A Guide to Mathematics Resources

Quick but very readable introduction to stochastic differential equations - those subject to additive "white noise" and related random disturbances. Topics include a quick survey of measure theoretic probability theory, an introduction to Brownian motion and the Ito stochastic calculus, and finally the theory of stochastic differential equations.

Introduction to ordinary differential equations and dynamical systems for beginning graduate students. Topics include: linear equations (including the Floquet theorem and the Frobenius method), the Sturm-Liouville boundary value problems, the Poincare-Bendixson theorem; attractors, Hamiltonian systems, the KAM theorem; stability, the Hartman-Grobman theorem; chaos, the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.

First published in 1963 and issues in a second edition in 1983, this book has been a basic reference for researchers in many applied sciences who seek practical information about the representation and manipulation of stationary stochastic processes.

Presents modern theoretical results on kernel-based approximation methods and demonstrates their implementation in various settings. Examples are drawn from fields as diverse as function approximation, spatial statistics, boundary value problems, machine learning, surrogate modeling and finance.

Provides the salient theoretical results needed for a basic understanding of meshfree approximation methods. Meshfree approximation methods, such as radial basis function and moving least squares method, are discussed from a scattered data approximation and partial differential equations point of view.

Collection of four short courses in group representations, graph spectra, statistical optimality, and symbolic dynamics, which highlights their common roots in linear algebra. Leads to high-level applications: representations of finite groups, leading to probability models and harmonic analysis; eigenvalues of growing graphs from quantum probability techniques; statistical optimality of designs from Laplacian eigenvalues of graphs; and symbolic dynamics, applying matrix stability and K-theory.

This updated 9th Edition of the world's top-selling Calculus textbook provides the strongest foundation for a future in STEM because of its problem-solving focus, mathematical precision and accuracy, and outstanding examples and problem sets. Selected and mentored by Stewart, coauthors Daniel Clegg and Saleem Watson continue his legacy.

Accessible to all students with a sound background in high school mathematics, this book covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Euler's formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions.

Presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering.

Rapid introduction to the Christoffel-Darboux kernel, a central object in approximation theory, which has any potential uses in modern data analysis and machine learning.

Popular science tour of Bayes's theorem - which describes the probability of an event based on prior knowledge of conditions that might be related to the event - and its global impact on modern life.

Shows how analysis and design tools can be adapted for control applications that combine the effects of network-induced interactions and hybrid dynamics with complex results.

Introduces the new features of the Mathematica software and the Wolfram language using real-world examples and based on the experience of the author as a consultant.

Explored models of card shuffles, including probability (Markov chains), combinatorics (symmetric function theory), algebra (Hopf algebras), and more. Shows applications to magic tricks and gambling and higher mathematics--Lie theory, algebraic topology, the geometry of hyperplane arrangements, stochastic calculus, etc.

Studies the solutions and optimizations of polynomial equations and their inequalities over the real numbers. For students at an advanced undergraduate or graduate level.

Summarizes the main ideas and theorems of Separate and Joint Continuity, which lies at the interface between General Topology and Functional Analysis (and the geometry of Banach spaces in particular).

Introductory textbook with a focus on the big ideas: ergodic properties of translation flows, counting problems for saddle connections, and associated renormalization techniques. Suitable for a one- or two-semester graduate course.

Focus on asymptotic behavior of problems formulated within cylindrical structures, including fluid flows in channels and the relevance of elasticity in the context of cylindrical bodies.

Comprehensive work that includes topics such as well-posedness, regularity, approximation, and optimal control. The book delves into the intricacies of fractional Navier-Stokes equations, fractional Rayleigh-Stokes equations, fractional Fokker-Planck equations, and fractional Schrödinger equations.

Higher special functions emerge from boundary eigenvalue problems of Fuchsian differential equations with more than three singularities. This detailed reference provides solutions for singular boundary eigenvalue problems of linear ordinary differential equations of second order, exploring previously unknown methods for finding higher special functions.

Explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof.

Addresses the mathematical question of population extinction or persistence. It explores structured population dynamics using discrete-time dynamical systems, emphasizing spatial structure.

Delves into Iwasawa theory, which originated in the late 1950s with Kenkichi Iwasawa’s work on ideal class groups in cyclotomic number fields. The theory was later extended to include elliptic curves and modular forms.

Explores regression modeling, emphasizing cases (people, organizations, and nations) as the focal point. By re-analyzing influential research, the authors offer nuanced interpretations, emphasizing intuition and reproducible examples. Bridges the gap between quantitative and qualitative research.

Focuses on applying type-3 fuzzy logic to solve complex prediction problems. By employing higher-order fuzzy logic, neural networks, and fractal theory, the book enhances prediction results and explores hybrid intelligent methods and tests them on various prediction scenarios, including COVID-19 and stock market predictions.

Discover the fascinating world of ‘C∞-schemes with corners’—singular spaces in differential geometry equipped with well-defined notions of boundary and corners. This groundbreaking work lays the foundation for theories of derived manifolds and derived orbifolds with corners, with applications spanning symplectic geometry and moduli spaces of J-holomorphic curves.

The authors explore group-based cryptography, bridging the gap between mathematicians and computer scientists. Focusing on recent developments, the book delves into families of groups—such as polycyclic groups and graph groups—highlighting their algorithmic properties and cryptographic applications.

Explore the fascinating world of automorphic forms beyond the GL₂ case in this insightful volume. Based on lecture notes from the twenty-fifth Arizona Winter School, the book covers topics such as rationality questions on unitary groups, theta lifts, quaternionic modular forms, and automorphic forms over function fields.

This book presents an extensive survey of fundamental results, essential techniques, and phenomena in the solvability, geometry, and regularity of Monge-Ampere equations. The modern treatment of boundary behaviors of solutions is thoroughly discussed.

The book includes results not previously found in other books, such as: edge connectivity; characterization of the cycle space; Minkowski decomposition; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem.

This book provides a comprehensive review on tensor algebra, including products, unfolding, eigenvalues, and decompositions. The author provides an overview of existing literature on the topic.

Krasner hyperrings are a generalization of hyperfields, introduced by Marc Krasner (1912-1985) in order to study complete valued fields. This book presents an elaborate study on hyperstructures in algebra.

This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry.

Andrea Malchiodi tackles the classical problem posed by Kazdan and Warner: prescribing a given function on a closed manifold as the scalar curvature of a metric within a conformal class. With a focus on understanding the matter, the book combines variational techniques, Liouville-type theorems, blow-up analysis, elliptic regularity theory, and topological arguments to address both existence and compactness issues.

This book introduces the central concepts of modern data science. The book demands no particular software familiarity, since the methods presented can be implemented in Python, R (used by the authors), or any other statistical software. Introduces the different types of data, the distinction between sample and population, and methods for sampling populations. Introduces summary statistics concerning the location and variability of a set of data. Additionally introduces commonly used summary diagrams.

This book treats the elements of discrete mathematics that have important applications in computer science, thus providing the necessary tools for the reader to come to a competent mathematical judgement of modern developments in the age of information.

Explores psychophysics, a branch of psychology that examines how physical objects are perceived by individuals, and highlights how psychophysics insights are relevant in economics, particularly in understanding risk aversion and decision-making under uncertainty and techniques for uncovering hidden patterns within noisy data.

Presents recent findings and results concerning the solutions of finite state-space Markov decision problems and determining Nash equilibria for related stochastic games with average and total expected discounted reward payoffs. Covers applications in areas such as ecology, economics, engineering, telecommunication, and biology.

Presents a unified viewpoint on the study of Hamilton-Jacobi equations and control problems with discontinuities. Covers both codimension-1 discontinuities and the stratified case, emphasizing connections with deterministic control problems and providing numerous examples and applications.

Comprehensive one-semester course in probability with measure theory, covering essential topics such as probability measures, random variables, and their laws, along with applications to statistics. The book includes over 150 exercises with full solutions, making it suitable for guided self-study.

Shows the role of spin- and pin-structures in orientability considerations in symplectic topology, and explores their applications in enumerative geometry. Establishes the equivalence of Spin- and Pin-structures from different perspectives, details constructions of orientations on determinants of real Cauchy-Riemann operators and applies these results to the enumerative geometry of real curves.

Presents a unified framework that generalizes Witten zeta-functions, Mordell-Tornheim multiple zeta-functions, and Euler-Zagier multiple zeta-functions. Covers topics such as meromorphic continuation, recursive analytic structure, special values at integer points, and functional relations. Notably, the authors connect these zeta-functions to Witten’s volume formula and demonstrate their relevance to multiple zeta values (MZVs) and Zagier’s conjecture.

Introduces manifolds, submanifolds, semi-Riemannian geometry, and their applications in geometry and physics. It explores conformal vector fields, their properties, and their interplay with Ricci solitons.

Compilation of research and surveys focuses on mathematical analysis, approximation theory, and optimization. The contributions are dedicated to branches of mathematics influenced by A.-M. Legendre. Topics covered include inequalities, dynamic contact problems, polynomial methods, and more.

Covers geometry and topology of symplectic manifolds equipped with pairs of complementary Lagrangian foliations. Notably, it systematically develops the foundations of Lagrangian foliations, and explores Künneth geometry from a basic differential topology.perspective

Presents schemes, simplicial sets, higher categories, model categories, derived algebraic geometry, and spectral algebraic geometry in a self-contained manner. It discusses Motives, Goodwillie Calculus, Higher Galois, Supersymmetry, and topics in physical mathematics.

Comprehensive work that delves into the mathematical foundations of linear models. It covers topics such as matrix algebra, statistical distributions, and their relevance in the context of linear modeling.

Exposition of the construction of normal forms and smooth conjugacies for perturbations of tempered exponential dichotomies. These dichotomies exhibit nonzero Lyapunov exponents, and the book covers both difference and differential equations.

Explores the applications of various techniques in number theory and mathematical analysis across diverse fields, including engineering, mathematics, physics, biology, and statistics. Features contributions from renowned scientists and mathematicians worldwide, presenting recent developments and original research.

Comprehensive exploration of enumeration techniques, covering topics such as recursion, generating functions, sieve methods, and inversion formulas. The book delves into counting problems related to sets, multisets, permutations, partitions, graphs, and trees. It’s an essential resource for anyone interested in discrete mathematics

Early topics cover set-theoretic topology from a filter convergence viewpoint, while later sections reveal these concepts as consequences of advanced convergence theory. The approach’s strength lies in its closure under natural operations, unlike topologies. Convergence theory complements topology similarly to how complex numbers complete the real numbers.

This book casts a new light on Bertrand's Paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability.

r. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.

Introduction to Cox rings, emphasizing concrete aspects of the theory. Key topics covered include finitely generated Cox rings, their relation to toric geometry, various classes of varieties with group actions, and applications in arithmetic problems (such as Manin’s conjecture).

Explores problems of communication, compression, and learning, emphasizing efficient algorithms for extracting information in its most compact form. Through information theory, it models and analyzes communication and compression problems, viewing learning as compression with side information. Delves into topics such as universal codes, minimax redundancy, and Markov sources.

Highlights the probabilistic approach, which predicts the performance of pattern matching algorithms using analytic combinatorics and information theory. Part I compiles known results via analytic methods, while Part II focuses on applications to data structures like digital trees and string-based compression.

Explores both algebraic and algorithmic perspectives on residuated structures that serve as a bridge between algebra and logic, providing insights into their properties and relationships.

Explores constructing p-energy on compact metric spaces. By considering a scaling limit of discrete p-energies on graphs approximating the original space, Kigami introduces the concept of conductive homogeneity, which allows for the construction of reasonable p-energy when p exceeds the Ahlfors regular conformal dimension of the space.

Detailed investigation of the spectral properties of multi-dimensional, massless Dirac operators. The authors derive a limiting absorption principle for the free massless Dirac operator on any compact interval in ℝ(\setminus){0}. They also prove the absence of singular continuous spectrum for interacting massless Dirac operators.

Focuses on self-similar and self-affine sets, specific types of fractals where this resemblance is precise and given by a contracting similitude or affine transformation. Topics include self-similar sets, measures with overlaps (including infinite Bernoulli convolutions), and the study of self-affine systems using ergodic theory and dynamical systems methods.

Focused on the process and applications of 3D printing for mathematical education, research, and visualization, alongside a discussion of the challenges and open mathematical problems that arise in the design and algorithmic aspects of 3D printing.

Develops the general abstract theory of Euclidean buildings, which are structures with Euclidean affine spaces as apartments. The book emphasizes the metric aspects of these objects, viewing them as CAT(0) spaces similar to Riemannian symmetric spaces of non-compact type

English translation of Adolf Hurwitz’s 1919 textbook on quaternions, along with his famous 1-2-3-4 theorem on composition algebras. This comprehensive work sheds historical light on the development of quaternion number theory, including its connections to classical preparatory works and analogies to algebraic numbers.