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Groups of divisibility

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One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980's in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field. Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations. The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Algebraic number theory. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Galois cohomology --- Cohomologie galoisienne. --- Algebraic number theory --- p-adic numbers --- Abelian extension. --- Abelian variety. --- Absolute Galois group. --- Algebraic closure. --- Barry Mazur. --- Big O notation. --- Birch and Swinnerton-Dyer conjecture. --- Cardinality. --- Class field theory. --- Coefficient. --- Cohomology. --- Complex multiplication. --- Conjecture. --- Corollary. --- Cyclotomic field. --- Dimension (vector space). --- Divisibility rule. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Error term. --- Euler product. --- Euler system. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Finite set. --- Functional equation. --- Galois cohomology. --- Galois group. --- Galois module. --- Gauss sum. --- Global field. --- Heegner point. --- Ideal class group. --- Integer. --- Inverse limit. --- Inverse system. --- Karl Rubin. --- Local field. --- Mathematical induction. --- Maximal ideal. --- Modular curve. --- Modular elliptic curve. --- Natural number. --- Orthogonality. --- P-adic number. --- Pairing. --- Principal ideal. --- R-factor (crystallography). --- Ralph Greenberg. --- Remainder. --- Residue field. --- Ring of integers. --- Scientific notation. --- Selmer group. --- Subgroup. --- Tate module. --- Taylor series. --- Tensor product. --- Theorem. --- Upper and lower bounds. --- Victor Kolyvagin. --- Courbes elliptiques --- Nombres, Théorie des

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Why do some civil wars end in successfully implemented peace settlements while others are fought to the finish? Numerous competing theories address this question. Yet not until now has a study combined the historical sweep, empirical richness, and conceptual rigor necessary to put them thoroughly to the test and draw lessons invaluable to students, scholars, and policymakers. Wallter examens conflicts from Greece to Laos, China to Columbia, Bosnia to Rwanda

Politique mondiale --- Conflits de basse intensite --- Guerre civile. --- Addis Ababa agreement (1993). --- Afghanistan. --- American Civil War. --- Angola. --- Arusha accords. --- Baumhoegger, Goswin. --- Boutros-Ghali, Boutros. --- Callaghan, James. --- El Salvador. --- Geneva peace agreement. --- Habyarimana, Juvénal. --- Johnson, Phyllis. --- Karker, Hassame. --- King, Charles. --- Lancaster House negotiations. --- Laos. --- Lebanon. --- Nicaragua. --- Rhodesian Front. --- Tela agreement. --- Touval, Saadia. --- United Nations. --- University of Michigan. --- Victoria Falls Conference. --- Young, Andrew. --- Zambia. --- autonomous regions. --- balance of power. --- cease-fire agreements. --- costs of war. --- credible commitment theory. --- democracy-autocracy scale. --- divisibility of stakes. --- ethics of intervention. --- federalism. --- hegemony. --- hypotheses. --- logit analysis. --- mediation. --- military consolidation. --- monitoring and verification. --- multivariate regression analysis. --- one-party rule. --- parliamentary system. --- partition. --- political pacts. --- power-sharing pacts. --- presidential system. --- proportional representation. --- selection bias probit. --- simulation procedure. --- three-stage process. --- Civil war --- Low-intensity conflicts (Military science) --- World politics

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This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- 512.64 --- 512.64 Linear and multilinear algebra. Matrix theory --- Linear and multilinear algebra. Matrix theory --- Addition. --- Analytic continuation. --- Arithmetic mean. --- Banach space. --- Binomial theorem. --- Block matrix. --- Bochner's theorem. --- Calculation. --- Cauchy matrix. --- Cauchy–Schwarz inequality. --- Characteristic polynomial. --- Coefficient. --- Commutative property. --- Compact space. --- Completely positive map. --- Complex number. --- Computation. --- Continuous function. --- Convex combination. --- Convex function. --- Convex set. --- Corollary. --- Density matrix. --- Diagonal matrix. --- Differential geometry. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence relation. --- Existential quantification. --- Extreme point. --- Fourier transform. --- Functional analysis. --- Fundamental theorem. --- G. H. Hardy. --- Gamma function. --- Geometric mean. --- Geometry. --- Hadamard product (matrices). --- Hahn–Banach theorem. --- Harmonic analysis. --- Hermitian matrix. --- Hilbert space. --- Hyperbolic function. --- Infimum and supremum. --- Infinite divisibility (probability). --- Invertible matrix. --- Lecture. --- Linear algebra. --- Linear map. --- Logarithm. --- Logarithmic mean. --- Mathematics. --- Matrix (mathematics). --- Matrix analysis. --- Matrix unit. --- Metric space. --- Monotonic function. --- Natural number. --- Open set. --- Operator algebra. --- Operator system. --- Orthonormal basis. --- Partial trace. --- Positive definiteness. --- Positive element. --- Positive map. --- Positive semidefinite. --- Positive-definite function. --- Positive-definite matrix. --- Probability measure. --- Probability. --- Projection (linear algebra). --- Quantity. --- Quantum computing. --- Quantum information. --- Quantum statistical mechanics. --- Real number. --- Riccati equation. --- Riemannian geometry. --- Riemannian manifold. --- Riesz representation theorem. --- Right half-plane. --- Schur complement. --- Schur's theorem. --- Scientific notation. --- Self-adjoint operator. --- Sign (mathematics). --- Special case. --- Spectral theorem. --- Square root. --- Standard basis. --- Summation. --- Tensor product. --- Theorem. --- Toeplitz matrix. --- Unit vector. --- Unitary matrix. --- Unitary operator. --- Upper half-plane. --- Variable (mathematics).

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Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics-but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits.From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics.Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Mathematics --- Math --- Science --- Study and teaching (Higher) --- Abstract algebra. --- Addition. --- Algebra. --- Algebraic equation. --- Algebraic number. --- Algorithm. --- Arbitrarily large. --- Arithmetic. --- Axiom. --- Binomial coefficient. --- Bolzano–Weierstrass theorem. --- Calculation. --- Cantor's diagonal argument. --- Church–Turing thesis. --- Closure (mathematics). --- Coefficient. --- Combination. --- Combinatorics. --- Commutative property. --- Complex number. --- Computable number. --- Computation. --- Constructible number. --- Continuous function (set theory). --- Continuous function. --- Continuum hypothesis. --- Dedekind cut. --- Dirichlet's approximation theorem. --- Divisibility rule. --- Elementary function. --- Elementary mathematics. --- Equation. --- Euclidean division. --- Euclidean geometry. --- Exponentiation. --- Extended Euclidean algorithm. --- Factorization. --- Fibonacci number. --- Floor and ceiling functions. --- Fundamental theorem of algebra. --- Fundamental theorem. --- Gaussian integer. --- Geometric series. --- Geometry. --- Gödel's incompleteness theorems. --- Halting problem. --- Infimum and supremum. --- Integer factorization. --- Integer. --- Least-upper-bound property. --- Line segment. --- Linear algebra. --- Logic. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Method of exhaustion. --- Modular arithmetic. --- Natural number. --- Non-Euclidean geometry. --- Number theory. --- Pascal's triangle. --- Peano axioms. --- Pigeonhole principle. --- Polynomial. --- Predicate logic. --- Prime factor. --- Prime number. --- Probability theory. --- Probability. --- Projective line. --- Pure mathematics. --- Pythagorean theorem. --- Ramsey theory. --- Ramsey's theorem. --- Rational number. --- Real number. --- Real projective line. --- Rectangle. --- Reverse mathematics. --- Robinson arithmetic. --- Scientific notation. --- Series (mathematics). --- Set theory. --- Sign (mathematics). --- Significant figures. --- Special case. --- Sperner's lemma. --- Subset. --- Successor function. --- Summation. --- Symbolic computation. --- Theorem. --- Time complexity. --- Turing machine. --- Variable (mathematics). --- Vector space. --- Word problem (mathematics). --- Word problem for groups. --- Zermelo–Fraenkel set theory.