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https://en.wikipedia.org/wiki/Blancmange%20curve
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In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a Blancmange pudding. It is a special case of the more general de Rham curve; see also fractal curve.
Definition
The blancmange function is defined on the unit interval by
where is the triangle wave, defined by ,
that is, is the distance from x to the nearest integer.
The Takagi–Landsberg curve is a slight generalization, given by
for a parameter ; thus the blancmange curve is the case . The value is known as the Hurst parameter.
The function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.
The function could also be defined by the series in the section Fourier series expansion.
Functional equation definition
The periodic version of the Takagi curve can also be defined as the unique bounded solution to the functional equation
Indeed, the blancmange function is certainly bounded, and solves the functional equation, since
Conversely, if is a bounded solution of the functional equation, iterating the equality one has for any N
whence . Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.
Graphical construction
The blancmange curve can be visually built up out of triangle wave functions if the infinite sum is approximated by finite sums of the first few terms. In the illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.
Properties
Convergence and continuity
The infinite sum defining converges absolutely for all : since for all , we have:
if
Therefore, the Takagi curve of parameter is defined on the unit interval (or ) if .
The Takagi function of parameter is continuous. Indeed, the functions defined by the partial sums are continuous and converge uniformly toward , since:
for all x when
This value can be made as small as we want by selecting a big enough value of n. Therefore, by the uniform limit theorem, is continuous if |w| < 1.
Subadditivity
Since the absolute value is a subadditive function so is the function , and its dilations ; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter .
The special case of the parabola
For , one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.
Differentiability
For values of the parameter the Takagi function is differentiable in classical sense at any which is not a dyadic rational. Precisely,
by derivation under the sign of series, for any non dyadic ratio
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https://en.wikipedia.org/wiki/Roefie%20Hueting
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Roelof (Roefie) Hueting (16 December 1929 – 24 June 2023) was a Dutch economist, former Head of the Department for Environmental Statistics of Statistics Netherlands, pianist and leader of the Down Town Jazz Band, and known for the development of the concept of Sustainable National Income (SNI).
Biography
Hueting was born in The Hague, son of Bernardus Hueting and Elisabeth Hueting-Steinvoorte. In 1949 he founded the Down Town Jazz Band, and earned his living as musician during his studies at the University of Amsterdam which he started in 1951 and received his MA in Economics in 1959. In 1974 he obtained his Ph.D. in economics (cum laude) at the University of Groningen with the thesis "New scarcity and economic growth: More welfare through less production?" under supervision of Jan Pen.
In 1959 he started as assistant public accountant. From 1962 till 1969 he was labour market researcher at the Ministry of Social Affairs, and from 1965 till 1968 at the Ministry of Housing and Physical Planning. After joining the Statistics Netherlands in 1969 he founded its Department of Environmental Statistics. Until his retirement in 1994 he chaired the Department for Environmental Statistics.
In 1991 he was decorated Officer of the Order of Orange-Nassau, and in 1994 awarded the United Nations Global 500 award.
Hueting died in The Hague on 24 June 2023, at the age of 93.
Work
Hueting developed the theoretical and practical framework of the Sustainable national income (SNI). Already in 1970 he published a collection of articles over the years 1967-1970 titled: “What is nature worth to us?”. He has analyzed the environment from the neoclassical point of view of scarcity and developed the concept of Sustainable National Income (SNI). The implication of the SNI is that the statistical measure of economic growth is revised.
Economics and the environment
The concept of sustainability was presented for the first time at The World Conservation Strategy, IUCN, 1980: "This is the kind of development that provides real improvements in the quality of human life and at the same time conserves the vitality and diversity of the Earth. The goal is development that will be sustainable. Today it may seem visionary but it is attainable. To more and more people it also appears our only rational option". (UNEP, IUCN, WWF)
There are various possible descriptions of this area of research but a good one is provided as follows:
"The increase in human numbers and economic activity has put Homo Sapiens in a position to influence nearly every flow of energy and matter on Earth. Explaining the extent and impacts of this influence is well beyond the theory and analytical tools of individual disciplines, such as economics or ecology. A new interdisciplinary approach is needed, one that unites the relevant aspects of different disciplines...
The theory and tools necessary to understand the relation among human populations, natural resources, the environment, and economic growth
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https://en.wikipedia.org/wiki/Group%20with%20operators
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In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way.
Groups with operators were extensively studied by Emmy Noether and her school in the 1920s. She employed the concept in her original formulation of the three Noether isomorphism theorems.
Definition
A group with operators can be defined as a group together with an action of a set on :
that is distributive relative to the group law:
For each , the application is then an endomorphism of G. From this, it results that a Ω-group can also be viewed as a group G with an indexed family of endomorphisms of G.
is called the operator domain. The associate endomorphisms are called the homotheties of G.
Given two groups G, H with same operator domain , a homomorphism of groups with operators is a group homomorphism satisfying
for all and
A subgroup S of G is called a stable subgroup, -subgroup or -invariant subgroup if it respects the homotheties, that is
for all and
Category-theoretic remarks
In category theory, a group with operators can be defined as an object of a functor category GrpM where M is a monoid (i.e. a category with one object) and Grp denotes the category of groups. This definition is equivalent to the previous one, provided is a monoid (otherwise we may expand it to include the identity and all compositions).
A morphism in this category is a natural transformation between two functors (i.e., two groups with operators sharing same operator domain M). Again we recover the definition above of a homomorphism of groups with operators (with f the component of the natural transformation).
A group with operators is also a mapping
where is the set of group endomorphisms of G.
Examples
Given any group G, (G, ∅) is trivially a group with operators
Given a module M over a ring R, R acts by scalar multiplication on the underlying abelian group of M, so (M, R) is a group with operators.
As a special case of the above, every vector space over a field k is a group with operators (V, k).
Applications
The Jordan–Hölder theorem also holds in the context of operator groups. The requirement that a group have a composition series is analogous to that of compactness in topology, and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each (normal) subgroup is an operator-subgroup relative to the operator set X, of the group in question.
See also
Group action
Notes
References
Group actions (mathematics)
Universal algebra
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https://en.wikipedia.org/wiki/Total%20ring%20of%20fractions
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In abstract algebra, the total quotient ring or total ring of fractions is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.
Definition
Let be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed set. Hence we may localize the ring at the set to obtain the total quotient ring .
If is a domain, then and the total quotient ring is the same as the field of fractions. This justifies the notation , which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of .
Examples
For a product ring , the total quotient ring is the product of total quotient rings . In particular, if A and B are integral domains, it is the product of quotient fields.
For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring is the ring of meromorphic functions on D, even if D is not connected.
In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero-divisors is the group of units of the ring, , and so . But since all these elements already have inverses, .
In a commutative von Neumann regular ring R, the same thing happens. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again, .
In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give the definition of a Cartier divisor.
The total ring of fractions of a reduced ring
Proof: Every element of Q(A) is either a unit or a zero divisor. Thus, any proper ideal I of Q(A) is contained in the set of zero divisors of Q(A); that set equals the union of the minimal prime ideals since Q(A) is reduced. By prime avoidance, I must be contained in some . Hence, the ideals are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A),
.
Let S be the multiplicatively closed set of non-zero-divisors of A. By exactness of localization,
,
which is already a field and so must be .
Generalization
If is a commutative ring and is any multiplicatively closed set in , the localization can still be constructed, but the ring homomorphism from to might fail to be injective. For example, if , then is the trivial ring.
Citations
References
Commutative algebra
Ring theory
de:Lokalisierung_(Algebra)#Totalquotientenring
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https://en.wikipedia.org/wiki/Milliken%E2%80%93Taylor%20theorem
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In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor.
Let denote the set of finite subsets of , and define a partial order on by α<β if and only if max α<min β. Given a sequence of integers and , let
Let denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition , there exist some and a sequence such that .
For each , call an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.
References
.
.
Ramsey theory
Theorems in discrete mathematics
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https://en.wikipedia.org/wiki/Schwartz%20space
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In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of , particulary, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.
Schwartz space is named after French mathematician Laurent Schwartz.
Definition
Let be the set of non-negative integers, and for any , let be the n-fold Cartesian product. The Schwartz space or space of rapidly decreasing functions on is the function spacewhere is the function space of smooth functions from into , and Here, denotes the supremum, and we used multi-index notation, i.e. and .
To put common language to this definition, one could consider a rapidly decreasing function as essentially a function such that , , , ... all exist everywhere on and go to zero as faster than any reciprocal power of . In particular, (, ) is a subspace of the function space (, ) of smooth functions from into .
Examples of functions in the Schwartz space
If α is a multi-index, and a is a positive real number, then
Any smooth function f with compact support is in S(Rn). This is clear since any derivative of f is continuous and supported in the support of f, so (xαDβ) f has a maximum in Rn by the extreme value theorem.
Because the Schwartz space is a vector space, any polynomial can by multiplied by a factor for a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.
Properties
Analytic properties
From Leibniz's rule, it follows that is also closed under pointwise multiplication:
If then the product .
The Fourier transform is a linear isomorphism .
If then is uniformly continuous on .
is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
Both and its strong dual space are also:
complete Hausdorff locally convex spaces,
nuclear Montel spaces,
It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,
Ultrabornological spaces,
reflexive barrelled Mackey spaces.
Relation of Schwartz spaces with other topological vector spaces
If , then .
If , then is dense in .
The space of all bump functions, , is included in .
See also
Bump function
Schwartz–Bruhat function
Nuclear space
References
Sources
Topological vector spaces
Smooth functions
Fourier analysis
Function spaces
Schwartz distributions
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https://en.wikipedia.org/wiki/Noncoherent%20STC
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Non-coherent space time codes are a way of transmitting data in wireless communications.
In this multiple antenna scheme, it is assumed that the receiver only has knowledge of the statistics of channel.
Non-coherent space-time transmission schemes were proposed by Tom Marzetta and Bertrand Hochwald in 1999, but these schemes are complex in terms of implementation.
References
Data transmission
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https://en.wikipedia.org/wiki/Vertex%20pipeline
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The function of the vertex pipeline in any GPU is to take geometry data (usually supplied as vector points), work with it if needed with either fixed function processes (earlier DirectX), or a vertex shader program (later DirectX), and create all of the 3D data points in a scene to a 2D plane for display on a computer monitor.
It is possible to eliminate unneeded data from going through the rendering pipeline to cut out extraneous work (called view volume clipping and backface culling). After the vertex engine is done working with the geometry, all the 2D calculated data is sent to the pixel engine for further processing such as texturing and fragment shading.
As of DirectX 9c, the vertex processor is able to do the following by programming the vertex processing under the Direct X API:
Displacement mapping
Geometry blending
Higher-order primitives
Point sprites
Matrix stacks
External links
Anandtech Article
3D computer graphics
Graphics standards
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https://en.wikipedia.org/wiki/166%20%28number%29
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166 (one hundred [and] sixty-six) is the natural number following 165 and preceding 167.
In mathematics
166 is an even number and a composite number. It is a centered triangular number.
Given 166, the Mertens function returns 0. 166 is a Smith number in base 10.
In astronomy
166 Rhodope is a dark main belt asteroid, in the Adeona family of asteroids
166P/NEAT is a periodic comet and centaur in the outer Solar System
HD 166 is the 6th magnitude star in the constellation Andromeda
In the military
166th Signal Photo Company was the official photo unit in the 89th Division of George Patton's Third Army in World War II
Convoy ON-166 was the 166th of the numbered ON series of merchant ship convoys outbound from the British Isles to North America departing February 11, 1943
Marine Medium Helicopter Squadron 166 is a United States Marine Corps helicopter
was a United States Coast Guard cutter during World War II
was a United States Navy yacht. She was the first American vessel lost in World War I
was a United States Navy during World War II
was a United States Navy during the World War I
was a United States Navy during World War II
was a United States Navy ship during World War II
USS Jamestown (AGTR-3/AG-166) was a United States Navy Oxford-class technical research ship following World War II
In sports
Sam Thompson’s 166 RBIs in 1887 stood as a Major League Baseball record until Babe Ruth broke the record in 1921
In transportation
British Rail Class 166
The now-defunct elevated IRT Third Avenue Line, 166th Street station in the Bronx, New York
London Buses route 166
Piaggio P.166 is a twin-engined push prop-driven utility aircraft developed by the Italian aircraft manufacturer Piaggio
Banat Air Flight 166 crashed on take-off en route from Romania on December 13, 1995
Alfa Romeo 166 and 166 2.4 JTD produced from 1998 to 2007
Ferrari 166 model cars produced from 1948 to 1953
Ferrari 166 Inter (1949) Coachbuilt street coupe and cabriolet
In other fields
166 is also:
The year AD 166 or 166 BC
The atomic number of an element temporarily called Unhexhexium
See also
List of highways numbered 166
United States Supreme Court cases, Volume 166
United Nations Security Council Resolution 166
External links
Number Facts and Trivia: 166
The Number 166
VirtueScience: 166
166th Street (3rd Avenue El)
References
Integers
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https://en.wikipedia.org/wiki/Correlation%20immunity
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In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in is statistically independent of the value of .
Definition
A function is -th order correlation immune if for any independent binary random variables , the random variable is independent from any random vector with .
Results in cryptography
When used in a stream cipher as a combining function for linear feedback shift registers, a Boolean function with low-order correlation-immunity is more susceptible to a correlation attack than a function with correlation immunity of high order.
Siegenthaler showed that the correlation immunity m of a Boolean function of algebraic degree d of n variables satisfies m + d ≤ n; for a given set of input variables, this means that a high algebraic degree will restrict the maximum possible correlation immunity. Furthermore, if the function is balanced then m + d ≤ n − 1.
References
Further reading
Cusick, Thomas W. & Stanica, Pantelimon (2009). "Cryptographic Boolean functions and applications". Academic Press. .
Cryptography
Boolean algebra
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https://en.wikipedia.org/wiki/Chow%20group
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In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
Rational equivalence and Chow groups
For what follows, define a variety over a field to be an integral scheme of finite type over . For any scheme of finite type over , an algebraic cycle on means a finite linear combination of subvarieties of with integer coefficients. (Here and below, subvarieties are understood to be closed in , unless stated otherwise.) For a natural number , the group of -dimensional cycles (or -cycles, for short) on is the free abelian group on the set of -dimensional subvarieties of .
For a variety of dimension and any rational function on which is not identically zero, the divisor of is the -cycle
where the sum runs over all -dimensional subvarieties of and the integer denotes the order of vanishing of along . (Thus is negative if has a pole along .) The definition of the order of vanishing requires some care for singular.
For a scheme of finite type over , the group of -cycles rationally equivalent to zero is the subgroup of generated by the cycles for all -dimensional subvarieties of and all nonzero rational functions on . The Chow group of -dimensional cycles on is the quotient group of by the subgroup of cycles rationally equivalent to zero. Sometimes one writes for the class of a subvariety in the Chow group, and if two subvarieties and have , then and are said to be rationally equivalent.
For example, when is a variety of dimension , the Chow group is the divisor class group of . When is smooth over (or more generally, a locally Noetherian normal scheme ), this is isomorphic to the Picard group of line bundles on .
Examples of Rational Equivalence
Rational Equivalence on Projective Space
Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree , so , we can construct a family of hypersurfaces defined as the vanishing locus of . Schematically, this can be constructed as
using the projection we can see the fiber over a point is the projective hypersurface defined by . This can be used to show that the cycle class of every hypersurface of degree is rationally equivalent to , since can be used to establish a rational equivalence. Notice that t
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https://en.wikipedia.org/wiki/Inventiones%20Mathematicae
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Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University).
Abstracting and indexing
The journal is abstracted and indexed in:
References
External links
Mathematics journals
Academic journals established in 1966
English-language journals
Springer Science+Business Media academic journals
Monthly journals
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https://en.wikipedia.org/wiki/MicroWorlds
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MicroWorlds is a program that uses the Logo programming language to teach language, mathematics, programming, and robotics concepts in primary and secondary education. It features an object in the shape of a turtle that can be given commands to move around the screen drawing shapes, creating animations, and playing games. The program's use of Logo is part of a large set of dialects and implementations created by Seymour Papert aimed at triggering the development of abstract ideas by children through experimentation. MicroWorlds is developed by Logo Computer Systems Inc. (LCSI) and released for Windows and Mac computers.
Release History
The precursors to MicroWorlds were the programs Apple Logo, Atari Logo, and LogoWriter released by LCSI for the Macintosh, Atari 8-bit family, and IBM Personal Computer in the 1980s. The first version to bear the MicroWorlds name was released in 1993 for DOS and Mac called MicroWorlds Project Builder. Two modules were released to accompany the software called "Math Links" and "Language Arts."
MicroWorlds 2.0 was released in 1996 for Windows 95 and in 1998 for Mac. Modules for weather and plants were released in 1997, as well as an internet browser plugin to view projects in Internet Explorer and Netscape Navigator without the full software installed. Spanish and Portuguese editions were released under the name MicroMundos.
MicroWorlds Pro, an advanced version intended for high school students, was released in 1999 for Windows 95/98/NT and in 2000 for Mac.
MicroWorlds EX, the current iteration of the software, was released in 2003 for Windows 98 and up (currently supported for Windows 7 and up), and in 2004 for Mac OS X. A “Robotics edition” was released for both platforms that worked with Lego RCX programmable bricks and the Handy Cricket microcontroller system. An "Exploring Math" module intended for Grades 4-7 was released in 2005 and a "Computer Science" module released in 2013. The program has been made available in French, Spanish, Russian, Chinese, Portuguese, Italian, Armenian, and Greek. MicroWorlds EX will not work with macOS 10.15 or higher due to requiring 32-bit support.
MicroWorlds JR, a derivative product teaching coding to young children who cannot read, was released in 2004 for Windows XP and 2005 for Mac OS X.
Features
MicroWorlds relies on Logo, a computer programming language based on words and syntax that are intended to be easy to learn and remember. The software is able to execute multiple tasks independently, can import pictures, and create multimedia projects like games and simulations.
Users write code in a dialect of the Logo programming language to move a customizable cursor (initially in the shape of a turtle), draw shapes, or to make dialog boxes appear. The user may write code in one of two areas of the program, using the program's "command module" to execute short commands immediately or the "procedure page" for more complex sets of instructions that can be stored and referenc
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https://en.wikipedia.org/wiki/Replicate
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Replicate may refer to:
Replicate (biology), the exact copy resulting from self-replication of genetic material, a cell, or an organism
Replicate (statistics), a fully repeated experiment or set of test conditions.
See also
Replication (disambiguation)
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https://en.wikipedia.org/wiki/Algebraic%20expression
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In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, is an algebraic expression. Since taking the square root is the same as raising to the power , the following is also an algebraic expression:
An algebraic equation is an equation involving only algebraic expressions.
By contrast, transcendental numbers like and are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, is constructed as a geometric relationship, and the definition of requires an infinite number of algebraic operations.
A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic. Thus,
is a rational expression, whereas
is not.
A rational equation is an equation in which two rational fractions (or rational expressions) of the form
are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.
Terminology
Algebra has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, - variables
In roots of polynomials
The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n 5.
Conventions
Variables
By convention, letters at the beginning of the alphabet (e.g. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and ) are used to represent variables. They are usually written in italics.
Exponents
By convention, terms with the highest power (exponent), are written on the left, for example, is written to the left of . When a coefficient is one, it is usually omitted (e.g. is written ). Likewise when the exponent (power) is one, (e.g. is written ), and, when the exponent is zero, the result is always 1 (e.g. is written , since is always ).
Algebraic and other mathematical expressions
The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the
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https://en.wikipedia.org/wiki/Poisson%20ring
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In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring.
Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics—the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.
Definition
The Poisson bracket must satisfy the identities
(skew symmetry)
(distributivity)
(derivation)
(Jacobi identity)
for all in the ring.
A Poisson algebra is a Poisson ring that is also an algebra over a field. In this case, add the extra requirement
for all scalars s.
For each g in a Poisson ring A, the operation defined as is a derivation. If the set generates the set of derivations of A, then A is said to be non-degenerate.
If a non-degenerate Poisson ring is isomorphic as a commutative ring to the algebra of smooth functions on a manifold M, then M must be a symplectic manifold and is the Poisson bracket defined by the symplectic form.
References
Ring theory
Symplectic geometry
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https://en.wikipedia.org/wiki/Biconjugate%20gradient%20method
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In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations
Unlike the conjugate gradient method, this algorithm does not require the matrix to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose .
The Algorithm
Choose initial guess , two other vectors and and a preconditioner
for do
In the above formulation, the computed and satisfy
and thus are the respective residuals corresponding to and , as approximate solutions to the systems
is the adjoint, and is the complex conjugate.
Unpreconditioned version of the algorithm
Choose initial guess ,
for do
Discussion
The biconjugate gradient method is numerically unstable (compare to the biconjugate gradient stabilized method), but very important from a theoretical point of view. Define the iteration steps by
where using the related projection
with
These related projections may be iterated themselves as
A relation to Quasi-Newton methods is given by and , where
The new directions
are then orthogonal to the residuals:
which themselves satisfy
where .
The biconjugate gradient method now makes a special choice and uses the setting
With this particular choice, explicit evaluations of and are avoided, and the algorithm takes the form stated above.
Properties
If is self-adjoint, and , then , , and the conjugate gradient method produces the same sequence at half the computational cost.
The sequences produced by the algorithm are biorthogonal, i.e., for .
if is a polynomial with , then . The algorithm thus produces projections onto the Krylov subspace.
if is a polynomial with , then .
See also
Biconjugate gradient stabilized method (BiCG-Stab)
Conjugate gradient method (CG)
Conjugate gradient squared method (CGS)
References
Numerical linear algebra
Gradient methods
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https://en.wikipedia.org/wiki/Lie%20bialgebra
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In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group.
Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
Definition
A vector space is a Lie bialgebra if it is a Lie algebra,
and there is the structure of Lie algebra also on the dual vector space which is compatible.
More precisely the Lie algebra structure on is given
by a Lie bracket
and the Lie algebra structure on is given by a Lie
bracket .
Then the map dual to is called the cocommutator,
and the compatibility condition is the following cocycle relation:
where is the adjoint.
Note that this definition is symmetric and is also a Lie bialgebra, the dual Lie bialgebra.
Example
Let be any semisimple Lie algebra.
To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space.
Choose a Cartan subalgebra and a choice of positive roots.
Let be the corresponding opposite Borel subalgebras, so that and there is a natural projection .
Then define a Lie algebra
which is a subalgebra of the product , and has the same dimension as .
Now identify with dual of via the pairing
where and is the Killing form.
This defines a Lie bialgebra structure on , and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group.
Note that is solvable, whereas is semisimple.
Relation to Poisson–Lie groups
The Lie algebra of a Poisson–Lie group G has a natural structure of Lie bialgebra.
In brief the Lie group structure gives the Lie bracket on as usual, and the linearisation of the Poisson structure on G
gives the Lie bracket on
(recalling that a linear Poisson structure on a vector space is the same thing as a Lie bracket on the dual vector space).
In more detail, let G be a Poisson–Lie group, with being two smooth functions on the group manifold. Let be the differential at the identity element. Clearly, . The Poisson structure on the group then induces a bracket on , as
where is the Poisson bracket. Given be the Poisson bivector on the manifold, define to be the right-translate of the bivector to the identity element in G. Then one has that
The cocommutator is then the tangent map:
so that
is the dual of the cocommutator.
See also
Lie coalgebra
Manin triple
References
H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics,
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https://en.wikipedia.org/wiki/Lefschetz%20manifold
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In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for , the cup product
be an isomorphism.
The topology of these symplectic manifolds is severely constrained, for example their odd Betti numbers are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to William Thurston.
Lefschetz maps
Let be a ()-dimensional smooth manifold. Each element
of the second de Rham cohomology space of induces a map
called the Lefschetz map of . Letting be the th iteration of , we have for each a map
If is compact and oriented, then Poincaré duality tells us that and are vector spaces of the same dimension, so in these cases it is natural to ask whether or not the various iterations of Lefschetz maps are isomorphisms.
The Hard Lefschetz theorem states that this is the case for the symplectic form of a compact Kähler manifold.
Definitions
If
and
are isomorphisms, then is a Lefschetz element, or Lefschetz class. If
is an isomorphism for all , then is a strong Lefschetz element, or a strong Lefschetz class.
Let be a -dimensional symplectic manifold. Then it is orientable, but maybe not compact. One says that is a Lefschetz manifold if is a Lefschetz element, and is a strong Lefschetz manifold if is a strong Lefschetz element.
Where to find Lefschetz manifolds
The real manifold underlying any Kähler manifold is a symplectic manifold. The strong Lefschetz theorem tells us that it is also a strong Lefschetz manifold, and hence a Lefschetz manifold. Therefore we have the following chain of inclusions.
{Kähler manifolds} {strong Lefschetz manifolds} {Lefschetz manifolds} {symplectic manifolds}
Chal Benson and Carolyn S. Gordon proved in 1988 that if a compact nilmanifold is a Lefschetz manifold, then it is diffeomorphic to a torus. The fact that there are nilmanifolds that are not diffeomorphic to a torus shows that there is some space between Kähler manifolds and symplectic manifolds, but the class of nilmanifolds fails to show any differences between Kähler manifolds, Lefschetz manifolds, and strong Lefschetz manifolds.
Gordan and Benson conjectured that if a compact complete solvmanifold admits a Kähler structure, then it is diffeomorphic to a torus. This has been proved. Furthermore, many examples have been found of solvmanifolds that are strong Lefschetz but not Kähler, and solvmanifolds that are Lefschetz but not strong Lefschetz. Such examples were given by Takumi Yamada in 2002.
Notes
Symplectic geometry
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https://en.wikipedia.org/wiki/Solvmanifold
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In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.)
A special class of solvmanifolds, nilmanifolds, was introduced by Anatoly Maltsev, who proved the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated.
Examples
A solvable Lie group is trivially a solvmanifold.
Every nilpotent group is solvable, therefore, every nilmanifold is a solvmanifold. This class of examples includes n-dimensional tori and the quotient of the 3-dimensional real Heisenberg group by its integral Heisenberg subgroup.
The Möbius band and the Klein bottle are solvmanifolds that are not nilmanifolds.
The mapping torus of an Anosov diffeomorphism of the n-torus is a solvmanifold. For , these manifolds belong to Sol, one of the eight Thurston geometries.
Properties
A solvmanifold is diffeomorphic to the total space of a vector bundle over some compact solvmanifold. This statement was conjectured by George Mostow and proved by Louis Auslander and Richard Tolimieri.
The fundamental group of an arbitrary solvmanifold is polycyclic.
A compact solvmanifold is determined up to diffeomorphism by its fundamental group.
Fundamental groups of compact solvmanifolds may be characterized as group extensions of free abelian groups of finite rank by finitely generated torsion-free nilpotent groups.
Every solvmanifold is aspherical. Among all compact homogeneous spaces, solvmanifolds may be characterized by the properties of being aspherical and having a solvable fundamental group.
Completeness
Let be a real Lie algebra. It is called a complete Lie algebra if each map
in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let G be a solvable Lie group whose Lie algebra is complete. Then for any closed subgroup of G, the solvmanifold is a complete solvmanifold.
References
Lie algebras
Structures on manifolds
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https://en.wikipedia.org/wiki/Richard%20Jeffrey
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Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of probability kinematics, also known as Jeffrey conditioning.
Life and career
Born in Boston, Massachusetts, Jeffrey served in the U.S. Navy during World War II. As a graduate student he studied under Rudolf Carnap and Carl Hempel. He received his M.A. from the University of Chicago in 1952 and his Ph.D. from Princeton in 1957. After holding academic positions at MIT, City College of New York, Stanford University, and the University of Pennsylvania, he joined the faculty of Princeton in 1974 and became a professor emeritus there in 1999. He was also a visiting professor at the University of California, Irvine.
Jeffrey, who died of lung cancer at the age of 76, was known for his sense of humor, which often came through in his breezy writing style. In the preface of his posthumously published Subjective Probability, he refers to himself as "a fond foolish old fart dying of a surfeit of Pall Malls".
Philosophical work
As a philosopher, Jeffrey specialized in epistemology and decision theory. He is perhaps best known for defending and developing the Bayesian approach to probability.
Jeffrey also wrote, or co-wrote, two widely used and influential logic textbooks: Formal Logic: Its Scope and Limits, a basic introduction to logic, and Computability and Logic, a more advanced text dealing with, among other things, the famous negative results of twentieth-century logic such as Gödel's incompleteness theorems and Tarski's indefinability theorem.
Radical probabilism
In Bayesian statistics, Bayes' theorem provides a useful rule for updating a probability when new frequency data becomes available. In Bayesian statistics, the theorem itself plays a more limited role. Bayes' theorem connects probabilities that are held simultaneously. It does not tell the learner how to update probabilities when new evidence becomes available over time. This subtlety was first pointed out in terms by Ian Hacking in 1967.
However, adapting Bayes' theorem, and adopting it as a rule of updating, is a temptation. Suppose that a learner forms probabilities Pold(A&B)=p and Pold(B)=q.
If the learner subsequently learns that B is true, nothing in the axioms of probability or the results derived therefrom tells him how to behave. He might be tempted to adopt Bayes' theorem by analogy and set his Pnew(A) = Pold(A | B) = p/q.
In fact, that step, Bayes' rule of updating, can be justified, as necessary and sufficient, through a dynamic Dutch book argument that is additional to the arguments used to justify the axioms. This argument was first put forward by David Lewis in the 1970s though he never published it.
That works when the new data is certain. C. I. Lewis had argued that "If anything is to be probable then something must be certain". There
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https://en.wikipedia.org/wiki/COGO
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COGO is a suite of programs used in civil engineering for modelling horizontal and vertical alignments and solving coordinate geometry problems. Cogo alignments are used as controls for the geometric design of roads, railways, and stream relocations or restorations.
COGO was originally a subsystem of MIT's Integrated Civil Engineering System (ICES), developed in the 1960s. Other ICES subsystems included STRUDL, BRIDGE, LEASE, PROJECT, ROADS and TRANSET, and the internal languages ICETRAN and CDL. Evolved versions of COGO are still widely used.
Some basic types of elements of COGO are points, Euler spirals, lines and horizontal curves (circular arcs).
More complex elements can be developed such as alignments or chains which are made up of a combination of points, curves or spirals.
See also
Civil engineering software
References
"Engineer's Guide to ICES COGO I", R67-46, Civil Engineering Dept MIT (Aug 1967)
"An Integrated Computer System for Engineering Problem Solving", D. Roos, Proc SJCC 27(2), AFIPS (Spring 1965). Sammet 1969, pp.615-620.
Mathematical software
Surveying
History of software
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https://en.wikipedia.org/wiki/Infinite%20dihedral%20group
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In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.
In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis.
Definition
Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rn is the identity, and we have a finite dihedral group of order 2n. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih∞. It has presentations
and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension), the group of permutations α: Z → Z satisfying |i − j| = |α(i) − α(j)|, for all i', j in Z.
The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group.
Aliasing
An example of infinite dihedral symmetry is in aliasing of real-valued signals.
When sampling a function at frequency (intervals ), the following functions yield identical sets of samples: }. Thus, the detected value of frequency is periodic, which gives the translation element . The functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity:
we can write all the alias frequencies as positive values: . This gives the reflection () element, namely ↦ . For example, with and , reflects to , resulting in the two left-most black dots in the figure. The other two dots correspond to and . As the figure depicts, there are reflection symmetries, at 0.5, , 1.5, etc. Formally, the quotient under aliasing is the orbifold'' [0, 0.5], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection.
See also
The orthogonal group O(2), another infinite generalization of the finite dihedral groups
The affine symmetric group, a family of groups including the infinite dihedral group
Notes
References
Infinite group theory
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https://en.wikipedia.org/wiki/Twisted%20K-theory
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In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory.
More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View.
In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics).
In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven. It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
Definition
To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that
the Fredholm operators on Hilbert space , is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space consists of the homotopy classes of maps
from to
A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of over , that is, the Cartesian product of and . Then the K-theory of consists of the homotopy classes of sections of this bundle.
We can make this yet more complicated by introducing a trivial
bundle over , where is the group of projective unitary operators on the Hilbert space . Then the group of maps
from to which are equivariant under an action of is equivalent to the original groups of maps
This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that bundles on are classified by elements of the third integral cohomology group of . This is a consequence of the fact that topologically is a representative Eilenberg–MacLane space
.
The generalization is then straightforward. Rosenberg has defined
,
the twisted K-theory of with twist given by the 3-class , to be the space of homotopy classes of sections of the trivial bundle over that are covariant with respect to a bundle fibered over with 3-class , that is
Equivalently, it is the space of homotopy classes of sections of the bundles associated to a bundle with class .
Relation to K-theory
When is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when is nontrivial this theory is no longer a rin
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https://en.wikipedia.org/wiki/Raymond%20Louis%20Wilder
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Raymond Louis Wilder (3 November 1896 in Palmer, Massachusetts – 7 July 1982 in Santa Barbara, California) was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests.
Life
Wilder's father was a printer. Raymond was musically inclined. He played cornet in the family orchestra, which performed at dances and fairs, and accompanied silent films on the piano.
He entered Brown University in 1914, intending to become an actuary. During World War I, he served in the U.S. Navy as an ensign. Brown awarded him his first degree in 1920, and a master's degree in actuarial mathematics in 1921. That year, he married Una Maude Greene; they had four children, thanks to whom they have ample descent.
Wilder chose to do his Ph.D. at the University of Texas at Austin, the most fateful decision of his life. At Texas, Wilder discovered pure mathematics and topology, thanks to the remarkable influence of Robert Lee Moore, the founder of topology in the US and the inventor of the Moore method for teaching mathematical proof. Moore was initially unimpressed by the young actuary, but Wilder went on to solve a difficult open problem that Moore had posed to his class. Moore suggested Wilder write up the solution for his Ph.D. thesis, which he did in 1923, titling it Concerning Continuous Curves. Wilder thus became the first of Moore's many doctoral students at the University of Texas.
After a year as an instructor at Texas, Wilder was appointed assistant professor at the Ohio State University in 1924. That university required that its academic employees sign a loyalty oath, which Wilder was very reluctant to sign because doing so was inconsistent with his lifelong progressive political and moral views.
In 1926, Wilder joined the faculty of the University of Michigan at Ann Arbor, where he supervised 26 Ph.Ds and became a research professor in 1947. During the 1930s, he helped settle European refugee mathematicians in the United States. Mathematicians who rubbed shoulders with Wilder at Michigan and who later proved prominent included Samuel Eilenberg, the cofounder of category theory, and the topologist Norman Steenrod. After his 1967 retirement from Michigan at the rather advanced age of 71, Wilder became a research associate and occasional lecturer at the University of California at Santa Barbara.
Wilder was vice president of the American Mathematical Society, 1950–1951, president 1955–1956, and the Society's Josiah Willard Gibbs Lecturer in 1969. He was president of the Mathematical Association of America, 1965–1966, which awarded him its Distinguished Service Medal in 1973. He was elected to the American National Academy of Sciences in 1963. Brown University (1958) and the University of Michigan (1980) awarded him honorary doctorates. The mathematics department at the University of California annually bestows one or more graduating seniors with an award in Wilder's name.
The historical, philosophi
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https://en.wikipedia.org/wiki/Hyperflex
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Hyperflex may refer to:
Flexion, in anatomy
Inflection point of a curve where the tangent meets to order at least 4, in mathematics
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https://en.wikipedia.org/wiki/Denmark%20national%20football%20team%20records%20and%20statistics
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The Denmark national football team statistics show the accomplishments of the players and coaches of the Danish men's ever since the controlling organ of the team, the Danish Football Association (DBU), started registering official games at the 1908 Summer Olympics.
Key
Most appearances
The 25 most capped players for Denmark are:
* denotes members of the 1992 European Championship-winning team.
Goalscorers
Top goalscorers
With 52 goals, Poul "Tist" Nielsen is currently the men's number one goalscorer in Danish football history. The players with 10 goals or more for Denmark are:
denotes members of the 1992 European Championship-winning team.
Highest goal average
Only people with at least five goals have been included.
Team captains
The ten players with the most caps as Danish team captains are:
Players still playing or available for selection are in bold. * denotes members of the 1992 European Championship-winning team.
Cards
As of 29 March 2015, the caution and sending-off statistics are:
Cautions
Players still playing or available for selection are in bold. * denotes members of the 1992 European Championship-winning team.
Sending-offs
Players still playing or available for selection are in bold. * denotes members of the 1992 European Championship-winning team.
Managers
The management statistics are:
Note that the Denmark national football team has not had a designated team manager for every match.
Matches
Note that Average points per game is calculated by using 3 points for a win and 1 point for a draw. Up to date as of 22 September 2021.
Footnotes
GF = Goals for
GA = Goals against
Richard Møller Nielsen coached 8 games in the 1988 Summer Olympics qualifications campaign, winning 6, drawing 1 and losing 1, with the combined score 25–3.
Last updated 30 June 2014.
In all 169 matches, played through 1911-12 (5 matches), 1916 (4), 1919-20 (6), 1920-38 (77), 1939 (4), 1940-45 (12), 1946-47 (10), 1948-52 (24), 1952-56 (24), 1957 (1), 1959 (1) and 1961 (1).
References
External links
Landsholdsspillere med mere end 50 kampe at Danish Football Association
Palle "Banks" Jørgensen, "Landsholdets 681 profiler fra 1908 til i dag", TIPS-Bladet, 2002,
Peders Fodboldstatistik
Haslunds fodboldsider
Records and statistics
Denmark
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https://en.wikipedia.org/wiki/Reciprocal%20rule
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In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule.
The reciprocal rule states that if f is differentiable at a point x and f(x) ≠ 0 then g(x) = 1/f(x) is also differentiable at x and
Proof
This proof relies on the premise that is differentiable at and on the theorem that is then also necessarily continuous there. Applying the definition of the derivative of at with gives
The limit of this product exists and is equal to the product of the existing limits of its factors:
Because of the differentiability of at the first limit equals and because of and the continuity of at the second limit thus yielding
A weak reciprocal rule that follows algebraically from the product rule
It may be argued that since
an application of the product rule says that
and this may be algebraically rearranged to say
However, this fails to prove that 1/f is differentiable at x; it is valid only when differentiability of 1/f at x is already established. In that way, it is a weaker result than the reciprocal rule proved above. However, in the context of differential algebra, in which there is nothing that is not differentiable and in which derivatives are not defined by limits, it is in this way that the reciprocal rule and the more general quotient rule are established.
Application to generalization of the power rule
Often the power rule, stating that , is proved by methods that are valid only when n is a nonnegative integer. This can be extended to negative integers n by letting , where m is a positive integer.
Application to a proof of the quotient rule
The reciprocal rule is a special case of the quotient rule, which states that if f and g are differentiable at x and g(x) ≠ 0 then
The quotient rule can be proved by writing
and then first applying the product rule, and then applying the reciprocal rule to the second factor.
Application to differentiation of trigonometric functions
By using the reciprocal rule one can find the derivative of the secant and cosecant functions.
For the secant function:
The cosecant is treated similarly:
See also
References
Articles containing proofs
Differentiation rules
Theorems in analysis
Theorems in calculus
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https://en.wikipedia.org/wiki/Singular%20measure
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In mathematics, two positive (or signed or complex) measures and defined on a measurable space are called singular if there exist two disjoint measurable sets whose union is such that is zero on all measurable subsets of while is zero on all measurable subsets of This is denoted by
A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.
Examples on Rn
As a particular case, a measure defined on the Euclidean space is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.
Example. A discrete measure.
The Heaviside step function on the real line,
has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure nor is absolutely continuous with respect to but if is any open set not containing 0, then but
Example. A singular continuous measure.
The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous.
Example. A singular continuous measure on
The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
See also
References
Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .
J Taylor, An Introduction to Measure and Probability, Springer, 1996. .
Integral calculus
Measures (measure theory)
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https://en.wikipedia.org/wiki/Howard%20Eves
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Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics.
Eves received his B.S. from the University of Virginia, an M.A. from Harvard University, and a Ph.D. in mathematics from Oregon State University in 1948, the last with a dissertation titled A Class of Projective Space Curves written under Ingomar Hostetter. He then spent most of his career at the University of Maine, 1954–1976. In later life, he occasionally taught at University of Central Florida.
Eves was a strong spokesman for the Mathematical Association of America, which he joined in 1942, and whose Northeast Section he founded. For 25 years he edited the Elementary Problems section of the American Mathematical Monthly. He solved over 300 problems proposed in various mathematical journals. His six volume Mathematical Circles series, collecting humorous and interesting anecdotes about mathematicians, was recently reprinted by the MAA, who also published his two volume Great Moments in the History of Mathematics, and his autobiographical Mathematical Reminiscences in 2001.
Eves had six children.
Books by Eves
1953. Introduction to the History of Mathematics, New York, Rinehart
1966. Functions of a Complex Variable, v. 1, Boston: Prindle, Weber & Schmidt
1966. Elementary matrix theory, Boston: Allyn and Bacon [Reprint: 1980. Dover Publications.]
1972. Survey of Geometry in 2 vols, 2nd ed. Boston: Allyn and Bacon.
1990. Foundations and Fundamental Concepts of Mathematics. 3rd. ed. Boston: PWS-Kent. [Reprint: 1997. Dover Publications.]
Mathematical Circles series
1969. In Mathematical Circles in 2 vols, slipcased. Boston: Prindle, Weber & Schmidt, Inc.
1971. Mathematical Circles Revisited, slipcased. Boston: Prindle, Weber & Schmidt, Inc.
1972. Mathematical Circles Squared, slipcased. Boston: Prindle, Weber & Schmidt, Inc.
1977. Mathematical Circles Adieu, slipcased. Boston: Prindle, Weber & Schmidt, Inc.
1988. Return to Mathematical Circles. Boston: PWS-Kent Publishing Company.
References
Cindy Eves-Thomas & Clayton W. Dodge (2004) Obituary of Howard Eves from Mathematical Association of America.
Clayton Dodge (2011) Tribute to Howard Eves from Providence College.
1911 births
2004 deaths
20th-century American mathematicians
21st-century American mathematicians
Geometers
University of Virginia alumni
Harvard University alumni
Oregon State University alumni
University of Maine faculty
University of Central Florida faculty
American historians of mathematics
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https://en.wikipedia.org/wiki/David%20Schweickart
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David Schweickart (born 1942) is an American mathematician and philosopher. He holds a BS in Mathematics from the University of Dayton, a PhD in Mathematics from the University of Virginia, and a PhD in Philosophy from Ohio State University. He currently is Professor of Philosophy at Loyola University Chicago.
He has taught at Loyola since 1975. He was a visiting professor of mathematics at the University of Kentucky from 1969 to 1970, and a visiting professor of philosophy at the University of New Hampshire from 1986 to 1987. He has also lectured in Spain, Cuba, El Salvador, Italy, the Czech Republic, and throughout the United States. In 1999, Schweickart was named Faculty Member of the Year at Loyola University Chicago.
He is an editor and contributing writer to SolidarityEconomy.net, an online journal dedicated to economic democracy.
Economic democracy
In After Capitalism and other works, Schweickart has developed the model of market socialism he refers to as "economic democracy". In his own words, "Economic Democracy is a market economy." It embodies several key ideas:
Workplace self-management, including election of supervisors
Management of capital investment by a form of public banking
A market for goods, raw materials, instruments of production, etc.
Protectionism to enforce trade equality between nations
The firms and factories are owned by society and managed by the workers. These enterprises, so managed, compete in markets to sell their goods. Profit is shared by the workers. Each enterprise is taxed for the capital they employ, and that tax is distributed to public banks, who fund expansion of existing and new industry.
Critiques
In 2006, Schweickart wrote a detailed critique of participatory economics, called Nonsense on Stilts: Michael Albert's Parecon. He claimed three fundamental features of the economic system are flawed.
Published works
After Capitalism (Rowman and Littlefield, 2002) -
Market Socialism: The Debate Among Socialists, with Bertell Ollman, Hillel Ticktin and James Lawler (Routledge, 1998)
Against Capitalism (Cambridge University Press, 1993; Spanish translation, 1997; Chinese translation, 2003)
Capitalism or Worker Control? An Ethical and Economic Appraisal (Praeger, 1980)
See also
American philosophy
List of American philosophers
References
External links
Schweickart's faculty homepage at Loyola University Chicago
The National Cooperative Grocers Association—An organization of cooperatively owned food stores. Cooperators hold economic democracy to be a key element of their movement.
SolidarityEconomy.net—An online journal of radical social change that features David Schweickart's works and other writings on economic democracy and leftist politics.
"After Capitalism" in Facebook
American economics writers
American male non-fiction writers
20th-century American mathematicians
21st-century American mathematicians
American philosophers
20th-century American philosophers
American socialists
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https://en.wikipedia.org/wiki/Neighbourhood%20%28disambiguation%29
|
A neighbourhood (also spelled neighborhood) is a geographically localised community within a larger city, town, suburb or rural area.
Neighbo(u)rhood(s) may also refer to:
Mathematics
Neighbourhood (mathematics), a concept in topology
Neighbourhood (graph theory), a grouping in graph theory
the Moore neighborhood and Von Neumann neighborhood, used in describing cellular automata
Music
Neighbourhood (album), a 2005 album by Manu Katché
Neighborhoods (Ernest Hood album), 1975
Neighborhoods (Olu Dara album), 2001
Neighborhoods (Blink-182 album), 2011
"Neighbourhood" (song), a 1995 song by British indie rock band Space
Four songs by Arcade Fire from their 2004 album Funeral:
"Neighborhood #1 (Tunnels)"
"Neighborhood #2 (Laika)"
"Neighborhood #3 (Power Out)"
"Neighbourhood", a 2000 song by Zed Bias
Neighborhood Records, a record label
The Neighborhood (album), 1990 album by Los Lobos
The Neighbourhood, an American rock band
The Neighbourhood (album), the band's self-titled album
Other uses
Neighbourhood (TV series), a Chinese TV series
Neighborhood Channel, a subchannel of WQED-TV in Pittsburgh, Pennsylvania, United States
Neighborhood (role-playing game), a 1982 role-playing game
The Neighborhood (novel), 2016 novel by Mario Vargas Llosa
The Neighborhood (TV series), American comedy series
The Neighborhood (film), 2017 Canadian film
The Neighborhood, comic strip by Jerry Van Amerongen
See also
Hood (disambiguation)
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https://en.wikipedia.org/wiki/Robert%20Moody
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Robert Vaughan Moody, (; born November 28, 1941) is a Canadian mathematician. He is the co-discover of Kac–Moody algebra, a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system.
"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." - A. J. Coleman
Born in Great Britain, he received a Bachelor of Arts in Mathematics in 1962 from the University of Saskatchewan, a Master of Arts in Mathematics in 1964 from the University of Toronto, and a Ph.D. in Mathematics in 1966 from the University of Toronto.
In 1966, he joined the Department of Mathematics as an assistant professor in the University of Saskatchewan. In 1970, he was appointed an associate professor and a professor in 1976. In 1989, he joined the University of Alberta as a professor in the Department of Mathematics.
In 1999, he was made an Officer of the Order of Canada. In 1980, he was made a fellow of the Royal Society of Canada. In 1996 Moody and Kac were co-winners of the Wigner Medal.
Selected works
with S. Berman:
with J. Patera:
with Bremner & Patera: Tables of weight space multiplicities, Marcel Dekker 1983
with A. Pianzola:
with S. Kass, J. Patera, & R. Slansky: Affine Lie Algebras, weight multiplicities and branching rules, 2 vols., University of California Press 1991 vol. 1 books.google
with Pianzola: Lie algebras with triangular decompositions, Canadian Mathematical Society Series, John Wiley 1995
with Baake & Grimm: Die verborgene Ordnung der Quasikristalle, Spektrum, Feb. 2002; What is Aperiodic Order?, Eng. trans. on arxiv.org
Notes
References
1941 births
Living people
British emigrants to Canada
Canadian mathematicians
Fellows of the Royal Society of Canada
Officers of the Order of Canada
University of Toronto alumni
Academic staff of the University of Alberta
Academic staff of the University of Saskatchewan
University of Saskatchewan alumni
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https://en.wikipedia.org/wiki/Born%20rule
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The Born rule (also called Born's rule) is a postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated and published by German physicist Max Born in July, 1926.
Details
The Born rule states that if an observable corresponding to a self-adjoint operator with discrete spectrum is measured in a system with normalized wave function (see Bra–ket notation), then:
the measured result will be one of the eigenvalues of , and
the probability of measuring a given eigenvalue will equal , where is the projection onto the eigenspace of corresponding to .
(In the case where the eigenspace of corresponding to is one-dimensional and spanned by the normalized eigenvector , is equal to , so the probability is equal to . Since the complex number is known as the probability amplitude that the state vector assigns to the eigenvector , it is common to describe the Born rule as saying that probability is equal to the amplitude-squared (really the amplitude times its own complex conjugate). Equivalently, the probability can be written as .)
In the case where the spectrum of is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure , the spectral measure of . In this case:
the probability that the result of the measurement lies in a measurable set is given by .
A wave function for a single structureless particle in space position implies that the probability density function for a measurement of the particles's position at time is:
In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures. A POVM is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by self-adjoint observables. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in quantum field theory. They are extensively used in the field of quantum information.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices on a Hilbert space that sum to the identity matrix,:
The POVM element is associated with the measurement outcome , such that the probab
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https://en.wikipedia.org/wiki/Round%20function
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In topology and in calculus, a round function is a scalar function ,
over a manifold , whose critical points form one or several connected components, each homeomorphic to the circle
, also called critical loops. They are special cases of Morse-Bott functions.
For instance
For example, let be the torus. Let
Then we know that a map
given by
is a parametrization for almost all of . Now, via the projection
we get the restriction
is a function whose critical sets are determined by
this is if and only if .
These two values for give the critical sets
which represent two extremal circles over the torus .
Observe that the Hessian for this function is
which clearly it reveals itself as rank of equal to one
at the tagged circles, making the critical point degenerate, that is, showing that the critical points are not isolated.
Round complexity
Mimicking the L–S category theory one can define the round complexity asking for whether or not exist round functions on manifolds and/or for the minimum number of critical loops.
References
Siersma and Khimshiasvili, On minimal round functions, Preprint 1118, Department of Mathematics, Utrecht University, 1999, pp. 18.. An update at
Differential geometry
Geometric topology
Types of functions
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https://en.wikipedia.org/wiki/Geometry%20instancing
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In real-time computer graphics, geometry instancing is the practice of rendering multiple copies of the same mesh in a scene at once. This technique is primarily used for objects such as trees, grass, or buildings which can be represented as repeated geometry without appearing unduly repetitive, but may also be used for characters. Although vertex data is duplicated across all instanced meshes, each instance may have other differentiating parameters (such as color, or skeletal animation pose) changed in order to reduce the appearance of repetition.
API support
Starting in Direct3D version 9, Microsoft included support for geometry instancing. This method improves the potential runtime performance of rendering instanced geometry by explicitly allowing multiple copies of a mesh to be rendered sequentially by specifying the differentiating parameters for each in a separate stream. The same functionality is available in Vulkan core, and the OpenGL core in versions 3.1 and up but may be accessed in some earlier implementations using the EXT_draw_instanced extension.
In offline rendering
Geometry instancing in Houdini, Maya or other 3D packages usually involves mapping a static or pre-animated object or geometry to particles or arbitrary points in space, which can then be rendered by almost any offline renderer. Geometry instancing in offline rendering is useful for creating things like swarms of insects, in which each one can be detailed, but still behaves in a realistic way that does not have to be determined by the animator. Most packages allow variation of the material or material parameters on a per instance basis, which helps ensure that instances do not appear to be exact copies of each other. In Houdini, many object level attributes (e.g. such as scale) can also be varied on a per instance basis. Because instancing geometry in most 3D packages only references the original object, file sizes are kept very small and changing the original changes all of the instances.
In many offline renderers, such as Pixar's PhotoRealistic RenderMan, instancing is achieved by using delayed load render procedurals to only load geometry when the bucket containing the instance is actually being rendered. This means that the geometry for all the instances does not have to be in memory at once.
Video cards that support geometry instancing
GeForce 6000 and up (NV40 GPU or later)
ATI Radeon 9500 and up (R300 GPU or later).
PowerVR SGX535 and up (found in Apple iPhone 3GS and later)
References
External links
EXT_draw_instanced documentation
A quick overview on D3D9 instancing on MSDN
VkVertexInputRate specifies vertex or instance rate
3D computer graphics
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https://en.wikipedia.org/wiki/%CE%A3-finite%20measure
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In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞). A set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.
A different but related notion that should not be confused with σ-finiteness is s-finiteness.
Definition
Let be a measurable space and a measure on it.
The measure is called a σ-finite measure, if it satisfies one of the four following equivalent criteria:
the set can be covered with at most countably many measurable sets with finite measure. This means that there are sets with for all that satisfy .
the set can be covered with at most countably many measurable disjoint sets with finite measure. This means that there are sets with for all and for that satisfy .
the set can be covered with a monotone sequence of measurable sets with finite measure. This means that there are sets with and for all that satisfy .
there exists a strictly positive measurable function whose integral is finite. This means that for all and .
If is a -finite measure, the measure space is called a -finite measure space.
Examples
Lebesgue measure
For example, Lebesgue measure on the real numbers is not finite, but it is σ-finite. Indeed, consider the intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line.
Counting measure
Alternatively, consider the real numbers with the counting measure; the measure of any finite set is the number of elements in the set, and the measure of any infinite set is infinity. This measure is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. But, the set of natural numbers with the counting measure is σ -finite.
Locally compact groups
Locally compact groups which are σ-compact are σ-finite under the Haar measure. For example, all connected, locally compact groups G are σ-compact. To see this, let V be a relatively compact, symmetric (that is V = V−1) open neighborhood of the identity. Then
is an open subgroup of G. Therefore H is also closed since its complement is a union of open sets and by connectivity of G, must be G itself. Thus all connected Lie groups are σ-finite under Haar measure.
Nonexamples
Any non-trivial measure taking only the two values 0 and is clearly non σ-finite. One example in is: for all , if and only if A is not empty; another one is: for all , if and only if A is uncountable, 0 otherwis
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https://en.wikipedia.org/wiki/Finite%20measure
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In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on.
Definition
A measure on measurable space is called a finite measure if it satisfies
By the monotonicity of measures, this implies
If is a finite measure, the measure space is called a finite measure space or a totally finite measure space.
Properties
General case
For any measurable space, the finite measures form a convex cone in the Banach space of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, which form a convex subset, and the probability measures, which are the intersection of the unit sphere in the normed space of signed measures and the finite measures.
Topological spaces
If is a Hausdorff space and contains the Borel -algebra then every finite measure is also a locally finite Borel measure.
Metric spaces
If is a metric space and the is again the Borel -algebra, the weak convergence of measures can be defined. The corresponding topology is called weak topology and is the initial topology of all bounded continuous functions on . The weak topology corresponds to the weak* topology in functional analysis. If is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.
Polish spaces
If is a Polish space and is the Borel -algebra, then every finite measure is a regular measure and therefore a Radon measure.
If is Polish, then the set of all finite measures with the weak topology is Polish too.
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Lebesgue%27s%20decomposition%20theorem
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In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures and such that:
(that is, is absolutely continuous with respect to )
(that is, and are singular).
These two measures are uniquely determined by and
Refinement
Lebesgue's decomposition theorem can be refined in a number of ways.
First, the decomposition of a regular Borel measure on the real line can be refined:
where
νcont is the absolutely continuous part
νsing is the singular continuous part
νpp is the pure point part (a discrete measure).
Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.
Related concepts
Lévy–Itō decomposition
The analogous decomposition for a stochastic processes is the Lévy–Itō decomposition: given a Lévy process X, it can be decomposed as a sum of three independent Lévy processes where:
is a Brownian motion with drift, corresponding to the absolutely continuous part;
is a compound Poisson process, corresponding to the pure point part;
is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.
See also
Decomposition of spectrum
Hahn decomposition theorem and the corresponding Jordan decomposition theorem
Citations
References
Integral calculus
Theorems in measure theory
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https://en.wikipedia.org/wiki/Milliken%27s%20tree%20theorem
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In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets.
Let T be a finitely splitting rooted tree of height ω, n a positive integer, and the collection of all strongly embedded subtrees of T of height n. In one of its simple forms, Milliken's tree theorem states that if then for some strongly embedded infinite subtree R of T, for some i ≤ r.
This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices.
Define where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is partition regular for each n < ω, but that the homogeneous subtree R guaranteed by the theorem is strongly embedded in T.
Strong embedding
Call T an α-tree if each branch of T has cardinality α. Define Succ(p, P)= , and to be the set of immediate successors of p in P. Suppose S is an α-tree and T is a β-tree, with 0 ≤ α ≤ β ≤ ω. S is strongly embedded in T if:
, and the partial order on S is induced from T,
if is nonmaximal in S and , then ,
there exists a strictly increasing function from to , such that
Intuitively, for S to be strongly embedded in T,
S must be a subset of T with the induced partial order
S must preserve the branching structure of T; i.e., if a nonmaximal node in S has n immediate successors in T, then it has n immediate successors in S
S preserves the level structure of T; all nodes on a common level of S must be on a common level in T.
References
Keith R. Milliken, A Ramsey Theorem for Trees J. Comb. Theory (Series A) 26 (1979), 215-237
Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, Trans. Amer. Math. Soc. 263 No.1 (1981), 137-148.
Ramsey theory
Theorems in discrete mathematics
Trees (set theory)
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https://en.wikipedia.org/wiki/Peter%20Mosses
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Peter David Mosses (born 1948) is a British computer scientist.
Peter Mosses studied mathematics as an undergraduate at Trinity College, Oxford, and went on to undertake a DPhil supervised by Christopher Strachey in the Programming Research Group while at Wolfson College, Oxford in the early 1970s. He was the last student to submit his thesis under Strachey before Strachey's death.
In 1978, Mosses published his compiler-compiler, the Semantic Implementation System (SIS), which uses a denotational semantics description of the input language.
Mosses has spent most of his career at BRICS in Denmark. He returned to a chair at Swansea University, Wales. His main contribution has been in the area of formal program semantics. In particular, with David Watt he developed action semantics, a combination of denotational, operational and algebraic semantics.
Currently, Mosses is a visitor at TU Delft, working with the Programming Languages Group.
References
External links
Home page
Living people
Alumni of Trinity College, Oxford
Alumni of Wolfson College, Oxford
Members of the Department of Computer Science, University of Oxford
British computer scientists
Academics of Swansea University
Formal methods people
1948 births
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https://en.wikipedia.org/wiki/Martingale
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Martingale may refer to:
Martingale (probability theory), a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value
Martingale (tack) for horses
Martingale (collar) for dogs and other animals
Martingale (betting system), in 18th century France
a dolphin striker, a spar aboard a sailing ship
In the sport of fencing, a martingale is a strap attached to the sword handle to prevent a sword from being dropped if disarmed
In the theatrical lighting industry, martingale is an obsolete term for a twofer, or occasionally a threefer
Martingale (clothing), a strap controlling the fullness of the clothing material.
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https://en.wikipedia.org/wiki/University%20of%20Waterloo%20Faculty%20of%20Mathematics
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The Faculty of Mathematics is one of six faculties of the University of Waterloo in Waterloo, Ontario, offering more than 500 courses in mathematics, statistics and computer science. The faculty also houses the David R. Cheriton School of Computer Science, formerly the faculty's computer science department. There are more than 31,000 alumni.
History
The faculty was founded on January 1, 1967, a successor to the University of Waterloo's Department of Mathematics, which had grown to be the largest department in the Faculty of Arts under the chairmanship of Ralph Stanton (and included such influential professors as W. T. Tutte). Initially located in the Physics building, the faculty was moved in May 1968 into the newly constructed Mathematics and Computing (MC) Building. Inspired by Stanton's famously gaudy ties, the students draped a large pink tie over the MC Building on the occasion of its opening, which later became a symbol of the faculty.
At the time of its founding, the faculty included five departments: Applied Analysis and Computer Science, Applied Mathematics, Combinatorics and Optimization, Pure Mathematics, and Statistics. In 1975 the Department of Applied Analysis and Computer Science became simply the Department of Computer Science; in 2005 it became the David R. Cheriton School of Computer Science. The Statistics Department also was later renamed the Department of Statistics and Actuarial Science. The Department of Combinatorics and Optimization is the only academic department in the world devoted to combinatorics.
The second building occupied by the Mathematics faculty was the Davis Centre, which was completed in 1988. This building includes a plethora of offices, along with various lecture halls and meeting rooms. (The Davis Centre is also home to the library originally known as the Engineering, Math, and Science [EMS] Library, which was originally housed on the fourth floor of the MC building.)
The Faculty of Mathematics finished construction of a third building, Mathematics 3 (M3), in 2011. This building now houses the Department of Statistics and Actuarial Science and a large lecture hall. An additional building, M4, has been proposed but has yet to be built.
Academics
Degrees
The Faculty of Mathematics grants the BMath (Bachelor of Mathematics) degree for most of its undergraduate programs. Computer Science undergraduates can generally choose between graduating with a BMath or a BCS (Bachelor of Computer Science) degree. The former requires more coursework in mathematics. Specialized degrees exist for the Software Engineering program (the BSE, or Bachelor of Software Engineering) and Computing and Financial Management (BCFM, or Bachelor of Computing and Financial Management). Postgraduate students are generally awarded an MMath (Master of Mathematics) or PhD.
Rankings
In the 2018 QS World University Rankings, the University of Waterloo was ranked 39th globally for Mathematics (and 3rd in Canada) and 31st globally for Com
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https://en.wikipedia.org/wiki/Geometry%20Center
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The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the center's work was the use of computer graphics and visualization for research and education in pure mathematics and geometry.
The center's founding director was Al Marden. Richard McGehee directed the center during its final years. The center's governing board was chaired by David P. Dobkin.
Geomview
Much of the work done at the center was for the development of Geomview, a three-dimensional interactive geometry program. This focused on mathematical visualization with options to allow hyperbolic space to be visualised. It was originally written for Silicon Graphics workstations, and has been ported to run on Linux systems; it is available for installation in most Linux distributions through the package management system. Geomview can run under Windows using Cygwin and under Mac OS X. Geomview has a web site at .
Geomview is built on the Object Oriented Graphics Library (OOGL). The displayed scene and the attributes of the objects in it may be manipulated by the graphical command language (GCL) of Geomview. Geomview may be set as a default 3-D viewer for Mathematica.
Videos
Geomview was used in the construction of several mathematical movies including:
Not Knot, exploring hyperbolic space rendering of knot complements.
Outside In, a movie about sphere eversion.
The shape of space, exploring possible three dimensional spaces.
Other software
Other programs developed at the Center included:
WebEQ, a web browser plugin allowing mathematical equations to be viewed and edited.
Kali, to explore plane symmetry groups.
The Orrery, a Solar System visualizer.
SaVi, a satellite visualisation tool for examining the orbits and coverage of satellite constellations.
Crafter, for structural design of spacecraft.
Surface Evolver, to explore minimal surfaces.
SnapPea, a hyperbolic 3-manifold analyzer.
qhull, to explore convex hulls.
KaleidoTile, to explore tessellations of the sphere, Euclidean plane, and hyperbolic plane.
Website
Richard McGehee, the center's director, has stated that the website was one of the first one hundred websites ever published.
Despite the Center being closed, its website is still online at as an archive of a wide range of geometric topics, including:
Geometry and the Imagination handouts for a two-week course by John Horton Conway, William Thurston and others.
Science U, a collection of interactive exhibits.
The Geometry Forum, an electronic community focused on geometry and math education.
Preprints, 99 preprints from the center.
The Topological Zoo, a collection of curves and surfaces.
Geomview is supported through the dedicated Geomview website.
Research
During its time of operation, a large number of mathematical workshops were held at the center. Many well-known mathematicians visited the center, inclu
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https://en.wikipedia.org/wiki/Kulkarni%E2%80%93Nomizu%20product
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In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two -tensors and gives as a result a -tensor.
Definition
If h and k are symmetric -tensors, then the product is defined via:
where the Xj are tangent vectors and is the matrix determinant. Note that , as it is clear from the second expression.
With respect to a basis of the tangent space, it takes the compact form
where denotes the total antisymmetrisation symbol.
The Kulkarni–Nomizu product is a special case of the product in the graded algebra
where, on simple elements,
( denotes the symmetric product).
Properties
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. For instance, on space forms (i.e. spaces of constant sectional curvature) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric with itself; namely, if we denote by
the -curvature tensor and by
the Riemann curvature tensor with , then
where is the scalar curvature and
is the Ricci tensor, which in components reads .
Expanding the Kulkarni–Nomizu product using the definition from above, one obtains
This is the same expression as stated in the article on the Riemann curvature tensor.
For this very reason, it is commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor each makes to the curvature of a Riemannian manifold. This so-called Ricci decomposition is useful in differential geometry.
When there is a metric tensor g, the Kulkarni–Nomizu product of g with itself is the identity endomorphism of the space of 2-forms, Ω2(M), under the identification (using the metric) of the endomorphism ring End(Ω2(M)) with the tensor product Ω2(M) ⊗ Ω2(M).
A Riemannian manifold has constant sectional curvature k if and only if the Riemann tensor has the form
where g is the metric tensor.
Notes
References
.
Differential geometry
Tensors
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https://en.wikipedia.org/wiki/Arne%20Skaug
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Arne Skaug (6 November 1906 – 4 March 1974) was a Norwegian economist, civil servant, diplomat and politician for the Labour Party. He is known as director of Statistics Norway from 1946 to 1948, Norwegian Minister of Trade and Shipping from 1955 to 1962 and later ambassador.
Early life and career
He was born in Horten as a son of Johan Anton Skaug (1881–1956) and Jenny Lovise Olsen (1882–1917). He graduated with the cand.oecon. degree in 1930, and was hired as a secretary in Statistics Norway in the same year. He continued to work there, except for study leaves from 1935 to 1936 in London. In 1935 and 1936 he released two books: Tidens sosialøkonomi and Dør vi ut? Befolkningsspørsmålet og arbeiderbevegelsen, both written together with Aase Lionæs. In 1939 he was hired in the Ministry of Provisioning, and was a research fellow in economics. He then spent the war years in the United States; first with studies from 1939 to 1941 and as assisting professor at the University of Wisconsin, Madison from 1941. From 1942 to 1946 he worked for Norwegian government in New York City and Washington DC.
Post-war career
In 1946 he returned to Norway to become the director of Statistics Norway. In May 1948 he left to become State Secretary in the Ministry of Foreign Affairs, becoming the first in the Ministry of Foreign Affairs to hold the State Secretary position, which had been introduced in 1947. He left this position in February 1949 to become the Norwegian ambassador to the Organisation for Economic Co-operation and Development in Paris. From January 1955 to January 1962 he served as the Minister of Trade and Shipping in Gerhardsen's Third Cabinet. He was also the acting Minister of Foreign Affairs from August to October 1957 and December 1960 to February 1961, when Halvard Lange had absences of leave. He was then the ambassador to the United Kingdom and Ireland from 1962 to 1968, and to Denmark from 1968 to 1974. He died in March 1974.
Skaug was heavily decorated. He was a Commander of the Greek Order of the Phoenix, and held the Grand Cross of the British Royal Victorian Order, the Swedish Order of the Polar Star, the Portuguese Order of Christ, the Siamese Order of the White Elephant, the Persian Order of the Lion and the Sun, the Danish Order of the Dannebrog, the Belgian Order of the Crown, the Icelandic Order of the Falcon
, the Finnish Order of the Lion and the Royal Norwegian Order of St. Olav.
References
1906 births
1974 deaths
People from Horten
Norwegian civil servants
University of Wisconsin–Madison faculty
Directors of government agencies of Norway
Labour Party (Norway) politicians
Norwegian state secretaries
Ministers of Trade and Shipping of Norway
Ambassadors of Norway to the United Kingdom
Ambassadors of Norway to Ireland
Ambassadors of Norway to Denmark
Norwegian expatriates in the United Kingdom
Norwegian expatriates in the United States
Norwegian expatriates in France
Norwegian expatriates in Denmark
Commanders of the Order of th
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https://en.wikipedia.org/wiki/Joseph%20Mundy
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Joseph Mundy did early work in computer vision and projective geometry using LISP, when computer vision still was a new area of research. In 1987 he presented his work in a video, which now is available for free at archive.org.
Here is an extract of the interview, which took place in the end of the video.
"What do students need to learn to be prepared to meet the challenges?" -
"I would like to comment on the necessary courses a student should take to really be prepared to carry out research in model-based vision. As we can see the geometry of image projection and the mathematics of transformation is a very key element in studying this field, but there are many other issues the student has to be prepared for. If we are going to talk about segmenting images and getting good geometric clues, we have to understand the relationship between the intensity of image data and its underlying geometry. And this would lead the student into such areas as optics, illumination theory, theory of shadows and the like. And also the mathematics underlying this kind of computations would of course require signal processing theory, fourier transform theory and the like. And in dealing with algebraic surfaces such as this curved surfaces as we talked about here, courses in algebraic geometry and higher pure forms of algebra will prove to be necessary in order to make any kind of progress in research to handle curved surfaces. So, I guess the bottom line of what I'm saying is: math courses, particularly those associated with geometric aspects will be key in all of this."
See also
Computer Vision
External links
University video communication on model-based computer vision
Machine Perception of Three-Dimensional Solids - the paper mentioned by Joseph Mundy in the video
A biography
CV
Computer vision researchers
Living people
Rensselaer Polytechnic Institute alumni
Rensselaer Polytechnic Institute faculty
General Electric people
Brown University faculty
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Benedict%20Gross
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Benedict Hyman Gross is an American mathematician who is a professor at the University of California San Diego, the George Vasmer Leverett Professor of Mathematics Emeritus at Harvard University, and former Dean of Harvard College.
He is known for his work in number theory, particularly the Gross–Zagier theorem on L-functions of elliptic curves, which he researched with Don Zagier.
Education and Professional career
Gross graduated from The Pingry School, a leading independent school in New Jersey, in 1967 as the valedictorian. In 1971, he graduated Phi Beta Kappa from Harvard University. He then received an M.Sc. from Oxford University as a Marshall Scholar in 1974 before returning to Harvard and completing his Ph.D. in 1978, under John Tate.
After holding faculty positions at Princeton University and Brown University, Gross became a tenured professor at Harvard in 1985 and remained there subsequently, as Dean of Harvard College from 2003 to 2007.
Benedict Gross was the mathematical consultant for the 1980 film It's My Turn containing the famous scene in which actress Jill Clayburgh, portraying a mathematics professor, impeccably proves the snake lemma.
Awards and honors
Gross is a 1986 MacArthur Fellow.
Gross, Zagier, and Dorian M. Goldfeld won the Cole Prize of the American Mathematical Society in 1987 for their work on the Gross–Zagier theorem. In 2012 he became a fellow of the American Mathematical Society.
Gross was elected as a fellow of the American Academy of Arts and Sciences in 1992 and as a member of the National Academy of Sciences in 2004.
He was elected to the American Philosophical Society in 2017.
He was named as a Harvard University Professor from 2011 to 2016 for his distinguished scholarship and professional work.
Major publications
Gross, Benedict H.; Harris, Joe. Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 157–182.
Gross, Benedict H. Heights and the special values of L-series. Number theory (Montreal, Que., 1985), 115–187, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987.
Gross, Benedict H. A tameness criterion for Galois representations associated to modular forms (mod p). Duke Math. J. 61 (1990), no. 2, 445–517.
Gross, Benedict H.; Prasad, Dipendra. On the decomposition of a representation of SOn when restricted to SOn−1. Canad. J. Math. 44 (1992), no. 5, 974–1002.
Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, 225–320.
Gross, B.; Kohnen, W.; Zagier, D. Heegner points and derivatives of L-series. II. Math. Ann. 278 (1987), no. 1-4, 497–562.
Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra. Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I. Astérisque No. 346 (2012), 1–109.
See also
Fat Chance: Probability from 0 to 1
Gross–Koblitz formula
References
External links
Benedict Gro
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https://en.wikipedia.org/wiki/Dodgem
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Dodgem is a simple abstract strategy game invented by Colin Vout in 1972 while he was a mathematics student at the University of Cambridge as described in the book Winning Ways. It is played on an n×n board with n-1 cars for each player—two cars each on a 3×3 board is enough for an interesting game, but larger sizes are also possible.
Play
The board is initially set up with n-1 blue cars along the left edge and n-1 red cars along the bottom edge, the bottom left square remaining empty. Turns alternate: player 1 ("Left")'s turn is to move any one of the blue cars one space forwards (right) or sideways (up or down). Player 2 ("Right")'s turn is to move any one of the red cars one space forwards (up) or sideways (left or right).
Cars may not move onto occupied spaces. They may leave the board, but only by a forward move. A car which leaves the board is out of the game. There are no captures. A player must always leave their opponent a legal move or else forfeit the game.
The winner is the player who first gets all their pieces off the board, or has all their cars blocked in by their opponent.
The game can also be played in Misere, where you force your opponent to move their pieces off the board.
Theory
The 3×3 game can be completely analyzed (strongly solved) and is a win for the first player—a table showing who wins from every possible position is given in Winning Ways, and given this information it is easy to read off a winning strategy.
David des Jardins showed in 1996 that the 4×4 and 5×5 games never end with perfect play—both players get stuck shuffling their cars from side to side to prevent the other from winning. He conjectures that this is true for all larger boards.
For a 3x3 board, there are 56 reachable positions. Out of the 56 reachable positions, 8 of them are winning, 4 of them are losing, and 44 are draws.
References
.
.
.
External links
"Dodgem" . . . any info? Thread from discussion group rec.games.abstract, 1996, containing David desJardins' analysis of the 4x4 and 5x5 games
Mathematical games
Abstract strategy games
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https://en.wikipedia.org/wiki/Generalized%20minimal%20residual%20method
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In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.
The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986. It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems.
The method
Denote the Euclidean norm of any vector v by . Denote the (square) system of linear equations to be solved by
The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., that .
The n-th Krylov subspace for this problem is
where is the initial error given an initial guess . Clearly if .
GMRES approximates the exact solution of by the vector that minimizes the Euclidean norm of the residual .
The vectors might be close to linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors which form a basis for . In particular, .
Therefore, the vector can be written as with , where is the m-by-n matrix formed by . In other words, finding the n-th approximation of the solution (i.e., ) is reduced to finding the vector , which is determined via minimizing the residue as described below.
The Arnoldi process also constructs , an ()-by- upper Hessenberg matrix which satisfies
an equality which is used to simplify the calculation of (see below). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method.
Because columns of are orthonormal, we have
where
is the first vector in the standard basis of , and
being the first trial vector (usually zero). Hence, can be found by minimizing the Euclidean norm of the residual
This is a linear least squares problem of size n.
This yields the GMRES method. On the -th iteration:
calculate with the Arnoldi method;
find the which minimizes ;
compute ;
repeat if the residual is not yet small enough.
At every iteration, a matrix-vector product must be computed. This costs about floating-point operations for general dense matrices of size , but the cost can decrease to for sparse matrices. In addition to the matrix-vector product, floating-point operations must be computed at the n -th iteration.
Convergence
The nth iterate minimizes the residual in the Krylov subspace . Since every subspace is contained in the next subspace, the residual does not increase. After m iterations, where m is the size of the matrix A, the Krylov space Km is the whole of Rm and hence the GMRES method arrives at the exact
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https://en.wikipedia.org/wiki/Lowell%20Schoenfeld
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Lowell Schoenfeld (April 1, 1920 – February 6, 2002) was an American mathematician known for his work in analytic number theory.
Career
Schoenfeld received his Ph.D. in 1944 from University of Pennsylvania under the direction of Hans Rademacher.
In 1953, as an assistant professor at the University of Illinois Urbana-Champaign, he married (as his second wife) associate professor Josephine M. Mitchell, causing the university to fire her from her tenured position under its anti-nepotism rules while allowing him to keep his more junior tenure-track job. They both resigned in protest, and after several short-term positions they were both able to obtain faculty positions at Pennsylvania State University in 1958. They were both promoted to full professor in 1961, and moved to the University at Buffalo in 1968.
Contributions
Schoenfeld is known for obtaining the following results in 1976, assuming the Riemann hypothesis:
for all x ≥ 2657, based on the prime-counting function π(x) and the logarithmic integral function li(x), and
for all x ≥ 73.2, based on the second Chebyshev function ψ(x).
References
External links
1920 births
2002 deaths
20th-century American mathematicians
21st-century American mathematicians
Number theorists
University of Pennsylvania alumni
University of Illinois Urbana-Champaign faculty
Pennsylvania State University faculty
University at Buffalo faculty
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https://en.wikipedia.org/wiki/Torsion%20tensor
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In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".
More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued 2-form on the manifold. If ∇ is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields X and Y, by
where [X,Y] is the Lie bracket of vector fields.
Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.
The torsion tensor
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes called the Cartan (torsion) tensor) of ∇ is the vector-valued 2-form defined on vector fields X and Y by
where is the Lie bracket of two vector fields. By the Leibniz rule, T(fX, Y) = T(X, fY) = fT(X, Y) for any smooth function f. So T is tensorial, despite being defined in terms of the connection which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.
Components of the torsion tensor
The components of the torsion tensor in terms of a local basis of sections of the tangent bundle can be derived by setting , and by introducing the commutator coefficients . The components of the torsion are then
Here are t
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https://en.wikipedia.org/wiki/Surface%20%28mathematics%29
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In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.
A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (ideally) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
Definitions
Often, a surface is defined by equations that are satisfied by the coordinates of its points. This is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface. If the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation
A surface may also be defined as the image, in some space of dimension at least 3, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. For example, the unit sphere may be parametrized by the Euler angles, also called longitude and latitude by
Parametric equations of surfaces are often irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles (modulo ). For the remaining two points (the north and south poles), one has , and the longitude may take any values. Also, there are surfaces for which there cannot exist a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations, whose images cover the surface. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane (see Surface (topology) and Surface (differential geometry)). This
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https://en.wikipedia.org/wiki/De%20Finetti
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de Finetti usually refers to the Italian statistician Bruno de Finetti, noted for the "operational subjective" conception of probability. His works include:
de Finetti's theorem, which explains why exchangeable observations are conditionally independent given some (usually) unobservable quantity
de Finetti diagram, used to graph the genotype frequencies of populations
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https://en.wikipedia.org/wiki/ITest
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The iTest (formerly known as the American High School Internet Mathematics Competition (AHSIMC)), was founded in 2004 by Bradley Metrock and takes place each September, offering students from across the country to compete against the best and brightest high school students in a highly competitive environment.
Guidelines
Any American high school student (or middle school, though the competition may be too challenging for all but the brightest of these) may compete in a team of up to five people. Students are not required to be students of the same school, or even residents of the same state. Because of the decentralized nature of the competition, an advisor such as a teacher of a guardian is required for each team to monitor the team and vouch for compliance with the competition's rules. No computer or calculator programs or applications are allowed on the iTest, though graphing calculator use is allowed.
Format
Prior to 2008, it had consisted of 25 multiple-choice questions (with 1 answer choice on the first problem, 2 on the second, etc.), 25 short-answer questions, and 10 "Ultimate" questions, which are much like relay questions in that each Ultimate question depends on the answer to the previous ones. In 2008, it consisted only of 100 multiple choice or short answer problems. There were 4 Tiebreaker proof questions up until 2008 - in the 2008 competition, ties were solely broken by submission time.
Participation
Students from 44 states have participated in the iTest, including home school students. In 2005, the iTest admitted the American School of Warsaw in Poland as part of a pilot program to open the competition to international schools.
Naming
When the iTest was founded in 2004, it was named the American High School Internet Mathematics Competition, as mentioned before. However, the iTest soon became its nickname, and the competition officially adopted the name in 2007.
Prizes
The iTest offers a different prize each year, attempting to reward the top students with cutting edge technology.
In 2004, each student on the winning team was awarded their choice of a Microsoft Xbox, Sony PlayStation 2, or GameCube.
In 2005, each participant of the winning team was given an Apple iPod nano.
In 2006, each member of the winning team received a Wii, Nintendo DS Lite, and a copy of Brain Age. (Wiis were not given due to availability issues).
In 2007, each member of the winning team was awarded a $200 Best Buy gift card.
Prizes are also offered for other criteria, including State Champion, Sponsor's Award for Leadership, and Best Team Name.
Winners
2004: Gangsta Feedbags
2005: Team Dominators
2006: Quagga
2007: North Carolina
2008: Purchase Cellophane (AAST)
iTest logos
References
External links
AHSMIC Forum moderated by iTest creator Bradley Metrock
iTest Blog
http://itest.sourceforge.net/ official site for iTest
https://www.bradleymetrock.com/ official site for Bradley Metrock
Mathematics competitions
Recurring events est
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https://en.wikipedia.org/wiki/Peter%20Phillips%20%28economist%29
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Peter Charles Bonest Phillips (born 23 March 1948) is an econometrician. Since 1979 he has been Professor of Economics and Statistics at Yale University. He also holds positions at the University of Auckland, Singapore Management University and the University of Southampton. He is currently the co-director of Center for Financial Econometrics of Sim Kee Boon Institute for Financial Economics at Singapore Management University and is an adjunct professor of econometrics at the University of Southampton.
Education
During his schooling, Phillips was the dux of Mount Albert Grammar School in New Zealand. He received a B.A. and M.A. from the University of Auckland and won prizes in both mathematics and economics. He received his PhD from London School of Economics under the supervision of John Denis Sargan in 1974.
Research
He is a founding editor of the journal Econometric Theory. Peter Phillips has published many theoretical articles and advanced many research areas in econometrics. He has published important articles on continuous time econometrics, finite-sample theory, asymptotic expansions, unit root and cointegration, long-range dependent time series, and panel data econometrics. He also introduced the use of the functional central limit theorem to derive asymptotic distributions of unit roots tests. Phillips mainly used frequentist statistical methods. Phillips has also supervised numerous Ph.D. students, including Steve Durlauf. In 1993 he was elected as a Fellow of the American Statistical Association.
According to the November 2015 ranking of economists by Research Papers in Economics, he is the 5th most influential economist.
Festschrift
In 2012, The Journal of Econometrics dedicated two Festschrifts to Phillips under the title Recent Advances in Nonstationary Time Series: A Festschrift in honor of Peter C.B. Phillips.
Selected publications
References
External links
Homepage
1948 births
Living people
New Zealand economists
Time series econometricians
Yale University faculty
Academic staff of the University of Auckland
Yale Sterling Professors
People educated at Mount Albert Grammar School
Fellows of the Econometric Society
Fellows of the American Statistical Association
Fellows of the American Academy of Arts and Sciences
Corresponding Fellows of the British Academy
University of Auckland alumni
Alumni of the London School of Economics
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https://en.wikipedia.org/wiki/Werner%20Ploberger
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Werner Ploberger (born 5 August 1956 in Vienna) is an Austrian economist. He graduated in mathematics from the Vienna University of Technology. Beginning in 1997, he was a professor of economics at the University of Rochester. Effective July 1, 2006, he is professor of economics at Washington University in St. Louis. He is married to Gabriele Ploberger, and has a son.
Literature
Testing for Structural Change in Dynamic Models (with W. Krämer and R. Alt), Econometrica, Vol. 56, No.6, 1988, pp. 1355–1369.
A New Test for Structural Stability in the Linear Regression Model (with W. Kraemer and K. Kontrus), Journal of Econometrics, vol. 40, 1989, pp. 307–318.
The CUSUM-Test with OLS-Residuals (with W. Krämer), Econometrica, Vol. 60, No. 2, 1992, pp. 271–285.
Posterior Odds Testing for a Unit Root with Data-Based Model Selection (with Peter C. B. Phillips), Econometric Theory, Vol.10, No. 3-4, 1994, pp 771–808.
Optimal Tests When a Nuisance Parametere is Present Only Under the Alternative (with Donald Andrews), Econometrica, Vol. 62, No. 6, 1994, pp. 1383–1414
An Asymptotic Theory of Bayesian Inference for Time Series (with Peter C.B. Phillips), Econometrica Vol. 64, No.2, 1996, pp 381–412
Asymptotic Theory of Integrated Conditional Moments Tests (with Herman J. Bierens), Econometrica vol 65 no. 5, 1997, pp. 1129–1145
Empirical Limits for Time Series Econometric Models (with Peter C. B. Phillips), Econometrica, Vol. 71(2), 2003, pp. 627–673
External links
Homepage
Washington University in St. Louis
Living people
Washington University in St. Louis faculty
University of Rochester faculty
Writers from Vienna
TU Wien alumni
1956 births
Austrian emigrants to the United States
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https://en.wikipedia.org/wiki/171%20%28number%29
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171 (one hundred [and] seventy-one) is the natural number following 170 and preceding 172.
In mathematics
171 is a triangular number and a Jacobsthal number.
There are 171 transitive relations on three labeled elements, and 171 combinatorially distinct ways of subdividing a cuboid by flat cuts into a mesh of tetrahedra, without adding extra vertices.
The diagonals of a regular decagon meet at 171 points, including both crossings and the vertices of the decagon.
There are 171 faces and edges in the 57-cell, an abstract 4-polytope with hemi-dodecahedral cells that is its own dual polytope.
Within moonshine theory of sporadic groups, the friendly giant is defined as having cyclic groups ⟨ ⟩ that are linked with the function,
∈ where is the character of at .
This generates 171 moonshine groups within associated with that are principal moduli for different genus zero congruence groups commensurable with the projective linear group .
See also
The year AD 171 or 171 BC
List of highways numbered 171
References
Integers
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https://en.wikipedia.org/wiki/174%20%28number%29
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174 (one hundred [and] seventy-four) is the natural number following 173 and preceding 175.
In mathematics
There are 174 7-crossing semi-meanders, ways of arranging a semi-infinite curve in the plane so that it crosses a straight line seven times. There are 174 invertible (0,1)-matrices. There are also 174 combinatorially distinct ways of subdividing a topological cuboid into a mesh of tetrahedra, without adding extra vertices, although not all can be represented geometrically by flat-sided polyhedra.
The Mordell curve has rank three, and 174 is the smallest positive integer for which has this rank. The corresponding number for curves is 113.
In other fields
In English draughts or checkers, a common variation is the "three-move restriction", in which the first three moves by both players are chosen at random. There are 174 different choices for these moves, although some systems for choosing these moves further restrict them to a subset that is believed to lead to an even position.
See also
The year AD 174 or 174 BC
List of highways numbered 174
References
Integers
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https://en.wikipedia.org/wiki/Alternation%20%28geometry%29
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In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.
Coxeter labels an alternation by a prefixed h, standing for hemi or half. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge.
More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation of a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cube 4.4.4 is alternated as 2.3.2.3.2.3 which is reduced to 3.3.3, being the tetrahedron, and all the 6 edges of the tetrahedra can also be seen as the degenerate faces of the original cube.
Snub
A snub (in Coxeter's terminology) can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra.
The snub square antiprism is an example of a general snub, and can be represented by ss{2,4}, with the square antiprism, s{2,4}.
Alternated polytopes
This alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the deleted vertices will not in general create uniform facets, and there are typically not enough degrees of freedom to allow an appropriate rescaling of the new edges. Exceptions do exist, however, such as the derivation of the snub 24-cell from the truncated 24-cell.
Examples:
Honeycombs
An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb.
An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
4-polytope
An alternated truncated 24-cell is the snub 24-cell.
4-honeycombs:
An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb.
A hypercube can always be alternated into a uniform demihypercube.
Cube → Tetrahedron (regular)
→
Tesseract (8-cell) → 16-cell (regular)
→
Penteract → demipenteract (semiregular)
Hexeract → demihexeract (uniform)
...
Altered polyhedra
Coxeter also used the operator a, which contains both halves, so retains the original symmetry. For even-sided regular polyhedra, a{2p,q} represents a compound polyhedron with two opposite copies of h{2p,q}. For odd-sided, greater than 3, regular polyhedra a{p,q}, becomes a star polyhedron.
Norman Johnson extended the use of the altered operator a{p,q}, b{p,q} for blended, and c{p,q} for converted, as , , and respectively.
The compound polyh
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https://en.wikipedia.org/wiki/Loop%20theorem
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In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.
A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M with boundary ∂M there is a map
with not nullhomotopic in , then there is an embedding with the same property.
The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco):
Let be a 3-manifold and let
be a connected surface in . Let be a normal subgroup such that .
Let be a continuous map such that and Then there exists an embedding such that and
Furthermore if one starts with a map f in general position, then for any neighborhood U of the singularity set of f, we can find such a g with image lying inside the union of image of f and U.
Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the sphere theorem (3-manifolds), which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial embedding of a sphere. There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction.
A proof not utilizing the tower construction exists of the first version of the loop theorem. This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by Klaus Johannson, Marc Lackenby, and Iain Aitchison with Hyam Rubinstein.
Corollary
One easy corollary of the loop theorem is a following: Let be a compact orientable irreducible 3-manifold. Then is incompressible if and only if is injective for each component of .
References
W. Jaco, Lectures on 3-manifolds topology, A.M.S. regional conference series in Math 43.
J. Hempel, 3-manifolds, Princeton University Press 1976.
Hatcher, Notes on basic 3-manifold topology, available online
Geometric topology
3-manifolds
Theory of continuous functions
Theorems in topology
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https://en.wikipedia.org/wiki/Conjugate%20index
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In mathematics, two real numbers are called conjugate indices (or Hölder conjugates) if
Formally, we also define as conjugate to and vice versa.
Conjugate indices are used in Hölder's inequality, as well as Young's inequality for products; the latter can be used to prove the former. If are conjugate indices, the spaces Lp and Lq are dual to each other (see Lp space).
See also
Beatty's theorem
References
Antonevich, A. Linear Functional Equations, Birkhäuser, 1999. .
Functional analysis
Linear functionals
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https://en.wikipedia.org/wiki/QQ%20%28disambiguation%29
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QQ refers to Tencent QQ, a Chinese instant messaging program.
QQ may also refer to:
Q–Q plot, a plot to compare distributions in statistics
Chery QQ, two compact Chinese cars models
Alliance Airlines (IATA code QQ)
Qinetiq (LSE stock symbol QQ)
Reno Air, formerly IATA code QQ
Q. texture, an originally Taiwanese term for the ideal texture of many foods
QQ, the production code for the 1968 Doctor Who serial The Web of Fear
QQ, an emoticon referring to a pair of tearing eyes
QQ Magazine, a gay lifestyle magazine, (1969 – ca. 1982)
QQ, a map showing an area of a quarter quadrangle
qq may refer to:
a Perl operator
See also
Q (disambiguation)
QQQ (disambiguation)
QQQQ (disambiguation)
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https://en.wikipedia.org/wiki/List%20of%20FC%20Bayern%20Munich%20records%20and%20statistics
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This list has details on FC Bayern Munich records and statistics.
Coaches
Until 1963
Information on the club's coaches before the Bundesliga era is hard to come by. The information as given in the following table is from the club's website.
Since 1963
In contrast to the pre-Bundesliga era, a list of coaches since the inception of the national league (Bundesliga) in 1963 is readily available on the club's website. Felix Magath (in 2005), Ottmar Hitzfeld (in 2008), Louis van Gaal (in 2010), Jupp Heynckes (in 2013 and 2018) and Hansi Flick (in 2020) were all awarded Germany's Football Manager of the Year title for their work at Bayern. Both Hitzfeld (in 2001) and Flick (in 2020) were also awarded the UEFA Coach of the Year and the IFFHS World's Best Club Coach, while Heynckes won both the FIFA World Coach of the Year and the IFFHS World's Best Club Coach title in 2013.
Presidents
At the club's founding Franz John was appointed as the first president. The current president, Herbert Hainer, is Bayern's 38th president with several presidents having multiple spells in office (counted separately.)
Honorary presidents
The club has six honorary presidents, Franz John, Siegfried Herrmann, Kurt Landauer, Wilhelm Neudecker, Franz Beckenbauer, and Uli Hoeneß, the only living ones being Beckenbauer and Hoeneß. Bayern has also designated honorary vice presidents: Hans Schiefele, Karl Pfab, Bernd Rauch, and Fritz Scherer.
Honours
Bayern have won 83 major trophies: 69 national titles and 14 international titles.
National titles
Official
German Champions/Bundesliga
Champions: (33) 1932, 1968–69, 1971–72, 1972–73, 1973–74, 1979–80, 1980–81, 1984–85, 1985–86, 1986–87, 1988–89, 1989–90, 1993–94, 1996–97, 1998–99, 1999–2000, 2000–01, 2002–03, 2004–05, 2005–06, 2007–08, 2009–10, 2012–13, 2013–14, 2014–15, 2015–16, 2016–17, 2017–18, 2018–19, 2019–20, 2020–21, 2021–22, 2022–23 (record)
Runners-up: (10) 1969–70, 1970–71, 1987–88, 1990–91, 1992–93, 1995–96, 1997–98, 2003–04, 2008–09, 2011–12 (record)
DFB-Pokal
Champions: (20) 1956–57, 1965–66, 1966–67, 1968–69, 1970–71, 1981–82, 1983–84, 1985–86, 1997–98, 1999–2000, 2002–03, 2004–05, 2005–06, 2007–08, 2009–10, 2012–13, 2013–14, 2015–16, 2018–19, 2019–20 (record)
Runners-up: 1984–85, 1998–99, 2011–12, 2017–18
Semi-finals: 1967–68, 1973–74, 1975–76, 2001–02, 2010–11, 2014–15, 2016–17
Quarter-finals: 1969–70, 1971–72, 1972–73, 1976–77, 1987–88, 1996–97, 2003–04, 2008–09, 2022–23
Round of 16: 1986–87, 1988–89, 1989–90, 1993–94, 2006–07
Round 3: 1974–75, 1977–78, 1979–80, 1980–81
Round 2: 1938–39, 1978–79, 1982–83, 1991–92, 1992–93, 1995–96, 2000–01, 2020–21, 2021–22
Round 1: 1935–36, 1936–37, 1940–41, 1943–44, 1990–91, 1994–95
Did not enter: (16) 1937–38, 1939–40, 1941–42, 1942–43, 1952–53, 1953–54, 1954–55, 1955–56, 1957–58, 1958–59, 1959–60, 1960–61, 1961–62, 1962–63, 1963–64, 1964–65
DFB/DFL-Supercup (1987–present; inactive 1997–2009)
Champions: (10) 1983, 1987, 1990, 2010, 2012, 2016, 2017, 2018, 2020, 2021 (
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https://en.wikipedia.org/wiki/Fredholm%20theory
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In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. The theory is named in honour of Erik Ivar Fredholm.
Overview
The following sections provide a casual sketch of the place of Fredholm theory in the broader context of operator theory and functional analysis. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details.
Fredholm equation of the first kind
Much of Fredholm theory concerns itself with the following integral equation for f when g and K are given:
This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation
where the function is given and is unknown. Here, stands for a linear differential operator.
For example, one might take to be an elliptic operator, such as
in which case the equation to be solved becomes the Poisson equation.
A general method of solving such equations is by means of Green's functions, namely, rather than a direct attack, one first finds the function such that for a given pair ,
where is the Dirac delta function.
The desired solution to the above differential equation is then written as an integral in the form of a Fredholm integral equation,
The function is variously known as a Green's function, or the kernel of an integral. It is sometimes called the nucleus of the integral, whence the term nuclear operator arises.
In the general theory, and may be points on any manifold; the real number line or -dimensional Euclidean space in the simplest cases. The general theory also often requires that the functions belong to some given function space: often, the space of square-integrable functions is studied, and Sobolev spaces appear often.
The actual function space used is often determined by the solutions of the eigenvalue problem of the differential operator; that is, by the solutions to
where the are the eigenvalues, and the are the eigenvectors. The set of eigenvectors span a Banach space, and, when there is a natural inner product, then the eigenvectors span a Hilbert space, at which point the Riesz representation theorem is applied. Examples of such spaces are the orthogonal polynomials that occur as the solutions to a class of second-order ordinary differential equations.
Given a Hilbert space as above, the kernel may be written in the form
In this form, the object is often called the Fredholm operator or the Fredholm kernel. That this is the same kernel as before follows from the completeness of the basis of the Hilbert space, namely, that one has
Since the are generally increasing, the resulting eigenvalues o
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https://en.wikipedia.org/wiki/Otto%20Schreier
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Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
Life
His parents were the architect Theodor Schreier (1873-1943) and his wife Anna (b. Turnau) (1878-1942). From 1920 Otto Schreier studied at the University of Vienna and took classes with Wilhelm Wirtinger, Philipp Furtwängler, Hans Hahn, Kurt Reidemeister, Leopold Vietoris, and Josef Lense. In 1923 he obtained his doctorate, under the supervision of Philipp Furtwängler, entitled On the expansion of groups (Über die Erweiterung von Gruppen). In 1926 he completed his habilitation with Emil Artin at the University of Hamburg (Die Untergruppen der freien Gruppe. Abhandlungen des Mathematischen Seminars der Universität Hamburg, Band 5, 1927, Seiten 172–179), where he had also given lectures before.
In 1928 he became a professor at the University of Rostock. He gave lectures in Hamburg and Rostock at the same time in the winter semester but fell seriously ill from sepsis in December 1928, of which he died six months later.
His daughter Irene was born a month after his death. His wife Edith (née Jakoby) and daughter were able to flee to the United States in January 1939. His daughter became a pianist and married the American mathematician Dana Scott (born 1932), whom she had met in Princeton. Otto Schreier's parents were murdered in the Theresienstadt concentration camp during the Holocaust.
Scientific contributions
Schreier was introduced to group theory by Kurt Reidemeister and first examined knot groups in 1924 following work by Max Dehn. His best-known work is his habilitation thesis on the subgroups of free groups, in which he generalizes the results of Reidemeister about normal subgroups. He proved that subgroups of free groups themselves are free, generalizing a theorem by Jakob Nielsen (1921).
In 1927 he showed that the topological fundamental group of a classical Lie group is abelian. In 1928 he improved Jordan-Hölder's theorem. With Emil Artin, he proved the Artin-Schreier theorem characterizing Real closed fields.
The Schreier conjecture of group theory states that the group of external automorphisms of any finite simple group is solvable (the conjecture follows from the classification theorem of finite simple groups, which is generally accepted).
With Emanuel Sperner, he wrote an introductory textbook on linear algebra, which was well-known in German-speaking countries for a long time.
Significance of the Artin–Schreier theorem
According to Hans Zassenhaus:
Results and concepts named after Otto Schreier
Nielsen–Schreier theorem
Schreier refinement theorem
Artin–Schreier theorem
Artin–Schreier theory
Schreier's subgroup lemma
Schreier–Sims algorithm
Schreier coset graph
Schreier conjecture
Schreier domain
References
External links
1901 births
1929 deaths
20th-century Austrian mathematicians
Austrian Je
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https://en.wikipedia.org/wiki/176%20%28number%29
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176 (one hundred [and] seventy-six) is the natural number following 175 and preceding 177.
In mathematics
176 is an even number and an abundant number. It is an odious number, a self number, a semiperfect number, and a practical number.
176 is a cake number, a happy number, a pentagonal number, and an octagonal number. 15 can be partitioned in 176 ways.
The Higman–Sims group can be constructed as a doubly transitive permutation group acting on a geometry containing 176 points, and it is also the symmetry group of the largest possible set of equiangular lines in 22 dimensions, which contains 176 lines.
In astronomy
176 Iduna is a large main belt asteroid with a composition similar to that of the largest main belt asteroid, 1 Ceres
Gliese 176 is a red dwarf star in the constellation of Taurus
Gliese 176 b is a super-Earth exoplanet in the constellation of Taurus. This planet orbits close to its parent star Gliese 176
In the Bible
Minuscule 176 (in the Gregory-Aland numbering), a Greek minuscule manuscript of the New Testament
176 is the highest verse number in the Bible. Found in Psalm 119.
In the military
Attack Squadron 176 United States Navy squadron during the Vietnam War
was a United States Navy troop transport during World War II, the Korean War and Vietnam War
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy Porpoise-class submarine during World War II
was a United States Navy during World War II
was a United States Navy following World War I
was a United States Navy Sonoma-class fleet tug during World War II
176th Wing is the largest unit of the Alaska Air National Guard
In transportation
Heinkel He 176 was a German rocket-powered aircraft
London Buses route 176
176th Street, Bronx elevated station on the IRT Jerome Avenue Line of the New York City Subway
In other fields
176 is also:
The year AD 176 or 176 BC
176 AH is a year in the Islamic calendar that corresponds to 792–793 CE
The atomic number of an element temporarily called Unsepthexium
See also
List of highways numbered 176
United Nations Security Council Resolution 176
United States Supreme Court cases, Volume 176
References
External links
Number Facts and Trivia: 176
The Number 176
The Positive Integer 176
VirtueScience: 176
Number Gossip: 176
Integers
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https://en.wikipedia.org/wiki/Alfred%20Menezes
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Alfred Menezes is co-author of several books on cryptography, including the Handbook of Applied Cryptography, and is a professor of mathematics at the University of Waterloo in Canada.
Education
Alfred Menezes' family is from Goa, a state in western India, but he was born in Tanzania and grew up in Kuwait except for a few years at a boarding school in India. His undergraduate and post-graduate degrees are from the University of Waterloo.
Academic career
After five years teaching at Auburn University, in 1997 he returned to the University of Waterloo, where he is now a professor of mathematics in the Department of Combinatorics and Optimization. He co-founded and is a member of the Centre for Applied Cryptographic Research, and has served as its Managing Director. Menezes' main areas of research are Elliptic Curve Cryptography (ECC), provable security, and related areas. He is a Canadian citizen.
Menezes' book Elliptic Curve Public Key Cryptosystems, published in 1993, was the first book devoted entirely to ECC. He co-authored the widely-used reference book Handbook of Applied Cryptography. Menezes has been a conference organizer or program committee member for approximately fifty conferences on Cryptography. He was Program Chair for Crypto 2007, and in 2012 he was an invited speaker at Eurocrypt.
In 2001 Menezes won the Hall Medal of the Institute of Combinatorics and its Applications.
Books
Selected publications
"Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields" (with G. Adj, I. Canales-Martinez, N. Cruz-Cortes, T. Oliveira, L. Rivera-Zamarripa and F. Rodriguez-Henriquez), Cryptology ePrint Archive: Report 2016/914. https://eprint.iacr.org/2016/914
"Another look at tightness II: Practical issues in cryptography" (with S. Chatterjee, N. Koblitz and P. Sarkar), Mycrypt 2016, Lecture Notes in Computer Science, 10311 (2017), 21–55.
"Another look at HMAC" (with N. Koblitz), Journal of Mathematical Cryptology, 7 (2013), 225–251.
"Elliptic curve cryptography: The serpentine course of a paradigm shift" (with A. H. Koblitz and N. Koblitz), Journal of Number Theory, 131 (2011), 781–814.
"Another look at 'provable security (with N. Koblitz), Journal of Cryptology, 20 (2007), 3–37.
"An efficient protocol for authenticated key agreement" (with L. Law, M. Qu, J. Solinas and S. Vanstone), Designs, Codes and Cryptography, 28 (2003), 119–134.
"Solving elliptic curve discrete logarithm problems using Weil descent" (with M. Jacobson and A. Stein), Journal of the Ramanujan Mathematical Society, 16 (2001), 231–260.
"The elliptic curve digital signature algorithm (ECDSA)" (with D. Johnson and S. Vanstone), International Journal on Information Security, 1 (2001), 36–63.
"Analysis of the Weil descent attack of Gaudry, Hess and Smart" (with M. Qu), Topics in Cryptology – CT-RSA 2001, Lecture Notes in Computer Science, 2020 (2001), 308–318.
"Unknown key-share attacks on the station-to-station (
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https://en.wikipedia.org/wiki/177%20%28number%29
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177 (one hundred [and] seventy-seven) is the natural number following 176 and preceding 178.
In mathematics
It is a Leyland number since .
It is a 60-gonal number, and an arithmetic number, since the mean of its divisors (1, 3, 59 and 177) is equal to 60, an integer.
177 is a Leonardo number, part of a sequence of numbers closely related to the Fibonacci numbers. In graph enumeration, there are 177 undirected graphs (not necessarily connected) that have seven edges and no isolated vertices, and 177 rooted trees with ten nodes and height at most three. There are 177 ways of re-connecting the (labeled) vertices of a regular octagon into a star polygon that does not use any of the octagon edges.
In other fields
177 is the second highest score for a flight of three darts, below the highest score of 180.
See also
The year AD 177 or 177 BC
List of highways numbered 177
References
Integers
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https://en.wikipedia.org/wiki/Minkowski%20geometry
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Minkowski geometry may refer to:
The geometry of a finite-dimensional normed space
The geometry of Minkowski space
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https://en.wikipedia.org/wiki/Fat-tailed%20distribution
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A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. Different research communities favor one or the other largely for historical reasons, and may have differences in the precise definition of either.
Fat-tailed distributions have been empirically encountered in a variety of areas: physics, earth sciences, economics and political science. The class of fat-tailed distributions includes those whose tails decay like a power law, which is a common point of reference in their use in the scientific literature. However, fat-tailed distributions also include other slowly-decaying distributions, such as the log-normal.
The extreme case: a power-law distribution
The most extreme case of a fat tail is given by a distribution whose tail decays like a power law.
That is, if the complementary cumulative distribution of a random variable X can be expressed as
then the distribution is said to have a fat tail if . For such values the variance and the skewness of the tail are mathematically undefined (a special property of the power-law distribution), and hence larger than any normal or exponential distribution. For values of , the claim of a fat tail is more ambiguous, because in this parameter range, the variance, skewness, and kurtosis can be finite, depending on the precise value of , and thus potentially smaller than a high-variance normal or exponential tail. This ambiguity often leads to disagreements about precisely what is or is not a fat-tailed distribution. For , the moment is infinite, so for every power law distribution, some moments are undefined.
Note: here the tilde notation "" refers to the asymptotic equivalence of functions, meaning that their ratio tends to a constant. In other words, asymptotically, the tail of the distribution decays like a power law.
Fat tails and risk estimate distortions
Compared to fat-tailed distributions, in the normal distribution, events that deviate from the mean by five or more standard deviations ("5-sigma events") have lower probability, meaning that in the normal distribution extreme events are less likely than for fat-tailed distributions. Fat-tailed distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) have "undefined sigma" (more technically, the variance is undefined).
As a consequence, when data arise from an underlying fat-tailed distribution, shoehorning in the "normal distribution" model of risk—and estimating sigma based (necessarily) on a finite sample size—would understate the true degree of predictive difficulty (and of risk). Many—notably Benoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and h
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https://en.wikipedia.org/wiki/Halpern%E2%80%93L%C3%A4uchli%20theorem
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In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following .
Let d,r < ω, be a sequence of finitely splitting trees of height ω. Let
then there exists a sequence of subtrees strongly embedded in such that
Alternatively, let
and
.
The HLLP theorem says that not only is the collection partition regular for each d < ω, but that the homogeneous subtree guaranteed by the theorem is strongly embedded in
References
Ramsey theory
Theorems in the foundations of mathematics
Trees (set theory)
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https://en.wikipedia.org/wiki/65%2C535
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65535 is the integer after 65534 and before 65536.
It is the maximum value of an unsigned 16-bit integer.
In mathematics
65535 is the sum of 20 through 215 (20 + 21 + 22 + ... + 215) and is therefore a repdigit in base 2 (1111111111111111), in base 4 (33333333), and in base 16 (FFFF).
It is the ninth number whose Euler totient has an aliquot sum that is : , and the twenty-eighth perfect totient number equal to the sum of its iterated totients.
65535 is the fifteenth 626-gonal number, the fifth 6555-gonal number, and the third 21846-gonal number.
65535 is the product of the first four Fermat primes: 65535 = (2 + 1)(4 + 1)(16 + 1)(256 + 1). Because of this property, it is possible to construct with compass and straightedge a regular polygon with 65535 sides (see, constructible polygon).
In computing
65535 occurs frequently in the field of computing because it is (one less than 2 to the 16th power), which is the highest number that can be represented by an unsigned 16-bit binary number. Some computer programming environments may have predefined constant values representing 65535, with names like .
In older computers with processors having a 16-bit address bus such as the MOS Technology 6502 popular in the 1970s and the Zilog Z80, 65535 (FFFF16) is the highest addressable memory location, with 0 (000016) being the lowest. Such processors thus support at most 64 KiB of total byte-addressable memory.
In Internet protocols, 65535 is also the number of TCP and UDP ports available for use, since port 0 is reserved.
In some implementations of Tiny BASIC, entering a command that divides any number by zero will return 65535.
In Microsoft Word 2011 for Mac, 65535 is the highest line number that will be displayed.
In HTML, 65535 is the decimal value of the web color Aqua (#00FFFF) .
See also
4,294,967,295
255 (number)
16-bit computing
References
Integers
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https://en.wikipedia.org/wiki/178%20%28number%29
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178 (one hundred [and] seventy-eight) is the natural number following 177 and preceding 179.
In mathematics
There are 178 biconnected graphs with six vertices, among which one is designated as the root and the rest are unlabeled. There are also 178 median graphs on nine vertices.
178 is one of the indexes of the smallest triple of dodecahedral numbers where one is the sum of the other two: the sum of the 46th and the 178th dodecahedral numbers is the 179th.
See also
The year 178 AD or 178 BC
List of highways numbered 178
References
Integers
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https://en.wikipedia.org/wiki/Dehn%27s%20lemma
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In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk.
This theorem was thought to be proven by , but found a gap in the proof. The status of Dehn's lemma remained in doubt until using work by Johansson (1938) proved it using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem.
Tower construction
Papakyriakopoulos proved Dehn's lemma using a tower of covering spaces. Soon afterwards gave a substantially simpler proof, proving a more powerful result. Their proof used Papakyriakopoulos' tower construction, but with double covers, as follows:
Step 1: Repeatedly take a connected double cover of a regular neighborhood of the image of the disk to produce a tower of spaces, each a connected double cover of the one below it. The map from the disk can be lifted to all stages of this tower. Each double cover simplifies the singularities of the embedding of the disk, so it is only possible to take a finite number of such double covers, and the top level of this tower has no connected double covers.
Step 2. If the 3-manifold has no connected double covers then all its boundary components are 2-spheres. In particular the top level of the tower has this property, and in this case it is easy to modify the map from the disk so that it is an embedding.
Step 3. The embedding of the disk can now be pushed down the tower of double covers one step at a time, by cutting and pasting the 2-disk.
References
External links
proof by Papakyriakopoulos from 1957
3-manifolds
Lemmas
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https://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf%20method
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The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform.
In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane, and Liouville's theorem implies that this function is an unknown polynomial, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.
Wiener–Hopf decomposition
The key step in many Wiener–Hopf problems is to decompose an arbitrary function into two functions with the desired properties outlined above. In general, this can be done by writing
and
where the contours and are parallel to the real line, but pass above and below the point , respectively.
Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. , by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.
Example
Consider the linear partial differential equation
where is a linear operator which contains derivatives with respect to and , subject to the mixed conditions on = 0, for some prescribed function ,
and decay at infinity i.e. → 0 as .
Taking a Fourier transform with respect to results in the following ordinary differential equation
where is a linear operator containing derivatives only, is a known function of and and
If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted , a general solution can be written as
where is an unknown function to be determined by the boundary conditions on =0.
The key idea is to split into two
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https://en.wikipedia.org/wiki/List%20of%20cities%20in%20Australia%20by%20population
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These lists of Australian cities by population provide rankings of Australian cities and towns according to various systems defined by the Australian Bureau of Statistics.
The eight Greater Capital City Statistical Areas are listed for the state and territory capital cities. All Significant Urban Areas (SUA), representing urban agglomerations of over 10,000 population, are listed next. The fifty largest Urban Centres (built-up area) are then ranked and, lastly, the fifty largest Local Government Areas (the units of local government below the states and territories) are also ranked.
Greater capital city statistical areas by population
Each capital city forms its own Greater Capital City Statistical Area (GCCSA), which according to the Australian Bureau of Statistics (ABS) represents a broad functional definition of each of the eight state and territory capital cities. In Australia, the population of the GCCSA is the most-often quoted figure for the population of capital cities. These units correspond broadly to the international concept of metropolitan areas.
Notes
Significant urban areas by population
The following table ranks the SUAs, including those of the capital cities (which are smaller than their respective GCCSAs, except for Canberra's, which includes adjacent Queanbeyan, in New South Wales). Capitals are in bold. Significant Urban Areas are defined to represent significant towns and cities, or agglomerations of smaller towns, that have at least 10,000 total population. Significant Urban Areas may contain more than one distinct Urban Centre.
70% of the Australian population live in the top eight most populous cities (images below).
50 largest urban centres by population
Urban centres are defined by the Australian Bureau of Statistics as being a population cluster of 1,000 or more people. For statistical purposes, people living in urban centres are classified as urban. The figures below represent the populations of the contiguous built-up areas of each city; with state and territory capitals in bold. These figures are only updated every census, as the ABS does not render population projections for Urban Centres, and as such can only be as up-to-date as the most recent census year.
List of local government areas by population
Local government areas (LGAs) are the main units of local government in Australia. They may be termed cities, councils, regions, shires, towns, or other names, and all function similarly. Local government areas cover around 90 per cent of the nation. Significant sections of South Australia and New South Wales are unincorporated, that is, have no defined local government, along with the ACT and smaller sections of Northern Territory and Victoria. Brisbane is the only state capital city with its respective LGA (City of Brisbane) covering a significant portion of its urban area. In other capital cities, the central LGA covers a much smaller proportion of the total urban area.
The populations of the central local g
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https://en.wikipedia.org/wiki/Relation%20algebra
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In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition of binary relations R and S, and with the converse of R as the converse relation.
Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminated in the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed by Alfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself be conducted without variables.
Definition
A relation algebra is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x−, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, and the relational constant , such that these operations and constants satisfy certain equations constituting an axiomatization of a calculus of relations. Roughly, a relation algebra is to a system of binary relations on a set containing the empty (0), universal (1), and identity relations and closed under these five operations as a group is to a system of permutations of a set containing the identity permutation and closed under composition and inverse. However, the first-order theory of relation algebras is not complete for such systems of binary relations.
Following Jónsson and Tsinakis (1993) it is convenient to define additional operations x ◁ y = x • y˘, and, dually, x ▷ y = x˘ • y. Jónsson and Tsinakis showed that , and that both were equal to x˘. Hence a relation algebra can equally well be defined as an algebraic structure . The advantage of this signature over the usual one is that a relation algebra can then be defined in full simply as a residuated Boolean algebra for which is an involution, that is, . The latter condition can be thought of as the relational counterpart of the equation 1/(1/x) = x for ordinary arithmetic reciprocal, and some authors use reciprocal as a synonym for converse.
Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence the latter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields the following finite axiomatization.
Axioms
The axioms B1-B10 below are adapted from Givant (2006: 283), and were first set out by Tarski in 1948.
L is a Boolean algebra under binary disjunction, ∨, and unary complementation ()−:
B1:
B2:
B3:
This axiomatization of Boolean algebra is due to Huntington (1933). Note that the meet of the implied Boolean algebr
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https://en.wikipedia.org/wiki/Gap%20theorem%20%28disambiguation%29
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In mathematics, gap theorem may refer to:
The Weierstrass gap theorem in algebraic geometry
The Ostrowski–Hadamard gap theorem on lacunary functions
The Fabry gap theorem on lacunary functions
The gap theorem of Fourier analysis, a statement about the vanishing of discrete Fourier coefficients for functions that are identically zero on an interval shorter than 2π
The gap theorem in computational complexity theory
Saharon Shelah's Main Gap Theorem which solved Morley's problem in model theory
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https://en.wikipedia.org/wiki/Singmaster%27s%20conjecture
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Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in Pascal's triangle is 1, because any other number x can appear only within the first x + 1 rows of the triangle.
Statement
Let N(a) be the number of times the number a > 1 appears in Pascal's triangle. In big O notation, the conjecture is:
Known bound
Singmaster (1971) showed that
Abbot, Erdős, and Hanson (1974) (see References) refined the estimate to:
The best currently known (unconditional) bound is
and is due to Kane (2007). Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that
holds for every .
Singmaster (1975) showed that the Diophantine equation
has infinitely many solutions for the two variables n, k. It follows that there are infinitely many triangle entries of multiplicity at least 6: For any non-negative i, a number a with six appearances in Pascal's triangle is given by either of the above two expressions with
where Fj is the jth Fibonacci number (indexed according to the convention that F0 = 0 and F1 = 1). The above two expressions locate two of the appearances; two others appear symmetrically in the triangle with respect to those two; and the other two appearances are at and
Elementary examples
2 appears just once; all larger positive integers appear more than once;
3, 4, 5 each appear two times; infinitely many appear exactly twice;
all odd prime numbers appear two times;
6 appears three times, as do all central binomial coefficients except for 1 and 2; (it is in principle not excluded that such a coefficient would appear 5, 7 or more times, but no such example is known)
all numbers of the form for prime appear four times;
Infinitely many appear exactly six times, including each of the following:
The next number in Singmaster's infinite family (given in terms of Fibonacci numbers), and the next smallest number known to occur six or more times, is :
The smallest number to appear eight times – indeed, the only number known to appear eight times – is 3003, which is also a member of Singmaster's infinite family of numbers with multiplicity at least 6:
It is not known whether infinitely many numbers appear eight times, nor even whether any other numbers than 3003 appear eight times.
The number of times n appears in Pascal's triangle is
∞, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2,
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https://en.wikipedia.org/wiki/Sphere%20theorem%20%283-manifolds%29
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In mathematics, in the topology of 3-manifolds, the sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.
One example is the following:
Let be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding .
The proof of this version of the theorem can be based on transversality methods, see .
Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is:
Let be any 3-manifold and a -invariant subgroup of . If is a general position map such that and is any neighborhood of the singular set , then there is a map satisfying
,
,
is a covering map, and
is a 2-sided submanifold (2-sphere or projective plane) of .
quoted in .
References
Geometric topology
3-manifolds
Theorems in topology
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https://en.wikipedia.org/wiki/John%20R.%20Taylor
|
John Robert Taylor is British-born emeritus professor of physics at the University of Colorado, Boulder.
He received his B.A. in mathematics at Cambridge University, and his Ph.D. from the University of California, Berkeley in 1963 with thesis advisor Geoffrey Chew. Taylor has written several college-level physics textbooks. His bestselling book is An Introduction to Error Analysis, which has been translated into nine languages. His intermediate-level undergraduate textbook, Classical Mechanics, was well-reviewed.
Awards
Taylor was designated a Presidential Teaching Scholar in 1991. He has also received an Emmy Award for his television series Physics 4 Fun (1988–1990).
References
Year of birth missing (living people)
Living people
21st-century American physicists
American textbook writers
American male non-fiction writers
Science teachers
University of Colorado faculty
University of Colorado Boulder faculty
Alumni of the University of Cambridge
University of California, Berkeley alumni
American science writers
Emmy Award winners
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https://en.wikipedia.org/wiki/Masaki%20Kashiwara
|
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module theory, Hodge theory, sheaf theory and representation theory.
Kashiwara and Sato established the foundations of the theory of systems of linear partial differential equations with analytic coefficients, introducing a cohomological approach that follows the spirit of Grothendieck's theory of schemes. Bernstein introduced a similar approach in the polynomial coefficients case. Kashiwara's master thesis states the foundations of D-module theory. His PhD thesis proves the rationality of the roots of b-functions (Bernstein–Sato polynomials), using D-module theory and resolution of singularities. He was a plenary speaker at International Congress of Mathematicians, 1978, Helsinki and an invited speaker, 1990, Kyoto.
He is a member of the French Academy of Sciences and of the Japan Academy.
Concepts and Theorems named after Kashiwara
Kashiwara constructibility theorem
Kashiwara index theorem
Kashiwara–Malgrange filtration (after Kashiwara and Bernard Malgrange)
Cauchy-Kowalevsky-Kashiwara theorem (after Kashiwara, Augustin-Louis Cauchy and Sofya Kovalevskaya )
Kashiwara operators
Kashiwara crystal basis
List of books available in English
Seminar on micro-local analysis, by Victor Guillemin, Masaki Kashiwara, and Takahiro Kawai (1979),
Systems of microdifferential equations, by Masaki Kashiwara; notes and translation by Teresa Monteiro Fernandes; introduction by Jean-Luc Brylinski (1983),
Introduction to microlocal analysis, by Masaki Kashiwara (1986)
Foundations of algebraic analysis, by Masaki Kashiwara, Takahiro Kawai, and Tatsuo Kimura; translated by Goro Kato (1986),
Algebraic analysis : papers dedicated to Professor Mikio Sato on the occasion of his sixtieth birthday, edited by Masaki Kashiwara, Takahiro Kawai (1988),
Sheaves on manifolds : with a short history <Les débuts de la théorie des faisceaux> by Christian Houzel, by Masaki Kashiwara, Pierre Schapira (1990),
Topological field theory, primitive forms and related topics, by Masaki Kashiwara et al.(1998),
Physical combinatorics, Masaki Kashiwara, Tetsuji Miwa, editors (2000),
MathPhys Odyssey 2001: integrable models and beyond: in honor of Barry M. McCoy, Masaki Kashiwara, Tetsuji Miwa, editors (2002),
D-modules and microlocal calculus, Masaki Kashiwara; translated by Mutsumi Saito (2003),
Categories and sheaves, Masaki Kashiwara and Pierre Schapira (2006),
List of books available in French
Bases cristallines des groupes quantiques, by Masaki Kashiwara (rédigé par Charles Cochet); Cours Spécialisés 9 (2002), viii+115 pages,
Notes
External links
Fifty years of Mathematics with Masaki Kashiwara, by Pierre Schapira
List of Publications
Photo
Videos of Masaki Kashiwara in the AV-Portal of the German National Library of Science and Technology
2018 Kyoto Prize in Basic Sciences
1947 births
Liv
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https://en.wikipedia.org/wiki/Large%20countable%20ordinal
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In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.
Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called Church–Kleene ω1 or ω (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω are the recursive ordinals (see below). Countable ordinals larger than this may still be defined, but do not have notations.
Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. The ordinals described here are not as large as the ones described in large cardinals, but they are large among those that have constructive notations (descriptions). Larger and larger ordinals can be defined, but they become more and more difficult to describe.
Generalities on recursive ordinals
Ordinal notations
Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) can manipulate them (and, essentially, compare them).
A different definition uses Kleene's system of ordinal notations. Briefly, an ordinal notation is either the name zero (describing the ordinal 0), or the successor of an ordinal notation (describing the successor of the ordinal described by that notation), or a Turing machine (computable function) that produces an increasing sequence of ordinal notations (that describe the ordinal that is the limit of the sequence), and ordinal notations are (partially) ordered so as to make the successor of o greater than o and to make the limit greater than any term of the sequence (this order is computable; however, the set O of ordinal notations itself is highly non-recursive, owing to the impossibility of deciding whether a given Turing machine does indeed produce a sequence of notations); a recursive ordinal is then an ordinal described by some ordinal notation.
Any ordinal smaller than a recursive ordinal is itself recursive, so the set of all recursive ordinals forms a certain (countable) ordinal, the Church–K
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https://en.wikipedia.org/wiki/Julian%20Coolidge
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Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian, a professor and chairman of the Harvard University Mathematics Department.
Biography
Born in Brookline, Massachusetts, he graduated from Harvard University and Oxford University.
Between 1897 and 1899, Julian Coolidge taught at the Groton School, where one of his students was Franklin D. Roosevelt. He left the private school to accept a teaching position at Harvard and in 1902 was given an assistant professorship, but took two years off to further his education with studies in Turin, Italy before receiving his doctorate from the University of Bonn. Julian Coolidge then returned to teach at Harvard where he remained for his entire academic career, interrupted only by a year at the Sorbonne in Paris as an exchange professor.
During World War I, he served with the U.S. Army's Overseas Expeditionary Force in France, rising to the rank of major. In 1919, he was awarded a Knight of France's Legion of Honor.
Coolidge returned to teach at Harvard where he was awarded a full professorship. In 1927 he was appointed chairman of the Mathematics Department at Harvard, a position he held until his retirement in 1940. A Fellow of the American Academy of Arts and Sciences, Coolidge served as president of the Mathematical Association of America and vice-president of the American Mathematical Society. He authored several books on mathematics and on the history of mathematics.
He was Master of Lowell House (one of Harvard's undergraduate residences) from 1930 to 1940.
Coolidge died in 1954 in Cambridge, Massachusetts, aged 80.
Writings
J. L. Coolidge (1909) The elements of non-Euclidean geometry, Oxford University Press.
J. L. Coolidge (1916) A Treatise on the Circle and the Sphere, Oxford University Press.
J. L. Coolidge (1924) The geometry of the complex domain, The Clarendon Press.
J. L. Coolidge (1925) An introduction to mathematical probability, Oxford University Press.
J. L. Coolidge (1931) A Treatise on Algebraic Plane Curves, Oxford University Press (Dover Publications 2004).
J. L. Coolidge (1940) A history of geometrical methods, Oxford University Press (Dover Publications 2003).
J. L. Coolidge (1945) History of the conic sections and quadric surfaces, The Clarendon Press.
J. L. Coolidge (1949) The Mathematics Of Great Amateurs, Oxford University Press (Dover Publications 1963).
See also
Commandino's theorem
Projective range
Screw theory
Spieker circle
References
External links
Coolidge: "Origin of Polar Coordinates" (from MacTutor)
1873 births
1954 deaths
20th-century American mathematicians
American historians of mathematics
United States Army personnel of World War I
University of Paris alumni
University of Bonn alumni
Alumni of Balliol College, Oxford
Harvard University alumni
Harvard University Department of Mathematics faculty
Harvard University faculty
Recipients of the Legion of Honour
Presidents of the Mathematical Association of Am
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https://en.wikipedia.org/wiki/Capacity
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Capacity or capacities may
refer to:
Mathematics, science, and engineering
Capacity of a container, closely related to the volume of the container
Capacity of a set, in Euclidean space, the total charge a set can hold while maintaining a given potential energy
Capacity factor, the ratio of the actual output of a power plant to its theoretical potential output
Storage capacity (energy), the amount of energy that the storage system of a power plant can hold
Nameplate capacity, the intended full-load sustained output of a facility such as a power plant
Heat capacity, a measurement of changes in a system's internal energy
Combining capacity, another term for valence in chemistry
Battery capacity, the amount of electric charge a battery can deliver at the rated voltage
Computer
Data storage capacity, amount of stored information that a storage device or medium can hold
Channel capacity, the highest rate at which information can be reliably transmitted
Social
Carrying capacity, the population size of a species that its environment can sustain
Capacity planning, the process of determining the production resources needed to meet product demand
Capacity building, strengthening the skills, competencies and abilities of developing societies
Productive capacity, the maximum possible output of an economy
Capacity management, a process used to manage information technology in business
Capacity utilization, the extent to which an enterprise or a nation uses its theoretical productive capacity
Road capacity, the maximum traffic flow rate that theoretically may be attained on a given road
Seating capacity, the number of people who can be seated in a specific space
Legal
Capacity (law), the capability and authority to undertake a legal action
Arts
Capacities (album), an album by Up Dharma Down
Capacity (album), an album by Big Thief
See also
Capacitance, in physics
Ability (disambiguation)
Cability (disambiguation)
Capacity (disambiguation)
Incapacitation (disambiguation)
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https://en.wikipedia.org/wiki/Degree%20of%20a%20field%20extension
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In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.
Definition and notation
Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F].
The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements).
The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rational functions has infinite degree over Q, but transcendence degree only equal to 1.
The multiplicativity formula for degrees
Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, there is a simple relation between the degrees of the three extensions L/K, M/L and M/K:
In other words, the degree going from the "bottom" to the "top" field is just the product of the degrees going from the "bottom" to the "middle" and then from the "middle" to the "top". It is quite analogous to Lagrange's theorem in group theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence.
The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in the sense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K are finite.
If M/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between M and K, via simple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediate field L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and [L:K] = p, in which case L is equal to M. Therefore, there are no intermediate fields (apart from M and K themselves).
Proof of the multiplicativity formula in the finite case
Suppose that K, L and M form a tower of fields as in the degree formula above, and that both d = [L:K] and e = [M:L] are finite. This means that we may select a basis {u1, ..., ud} for L over K, and a basis {w1, ..., we} for M over L. We will show that the elements umwn, for m ranging through 1, 2, ..., d and n ranging through 1, 2, ..., e, form a basis for M/K; since there are precisely de of them, this proves that the dimension of M/K is de, which is the desired result.
First we check that they span M/K. If x is any element
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https://en.wikipedia.org/wiki/Hiraku%20Nakajima
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Hiraku Nakajima (Japanese: 中島 啓 Nakajima Hiraku; born November 30, 1962) is a Japanese mathematician, and a professor of the Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo. He is International Mathematical Union president for the 2023–2026 term.
He obtained his Ph.D. from the University of Tokyo in 1991. In 2002 he was plenary speaker at the International Congress of Mathematicians in Beijing. He won the 2003 Cole Prize in algebra for his work on representation theory and geometry. He proved Nekrasov's conjecture.
Biography
1985 - BA from the University of Tokyo
1987 - MA from the University of Tokyo, and became a research associate at the University of Tokyo
1991 - PhD from the University of Tokyo
1992 - Associate professor at Tohoku University
1995 - Associate professor at the University of Tokyo
1997 - Associate professor at Kyoto University
2000 - Full professor at Kyoto University
2018 - Full professor at Kavli Institute for the Physics and Mathematics of the Universe
Awards and prizes
1997 - Geometry Prize of the Mathematical Society of Japan
2000 - Spring Prize of the Mathematical Society of Japan
2003 - Cole Prize in algebra of the American Mathematical Society
2005 - JSPS prize of the Japan Society for the Promotion of Science
2014 - Japan Academy Prize
Notable publications
Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97 (1989), no. 2, 313–349.
Hiraku Nakajima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365–416.
Hiraku Nakajima. Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. of Math. (2) 145 (1997), no. 2, 379–388. ,
Hiraku Nakajima. Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 (1998), no. 3, 515–560.
Hiraku Nakajima. Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14 (2001), no. 1, 145–238.
External links
Nakajima's homepage
1962 births
Living people
Academics from Tokyo
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Institute for Advanced Study visiting scholars
Azabu High School alumni
University of Tokyo alumni
Academic staff of Tohoku University
Academic staff of the University of Tokyo
Academic staff of Kyoto University
Presidents of the International Mathematical Union
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https://en.wikipedia.org/wiki/Robert%20S.%20Boyer
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Robert Stephen Boyer is an American retired professor of computer science, mathematics, and philosophy at The University of Texas at Austin. He and J Strother Moore invented the Boyer–Moore string-search algorithm, a particularly efficient string searching algorithm, in 1977. He and Moore also collaborated on the Boyer–Moore automated theorem prover, Nqthm, in 1992. Following this, he worked with Moore and Matt Kaufmann on another theorem prover called ACL2.
Publications
Boyer has published extensively, including the following books:
A Computational Logic Handbook, with J S. Moore. Second Edition. Academic Press, London, 1998.
Automated Reasoning: Essays in Honor of Woody Bledsoe, editor. Kluwer Academic, Dordrecht, The Netherlands, 1991.
A Computational Logic Handbook, with J S. Moore. Academic Press, New York, 1988.
The Correctness Problem in Computer Science, editor, with J S. Moore. Academic Press, London, 1981.
A Computational Logic, with J S. Moore. Academic Press, New York, 1979.
See also
Boyer–Moore majority vote algorithm
QED manifesto
References
External links
Home page of Robert S. Boyer. Accessed February 18, 2016.
University of Texas, College of Liberal Arts Honors Retired Faculty - 2008. Accessed March 21, 2009.
Robert Stephen Boyer at the Mathematics Genealogy Project
Living people
Alumni of the University of Edinburgh
University of Texas at Austin faculty
Fellows of the Association for the Advancement of Artificial Intelligence
Year of birth missing (living people)
Formal methods people
Lisp (programming language) people
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https://en.wikipedia.org/wiki/Karen%20Uhlenbeck
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Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richardson Foundation Regents Chair. She is currently a distinguished visiting professor at the Institute for Advanced Study and a visiting senior research scholar at Princeton University.
Uhlenbeck was elected to the American Philosophical Society in 2007. She won the 2019 Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." She is the first, and so far only, woman to win the prize since its inception in 2003. She donated half of the prize money to organizations which promote more engagement of women in research mathematics.
Life and career
Uhlenbeck was born in Cleveland, Ohio, to engineer Arnold Keskulla and schoolteacher and artist Carolyn Windeler Keskulla. While she was a child, the family moved to New Jersey. Uhlenbeck's maiden name, Keskulla, comes from Keskküla and from her grandfather who was Estonian. Uhlenbeck received her B.A. (1964) from the University of Michigan. She began her graduate studies at the Courant Institute of Mathematical Sciences at New York University, and married biophysicist Olke C. Uhlenbeck (the son of physicist George Uhlenbeck) in 1965. When her husband went to Harvard, she moved with him and restarted her studies at Brandeis University, where she earned an MA (1966) and PhD (1968) under the supervision of Richard Palais. Her doctoral dissertation was titled The Calculus of Variations and Global Analysis.
After temporary jobs at the Massachusetts Institute of Technology and University of California, Berkeley, and having difficulty finding a permanent position with her husband because of the "anti-nepotism" rules then in place that prevented hiring both a husband and wife even in distinct departments of a university, she took a faculty position at the University of Illinois at Urbana–Champaign in 1971. However, she disliked Urbana and moved to the University of Illinois at Chicago in 1976 as well as separating from her first husband Olke Uhlenbeck in the same year. From 1979 to 1981 Uhlenbeck served on the Council of the AMS as a Member at Large. She moved again to the University of Chicago in 1983. In 1988, by which time she had married mathematician Robert F. Williams, she moved to the University of Texas at Austin as the Sid W. Richardson Foundation Regents Chairholder. Uhlenbeck is currently a professor emeritus at the University of Texas at Austin, a visiting associate at the Institute for Advanced Study and a visiting senior research scholar at Princeton University.
Research
Uhlenbeck is one of the founders of the field of geometric analysis, a discipline that uses differential geometry to stud
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https://en.wikipedia.org/wiki/Factorial%20moment%20generating%20function
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In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as
for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle , see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then is also called probability-generating function (PGF) of X and is well-defined at least for all t on the closed unit disk .
The factorial moment generating function generates the factorial moments of the probability distribution.
Provided exists in a neighbourhood of t = 1, the nth factorial moment is given by
where the Pochhammer symbol (x)n is the falling factorial
(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)
Examples
Poisson distribution
Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is
(use the definition of the exponential function) and thus we have
See also
Moment (mathematics)
Moment-generating function
Cumulant-generating function
References
Factorial and binomial topics
Moment (mathematics)
Generating functions
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https://en.wikipedia.org/wiki/Doob%20martingale
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In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time.
When analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, Chernoff's inequality, Chebyshev's inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are martingales and Azuma's inequality.
Definition
Let be any random variable with . Suppose is a filtration, i.e. when . Define
then is a martingale, namely Doob martingale, with respect to filtration .
To see this, note that
;
as .
In particular, for any sequence of random variables on probability space and function such that , one could choose
and filtration such that
i.e. -algebra generated by . Then, by definition of Doob martingale, process where
forms a Doob martingale. Note that . This martingale can be used to prove McDiarmid's inequality.
McDiarmid's inequality
The Doob martingale was introduced by Joseph L. Doob in 1940 to establish concentration inequalities such as McDiarmid's inequality, which applies to functions that satisfy a bounded differences property (defined below) when they are evaluated on random independent function arguments.
A function satisfies the bounded differences property if substituting the value of the th coordinate changes the value of by at most . More formally, if there are constants such that for all , and all ,
See also
References
Probabilistic inequalities
Statistical inequalities
Martingale theory
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https://en.wikipedia.org/wiki/List%20of%20New%20Zealand%20urban%20areas%20by%20population
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This article lists urban areas of New Zealand—as defined by Statistics New Zealand—ranked by population. Only the 150 largest urban areas are listed.
Urban areas are defined by the Statistical Standard for Geographic Areas 2018 (SSGA18).
See also
List of cities in New Zealand
List of towns in New Zealand
References
Lists of urban areas
Urban areas
Urban areas
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https://en.wikipedia.org/wiki/Kapiti%20Urban%20Area
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The Kapiti Urban Area is a statistical area that was defined by Statistics New Zealand to cover a group of urban settlements of the Kāpiti Coast District, in the Wellington Region. It was classified as a main urban area under the New Zealand Standard Areas Classification 1992 because its population exceeded 30,000.
The settlements comprise (north to south):
Waikanae
Paraparaumu (including Otaihanga, Raumati Beach and Raumati South)
Paekākāriki
The largest settlement is Paraparaumu. Raumati may be considered a suburb of Paraparaumu or a separate town in its own right – there are no legal definitions for towns in New Zealand. Kapiti Urban Area is better described as a commuter area of Wellington than an independent city.
Under Statistical Standard for Geographic Areas 2018, Kapiti Urban Area was split into separate urban areas for the three settlements.
The Kāpiti Coast District also includes the settlements of Te Horo and Ōtaki, which are outside Kapiti Urban Area.
References
Kāpiti Coast District
Main urban areas in New Zealand
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https://en.wikipedia.org/wiki/Population%20census%20in%20Hong%20Kong
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Population censuses / by-censuses in Hong Kong are conducted by the Census and Statistics Department (C&SD) of the Hong Kong SAR Government. The aim is to provide up-to-date benchmark statistics on the demographic and socio-economic characteristics of the population and on its geographical distribution. Since 1961, a population census has been conducted in Hong Kong every 10 years and a by-census in the middle of the intercensal period. The last census, 2021 population census in Hong Kong was conducted by C&SD from 23 June to 4 August 2021.
Objectives
It is an established practice in Hong Kong to conduct a population census every 10 years and a population by-census in the middle of the intercensal period. The next population census will be conducted in 2021. The aim is to provide up-to-date benchmark statistics on the demographic and socio-economic characteristics of the population and on its geographical distribution. Such statistics are vital to the Government for planning and policy formulation. It is also important to the private sector and academia for business and research purposes.
Census and Statistics Ordinance
The Census and Statistics Ordinance, which was first effective in 1978, is the main law governing the work of the C&SD. Under Section 9 of the Census and Statistics Ordinance, the Chief Executive in Council (Cap 316) makes a Census Order for each round of population census / by-census.
Apart from the dates for conducting the population census, the Order specifies also the purpose of the census, the persons, premises and vessels in respect of which particulars are to be obtained, the persons who are required to give the requisite information, and the deadline for destroying the completed questionnaires. A schedule specifying the matters in respect of which respondents are required to give particulars is included in the Order.
According to the Census and Statistics (2021 Population Census) Order, the Commissioner will take a census of population from 23 June 2021 to 4 August 2021 (both dates inclusive) and the Commissioner must destroy all completed schedules collected or received by census officers for the 2021 census, and all copies of the schedules, on or before 22 June 2022.
Data collection methods
Two full censuses were held in Hong Kong in 1961 and 1971, and two sample censuses in 1966 and 1976.
In population censuses conducted since 1981, a simple enumeration on all households in Hong Kong has been carried out to provide basic information (e.g. year and month of birth and sex) and a detailed enquiry on a certain proportion of households to provide a broad range of demographic and socio-economic characteristics of the population. Therefore, two types of questionnaire, namely the "Short Form" and the "Long Form", are used. In the 2001 population census, one-seventh of the households completed the "Long Form". Since the 2011 population census, around one-tenth of the households completed the "Long Form".
The
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https://en.wikipedia.org/wiki/FCT
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FCT may refer to:
Mathematics
Flux-corrected transport
Fast cosine transform
International Symposium on Fundamentals of Computation Theory
Places
Australian Capital Territory, formerly the Federal Capital Territory
Claremont railway station, Perth, in Western Australia
Federal Capital Territory (Nigeria)
Federal Capital Territory (Pakistan), around Karachi, now defunct
Fort Canning Tunnel, in Singapore
Sport
FC Trollhättan, a Swedish football club
FC Twente, a Dutch football club
Feminine Cycling Team, a German cycling team
Other uses
2001 Sino-Russian Treaty of Friendship
Faculdade de Ciências e Tecnologia (disambiguation)
Fellow of the Association of Corporate Treasurers, a professional organisation
Florida Communities Trust
Fundação para a Ciência e Tecnologia, the main funding agency for scientific research in Portugal
Functional testing
Fukushima Central Television, a Japanese television company
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https://en.wikipedia.org/wiki/SO%285%29
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In mathematics, SO(5), also denoted SO5(R) or SO(5,R), is the special orthogonal group of degree 5 over the field R of real numbers, i.e. (isomorphic to) the group of orthogonal 5×5 matrices of determinant 1.
Geometric interpretation
SO(5) is a subgroup of the direct Euclidean group E+(5), the group of direct isometries, i.e., isometries preserving orientation, of R5, consisting of elements which leave the origin fixed.
More precisely, we have:
SO(5) E+(5) / T
where T is the translational group of R5.
Lie group
SO(5) is a simple Lie group of dimension 10.
See also
Orthogonal matrix
Orthogonal group
Rotation group SO(3)
List of simple Lie groups
Lie groups
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https://en.wikipedia.org/wiki/Projective%20unitary%20group
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In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars.
Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space.
In terms of matrices, elements of are complex unitary matrices, and elements of the center are diagonal matrices equal to , where is the identity matrix. Thus, elements of correspond to equivalence classes of unitary matrices under multiplication by a constant phase .
This space is not (which only requires the determinant to be one), because still contains elements where is an -th root of unity (since then ).
Abstractly, given a Hermitian space , the group is the image of the unitary group in the automorphism group of the projective space .
Projective special unitary group
The projective special unitary group PSU() is equal to the projective unitary group, in contrast to the orthogonal case.
The connections between the U(), SU(), their centers, and the projective unitary groups is shown in the Figure on the right (notice that in the figure the integers are denoted instead of ).
The center of the special unitary group is the scalar matrices of the th roots of unity:
The natural map
is an isomorphism, by the second isomorphism theorem, thus
and the special unitary group SU() is an -fold cover of the projective unitary group.
Examples
At n = 1, U(1) is abelian and so is equal to its center. Therefore PU(1) = U(1)/U(1) is a trivial group.
At n = 2, , all being representable by unit norm quaternions, and via:
Finite fields
One can also define unitary groups over finite fields: given a field of order q, there is a non-degenerate Hermitian structure on vector spaces over unique up to unitary congruence, and correspondingly a matrix group denoted or and likewise special and projective unitary groups. For convenience, this article uses the convention.
Recall that the group of units of a finite field is cyclic, so the group of units of and thus the group of invertible scalar matrices in is the cyclic group of order The center of has order q + 1 and consists of the scalar matrices which are unitary, that is those matrices with The center of the special unitary group has order gcd(n, q + 1) and consists of those unitary scalars which also have order dividing n.
The quotient of the unitary group by its center is the projective unitary group, and the quotient of the special unitary group by its center is the projective special unitary group In most cases (n ≥ 2 and ), is a perfect group and is a finite simple group, .
The topology of PU(H)
PU(H) is a classifying space for circle bundles
The same construction may be applied to matrices acting on an infinite-dimensional Hilbert space .
Let U(H) denote the space of unitary operators on an infinite-dimensional Hilbert space. When f: X → U(H) is a continuous
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https://en.wikipedia.org/wiki/Labour%20Statistics%20Convention%2C%201985
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Labour Statistics Convention, 1985 is an International Labour Organization Convention.
It was established in 1985, with the preamble stating:
Ratifications
As of 2023, the convention had been ratified by 51 states.
External links
Text.
Ratifications.
International Labour Organization conventions
Statistical data agreements
Treaties concluded in 1985
Treaties entered into force in 1988
Treaties of Armenia
Treaties of Australia
Treaties of Austria
Treaties of the Byelorussian Soviet Socialist Republic
Treaties of Benin
Treaties of Bolivia
Treaties of Brazil
Treaties of Canada
Treaties of Colombia
Treaties of Costa Rica
Treaties of Cyprus
Treaties of Czechoslovakia
Treaties of the Czech Republic
Treaties of Denmark
Treaties of El Salvador
Treaties of Finland
Treaties of Germany
Treaties of Greece
Treaties of Guatemala
Treaties of Hungary
Treaties of India
Treaties of Ireland
Treaties of Israel
Treaties of Italy
Treaties of South Korea
Treaties of Kyrgyzstan
Treaties of Latvia
Treaties of Lithuania
Treaties of Mauritius
Treaties of Mexico
Treaties of Moldova
Treaties of the Netherlands
Treaties of New Zealand
Treaties of Norway
Treaties of Panama
Treaties of Poland
Treaties of Portugal
Treaties of the Soviet Union
Treaties of San Marino
Treaties of Slovakia
Treaties of Spain
Treaties of Sri Lanka
Treaties of Eswatini
Treaties of Sweden
Treaties of Switzerland
Treaties of Tajikistan
Treaties of the Ukrainian Soviet Socialist Republic
Treaties of the United Kingdom
Treaties extended to the Isle of Man
Treaties extended to Gibraltar
Treaties of the United States
Treaties extended to Jersey
Treaties extended to Guernsey
Treaties extended to the British Virgin Islands
Treaties extended to Bermuda
Treaties extended to Norfolk Island
1985 in labor relations
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https://en.wikipedia.org/wiki/Profunctor
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In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
Definition
A profunctor (also named distributor by the French school and module by the Sydney school) from a category to a category , written
,
is defined to be a functor
where denotes the opposite category of and denotes the category of sets. Given morphisms respectively in and an element , we write to denote the actions.
Using the cartesian closure of , the category of small categories, the profunctor can be seen as a functor
where denotes the category of presheaves over .
A correspondence from to is a profunctor .
Profunctors as categories
An equivalent definition of a profunctor is a category whose objects are the disjoint union of the objects of and the objects of , and whose morphisms are the morphisms of and the morphisms of , plus zero or more additional morphisms from objects of to objects of . The sets in the formal definition above are the hom-sets between objects of and objects of . (These are also known as het-sets, since the corresponding morphisms can be called heteromorphisms.) The previous definition can be recovered by the restriction of the hom-functor to .
This also makes it clear that a profunctor can be thought of as a relation between the objects of and the objects of , where each member of the relation is associated with a set of morphisms. A functor is a special case of a profunctor in the same way that a function is a special case of a relation.
Composition of profunctors
The composite of two profunctors
and
is given by
where is the left Kan extension of the functor along the Yoneda functor of (which to every object of associates the functor ).
It can be shown that
where is the least equivalence relation such that whenever there exists a morphism in such that
and .
Equivalently, profunctor composition can be written using a coend
The bicategory of profunctors
Composition of profunctors is associative only up to isomorphism (because the product is not strictly associative in Set). The best one can hope is therefore to build a bicategory Prof whose
0-cells are small categories,
1-cells between two small categories are the profunctors between those categories,
2-cells between two profunctors are the natural transformations between those profunctors.
Properties
Lifting functors to profunctors
A functor can be seen as a profunctor by postcomposing with the Yoneda functor:
.
It can be shown that such a profunctor has a right adjoint. Moreover, this is a characterization: a profunctor has a right adjoint if and only if factors through the Cauchy completion of , i.e. there exists a functor such that .
References
Functors
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