valued measures generalizing the projection-valued measures treated
above.
## 8 Conclusion
The mean, as we have seen, is ubiquitous in scientific explanation. Not only
does it provide a summary of sample data and, when it exists, of data from the
entire population, but it establishes a connection between samples and the
whole population. Furthermore, it facilitates generalization of deterministic
observables that are functions of the deterministic state to probabilistic
observables that are functions of the probabilistic state. The Maximum Entropy
Principle then makes use of constrained means to identify macroscopic
distribution of physical importance parametrized by these means. While quantum
mechanics abandons determinism, it retains the notion of mean to summarize the
possible results of experiments and the measurement of quantum observables.
Although not all observables have finite means, weak and doubly weak means and
the alternatives identified in Theorem 3.1 provide an enumeration of possible
behaviors of variables and associated probability distributions, and give
further insight into potentialities associated with large data sets.
## References
* [1]
* [2] de Finetti, B. (1972). Probability, Induction, and Statistics, Wiley, New York.
* [3] Gumbel, E. J. (2004). Statistics of Extremes, Dover, Mineola, New York.
* [4] Hewitt, E. and Stromberg, K. (1965). Real and Abstract Analysis, Springer-Verlag, New York.
* [5] Jaynes, E. T. (2004). Probability Theory: The Logic of Science, ed. by G. Larry Bretthorst, Cambridge University Press, Cambridge, U. K.
* [6] Kolmogorov, A. N. (1950). Foundations of the Theory of Probability, Chelsea Publishing Company, New York.
* [7] Kolmogorov, A. N. (1928) Über die Summen durch den Zufall bestimmter anabhängiger Grössen. Math. Ann. 99 309–319.
* [8] Kolmogorov, A. N. (1929) Bemerkung zu meiner Arbeit “Über die Summen zufälliger Grössen.” Math. Ann. 102 484–488.
* [9] Kosko, B. (2008) The Sample Mean The Edge: What have you changed your mind about? Why? http://www.edge.org/responses/what-have-you-changed-your-mind-about-why.
* [10] Mackey, G. W. (1963). Mathematical Foundations of Quantum Mechanics, Benjamin, New York.
* [11] Mathews, J. and Walker, R. L. (1970). Mathematical Methods of Physics, Benjamin, New York.
* [12] Pollard, D. (2002). A User’s Guide to Measure Theoretic Probability, Cambridge University Press, Cambridge, U. K.
* [13] Taleb, N. N. (2010). The Black Swan: The Impact of the Highly Improbable, Random House, New York.
* [14] Whittle, P. (1992). Probability via Expectation, Springer-Verlag, New York, 3rd edition.
|
arxiv-papers
| 2012-10-15T06:07:23 |
2024-09-04T02:49:36.537354
|
{
"license": "Public Domain",
"authors": "John E. Gray and Andrew Vogt",
"submitter": "John E Gray Mr",
"url": "https://arxiv.org/abs/1210.3908"
}
|
1210.3935
|
# Bloch’s conjecture for Campedelli and Barlow surfaces
Claire Voisin
CNRS and École Polytechnique
###### Abstract
Campedelli surfaces are regular surfaces of general type with $p_{g}=0$. They
specialize to double covers of Barlow surfaces. We prove that the $CH_{0}$
group of a Campedelli surface is equal to $\mathbb{Z}$, which implies the same
result for the Barlow surfaces.
À la mémoire de Friedrich Hirzebruch
## 0 Introduction
In this paper, we establish an improved version of the main theorem of [27]
and use it in order to prove the Bloch conjecture for Campedelli surfaces. We
will also give a conditional application (more precisely, assuming the
variational Hodge conjecture) of the same method to the Chow motive of low
degree $K3$ surfaces.
Bloch’s conjecture for $0$-cycles on surfaces states the following:
###### Conjecture 0.1.
(cf. [7]) Let $\Gamma\in CH^{2}(Y\times X)$, where $Y$ is smooth projective
and $X$ is a smooth projective surface. Assume that
$[\Gamma]^{*}:H^{2,0}(X)\rightarrow H^{2,0}(Y)$ vanishes. Then
$\Gamma_{*}:CH_{0}(Y)_{alb}\rightarrow CH_{0}(X)_{alb}$
vanishes.
Here $[\Gamma]\in H^{4}(X\times X,\mathbb{Q})$ is the cohomology class of
$\Gamma$ and
$CH_{0}(Y)_{alb}:={\rm Ker}\,(CH_{0}(Y)_{hom}\stackrel{{\scriptstyle
alb_{Y}}}{{\rightarrow}}Alb(Y)).$
Particular cases concern the case where $\Gamma$ is the diagonal of a surface
$X$ with $q=p_{g}=0$. Then $\Gamma_{*}=Id_{CH_{0}(X)_{alb}}$, so that the
conjecture predicts $CH_{0}(X)_{alb}=0$. This statement is known to hold for
surfaces which are not of general type by [9], and for surfaces of general
type, it is known to hold by Kimura [17] if the surface $X$ is furthermore
rationally dominated by a product of curves (cf. [5] for many such examples).
Furthermore, for several other families of surfaces of general type with
$q=p_{g}=0$, it is known to hold either for the general member of the family
(eg. the Godeaux surfaces, cf. [28]), or for specific members of the family
(for example the Barlow surface [4]).
A slightly more general situation concerns surfaces equipped with the action
of a finite group $G$. This has been considered in the paper [27], where the
following theorem concerning group actions on complete intersection surfaces
is proved: Let $X$ be a smooth projective variety with trivial Chow groups
(i.e. the cycle class map $CH_{i}(X)_{\mathbb{Q}}\rightarrow
H^{2n-2i}(X,\mathbb{Q}),\,n={\rm dim}\,X$ is injective for all $i$). Let $G$
be a finite group acting on $X$ and let $E$ be a $G$-equivariant rank $n-2$
vector bundle on $X$ which has “enough” $G$-invariant sections (for example,
if the group action is trivial, one asks that $E$ is very ample). Let
$\pi\in\mathbb{Q}[G]$ be a projector. Then $\pi$ gives a self-correspondence
$\Gamma^{\pi}$ with $\mathbb{Q}$-coefficients (which is a projector) of the
$G$-invariant surfaces $S=V(\sigma),\,\sigma\in H^{0}(X,E)^{G}$ (cf. Section
1). We use the notation
$H^{2,0}(S)^{\pi}:={\rm Im}\,([\Gamma^{\pi}]^{*}:H^{2,0}(S)\rightarrow
H^{2,0}(S)),$ $CH_{0}(S)_{\mathbb{Q},hom}^{\pi}:={\rm
Im}\,([\Gamma^{\pi}]_{*}:CH_{0}(S)_{\mathbb{Q},hom}\rightarrow
CH_{0}(S)_{\mathbb{Q},hom})).$
###### Theorem 0.2.
Assume that the smooth surfaces $S=V(\sigma),\,\sigma\in H^{0}(E)^{G}$,
satisfy $H^{2,0}(S)^{\pi}=0$. Then we have
$CH_{0}(S)_{\mathbb{Q},hom}^{\pi}=0$.
Note that the Bloch conjecture for finite group actions on surfaces which do
not fit at all in the above geometric setting, namely finite order symplectic
automorphisms of $K3$ surfaces, has been recently proved in [15] and [30] by
completely different methods. For these symplectic automorphisms, one
considers the cycle $\Delta_{X}-\frac{1}{|G|}\sum_{g\in G}{\rm Graph}\,g$,
which acts as the identity minus the projector onto the $G$-invariant part,
and one proves that it acts as $0$ on $CH_{0}(X)_{hom}$ (in fact on the whole
of $CH_{0}$) according to Conjecture 0.1.
Theorem 0.2 is rather restrictive geometrically, due to the fact that not only
we consider $0$-sets of sections of a vector bundle, but also we impose this
very ampleness assumption on the vector bundle. Our first result in this paper
is a relaxed version of this theorem, which works in a much more general
geometric context and will be applicable in particular to the case of
Campedelli surfaces.
Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Let $\Gamma\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$ be a relative $0$-self
correspondence. Let
$\Gamma_{t}:=\Gamma_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}$ be the
restricted cycle, with cohomology class $[\Gamma_{t}]\in
H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q})$. We have the actions
$\Gamma_{t*}:CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}}\rightarrow
CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}},\,\,[\Gamma_{t}]^{*}:H^{i,0}(\mathcal{S}_{t})\rightarrow
H^{i,0}(\mathcal{S}_{t}).$
###### Theorem 0.3.
Assume the following:
(1) The fibers $\mathcal{S}_{t}$ satisfy $h^{1,0}(\mathcal{S}_{t})=0$ and
$[\Gamma_{t}]^{*}:H^{2,0}(\mathcal{S}_{t})\rightarrow
H^{2,0}(\mathcal{S}_{t})$ is equal to zero.
(2) A smooth projective (equivalently any smooth projective) completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.
Then $\Gamma_{t*}:CH_{0}(\mathcal{S}_{t})_{hom}\rightarrow
CH_{0}(\mathcal{S}_{t})_{hom}$ is nilpotent for any $t\in B$.
This statement is both weaker and stronger than Theorem 0.2 since on the one
hand, the conclusion only states the nilpotence of $\Gamma_{t*}$, and not its
vanishing, while on the other hand the geometric context is much more flexible
and the assumption on the total space of the family is much weaker.
In fact the nilpotence property is sufficient to imply the vanishing in a
number of situations which we describe below. The first situation is the case
where we consider a family of surfaces with $h^{2,0}=h^{1,0}=0$. Then we get
the following consequence (the Bloch conjecture for surfaces with $q=p_{g}=0$
under assumption (2) below):
###### Corollary 0.4.
Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Assume the
following:
(1) The fibers $\mathcal{S}_{t}$ satisfy
$H^{1,0}(\mathcal{S}_{t})=H^{2,0}(\mathcal{S}_{t})=0$.
(2) A projective completion (or, equivalently, any projective completion)
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.
Then $CH_{0}(\mathcal{S}_{t})_{hom}=0$ for any $t\in B$.
We refer to Section 1, Theorem 1.6 for a useful variant involving group
actions, which allows to consider many more situations (cf. [27] and Section 2
for examples).
###### Remark 0.5.
The proof will show as well that we can replace assumption (2) in these
statements by the following one:
(2’) A (equivalently any) smooth projective completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ has trivial $CH_{0}$ group.
However, it seems more natural to put a geometric assumption on the total
space since this is in practice much easier to check.
In the second section of this paper, we will apply these results to prove
Bloch’s conjecture 0.1 for Campedelli surfaces (cf. [11], [25], [10]).
Campedelli surfaces can be constructed starting from a $5\times 5$ symmetric
matrix $M(a),\,a\in\mathbb{P}^{11}$, of linear forms on $\mathbb{P}^{3}$
satisfying certain conditions (cf. (3)) making their discriminant invariant
under the Godeaux action (4) of $\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{P}^{3}$.
The general quintic surface $V(a)$ defined by the determinant of $M(a)$ has
$20$ nodes corresponding to the points $x\in\mathbb{P}^{3}$ where the matrix
$M(a,x)$ has rank $3$, and it admits a double cover $S(a)$ which is étale away
from the nodes, and to which the $\mathbb{Z}/5\mathbb{Z}$-action lifts. Then
the Campedelli surface $\Sigma(a)$ is the quotient of $S(a)$ by this lifted
action.
Campedelli surfaces have a $4$-dimensional moduli space. For our purpose, the
geometry of the explicit $11$-dimensional parameter space described in [25] is
in fact more important than the structure of the moduli space.
###### Theorem 0.6.
Let $S$ be a Campedelli surface. Then $CH_{0}(S)=\mathbb{Z}$.
The starting point of this work was a question asked by the authors of [10]:
They needed to know that the Bloch conjecture holds for a simply connected
surface with $p_{g}=0$ (eg a Barlow surface [3]) and furthermore, they needed
it for a general deformation of this surface. The Bloch conjecture was proved
by Barlow [4] for some Barlow surfaces admitting an extra group action
allowing to play on group theoretic arguments as in [16], but it was not known
for the general Barlow surface.
Theorem 0.6 implies as well the Bloch conjecture for the Barlow surfaces since
the Barlow surfaces can be constructed as quotients of certain Campedelli
surfaces admitting an extra involution, namely the determinantal equation
defining $V(a)$ has to be invariant under the action of the dihedral group of
order $10$ (cf. [25]). The Campedelli surfaces appearing in this construction
of Barlow surfaces have only a $2$-dimensional moduli space. It is interesting
to note that we get the Bloch conjecture for the general Barlow surface via
the Bloch conjecture for the general Campedelli surface, but that our strategy
does not work directly for the Barlow surface, which has a too small parameter
space (cf. [11], [25]).
The third section of this paper applies Theorem 0.3 to prove a conditional
result on the Chow motive of $K3$ surfaces which can be realized as $0$-sets
of sections of a vector bundle on a rationally connected variety (cf. [20],
[21]). Recall that the Kuga-Satake construction (cf. [18], [13]) associates to
a polarized $K3$ surface $S$ an abelian variety $K(S)$ with the property that
the Hodge structure on $H^{2}(S,\mathbb{Q})$ is a direct summand of the Hodge
structure of $H^{2}(K(S),\mathbb{Q})$. The Hodge conjecture predicts that the
corresponding degree $4$ Hodge class on $S\times K(S)$ is algebraic. This is
not known in general, but this is established for $K3$ surfaces with large
Picard number (cf. [19], [23]). The next question concerns the Chow motive (as
opposed to the numerical motive) of these $K3$ surfaces. The Kuga-Satake
construction combined with the Bloch conjecture implies that the Chow motive
of a $K3$ surface is a direct summand of the Chow motive of its Kuga-Satake
variety. In this direction, we prove the following Theorem 0.7: Let $X$ be a
rationally connected variety of dimension $n$ and let $E\rightarrow X$ be a
rank $n-2$ globally generated vector bundle satisfying the following
properties:
(i) The restriction map $H^{0}(X,E)\rightarrow H^{0}(z,E_{\mid z})$ is
surjective for general $z=\\{x,\,y\\}\subset X$.
(ii) The general section $\sigma$ vanishing at two general points $x,\,y$
determines a smooth surface $V(\sigma)$.
We consider the case where the surfaces $S=V(\sigma)$ are algebraic $K3$
surfaces. For example, this is the case if ${\rm det}\,E=-K_{X}$, and the
surfaces $S=V(\sigma)$ for general $\sigma\in H^{0}(X,E)$ have irregularity
$0$. Almost all general algebraic $K3$ surfaces of genus $\leq 20$ have been
described this way by Mukai (cf. [20], [21]), where $X$ is a homogeneous
variety with Picard number $1$. Many more examples can be constructed starting
from an $X$ with Picard number $\geq 2$.
###### Theorem 0.7.
Assume the variational Hodge conjecture in degree $4$. Then the Chow motive of
a $K3$ surface $S$ as above is a direct summand of the Chow motive of an
abelian variety.
###### Remark 0.8.
The variational Hodge conjecture for degree $4$ Hodge classes is implied by
the Lefschetz standard conjecture in degrees $2$ and $4$. It is used here only
to conclude that the Kuga-Satake correspondence is algebraic for any $S$ as
above. Hence we could replace the variational Hodge conjecture by the
Lefschetz standard conjecture or by the assumption that the cohomological
motive of a general $K3$ surfaces $S$ in our family is a direct summand of the
cohomological motive of an abelian variety. The contents of the theorem is
that we then have the same result for the Chow motive.
As a consequence of this result, we get the following (conditional)
corollaries.
###### Corollary 0.9.
With the same assumptions as in Theorem 0.7, let $S$ be a member of the family
of $K3$ surfaces parameterized by $\mathbb{P}(H^{0}(X,E))$, and let $\Gamma\in
CH^{2}(S\times S)$ be a correspondence such that
$[\Gamma]^{*}:H^{2,0}(S)\rightarrow H^{2,0}(S)$ is zero. Then
$Z_{*}:CH_{0}(S)_{hom}\rightarrow CH_{0}(S)_{hom}$ is nilpotent.
###### Remark 0.10.
Note that there is a crucial difference between Theorem 0.3 and Corollary 0.9:
In Corollary 0.9, the cycle $\Gamma$ is not supposed to exist on the general
deformation $S_{t}$ of $S$. (Note also that the result in Corollary 0.9 is
only conditional since we need the Lefschetz standard conjecture, or at least
to know that the Kuga-Satake correspondence is algebraic for general $S_{t}$,
while Theorem 0.3 is unconditional!)
###### Corollary 0.11.
With the same assumptions as in Theorem 0.7, the transcendental part of the
Chow motive of any member of the family of $K3$ surfaces parameterized by
$\mathbb{P}(H^{0}(X,E))$ is indecomposable, that is, any submotive of it is
either the whole motive or the $0$-motive.
Thanks. I thank Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil
Katzarkov and Pavel Sosna for asking me the question whether general Barlow
surfaces satisfy the Bloch conjecture, and for providing references on Barlow
versus Campedelli surfaces.
## 1 Proof of Theorem 0.3 and some consequences
This section is devoted to the proof of Theorem 0.3 and its consequences
(Corollary 0.4 or its more general form Theorem 1.6).
The proof will follow essentially the idea of [27]. The main novelty in the
proof lies in the use of Proposition 1.3. For completeness, we also outline
the arguments of [27], restricted to the surface case.
Consider the codimension $2$-cycle
$\Gamma\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}.$
Assumption (1) tells us that the restricted cycle
$\Gamma_{t}:=\Gamma_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}$ is
cohomologous to the sum of a cycle supported on a product of (nonnecessarily
irreducible curves) in $\mathcal{S}_{t}$ and of cycles pulled-back from
$CH_{0}(\mathcal{S}_{t})$ via the two projections. We deduce from this (cf.
[27, Prop. 2.7]):
###### Lemma 1.1.
There exist a codimension $1$ closed algebraic subset
$\mathcal{C}\subset\mathcal{S}$, a codimension $2$ cycle $\mathcal{Z}$ on
$\mathcal{S}\times_{B}\mathcal{S}$ with $\mathbb{Q}$-coefficients supported on
$\mathcal{C}\times_{B}\mathcal{C}$, and two codimension $2$ cycles
$\mathcal{Z}_{1},\,\mathcal{Z}_{2}$ with $\mathbb{Q}$-coefficients on
$\mathcal{S}$, such that the cycle
$\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}$
has its restriction to each fiber $\mathcal{S}_{t}\times\mathcal{S}_{t}$
cohomologous to $0$, where
$p_{1},\,p_{2}:\mathcal{S}\times_{B}\mathcal{S}\rightarrow\mathcal{S}$ are the
two projections.
This lemma is one of the key observations in [27]. The existence of the data
above is rather clear after a generically finite base change
$B^{\prime}\rightarrow B$ because it is true fiberwise. The key point is that,
working with cycles with $\mathbb{Q}$-coefficients, we can descend to $B$ and
hence, do not need in fact this base change which would ruin assumption (2).
The next step consists in passing from the fiberwise cohomological equality
$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}]_{\mid\mathcal{S}_{t}\times\mathcal{S}_{t}}=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q})$
to the following global vanishing:
###### Lemma 1.2.
(cf. [27, Lemma 2.12]) There exist codimension $2$ algebraic cycles
$\mathcal{Z}^{\prime}_{1}$, $\mathcal{Z}^{\prime}_{2}$ with
$\mathbb{Q}$-coefficients on $\mathcal{S}$ such that
$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}]=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}\times_{B}\mathcal{S},\mathbb{Q}).$
The proof of this lemma consists in the study of the Leray spectral sequence
of the fibration $p:\mathcal{S}\times_{B}\mathcal{S}\rightarrow B$. We know
that the class
$[\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}_{1}-p_{2}^{*}\mathcal{Z}_{2}]$
vanishes in the Leray quotient $H^{0}(B,R^{4}p_{*}\mathbb{Q})$ of
$H^{4}(\mathcal{S}\times_{B}\mathcal{S},\mathbb{Q})$. It follows that it is of
the form $p_{1}^{*}\alpha_{1}+p_{2}^{*}\alpha_{2}$, for some rational
cohomology classes $\alpha_{1},\,\alpha_{2}$ on $\mathcal{S}$. One then proves
that $\alpha_{i}$ can be chosen to be algebraic on $\mathcal{S}$.
The new part of the argument appears in the following proposition:
###### Proposition 1.3.
Under assumption (2), the following hold :
(i) The cycle
$\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}$
is algebraically equivalent to $0$ on $\mathcal{S}\times_{B}\mathcal{S}$.
(ii) The restriction to the fibers $\mathcal{S}_{t}\times\mathcal{S}_{t}$ of
the codimension $2$ cycle
$\mathcal{Z}^{\prime}=\Gamma-\mathcal{Z}-p_{1}^{*}\mathcal{Z}^{\prime}_{1}-p_{2}^{*}\mathcal{Z}^{\prime}_{2}$
is a nilpotent element (with respect to the composition of self-
correspondences) of
$CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$.
Proof. We work now with a smooth projective completion
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$. Let
$D:=\overline{\mathcal{S}\times_{B}\mathcal{S}}\setminus{\mathcal{S}\times_{B}\mathcal{S}}$
be the divisor at infinity. Let $\widetilde{D}\stackrel{{\scriptstyle
j}}{{\rightarrow}}\overline{\mathcal{S}\times_{B}\mathcal{S}}$ be a
desingularization of $D$. The codimension $2$ cycle $\mathcal{Z}^{\prime}$
extends to a cycle $\overline{\mathcal{Z}^{\prime}}$ over
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$. We know from Lemma 1.2 that
$[\overline{\mathcal{Z}^{\prime}}]=0\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q})$
and this implies by [31, Prop. 3] that there is a degree $2$ Hodge class
$\alpha$ on $\widetilde{D}$ such that
$j_{*}\alpha=[\overline{\mathcal{Z}^{\prime}}]\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q}).$
By the Lefschetz theorem on $(1,1)$-classes, $\alpha$ is the class of a
codimension $1$ cycle $\mathcal{Z}^{\prime\prime}$ of $\widetilde{D}$ and we
conclude that
$[\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime}]=0\,\,{\rm
in}\,\,H^{4}(\overline{\mathcal{S}\times_{B}\mathcal{S}},\mathbb{Q}).$
We use now assumption (2) which says that the variety
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ is rationally connected. It has
then trivial $CH_{0}$, and so any codimension $2$ cycle homologous to $0$ on
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$ is algebraically equivalent to
$0$ by the following result to to Bloch and Srinivas:
###### Theorem 1.4.
[8] On a smooth projective variety $X$ with $CH_{0}(X)$ supported on a
surface, homological equivalence and algebraic equivalence coincide for
codimension $2$ cycles.
We thus conclude that
$\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime}$ is
algebraically equivalent to $0$ on
$\overline{\mathcal{S}\times_{B}\mathcal{S}}$, hence that
$\mathcal{Z}^{\prime}=(\overline{\mathcal{Z}^{\prime}}-j_{*}\mathcal{Z}^{\prime\prime})_{\mid\mathcal{S}\times_{B}\mathcal{S}}$
is algebraically equivalent to $0$ on ${\mathcal{S}\times_{B}\mathcal{S}}$.
(ii) This is a direct consequence of (i), using the following nilpotence
result proved independently in [26] and [29]:
###### Theorem 1.5.
On any smooth projective variety, self-correspondences algebraically
equivalent to $0$ are nilpotent for the composition of correspondences.
Proof of Theorem 0.3. Using the same notations as in the previous steps, we
know by Proposition 1.3 that under assumptions (1) and (2), the self-
correspondence
$\displaystyle\Gamma_{t}-\mathcal{Z}_{t}-p_{1}^{*}\mathcal{Z}^{\prime}_{1,t}-p_{2}^{*}\mathcal{Z}^{\prime}_{2,t}$
(1)
on $\mathcal{S}_{t}$ with $\mathbb{Q}$-coefficients is nilpotent. In
particular, the morphism it induces at the level of Chow groups is nilpotent.
On the other hand, recall that $\mathcal{Z}_{t}$ is supported on a product of
curves on $\mathcal{S}_{t}$, hence acts trivially on
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q}}$. Obviously, both cycles
$p_{1}^{*}\mathcal{Z}^{\prime}_{1,t},\,p_{2}^{*}\mathcal{Z}^{\prime}_{2,t}$
act trivially on $CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}$. Hence the self-
correspondence (1) acts as $\Gamma_{t*}$ on
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}$.
Let us now turn to our main application, namely Corollary 0.4, or a more
general form of it which involves a family of surfaces $S_{t}$ with an action
of a finite group $G$ and a projector $\pi\in\mathbb{Q}[G]$. Writing such a
projector as $\pi=\sum_{g\in G}a_{g}g$, $a_{g}\in\mathbb{Q}$, such a projector
provides a codimension $2$ cycle
$\displaystyle\Gamma^{\pi}_{t}=\sum_{g}a_{g}{\rm Graph}\,g\in
CH^{2}(S_{t}\times S_{t})_{\mathbb{Q}},$ (2)
with actions
${\Gamma^{\pi}_{t}}^{*}=\sum_{g}a_{g}g^{*}$
on the holomorphic forms of $S$, and
${\Gamma^{\pi}_{t}}_{*}=\sum_{g}a_{g}g_{*}$
on $CH_{0}(S_{t})_{\mathbb{Q}}$. Denote respectively
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}$ the image of the projector
${\Gamma^{\pi}_{t}}_{*}$ acting on $CH_{0}(S_{t})_{\mathbb{Q},hom}$ and
$H^{2,0}(S_{t})^{\pi}$ the image of the projector ${\Gamma^{\pi}_{t}}^{*}$
acting on $H^{2,0}(S_{t})$.
###### Theorem 1.6.
Let $\mathcal{S}\rightarrow B$ be a smooth projective morphism with two
dimensional connected fibers, where $B$ is quasi-projective. Let $G$ a finite
group acting in a fiberwise way on $\mathcal{S}$ and let $\pi\in\mathbb{Q}[G]$
be a projector. Assume the following:
(1) The fibers $\mathcal{S}_{t}$ satisfy $H^{1,0}(\mathcal{S}_{t})=0$ and
$H^{2,0}(\mathcal{S}_{t})^{\pi}=0$.
(2) A smooth projective completion (or, equivalently, any smooth projective
completion) $\overline{\mathcal{S}\times_{B}\mathcal{S}}$ of the fibered self-
product $\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected.
Then $CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}=0$ for any $t\in B$.
Proof. The group $G$ acts fiberwise on $\mathcal{S}\rightarrow B$. Thus we
have the universal cycle
$\Gamma^{\pi}\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$
defined as $\sum_{g\in G}a_{g}{\rm Graph}\,g$, where the graph is taken over
$B$. Since by assumption the action of $[\Gamma^{\pi}]^{*}$ on
$H^{2,0}(\mathcal{S}_{t})$ is $0$, we can apply Theorem 0.3, and conclude that
$\Gamma^{\pi}_{t*}$ is nilpotent on $CH_{0}(S_{t})_{\mathbb{Q,}hom}$. On the
other hand, $\Gamma^{\pi}_{t*}$ is a projector onto
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}$. The fact that it is
nilpotent implies thus that it is $0$, hence that
$CH_{0}(\mathcal{S}_{t})_{\mathbb{Q},hom}^{\pi}=0$.
## 2 Campedelli and Godeaux surfaces
Our main goal in this section is to check the main assumption of Theorem 1.6,
namely the rational connectedness of the fibered self-product of the universal
Campedelli surface, in order to prove Theorem 0.6.
We follow [11], [25]. Campedelli surfaces can be described as follows:
Consider the following symmetric $5\times 5$ matrix
$\displaystyle
M_{a}=\begin{pmatrix}a_{1}x_{1}&a_{2}x_{2}&a_{3}x_{3}&a_{4}x_{4}&0\\\
a_{2}x_{2}&a_{5}x_{3}&a_{6}x_{4}&0&a_{7}x_{1}\\\
a_{3}x_{3}&a_{6}x_{4}&0&a_{8}x_{1}&a_{9}x_{2}\\\
a_{4}x_{4}&0&a_{8}x_{1}&a_{10}x_{2}&a_{11}x_{3}\\\
0&a_{7}x_{1}&a_{9}x_{2}&a_{11}x_{3}&a_{12}x_{4}\end{pmatrix}$ (3)
depending on $12$ parameters $a_{1},\ldots,a_{12}$ and defining a symmetric
bilinear (or quadratic) form $q(a,x)$ on $\mathbb{C}^{5}$ depending on
$x\in\mathbb{P}^{3}$. This is a homogeneous degree $1$ matrix in the variables
$x_{1},\ldots,x_{4}$, and the vanishing of its determinant gives a degree $5$
surface $V(a)$ in $\mathbb{P}^{3}$ with nodes at those points
$x=(x_{1},\ldots,x_{4})$ where the matrix has rank only $3$. We will denote by
$T$ the vector space generated by $a_{1},\ldots,a_{12}$ and
$B\subset\mathbb{P}(T)$ the open set of parameters $a$ satisfying this last
condition.
The surface $V(a)$ is invariant under the Godeaux action of
$\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{P}^{3}$, given by
$g^{*}x_{i}=\zeta^{i}x_{i},\,i=1,\ldots,\,4,$
where $g$ is a generator of $\mathbb{Z}/5\mathbb{Z}$ and $\zeta$ is a fifth
primitive root of unity. This follows either from the explicit development of
the determinant (see [25], where one monomial is incorrectly written:
$x_{1}x_{2}^{2}x_{4}^{2}$ should be $x_{1}x_{3}^{2}x_{4}^{2}$) or from the
following argument that we will need later on: Consider the following linear
action of $\mathbb{Z}/5\mathbb{Z}$ on $\mathbb{C}^{5}$:
$\displaystyle g(y_{1},\ldots,y_{5})=(\zeta
y_{1},\zeta^{2}y_{2},\ldots,\zeta^{5}y_{5}).$ (4)
Then one checks immediately that
$\displaystyle q(a,x)(gy,gy^{\prime})=\zeta q(a,gx)(y,y^{\prime}),$ (5)
so that the discriminant of $q(a,x)$, as a function of $x$, is invariant under
the action of $g$.
The Campedelli surface $\Sigma(a)$ is obtained as follows: there is a natural
double cover $S(a)$ of $V(a)$, which is étale away from the nodes, and
parameterizes the rulings in the rank four quadric $Q(a,x)$ defined by
$q(a,x)$ for $x\in V(a)$. This double cover is naturally equipped with a lift
of the $\mathbb{Z}/5\mathbb{Z}$-action on $V(a)$, which is explicitly
described as follows: Note first of all that the quadrics $Q(a,x)$ pass
through the point $y_{0}=(0,0,1,0,0)$ of $\mathbb{P}^{4}$.
###### Lemma 2.1.
For the general point $(a,x)\in\mathbb{P}(T)\times\mathbb{P}^{3}$ such that
$x\in V(a)$, the quadric $Q(a,x)$ is not singular at the point $y_{0}$.
Proof. It suffices to exhibit one pair $(a,x)$ satisfying this condition, and
such that the surface $V(a)$ is well-defined, that is the discriminant of
$q(a,x)$, seen as a function of $x$, is not identically $0$. Indeed, the
family of surfaces $V(a)$ is flat over the base near such a point, and the
result for the generic pair $(a,x)$ will then follow because the considered
property is open on the total space of this family.
We choose $(a,x)$ in such a way that the first column of the matrix (3) is
$0$, so that $x\in V(a)$. For example, we impose the conditions:
$a_{1}=0,\,a_{2}=0,\,x_{3}=0,\,x_{4}=0.$
Then the quadratic form $q(a,x)$ has for matrix
$\begin{pmatrix}0&0&0&0&0\\\ 0&0&0&0&a_{7}x_{1}\\\
0&0&0&a_{8}x_{1}&a_{9}x_{2}\\\ 0&0&a_{8}x_{1}&a_{10}x_{2}&0\\\
0&a_{7}x_{1}&a_{9}x_{2}&0&0\end{pmatrix}$
It is clear that $y_{0}$ is not a singular point of the corresponding quadric
if $a_{8}x_{1}\not=0$. On the other hand, for $a$ satisfying
$a_{1}=0,\,a_{2}=0$, and for a general point $x=(x_{1},\ldots,x_{4})$, the
matrix of $q(a,x)$ takes the form
$\begin{pmatrix}0&0&a_{3}x_{3}&a_{4}x_{4}&0\\\
0&a_{5}x_{3}&a_{6}x_{4}&0&a_{7}x_{1}\\\
a_{3}x_{3}&a_{6}x_{4}&0&a_{8}x_{1}&a_{9}x_{2}\\\
a_{4}x_{4}&0&a_{8}x_{1}&a_{10}x_{2}&a_{11}x_{3}\\\
0&a_{7}x_{1}&a_{9}x_{2}&a_{11}x_{3}&a_{12}x_{4}.\end{pmatrix}$
It is elementary to check that this matrix is generically of maximal rank.
It follows from this lemma that for generic $a$ and generic $x\in V(a)$, the
two rulings of $Q(a,x)$ correspond bijectively to the two planes through
$y_{0}$ contained in $Q(a,x)$. Hence for generic $a$, the surface $S(a)$ is
birationally equivalent to the surface $S^{\prime}(a)$ parameterizing planes
passing through $y_{0}$ and contained in $Q(a,x)$ for some
$x\in\mathbb{P}^{3}$ (which lies then necessarily in $V(a)$). It follows from
formula (5) that $S^{\prime}(a)$, which is a surface contained in the
Grassmannian $G_{0}$ of planes in $\mathbb{P}^{4}$ passing through $y_{0}$, is
invariant under the $\mathbb{Z}/5\mathbb{Z}$-action on $G_{0}$ induced by (4).
Hence there is a canonical $\mathbb{Z}/5\mathbb{Z}$-action on $S^{\prime}(a)$
which is compatible with the $\mathbb{Z}/5\mathbb{Z}$-action on $V(a)$, and
this immediately implies that the latter lifts to an action on the surface
$S(a)$ for any $a\in B$.
The Campedelli surface is defined as the quotient of $S(a)$ by
$\mathbb{Z}/5\mathbb{Z}$. In the following, we are going to apply Theorem 1.6,
and will thus work with the universal family $\mathcal{S}\rightarrow B$ of
double covers, with its $\mathbb{Z}/5\mathbb{Z}$-action defined above, where
$B\subset\mathbb{P}^{11}$ is the Zariski open set parameterizing smooth
surfaces $S(a)$.
We prove now:
###### Proposition 2.2.
The universal family $\mathcal{S}\rightarrow B$ has the property that the
fibered self-product $\mathcal{S}\times_{B}\mathcal{S}$ has a rationally
connected smooth projective compactification.
Proof. By the description given above, the family $\mathcal{S}\rightarrow B$
of surfaces $S(a),\,a\in B$, maps birationally to an irreducible component of
the following variety
$\displaystyle W=\\{(a,x,[P])\in\mathbb{P}(T)\times\mathbb{P}^{3}\times
G_{0},\,q(a,x)_{\mid P}=0\\},$ (6)
by the rational map which to a general point $(a,x),x\in V(a)$ and a choice of
ruling in the quadric $Q(a,x)$ associates the unique plane $P$ passing through
$y_{0}$, contained in $Q(a,x)$ and belonging to the chosen ruling. It follows
that $\mathcal{S}\times_{B}\mathcal{S}$ maps birationally by the same map
(which we will call $\Psi$) onto an irreducible component $W_{2}^{0}$ of the
following variety
$\displaystyle
W_{2}:=\\{(a,x,y,[P],[P^{\prime}])\in\mathbb{P}(T)\times\mathbb{P}^{3}\times\mathbb{P}^{3}\times
G_{0}\times G_{0},\,q(a,x)_{\mid P}=0,\,q(a,y)_{\mid P^{\prime}}=0\\}.$ (7)
Let $\mathcal{E}$ be the vector bundle of rank $5$ on $G_{0}$ whose fiber at a
point $[P]$ parameterizing a plane $P\subset\mathbb{P}^{4}$ passing through
$y_{0}$ is the space
$H^{0}(P,\mathcal{O}_{P}(2)\otimes\mathcal{I}_{y_{0}}).$
The family of quadrics $q(a,x)$ on $\mathbb{P}^{4}$ provides a $12$
dimensional linear space $T$ of sections of the bundle
$\mathcal{F}_{0}:=pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{3}^{*}\mathcal{E}\oplus pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{4}^{*}\mathcal{E}$
on
$Y_{0}:=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0},$
where as usual the $pr_{i}$’s denote the various projections from $Y_{0}$ to
its factors. For a point $a\in T$, the corresponding section of
$\mathcal{F}_{0}$ is equal to $(q(a,x)_{\mid P},q(a,x^{\prime})_{\mid
P^{\prime}})$ at the point $(x,x^{\prime},[P],[P^{\prime}])$ of $Y_{0}$.
Formula (7) tells us that $W_{2}$ is the zero set of the corresponding
universal section of the bundle
${pr^{\prime}_{1}}^{*}\mathcal{O}_{\mathbb{P}(T)}(1)\otimes{pr^{\prime}_{2}}^{*}\mathcal{F}_{0}$
on $\mathbb{P}(T)\times Y_{0}$, where the $pr^{\prime}_{i}$ are now the two
natural projections from $\mathbb{P}(T)\times Y_{0}$ to its summands. Note
that $W_{2}^{0}$ has dimension $15$, hence has the expected codimension $10$,
since ${\rm dim}\,\mathbb{P}(T)\times Y_{0}=25$.
There is now a subtlety in our situation: It is not hard to see that $T$
generates generically the bundle $\mathcal{F}_{0}$ on $Y_{0}$. Hence there is
a “main” component of $W_{2}$ which is also of dimension $15$, and is
generically fibered into $\mathbb{P}^{1}$’s over $Y_{0}$. This component is
not equal to $W_{2}^{0}$ for the following reason: If one takes two general
planes $P,\,P^{\prime}$ through $y_{0}$, and two general points
$x,\,x^{\prime}\in\mathbb{P}^{3}$, the conditions that $q(a,x)$ vanishes
identically on $P$ and $q(a,x^{\prime})$ vanishes identically on $P^{\prime}$
implies that the third column of the matrix $M(a)$ is identically $0$, hence
that the point $y_{0}$ generates in fact the kernel of both matrices $M(a,x)$
and $M(a,x^{\prime})$. On the other hand, by construction of the map $\Psi$,
generically along ${\rm Im}\,\Psi\subset W_{2}^{0}$, the point $y_{0}$ is a
smooth point of the quadrics $Q(a,x)$ and $Q(a,x^{\prime})$.
The following lemma describes the component $W_{2}^{0}$.
###### Lemma 2.3.
Let $\Phi:W_{2}^{0}\rightarrow Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times
G_{0}\times G_{0}$ be the restriction to $W_{2}^{0}$ of the second projection
$\mathbb{P}(T)\times Y_{0}\rightarrow Y_{0}$.
(i) The image ${\rm Im}\,\Phi$ is a hypersurface $\mathcal{D}$ which admits a
rationally connected desingularization.
(ii) The generic rank of the evaluation map restricted to $\mathcal{D}$:
$T\otimes\mathcal{O}_{\mathcal{D}}\rightarrow{\mathcal{F}_{0}}_{\mid\mathcal{D}}$
is $9$. (Note that the generic rank of the evaluation map
$T\otimes\mathcal{O}_{Y_{0}}\rightarrow{\mathcal{F}_{0}}$ is $10$.)
Proof. We already mentioned that the map $\Phi\circ\Psi$ cannot be dominating.
Let us explain more precisely why, as this will provide the equation for
$\mathcal{D}$: Let $[P],\,[P^{\prime}]\in G_{0}$ and
$x,\,x^{\prime}\in\mathbb{P}^{3}$. The condition that
$(x,x^{\prime},P,P^{\prime})\in\Phi(W_{2}^{0})={\rm Im}\,(\Phi\circ\Psi)$ is
that for some $a\in\mathbb{P}(V)$,
$\displaystyle q(a,x)_{\mid P}=0,\,q(a,x^{\prime})_{\mid P^{\prime}}=0$ (8)
and furthermore that $y_{0}$ is a smooth point of both quadrics $Q(a,x)$ and
$Q(a,x^{\prime})$.
Let $e,\,f\in P$ be vectors such that $y_{0},e,f$ form a basis of $P$, and
choose similarly $e^{\prime},f^{\prime}\in P^{\prime}$ in order to get a basis
of $P^{\prime}$. Then among equations (8), we get
$\displaystyle
q(a,x)(y_{0},e)=0,\,q(a,x)(y_{0},f)=0,\,q(a,x^{\prime})(y_{0},x^{\prime})(y_{0},e^{\prime})=0,\,q(a,x^{\prime})(y_{0},f^{\prime})=0.$
(9)
For fixed $P,\,P^{\prime},\,x,\,x^{\prime}$, these equations are linear forms
in the variables $a_{3},\,a_{6},\,a_{8},\,a_{9}$ and it is not hard to see
that they are linearly independent for a generic choice of
$P,\,P^{\prime},\,x,\,x^{\prime}$, so that (9) implies that
$a_{3}=a_{6}=a_{8}=a_{9}=0$. But then, looking at the matrix $M(a)$ of (3) we
see that $y_{0}$ is in the kernel of $Q(a,x)$ and $Q(a,x^{\prime})$. As
already mentioned, the latter does not happen generically along ${\rm
Im}\,\Psi$, and we deduce that ${\rm Im}\,(\Phi\circ\Psi)$ is contained in the
hypersurface $\mathcal{D}$ where the four linear forms (9) in the four
variables $a_{3},\,a_{6},\,a_{8},\,a_{9}$ are not independent.
We claim that $\mathcal{D}$ is rationally connected. To see this we first
prove that it is irreducible, which is done by restricting the equation $f$ of
$\mathcal{D}$ (which is the determinant of the $(4\times 4)$ matrix whose
columns are the linear forms (9) written in the basis
$a_{3},\,a_{6},\,a_{8},\,a_{9}$), to a subvariety $Z$ of $Y_{0}$ of the form
$Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times C\times C^{\prime}\subset Y_{0}$,
where $C,\,C^{\prime}$ are curves in $G_{0}$.
Consider the following $1$ dimensional families $C\cong\mathbb{P}^{1}$,
$C^{\prime}\cong\mathbb{P}^{1}$ of planes passing through $y_{0}$:
$P_{t}=<e_{1},\lambda e_{2}+\mu
e_{4},e_{3}>,\,t=(\lambda,\mu)\in\mathbb{P}^{1}\cong C,$
$P^{\prime}_{t^{\prime}}=<e_{5},\lambda^{\prime}e_{2}+\mu^{\prime}e_{4},e_{3}>,\,t^{\prime}=(\lambda^{\prime},\mu^{\prime})\in\mathbb{P}^{1}\cong
C^{\prime}.$
The equations (9) restricted to the parameters $t,t^{\prime},x,x^{\prime}$
give the following four combinations of $a_{3},\,a_{6},\,a_{8},\,a_{9}$
depending on $\lambda,\,\mu,\,x,x^{\prime}$:
$a_{3}x_{3},\,\,\lambda a_{6}x_{4}+\mu
a_{8}x_{1},\,\,\,a_{9}x^{\prime}_{2},\,\,\lambda^{\prime}a_{6}x^{\prime}_{4}+\mu^{\prime}a_{8}x^{\prime}_{1}.$
Taking the determinant of this family gives
$f_{\mid
Z}=x_{3}x^{\prime}_{2}(\lambda\mu^{\prime}x_{4}x^{\prime}_{1}-\lambda^{\prime}\mu
x^{\prime}_{4}x_{1}).$
The hypersurface in $Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times C\times
C^{\prime}$ defined by $f_{\mid Z}$ has three irreducible components, which
belong respectively to the linear systems
$|pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|pr_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
pr_{3}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes
pr_{4}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,$
where the $pr_{i}$’s are now the projections from
$Z=\mathbb{P}^{3}\times\mathbb{P}^{3}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
to its factors.
As the restriction map from ${\rm Pic}\,Y_{0}$ to ${\rm Pic}\,Z$ is injective,
we see that if $\mathcal{D}$ was reducible, it would have an irreducible
component in one of the linear systems
$|p_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|p_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,\,|p_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
p_{2}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)\otimes
p_{3}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes
p_{4}^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)|,$
where the $p_{i}$’s are now the projections from
$Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0}$ to its
factors. In particular, its restriction to either any
$\mathbb{P}^{3}\times\\{x^{\prime}\\}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
or any $\\{x\\}\times\mathbb{P}^{3}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
would have an irreducible component of degree $1$.
That this is not the case is proved by considering now the case where
$P=<e_{1}+e_{2},\,e_{4}+e_{5},\,e_{3}>,\,\,P^{\prime}=<e_{1}+e_{4},\,e_{2}+e_{5},\,e_{3}>.$
Then the four linear forms in (9) become
$a_{3}x_{3}+a_{6}x_{4},\,a_{8}x_{1}+a_{9}x_{2},\,a_{3}x^{\prime}_{3}+a_{8}x^{\prime}_{1},\,a_{6}x^{\prime}_{4}+a_{9}x^{\prime}_{2}.$
It is immediate to check that for generic choice of $x^{\prime}$, the
quadratic equation in $x$ given by this determinant has rank at least $4$, and
in particular is irreducible. Hence the restriction of $f$ to
$\mathbb{P}^{3}\times\\{x^{\prime}\\}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$
has no irreducible component of degree $1$, and similarly for
$\\{x\\}\times\mathbb{P}^{3}\times\\{[P]\\}\times\\{[P^{\prime}]\\}$.
Rational connectedness of $\mathcal{D}$ now follows from the fact that the
projection on the last three summands of $Y_{0}$, restricted to $\mathcal{D}$,
$p:\mathcal{D}\hookrightarrow Y_{0}\rightarrow\mathbb{P}^{3}\times G_{0}\times
G_{0},$
has for general fibers smooth two-dimensional quadrics, as shows the
computation we just made. Hence $\mathcal{D}$ admits a rationally connected
smooth projective model, for example by [14].
We already proved that ${\rm Im}\,(\Phi\circ\Psi)\subset\mathcal{D}$. We will
now prove that ${\rm Im}\,(\Phi\circ\Psi)$ contains a Zariski open subset of
$\mathcal{D}$ and also statement ii). This is done by the following argument
which involves an explicit computation at a point $a_{0}\in\mathbb{P}(T)$
where the surface $V(a)$ becomes very singular but has enough smooth points to
conclude.
We start from the following specific matrix :
$\displaystyle M_{0}=\begin{pmatrix}x_{1}&0&x_{3}&0&0\\\ 0&x_{3}&x_{4}&0&0\\\
x_{3}&x_{4}&0&x_{1}&0\\\ 0&0&x_{1}&x_{2}&0\\\ 0&0&0&0&x_{4}\end{pmatrix}$ (10)
We compute ${\rm
det}\,M_{0}=-x_{4}x_{3}x_{1}^{3}-x_{1}x_{2}x_{4}^{3}-x_{4}x_{2}x_{3}^{3}$. The
surface $V(a_{0})$ defined by ${\rm det}\,M_{0}$ is rational and smooth at the
points
$\displaystyle x=(1,-\frac{1}{2},1,1),\,\,\,x^{\prime}=(1,-\frac{2}{9},2,-1).$
(11)
For each of the corresponding planes $P$ and $P^{\prime}$ contained in
$Q(a_{0},x)$, $Q(a_{0},x^{\prime})$ respectively and passing through $y_{0}$,
we have two choices. We choose the following:
$\displaystyle
P:=<y_{0},\,e,\,f>,\,e=e_{1}+e_{2}-2e_{4},\,f=e_{2}-e_{4}+\sqrt{-\frac{1}{2}}e_{5},$
(12) $\displaystyle
P^{\prime}=<y_{0},\,e^{\prime},\,f>,\,e^{\prime}=4e_{1}-e_{2}-9e_{4},\,f^{\prime}=e_{2}+e_{4}+\frac{4}{3}e_{5}.$
(We use here the standard basis $(e_{1},\ldots,e_{5})$ of $\mathbb{C}^{5}$, so
that $y_{0}=e_{3}$.) The vector $e-e_{3}$ (resp. $e^{\prime}-2e_{3}$)
generates the kernel of $q(a_{0},x)$ (resp. $q(a_{0},x^{\prime})$). As the
quadrics $q(a_{0},x)$ and $q(a_{0},x^{\prime})$ have rank $3$ and the surface
$V(a_{0})$ is smooth at $x$ and $x^{\prime}$, the points $(x,[P])$,
$(x^{\prime},[P^{\prime}])$ determine smooth points of the surface $S(a_{0})$.
Even if the surface $S(a_{0})$ is not smooth, so that $a_{0}\not\in B$, the
universal family $\mathcal{S}\rightarrow B$ extends to a smooth non proper map
$p:\mathcal{S}_{e}\rightarrow B_{e}\subset\mathbb{P}(T)$ near the points
$(a_{0},x,[P])$ and $(a_{0},x^{\prime},[P^{\prime}])$. Furthermore the
morphism $\Psi$ is well defined at the points $(a_{0},x,P)$,
$(a_{0},x^{\prime},P^{\prime})$ since the point $y_{0}$ is not a singular
point of the quadric $Q(a_{0},x)$ or $Q(a_{0},x^{\prime})$.
We claim that the map
$\Phi\circ\Psi:\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}\rightarrow
Y_{0}=\mathbb{P}^{3}\times\mathbb{P}^{3}\times G_{0}\times G_{0}$
has constant rank $13$ near the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))\in\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}.$
This implies that the image of $\Phi\circ\Psi$ is of dimension $13$, which is
the dimension of $\mathcal{D}$, so that ${\rm Im}\,\Phi\circ\Psi$ has to
contain a Zariski open set of $\mathcal{D}$ since $\mathcal{D}$ is
irreducible. This implies that ${\rm Im}\,\Phi=\mathcal{D}$ as desired.
To prove the claim, we observe that it suffices to show the weaker claim that
the rank of
$\displaystyle(\Phi\circ\Psi)_{*}:T_{\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}}\rightarrow
T_{Y_{0}}$ (13)
is equal to $13$ at the given point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$. Indeed, as
$\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}$ is smooth at the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$, the rank of the
differential in (13) can only increase in a neighborhood of
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$ and on the other hand, we
know that it is not of maximal rank at any point of
$\mathcal{S}^{\prime}\times_{B^{\prime}}\mathcal{S}^{\prime}$, since ${\rm
Im}\,(\Phi\circ\Psi)\subset\mathcal{D}$. Hence it must stay constant near
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$.
We now compute the rank of $(\Phi\circ\Psi)_{*}$ at the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$. Note that the birational
map $\Psi:\mathcal{S}_{e}\times_{B_{e}}\mathcal{S}_{e}\rightarrow W_{2}$ is a
local isomorphism near the point
$((a_{0},x,[P]),(a_{0},x^{\prime},[P^{\prime}]))$, because it is well defined
at this point and its inverse too. This argument proves that not only
$W_{2}^{0}$ but also $W_{2}$ is smooth of dimension $15$ near the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$. Hence it suffices to compute the rank
of the differential
$\displaystyle\Phi_{*}:T_{W_{2},(a_{0},x,x^{\prime},[P],[P^{\prime}])}\rightarrow
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}$ (14)
of the map $\Phi:W_{2}\rightarrow Y_{0}$ at the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$.
Recalling that $W_{2}$ is defined as the zero set of the universal section
$\sigma_{univ}$ of the bundle
${pr^{\prime}_{1}}^{*}\mathcal{O}_{\mathbb{P}(T)}(1)\otimes{pr^{\prime}_{2}}^{*}\mathcal{F}_{0}$
on $\mathbb{P}(T)\times Y_{0}$, the tangent space of $W_{2}$ at the point
$(a_{0},x,x^{\prime},[P],[P^{\prime}])$ is equal to
${\rm Ker}\,(d\sigma_{univ}:T_{\mathbb{P}(T),a_{0}}\times
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])}).$
Clearly the differential $d\sigma_{univ}$ restricted to
$T_{\mathbb{P}(T),a_{0}}$ is induced by the evaluation map
$\displaystyle{\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}:T\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])}$
(15)
at the point $(x,x^{\prime},[P],[P^{\prime}])$ of $Y_{0}$. On the other hand,
the differential $d\sigma_{univ}$ restricted to the tangent space
$T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}$ is surjective because the variety
$S(a_{0})\times S(a_{0})$ is smooth of codimension $4$ and isomorphic via
$\Phi$ to $V(\sigma_{univ}(a_{0}))$ near $(x,x^{\prime},[P],[P^{\prime}])$. It
follows from this that the corank of the second projection
$\Phi_{*}:{\rm
Ker}\,d\sigma_{univ}=T_{W_{2},(a_{0},x,x^{\prime},[P],[P^{\prime}])}\subset
T_{\mathbb{P}(T),a_{0}}\oplus
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])}\rightarrow
T_{Y_{0},(x,x^{\prime},[P],[P^{\prime}])},$
is equal to the corank of the map
$d\sigma_{univ}:T_{\mathbb{P}(T),a_{0}}\rightarrow\mathcal{F}_{0,(x,x^{\prime},[P],[P^{\prime}])})$,
that is to the corank of the evaluation map ${\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}$ of (15).
In particular, the rank of $\Phi_{*}$ in (14) is equal to $13$ if and only if
the rank of the evaluation map ${\rm ev}_{x,x^{\prime},[P],[P^{\prime}]}$ is
equal to $9$, which is our statement (ii).
In conclusion, we proved that (i) is implied by (ii) and that (ii) itself
implied by (ii) at the given point $(x,x^{\prime},[P],[P^{\prime}])$ of (11),
(12). It just remains to prove that the rank of ${\rm
ev}_{(x,x^{\prime},[P],[P^{\prime}])}$ is equal to $9$ at this point, which is
done by the explicit computation of the rank of the family of linear forms in
the $a_{i}$ given by
$q(a,x)(y_{0},e),\,\,\,q(a,x)(y_{0},f),\,\,\,q(a,x)(e,e),\,\,\,q(a,x)(f,f),\,\,\,q(a,x)(e,f),$
$q(a,x^{\prime})(y_{0},e^{\prime}),\,\,\,q(a,x^{\prime})(y_{0},f^{\prime}),\,\,\,q(a,x^{\prime})(e^{\prime},e^{\prime}),\,\,\,q(a,x^{\prime})(f^{\prime},f^{\prime}),\,\,\,q(a,x^{\prime})(e^{\prime},f^{\prime}).$
These forms are the following:
$a_{3}+a_{6}-2a_{8},\,\,\,\,\,\,a_{6}-a_{8}-\frac{1}{2}\sqrt{-\frac{1}{2}}a_{9},\,\,\,\,\,\,a_{1}+a_{5}-2a_{10}-a_{2}-4a_{4},$
$a_{5}-\frac{a_{10}}{2}-\frac{a_{12}}{2}+2\sqrt{-\frac{1}{2}}a_{7}-2\sqrt{-\frac{1}{2}}a_{11},\,\,\,\,\,\,-\frac{a_{2}}{2}-a_{4}+a_{5}+\sqrt{-\frac{1}{2}}a_{7}-a_{10}-2\sqrt{-\frac{1}{2}}a_{11}$
$8a_{3}+a_{6}-9a_{8},\,\,\,\,\,\,-a_{6}+a_{8}-\frac{8}{27}a_{9},\,\,\,\,\,\,16a_{1}+2a_{5}-18a_{10}+\frac{16}{9}a_{2}+72a_{4},$
$2a_{5}-\frac{2}{9}a_{10}-\frac{16}{9}a_{12}+\frac{8}{3}a_{7}+\frac{16}{3}a_{11},\,\,\,-\frac{8}{9}a_{2}-4a_{4}-2a_{5}-\frac{4}{3}a_{7}+2a_{10}-24a_{11}.$
It is immediate to check that the rank of this family is $9$.
It follows from Lemma 2.3 that $W_{2}^{0}$ (or rather a smooth projective
birational model of $W_{2}^{0}$) is rationally connected, since a Zariski open
set of $W_{2}^{0}$ is a $\mathbb{P}^{2}$-bundle over a Zariski open set
$\mathcal{D}^{0}$ of $\mathcal{D}$ which is smooth and admits a rationally
connected completion. Hence $\mathcal{S}\times_{B}\mathcal{S}$ is also
rationally connected, as it is birationally equivalent to $W_{2}^{0}$.
We get by application of Theorem 1.6 the following statement (cf. Theorem
0.6):
###### Corollary 2.4.
The Campedelli surface $\Sigma(a)$ has $CH_{0}$ equal to $\mathbb{Z}$.
Proof. Indeed, the surface $\Sigma(a)$ has $h^{1,0}=h^{2,0}=0$ (cf. [11])
which means equivalently that the surface $S(a)$ introduced above has
$h^{2,0}_{inv}=0$ where “inv” means invariant under the
$\mathbb{Z}/5\mathbb{Z}$-action, which by the above description is defined on
$\mathcal{S}$. The result is then a consequence of Theorem 1.6 applied to the
projector $\pi_{inv}$ onto the $\mathbb{Z}/5\mathbb{Z}$-invariant part, which
shows that
$CH_{0}(\Sigma(a))_{\mathbb{Q},hom}=CH_{0}(S(a))^{\pi_{inv}}_{\mathbb{Q},hom}=0$
and from Roitman’s theorem [24] which says that $CH_{0}(\Sigma(a))$ has no
torsion.
###### Corollary 2.5.
The Barlow surface $\Sigma^{\prime}(b)$ has $CH_{0}$ equal to $\mathbb{Z}$.
Proof. Indeed, the Barlow surface $\Sigma^{\prime}(b)$ is a quotient of the
Campedelli surface $\Sigma(b)$ by an involution, hence
$CH_{0}(\Sigma^{\prime}(b))\hookrightarrow CH_{0}(\Sigma(b))$ since
$CH_{0}(\Sigma^{\prime}(b))$ has no torsion by [24].
## 3 On the Chow motive of complete intersections $K3$ surfaces
Let $S$ be a $K3$ surface. The Hodge structure on $H^{2}(S,\mathbb{Z})$ is a
weight $2$ polarized Hodge structure with $h^{2,0}=1$. In [18], Kuga and
Satake construct an abelian variety $K(S)$ associated to this Hodge structure.
Its main property is the fact that there is a natural injective morphism of
weight $2$ Hodge structures :
$H^{2}(S,\mathbb{Z})\rightarrow H^{2}(K(S),\mathbb{Z}).$
Such a morphism of Hodge structures in turn provides, using Künneth
decomposition and Poincaré duality a Hodge class
$\alpha_{S}\in{\rm Hdg}^{4}(S\times K(S))$
where ${\rm Hdg}^{4}(X)$ denotes the space of rational Hodge classes on $X$.
This class is not known in general to satisfy the Hodge conjecture, that is,
to be the class of an algebraic cycle. This is known to hold for Kummer
surfaces (see [19]), and for some $K3$ surfaces with Picard number $16$ (see
[23]). The deformation theory of $K3$ surfaces, and more particularly the fact
that any projective $K3$ surface deforms to a Kummer surface, combined with
the global invariant cycle theorem of Deligne [12] imply the following result,
(cf. [13], [2]):
###### Theorem 3.1.
Let $S$ be a projective $K3$ surface. There exist a connected quasi-projective
variety $B$, a family of projective $K3$ surfaces $\mathcal{S}\rightarrow B$,
a family of abelian varieties $\mathcal{K}\rightarrow B$, (where all varieties
are quasi-projective and all morphisms are smooth projective), and a Hodge
class
$\eta\in{\rm Hdg}^{4}(\mathcal{S}\times_{B}\mathcal{K}),$
such that :
1. 1.
for some point $t_{0}\in B$, $\mathcal{S}_{t_{0}}\cong S$;
2. 2.
For any point $t\in B$, $\mathcal{K}_{t}\cong K(S_{t})$ and
$\eta_{t}=\alpha_{S_{t}}$.
3. 3.
For some point $t_{1}\in B$, the class $\eta_{t_{1}}$ is algebraic.
Here by “Hodge class $\eta$”, we mean that the class $\eta$ comes from a Hodge
class on some smooth projective compactification of
$\mathcal{S}\times_{B}\mathcal{K}$.
The existence of the family is an algebraicity statement for the Kuga-Satake
construction (see [13]), which can then be done in family for $K3$ surfaces
with given polarization, while the last item follows from the fact that the
locally complete such families always contain Kummer fibers $S^{\prime}$ for
which the class $\alpha_{S^{\prime}}$ is known to be algebraic.
###### Corollary 3.2.
[2] The Hodge class $\alpha_{S}$ is “motivated”.
This means (cf. [2]) that this Hodge class can be constructed via algebraic
correspondences from Hodge classes on auxiliary varieties, which are either
algebraic or obtained by inverting Lefschetz operators. In particular the
class $\alpha_{S}$ is algebraic if the standard Lefschetz conjecture holds.
###### Corollary 3.3.
Assume the variational form of the Hodge conjecture holds, or assume the
Lefschetz standard conjecture in degrees $2$ and $4$. Then the class
$\alpha_{S}$ is algebraic for any projective $K3$ surface $S$.
Indeed, the variational Hodge conjecture states that in the situation of
Theorem 3.1, if a Hodge class $\eta$ on the total space has an algebraic
restriction on one fiber, then its restriction to any fiber is algebraic. In
our situation, it will be implied by the Lefschetz conjecture for degree $2$
and degree $4$ cohomology on a smooth projective compactification of
$\mathcal{S}\times_{B}\mathcal{K}$.
From now on, we assume that the Kuga-Satake correspondence $\alpha_{S}$ is
algebraic for a general projective $K3$ surface $S$ of genus $g$ (which means
by definition that $S$ comes equipped with an ample line bundle of self-
intersection $2g-2$). We view the class $\alpha_{S}$ as an injective morphism
of Hodge structures
$\displaystyle\alpha_{S}:H^{2}(S,\mathbb{Q})\rightarrow
H^{2}(K(S),\mathbb{Q})$ (16)
and use a polarization $h$ of $K(S)$ to construct an inverse of $\alpha_{S}$
by the following lemma (which can be proved as well by the explicit
description $K(S)$ as a complex torus and its polarization, cf. [18]). In the
following, $H^{2}(S,\mathbb{Q})_{tr}$ denotes the transcendental cohomology of
$S$, which is defined as the orthogonal complement of the Néron-Severi group
of $S$.
###### Lemma 3.4.
Let $h=c_{1}(H)\in H^{2}(K(S),\mathbb{Q})$ be the class of an ample line
bundle, where $S$ is a very general $K3$ surface of genus $g$. Then there
exists a nonzero rational number $\lambda_{g}$ such that the endomorphism
${}^{t}\alpha_{S}\circ(h^{N-2}\cup)\circ\alpha_{S}:H^{2}(S,\mathbb{Q})\rightarrow
H^{2}(S,\mathbb{Q})$
restricts to $\lambda_{g}Id$ on $H^{2}(S,\mathbb{Q})_{tr}$, where $N={\rm
dim}\,K(S)$ and
${}^{t}\alpha_{S}:H^{2N-2}(K(S),\mathbb{Q})\rightarrow H^{2}(S,\mathbb{Q})$
is the transpose of the map $\alpha_{S}$ of (16) with respect to Poincaré
duality.
Proof. The composite ${}^{t}\alpha_{S}\circ h^{N-2}\circ\alpha_{S}$ is an
endomorphism of the Hodge structure on $H^{2}(S,\mathbb{Q})$. As $S$ is very
general, this Hodge structure has only a one dimensional $\mathbb{Q}$-vector
space of algebraic classes, generated by the polarization, and its orthogonal
is a simple Hodge structure with only the homotheties as endomorphisms. We
conclude that ${}^{t}\alpha_{S}\circ h^{N-2}\circ\alpha_{S}$ preserves
$H^{2}(S,\mathbb{Q})_{tr}$ and acts on it as an homothety with rational
coefficient (which is thus independent of $S$). It just remains to show that
it does not act as zero on $H^{2}(S,\mathbb{Q})_{tr}$. This follows from the
second Hodge-Riemmann bilinear relations which say that for $\omega\in
H^{2,0}(S),\,\omega\not=0$, we have
$\langle\omega,\,{{}^{t}\alpha_{S}}(h^{N-2}\cup\alpha_{S}(\overline{\omega}))\rangle_{S}=\langle\alpha_{S}(\omega),h^{N-2}\cup\alpha_{S}(\overline{\omega})\rangle_{K(S)}$
which is $>0$ because $\alpha_{S}(\omega)\not=0$ in $H^{2,0}(K(S))$. Hence
${}^{t}\alpha_{S}(h^{N-2}\cup\alpha_{S}(\overline{\omega}))\not=0$.
We now start from a rationally connected variety $X$ of dimension $n$, with a
vector bundle $E$ of rank $n-2$ on $X$, such that $-K_{X}={\rm det}\,E$ and
the following properties hold:
(*) For general $x,\,y\in X$, and for general $\sigma\in
H^{0}(X,E\otimes\mathcal{I}_{x}\otimes\mathcal{I}_{y})$, the zero locus
$V(\sigma)$ is a smooth connected surface with $0$ irregularity (hence a
smooth $K3$ surface).
Let $L$ be an ample line bundle on $X$, inducing a polarization of genus $g$
on the $K3$ surface $S_{\sigma}:=V(\sigma)$ for $\sigma\in
B\subset\mathbb{P}(H^{0}(X,E))$.
We now prove:
###### Theorem 3.5.
Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic, for the
general $K3$ surface $S$ with such a polarization. Then, for any $\sigma\in
B$, the Chow motive of $S_{\sigma}$ is a direct summand of the motive of an
abelian variety.
Let us explain the precise meaning of this statement. The algebraicity of the
Kuga-Satake correspondence combined with Lemma 3.4 implies that there is a
codimension $2$ algebraic cycle
$Z_{S}\in CH^{2}(S\times K(S))_{\mathbb{Q}}$
with the property that the cycle $\Gamma_{S}$ defined by
$\Gamma_{S}={{}^{t}Z_{S}}\circ h^{N-2}\circ Z_{S}\in CH^{2}(S\times
S)_{\mathbb{Q}}$
has the property that its cohomology class $[\Gamma]\in H^{4}(S\times
S,\mathbb{Q})$ induces a nonzero homothety
$\displaystyle[\Gamma]_{*}=\lambda Id:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr},$ (17)
which can be equivalently formulated as follows: Let us introduce the cycle
$\Delta_{S,tr}$, which in the case of a $K3$ surface is canonically defined,
by the formula
$\displaystyle\Delta_{S,tr}=\Delta_{S}-o_{S}\times S-S\times
o_{S}-\sum_{ij}\alpha_{ij}C_{i}\times C_{j},$ (18)
where $\Delta_{S}$ is the diagonal of $S$, $o_{S}$ is the canonical $0$-cycle
of degree $1$ on $S$ introduced in [6], the $C_{i}$ form a basis of $({\rm
Pic}\,S)\otimes\mathbb{Q}={\rm NS}(S)_{\mathbb{Q}}$ and the $\alpha_{ij}$ are
the coefficients of the inverse of the matrix of the intersection form of $S$
restricted to $NS(S)$. This corrected diagonal cycle is a projector and it has
the property that its action on cohomology is the orthogonal projector
$H^{*}(S,\mathbb{Q})\rightarrow H^{2}(S,\mathbb{Q})_{tr}$. Formula (17) says
that we have the cohomological equality
$\displaystyle[\Gamma\circ\Delta_{S,tr}]=\lambda[\Delta_{S,tr}]\,\,{\rm
in}\,\,H^{4}(S\times S,\mathbb{Q}).$ (19)
A more precise form of Theorem 3.5 says that we can get in fact such an
equality at the level of Chow groups:
###### Theorem 3.6.
Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic, for a general
$K3$ surface with such a polarization. Then, for any $\sigma\in B$, there is
an abelian variety $A_{\sigma}$, and cycles $Z\in CH^{2}(S_{\sigma}\times
A_{\sigma})_{\mathbb{Q}}$, $Z^{\prime}\in CH^{N^{\prime}}(A_{\sigma}\times
S_{\sigma})_{\mathbb{Q}}$, $N^{\prime}={\rm dim}\,A_{\sigma}$, with the
property that
$\displaystyle Z^{\prime}\circ Z\circ\Delta_{S,tr}=\lambda\Delta_{S,tr}$ (20)
for a nonzero rational number $\lambda$.
The proof will use two preparatory lemmas. Let
$B\subset\mathbb{P}(H^{0}(X,E))$ be the open set parameterizing smooth
surfaces $S_{\sigma}=V(\sigma)\subset X$. Let $\pi:\mathcal{S}\rightarrow B$
be the universal family, that is:
$\displaystyle\mathcal{S}=\\{(\sigma,x)\in B\times X,\,\sigma(x)=0\\}.$ (21)
The first observation is the following:
###### Lemma 3.7.
Under assumption (*), the fibered self-product
$\mathcal{S}\times_{B}\mathcal{S}$ is rationally connected (or rather, admits
a smooth projective rationally connected completion).
Proof. From (21), we deduce that
$\displaystyle\mathcal{S}\times_{B}\mathcal{S}=\\{(\sigma,x,y)\in B\times
X\times X,\,\sigma(x)=\sigma(y)=0\\}.$ (22)
Hence $\mathcal{S}\times_{B}\mathcal{S}$ is Zariski open in the following
variety:
$W:=\\{(\sigma,x,y)\in\mathbb{P}(H^{0}(X,E))\times X\times
X,\,\sigma(x)=\sigma(y)=0\\}.$
In particular, as it is irreducible, it is Zariski open in one irreducible
component $W^{0}$ of $W$.
Consider the projection on the two last factors:
$(p_{2},p_{3}):W\rightarrow X\times X.$
Its fibers are projective spaces, so that there is only one “main” irreducible
component $W^{1}$ of $W$ dominating $X\times X$ and it admits a smooth
rationally connected completion since $X\times X$ is rationally connected.
Assumption (*) now tells us that at a general point of $W^{1}$, the first
projection $p_{1}:W\rightarrow B$ is smooth of relative dimension $4$. It
follows that $W$ is smooth at this point which belongs to both components
$W^{0}$ and $W^{1}$. Thus $W^{0}=W^{1}$ and
$\mathcal{S}\times_{B}\mathcal{S}\cong_{birat}W^{0}$ admits a smooth
rationally connected completion.
The next step is the following lemma:
###### Lemma 3.8.
Assume the Kuga-Satake correspondence $\alpha_{S}$ is algebraic for the
general polarized $K3$ surface of genus $g$. Then there exist a rational
number $\lambda\not=0$, a family $\mathcal{A}\rightarrow B$ of polarized
$N^{\prime}$-dimensional abelian varieties, with relative polarization
$\mathcal{L}$ and a codimension $2$ cycle
$\mathcal{Z}\in CH^{2}(\mathcal{S}\times_{B}\mathcal{A})_{\mathbb{Q}}$
such that for very general $t\in B$, the cycle
$\Gamma_{t}:={{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}\circ\mathcal{Z}_{t}\circ\Delta_{tr,t}$
satisfies:
$\displaystyle[\Gamma_{t}]=\lambda[\Delta_{tr,t}]\in H^{4}(S_{t}\times
S_{t},\mathbb{Q}).$ (23)
In this formula, the term $c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}$ is defined
as the self-correspondence of $\mathcal{A}_{t}$ which consists of the cycle
$c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}$ supported on the diagonal of
$\mathcal{A}_{t}$. We also recall that the codimension $2$-cycle
$\Delta_{tr,t}\in CH^{2}(S_{t}\times S_{t})_{\mathbb{Q}}$ is the projector
onto the transcendental part of the motive of $S_{t}$. The reason why the
result is stated only for the very general point $t$ is the fact that due to
the possible jump of the Picard group of $S_{t}$, the generic cycle
$\Delta_{tr,\overline{\eta}}$ does not specialize to the cycle $\Delta_{tr,t}$
at any closed point $t\in B$, but only at the very general one. In fact, the
statement is true at any point, but the cycle $\Delta_{tr,t}$ has to be
modified when the Picard group jumps.
Proof of Lemma 3.8. By our assumption, using the countability of relative
Hilbert schemes and the existence of universal objects parameterized by them,
there exist a generically finite cover $r:B^{\prime}\rightarrow B$, a
universal family of polarized abelian varieties
$\mathcal{K}\rightarrow B^{\prime},\,\mathcal{L}_{K}\in{\rm Pic}\,\mathcal{K}$
and a codimension $2$ cycle $\mathcal{Z}^{\prime}\in
CH^{2}(\mathcal{S}^{\prime}\times_{B^{\prime}}\mathcal{K})_{\mathbb{Q}}$,
where $\mathcal{S}^{\prime}:=\mathcal{S}\times_{B}B^{\prime}$, with the
property that
$\displaystyle[\mathcal{Z}_{t}]=\alpha_{\mathcal{S}_{t}}\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times K(\mathcal{S}_{t}),\mathbb{Q})$ (24)
for any $t\in B^{\prime}$. Furthermore, by Lemma 3.4, we know that there
exists a nonzero rational number $\lambda_{g}$ such that for any $t\in
B^{\prime}$, we have
${}^{t}\alpha_{\mathcal{S}_{t}}\circ(h^{N-2}\cup)\circ\alpha_{\mathcal{S}_{t}}=\lambda_{g}Id:H^{2}(\mathcal{S}_{t},\mathbb{Q})_{tr}\rightarrow
H^{2}(\mathcal{S}_{t},\mathbb{Q})_{tr},$ (25)
where as before $N$ is the dimension of $K(\mathcal{S}_{t})$, and
$h_{t}=c_{1}(\mathcal{L}_{K\mid\mathcal{K}_{t}})$. We now construct the
following family of abelian varieties on $B$ (or a Zariski open set of it)
$\mathcal{A}_{t}=\prod_{t^{\prime}\in r^{-1}(t)}\mathcal{K}_{t^{\prime}},$
with polarization given by
$\displaystyle\mathcal{L}_{t}=\sum_{t^{\prime}\in
r^{-1}(t)}pr_{t^{\prime}}^{*}(\mathcal{L}_{K,t^{\prime}}),$ (26)
where $pr_{t^{\prime}}$ is the obvious projection from
$\mathcal{A}_{t}=\prod_{t^{\prime}\in r^{-1}(t)}K_{t^{\prime}}$ to its factor
$\mathcal{K}_{t^{\prime}}$, and the following cycle $\mathcal{Z}\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{K})_{\mathbb{Q}}$, with fiber at $t\in B$
given by
$\displaystyle\mathcal{Z}_{t}=\sum_{t^{\prime}\in
r^{-1}(t)}(Id_{\mathcal{S}_{t}},pr_{t^{\prime}})^{*}\mathcal{Z}^{\prime}_{t^{\prime}}.$
(27)
In the last formula, we use of course the identification
$\mathcal{S}^{\prime}_{t^{\prime}}=\mathcal{S}_{t},\,r(t^{\prime})=t.$
It just remains to prove formula (23). Combining (26) and (27), we get, using
again the notation $h_{t}=c_{1}(\mathcal{L}_{t})\in H^{2}(A_{t},\mathbb{Q})$,
$h_{t^{\prime}}=c_{1}(\mathcal{L}_{K,t^{\prime}})\in
H^{2}(K_{t^{\prime}},\mathbb{Q})$:
$[\Gamma_{t}]^{*}=$ $(\sum_{t^{\prime}\in
r^{-1}(t)}[(pr_{t^{\prime}},Id_{\mathcal{S}_{t}})^{*}(^{t}\mathcal{Z}^{\prime}_{t^{\prime}})]^{*})\circ(\sum_{t^{\prime\prime}\in
r^{-1}(t)}pr_{t^{\prime\prime}}^{*}(h_{t^{\prime\prime}}))^{N^{\prime}-2}\cup\circ(\sum_{t^{\prime\prime\prime}\in
r^{-1}(t)}(Id_{\mathcal{S}_{t}},pr_{t^{\prime\prime\prime}})^{*}[\mathcal{Z}^{\prime}_{t^{\prime\prime\prime}}]^{*})\circ\pi_{tr,t}:$
$H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}).$
Note that $N^{\prime}={\rm dim}\,\mathcal{A}_{t}=\natural(r^{-1}(t))\,N={\rm
deg}\,(B^{\prime}/B)\,\,N$. We develop the product above, and observe that the
only nonzero terms appearing in this development come from by taking
$t^{\prime}=t^{\prime\prime\prime}$ and putting the monomial
$pr_{t^{\prime}}^{*}(h_{t^{\prime}}))^{N-2}\cup_{t^{\prime\prime}\not=t^{\prime}}pr_{t^{\prime\prime}}^{*}(h_{t^{\prime\prime}}))^{N}$
in the middle term. The other terms are $0$ due to the projection formula. Let
us explain this in the case of only two summands $K_{1},\,K_{2}$ of dimension
$r$ with polarizations $l_{1},\,l_{2}$ and two cycles $Z_{i}\in CH^{2}(S\times
K_{i})$ giving rise to a cycle of the form
$(Id_{S},pr_{1})^{*}Z_{1}+(Id_{S},pr_{2})^{*}Z_{2}\in CH^{2}(S\times
K_{1}\times K_{2}),$
where the $pr_{i}$’s are the projections from $S\times K_{1}\times K_{2}$ to
$K_{i}$: Then we have
$[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ(l_{1}+l_{2})^{2r-2}\cup\circ[{(Id_{S},pr_{2})}^{*}Z_{2}]^{*}=0:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr}$
by the projection formula and for the same reason
$[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ(l_{1}+l_{2})^{2r-2}\cup\circ[{(Id_{S},pr_{1})}^{*}Z_{1}]^{*}$
$=[{(pr_{1},Id_{S})^{*}}^{t}Z_{1}]^{*}\circ\sum_{0\leq k\leq
r}\binom{2r-2}{k}(l_{1}^{k}l_{2}^{2r-2-k})\cup\circ[{(Id_{S},pr_{1})}^{*}Z_{1}]^{*}$
$\underset{proj.\,formula}{=}\binom{2r-2}{r}{\rm
deg}\,(l_{2}^{r})\,\,[Z_{1}]^{*}\circ
l_{1}^{r-2}\cup\circ[^{t}Z_{1}]^{*}:H^{2}(S,\mathbb{Q})_{tr}\rightarrow
H^{2}(S,\mathbb{Q})_{tr}.$
We thus get (as the degrees ${\rm deg}\,(h_{t^{\prime}}^{N})$ of the
polarizations $h_{t^{\prime}}$ on the abelian varieties
$\mathcal{K}_{t^{\prime}}$ are all equal):
$[\Gamma_{t}]^{*}=M({\rm deg}\,(h_{t^{\prime}}^{N}))^{{\rm
deg}\,(B^{\prime}/B)-1}(\sum_{t^{\prime}\in
r^{-1}(t)}[^{t}\mathcal{Z}^{\prime}_{t^{\prime}})]^{*})\circ
h_{t^{\prime}}^{N-2}\cup\circ[\mathcal{Z}^{\prime}_{t^{\prime}}]^{*})\circ\pi_{tr,t}:$
$H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}),$
where $M$ is the multinomial coefficient appearing in front of the monomial
$x_{1}^{N-2}x_{2}^{N}\ldots x_{{\rm deg}\,(B^{\prime}/B)}^{N}$ in the
development of $(x_{1}+\ldots+x_{{\rm deg}\,(B^{\prime}/B)})^{N{\rm
deg}\,(B^{\prime}/B)-2}$.
By (24) and (25), we conclude that
$[\Gamma_{t}]^{*}=M({\rm deg}\,(h_{t^{\prime}}^{N}))^{{\rm
deg}\,(B^{\prime}/B)-1}{{\rm
deg}\,(B^{\prime}/B)}\lambda_{g}\pi_{tr,t}:H^{*}(\mathcal{S}_{t},\mathbb{Q})\rightarrow
H^{*}(\mathcal{S}_{t},\mathbb{Q}),$
which proves formula (23) with $\lambda=M({\rm
deg}\,(h_{t^{\prime}}^{N}))^{{\rm deg}\,(B^{\prime}/B)-1}{{\rm
deg}\,(B^{\prime}/B)}\lambda_{g}$.
Proof of Theorem 3.6. Consider the cycle $\mathcal{Z}\in
CH^{2}(\mathcal{S}\times_{B}\mathcal{A})_{\mathbb{Q}}$ of Lemma 3.8, and the
cycle $\Gamma\in CH^{2}(\mathcal{S}\times_{B}\mathcal{S})_{\mathbb{Q}}$
$\displaystyle\Gamma:=\Delta_{tr}\circ{{}^{t}\mathcal{Z}}\circ
c_{1}(\mathcal{L})^{N^{\prime}-2}\circ\mathcal{Z}\circ\Delta_{tr}$ (28)
where now $c_{1}(\mathcal{L})$ is the class of $\mathcal{L}$ in
$CH^{1}(\mathcal{A})$ and we denote by $c_{1}(\mathcal{L})^{N-2}$ the relative
self-correspondence of $\mathcal{A}$ given by the cycle
$c_{1}(\mathcal{L})^{N-2}$ supported on the relative diagonal of $\mathcal{A}$
over $B$ (it thus induces the intersection product with
$c_{1}(\mathcal{L})^{N-2}$ on Chow groups). Furthermore the composition of
correspondences is the relative composition over $B$, and $\Delta_{tr}$ is the
generic transcendental motive, (which is canonically defined in our case, at
least after restricting to a Zariski open set of $B$) obtained as follows:
Choose a $0$-cycle $o_{\overline{\eta}}$ of degree $1$ on the generic
geometric fiber $\mathcal{S}_{\overline{\eta}}$ and choose a basis
$L_{1},\ldots,L_{k}$ of ${\rm Pic}\,\mathcal{S}_{\overline{\eta}}$. We have
then the projector $\Delta_{alg,\overline{\eta}}\in
CH^{2}(\mathcal{S}_{\overline{\eta}})_{\mathbb{Q}}$ defined as in (18) using
the fact that the intersection pairing on the group of cycles
$<o_{\overline{\eta}},\,\mathcal{S}_{\overline{\eta}},\,L_{i}>$ is
nondegenerate. The transcendental projector $\Delta_{tr,\overline{\eta}}$ is
defined as
$\Delta_{\mathcal{S}_{\overline{\eta}}}-\Delta_{alg,\overline{\eta}}$. This is
a codimension $2$ cycle on
$\mathcal{S}_{\overline{\eta}}\times_{\overline{\mathbb{C}(B)}}\mathcal{S}_{\overline{\eta}}$
but it comes from a cycle on
$\mathcal{S}^{\prime\prime}\times_{B^{\prime\prime}}\mathcal{S}^{\prime\prime}$
for some generically finite covers $B^{\prime\prime}\rightarrow
B,\,\mathcal{S}^{\prime\prime}=\mathcal{S}\times_{B}B^{\prime\prime}$, and the
latter descends finally to a codimension $2$-cycle on
$\mathcal{S}\times_{B}\mathcal{S}$, which one checks to be a multiple of a
projector, at least over a Zariski open set of $B$.
The cycle $\Gamma$ satisfies (23), which we rewrite as
$\displaystyle[\Gamma^{\prime}_{t}]=0\,\,{\rm
in}\,\,H^{4}(\mathcal{S}_{t}\times\mathcal{S}_{t},\mathbb{Q}),$ (29)
where $\Gamma^{\prime}:=\Gamma-\lambda\Delta_{tr}$.
Lemma 3.7 tells us that the fibered self-product
$\mathcal{S}\times\mathcal{S}$ is rationally connected. We can thus apply
Theorem 0.3 and conclude that $\Gamma^{\prime}_{t}\in
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$ is nilpotent. It
follows that $\Gamma_{t}=\lambda\Delta_{tr,t}+N_{t}$, where $\lambda\not=0$
and $N_{t}$ is a nilpotent cycle in $\mathcal{S}_{t}\times\mathcal{S}_{t}$
having the property that
$\Delta_{tr,t}\circ N_{t}=N_{t}\circ\Delta_{tr,t}=N_{t}.$
We conclude immediately from the standard inversion formula for
$\lambda\,\mathbb{I}+N$, with $N$ nilpotent and $\lambda\not=0$, that there
exists a correspondence $\Phi_{t}\in
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}$ such that
$\Phi_{t}\circ\Gamma_{t}=\Delta_{tr,t}\,\,{\rm in}\,\
CH^{2}(\mathcal{S}_{t}\times\mathcal{S}_{t})_{\mathbb{Q}}.$
Recalling now formula (28):
$\Gamma_{t}:=\Delta_{tr,t}\circ{{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}\circ\mathcal{Z}_{t}\circ\Delta_{tr,t},$
we proved (20) with
$Z^{\prime}=\Phi_{t}\circ\Delta_{tr,t}\circ{{}^{t}\mathcal{Z}_{t}}\circ
c_{1}(\mathcal{L}_{t})^{N^{\prime}-2}.$
Let us finish by explaining the following corollaries. They all follow from
Kimura’s theory of finite dimensionality and are a strong motivation to
establish the Kuga-Satake correspondence at a Chow theoretic level rather than
cohomological one.
###### Corollary 3.9.
With the same assumptions as in Theorem 3.5, for any $\sigma\in B$, any self-
correspondence of $S_{\sigma}$ which is homologous to $0$ is nilpotent. In
particular, for any finite group action $G$ on $S_{\sigma}$ and any projector
$\pi\in\mathbb{Q}[G]$, if $H^{2,0}(S_{\sigma})^{\pi}=0$, then
$CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}=0$.
Proof. The first statement follows from Theorem 3.5 Kimura’s work (cf. [17])
which says that abelian varieties have a finite dimensional motive and that
for any finite dimensional motive, self-correspondences homologous to $0$ are
nilpotent.
In the case of a finite group action, we consider as in the previous section
the self-correspondence $\Gamma^{\pi}$. It is nonnecessarily homologous to
$0$, but as it acts as $0$ on $H^{2,0}(S)$, its class can be written as
$[\Gamma^{\pi}]=\sum_{i}\alpha_{i}[C_{i}]\times[C^{\prime}_{i}]\,\,{\rm
in}\,\,H^{4}(S_{\sigma}\times S_{\sigma},\mathbb{Q})$
for some rational numbers $\alpha_{i}$ and curves $C_{i},\,C^{\prime}_{j}$ on
$S_{\sigma}$. Then we have
$[\Gamma^{\pi}-\sum_{i}\alpha_{i}C_{i}\times C^{\prime}_{i}]=0\,\,{\rm
in}\,\,H^{4}(S_{\sigma}\times S_{\sigma},\mathbb{Q}),$
from which we conclude by Kimura’s theorem that the self-correspondence
$Z:=\Gamma^{\pi}-\sum_{i}\alpha_{i}C_{i}\times C^{\prime}_{i}\in
CH^{2}(S_{\sigma}\times S_{\sigma})_{\mathbb{Q}}$
is nilpotent. It follows that
$Z_{*}:CH_{0}(S_{\sigma})_{\mathbb{Q},hom}\rightarrow
CH_{0}(S_{\sigma})_{\mathbb{Q},hom}$ is nilpotent. As it is equal to
$\Gamma^{\pi}_{*}:CH_{0}(S_{\sigma})_{\mathbb{Q},hom}\rightarrow
CH_{0}(S_{\sigma})_{\mathbb{Q},hom},$
which is the projector on $CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}$, we
conclude that $CH_{0}(S_{\sigma})_{\mathbb{Q},hom}^{\pi}=0$.
###### Corollary 3.10.
With the same assumptions as in Theorem 3.5, the transcendental part of the
Chow motive of any member of the family of $K3$ surfaces parameterized by
$\mathbb{P}(H^{0}(X,E))$ is indecomposable, that is, any submotive of it is
either the whole motive or the $0$-motive.
Proof. Recall that the transcendental motive of $S_{\sigma}$ is $S_{\sigma}$
equipped with the projector $\pi_{tr}$ defined in (18). Let now $\pi\in
CH^{2}(S_{\sigma}\times S_{\Sigma})_{\mathbb{Q}}$ be a projector of the
transcendental motive of $S_{\sigma}$, that is
$\pi\circ\pi_{tr}=\pi_{tr}\circ\pi=\pi$. Since $h^{2,0}(S_{\sigma})=1$,
$\pi_{*}$ acts either as $0$ or as $Id$ on $H^{2,0}(S_{\sigma})$. In the first
case, ${\rm Ker}\,(\pi_{*})_{\mid H^{2}(S,\mathbb{Q})_{tr}}$ is a sub-Hodge
structure with $(2,0)$-component equal to $H^{2,0}(S)$. Its orthogonal
complement is then contained in $NS(S_{\sigma})_{\mathbb{Q}}$ which implies
that $\pi_{*}=0$ on $H^{2}(S,\mathbb{Q})_{tr}$. In the second case, we find
similarly that $\pi_{*}=Id$ on $H^{2}(S,\mathbb{Q})_{tr}$. Since
$\pi=\pi_{tr}\circ\pi=\pi\circ\pi_{tr}$, it follows that $\pi_{*}$ acts either
by $0$ or as $\pi_{tr}$ on $H^{*}(S,\mathbb{Q})$. Hence the cohomology class
of either $\pi$ or $\pi_{tr}-\pi$ is equal to $0$, from which we conclude by
theorem 3.5 that $\pi$ or $\pi_{tr}-\pi$ is nilpotent. As both are projectors,
we find that $\pi=0$ or $\pi_{tr}=0$ in $CH^{2}(S_{\sigma}\times
S_{\sigma})_{\mathbb{Q}}$.
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Centre de mathématiques Laurent Schwartz
91128 Palaiseau Cédex, France
voisin@math.jussieu.fr
|
arxiv-papers
| 2012-10-15T08:19:45 |
2024-09-04T02:49:36.550797
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claire Voisin",
"submitter": "Claire Voisin",
"url": "https://arxiv.org/abs/1210.3935"
}
|
1210.4112
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-305 LHCb-PAPER-2012-026 October 15, 2012
Measurement of the $D^{\pm}$ production asymmetry in
7$\mathrm{\,Te\kern-2.07413ptV}$ $pp$ collisions
The LHCb collaboration†††Authors are listed on the following pages.
The asymmetry in the production cross-section $\sigma$ of $D^{\pm}$ mesons,
$A_{\mathrm{P}}=\frac{\sigma(D^{+})-\sigma(D^{-})}{\sigma(D^{+})+\sigma(D^{-})},$
is measured in bins of pseudorapidity $\eta$ and transverse momentum $p_{\rm
T}$ within the acceptance of the LHCb detector. The result is obtained with a
sample of $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays
corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected
in $pp$ collisions at a centre of mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ at the Large Hadron Collider. When integrated
over the kinematic range $2.0<\mbox{$p_{\rm
T}$}<18.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.20<\eta<4.75$, the
production asymmetry is $A_{\mathrm{P}}=(-0.96\pm 0.26\pm 0.18)\%$. The
uncertainties quoted are statistical and systematic, respectively. The result
assumes that any direct $C\\!P$ violation in the $D^{+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decay is negligible. No significant
dependence on $\eta$ or $p_{\rm T}$ is observed.
Submitted to Physics Letters B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, H. Carranza-
Mejia47, L. Carson50, K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch.
Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N. Chiapolini37, M.
Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M.
Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins35, A. Comerma-Montells33, A. Contu52,15, A. Cook43, M.
Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, D. Craik45, S.
Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F. Dettori39, A. Di
Canto11, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, M. Dogaru26, F.
Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, A. Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S.
Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-
Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R.
Forty35, O. Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A.
Gallas Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35,
I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20, Y.M. Kim47,
O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P. Koppenburg38, M. Korolev29,
A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V. Kudryavtsev31, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18,35, C.
Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11, B. Liu3,
G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36,
H. Lu3, J. Luisier36, H. Luo47, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, R. Matev35,
Z. Mathe35, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e,
J. McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J.
Merkel9, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D.
Moran51, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, T.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N.
Nikitin29, T. Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53,
A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33,
B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo
Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D.
Popov10, B. Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V.
Pugatch41, A. Puig Navarro36, W. Qian3, J.H. Rademacker43, B.
Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39,
S. Redford52, M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K.
Rinnert49, V. Rives Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51,
P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero
Vidal34, J. Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido
Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin
Sedes34, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R.
Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, P. Schaack50, M. Schiller39, H. Schindler35, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A.
Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M.
Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6,
P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27,
T. Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50,
R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51,
K. Sobczak5, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2,
B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
D. Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M.
Ubeda Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, H. Voss10, C.
Voß55, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K.
Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
## 1 Introduction
The Large Hadron Collider (LHC) offers an excellent opportunity to study heavy
flavour physics. The rate of production of $c\overline{}c$ and $b\overline{}b$
pairs is substantial in the forward region close to the beam direction. The
associated cross-sections were measured at the LHCb experiment in the forward
region to be $\sigma_{c\overline{}c}=1230\pm 190$$\rm\,\upmu b$ and
$\sigma_{b\overline{}b}=74\pm 14$$\rm\,\upmu b$ at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ [1, 2].
Direct production of $c\overline{}c$ pairs at the LHC occurs almost entirely
via QCD and electroweak processes that do not discriminate between $c$ and
$\overline{}c$ quarks. However, in hadronization the symmetry is broken by the
presence of valence quarks, which introduce several processes that distinguish
between $c$ and $\overline{}c$ quarks [3, 4, 5]. For example, a $c$ quark
could couple to valence quarks to form a charmed baryon, leaving an excess of
$\overline{}c$ quarks. These would hadronize to create an excess of $D^{-}$
mesons over $D^{+}$ mesons. Furthermore, the kinematic distributions of
charmed hadrons and their antiparticles can differ, introducing production
asymmetries in local kinematic regions. Analogous production asymmetries in
the strange sector are well-established at the LHC, and are seen to be large
at high rapidity [6]. However, no evidence for a $D^{+}_{s}$ production
asymmetry was found in a recent study [7].
Searches for $C\\!P$ violation (CPV) in charmed hadron decays can be used to
probe for evidence of physics beyond the Standard Model [8]. Direct CPV is
measured using time-integrated observables, and is of particular interest
following evidence for CPV in two-body $D^{0}$ decays reported by LHCb [9] and
subsequently by CDF [10]. In order to understand the origin of this effect,
more precise measurements of $C\\!P$ asymmetries in a suite of decay modes are
required. Production asymmetries have the same experimental signature as
direct CPV effects and are potentially much larger than the $C\\!P$
asymmetries to be determined. This problem can sometimes be avoided by taking
the difference in asymmetry between two decay modes with a common production
asymmetry [9] or by studying the difference in kinematic distributions of
multi-body decays [11]. However, these methods result in a reduction in
statistical power and are not applicable to all final states. It is therefore
important to measure production asymmetries directly.
In this Letter, the $D^{\pm}$ production asymmetry, defined as
$A_{\mathrm{P}}=\frac{\sigma(D^{+})-\sigma(D^{-})}{\sigma(D^{+})+\sigma(D^{-})},$
(1)
for cross sections $\sigma(D^{\pm})$, is determined with a sample of
$D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$,
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays.111Charge
conjugate decays are implied throughout this letter unless stated otherwise.
As there are no charged kaons in the final state, the detector biases in this
decay are simpler to understand than those in other $D^{+}$ decays with higher
branching fractions. The $K^{0}_{\rm\scriptscriptstyle S}$, a pseudoscalar
particle, has a charge-symmetric decay, and the charge asymmetry in the pion
efficiency at LHCb has been measured previously for the 2011 data sample [7].
However, there is the possibility of CPV in the decay. The expected CPV in the
$D^{+}$ decay, due to the interference of the Cabibbo-favoured and doubly
Cabibbo-suppressed amplitudes, is defined by the charge asymmetry in the
partial widths $\Gamma(D^{\pm})$,
$A_{CP}=\frac{\Gamma(D^{+})-\Gamma(D^{-})}{\Gamma(D^{+})+\Gamma(D^{-})}.$ (2)
$A_{CP}$ is negligible in the Standard Model: a simple consideration of the
CKM matrix leads to a value of at most $1\times 10^{-4}$ depending on the
strong phase difference between the two amplitudes [12]. Since both amplitudes
are at tree level, no enhancement of CPV due to new physics is expected. The
current world-best measurement of $A_{CP}$, by the Belle collaboration, is
consistent with zero: $(0.024\pm 0.094\pm 0.067)\%$ [13,
*PhysRevLett.109.119903]. On the other hand, CPV in the neutral kaon system
induces an asymmetry which must be considered. This will be discussed further
in Sect. 5.
## 2 Detector description
The LHCb detector [15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet of reversible polarity with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift-
tubes placed downstream. The combined tracking system has a momentum
resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
$p_{\rm T}$. Charged hadrons are identified using two ring-imaging Cherenkov
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers. The trigger consists of a hardware stage, based on
information from the calorimeter and muon systems, an inclusive software
stage, which uses the tracking system, and a second software stage that
exploits the full event information.
## 3 Dataset and selection
The data sample used in this analysis corresponds to 1.0$\mbox{\,fb}^{-1}$ of
$pp$ collisions taken at a centre of mass energy of
$7$$\mathrm{\,Te\kern-1.00006ptV}$ at the Large Hadron Collider in 2011. The
polarity of the LHCb magnetic field was changed several times during the run,
and approximately half of the data were taken with each polarity, referred to
as ‘magnet-up’ and ‘magnet-down’ data hereafter. To optimise the event
selection and estimate efficiencies, 12.5 million $pp$ collision events
containing $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$,
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{-}\pi^{+}$ decays were
simulated with Pythia 6.4 [16] with a specific LHCb configuration [17]. Decays
of hadronic particles are described by EvtGen [18]. The interactions of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].
Pairs of oppositely charged tracks with a pion mass hypothesis are combined to
form $K^{0}_{\rm\scriptscriptstyle S}$ candidates. Only those
$K^{0}_{\rm\scriptscriptstyle S}$ candidates with $\mbox{$p_{\rm
T}$}>700$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and invariant mass within
35${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal value [22] are
retained. Surviving candidates are then combined with a third charged track,
the bachelor pion, to form a $D^{+}$ candidate, with the mass of the
$K^{0}_{\rm\scriptscriptstyle S}$ candidate constrained to its nominal value
in a kinematic fit. Each of the three pion tracks must be detected in the
VELO, so only those $K^{0}_{\rm\scriptscriptstyle S}$ mesons that decay well
within the VELO are used. This creates a bias towards short
$K^{0}_{\rm\scriptscriptstyle S}$ decay times. Both the
$K^{0}_{\rm\scriptscriptstyle S}$ and $D^{+}$ candidates are required to have
acceptable vertex fit quality.
Further requirements are applied in order to reduce the background and to
align the selection of bachelor pions with the dataset used to determine the
charge asymmetry in the tracking efficiency (see Sect. 6). The daughters of
the $K^{0}_{\rm\scriptscriptstyle S}$ must have
$p>2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\mbox{$p_{\rm
T}$}>250$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Impact parameter requirements
are used to ensure that both the $K^{0}_{\rm\scriptscriptstyle S}$ candidate
and its daughter tracks do not originate at any primary vertex (PV) in the
event, and the $K^{0}_{\rm\scriptscriptstyle S}$ decay vertex must be at least
10$\rm\,mm$ downstream of the PV with which it is associated. The bachelor
pion must have $p>5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\mbox{$p_{\rm
T}$}>500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, be positively identified as a
pion rather than as a kaon, electron or muon, and must not come from any PV.
In addition, fiducial requirements are applied as in Ref. [9] to exclude
regions with large tracking efficiency asymmetry. All three tracks must have
an acceptable track fit quality. The $D^{+}$ candidate is required to have
$\mbox{$p_{\rm T}$}>1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, to point to a PV
(suppressing $D$ from $B$ decays), and to have a decay time significantly
greater than zero. After these criteria are applied, the remaining background
is mostly from random combinations of tracks. The invariant mass distribution
of selected candidates is shown in Fig. 1.
Figure 1: Mass distribution of selected $K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$ candidates. The data are represented by symbols with error bars.
The dashed curves indicate the signal and the $D^{+}_{s}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays, the lower solid line
represents the background shape, and the upper solid line shows the sum of all
fit components.
In selected events, a trigger decision may be based on part or all of the
$D^{+}$ signal candidate, on other particles in the event, or both. The second
stage of the software trigger is required to find a fully reconstructed
candidate which meets the criteria to be a signal $D^{+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decay. To control potential charge
asymmetries introduced by the hardware trigger, two possibilities, not
mutually exclusive, are allowed. The hardware trigger decision must be based
on one or both of the $K^{0}_{\rm\scriptscriptstyle S}$ daughter tracks, or on
a particle other than the decay products of the $D^{+}$ candidate. In both
cases, the inclusive software trigger must make a decision based on one of the
three tracks that form the $D^{+}$. For the first case, it is explicitly
required that the same track activated the hardware trigger, and therefore
this is independent of the $D^{+}$ charge. The second possibility does not
depend directly on the $D^{+}$ charge, but an indirect dependence could be
introduced if the probability for particles produced in association with the
signal candidate to activate the trigger differs between $D^{+}$ and $D^{-}$.
This will be discussed further in Sect. 7. After applying the selection and
trigger requirements, 1,031,068 $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$
candidates remain.
## 4 Yield determination
Figure 2: Background-subtracted distribution of transverse momenta $p_{\rm
T}$ versus pseudorapidity $\eta$ for selected $D^{+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ candidates in a signal region of
$1845<m<1890$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The bin marked with
an asterisk is excluded from the weighted average over the production
asymmetries in the bins used to obtain the final result.
The signal yields are measured in 48 bins of $p_{\rm T}$ and $\eta$ using
binned likelihood fits to the distribution of the
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ mass $m$. The bins are shown in Fig.
2. The shapes of the $D^{+}_{(s)}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$ mass peaks are described by ‘Cruijff’ functions [23],
$f(m)\propto\exp\left(\frac{-(m-\mu)^{2}}{2\sigma_{L,R}^{2}+(m-\mu)^{2}\alpha_{L,R}}\right)$
(3)
with the measured masses defined by the free parameter $\mu$, the widths by
$\sigma_{L}$ and $\sigma_{R}$, and the tails by $\alpha_{L}$ and $\alpha_{R}$.
The parameters $\alpha_{L}$ and $\sigma_{L}$ are used for $m<\mu$ and
$\alpha_{R}$ and $\sigma_{R}$ for $m>\mu$. The background is fitted with a
straight line plus an additional Gaussian component to account for background
from $D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{0}$
decays. The yield of the latter is consistent with zero in most $p_{\rm T}$,
$\eta$ bins. The fit is performed simultaneously over four subsamples ($D^{+}$
magnet-up, $D^{+}$ magnet-down, $D^{-}$ magnet-up, and $D^{-}$ magnet-down
data) with the masses and yields of the $D^{\pm}_{(s)}$, and the yield of
background, allowed to vary independently in the four subsamples. All other
parameters are shared. The charge asymmetries are then determined from the
yields. The results are cross-checked with a sideband subtraction procedure
under the assumption of a linear background.
## 5 Effect of _CP_ violation in the neutral kaon system
$C\\!P$ violation in the neutral kaon system can affect the observed asymmetry
in the $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decay [24].
The bias on $A_{\mathrm{P}}$ due to the CPV depends on the decay time
acceptance $F(t)$ of the $K^{0}_{\rm\scriptscriptstyle S}$ meson, according to
$A_{\epsilon}\sim
2\Re(\epsilon)\left[1-\frac{\int_{0}^{\infty}F(t)e^{-\frac{1}{2}(\Gamma_{\mathrm{S}}+\Gamma_{\mathrm{L}})t}\left(\cos\Delta
mt-\frac{\Im(\epsilon)}{\Re(\epsilon)}\sin\Delta
mt\right)dt}{\int_{0}^{\infty}F(t)e^{-\Gamma_{\mathrm{S}}t}dt}\right],$ (4)
where $\epsilon$ parameterises the indirect CPV in neutral kaon mixing,
$\Gamma_{\mathrm{S}}$ and $\Gamma_{\mathrm{L}}$ are the decay widths of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $K^{0}_{\rm\scriptscriptstyle L}$
respectively, and $\Delta m$ is their mass difference [25, 26]. Direct CPV and
terms of order $\epsilon^{2}$ are neglected. To determine the decay time
acceptance, the $K^{0}_{\rm\scriptscriptstyle S}$ decay time is fitted with an
empirical function shown in Fig. 3. All of the $K^{0}_{\rm\scriptscriptstyle
S}$ candidates used in this analysis decay inside the VELO with an average
measured lifetime of $6.97\pm 0.02$ ps, which is much shorter than the nominal
$K^{0}_{\rm\scriptscriptstyle S}$ lifetime of 89.5 ps. Using
$\Re(\epsilon)=1.65\times 10^{-3}$ [22] in Eq. 4, we obtain
$A_{\epsilon}=(2.831^{+0.003}_{-0.004})\times 10^{-4}$ for the CPV in the
neutral kaon system, where the uncertainty quoted is statistical only. This
value is subtracted from the measured production asymmetry and a systematic
uncertainty equal to its central value is assigned.
Figure 3: Observed $K^{0}_{\rm\scriptscriptstyle S}$ decay time distribution
within the LHCb acceptance. The data points are fitted with an empirical
function (solid curve). This contains a component for the upper decay time
acceptance, due mainly to the requirement that the
$K^{0}_{\rm\scriptscriptstyle S}$ decays inside the VELO (dashed curve) and a
component for the lower decay time acceptance, due to the selection cuts
(dotted curve). These are shown scaled by arbitrary factors. The CPV is not
sensitive to the fine details of the distribution, so the fit quality is not
important.
## 6 Results
In order to convert the measured charge asymmetries in the 48 bins of $p_{\rm
T}$ and $\eta$ into production asymmetries, a correction for the asymmetry in
the pion reconstruction efficiency is made. This asymmetry was evaluated
previously in eight bins of pion azimuthal angle $\phi$ and two bins of pion
momentum with a control sample of $D^{*+}\rightarrow D^{0}\pi^{+}$,
$D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ decays in the same dataset [7].
The average efficiency asymmetry ratios
$\epsilon_{\pi^{+}}/\epsilon_{\pi^{-}}$ in that sample were found to be
$0.9914\pm 0.0040$ for magnet-up data and $1.0045\pm 0.0034$ for magnet-down
data.
After the correction is applied, the resulting asymmetries for magnet-up and
magnet-down data in each $D^{+}$ $p_{\rm T}$ and $\eta$ bin are averaged with
equal weights to obtain the production asymmetries in two-dimensional bins of
$p_{\rm T}$ and $\eta$, given in Table 6. Any left-right asymmetries that
differ between the signal $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$ decay and the $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ control
channel will cancel in this average.
Figure 4: Production asymmetry as a function of (a) transverse momentum
$p_{\rm T}$ and (b) pseudorapidity $\eta$. The straight line fits have slopes
of $(0.09\pm 0.07)\times 10^{-2}$ (${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)-1
and $(-0.36\pm 0.28)$%, and values of $\chi^{2}$ per degree of freedom of
$5.5/6$ and $2.2/4$, respectively. The error bars include only the statistical
uncertainty on the $D^{+}$ signal sample and are uncorrelated within a given
plot.
Reconstruction and selection efficiencies from the simulation are used to
calculate binned efficiency-corrected yields. These are used to weight the
production asymmetries in the average over the $p_{\rm T}$ and $\eta$ bins.
The result is an asymmetry for $D^{+}$ produced in the LHCb acceptance. The
same weighting technique is applied to obtain production asymmetries as one-
dimensional functions of $p_{\rm T}$ and $\eta$. The bin marked with an
asterisk in Fig. 2 has a high cross section but is mostly outside the
acceptance and so it is excluded from the average. After subtracting the
contribution from CPV in the kaon system, the production asymmetry is
$(-0.96\pm 0.19\pm 0.18)$%. The uncertainties are the statistical errors on
the $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ yields and that
due to the tagged $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ sample used to
calculate the pion efficiencies. Summing these in quadrature, we obtain
$A_{\mathrm{P}}=\left(-0.96\pm 0.26\,(\mathrm{stat.})\right)\%.$
The production asymmetry as a function of $p_{\rm T}$ and $\eta$ is given in
Fig. 4. No significant dependence of the asymmetry on these variables is
observed. As a cross-check, the average production asymmetry is calculated for
magnet-up and magnet-down data separately, and found to be fully consistent:
$(-1.07\pm 0.41)\%$ and $(-0.85\pm 0.34)\%$, respectively.
Table 1: Production asymmetry for $D^{+}$ mesons, in percent, in
$(\mbox{$p_{\rm T}$},\eta)$ bins, for $2.0<\mbox{$p_{\rm
T}$}<18.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.20<\eta<4.75$. The
uncertainties shown are statistical only; the systematic uncertainty is
$0.17\%$ (see Table 2).
| $\eta$
---|---
$p_{\rm T}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | $(2.20,2.80)$ | $(2.80,3.00)$ | $(3.00,3.25)$ | $(3.25,3.50)$ | $(3.50,3.80)$ | $(3.80,4.75)$
$(2.00,3.20)$ | $-0.0\pm 2.5$ | $-2.2\pm 1.2$ | $-0.4\pm 0.8$ | $-0.4\pm 0.7$ | $-1.2\pm 0.6$ | $-1.2\pm 0.5$
$(3.20,4.00)$ | $-0.4\pm 0.9$ | $-0.4\pm 0.7$ | $-0.4\pm 0.5$ | $-1.1\pm 0.5$ | $+0.1\pm 0.5$ | $-1.2\pm 0.5$
$(4.00,4.55)$ | $+0.1\pm 0.8$ | $-1.0\pm 0.8$ | $-1.3\pm 0.6$ | $-2.0\pm 0.6$ | $-0.1\pm 0.6$ | $-2.1\pm 0.7$
$(4.55,5.20)$ | $-1.6\pm 0.7$ | $-0.6\pm 0.8$ | $-0.5\pm 0.6$ | $-0.7\pm 0.6$ | $-1.6\pm 0.6$ | $-2.0\pm 0.8$
$(5.20,6.00)$ | $-0.5\pm 0.7$ | $-0.8\pm 0.8$ | $+0.2\pm 0.7$ | $-0.3\pm 0.7$ | $-0.6\pm 0.7$ | $-1.2\pm 0.9$
$(6.00,7.00)$ | $-1.4\pm 0.8$ | $+0.5\pm 1.0$ | $-0.9\pm 0.9$ | $-0.6\pm 0.9$ | $-0.7\pm 0.9$ | $-1.6\pm 1.2$
$(7.00,9.50)$ | $-0.4\pm 0.8$ | $-0.4\pm 1.1$ | $-0.2\pm 1.1$ | $+1.7\pm 1.1$ | $-1.4\pm 1.1$ | $+1.2\pm 1.4$
$(9.50,18.00)$ | $-0.6\pm 1.3$ | $+1.8\pm 2.3$ | $-2.5\pm 2.2$ | $+1.8\pm 2.4$ | $+1.1\pm 2.5$ | $\phantom{0}\mathord{-}7\pm 11$
## 7 Systematic uncertainties
The sources of systematic uncertainty are summarised in Table 2. The dominant
uncertainty of $1.5\times 10^{-3}$ is due to asymmetries introduced by the
trigger. Events which are triggered independently of the signal decay, i.e. by
a track that does not form part of the signal candidate, could be triggered by
particles produced in association with the $D^{+}$ meson. If this occurs, the
asymmetry in this sample would be correlated with the production asymmetry,
and would bias the measurement of it. This was studied with a control sample
of the abundant $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ decay. To mimic the
charge-unbiased sample of $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$ decays which are triggered by a $K^{0}_{\rm\scriptscriptstyle S}$
daughter, we choose the kaon and one pion at random and require that the
trigger decision is based on one of these tracks. This is close to being
charge-symmetric between $D^{+}$ and $D^{-}$ candidates, with some residual
effects due to differences in material interaction between $K^{+}$ and $K^{-}$
mesons. The raw asymmetry in this subsample of $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ decays is then compared to that in the much larger sample
of candidates that are triggered independently of the signal decay. The
difference in raw charge asymmetry between these two samples, $(1.5\pm
0.4)\times 10^{-3}$, is a measure of the scale of the bias. Unlike the signal,
the $K^{-}\pi^{+}\pi^{+}$ decay also includes a component due to the
$K^{+}/K^{-}$ asymmetry, and therefore this is treated as a systematic
uncertainty rather than a correction. This is cross checked with other control
samples such as $D^{+}_{s}\rightarrow\phi\pi^{+}$ and the uncertainty is found
to be conservative.
Table 2: Summary of absolute values of systematic uncertainties on $A_{\mathrm{P}}$. For the binned production asymmetries given in Table 6, all uncertainties except that on the reconstruction efficiency apply, giving a combined systematic uncertainty of 0.17%. Systematic effect | Uncertainty (%)
---|---
Trigger asymmetries | 0.15
$D$ from $B$ | 0.04
Selection criteria | 0.05
Running conditions | 0.04
Pion efficiency | 0.02
Fitting | 0.04
Kaon $C\\!P$ violation | 0.03
Weights (reconstruction efficiency) | 0.05
Total including uncertainty on weights | 0.18
Further systematic uncertainties arise from the contamination of the prompt
sample by $D$ candidates that originate from $B$ decays. The yield of these is
calculated using the measured cross-sections [1, 2], branching ratios, and
efficiencies determined from the simulation. The fraction of $D$ candidates
from $B$ decays is found to be $(1.2\pm 0.3)\%$. This quantity is combined
with the $B^{0}$ production asymmetry, which is estimated to be $(-1.0\pm
1.3)\%$ [27], to determine the systematic uncertainty.
Certain selection criteria differ between the $D^{+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ signal sample and the
$D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ decays used to determine the
asymmetry in the pion efficiencies. The charge asymmetry is found to depend
weakly on the value of the requirement on the pion $p_{\rm T}$. Pions in the
signal sample must have $p_{\rm T}$
$>500$${\mathrm{\,Me\kern-0.90005ptV\\!/}c}$ while those in the control sample
must have $p_{\rm T}$ $>300$${\mathrm{\,Me\kern-0.90005ptV\\!/}c}$. A
systematic uncertainty is calculated by estimating the proportion of signal
candidates with $300<\mbox{$p_{\rm
T}$}<500$${\mathrm{\,Me\kern-0.90005ptV\\!/}c}$ and multiplying this fraction
by the difference between the charge asymmetries in the low $p_{\rm T}$ region
and the average.
The difference in signal yields per pb-1 of integrated luminosity between
magnet-up and magnet-down data is used to determine a systematic uncertainty
for changes in running conditions that could impair the cancellation of
detector asymmetries achieved by averaging over the magnet polarities. There
is also a systematic uncertainty on the pion efficiency asymmetry associated
with the determination of the yields of $D^{0}\rightarrow
K^{-}\pi^{+}\pi^{-}\pi^{+}$ decays. The error associated with the mass fit is
determined by comparing fitted and sideband-subtracted results. The CPV in the
neutral kaon decay, discussed in Sect. 5, is also included as a systematic
uncertainty.
Other systematic effects such as regeneration in the neutral kaon system [28],
second order effects due to the kinematic binning of the $D^{+}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ sample, and asymmetric backgrounds
such as that from $D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$
with the kaon misidentified as a pion, were considered but found to be
negligible. When taking the average asymmetry weighted by the efficiency-
corrected yield in each bin (Eq. 6), the limited number of simulated events
leads to an uncertainty on the reconstruction efficiency and hence on the per-
bin weights. This does not contribute to the uncertainty on the individual
asymmetries given in Table 6, which are calculated without using the
simulation. A quadratic sum yields an overall systematic uncertainty of
$1.8\times 10^{-3}$.
In principle, CPV in the charm decay could occur via the interference of
Cabibbo-favoured and doubly Cabibbo-suppressed amplitudes, but this is
strongly suppressed by the CKM matrix and no evidence for it has been observed
at the $B$-factories [29, 13]. If we allowed for the possibility of new
physics or large unexpected enhancements of the Standard Model CPV in these
tree-level $D^{+}$ decays, the uncertainty on the null result found at Belle
[13] would increase the total systematic uncertainty to $2.1\times 10^{-3}$.
## 8 Conclusions
Evidence for a charge asymmetry in the production of $D^{+}$ decays is
observed at LHCb. In the kinematic range $2.0<\mbox{$p_{\rm
T}$}<18.0$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and $2.20<\eta<4.75$,
excluding the region with $2.0<\mbox{$p_{\rm
T}$}<3.2$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$, $2.20<\eta<2.80$, the average
asymmetry is
$A_{\mathrm{P}}=(-0.96\pm 0.26\pm 0.18)\%,$
where the first uncertainty is statistical and the second is systematic. The
result is inconsistent with zero at approximately three standard deviations.
There is no evidence for a significant dependence on $p_{\rm T}$ or
pseudorapidity at the present level of precision. The bias on the measured
asymmetry due to $C\\!P$ violation in kaon decays has been calculated and
found to be almost negligible for this dataset. These results are consistent
with expectations [5] and lay the foundations for searches for $C\\!P$
violation in Cabibbo suppressed $D^{+}$ decays.
## 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), CIEMAT, IFAE and UAB (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
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|
arxiv-papers
| 2012-10-15T17:19:39 |
2024-09-04T02:49:36.578252
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, C. Baesso, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles,\n Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal,\n G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, A. Di Canto, J. Dickens, H. Dijkstra, P. Diniz\n Batista, M. Dogaru, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, 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, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, I. R. Kenyon, U. Kerzel, T. Ketel,\n A. Keune, B. Khanji, Y. M. Kim, O. Kochebina, V. Komarov, R. F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, Y. Li, L. Li Gioi, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, H. Luo, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S. Malde, G. Manca,\n G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, T. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D. A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, H. Voss, C. Vo{\\ss}, R. Waldi, R. Wallace,\n S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale,\n M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams,\n M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton,\n S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Hamish Gordon",
"url": "https://arxiv.org/abs/1210.4112"
}
|
1210.4209
|
The short life of a drop
Guillermo Hernández-Cruz1, Minerva Vargas2, Heberto Pérez1
J. Arturo Pimentel3, Gabriel Corkidi3 and Eduardo Ramos1
1Center for Energy Research, Universidad Nacional Autónoma de México
2 Instituto Tecnológico de Zacatepec
3 Biotechnology Institute, Universidad Nacional Autónoma de México
###### Abstract
This is the companion paper of the Fluid Dynamics Video ”The short life of a
drop” where it is argued that the geometry of the sediment of a drop of water
with particles in suspension can be correlated with the dynamics of the fluid
inside the drop during the drying process.
We study the dynamics of the flow inside a sessile drop of water with a
suspension of polystyrene spheres of 1$\mu$m in diameter. According to our
observations it is possible to correlate the motion of the fluid inside the
drop with the pattern of sediments. The initial volume of the drops is 1$\mu$l
and the motion is recorded with a micro PIV equipment that comprises an
Inverted Optical Microscope (Olympus IX71) and a video camera (Optronis
CR5000x2) that captures 512 $\times$ 512 images at a rate of 60 fps. The
microscope has an amplification of 4X and the objective has a nominal depth of
field of 40 $\mu$m. The drop sits on a soda lime glass substrate that is kept
at 20 ∘C. The initial footprint of the drop is 2 mm in diameter and the
suspension has 5.67 $\times 10^{6}$ spheres per $\mu$l of suspension. The
evaporation process lasts typically several minutes with the exact duration
depending on the relative humidity in the room. The evaporation process can be
divided in four parts. In the first, the drop takes the shape of a spherical
cap and the velocity of the fluid inside the drop is very slow and difficult
to resolve but particles emigrate steadily towards the contact line and build
up to start forming the coffee ring. In the second part, the drop is almost
flat and the velocity field inside the drop is mostly radial to compensate for
the preferential evaporation near the contact line. In this part, the radial
velocity near the edge diverges as the thickness of the liquid layer is
reduced and the rapid accumulation of the particles near the contact line
result in a quick thickening of the coffee ring. In these two first stages,
the edge of the drop remains pinned and can be well approximated by a circle.
The third part of the process starts when the liquid film is sufficiently thin
and rips off from the coffee ring and a complicated interaction, of surface
tension (which pulls the liquid to reduce the area) and evaporation, determine
the internal flow. In the initial stages of this part a transient formation of
a secondary coffee ring slows down the motion of the retracting contact line
forming relatively thick sediment segments with the shape of short circular
arcs. Subsequently, the geometry of the outer boundary of the drop becomes
unstable and small perturbations modify its local curvature to generate a time
dependent non-circular edge. During the fourth part of the process, local
irregular features of the drop edge are smoothed out by surface tension. This
effect also generates fast inward radial flows which sweep particles forming
thin radial segments of sediments. This situation prevails until the liquid
totally evaporates.
|
arxiv-papers
| 2012-10-15T22:18:33 |
2024-09-04T02:49:36.590931
|
{
"license": "Public Domain",
"authors": "Guillermo Hernandez-Cruz, Minerva Vargas, Heberto Perez, Arturo\n Pimentel, Gabriel Corkidi, Eduardo Ramos",
"submitter": "Minerva Vargas",
"url": "https://arxiv.org/abs/1210.4209"
}
|
1210.4259
|
119–126
# Deriving fundamental parameters of millisecond pulsars via AIC in white
dwarfs
A. Taani1† Y.H. Zhao1 A. Moraghan2 1National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, China †email: alitaani@bao.ac.cn
2Center for Galaxy Evolution Research and Department of Astronomy, Yonsei
University, Seoul 120-749, Republic of Korea
(2012)
###### Abstract
We present a study of the observational properties of Millisecond Pulsars
(MSPs) by way of their magnetic fields, spin periods and masses. These
measurements are derived through the scenario of Accretion Induced Collapse
(AIC) of white dwarfs (WDs) in stellar binary systems, in order to provide a
greater understanding of the characteristics of MSP populations. In addition,
we demonstrate a strong evolutionary connection between neutron stars and WDs
with binary companions from a stellar binary evolution perspective via the AIC
process.
###### keywords:
Neutron stars, white dwarfs, cataclysmic variables, fundamental parameters.
††volume: 290††journal: Feeding compact objects: Accretion on all
scales††editors: C.M. Zhang, T. Belloni, M. Mendez, and S.N Zhang, eds.
## 1 Introduction
Observable parameters of binary Millisecond Pulsars (MSPs), e.g. mass of the
pulsar, mass of the companion, spin period, orbital period, eccentricity,
etc., are used to probe the past accretion history of the MSPs. The purpose of
this proceeding is to demonstrate how to infer some of the observable
quantities (spin period, magnetic field and mass) during the Accretion Induced
Collapse (AIC) of a white dwarf (WD) on its way to become a member of the MSP
family.
Figure 1: Left: The distribution of MSPs in terms of spin-period (observed
data from [Manchester et al.(2005)]. Right: Mass distribution of massive CVs
[Ritter & Kolb (2011), (Ritter & Kolb 2011)].
## 2 Evolution of spin period
We are able to determine the spin period of MSPs originating from WDs. The
process begins with simple Keplerian frequency. We assume that the angular
velocity of the NS is equal to the Keplerian angular velocity, $v_{K}$, of the
magnetosphere, at roughly the Alfv$\acute{\textrm{e}}$n surface,
$v_{K}\propto R_{NS}^{-3/2}\rightarrow P_{MSP}\sim R_{NS}^{3/2}$ (1)
from which we obtain the spin of the MSP, $P_{MSP}$, as a function of the WD
spin, $P_{WD}$.
$P_{MSP}\sim P_{WD,min}\left(\frac{R_{NS}}{R_{WD}}\right)^{3/2}$ (2)
where $P_{WD,min}$ is the minimum spin period of the standard WD [Warner
(1995), (Warner 1995)]. Assuming $P_{WD,min}$ $\sim 30$ s, $R_{NS}=10$ km and
$R_{WD}=1000$ km, we thus obtain, $P_{MSP}\sim 1ms.$ Fig. 1 (left) shows the
observed and derived spin period distributions of MSPs. As for the observed
MSPs, they show a relatively Gaussian distribution. According to these
distributions, the ratio of MSPs originating from CVs is about $\sim 10\%$.
This result agrees with some theoretical predictions such as those by [Warner
(1995)] and [Warner & Woudt (2002)].
## 3 Magnetic field
To investigate the correlation between the magnetic fields of MSPs with bottom
fields of CVs (where the field is partially restructured due to accretion), we
follow the model of [Zhang et al. (2009)]. The magnetic flux is assumed to be
conserved. This corresponds to the magnetic fields produced by AIC.
$B_{NS}=B_{f,WD}\times\left(\frac{R_{WD}}{R_{NS}}\right)^{2}$ (3)
If we adopt $R_{NS}$ = $15\times 10^{5}$ cm and $B_{f,WD}\sim 10^{3}$G in CVs,
the minimum value, $B_{NS}\sim 10^{8.5-9}G.$
## 4 Mass
The sample of CVs whose masses we have considered is the set of binary systems
collected by [Ritter & Kolb (2011)]. Among them we have 26 massive CVs in the
range ${\rm~{}1.0-1.3M_{\odot}}$. Fig. 1 (right) shows the relatively Gaussian
distribution, with mean at ${\rm~{}M_{CV}\sim 1.1M_{\odot}}$. A summary of the
known properties of these systems is given in Table 1 of [Taani et al.
(2012)]. Note that the AIC process leads to a MSP with mass less than
Chandrasekhar limit [Zhang et al. (2011)]. This provides evidence for the AIC
in massive CVs and evolutionary hypotheses of MSP birthrate.
## 5 Summary and Conclusions
1. 1.
CVs would be invoked via the capability of producing a significant portion of
the MSPs via the AIC process, a regime which may be unattainable by normal
channels.
2. 2.
We find that the quantitative implications of our calculations are that we
estimate the expected $P$ in the observed of MSPs which could have originated
from CVs to be $\sim 10\%$.
3. 3.
We further find that the predictions of some parameters after AIC process for
the average levels are consistent with the observed MSP population. Future
work will consider other quantities e.g. orbital period, eccentricity, and
mass ratio (q) using more data sets from SDSS.
We are grateful for the discussion with Cole Miller. This research has been
supported by NBRPC (2009CB824800, 2012CB821800) and the NSFC (10773017,
11173034).
## References
* [Manchester et al.(2005)] Manchester, R.N., Hobbs, G.B., & Teoh, A., et al. 2005 AJ 129, 1993
* [Ritter & Kolb (2011)] Ritter, H. & Kolb, U., 2011, VizieR Online Data Catalog 1, 2018
* [Taani et al. (2012)] Taani, A., Zhang, C.M., & Al-Wardat, M., et al. 2012 Ap&SS 340, 147
* [Warner (1995)] Warner, B., 1995, Cataclysmic Variable Stars. Cambridge Astrophysics Series vol. 28
* [Warner & Woudt (2002)] Warner, B. & Woudt, P.A., 2002 The Physics of Cataclysmic Variables and Related Objects. Astronomical Society of the Pacific Conference Series vol. 261
* [Zhang et al. (2009)] Zhang, C.M., Wickramasinghe, D.T., & Ferrario, L., et al. 2009 MNRAS 397, 2208
* [Zhang et al. (2011)] Zhang, C.M., Wang, J., & Zhao, Y. H., et al. 2011 A&A 527, 83
|
arxiv-papers
| 2012-10-16T06:23:41 |
2024-09-04T02:49:36.598368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Taani, Y. H. Zhao and A. Moraghan",
"submitter": "Ali Abed El-Kareem Taani",
"url": "https://arxiv.org/abs/1210.4259"
}
|
1210.4285
|
# Absolute Properties of An Overcontact Binary HH Boo
H.A. Dal ali.dal@ege.edu.tr E. Sipahi Ege University, Science Faculty,
Department of Astronomy and Space Sciences, 35100 Bornova, İzmir, Turkey
###### Abstract
We obtained multi-colour light curves of HH Boo. We analysed the orbital
period variation of the system. The analysis indicated that there is possible
mass transfer from the second component to the primary or mass loss with
$-5.04\times 10^{-7}$ $M_{\odot}$ per year. Re-analysing the available radial
velocity curve, we analysed the light curves. The inclination ($i$) of the
system was found to be 69∘.71$\pm$0∘.16, while the semi-major axis ($a$) was
computed as 2.246$\pm$0.064 $R_{\odot}$. The mass of the primary component was
found to be 0.92$\pm$0.08 $M_{\odot}$, while it was obtained as 0.58$\pm$0.06
$M_{\odot}$ for the secondary component. The radius of the primary component
was computed as 0.98$\pm$0.03 $R_{\odot}$, while it was computed as
0.80$\pm$0.02 $R_{\odot}$ for the secondary component. We demonstrated that HH
Boo should be a member of the A-type subclass of W UMa binaries.
###### keywords:
techniques: photometric — (stars:) binaries: eclipsing — stars: late-type —
stars: individual: (HH Boo)
††journal: New Astronomy
## 1 Introduction
HH Boo (GSC 03472-00641) is classified as an eclipsing binary of W UMa type (a
contact binary) in the SIMBAD Database. For the first time, the system was
listed in the TYCHO-2 Catalogue by Høg et al. (2000), who gave $V=11^{m}.32$
and $(B-V)=0^{m}.45$ for the system. Its variability nature was found by
Maciejewski et al. (2003), who obtained the first light curve and gave the
light elements as following: $T_{0}=2452764.50965$ and $P=0^{d}.318618$.
Considering the spectra of the system, Maciejewski et al. (2003) indicated
that the system should be from the spectral type G5III. Then, Maciejewski &
Ligeza (2004) obtained the radial velocity curve of the system. They found
that mass ratio is 0.633$\pm$0.042, while $M_{1}sin^{3}i=0.78\pm 0.08$
$M_{\odot}$ and $M_{2}sin^{3}i=0.49\pm 0.05$ $M_{\odot}$. Although Maciejewski
et al. (2003) have obtained a light curve, there is no light curve analysis
given in the literature. However, many minima times have been obtained along
the years. Most of them are listed in the modern database of O-C Gateway
(Paschke & Brát, 2006), while some others are given by Maciejewski & Karska
(2004), Maciejewski & Niedzielski (2004), Brát et al. (2008), Hubscher et al.
(2009), Hubscher et al. (2010), Walter (2010).
According to Ammons et al. (2006), who determined new temperatures and
metallicities for more than 100,000 FGK dwarfs, the temperature of the system
is 5699 K, while $[Fe/H]$ was found to be -0.57. The distance is 54 pc.
In the literature, there are several systems similar to HH Boo (Essam et al.,
2010). In this study, we obtained multi-colour light curves. Analysing the
orbital period variation, we adjusted the light elements of the system and
derived the mass transfer rate between the components. Then, re-analysing the
radial velocity obtained by Maciejewski & Ligeza (2004), we analysed the
multi-colour light curve using the parameters such as mass ratio and semi-
major axis.
## 2 Observations
Observations of the system were acquired with a thermoelectrically cooled ALTA
$U+47$ 1024$\times$1024 pixel CCD camera attached to a 35 cm - Schmidt -
Cassegrains - type MEADE telescope at Ege University Observatory. The
observations were continued in BVR bands in the observing season 2011. Some
basic parameters of program stars are listed in Table 1, in which the
brightness and colours were taken from the General Catalogue of Variable Stars
(Kholopov et al., 1988).
Although the program and comparison stars are very close on the sky,
differential atmospheric extinction corrections were applied. The atmospheric
extinction coefficients were obtained from observations of the comparison
stars on each night. Heliocentric corrections were also applied to the times
of the observations. The mean averages of the standard deviations are 0m.023,
0m.011, and 0m.010 for observations acquired in the BVR bands, respectively.
To compute the standard deviations of observations, we used the standard
deviations of the reduced differential magnitudes in the sense comparisons
(GSC 03472-00043) minus check (GSC 03472-01201) stars for each night. There
was no variation observed in the standard brightness comparison stars.
## 3 Analysis of Orbital Period Variation
There are several minima times of the system in the literature (Paschke &
Brát, 2006; Hubscher et al., 2010). In the analysis, we used nor visual
observations neither the minima times with large error. We also obtained new
15 minima times. All the minima times used in this study are listed in Table
2. The standard deviation of each one obtained in this study is given in the
brackets near its related digits in Table 2.
Using the regression calculation, investigations demonstrated that the main
variation could be described by a downward parabolic curve, which must be
caused by possible mass transfer from the second component to the primary or
mass loss. Therefore, the main period variation was represented by the
quadratic light elements, which are given by Equation (1).
$Min~{}I~{}(Hel.)~{}=~{}24~{}54912.3236(2)~{}+~{}0^{d}.318666(1)~{}\times~{}E-1.3349(1)\times
10^{-11}~{}\times~{}E^{2}$ (1)
where the standard deviations of each coefficient and each constant are given
in the brackets near their related digits.
The period variation and its quadratic fit are shown in Figure 1. In the
analyses, the weighted sum of the squared residuals, $\Sigma w(O-C)^{2}$, was
found to be 0.000038 $day^{2}$. Considering the quadratic term ($Q$), the
parameter of the period variation ($dP/dt$) was found to be -9.60$\times
10^{-7}$ $yr^{-1}$. The relation between the parameter of the period variation
($dP/dt$) and the rate of mass transfer was determined with Equation (2) by
Pringle & Wade (1985):
$\dot{m}~{}=~{}\frac{1}{3}~{}\times~{}\frac{M_{1}~{}\times~{}M_{2}}{M_{1}~{}-~{}M_{2}}~{}\times~{}\frac{\dot{P}}{P}$
(2)
where $\dot{m}$ is mass transfer rate per year. Using this equation, the mass
transferring from the secondary component to the primary was found to be
$-5.04\times 10^{-7}$ $M_{\odot}~{}yr^{-1}$.
## 4 Radial Velocity and Light Curve Analyses
We took the radial velocity curve of the system from Maciejewski & Ligeza
(2004). In order to adjust their radial velocity solution, we used the
Spectroscopic Binary Solver software (Johnson, 2004) and re-analysed the
radial velocity curve. In the analysis, we used the orbital period adjusted by
the $O-C$ analyses in this study, and we fixed it. The results are listed in
Table 3. The solution generally gave the same values obtained by Maciejewski &
Ligeza (2004) with the little differences. The theoretical radial velocity
curves derived by the Spectroscopic Binary Solver software are shown in Figure
2.
HH Boo is classified as an eclipsing binary of W UMa type in the SIMBAD
Database, and it is more likely a contact binary system. In this respect, we
analysed the light curves obtained in the BVR bands with using the PHOEBE
V.0.31a software (Prša & Zwitter, 2005), which is used in the version 2003 of
the Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson, 1990). We tried to
analyse the light curves with different modes, such as the ”overcontact binary
not in thermal contact”, ”semi-detached system with the primary component
filling its Roche-Lobe”, and ”double contact binary” modes. The initial
analyses demonstrated that an astrophysical acceptable result can be obtained
if the analysis is carried out in the ”overcontact binary not in thermal
contact” mode, while no acceptable results in the astrophysical sense could be
statistically obtained in all the others modes.
According to the radial velocity analysis, the mass ratio of the system should
be 0.632$\pm$0.038. Ammons et al. (2006) determined the temperature of the
system as 5699 K. Thus, the temperature of the primary component was fixed to
5699 K in the analyses, and the temperature of the secondary was taken as a
free parameter. Considering the spectral type corresponding to this
temperature, the albedos ($A_{1}$ and $A_{2}$) and the gravity darkening
coefficients ($g_{1}$ and $g_{2}$) of the components were adopted for the
stars with the convective envelopes (Lucy, 1967; Rucinski, 1969). The non-
linear limb-darkening coefficients ($x_{1}$ and $x_{2}$) of the components
were taken from Van Hamme (1993). In the analyses, the fractional luminosity
($L_{1}$) of the primary component and the inclination ($i$) of the system
were also taken as the adjustable free parameters.
The parameters derived from the analyses are listed in Table 4, while the
synthetic light curves are shown in Figure 3. In addition, using the
parameters obtained from the light curve analysis, we also derived the Roche
geometry of the system that is shown in Figure 4.
Using the radial velocity curves, the mass ratio of the system ($q$) was
obtained as 0.632$\pm$0.038, and the semi-major axis ($a$) was found to be
2.246$\pm$0.064 $R_{\odot}$. Considering both the radial velocity curve
solution and the inclination ($i$) of the system found from the light curve
analysis, the masses were found to be 0.92$\pm$0.08 $M_{\odot}$ for the
primary component and 0.58$\pm$0.06 $M_{\odot}$ for the secondary component.
Considering the semi-major axis, the radius of the primary component was
computed as 0.98$\pm$0.03 $R_{\odot}$, while it was computed as 0.80$\pm$0.02
$R_{\odot}$ for the secondary component. In addition, the luminosity of the
primary component was computed as 0.91$\pm$0.08 $L_{\odot}$, and it was
computed as 0.47$\pm$0.03 $L_{\odot}$ for the secondary component.
In order to test whether the absolute parameters are generally acceptable in
the astrophysical sense, or not, we compared the components with other systems
in the mass-radius ($M-R$), mass-luminosity ($M-L$), and luminosity-effective
temperature ($T_{eff}-L$) planes. All the comparisons are shown in Figure 5.
In the figure, the lines represent the ZAMS theoretical model developed for
the stars with $Z=0.02$ by Girardi et al. (2000), while dashed lines represent
the TAMS theoretical model. The filled circles represent the primary
components, while the open circles represents the secondary ones. The
components of HH Boo are located together with some samples of its analogues,
such as YY CrB, DN Boo, CK Boo, $\epsilon$ CrA, FG Hya, TV Mus, AW UMa, GR
Vir, V776 Cas. The sample systems were taken from Essam et al. (2010), and
they are shown in purple colour, while HH Boo is shown in black colour in
Figure 5.
## 5 Discussion
In this study, we tried to determine the nature of an eclipsing binary system
HH Boo. The analysis of orbital period variation indicates possible mass
transfer from the second component to the primary and/or possible mass loss
from the system. The size of transferring or loosing mass was computed as
$-5.04\times 10^{-7}$ $M_{\odot}$ per year, which causes a decreasing in the
orbital period due to the angular momentum loss. In fact, the parameter of the
period variation was found to be -9.60$\times 10^{-7}$ $yr^{-1}$. We adjusted
the orbital period as $0^{d}.318666$. It is well known that several binaries
of W UMa type such as YY CrB (Essam et al., 2010), BS Cas (Yang et al., 2008),
VZ Tri (Yang, 2010), exhibit large the period variation due to the large mass
transfer.
We also re-analysed available radial velocity curve. Our solution seems to be
a little bit different according to the results found by Maciejewski & Ligeza
(2004). The differences should be caused due to the orbital period used in
this study. Here, the adjusted period is a bit different from the one used by
Maciejewski & Ligeza (2004).
For the first time in the literature, we analysed light curves of the system.
The inclination ($i$) of the system was found to be 69∘.71$\pm$0∘.16, while
the temperature of the secondary component was found to be 5352$\pm$15 K from
the analysis. The fractional radii were found to be $r_{1}=0.435\pm 0.001$ for
the primary component and $r_{2}=0.354\pm 0.001$ for the secondary one. In
this case, the sum of fractional radii was computed as $r_{1}+r_{2}\simeq
0.80$. Thus, HH Boo seems to be in agreement with Kopal (1956)’s criteria for
overcontact systems. The $O-C$ analysis indicates that the orbital period as
$0^{d}.318666$. In addition, the temperature of the primary component is 5699
K, while the secondary one is 5352 K. Although some W UMa type binaries have
components with some different surface temperature, they generally have the
same surface temperature. Here, the primary component of HH Boo is a little
bit hotter than the secondary one. The $O-C$ analysis indicates possible mass
transfer from the second component to the primary. This case should be the
reason of the hotter primary component. Considering some characteristics of
the system such as the short orbital period, small mass ratio, hotter primary
component and ect., HH Boo seems to be in agreement with the members of the
A-type subclass of W UMa binaries (Berdyugina, 2005; Rucinski, 1985).
As it is seen from Figure 5, comparing HH Boo with its analogues in some
planes, such as $M-R$, $M-L$, and $T_{eff}-L$ planes, demonstrated that the
components of the system are in agreement with their analogues. On the other
hand, both components are seen closer to each other in the figures. If the
result obtained from the $O-C$ analysis is taken into account, it is possible
that the secondary component will continue to lose its mass, while the primary
will collect. Therefore, both components will have been separated from each
other in the planes shown in Figure 5. However, they will get closer to each
other in their orbits as it is in the models of Rocha-Pinto et al. (2002).
## Acknowledgments
The author acknowledges the generous observing time awarded to the Ege
University Observatory. We also thank the referee for useful comments that
have contributed to the improvement of the paper.
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Figure 1: The O-C variation of HH Boo. Figure 2: Radial velocity curve of HH
Boo. Open circles represent the observations of the primary, while filled
circles represent the secondary components. Solid curves are the theoretical
radial velocity curves derived by the Spectroscopic Binary Solver software.
Figure 3: The BVR light curves of HH Boo and the synthetic solutions for the
observations in each band. Figure 4: The Roche geometry of HH Boo. Figure 5:
The places of the components of HH Boo in the planes of (upper panel) the
mass-radius ($M-R$), (middle panel) mass-luminosity ($M-L$), and (bottom
panel) luminosity-effective temperature ($T_{eff}-L$). In the panels, the
continuous and dashed lines represent the ZAMS and TAMS theoretical models
developed by Girardi et al. (2000), respectively. The filled circles represent
the primary components, while the open circles represent the secondary ones.
The dark circles represent the HH Boo components, while the purple coloured
circles represent the components of other contact binaries.
Table 1: Basic parameters for the observed stars. Star | Alpha (J2000) | Delta (J2000) | V | B-V
---|---|---|---|---
| (h m s) | (∘ ′ ′′) | (mag) | (mag)
HH Boo | 14 21 44.06 | 46 41 59.40 | 11.272 | 0.654
GSC 03472-00043 | 14 21 46.59 | 46 43 49.40 | 12.906 | 0.595
GSC 03472-01201 | 14 21 37.30 | 46 46 18.90 | 13.701 | 0.662
Table 2: The minima times and ($O-C$) residuals (In the first column, the standard deviations of obtained minima times are given in the brackets near themselves). O | E | $(O-C)_{II}$ | Type | Method | REF
---|---|---|---|---|---
52749.5328 | -9420.0 | -0.00179 | I | V | Paschke & Brát (2006)
52764.5096 | -9373.0 | -0.00230 | I | V | Paschke & Brát (2006)
54148.6406 | -5029.5 | 0.00192 | II | Ir | Paschke & Brát (2006)
54513.5136 | -3884.5 | 0.00209 | II | Ir | Hubscher et al. (2010)
54599.3935 | -3615.0 | 0.00146 | I | CCD+R | Paschke & Brát (2006)
54912.3233 | -2633.0 | 0.00100 | I | Ir | Paschke & Brát (2006)
54912.4835 | -2632.5 | 0.00187 | II | Ir | Paschke & Brát (2006)
54937.3392 | -2554.5 | 0.00160 | II | Ir | Paschke & Brát (2006)
54958.6887 | -2487.5 | 0.00046 | II | V | Paschke & Brát (2006)
54958.8490 | -2487.0 | 0.00143 | I | V | Paschke & Brát (2006)
55644.9336 | -334.0 | -0.00237 | I | V | Paschke & Brát (2006)
55680.7853 | -221.5 | -0.00062 | II | V | Paschke & Brát (2006)
55716.4767(3) | -109.5 | 0.00013 | II | R | This Study
55716.4768(5) | -109.5 | 0.00023 | II | B | This Study
55716.4768(2) | -109.5 | 0.00025 | II | V | This Study
55743.4026(3) | -25.0 | -0.00128 | I | R | This Study
55743.4032(4) | -25.0 | -0.00060 | I | V | This Study
55743.4038(2) | -25.0 | -0.00008 | I | B | This Study
55747.3863(5) | -12.5 | -0.00083 | II | R | This Study
55747.3871(3) | -12.5 | -0.00003 | II | V | This Study
55747.3874(4) | -12.5 | 0.00022 | II | B | This Study
55751.3696(3) | 0.0 | -0.00087 | I | V | This Study
55751.3700(3) | 0.0 | -0.00051 | I | R | This Study
55751.3700(6) | 0.0 | -0.00050 | I | B | This Study
55751.3701(2) | 0.0 | -0.00037 | I | R | This Study
55751.3702(3) | 0.0 | -0.00027 | I | V | This Study
55751.3702(4) | 0.0 | -0.00026 | I | B | This Study
Table 3: The results of the analysis of Radial Velocity curve. Parameter | Value | Error
---|---|---
Long. of Periastron1 (∘) | 238.669 | ${}^{+49.884}_{-72.41}$
Long. of Periastron2 (∘) | 58.669 | ${}^{+49.884}_{-72.41}$
Eccentricity ($e$) | 0.048 | ${}^{+0.043}_{-0.042}$
Semi-Amplitude1 ($kms^{-1}$) | 129.647 | ${}^{+6.4243}_{-6.2471}$
Semi-Amplitude2 ($kms^{-1}$) | 205.078 | ${}^{+7.0675}_{-6.7157}$
Systemic Velocity ($kms^{-1}$) | 2.9209 | ${}^{+4.0328}_{-4.0429}$
$a_{1}sini$ ($km$) | $5.67\times 10^{5}$ | $\pm 3.68\times 10^{4}$
$a_{2}sini$ ($km$) | $8.98\times 10^{5}$ | $\pm 4.86\times 10^{4}$
$m_{1}sin^{3}i$ ($M_{\odot}$) | $7.56\times 10^{-1}$ | $\pm 9.99\times 10^{-2}$
$m_{2}sin^{3}i$ ($M_{\odot}$) | $4.78\times 10^{-1}$ | $\pm 6.54\times 10^{-2}$
Table 4: The parameters of components obtained from the light curve analysis. Parameter | Value
---|---
$i$ (∘) | 69.71$\pm$0.16
$T_{1}$ (K) | 5699
$T_{2}$ (K) | 5352$\pm$15
$\Omega_{1}$ | 3.0492
$\Omega_{2}$ | 3.0402$\pm$0.0051
$L_{1}/L_{T}$ (B) | 0.691$\pm$0.046
$L_{1}/L_{T}$ (V) | 0.669$\pm$0.037
$L_{1}/L_{T}$ (R) | 0.657$\pm$0.031
$g_{1}$, $g_{2}$ | 0.32, 0.32
$A_{1}$, $A_{2}$ | 0.50, 0.50
$x_{1,bol}$, $x_{2,bol}$ | 0.619, 0.619
$x_{1,B}$, $x_{2,B}$ | 0.765, 0.765
$x_{1,V}$, $x_{2,V}$ | 0.732, 0.732
$x_{1,R}$, $x_{2,R}$ | 0.655, 0.655
$<r_{1}>$ | 0.435$\pm$0.001
$<r_{2}>$ | 0.354$\pm$0.001
|
arxiv-papers
| 2012-10-16T07:52:43 |
2024-09-04T02:49:36.606011
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, E. Sipahi",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1210.4285"
}
|
1210.4291
|
# Nonlinear mirror modes in the presence of hot electrons
E.A. Kuznetsov(a,b), T. Passot (c) and P.L. Sulem (c)
e-mail:kuznetso@itp.ac.ru (a)P.N. Lebedev Physical Institute RAS, 53 Leninsky
Ave., 119991 Moscow, Russia
(b)Space Research Institute RAS, 84/32 Profsoyuznaya str., 117997, Moscow,
Russia
(c)Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur,
PB 4229, 06304 Nice Cedex 4, France
###### Abstract
A non-perturbative calculation of the gyrotropic pressures associated with
large-scale mirror modes is performed, taking into account a finite, possibly
anisotropic electron temperature. In the small-amplitude limit, this leads to
an extension of an asymptotic model previously derived for cold electrons. A
model equation for the profile of subcritical finite-amplitude large-scale
structures is also presented.
PACS: 52.35.Py, 52.25.Xz, 94.30.cj, 94.05.-a
## I Introduction
Pressure-balanced magnetic structures in the form of strong magnetic
enhancements (humps) and depressions (holes) that are quasi-stationary in the
plasma frame, with no or little change in the magnetic field direction, are
commonly observed in regions of the solar wind and of planetary magnetosheaths
with relatively large $\beta$ and a dominant (generally ion) temperature in
the transverse direction ( see, for instance, [1, 2] and references therein).
The origin of these structures is still not fully understood, but they are
usually viewed as nonlinearly saturated states of the mirror instability (MI)
discovered by Vedenov and Sagdeev [3]. It is a kinetic instability whose
growth rate was first obtained under the assumption of cold electrons, a
regime where the contributions of the parallel electric field $E_{\|}$ can be
neglected. However, in realistic space plasmas, the electron temperature can
hardly be ignored [4]. The linear theory retaining the electron temperature
and its possible anisotropy, in the quasi-hydrodynamic limit (which neglects
finite Larmor radius corrections), was developed in the case of bi-Maxwellian
distribution functions by several authors [5]–[9]. A general estimate of the
growth rate under the sole condition that it is small compared with the ion
gyrofrequency (a condition reflecting close vicinity to threshold) is
presented in [10]. The instability then develops in quasi-perpendicular
directions, making the parallel magnetic perturbation dominant. This analysis
includes in particular regimes with a significant electron temperature
anisotropy for which the instability extends beyond the ion Larmor radius. In
the limit where the instability is limited to scales large compared with the
ion Larmor radius, only the leading order contribution in terms of the small
parameter $\gamma/(|k|_{z}v_{\|i})$ is to be retained in estimating Landau
damping, and the growth rate is given by
$\displaystyle\gamma=\frac{2}{\sqrt{\pi}}\frac{T_{\|i}}{T_{\perp
i}}\frac{|k_{z}|v_{\|i}}{E}\Big{\\{}\Gamma-\frac{1}{\beta_{\perp}}\Big{(}1+\frac{\beta_{\perp}-\beta_{\|}}{2}\Big{)}\frac{k_{z}^{2}}{k_{\perp}^{2}}$
$\displaystyle\qquad-\frac{3}{4(1+\theta_{\perp})}\Big{(}\frac{T_{\perp
i}}{T_{\|i}}-1\Big{)}(1+F)k_{\perp}^{2}r_{L}^{2}\Big{\\}},$ (1)
where
$\Gamma=\frac{T_{\perp i}}{T_{\parallel
i}}\frac{(\theta_{\parallel}+\theta_{\perp})^{2}+2\theta_{\parallel}(\theta_{\perp}^{2}+1)}{2\theta_{\parallel}(1+\theta_{\perp})(\theta_{\parallel}+1)}-1-\frac{1}{\beta_{\perp}}$
(2)
measures the distance to threshold and
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1+\theta_{\perp}}{(1+\theta_{\parallel})^{2}}\left[2+\theta_{\perp}(4+\theta_{\perp})+\theta_{\parallel}^{2}\right]$
$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{T_{\|e}}{T_{\|e}+T_{\|i}}\Big{\\{}-1+\frac{\theta_{\perp}}{\theta_{\|}}$
$\displaystyle-\frac{2}{3}\frac{T_{\|i}}{T_{\perp
i}}\Big{[}\Big{(}\frac{T_{\|i}}{T_{\perp i}}-1\Big{)}\frac{1}{\beta_{\perp
i}}-\theta_{\perp}\Big{(}\frac{T_{\perp
e}}{T_{\|e}}-1\Big{)}\Big{]}\Big{\\}}.$
Here, ${T_{\perp\alpha}}$ and ${T_{\|\alpha}}$ are the perpendicular and
parallel (relative to the ambient magnetic field $\mathbf{B}_{0}$ taken in the
$z$ direction) temperatures of the species $\alpha$ ($\alpha=i$ for ions and
$\alpha=e$ for electrons ), $\theta_{\perp}={T_{\perp e}}/{T_{\perp i}}$,
$\theta_{\|}={T_{\|e}}/{T_{\|i}}$ and $\beta_{\perp}=\beta_{\perp
i}+\beta_{\perp e}$ with $\beta_{\perp\alpha}=8\pi
p{{}_{\perp\alpha}/B}_{0}^{2}$ where $p{{}_{\perp\alpha}}$ is the
perpendicular thermal pressure (similar definition for $\beta_{\|}$).
Furthermore, the parallel thermal velocity is defined as
$v_{\|\alpha}=\sqrt{{2T_{\|\alpha}}/{m_{\alpha}}}$, and $r_{L}=({2{T_{\perp
i}/m_{p})^{1/2}/\Omega_{i}}}$ denotes the ion Larmor radius
($\Omega_{i}=eB_{0}/m_{i}c$ is the ion gyrofrequency).
The growth rate given by Eq. (1) has the same structure as in the cold
electron regime considered in [11] in the case of bi-Maxwellian ions and
generalized in [12] and [9] to an arbitrary distribution function. The first
term within the curly brackets provides the threshold condition which
coincides with that given in [5]–[11]. The second one reflects the magnetic
field line elasticity and the third one (where $F$ depends on the electron
temperatures due to the coupling between the species induced by the parallel
electric field which is relevant for hot electrons) provides the arrest of the
instability at small scales by finite Larmor radius (FLR) effects.
An aim of this letter is to extend to hot electrons the weakly nonlinear
analysis previously developed for cold electrons [13, 14]. Since in this
asymptotics, FLR contributions appear only at the linear level, the idea is to
use the drift kinetic formalism to calculate the nonlinear terms. We show that
the equation governing the evolution of weakly nonlinear mirror modes has the
same form as in the case of cold electrons. In particular, the sign of the
nonlinear coupling coefficient that prescribes the shape of mirror structures,
is not changed. This equation is of gradient type equation with a free energy
(or a Lyapunov functional) which is unbounded from below. This leads to
finite-time blowing-up solutions [15], associated with the existence of a
subcritical bifurcation [13, 14]. To describe subcritical stationary mirror
structures in the strongly nonlinear regime, we present an anisotropic MHD
model where the perpendicular and parallel pressures are determined from the
drift kinetic equations in the adiabatic approximation, in the form of
prescribed functions of the magnetic field amplitude.
## II Basic equations
A main condition characterizing mirror modes, at least near threshold, is
provided by the force balance equation
$\displaystyle-\nabla\Big{(}p_{\perp}+\frac{B^{2}}{8\pi}\Big{)}+\Big{[}1+\frac{4\pi}{B^{2}}(p_{\perp}-p_{\|})\Big{]}\frac{(\mathbf{B}\cdot\nabla)\mathbf{B}}{4\pi}$
$\displaystyle+\mathbf{B}(\mathbf{B}\cdot\nabla)\Big{(}\frac{p_{\perp}-p_{\|}}{B^{2}}\Big{)}-\nabla\cdot\mathbf{\Pi}=0,$
(3)
where the pressure tensor, viewed as the the sum of the contributions of the
various species, has been written as the sum of a gyrotropic part
characterized by the parallel ($p_{\|}=\sum_{\alpha}p_{\|\alpha}$) and
perpendicular ($p_{\perp}=\sum_{\alpha}p_{\perp\alpha}$) pressures, and of a
gyroviscous contribution $\Pi$ originating from the sole ion FLR effects when
concentrating on scales large compared with the electron Larmor radius. As
mentioned above, FLR effects arising only at the linear level with respect to
the amplitude of the perturbations, the other linear and nonlinear
contributions can be evaluated from the drift kinetic equation for each
particle species
$\frac{\partial f_{\alpha}}{\partial t}+v_{\|}\mathbf{b}\cdot\nabla
f_{\alpha}+\Big{[}-\mu\mathbf{b}\cdot\nabla
B+\frac{e_{\alpha}}{m_{\alpha}}E_{\|}\Big{]}\frac{\partial
f_{\alpha}}{\partial v_{\|}}=0$ (4)
We ignore the transverse electric drift which is subdominant for mirror modes.
In this approximation, both ions and electrons move in the direction of the
magnetic field (defined by the unit vector $\mathbf{b}=\mathbf{B}/B$) under
the effect of the magnetic force $\mu\mathbf{\ b}\cdot\nabla B$ and the
parallel electric field $E_{\|}=-\mathbf{b}\cdot\nabla\phi$ where the magnetic
moment $\mu=v_{\perp}^{2}/(2B)$ is an adiabatic invariant which plays the role
of a parameter in Eq. (4). Here $\phi$ is the electric potential. The quasi-
neutrality condition $n_{e}=n_{i}\equiv n$, where $n_{\alpha}=B\int
f_{\alpha}d\mu dv_{\|}d\varphi\equiv\int f_{\alpha}d^{3}v$, is used to close
the system and eliminate $E_{\|}$.
In this framework where FLR effects are neglected, the gyrotropic pressures
are given by $p_{\alpha\|}\equiv m_{\alpha}\int
v_{\|}^{2}f_{\alpha}d^{3}v=m_{\alpha}B\int v_{\|}^{2}f_{\alpha}d\mu
dv_{\|}d\varphi$, and $p_{\alpha\perp}\equiv\frac{1}{2}m_{\alpha}\int
v_{\perp}^{2}f_{\alpha}d^{3}v=m_{\alpha}B^{2}\int\mu f_{\alpha}d\mu
dv_{\|}d\varphi$.
The asymptotic equation governing the mirror dynamics near threshold is
obtained by expanding Eqs. (3), (4) and the quasi-neutrality condition, with
the pressure tensor elements for each species computed near a bi-Maxwellian
equilibrium state characterized by the temperatures $T_{\perp\alpha}$ and
$T_{\|\alpha}$.
## III Linear instability
Before turning to the nonlinear regime, we briefly review the derivation of
the MI linear growth rate in the simplified framework provided by the drift
kinetic approximation which is only valid at scales large enough for FLR
effects to be subdominant.
Linearizing Eq. (3) about the background field $\mathbf{B_{0}}$ and
equilibrium pressures $p_{\perp}^{(0)}$ and $p_{\|}^{(0)}$, and considering
perturbations $\widetilde{\mathbf{B}}$ and $p_{\perp}^{(1)}\propto e^{-i\omega
t+i\mathbf{k\cdot r}}$, we get
$p_{\perp}^{(1)}+\frac{B_{0}\widetilde{B}_{z}}{4\pi}=-\frac{k_{z}^{2}}{k_{\perp}^{2}}\Big{(}1+\frac{\beta_{\perp}-\beta_{\|}}{2}\Big{)}\frac{B_{0}\widetilde{B}_{z}}{4\pi}.$
(5)
Here, $p_{\perp}^{(1)}$ has to be calculated from the linearized drift kinetic
equation
$\frac{\partial f_{\alpha}^{(1)}}{\partial t}+v_{\|}\frac{\partial
f_{\alpha}^{(1)}}{\partial
z}+\Big{[}-\mu\frac{\partial\widetilde{B}_{z}}{\partial
z}+\frac{e_{\alpha}}{m_{\alpha}}E_{\|}\Big{]}\frac{\partial
f_{\alpha}^{(0)}}{\partial v_{\|}}=0,$ (6)
where we assume each $f_{\alpha}^{(0)}$ to be a bi-Maxwellian distribution
function
$f_{\alpha}^{(0)}=A_{\alpha}\exp\Big{[}-\frac{v_{\parallel}^{2}}{v_{\parallel\alpha}^{2}}-\frac{\mu
B_{0}m_{\alpha}}{T_{\perp\alpha}}\Big{]},$ (7)
with
$A_{\alpha}~{}=~{}n_{0}m_{\alpha}/(2\pi\sqrt{\pi}v_{\parallel\alpha}T_{\perp\alpha})$.
Equation (6) is solved in Fourier representation, as
$f_{\alpha}^{(1)}=-\frac{\mu\widetilde{B}_{z}+\frac{e_{\alpha}}{m_{\alpha}}\phi}{\omega-
k_{z}v_{\|}}k_{z}\frac{\partial f_{\alpha}^{(0)}}{\partial v_{\|}}.$ (8)
The neutrality condition allows one to express the potential $\phi$ in terms
of $\widetilde{B}_{z}$. Indeed, assuming
$\displaystyle{\zeta={\sqrt{\pi}\omega}/(|k_{z}|v_{\parallel i})\ll 1}$ (so
that the contribution from the Landau pole is small),
$\int f_{i}^{(1)}dv_{z}d\mu d\varphi=-\frac{n_{0}}{B_{0}T_{\parallel
i}}\Big{[}T_{\perp
i}\frac{\widetilde{B}_{z}}{B_{0}}+e\phi\Big{]}\Big{[}1+i\zeta\Big{]}.$
Similarly, neglecting the electron Landau resonance contribution because of
the small mass ratio,
$\int f_{e}^{(1)}dv_{\|}d\mu
d\varphi=-\frac{n_{0}}{B_{0}T_{\|e}}\Big{[}T_{\perp
e}\frac{\widetilde{B}_{z}}{B_{0}}-e\phi\Big{]}.$
Consequently,
$e\phi\approx\frac{T_{\perp
i}}{1+\theta_{\parallel}}\Big{[}(\theta_{\perp}-\theta_{\parallel})-\frac{\theta_{\parallel}(1+\theta_{\perp})}{1+\theta_{\parallel}}i\zeta\Big{]}\frac{\widetilde{B}_{z}}{B_{0}}.$
(9)
We thus recover that for mirror modes, the parallel electric field vanishes
when the electrons are cold ($\theta_{\perp}=\theta_{\|}=0$). Interestingly,
when $\theta_{\perp}=\theta_{\parallel}$, only the Landau pole contributes to
$\phi$.
It is now necessary to evaluate
$p_{\perp}^{(1)}=2\frac{\widetilde{B}_{z}}{B_{0}}p_{\perp}^{(0)}+B_{0}^{2}\sum_{\alpha}m_{\alpha}\int\mu
f_{\alpha}^{(1)}d\mu dv_{\|}d\varphi.$
Using
$\displaystyle\int\frac{k_{z}v_{\|}}{\omega-k_{z}v_{\|}}f_{i}^{(0)}d\mu
dv_{\|}d\varphi$ $\displaystyle=$
$\displaystyle-\frac{n_{0}}{B_{0}}(1+i\zeta)$
$\displaystyle\int\frac{k_{z}v_{\|}}{\omega-k_{z}v_{\|}}f_{e}^{(0)}d\mu
dv_{\|}d\varphi$ $\displaystyle=$ $\displaystyle-\frac{n_{0}}{B_{0}},$
we get
$p_{\perp}^{(1)}=-\beta_{\perp}\frac{B_{0}^{2}}{4\pi}\Big{[}\frac{1}{\beta_{\perp}}+\Gamma+\frac{T_{\perp
i}}{T_{\|i}}\frac{i\zeta
D}{2(1+\theta_{\perp})}\Big{]}\frac{\widetilde{B}_{z}}{B_{0}}.$
Substituting this expression into the linearized force balance equation yields
the linear instability growth rate given by Eq. (1), up to the FLR term which
is not captured by the drift kinetic approximation. Note that the growth rate
given by Eq. (1) is consistent with the applicability condition
$\gamma/|k_{z}|\ll v_{{\|}i}$ near threshold ($\Gamma\ll 1$), as $k_{z}$ and
$(k_{z}/k_{\perp})^{2}$ scale like $\Gamma$, while $\gamma$ like $\Gamma^{2}$.
## IV General pressure estimates
As demonstrated in [13, 14], the scalings resulting from the linear theory
near threshold imply an adiabaticity condition to leading order. It is thus
enough to consider the stationary kinetic equation
$v_{\|}\mathbf{b}\cdot\nabla f_{\alpha}-(\mathbf{b}\cdot\nabla)\left[\mu
B+\frac{e_{\alpha}}{m_{\alpha}}\phi\right]\frac{\partial f_{\alpha}}{\partial
v_{\|}}=0.$ (10)
It turns out that Eq. (10) is exactly solvable, the general solution being an
arbitrary function $f_{\alpha}=g_{\alpha}(\mu,W_{\alpha})$ of the particle
energy $\displaystyle{\ W_{\alpha}={v_{\|}^{2}}/{2}+\mu
B+\frac{e_{\alpha}}{m_{\alpha}}\phi}$, and of $\mu$. To find the function
$g_{\alpha}(\mu,W_{\alpha})$, we use the adiabaticity argument which means
that, to leading order, $g_{\alpha}$ as a function of $\mu$ and $W_{\alpha}$
retains its form during the evolution. Therefore, the function
$g_{\alpha}(\mu,W_{\alpha})$ is found by matching with the initial
distribution function $f_{\alpha}^{(0)}$ given by Eq. (7) which corresponds to
$\phi=0$ and $W_{\alpha}=\frac{v_{\|}^{2}}{2}+\mu B_{0}$. We get
$\displaystyle
g_{\alpha}(\mu,W_{\alpha})=A_{\alpha}\exp\Big{[}-\frac{v_{\parallel}^{2}}{v_{\parallel\alpha}^{2}}-\frac{\mu
B_{0}m_{\alpha}}{T_{\perp\alpha}}\Big{]}$
$\displaystyle\quad=A_{\alpha}\exp\Big{[}-\frac{2W_{\alpha}}{v_{\parallel\alpha}^{2}}+\mu
B_{0}m_{\alpha}\Big{(}\frac{1}{T_{\parallel\alpha}}-\frac{1}{T_{\perp\alpha}}\Big{)}\Big{]}.$
(11)
Thus, $g_{\alpha}(\mu,W_{\alpha})$ is a Boltzmann distribution function with
respect to $W_{\alpha}$ but, at fixed $W_{\alpha}$, it displays an exponential
growth relatively to $\mu$ if $T_{\perp\alpha}>$ $T_{\parallel\alpha}$. This
effect can however be compensated by the dependence of $W_{\alpha}$ in $\mu$.
This means that only a fraction of the phase space $(\mu,W_{\alpha})$ is
accessible, a property possibly related with the existence of trapped and
untrapped particles.
Note that expanding Eq. (11) relatively to $\widetilde{B}_{z}/B_{0}$ and
$e\phi^{(1)}/T_{\perp i}$ reproduces the first order contribution to the
distribution function given by Eq. (8) with $\omega=0$, and also the second
order correction found in [13, 14] in the case of cold electrons. It should be
emphasized that Eq. (11) only assumes adiabaticity and remains valid for
finite perturbations.
The function $g_{\alpha}$ can also be rewritten in terms of $v_{\|}$,
$v_{\perp}$ and $\phi$ as
$\displaystyle
g_{\alpha}=A_{\alpha}\exp\Big{[}-\frac{m_{\alpha}v_{\|}^{2}}{2T_{\parallel\alpha}}-\frac{e_{\alpha}\phi}{T_{\parallel\alpha}}\Big{]}\times$
$\displaystyle\qquad\exp\left\\{-\frac{m_{\alpha}v_{\perp}^{2}}{2T_{\perp\alpha}}\Big{(}\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}-\frac{B_{0}}{B}\Big{[}\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}-1\Big{]}\Big{)}\right\\},$
which can be viewed as the bi-Maxwellian distribution function with the
renormalized transverse temperature
$T_{\perp\alpha}^{(eff)}=T_{\perp\alpha}\left[\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}-\frac{B_{0}}{B}\Big{(}\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}-1\Big{)}\right]^{-1}.$
Note the Boltzmann factor $\exp{-[e_{\alpha}\phi/T_{\parallel\alpha}]}$ in the
expression of $g_{\alpha}$. For cold electrons, the ion distribution function
was obtained in [16] by assuming that it remains bi-Maxwellian, and owing to
the invariance of the kinetic energy and of the magnetic moment. This
estimate, obtained by neglecting both time dependency (and consequently Landau
resonance) and finite Larmor radius corrections, reproduces the closure
condition given in [17].
After rewriting Eq. (11) in the form
$g_{\alpha}=A_{\alpha}\exp\Big{[}-\frac{e_{\alpha}\phi}{T_{\parallel\alpha}}-\frac{v_{\|}^{2}}{v_{\parallel\alpha}^{2}}-\frac{\mu
B_{0}m_{\alpha}}{T_{\perp\alpha}}\Big{(}1+\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}\frac{B-B_{0}}{B_{0}}\Big{)}\Big{]},$
the quasi-neutrality condition gives
$\displaystyle\left(1+\frac{T_{\perp i}}{T_{\parallel
i}}\frac{B-B_{0}}{B_{0}}\right)^{-1}\exp\left(-\frac{e\phi}{T_{\parallel
i}}\right)=$ $\displaystyle\left(1+\frac{T_{\perp e}}{T_{\parallel
e}}\frac{B-B_{0}}{B_{0}}\right)^{-1}\exp\left(\frac{e\phi}{T_{\parallel
e}}\right)$
or
$\displaystyle e\phi=(T_{\parallel i}^{-1}+T_{\parallel e}^{-1})^{-1}\times$
$\displaystyle\log\left[\left(1+\frac{T_{\perp e}}{T_{\parallel
e}}\frac{B-B_{0}}{B_{0}}\right)\left(1+\frac{T_{\perp i}}{T_{\parallel
i}}\frac{B-B_{0}}{B_{0}}\right)^{-1}\right].$ (12)
Interestingly, the electron density (and thus also that of the ions)
$n_{e}=n_{0}\frac{B}{B_{0}}\left(1+\frac{T_{\perp e}}{T_{\parallel
e}}\frac{B-B_{0}}{B_{0}}\right)^{-1}\exp\left[\frac{e\phi}{T_{\parallel
e}}\right]$
has the usual Boltzmann factor $\exp\left[e\phi/T_{\parallel e}\right]$ and
also an algebraic prefactor depending on the magnetic field $B$. In the case
of isotropic electron temperature ($T_{\perp e}=T_{\parallel e}\equiv T_{e}$),
the electron density has the usual Boltzmann form
$n_{e}=n_{0}\exp\left[e\phi/T_{e}\right]$.
Equation (12) shows that the potential vanishes in two cases: for cold
electrons and also when electron and ion temperature anisotropies $a_{e}$ and
$a_{i}$ (with $a_{\alpha}=T_{\perp\alpha}/T_{\|\alpha}$) are equal, a case
considered in the linear theory of the mirror instability [5, 18, 11].
In order to evaluate explicitly the perpendicular pressure for each species
$\displaystyle p_{\perp\alpha}=m_{\alpha}B^{2}\int\mu g_{\alpha}d\mu
dv_{\|}d\varphi$ $\displaystyle\ \
=n_{0}T_{\perp\alpha}\frac{B^{2}}{B_{0}^{2}}\Big{(}1+\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}\frac{B-B_{0}}{B_{0}}\Big{)}^{-2}\exp\Big{(}-\frac{e_{\alpha}\phi}{T_{\parallel\alpha}}\Big{)},$
where $e\phi$ is given by Eq. (12), it is convenient to introduce the
functions
$\displaystyle S_{\perp i}(u)$ $\displaystyle=$
$\displaystyle\left(\frac{1+u}{1+a_{i}u}\right)^{2}\left(\frac{1+a_{i}u}{1+a_{e}u}\right)^{c_{i}}$
(13) $\displaystyle S_{\perp e}(u)$ $\displaystyle=$
$\displaystyle\left(\frac{1+u}{1+a_{e}u}\right)^{2}\left(\frac{1+a_{e}u}{1+a_{i}u}\right)^{c_{e}},$
(14)
with the notations $u=(B-B_{0})/B_{0}$ and
$c_{\alpha}=T_{\parallel\alpha}^{-1}/(T_{\parallel i}^{-1}+T_{\parallel
e}^{-1})$. The two latter functions transform one into the other by exchanging
the subscripts $i$ and $e$. The ion and electron perpendicular pressures are
then written as $p_{\perp\alpha}=n_{0}T_{\perp\alpha}S_{\perp\alpha}(u)$. In
the special case of cold electrons,
$p_{\perp}=n_{0}T_{\perp i}\frac{B^{2}}{B_{0}^{2}}\Big{(}1+\frac{T_{\perp
i}}{T_{\parallel i}}\frac{B-B_{0}}{B_{0}}\Big{)}^{-2},$ (15)
which is algebraic relatively to $B$. From this expression as well as from the
general formula for $p_{\perp}=p_{\perp i}+p_{\perp e}$ given by Eqs. (13) and
(14) it follows that the perpendicular and magnetic pressures are
anticorrelated. When $B$ increases (decreases), the ratio of the perpendicular
to the magnetic pressure, i.e. the local $\beta_{\perp}$, decreases
(increases), which corresponds to a reduction (an increase) of the distance to
threshold. This implies that the instability cannot saturate at small
amplitudes.
Similarly, for the parallel pressure, we have
$p_{{}_{\parallel\alpha}}=n_{0}T_{{}_{\parallel\alpha}}\frac{B}{B_{0}}\Big{(}1+\frac{T_{\perp\alpha}}{T_{\parallel\alpha}}\frac{B-B_{0}}{B_{0}}\Big{)}^{-1}\exp\Big{(}-\frac{e_{\alpha}\phi}{T_{\parallel\alpha}}\Big{)}.$
that rewrites $p_{\|\alpha}=n_{0}T_{\|\alpha}S_{\|\alpha}(u)$ with
$\displaystyle S_{\|i}(u)$ $\displaystyle=$
$\displaystyle\left(\frac{1+u}{1+a_{i}u}\right)\left(\frac{1+a_{i}u}{1+a_{e}u}\right)^{c_{i}}$
(16) $\displaystyle S_{\|e}(u)$ $\displaystyle=$
$\displaystyle\left(\frac{1+u}{1+a_{e}u}\right)\left(\frac{1+a_{e}u}{1+a_{i}u}\right)^{c_{e}}.$
(17)
## V The weakly nonlinear regime
As it follows from Eq. (5), in the linear regime near threshold, the
fluctuations of perpendicular and magnetic pressures almost compensate each
other. In the weakly nonlinear regime, the second order correction to the
total (perpendicular plus magnetic) pressure is thus relevant and leads to a
local shift of $\Gamma$. To find this correction, we consider the expansions
of the perpendicular pressures of the ions and electrons in the $u$ variable.
Because of the symmetry between the functions $S_{\perp i}(u)$ and $S_{\perp
e}(u)$, it is enough to consider the expansion
$\displaystyle S_{\perp i}(u)$ $\displaystyle=$ $\displaystyle
1+u\Big{(}2-2a_{i}-c_{i}(a_{e}-a_{i})\Big{)}$
$\displaystyle+u^{2}\Big{[}c_{i}\Big{(}a_{e}a_{i}-a_{i}^{2}+\frac{1}{2}(a_{e}-a_{i})^{2}\Big{)}$
$\displaystyle-4a_{i}+3a_{i}^{2}+\frac{1}{2}c_{i}^{2}(a_{e}-a_{i})^{2}-2c_{i}(a_{e}-a_{i})$
$\displaystyle+2\alpha_{i}c_{i}(a_{e}-a_{i})+1\Big{]}+O\left(u^{3}\right)$
As a result, the second order contributions to the perpendicular ion pressure
is given by
$\displaystyle p_{i\perp}^{(2)}=n_{0}T_{\perp
i}\Big{[}3a_{i}^{2}-4a_{i}+1+c_{i}(a_{e}-a_{i})$
$\displaystyle\qquad\times\Big{(}\frac{1}{2}(c_{i}+1)(a_{e}-a_{i})-2+3a_{i}\Big{)}\Big{]}u^{2},$
with an analogous formula for the perpendicular electron pressure, obtained by
exchanging the $i$ and $e$ indices. Furthermore, the threshold condition
rewrites
$\displaystyle\frac{B_{0}^{2}}{4\pi}+n_{0}\Big{\\{}T_{\perp
i}\left[2-2a_{i}-c_{i}\left(a_{e}-a_{i}\right)\right]$
$\displaystyle\qquad+T_{\perp
e}\left[2-2a_{e}+c_{e}\left(a_{e}-a_{i}\right)\right]\Big{\\}}=0.$ (18)
The quadratic contributions to the pressure balance (3), originating from
$p_{i\perp}^{(2)}+p_{e\perp}^{(2)}+\left(B-B_{0}\right)^{2}/(8\pi)$, are
collected in a term $\Lambda\left(\frac{B-B_{0}}{B_{0}}\right)^{2}$ with
$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle n_{0}\Big{\\{}T_{\perp
i}\Big{(}3a_{i}^{2}-4a_{i}+1$ (19)
$\displaystyle+c_{i}(a_{e}-a_{i})\Big{[}\frac{1}{2}(1+c_{i})(a_{e}-a_{i})-2+3a_{i}\Big{]}\Big{)}$
$\displaystyle+T_{\perp e}\Big{(}3a_{e}^{2}-4a_{e}+1+c_{e}(a_{e}-a_{i})$
$\displaystyle\times\Big{[}\frac{1}{2}(1+c_{e})(a_{e}-a_{i})+2-3a_{e}\Big{]}\Big{)}\Big{\\}}+\frac{B_{0}^{2}}{8\pi}.$
The value $\Lambda_{c}$ of $\Lambda$ at threshold is obtained by expressing
${B_{0}^{2}}/{8\pi}$ by means of Eq. (18), which gives
$\displaystyle\Lambda_{c}$ $\displaystyle=$ $\displaystyle
n_{0}\Big{\\{}T_{\perp i}\Big{[}3a_{i}^{2}-4a_{i}+1$
$\displaystyle+c_{i}(a_{e}-a_{i})\Big{(}\frac{1}{2}(1+c_{i})(a_{e}-a_{i})-2+3a_{i}\Big{)}$
$\displaystyle-\frac{1}{2}\Big{(}2-2a_{i}-c_{i}(a_{e}-a_{i})\Big{)}\Big{]}$
$\displaystyle+T_{\perp e}\Big{[}3a_{e}^{2}-4a_{e}+1+c_{e}(a_{e}-a_{i})$
$\displaystyle\times\Big{(}\frac{1}{2}(1+c_{e})(a_{e}-a_{i})+2-3a_{e}]$
$\displaystyle-\frac{1}{2}\Big{(}2-2a_{e}+c_{e}(a_{e}-a_{i})\Big{)}\Big{]}\Big{\\}}.$
After some algebra, defining $\lambda_{c}=\Lambda_{c}/(n_{0}T_{\perp i})$, one
gets
$\displaystyle\frac{\lambda_{c}}{\alpha_{i}}=\frac{T_{\perp i}}{T_{\parallel
i}}\Big{[}3+3\frac{\theta_{\perp}^{3}}{\theta_{\parallel}^{2}}-\frac{1}{2}\frac{\left(\theta_{\perp}-\theta_{\parallel}\right)^{2}}{\theta_{\parallel}^{2}\left(1+\theta_{\parallel}\right)^{2}}$
$\displaystyle\qquad\times\left(4\theta_{\perp}+4\theta_{\parallel}^{2}+\allowbreak
5\left(\theta_{\perp}+1\right)\theta_{\parallel}\right)\Big{]}$
$\displaystyle\qquad-\frac{3}{2\theta_{\parallel}\left(1+\theta_{\parallel}\right)}\left[\left(\theta_{\perp}+\theta_{\parallel}\right)^{2}+2\theta_{\parallel}(1+\theta_{\perp}^{2})\right].$
(20)
Proceeding as in [13], retaining the contribution of the above quadratic terms
to the pressure balance, leads one to supplement a nonlinear contribution to
Eq. (1) that becomes
$\displaystyle\frac{\partial u}{\partial
t}=\frac{2}{\sqrt{\pi}}\frac{T_{\|i}}{T_{\perp
i}}\frac{v_{\|i}}{D}{\widehat{{\cal K}_{z}}}\Big{\\{}\Gamma
u-\frac{\chi}{\beta_{\perp}}(\Delta_{\perp})^{-1}\partial_{zz}u$
$\displaystyle+\frac{3}{4}\Big{(}\frac{T_{\perp
i}}{T_{\|i}}-1\Big{)}\frac{1+F}{1+\theta_{\perp}}r_{L}^{2}\Delta_{\perp}u-\frac{\lambda_{c}}{2(1+\theta_{\perp})}u^{2}\Big{\\}}$
(21)
Here the integral operator ${\widehat{{\cal K}}_{z}}$ reduces in Fourier
representation to $|k_{z}|$ and $\chi=1+\frac{\beta_{\perp}-\beta_{\|}}{2}$.
Furthermore, within the present approximation, $u$ coincides with
${\widetilde{B}}_{z}/B_{0}$.
Equation (21) extends the result of [13, 21] valid for cold electrons. As in
the latter case, this equation is a gradient type equation,
$\frac{\partial u}{\partial t}=-{\widehat{{\cal K}_{z}}}\frac{\delta F}{\delta
u},$
for which the free energy (written in dimensionless variables)
$F=\int\left\\{\frac{1}{2}\left[-\Gamma
u^{2}+(\partial_{z}u)^{2}+u\Delta_{\perp}^{-1}\partial_{zz}u\right]+\frac{1}{3}\lambda_{c}u^{3}\right\\}d{\bf
r}$
is unbounded from below due to the integral $\int\lambda_{c}u^{3}d{\bf r}$.
This leads to a blow-up behavior, associated with a subcritical bifurcation
[13], [14]. Saturation at large values of the amplitude and formation of
stationary structures requires additional nonlinear effects such as the
influence of resonant particles on the nonlinear coupling [22]. Equation (21)
that does not include saturation processes is not suitable to address the
question of the reduction of the temperature anisotropy by the development of
the mirror instability mentioned in [3]. This effect is reproduced by the
quasi-linear theory [19], and was also studied in the context of the so-called
FLR-Landau fluid model that, like the present asymptotics, retains a linear
description of the Landau resonance and of FLR effects, but includes all the
hydrodynamic nonlinearities and does not a priori prescribe a pressure balance
condition. It was observed in this case that during the saturation phase, the
mean temperatures rapidly evolve in a way as to reduce the distance to
threshold [20].
As demonstrated in [13, 14], the sign of the nonlinear coupling $\lambda_{c}$
defines the type of the mirror structures, namely holes ($\lambda_{c}>0$) or
humps ($\lambda_{c}<0$), near threshold. This sign is strongly dependent on
the equilibium distribution function [23] . It is nevertheless of interest to
consider the case where both ions and electrons have a bi-Maxwellian
distribution function. It turns out that the sign of $\lambda_{c}$ can then be
determined analytically in a few special cases.
(i) Limit $\theta_{\|}\ll\theta_{\perp}$:
$\frac{\Lambda_{c}}{n_{0}T_{\perp
i}a_{i}}=\frac{\theta_{\perp}^{2}}{\theta_{\parallel}}\left(\frac{T_{\perp
e}}{T_{\parallel e}}-\frac{3}{2}\right)>0.$
(ii) Equal anisotropies ($\theta_{\perp}=\theta_{\parallel}$)
$\displaystyle\Lambda_{c}$ $\displaystyle=$ $\displaystyle n_{0}(T_{\perp
i}+T_{\perp e})\left(3a^{2}-4a+1\right)$ $\displaystyle-n_{0}(T_{\perp
i}+T_{\perp e})\left(1-a\right)=3a\frac{B_{0}^{2}}{8\pi}>0.$
(iii) Isotropic electron temperature: The coefficient $\Lambda_{c}$ can be
rewritten in the form
$\displaystyle\Lambda_{c}$ $\displaystyle=$ $\displaystyle
n_{0}(a_{i}-1)\\{T_{\perp i}\Big{(}(3a_{i}-1)$
$\displaystyle+c_{i}\Big{[}\frac{1}{2}(1+c_{i})\left(\alpha_{i}-1\right)+2-3a_{i}\Big{]}\Big{)}$
$\displaystyle+T_{e}c_{e}\Big{[}\frac{1}{2}\left(1+c_{e}\right)\left(a_{i}-1\right)+1\Big{]}\\}+\frac{B_{0}^{2}}{8\pi}.$
Furthermore, at threshold,
$\frac{1}{2}n_{0}(a_{i}-1)\left[T_{\perp i}\left(2-c_{i}\right)+T_{\perp
e}c_{e}\right]=\frac{B_{0}^{2}}{8\pi}>0.$
Hence, we simultaneously have two inequalities $a_{i}>1$ and $T_{\perp
e}c_{e}>T_{\perp i}(c_{i}-2)$. Therefore,
$\displaystyle\Lambda_{c}$ $\displaystyle=$ $\displaystyle
n_{0}(a_{i}-1)\Big{\\{}T_{\perp i}\Big{(}(3a_{i}-1)$
$\displaystyle+c_{i}\Big{[}\frac{1}{2}(1+c_{i})\Big{(}a_{i}-1\Big{)}+2-3a_{i}\Big{]}\Big{)}$
$\displaystyle+T_{e}c_{e}\Big{[}\frac{1}{2}(1+c_{e})(a_{i}-1)+1\Big{]}\\}$
$\displaystyle+\frac{1}{2}n_{0}(a_{i}-1)\left[T_{\perp
i}\left(2-c_{i}\right)+T_{\perp e}c_{e}\right]$ $\displaystyle=$
$\displaystyle n_{0}(a_{i}-1)\Big{\\{}T_{\perp i}\Big{(}3a_{i}(1-c_{i})$
$\displaystyle+c_{i}\Big{[}\frac{1}{2}(1+c_{i})\left(a_{i}-1\right)+\frac{3}{2}\Big{]}\Big{)}$
$\displaystyle+T_{e}c_{e}\Big{[}2+\frac{1}{2}\left(1+c_{e}\right)\left(a_{i}-1\right)\Big{]}\Big{\\}},$
which is positive, because $1-c_{i}=c_{e}=(1+\theta_{\parallel})^{-1}>0$ and
$a_{i}>1$.
Figure 1: Fig. 1. Variation with $\theta_{\|}$ of the distance to threshold
$\Gamma$ given by Eq. (2) (dashed line) and of the normalized nonlinear
coupling coefficient $\lambda$ (solid line) evaluated from Eq. (19) for
$\theta_{\perp}=1$ , $a_{i}=1.1$ and $\beta_{\perp i}=10$.
Figure 2: Fig. 2. Variation with $\beta_{\perp i}$ of the minimum
$\mathrm{min}\,(\lambda_{c})$ of the normalized nonlinear coupling coefficient
taken in an interval of values of $a_{p}$ between $0$ and $a_{p1}(\beta_{\perp
i})$, defined such that the threshold is obtained for a value of $\theta_{\|}$
equal to $100$, for $\theta_{\perp}=0.2$ (solid line), $\theta_{\perp}=1$
(dotted line) and $\theta_{\perp}=5$ (dashed line).
(iv) More general conditions: A numerical approach was used in this case.
Figure 1 displays, for typical values of the parameters (taken here as
$\theta_{\perp}=1$, $a_{i}=1.1$ and $\beta_{\perp i}=10$), the distance to
threshold $\Gamma$ (dashed line) given by Eq. (2) and the non-dimensional
nonlinear coupling coefficient $\lambda=\Lambda/(n_{0}T_{\perp i})$ (solid
line), where $\Lambda$ is given by Eq. (19), as a function of $\theta_{\|}$.
This graph is typical of the general behavior of these functions and shows
that they are both decreasing as $\theta_{\|}$ increases, with $\lambda$
possibly reaching negative values, but only below threshold. In order to show
that the value $\lambda_{c}$, given by Eq. (20), of $\lambda$ at threshold is
positive in a wider range of parameters, we display in Fig. 2, as a function
of $\beta_{\perp i}$ for $\theta_{\perp}=0.2$ (solid line), $\theta_{\perp}=1$
(dotted line) and $\theta_{\perp}=5$ (dashed line), the quantity
$\mathrm{{min}\,(\lambda_{c})}$ obtained after minimizing $\lambda_{c}$ in an
interval of values of $a_{p}$ between $0$ and $a_{p1}(\beta_{\perp i})$. The
latter quantity is arbitrarily defined such that the threshold is obtained for
a value of $\theta_{\|}$ equal to $100$. This graph shows that
$\mathrm{min}(\lambda_{c})$ varies little with $\theta_{\perp}$ but is very
sensitive to $\beta_{\perp i}$. As the latter parameter is increased,
$\mathrm{{min}\,(\lambda_{c})}$ decreases but remains always positive.
Although this numerical observation is not a rigorous proof, it convincingly
shows that $\Lambda>0$ in the parameter range of physical interest.
## VI Stationary nonlinear structures
Substituting the explicit expressions of the gyrotropic pressures in terms of
the magnetic field amplitude given in the Section IV, within the equation for
the balance of forces
$\displaystyle-\nabla\Big{(}p_{\perp}+\frac{B^{2}}{8\pi}\Big{)}+\Big{[}1+\frac{4\pi}{B^{2}}(p_{\perp}-p_{\|})\Big{]}\frac{(\mathbf{B}\cdot\nabla)\mathbf{B}}{4\pi}$
$\displaystyle\qquad+\mathbf{B}(\mathbf{B}\cdot\nabla)\Big{(}\frac{p_{\perp}-p_{\|}}{B^{2}}\Big{)}\,=0,$
(22)
leads to a closed system that seems overdetermined due to the divergenceless
condition $\nabla\cdot{\bf B}=0$. In fact, it can be checked, after some
algebra using the explicit expressions (13,14) and (16,17), that the
projection of Eq. (22) on the magnetic field vanishes identically, thus
reducing the system to three equations for three unknowns. These equations can
be useful for finding, possibly numerically, stationary profiles of three-
dimensional finite-amplitude stationary mirror structures. Note that Eq. (22)
differs from the Grad-Shafranov equation [24, 25] in that the parallel and
perpendicular pressures are here prescribed functions of the magnetic field
amplitude. A main issue concerns the existence of stable subcritical
solutions, a question that is beyond the scope of this letter and will be
addressed in forthcoming works. Such structures are reported by satellite
observations [26, 27] and are also expected from the subcritical character of
the mirror instability [14]. Equilibrium solutions were computed in one-space
dimension in [17], where they lead to discontinuous profiles. Their
regularization would require that FLR corrections be retained. These
additional contributions are known from the linear kinetic theory but their
extension to the finite-amplitude case remains a challenging problem.
This work was supported by the CNRS PICS programme 6073 and RFBR grant
12-02-91062-CNRS_a. T.P. and P.L.S. benefited from support from INSU-CNRS
Programme National PNST. The work of E.K. was also supported by the RAS
Presidium Program ”Fundamental problems of nonlinear dynamics in mathematical
and physical sciences”, Grant NSh 7550.2006.2 and by the French Ministère de
l’Enseignement Supérieur et de la Recherche.
## References
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* [9] P. Hellinger, Phys. Plasmas 14, 082105 (2007).
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* [15] V.E. Zakharov and E.A. Kuznetsov, Physics - Uspekhi 55, 535 (2012).
* [16] O.D. Constantinescu, J. Atm. Solar-Terrestrial Phys. 64, 645-649 (2002).
* [17] T. Passot, V. Ruban and P. L. Sulem, Phys. Plasmas 13, 10231 (2005).
* [18] A. Hasegawa, Phys. Fluids 12, 2642 ,(1969).
* [19] V.D. Shapiro and V.I. Shevchenko, Zh. Eksp. Teor. Fiz. 45, 1612 (1963) [J. Sov. Phys. JETP 18, 1109 (1964)].
* [20] D. Borgogno, T. Passot, and P.L. Sulem, Nonlin. Process. Geophys., 14, 373–383 (2007).
* [21] F. Califano, P. Hellinger, E. Kuznetsov, T. Passot, P.L. Sulem, and P. Travnicek, J. Geophys. Res. 113, A08219, (2008).
* [22] T. Passot, P.L. Sulem, E. Kuznetsov, and P. Hellinger, AIP Conf. Proc., 1188, 205 (2009).
* [23] P. Hellinger, E. Kuznetsov, T. Passot, P.L. Sulem, and P. Trávníček, Geophys. Res. Lett. 36, L06103 (2009).
* [24] H. Grad and H. Rubin, Proc. 2nd Conf. on the Peaceful Use of Atomic Energy, 31, 190 (IAEA, Geneva, 1958).
* [25] V.D. Shafranov, Zh. Eksp. Teor. Fiz. 33, 710 (1957) [J. Sov. Phys. JETP 8, 545 (1958)]; Rev. Plasma Phys. 2, 103 (Consultant Bureau, New York, 1966).
* [26] J. Soucek, E. Lucek and I. Dandouras, J. Geophys. Res. 113, A04203 (2007).
* [27] V. Génot, E. Budnik, P. Hellinger, T. Passot, G. Belmont, P.M. Trévníc̆ek, P.L. Sulem, E. Lucek and I. Dandouras, Ann. Geophys. 27, 601-615 (2009).
|
arxiv-papers
| 2012-10-16T08:20:02 |
2024-09-04T02:49:36.614582
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E.A. Kuznetsov, T. Passot and P.L. Sulem",
"submitter": "Evgenii A. Kuznetsov",
"url": "https://arxiv.org/abs/1210.4291"
}
|
1210.4348
|
* # The Nature of V1464 Aql: A New Ellipsoidal Variable with a $\delta$ Scuti Component
DAL, H.A. 1,2, SİPAHİ, E. 1 1 Department of Astronomy and Space Sciences,
University of Ege, Bornova, 35100 İzmir, Turkey
2 Corresponding Author, Email: ali.dal@ege.edu.tr
Abstract: Taking into account the results obtained from the models and
analyses of the BVRI light curves, we discuss the nature of V1464 Aql. The
analyses indicated that the mass ratio of the system is $q=0.71\pm 0.02$,
while the inclination of the system ($i$) is 38∘.45$\pm$0∘.22. Taking the
primary component’s temperature as 7420$\pm$192 K, we found that the
temperature of the secondary is 6232$\pm$161 K. The mass of the primary
component was found to be 1.74$\pm$0.05 $M_{\odot}$, while it is 1.23$\pm$0.01
$M_{\odot}$ for the secondary. The primary component’s radius was found to be
2.10$\pm$0.05 $R_{\odot}$, while it was found as 1.80$\pm$0.01 $R_{\odot}$ for
the secondary. Revealing that the system should not exhibit any eclipses, we
demonstrated that the main variation with large amplitude should be caused due
to the ellipsoidal effect. Indeed, the Fourier analysis also supported the
result. For the first time in the literature, we revealed that the primary
component is a $\delta$ Scuti star. The period of pulsation was found to be
58.482$\pm$0.002, 58.482$\pm$0.001, 60.966$\pm$0.002, 60.964$\pm$0.003 minutes
in BVRI bands, respectively. We plotted V1464 Aql in the plane of
$log(P_{orb})$-$log(P_{pulse})$. Using more than 160 binaries, whose one or
both components are pulsating, we derived a new linear fit in the plane of
$log(P_{orb})$-$log(P_{pulse})$ for each type binary. Using the linear fit of
each group, we obtained new calibrations between $log(P_{orb})$ and
$log(P_{pulse})$ for different type pulsating stars. In addition, a
calibration has been obtained for the first time for the pulsating stars from
the spectral types O and B. V1464 Aql seems to be located near the other
ellipsoidal and close binaries. Thus, we listed V1464 Aql as a new candidate
for the ellipsoidal variables with a $\delta$ Scuti component.
Keywords: techniques: photometric — methods: data analysis — (stars:) binaries
(including multiple): close — (stars:) binaries: spectroscopic — stars:
variables: $\delta$ Scuti — stars: individual(V1464 Aql)
## 1 Introduction
The pulsation of a star, as well as being a component of eclipsing binary
itself, is very important nature to understand its evolution. There are
several type pulsating stars such as Cepheid-type pulsating stars, $\gamma$
Doradus-type pulsating stars, and $\delta$ Scuti-type pulsating stars and etc.
All these types are in the Instability Strip on the main-sequence in the
Hertzsprung-Russell diagram. Investigating the frequencies of oscillations,
asteroseismology known as stellar seismology tryies to identify the physical
processes behind the pulsating nature and so the stellar interiors. Pulsating
stars with their key roles are important objects to understand stellar
evolution (Cunha et al., 2007; Aerts et al., 2010). However, in some cases,
the pulsation features can not be enough to solve all the stellar interior and
its nature. In this point, being a component of a binary can take an important
role to get down the problems. Especially, analysis of a light curve together
with a radial velocity gives us lots of physical parameters such as mass,
radius, $log~{}g$, and ect., for the components of the eclipsing binaries
(Wilson & Devinney, 1971; Wilson, 1990). Considering the both pulsating and
eclipsing behaviour, the physical natures of stars can be easily identifiable
(Lampens, 2006; Pigulski, 2006). Although there are many pulsating-single
stars, a few of them are a component in a binary system, especially eclipsing
system. As seen from the literature, the number of pulsating stars is over
635, while the incidence of eclipsing binary component all among them is
$\sim$5.5$\%$ (Kim et al., 2010).
In this study, we introduce V1464 Aql as a new candidate for a binary with
pulsating component. In the SIMBAD database, V1464 Aql was classified as a
spectroscopic binary system from the spectral type of A2V. In the ASAS
Database (Pojmánski, 1997), V1464 Aql’s variability type was suggested as
possibly RR Lyr (RRc), or an eclipsing binary, possibly contact (EC) or semi-
detached system (ESD). The system was listed as a contact binary by Duerbeck,
(1997) and Pribulla & Rucinski, (2006) and a spectroscopic binary by Pourbaix
et al., (2004). Duerbeck, (1997) listed the period of V1464 Aql as 0d.6978,
while Rucinski & Duerbeck, (2006) stated it as 0d.697822. The detailed
spectroscopic observations were made by Rucinski & Duerbeck, (2006). They
obtained a single-lined radial velocity with amplitude of 30.62$\pm$2.35
$kms^{-1}$. The authors noted that the lines come from visible component are
largely broadened. The value of $v\sin i$ was found to be 94$\pm$4 $kms^{-1}$.
They also claim that the spectral type of the system should be F1-2 rather
than early A.
In this study, we investigate the nature of this interesting system. For this
aim, we observed system photometricaly, and obtained the multi-band light
curves, in which the short-term intrinsic light variations are seen, for the
first time. We analysed the light curves, and also tested the results with the
method based on the Fourier transform described by Morris, (1985) and Morris
& Naftilan, (1993). Apart from light curve analyses, we also detected some
large-amplitude pulsation with short period. We analysed the pulsating
behaviour.
## 2 Observations
Observations were acquired with a thermoelectrically cooled ALTA U+42
2048$\times$2048 pixel CCD camera attached to a 40 cm - Schmidt - Cassegrains
- type MEADE telescope at Ege University Observatory. The observations were
made in BVRI bands during 11 nights in 03, 06, 07, 09, 16, 17 August 2011 and
04, 08, 15, 19, 30 September 2011. V1464 Aql was observed during all the night
in each observations. Some basic parameters of the program stars are listed in
Table 1. The names of the stars are listed in the first column, while J2000
coordinates are listed in the second column. The V magnitudes are in the third
column, and B-V colours are listed in the last column.
Although the program and comparison stars are very close on the sky, using the
calibration described by Hardie, (1962), the differential atmospheric
extinction corrections were applied. The atmospheric extinction coefficients
were obtained from observations of the comparison stars on each night.
Heliocentric corrections were also applied to the times of the observations.
The mean averages of the standard deviations are 0m.023, 0m.011, 0m.010, and
0m.013 for observations acquired in the BVRI bands, respectively. To compute
the standard deviations of observations, we used the standard deviations of
the reduced differential magnitudes in the sense comparisons minus check stars
for each night. There is no variation observed in the brightness of comparison
stars.
The obtained BVRI data indicated that the system exhibits two main variations
in each band. One of them has large amplitude with longer period, while the
second one has relatively smaller amplitude with shorter period. We firstly
investigated the larger variation. For this aim, we determine the most
symmetric and deeper minimum. The light curve obtained in the observations on
September 04, 2011 has such a minima. Using two different methods described by
Kwee & van Woerden, (1956) and Winkler, (1967), we computed a minimum time.
Both methods gave the same minimum time with almost the same error. Then,
using the method of the Discrete Fourier Transform (DFT) (Scargle, 1982), we
analysed its period by the PERIOD04 software (Lenz & Breger, 2005). The
results obtained from DFT were tested by two other methods. One of them is
CLEANest, which is another Fourier method (Foster, 1995). The second method
is the Phase Dispersion Minimization (PDM), which is a statistical method
(Stellingwerf, 1978). These methods confirmed the results obtained by DFT.
$JD~{}(Hel.)~{}=~{}24~{}55809.3467(1)~{}+~{}0^{d}.697822(3)~{}\times~{}E.$ (1)
All the found ephemeris of the system are given by Equation (1). The analyses
reveal that the period of the larger variation is close to the ones given by
Duerbeck, (1997) and Rucinski & Duerbeck, (2006). All the observation were
phased with using the ephemeris to prepare the light curve analyses.
## 3 Light Curve Analyses
### 3.1 Eclipsing Binary Model
In the literature, V1464 Aql was classified as a contact binary (Duerbeck,
1997; Pribulla & Rucinski, 2006). In addition, it was noted in the ASAS
Database (Pojmánski, 1997) that the system could also be an eclipsing binary,
especially a contact binary. Considering this knowledge, we tried to model the
light curves with the Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson,
1990). For this aim, we analysed the light curves obtained in the BVRI bands
together with the available single-lined radial velocity curve simultaneously,
using the PHOEBE V.0.31a software (Prša & Zwitter, 2005), which is used in
the version 2003 of the Wilson-Devinney Code (Wilson & Devinney, 1971;
Wilson, 1990). We tried to analyse the light curves with several different
modes, such as the ”overcontact binary not in thermal contact” and ”double
contact binary” modes. The initial analyses demonstrated that an astrophysical
acceptable result can be obtained if the analysis is carried out in the
”double contact binary” mode, while no acceptable results could be obtained in
the other modes.
In the literature, the temperature is given in a range from 6843 K (Ammons et
al., 2006) to 8970 K (Wright et al., 2003) for V1464 Aql. We took JHK
brightness of the system ($J=7^{m}.963\pm 0^{m}.027$, $H=7^{m}.830\pm
0^{m}.027$, $K=7^{m}.788\pm 0^{m}.020$) from the NOMAD Catalogue (Zacharias et
al., 2005). Using these brightnesses, we derived de-reddened colours as a
$(J-H)_{\circ}$=$0^{m}.110\pm 0^{m}.038$ and $(H-K)_{\circ}$=$0^{m}.030\pm
0^{m}.034$ for the system. Using the calibrations given by Tokunaga, (2000),
we derived the temperature of the primary component as 7420$\pm$192 K
depending on these de-reddened colours. Taking into account the standard
deviation of the brightness in JHK bands, we have computed the standard
deviation of the temperature as 192 K. Thus, the temperature was found to be
$7420\pm 192$ K. Thus, the temperature of the primary component was fixed to
7420$\pm$192 K in the analyses, and the temperature of the secondary was taken
as a free parameter. Considering the spectral type corresponding to this
temperature, the albedos ($A_{1}$ and $A_{2}$) and the gravity darkening
coefficients ($g_{1}$ and $g_{2}$) of the components were adopted for the
stars with the convective envelopes (Lucy, 1967; Rucinski, 1969). The non-
linear limb-darkening coefficients ($x_{1}$ and $x_{2}$) of the components
were taken from van Hamme, (1993). In the analyses, their dimensionless
potentials ($\Omega_{1}$ and $\Omega_{2}$), the fractional luminosity
($L_{1}$) of the primary component, the inclination ($i$) of the system, the
mass ratio of the system ($q$) and the semi-major axis ($a$) were taken as the
adjustable free parameters.
Because of being single-lined, the available radial velocity curve does not
give any mass ratio for the system. Therefore, considering general sum of
weighted squared residuals ($\Sigma res^{2}$), we tried to find the mass ratio
for the system. In analyses, the $\Sigma res^{2}$ values indicated that the
photometric mass ratio of the components is $q=0.71\pm 0.02$. According to
this result, we assume that the mass ratio of the system is $q=0.71\pm 0.02$.
The temperature of the secondary component was found to be 6232 K and its
error was found to be 27 K. However, the found error seems to be unreal in the
sense of statistical. Taking into account the standard deviation of the
brightness in JHK bands, we have computed the standard deviation of the
temperature as 161 K. Finally, the temperature was found to be $6232\pm 161$ K
for the secondary component. All the parameters derived from the analyses are
listed in Tables 2, while the synthetic light curves are shown in Figure 1. In
addition, the radial velocity obtained by Rucinski & Duerbeck, (2006) and the
synthetic curve is shown in Figure 2. Finally, we also derived the 3D model of
Roche geometry and the geometric configurations at four special phases 0.00,
0.25, 0.50 and 0.75 for the system, using the parameters obtained from the
light curve analysis. The derived 3D model of Roche geometry and the geometric
configurations are shown in Figure 3.
Although there is not any available double lined-radial velocity curve, we
tried to estimate the absolute parameters of the components. According to
Tokunaga, (2000), the mass of the primary component must be 1.74$\pm$0.05
$M_{\odot}$ corresponding to its surface temperature. Considering possible
mass ratio of the system, the mass of the secondary component was found to be
1.23$\pm$0.01 $M_{\odot}$.
Using Kepler’s third law, we calculated possible the semi-major axis as a
4.76$\pm$0.03 $R_{\odot}$. Considering this estimated semi-major axis, the
radius of the primary component was computed as 2.10$\pm$0.05 $R_{\odot}$,
while it was found as 1.80$\pm$0.01 $R_{\odot}$ for the secondary component.
Using the estimated radii and the obtained temperatures of the components, the
luminosity of the primary component was estimated to be 11.97$\pm$0.31
$L_{\odot}$, and it was found as 4.41$\pm$0.02 $L_{\odot}$ for the secondary
component. We plotted the distribution of the radii versus the masses and also
the luminosity versus the temperature for some stars in Figure 4. The lines
represent the ZAMS theoretical model developed for the stars with $Z=0.02$ by
Girardi et al., (2000), while dashed lines represent the TAMS theoretical
model. According to the obtained results, the absolute parameters are
generally an acceptable in the astrophysical sense.
### 3.2 Elipsoidal Binary Model
As seen from the Roche geometry shown in Figure 3, and the results of the
light curve analysis, the system does not exhibit any eclipses. However, the
shape of the components should causes some light variation known as the
ellipsoidal variation. The ellipsoidal variable stars are non-eclipsing binary
stars and especially close binaries (Morris, 1985; Beech, 1985). In the case
of the ellipsoidal variables, the inclination angle ($i$) of the binary is so
small that the system does not exhibit any eclipses. The main variation is due
to non-spherical shapes of the components. According to Morris, (1985),
Beech, (1985) and Morris & Naftilan, (1993), if the main variation is caused
due to the ellipsoidal effect indeed, the light curves of the ellipsoidal
variables can be modelled by the Fourier analysis given by Equation (2), and
also one will expect that the $\cos(2\theta)$term must be dominant among all
other terms in the results of the Fourier analysis. This is the base of the
modern methods used in recent studies, such as Faigler & Mazeh, (2011) and
Faigler et al., (2012).
$L(\theta)=A_{0}~{}+~{}\sum_{\mbox{\scriptsize\
i=1}}^{N}~{}A_{i}~{}cos(i\theta)~{}+~{}\sum_{\mbox{\scriptsize\
i=1}}^{N}~{}B_{i}~{}sin(i\theta)$ (2)
All the BVRI light curves were modelled with the Fourier series. The derived
Fourier models are shown in Figure 1, and the Fourier Coefficients are listed
in Table 3. The $A_{i}$ coefficients listed in the table are the coefficients
of the $\cos(i\theta)$ terms, while $B_{i}$ parameters are the coefficients of
the $\sin(i\theta)$ terms given in Equation (2). In fact, the most dominant
one is $\cos(2\theta)$ term for each of the BVRI-bands. Thus, it is obvious
that the main effect seen in the light variations is the ellipticity effect.
However, if the most dominant one was $\cos(\theta)$ term, we would consider
the other effects such as magnetic activity occurring on the surface of cool
stars. There are many systems, which are similar to V1464 Aql, such 75 Pegasi
and 42 Persei (Martin et al., 1990, 1991), and several other systems (Faigler
& Mazeh, 2011; Faigler et al., 2012).
## 4 The Pulsation
Apart from the main variation, there is a shorter period oscillation. In order
to understand the shorter period oscillation, we firstly obtained all the pre-
whitened light curves. To obtain the pre-whitened data, we extracted synthetic
light curves from all the observation in each band. In the second step, all
the pre-whitened data were analysed with both the Discrete Fourier Transform
(DFT, Scargle, 1982) and the Phase Dispersion Minimization (PDM), which is a
statistical method (Stellingwerf, 1978).
The periods found from the pre-whitened data are listed in Table 4. The
normalized power-spectrums, which exhibit the quality of the period analysis,
are shown in Figure 5. Using the light elements given with Equation (1) and
the parameters found from the pre-whitened data, all the nightly light
variations were modelled for each band. The modelled light variations are seen
in Figure 6.
Soydugan et al., (2006) listed some eclipsing binaries, whose primary and/or
secondary components are pulsating. Using the physical parameters found in
this study, the location of V1464 Aql’s primary component is shown in Figure
7. In this figure, following Soydugan et al., (2006), the ZAMS (broad line)
and TAMS (dashed line) were taken from Girardi et al., (2000). The borders of
the Instability Strip on the main-sequence were computed from Rolland et al.,
(2002). We show that V1464 Aql’s primary component is located among the
eclipsing binaries, whose one or two component(s) are in the Instability Strip
on the main-sequence. The primary component of V1464 Aql is a pulsating
variable for the case at least. In the literature, there are also four
ellipsoidal variables with a $\delta$ Scuti component. As it is listed in
Table 5, these are XX Pyx (Aerts et al., 2002), HD 173977 (Chapellier et al.,
2004), HD 207651 (Henry et al., 2004) and HD 149420 (Fekel & Henry, 2006).
In this case, V1464 Aql is fifth ellipsoidal variable binary with a $\delta$
Scuti component.
As it is known from Turner, (2011); Liakos et al., (2012), the close
binaries with a pulsating component exhibit some calibrations in the plane of
$log(P_{orb})$-$log(P_{pulse})$. For this aim, we took the data of all listed
close binaries, whose components are from A or F spectral types, from Turner,
(2011); Liakos et al., (2012). We also took the data of listed binaries,
whose components are generally from O or B spectral types, from Aerts &
Harmanec, (2004). In addition, we plotted all the Long Secondary Period
Variables (hereafter LSPVs) listed by Kiss et al., (1999); Olivier & Wood,
(2003) in the same plane. In the case of the binaries, whose components are
from A and F spectral types, the pulsating components are $\delta$ Scuti
stars, while the components are the other pulsating stars excepted $\delta$
Scuti star for binaries, whose components are generally from the spectral
types O or B. Although the LSPVs are also debated subject in the literature,
there is a relation between their periods (Kiss et al., 1999; Olivier & Wood,
2003). In Figure 8, we plotted all the binaries in the plane of
$log(P_{orb})$-$log(P_{pulse})$. All the components plotted in this figure are
pulsating stars. Moreover, we plotted the pulsating components of all known
ellipsoidal variables, whose one component is a $\delta$ Scuti star.
$log(P_{pulse})~{}=~{}-0.040(0.070)~{}\times~{}log(P_{orb})~{}-~{}0.337(0.025)$
(3)
$log(P_{pulse})~{}=~{}0.593(0.041)~{}\times~{}log(P_{orb})~{}-~{}1.545(0.021)$
(4)
$log(P_{pulse})~{}=~{}1.083(0.024)~{}\times~{}log(P_{orb})~{}-~{}1.449(0.059)$
(5)
Using GraphPad Prism V5.02 software (Motulsky, 2007), we re-modelled the
distribution of each group with the linear fit. Here, the standard deviations
of each coefficient and each constant are given in the brackets near
themselves. The derived linear fits are also plotted in Figure 8. To test
whether the derived linear fits are statistically acceptable, we computed the
probability value (hereafter p-value). The threshold level of significance
(hereafter $\alpha$ value) was taken as 0.005 for the p-value, which allowed
us to test whether the p-value are statistically acceptable or not (Wall &
Jenkins, 2003). The derived linear fit is given with Equation (3) for the
binaries, whose components are from spectral types O and B. The p-value was
found to be $p-value$ $<$ 0.0027. The linear fit of the A-F binaries is given
with Equation (4), while it is given with Equation (5) for the ellipsoidal
variables. The computed $p-values$ are $<$ 0.0001 for both groups. Considering
the $\alpha$ value, the derived linear fits are statistically acceptable.
As seen from Figure 8, in the plane of $log(P_{orb})$-$log(P_{pulse})$
distribution, V1464 Aql locates near the close binaries and ellipsoidal
variables from the spectral types A and F.
## 5 Results and Discussion
In the literature, there are several stars or systems, which exhibits
combinations of a few different variations. Some of them are eclipsing
binaries with a pulsating component, and some semi-regular variable with
unknown short-term variations, or some pulsating ellipsoidal variables, ect
(Derekas et al., 2006; Nie et al., 2010; Faigler & Mazeh, 2011; Faigler et
al., 2012).
In several studies, V1464 Aql was classified as a spectroscopic binary system,
RR Lyr (RRc), an eclipsing binary, possibly contact (EC) or semi-detached
system (ESD) (Duerbeck, 1997; Pribulla & Rucinski, 2006; Pourbaix et al.,
2004; Rucinski & Duerbeck, 2006). We analysed the BVRI light curves. As seen
from Table 2, the inclination of the system ($i$) is found to be
38∘.45$\pm$0∘.22. In the case of this ($i$) value, the system does not exhibit
any eclipses. As shown in Figure 3, the 3D geometric configurations derived at
four special phases for the system also demonstrated the non-eclipsing cases.
However, these configurations indicate that the system is on the edge of
eclipsing. Although the system may exhibit the grazing eclipses, these
eclipses become unseen among the other small variability. As it is estimated
in the literature, the 3D model of Roche geometry reveals that the components
of the system are near the filling their Roche Lobes. According to these
results, V1464 Aql should be a candidate of contact binary system, but non-
eclipsing due to the inclination of the system ($i$). The Fourier analysis
also supported this result. The coefficients of the $\cos(2\theta)$ term
($A_{2}$) were found to be 0.0492$\pm$0.0008, 0.0372$\pm$0.0007,
0.0355$\pm$0.0006 and 0.0327$\pm$0.0005 for the BVRI bands, respectively.
According to these values, the main effect in the light variation of the
system is come from the shape of the components, known as ellipsoidal effect.
The calibration given by Tokunaga, (2000), the mass of the primary component
should be 1.74$\pm$0.05 $M_{\odot}$ corresponding to its surface temperature.
Under this assumption, the light curve analysis reveals that the temperature
of the secondary component was found to be 6232$\pm$161 K. The light curve
analysis indicate that the mass ratio of the system should be $q=0.71\pm
0.02$. In this point it should be noted that the standard deviation of the
primary component temperature and absolute parameters were computed depending
on the standard deviations given for the colours in the NOMAD Catalogue
(Zacharias et al., 2005). However, the error of temperature was found from
the light curve analysis for the secondary component. Considering the standard
deviations of the other parameters, it seem a bit lower than it should be. The
places of the components plotted in Figure 4 demonstrated that the results
found from the analysis are acceptable in the astrophysical sense. On the
other hand, the secondary component is seen more closer to the TAMS than the
primary in Figure 4. The stars seem not to be coeval to each other. However,
it is well known that there are several binaries such as YY CrB (Essam et al.,
2010), BS Cas (Yang et al., 2008), VZ Tri (Yang, 2010). All these samples
exhibits the same behaviour. Their common properties are that all of them are
contact binaries with a large the period variation due to the large mass
transfer. Indeed, if V1464 Aql is an analogue of these systems, this makes
V1464 Aql very important for the future studies.
The fractional radii were found to be $r_{1}=0.441\pm 0.002$ for the primary
component and $r_{2}=0.378\pm 0.002$ for the secondary one. In this case, the
sum of fractional radii was computed as $r_{1}+r_{2}\simeq 0.80$. Thus, V1464
Aql seems to be in agreement with Kopal, (1956)’s criteria for overcontact
systems. The period analysis indicates that the orbital period as
$0^{d}.697822$. In addition, the temperature of the primary component is
7420$\pm$192 K, while the secondary one is 6232$\pm$161 K. Although some
contact binaries have components with some different surface temperature, they
generally have the same surface temperature. Here, the primary component of
V1464 Aql is hotter than the secondary one. Considering some characteristics
of the system such as the short orbital period, small mass ratio, hotter
primary component and ect., V1464 Aql seems to be in agreement with those of
A-type W UMa binaries (Berdyugina, 2005; Rucinski, 1985). The period
analyses reveal that the period of the short-term variation was found to be
58.482$\pm$0.002, 58.482$\pm$0.001, 60.966$\pm$0.002, 60.964$\pm$0.003 minutes
in BVRI bands, respectively. However, it must be noted that we did not find
any secondary frequency for the short-term variation. The period differences
between each band should be caused by the different sensitivity of each band.
The sensitivity is decreasing from B band to I band, because the amplitude of
the pulsation is decreasing from B band to I band. As it is seen from the
standard deviations given in Table 4 and also from the light curves shown in
Figure 6, the scattering in the light curves is increasing from B band to I
band. The period analysing methods we used are depend on the statistical
method. The analyses gave the best period statistically for the pulsation in
each band. In this case, the most reliable periods are ones found from B and V
bands.
Figure 5 indicates that the short-term variation dominately exhibits itself in
shorter wavelength rather than longer ones. In fact, this is also seen clearly
in Figure 6. The amplitude of the short-term variation get larger from the
I-band to the B-band. Moreover, the primary component is located in the
Instability Strip on the main-sequence, as seen in Figure 7. Considering the
mass of the primary component and also the period of this short-term
variation, this second variation should be caused due to the pulsation of the
primary component. According to Aerts et al., (2010), the periods of $\delta$
Scuti stars are in the range 18 min to 8 hr. Moreover, the mass of them is in
the range 1.50 to 2.50 $M_{\odot}$. The mass value of the component and the
period of the short-term variation are agreement with these values. In this
case, this variation should be caused by the pulsation of a $\delta$ Scuti
star.
As it is seen from Figure 1, the light curves seem to be scattered. There are
also a few light curves in the ASAS Database (Pojmánski, 1997). They are also
very scattered. However, the models of pulsation shown in Figure 6
demonstrates that the scattering in the light curves is caused due to the
pulsation.
In the literature, there are four systems, whose light curves have the same
properties. All of them are also ellipsoidal variables, which have a $\delta$
Scuti component. Aerts et al., (2002) found more than 10 frequencies due to
the $\delta$ Scuti component in XX Pyx, whose orbital period is 1d.15. HD
207651 is a triple system. Henry et al., (2004) demonstrated that its
component A is an ellipsoidal binary with a $\delta$ Scuti component. They
found a few pulsation frequencies, and the more possible one is 0d.7354.
Chapellier et al., (2004) found $\sim$0d.1169 for the pulsation in the case
of HD 173977. According to the light curve analysis given by Fekel & Henry,
(2006), HD 149420 should be an eclipsing binary. However, it is likely to be
an ellipsoidal variable. They found the pulsating periods of $\delta$ Scuti
component as 0d.076082 and 0d.059256. In the case of V1464 Aql, we found
pulsating period $\sim$0d.040641.
On the other hand, in the literature, classifying of V1464 Aql is a debated
subject (Pojmánski, 1997; Duerbeck, 1997; Pribulla & Rucinski, 2006;
Pourbaix et al., 2004). Because of this, we plotted V1464 Aql among the
different type binaries and variables in the plane of
$log(P_{orb})$-$log(P_{pulse})$ shown in Figure 8. As it is seen in the
figure, indeed, V1464 Aql locates near the close and the ellipsoidal binaries,
whose components are from the spectral types A and F. This case also supports
that the effective temperature found for the primary component of V1464 Aql is
generally an acceptable in the astrophysical sense. In addition, it this
point, we also derived some new calibrations for different type binaries. Both
Turner, (2011) and Liakos et al., (2012) have already derived the
calibration for the close binaries. Especially, Liakos et al., (2012) derived
the calibrations separately for the detached binaries, semi-detached binaries.
As given with Equation (3), combining all the close binaries listed by both
Turner, (2011) and Liakos et al., (2012), we derived it together for all the
close binary from the spectral types A and F. Thus, we obtained more reliable
calibrations, using more larger data set. In addition, in the literature, the
calibration has not been derived for binary from the spectral types O and B.
Although one of their components is pulsating in these binaries, they are
generally different class instead of the $\delta$ Scuti stars. As given with
Equation (4), we tried to derive a similar calibration for these stars for the
first time in the literature. A similar calibration was derived for the
ellipsoidal binaries, as given with Equation (5).
Rucinski & Duerbeck, (2006) noted that there are largely broadened lines come
from the visible component. They also found that the $v\sin i$ value of the
component is 94$\pm$4 $kms^{-1}$. Although this rapid rotation can cause the
broadening in the lines, the pulsation can also cause it. Moreover, the
surface temperature of the primary component is 7420$\pm$192 K, while it is
6232$\pm$161 K for the secondary. In this case, it is expected that we see
double lines in the spectrum, though the inclination of the system ($i$) is
38∘.45. However, Rucinski & Duerbeck, (2006) stated that V1464 Aql is a
single-lined system, and the amplitude of the radial velocity is
30.62$\pm$2.35 $kms^{-1}$. The given amplitude of the radial velocity can
easily confuse someone’s mind, because it is very large enough. In this case,
we also expected to see the lines of the secondary component, and thus, the
radial velocity of the secondary component. On the other hand, considering
both the spectral and the photometrical results together, rapid rotation and
pulsation nature of the primary component can cause some effects on its
spectrum in the case of lower inclination ($i$) of 38∘.45. The broadening of
the lines should make the lines come from the secondary component disappear.
Consequently, V1464 Aql is a new candidate for the ellipsoidal system with a
pulsating component. The new studies will reveal its pulsating nature very
well, thus they will reveal the absolute nature of the system.
## Acknowledgments
The author acknowledges the generous observing time awarded to the Ege
University Observatory. We also thank the referee for useful comments that
have contributed to the improvement of the paper.
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Figure 1: The observed BVRI light curves and the synthetic curves obtained
from the light curve analysis and Fourier models. In the figure, the filled
circles represent the observations, while the dashed (blue) lines represent
the synthetic curves obtained from the light curve analysis and the (red) line
represent the derived Fourier fits. Figure 2: The radial velocity (filled
circles) obtained by Rucinski & Duerbeck, (2006) and the synthetic curve
(blue line) derived by using the parameters found from the light curve
analysis. In the figure, the dotted line represents the center-of-mass
velocity. Figure 3: The 3D model of Roche geometry and the geometric
configurations at four special phases 0.00, 0.25, 0.50 and 0.75, illustrated
for V1464 Aql, using the parameters obtained from the light curve analysis.
Figure 4: The places of the components of V1464 Aql in the Mass-Radius (upper
panel) and Temperature-Luminosity (bottom panel) distributions. In the figure,
the lines represent the ZAMS theoretical model developed by Girardi et al.
(2000), while the dashed lines represent the theoretical TAMS model. The grey
lines represent the theoretical evolutional tracks. The filled circles
represent the primary component, while the open circles represent the
secondary component. Figure 5: V1464 Aql’s normalized power-spectrums obtained
from the period analyses of the pre-whitened data with DPT method in each
band. The power-spectrum obtained from BVRI-bands are given in the panels a,
b, c, d, respectively. Figure 6: All the models of nightly light variations.
Figure 7: The location of V1464 Aql’s primary component in the Instability
Strip on the main-sequence. The symbol of asterisk represents the primary
component of V1464 Aql, while the dim circles represent some semi- and un-
detached binaries taken from Soydugan et al. (2006) and and references
therein. In the figure, the ZAMS and TAMS were taken from Girardi et al.
(2000), while the borders of the Instability Strip were computed from Rolland
et al. (2002). Figure 8: The location of V1464 Aql in the plane of
$log(P_{orb})$-$log(P_{pulse})$ among different type binary stars. In the
figure, filled circles represent the observed values, while the lines
represent the theoretical linear fit of each group.
Table 1: Basic parameters for the observed stars. The coordinates were taken from the SIMBAD database, the brightness and colours were taken from the All-sky Compiled Catalogue of 2.5 million stars (Kharchenko et al., 2009). Star | $\alpha$ / $\delta$ (J2000) | V (mag) | B-V (mag)
---|---|---|---
V1464 Aql | 19h 50m 15s.473 / -08∘ 36′ 06′′.26 | 8.661 | 0.276
GSC 5725 2283 (Comparison) | 19h 49m 59s.290 / -08∘ 35′ 05′′.86 | 10.137 | 1.123
GSC 5725 2387 (Check) | 19h 50m 15s.473 / -08∘ 36′ 06′′.26 | 9.860 | 1.272
Table 2: The parameters of the components obtained from the light curve
analysis.
Parameter | Value
---|---
Orbital Period ($P$) | 0d.697822
Mass Ratio ($M_{2}/M_{1}=q$) | 0.71$\pm$0.02
Inclination ($i$) | 38∘.45$\pm$0∘.22
Temperature ($T_{1}$) | 7420$\pm$192∗ K
Temperature ($T_{2}$) | 6232$\pm$161 K
Dimensionless Potential ($\Omega_{1}$) | 3.2592$\pm$0.0006
Dimensionless Potential ($\Omega_{2}$) | 3.2582$\pm$0.0006
Fractional Luminosity (L1/LT, $B$) | 0.673$\pm$0.049
Fractional Luminosity (L1/LT, $V$) | 0.646$\pm$0.042
Fractional Luminosity (L1/LT, $R$) | 0.623$\pm$0.037
Fractional Luminosity (L1/LT, $I$) | 0.605$\pm$0.031
Gravity-darkening Coefficients ($g_{1}$, $g_{2}$) | 1.00, 0.32
Albedo ($A_{1}$, $A_{2}$) | 1.00, 0.50
Limb-darkening Coefficients ($x_{1,bol}$, $x_{2,bol}$) | 0.657, 0.657
Limb-darkening Coefficients ($x_{1,B}$, $x_{2,B}$) | 0.808, 0.809
Limb-darkening Coefficients ($x_{1,V}$, $x_{2,V}$) | 0.703, 0.704
Limb-darkening Coefficients ($x_{1,R}$, $x_{2,R}$) | 0.592, 0.592
Limb-darkening Coefficients ($x_{1,I}$, $x_{2,I}$) | 0.489, 0.489
Fractional Radius ($<r_{1}>$) | 0.441$\pm$0.002
Fractional Radius ($<r_{2}>$) | 0.378$\pm$0.002
Absolute Luminosity (L1) | 11.97$\pm$0.31 $L_{\odot}$
Absolute Luminosity (L2) | 4.41$\pm$0.02 $L_{\odot}$
Semi-Amplitude of Radial Velocity ($kms^{-1}$) | 30.8132${}^{+0.3705}_{-0.3615}$
Systemic Velocity ($kms^{-1}$) | 16.2560${}^{+0.2434}_{-0.2469}$
∗ The error is computed depending on the standard deviations of de-reddened
colours.
Table 3: The coefficients derived from the Fourier model. Filter | $A_{0}$ | $A_{1}$ | $A_{2}$ | $B_{1}$ | $B_{2}$
---|---|---|---|---|---
B | -2.0271$\pm$0.0006 | -0.0075$\pm$0.0008 | 0.0492$\pm$0.0008 | 0.0078$\pm$0.0008 | 0.0091$\pm$0.0008
V | -1.1129$\pm$0.0005 | 0.0052$\pm$0.0006 | 0.0372$\pm$0.0007 | 0.0032$\pm$0.0006 | 0.0041$\pm$0.0006
R | -0.6554$\pm$0.0004 | 0.0064$\pm$0.0006 | 0.0355$\pm$0.0006 | 0.0008$\pm$0.0006 | 0.0052$\pm$0.0005
I | -0.2349$\pm$0.0004 | 0.0092$\pm$0.0005 | 0.0327$\pm$0.0005 | 0.0000$\pm$0.0005 | 0.0041$\pm$0.0005
Table 4: The pulsation period found from period analyses in each filter. Filter | Period (minute) | Amplitude (mag)
---|---|---
B | 58.482$\pm$0.002 | 0.030$\pm$0.002
V | 58.482$\pm$0.001 | 0.024$\pm$0.003
R | 60.966$\pm$0.002 | 0.017$\pm$0.006
I | 60.964$\pm$0.003 | 0.011$\pm$0.006
Table 5: The known ellipsoidal binaries having a pulsating component in the
literature.
Stars | Spectral | V | $P_{orb}$ | $P_{pulse}$ | $M_{1}$ | $M_{2}$ | $R_{1}$ | $R_{2}$ | Ref.
---|---|---|---|---|---|---|---|---|---
| Type1 | (mag)1 | (day) | (day) | ($M_{\odot}$) | ($M_{\odot}$) | ($R_{\odot}$) | ($R_{\odot}$) |
XX Pyx | A4 | 11.490 | 1.150000 | 0.02624 | 1.85 | - | 1.95 | - | 2
HD207651 | A5 | 7.210 | 1.470800 | 0.73540 | - | - | - | - | 3
HD173977 | F2 | 8.070 | 1.800745 | 0.06889 | 1.87 | 1.30 | 2.87 | 1.42 | 4
HD149420 | A8 | 6.874 | 3.394306 | 0.07608 | - | - | - | - | 5
1 Taken from the SIMBAD data base
2 Aerts et al., (2002)
3 Henry et al., (2004)
4 Chapellier et al., (2004)
5 Fekel & Henry, (2006)
|
arxiv-papers
| 2012-10-16T10:31:59 |
2024-09-04T02:49:36.625765
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, E. Sipahi",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1210.4348"
}
|
1210.4440
|
# On the convergence of multiple Fourier series of functions of bounded
partial generalized variation
Ushangi Goginava and Artur Sahakian U. Goginava, Department of Mathematics,
Faculty of Exact and Natural Sciences, Iv. Javakhishvili Tbilisi State
University, Chavchavadze str. 1, Tbilisi 0128, Georgia zazagoginava@gmail.com
A. Sahakian, Yerevan State University, Faculty of Mathematics and Mechanics,
Alex Manoukian str. 1, Yerevan 0025, Armenia sart@ysu.am
###### Abstract.
The convergence of multiple Fourier series of functions of bounded partial
$\Lambda$-variation is investigated. The sufficient and necessary conditions
on the sequence $\Lambda=\\{\lambda_{n}\\}$ are found for the convergence of
multiple Fourier series of functions of bounded partial $\Lambda$-variation.
00footnotetext: 2010 Mathematics Subject Classification: 42B08 Key words and
phrases: Fourier series, Bounded $\Lambda$-variation .
## 1\. Classes of Functions of Bounded Generalized Variation
In 1881 Jordan [10] introduced a class of functions of bounded variation and
applied it to the theory of Fourier series. Hereafter this notion was
generalized by many authors (quadratic variation, $\Phi$-variation,
$\Lambda$-variation ets., see [10, 15, 14, 11]). In two dimensional case the
class BV of functions of bounded variation was introduced by Hardy [9].
Let $T:=[0,2\pi]$ and
$J^{k}=\left(a^{k},b^{k}\right)\subset T,\qquad k=1,2,\ldots d.$
Consider a measurable function $f\left(x\right)$ defined on $R^{d}$ and
$2\pi$-periodic with respect to each variable. For $d=1$ we set
$f\left(J^{1}\right):=f\left(b^{1}\right)-f\left(a^{1}\right).$
If for any function of $d-1$ variables the expression
$f\left(I^{1}\times\cdots\times I^{d-1}\right)$ is already defined, then for a
function of $d$ variables the mixed difference is defined as follows:
$f\left(J^{1}\times\cdots\times J^{d}\right):=f\left(J^{1}\times\cdots\times
J^{d-1},b^{d}\right)-f\left(J^{1}\times\cdots\times J^{d-1},a^{d}\right).$
Let $E=\\{I_{k}\\}$ be a collection of nonoverlapping intervals from $T$
ordered in arbitrary way and let $\Omega$ be the set of all such collections
$E$. We denote by $\Omega_{n}$ the set of all collections of $n$
nonoverlapping intervals $I_{k}\subset T.$
For sequences of positive numbers
$\Lambda^{j}=\\{\lambda^{j}_{n}\\}_{n=1}^{\infty}$, $j=1,2,\ldots,d$, the
$\left(\Lambda^{1},\ldots,\Lambda^{d}\right)$-variation of $f$ with respect to
index set $D:=\\{1,2,...,d\\}$ is defined as follows:
$V_{\Lambda^{1},\ldots,\Lambda^{d}}^{D}\left(f\right):=\sup\limits_{\\{I_{i_{j}}^{j}\\}_{i_{j}=1}^{k_{j}}\in\Omega}\
\sum\limits_{i_{1},...,i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{\lambda_{i_{1}}\cdots\lambda_{i_{d}}}.$
For an index set $\alpha=\\{j_{1},...,j_{p}\\}\subset D$ and any
$x=\left(x^{1},...,x^{d}\right)\in R^{d}$ we set
${\widetilde{\alpha}}:=D\setminus\alpha$ and denote by $x^{\alpha}$ the vector
of $R^{p}$ consisting of components $x^{j},j\in\alpha$, i.e.
$x^{\alpha}=\left(x^{j_{1}},...,x^{j_{p}}\right)\in R^{p}.$
By
$V_{\Lambda^{j_{1}},...,\Lambda^{j_{p}}}^{{\alpha}}\left(f,x^{\widetilde{\alpha}}\right)$
and $f\left(I_{i_{j_{1}}}^{1}\times\cdots\times
I_{i_{j_{p}}}^{p},x^{\widetilde{\alpha}}\right)$ we denote respectively the
$\left(\Lambda^{j_{1}},...,\Lambda^{j_{p}}\right)$-variation and the mixed
difference of $f$ as a function of variables $x^{j_{1}},...,x^{j_{p}}$ over
the $p$-dimensional cube $T^{p}$ with fixed values $x^{\widetilde{\alpha}}$ of
other variables. The
$\left(\Lambda^{j_{1}},...,\Lambda^{j_{p}}\right)$-variation of $f$ with
respect to index set ${\alpha}$ is defined as follows:
$V_{\Lambda^{j_{1}},...,\Lambda^{j_{p}}}^{{\alpha}}\left(f\right)=\sup\limits_{x^{{\widetilde{\alpha}}}\in
T^{d-p}}V_{\Lambda^{j_{1}},...,\Lambda^{j_{p}}}^{{\alpha}}\left(f,x^{\widetilde{\alpha}}\right).$
###### Definition 1.
We say that the function $f$ has total Bounded
$\left(\Lambda^{1},...,\Lambda^{d}\right)$-variation on $T^{d}=[0,2\pi]^{d}$
and write $f\in BV_{\Lambda^{1},...,\Lambda^{d}}$, if
$V_{\Lambda^{1},...,\Lambda^{d}}(f):=\sum\limits_{\alpha\subset
D}V_{\Lambda^{j_{1}},...,\Lambda^{j_{p}}}^{{\alpha}}\left(f\right)<\infty.$
###### Definition 2.
We say that the function $f$ is continuous in
$\left(\Lambda^{1},...,\Lambda^{d}\right)$-variation on $T^{d}=[0,2\pi]^{d}$
and write $f\in CV_{\Lambda^{1},...,\Lambda^{d}}$, if
$\lim\limits_{n\rightarrow\infty}V_{\Lambda^{j_{1}},...,\Lambda^{j_{k-1}},\Lambda_{n}^{j_{k}},\Lambda^{j_{k+1}},...,\Lambda^{j_{p}}}^{{\alpha}}\left(f\right)=0,\qquad
k=1,2,\ldots,p$
for any $\alpha\subset D,\ \alpha:=\\{j_{1},...,j_{p}\\}$, where
$\Lambda_{n}^{j_{k}}:=\left\\{\lambda_{s}^{j_{k}}\right\\}_{s=n}^{\infty}$.
###### Definition 3.
We say that the function $f$ has Bounded Partial
$\left(\Lambda^{1},...,\Lambda^{d}\right)$-variation and write $f\in
PBV_{\Lambda^{1},...,\Lambda^{d}}$ if
$PV_{\Lambda^{1},...,\Lambda^{d}}(f):=\sum\limits_{i=1}^{d}V_{\Lambda^{i}}^{\\{i\\}}\left(f\right)<\infty.$
In the case $\Lambda^{1}=\cdots=\Lambda^{d}=\Lambda$ we denote
$BV_{\Lambda}:=BV_{\Lambda^{1},...,\Lambda^{d}},\quad
CV_{\Lambda}:=CV_{\Lambda^{1},...,\Lambda^{d}},\quad
PBV_{\Lambda}:=PBV_{\Lambda^{1},...,\Lambda^{d}}$
and
$CV_{\Lambda}:=V_{\Lambda^{1},...,\Lambda^{d}}(f)CV_{\Lambda},\qquad
PV_{\Lambda}(f):=PV_{\Lambda^{1},...,\Lambda^{d}}(f).$
If $\lambda_{n}\equiv 1$ (or if $0<c<\lambda_{n}<C<\infty,\ n=1,2,\ldots$) the
classes $BV_{\Lambda}$ and $PBV_{\Lambda}$ coincide with the Hardy class $BV$
and $PBV$ respectively. Hence it is reasonable to assume that
$\lambda_{n}\rightarrow\infty$ and since the intervals in $E=\\{I_{i}\\}$ are
ordered arbitrarily, we suppose, without loss of generality, that the sequence
$\\{\lambda_{n}\\}$ is increasing. Thus,
(1)
$1<\lambda_{1}\leq\lambda_{2}\leq\ldots,\qquad\lim_{n\rightarrow\infty}\lambda_{n}=\infty.$
When $\lambda_{n}=n$ for all $n=1,2\ldots$ we say Harmonic Variation instead
of $\Lambda$-variation and write $H$ instead of $\Lambda$ ($BV_{H}$,
$PBV_{H}$, $CV_{H}$, ets).
###### Remark 1.
The notion of $\Lambda$-variation was introduced by Waterman [14] in one
dimensional case, by Sahakian [13] in two dimensional case and by Sablin [12]
in the case of higher dimensions. The notion of bounded partial variation
(class $PBV$) was introduced by Goginava in [6, 7]. These classes of functions
of generalized bounded variation play an important role in the theory Fourier
series.
Observe, that the number of variations in Definition 1 of total variation is
$2^{d}-1$, while the number of variations in Definition 2 of partial variation
is only $d$.
The statements of the following theorem are known.
###### Theorem A.
1) (Dragoshanski [5]) If $d=2$, then $BV_{H}=CV_{H}$.
2) (Bakhvalov [1]) $CV_{H}=\bigcup_{\Gamma}BV_{\Gamma}$ for any $d$, where the
union is taken over all sequences $\Gamma=\\{\gamma_{n}\\}_{n=1}^{\infty}$
with $\gamma_{n}=o(n)$ as $n\rightarrow\infty$.
3) (Goginava, Sahakian [8]) If $d=2$, then $PBV_{\Lambda}\subset BV_{H}$,
provided that
(2) $\frac{\lambda_{n}}{n}\downarrow
0\quad\mathrm{{and}\quad\sum\limits_{n=1}^{\infty}\frac{\lambda_{n}}{n^{2}}<\infty,}$
Using the third statement of Theorem A, we have proved in [8] the convergence
of double Fourier series of functions of any class $PBV_{\Lambda}$ with (2).
To obtain similar result for higher dimensions we need stronger result, since
the inclusion $PBV_{\Lambda}\subset BV_{H}$ is not enough in this case (see
next section for details).
###### Theorem 1.
Let $\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ and $d\geq 2$. If
(3) $\frac{\lambda_{n}}{n}\downarrow
0\quad\mathrm{{and}\quad\sum\limits_{n=1}^{\infty}\frac{\lambda_{n}\log^{d-2}n}{n^{2}}<\infty,}$
then there exists a sequence $\Gamma=\\{\gamma_{n}\\}_{n=1}^{\infty}$ with
(4) $\gamma_{n}=o(n)\quad\mathrm{as}\quad n\rightarrow\infty,$
such that $PBV_{\Lambda}\subset BV_{\Gamma}$.
###### Proof.
Choosing the sequence $\\{A_{n}\\}_{n=1}^{\infty}$ such that
(5) $A_{n}\uparrow\infty,\qquad\frac{\lambda_{n}A_{n}}{n}\downarrow
0,\qquad\sum\limits_{n=1}^{\infty}\frac{\lambda_{n}\log^{d-2}nA_{n}^{d}}{n^{2}}<\infty,$
we set
(6) $\gamma_{n}=\frac{n}{A_{n}},\qquad n=1,2,\ldots$
We prove that there is a constant $C>0$ such that
(7)
$\sum_{i_{1},\ldots,i_{p}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{p}}^{p},x^{\widetilde{\alpha}}\right)\right|}{\gamma_{i_{1}}\cdots\gamma_{i_{p}}}<C\cdot
PV_{\Lambda}(f),$
for any $f\in PBV_{\Lambda}$ and $\alpha:=\\{i_{1},...,i_{p}\\}\subset D$,
${\\{I_{i_{j}}^{j}\\}_{i_{j}=1}^{k_{j}}\in\Omega}$.
To prove (7) observe, that
(8)
$\displaystyle\sum_{i_{1},\ldots,\,i_{p}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{p}}^{p},x^{\widetilde{\alpha}}\right)\right|}{\gamma_{i_{1}}\cdots\gamma_{i_{p}}}$
$\displaystyle=\sum_{\sigma}\sum_{i_{\sigma(1)}\leq\cdots\leq
i_{\sigma(p)}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{p}}^{p},x^{\widetilde{\alpha}}\right)\right|}{\gamma_{i_{1}}\cdots\gamma_{i_{p}}}<\infty,$
where the sum is taken over all rearrangements
$\sigma=\\{\sigma(k)\\}_{k=1}^{p}$ of the set $\\{1,2,\ldots,p\\}$.
Denoting $M=PV_{\Lambda}(f)$ and using (6), (5) and (3) we obtain:
$\displaystyle\sum\limits_{i_{1}\leq i_{2}\leq\cdots\leq
i_{p}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{p}}^{p},x^{\widetilde{\alpha}}\right)\right|}{\gamma_{i_{1}}\cdots\gamma_{i_{p}}}$
$\displaystyle=$ $\displaystyle\sum\limits_{i_{1}\leq i_{2}\leq\cdots\leq
i_{p-1}}\frac{A_{i_{1}}\cdots
A_{i_{p-1}}}{{i_{1}}\cdots{i_{p-1}}}\sum\limits_{i_{p}\geq
i_{p-1}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{p}}^{p},x^{\widetilde{\alpha}}\right)\right|}{\lambda_{i_{p}}}\cdot\frac{\lambda_{i_{p}}A_{i_{p}}}{i_{p}}$
$\displaystyle\leq$ $\displaystyle M\sum\limits_{i_{1}\leq i_{2}\leq\cdots\leq
i_{p-1}}\frac{A_{i_{p-1}}^{p}\lambda_{i_{p-1}}}{i_{p-1}^{2}}\cdot\frac{1}{{i_{1}}\cdots
i_{p-2}}$ $\displaystyle=$ $\displaystyle
M\sum\limits_{i_{i_{p}-1}=1}^{\infty}\frac{A_{i_{p-1}}^{p}\lambda_{i_{p-1}}}{i_{p-1}^{2}}\sum\limits_{i_{p-2}=1}^{i_{p-1}}\frac{1}{i_{p-2}}\sum\limits_{i_{p-3}=1}^{i_{p-2}}\frac{1}{i_{p-3}}\cdots\sum\limits_{i_{1}=1}^{i_{2}}\frac{1}{{i_{1}}}$
$\displaystyle\leq$ $\displaystyle
M\sum\limits_{i_{p-1}=1}^{\infty}\frac{A_{i_{p-1}}^{p}\lambda_{i_{p-1}}}{i_{p-1}^{2}}\left(\sum\limits_{i_{=}1}^{i_{p-1}}\frac{1}{i}\right)^{p-2}\leq
C\cdot
M\sum\limits_{n=1}^{\infty}\frac{A_{n}^{p}\lambda_{n}\log^{d-2}n}{n^{2}}<\infty.$
Similarly we can prove that all other summands in the right hind side of (8)
are finite. Theorem 1 is proved. ∎
In view of Theorem A, Theorem 1 implies
###### Corollary 1.
If the sequence $\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ satisfies (3), then
$PBV_{\Lambda}\subset CV_{H}$.
###### Definition 4.
The partial modulus of variation $v_{i}\left(n,f\right)$, $i=1,\ldots,d$ of a
function $f$ are defined by
$v_{i}\left(n,f\right):=\sup\limits_{x^{\beta}}\sup\limits_{\\{I_{j}\\}\in\Omega_{n}}\sum\limits_{j=1}^{n}\left|f\left(I_{j},x^{\beta}\right)\right|,\quad\beta=D\setminus\\{i\\},\quad
n=1,2,\ldots.$
For functions of one variable the concept of modulus of variation was
introduced by Chanturia [2].
###### Theorem 2.
Let $f$ be defined on $T^{d}$ and
(9)
$\sum\limits_{j=1}^{\infty}\frac{\sqrt[d]{v_{i}\left(2^{j},f\right)}}{2^{j/d}}<\infty,\qquad
i=1,...,d.$
Then there exists a sequence $\Delta=\\{\delta_{n}\\}_{n=1}^{\infty}$ with
$\delta_{n}=o\left(n\right)\quad\mathrm{as}\quad n\to\infty,$
such that $f\in BV_{\Delta}.$
###### Proof.
We use induction on dimension $d$. We have proved in [8], that in the case
$d=2$ the condition (9) implies $f\in BV_{H}$, which combined with Theorem A
proves Theorem 2 for $d=2$.
Supposing Theorem 2 is true if the dimension is less than $d$, we prove it for
the dimension $d>2$.
According to induction hypothesis it is enough to prove that there exists a
sequence $\delta_{n}=o(n)$ such that
$\sup_{\\{I_{i_{j}}^{j}\\}_{i_{j}=1}^{k_{j}}\in\Omega}\
\sum_{i_{1},\ldots,i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{\delta_{i_{1}}\cdots\delta_{i_{d}}}<\infty.$
Let the sequence $\\{B_{2^{j}}\\}_{j=1}^{\infty}$ be chosen so that
$B_{2^{j}}\uparrow\infty,\qquad\sum\limits_{j=1}^{\infty}\frac{B_{2^{j}}\sqrt[d]{v_{i}\left(2^{j},f\right)}}{2^{j/d}}<\infty,\qquad
i=1,...,d.$
Defining
$B_{n}=B_{2^{N}},\quad\mathrm{for}\quad 2^{N}\leq n<2^{N+1},\qquad
N=0,1,.....,$
we set
(10) $\delta_{n}=\frac{n}{B_{n}},\qquad n=1,2\ldots$
Then we can write
$\displaystyle\sum_{i_{1},\ldots,i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{\delta_{i_{1}}\cdots\delta_{i_{d}}}$
$\displaystyle=$ $\displaystyle\sum_{i_{1},\ldots,i_{d}}B_{i_{1}}\cdots
B_{i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{i_{1}\cdots i_{d}}$ $\displaystyle=$
$\displaystyle\sum\limits_{r_{1}=0}^{\infty}\cdots\sum\limits_{r_{d}=0}^{\infty}\sum\limits_{i_{1}=2^{r_{1}}}^{2^{r_{1}+1}-1}\cdots\sum\limits_{i_{d}=2^{r_{d}}}^{2^{r_{d}+1}-1}B_{i_{1}}\cdots
B_{i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{i_{1}\cdots i_{d}}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{r_{1}=0}^{\infty}\cdots\sum\limits_{r_{d}=0}^{\infty}\frac{B_{2^{r_{1}}}}{2^{r_{1}}}\cdots\frac{B_{2^{r_{d}}}}{2^{r_{d}}}\sum\limits_{i_{1}=2^{r_{1}}}^{2^{r_{1}+1}-1}\cdots\sum\limits_{i_{d}=2^{r_{d}}}^{2^{r_{d}+1}-1}\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|.$
It is easy to show that
$\displaystyle\sum\limits_{i_{1}=2^{r_{1}}}^{2^{r_{1}+1}-1}\cdots\sum\limits_{i_{d}=2^{r_{d}}}^{2^{r_{d}+1}-1}\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|$ $\displaystyle\leq$ $\displaystyle
c\left(d\right)\prod\limits_{i\in\beta}2^{r_{i}}\sup\limits_{x^{\beta}}\sup\limits_{\\{I_{i_{k}}^{k}\\}\in\Omega_{2^{r_{k}}}}\sum\limits_{i_{k}=2^{r_{k}}}^{2^{r_{k}+1}-1}\left|f\left(I_{i_{k}}^{k},x^{\beta}\right)\right|,$
where $\beta:=D\setminus\\{k\\}$, $k=1,...,d$.
Consequently,
$\displaystyle\sum\limits_{i_{1}=2^{r_{1}}}^{2^{r_{1}+1}-1}\cdots\sum\limits_{i_{d}=2^{r_{d}}}^{2^{r_{d}+1}-1}\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|$ $\displaystyle=$
$\displaystyle\left[\left(\sum\limits_{i_{1}=2^{r_{1}}}^{2^{r_{1}+1}-1}\cdots\sum\limits_{i_{d}=2^{r_{d}}}^{2^{r_{d}+1}-1}\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|\right)^{1/d}\right]^{d}$ $\displaystyle\leq$
$\displaystyle
c\left(d\right)\prod\limits_{k=1}^{d}2^{r_{k}\left(1-1/d\right)}\left(\sup\limits_{x^{\beta}}\sup\limits_{\\{I_{i_{k}}^{k}\\}\in\Omega_{2^{r_{k}}}}\sum\limits_{i_{k}=2^{r_{k}}}^{2^{r_{k}+1}-1}\left|f\left(I_{i_{k}}^{k},x^{\beta}\right)\right|\right)^{1/d}$
$\displaystyle=$ $\displaystyle
c\left(d\right)\prod\limits_{k=1}^{d}2^{r_{k}\left(1-1/d\right)}\sqrt[d]{v_{k}\left(2^{r_{k}},f\right)}.$
Combining (1) and (1) we obtain
$\displaystyle\sum_{i_{1},\ldots,i_{d}}\frac{\left|f\left(I_{i_{1}}^{1}\times\cdots\times
I_{i_{d}}^{d}\right)\right|}{\delta_{i_{1}}\cdots\delta_{i_{d}}}$
$\displaystyle\leq$ $\displaystyle
c\left(d\right)\sum\limits_{r_{1}=0}^{\infty}\cdots\sum\limits_{r_{d}=0}^{\infty}\frac{B_{2^{r_{1}}}v_{1}\left(2^{r_{1}},f\right)}{2^{r_{1}/d}}\cdots\frac{B_{2^{r_{d}}}v_{d}\left(2^{r_{d}},f\right)}{2^{r_{d}}/d}<\infty.$
Theorem 2 is proved. ∎
## 2\. Convergence of multiple Fourier series
The Fourier series of function $f\in L^{1}\left(T^{d}\right)$ with respect to
the trigonometric system is the series
$S\left[f\right]:=\sum_{n_{1},...,n_{d}=-\infty}^{+\infty}\widehat{f}\left(n_{1},....,n_{d}\right)e^{i\left(n_{1}x+\cdots+n_{d}x_{d}\right)},$
where
$\widehat{f}\left(n_{1},....,n_{d}\right)=\frac{1}{\left(2\pi\right)^{d}}\int_{T^{d}}f(x^{1},...,x^{d})e^{-i\left(n_{1}x^{1}+\cdots+n_{d}x^{d}\right)}dx^{1}\cdots
dx^{d}$
are the Fourier coefficients of $f$. The rectangular partial sums are defined
as follows:
$\displaystyle S_{N_{1},...,N_{d}}\left(f;x^{1},...,x^{d}\right)$
$\displaystyle:=\sum_{n_{1}=-N_{1}}^{N_{1}}\cdots\sum_{n_{d}=-N_{d}}^{N_{d}}\widehat{f}\left(n_{1},....,n_{d}\right)e^{i\left(n_{1}x^{1}+\cdots+n_{d}x^{d}\right)}$
$\displaystyle=\frac{1}{\pi^{d}}\int\limits_{T^{d}}f\left(x_{1},\cdots,x_{d}\right)\prod\limits_{s=1}^{d}D_{N_{s}}\left(x_{s}\right)dx_{1}\cdots
dx_{d},$
where $D_{N}(t)=\frac{sin\left(N+\frac{1}{2}\right)t}{2sin\frac{t}{2}}$ is the
Dirichlet kernel.
In this paper we consider convergence of only rectangular partial sums
(convergence in the sense of Pringsheim) of $d$-dimensional Fourier series.
We denote by $C(T^{d})$ the space of continuous and $2\pi$-periodic with
respect to each variable functions with the norm
$\|f\|_{C}:=\sup_{\left(x^{1},\ldots,\,x^{d}\right)\in
T^{d}}|f(x^{1},\ldots,x^{d})|.$
We say that the point $x:=\left(x^{1},\ldots,x^{d}\right)$ is a regular point
of function $f$ if the following limits exist
$f\left(x^{1}\pm 0,...,x^{d}\pm
0\right):=\lim\limits_{t^{1},\ldots,\,t^{d}\downarrow 0}f\left(x^{1}\pm
t^{1},\ldots,x^{d}\pm t^{d}\right).$
For the regular point $x:=\left(x^{1},\ldots,x^{d}\right)$ we denote
(13) $f^{\ast}\left(x^{1},\ldots,x^{d}\right):=\frac{1}{2^{d}}\sum
f\left(x^{1}\pm 0,\ldots,x^{d}\pm 0\right).$
###### Definition 5.
We say that the class of functions $V\subset L^{1}(T^{d})$ is a class of
convergence on $T^{d}$, if for any function $f\in V$
1) the Fourier series of $f$ converges to $f^{\ast}({x})$ at any regular point
${x}\in T^{d}$,
2) the convergence is uniform on any compact $K\subset T^{d}$, if $f$ is
continuous on the neighborhood of $K$.
The well known Dirichlet-Jordan theorem (see [16]) states that the Fourier
series of a function $f(x),\ x\in T$ of bounded variation converges at every
point $x$ to the value $\left[f\left(x+0\right)+f\left(x-0\right)\right]/2$.
If $f$ is in addition continuous on $T$, the Fourier series converges
uniformly on $T$.
Hardy [9] generalized the Dirichlet-Jordan theorem to the double Fourier
series and proved that $BV$ is a class of convergence on $T^{2}$.
The following theorem was proved by Waterman (for $d=1$) and Sahakian (for
$d=2$).
###### Theorem WS (Waterman [14], Sahakian [13]).
If $d=1$ or $d=2$, then the class $BV_{H}$ is a class of convergence on
$T^{d}$.
In [1] Bakhvalov showed that the class $BV_{H}$ is not a class of convergence
on $T^{d}$, if $d>2$. On the other hand, he proved the following
###### Theorem B (Bakhvalov [1]).
The class $CV_{H}$ is a class of convergence on $T^{d}$ for any $d=1,2,\ldots$
Convergence of spherical and other partial sums of double Fourier series of
functions of bounded $\Lambda$-variation was investigated in deatails by
Dyachenko [3, 4].
The main result of this paper is the following theorem, that we have proved in
[8] for $d=2$.
###### Theorem 3.
Let $\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ and $d\geq 2$.
a) If
(14) $\sum\limits_{n=1}^{\infty}\frac{\lambda_{n}\log^{d-2}n}{n^{2}}<\infty,$
then $PBV_{\Lambda}$ is a class of convergence on $T^{d}$.
b) If
(15)
$\frac{\lambda_{n}}{n}=O\left(\frac{\lambda_{[n^{\delta}]}}{[n^{\delta}]}\right)$
for some $\delta>1$, and
(16) $\sum\limits_{n=1}^{\infty}\frac{\lambda_{n}\log^{d-2}n}{n^{2}}=\infty,$
then there exists a continuous function $f\in PBV_{\Lambda}$, the Fourier
series of which diverges at $\left(0,\ldots,0\right).$
###### Proof of Theorem 2.
Part a) immediately follows from Corollary 1 and Theorem B.
To prove part b) we denote
$A_{i_{1},\ldots,i_{d}}:=\left[\frac{\pi
i_{1}}{N+1/2},\frac{\pi\left(i_{1}+1\right)}{N+1/2}\right)\times\cdots\times\left[\frac{\pi
i_{d}}{N+1/2},\frac{\pi\left(i_{d}+1\right)}{N+1/2}\right),$
$W:=\left\\{(i_{1},\ldots,i_{d}):i_{d}<i_{s}<i_{d}+m_{i_{d}},\ 1\leq s<d,\
1\leq i_{d}\leq N_{\delta}\right\\},\quad$
$N_{\delta}=\left[\left(\frac{N}{2}\right)^{\frac{1}{\delta}}\right],\qquad
t_{j}:=\left(\sum\limits_{i=1}^{m_{j}}\frac{1}{\lambda_{i}}\right)^{-1},\qquad
m_{j}:=\left[j^{\delta}\right],$
where $[x]$ is the integer part of $x$.
It is not hard to see, that for any sequence $\Lambda=\\{\lambda_{n}\\}$
satisfying (1) the class $C(T^{d})\cap PBV_{\Lambda}$ is a Banach space with
the norm
$\|f\|_{PBV_{\Lambda}}:=\|f\|_{C}+PV_{\Lambda}(f).$
Consider the following function
$f_{N}\left(x_{1},\ldots,x_{d}\right):=\sum\limits_{\left(i_{1},\ldots,\,i_{d}\right)\in
W}t_{i_{d}}1_{A_{i_{1},\ldots,i_{d}}}\left(x_{1},\ldots,x_{d}\right)\prod\limits_{s=1}^{d}\sin\left(N+1/2\right)x_{s},$
where $1_{A}\left(x_{1},\ldots,x_{d}\right)$ is the characteristic function of
the set $A\subset T^{d}$.
Let $\left(i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{d}\right)$ be fixed
$(k=1,\ldots,d-1)$. Then it is easy to show that
$V_{\Lambda}^{{k}}\left(f_{N}\right)\leq C\cdot
t_{i_{d}}\left(\sum\limits_{i_{k}=i_{d}+1}^{i_{d}+m_{i_{d}}}\frac{1}{\lambda_{i_{k}-i_{d}}}\right)\leq
C\cdot
t_{i_{d}}\left(\sum\limits_{i_{k}=1}^{m_{i_{d}}}\frac{1}{\lambda_{i_{k}}}\right)\leq
C<\infty.$
If $\left(i_{1},\ldots,i_{d-1}\right)$ is fixed, the condition
$\left(i_{1},\ldots,i_{d}\right)\in W$ implies
$\max\left\\{i_{d}\left(i_{s}\right):1\leq s\leq
d-1\right\\}<i_{d}<\min\left\\{i_{s}:1\leq s\leq d-1\right\\},$
where
$i_{d}\left(i_{s}\right):=\min\left\\{i_{d}:i_{d}+m_{i_{d}}>i_{s}\right\\}.$
Consequently, by the definition of the function $f_{N}$ we obtain that for any
$s=1,...,d-1$
$\displaystyle V_{\Lambda}^{{d}}\left(f_{N}\right)$ $\displaystyle\leq$
$\displaystyle
C\sum\limits_{i_{d}=i_{d}\left(i_{s}\right)+1}^{i_{s}}\frac{t_{i_{d}}}{\lambda_{i_{d}-i_{d}\left(i_{s}\right)}}$
$\displaystyle\leq$ $\displaystyle C\cdot
t_{i_{d}\left(i_{s}\right)}\sum\limits_{i_{d}=i_{d}\left(i_{s}\right)+1}^{i_{s}}\frac{1}{\lambda_{i_{d}-i_{d}\left(i_{s}\right)}}$
$\displaystyle=$ $\displaystyle C\cdot
t_{i_{d}\left(i_{s}\right)}\sum\limits_{i_{d}=1}^{i_{s}-i_{d}\left(i_{s}\right)}\frac{1}{\lambda_{i_{d}}}\leq
C\cdot
t_{i_{d}\left(i_{s}\right)}\sum\limits_{i_{d}=1}^{m_{i_{d}\left(i_{s}\right)}}\frac{1}{\lambda_{i_{d}}}=C<\infty.$
Hence $f_{N}\in PBV_{\Lambda}$ and
(17) $\|f_{N}\|_{PV_{\Lambda}}\leq C,\quad N=1,2,\ldots.$
Observe, that by (15) we have
$\frac{1}{t_{j}}=\sum_{i=1}^{m_{j}}\frac{1}{\lambda_{i}}=\sum_{i=1}^{m_{j}}\frac{1}{i}\cdot\frac{i}{\lambda_{i}}\leq
C\frac{m_{j}}{\lambda_{m_{j}}}\log m_{j}\leq C\frac{j\log j}{\lambda_{j}}.$
Hence
$t_{j}\log j\geq c\frac{\lambda_{j}}{j}.$
Consequently,
$\displaystyle\pi^{d}S_{N,\cdots,N}\left(f_{N};0,\cdots,0\right)$
$\displaystyle=$
$\displaystyle\int\limits_{T^{d}}f_{N}\left(x^{1},\cdots,x^{d}\right)\prod\limits_{s=1}^{d}D_{N}\left(x^{s}\right)dx^{1}\cdots
dx^{d}$ $\displaystyle=$
$\displaystyle\sum\limits_{\left(i_{1},\cdots,i_{d}\right)\in
W}t_{i_{d}}\int\limits_{A_{i_{1},\cdots,\,i_{d}}}\prod\limits_{s=1}^{d}\frac{\sin^{2}\left(N+1/2\right)x^{s}}{2\sin\left(x^{s}/2\right)}dx^{1}\cdots
dx^{d}$ $\displaystyle\geq$ $\displaystyle
c\sum\limits_{\left(i_{1},\cdots,\,i_{d}\right)\in
W}t_{i_{d}}\frac{1}{i_{1}\cdots i_{d}}$ $\displaystyle\geq$ $\displaystyle
c\sum\limits_{i_{d}=1}^{N_{\delta}}\frac{t_{i_{d}}}{i_{d}}\sum\limits_{i_{1}=i_{d}}^{i_{d}+m_{i_{d}}}\cdots\sum\limits_{i_{d-1}=i_{d}}^{i_{d}+m_{i_{d}}}\frac{1}{i_{1}\cdots
i_{d-1}}$ $\displaystyle\geq$ $\displaystyle
c\sum\limits_{i_{d}=1}^{N_{\delta}}\frac{t_{i_{d}}}{i_{d}}\log^{d-1}\left(\frac{i_{d}+m_{i_{d}}}{i_{d}}\right)$
$\displaystyle\geq$ $\displaystyle
c(\delta-1)^{d-1}\sum\limits_{i_{d}=1}^{N_{\delta}}\frac{t_{i_{d}}\log
i_{d}}{i_{d}}\log^{d-2}i_{d}$ $\displaystyle\geq$ $\displaystyle
c(\delta-1)^{d-1}\sum\limits_{n=1}^{N_{\delta}}\frac{\lambda_{n}\log^{d-2}n}{n^{2}}\rightarrow\infty,$
as $N\rightarrow\infty$, according to (16).
By Banach-Steinhaus Theorem, (17) and (2) imply the existence of a continuous
function $f\in PBV_{\Lambda}$ such that
$\sup_{N}|S_{N,\cdots,N}[f,(0,\cdots,0)]|=\infty.$
∎
###### Corollary 2.
a) If $\Lambda=\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ with
$\lambda_{n}=\frac{n}{\log^{d-1+\varepsilon}n},\qquad n=1,2,\ldots$
for some $\varepsilon>0$, then the class $PBV_{\Lambda}$ is a class of
convergence on $T^{d}$.
b) If $\Lambda=\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ with
$\lambda_{n}=\frac{n}{\log^{d-1}n},\qquad n=1,2,\ldots,$
then the class $PBV_{\Lambda}$ is not a class of convergence on $T^{d}$.
The second part of Theorem 2 and Corollary 1 imply
###### Corollary 3.
If the sequence $\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ satisfies (15) and
(16), then $PBV_{\Lambda}\not\subset CV_{H}$.
Theorem 2 and Theorems A and B imply
###### Theorem 4.
The set of functions
$\left\\{f:\sum\limits_{j=0}^{\infty}\frac{\sqrt[d]{v_{i}\left(2^{j},f\right)}}{2^{j/d}}<\infty,\
i=1,...,d\right\\}$
is a class of convergence on $T^{d}$.
###### Corollary 4.
The set of functions
$\left\\{f:v_{i}\left(n,f\right)=O\left(n^{\alpha}\right),\
i=1,...,d\right\\}$
is a class of convergence on $T^{d}$ for any $\alpha\in(0,1)$.
## References
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|
arxiv-papers
| 2012-10-16T14:48:29 |
2024-09-04T02:49:36.637916
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ushangi Goginava and Artur Sahakian",
"submitter": "Ushangi Goginava",
"url": "https://arxiv.org/abs/1210.4440"
}
|
1210.4492
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-302 LHCb-PAPER-2012-021 October 15, 2012
Measurement of the $C\\!P$ asymmetry in $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
A measurement of the $C\\!P$ asymmetry in $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays is presented, based on 1.0$\mbox{\,fb}^{-1}$ of
$pp$ collision data recorded by the LHCb experiment during 2011. The
measurement is performed in six bins of invariant mass squared of the
$\mu^{+}\mu^{-}$ pair, excluding the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\psi{(2S)}$ resonance regions. Production and detection
asymmetries are removed using the $B^{0}\rightarrow J/\psi K^{*0}$ decay as a
control mode. The integrated $C\\!P$ asymmetry is found to be $-0.072\pm
0.040\,(\mbox{stat.})\pm 0.005\,(\mbox{syst.})$, consistent with the Standard
Model.
Submitted to Physical Review Letters
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, V.
Balagura36,28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S.
Belogurov28, K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson47, J. Benton43, A. Berezhnoy29, R. Bernet37, M.-O.
Bettler44, M. van Beuzekom38, A. Bien11, S. Bifani12, T. Bird51, A.
Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50, J.
Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14,c, G. Carboni21,k, R.
Cardinale19,i, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49, M.
Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, P. Chen3,36, N.
Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-Montells33, A.
Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36,
D. Craik45, S. Cunliffe50, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14, O. Deschamps5, F.
Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch. Elsasser37, D.
Elsby42, D. Esperante Pereira34, A. Falabella14,e, C. Färber11, G. Fardell47,
C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F. Ferreira
Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick35, M. Fontana10,
F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, C. Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T.
Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O. Grünberg55, B. Gui53,
E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C.
Haen35, S.C. Haines44, S. Hall50, T. Hampson43, S. Hansmann-Menzemer11, N.
Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, R.S. Huston12, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, M. Karbach35, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20,
Y.M. Kim47, O. Kochebina7, I. Komarov29, V. Komarov36, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C.
Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N.
Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J. McCarthy42,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, D.A.
Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51, P.
Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R. Muresan26,
B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T. Nakada36, R.
Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D. Nguyen36, C.
Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
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Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
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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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25National Center for Nuclear Research (NCBJ), Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
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
oHanoi University of Science, Hanoi, Viet Nam
pMassachusetts Institute of Technology, Cambridge, MA, United States
The decay $B^{0}\rightarrow K^{*0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ is
a flavour changing neutral current process which proceeds via electroweak loop
and box diagrams in the Standard Model (SM) [1]. The decay is highly
suppressed in the SM and therefore physics beyond the SM such as supersymmetry
[2] can contribute with a comparable amplitude via gluino or chargino loop
diagrams. A number of observables are sensitive to such contributions,
including the partial rate of the decay, the $\mu^{+}\mu^{-}$ forward-backward
asymmetry ($A_{\mathrm{FB}}$) and the $C\\!P$ asymmetry (${\cal A}_{C\\!P}$).
The $C\\!P$ asymmetry for $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ is defined
as
${\cal A}_{C\\!P}=\frac{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-})-\Gamma(B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-})}{\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-})+\Gamma(B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-})}$ (1)
where $\Gamma$ is the decay rate and the initial flavour of the $B$ meson is
tagged by the charge of the kaon from the $K^{*}$ decay. The $C\\!P$ asymmetry
is predicted to be of the order $10^{-3}$ in the SM [3, 4], but is sensitive
to physics beyond the SM that changes the operator basis by modifying the
mixture of the vector and axial-vector components [5, 6]. Some models that
include new phenomena enhance the observed $C\\!P$ asymmetry up to $\pm 0.15$
[7]. The theoretical prediction within a given model has a small error as the
form factor uncertainties, which are the dominant theoretical errors for the
decay rate, cancel in the ratio.
The $C\\!P$ asymmetry in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays has
previously been measured by the Belle [8] and BaBar [9] collaborations, with
both results consistent with the SM. The LHCb collaboration has recently
demonstrated its potential in this area with the most precise measurement of
$A_{\mathrm{FB}}$ [10], and in this Letter, the measurement of the $C\\!P$
asymmetry by LHCb is presented.
The LHCb detector [11] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers. The trigger consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage which makes use of a full event reconstruction.
The simulated events used in this analysis are produced using the Pythia 6.4
generator [12], with a choice of parameters specifically configured for LHCb
[13]. The EvtGen package [14] describes the decay of the particles and the
Geant4 toolkit [15, *Agostinelli:2002hh] simulates the detector response,
implemented as described in Ref. [17]. QED radiative corrections are generated
with the Photos package [18].
The events used in the analysis are selected by a dedicated muon hardware
trigger and then by one or more of a set of different muon and topological
software triggers [19, 20]. The hardware trigger requires the muons have
$p_{\rm T}$ greater than 1.48${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the
software trigger requires that one of the final state particles to have both
$\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP with
respect to all $pp$ interaction vertices $>100\,\upmu\rm m$ [20]. Triggered
candidates are subject to the same two-stage selection as that used in Ref.
[10]. The first stage is a cut-based selection, which includes requirements on
the $B^{0}$ candidate’s vertex fit $\chi^{2}$, flight distance and invariant
mass, and each track’s impact parameters with respect to any interaction
vertex, $p_{\rm T}$ and polar angle. Background from misidentified kaon and
pion tracks is removed using information from the particle identification
(PID) system, and muon tracks are required to have hits in the muon system.
Finally, the production vertex of the $B^{0}$ candidate must lie within 5 mm
of the beam axis in the transverse directions, and within 200 mm of the
average interaction position in the beam ($z$) direction.
In the second stage, the candidates must pass a multivariate selection that
uses a boosted decision tree (BDT) [21] that implements the AdaBoost algorithm
[22]. This is a tighter selection which takes into account other variables
including the decay time and flight direction of the $B^{0}$ candidates, the
$p_{\rm T}$ of the hadrons, measures of the track and vertex quality, and PID
information for the daughter tracks. For the rest of the Letter, the inclusion
of charge conjugate modes is implied unless explicitly stated.
In order to obtain a clean sample of $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$
decays, the $c\bar{c}$ resonant decays $B^{0}\rightarrow J/\psi K^{*0}$ and
$B^{0}\rightarrow\psi{(2S)}K^{*0}$ are removed by excluding events with
$\mu^{+}\mu^{-}$ invariant mass, $m_{\mu^{+}\mu^{-}}$, satisfying
$2.95<m_{\mu^{+}\mu^{-}}<3.18{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ or
$3.59<m_{\mu^{+}\mu^{-}}<3.77{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. If
$m_{K^{+}\pi^{-}\mu^{+}\mu^{-}}<5.23{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
then the vetoes are extended downwards by
$0.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to remove the radiative tails
of the resonances. Backgrounds involving misidentified particles are vetoed
using cuts on the masses of the $B^{0}$ and $K^{*0}$ mesons and the
$\mu^{+}\mu^{-}$ pair, as well as using the PID information for the daughter
particles. These include $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ candidates
in which a kaon has been misidentified as a pion, $B^{0}\rightarrow J/\psi
K^{*0}$ candidates where a hadron is swapped with a muon and $B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates which combine with a random low momentum pion.
The vetoes are described fully in Ref. [10]. ${\cal A}_{C\\!P}$ may be diluted
by $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates with the kaon and pion
misidentified as each other, which is estimated as 0.8% of the
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ yield using simulated events. All
$B^{0}$ candidates must have a mass in the range
$5.15-5.80{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the tight low mass edge
of this window serves to remove background from partially reconstructed $B$
meson decays. All $K^{*0}$ candidates must have an invariant mass of the kaon-
pion pair within $0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ of the nominal
$K^{*0}$($892$) mass. A proton veto, using PID information from a neural
network, is also applied to remove background from $\Lambda_{b}$ decays, where
a proton in the final state is misidentified as a kaon or pion in the
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay.
Approximately 2% of selected events contain two $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ candidates which have tracks in common. The majority of
these candidates arise from swapping the assignment of the kaon and pion
hypothesis. As the charges of the kaon and pion tag the flavour of the $B$
meson these duplicate candidates can bias the measured value of ${\cal
A}_{C\\!P}$. This is accounted for by randomly removing one of the two
candidates from the sample. This process is repeated many times over the full
sample with a different random seed in each case and the average measured
value of ${\cal A}_{C\\!P}$ is taken as the result.
An accurate measurement of ${\cal A}_{C\\!P}$ requires that the differences in
the production rates ($R$) of $B^{0}$/$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons and detection efficiencies
($\epsilon$) between the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ modes be accounted
for. Assuming all asymmetries are small the raw measured asymmetry may be
expressed as
$\mathcal{A}_{\textrm{RAW}}={\cal
A}_{C\\!P}+\kappa\mathcal{A}_{\textrm{P}}+\mathcal{A}_{\textrm{D}},$ (2)
where the production asymmetry, which is of the order of 1% [23], is defined
as ${\mathcal{A}_{\textrm{P}}\equiv\left[R\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\right)-R\left(B^{0}\right)\right]/\left[R\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\right)+R\left(B^{0}\right)\right]}$
and the detection asymmetry is
${\mathcal{A}_{\textrm{D}}\equiv\left[\epsilon\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\right)-\epsilon\left(B^{0}\right)\right]/\left[\epsilon\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\right)+\epsilon\left(B^{0}\right)\right]}$.
The production asymmetry is diluted through $B^{0}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ oscillations by a factor $\kappa$
$\kappa\equiv\frac{\int_{0}^{\infty}\epsilon(t)e^{-\Gamma t}\cos\Delta
mt\,\mathrm{d}t}{\int_{0}^{\infty}\epsilon(t)e^{-\Gamma t}\,\mathrm{d}t},$ (3)
where $t$, $\Gamma$, and $\Delta m$ are the decay time, mean decay rate, and
mass difference between the light and heavy eigenstates of the $B^{0}$ meson
respectively. The quantity $\mathcal{A}_{\textrm{D}}$ is dominated by the
$K^{+}\pi^{-}/K^{-}\pi^{+}$ detection asymmetry which arises due to left-right
asymmetries in the LHCb detector and different interactions of positively and
negatively charged tracks with the detector material. The left-right asymmetry
is cancelled by taking an average with equal weights of the $C\\!P$
asymmetries measured in two independent data samples with opposite polarities
of the LHCb dipole magnet. These data samples correspond to 61% and 39% of the
total data sample.
The production and interaction asymmetries are corrected for using the
$B^{0}\rightarrow J/\psi K^{*0}$ decay mode as a control channel. Since
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi K^{*0}$
decays have the same final state and similar kinematics, the measured raw
asymmetry for ${B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ decays may be simply expressed as
$\mathcal{A}_{\textrm{RAW}}\left(B^{0}\rightarrow J/\psi
K^{*0}\right)=\kappa\mathcal{A}_{\textrm{P}}+\mathcal{A}_{\textrm{D}}$, in the
absence of a $C\\!P$ asymmetry. $B^{0}\rightarrow J/\psi K^{*0}$ proceeds via
a $b\rightarrow c\overline{}cs$ transition, as does the decay mode
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, and hence
should have a $C\\!P$ asymmetry similar to ${\cal
A}_{C\\!P}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=(1\pm 7)\times 10^{-3}$ [24, 25]. For this analysis, it is
assumed that ${\cal A}_{C\\!P}(B^{0}\rightarrow J/\psi K^{*0})=0$. The $C\\!P$
asymmetry in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays is then calculated
as
${\cal A}_{C\\!P}=\mathcal{A}_{\textrm{RAW}}\left(B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}\right)-\mathcal{A}_{\textrm{RAW}}\left(B^{0}\rightarrow
J/\psi K^{*0}\right).$ (4)
Non-cancelling asymmetries due to differences between the kinematics of
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi K^{*0}$
decays are considered as systematic effects.
The full data sample, containing approximately 900 $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal decays, is split into the six bins of
$\mu^{+}\mu^{-}$ invariant mass squared ($q^{2}$) used by the LHCb, Belle, and
CDF angular analyses [10, 8, 26]. An additional bin of
$1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is used, to be
compared to the theoretical prediction in Ref. [4]. The $B^{0}\rightarrow
J/\psi K^{*0}$ data sample contains approximately 104000 signal decays with
$3.04<q^{2}<3.16{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The values of
${\cal A}_{C\\!P}$ are measured using a simultaneous unbinned maximum-
likelihood fit to the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and
$B^{0}\rightarrow J/\psi K^{*0}$ invariant mass distributions in the range
$5.15-5.80{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The simultaneous fit in
each $q^{2}$ bin spans eight data samples, split between the initial particles
$B^{0}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, the decay
modes $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi
K^{*0}$, and magnet polarity, where the $B^{0}\rightarrow J/\psi K^{*0}$
sample is common to all $q^{2}$ bins. This fit returns two values of ${\cal
A}_{C\\!P}$, one for each magnet polarity, and an average with equal weights
is made to find the value of ${\cal A}_{C\\!P}$ in that $q^{2}$ bin. An
integrated value of ${\cal A}_{C\\!P}$ over all $q^{2}$ is also calculated.
The signal invariant mass distributions for the $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi K^{*0}$ decays are modelled
using the sum of two Crystal Ball functions [27] with common peak and tail
parameters but different widths. The values of the tail parameters are
determined from fits to simulated events and fixed in the fit. Combinatorial
background arising from the random misassociation of tracks to form a $B^{0}$
candidate is modelled using an exponential function. The $B^{0}\rightarrow
J/\psi K^{*0}$ fit also accounts for a peaking $B^{0}_{s}\rightarrow
J/\psi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ contribution, which
has the same shape as the signal and an expected yield which is $(0.7\pm
0.2)\%$ of that of $B^{0}\rightarrow J/\psi K^{*0}$ [28]. In the simultaneous
fit, the signal shape is the same for the two modes, but the signal and
background yields and the exponential background parameter may vary. Figure 1
shows the mass fit to the $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay in the
full $q^{2}$ range.
Figure 1: Mass fits for $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays used to
extract the integrated $C\\!P$ asymmetry. The curves displayed are the full
mass fit (blue, solid), the signal peak (red, short-dashed), and the
background (grey, long-dashed). The mass fits on the top row correspond to the
(a) $B^{0}$ and (b) $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays
for one magnet polarity, while the bottom row shows the mass fits for (c)
$B^{0}$ and (d) $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ for the
reverse polarity.
Many sources of systematic uncertainty cancel in the difference of the raw
asymmetries between $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and
$B^{0}\rightarrow J/\psi K^{*0}$ decays and in the average of $C\\!P$
asymmetries measured using data recorded with opposite magnet polarities.
However, systematic uncertainties can arise from residual non-cancelling
asymmetries due to the different kinematic behaviour of $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi K^{*0}$ decays. The effect
is estimated by reweighting $B^{0}\rightarrow J/\psi K^{*0}$ candidates so
that their kinematic variables are distributed in the same way as for
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates. The value of
$\mathcal{A}_{\textrm{RAW}}\left(B^{0}\rightarrow J/\psi K^{*0}\right)$ is
then calculated for these reweighted events and the difference from the
default value is taken as the systematic uncertainty. This procedure is
carried out separately for a number of quantities including the $p$, $p_{\rm
T}$, and pseudorapidity of the $B^{0}$ and the $K^{*0}$ mesons. The total
systematic uncertainty associated to the different kinematic behaviour of the
two decays is calculated by adding each individual contribution in quadrature.
This is conservative, as many of the variables are correlated.
The random removal of multiple candidates discussed above also introduces a
systematic uncertainty on ${\cal A}_{C\\!P}$. The uncertainty on the mean
value of ${\cal A}_{C\\!P}$ from the ten different random removals is taken as
the systematic uncertainty.
The forward-backward asymmetry in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$
decays [10], which varies as a function of $q^{2}$, causes positive and
negative muons to have different momentum distributions. Different detection
efficiencies for positive and negative muons introduce an asymmetry that
cannot be accounted for by the $B^{0}\rightarrow J/\psi K^{*0}$ decay, which
does not have a comparable forward-backward asymmetry. The selection
efficiencies for positive and negative muons are evaluated using muons from
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay in data and the resulting
asymmetry in the selected $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ sample is
calculated in each $q^{2}$ bin.
A number of possible effects due to the choice of model for the mass fit are
considered. The signal model is replaced with a sum of two Gaussian
distributions and a possible difference in the mass resolution for
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ and $B^{0}\rightarrow J/\psi K^{*0}$
decays is investigated by allowing the width of the $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ signal peak to vary in a range of 0.7$-$1.3 times that
of the $B^{0}\rightarrow J/\psi K^{*0}$ model. As a further cross-check,
${\cal A}_{C\\!P}$ is calculated using a weighted average of the measurements
from the six $q^{2}$ bins and the result is found to be consistent with that
obtained from the integrated dataset.
Table 1: Systematic uncertainties on ${\cal A}_{C\\!P}$, from residual kinematic asymmetries, muon asymmetry, choice of signal model, and the modelling of the mass resolution, for each $q^{2}$ bin. The total uncertainty is calculated by adding the individual uncertainties in quadrature. | Sources of systematic uncertainties |
---|---|---
| multiple | residual | $\mu^{\pm}$ detection | signal | mass |
$q^{2}$ region (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | cands. | asymmetries | asymmetry | model | resol. | Total
$0.05<~{}$$q^{2}$ | $<2.00$ | $0.002$ | $0.007$ | $0.005$ | $0.005$ | $0.001$ | $0.010$
$2.00<~{}$$q^{2}$ | $<4.30$ | $0.006$ | $0.007$ | $0.006$ | $0.007$ | $0.010$ | $0.016$
$4.30<~{}$$q^{2}$ | $<8.68$ | $0.004$ | $0.003$ | $0.006$ | $0.004$ | $0.003$ | $0.010$
$10.09<~{}$$q^{2}$ | $<12.86$ | $0.003$ | $0.007$ | $0.009$ | $0.001$ | $0.002$ | $0.011$
$14.18<~{}$$q^{2}$ | $<16.00$ | $0.001$ | $0.006$ | $0.007$ | $0.001$ | $0.001$ | $0.009$
$16.00<~{}$$q^{2}$ | $<20.00$ | $0.003$ | $0.005$ | $0.003$ | $0.003$ | $0.009$ | $0.012$
$1.00<~{}$$q^{2}$ | $<6.00$ | $0.001$ | $0.006$ | $0.005$ | $0.002$ | $0.003$ | $0.009$
$0.05<~{}$$q^{2}$ | $<20.00$ | $0.002$ | $0.002$ | $0.005$ | $0.001$ | $0.001$ | $0.005$
The results of the full ${\cal A}_{C\\!P}$ fit are presented in Table 2 and
Figure 2. The raw asymmetry in $B^{0}\rightarrow J/\psi K^{*0}$ decays is
measured as
$\mathcal{A}_{\textrm{RAW}}\left(B^{0}\rightarrow J/\psi
K^{*0}\right)=-0.0110\pm 0.0032\pm 0.0006.$
where the first uncertainty is statistical and the second is systematic. The
$C\\!P$ asymmetry integrated over the full $q^{2}$ range is calculated and
found to be
${\cal A}_{C\\!P}\left(B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}\right)=-0.072\pm
0.040\pm 0.005.$
The result is consistent with previous measurements made by Belle [8], ${\cal
A}_{C\\!P}$ $(B\rightarrow K^{*}l^{+}l^{-})=-0.10\pm 0.10\pm 0.01$, and BaBar
[9], ${\cal A}_{C\\!P}$ $(B\rightarrow K^{*}l^{+}l^{-})=0.03\pm 0.13\pm 0.01$.
This measurement is significantly more precise than all other measurements of
${\cal A}_{C\\!P}$ in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays to date.
Table 2: Values of ${\cal A}_{C\\!P}$ for $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ in the $q^{2}$ bins used in the analysis. | signal | | statistical | systematic | total
---|---|---|---|---|---
$q^{2}$ region (${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) | yield | $~{}~{}~{}{\cal A}_{C\\!P}~{}~{}~{}$ | uncertainty | uncertainty | uncertainty
$0.05<~{}$$q^{2}$ | $<2.00$ | $168$$\pm$ | $15$ | $-0.196$ | $0.094$ | $0.010$ | $0.095$
$2.00<~{}$$q^{2}$ | $<4.30$ | $72$$\pm$ | $11$ | $-0.098$ | $0.153$ | $0.016$ | $0.154$
$4.30<~{}$$q^{2}$ | $<8.68$ | $266$$\pm$ | $19$ | $-0.021$ | $0.073$ | $0.010$ | $0.075$
$10.09<~{}$$q^{2}$ | $<12.86$ | $157$$\pm$ | $15$ | $-0.054$ | $0.097$ | $0.011$ | $0.098$
$14.18<~{}$$q^{2}$ | $<16.00$ | $116$$\pm$ | $12$ | $-0.201$ | $0.104$ | $0.009$ | $0.104$
$16.00<~{}$$q^{2}$ | $<20.00$ | $128$$\pm$ | $13$ | $0.089$ | $0.100$ | $0.012$ | $0.101$
$1.00<~{}$$q^{2}$ | $<6.00$ | $194$$\pm$ | $17$ | $-0.058$ | $0.064$ | $0.009$ | $0.064$
$0.05<~{}$$q^{2}$ | $<20.00$ | $904$$\pm$ | $35$ | $-0.072$ | $0.040$ | $0.005$ | $0.040$
Figure 2: Fitted value of ${\cal A}_{C\\!P}$ in $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays in bins of the $\mu^{+}\mu^{-}$ invariant mass
squared ($q^{2}$). The red vertical lines mark the charmonium vetoes. The
points are plotted at the mean value of $q^{2}$ in each bin. The uncertainties
on each ${\cal A}_{C\\!P}$ value are the statistical and systematic
uncertainties added in quadrature. The dashed line corresponds to the $q^{2}$
integrated value, and the grey band is the 1$\sigma$ uncertainty on this
value.
## 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 the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, 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. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), CIEMAT, IFAE and UAB (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
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|
arxiv-papers
| 2012-10-16T17:05:29 |
2024-09-04T02:49:36.648279
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, A. Bates, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n 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, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche,\n J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, 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, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F.\n Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, C.\n Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V.\n Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T.\n Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, D.\n Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\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, M. Kaballo, S.\n Kandybei, M. Karacson, M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, O. Kochebina, I. Komarov, V. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi,\n M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E.\n Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, M. Maino, S.\n Malde, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, D.A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, 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.K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, K. Sobczak, F.J.P.\n Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, D.\n Urner, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen,\n B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, C.\n Vo{\\ss}, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K.\n Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Simon Wright",
"url": "https://arxiv.org/abs/1210.4492"
}
|
1210.4540
|
# Spectral Energy Distributions of low-luminosity radio galaxies at z$\sim
1-3$:
a high-zview of the host/AGN connection.
Ranieri D. Baldi11affiliation: SISSA, via Bonomea 265, 34136 Trieste, Italy
22affiliation: Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA , Marco Chiaberge22affiliation: Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
33affiliation: INAF-Istituto di Radio Astronomia, via P. Gobetti 101, I-40129
Bologna, Italy 44affiliation: Center for Astrophysical Sciences, Johns Hopkins
University, 3400 N. Charles Street Baltimore, MD 21218 , Alessandro
Capetti55affiliation: INAF-Osservatorio Astronomico di Torino, Strada
Osservatorio 20, 10025 Pino Torinese, Italy , Javier Rodriguez-
Zaurin66affiliation: Insituto de Astrofisica de Canarias, Via Lactea s/n, La
Laguna 38200, Spain; Departamento de Astrofisica, Universidad de La Laguna, La
Laguna 38206, Spain 22affiliation: Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA , Susana Deustua22affiliation: Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
,William B. Sparks22affiliation: Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA rbaldi@sissa.it
###### Abstract
We study the Spectral Energy Distributions, SEDs, (from FUV to MIR bands) of
the first sizeable sample of 34 low-luminosity radio galaxies at high
redshifts, selected in the COSMOS field. To model the SEDs we use two
different template-fitting techniques: i) the Hyperz code that only considers
single stellar templates and ii) our own developed technique 2SPD that also
includes the contribution from a young stellar population and dust emission.
The resulting photometric redshifts range from z$\sim$0.7 to 3 and are in
substantial agreement with measurements from earlier work, but significantly
more accurate. The SED of most objects is consistent with a dominant
contribution from an old stellar population with an age $\sim 1-3\times$ 109
years. The inferred total stellar mass range is $\sim$10${}^{10}-10^{12}$ M⊙.
Dust emission is needed to account for the 24 $\mu m$ emission in 15 objects.
Estimates of the dust luminosity yield values in the range $L_{\rm dust}\sim
10^{43.5}-10^{45.5}$ erg s-1. The global dust temperature, crudely estimated
for the sources with a MIR excess, is $\sim$ 300-850 K. A UV excess is often
observed with a luminosity in the range $\sim 10^{42}-10^{44}$ erg s-1 at 2000
Å rest frame.
Our results show that the hosts of these high-z low-luminosity radio sources
are old massive galaxies, similarly to the local FRIs. However, the UV and MIR
excesses indicate the possible significant contribution from star formation
and/or nuclear activity in such bands, not seen in low-z FRIs. Our sources
display a wide variety of properties: from possible quasars at the highest
luminosities, to low-luminosity old galaxies.
###### Subject headings:
Galaxies: active – galaxies: high-redshift – Galaxies: nuclei – Galaxies:
evolution – Galaxies: photometry – Infrared: galaxies
## 1\. Introduction
The search for radio-loud Active Galactic Nuclei (AGN) is one of the widely
used tools to study the distant universe (z $>$ 1). Indeed the first objects
found at z $>$ 1 were radio-loud quasars (see Stern & Spinrad 1999 and
references therein) and a large number of radio galaxies ($\sim$300) are known
to exist at high redshifts (e.g., Miley & De Breuck 2008).
Flux-limited samples of radio galaxies such as the 3CR and its deeper
successors 6C and 7C catalogs are affected by a tight redshift-luminosity
correlation. This, alongside with the steep luminosity function of radio
sources, gives rise to a selection bias which results in the presence of high
power sources predominantly at high redshifts only and low power sources
exclusively at low redshift. Therefore, our knowledge of radio galaxies and
their hosts at high redshifts is exclusively based on studies of powerful
(edge-brightened, Fanaroff & Riley 1974) FR IIs. Thus, the properties of the
low-luminosity AGN population in the early Universe is essentially unknown.
The missing pieces of the puzzle might be obtained by studying a sample of low
luminosity (’edge-darkened’, Fanaroff & Riley 1974) FR Is at high redshifts.
The lower power of these objects with respect to that of high-z FR IIs might
allow us to analyze with more clarity the properties of the hosts at high
redshifts. Furthermore, in terms of host galaxy properties, low power sources
are both more abundant and most likely more similar to quiescent galaxies (in
terms of host galaxy properties) than FR IIs.
Only a few FR Is at high redshift are present in the 7C sample (Heywood et
al., 2007) and two were possibly found in HDF North (Snellen & Best, 2001).
Chiaberge et al. (2009) obtained the first sizeable sample of 37 FR I
candidates at z $\gtrsim 1$, located in the 2 deg2 area of the sky observed by
the COSMOS survey (Scoville et al., 2007). In this paper we perform a detailed
analysis of the photometric properties of the host galaxies of these radio
sources, by taking advantage of the large multi-wavelength coverage provided
by the COSMOS collaboration. This allows us to derive their spectral energy
distributions (SEDs) from $\sim$ 0.15 $\mu$m to 24 $\mu$m. We then model the
obtained SEDs with synthetic stellar population templates with the aim of both
inferring the properties of each galaxy and of deriving an accurate estimate
of the photometric redshifts.
Mobasher et al. (2007) and Ilbert et al. (2009) performed a similar study with
the aim of determining the photo-z for the whole COSMOS sample. However, as
already noted by Chiaberge et al. (2009), the faint counterparts of these
radiogalaxies might be easily misidentified or be missed from the COSMOS
catalog. Therefore a careful object-by-object study of the sample is necessary
to correctly identify the genuine counterparts of the radio sources at all
wavelengths. This allows us to obtain reliable SEDs and perform a robust study
of the hosts of these objects. In this work, using the multi-band data
provided by the COSMOS survey (described in Section 3), we obtain the
photometric data from the catalog checking visually the correct identification
of each object of the sample (Section 4). In Section 5 we describe the codes
used to model the SEDs, Hyperz and 2SPD. We present the results obtained from
the SED modeling in Section 6: the photometric redshifts, the radio power
distribution, and the host properties, such as stellar ages, masses and dust
and UV components. In Section 7 we summarize the results and we discuss our
preliminary findings.
We adopt a Hubble constant of H0 = 71 km s-1 Mpc-1, $\Omega_{m}$ = 0.27 and
$\Omega_{vac}$ = 0.73, as given by the WMAP cosmology (e.g., Spergel et al.
2003; Jarosik et al. 2011). All the magnitudes are in AB mag system, if not
otherwise specified.
## 2\. Sample
Chiaberge et al. (2009) selected a sample of 37 high-z low-power radio
galaxies at z $\sim$ 1-2 in the Cosmic Evolution Survey (COSMOS) field
(Scoville et al., 2007). This is the first sizeable sample of low-power radio
sources at mid- to high-redshifts. This sample has been obtained with a four-
steps multi-wavelength selection process, which is described in detail in
Chiaberge et al. (2009). Here we briefly summarize the selection procedure
which depends on the following assumptions: i) the FR I/FR II break luminosity
at 1.4 GHz does not change with redshift, and ii) the host properties of
distant low-luminosity sources are similar to those of high-power FR IIs in
the same redshift range.
The first step consists of selecting radio sources from the FIRST catalog
(Becker et al., 1995) whose 1.4 GHz total flux corresponds to the luminosities
expected by FR Is at 1$<z<$2\. The second step is based on a radio
morphological classification: the radio sources which show clear “edge-
brightened” structures are rejected in order to exclude the bona-fide FR IIs
from the sample. The third step implies the optical identification of the
radio sources in the COSMOS optical images. The objects associated with host
galaxies brighter than $i$ $<$ 22 were rejected to exclude lower redshifts
starburst galaxies. As a final step, u-band dropouts are also rejected since
these are objects most likely located at redshift higher than z=2. The
resulting sample consists of 37 FR I candidates. A posteriori, the photometric
redshifts range of most of them is between $\sim$1 and $\sim$2 (Mobasher et
al., 2007; Ilbert et al., 2009), with the exception of 3 objects (namely, 7,
27, and 66111The ’naming convention’ of the sources used in the paper is based
on the listing number in the COSMOS catalog.) that we exclude from any further
analysis because out of the redshift range of interest. We are then left with
a final sample of 34 objects. The list of the radio sources is presented in
Table 1.
Table 1Radio positions, redshifts and radio flux of the sample ID | RA | DEC | zphot,Ilbert | zspec | FNVSS | FFIRST
---|---|---|---|---|---|---
1 | 150.20744 | 2.2818749 | 0.92${}^{+0.02}_{-0.06}$ | 0.8827a-0.8823b | $<$2.5 | 1.79
2 | 150.46751 | 2.7598829 | | | 2.6 | 1.08
3 | 150.00253 | 2.2586310 | 1.96${}^{+0.36}_{-0.41}$ | | 5.2 | 4.21
4 | 149.99153 | 2.3027799 | 1.45${}^{+0.08}_{-0.15}$ | | 7.5 | 5.99
5 | 150.10612 | 2.0144780 | 1.84${}^{+0.22}_{-0.12}$ | | 3.4 | 1.30
11 | 150.07816 | 1.8985500 | 1.31${}^{+0.60}_{-0.24}$ | | $<$2.5 | 1.13
13 | 149.97784 | 2.5042069 | 1.09${}^{+0.06}_{-0.06}$ | | 2.4 | 1.51
16 | 150.53772 | 2.2673550 | 0.95${}^{+0.05}_{-0.02}$ | 0.9687b | 4.4 | 5.70
18 | 149.69325 | 2.2674670 | 0.92${}^{+0.01}_{-0.09}$ | | 5.1 | 4.39
20 | 149.83209 | 2.5695460 | 1.00${}^{+0.02}_{-0.05}$ | | $<$2.5 | 1.33
22 | 149.89508 | 2.6292144 | | | $<$2.5 | 2.74
25 | 150.45673 | 2.5597000 | 1.12${}^{+0.29}_{-0.03}$ | 0.7917b∗ | 2.7 | 2.18
26 | 149.62114 | 2.0919881 | 1.20${}^{+0.06}_{-0.02}$ | | 3.2 | 1.88
28 | 149.60064 | 2.0918673 | | | 2.4 | 1.77
29 | 149.64587 | 1.9529760 | 1.59${}^{+0.45}_{-0.30}$ | | 2.3 | 2.12
30 | 149.61542 | 1.9910541 | 0.90${}^{+0.31}_{-0.03}$ | | 2.4 | 1.26
31 | 149.61916 | 1.9163600 | 0.91${}^{+0.02}_{-0.05}$ | 0.9132a-0.9123b | 4.1 | 3.71
32 | 149.66830 | 1.8379777 | | | 3.1 | 1.31
34 | 150.56023 | 2.5861051 | 1.42${}^{+0.65}_{-0.32}$ | | 4.5 | 5.25
36 | 150.55662 | 1.7913361 | | | 3.3 | 3.19
37 | 150.74336 | 2.1705379 | 1.27${}^{+0.09}_{-0.02}$ | | 2.2 | 1.87
38 | 150.53645 | 2.6842549 | 1.12${}^{+0.10}_{-0.05}$ | | 11.6 | 10.01
39 | 149.95804 | 2.8288901 | 1.08${}^{+0.03}_{-0.03}$ | | $<$2.5 | 1.37
52 | 149.90590 | 2.3964710 | 0.74${}^{+0.02}_{-0.03}$ | 0.7417b | $<$2.5 | 1.54
70 | 150.61987 | 2.2894360 | 2.21${}^{+0.57}_{-0.37}$ | | 4.5 | 3.90
202 | 149.99506 | 1.6324950 | 1.19${}^{+0.24}_{-0.14}$ | | 3.3 | 1.08
219 | 150.06444 | 2.8754051 | 1.04${}^{+0.01}_{-0.09}$ | | $<$2.5 | 1.85
224 | 150.28999 | 1.5408180 | 1.10${}^{+0.07}_{-0.03}$ | | 3.2 | 3.31
226 | 150.43864 | 1.5934480 | 1.76${}^{+0.61}_{-0.14}$ | | $<$2.5 | 1.19
228 | 149.49455 | 2.5052481 | 1.89${}^{+0.74}_{-0.59}$ | | 3.7 | 2.04
234 | 150.78925 | 2.4539680 | 1.10${}^{+0.12}_{-0.03}$ | | 5.2 | 4.43
236 | 150.70554 | 2.6296339 | | 2.132c | 7.0 | 7.10
258 | 149.55934 | 1.6310670 | 0.90${}^{+0.02}_{-0.02}$ | 0.9009a | 3.7 | 2.24
285 | 150.72131 | 1.5823840 | 1.21${}^{+0.06}_{-0.08}$ | | 3.5 | 2.95
## 3\. Data
### 3.1. UV, optical, and IR photometric data
The photometric data used to build the SEDs of our sources are taken from the
COSMOS survey (Scoville et al., 2007). The survey comprises ground based as
well as imaging and spectroscopic observations from radio to X-rays
wavelengths, covering a 2 deg2 area. Given the high sensitivity and resolution
of these data, COSMOS provides samples of high redshift objects with greatly
reduced cosmic variance as compared to earlier surveys.
Ground-based UV, optical, and IR observations and data reduction are presented
in Capak et al. (2007), Capak et al. (2008) and Taniguchi et al. (2007). A
multiwavelength photometric catalog was generated using SExtractor (Bertin &
Arnouts, 1996). The COSMOS catalog is derived from a combination of the CFHT
$i^{*}$ and Subaru $i^{+}$ images, to which the authors refer as ’I-band
images. The catalog includes objects with total (”mag-auto”) I $<$ 25 and
searches for counterparts in a radius of 1″around the I-band detection. At
fainter magnitudes the catalog begins to be incomplete and have more spurious
detections, and photometric redshifts are poorly constrained.
For this study we use the COSMOS Intermediate and Broad Band Photometry
Catalog 2008 (Capak et al., 2008)222http://irsa.ipac.caltech.edu/cgi-
bin/Gator/nph-dd which provides the multiwavelength magnitudes of our sources
from FUV to K bands. Narrow band filters are not considered due to the
possible strong contamination of emission lines from the AGN and their low
signal-to-noise ratio.
HST (Koekemoer et al., 2007) and Spitzer data (both IRAC and MIPS, Sanders et
al. 2007) are also included in the COSMOS survey. The later is presented in
two separate catalogs. S-COSMOS IRAC 4-channel Photometry Catalog June 2007 is
for IRAC data. S-COSMOS MIPS 24 Photometry Catalog October 2008 and S-COSMOS
MIPS 24 $\mu$m DEEP Photometry Catalog June 2007 are for MIPS data at 24
$\mu$m with two different flux limits, 0.15 and 0.08 mJy, respectively.
For the sake of clarity, we summarize here the available datasets from the
COSMOS survey used in this work (Table 2):
1. 1.
GALEX: UV data were taken using GALEX (Martin et al., 2005). The observations,
performed as part of the GALEX Deep Imaging Survey, are in the NUV and FUV
bands, respectively with an angular resolution of 5$\farcs$6 (NUV) and
4$\farcs$2 (FUV) (Morrissey et al., 2007).
2. 2.
HST: the COSMOS HST data are single-orbit F814W ACS images. They have the
highest angular resolution ($\sim 0\farcs 09$, Koekemoer et al. 2007) among
the COSMOS images.
3. 3.
SUBARU: The Suprime-Cam instrument on the Subaru telescope observed the COSMOS
field in six broad bands ($B_{J}$, $g^{+}$, $V_{J}$, $r^{+}$, $i^{+}$,
$z^{+}$) with an angular resolution of $\sim$0$\farcs$2 (Taniguchi et al.,
2007).
4. 4.
CFHT: the Canada-France-Hawaii Telescope (CFHT) provides $u^{*}$ and $i^{*}$
images using Megacam (Boulade et al., 2003) and $K$ images with Wircam.
5. 5.
UKIRT: near infrared Wide Field camera (WFCAM) on the United Kingdom Infrared
Telescope (UKIRT) provides the $J$-band images (Casali et al., 2007).
6. 6.
NOAO: $K_{S}$ data are taken at the Kitt Peak National Observatory (KPNO)
telescope with FLAMINGOS and The Cerro Tololo Inter-American Observatory
(CTIO) telescope (Capak et al., 2007). These telescopes belong to the National
Optical Astronomy Observatory (NOAO).
7. 7.
Spitzer: Spitzer cycle-2 S-COSMOS is an infrared imaging survey of the COSMOS
field (Sanders et al., 2007). They obtained observations with the IRAC camera
in four channels, at 3.6, 4.5, 5.6, and 8 $\mu$m, and with MIPS in the 24, 70,
160 $\mu$m band. We only consider the data at 24 $\mu$m, since no object is
detected at longer wavelengths (with the exception of object 37 which is also
detected at 70 $\mu$m).
Table 2COSMOS broad bands and their properties. Filter | Telescope | $\lambda_{eff}$ | FWHM | sensitivity
---|---|---|---|---
$FUV$ | GALEX | 1538.6Å | 230.8Å | 25.7
$NUV$ | GALEX | 2315.7Å | 789.1Å | 26.0
$u^{*}$ | CFHT | 3911.0Å | 538.0Å | 26.5
$B_{J}$ | Subaru | 4439.6Å | 806.7Å | 27.0
$g^{+}$ | Subaru | 4728.3Å | 1162.9Å | 27.0
$V_{J}$ | Subaru | 5448.9Å | 934.8Å | 26.6
$r^{+}$ | Subaru | 6231.8Å | 1348.8Å | 26.8
$i^{*}$ | CFHT | 7628.9Å | 1460.0Å | 24.0
$i^{+}$ | Subaru | 7629.1Å | 1489.4Å | 26.2
$F814W$ | HST | 8037.2Å | 1539.0Å | 27.2
$z^{+}$ | Subaru | 9021.6Å | 9021.6Å | 25.2
$J$ | UKIRT | 12444.1Å | 1558.0Å | 23.7
$K_{S}$ | NOAO | 21434.8Å | 3115.0Å | 21.6
$K$ | CFHT | 21480.2Å | 3250.0Å | 23.7
IRAC1 | Spitzer | 35262.5Å | 7412.0Å | 23.9
IRAC2 | Spitzer | 44606.7Å | 10113.0Å | 23.3
IRAC3 | Spitzer | 56764.4Å | 13499.0Å | 21.3
IRAC4 | Spitzer | 77030.1Å | 28397.0Å | 21.0
MIPS1 | Spitzer | 23.68$\mu$m | 4.7$\mu$m | 29.6
### 3.2. Radio data
Chiaberge et al. (2009) selected the sample in the COSMOS field using radio
fluxes at 1.4 GHz from the FIRST survey (Becker et al., 1995). The data are
obtained with the VLA in B configuration with an angular resolution of
$\sim$5″and reach a flux limit of $\sim$1 mJy and are listed in Chiaberge et
al. (2009).
In addition, the NVSS survey (Condon et al., 1998) provides 1.4 GHz radio data
for our sample, but at lower resolution ($\sim$45 ″) and with a higher flux
density limit ($\sim$2.5 mJy) than the FIRST survey. These differences imply
that seven of our objects are missed in the NVSS catalog. Nonetheless, the
NVSS data are useful since they are more sensitive to diffuse low surface
brightness radio emission that the FIRST data. The radio fluxes are taken from
the NVSS archive333http://www.cv.nrao.edu/nvss/NVSSlist.shtml. A NVSS/FIRST
comparison indicates that the flux ratio between the two catalogs is usually
between $\sim 1$ and 2, and never larger than $\sim$ 3\. Table 1 shows the
NVSS and FIRST radio fluxes of the objects.
### 3.3. Spectroscopy
The COSMOS survey provides spectroscopic data from the Very Large Telescope
(VLT) (zCOSMOS, Lilly et al. 2007) and from the Magellan (Baade) telescope
(Trump et al., 2007)
zCOSMOS is a large-area redshift survey which consists of two parts. The
first, namely zCOSMOS-bright, considers a magnitude-limited (I-band mag
$<$22.5) sample of $\sim$20,000 galaxies located within the central 1 deg2.
Spectra cover the wavelength range 5500 Å $<\lambda<$ 9000 Å. The second,
namely zCOSMOS-deep, is an ongoing survey (not yet public) of $\sim$10,000
blue galaxies in the same field filtered with a color selection to be in the
range of 1.4 $<$ z $<$ 3.0. The spectra cover the wavelength range of 3600 Å
$<\lambda<$ 6800 Å.
The Magellan survey presents spectroscopic redshifts for the first 466 X-ray
and radio-selected AGN candidates. The wavelength coverage of these spectra is
$\sim$ 5500-9200Å. Their redshift yield is 72% for iAB $<$ 24 and $>$90% for
i${}_{AB}<$22.
In this work we use only spectroscopic measurements with a confidence level
greater than 99%. Six objects of the 34 objects are included in the
spectroscopic surveys described above with the required quality (Table 1).
## 4\. Multi-band counterparts identification
The process of counterparts identification of our radio galaxies in the COSMOS
catalog suffers from the limitations typical of any multiband survey (such as
misidentification of targets with close neighbor or the contamination by
nearby bright sources). We then prefer to perform our multi-band counterpart
identification on each source by visually inspecting its multiband images,
rather than blindly use the data provided by the COSMOS catalog.
We start the process looking for a I-band counterpart to the radio source in
the COSMOS catalog by adopting 0$\farcs$3 as search radius. 29 identifications
are found in our sample, most at distances smaller than 0$\farcs$1\. Three
objects (22, 28, and 32) are clearly visible in the I-band images, but they
are not found in the COSMOS catalog, since they are below its detection
threshold. In addition, there are two objects (2 and 36) for which no I-band
counter part is found.
For the I-band detected sources, we search in the COSMOS broadband catalog
that provides photometry (from the FUV to the K band) over an aperture of 3″
diameter. The COSMOS catalog associates its counterparts in the remaining
bands with the brightest and closest sources to its I-band position within a
radius of 1″. We extend this method including also the Spitzer/IRAC and MIPS
catalogs, by using a larger search radius (2″) due to their coarser
resolutions with respect to the COSMOS broadband images. For the 5 sources not
present in the COSMOS catalog (because they are too faint in the I-band) we
note that in all cases a clear counterpart is instead present at longer
wavelength providing us with a robust multiband identification.
Figure 1.— Images of object 28 taken from the COSMOS survey. From upper left
to lower right panels we show the images from Subaru $i^{+}$, Subaru $z^{+}$,
UKIRT J, CFHT K and the four channels of Spitzer/IRAC. The size of the image
is 5$\farcs$5 $\times$ 5$\farcs$5\. We overlap the radio contours (green
lines) at 1.4 GHz from VLA COSMOS data on the frames to show the position of
the radio source.
During the visual check of the individual counterparts, two main problems
emerge in the COSMOS broadband catalog: i) the presence of nearby sources,
within the 3″ radius used for the aperture photometry, contaminating the broad
band measurements of the genuine emission from the radio-galaxies of our
sample; ii) the counterparts to the i-band object does not always correspond
to the same object over the various bands. Object 28 is a clear example of
this situation (see Fig. 1): from the J band to 8$\mu$m the brightest source
in the field is coincident with the radio source, while in the $i^{+}$ and
$z^{+}$ bands this is out-shined by a nearby object, causing an erroneous
identification in the COSMOS catalog across the various bands. Furthermore,
the presence of this neighbor also causes a strong (and dominant short-ward of
the J band) contamination to the genuine flux of the radio-galaxy. Other
examples of contamination and mis-identification are reported in Fig.2.
In order to amend these problems, we first perform a new 3″ aperture
photometry properly centered on the position of the radio source, to isolate
the genuine emission of the counterparts. In case of contamination from nearby
source(s), we subtract from the flux resulting from the photometry centered on
the radio source the emission from the neighbor(s), limiting to the fraction
that falls into the 3″ aperture. When we cannot separate a source from a close
source or when the counterpart is not visible in a given band, we measure an
1-$\sigma$ upper limit to the flux.
At 24 $\mu$m, when the counterpart of a source is not seen at the radio
position, we use the detection limit of MIPS catalogs as upper-limit to the
source flux. However, when a nearby source contaminates the emission of our
target, we prefer to measure the upper-limit directly on the image at the
radio position. In addition, in the NUV/FUV band, if the source is not
detected in GALEX we prefer not to include upper limits in our analysis,
because its corresponding flux is substantially higher than those at larger
wavelengths.
Figure 2.— Left panel: Subaru $i^{+}$ image of object 34 with superposed the
radio contours (green) and the 3″ aperture centered on the radio source
(blue). This is a case in which, in addition to the radio-galaxy, other
objects lie inside the integration aperture of the target, contaminating the
COSMOS catalog flux measurement. Middle and right panels: Subaru $z^{+}$ and
CFHT $u^{*}$ images of object 5 with superposed the radio contours. This is a
case of misidentification of a source with a neighbor included within the 1″
from the target) in different bands. The size of the images is 2″$\times$ 2″.
In some cases the GALEX photometry returns apparently incorrect results. In
Fig. 3 we show as an example the HST and GALEX images of object 13; this
source is labeled as detected by the COSMOS catalog in both the FUV and NUV
GALEX bands, while clearly there is no emission above the background level.
The opposite is seen in MIPS measurements, when clearly visible objects are
missing from the catalog (see Fig. 4). In these cases, we do not consider the
GALEX data, while for MIPS we provide a new estimate to the 24 $\mu$m flux.
We also find an apparent error in the reported UKIRT J-band photometric
points. We obtain our own measurements on a sample of stars present in the
COSMOS field (Wright et al., 2010) using the PSF-matched and PSF-original
J-band images444The PSF-matched images are obtained by convolving each PSF-
homogenized image with a Gaussian kernel that produced the same flux ratio
between a 3″and 10″aperture for a point source to avoid PSF-matching problems
in the multiband catalog. The PSF-original images are the pure images without
any PSF matching. Both the images are provided by the COSMOS catalog. and we
compare them to the magnitudes given by the catalog. This test reveals that
the zero-point mag of the PSF-matched J-band images is higher than the
corresponding value of the PSF-original J-band images, taking into account of
the effect of the PSF convolution. This systematic difference of the zero-
point magnitudes is $\Delta$J = 0.90$\pm$0.06, intriguingly similar to the
offset between Vega and AB mag system for the UKIRT J-band (JAB = JVega \+
0.94 for UKIRT, Hewett et al. 2006). We obviate this problem by performing
3″-aperture photometry on the J-band counterparts on the UKIRT images smoothed
with a Gaussian kernel to a 1.2-1.5″ FWHM to reproduce the effect of the PSF
matching.
Figure 3.— From left to right: HST/ACS F814W, GALEX NUV and FUV bands images
of object 13. The radio source (whose position is marked by the circles at the
images center) does not apparently correspond to any UV emission, contrarily
to the detection found in the catalog. The size of the images is
7$\farcs$5$\times$7$\farcs$5\.
Figure 4.— Images of object 70 from ACS (left) and MIPS 24$\mu$m (right). This
provides an example of a source clearly emitting at 24$\mu$m, but not present
in the COSMOS Spitzer/MIPS catalog. The position of the radio source is
represented by the green radio contours from VLA COSMOS. The size of the
images is 28″$\times$28″.
Once we obtain the correct aperture photometry of the various multi-band
counterparts of the radio galaxies, we apply the appropriate aperture
corrections. While this is not needed for the Subaru, CFHT, and UKIRT, for
GALEX data we apply an aperture correction by multiplying the total flux by a
factor of 0.759 (Capak et al., 2007). Similarly, since we select from the IRAC
catalog the 2$\farcs$9 aperture photometry (because it is closest to the 3″
aperture of the COSMOS broadband catalog among the different optional
apertures), the IRAC aperture flux was converted to total source flux by using
the aperture correction factors (from the IRAC manual) of 1.19, 1.27, 1.48,
and 1.27 for the four channels. The MIPS data instead already include the
aperture correction. Since the CFHT telescope is more sensitive and with
higher resolution than the images from the NOAO telescopes, this usually
results in far smaller errors for the K-band magnitudes. In such cases, we
prefer to use only the CFHT K-band data. The results of the corrected
3″-aperture photometric measurements are tabulated in Tables 6 and 7.
## 5\. SED fitting
The SEDs are derived by collecting multiband data from the FUV to the MIR.
Since not all of the objects are detected in the entire set of available
bands, the number of detections used to constrain the SED fitting ranges from
15 to 19.
The synthetic stellar templates used to model the observed SEDs are the
Bruzual & Charlot (2009) (priv. comm.) and Maraston (2005) templates. These
templates are defined with different Initial Mass Function (IMF) (Salpeter,
1955; Kroupa, 2001; Chabrier, 2003) with solar metallicity (Z⊙ = 0.02). These
libraries contain composite stellar population (CSP) computed with different
star formation histories: a constant star-forming system (with constant star
formation rate of 1 M⊙/yr); a single burst of star formation; and ten
$\mu$-models555These templates correspond to synthetic spectra computed with
exponentially decaying star formation rate, $\psi(t)$ $\propto$ e(-t/τ) where
$\tau$ is the star formation timescale. with time-decays of 0.1, 0.2, 0.3,
0.6, 1.0, 2.0, 3.0, 5.0, 10.0, and 15.0 Gyr. These templates include 221
tracks of ages from 0.1 Myr to 20 Gyr and cover the wavelength range from 91 Å
to 160 $\mu$m. Fig. 5 shows the stellar templates used for SED modeling.
Figure 5.— Examples of SED templates from BC09 (Bruzual & Charlot, 2009) (left
panel) and MA05 (Maraston, 2005) (right panel) in units of L☉ $\AA^{-1}$. The
templates correspond to different ages and different star formation histories.
Arranging the BC09 and MA05 templates with increasing luminosity at 3550 Å
(following the dashed line), the different plotted models correspond to
$\mu$=15 Gyr decaying CSP of 1.1 Myr (red line); single stellar population of
20 Gyr (gray line); $\mu$=0.1 Gyr decaying CSP of 10 Gyr (dark green line);
$\mu$=10 Gyr decaying CSP of 2.8 Myr (light green line); $\mu$=0.3 Gyr
decaying CSP 5.25 Gyr (orange line); $\mu$=5 Gyr decaying CSP of 6.9 Myr (blue
line); $\mu$=3 Gyr decaying CSP of 17 Myr (yellow line); $\mu$=0.6 Gyr
decaying CSP of 1.7 Gyr (pink line); $\mu$=2 Gyr decaying CSP of 38 Myr
(fuchsia line); and $\mu$=1 Gyr decaying CSP of 0.2 Gyr (light blue line). We
invite the reader to download the color version of the plot to distinguish the
different lines.
The Bruzual & Charlot (2009) models differs from those of Maraston (2005)
mainly because of a different recipe for the inclusion of thermally pulsing
asymptotic giant branch (TP-AGB) phase. These stars significantly contribute
to the IR emission for ages higher than $\sim$1 Gyr. The resulting effect is
that the Bruzual & Charlot templates are bluer and give slightly older ages
than Maraston (2005) models.
### 5.1. SED fitting method: Hyperz
We firstly estimate the photometric redshift for our sample by using the
template fitting technique, Hyperz (Version 1.3 , obtained from M. Bolzonella
in priv. comm.), described in detail by Bolzonella et al. (2000). This SED
fitting procedure is based on reproducing the overall shape of the SEDs and
recognizing strong spectral properties, such as the 4000 Å Ca break or Lyman
break at 912 Å. This code consists of a convolution of templates, which
represent the rest-frame SEDs for galaxies with different star formation
histories, with the filter response functions given by the COSMOS survey.
The basic assumption of this SED modeling is that the host galaxies of the
radio sources in our sample are dominated by stellar emission. Thus the MIPS
24 $\mu$m photometric point is excluded because this emission is obviously not
of stellar origin. Furthermore, object 236, a spectroscopically-confirmed QSO
(see Sect. 5.3), is excluded from this analysis. The code also takes into
account extinction (by using the reddening law from Calzetti et al. 2000)
which is applied to the templates. The reddened templates are then shifted in
wavelength searching for the correct redshift. The best fit is obtained
through $\chi^{2}$ minimization. The grid spacings in redshift and in $A_{V}$
are 0.01 and 0.1, respectively. The Hyperz code gives the probability
P($\chi_{rid}^{2}$) associated with $\chi_{rid}^{2}$ (i.e., the reduced
$\chi^{2}$, which is defined as $\chi^{2}_{min}$/$\nu$, where $\nu$ is the
number of degrees of freedom). Because we are essentially interested in the
redshift (the other parameters are affected by the degeneracy), the degrees of
freedom are Nfilters \- 1.
Similarly to Ilbert et al. (2009), we increase the flux errors by 10% when the
probability associated with the $\chi_{rid}^{2}$ is less than 0.01. This
factor does not shift the best-fit photo-z value but broadens the $\chi^{2}$
peak and derived redshift uncertainty. The program provides the redshift of
the object measured at the different confidence intervals defined by the
values of $\chi_{rid}^{2}$. We conservatively choose the solution at a 99%
confidence level. Bolzonella et al. (2000) discussed the effects of degeneracy
in the parameter space defined by star formation history, age, and reddening.
However, they show that this does not strongly affect the value of the
photometric redshift. Hyperzmass, a code which works in a similar way to
Hyperz, returns the mass of the stellar population corresponding to the best
fit and we adopt again the 99%-confidence level to quote the errors on the
mass.
Fig. 15 (left panels) shows the SED fit obtained with Hyperz. The resulting
properties from the modeling are listed in Table 3. Inspection of this figure
indicates that Hyperz does not always return satisfactory results; indeed, in
several cases, it cannot reproduce the bluest part of the spectrum (possibly
due to a young stellar component) or the substantial SED “bump” at long
wavelengths, clearly suggestive of dust emission and often extending even to
the IRAC 3.6 $\mu$m measurements.
### 5.2. ‘2SPD’ fitting technique
In order to improve the quality of the SED fitting we developed a code that
includes 2 Stellar Populations and Dust component (2SPD). More precisely, we
take into account two different stellar populations, typically, one younger
and one older (YSP and OSP, respectively). We model the dust component with a
single (or, in a few cases, two) temperature black-body emission. Thus, at
this stage, we can include the 24 $\mu$m MIPS flux in the fitting process.
The synthetic models used are the same as described in the previous section,
but limiting to those with single stellar population with ages ranging from 1
Myr to 12.5 Gyr. We adopt a dust-screen model for the extinction normalized
with the free parameter $A_{V}$, and the Calzetti et al. (2000) law.
The code searches for the best match between the sum of the different
components (young and old stellar templates and dust emission) and the
photometric points minimizing the appropriate $\chi^{2}$ function. 2SPD
returns the following free parameters: $z$, $A_{V}$, the age of the two
stellar populations, the temperature of the dust component(s), and the various
normalization factors. From these quantities we measure the stellar mass
content of the two stellar populations at 4800 Å rest frame. Similarly to the
case of Hyperz, caution should be exerted before associating these values to
physical quantities because of degeneracy in the parameter space, apart from
the photometric redshifts.
To estimate the errors on the photo-z and mass derivations, we measure the
99%- confident solutions for these quantities. This is computed by varying the
value the parameter of interest until the $\chi^{2}$ value increases by
$\Delta\chi^{2}$ = 6.63, corresponding to a confidence level of 99% for that
parameter.
Fig. 15 (righ panels) shows the outputs of 2SPD code, while Table 4 presents
the resulting parameters of the fit.
The dust emission is usually poorly constrained. In many cases no excess above
the stellar component is required at $\lambda<24\mu$m. In this case dust is
needed only to account for the 24 $\mu$m data-point and often this measurement
is an upper limit. The values reported in Table 4 correspond to the best
fitting model for each galaxy and must be interpreted as approximate rather
than real measurements of the dust emission properties. We will return to the
dust properties in more detail in Sect. 6.4.
### 5.3. COSMOS photometric redshifts
As a comparison with our results we also collect the photometric redshifts
from the COSMOS redshift catalog. Mobasher et al. (2007) provide photometric
redshifts for 860,000 galaxies in the COSMOS field with $i^{+}$ $<$ 25\. Their
technique is based on a $\chi^{2}$ template-fitting procedure applied to the
SEDs derived with up to 16 photometric points from the $u$ to the K band. To
evaluate the reliability of the derived photometric redshifts, they compare
them with spectroscopic redshifts from a sample of 868 galaxies in zCOSMOS
with z $<$ 1.2. The rms scatter between photometric and spectroscopic
redshifts is $\sigma_{(zphot-zspec)/(1+zspec)}$ = 0.031 with a small fraction
of outliers (2.5%).
Ilbert et al. (2009) substantially improve this analysis by including data
from 30 broad, intermediate, and narrow bands covering the wavelength range
from UV to MIR. Redshifts are computed for 607,617 sources (with $i^{+}<$ 26).
The method used by these authors accounts for the contribution of emission
lines to the SEDs. Comparison with 4,148 zCOSMOS spectroscopic redshifts
indicates a dispersion of $\sigma_{(zphot-zspec)/(1+zspec)}$ = 0.007 at
$i^{+}_{AB}<$ 22.5. Nevertheless, the accuracy is strongly degraded at
$i^{+}>$ 25.5. Table 1 shows the photometric redshifts measured by Ilbert et
al. (2009).
Most of the radio galaxies of our sample are included in the COSMOS
Photometric Redshift Catalog Fall 2008, which provides the photometric
redshifts measured by Ilbert et al. (2009). We adopt again a 0$\farcs$3 search
radius and find 29 objects. We report the results in Table 3. One of these
objects (236) turns out to be a spectroscopically-confirmed QSO at z = 2.132
(Prescott et al., 2006). The attempt of Ilbert et al. (2009) to estimate its
photometric redshift failed because the templates used by these authors were
not suitable to fit its AGN-dominated SED. Similarly, Mobasher et al. (2007)
provided for this object a tentative photometric redshift, which is
z=1.23${}^{+0.03}_{-0.17}$, which is clearly inconsistent with the
spectroscopic value.
## 6\. RESUTS
The SED modeling process has been performed for all the objects (except for
the spectroscopically-confirmed QSO, object 236), by using the two template-
fitting techniques (Fig 15). As expected, 2SPD code has turned out to more
reliably model the SEDs of our sources than Hyperz. This is crucial in UV
band. In fact, when significant UV emission appears in the SED, such as, in
objects 31, 258 and 285, Hyperz technique struggles to fit simultaneously such
a UV component and the remaining part of SED, which is well represented by an
OSP. In this case, the best fit result from Hyperz is obtained using an old
composite stellar population. This is due to the fact that the $\chi^{2}$ is
lower if only the the old component of the SED is fitted, rather than the UV
data points which are less well represented. Conversely, as expected, 2SPD
technique is more efficient to treat this case, since it just adds a small
fraction of YSP to a dominant OSP to reproduce the UV emission, by departing
from the concept of single star formation history.
Table 3Hyperz SED fitting ID | redshift | CSP Template | Log M∗
---|---|---|---
| zphot,Hyperz | zphot,2SPD | type | SFH | Age | AV | $\chi_{rid}^{2}$ | Log M⊙
1 | 0.85${}^{+0.05}_{-0.08}$ | 0.88${}^{+0.04}_{-0.05}$ | BC | ssp | 1.434 | 1.00 | 0.73 | 10.98${}^{+0.07}_{-0.04}$
2 | 1.31${}^{+0.14}_{-0.07}$ | 1.33${}^{+0.10}_{-0.09}$ | BC | $\tau$=0.3 | 0.5088 | 2.9 | 1.43 | 11.00${}^{+0.09}_{-0.12}$
3 | 2.33${}^{+0.18}_{-0.18}$ | 2.20${}^{+0.32}_{-0.44}$ | Ma | $\tau$=0.3 | 1.0152 | 1.0 | 11.50 | 10.43${}^{+0.07}_{-0.02}$
4 | 1.44${}^{+0.14}_{-0.30}$ | 1.37${}^{+0.10}_{-0.06}$ | BC | $\tau$=0.1 | 0.3602 | 2.4 | 0.30 | 11.10${}^{+0.16}_{-0.09}$
5 | 1.93${}^{+0.06}_{-0.11}$ | 2.01${}^{+0.22}_{-0.35}$ | Ma | const | 2.0 | 2.3 | 9.57 | 11.68${}^{+0.21}_{-0.15}$
11 | 1.55${}^{+0.28}_{-0.15}$ | 1.57${}^{+0.14}_{-0.09}$ | Ma | $\tau$=1.0 | 1.0152 | 0.8 | 0.51 | 11.05${}^{+0.11}_{-0.06}$
13 | 1.12${}^{+0.02}_{-0.01}$ | 1.19${}^{+0.08}_{-0.11}$ | BC | ssp | 0.1805 | 2.4 | 12.60 | 10.91${}^{+0.03}_{-0.03}$
16 | 1.04${}^{+0.06}_{-0.15}$ | 0.97${}^{+0.12}_{-0.07}$ | BC | ssp | 0.1805 | 2.2 | 0.78 | 10.58${}^{+0.10}_{-0.05}$
18 | 0.92${}^{+0.02}_{-0.02}$ | 0.92${}^{+0.14}_{-0.11}$ | BC | ssp | 0.1805 | 2.1 | 6.63 | 10.48${}^{+0.03}_{-0.04}$
20 | 0.80${}^{+0.20}_{-0.05}$ | 0.88${}^{+0.02}_{-0.02}$ | BC | $\tau$=0.3 | 1.7 | 1.4 | 0.76 | 10.85${}^{+0.15}_{-0.08}$
22 | 1.21${}^{+0.09}_{-0.05}$ | 1.30${}^{+0.05}_{-0.04}$ | BC | ssp | 0.5088 | 3.8 | 3.83 | 10.52${}^{+0.87}_{-0.08}$
25 | 1.37${}^{+0.09}_{-0.12}$ | 1.33${}^{+0.11}_{-0.13}$ | BC | ssp | 0.1805 | 2.2 | 2.21 | 11.08${}^{+0.04}_{-0.11}$
26 | 1.09${}^{+0.05}_{-0.10}$ | 1.09${}^{+0.12}_{-0.07}$ | BC | $\tau$=0.1 | 0.7187 | 1.8 | 1.75 | 11.29${}^{+0.01}_{-0.05}$
28 | 2.61${}^{+0.37}_{-0.22}$ | 2.90${}^{+0.20}_{-0.26}$ | Ma | ssp | 1.0152 | 0.64 | 3.31 | 11.73${}^{+0.03}_{-0.04}$
29 | 1.58${}^{+0.37}_{-0.17}$ | 1.32${}^{+0.23}_{-0.24}$ | Ma | $\tau$=0.3 | 0.7187 | 1.0 | 0.89 | 10.36${}^{+0.14}_{-0.04}$
30 | 0.99${}^{+0.37}_{-0.11}$ | 1.06${}^{+0.11}_{-0.07}$ | BC | $\tau$=0.3 | 1.7 | 1.8 | 1.14 | 11.10${}^{+0.14}_{-0.18}$
31 | 0.81${}^{+0.21}_{-0.18}$ | 0.88${}^{+0.03}_{-0.05}$ | BC | $\tau$=0.3 | 0.7187 | 2.0 | 1.40 | 10.66${}^{+0.09}_{-0.04}$
32 | 3.11${}^{+0.21}_{-0.14}$ | 2.71${}^{+0.38}_{-0.34}$ | BC | ssp | 0.01 | 3.0 | 2.98 | 10.83${}^{+0.03}_{-0.08}$
34 | 1.50${}^{+0.56}_{-0.29}$ | 1.55${}^{+0.41}_{-0.19}$ | Ma | $\tau$=15 | 2.0 | 2.0 | 0.60 | 10.85${}^{+0.64}_{-0.37}$
36 | 0.91${}^{+0.13}_{-0.08}$ | 1.07${}^{+0.10}_{-0.04}$ | BC | $\tau$=0.6 | 0.0151 | 4.6 | 1.82 | 10.05${}^{+0.11}_{-0.21}$
37 | 2.04${}^{+0.15}_{-0.24}$ | 1.38${}^{+0.43}_{-0.42}$ | Ma | ssp | 0.0132 | 1.6 | 11.73 | 11.86${}^{+0.13}_{-0.03}$
38 | 1.34${}^{+0.14}_{-0.42}$ | 1.30${}^{+0.17}_{-0.28}$ | Ma | ssp | 0.01995 | 2.8 | 1.23 | 10.82${}^{+0.13}_{-0.17}$
39 | 0.80${}^{+0.23}_{-0.05}$ | 1.10${}^{+0.05}_{-0.05}$ | BC | const | 1.7 | 3.7 | 8.72 | 10.80${}^{+0.15}_{-0.12}$
52 | 0.75${}^{+0.12}_{-0.11}$ | 0.74${}^{+0.18}_{-0.19}$ | BC | $\tau$=0.3 | 0.5088 | 2.0 | 1.60 | 10.54${}^{+0.19}_{-0.09}$
70 | 2.39${}^{+0.52}_{-0.14}$ | 2.32${}^{+0.53}_{-0.20}$ | Ma | $\tau$=15.0 | 1.434 | 1.4 | 1.16 | 11.17${}^{+0.19}_{-0.12}$
202 | 0.95${}^{+0.44}_{-0.24}$ | 1.31${}^{+0.09}_{-0.12}$ | BC | $\tau$=0.3 | 1.0152 | 2.6 | 0.83 | 10.60${}^{+0.28}_{-0.14}$
219 | 1.04${}^{+0.07}_{-0.15}$ | 1.03${}^{+0.02}_{-0.04}$ | BC | ssp | 0.1278 | 2.6 | 1.98 | 10.96${}^{+0.12}_{-0.01}$
224 | 1.07${}^{+0.11}_{-0.14}$ | 1.10${}^{+0.10}_{-0.04}$ | BC | $\tau$=0.3 | 1.0152 | 2.0 | 0.85 | 10.92${}^{+0.12}_{-0.21}$
226 | 1.98${}^{+0.16}_{-0.27}$ | 2.35${}^{+0.63}_{-0.31}$ | Ma | ssp | 0.01 | 2.0 | 9.57 | 10.08${}^{+0.02}_{-0.05}$
228 | 1.30${}^{+0.09}_{-0.06}$ | 1.31${}^{+0.05}_{-0.07}$ | BC | $\tau$=0.1 | 0.0263 | 4.2 | 1.25 | 10.35${}^{+0.20}_{-0.40}$
234 | 1.02${}^{+0.08}_{-0.04}$ | 1.10${}^{+0.14}_{-0.08}$ | BC | ssp | 0.1278 | 3.0 | 1.66 | 10.74${}^{+0.08}_{-0.01}$
258 | 0.83${}^{+0.11}_{-0.06}$ | 0.96${}^{+0.19}_{-0.13}$ | BC | $\tau$=3.0 | 6.5 | 1.0 | 5.27 | 10.90${}^{+0.09}_{-0.07}$
285 | 1.22${}^{+0.13}_{-0.06}$ | 1.10${}^{+0.13}_{-0.08}$ | BC | const | 0.0151 | 2.8 | 4.56 | 10.03${}^{+0.18}_{-0.01}$
Table 42SPD SED fitting ID | redshift | YSP | OSP | log M∗ | Dust | IR excess | UV
---|---|---|---|---|---|---|---
| zphot | Age | AV | $f_{YSP}$ | Log M∗ | Age | AV | | Tdust | Ldust | LIRexc. | $\alpha_{8-24}$ | LUV
1 | 0.88${}^{+0.04}_{-0.05}$ | 0.02 | 0.80 | 1.5% | 0.49% | 4.0 | 0.15 | 10.08${}^{+0.04}_{-0.04}$ | 140 | 2.4 | $<$43.31 | |
2 | 1.33${}^{+0.10}_{-0.09}$ | 0.05 | 0.90 | 22.1% | 1.3% | 2.0 | 0.98 | 11.00${}^{+0.04}_{-0.04}$ | 156 | 7.4 | $<$43.65 | | 42.53
3 | 2.20${}^{+0.32}_{-0.44}$ | 0.03 | 0.12 | 6.5% | 0.69% | 1.0 | 0.00 | 10.59${}^{+0.08}_{-0.10}$ | 201-347 | 250-59 | 45.27 | -0.01 | 43.21
4 | 1.37${}^{+0.10}_{-0.06}$ | 0.009 | 1.00 | 6.9% | 0.24% | 3.0 | 0.57 | 11.16${}^{+0.04}_{-0.03}$ | 100 | 5.2 | $<$43.55 | | 42.71
5 | 2.01${}^{+0.22}_{-0.35}$ | 0.02 | 1.80 | 53.1% | 5.9% | 3.0 | 0.93 | 11.49${}^{+0.04}_{-0.03}$ | 121 | 144.0 | 44.76 | $>$0.49 |
11 | 1.57${}^{+0.14}_{-0.09}$ | 0.007 | 1.87 | 17.2% | 0.62% | 3.0 | 0.30 | 10.98${}^{+0.10}_{-0.05}$ | 98 | 15.4 | $<$43.80 | |
13 | 1.19${}^{+0.08}_{-0.11}$ | 0.03 | 1.17 | 16.2% | 3.2% | 1.0 | 0.58 | 10.72${}^{+0.04}_{-0.03}$ | 155-406 | 7.0-6.2 | 44.23 | -0.41 | 42.65m
16 | 0.97${}^{+0.12}_{-0.07}$ | 0.006 | 1.16 | 14.8% | 0.23% | 2.0 | 0.52 | 10.74${}^{+0.06}_{-0.06}$ | 158 | 4.8 | 43.60 | $>$-0.12 | 42.42m
18 | 0.92${}^{+0.14}_{-0.11}$ | 0.004 | 2.55 | 50.3% | 13.5% | 0.2 | 1.35 | 10.02${}^{+0.08}_{-0.08}$ | 116 | 13.9 | $<$43.96 | |
20 | 0.88${}^{+0.02}_{-0.02}$ | 0.002 | 1.26 | 3.7% | 0.07% | 3.0 | 0.42 | 11.03${}^{+0.02}_{-0.03}$ | 173 | 1.4 | $<$43.02 | | 42.25m
22 | 1.30${}^{+0.05}_{-0.04}$ | 0.04 | 1.18 | 10.1% | 0.35% | 2.0 | 1.50 | 11.16${}^{+0.02}_{-0.03}$ | 150 | 11.8 | 43.87 | $>$0.66 |
25 | 1.33${}^{+0.11}_{-0.13}$ | 0.002 | 1.17 | 4.1% | 0.28% | 0.4 | 1.05 | 10.75${}^{+0.04}_{-0.05}$ | 138 | 8.8 | 43.75 | $>$0.26 | 42.87m
26 | 1.09${}^{+0.12}_{-0.07}$ | 0.006 | 1.36 | 7.4% | 0.17% | 1.0 | 0.98 | 11.12${}^{+0.04}_{-0.04}$ | 128 | 3.9 | $<$43.45 | | 42.50m
28 | 2.90${}^{+0.20}_{-0.26}$ | 0.001 | 1.97 | 24.7% | 3.9% | 0.9 | 0.00 | 11.38${}^{+0.04}_{-0.04}$ | 146 | 47.4 | 44.28 | $>$0.22 |
29 | 1.32${}^{+0.23}_{-0.24}$ | 0.004 | 0.38 | 11.9% | 0.18% | 0.4 | 0.94 | 10.03${}^{+0.05}_{-0.05}$ | 83 | 16.4 | $<$43.68 | | 42.83
30 | 1.06${}^{+0.11}_{-0.07}$ | 0.001 | 1.77 | 3.0% | 0.13% | 2.0 | 0.73 | 11.03${}^{+0.05}_{-0.05}$ | 130 | 3.5 | $<$43.47 | |
31 | 0.88${}^{+0.03}_{-0.05}$ | 0.007 | 0.23 | 7.5% | 0.06% | 2.0 | 0.23 | 10.75${}^{+0.03}_{-0.03}$ | 164 | 2.6 | $<$43.35 | | 42.86
32 | 2.71${}^{+0.38}_{-0.34}$ | 0.002 | 0.98 | 10.6% | 0.26% | 2.0 | 0.39 | 10.98${}^{+0.04}_{-0.04}$ | 205-452 | 51-58 | 44.96 | -0.98 | 43.05
34 | 1.55${}^{+0.41}_{-0.19}$ | 0.005 | 0.55 | 11.0% | 0.05% | 3.0 | 0.51 | 10.99${}^{+0.07}_{-0.07}$ | 140 | 19.7 | $<$44.04 | | 42.82
36 | 1.07${}^{+0.10}_{-0.04}$ | 0.008 | 2.30 | 42.4% | 2.0% | 3.0 | 1.07 | 10.83${}^{+0.02}_{-0.03}$ | 186 | 4.4 | 43.46 | $>$0.20 |
37 | 1.38${}^{+0.43}_{-0.42}$ | 0.001 | 0.00 | 1.0% | 0.09% | 0.3 | 0.09 | 10.61${}^{+0.11}_{-0.11}$ | 213-614 | 84-22 | 45.03 | -0.25 | 44.10
38 | 1.30${}^{+0.17}_{-0.28}$ | 0.005 | 0.95 | 27.0% | 0.58% | 0.9 | 0.86 | 10.65${}^{+0.07}_{-0.07}$ | 260 | 6.2 | 43.77 | 0.45 | 43.03
39 | 1.10${}^{+0.05}_{-0.05}$ | 0.007 | 1.86 | 15.0% | 0.83% | 1.0 | 0.97 | 10.88${}^{+0.03}_{-0.03}$ | 80 | 5.3 | $<$43.20 | |
52 | 0.74${}^{+0.18}_{-0.19}$ | 0.004 | 0.66 | 13.6% | 0.11% | 2.0 | 0.34 | 10.78${}^{+0.10}_{-0.10}$ | 189 | 2.6 | 43.33 | 0.61 | 42.96
70 | 2.32${}^{+0.53}_{-0.20}$ | 0.009 | 0.48 | 12.8% | 0.87% | 0.4 | 0.74 | 10.65${}^{+0.07}_{-0.03}$ | 180 | 27.5 | 44.07 | $>$-0.50 | 43.54
202 | 1.31${}^{+0.09}_{-0.12}$ | 0.005 | 0.14 | 1.2% | 0.004% | 3.0 | 0.14 | 10.86${}^{+0.03}_{-0.06}$ | 124 | 7.4 | $<$43.66 | | 42.21
219 | 1.03${}^{+0.02}_{-0.04}$ | 0.02 | 0.83 | 6.2% | 0.69% | 0.3 | 1.47 | 10.67${}^{+0.03}_{-0.03}$ | 112 | 2.8 | $<$43.32 | | 42.69m
224 | 1.10${}^{+0.10}_{-0.04}$ | 0.004 | 0.63 | 3.8% | 0.04% | 1.0 | 0.76 | 10.71${}^{+0.03}_{-0.04}$ | 155 | 4.5 | $<$43.53 | | 42.53m
226 | 2.35${}^{+0.63}_{-0.31}$ | 0.001 | 0.23 | 2.8% | 0.12% | 0.9 | 0.00 | 10.59${}^{+0.07}_{-0.09}$ | 212-467 | 88-36 | 45.01 | -0.46 | 43.34
228 | 1.31${}^{+0.05}_{-0.07}$ | 0.002 | 2.29 | 37.1% | 0.72% | 2.0 | 2.21 | 11.06${}^{+0.03}_{-0.03}$ | 133 | 6.9 | $<$43.64 | |
234 | 1.10${}^{+0.14}_{-0.08}$ | 0.006 | 2.30 | 46.3% | 2.2% | 2.0 | 1.05 | 10.83${}^{+0.04}_{-0.04}$ | 104 | 5.8 | $<$43.52 | |
258 | 0.96${}^{+0.19}_{-0.13}$ | 0.001 | 0.08 | 1.6% | 0.05% | 0.08 | 1.72 | 10.64${}^{+0.08}_{-0.09}$ | 134 | 3.0 | $<$43.41 | | 43.27
285 | 1.10${}^{+0.13}_{-0.08}$ | 0.001 | 0.13 | 2.1% | 0.05% | 0.04 | 1.56 | 10.43${}^{+0.04}_{-0.04}$ | 135-448 | 1.9-1.8 | 43.73 | -0.22 | 43.08
Table 5’Final’ redshifts and radio properties of the sample ID | z | LNVSS | LFIRST | class
---|---|---|---|---
1 | 0.88s | $<$31.78 | 31.93 | LP
2 | 1.33 | 32.02 | 32.4 | LP
3 | 2.20 | 33.1 | 33.19 | HP
4 | 1.37 | 32.77 | 32.86 | HP
5 | 2.01 | 32.47 | 32.89 | HP
11 | 1.57 | $<$32.18 | 32.52 | LP
13 | 1.19 | 32.02 | 32.22 | LP
16 | 0.97s | 32.38 | 32.27 | LP
18 | 0.92 | 32.22 | 32.28 | LP
20 | 0.88 | $<$31.66 | 31.93 | LP
22 | 1.30 | $<$32.38 | 32.34 | LP
25 | 1.33 | 32.29 | 32.39 | LP
26 | 1.09 | 32.02 | 32.25 | LP
28 | 2.90 | 32.99 | 33.13 | HP
29 | 1.32 | 32.28 | 32.31 | LP
30 | 1.06 | 31.83 | 32.11 | LP
31 | 0.91s | 32.14 | 32.18 | LP
32 | 2.71 | 32.8 | 33.17 | HP
34 | 1.55 | 32.84 | 32.77 | HP
36 | 1.07 | 32.23 | 32.25 | LP
37 | 1.38 | 32.26 | 32.33 | LP
38 | 1.30 | 32.94 | 33.00 | HP
39 | 1.10 | $<$31.90 | 32.16 | LP
52 | 0.74s | $<$31.54 | 31.73 | LP
70 | 2.32 | 33.12 | 33.18 | HP
202 | 1.31 | 31.98 | 32.46 | LP
219 | 1.03 | $<$31.96 | 32.09 | LP
224 | 1.10 | 32.28 | 32.27 | LP
226 | 2.35 | $<$32.61 | 32.94 | HP
228 | 1.31 | 32.25 | 32.51 | LP
234 | 1.10 | 32.41 | 32.48 | LP
236 | 2.13 | 33.29 | 33.29 | HP
258 | 0.90s | 31.90 | 32.12 | LP
285 | 1.10 | 32.23 | 32.31 | LP
### 6.1. Photometric redshifts
The first test on the accuracy of our photo-z derivation is a comparison with
the spectroscopic redshifts, available for 6 objects (namely 1, 16, 25, 31,
52, and 258). They are all compatible with each other within the errors with
the only exception of object 25. For this object the photometric redshifts
measured with Hyperz and 2SPD (z = 1.37${}^{+0.09}_{-0.12}$ and z =
1.33${}^{+0.11}_{-0.13}$, respectively) are significantly different from the
redshift inferred from its Magellan spectrum (z = 0.7917). However, the
Magellan spectrum does not match with the COSMOS measurements, showing a large
offset, of $\sim$0.8 dex. Apparently, the object observed by Magellan is not
the radio galaxy 25 and we do not consider its spectroscopic-z as reliable.
Reassuringly, we checked that the spectra and photometric data-points agree
for the other 5 objects with available spectra.
Figure 6.— Spectrum of object 25 (red solid line) from the Magellan survey
(Trump et al., 2007) which provides a spectroscopic redshift of 0.7917. We
overlap the synthetic SED (green dashed line) derived from Hyperz at redshift
z = 1.37, fitting the photometric points. The wavelengths on the top
corresponds to the observed wavelengths, and at the bottom to rest frame.
We then compare the photometric redshifts derived with the two SED fitting
techniques. The overall range of $z$ is in both cases from $\sim$0.7 to $\sim$
3\. The median photo-z are 1.21 and 1.30 for Hyperz and 2SPD respectively.
Generally, the photometric redshifts derived with the two methods are
consistent with each other within the errors (Fig. 7, left panel). The
normalized redshift differences ($\Delta z/(1+z)$) are smaller than 0.08 for
all but 5 objects that reach $\Delta z/z\sim 0.12-0.15.$, and a single strong
outlier (object 37, $\Delta z/z\sim 0.28$). For this galaxy, the Hyperz fit to
the SED is particularly weak as it is not able to fit its photometric points
at both FUV and MIR wavelengths.
Figure 7.— Comparison of the photometric-z measured with 2SPD with those
obtained with Hyperz (left) and by Ilbert et al. (2009) (right).
Furthermore, we compare the photometric redshifts measured with our method and
with the template-fitting technique of Ilbert et al. (2009). Objects 2, 22,
28, 32, and 36 are not involved in this comparison since they are not included
in the Ilbert et al. (2009) sample. Fig. 7 (right panel) shows the comparison
between the photo-z resulting from 2SPD and from Ilbert et al. (2009). Again,
the two methods yield similar photometric redshifts with $\Delta
z/(1+z)\lesssim 0.11$ for all but 2 objects, namely 226 and 228 for which we
obtain $\Delta z/(1+z)=0.18$ and 0.25 respectively. In the case of object 226,
similarly to the object 37 discussed above, the SED obtained with 2SPD reveals
a strong excess at both UV and MIR wavelengths. Object 228 is instead a case
of misidentification in the COSMOS multiband catalog. This result stresses the
importance of our detailed work of identification to infer the genuine
properties of the galaxies.
Therefore, unless a spectroscopic redshift is available, we generally prefer
to use the photometric redshift obtained with 2SPD because our code also
includes the dust component which turns out to be important in the SED
modeling. Table 5 shows the redshifts used throughout all the paper.
### 6.2. Radio power distribution
Chiaberge et al. (2009) selected the sample in the COSMOS field looking for
radio sources with radio luminosities below the FR I/FR II break. Since we now
have improved estimates of the photometric redshifts of the objects, we can
derive a more accurate value for their radio power. We show in Fig. 8 the
histograms of K-corrected (using a radio spectral index $\alpha$ = 0.8) radio
powers at 1.4 GHz measured with NVSS and FIRST. The resulting 1.4 GHz
luminosities are in the range 10${}^{31.5}-10^{33.3}$ erg s-1 Hz-1. The local
FR I/FR II break luminosity (Fanaroff & Riley, 1974), converted from 178 MHz
to 1.4 GHz adopting $\alpha$ = 0.8, is L${}_{1.4\,\rm GHz}\sim 10^{32.6}$ erg
s-1 Hz-1. Therefore, the radio distribution of our sample actually straddles
the FR I/FR II break. Nonetheless we remind the reader that radio sources with
a clear FR II morphologies were previously excluded by Chiaberge et al.
(2009).
Figure 8.— Distribution of the K-corrected total radio luminosity (in erg s-1
Hz-1) at 1.4 GHz from the FIRST data (upper panel) and NVSS data (lower
panel). The dashed lines corresponds to the local FR I/FR II break used to
separate the sample in high power (HP, shaded histograms) and low power (LP)
sources.
As already discussed by Chiaberge et al. (2009), the high-frequency radio
observations available for this sample (their rest frame frequency are in the
$\sim$ 3 - 5 GHz range) are not ideal to observe and detect their total radio
structures because they miss diffuse/steep-spectrum radio emission. Lower
frequency radio observations are needed for a more accurate measurement of
their intrinsic total radio power distribution.
Since the radio power distribution is rather broad (covering two orders of
magnitudes), we separate the sample into two groups, including sources of high
and low power (HP and LP, respectively) using the local FR I/FR II break as
divide (see Table 5). This operative definition enables us to explore the role
of radio power in defining the overall properties of these galaxies. Two third
of the sample shows radio luminosities below the break. The LP and HP classes
are, not surprisingly, roughly separated also in redshifts (Fig. 9), since our
sample is limited in radio flux, and indeed the HP sources generally lie at z
$\gtrsim$ 1.5.
Figure 9.— Redshifts versus the K-corrected FIRST radio luminosity at 1.4 GHz
(in erg s-1 Hz-1). The empty points are the LPs and the full points are the
HPs.
### 6.3. Host galaxy properties inferred from SED modeling
We now focus on the properties of the host galaxies inferred from SED modeling
and in particular on their stellar populations.
Figure 10.— Left panel: distribution of stellar masses (in M⊙) of our sample
obtained with 2SPD (upper panel) and with Hyperz (lower panel). Right panel:
comparison between the stellar masses (in M⊙) inferred from SED modeling using
Hyperz and 2SPD techniques. The black histogram (points) represents the HPs,
while the empty one the LPs.
One of the key and more robust result is the measurement of the stellar
content, M∗. The inferred mass range is $\sim$10${}^{10}-10^{12}$ M⊙. The
masses measured with Hyperzmass and with 2SPD are compared in Fig. 10. The two
techniques return generally consistent values of the stellar masses for most
sources (the median values are in both cases $\sim$ 7 $\times$ 1010 M⊙), but
with a few evident outliers. The stellar masses derived from single- and two-
stellar components are know to differ significantly (e.g. Papovich et al.
2006). Furthermore, the inclusion of dust components by 2SPD has two effects
by changing i) the stellar mass since thermal emission largely contributes to
observed infrared emission, and ii) the photometric redshift. The largest
outlier is object 37 whose SED is poorly fitted by Hyperz with a very high
mass 7.0 $\times$ 1011 M⊙, while a strong dust component and a mass 10 times
lower are required by 2SPD. At the opposite end of the mass distribution, we
find two objects (36 and 226) in which again 2SPD finds a strong dust
component, but where the Hyperz is a factor 3 to 7 lower than that obtained
with 2SPD. Apparently, the presence of dust components has a stronger impact
on the stellar mass estimate than the inclusion of a young stellar population.
Considering the radio power of the sources (see Fig. 11, left panel), HPs show
slightly larger stellar mass content than LPs. The median values (obtained
from 2SPD code) for LPs is 5.8 $\times$ 1010 M⊙, while for HPs is 8.7 $\times$
1010 M⊙. However, the two distributions are very broad and show an almost
complete overlap. A very similar result is found looking for differences in
mass between objects at different redshifts (see Fig. 11, right panel).
Figure 11.— Stellar masses (in M⊙) measured with 2SPD in relation with the
K-corrected FIRST radio powers (in erg s-1 Hz-1) (left panel) and redshifts
(right panel) of the sample. Empty points are LPs, while filled points are
HPs.
Due to the degeneration inherent to the modeling of SEDs the remaining results
of the stellar populations are relatively unconstrained. Overall, if we
concentrate in the whole sample, the SED fitting shows that the hosts are
mainly dominated by a massive OSP with an age of $\sim$ 1$-$3 $\times$ 109
years. Although less reliable, the YSPs required to fit the UV excesses show
ages of $1-30$ Myr, with a contribution to the total mass of the galaxy of
$\lesssim 1\%$ for most sources. The flux contribution at 4800 Å (rest frame)
of the YSP is $<$30% for most of the objects.
### 6.4. Dust emission
As discussed in Section 5.2, the parameters related to the dust emission must
be taken as approximate rather than a real measurements of the dust component.
Nonetheless, it is still significant that dust emission is required to
adequately model the SEDs of 15 objects due to the detection of emission at 24
$\mu$m. In addition significant excesses above the stellar emission are
observed also at shorter infrared wavelengths in 8 of these galaxies. In order
to explore the dust properties we estimated the residuals between the best
fitting model including only the stellar component and the data-points,
looking for an excess at long wavelengths due to dust. We then integrated the
residuals to obtain the total dust luminosity in the range covered by the
Spitzer data, i.e. $\sim$ 3 - 26 $\mu$m. The integration is performed by
assuming that the spectrum is represented by a multiple step function (see
Fig. 12).
Figure 12.— Residuals obtained by subtracting the stellar component from the
SED in the 5-bands Spitzer data which we ascribe to the dust emission. We show
the results for object 3 (black) and 5 (red) as an example. The step function
connecting the data-points is used to obtain the IR excess flux, i.e., the
total dust flux.
The resulting dust luminosities (see Table 4), estimated as IR excess
luminosities, are in the range $L_{\rm dust}\sim 10^{43.5}-10^{45.5}$ erg s-1.
A trend links radio and dust emission (see Fig. 13, left panel) with most
(7/9) HP radio-sources showing a significant dust emission with luminosities
larger than $L_{\rm dust}\sim 10^{44}$ erg s-1. The LPs are instead (with only
2 exceptions) of lower Ldust and in many cases (16 galaxies) only upper limits
can be derived.
We check the statistical significance of such a trend, by using a censored
statistical analysis (ASURV, Lavalley et al. 1992) which takes into account of
the presence of upper limits. Using the generalized Kendall’s $\tau$ test
(Kendall, 1983), the probability that a fortuitous correlation appears is
0.0003. However, the common dependence of the two luminosities on redshift
might play an important role. Therefore, we perform a partial correlation
analysis (Akritas & Siebert, 1996) to examine the linear relation between the
luminosities excluding the dependence on redshift. Operatively, we use the
partial Kendall’s $\tau$ test, whose null hypothesis is the absence of the
correlation excluding the redshift variable. The partial Kendall’s coefficient
$\tau$ is 0.20 and the standard deviation is 0.081. The null hypothesis is
then rejected at the level of 0.05.
We also measured the spectral index of the residuals between 8 $\mu$m and 24
$\mu$m, $\alpha_{(8-24)}$. Taking into account only significant excesses at 8
$\mu$m (residuals larger than 3 $\sigma$), this value can be estimated in 8
cases, with values spanning between $\alpha_{(8-24)}\sim$ 1 and -1. For the
remaining objects with only a 24 $\mu$m detection, the limit to the 8 $\mu$m
flux translates into a lower limit of $\alpha_{(8-24)}\gtrsim 1$. This
parameter can be crudely related to the overall dust temperature. By assuming
a single black-body dust component, the values of $\alpha_{(8-24)}$ translate
into a temperature range of 500-850 K and 300-550 K for $\alpha_{(8-24)}=-1$
and $1$, respectively. The derived temperature depends on redshift, with the
lower (upper) values of T being derived for $z=0.75$ ($z=2$).
Fig. 13 (middle panel) shows a broad relation between the IR spectral index
and the IR excess luminosities. This would mean that the decrease of
$\alpha_{(8-24)}$ is ascribable to the increase of the high-temperature dust
component. Since a large radio power also implies a large dust luminosity,
this may suggest a possible AGN nature of the dust heating source for the
objects with largest IR excess.
Figure 13.— Infrared excess luminosity ($\>{\rm erg}\,{\rm s}^{-1}$) versus:
(left panel) K-corrected FIRST radio luminosity ($\>{\rm erg}\,{\rm s}^{-1}$),
and (right panel) spectral index from 8 to 24 $\mu$m estimated from the IR
excess in the SED above the stellar emission. Empty points are the LPs and
full points are the HPs.
### 6.5. UV excess
As noted in Sect. 5.1, the Hyperz code often does not reproduce satisfactorily
the bluest part of the SED and we chose to model such part of spectrum with a
young stellar population, included by our 2SPD. The inclusion of this
component does not alter significantly the photometric redshift or the
galaxies mass, but it is clearly of great importance to understand the nature
of these radio galaxies. Nonetheless, inspection of the SEDs obtained with
2SPD indicates that the UV excesses (above the contribution of the old stellar
component) are usually very poorly constrained. Furthermore, the very stellar
origin is not granted and the UV excess might hide an AGN contribution.
It is necessary to introduce a model-independent criterion to assess which
sources actually show an UV excess and to estimate its luminosity. We visually
inspected all SEDs, searching for sources with a substantial flattening in the
SED at short wavelengths or with a change of the slope between the OSP and the
emission in the UV band. Fifteen sources show a clear UV excess (namely object
2, 3, 4, 29, 31, 32, 34, 37, 38, 52, 70, 202, 226, 258, and 285) and
additional 7 objects show a marginal UV excess (namely object 13, 16, 20, 25,
26, 219, and 224). In the remaining galaxies the SEDs in UV bands drop sharply
and are well reproduced by the emission from OSPs. In order to quantify the UV
contribution, we measure the flux at 2000 Å in the rest frame, LUV, from the
best fitting model, for those objects showing the UV excess. The UV
luminosities range is $10^{42}-10^{44}$ erg s-1. HPs show larger UV
luminosities than LPs by a factor 2.5, even though the strongest UV excess is
seen in the LP radio galaxy 37. If we concentrate the sources which do not
show an UV excess, the 2000 Å luminosity is $\lesssim 10^{43}$ erg s-1
(mostly, $\sim 10^{42}$ erg s-1).
In order to explore the nature of the UV emission, it is useful to compare the
UV luminosity with the multiband properties obtained in the previous sections.
No relations appear to link the UV excess to the redshift and the mass of the
galaxy. Conversely, the UV luminosity appears to be linked to the radio power
and IR excess luminosity (Fig. 14).
For the radio and UV luminosities the linear correlation coefficient is
$r=0.404$ and the probability of the presence of a fortuitous relation is P=
0.062. The partial Kendall’s coefficient $\tau$ is 0.17 and the standard
deviation is 0.15: the null hypothesis, i.e. the absence of the correlation,
can not be rejected.
For the IR-UV relation, the generalized Kendall’s $\tau$ test returns a
probability of no correlation of P= 0.0072. We obtain a partial Kendall’s
coefficient $\tau$ = 0.41 and a standard deviation of 0.069: the null
hypothesis, i.e. the absence of the correlation, is rejected at the level of
0.05.
Figure 14.— UV luminosity ($\>{\rm erg}\,{\rm s}^{-1}$) measured at 2000 Å
versus: (left panel) K-corrected FIRST radio luminosity ($\>{\rm erg}\,{\rm
s}^{-1}$) , and (right panel) infrared excess luminosity ($\>{\rm erg}\,{\rm
s}^{-1}$) . Empty points are the LPs and full points are the HPs.
## 7\. Discussion and Conclusions
We used the multiwavelength data provided by the COSMOS survey to analyze the
first sizeable sample of 34 low-power radio galaxies at high redshifts (z
$\gtrsim$ 1), selected by Chiaberge et al. (2009). We performed a careful
visual inspection of all their multiwavelength counterparts to identify their
genuine emission and to infer their SEDs. Those would have been compromised in
case of a blind use of the COSMOS catalog. Taking advantage of the FUV-MIR
spectral coverage, we modeled their SEDs using two different template fitting
techniques: i) the Hyperz code that uses a single stellar population and ii)
our own code 2SPD that includes also dust component(s) and a young stellar
population. We analyzed the properties of the SEDs of the sample. We here
summarize the main results and briefly discuss them:
1. (i)
The photometric redshifts of these radio sources range from $\sim$0.7 to 3\.
The photo-z measured with the two techniques are generally consistent with
each other, with those measured by Ilbert et al. (2009), and with the
available spectroscopic redshifts. In addition, we measured the photometric
redshifts for five objects which were not included in the Ilbert et al. (2009)
sample.
2. (ii)
The new and accurate measurements of the photometric redshifts enable us to
infer the radio power distribution of these objects. The resulting K-corrected
1.4 GHz luminosities are in the range 10${}^{31.5}-10^{33.3}$ erg s-1 Hz-1,
straddling the FR I/FR II break luminosity (L${}_{1.4\,\rm GHz}\sim 10^{32.6}$
erg s-1 Hz-1) as defined for low redshifts galaxies.
3. (iii)
The resulting stellar masses are mostly confined in the range
$\sim$1010.5-11.5 M⊙, with a median value of 7 $\times$ 1010 M⊙.
4. (iv)
The SED of most objects of the sample is consistent with the presence of a
dominant contribution from an old stellar population with an age $\sim
1-3\times$ 109 years. However, significant excesses are often observed at the
shortest and/or longest wavelengths.
5. (v)
A dust component is needed to account for the 24 $\mu$m emission in 15
galaxies and significant excesses above the stellar emission are observed also
at shorter infrared wavelengths in 8 of these galaxies. Estimates of the dust
luminosity yield values in the range $L_{\rm dust}\sim 10^{43.5}-10^{45.5}$
erg s-1. The overall dust temperature, estimated for the 8 radio-galaxies with
a substantial dust excess at $\lambda\lesssim 8$$\mu$m, is in the range
300-850 K.
6. (vi)
Inspection of the SED obtained with 2SPD indicates that the UV excesses (above
the contribution of the old stellar component) are often present
(significantly in 15 sources and marginally in 7 sources), but they are
usually weakly constrained. The UV luminosities measured at 2000 Å (rest
frame) is in the range $10^{42}-10^{44}$ erg s-1.
7. (vii)
Although the censored analysis does not provide significantly high statistical
parameters, we can tentatively confirm the presence of positive links between
the dust emission with both the radio and UV luminosities. For these
relations, the possibility of a common luminosity dependence on distances is
rejected at the level of 0.05.
The selection performed by Chiaberge et al. (2009) with the aim of searching
for FR I candidates at z $\sim 1-2$ turned out to be successful. In fact, our
work confirms i ) their location in the range of redshifts aimed with the
selection process, although extending slightly below 1 and up to z $\sim$ 3
and ii) the low radio luminosity of the sample, generally consistent with
those of the local FR I, although 1/3 of the sample exceeds the local FR I/FR
II luminosity break. The extension of their radio power to larger luminosities
does not necessarily imply a FR II nature for those sources. In fact the
sources with a clear FR II morphology were excluded from the analysis. In
addition, in the local Universe, the radio distribution of FR Is is broad and
overlaps with the FR II distribution (Zirbel & Baum, 1995). Furthermore, this
overlap increases at higher radio frequencies.
Overall, the hosts of these high-z low-luminosity radio-sources are similar to
those of the local FR I which usually live in red massive early-type galaxies
(e.g., Zirbel 1996; Best et al. 2005; Baldi & Capetti 2008; Smolčić 2009;
Baldi & Capetti 2010).
However, the SED modeling reveals that additional components to the old
stellar population have to be included to account for the emission at the SED
extremes, i.e. in either the UV or in the MIR band, in most of the sources of
the sample. This behavior is not seen in the low-z FR I that are generally
faint in UV (both from the point of view of star formation and nuclear
emission, Chiaberge et al. 2002; Baldi & Capetti 2008). Similarly, their MIR
luminosities exceed by a very large factor (between 30 and 3000 for the MIR
detected sources) the typical low-z FR I luminosities ($\sim 10^{42}\>{\rm
erg}\,{\rm s}^{-1}$, Hardcastle et al. 2009). Conversely, the UV and MIR
properties are somewhat similar to those of local FR IIs, which show bluer
color (e.g., Baldi & Capetti 2008; Smolčić 2009) and large dust amount (e.g.,
de Koff et al. 2000; Dicken et al. 2010) than FR Is.
The origin of the MIR and UV emission can not be firmly established based on
the available data. The estimate of the dust temperature (possible for only 8
objects) is in the range expected from dust heating from a quasar-like nucleus
($T\gtrsim 300$ K, e.g. Siebenmorgen et al. 2004; Ogle et al. 2006) and far
larger than the dust associated with star formation (e.g., Hwang et al. 2010;
Sreenilayam & Fich 2011; Boquien et al. 2011; Patel et al. 2011). The high end
of the MIR luminosity reaches values similar to those found in high power
radio-galaxies and QSOs. However, dust emission is seen only in less than half
the sources of the sample.
Similarly, a UV component in excess to the old stellar population is firmly
detected in the same number of sources (but there is not a one-to-one
correspondence between UV and MIR emission). The observed UV luminosities are
much larger that the faint non-thermal UV-nuclei seen in FR I and, instead,
similar to those FR IIs (Chiaberge et al., 1999, 2002), dominated by the
accretion disk emission.
Summing up, we find that the sources of the sample display a wide variety of
properties, despite the relatively narrow range in radio luminosities. The 8
objects with the strongest MIR excess, and with high dust temperature
indicative of a quasar-like nature, also show a significant UV excess. To this
group we must obviously add the object 236, the spectroscopically confirmed
QSO. We must note, however, that the SED of these 8 sources is much redder
than that of object 236 and that their HST images do not provide any evidence
for the presence of bright unresolved nuclei, as described by Chiaberge et al.
(2009) based on visual inspection of the data.
At the opposite end of the radio luminosity distribution, there are 7 galaxies
not showing any UV excess nor any 24 $\mu$m emission (and this number raises
to 11 by including also the objects with only marginal UV excesses).
The remaining galaxies, amounting to about half of the sample, show an excess
only in either the UV or MIR band. For those objects which do not show an UV
excess or are not detected at 24 $\mu$m, the limits to the UV/MIR ratio are
broadly consistent with the values obtained for the remaining galaxies of the
sample. In particular we do not find objects with a high MIR luminosity
without an UV excess that might be expected in the case of an obscured QSO.
Unfortunately, with the available information we cannot distinguish between an
origin related to star formation or to an active nucleus. In addition,
relatively large amounts of dust, suggested by the dust luminosities, indicate
that possible obscuration may prevent us from detecting UV emission.
A further detailed analysis of the nuclear properties is needed to understand
which type of AGN are associated with these high-z radio galaxies. This will
include their X-ray emission (Tundo et al., 2011) and the radio core flux,
available from the COSMOS/VLA data, but which we defer to a future study. This
might also provide new insights on the controversial FR I-QSO association
(e.g., Falcke et al. 1995; Baum et al. 1995; Cao & Rawlings 2004; Blundell &
Rawlings 2001, see also Blundell 2003 for review on this subject).
Summarizing, this work validates the first sizeable sample of low-luminosity
radio galaxies at high redshifts, 0.7 $\lesssim$ z $\lesssim$ 3\. This opens
the possibility to perform a detailed comparison of the host and nuclear
properties of these sources with those of i) the local low-luminosity radio-
galaxies, ii) the powerful radio-galaxies in the same redshift range, and iii)
the population of non active galaxies at $z\sim 1-3$. These issues will be
addressed in a forthcoming paper.
R.D.B. acknowledges the financial support (grant DDRF D0001.82439) from Space
Telescope Science Institute, Baltimore. We are grateful to M. Bolzonella, C.
Maraston, and J. Pforr. which significantly help with the SED modeling. We
also thank the referee and A. Celotti for their contributions to improve the
paper.
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$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{1cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed1_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{2cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed2_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{3cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed3_2sp_2dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{4cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed4_2sp_dustnew.ps}\end{array}$
Figure 15.— Left panels: SEDs of the sample (green line) fitting the
photometric points, as result of Hyperz. We plot the photometric point at 24
$\mu$m (Spitzer/MIPS), if available, but is not included in the modeling.
Right panels: SEDs of the objects fitted with 2SPD. The total model is the
green line, the YSP is the blue line, the OSP is the red line, and the dust
component(s) is the light blue line. For both the panels, the wavelengths on
top of the plots correspond to observed wavelengths, while those on bottom are
at rest frame. For object 236 (QSO) we only show the photometric points.
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{5cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed5_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{11cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed11_2sp_dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{13cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed13_2sp_2dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{16cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed16_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{18cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed18_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{20cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed20_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{22cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed22_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{25cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed25_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{26cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed26_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{28cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed28_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{29cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed29_2sp_dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{30cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed30_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{31cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed31_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{32cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed32_2sp_2dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{34cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed34_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{36cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed36_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{37cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed37_2sp_2dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{38cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed38_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{39cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed39_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{52cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed52_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{ccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{70cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed70_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{202cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed202_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{219cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed219_2sp_dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{224cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed224_2sp_dustnew.ps}\end{array}$
Figure 15.— Continued
$\begin{array}[]{cccc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{226cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed226_2sp_2dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{228cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed228_2sp_dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{234cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed234_2sp_dustnew.ps}\\\
\vspace{1em}\includegraphics[scale={0.33},angle={90}]{236cspnew.ps}&\end{array}$
Figure 15.— Continued
$\begin{array}[]{cc}\vspace{2em}\includegraphics[scale={0.33},angle={90}]{258cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed258_2sp_dustnew.ps}\\\
\vspace{2em}\includegraphics[scale={0.33},angle={90}]{285cspnew.ps}&\includegraphics[scale={0.33},angle={90}]{sed285_2sp_2dustnew.ps}\end{array}$
Figure 15.— Left panels: SEDs of the sample (green line) fitting the
photometric points, as result of Hyperz. We plot the photometric point at 24
$\mu$m (Spitzer/MIPS), if available, but is not included in the modeling.
Right panels: SEDs of the objects fitted with 2SPD. The total model is the
green line, the YSP is the blue line, the OSP is the red line, and the dust
component(s) is the light blue line. For both the panels, the wavelengths on
top of the plots correspond to observed wavelengths, while those on bottom are
at rest frame. For object 236 (QSO) we only show the photometric points
Table 6COSMOS multiwavelength counterparts of the sample
ID | $u^{*}$ | $B_{J}$ | $g^{+}$ | $V_{J}$ | $r^{+}$ | $i^{*}$ | $i^{+}$ | $F814W$ | $z^{+}$ | $J$ | $K_{S}$ | $K$
---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
1 | 26.56$\pm$0.15 | 25.89$\pm$0.13 | 25.61$\pm$0.10 | 24.42$\pm$0.05 | 23.55$\pm$0.03 | 22.24$\pm$0.05 | 22.29$\pm$0.11∗ | 21.87$\pm$0.06 | 21.32$\pm$0.01 | 20.46$\pm$0.10∗ | 19.59$\pm$0.09∗ | 19.57$\pm$0.03
2 | 26.79$\pm$0.20∗ | 26.43$\pm$0.11∗ | 26.29$\pm$0.11∗ | 25.93$\pm$0.10∗ | 25.26$\pm$0.08∗ | 24.35$\pm$0.45∗ | 24.43$\pm$0.10∗ | 24.49$\pm$0.34∗ | 23.44$\pm$0.07∗ | 22.18$\pm$0.15∗ | 20.85$\pm$0.02 | 20.90$\pm$0.15
3 | 26.69$\pm$0.17 | 25.58$\pm$0.10 | 25.81$\pm$0.12 | 25.60$\pm$0.11 | 25.53$\pm$0.10 | 25.31$\pm$0.78 | 25.10$\pm$0.08 | 24.96$\pm$0.79 | 24.48$\pm$0.12 | 23.39$\pm$0.25∗ | 21.57$\pm$0.03 | 22.84$\pm$0.96
4 | 26.64$\pm$0.17 | 25.93$\pm$0.13 | 25.88$\pm$0.13 | 25.46$\pm$0.10 | 25.12$\pm$0.08 | 24.27$\pm$0.36 | 24.20$\pm$0.05 | 23.82$\pm$0.29 | 23.27$\pm$0.04 | 21.93$\pm$0.10∗ | 20.53$\pm$0.01 | 20.74$\pm$0.16
5 | $<$25.77∗ | $<$25.68∗ | $<$25.53∗ | $<$25.17∗ | $<$25.23∗ | 24.12$\pm$0.27 | 24.59$\pm$0.41∗ | 23.98$\pm$0.33 | 23.41$\pm$0.25 | 22.24$\pm$0.13∗ | 20.65$\pm$0.12 | 20.82$\pm$0.15
11 | $<$29.40∗ | 28.75$\pm$1.14 | 28.00$\pm$0.76 | 27.11$\pm$0.35 | 26.27$\pm$0.17 | 25.16$\pm$0.84 | 24.96$\pm$0.07 | 24.64$\pm$0.60 | 24.09$\pm$0.08 | 22.28$\pm$0.07∗ | 21.18$\pm$0.16 | 21.75$\pm$0.49
13 | 26.14$\pm$0.15∗ | 25.75$\pm$0.13∗ | 25.74$\pm$0.13 | 25.02$\pm$0.09∗ | 24.38$\pm$0.06∗ | 23.20$\pm$0.17 | 23.29$\pm$0.04∗ | 22.90$\pm$0.13 | 22.21$\pm$0.02 | 21.44$\pm$0.09∗ | 20.46$\pm$0.10 | 20.54$\pm$0.16
16 | 26.07$\pm$0.12 | 25.71$\pm$0.12 | 25.32$\pm$0.11 | 24.84$\pm$0.08 | 24.23$\pm$0.05 | 23.10$\pm$0.14 | 23.15$\pm$0.03 | 22.85$\pm$0.10 | 22.16$\pm$0.02 | 21.47$\pm$0.08∗ | 20.40$\pm$0.08 | 20.50$\pm$0.11
18 | 26.16$\pm$0.13∗ | 25.76$\pm$0.10∗ | 25.51$\pm$0.10 | 24.61$\pm$0.08∗ | 23.99$\pm$0.06∗ | 22.91$\pm$0.11∗ | 22.80$\pm$0.04∗ | 22.51$\pm$0.10∗ | 22.04$\pm$0.03∗ | 21.54$\pm$0.07∗ | 20.71$\pm$0.04 | 20.92$\pm$0.18
20 | 26.11$\pm$0.15 | 25.74$\pm$0.15 | 25.51$\pm$0.10 | 24.57$\pm$0.06 | 23.67$\pm$0.04 | 22.35$\pm$0.09 | 22.40$\pm$0.02 | 22.19$\pm$0.08 | 21.53$\pm$0.02 | 20.79$\pm$0.06∗ | 19.91$\pm$0.07 | 20.10$\pm$0.07
22 | $<$27.38∗ | $<$27.44∗ | $<$26.97∗ | $<$26.85∗ | $<$26.00∗ | | $<$25.13∗ | | 23.93$\pm$0.06 | 22.14$\pm$0.17∗ | 20.79$\pm$0.10 | 21.24$\pm$0.15
25 | 26.66$\pm$0.22 | 25.36$\pm$0.09 | 25.44$\pm$0.10 | 24.97$\pm$0.07 | 24.55$\pm$0.05 | 23.64$\pm$0.32 | 23.52$\pm$0.03 | 23.32$\pm$0.19 | 22.57$\pm$0.03 | 21.43$\pm$0.09∗ | 20.57$\pm$0.05 | 20.59$\pm$0.17
26 | 26.58$\pm$0.20 | 25.44$\pm$0.10 | 25.56$\pm$0.10 | 24.92$\pm$0.07 | 23.90$\pm$0.04 | 22.68$\pm$0.10 | 22.60$\pm$0.02 | 22.32$\pm$0.09 | 21.57$\pm$0.01 | 20.42$\pm$0.04∗ | 19.42$\pm$0.03 | 19.48$\pm$0.04
28 | $<$27.39∗ | $<$27.50∗ | $<$27.20∗ | $<$26.90∗ | $<$26.80∗ | | 25.55$\pm$0.30∗ | | 24.27$\pm$0.20∗ | 22.48$\pm$0.20∗ | 20.99$\pm$0.08∗ |
29 | 25.81$\pm$0.12 | 25.51$\pm$0.10 | 25.49$\pm$0.09 | 25.31$\pm$0.09 | 25.18$\pm$0.08 | 25.00$\pm$0.87 | 24.78$\pm$0.07 | 24.92$\pm$0.74 | 24.30$\pm$0.11 | 22.86$\pm$0.27∗ | 22.19$\pm$0.10 | 22.89$\pm$0.80
30 | 27.85$\pm$0.63 | 26.77$\pm$0.22 | 26.82$\pm$0.23 | 25.90$\pm$0.13 | 24.69$\pm$0.06 | 23.41$\pm$0.19 | 23.41$\pm$0.03 | 23.01$\pm$0.15 | 22.32$\pm$0.02 | 21.25$\pm$0.08∗ | 20.02$\pm$0.04 | 20.19$\pm$0.09
31 | 24.75$\pm$0.08 | 24.50$\pm$0.06 | 24.51$\pm$0.05 | 23.94$\pm$0.04 | 23.42$\pm$0.03 | 22.33$\pm$0.08 | 22.33$\pm$0.02 | 22.07$\pm$0.07 | 21.61$\pm$0.02 | 20.97$\pm$0.08∗ | 20.12$\pm$0.05 | 20.32$\pm$0.09
32 | $<$27.54∗ | 27.85$\pm$0.48∗ | $<$27.19∗ | $<$27.00∗ | 26.44$\pm$0.30∗ | | 26.07$\pm$0.25∗ | | 25.15$\pm$0.28∗ | 24.13$\pm$0.50∗ | 22.43$\pm$0.17∗ | 23.24$\pm$1.44
34 | 26.09$\pm$0.40∗ | 26.19$\pm$0.20∗ | 26.00$\pm$0.19∗ | 25.58$\pm$0.12∗ | 25.44$\pm$0.10∗ | 25.09$\pm$0.66∗ | 24.88$\pm$0.07∗ | 24.95$\pm$0.25∗ | 24.12$\pm$0.11∗ | 23.10$\pm$0.51∗ | 21.10$\pm$0.29∗ | 20.96$\pm$0.14
36 | $<$27.8∗ | $<$26.80∗ | $<$26.40∗ | 26.18$\pm$0.45∗ | 25.50$\pm$0.31∗ | 24.31$\pm$0.33∗ | 24.15$\pm$0.15∗ | 24.00$\pm$0.25∗ | 23.20$\pm$0.24∗ | 22.04$\pm$0.15∗ | 20.62$\pm$0.07 | 20.44$\pm$0.10
37 | 22.70$\pm$0.02 | 22.35$\pm$0.02 | 22.63$\pm$0.02 | 22.38$\pm$0.02 | 22.18$\pm$0.02 | 21.93$\pm$0.05 | 21.93$\pm$0.01 | | 21.17$\pm$0.01 | 20.31$\pm$0.05∗ | 20.14$\pm$0.11∗ | 20.06$\pm$0.08
38 | 25.50$\pm$0.10 | 25.00$\pm$0.08 | 24.96$\pm$0.07 | 24.66$\pm$0.06 | 24.20$\pm$0.04 | 23.58$\pm$0.32 | 23.49$\pm$0.03 | 23.47$\pm$0.22 | 22.69$\pm$0.03 | 22.08$\pm$0.16∗ | 20.89$\pm$0.05 | 21.34$\pm$0.22
39 | $<$27.07∗ | 26.05$\pm$0.15∗ | $<$26.00∗ | 25.28$\pm$0.09 | 24.42$\pm$0.05 | 23.13$\pm$0.18 | 23.13$\pm$0.03 | 22.86$\pm$0.15 | 22.18$\pm$0.02 | 21.06$\pm$0.08∗ | 19.99$\pm$0.04 | 20.27$\pm$0.09
52 | 24.24$\pm$0.04 | 23.64$\pm$0.04 | 23.76$\pm$0.04 | 23.13$\pm$0.03 | 22.62$\pm$0.02 | 21.62$\pm$0.05 | 21.67$\pm$0.01 | 21.52$\pm$0.05 | 21.18$\pm$0.01 | 20.58$\pm$0.05∗ | 19.79$\pm$0.04 | 19.78$\pm$0.06
70 | 26.16$\pm$0.14∗ | 25.14$\pm$0.09∗ | 25.27$\pm$0.14∗ | 24.79$\pm$0.08∗ | 24.51$\pm$0.06 | 24.34$\pm$0.58 | 24.35$\pm$0.06 | 24.13$\pm$0.43 | 24.01$\pm$0.09 | 23.42$\pm$0.50∗ | 21.76$\pm$0.09 | 21.40$\pm$0.27
202 | 27.03$\pm$0.28 | 27.11$\pm$0.29 | 27.40$\pm$0.42 | 26.92$\pm$0.29 | 25.86$\pm$0.12 | 24.44$\pm$0.53 | 24.41$\pm$0.06 | 24.29$\pm$0.45 | 23.48$\pm$0.05 | 22.19$\pm$0.17∗ | 21.07$\pm$0.05 | 21.62$\pm$0.30
219 | 25.68$\pm$0.10 | 25.09$\pm$0.08 | 25.23$\pm$0.08 | 24.49$\pm$0.05 | 23.81$\pm$0.04 | 22.74$\pm$0.09 | 22.71$\pm$0.02 | 22.37$\pm$0.09 | 21.70$\pm$0.02 | 21.02$\pm$0.06∗ | 19.93$\pm$0.04 | 20.15$\pm$0.37
224 | 26.40$\pm$0.18 | 25.75$\pm$0.13 | 25.52$\pm$0.10 | 25.36$\pm$0.09 | 24.62$\pm$0.06 | 23.52$\pm$0.27 | 23.49$\pm$0.03 | | 22.50$\pm$0.03 | 21.31$\pm$0.08∗ | 20.39$\pm$0.04 | 20.49$\pm$0.18
226 | 25.40$\pm$0.11 | 25.30$\pm$0.09 | 25.48$\pm$0.11 | 25.23$\pm$0.09 | 25.17$\pm$0.08 | 24.59$\pm$1.00 | 24.75$\pm$0.08 | 24.78$\pm$0.68 | 24.37$\pm$0.11 | 23.41$\pm$0.46∗ | 21.90$\pm$0.10 | 21.75$\pm$0.36
228 | $<$28.50∗ | $<$27.70∗ | $<$27.70∗ | $<$26.80∗ | 26.17$\pm$0.16 | | 25.17$\pm$0.10 | 25.17$\pm$0.67 | 24.38$\pm$0.11 | 23.10$\pm$0.40∗ | 21.14$\pm$0.06 | 21.23$\pm$0.22
234 | 27.40$\pm$0.51∗ | 26.43$\pm$0.22∗ | 26.65$\pm$0.28∗ | 25.50$\pm$0.12∗ | 24.88$\pm$0.08∗ | 23.76$\pm$0.30 | 23.52$\pm$0.08∗ | | 22.76$\pm$0.03 | 21.38$\pm$0.20∗ | 20.31$\pm$0.11∗ | 20.57$\pm$0.39
236 | 20.92$\pm$0.01 | 20.73$\pm$0.01 | 21.02$\pm$0.01 | 20.59$\pm$0.01 | 20.65$\pm$0.01 | 20.32$\pm$0.02 | | 20.26$\pm$0.02 | 19.94$\pm$0.01 | 20.05$\pm$0.04∗ | 19.27$\pm$0.02 | 19.32$\pm$0.04
258 | 24.01$\pm$0.04 | 23.54$\pm$0.04 | 23.70$\pm$0.03 | 23.19$\pm$0.03 | 22.68$\pm$0.02 | 21.87$\pm$0.06 | 21.85$\pm$0.01 | 21.56$\pm$0.04 | 21.11$\pm$0.01 | 20.61$\pm$0.05∗ | 19.62$\pm$0.04 | 19.71$\pm$0.04
285 | 24.60$\pm$0.18∗ | 24.65$\pm$0.15∗ | 24.63$\pm$0.24∗ | 23.88$\pm$0.14∗ | 23.70$\pm$0.12∗ | 23.32$\pm$0.28 | 23.06$\pm$0.10∗ | | 22.36$\pm$0.11∗ | 21.80$\pm$0.13∗ | 20.78$\pm$0.10∗ | 20.84$\pm$0.10
Note. — Column description: (1) ID number of the object; (2)-(3) right
ascension and declination of radio source; (4) photometric redshifts found by
Ilbert et al. (2009); (5) spectroscopic redshift: $a$ from zCOSMOS survey
(Lilly et al., 2007), $b$ form Magellan survey (Trump et al., 2007), and $c$
from Prescott et al. (2006). The spectroscopic redshift of object 25 (marked
with $*$) is considered incorrect (see Section 6.1); (6)-(7) flux at 1.4 GHz
from NVSS (from http://www.cv.nrao.edu/nvss/NVSSlist.shtml) and FIRST (taken
from Chiaberge et al. 2009) catalogs in mJy.
Note. — List of the filters used for our analysis, the associated telescopes,
the effective wavelengths $\lambda_{eff}$, the PSF full-width half maximum
(FWHM) of the images in each band, and their sensitivities at 5$\sigma$ (mag).
Note. — Results from the analysis of the SEDs of the sample with Hyperz.
Column description: (1) ID number of the object; (2) photometric redshift as
result of Hyperz; all errors are quoted at at 99% of confidence level; (3)
photometric redshifts obtained with 2SPD; (4) source library of the best
template which fits the observed SED: BC from Bruzual & Charlot (2009) and MA
from Maraston (2005); (5) star formation history of the best template: $ssp$
corresponds to a single stellar population and $\tau$ = ’N’ corresponds to the
$\mu$-models with exponentially decaying star formation rate with star
formation timescale $\tau$ = N Gyr; (6) age of the best template in Gyr; (7)
$A_{V}$ associated with the best template; (8) $\chi_{rid}^{2}$ of the best
template; (9) stellar mass and its error of the galaxy in $Log\,M_{\odot}$ as
result of Hyperzmass.
Note. — Results from the analysis of the SEDs with 2SPD. Column description:
(1) ID number of the object; (2) photometric redshift measured with 2SPD;
(3)-(4)-(5)-(6) age in Gyr, AV, flux fraction and mass fraction of the young
stellar population (YSP) at 4800 Årest frame; (7)-(8) age in Gyr and AV of the
old stellar population (OSP); (9) the total stellar mass of the galaxy in M⊙;
(10)-(11) the effective temperature (in K) of the one or two dust components
and their luminosities, Ldust (in units of 109 L⊙); (12)-(13) the infrared
excess luminosity (in erg s-1) defined in the text (Section 6.4) and the
spectral index measured on the infrared excess at 8 and 24$\mu$m; (14) UV
luminosity at 2000 Å in the rest frame in erg s-1. The marginal UV excesses
are marked with a m.
Note. — Column description: (1) ID number of the object; (2) redshift of the
object used throughout the work, with the spectroscopic redshifts marked with
a s; (3)-(4) K-corrected radio luminoisty at 1.4 GHz from NVSS (from
http://www.cv.nrao.edu/nvss/NVSSlist.shtml) and FIRST (taken from Chiaberge et
al. 2009) in erg s-1 Hz-1; (5) classification based on the radio power: low or
high power (LP or HP) radio sources.
Note. — Column description: (1) ID number of the object; (2) CFHT $u^{*}$
magnitude with its error; (3)-(4)-(5)-(6) Subaru $B_{J}$, $g^{+}$, $V_{J}$,
$r^{+}$ magnitudes with their errors; (7) CFHT $i^{*}$ magnitude with its
error; (8) Subaru $i^{+}$ magnitude with its error; (9) HST/ACS $F814W$
magnitude with its error; (10) Subaru $z^{+}$ magnitude with its error; (11)
UKIRT $J$ magnitude with its error; (12) CFHT $K$ magnitude with its error;
(13) NOAO$K_{S}$ with its error. The values marked by $*$ are measured by our
3″-aperture photometry on the images.
Table 7COSMOS multiwavelength counterparts of the sample
ID | $FUV$ | $NUV$ | $IRAC1$ | $IRAC2$ | $IRAC3$ | $IRAC4$ | $MIPS$
---|---|---|---|---|---|---|---
1 | | | 54.89$\pm$0.23∗ | 36.50$\pm$0.29∗ | 23.87$\pm$0.97∗ | 14.73$\pm$2.07∗ | $<$0.15
2 | | | 41.44$\pm$0.19 | 45.75$\pm$0.30 | 28.07$\pm$0.99 | 21.8$\pm$2.49 | $<$0.15
3 | | | 13.65$\pm$0.15 | 23.62$\pm$0.25 | 55.88$\pm$0.94 | 166.89$\pm$1.98 | 1.47$\pm$0.02
4 | | | 42.97$\pm$0.19 | 48.78$\pm$0.29 | 37.71$\pm$1.04 | 29.34$\pm$2.19 | $<$0.08
5 | | | 66.44$\pm$0.19 | 89.03$\pm$0.29 | 91.57$\pm$0.99 | 63.65$\pm$1.97 | 0.87$\pm$0.02
11 | | | 25.34$\pm$0.17 | 29.34$\pm$0.26 | 25.07$\pm$1.00 | 9.64$\pm$1.98 | $<$0.15
13 | | | 49.51$\pm$0.18 | 51.39$\pm$0.26 | 53.24$\pm$0.98 | 75.84$\pm$2.04 | 0.32$\pm$0.03
16 | | | 44.29$\pm$0.18 | 32.54$\pm$0.28 | 26.19$\pm$0.94 | 18.39$\pm$2.26 | 0.22$\pm$0.02
18 | | | 31.03$\pm$0.17 | 24.94$\pm$0.28 | 18.13$\pm$0.99 | 14.03$\pm$2.15 | $<$0.61∗
20 | | | 63.11$\pm$0.18 | 44.73$\pm$0.27 | 25.12$\pm$0.90 | 17.46$\pm$2.06 | $<$0.08
22 | | | 57.27$\pm$ 0.17 | 64.90$\pm$0.29 | 41.55$\pm$0.89 | 32.84$\pm$2.16 | 0.25$\pm$0.03∗
25 | | | 49.17$\pm$0.19 | 55.52$\pm$0.28 | 38.82$\pm$0.99 | 25.08$\pm$2.19 | 0.18$\pm$0.05∗
26 | | | 126.74$\pm$0.25 | 108.17$\pm$0.29 | 59.54$\pm$0.98 | 44.75$\pm$1.92 | $<$0.15
28 | | | 32.46$\pm$0.16 | 39.98$\pm$0.26 | 46.73$\pm$0.91 | 40.34$\pm$1.98 | 0.15$\pm$0.02
29 | | | 9.81$\pm$0.16 | 10.17$\pm$0.25 | 6.50$\pm$1.00∗ | $<$7.38∗ | $<$0.15
30 | | | 74.26$\pm$0.19 | 58.67$\pm$0.28 | 33.32$\pm$0.91 | 22.76$\pm$2.05 | $<$0.15
31 | | 25.15$\pm$0.24 | 47.86$\pm$0.16 | 34.60$\pm$0.25 | 24.98$\pm$0.81 | 19.36$\pm$1.78 | $<$0.15
32 | | | 16.97$\pm$0.15 | 32.41$\pm$0.24 | 58.17$\pm$0.95 | 98.81$\pm$1.81 | 0.26$\pm$0.02
34 | | | 22.80$\pm$0.44∗ | 26.73$\pm$0.48∗ | 25.44$\pm$1.10∗ | 12.07$\pm$2.40∗ | $<$0.26∗
36 | | | 49.50$\pm$0.16 | 43.80$\pm$0.27 | 30.24$\pm$0.87 | 21.89$\pm$1.97 | 0.15$\pm$0.04∗
37 | | 23.55$\pm$0.15 | 88.32$\pm$0.22 | 123.39$\pm$0.32 | 158.41$\pm$1.10 | 249.52$\pm$2.18 | 1.58$\pm$0.39
38 | | | 30.47$\pm$0.17 | 28.00$\pm$0.28 | 29.99$\pm$0.95 | 24.94$\pm$2.35 | 0.13$\pm$0.06∗
39 | | | 74.19$\pm$0.21 | 66.35$\pm$0.32 | 34.74$\pm$0.96 | 25.29$\pm$2.40 | $<$0.08
52 | | 24.97$\pm$0.21 | 59.84$\pm$0.21 | 41.39$\pm$0.28 | 36.37$\pm$1.09 | 28.67$\pm$2.04 | 0.16$\pm$0.02
70 | | | 17.58$\pm$0.16 | 20.57$\pm$0.26 | 29.83$\pm$1.04 | 21.32$\pm$2.17 | 0.13$\pm$0.03∗
202 | | | 24.81$\pm$0.14 | 22.02$\pm$0.24 | 15.90$\pm$0.84 | 6.46$\pm$1.87 | $<$0.15
219 | | | 98.71$\pm$0.23 | 86.12$\pm$0.32 | 48.50$\pm$1.02 | 38.60$\pm$2.26 | 0.12$\pm$0.01
224 | | | 48.09$\pm$0.17 | 41.65$\pm$0.26 | 25.56$\pm$0.88 | 18.53$\pm$2.07 | $<$0.15
226 | | | 16.94$\pm$0.15 | 30.93$\pm$0.28 | 53.89$\pm$0.89 | 104.92$\pm$2.16 | 0.54$\pm$0.02
228 | | | 39.72$\pm$0.18 | 47.82$\pm$0.27 | 36.06$\pm$0.99 | 23.46$\pm$2.03 | $<$0.15
234 | | | 50.19$\pm$0.19 | 45.78$\pm$0.26 | 33.95$\pm$1.03 | 14.69$\pm$2.00 | $<$0.15
236 | | | 99.23$\pm$0.20 | 146.61$\pm$0.36 | 224.02$\pm$1.04 | 324.18$\pm$2.50 | 0.98$\pm$0.04
258 | 24.97$\pm$0.15 | 24.32$\pm$0.09 | 84.11$\pm$0.23 | 59.75$\pm$0.30 | 37.90$\pm$1.06 | 31.78$\pm$2.12 | $<$0.15
285 | | 24.85$\pm$0.18 | 34.06$\pm$0.17 | 33.78$\pm$0.26 | 33.68$\pm$0.95 | 32.70$\pm$2.03 | 0.13$\pm$0.03∗
Note. — Column description: (1) ID number of the object; (2)-(3) GALEX FUV and
NUV magnitudes with their errors; (4)-(5)-(6)-(7) Spitzer/IRAC 4-channel (3.6,
4.5, 5.8, and 8.0 $\mu$m) fluxes with their errors; (8) Spitzer/MIPS flux at
24$\mu$m with its error. The values marked by $*$ are measured by our
3″-aperture photometry on the images.
|
arxiv-papers
| 2012-10-16T19:54:29 |
2024-09-04T02:49:36.660145
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ranieri D. Baldi (1,2), Marco Chiaberge (2,3,4), Alessandro Capetti\n (5), Javier Rodriguez-Zaurin (2,6), Susana Deustua (2), William B. Sparks\n (2), ((1) SISSA, Trieste, Italy, (2) Space Telescope Science Institute,\n Baltimore, USA, (3) INAF-Istituto di Radio Astronomia, Bologna, Italy, (4)\n Johns Hopkins University, Baltimore, USA, (5) INAF- Osservatorio Astronomico\n di Torino, Italy, (6) Insituto de Astrofisica de Canarias, La Laguna, Spain)",
"submitter": "Ranieri Diego Baldi",
"url": "https://arxiv.org/abs/1210.4540"
}
|
1210.4545
|
# On the convergence of Cesàro means of negative order of double trigonometric
Fourier series of functions of bounded partial generalized variation
Ushangi Goginava and Artur Sahakian U. Goginava, Institute of Mathematics,
Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze
str. 1, Tbilisi 0128, Georgia z_goginava@hotmail.com A. Sahakian, Yerevan
State University, Faculty of Mathematics and Mechanics, Alex Manoukian str. 1,
Yerevan 0025, Armenia sart@ysu.am
###### Abstract.
The convergence of Cesàro means of negative order of double trigonometric
Fourier series of functions of bounded partial $\Lambda$-variation is
investigated. The sufficient and neccessary conditions on the sequence
$\Lambda=\\{\lambda_{n}\\}$ are found for the convergence of Cesàro means of
Fourier series of functions of bounded partial $\Lambda$-variation.
00footnotetext: 2000 Mathematics Subject Classification 42B08 Key words and
phrases: Fourier series, $\Lambda$-variation, Generalized variation, Cesàro
means.
## 1\. Classes of Functions of Bounded Generalized Variation
In 1881 Jordan [9] introduced the class of functions of bounded variation and
applied it to the theory of Fourier series. Hereinafter this notion was
generalized by many authors (quadratic variation, $\Phi$-variation,
$\Lambda$-variation ets., see [2], [10], [13], [15]). In two dimensional case
the class BV of functions of bounded variation was introduced by Hardy [8].
Let $f$ be a real function of two variable of period $2\pi$ with respect to
each variable. Given intervals $I=(a,b)$, $J=(c,d)$ and points $x,y$ from
$T:=[0,2\pi]$ we denote
$f(I,y):=f(b,y)-f(a,y),\qquad f(x,J)=f(x,d)-f(x,c)$
and
$f(I,J):=f(a,c)-f(a,d)-f(b,c)+f(b,d).$
Let $E=\\{I_{i}\\}$ be a collection of nonoverlapping intervals from $T$
ordered in arbitrary way and let $\Omega$ be the set of all such collections
$E$.
The Hardy class BV consists of functions $f$ satisfying the condition
$\sup_{E\in\Omega}\sum_{i}{|f(I_{i},0)|}+\sup_{x}\sup_{F\in\Omega}\sum_{j}{|f(0,J_{j})|}+\sup_{F,\,E\in\Omega}\sum_{i}\sum_{j}{|f(I_{i},J_{j})|}<\infty,$
where $E=\\{I_{i}\\}$ and $F=\\{J_{j}\\}$.
In [6] U. Goginava introduced the class $PBV$ of functions of bounded partial
bounded variation, i.e. functions $f$ having uniformly bounded variation with
respect to each variable:
$\sup_{y}\sup_{E\in\Omega}\sum_{i}{|f(I_{i},y)|}+\sup_{x}\sup_{F\in\Omega}\sum_{j}{|f(x,J_{j})|}<\infty.$
For the sequence of positive numbers
$\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ we denote
$\Lambda
V_{1}(f)=\sup_{y}\sup_{E\in\Omega}\sum_{i}\frac{|f(I_{i},y)|}{\lambda_{i}}\,\,\,\,\,\,\left(E=\\{I_{i}\\}\right),$
$\Lambda
V_{2}(f)=\sup_{x}\sup_{F\in\Omega}\sum_{j}\frac{|f(x,J_{j})|}{\lambda_{j}}\qquad(F=\\{J_{j}\\}),$
$\Lambda
V_{1,2}(f)=\sup_{F,\,E\in\Omega}\sum_{i}\sum_{j}\frac{|f(I_{i},J_{j})|}{\lambda_{i}\lambda_{j}}.$
###### Definition 1.
We say that the function $f$ has Bounded $\Lambda$-variation on
$T^{2}=[0,2\pi]^{2}$ and write $f\in\Lambda BV$, if
$\Lambda V(f):=\Lambda V_{1}(f)+\Lambda V_{2}(f)+\Lambda V_{1,2}(f)<\infty.$
We say that the function $f$ has Bounded Partial $\Lambda$-variation and write
$f\in P\Lambda BV$ if
$P\Lambda V(f):=\Lambda V_{1}(f)+\Lambda V_{2}(f)<\infty.$
If $\lambda_{n}\equiv 1$ (or if $0<c<\lambda_{n}<C<\infty,\ n=1,2,\ldots$) the
classes $\Lambda BV$ and $P\Lambda BV$ coincide with the Hardy class $BV$ and
PBV respectively. Hence it is reasonable to assume that $\lambda_{n}\to\infty$
and since the intervals in $E=\\{I_{i}\\}$ are ordered arbitrarily, we will
suppose, without loss of generality, that the sequence $\\{\lambda_{n}\\}$ is
increasing. Thus,
(1)
$1<\lambda_{1}\leq\lambda_{2}\leq\ldots,\qquad\lim_{n\to\infty}\lambda_{n}=\infty.$
In the case when $\lambda_{n}=n,\ n=1,2\ldots$ we say Harmonic Variation
instead of $\Lambda$-variation and write $H$ instead of $\Lambda$ ($HBV$,
$PHBV$, $HV(f)$, etc).
The notion of $\Lambda$-variation was introduced by D. Waterman [13] in one
dimensional case and A. Sahakian [12] in two dimensional case.
###### Definition 2 (Waterman [14]).
Let $\Lambda=\\{\lambda_{n}\\}_{n=1}^{\infty}$ and
$\Lambda_{k}=\\{\lambda_{n}\\}_{n=k}^{\infty}$, $k=1,2,\ldots$. We say that
the function $f$ is continuous in $\Lambda$-variation and write $f\in C\Lambda
BV$, if
$\lim_{k\to\infty}\Lambda_{k}V(f)=0.$
## 2\. $\left(C;\alpha,\beta\right)\,\left(-1<\alpha,\beta<0\right)$
Summability
Let $f\in L^{1}\left(T^{2}\right).$ The Fourier series of $f$ with respect to
the trigonometric system is the series
$S\left[f,\left(x,y\right)\right]:=\sum_{m,n=-\infty}^{+\infty}\widehat{f}\left(m,n\right)e^{imx}e^{iny},$
where
$\widehat{f}\left(m,n\right)=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}f(x,y)e^{-imx}e^{-iny}dxdy$
are the Fourier coefficients of the function $f$. The rectangular partial sums
are defined as follows:
$S_{M,N}\left[f,(x,y)\right]:=\sum_{m=-M}^{M}\sum_{n=-N}^{N}\widehat{f}\left(m,n\right)e^{imx}e^{iny},$
The Cesàro $(C;\alpha,\beta),$ $\alpha,\beta>-1$, means of two-dimensional
Fourier series are defined by
$\sigma_{n,m}^{\alpha,\beta}f(x,y):=\frac{1}{A_{n}^{\alpha}}\frac{1}{A_{m}^{\beta}}\sum_{i=0}^{n}\sum_{j=0}^{m}A_{n-i}^{\alpha-1}A_{m-j}^{\beta-1}S_{i,j}\left[f,(x,y)\right]$
where
$A_{0}^{\alpha}=1,\,\,\,A_{k}^{\alpha}=\frac{(\alpha+1)\cdots(\alpha+k)}{k!},\quad
k=1,2,....$
We say that the double trigonometric Fourier series of the function $f$ is
$(C;-\alpha,-\beta)$ summable to $f$, if
$\lim_{n,\,m\to\infty}\sigma_{n,m}^{\alpha,\beta}f(x,y)=f(x,y).$
It is well-known that (see [17], p. 157 )
$\sigma_{mn}^{\left(\alpha,\beta\right)}f\left(x,y\right)=\frac{1}{\pi^{2}}\int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi}f\left(x+t,y+s\right)K_{m}^{\alpha}\left(s\right)K_{n}^{\beta}\left(t\right)dsdt,$
where the kernel $K_{n}^{\alpha}$, $-1<\alpha<0$ satisfies the following
conditions:
(2) $\left|K_{n}^{-\alpha}(u)\right|\leq 2n,\quad u\in T,$ (3)
$K_{n}^{\alpha}\left(u\right)=\varphi_{n}^{\alpha}\left(u\right)+O\left(1/nt^{2}\right),\,\,\,0\leq|u|\leq\pi,$
where
(4)
$\varphi_{n}^{\alpha}\left(u\right)=\frac{\sin\left[\left(n+1/2+\alpha/2\right)u-\alpha\pi/2\right]}{A_{n}^{\alpha}\left[2\sin
u/2\right]^{1+\alpha}},$
The coefficients $A_{n}^{\alpha}$ have following bounds:
(5) $c_{1}(\alpha)n^{\alpha}\leq A_{n}^{\alpha}\leq c_{2}(\alpha)n^{\alpha}.$
Denote
${}_{1}\Delta_{i}^{m}f\left(x,y\right):=f\left(x+\frac{2i\pi}{m},\
y\right)-f\left(x+\frac{\left(2i+1\right)\pi}{m},y\right),$
${}_{2}\Delta_{j}^{n}f\left(x,y\right):=f\left(x,\
y+\frac{2j\pi}{n}\right)-f\left(x,\ y+\frac{\left(2j+1\right)\pi}{n}\right),$
$\Delta_{ij}^{mn}f\left(x,y\right)=f\left(x+\frac{2i\pi}{m},\
y+\frac{2j\pi}{n}\right)-f\left(x+\frac{\left(2i+1\right)\pi}{m},\
y+\frac{2j\pi}{n}\right)$ $-f\left(x+\frac{2i\pi}{m},\
y+\frac{\left(2j+1\right)\pi}{n}\right)+f\left(x+\frac{\left(2i+1\right)\pi}{m},\
y+\frac{\left(2j+1\right)\pi}{n}\right).$
## 3\. Formulation of Problems
Let $C(T^{2})$ be the space of $2\pi$-periodic with respect to each variable
continuous functions with the norm
$\|f\|_{C}:=\sup_{x,y\in T^{2}}|f(x,y)|.$
For the function $f(x,y)$ we denote by $f\left(x\pm 0,y\pm 0\right)$ the open
coordinate quadrant limits (if exist) at the point $\left(x,y\right)$ and set
$\displaystyle\qquad\sum f\left(x\pm 0,y\pm 0\right)$ $\displaystyle=$
$\displaystyle\hskip-5.69054pt\big{\\{}f\left(x+0,y+0\right)+f\left(x+0,y-0\right)+f\left(x-0,y+0\right)+f\left(x-0,y-0\right)\big{\\}}.$
The well known Dirichlet-Jordan theorem (see [17]) states that the Fourier
series of a function $f(x),\ x\in T$ of bounded variation converges at every
point $x$ to the value $\left[f\left(x+0\right)+f\left(x-0\right)\right]/2$.
If $f$ is in addition continuous on $T$ the Fourier series converges uniformly
on $T$. This result was generalized by Waterman [13].
###### Theorem W1 (Waterman [13]).
If $f\in HBV$, then $S[f]$ converges at every point $x$ to the value
$\left[f\left(x+0\right)+f\left(x-0\right)\right]/2$. If $f$ is in addition
continuous on $T$, then $S[f]$ converges uniformly on $T$.
Hardy [8] generalized the Dirichlet-Jordan theorem to the double Fourier
series. He proved that if function $f(x,y)$ has bounded variation in the sense
of Hardy ($f\in BV$), then $S\left[f\right]$ converges at any point
$\left(x,y\right)$ to the value $\frac{1}{4}\sum f\left(x\pm 0,y\pm 0\right)$.
If $f$ is in addition continuous on $T^{2}$ then $S\left[f\right]$ converges
uniformly on $T^{2}$.
###### Theorem S (Sahakian [12]).
The Fourier series of a function $f\left(x,y\right)\in HBV$ converges to
$\frac{1}{4}\sum f\left(x\pm 0,y\pm 0\right)$ at any point $\left(x,y\right)$,
where the quadrant limits (3) exist. The convergence is uniformly on any
compact $K$, where the function $f$ is continuous.
Analogs of Theorem S for higher dimensions can be found in [11] and [1].
Convergence of spherical and other partial sums of double Fourier series of
functions of bounded $\Lambda$-variation was investigated in details by
Dyachenko (see [3], [4], [5] and references therein).
The first author [6] has proved that in Hardy’s theorem there is no need to
require the boundedness of mixed variation. In particular, the following is
true
###### Theorem G1 (Goginava [6]).
Let $f\in C\left(T^{2}\right)\bigcap PBV$. Then $S\left[f\right]$ converges
uniformly on $T^{2}$.
For one-dimensional Fourier series Waterman [14] proved the following
###### Theorem W2 (Waterman [14]).
Let $0<\alpha<1$ and $f\in C\\{n^{1-\alpha}\\}BV$. Then $S[f]$ is everywhere
$\left(C,-\alpha\right)$ summable to the value
$\left[f\left(x+0\right)+f\left(x-0\right)\right]/2$ and the summability is
uniform on each closed interval of continuity.
Later Sablin proved in [11], that for $0<\alpha<1$ the classes
$\\{n^{1-\alpha}\\}BV$ and $C\\{n^{1-\alpha}\\}BV$ coincide.
Zhizhiashvili [16] has inverstigated the convergence of Cesàro means of double
trigonometric Fourier series. In particular, the following theorem was proved.
###### Theorem Zh (Zhizhiashvili [16]).
Let $\alpha,\beta>0$ and $\alpha+\beta<1$. If $f\in BV$, then the double
Fourier series of $f$ is $\left(C;-\alpha,-\beta\right)$ summable to
$\frac{1}{4}\sum f\left(x\pm 0,y\pm 0\right)$ in any point $\left(x,y\right)$
. The convergence is uniformly on any compact $K$, where the function $f$ is
continuous.
For functions of partial bounded variation the problem was solved by the first
author in [7].
###### Theorem G2 (Goginava [7]).
Let $f\in C\left(T^{2}\right)\cap PBV$ and $\alpha+\beta<1,\ \alpha,\beta>0.$
Then the double trigonometric Fourier series of the function $f$ is uniformly
$(C;-\alpha,-\beta)$ summable to $f$.
###### Theorem G3 (Goginava [7]).
Let $\alpha+\beta\geq 1,\ \alpha,\beta>0.$ Then there exists a continuous
function $f_{0}\in PBV$ such that the Cesàro $(C;-\alpha,-\beta)$ means
$\sigma_{n,m}^{-\alpha,-\beta}\left(f_{0};0,0\right)\,$of the double
trigonometric Fourier series of $f_{0}$ diverge over cubes.
In this paper we consider the following problem. Let
$\alpha,\beta\in\left(0,1\right),\,\alpha+\beta<1.$ Under what conditions on
the sequence $\Lambda=\\{\lambda_{n}\\}$ the double Fourier series of the
function $f\in P\Lambda BV$ is $(C;-\alpha,-\beta)$ summable.
The solution is given in Theorems 1 and 2 bellow.
## 4\. Main Results
###### Theorem 1.
Let $\alpha,\beta\in\left(0,1\right),\ \alpha+\beta<1$ and the sequence
$\Lambda=\\{\lambda_{k}\\}$ satisfies the conditions:
$\frac{\lambda_{k}}{k^{1-\left(\alpha+\beta\right)}}\downarrow
0,\qquad\sum\limits_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2-\left(\alpha+\beta\right)}}<\infty.$
Then the double Fourier series of the function $f\in P\Lambda BV$ is
$\left(C;-\alpha,-\beta\right)$ summable to $\frac{1}{4}\sum f\left(x\pm
0,y\pm 0\right)$ at any point $\left(x,y\right)$, where the quadrant limits
(3) exist. The convergence is uniform on any compact $K$, where the function
$f$ is continuous.
###### Theorem 2.
Let $\alpha,\beta\in\left(0,1\right),\,\alpha+\beta<1$ and the sequence
$\Lambda=\\{\lambda_{k}\\}$ satisfies the conditions:
$\frac{\lambda_{k}}{k^{1-\left(\alpha+\beta\right)}}\downarrow
0,\qquad\sum\limits_{k=1}^{\infty}\frac{\lambda_{k}}{k^{2-\left(\alpha+\beta\right)}}=\infty.$
Then there exists a continuous function $f\in P\Lambda BV$ for which
$\left(C;-\alpha,-\beta\right)$ means of two-dimensional Fourier series
diverges over cubes at $\left(0,0\right).$
###### Corollary 1.
Let $\alpha,\beta\in\left(0,1\right),\,\alpha+\beta<1$.
a)If $f\in
P\left\\{\frac{n^{1-\left(\alpha+\beta\right)}}{\log^{1+\varepsilon}\left(n+1\right)}\right\\}BV$
for some $\varepsilon>0$, then the double Fourier series of the function $f$
is $\left(C;-\alpha,-\beta\right)$ summable to $\frac{1}{4}\sum f\left(x\pm
0,y\pm 0\right)$ in any point $\left(x,y\right)$ , where the quadrant limits
(3) exist. The convergence is uniform on any compact $K$, where the function
$f$ is continuous.
b) There exists a continuous function $f\in
P\left\\{\frac{n^{1-\left(\alpha+\beta\right)}}{\log\left(n+1\right)}\right\\}BV$
such that $\left(C;-\alpha,-\beta\right)$ means of two-dimensional Fourier
series of $f$ diverges over cubes at $\left(0,0\right).$
###### Corollary 2.
Let $\alpha,\beta\in\left(0,1\right),\,\alpha+\beta<1$ and$\,\,f\in PBV$ .
Then the double Fourier series of the function $f$ is
$\left(C;-\alpha,-\beta\right)$ summable to $\frac{1}{4}\sum f\left(x\pm
0,y\pm 0\right)$ in any point $\left(x,y\right)$ , where the quadratic limits
(3) exist. The convergence is uniform on any compact $K$, where the function
$f$ is continuous.
## 5\. Proofs
###### Proof of Theorem 1.
It is easy to show that
$\sigma_{mn}^{\left(-\alpha,-\beta\right)}f\left(x,y\right)-\frac{1}{4}\sum
f\left(x\pm 0,y\pm 0\right)$
$=\frac{1}{\pi^{2}}\sum\limits_{i=1}^{4}\int\limits_{0}^{\pi}\int\limits_{0}^{\pi}\varphi_{i}\left(x,y,s,t\right)K_{m}^{-\alpha}\left(s\right)K_{n}^{-\beta}\left(t\right)dsdt$
$=:\sum\limits_{i=1}^{4}I_{mn}^{\left(k\right)}\left(x,y\right).$
where
$\varphi_{1}\left(x,y,s,t\right):=f\left(x+s,y+t\right)-f\left(x+0,y+0\right),$
$\varphi_{2}\left(x,y,s,t\right):=f\left(x-s,y+t\right)-f\left(x-0,y+0\right),$
$\varphi_{3}\left(x,y,s,t\right):=f\left(x+s,y-t\right)-f\left(x+0),y-0\right),$
$\varphi_{4}\left(x,y,s,t\right):=f\left(x-s,y-t\right)-f\left(x-0,y-0\right).$
For $I_{mn}^{\left(1\right)}\left(x,y\right)$ we can write
(7) $\pi^{2}I_{mn}^{\left(1\right)}\left(x,y\right)$
$=\left(\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}+\int\limits_{0}^{\pi/m}\int\limits_{\pi/n}^{\pi}+\int\limits_{\pi/m}^{\pi}\int\limits_{0}^{\pi/n}+\int\limits_{\pi/m}^{\pi}\int\limits_{\pi/n}^{\pi}\right)\left(\varphi_{1}\left(x,y,s,t\right)K_{m}^{-\alpha}\left(s\right)K_{n}^{-\beta}\left(t\right)dsdt\right)$
$=:\sum\limits_{k=1}^{4}I_{mn}^{\left(1k\right)}\left(x,y\right).$
From (2) we have
(8) $\left|I_{mn}^{\left(11\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)mn\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|dsdt$
$\leq
c\left(\alpha,\beta\right)\sup\limits_{0<s<\pi/m,0<t<\pi/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|=o\left(1\right)\,\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
Using (3), we obtain
(9)
$I_{mn}^{\left(12\right)}\left(x,y\right)=\int\limits_{0}^{\pi/m}\int\limits_{\pi/n}^{\pi}\varphi_{1}\left(x,y,s,t\right)K_{m}^{-\alpha}\left(s\right)\varphi_{n}^{-\beta}\left(t\right)dsdt$
$+\int\limits_{0}^{\pi/m}\int\limits_{\pi/n}^{\pi}\varphi_{1}\left(x,y,s,t\right)K_{m}^{-\alpha}\left(s\right)O\left(\frac{1}{nt^{2}}\right)dsdt$
$=:I_{mn}^{\left(121\right)}\left(x,y\right)+I_{mn}^{\left(122\right)}\left(x,y\right).$
We can write
$\displaystyle\left|I_{mn}^{\left(122\right)}\left(x,y\right)\right|$
$\displaystyle\leq$
$\displaystyle\int\limits_{0}^{\pi/m}\int\limits_{\pi/n}^{\pi/\sqrt{n}}\left|\varphi_{1}\left(x,y,s,t\right)\right||K_{m}^{-\alpha}\left(s\right)|O\left(\frac{1}{nt^{2}}\right)dsdt$
$\displaystyle+\int\limits_{0}^{\pi/m}\int\limits_{\pi/\sqrt{n}}^{\pi}\left|\varphi_{1}\left(x,y,s,t\right)\right||K_{m}^{-\alpha}\left(s\right)|O\left(\frac{1}{nt^{2}}\right)dsdt$
$\displaystyle\leq$ $\displaystyle
c\left(\alpha,\beta,f\right)\left\\{\sup\limits_{0<s<\pi/m,0<t<\pi/\sqrt{n}}\left|\varphi_{1}\left(x,y,s,t\right)\right|+\int\limits_{\pi/\sqrt{n}}^{\pi}\frac{dt}{nt^{2}}\right\\}$
$\displaystyle=$ $\displaystyle
o\left(1\right)\,\,\,\text{as\thinspace\thinspace\thinspace}n,m\rightarrow\infty.$
In order to estimate $I_{mn}^{\left(121\right)}\left(x,y\right)$ it is enough
to estimate the following expression
$J_{mn}\left(x,y\right):=n^{\beta}\int\limits_{0}^{\pi/m}\int\limits_{\pi/n}^{\pi}\varphi_{1}\left(x,y,s,t\right)K_{m}^{-\alpha}\left(s\right)w_{\beta}\left(t\right)\sin
ntdsdt,$
where
$w_{\beta}\left(t\right)=\frac{\cos\frac{1-\beta}{2}t}{\left(\sin
t/2\right)^{1-\beta}}.$
We have
$J_{mn}\left(x,y\right)=n^{\beta}\sum\limits_{i=1}^{n-1}\int\limits_{0}^{\pi/m}K_{m}^{-\alpha}\left(s\right)\left(\int\limits_{i\pi/n}^{\left(i+1\right)\pi/n}\varphi_{1}\left(x,y,s,t\right)w_{\beta}\left(t\right)\sin
ntdt\right)ds$
$=n^{\beta}\sum\limits_{i=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}K_{m}^{-\alpha}\left(s\right)\left(\int\limits_{0}^{\pi/n}\left[\varphi_{1}\left(x,y,s,t+\frac{2i\pi}{n}\right)-\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right]\right.$
$\left.w_{\beta}\left(t+\frac{2i\pi}{n}\right)\sin ntdt\right)ds$
$+n^{\beta}\sum\limits_{i=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}K_{m}^{-\alpha}\left(s\right)\left(\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right.$
$\left.\left[w_{\beta}\left(t+\frac{2i\pi}{n}\right)-w_{\beta}\left(t+\frac{\left(2i+1\right)\pi}{n}\right)\right]\sin
ntdt\right)ds$
$=:J_{mn}^{\left(1\right)}\left(x,y\right)+J_{mn}^{\left(2\right)}\left(x,y\right).$
Using the following inequality:
(11)
$\left|w_{\beta}\left(t+\frac{2i\pi}{n}\right)-w_{\beta}\left(t+\frac{\left(2i+1\right)\pi}{n}\right)\right|\leq\frac{c\left(\beta\right)n^{1-\beta}}{i^{2-\beta}},$
for $J_{mn}^{\left(2\right)}\left(x,y\right)$ we can write
(12) $\left|J_{mn}^{\left(2\right)}\left(x,y\right)\right|$ $\leq
c\left(\beta\right)mn\sum\limits_{i=1}^{\left(n-1\right)/2}\frac{1}{i^{2-\beta}}\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dt\right)ds$
$\leq
c\left(\beta\right)nm\sum\limits_{i\leq\sqrt{n}}\frac{1}{i^{2-\beta}}\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dt\right)ds$
$+c\left(\beta\right)nm\sum\limits_{\sqrt{n}<i\leq\left(n-1\right)/2}\frac{1}{i^{2-\beta}}\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dt\right)ds$
$\leq
c\left(\beta\right)\sup\limits_{0<s<\pi/n,0<s<4\pi/\sqrt{n}}\left|\varphi_{1}\left(x,y,s,t\right)\right|+c\left(\beta,f\right)\left(\frac{1}{\sqrt{n}}\right)^{1-\beta}=o\left(1\right),$
as $n,m\rightarrow\infty$.
To estimate $J_{mn}^{\left(1\right)}\left(x,y\right)$, we denote
(13) $\mu\left(n,m\right):=\left[\min\left\\{\frac{1}{2}\ln n-1,\
\left(s\left(n,m\right)\right)^{-1}\right\\}\right],$
where $[a]$ is the integer part of $a$ and
(14) $s\left(n,m\right):=\sup\limits_{0<s<\pi/m,\ 0<t<\pi\ln
n/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|.$
Then we have
(15) $\left|J_{mn}^{\left(1\right)}\left(x,y\right)\right|\leq
c\left(\beta\right)nm\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\sum\limits_{i=1}^{\mu\left(n,m\right)}\frac{1}{i^{1-\beta}}\left|\varphi_{1}\left(x,y,s,t+\frac{2i\pi}{n}\right)\right.\right.$
$\left.\left.-\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dtds\right)$
$+c\left(\beta\right)nm\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\sum\limits_{i=\mu\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{i^{1-\beta}}\left|\varphi_{1}\left(x,y,s,t+\frac{2i\pi}{n}\right)\right.\right.$
$\left.\left.-\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dt\right)ds$
$\leq c\left(\beta\right)\sup\limits_{0<s<\pi/m,\
0<t<\left(2\mu\left(n,\,m\right)+1\right)\pi/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|\left(\mu\left(n,m\right)\right)^{\beta}$
$+c\left(\beta\right)nm\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\sum\limits_{i=\mu\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{\lambda_{i}}\left|\varphi_{1}\left(x,y,s,t+\frac{2i\pi}{n}\right)\right.\right.$
$\left.\left.-\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|\frac{\lambda_{i}}{i^{1-\beta}}dt\right)ds$
$\leq c\left(\beta\right)\sup\limits_{0<s<\pi/m,\ 0<t<\pi\ln
n/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|\left(\mu\left(n,m\right)\right)^{\beta}$
$+c\left(\beta\right)nm\frac{\lambda_{\mu\left(n,m\right)}}{\left(\mu\left(n,m\right)\right)^{1-\beta}}\int\limits_{0}^{\pi/m}\left(\int\limits_{0}^{\pi/n}\sum\limits_{i=\mu\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{\lambda_{i}}\left|\varphi_{1}\left(x,y,s,t+\frac{2i\pi}{n}\right)\right.\right.$
$\left.\left.-\varphi_{1}\left(x,y,s,t+\frac{\left(2i+1\right)\pi}{n}\right)\right|dt\right)ds$
$\leq
c\left(\beta\right)s\left(n,m\right)\left(\mu\left(n,m\right)\right)^{\beta}+c\left(\beta\right)\frac{\lambda_{\mu\left(n,m\right)}}{\left(\mu\left(n,m\right)\right)^{1-\beta}}V_{2}\Lambda\left(f\right)=o\left(1\right)\,\,\,\,\text{as\thinspace\thinspace\thinspace}n,m\rightarrow\infty.$
Combining (9), (10), (12) and (15) we conclude that
(16) $I_{mn}^{\left(12\right)}\left(x,y\right)\rightarrow
0\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
Analogously, we can prove that
(17) $I_{mn}^{\left(13\right)}\left(x,y\right)\rightarrow
0\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
In order to estimate $I_{mn}^{\left(14\right)}\left(x,y\right)$ it is enough
to estimate the following expression
$L_{mn}\left(x,y\right):=m^{\alpha}n^{\beta}\int\limits_{\pi/m}^{\pi}\int\limits_{\pi/n}^{\pi}\varphi_{1}\left(x,y,s,t\right)w_{\alpha}\left(s\right)w_{\beta}\left(t\right)\sin
ms\sin ntdtds.$
We have
(18)
$L_{mn}\left(x,y\right)=m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{2j\pi}{n}\right)$
$\times
w_{\alpha}\left(s+\frac{2i\pi}{m}\right)w_{\beta}\left(t+\frac{2j\pi}{n}\right)\sin
ms\sin ntdtds$
$-m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{2j\pi}{n}\right)$
$\times
w_{\alpha}\left(s+\frac{\left(2i+1\right)\pi}{m}\right)w_{\beta}\left(t+\frac{2j\pi}{n}\right)\sin
ms\sin ntdtds$
$-m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)$
$\times
w_{\alpha}\left(s+\frac{2i\pi}{m}\right)w_{\beta}\left(t+\frac{\left(2j+1\right)\pi}{n}\right)\sin
ms\sin ntdtds$
$+m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)$
$\times
w_{\alpha}\left(s+\frac{\left(2i+1\right)\pi}{m}\right)w_{\beta}\left(t+\frac{\left(2j+1\right)\pi}{n}\right)\sin
ms\sin ntdtds$
$=m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left[\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{2j\pi}{n}\right)\right.$
$-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{2j\pi}{n}\right)-\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)$
$\left.+\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right]$
$\times
w_{\alpha}\left(s+\frac{2i\pi}{m}\right)w_{\beta}\left(t+\frac{2j\pi}{n}\right)\sin
ms\sin ntdtds$
$+m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left[\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{2j\pi}{n}\right)\right.$
$\left.-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right]$
$\times\left[w_{\alpha}\left(s+\frac{2i\pi}{m}\right)-w_{\alpha}\left(s+\frac{\left(2i+1\right)\pi}{m}\right)\right]w_{\beta}\left(t+\frac{2j\pi}{n}\right)\sin
ms\sin ntdtds$
$+m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left[\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right.$
$\left.-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+\right)1\pi}{n}\right)\right]$
$\times\left[w_{\beta}\left(t+\frac{2j\pi}{n}\right)-w_{\beta}\left(t+\frac{\left(2j+1\right)\pi}{n}\right)\right]w_{\alpha}\left(s+\frac{2i\pi}{m}\right)\sin
ms\sin ntdtds$
$+m^{\alpha}n^{\beta}\sum\limits_{i=1}^{\left(m-1\right)/2}\sum\limits_{j=1}^{\left(n-1\right)/2}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)$
$\times\left[w_{\beta}\left(t+\frac{2j\pi}{n}\right)-w_{\beta}\left(t+\frac{\left(2j+1\right)\pi}{n}\right)\right]$
$\left[w_{\alpha}\left(s+\frac{2i\pi}{m}\right)-w_{\alpha}\left(s+\frac{\left(2i+1\right)\pi}{m}\right)\right]\sin
ms\sin ntdtds$
$=:\sum\limits_{k=1}^{4}L_{mn}^{\left(k\right)}\left(x,y\right).$
By (11) we obtain
(19) $\left|L_{mn}^{\left(4\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)mn\sum\limits_{i=1}^{\left[\sqrt{m}\right]}\frac{1}{i^{2-\alpha}}\sum\limits_{j=1}^{\left[\sqrt{n}\right]}\frac{1}{j^{2-\beta}}$
$\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right|$
$+c\left(\alpha,\beta,f\right)\sum\limits_{i=1}^{\infty}\frac{1}{i^{2-\alpha}}\sum\limits_{j=\left[\sqrt{n}\right]}^{\infty}\frac{1}{j^{2-\beta}}$
$+c\left(\alpha,\beta,f\right)mn\sum\limits_{i=\left[\sqrt{m}\right]}^{\infty}\frac{1}{i^{2-\alpha}}\sum\limits_{j=1}^{\infty}\frac{1}{j^{2-\beta}}$
$\leq
c\left(\alpha,\beta\right)\sup\limits_{0<s<4\pi/\sqrt{m},0<t<4\pi/\sqrt{n}}\left|\varphi_{1}\left(x,y,s,t\right)\right|+o\left(1\right)$
$=o\left(1\right)\text{\thinspace\thinspace\thinspace\thinspace as
}n,m\rightarrow\infty.$
Let
(20) $\tau\left(n,m\right):=\left[\min\left\\{\frac{1}{2}\ln
n-1,\frac{1}{2}\ln m-1,\left(l\left(n,m\right)\right)^{-1}\right\\}\right],$
where
$l\left(n,m\right):=\sup\limits_{0<s<\pi\ln m/m,\ 0<t<\pi\ln
n/n}\left|\varphi_{1}\left(x,y,s,t\right)\right|$
Then we can write
(21) $\left|L_{mn}^{\left(3\right)}\left(x,y\right)\right|$ $\leq
c\left(\alpha,\beta\right)mn\sum\limits_{i=1}^{\tau\left(n,m\right)}\frac{1}{i^{1-\alpha}}\sum\limits_{j=1}^{\tau\left(n,m\right)}\frac{1}{j^{2-\beta}}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right.$
$\left.-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right|dtds$
$+c\left(\alpha,\beta\right)mn\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{1}{\lambda_{i}}\frac{\lambda_{i}}{i^{1-\alpha}}\sum\limits_{j=1}^{\tau\left(n,m\right)}\frac{1}{j^{2-\beta}}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right.$
$\left.-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right|dtds$
$+c\left(\alpha,\beta\right)mn\sum\limits_{i=1}^{\left(m-1\right)/2}\frac{1}{\lambda_{i}}\frac{\lambda_{i}}{i^{1-\alpha}}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{2-\beta}}\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}\left|\varphi_{1}\left(x,y,s+\frac{2i\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right.$
$\left.-\varphi_{1}\left(x,y,s+\frac{\left(2i+1\right)\pi}{m},t+\frac{\left(2j+1\right)\pi}{n}\right)\right|dtds$
$\leq
c\left(\alpha,\beta\right)l\left(n,m\right)\left(\tau\left(n,m\right)\right)^{\alpha+\beta}$
$+c\left(\alpha,\beta\right)\frac{\lambda_{\tau\left(n,m\right)}}{\left(\tau\left(n,m\right)\right)^{1-\alpha}}V_{1}\Lambda\left(f\right)+c\left(\alpha,\beta\right)\frac{1}{\left(\tau\left(n,m\right)\right)^{1-\beta}}V_{1}\Lambda\left(f\right)$
$=o\left(1\right)\,\,\,\text{as\thinspace\thinspace\thinspace}n,m\rightarrow\infty.$
Analogously, we can prove that
(22)
$\left|L_{mn}^{\left(2\right)}\left(x,y\right)\right|=o\left(1\right)\,\,\,\text{as\thinspace\thinspace\thinspace}n,m\rightarrow\infty.$
For $L_{mn}^{\left(1\right)}\left(x,y\right)$ we can write
(23) $\left|L_{mn}^{\left(1\right)}\left(x,y\right)\right|$ $\leq
c\left(\alpha,\beta\right)mn\left\\{\sum\limits_{i=1}^{\tau\left(n,m\right)}\frac{1}{i^{1-\alpha}}\sum\limits_{j=1}^{\tau\left(n,m\right)}\frac{1}{j^{1-\beta}}+\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{1}{i^{1-\alpha}}\sum\limits_{j=1}^{\tau\left(n,m\right)}\frac{1}{j^{1-\beta}}\right.$
$\left.+\sum\limits_{i=1}^{\tau\left(n,m\right)}\frac{1}{i^{1-\alpha}}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{1-\beta}}+\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{1}{i^{1-\alpha}}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{1-\beta}}\right\\}$
$\left(\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}|\Delta_{ij}^{mn}f\left(x+s,y+t\right)|dsdt\right)=:\sum\limits_{k=1}^{4}L_{mn}^{\left(1k\right)}\left(x,y\right).$
From (20) we obtain that
(24) $\left|L_{mn}^{\left(11\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)\left(l\left(n,m\right)^{1-\alpha-\beta}\right)\rightarrow
0\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
Next, we have
(25) $\left|L_{mn}^{\left(13\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)mn\left\\{\sum\limits_{i=1}^{\tau\left(n,m\right)}\frac{1}{i^{1-\alpha}}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{1-\beta}}\right.$
$\left.\left(\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}|\Delta_{ij}^{mn}f\left(x+s,y+t\right)|dsdt\right)\right\\}$
$\leq
c\left(\alpha,\beta\right)n\left\\{\sum\limits_{i=1}^{\tau\left(n,m\right)}\frac{1}{i^{1-\alpha}}\left(\int\limits_{0}^{\pi/n}\sup\limits_{x}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{\lambda_{j}}{j^{1-\beta}}\frac{1}{\lambda_{j}}|_{2}\Delta_{j}^{n}f\left(x,y+t\right)|dt\right)\right\\}$
$\leq
c\left(\alpha,\beta\right)\frac{\tau\left(n,m\right)}{\left(\tau\left(n,m\right)\right)^{1-\beta-\alpha}}V_{2}\Lambda\left(f\right)\rightarrow
0,\quad\text{as}\ n,m\rightarrow\infty.$
Analogously, we can prove that
(26) $\left|L_{mn}^{\left(12\right)}\left(x,y\right)\right|\rightarrow
0,\quad\text{as}\ n,m\rightarrow\infty.$
From the condition of the Theorem 1 we can write
(27) $\left|L_{mn}^{\left(14\right)}\left(x,y\right)\right|$ $\leq
c\left(\alpha,\beta\right)nm\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{1}{i^{1-\alpha}}\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{1-\beta}}$
$\left(\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}|\Delta_{ij}^{mn}f\left(x+s,y+t\right)|dsdt\right)$
$\leq
c\left(\alpha,\beta\right)nm\left\\{\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{1}{i^{1-\alpha}}\sum\limits_{j=i}^{\left(n-1\right)/2}\frac{\lambda_{j}}{j^{1-\beta}}\frac{1}{\lambda_{j}}\right.$
$+\left.\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{1}{j^{1-\beta}}\sum\limits_{i=j}^{\left(m-1\right)/2}\frac{1}{\lambda_{i}}\frac{\lambda_{i}}{i^{1-\alpha}}\right\\}$
$\left(\int\limits_{0}^{\pi/m}\int\limits_{0}^{\pi/n}|\Delta_{ij}^{mn}f\left(x+s,y+t\right)|dsdt\right)$
$\leq
c\left(\alpha,\beta\right)n\sum\limits_{i=\tau\left(n,m\right)}^{\left(m-1\right)/2}\frac{\lambda_{i}}{i^{2-\left(\alpha+\beta\right)}}\left(\int\limits_{0}^{\pi/n}\sup\limits_{x}\sum\limits_{j=i}^{\left(n-1\right)/2}\frac{1}{\lambda_{j}}|_{2}\Delta_{j}^{n}f\left(x,y+t\right)|dt\right)$
$+c\left(\alpha,\beta\right)m\sum\limits_{j=\tau\left(n,m\right)}^{\left(n-1\right)/2}\frac{\lambda_{j}}{i^{2-\left(\alpha+\beta\right)}}\left(\int\limits_{0}^{\pi/m}\sup\limits_{y}\sum\limits_{i=j}^{\left(m-1\right)/2}\frac{1}{\lambda_{j}}|_{1}\Delta_{i}^{m}f\left(x+s,y\right)|ds\right)$
$\leq
c\left(\alpha,\beta\right)\left(V_{1}\Lambda\left(f\right)+V_{2}\Lambda\left(f\right)\right)\sum\limits_{j=\tau\left(n,m\right)}^{\infty}\frac{\lambda_{j}}{j^{2-\left(\alpha+\beta\right)}}\rightarrow
0,$
as $n,m\rightarrow\infty$.
Combining (23)-(27) we conclude that
(28) $L_{mn}^{\left(1\right)}\left(x,y\right)\rightarrow
0\,\,\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
From (18), (19), (21), (22) and (28) we obtain
(29) $L_{mn}\left(x,y\right)\rightarrow
0\,\,\,\,\,\text{as\thinspace\thinspace\thinspace}m,n\rightarrow\infty.$
Finally, combining (7), (8), (16), (17) and (29) we get
$I_{mn}^{\left(1\right)}\left(x,y\right)\rightarrow
0\,\,\,\,\,as\,\,\,m,n\rightarrow\infty.$
Analogously, we can prove that
$I_{mn}^{\left(k\right)}\left(x,y\right)\rightarrow
0\,\,\,\,\,as\,\,\,m,n\rightarrow\infty,\,\,k=2,3,4.$
To complete the proof of Theorem 1, note that if $f$ is continuous on some
compact $K$, then the relations
$\lim_{s,t\to 0}\varphi_{i}(x,y,s,t)=0,\quad i=1,2,3,4,$
hold uniformly on $(x,y)\in K$ and all estimates in the proof also hold
uniformly on $(x,y)\in K$. Hence the $(C;-\alpha,\beta)$ means
$\sigma_{n,m}^{\alpha,\beta}(f;x,y)$ will converge to$f$ uniformly on $K$. ∎
###### Proof of Theorem 2.
It is not hard to see, that for any sequence $\Lambda=\\{\lambda_{n}\\}$
satisfying (1) the class $C\left(T^{2}\right)\bigcap P\Lambda BV$ is a Banach
space with the norm
$\left\|f\right\|_{P\Lambda BV}:=\left\|f\right\|_{C}+P\Lambda
V\left(f\right).$
Denote
$A_{i,j}:=\left[\frac{\pi
i-\alpha\pi/2}{N+1/2-\alpha/2},\frac{\pi\left(i+1\right)-\alpha\pi/2}{N+1/2-\alpha/2}\right)\times\left[\frac{\pi
j-\beta\pi/2}{N+1/2-\beta/2},\frac{\pi\left(j+1\right)-\beta\pi/2}{N+1/2-\beta/2}\right)$
and
$W:=\left\\{\left(i,j\right):j<i<2j,1<j<\frac{N-1}{2}\right\\}.$
Let
$\displaystyle f_{N}\left(x,y\right)$ $\displaystyle:$
$\displaystyle=\sum\limits_{\left(i,j\right)\in
W}t_{j}\mathbf{1}_{A_{i,j}}\left(x,y\right)\sin\left[\left(N+1/2-\alpha/2\right)x+\alpha\pi/2\right]$
$\displaystyle\times\sin\left[\left(N+1/2-\beta/2\right)y+\beta\pi/2\right],$
where
$t_{j}:=\left(\sum\limits_{i=1}^{j}\frac{1}{\lambda_{i}}\right)^{-1}.$
First, we prove that $f\in P\Lambda BV.$ Indeed, let
$y\in\left[\frac{\pi
j-\beta\pi/2}{N+1/2-\beta/2},\frac{\pi\left(j+1\right)-\beta\pi/2}{N+1/2-\beta/2}\right).$
Then it is evident that
$\sum\limits_{i}\frac{\left|f\left(I_{i},y\right)\right|}{\lambda_{i}}\leq
c\left(\sum\limits_{i=j}^{2j-1}\frac{1}{\lambda_{2j-i}}\right)t_{j}\leq
c<\infty,$
consequently,
(30) $V_{1}\Lambda\left(f\right)<\infty.$
Let
$x\in\left[\frac{\pi
i-\alpha\pi/2}{N+1/2-\alpha/2},\frac{\pi\left(i+1\right)-\alpha\pi/2}{N+1/2-\alpha/2}\right)$
then from construction of the function $f$ we have
$\sum\limits_{j}\frac{\left|f\left(x,J_{j}\right)\right|}{\lambda_{j}}\leq
c\sum\limits_{j=\left[i/2\right]}^{i}\frac{t_{j}}{\lambda_{j-\left[i/2\right]+1}}\leq
ct_{\left[i/2\right]}\left(\sum\limits_{j=1}^{i-\left[i/2\right]+1}\frac{1}{\lambda_{j}}\right)\leq
c<\infty.$
Hence
(31) $V_{2}\Lambda\left(f\right)<\infty.$
Combining (30) and (31) and we conclude that $f\in P\Lambda BV.$
From (2)-(5) we can write
(32) $\pi^{2}\sigma_{N,N}^{\left(-\alpha,-\beta\right)}f_{N}\left(0,0\right)$
$=\int\limits_{T^{2}}f_{N}\left(x,y\right)K_{N}^{-\alpha}\left(x\right)K_{N}^{-\beta}\left(y\right)dxdy$
$=\sum\limits_{\left(i,j\right)\in
W}t_{j}\int\limits_{A_{i,j}}\sin\left[\left(N+1/2-\alpha/2\right)x+\alpha\pi/2\right]\sin\left[\left(N+1/2-\beta/2\right)y+\beta\pi/2\right]$
$\times O\left(\frac{1}{Nx^{2}}\right)O\left(\frac{1}{Ny^{2}}\right)dxdy$
$+\sum\limits_{\left(i,j\right)\in
W}t_{j}\int\limits_{A_{i,j}}\sin\left[\left(N+1/2-\alpha/2\right)x+\alpha\pi/2\right]\frac{\sin^{2}\left[\left(N+1/2-\beta/2\right)y+\beta\pi/2\right]}{A_{N}^{-\beta}\left(2\sin
y/2\right)^{1-\beta}}$ $\times O\left(\frac{1}{Nx^{2}}\right)dxdy$
$+\sum\limits_{\left(i,j\right)\in
W}t_{j}\int\limits_{A_{i,j}}\frac{\sin^{2}\left[\left(N+1/2-\alpha/2\right)x+\alpha\pi/2\right]}{A_{N}^{-\alpha}\left(2\sin
x/2\right)^{1-\alpha}}\sin\left[\left(N+1/2-\beta/2\right)y+\beta\pi/2\right]$
$\times O\left(\frac{1}{Ny^{2}}\right)dxdy$ $+\sum\limits_{\left(i,j\right)\in
W}t_{j}\int\limits_{A_{i,j}}\frac{\sin^{2}\left[\left(N+1/2-\alpha/2\right)x+\alpha\pi/2\right]}{A_{N}^{-\alpha}\left(2\sin
x/2\right)^{1-\alpha}}\frac{\sin^{2}\left[\left(N+1/2-\beta/2\right)y+\beta\pi/2\right]}{A_{N}^{-\beta}\left(2\sin
y/2\right)^{1-\beta}}dxdy$
$=:\sum\limits_{k=1}^{4}F_{N}^{\left(k\right)}\left(x,y\right)$
It is easy to show that
(33) $\left|F_{N}^{\left(1\right)}\left(x,y\right)\right|\leq
c\sum\limits_{\left(i,j\right)\in W}\frac{t_{j}}{ij}$
$=c\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j}\sum\limits_{i=j+1}^{2j-1}\frac{1}{i}\leq
c\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j},$ (34)
$\left|F_{N}^{\left(2\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)\sum\limits_{\left(i,j\right)\in
W}\frac{t_{j}}{ij^{1-\beta}}$
$=c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\beta}}\sum\limits_{i=j+1}^{2j-1}\frac{1}{i}$
$\leq
c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\beta}},$
(35) $\left|F_{N}^{\left(3\right)}\left(x,y\right)\right|\leq
c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j}\sum\limits_{i=j+1}^{2j-1}\frac{1}{i^{1-\alpha}}$
$\leq
c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\alpha}}.$
From the construction of the function $f_{N}$ we can write
(36) $\left|F_{N}^{\left(4\right)}\left(x,y\right)\right|$
$=\frac{1}{\left(N+1/2-\alpha/2\right)\left(N+1/2-\beta/2\right)}\sum\limits_{\left(i,j\right)\in
W}t_{j}$ $\int\limits_{\pi i}^{\pi\left(i+1\right)}\int\limits_{\pi
j}^{\pi\left(j+1\right)}\frac{\sin^{2}u}{A_{N}^{-\alpha}\left(2\sin\frac{u-\alpha\pi/2}{2\left(N+1/2-\alpha/2\right)}\right)^{1-\alpha}}\frac{\sin^{2}v}{A_{N}^{-\alpha}\left(2\sin\frac{v-\beta\pi/2}{2\left(N+1/2-\beta/2\right)}\right)^{1-\beta}}dudv$
$\geq\frac{c\left(\alpha,\beta\right)N^{\alpha+\beta}}{N^{2}}\sum\limits_{\left(i,j\right)\in
W}t_{j}\frac{N^{2-\left(\alpha+\beta\right)}}{i^{1-\alpha}j^{1-\beta}}\int\limits_{\pi
i}^{\pi\left(i+1\right)}\sin^{2}udu\int\limits_{\pi
j}^{\pi\left(j+1\right)}\sin^{2}vdv$ $\geq
c\left(\alpha,\beta\right)\sum\limits_{\left(i,j\right)\in
W}\frac{t_{j}}{j^{1-\beta}}\frac{1}{i^{1-\alpha}}\geq
c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\beta}}\sum\limits_{i=j+1}^{2j-1}\frac{1}{i^{1-\alpha}}$
$\geq
c\left(\alpha,\beta\right)\sum\limits_{j=1}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\left(\beta+\alpha\right)}}.$
Since
$\frac{1}{j^{1-\alpha}}+\frac{1}{j^{1-\beta}}=o\left(\frac{1}{j^{1-(\alpha+\beta)}}\right)$
as $j\to\infty$, from (32)-(36) we conclude that if $j_{0}$ is big enough and
$N>2j_{0}$, then
(37)
$\pi^{2}\left|\sigma_{N,N}^{\left(-\alpha,-\beta\right)}f_{N}\left(0,0\right)\right|\geq
c\left(\alpha,\beta\right)\sum\limits_{j=j_{0}}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{1-\left(\beta+\alpha\right)}}.$
Let
$\lambda_{j}=j^{1-\left(\alpha+\beta\right)}\gamma_{j},\,\,\,\gamma_{j}\geq\gamma_{j+1},\,\,j=1,2,....$
Then we can write
$\frac{1}{t_{j}}=\sum\limits_{i=1}^{j}\frac{1}{\lambda_{i}}=\sum\limits_{i=1}^{j}\frac{1}{i^{1-\left(\alpha+\beta\right)}\gamma_{i}}\leq
c\left(\alpha,\beta\right)\frac{j^{\alpha+\beta}}{\gamma_{j}}.$
Consequently,
(38) $t_{j}j^{\alpha+\beta}\geq c\left(\alpha,\beta\right)\gamma_{j}.$
Combining (37) and (38) we obtain
(39)
$\pi^{2}\left|\sigma_{N,N}^{\left(-\alpha,-\beta\right)}f_{N}\left(0,0\right)\right|\geq
c\left(\alpha,\beta\right)\sum\limits_{j=j_{0}}^{\left[\left(N-1\right)/2\right]}\frac{\gamma_{j}}{j}$
$=c\left(\alpha,\beta\right)\sum\limits_{j=j_{0}}^{\left[\left(N-1\right)/2\right]}\frac{t_{j}}{j^{2-\left(\beta+\alpha\right)}}\rightarrow\infty\text{\thinspace\thinspace\thinspace
as\thinspace\thinspace\thinspace}N\rightarrow\infty.$
Applying the Banach-Steinhaus Theorem, from (39) we obtain that there exists a
continuous function $f\in P\Lambda BV$ such that
$\sup\limits_{N}\left|\sigma_{N,N}^{\left(-\alpha,-\beta\right)}f_{N}\left(0,0\right)\right|=+\infty.$
∎
Acknowledgement. The authors would like to thank the referee for helpful
suggestions.
## References
* [1] Bakhvalov, A. N. Continuity in $\Lambda$-variation of functions of several variables and the convergence of multiple Fourier series (Russian). Mat. Sb. 193, 12(2002), 3–20; English transl.in Sb. Math. 193, 11-12(2002), no. 11-12, 1731–1748.
* [2] Chanturia, Z. A, The modulus of variation of a function and its application in the theory of Fourier series, Soviet. Math. Dokl. 15 (1974), 67-71.
* [3] Dyachenko M. I, Waterman classes and spherical partial sums of double Fourier series, Anal. Math. 21(1995), 3-21
* [4] Dyachenko M. I, Two-dimensional Waterman classes and $u$-convergence of Fourier series (Russian).Mat. Sb. 190, 7(1999), 23–40; English transl.in Sb. Math. 190, 7-8(1999), 955–972.
* [5] Dyachenko M. I.; Waterman D. , Convergence of double Fourier series and W-classes, Trans. Amer. Math. Soc. 357 (2005), 397-407.
* [6] Goginava U., On the uniform convergence of multiple trigonometric Fourier series. East J. Approx. 3, 5(1999), 253-266.
* [7] Goginava, U. On the uniform summability of two-dimensional trigonometric Fourier series. Proc. A. Razmadze Math. Inst. 124 (2000), 55–72.
* [8] Hardy G. H., On double Fourier series and especially which represent the double zeta function with real and incommensurable parameters. Quart. J. Math. Oxford Ser. 37(1906), 53-79.
* [9] Jordan C., Sur la series de Fourier. C.R. Acad. Sci. Paris. 92(1881), 228-230.
* [10] Marcinkiewicz J. On a class of functions and their Fourier series. Compt. Rend. Soc. Sci. Warsowie, 26( 1934), 71-77.
* [11] Sablin A. I., $\Lambda$-variation and Fourier series (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 10 (1987), 66–68; English transl. in Soviet Math. (Iz. VUZ) 31 (1987), 87-90.
* [12] Sahakian A. A., On the convergence of double Fourier series of functions of bounded harmonic variation (Russian). Izv. Akad. Nauk Armyan. SSR Ser. Mat. 21, 6(1986), 517-529; English transl. Soviet J. Contemp. Math. Anal. 21, 6(1986), 1-13.
* [13] Waterman D, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44(1972), 107-117.
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|
arxiv-papers
| 2012-10-16T14:45:04 |
2024-09-04T02:49:36.676655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ushangi Goginava and Artur Sahakian",
"submitter": "Ushangi Goginava",
"url": "https://arxiv.org/abs/1210.4545"
}
|
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