Absolute Values -- Valuations -- Extension of Valuations -- Henselian Fields -- Structure Theory -- Applications of Valuation Theory.Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valu…
The Integers -- Groups -- Rings -- Polynomials -- Vector Spaces and Modules -- Some Linear Groups -- Field Theory -- Finite Fields -- The Real and Complex Numbers -- Sets.Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theor…
Trigonometric Fundamentals -- Introductory Problems -- Advanced Problems -- Solutions to Introductory Problems -- Solutions to Advanced Problems.103 Trigonometry Problems contains highly-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Though many problems may initially appear impenetrable to the novice, most can be solv…
Algebraic Preliminaries -- Integration of Rational Functions -- Differential Fields -- The Order Function -- Integration of Transcendental Functions -- The Risch Differential Equation -- Parametric Problems -- The Coupled Differential System -- Structure Theorems -- Parallel Integration.This first volume in the series "Algorithms and Computation in Mathematics" is destined to become the standar…
Profinite semigroups and applications -- The structure of free algebras -- Completeness of automaton mappings with respect to equivalence relations -- Completeness of uniformly delayed operations -- Classification in finite model theory: counting finite algebras -- Syntactic semigroups and the finite basis problem -- Endoprimal algebras -- The complexity of constraint satisfaction: an algebraic…
to residues and resultants -- Solving equations via algebras -- Symbolic-numeric methods for solving polynomial equations and applications -- An algebraist’s view on border bases -- Tools for computing primary decompositions and applications to ideals associated to Bayesian networks -- Algorithms and their complexities -- Toric resultants and applications to geometric modelling -- to numerica…
Historical Introduction: Issai Schur and the Early Development of the Schur Complement -- Basic Properties of the Schur Complement -- Eigenvalue and Singular Value Inequalities of Schur Complements -- Block Matrix Techniques -- Closure Properties -- Schur Complements and Matrix Inequalities: Operator-Theoretic Approach -- Schur complements in statistics and probability -- Schur Complements and …
The statistics of asset allocation -- Univariate statistics -- Multivariate statistics -- Modeling the market -- Classical asset allocation -- Estimating the distribution of the market invariants -- Evaluating allocations -- Optimizing allocations -- Accounting for estimation risk -- Estimating the distribution of the market invariants -- Evaluating allocations -- Optimizing allocations.This en…
Supplementary Thoughts on Symplectic Groups -- Automorphisms of the Modular Curve -- Reducing the Fontaine-Mazur Conjecture to Group Theory -- Relating Two Genus 0 Problems of John Thompson -- Relatively Projective Groups as Absolute Galois Groups -- Invariants of Binary Forms -- Some Classical Views on the Parameters of the Grothendieck-Teichmüller Group -- The Image of a Hurwitz Space Under …
Invited Lectures -- Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications -- A General Frame for the Construction of Constrained Curves -- DMBVP for Tension Splines -- Robust Numerical Methods for the Singularly Perturbed Black-Scholes Equation -- Contributed Lectures -- On Certain Properties of Spaces of Locally Sobolev Functions -- On Some Properties of Hom…