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In mathematics, physics, and engineering, the imaginary unit allows the real number system, mathbb{R}, to be extended to the complex number system, mathbb{C}.It is denoted by i or the Latin j or the Greek iota (see Alternative notations below). Its precise definition is dependent upon the particular method of extension. The primary motivation for this extension is the fact that not every polynomial equation with real coefficients f(x) = 0 has a solution in the real number system. In particular, the equation x2 + 1 = 0 has no real solution (see Definition below). However, if complex numbers are allowed as solutions, then this equation, and indeed every non-zero degree polynomial equation f(x) = 0, has a solution (see Algebraic closure and Fundamental theorem of algebra). The imaginary unit is often loosely referred to as the "square root of ?1". However, there are in fact two square roots of ?1 (just as there are two square roots of every real number), namely i and ?i. Misuse of the imaginary unit may lead to difficulties. ... Read more

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The book concerns limit theorems and laws of large numbersfor scaled unionsof independent identically distributedrandom sets. These results generalizewell-known facts fromthe theory of extreme values. Limiting distributions(calledunion-stable) are characterized and found explicitly formany examples of random closed sets. The speed ofconvergence in the limit theorems for unions is estimated bymeans of the probability metrics method.It includes theevaluation of distances between distributions ofrandomsets constructed similarly to the well-known distancesbetween distributions of random variables. The techniquesinclude regularly varying functions, topological propertiesof the space of closed sets, Choquetcapacities, convexanalysis and multivalued functions.Moreover, theconcept of regular variation is elaborated formultivalued (set-valued) functions. Applications of thelimit theorems to simulation of random sets, statisticaltests, polygonal approximations of compacts, limittheoremsfor pointwise maxima of random functions are considered.Several open problems are mentioned.Addressed primarily to researchers in the theory of randomsets, stochastic geometry and extreme value theory, the bookwill also be of interest to applied mathematicians workingon applications of extremal processes and their spatialcounterparts. The book is self-contained, and nofamiliaritywith the theory of random sets is assumed. ... Read more

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This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another. Assuming familiarity with only the basics of Boolean algebras and set theory, through to simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 97 are formulated.

Based on Cardinal Functions on Boolean Algebras (1990) by the same author, the present work is nearly twice the size of the original work. It contains solutions to many of the open problems which are discussed in greater detail than before. Among the new topics considered are ultraproducts and Fedorchukís theorem, and there is a more complete treatment of the cellularity of free products. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including tree algebras and superatomic algebras.

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"This book is an indispensable tool for anyone working in Boolean algebra, and is also recommended for set-theoretic topologists." - Zentralblatt MATH

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Chapters: Banach Algebra, Corona Theorem, Abstract Index Group, Shilov Boundary, Gelfand-mazur Theorem, Approximate Identity, Noncommutative Topology, Uniform Algebra, Banach Function Algebra, Disk Algebra, Banach *-Algebra. Source: Wikipedia. Pages: 35. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related by the following inequality: (i.e., the norm of the product is less than or equal to the product of the norms.) This ensures that the multiplication operation is continuous. If in the above we relax Banach space to normed space the analogous structure is called a normed algebra. A Banach algebra is called "unital" if it has an identity element for the multiplication whose norm is 1, and "commutative" if its multiplication is commutative. Any Banach algebra (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of . Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity. The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements. Banach algebras can also be defined over fields of p...More: http://booksllc.net/?id=4665 ... Read more

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Chapters: Euler's Formula, Complex Number, Euler's Identity, Exponential Function, Polynomial, Fundamental Theorem of Algebra, Imaginary Number, Holomorphic Function, de Moivre's Formula, Cauchy's Integral Formula, Cauchy's Integral Theorem, List of Fourier-Related Transforms, Laurent Series, Residue Theorem, Riemann Mapping Theorem, Entire Function, Meromorphic Function, Cauchy-riemann Equations, Analytic Continuation, Casorati-weierstrass Theorem, Pole, Removable Singularity, Analytic Function, Winding Number, Riemann Surface, Essential Singularity, Generalized Continued Fraction, Methods of Contour Integration, Holomorphic Functional Calculus, Complex Plane, Complex Logarithm, Dessin D'enfant, Branch Point, Mellin Transform, Gauss's Continued Fraction, Almost Periodic Function, Line Integral, Periodic Points of Complex Quadratic Mappings, Mellin Inversion Theorem, Schramm-loewner Evolution, Bieberbach Conjecture, Complex Quadratic Polynomial, Kramers-kronig Relation, Radius of Convergence, Algebraic Geometry and Analytic Geometry, Cayley Transform, Argument, Partial Fractions in Complex Analysis, Riemann-hilbert Problem, Conformal Pictures, Motor Variable, Subharmonic Function, Liouville's Theorem, Edge-Of-The-Wedge Theorem, Monodromy Theorem, Regular Singular Point, Lacunary Function, Paley-wiener Theorem, Schwarz Lemma, Weierstrass Factorization Theorem, Conformal Radius, Lee-yang Theorem, Euler's Continued Fraction Formula, Logarithmic Derivative, Rouché's Theorem, Antiderivative, Cousin Problems, Filled Julia Set, Picard Theorem, Several Complex Variables, Morera's Theorem, Nachbin's Theorem, Ramification, Schwarzian Derivative, Analytic Capacity, Domain Coloring, Siegel Disc, Analyticity of Holomorphic Functions, Fatou's Theorem, Bombieri Norm, Hyperfunction, Complex Conjugate Root Theorem, Generalised Circle, Principal Value, Doubly-Periodic Function, Jordan's Lemma, Normal Family, Nörlund-rice Integral, Positive-Defi...More: http://booksllc.net/?id=5826 ... Read more

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Chapters: Analytic Geometry, Homological Algebra, Bézout's Theorem, Generalized Riemann Hypothesis, Motive, Dessin D'enfant, Glossary of Arithmetic and Diophantine Geometry, Hodge Structure, Smooth Function, Nakayama Lemma, Resolution of Singularities, Algebraic K-Theory, Grassmannian, Valuation, Gröbner Basis, Calabi-yau Manifold, Chow Ring, Cotangent Complex, Algebraic Variety, Crystalline Cohomology, Esquisse D'un Programme, Macdonald Polynomial, Hilbert Scheme, Flat Module, Algebraic Geometry and Analytic Geometry, Homogeneous Polynomial, Quantum Cohomology, Gromov-witten Invariant, Drinfel'd Module, General Position, Ample Line Bundle, Milnor Number, Flip, Generic Property, Minimal Model Program, Hilbert's Arithmetic of Ends, Regular Local Ring, Adjunction Formula, Néron-tate Height, Pseudoholomorphic Curve, Codimension, Injective Sheaf, Characteristic Set, Kähler Manifold, Conifold, Dévissage, Rational Mapping, Arakelov Theory, Hasse-weil Zeta Function, Cohen-macaulay Ring, Algebraic Stack, List of Algebraic Geometry Topics, Lefschetz Theorem on (1,1)-Classes, A¹ Homotopy Theory, Adequate Equivalence Relation, Italian School of Algebraic Geometry, Fondements de La Géometrie Algébrique, Standard Conjectures on Algebraic Cycles, Shimura Variety, Algebraic Space, Cone of Curves, Hyperkähler Manifold, Deformation Theory, Noncommutative Algebraic Geometry, Flat Topology, Descent, Excellent Ring, Stable Vector Bundle, Finite Morphism, Amoeba, Tropical Geometry, Canonical Singularity, Complete Intersection, Tangent Cone, N!-Conjecture, Conic Bundle, Irreducible Component, Canonical Ring, Residue Field, Abstract Algebraic Variety, Inverse Image Functor, Kähler Differential, Formal Scheme, Grothendieck's Galois Theory, Mnev's Universality Theorem, Jacobian Conjecture, Local Parameter, Period, Differential of the First Kind, Positive Form, Zariski Tangent Space, Schubert Variety, Generic Point, Noetherian Topological Space, Néron ...More: http://booksllc.net/?id=1997 ... Read more

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Chapters: Determinant, Power Set, Dimension, Isomorphism, Universal Algebra, Binomial Theorem, Algebraic Number, Identity Element, Automorphism, Union, Kernel, Congruence Relation, Operator, Algebraic Extension, Ratio, Factorization, Coefficient, Rational Root Theorem, Operand, Complement, Inverse Element, Recurrence Relation, Cyclotomic Polynomial, History of Elementary Algebra, Laws of Form, Partial Fraction, New Algebra, K-Theory, Timeline of Algebra, Derivative, Order of Operations, Ping-Pong Lemma, Grigorchuk Group, Timeline of Algebra and Geometry, Grushko Theorem, Infrastructure, Variable, Boundedly Generated Group, Completing the Square, Distributive Lattice, Association Scheme, Rank of a Group, the Compendious Book on Calculation by Completion and Balancing, Graph Dynamical System, Algebraic Function, Schur Algebra, Irreducible Polynomial, Multinomial Theorem, Variety, Filtration, Bose-mesner Algebra, Canonical Form, Permanent, Digital Root, Closed-Form Expression, Intersection, Nested Radical, List of Basic Algebra Topics, Symmetric Difference, Monomial, Ax-grothendieck Theorem, Field Arithmetic, Brahmagupta-fibonacci Identity, Cycle, Freshman's Dream, Goursat's Lemma, Transforming Polynomials, Stack, Median Algebra, Spherical Design, Degree, Algebraic Element, Haran's Diamond Theorem, Continuant, Unital, Absolutely Irreducible, Differential Equations of Addition, Conjugate, Tertiary Ideal, Constant, Reduct, Immanant of a Matrix, Distributive Homomorphism, Substitution, Polynomial Diophantine Equation, Prosolvable Group, Componendo and Dividendo, Temperley-lieb Algebra, Operation Theory, Indeterminate, Real Structure, Generalized Arithmetic Progression, Algebraic Biology, Plugging In, de Morgan Algebra, Invertible Module, Symbolic Method, Topological Module, Trinomial, Monomial Basis, Alternatization, Sylvester's Determinant Theorem, Theory of Equations, Multiplicative Digital Root, Content, Aperiodic Finite State Autom...http://booksllc.net/?id=9550030 ... Read more

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Chapters: Stokes' Theorem, Exterior Algebra, Differential Form, Hodge Dual, Volume Form, de Rham Cohomology, Exterior Derivative, Closed and Exact Differential Forms, Vector-Valued Differential Form, Hodge Theory, Berezin Integral, Complex Differential Form, Solder Form, Positive Form, Differential Ideal, One-Form, Dolbeault Cohomology, Interior Product, Two-Form, Exterior Bundle, Dolbeault Operator. Source: Wikipedia. Pages: 122. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. Also like the cross product, the exterior product is alternating, meaning that u u = 0 for all vectors u, or equivalently u v = v u for all vectors u and v. In linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a linear transformation that is basis-independent, and is fundamentally related to ideas of rank and linear independence. The exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann ) of a given vector space V over a field K is the unital associative algebra (V) generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. The universal construction allows the exterior algebra to be def...More: http://booksllc.net/?id=221537 ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Von Neumann Algebras, C*-Algebra, Approximately Finite Dimensional C*-Algebra, Commutation Theorem, Direct Integral, Spectrum of a C*-Algebra, Crossed Product, Hilbert C*-Module, Abelian Von Neumann Algebra, Compact Quantum Group, Tomita-takesaki Theory, Baum-connes Conjecture, Subfactor, Central Carrier, Wold Decomposition, Affiliated Operator, Multiplier Algebra, Dixmier Trace, Schröder-bernstein Theorems for Operator Algebras, Von Neumann Bicommutant Theorem, Cuntz Algebra, Hyperfinite Type Ii Factor, Partial Isometry, Finite Dimensional Von Neumann Algebra, Uniformly Hyperfinite Algebra, Nuclear C*-Algebra, Universal C*-Algebra, Baer Ring, Ultrastrong Topology, Russo-dye Theorem, Hereditary C*-Subalgebra, Calkin Algebra, Ultraweak Topology, Octacube, Temperley-lieb Algebra, Toeplitz Algebra, Naimark's Problem, Enveloping Von Neumann Algebra, Kaplansky Density Theorem, Dirichlet Algebra, Locally Compact Quantum Group, Approximately Finite-Dimensional, Projectionless Algebra, Hilbert Algebra. Excerpt:In operator algebras , an algebra is said to be approximately finite dimensional if it contains an increasing sequence of finite dimensional subalgebras that is dense. One can considerA hyperlinked version of this chapter is at In C*-algebras , an approximately finite dimensional , or AF , C*-algebra is one that is the inductive limit of a sequence of finite dimensional C*-algebras. Approximate finite dimensionality was first defined and described combinatorially by Bratteli . Elliott gave a complete classification of AF algebras using the K 0 functor whose range consists of certain type of abelian groups.Definition and basic properties Finite dimensional C*-algebras An arbitrary finite dimensional C*-algebra A takes the following form, up to isomorphism:where Mi denotes the ful... ... Read more

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This paperback reprint of one of the best in the field covers a broad range of limit theorems useful in mathematical statistics, along with methods of proof and techniques of application. The manipulation of "probability" theorems to obtain "statistical" theorems is emphasized. ... Read more

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Great Reference
As a graduate student in statistics, I have had plenty of opportunities to browse through the pages of this book in search of that theorem, technique, or tool that would be applicable to my own work.In many occasions I was succesful in such search.This title compiles and synthesizes a wealth of definitions, theorems, and techniques classically used in asymptotic theory of estimators.It is as much as possible a self-contained work with enough introductory material to be used independently of other references.A bit outdated now, and with minimal material on dependent random variables, this title is still a classic compendium on statistical approximation theory. ... Read more

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The noncommutative versions of fundamental classical results on the almost sure convergence in L2-spaces are discussed: individual ergodic theorems, strong laws of large numbers, theorems on convergence of orthogonal series, of martingales of powers of contractions etc. The proofs introduce new techniques in von Neumann algebras. The reader is assumed to master the fundamentals of functional analysis and probability. The book is written mainly for mathematicians and physicists familiar with probability theory and interested in applications of operator algebras to quantum statistical mechanics. ... Read more

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High Quality Content by WIKIPEDIA articles! The Schröder-Bernstein theorem, from set theory, has analogs in the context operator algebras. This article discusses such operator-algebraic results. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F. Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F. For closed subspaces M and N where projections PM and PN, onto M and N respectively, are elements of M, M « N if PM « PN. ... Read more

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Reprint of 1910 Cambridge University Press, Second edition. Cloth, [vi] 387 pp. With a Supplement Containing an Account of Fermat's Theorems and Problems Connected with Diophantine Analysis and Some Solutions of Diophantine. ... Read more