Format: Paperback / softback,
56 pages, height x width: 235x155 mm, weight: 157 g,
1 Illustrations, black and white; XIV, 56 p. 1 illus.,
Series: SpringerBriefs in Statistics
Pub. Date: 12-Mar-2022
ISBN-13: 9783030969318
This short book elaborates on selected aspects of stochastic-statistical dependencies in multivariate statistics. Each chapter provides a rigorous and self-contained treatment of one specific topic, poses a particular problem within its scope, and concludes by presenting its solution. The presented problems are not only relevant for research in mathematical statistics, but also entertaining, with elegant proofs and appealing solutions. The chapters cover correlation coefficients of bivariate normal distributions, empirical likelihood ratio tests for the population correlation, the rearrangement algorithm, covariances of order statistics, equi-correlation matrices, skew-normal distributions and the weighted bootstrap. This book is primarily intended for early-career researchers in mathematical statistics, but will also be interesting for lecturers in the field. Its goal is to rouse the readerfs interest, further their knowledge of the subject and provide them with some useful mathematical techniques.
Preface.- General preliminaries.- Correlation coefficients of bivariate normal distributions.- Empirical likelihood ratio tests for the population correlation coefficient.- The rearrangement algorithm.- On the covariances of order statistics.- On equi-correlation matrices.- Skew-normal distributions.- The weighted bootstrap.- index.
Format: Paperback / softback,
430 pages, height x width: 235x155 mm, weight: 676 g,
21 Illustrations, color; 3 Illustrations, black and white; XIII, 430 p. 24 illus.,
21 illus. in color.; 21 Illustrations, color; 3 Illustrations, black and white; XIII, 430 p. 24 illus., 21 illus. in color
Series: Springer Monographs in Mathematics
Pub. Date: 12-Mar-2022
ISBN-13: 9783030690588
This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure.
Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann?Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi?Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface.
Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.
1 Fundamentals.- 2 Basics on Minimal Surfaces.- 3 Approximation and Interpolations Theorems for Minimal Surfaces.- 4 Complete Minimal Surfaces of Finite Total Curvature.- 5 The Gauss Map of a Minimal Surface.- 6 The Riemann-Hilbert Problem for Minimal Surfaces.- 7 The Calabi-Yau Problem for Minimal Surfaces.- 8 Minimal Surfaces in Minimally Convex Domains.- 9 Minimal Hulls, Null Hulls, and Currents.- References.- Index.
Format: Paperback / softback,
109 pages, height x width: 235x155 mm,
20 Illustrations, color; 2 Illustrations, black and white; XI, 109 p. 22 illus., 20 illus. in color.
Series: SpringerBriefs in Mathematics
Pub. Date: 02-May-2022
ISBN-13: 9783030981907
This book aims the optimal design of a material (thermic or electrical) obtained as the mixture of a finite number of original materials, not necessarily isotropic. The problem is to place these materials in such a way that the solution of the corresponding state equation minimizes a certain functional that can depend nonlinearly on the gradient of the state function. This is the main novelty in the book.
It is well known that this type of problems has no solution in general and therefore that it is needed to work with a relaxed formulation. The main results in the book refer to how to obtain such formulation, the optimality conditions, and the numerical computation of the solutions. In the case of functionals that do not depend on the gradient of the state equation, it is known that a relaxed formulation consists of replacing the original materials with more general materials obtained via homogenization. This includes materials with different properties of the originals but whose behavior can be approximated by microscopic mixtures of them. In the case of a cost functional depending nonlinearly on the gradient, it is also necessary to extend the cost functional to the set of these more general materials. In general, we do not dispose of an explicit representation, and then, to numerically solve the problem, it is necessary to design strategies that allow the functional to be replaced by upper or lower approximations.
The book is divided in four chapters. The first is devoted to recalling some classical results related to the homogenization of a sequence of linear elliptic partial differential problems. In the second one, we define the control problem that we are mainly interested in solving in the book. We obtain a relaxed formulation and their main properties, including an explicit representation of the new cost functional, at least in the boundary of its domain. In the third chapter, we study the optimality conditions of the relaxed problem, and we describe some algorithms to numerically solve the problem. We also provide some numerical experiments carried out using such algorithms. Finally, the fourth chapter is devoted to briefly describe some extensions of the results obtained in Chapters 2 and 3 to the case of dealing with several state equations and the case of evolutive problems.
The problems covered in the book are interesting for mathematicians and engineers whose work is related to mathematical modeling and the numerical resolution of optimal design problems in material sciences. The contents extend some previous results obtained by the author in collaboration with other colleagues.
Chapter
1. Homogenization of Elliptic PDE with Varying Coefficients.-
Chapter
2. The Relaxed Formulation of an Optimal Design Problem via Homogenization Theory.
Chapter
3. Optimality Conditions and Numerical Resolution.
Chapter
4. Some Extesions: Multi-State and Evolutive Problems
Format: Hardback,
351 pages, height x width: 235x155 mm, 15 Tables,
color; 9 Illustrations, color; 2 Illustrations, black and white; XIII, 351 p. 11 illus., 9 illus. in color.
Series: Forum for Interdisciplinary Mathematics
Pub. Date: 04-Jun-2022
ISBN-13: 9789811906671
This book collects chapters on fixed-point theory and fractional calculus and their applications in science and engineering. It discusses state-of-the-art developments in these two areas through original new contributions from scientists across the world. It contains several useful tools and techniques to develop their skills and expertise in fixed-point theory and fractional calculus. New research directions are also indicated in chapters. This book is meant for graduate students and researchers willing to expand their knowledge in these areas. The minimum prerequisite for readers is the graduate-level knowledge of analysis, topology and functional analysis.
Best Proximity Points for Some Multivalued Contractive Mappings.- Best Proximity Point Theorems via Some Generalized Notions.- New Fixed-Figure Results on Metric Spaces.- Some Fixed Point Results for Suzuki F-Contractions Involving Quadratic Terms in Modular b-Metric Spaces.- Some Common Fixed Point Results via -Series for a Family of JS-Contraction-type Mappings.- Solution of Nonlinear First-Order Hybrid Integro-Differential Equations via Fixed Point Theorem.- Application of Darbo's Fixed Point Theorem for Existence Result of Generalized 2D Functional Integral Equations.- Results on Generalized Tripled Fuzzy b-Metric Spaces.- A Novel Controlled Picture Fuzzy Metric Space and Some Related Fixed Point Results.- Theoretical Analysis for a Generalized Fractional-Order Boundary Value Problem.- On Well-posed Variational Problems Involving Multidimensional Integral Functionals.- On the Coupled System of Tempered Fractional Differential Equations with Anti-periodic Boundary Conditions.- Application of Measure of Noncompactness on the Infinite System of Hadamard Fractional Iintegral Equations.- Observability, Reachability, Trajectory Reachability and Optimal Reachability of Fractional Dynamical Systems using Riemann-Liouville Fractional Derivative.- Fractional Calculus Approach to Logistic Equation and its Applications.- Hermite-Hadamard Type Inequalities for Coordinated Quasi-Convex Functions via Generalized Fractional Integrals.- Leray-Schauder Theorem for Implicit Fractional Differential Equation and Nonlocal Multi-Point Conditions.- The q-Deformed Hamiltonian, Lagrangian, Entropy and Fisher Information.
Format: Hardback,
240 pages, height x width: 235x155 mm,
3 Illustrations, black and white; X, 240 p. 3 illus.
Series: Springer Monographs in Mathematics
Pub. Date: 08-Jun-2022
ISBN-13: 9783030966652
In the first two chapters we review the theory developped by Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both for Stein algebras and for the algebra of real analytic functions on a C-analytic space. Here we find a relation between real Nullstellensatz and seventeenth Hilbertfs problem for positive semidefinite analytic functions. Namely, a positive answer to Hilbertfs problem implies a solution for the real Nullstellensatz more similar to the one for real polinomials. A chapter is devoted to the state of the art on this problem that is far from a complete answer.
In the last chapter we deal with inequalities. We describe a class of semianalytic sets defined by countably many global real analytic functions that is stable under topological properties and under proper holomorphic maps between Stein spaces, that is, verifies a direct image theorem. A smaller class admits also a decomposition into irreducible components as it happens for semialgebraic sets. During the redaction some proofs have been simplified with respect to the original ones.
Introduction
Chapter
1. The class of C-analytic spaces
Chapter
2. More on analytic sets
Chapter
3. Nullstellensatze
Chapter
4. The 17th Hilbert's Problem for real analytic functions
Chapter
5. Analytic inequalities References
Format: Paperback / softback,
500 pages, height x width: 235x155 mm,
18 Illustrations, black and white; X, 500 p. 18 illus.
Series: Universitext
Pub. Date: 15-May-2022
ISBN-13: 9781447175223
Algebraic Geometry is a fascinating branch of Mathematics that combines methods from both Algebra and Geometry. It transcends the limited scope of pure Algebra by means of geometric construction principles. Putting forward this idea, Grothendieck revolutionized Algebraic Geometry in the late 1950s by inventing schemes. Schemes now also play an important role in Algebraic Number Theory, a field that used to be far away from Geometry. The new point of view paved the way for spectacular progress, such as the proof of Fermat's Last Theorem by Wiles and Taylor.
This book explains the scheme-theoretic approach to Algebraic Geometry for non-experts, while more advanced readers can use it to broaden their view on the subject. A separate part presents the necessary prerequisites from Commutative Algebra, thereby providing an accessible and self-contained introduction to advanced Algebraic Geometry.
Every chapter of the book is preceded by a motivating introduction with an informal discussion of its contents and background. Typical examples, and an abundance of exercises illustrate each section. Therefore the book is an excellent companion for self-studying or for complementing skills that have already been acquired. It can just as well serve as a convenient source for (reading) course material and, in any case, as supplementary literature. The present edition is a critical revision of the earlier text.
Rings and Modules.- The Theory of Noetherian Rings.- Integral Extensions.- Extension of Coefficients and Descent.- Homological Methods: Ext and Tor.- Affine Schemes and Basic Constructions.- Techniques of Global Schemes.- Etale and Smooth Morphisms.- Projective Schemes and Proper Morphisms.