Format: Hardback, 639 pages, height x width: 254x178 mm, X, 639 p., 1 Hardback
Series: Springer Undergraduate Texts in Mathematics and Technology
Pub. Date: 04-Sep-2025
ISBN-13: 9783031937637
This text provides a mathematically rigorous introduction to modern methods of machine learning and data analysis at the advanced undergraduate/beginning graduate level. The book is self-contained and requires minimal mathematical prerequisites. There is a strong focus on learning how and why algorithms work, as well as developing facility with their practical applications. Apart from basic calculus, the underlying mathematics linear algebra, optimization, elementary probability, graph theory, and statistics is developed from scratch in a form best suited to the overall goals. In particular, the wide-ranging linear algebra components are unique in their ordering and choice of topics, emphasizing those parts of the theory and techniques that are used in contemporary machine learning and data analysis. The book will provide a firm foundation to the reader whose goal is to work on applications of machine learning and/or research into the further development of this highly active field of contemporary applied mathematics.
To introduce the reader to a broad range of machine learning algorithms and how they are used in real world applications, the programming language Python is employed and offers a platform for many of the computational exercises. Python notebooks complementing various topics in the book are available on a companion GitHub site specified in the Preface, and can be easily accessed by scanning the QR codes or clicking on the links provided within the text. Exercises appear at the end of each section, including basic ones designed to test comprehension and computational skills, while others range over proofs not supplied in the text, practical computations, additional theoretical results, and further developments in the subject. The Students Solutions Manual may be accessed from GitHub. Instructors may apply for access to the Instructors Solutions Manual from the link supplied on the texts Springer website.
The book can be used in a junior or senior level course for students majoring
in mathematics with a focus on applications as well as students from other
disciplines who desire to learn the tools of modern applied linear algebra
and optimization. It may also be used as an introduction to fundamental
techniques in data science and machine learning for advanced undergraduate
and graduate students or researchers from other areas, including statistics,
computer science, engineering, biology, economics and finance, and so on
Preface.- 1 Vectors.- 2 Inner Product, Orthogonality, Norm.- 3 Matrices.-
4. How Matrices Interact with Inner Products and Norms.- 5
Eigenvalues and Singular Values.- 6 Basics of Optimization.- 7 Introduction
to Machine Learning and Data.- 8 Principal Component Analysis.- 9 Graph
Theory and Graph-based Learning.- 10 Neural Networks and Deep Learning.- 11
Advanced Optimization.- Bibliography.- Index.
Format: Hardback, 662 pages, height x width: 235x155 mm, X, 662 p., 1 Hardback
Series: Undergraduate Texts in Mathematics
Pub. Date: 26-Aug-2025
ISBN-13: 9783031918407
This text covers topics in algebraic geometry and commutative algebra with careful attention to their practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometrythe elimination theorem, the extension theorem, the closure theorem and the Nullstellensatzthere are chapters on polynomial and rational functions between varieties, robotics and geometric theorem proving, invariant theory of finite groups, projective algebraic geometry, dimension theory, and progress made over the last decades in computing Gröbner bases.
The fifth edition builds on the fourth edition in two main ways. First, a number of typographical errors, found by readers and by the authors since 2018, have been corrected. Second, new material on toric varieties, monomial curves, and other topics of current interest in algebraic geometry has been added. This enhances the opportunities for active learning through new examples, new exercises, and new projects in Appendix D, all supplemented by additional references. The book also includes updated computer algebra material in Appendix C.
The book may be used for a first or second course in undergraduate abstract algebra and, with some augmentation perhaps, for beginning graduate courses in algebraic geometry or computational commutative algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple, Mathematica® and SageMath, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.
The book gives an introduction to Buchbergers algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. The book is well-written. The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.
Preface.- Notation for Sets and Functions.- 1 Geometry, Algebra, and
Algorithms.- 2 Groebner Bases.- 3 Elimination Theory.- 4 The Algebra-Geometry
Dictionary.- 5 Polynomial and Rational Functions on a Variety.- 6 Robotics
and Automatic Geometric Theorem Proving.- 7 Invariant Theory of Finite
Groups.- 8 Projective Algebraic Geometry.- 9 The Dimension of a Variety.- 10
Additional Groebner Basis Algorithms.- Appendix A Some Concepts from
Algebra.- Appendix B Pseudocode.- Appendix C Computer Algebra Systems.-
Appendix D Independent Projects.- References.- Index.
Format: Hardback, 553 pages, height x width: 235x155 mm, Approx. 555 p.,
1 Hardback
Series: Springer Series in Computational Mathematics 67
Pub. Date: 05-Sep-2025
ISBN-13: 9783031993152
This book presents a systematic approach to the numerical analysis of several carefully selected classes of optimal control problems governed by elliptic partial differential equations (PDEs). A priori error estimates for the discretization error between the optimal solutions of the continuous and discretized problems are derived, and numerical experiments are included to illustrate the results.
The proofs are presented in a structured and accessible manner, facilitating a clear understanding of the techniques used. The necessary results from functional analysis and elliptic PDE theory are provided in a self-contained way. Essential aspects of finite element theoryincluding some results newly established by the author(s)are also covered, with all relevant proofs provided in full detail.
Portions of the material have been successfully used in graduate-level courses and seminars, as well as in Masters and PhD theses. The book is intended for researchers and graduate students interested in finite element analysis and/or optimal control problems involving PDEs.
Chapter 1. Introduction.- Part I. Theory and Finite Elements for Elliptic
PDEs.
Chapter 2. Topics from the Theory of Elliptic PDEs.
Chapter 3. Topics from Finite Elements for Elliptic PDEs.- Part II. Distributed Control Problems.
Chapter 4. No Inequality Constraints.
Chapter 5. Control Constraints.
Chapter 6. Bang-Bang Controls.
Chapter 7. Semilinear State Equation.- Part III. Boundary Control Problems.
Chapter 8. Neumann Control.-
Chapter 9. Dirichlet Control.- Part IV. Problems Involving Dirac Measures.-
Chapter 10. Pointwise Control.
Chapter 11. Pointwise Tracking.
Chapter 12. Finitely Many Pointwise State Constraints.- Part V. Problems Involving General Measures.
Chapter 13. State Constraints.
Chapter 14. Sparse Controls.
Format: Paperback / softback, 116 pages, height x width: 235x155 mm, XI, 116 p., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2367
Pub. Date: 24-Jun-2025
ISBN-13: 9789819619597
A groundbreaking theory has emerged for spectral analysis of pseudo-Riemannian
locally symmetric spaces, extending beyond the traditional Riemannian framework.
The theory introduces innovative approaches to global analysis of locally
symmetric spaces endowed with an indefinite metric. Breakthrough methods
in this area are introduced through the development of the branching theory
of infinite-dimensional representations of reductive groups, which is based
on geometries with spherical hidden symmetries. The book elucidates the
foundational principles of the new theory, incorporating previously inaccessible
material in the literature.
The book covers three major topics.
(1) (Theory of Transferring Spectra) It presents a novel theory on transferring spectra along the natural fiber bundle structure of pseudo-Riemannian locally homogeneous spaces over Riemannian locally symmetric spaces.
(2) (Spectral Theory) It explores spectral theory for pseudo-Riemannian locally symmetric spaces, including the proof of the essential self-adjointness of the pseudo-Riemannian Laplacian, spectral decomposition of compactly supported smooth functions, and the Plancherel-type formula.
(3) (Analysis of the Pseudo-Riemannian Laplacian) It establishes the abundance
of real analytic joint eigenfunctions and the existence of an infinite
L2 spectrum under certain additional conditions.
1 Introduction.- 2 Method of proof.- Part I Generalities.- 3 Reminders:
spectral analysis on spherical homogeneous spaces.- 4 Discrete spectrum of
type I and II.- 5 Dierential operators coming from 𝑳 and from the
fiber 𝑭.- Part II Proof of the theorems of
Chapter 1.- 6 Essential
self-adjointness of the Laplacian.- 7 Transfer of Riemannian eigenfunctions
and spectral decomposition.- 8 Consequences of conditions (A) and (B) on
representations of G and 𝑳.- 9 The maps i𝝉,𝚪 and
p𝝉,𝚪 preserve type I and type II.- 10 Infinite discrete
spectrum of type II.- Part III Representation-theoretic description of the
discrete spectrum.- 11 A conjectural picture.- 12 The discrete spectrum in
terms of group representations.
Format: Hardback, 225 pages, height x width: 235x155 mm, 58 Illustrations, color;
8 Illustrations, black and white; X, 225 p. 66 illus., 58 illus. in color., 1 Hardback
Series: Trends in Mathematics
Pub. Date: 04-Oct-2025
ISBN-13: 9783031962776
This volume presents a timely overview of control theory and inverse problems, and highlights recent advances in these active research areas. The chapters are based on talks given at the spring school "Control Theory & Inverse Problems” held in Monastir, Tunisia in May 2024. In addition to providing a snapshot of these two areas, chapters also highlight breakthroughs on more specific topics, such as:
Neural network modeling
Optimal control of the Euler-Bernoulli equation
Traffic signal control
Spectral description of a cell growth and division equation
Control Theory and Inverse Problems will be a valuable resource for both established researchers as well as more junior members of the community.
A novel Optimization Process using Adaptive filling curves. Estimation
of the age-dependent viral hepatitis A infection force.- Neural network
modeling approaches for model predictive control: overview and challenges.-
Characteristic Real Roots and CRRID property of time-delay systems: Recent
results and prospective.- Optimal control of the Euler-Bernoulli equation:
Theoretical and Numerical results.- Long-time behavior of coupled wave-beam
systems: absence of exponential stability with kelvin-voigt damping.-
Modelling and Control of a Rigid and Flexible Airship with Suspended Load in
the XGZ plane.- Spectral description of a cell growth and division equation.-
Kalman Condition For Null Controllability For Parabolic Systems With Dynamic
Boundary Conditions.- Multi-Agent Reinforcement Learning for Tra c Signal
Control.- Finite-time stabilization of linear systems.- Efficient processing
of thermographic and ultrasonic data using topological derivative methods for
nondestructive evaluation of aluminum plates.- Further Insights on the
Active/Passive Postural Quiet Stance Regulation Model.
Format: Hardback, 128 pages, height x width: 240x168 mm, 11 Illustrations, black and white; XII, 128 p. 11 illus., 1 Hardback
Series: Synthesis Lectures on Engineering, Science, and Technology
Pub. Date: 20-Sep-2025
ISBN-13: 9783031958229
This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincaré group. The focus is on the construction of the representations that provide the basis for the formulation of current relativistic quantum field theories of scalar fields, the Dirac field, and the electromagnetic field. Such construction is tied to the use of the methods of operator theory that also provide the basis for the formulation of quantum mechanics, up to the interpretation of the measurement process. In addition, since representation spaces of primary interest in quantum theory are infinite dimensional, the use of these methods is essential. Consequently, the book also calculates the generators of relevant strongly continuous one-parameter groups that are associated with the representations and, where appropriate, the corresponding spectrum. Part II of Quantum Spin and Representations of the Poincaré Group specifically addresses: construction of a double cover of the restricted Lorentz Group; Weyl spinors; Weyl representation of SL(2, C); an extension to a strongly continuous representation of a semi-direct product of R^4 and SL(2, C); Dirac spinors; Dirac fields; Dirac equation; Spin 1 representations of SL(2, C); Maxwell fields; and Maxwell's equations.
Introduction.- Construction of a Double Cover of the Restricted Lorentz
Group.- Weyl Spinors.- Weyl Representation of SL(2, C).- An Extension to a
Strongly Continuous Representation of a Semi-direct Product of R^4 and SL(2,
C).- Dirac Spinors.- Dirac Fields.- Dirac Equation.- Spin 1 Representations
of SL(2, C).- Maxwell Fields.- Maxwell's Equations.- Appendix.-
Bibliography.- Index of Symbols.- Index.