Algebra Books

Book SynopsisThis book offers a user friendly, hands-on, and systematic introduction to applied and computational harmonic analysis: to Fourier analysis, signal processing and wavelets; and to their interplay and applications. The approach is novel, and the book can be used in undergraduate courses, for example, following a first course in linear algebra, but is also suitable for use in graduate level courses. The book will benefit anyone with a basic background in linear algebra. It defines fundamental concepts in signal processing and wavelet theory, assuming only a familiarity with elementary linear algebra. No background in signal processing is needed. Additionally, the book demonstrates in detail why linear algebra is often the best way to go. Those with only a signal processing background are also introduced to the world of linear algebra, although a full course is recommended. The book comes in two versions: one based on MATLAB, and one on Python, demonstrating the feasibility and applications of both approaches. Most of the MATLAB code is available interactively. The applications mainly involve sound and images. The book also includes a rich set of exercises, many of which are of a computational nature.Trade Review“A beginning student who is unfamiliar with the mathematics behind signal processing will find much here that explains the techniques and the issues associated with their use. Altogether the book presents a beautiful introduction to the uses of linear algebra in signal processing.” (MAA Reviews, July 18, 2020)“This book is a very useful textbook for undergraduate students in applied mathematics and engineering disciplines, but it is also suitable for graduate level courses.” (Manfred Tasche, zbMATH 1420.65001, 2019)Table of Contents1. Sound and Fourier series.- 2. Digital Sound and Discrete Fourier Analysis.- 3. Discrete Time Filters.- 4. Motivation for Wavelets and Some Simple Examples.- 5. The Filter Representation of Wavelets.- 6. Constructing Interesting Wavelets.- 7. The Polyphase Representation of Filter Bank Transforms.- 8. Digital Images.- 9. Using Tensor Products to Apply Wavelets to Images.- A Basic Linear Algebra.

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Book SynopsisDieses Trainingsbuch ist das ideale Begleitbuch für alle Bachelorstudierenden im Fach Mathematik und für die Grundlagenvorlesungen in ingenieur-, natur- und wirtschaftswissenschaftlichen Studiengängen. Es ist speziell geeignet zur Vorbereitung auf Assessmentprüfungen und Basisprüfungen im Themenbereich Lineare Algebra. In Band I werden die folgenden zentralen Themen behandelt: Matrizen, Determinanten Lineare Gleichungssysteme Vektorräume Lineare Abbildungen Eigenwerte und Eigenvektoren Der Stoff wird nicht in der klassischen Lehrbuch-Struktur von Definition, Satz und Beweis präsentiert, sondern kann anhand von mehr als 600 Aufgaben mit unterschiedlichen Schwierigkeitsgraden erlernt und trainiert werden. Alle Übungen werden Schritt für Schritt durchgerechnet, der Lösungsweg wird verständlich erklärt und es werden viele Rechentipps gezeigt. Dabei wird ein breites Spektrum von typischen (Prüfungs-)Aufgabentypen berücksichtigt. Am Ende geben 150 Multiple-Choice Testfragen und 4 konkrete Musterprüfungen, mit ausführlichen Lösungen, dem Leser die Möglichkeit sein Wissen final zu testen und dadurch den Stoff zu festigen.Table of ContentsMatrizen.- Lineare Gleichungssysteme.- Determinanten.- LR-Zerlegung.- Vektorräume und Unterräume.- Vektorraumstruktur: Basis, Dimension und Koordinaten.- Lineare Abbildungen I: Definition und Matrixdarstellung.- Lineare Abbildungen II: Kern, Bild und Basiswechsel.- Eigenwerte und Eigenvektoren.- Prüfungstrainer.- Anhang.- A Analytische Geometrie.- B Mengen, Gruppen und Körper.- C Die komplexen Zahlen.

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Book SynopsisThis textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach. Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.Table of Contents1. Algebraic Preliminaries.- 2. Algebraic Numbers and Their Polynomials.- 3. Extending Fields.- 4. Irreducible Polynomials.- 5. Straightedge and Compass Constructions.- 6. Proofs of the Geometric Impossibilities.- 7. Zeros of Polynomials of Degrees 2, 3, and 4.- 8. Quintic Equations 1: Symmetric Polynomials.- 9. Quintic Equations II: The Abel–Ruffini Theorem.- 10. Transcendence of e and π.- 11. An Algebraic Postscript.- 12. Other Impossibilities: Regular Polygons and Integration in Finite Terms.- References.- Index.

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Book SynopsisThe directed Cayley diameter and the Davenport constant.- Heaps and trusses.- Prüfer v-multiplication domains, a survey.- Some applications of a new approach to factorization.- Atomicity in integral domains.- Norms, normsets, and factorization.-On the Ratliff-Rush closure of an ideal of a one-dimensional ring.- Graded identities of infinite-dimensional Lie algebras.-On a class of quotients of Rees algebras: a survey.- P-adic approximation of algebraic integers and residue class rings of integer-valued polynomials.- On matrix superalgebras with pseudoautomorphism.- Krull Rings, Semistar Operations, and Bases of Regularity.- Descriptions of radicals in polynomial ring extensions.- Boolean inverse semigroups and their type monoids.-Generalized derivations characterized by their action on Jordan products.- Open problems on relations of numerical semigroups.- Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely.- The role of divisors in non-commutative ideal theory.- Essential-like properties for integer-valued polynomial rings: a survey with a note on valuation domains.- On the isomorphism problem for power semigroups.- Dimension-free matricial Nullstellensatze for noncommutative polynomials.

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Book Synopsis- 1. Basic Notions.- 2. Some Classes of Modules.- 3. Right Artinian Rings.- 4. Local Rings, Injective Modules, Flat Modules.- 5. Additive Categories, Abelian Categories.- 6. Appendices.

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Book SynopsisDistributed Fusion Estimation in the Presence of Measurement Quantization and Mixed Attacks.- p-Frobenius Numbers of Numerical Semigroups Generated by Three Consecutive Squares.- On Solutions of a Third Order Linear Difference Equation with Variable Coefficients Applications.- Binet-Fibonacci Calculus and N = 2 Supersymmetric Golden Quantum Oscillator.- Geometry and Entanglement of Super-Qubit Quantum States.- On the Bi-periodic Edouard and the Bi-periodic EdouardLucas Numbers.- The Investment Portfolio Selection with Social Network Decision-Making with Minimum Cost Consensus Model and Incomplete Fermatean Fuzzy Preference Relations.- Boundary Value Problems for the Bitsadze Equation on a Quarter Plane.- Galois Bundles and Automorphisms of the Principal Bundle Moduli Space.- Linear Algebra in Crystal Geometry, and Vice Versa.- On Dual Biquaternionic Sequence Involving Vietoris' Numbers.- A Quadratic Programming Model for Operating Rooms Scheduling based on Resources Availability.- Derivative method for solving cubic equations.- Improved Computational Techniques for Heat Sources Localization.- Open access fisheries model considering depensatory growth functions in the exploited resource.- An Empirical Comparison of Supervised Machine Learning Models in  Predicting Mathematics Performance in Somaliland.- Using SOLO Taxonomy to develop a structured design for mathematics exam questions.- Application of Supervised Machine Learning Algorithms to Identify the Prevalence and Determinants of Spontaneous Abortion among Ever-married Women in Somaliland: Insights from SLDHS Data 2020.- Modified Hungarian Method (MHM) in Optimizing Competency-Preference Scores in Lecturer-To-Course Assignment.- Effects of autonomous vehicles on particulate matter emissions.- Codimension-2 Bifurcation Analysis of a Modi?ed Non-degenerate Fisher Equation Introducing Two Unfolding Parameters.- Modelling Medfly Pest Management.- BETWEEN t*-CLOSED AND *IG-CLOSED SETS.- Fault Injection Attacks against RSA-CRT Digital Signature.- Analysis of Prognostic Factors in Prostate Cancer - A New Approach.- On k-order Jacobsthal Polynomials and Their Properties.- On Euclidean Norms Of Max On Euclidean Norms Of Max Matrices With Chebyshev Polynomials.- Bridging academic learning and community service.- Escape Rooms and Students Competencies.- Descriptive and Inferential Analysis of Renal Health and Patterns of Water Consumption: A case study of Ciudad Hidalgo.- Modeling Solar Energy Through Mathematics.- Problem-Based Learning, Teamwork and Entrepreneurship in the Numerical Methods Course.- Advanced Teaching Strategies for Enhancing STEM Education in Higher Institutions.- Numerical Methods: Artifacts in teaching in a mathematics degree.

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Book Synopsis1 Singularity Theory.- 2 Local Deformation Theory.- 3 Singularities in Arbitrary Characteristics.- Appendix A: Sheaves.- Appendix B: Commutative Algebra.- Appendix C: Formal Deformation Theory.

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Book Synopsis- 1. Basic Monoid Theory.- 2. The Formalism of Module and Ideal Systems.- 3. Prime and Primary Ideals and Noetherian Conditions.- 4. Invertibility, Cancellation and Integrality.- 5. Arithmetic of Cancellative Mori Monoids.- 6. Ideal Theory of Polynomial Rings.

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Book Synopsis- 1. Introduction.- 2. Review of Linear Algebra.- 3. Multilinear Forms.- 4. Inner Products.- 5. Tensors.- 6. Some Physical Tensors.- 7. Solutions to Exercises.

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Book SynopsisChapter 1. Finite Dimensional Division Algebras.- Chapter 2. Structure of Finite Dimensional Algebras.- Chapter 3. Modules and Semisimple Rings.- Chapter 4. Structure of Rings.- Chapter 5. Tensor Products in Noncommutative Algebra.- Chapter 6. Noncommutative Polynomials.- Chapter 7. Rings of Quotients and Structure of PI-Rings.

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Book Synopsis1 Sistemi lineari.- 2 Numeri interi e razionali.- 3 Numeri reali e complessi.- 4 Spazi vettoriali.- 5 Applicazioni lineari.- 6 Operazioni con le matrici.- 7 Riduzioni a scala ed applicazioni.- 8 Determinanti.- 9 Endomorfismi e polinomi caratteristici.- 10 Polinomi minimi.- 11 Forme canoniche di Jordan e razionale.- 12 Spazi duali.- 13 Spazi quoziente.- 14 Spazi vettoriali euclidei ed hermitiani.- 15 Forme bilineari e quadratiche.- 16 Numeri trascendenti.- 17 Soluzioni e suggerimenti di alcuni esercizi.

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Book SynopsisIntroduction.- Preliminaries.- The Jacobson Radical.- The Yang-Baxter Equation.- Nilpotent Groups.- Solvable Groups.- Complements.- Skew Braces.- The Structure Skew Brace of a Solution.- Bierbach Groups.- Garside Groups.- Left Nilpotent Skew Braces.- Right Nilpotent Skew Braces.- Multipermutation Solutions.- Factorizations.- Transitive Groups.- Involutive Solutions.- Simple Skew Braces.

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Book SynopsisQuotients of vector spaces.- Sequences and chain complexes.- Chain maps.- Abstract simplicial complexes.- Simplicial homology and homotopy.- Sequences and chain complexes of sequences.

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Book SynopsisThis textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.Review of the first edition:This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended.— The Mathematical GazetteTrade Review“The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended.” (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)Table of ContentsPart I: General Theory.-Matrix Lie Groups.- The Matrix Exponential.- Lie Algebras.- Basic Representation Theory.- The Baker–Campbell–Hausdorff Formula and its Consequences.- Part II: Semisimple Lie Algebras.- The Representations of sl(3;C).-Semisimple Lie Algebras.- Root Systems.- Representations of Semisimple Lie Algebras.- Further Properties of the Representations.- Part III: Compact lie Groups.- Compact Lie Groups and Maximal Tori.- The Compact Group Approach to Representation Theory.- Fundamental Groups of Compact Lie Groups.- Appendices.

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Book SynopsisThis is an English translation of the now classic "Algbre Locale - Multiplicits" originally published by Springer as LNM 11. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities. Many modifications to the original French text have been made for this English edition, making the text easier to read, without changing its intended informal character.Table of ContentsI. Prime Ideals and Localization.- §1. Notation and definitions.- §2. Nakayama’s lemma.- §3. Localization.- §4. Noetherian rings and modules.- §5. Spectrum.- §6. The noetherian case.- §7. Associated prime ideals.- §8. Primary decompositions.- II. Tools.- A: Filtrations and Gradings.- §1. Filtered rings and modules.- §2. Topology defined by a filtration.- §3. Completion of filtered modules.- §4. Graded rings and modules.- §5. Where everything becomes noetherian again — q -adic filtrations.- B: Hilbert-Samuel Polynomials.- §1. Review on integer-valued polynomials.- §2. Polynomial-like functions.- §3. The Hilbert polynomial.- §4. The Samuel polynomial.- III. Dimension Theory.- A: Dimension of Integral Extensions.- §1. Definitions.- §2. Cohen-Seidenberg first theorem.- §3. Cohen-Seidenberg second theorem.- B: Dimension in Noetherian Rings.- §1. Dimension of a module.- §2. The case of noetherian local rings.- §3. Systems of parameters.- C: Normal Rings.- §1. Characterization of normal rings.- §2. Properties of normal rings.- §3. Integral closure.- D: Polynomial Rings.- §1. Dimension of the ring A[X1,..., Xn].- §2. The normalization lemma.- §3. Applications. I. Dimension in polynomial algebras.- §4. Applications. II. Integral closure of a finitely generated algebra.- §5. Applications. III. Dimension of an intersection in affine space.- IV. Homological Dimension and Depth.- A: The Koszul Complex.- §1. The simple case.- §2. Acyclicity and functorial properties of the Koszul complex.- §3. Filtration of a Koszul complex.- §4. The depth of a module over a noetherian local ring.- B: Cohen-Macaulay Modules.- §1. Definition of Cohen-Macaulay modules.- §2. Several characterizations of Cohen-Macaulay modules.- §3. The support of a Cohen-Macaulay module.- §4. Prime ideals and completion.- C: Homological Dimension and Noetherian Modules.- §1. The homological dimension of a module.- §2. The noetherian case.- §3. The local case.- D: Regular Rings.- §1. Properties and characterizations of regular local rings.- §2. Permanence properties of regular local rings.- §3. Delocalization.- §4. A criterion for normality.- §5. Regularity in ring extensions.- Appendix I: Minimal Resolutions.- §1. Definition of minimal resolutions.- §2. Application.- §3. The case of the Koszul complex.- Appendix II: Positivity of Higher Euler-Poincaré Characteristics.- Appendix III: Graded-polynomial Algebras.- §1. Notation.- §2. Graded-polynomial algebras.- §3. A characterization of graded-polynomial algebras.- §4. Ring extensions.- §5. Application: the Shephard-Todd theorem.- V. Multiplicities.- A: Multiplicity of a Module.- §1. The group of cycles of a ring.- §2. Multiplicity of a module.- B: Intersection Multiplicity of Two Modules.- §1. Reduction to the diagonal.- §2. Completed tensor products.- §3. Regular rings of equal characteristic.- §4. Conjectures.- §5. Regular rings of unequal characteristic (unramified case).- §6. Arbitrary regular rings.- C: Connection with Algebraic Geometry.- §1. Tor-formula.- §2. Cycles on a non-singular affine variety.- §3. Basic formulae.- §4. Proof of theorem 1.- §5. Rationality of intersections.- §6. Direct images.- §7. Pull-backs.- §8. Extensions of intersection theory.- Index of Notation.

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Book SynopsisIntroduction.- Principle of deep cognitive engagement for learning with instructionalvideos.- Principle of conceptual focus for the case of understanding variables andalgebraic expressions.- Research questions and research approaches.- Design of a learning environment with interactive video in the knowledge organization phase.- Qualitative study on how interactive features can scaffold cognitive engagement and conceptual focus.- Randomized controlled trial on how interactive videos enhance understanding of generalizing.- Summary and discussion of the results.

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Book SynopsisEinführung in die mathematische Sprache.- Körper.- Vektorräume und lineare Abbildungen.- Basis und Dimension.- Lineare Abbildungen und Matrizen allgemeiner Fall.- Lineare Gleichungssysteme das Gauß-Verfahren.- Äquivalenzrelationen und Quotientenvektorräume.- Der Polynomring über einem Körper.- Die Determinante.- Eigenwerte.- Skalarprodukte, euklidische und unitäre Vektorräume.- Der Dualraum.- Hauptachsentransformation.

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Book Synopsis1. Zahlen.- 2. Algebra.

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Book Synopsis1. Fundamentals .- 2. Sets and Counting .- 3. Numbers and their Representations .- 4. Relations.- 5. Mappings.- 6. Graphs.- 7. Groupoid, Semigroup, Group.- 8. From Semirings to Fields.- 9. Act, Vector Space, Extension.- 10 Rings and Modules. 11 Matroids.- 12 Categories.- Literature.- Symbols.- Index.

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