This book presents a thorough discussion of the theory of abstract inverse linear problems on Hilbert space. Given an unknown vector f in a Hilbert space H, a linear operator A acting on H, and a vector g in H satisfying Af=g, one is interested in approximating f by finite linear combinations of g, Ag, A2g, A3g, ... The closed subspace generated by the latter vectors is called the Krylov subspace of H generated by g and A. The possibility of solving this inverse problem by means of projection methods on the Krylov subspace is the main focus of this text. After giving a broad introduction to the subject, examples and counterexamples of Krylov-solvable and non-solvable inverse problems are provided, together with results on uniqueness of solutions, classes of operators inducing Krylov-solvable inverse problems, and the behaviour of Krylov subspaces under small perturbations. An appendix collects material on weaker convergence phenomena in general projection methods. This subject of this book lies at the boundary of functional analysis/operator theory and numerical analysis/approximation theory and will be of interest to graduate students and researchers in any of these fields.
Introduction and motivation.- Krylov solvability of bounded linear inverse problems.- An analysis of conjugate-gradient based methods with unbounded operators.- Krylov solvability of unbounded inverse problems.- Krylov solvability in a perturbative framework.- Outlook on general projection methods and weaker convergence.- References.- Index.
Format: Hardback, 134 pages, height x width: 235x155 mm, XIV, 134 p.
Pub. Date: 16-Mar-2023
ISBN-13: 9783031235245
This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content of high school geometry. It begins with a set of points, a measure of distance between pairs of points and ten simple axioms. From there the notions of length, area and angle measure, along with congruence and similarity, are carefully defined and their properties proven as theorems. It concludes with a proof of the consistency of the axioms used and a full description of their models. It is provided in guided inquiry (inquiry-based) format with the intention that students will be active learners, proving the theorems and presenting their proofs to their class with the instructor as a mentor and a guide. The book is written for graduate and advanced undergraduate students interested in teaching secondary school mathematics, for pure math majors interested in learning about the foundations of geometry, for faculty preparing future secondary school teachers and as a reference for any professional mathematician. It is written with the hope of anchoring K-12 geometry in solid modern mathematics, thereby fortifying the teaching of secondary and tertiary geometry with a deep understanding of the subject.
Foundational Principles.- Neutral Geometry.- Similar Figures.- Area Measure.- Angle Measure.- Trigonometry.- Circle Measure.- Consistency and Models.- Appendix A: Axioms.- Appendix B: MCL #9.- Appendix C: Font Guide.- Appendix D: Theorem Index.- Appendix E: Notation Index.- Bibliography.- Index.
This volume includes articles spanning several research areas in number theory, such as arithmetic geometry, algebraic number theory, analytic number theory, and applications in cryptography and coding theory. Most of the articles are the results of collaborations started at the 3rd edition of the Women in Numbers Europe (WINE) conference between senior and mid-level faculty, junior faculty, postdocs, and graduate students. The contents of this book should be of interest to graduate students and researchers in number theory.
From modular to adic Langlands correspondences for U(1, 1)(Q2/Q): deformations in the non-supercuspidal case" (A. David).- Explicit connections between supersingular isogeny graphs and Bruhat-Tits trees (L. Amoros, A. Lezzi, K. Lauter, C. Martindale, and J. Sotakova).- Semi-regular sequences and other random systems of equations (E. Gorla).- Reduction types of genus-3 curves in a special stratum of their moduli space (A. Somoza).- Constructions of new matroids and designs over Fq (M. Ceria).- The Complexity of MinRank (E. Gorla).- Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces (C. Salgado).- Integers represented by ternary quadratic forms (D. Schindler).- Construction of Poincare-type series by generating kernels (L. Smajlovic).- The Hasse norm principle in global function fields (R. Newton).- Asymptotics of class numbers for real quadratic fields (N. Raulf).- Some split symbol algebras of prime degree (D. Savin).
Format: Paperback / softback, 219 pages, height x width: 235x155 mm, weight: 361 g, 15 Illustrations, black and white; XII, 219 p. 15 illus
Series: Applied Mathematical Sciences 205
Pub. Date: 05-Feb-2023
ISBN-13: 9783030799137
This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.
Preface.- Notation, Terminology and Some Functional Analysis.- Basics in Optimization.- Continuity of Set-valued Mappings.- Lipschitz Continuity of Polyhedral Mappings.- Metric Regularity.- Lyusternik-Graves Theorem.- Mappings with Convex Graphs.- Derivative Criteria for Metric Regularity.- Strong Regularity.- Variational Inequalities over Polyhedral Sets.- Nonsmooth Inverse Function Theorems.- Lipschitz Stability in Optimization.- Strong Subregularity.- Continuous Selections.- Radius of Regularity.- Newton Method for Generalized Equations.- The Constrained Linear-Quadratic Optimal Control Problem.- Regularity in Nonlinear Control.- Discrete Approximations.- Optimal Feedback Control.- Model Predictive Control.- Bibliographical Remarks and Further Reading.
Format: Paperback / softback, 239 pages, height x width: 235x155 mm, 3 Illustrations, color;
33 Illustrations, black and white; XVIII, 239 p. 36 illus., 3 illus. in color.,
Series: Undergraduate Lecture Notes in Physics
Pub. Date: 03-May-2023
ISBN-13: 9783031242274
Integrals and sums are not generally considered for evaluation using complex integration. This book proposes techniques that mainly use complex integration and are quite different from those in the existing texts. Such techniques, ostensibly taught in Complex Analysis courses to undergraduate students who have had two semesters of calculus, are usually limited to a very small set of problems. Few practitioners consider complex integration as a tool for computing difficult integrals. While there are a number of books on the market that provide tutorials on this subject, the existing texts in this field focus on real methods. Accordingly, this book offers an eye-opening experience for computation enthusiasts used to relying on clever substitutions and transformations to evaluate integrals and sums. The book is the result of nine years of providing solutions to difficult calculus problems on forums such as Math Stack Exchange or the author's website, residuetheorem.com. It serves to detail to the enthusiastic mathematics undergraduate, or the physics or engineering graduate student, the art and science of evaluating difficult integrals, sums, and products.
1. Review of foundational concepts1.1. Sequences and Series 1.1.1. Sequences of Real Numbers and their Series - sequences, limits, series, convergence, harmonic numbers, summation by parts, change in the order of summation 1.1.2. Power Series and Generating Functions - definitions, radius of convergence, generating function representations of sequences, convolution 1.2. Integrals 1.2.1. Riemann Sums - definition, direct evaluation of certain sums 1.2.2. Fundamental Theorem - definition of definite integral, statement of theorem, verifications 1.2.3. Multiple Integrals - double integrals, conditions for reversal or order of integration 1.3. Evaluation Techniques 1.3.1. Integration by Parts - review 1.3.2. Conversion to Multiple Integrals - "Feynman's Technique," replacing a portion of an integrand with an integral representation and reversing the order of integration 1.3.3. Green's Theorem - review, path integrals and parametrization, Stokes' Theorem, applications 1.3.4. Partial Fractions review 1.4. Problems
2. Complex Integration 2.1. Analytic Functions 2.1.1. Cauchy-Riemann Conditions - complex functions and their derivatives, defining analytic functions as a direction-independent derivative, harmonic functions 2.1.2. Evaluating Complex Integrals - numerical examples of parametrizations 2.1.3. Path Independence - demonstrate for analytic functions and demonstrate invalidity for nonanalytic integrands 2.2. Cauchy's Theorems 2.2.1. Winding Numbers - definition in terms of a complex integral 2.2.2. Cauchy's Integral Theorem - derivation and illustration for a wide variety of integrands and contours 2.2.3. Cauchy's Theorem - statement, examples, Liouville's Theorem, Morera's Theorem 2.3. Useful Results 2.3.1. Taylor Series - review, error analysis in complex plane, convergence 2.3.2. Laurent Series - regions of validity (e.g., annuli), analytic continuation 2.3.3. Argument Principle - derivation for zeroes and poles 2.3.4. Rouche's Theorem - derivation, illustration for determining poles within integration contours 2.4. Multivalued Functions - branch points, branch cuts, Riemann surfaces 2.5. Problems
3. Evaluation of Real Integrals and Sums 3.1. Preliminary Matters 3.1.1. Poles and Residue Theory - residue definition, residue computation 3.1.2. Essential Singularities - computation of residues of essential singularities 3.1.3. Branch Points - illustration of a unified approach to expressing an integral of a function in terms of its singularities 3.2. Definite Integrals 3.2.1. Integrands Having Both Poles and Branch Points - e.g., integrands featuring logs and exponents less than -1 3.2.2. Integrands Defined Over - insertion of one higher power of log(z) in the integrand, residue backpropagation 3.2.3. Integrands Having Rational Functions of Polynomials and Trigonometric Functions - integration over the unit circle, modifying the unit circle in the presence of singularities, replacing monomial with a branch point in constructing a contour integral 3.2.4. Alternative Contours: Wedges, Rectangles, and Others - reducing the number of singularities in a contour to simplify calculation 3.2.5. Integrands Having Algebraic Functions and the Residue At Infinity - whole new paradigm in evaluating definite integrals with finite limits of an integrand having branch points at the finite limits, defining the residue at infinity, branch point at infinity 3.3. Sums 3.3.1. Complex Integral Representations - selection of integrand and contour to produce sums, demonstration of convergence of complex integral as contour expands to infinity 3.3.2. Examples - rational summands, summands with trigonometric functions 3.4. Problems
4. Cauchy Principal Value 4.1. Integrands Having Poles On the Contour 4.1.1. Definition of a Cauchy Principal Value - definition as a limit, illustration with simple examples 4.1.2. Managing Divergent Terms of a Contour Integral - detailed illustrations of evaluating definite integrals via complex integrals having contributions with divergent terms that cancel 4.2. Analytic Signals and Hilbert Transforms - equivalence of Cauchy-Riemann equations and Hilbert transforms of real and imaginary parts of an analytic function, illustrations of analytic signals having harmonic real and imaginary parts, examples of deriving imaginary parts of analytic function from real part 4.3. Problems
5. Integral Transforms 5.1. Preliminary Matters 5.1.1. The Dirac Delta Function - derivation via self-transform in Hilbert transform integrals, review of properties 5.1.2. A General Discussion of Integral Transforms - integral transforms require a computable inverse to be of any use, conditions under which inverses exist, general format of integral transforms 5.2. The Fourier Transform 5.2.1. Definition and Plancherel's Theorem - mean square error, and inner product spaces, the Fourier Transform as a Principal Value 5.2.2. Jordan's Lemma - evaluating Fourier integrals using complex integration, convergence conditions 5.2.3. Parseval's Theorem - statement, examples of integral evaluations, Fourier series and application of theorem to sums 5.2.4. Convolution Theorem - statement and derivation, applications 5.2.5. Analyticity of the Fourier Transform In the Complex Plane - theorem relating rates of convergence of Fourier transforms and their inverses in the complex plane, strips of convergence, causality 5.2.6. Poisson Sum Formula - derivation, application to computation of error function to machine precision anywhere in the complex plane 5.3. The Laplace Transform 5.3.1. Definition - extending the discussion of analyticity of the Fourier transform with an exponentially decaying kernel rather than an oscillatory kernel, derivation of inverse as an integral in the complex plane 5.3.2. Convolution Theorem - derivation, examples, application to computing certain classes of definite integrals 5.3.3. Inversion Via Complex Integration 5.3.3.1. Solutions to Ordinary Differential Equations and Rational Transforms - initial conditions, homogeneous and inhomogeneous equations, inversion via the residue theorem 5.3.3.2. Solutions to Partial Differential Equations and Multivalued Transforms - heat equation produces multivalued transforms, evaluation of inverse Laplace transforms to derive solutions 5.4. The Mellin Transform 5.4.1. Definition discussion of strip of convergence, inverse Mellin transform 5.4.2. Convolution Theorem - derivation; NB this will be used in the next chapter 5.4.3. Scaling - expression of scaled integrals in terms of residues 5.5. Problems
6. Asymptotic Analysis 6.1. Definitions 6.1.1. Big-O, Little-O, and The Squiggle - i.e., definitions of asymptotic equivalence in specific limits 6.1.2. Asymptotic Series - definition, properties, numerical calculations, summation acceleration techniques 6.2. Integration by Parts - development of asymptotic series; limitations 6.2.1. Euler-Maclurin Formula - derivation of asymptotic series using integration by parts, application to evaluation of sums 6.3. Watson's Lemma and h-Transforms - asymptotic behavior of monotonic integrands, application of Mellin transforms in derivation 6.3.1. Application to Complex Integration - evaluation of integrals with branch points at infinity using h-transforms 6.4. Laplace's Method - asymptotic behavior of nonmonotomic, nonoscillatory integrals 6.5. The Method of Steepest Descents - deriving asymptotic behavior of complex integrals, derive behavior of real integrals by using Cauchy's Theorem 6.6. Problems