Student Mathematical Library Volume: 94
2021; 217 pp; Softcover
MSC: Primary 15; 26; 49;
Print ISBN: 978-1-4704-6332-8
This book provides an elementary analytically inclined journey to a fundamental result of linear algebra: the Singular Value Decomposition (SVD). SVD is a workhorse in many applications of linear algebra to data science. Four important applications relevant to data science are considered throughout the book: determining the subspace that gbesth approximates a given set (dimension reduction of a data set); finding the gbesth lower rank approximation of a given matrix (compression and general approximation problems); the Moore-Penrose pseudo-inverse (relevant to solving least squares problems); and the orthogonal Procrustes problem (finding the orthogonal transformation that most closely transforms a given collection to a given configuration), as well as its orientation-preserving version.
The point of view throughout is analytic. Readers are assumed to have had a rigorous introduction to sequences and continuity. These are generalized and applied to linear algebraic ideas. Along the way to the SVD, several important results relevant to a wide variety of fields (including random matrices and spectral graph theory) are explored: the Spectral Theorem; minimax characterizations of eigenvalues; and eigenvalue inequalities. By combining analytic and linear algebraic ideas, readers see seemingly disparate areas interacting in beautiful and applicable ways.
MAA Press: An Imprint of the American Mathematical Society
Dolciani Mathematical Expositions Volume: 57
2021; 207 pp; Softcover
MSC: Primary 11; 14;
Print ISBN: 978-1-4704-5622-1
Challenge: Can you find all the integers a, b, c satisfying 2a2+3b2=5c2? Looks simple, and there are in fact a number of easy solutions. But most of them turn out to be anything but obvious! There are infinitely many possibilities, and as any computer will tell you, each of a, b, c will usually be large. So the challenge remains c
Find all integers a, b, c satisfying 2a2+3b2=5c2.
A major advance in number theory means this book can give an easy answer to this and countless similar questions. The idea behind the approach is transforming a degree-two equation in integer variables a, b, c into a plane curve defined by a polynomial. Working with the curve makes obtaining solutions far easier, and the geometric solutions then get translated back into integers. This method morphs hard problems into routine ones and typically requires no more than high school math. (The complete solution to 2a2+3b2=5c2 is included in the book.)
In addition to equations of degree two, the book addresses degree-three equations?a branch of number theory that is today something of a cottage industry, and these problems translate into gelliptic curvesh. This important part of the book includes many pictures along with the exposition, making the material meaningful and easy to grasp.
This book will fit nicely into an introductory course on number theory. In addition, the many solved examples, illustrations, and exercises make self-studying the book an option for students, thus becoming a natural candidate for a capstone course.
Undergraduate and graduate students interested in elliptic curves.
2021; 348 pp; Softcover
MSC: Primary 97;
Print ISBN: 978-1-4704-6377-9
The purpose of this handbook is to help launch institutional transformations in mathematics departments to improve student success. We report findings from the Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL) study. SEMINAL's purpose is to help change agents, those looking to (or currently attempting to) enact change within mathematics departments and beyond?trying to reform the instruction of their lower division mathematics courses in order to promote high achievement for all students. SEMINAL specifically studies the change mechanisms that allow postsecondary institutions to incorporate and sustain active learning in Precalculus to Calculus 2 learning environments. Out of the approximately 2.5 million students enrolled in collegiate mathematics courses each year, over 90% are enrolled in Precalculus to Calculus 2 courses. Forty-four percent of mathematics departments think active learning mathematics strategies are important for Precalculus to Calculus 2 courses, but only 15 percnt state that they are very successful at implementing them. Therefore, insights into the following research question will help with institutional transformations: What conditions, strategies, interventions and actions at the departmental and classroom levels contribute to the initiation, implementation, and institutional sustainability of active learning in the undergraduate calculus sequence (Precalculus to Calculus 2) across varied institutions
Department chairs, deans, and mathematicians wishing to make changes in the teaching of pre-calculus and calculus at a variety of institutions.
MAA Press: An Imprint of the American Mathematical Society
AMS/MAA Textbooks, Volume: 66
2021; 434 pp; Softcover
MSC: Primary 15;
Print ISBN: 978-1-4704-6295-6
Linear Algebra: Gateway to Mathematics uses linear algebra as a vehicle to introduce students to the inner workings of mathematics. The structures and techniques of mathematics in turn provide an accessible framework to illustrate the powerful and beautiful results about vector spaces and linear transformations.
The unifying concepts of linear algebra reveal the analogies among three primary examples: Euclidean spaces, function spaces, and collections of matrices. Students are gently introduced to abstractions of higher mathematics through discussions of the logical structure of proofs, the need to translate terminology into notation, and efficient ways to discover and present proofs. Application of linear algebra and concrete examples tie the abstract concepts to familiar objects from algebra, geometry, calculus, and everyday life.
Students will finish a course using this text with an understanding of the basic results of linear algebra and an appreciation of the beauty and utility of mathematics. They will also be fortified with a degree of mathematical maturity required for subsequent courses in abstract algebra, real analysis, and elementary topology. Students who have prior background in dealing with the mechanical operations of vectors and matrices will benefit from seeing this material placed in a more general context.
Undergraduate students interested in learning linear algebra.
Paperback ISBN: 9780128241547
Imprint: Academic Press
Published Date: 23rd January 2021
Page Count: 502
Provides a historical overview for a family of the special polynomials
Presents a logical investigation of a family of the hypergeometric series
Proposes a new family of the hypergeometric supertrigonometric functions
Presents a new family of the hypergeometric superhyperbolic functions
Readership
Researchers in the fields of mathematics, physics, chemistry and engineering. It can also be used as a textbook for an introductory course on special functions
and applications for senior undergraduate and graduate students in the above- mentioned areas. Research scientists and students in the fields of Applied Mathematics,
Pure Mathematics, Mathematical Analysis, Special Functions
1 Euler Gamma function, Pochhammer symbols and Euler beta function
1.1 The Euler gamma function
1.2 Pochhammer symbols
1.3 Euler beta function
2 Hypergeometric supertrigonometric and superhyperbolic functions via Clausen hypergeometric series
2.1 Definitions, convergences, and properties for the Clausen hypergeometric series
2.2 The hypergeometric supertrigonometric functions of the Clausen hypergeometric series
2.3 The hypergeometric superhyperbolic functions for the Clausen hypergeometric series.
2.4 The hypergeometric supertrigonometric functions with three numerator parameters and two
denominator parameters
2.5 The hypergeometric superhyperbolic functions with three numerator parameters and two denominator parameters
2.6 The analytic number theory involving the Clausen hypergeometric functions
No. of pages: 502
Language: English
Copyright: c Academic Press 2021
Published: 23rd January 2021
Imprint: Academic Press
Paperback ISBN: 9780128241547
Paperback ISBN: 9780128219829
Imprint: Academic Press
Published Date: 20th February 2021
Page Count: 838
The Second Edition of Quantum Information Processing, Quantum Computing, and Quantum Error Correction: An Engineering Approach presents a self-contained introduction to all aspects of the area, teaching the essentials such as state vectors, operators, density operators, measurements, and dynamics of a quantum system. In additional to the fundamental principles of quantum computation, basic quantum gates, basic quantum algorithms, and quantum information processing, this edition has been brought fully up to date, outlining the latest research trends. These include:
A self-contained introduction to quantum information processing, and quantum error correction
Integrates quantum information processing, quantum computing, and quantum error correction
Describes the latest trends in the quantum information processing, quantum error correction and quantum computing
Presents the basic concepts of quantum mechanics
In-depth presentation of the design and realization of a quantum information processing and quantum error correction circuit
Readership
Engineers practicing currently, as well as engineering and science students who wish to learn the basic concepts of quantum computing, quantum information, and quantum error correction.
1. Introduction
2. Quantum Mechanics Fundamentals
3. Quantum Circuits and Modules
4. Quantum Information Processing Fundamentals
5. Quantum Algorithms and Methods
6. Information Theory and Classical Error Correcting Codes
7. Quantum Information Theory
8. Quantum Error Correction
9. Quantum Stabilizer Codes and Beyond
10. Quantum LDPC Codes
11. Fault-Tolerant Quantum Error-Correction and Fault-Tolerant Quantum Computing
12. Cluster State-Based Quantum Computing
13. Physical Implementations of Quantum Information Processing
14. Quantum Machine Learning
15. Quantum Key Distribution
No. of pages: 838
Language: English
Copyright: c Academic Press 2021
Published: 20th February 2021
Imprint: Academic Press
eBook ISBN: 9780128219874
Paperback ISBN: 9780128219829
Paperback ISBN: 9780323854481
Imprint: Academic Press
Published Date: 1st February 2021
Page Count: 380
New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications provides a detailed discussion on the underpinnings of the theory, methods and real-world applications of this numerical scheme. The book's authors explore how this efficient and accurate numerical scheme is useful for solving partial and ordinary differential equations, as well as systems of ordinary and partial differential equations with different types of integral operators. Content coverage includes the foundational layers of polynomial interpretation, Lagrange interpolation, and Newton interpolation, followed by new schemes for fractional calculus. Final sections include six chapters on the application of numerical scheme to a range of real-world applications.
Over the last several decades, many techniques have been suggested to model real-world problems across science, technology and engineering. New analytical methods have been suggested in order to provide exact solutions to real-world problems. Many real-world problems, however, cannot be solved using analytical methods. To handle these problems, researchers need to rely on numerical methods, hence the release of this important resource on the topic at hand.
Offers an overview of the field of numerical analysis and modeling real-world problems
Provides a deeper understanding and comparison of Adams-Bashforth and Newton polynomial numerical methods
Presents applications of local fractional calculus to a range of real-world problems
Explores new scheme for fractal functions and investigates numerical scheme for partial differential equations with integer and non-integer order
Includes codes and examples in MATLAB in all relevant chapters
Readership
Graduate students and researchers in mathematics (pure and applied), engineering, physics, economics
1 Polynomial Interpolation
1.1 Some Interpolation Polynomials
1.1.1 Bernstein Polynomial
1.1.2 The Newton Polynomial Interpolation
1.1.3 Hermite Interpolation
1.1.4 Cubic Polynomial
1.1.5 B-spline Polynomial
1.1.6 Legendre Polynomial
1.1.7 Chebyshev Polynomial
1.1.8 Lagrange-Sylvester interpolation
2 Lagrange Interpolation: Numerical Scheme
2.1 Classical Differential Equation
2.1.1 Numerical Illustrations
2.2 Fractal Differential Equation
2.2.1 Numerical Illustrations
2.3 Differential Equation with Caputo-Fabrizio Operator
2.3.1 Error Analysis with Exponential Kernel
2.3.2 Numerical Illustrations
2.4 Differential Equation with Caputo Fractional Operator
2.4.1 Error Analysis with Power-Law Kernel
2.4.2 Numerical Illustrations
2.5 Differential Equation with Atangana-Baleanu Operator
2.5.1 Error Analysis with Mittag-Leffler Kernel
2.5.2 Numerical Illustrations
2.6 Differential Equation with Fractal-Fractional with Power-Law Kernel
2.6.1 Error Analysis with Caputo Fractal-Fractional Derivative
2.6.2 Numerical Illustrations
2.7 Differential Equation with Fractal-Fractional with Exponential Decay Kernel
2.7.1 Error Analysis with Caputo-Fabrizio Fractal-Fractional Derivative
2.7.2 Numerical Illustrations
2.8 Differential Equation with Fractal-Fractional with Mittag-Leffler Kernel
2.8.1 Error Analysis with Atangana-Baleanu fractal-fractional derivative
2.8.2 Numerical Illustrations
2.9 Differential equation with Fractal-Fractional with Variable Order with Exponential Decay Kernel
2.9.1 Error Analysis with Fractal-Fractional with Variable Order with Exponential Decay Kernel
2.9.2 Numerical Illustrations
2.10 Differential Equation with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel
2.10.1 Error Analysis with Fractal-Fractional with Variable Order with Mittag-Leffler Kernel
2.10.2 Numerical Illustrations
2.11 Differential Equation with Fractal-Fractional with Variable Order with Power-Law Kernel
2.11.1 Error Analysis with Fractal-Fractional with Variable Order with Power-Law Kernel
2.11.2 Numerical Illustrations
No. of pages: 380
Language: English
Copyright: c Academic Press 2021
Published: 1st February 2021
Imprint: Academic Press
Paperback ISBN: 9780323854481
Paperback ISBN: 9780323852388
Imprint: Academic Press
Published Date: 1st September 2021
Page Count: 300
Fractal Vector Analysis: A Local Fractional Calculus Point of View provides an overview of fractal vector calculus, which includes local fractional line integrals, local fractional surface integrals, and local fractional volume integrals. The book presents an overview of key breakthroughs in classical calculus in vector spaces. Readers will gain a deeper understanding of some applications of local fractional calculus from the fractals point of view. Coverage will include double and triple local fractional integrals, as well as elliptic, parabolic and hyperbolic local fractional PDEs.
The potential audience includes, but is not limited to, researchers in the fields of mathematics, physics, and engineering. It could also be used as a textbook for an introductory course on fractal vector calculus and applications, for senior undergraduate and graduate students in the above-mentioned areas.
Provides a deeper understanding of many applications of local fractional calculus from the fractals point of view
Presents a historical overview of local fractional calculus and explores a range of potential applications for real-world problems in science and engineering
Explores a novel optimization method for fractal functions and investigates local fractional Fourier type integral transform
Readership
Graduate students and researchers in mathematics (pure and applied), engineering, physics, economics
Chapter 1 Preliminaries
1.1 Theory of fractional sets
1.2 Real line number system
1.3 Generalized point sets and intervals
1.4 Generalized neighborhood
1.7 Generalized functions
1.8 Fractal orthogonal systems
1.9 Generalized limit of fractal functions
1.10 Local fractional continuity of nondifferentiable functions
1.11 Fractal functions
Chapter 2 Local fractional calculus of one-variable functions defined on fractal sets
2.1 Historical investigation of local fractional calculus
2.2 Local fractional derivative
2.3 Local fractional integral
2.3 Local fractional Taylor type theorem for fractal functions
2.4 Local fractional Mc-Laurin type series for fractal functions
2.5 Local fractional indefinite integral
2.8 Local fractional differential equations
2.9 The extended mean value theorem
2.10 Local fractional improper integrals of first kind
2.11 The solutions for the local fractional linear and nonlinear ODEs
No. of pages: 300
Language: English
Copyright: c Academic Press 2021
Published: 1st September 2021
Imprint: Academic Press
Paperback ISBN: 9780323852388