Format: Hardback, 230 pages, height x width: 235x155 mm, 70 Illustrations, color; 5 Illustrations, black and white; XIV, 230 p. 75 illus., 70 illus. in color., 1 Hardback
Series: Interdisciplinary Applied Mathematics 60
Pub. Date: 30-Jul-2024
This monograph takes the reader through recent advances in data-driven methods and machine learning for problems in sciencespecifically in continuum physics. It develops the foundations and details a number of scientific machine learning approaches to enrich current computational models of continuum physics, or to use the data generated by these models to infer more information on these problems. The perspective presented here is drawn from recent research by the author and collaborators. Applications drawn from the physics of materials or from biophysics illustrate each topic. Some elements of the theoretical background in continuum physics that are essential to address these applications are developed first. These chapters focus on nonlinear elasticity and mass transport, with particular attention directed at descriptions of phase separation. This is followed by a brief treatment of the finite element method, since it is the most widely used approach to solve coupled partial differential equations in continuum physics.
With these foundations established, the treatment proceeds to a number of recent developments in data-driven methods and scientific machine learning in the context of the continuum physics of materials and biosystems. This part of the monograph begins by addressing numerical homogenization of microstructural response using feed-forward as well as convolutional neural networks. Next is surrogate optimization using multifidelity learning for problems of phase evolution. Graph theory bears many equivalences to partial differential equations in its properties of representation and avenues for analysis as well as reduced-order descriptions--all ideas that offer fruitful opportunities for exploration. Neural networks, by their capacity for representation of high-dimensional functions, are powerful for scale bridging in physics--an idea on which we present a particular perspective in the context of alloys.
One of the most compelling ideas in scientific machine learning is the identification of governing equations from dynamical data--another topic that we explore from the viewpoint of partial differential equations encoding mechanisms. This is followed by an examination of approaches to replace traditional, discretization-based solvers of partial differential equations with deterministic and probabilistic neural networks that generalize across boundary value problems. The monograph closes with a brief outlook on current emerging ideas in scientific machine learning.
Part I. Introduction and Background in Continuum Materials Physics.- ?Introduction.- Nonlinear Elasticity.- Phase Field Methods.- Part II. Solving Partial Differential Equations.- Finite Element Methods.- Part III. Data-driven Modelling and Scientific Machine Learning.- Reduced Order Models: Numerical Homogenization for the Elastic Response of Material Microstructures.- Surrogate Optimization.- Graph Theoretic Methods.- Scale Bridging.- Inverse Modeling and System Inference from Data.- Machine Learning Solvers of Partial Differential Equations.- An Outlook on Scientific Machine Learning in Continuum Physics.- References.
Format: Hardback, 597 pages, height x width: 235x155 mm, XVI, 594 p. 18 illus., 1 Hardback
Series: Grundlehren der mathematischen Wissenschaften 316
Pub. Date: 09-Oct-2024
ISBN-13: 9783031651328
This is the second edition of an influential monograph on logarithmic potentials with external fields, incorporating some of the numerous advancements made since the initial publication.
As the title implies, the book expands the classical theory of logarithmic potentials to encompass scenarios involving an external field. This external field manifests as a weight function in problems dealing with energy minimization and its associated equilibria. These weighted energies arise in diverse applications such as the study of electrostatics problems, orthogonal polynomials, approximation by polynomials and rational functions, as well as tools for analyzing the asymptotic behavior of eigenvalues for random matrices, all of which are explored in the book. The theory delves into diverse properties of the extremal measure and its logarithmic potentials, paving the way for various numerical methods.
This new, updated edition has been thoroughly revised and is reorganized into three parts, Fundamentals, Applications and Generalizations, followed by the Appendices. Additions to the new edition include
new material on the following topics: analytic and Cﾂｲ weights, differential and integral formulae for equilibrium measures, constrained energy problems, vector equilibrium problems, and a probabilistic approach to balayage and harmonic measures; a new chapter entitled Classical Logarithmic Potential Theory, which conveniently summarizes the main results for logarithmic potentials without external fields; several new proofs and sharpened forms of some main theorems; expanded bibliographic and historical notes with dozens of additional references.
Aimed at researchers and students studying extremal problems and their applications, particularly those arising from minimizing specific integrals in the presence of an external field, this book assumes a firm grasp of fundamental real and complex analysis. It meticulously develops classical logarithmic potential theory alongside the more comprehensive weighted theory.
Part 1 Fundamentals. I Weighted Potentials.- II Recovery of Measures,
Green Functions and Balayage.- III Weighted Polynomials.- IV Determination of
the Extremal Measure.- Part 2 Applications and Generalizations.- V Extremal
Point Methods.- VI Weights on the Real Line.- VII Applications Concerning
Orthogonal Polynomials.- VIII Signed Measures.- IX Some Problems from
Physics.- X Generalizations.- Part 3 Appendices.- A.I Basic Tools.- A.II The
Dirichlet Problem and Harmonic Measures.- A.III Weighted approximation in
.- A.IV Classical Logarithmic Potential Theory.
Format: Hardback, 668 pages, height x width: 235x155 mm, 93 Illustrations, color; 19 Illustrations, black and white; XIII, 668 p. 112 illus., 93 illus. in color., 1 Hardback
Series: Springer Proceedings in Mathematics & Statistics 460
Pub. Date: 13-Jul-2024
ISBN-13: 9783031597619
This book presents the refereed proceedings of the 15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held in Linz, Austria, and organized by the Johannes Kepler University Linz and the Austrian Academy of Sciences, in July 2022. These biennial conferences are major events for Monte Carlo and quasi-Monte Carlo researchers. The proceedings include articles based on invited lectures as well as carefully selected contributed papers on all theoretical aspects and applications of Monte Carlo and quasi-Monte Carlo methods. Offering information on the latest developments in these highly active areas, this book is an excellent reference resource for theoreticians and practitioners interested in solving high-dimensional computational problems, in particular arising in finance, statistics and computer graphics.
Part I Invited Articles: C. A. Beschle, A. Barth, Quasi Continuous Level
Monte Carlo for Random Elliptic PDEs.- M. B. Giles, MLMC Techniques for
Discontinuous Functions.- T. Helin, A. M. Stuart, A. L. Teckentrup, K. C.
Zygalakis, Introduction to Gaussian Process Regression in Bayesian Inverse
Problems, with new Results on Experimental Design for Weighted Error
Measures.- V. Kaarnioja, Frances Y. Kuo, Ian H. Sloan, Lattice-Based Kernel
Approximation and Serendipitous Weights for Parametric PDEs in Very High
Dimensions.- E. Novak, Optimal Algorithms for Numerical Integration: Recent
Results and Open Problems.- Chris. J. Oates, Minimum Kernel Discrepancy
Estimators.- Gabriel Stoltz, Error Estimates and Variance Reduction for
Nonequilibrium Stochastic Dynamics.- Part II Contributed Articles: F. Bernal,
A. Berridi, Heuristics for the Probabilistic Solution of BVPs with Mixed
Boundary Conditions.- Sou-Cheng T. Choi, Y. Ding, Fred J. Hickernell, J.
Rathinavel, Aleksei G. Sorokin, Challenges in Developing Great Quasi-Monte
Carlo Software.- L. Enzi, S. Thonhauser, Numerical Computation of Risk
Functionals in PDMP Risk Models.- J. Fiedler, M. Gnewuch, Christian Weiﾃ・ New
Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices.-
C. Garcﾄｷa-Pareja, F. Nobile, Unbiased Likelihood Estimation of Wright-Fisher
Diffusion Processes.- Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan, A.
Srikumar, Theory and Construction of Quasi-Monte Carlo Rules for Asian Option
Pricing and Density Estimation.- Philipp A. Guth, V. Kaarnioja, Application
of Dimension Truncation Error Analysis to High Dimensional Function
Approximation in Uncertainty Quantification.- Rami El Haddad, C. Lﾃｩcot,
Pierre LEcuyer, Simple Stratified Sampling for Simulating Multi-Dimensional
Markov Chains.- K. Harsha, M. Gnewuch, M. Wnuk, Infinite-Variate
請2-Approximation with Nested Subspace Sampling.- S. Heinrich,
Randomized Complexity of Vector-Valued Approximation.-
A. Keller, C. Wﾃ､chter, N. Binder, Quasi-Monte Carlo Algorithms (not only) for
Graphics Software.- S. Krumscheid, Per Pettersson, Sequential Estimation
using Hierarchically Stratified Domains with Latin Hypercube Sampling.-
Frances Y. Kuo, Weiwen Mo, D. Nuyens, Ian H. Sloan, A. Srikumar, Comparison
of Two Search Criteria for Lattice-based Kernel Approximation.- M. Longo, C.
Schwab, A. Stein, A-posteriori QMC-FEM Error Estimation for Bayesian
Inversion and Optimal Control with Entropic Risk Measure.- E. Lvbak, F.
Blondeel, A. Lee, L. Vanroye, Andreas Van Barel, G. Samaey, Reversible Random
Number Generation for Adjoint Monte Carlo Simulation of the Heat Equation.-
H. Maatouk, D. Rulliﾄe, X. Bay, Large Scale Gaussian Processes with
Matherons Update Rule and Karhunen-Loe Expansion.- A. Mickel, A.
Neuenkirch, The Order Barrier for the -approximation of the
Log-Heston SDE at a Single Point.- D. Nuyens, L. Wilkes, A Randomised Lattice
Rule Algorithm with Pre-determined Generating Vector and Random Number of
Points for Korobov Spaces with 0 < 1/2.- L. Paulin, D.
Coeurjolly, N. Bonneel, Jean-Claude Iehl, V. Ostromoukhov, A. Keller,
Generator Matrices by Solving Integer Linear Programs.- K. Ravi, T. Neckel,
Hans-Joachim Bungartz, Multi-fidelity No-U-Turn Sampling.- C. Reisinger, M.
Olympia Tsianni, Convergence of the EulerMaruyama Particle Scheme for a
Regularised McKeanVlasov Equation Arising from the Calibration of
Local-stochastic Volatility models.- A. G. Sorokin, J. Rathinavel, On
Bounding and Approximating Functions of Multiple Expectations using
Quasi-Monte Carlo.- K. Spendier, M. Szﾃｶlgyenyi, Convergence of the
Tamed-EulerMaruyama Method for SDEs with Discontinuous and Polynomially
Growing Drift.- Vﾄｷctor de la Torre, J. Marzo, QMC Strength for some Random
Configurations on the Sphere.- P. Vanmechelen, G. Lombaert, G. Samaey,
Multilevel MCMC with Level-Dependent Data in a Model Case of Structural
Damage Assessment.- M. Wnuk, A Note on Compact Embeddings of Reproducing
Kernel Hilbert Spaces in 2 and Infinite-variate Function
Approximation.
Format: Hardback, 302 pages, height x width: 235x155 mm, 13 Illustrations, black and white; Approx. 315 p., 1 Hardback
Pub. Date: 30-Sep-2024
ISBN-13: 9783031642166
Incompleteness is a fascinating phenomenon at the intersection of mathematical foundations, computer science, and epistemology that places a limit on what is provable. However, despite its importance, it is often overlooked in the mathematics curricula because it is difficult to teach. This book aims to help bridge this pedagogical gap by providing a complete and accessible technical exposition of incompleteness for a wide audience. The author accomplishes this by making conceptually difficult proofs more approachable by providing intuitive explanations of the main ideas. Care is taken to emphasize the different layers of the mathematical argument the layer within and the metalayer about an axiomatic system.
Structurally, the book efficiently examines key results and arrives at some of the most interesting concepts as quickly as possible. It begins with Gﾃｶdel's incompleteness theorems before continuing on to challenging concepts in the arithmetized completeness theorem, the Paris-Harrington theorem, and the independence of the continuum hypothesis. Other topics covered include the Lucas-Penrose arguments, ordinals and cardinals, and axiomatic set theory. Additionally, the authors coverage of forcing is a notable addition to the existing literature.
Introduction to Incompleteness will be of interest to researchers, students, and instructors looking for a resource to teach this topic. It may also be suitable for self-study. Knowledge of undergraduate-level theoretical mathematics or computer science is required, as well as a familiarity with abstract proofs.
Part 1: Godel's Theorems.- Formal Axiomatic Systems.- Peano Arithmetic
and Recursive Functions.- Godel's Incompleteness Theorems.- Structures,
Models, and Satisfaction.- Completeness and Compactness Theorems.-
Completeness and Peano Arithmetic.- The Lucas-Penrose Arguments.- Part II:
Incompleteness in arithmetic and set theory.- Incompleteness in Finite
Combinatorics.- Consistency of PA and E0 Induction.- Set Theory.-
Independence of CH--forcing.- Independence of CH--forcing CH and -CH.
Format: Hardback, 181 pages, height x width: 235x155 mm, XII, 181 p., 1 Hardback
Series: Coimbra Mathematical Texts 2
Pub. Date: 15-Sep-2024
ISBN-13: 9783031584596
This textbook arises from a masters course taught by the author at the University of Coimbra. It takes the reader from the very classical Galois theorem for fields to its generalization to the case of rings. Given a finite-dimensional Galois extension of fields, the classical bijection between the intermediate field extensions and the subgroups of the corresponding Galois group was extended by Grothendieck as an equivalence between finite-dimensional split algebras and finite sets on which the Galois group acts. Adding further profinite topologies on the Galois group and the sets on which it acts, these two theorems become valid in arbitrary dimension. Taking advantage of the power of category theory, the second part of the book generalizes this most general Galois theorem for fields to the case of commutative rings. This book should be of interest to field theorists and ring theorists wanting to discover new techniques which make it possible to liberate Galois theory from its traditional restricted context of field theory. It should also be of great interest to category theorists who want to apply their everyday techniques to produce deep results in other domains of mathematics.
Historical introduction.- Part I Some Galois theorems for fields.- 1 The
classical Galois theorem.- 2 The Galois theorem of Grothendieck.- 3 Profinite
topological spaces.- 4 The Galois theorems in arbitrary dimension.- Part II
The Galois theory of rings.- 5 Adjunctions and monads.- 6 Profinite groupoids
and presheaves.- 7 The descent theory of rings.- 8 The Pierce spectrum of a
ring.- 9 The Galois theorem for rings.- Further Reading.- Index.
Format: Hardback, 240 pages, height x width: 235x155 mm, 66 Illustrations, color; 11 Illustrations, black and white; X, 240 p. 77 illus., 66 illus. in color., 1 Hardback
Pub. Date: 04-Oct-2024
ISBN-13: 9783031663970
Prof. Pedro A. Morettin is a Distinguished Professor of Statistics at the Institute of Mathematics and Statistics of the University of Sﾄ㎜ Paulo (IME-USP), where he has built an academic career spanning almost six decades. His work has had a significant impact on Time Series Analysis and Wavelet Statistical Methods, as exemplified by the papers appearing in this Festschrift, which are authored by renowned researchers in both fields. Besides his long-term commitment to research, Prof. Morettin is very active in mentoring and serving the profession. Moreover, he has written several textbooks, which are still a leading source of knowledge and learning for undergraduate and graduate students, practitioners, and researchers.
Divided into two parts, the Festschrift presents a collection of papers that illustrate Prof. Morettins broad contributions to Time Series and Econometrics, and to Wavelets. The reader will be able to learn state-of-the-art statistical methodologies, from periodic ARMA models, fractional Brownian motion, and generalized Ornstein-Uhlenbeck processes to spatial models, passing through complex structures designed for high-dimensional data analysis, such as graph and dynamic models. The topics and data features discussed here include high-frequency sampling, fNRIS, forecasting, portfolio apportionment, volatility assessment, dairy production, and inflation, which are relevant to econometrics, medicine, and the food industry. The volume ends with a discussion of several very powerful tools based on wavelets, spectral analysis, dimensionality reduction, self-similarity, scaling, copulas, and other notions.
- Part I Time Series and Econometrics.- Analysis of High-Frequency
Seasonal Time Series.- Stochastic Volatility With Feedback.- Structural
Breaks and Common Factors.- A Note About Calibration Tests for VaR and ES.-
Dynamic Ordering Learning in Multivariate Forecasting.- A Generalization of
the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and
Estimation.- Does the Private Database Help to Explain Brazilian Inflation?.-
Identifiability and Whittle Estimation of Periodic ARMA Models.- Dynamic
Factor Copulas for Minimum-CVaR Portfolio Optimization.- Part II Wavelets.-
Does White Noise Dream of Square Waves?: A Matching Pursuit Conundrum.-
Robust Wavelet-based Assessment of Scaling with Applications.- An Overview of
Spectral Graph Wavelets.- Statistical Inferences on Brain Functional Networks
Using Graph Theory and Multivariate Wavestrapping: An fNIRS Hyperscanning
Illustration.- UtilizingWavelet Transform in the Analysis of Scaling Dynamics
for Milk Quality Evaluation.- Wavelet Estimation of Nonstationary Spatial
Covariance Function.
Format: Hardback, 786 pages, height x width: 235x155 mm, XIV, 794 p., 1 Hardback
Series: CMS/CAIMS Books in Mathematics 13
Pub. Date: 16-Oct-2024
ISBN-13: 9783031636646
This book offers a comprehensive introduction to various aspects of functional analysis and operator algebras.
In Part I, readers will find the foundational material suitable for a one-semester course on functional analysis and linear operators. Additionally, Part I includes enrichment topics that provide flexibility for instructors.
Part II covers the fundamentals of Banach algebras and C*-algebras, followed by more advanced material on C* and von Neumann algebras. This section is suitable for use in graduate courses, with instructors having the option to select specific topics.
Part III explores a range of important topics in operator theory and operator algebras. These include $H^p$ spaces, isometries and Toeplitz operators, nest algebras, dilation theory, applications to various classes of nonself-adjoint operator algebras, and noncommutative convexity and Choquet theory. This material is suitable for graduate courses and learning seminars, offering instructors flexibility in selecting topics.
Part I Functional Analysis.- 1 Set Theory and Topology.- 2 Banach
Spaces.- 3 LCTVSs and Weak Topologies.- 4 Linear Operators.- 5 Compact
Operators.- Part II Banach and C*-algebras.- 6 Banach Algebras.- 7
Commutative Banach Algebras.- 8 Noncommutative Banach Algebras.- 9
C*-Algebras.- 10 Von Neumann Algebras.- Part III Operator Theory.- 11 Hardy
Spaces.- 12 Isometries and Toeplitz Operators.- 13 Nest Algebras.- 14
Dilation Theory.- 15 Nonselfadjoint Operator Algebras.- 16 Noncommutative
Convexity.
Format: Hardback, 333 pages, height x width: 235x155 mm, 50 Illustrations, color; 29 Illustrations, black and white; VI, 294 p. 79 illus., 51 illus. in color., 1 Hardback
Pub. Date: 01-Oct-2024
ISBN-13: 9783031665004
This book delves into the realm of nonparametric estimations, offering insights into essential notions such as probability density, regression, Tsallis Entropy, Residual Tsallis Entropy, and intensity functions.
Through a series of carefully crafted chapters, the theoretical foundations of flexible nonparametric estimators are examined, complemented by comprehensive numerical studies. From theorem elucidation to practical applications, the text provides a deep dive into the intricacies of nonparametric curve estimation.
Tailored for postgraduate students and researchers seeking to expand their understanding of nonparametric statistics, this book will serve as a valuable resource for anyone who wishes to explore the applications of flexible nonparametric techniques.
- Tilted Nonparametric Regression Function Estimation.- Some Asymptotic
Properties of Kernel Density Estimation Under Length-Biased and
Right-Cencored Data.- Functional Data Analysis: Key Concepts and
Applications.- Convolution Process revisited in finite location mixtures and
GARFISMA long memory time series.- Non-parametric Estimation of Tsallis
Entropy and Residual Tsallis Entropy Under -mixing Dependent Data.-
Non-parametric intensity estimation for spatial point patterns with R.- A
Censored Semicontinuous Regression for Modeling Clustered /Longitudinal
Zero-Inflated Rates and Proportions: An Application to Colorectal Cancer.-
Singular Spectrum Analysis.- Hellinger-Bhattacharyya cross-validation for
shape-preserving multivariate wavelet thresholding.- Bayesian nonparametrics
and mixture modelling.- A kernel scale mixture of the skew-normal
distribution.- M-estimation of an intensity function and an underlying
population size under random right truncation.