1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Linear form
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In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. The set of all linear functionals from V to k, Homk, forms a space over k with the addition of the operations of addition. This space is called the space of V, or sometimes the algebraic dual space. It is often written V∗ or V′ when the field k is understood, if V is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If V is a Banach space, then so is its dual, to distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual. In finite dimensions, every linear functional is continuous, so the dual is the same as the algebraic dual. Suppose that vectors in the coordinate space Rn are represented as column vectors x → =. For each row there is a linear functional f defined by f = a 1 x 1 + ⋯ + a n x n. This is just the product of the row vector and the column vector x →, f =. Linear functionals first appeared in functional analysis, the study of spaces of functions. Let Pn denote the space of real-valued polynomial functions of degree ≤n defined on an interval. If c ∈, then let evc, Pn → R be the evaluation functional, the mapping f → f is linear since = f + g = α f. If x0, …, xn are n+1 distinct points in, then the evaluation functionals evxi, the integration functional I defined above defines a linear functional on the subspace Pn of polynomials of degree ≤ n. If x0, …, xn are n+1 distinct points in, then there are coefficients a0, … and this forms the foundation of the theory of numerical quadrature. This follows from the fact that the linear functionals evxi, f → f defined above form a basis of the space of Pn. Linear functionals are particularly important in quantum mechanics, quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a mechanical system can be identified with a linear functional. For more information see bra–ket notation, in the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions
3.
Israel Gelfand
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Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gelfand, or Izrail M. Gelfand was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and his legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, as well as his own son, Sergei Gelfand. A native of Kherson Governorate of the Russian Empire, Gelfand was born into a Jewish family in the small southern Ukrainian town of Okny, according to his own account, Gelfand was expelled from high school because his father had been a mill owner. Bypassing both high school and college, he proceeded to study at Moscow State University, where his advisor was the preeminent mathematician Andrei Kolmogorov. He nevertheless managed to attend lectures at the University and began study at the age of 19. The Gelfand–Tsetlin basis is a widely used tool in theoretical physics, Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and he worked extensively in mathematics education, particularly with correspondence education. In 1994, he was awarded a MacArthur Fellowship for this work, Gelfand was married to Zorya Shapiro, and their two sons, Sergei and Vladimir both live in the United States. A third son, Aleksandr, died of leukemia, following the divorce from his first wife, Gelfand married his second wife, Tatiana, together they had a daughter, Tatiana. The family also includes four grandchildren and three great-grandchildren, the memories about I. Gelfand are collected at the special site handled by his family. Gelfand held several degrees and was awarded the Order of Lenin three times for his research. In 1977 he was elected a Foreign Member of the Royal Society and he won the Wolf Prize in 1978, Kyoto Prize in 1989 and MacArthur Foundation Fellowship in 1994. Israel Gelfand died at the Robert Wood Johnson University Hospital near his home in Highland Park and he was less than five weeks past his 96th birthday. His death was first reported on the blog of his former collaborator Andrei Zelevinsky and confirmed a few hours later by an obituary in the Russian online newspaper Polit. ru. Gelfand, I. M. Lectures on linear algebra, Courier Dover Publications, ISBN 978-0-486-66082-0 Gelfand, I. M. Fomin, Sergei V. Silverman, Richard A. ed. Calculus of variations, Englewood Cliffs, ISBN 978-0-486-41448-5, MR0160139 Gelfand, I. Raikov, D. Shilov, G. Commutative normed rings, Translated from the Russian, with a chapter, New York. ISBN 978-0-8218-2022-3, MR0205105 Gelfand, I. M. Shilov, G. E. Generalized functions. Vol. I, Properties and operations, Translated by Eugene Saletan, Boston, MA, Academic Press, ISBN 978-0-12-279501-5, MR0166596 Gelfand, I. M. Shilov, G. E. Generalized functions
4.
Irving Segal
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Irving Ezra Segal was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for his developments in quantum theory and in functional and harmonic analysis. Irving Ezra Segal was born in the Bronx in 1918 to Jewish parents, in 1934 was admitted to Princeton University at the age of 16. He was then admitted to Yale, and in three years time had completed his doctorate, receiving his PhD in 1940. Segal taught at Harvard University, then he joined the Institute for Advanced Study in Princeton on a Guggenheim Memorial Fellowship, working from 1941–43 with Albert Einstein and Von Neumann. During World War II Segal served in the U. S. Army conducting research in ballistics at the Aberdeen Proving Ground in Maryland and he joined the mathematics department at the University of Chicago in 1948 where he served until 1960. In 1960 he joined the department at M. I. T. where he remained as a professor until his death in 1998. He won three Guggenheim Fellowships, in 1947,1951 and 1967, and received the Humboldt Award in 1981 and he was an Invited Speaker of the ICM in 1966 in Moscow and in 1970 in Nice. He was elected to the National Academy of Sciences in 1973, Segal died in Lexington, Massachusetts in 1998. Edward Nelsons obituary article about Segal concludes, metaplectic group Symplectic group Symplectic spinor bundle Segal, I. Robertson, Edmund F. Irving Segal, MacTutor History of Mathematics archive, Irving Segal at the Mathematics Genealogy Project
5.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist
6.
Map (mathematics)
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There are also a few, less common uses in logic and graph theory. In many branches of mathematics, the map is used to mean a function. For instance, a map is a function in topology. Some authors, such as Serge Lang, use only to refer to maps in which the codomain is a set of numbers. Sets of maps of special kinds are the subjects of many important theories, see for instance Lie group, mapping class group, in the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, all these usages can be applied to maps as general functions or as functions with special properties. In category theory, map is used as a synonym for morphism or arrow. In formal logic, the map is sometimes used for a functional predicate. In graph theory, a map is a drawing of a graph on a surface without overlapping edges, if the surface is a plane then a map is a planar graph, similar to a political map
7.
Involution (mathematics)
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In mathematics, an involution, or an involutory function, is a function f that is its own inverse, f = x for all x in the domain of f. The identity map is an example of an involution. Common examples in mathematics of nontrivial involutions include multiplication by −1 in arithmetic, other examples include circle inversion, rotation by a half-turn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The number of involutions, including the identity involution, on a set with n =0,1,2. Elements is given by a recurrence relation found by Heinrich August Rothe in 1800, a0 = a1 =1, an = an −1 + an −2, for n >1. The first few terms of this sequence are 1,1,2,4,10,26,76,232, these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The composition g ∘ f of two involutions f and g is an if and only if they commute, g ∘ f = f ∘ g. Every involution on an odd number of elements has at least one fixed point, more generally, for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity. Basic examples of involutions are the functions, f 1 = − x, or f 2 =1 x and these are not the only pre-calculus involutions. Another in R + is, f = ln , x >0, the graph of an involution is line-symmetric over the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the 45° line y = x and this can be seen by swapping x with y. If, in particular, the function is an involution, then it will serve as its own reflection, other elementary involutions are useful in solving functional equations. A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane, performing a reflection twice brings a point back to its original coordinates. Another is the reflection through the origin, this is an abuse of language as it is not a reflection. These transformations are examples of affine involutions, an involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Coxeter relates three theorems on involutions, Any projectivity that interchanges two points is an involution, the three pairs of opposite sides of a complete quadrangle meet any line in three pairs of an involution. If an involution has one fixed point, it has another, in this instance the involution is termed hyperbolic, while if there are no fixed points it is elliptic. Another type of involution occurring in geometry is a polarity which is a correlation of period 2
8.
Irreducible representation
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Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations. The structure analogous to a representation in the resulting theory is a simple module. Let ρ be a representation i. e. a homomorphism ρ, G → G L of a group G where V is a space over a field F. If we pick a basis B for V, ρ can be thought of as a function from a group into a set of invertible matrices, however, it simplifies things greatly if we think of the space V without a basis. A linear subspace W ⊂ V is called G -invariant if ρ w ∈ W for all g ∈ G, the restriction of ρ to a G -invariant subspace W ⊂ V is known as a subrepresentation. A representation ρ, G → G L is said to be if it has only trivial subrepresentations. If there is a proper non-trivial invariant subspace, ρ is said to be reducible, Group elements can be represented by matrices, although the term represented has a specific and precise meaning in this context. A representation of a group is a mapping from the elements to the general linear group of matrices. The representations D and D are said to be equivalent representations, K, although some authors just write the numerical label without brackets. The dimension of D is the sum of the dimensions of the blocks, d i m = d i m + d i m + … + d i m If this is not possible, then the representation is indecomposable. Identifying the irreducible representations therefore allows one to label the states, predict how they will split under perturbations and this allows them to derive relativistic wave equations. The theory of groups and quantum mechanics, a. D. Boardman, D. E. OConner, P. A. Young. Symmetry and its applications in science, Group theory in quantum mechanics, an introduction to its present usage. Group Theory in Quantum Mechanics, An Introduction to Its Present Usage, manchester Physics Series, John Wiley & Sons. Weinberg, S, The Quantum Theory of Fields,1, Cambridge university press, molecular Quantum Mechanics, An introduction to quantum chemistry. Group theoretical discussion of wave equations. On Unitary Representations Of The Inhomogeneous Lorentz Group, commission on Mathematical and Theoretical Crystallography, Summer Schools on Mathematical Crystallography. Some Notes on Young Tableaux as useful for irreps of su. Hunt, Representations of Lorentz and Poincaré groups
9.
Group action
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In mathematics, an action of a group is a way of interpreting the elements of the group as acting on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, actions of groups on vector spaces are called representations of the group. Some groups can be interpreted as acting on spaces in a canonical way, more generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. A common way of specifying non-canonical actions is to describe a homomorphism φ from a group G to the group of symmetries of a set X. The action of an element g ∈ G on a point x ∈ X is assumed to be identical to the action of its image φ ∈ Sym on the point x. The homomorphism φ is also called the action of G. Thus, if G is a group and X is a set, if X has additional structure, then φ is only called an action if for each g ∈ G, the permutation φ preserves the structure of X. The abstraction provided by group actions is a one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them, in particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, the group G is said to act on X. The set X is called a G-set. In complete analogy, one can define a group action of G on X as an operation X × G → X mapping to x. g. =. h for all g, h in G and all x in X, for a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula −1 = h−1g−1, one can construct an action from a right action by composing with the inverse operation of the group. Also, an action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality. The trivial action of any group G on any set X is defined by g. x = x for all g in G and all x in X, that is, every group element induces the identity permutation on X. In every group G, left multiplication is an action of G on G, g. x = gx for all g, x in G
10.
Universal representation (C*-algebra)
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The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. Let A be a C*-algebra with state space S, the representation Φ, = ∑ ρ ∈ S ⊕ π ρ on the Hilbert space H Φ is known as the universal representation of A. As the universal representation is faithful, A is *-isomorphic to the C*-subalgebra Φ of B, with τ a state of A, let πτ denote the corresponding GNS representation on the Hilbert space Hτ. Using the notation defined here, τ is ωx ∘ πτ for a unit vector x in Hτ. Thus τ is ωy ∘ Φ, where y is the unit vector ∑ρ∈S ⊕yρ in HΦ, defined by yτ=x, yρ=0. Since the mapping τ → τ ∘ Φ−1 takes the space of A onto the state space of Φ. Let Φ− denote the weak-operator closure of Φ in B, each bounded linear functional ρ on Φ is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional ρ on the von Neumann algebra Φ−. If ρ is hermitian, or positive, the same is true of ρ, the mapping ρ → ρ is an isometric isomorphism from the dual space Φ* onto the predual of Φ−. As the set of linear functionals determining the weak topologies coincide, thus the weak-operator and ultraweak topologies on Φ both coincide with the weak topology of Φ obtained from its norm-dual as a Banach space. If K is a subset of Φ, the ultraweak closure of K coincides with the strong-operator, weak-operator closures of K in B. The norm closure of K is Φ ∩ K−, one can give a description of norm-closed left ideals in Φ from the structure theory of ideals for von Neumann algebras, which is relatively much more simple. If π is a representation of A, there is a projection P in the center of Φ− and a *-isomorphism α from the von Neumann algebra Φ−P onto π− such that π = α for each a in A. This can be captured in the commutative diagram below, Here ψ is the map that sends a to aP, α0 denotes the restriction of α to ΦP. As α is ultraweakly bicontinuous, the same is true of α0, moreover, ψ is ultraweakly continuous, and is a *-isomorphism if π is a faithful representation. Let A be a C*-algebra acting on a Hilbert space H, for ρ in A* and S in Φ−, let Sρ in A* be defined by Sρ = ρ∘Φ−1 for all a in A. If P is the projection in the commutative diagram when π, A → B is the inclusion mapping, then ρ in A* is ultraweakly continuous if. A functional ρ in A* is said to be singular if Pρ =0, each ρ in A* can be uniquely expressed in the form ρ=ρu+ρs, with ρu ultraweakly continuous and ρs singular. Moreover, ||ρ||=||ρu||+||ρs|| and if ρ is positive, or hermitian, kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. II, Advanced Theory, American Mathematical Society
11.
Convex set
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In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the hull of A. It is the smallest convex set containing A, a convex function is a real-valued function defined on an interval with the property that its epigraph is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets, the branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
12.
Extreme point
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The Krein–Milman theorem states that if S is convex and compact in a locally convex space, then S is the closed convex hull of its extreme points, In particular, such a set has extreme points. The Krein–Milman theorem is stated for locally convex vector spaces. A theorem of Gerald Edgar states that, in a Banach space with the Radon–Nikodym property, if S is a polytope, then the k-extreme points are exactly the interior points of the k-dimensional faces of S. More generally, for any convex set S, the points are partitioned into k-dimensional open faces. The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be proved using the concept of k-extreme points. If S is closed, bounded, and n-dimensional, and if p is a point in S, the theorem asserts that p is a convex combination of extreme points. If k =0, then its trivially true, otherwise p lies on a line segment in S which can be maximally extended. If the endpoints of the segment are q and r, then their extreme rank must be less than that of p, choquet theory Paul E. Black, ed. extreme point. Dictionary of algorithms and data structures, US National institute of standards and technology. Borowski, Ephraim J. Borwein, Jonathan M. extreme point
13.
John von Neumann
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John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics, physics, economics, computing, and statistics. He published over 150 papers in his life, about 60 in pure mathematics,20 in physics, and 60 in applied mathematics and his last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded the discovery of the structure of DNA, also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939, on the ergodic theorem, Princeton, 1931–1932. During World War II he worked on the Manhattan Project, developing the mathematical models behind the lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. Von Neumann was born Neumann János Lajos to a wealthy, acculturated, Von Neumanns place of birth was Budapest in the Kingdom of Hungary which was then part of the Austro-Hungarian Empire. He was the eldest of three children and he had two younger brothers, Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa was a banker, who held a doctorate in law and he had moved to Budapest from Pécs at the end of the 1880s. Miksas father and grandfather were both born in Ond, Zemplén County, northern Hungary, johns mother was Kann Margit, her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in apartments above the Kann-Heller offices in Budapest. In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph, the Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town, the appellation was chosen in reference to Margaret, Neumann János became Margittai Neumann János, which he later changed to the German Johann von Neumann. Von Neumann was a child prodigy, as a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her, formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins, Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. A copy was contained in a private library Max purchased, One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911 and this was one of the best schools in Budapest, part of a brilliant education system designed for the elite
14.
William Arveson
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William Arveson was a mathematician specializing in operator algebras who worked as a professor of Mathematics at the University of California, Berkeley. Arveson obtained his Ph. D. from UCLA in 1964, of particular note is Arvesons work on completely positive maps. One of his results in this area is an extension theorem for completely positive maps with values in the algebra of all bounded operators on a Hilbert space. This theorem led naturally to the question of injectivity of von-Neumann algebras in general, one of the major features of Arvesons work was the use of algebras of operators to elucidate single operator theory. In the late 80s and 90s Arveson played a role in developing the theory of one-parameter semigoups of *-endomorphisms on von Neumann algebras - also known as E-semigroups. Among his achievements, he introduced product systems and proved that they are complete invariants of E-semigroups up to cocycle conjugacy, books Papers Markiewicz, Daniel, Jorgensen, Palle E. T. William B
15.
Richard Kadison
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Richard V. Kadison is an American mathematician known for his contributions to the study of operator algebras. Kuemmerle Professor in the Department of Mathematics of the University of Pennsylvania, Kadison is a member of the U. S. National Academy of Sciences, and a foreign member of the Royal Danish Academy of Sciences and Letters and of the Norwegian Academy of Science and Letters. He is a 1969 Guggenheim Fellow, Richard Kadison was awarded the 1999 Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society. In 2012 he became a fellow of the American Mathematical Society, dick Kadison was a skilled gymnast with a specialty in rings. He married Karen M. Holm on June 5,1956, an exercise approach, Birkhäuser, Basel, III,1991, xiv+273 pp. ISBN 0-8176-3497-5, IV,1992, xiv+586 pp. ISBN 0-8176-3498-3 On representations of finite type. Proc Natl Acad Sci U S A.95, 13392–6, with I. M. Singer, Some Remarks on Representations of Connected Groups. Proc Natl Acad Sci U S A.38, 419–23, with Bent Fuglede, On a Conjecture of Murray and von Neumann. Proc Natl Acad Sci U S A.37, 420–5, with Zhe Liu, A note on derivations of Murray–von Neumann algebras. Proc Natl Acad Sci U S A.111, 2087–93, proc Natl Acad Sci U S A.99, 5217–22. Proc Natl Acad Sci U S A.99, 4178–84, proc Natl Acad Sci U S A.43, 273–6. On the Additivity of the Trace in Finite Factors, proc Natl Acad Sci U S A.41, 385–7. Proc Natl Acad Sci U S A.41, 169–73, with Bent Fuglede, On Determinants and a Property of the Trace in Finite Factors. Proc Natl Acad Sci U S A.37, 425–31, kadison–Kaplansky conjecture Kadisons inequality Kadison–Singer problem Kadison transitivity theorem Kadison–Sakai theorem Kadison–Kastler metric Richard Kadison at the Mathematics Genealogy Project
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker