Format: Paperback / softback, 275 pages, height x width: 235x155 mm, 7 Tables, color;
7 Illustrations, color; 4 Illustrations, black and white; X, 275 p. 11 illus., 7 illus. in color.
Series: C.I.M.E. Foundation Subseries 2332
Pub. Date: 10-Oct-2023
ISBN-13: 9783031378829
This book collects the lecture notes of the Summer School on Convex Geometry, held in Cetraro, Italy, from August 30th to September 3rd, 2021.
Convex geometry is a very active area in mathematics with a solid tradition and a promising future. Its main objects of study are convex bodies, that is, compact and convex subsets of n-dimensional Euclidean space. The so-called Brunn--Minkowski theory currently represents the central part of convex geometry.
The Summer School provided an introduction to various aspects of convex geometry: The theory of valuations, including its recent developments concerning valuations on function spaces; geometric and analytic inequalities, including those which come from the Lp Brunn--Minkowski theory; geometric and analytic notions of duality, along with their interplay with mass transportation and concentration phenomena; symmetrizations, which provide one of the main tools to many variational problems (not only in convex geometry). Each of these parts is represented by one of the courses given during the Summer School and corresponds to one of the chapters of the present volume. The initial chapter contains some basic notions in convex geometry, which form a common background for the subsequent chapters.
The material of this book is essentially self-contained and, like the Summer School, is addressed to PhD and post-doctoral students and to all researchers approaching convex geometry for the first time.
1. Notation and Introductory Material.-
2. Valuations on Convex Bodies and Functions.-
3. Geometric and Functional Inequalities.-
4. Dualities, Measure Concentration and Transportation.-
5. Symmetrizations.
Format: Paperback / softback, 276 pages, height x width: 235x155 mm, X, 276 p.,
Series: Lecture Notes in Mathematics 2333
Pub. Date: 21-Oct-2023
ISBN-13: 9783031379123
This monograph provides some useful tools for performing global geometric analysis on real analytic manifolds. At the core of the methodology of the book is a variety of descriptions for the topologies for the space of real analytic sections of a real analytic vector bundle and for the space of real analytic mappings between real analytic manifolds. Among the various descriptions for these topologies is a development of geometric seminorms for the space of real analytic sections. To illustrate the techniques in the book, a number of fundamental constructions in differential geometry are shown to induce continuous mappings on spaces of real analytic sections and mappings.Aimed at researchers at the level of Doctoral students and above, the book introduces the reader to the challenges and opportunities of real analytic analysis and geometry.
Notation and background.- Topology for spaces of real analytic sections
and mappings.- Geometry: lifts and differentiation of tensors.- Analysis:
norm estimates for derivatives.- Continuity of some standard geometric
operations.
Format: Paperback / softback, 150 pages, height x width: 235x155 mm, X, 150 p
Series: Lecture Notes in Mathematics 2334
Pub. Date: 22-Sep-2023
ISBN-13: 9783031383830
The present book develops the mathematical and numerical analysis of linear, elliptic and parabolic partial differential equations (PDEs) with coefficients whose logarithms are modelled as Gaussian random fields (GRFs), in polygonal and polyhedral physical domains. Both, forward and Bayesian inverse PDE problems subject to GRF priors are considered.
Adopting a pathwise, affine-parametric representation of the GRFs, turns the random PDEs into equivalent, countably-parametric, deterministic PDEs, with nonuniform ellipticity constants. A detailed sparsity analysis of Wiener-Hermite polynomial chaos expansions of the corresponding parametric PDE solution families by analytic continuation into the complex domain is developed, in corner- and edge-weighted function spaces on the physical domain.
The presented Algorithms and results are relevant for the mathematical analysis of many approximation methods for PDEs with GRF inputs, such as model order reduction, neural network and tensor-formatted surrogates of parametric solution families. They are expected to impact computational uncertainty quantification subject to GRF models of uncertainty in PDEs, and are of interest for researchers and graduate students in both, applied and computational mathematics, as well as in computational science and engineering.
Introduction.- Preliminaries.- Elliptic divergence-form PDEs with
log-Gaussian coefficient.- Sparsity for holomorphic functions.- Parametric
posterior analyticity and sparsity in BIPs.- Interpolation and quadrature.-
Multilevel approximation.- Conclusions.
Format: Paperback / softback, 240 pages, height x width: 235x155 mm, 98 Tables, color;
98 Illustrations, color; 22 Illustrations, black and white; X, 240 p. 120 illus., 98 illus. in color.
Series: Lecture Notes in Mathematics 2335
Pub. Date: 02-Oct-2023
ISBN-13: 9783031368530
These Lecture Notes provide an introduction to the study of those discrete surfaces which are obtained by randomly gluing polygons along their sides in a plane. The focus is on the geometry of such random planar maps (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walks, self-avoiding random walks...).A "Markovian" approach is adopted to explore these random discrete surfaces, which is then related to the analogous one-dimensional random walk processes. This technique, known as "peeling exploration" in the literature, can be seen as a generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). It is revealed that different types of Markovian explorations can yield different types of information about a surface.
Based on an Ecole d'Ete de Probabilites de Saint-Flour course delivered by the author in 2019, the book is aimed at PhD students and researchers interested in graph theory, combinatorial probability and geometry. Featuring open problems and a wealth of interesting figures, it is the first book to be published on the theory of random planar maps.
Part I Planar) Maps.-
1. Discrete random surfaces in high genus.-
2. Why
are planar maps exceptional?.-
3. The miraculous enumeration of bipartite
maps.- Part II Peeling explorations.-
4. Peeling of finite Boltzmann maps.-
5. Classification of weight sequences.- Part III Infinite Boltzmann maps.-
6.
Infinite Boltzmann maps of the half-plane.-
7. Infinite Boltzmann maps of the
plane.-
8. Hyperbolic random maps.-
9. Simple boundary, yet a bit more
complicated.-
10. Scaling limit for the peeling process.- Part IV
Percolation(s).-
11. Percolation thresholds in the half-plane.-
12. More on
bond percolation.- Part V Large scale geometry.-
13. Metric growths.-
14. A
taste of scaling limit.- Part VI Simple random walk.-
15. Recurrence,
Transience, Liouville and speed.-
16. Subdiffusivity and pioneer points.
Format: Hardback, 365 pages, height x width: 235x155 mm, 61 Tables, color;
61 Illustrations, color; 8 Illustrations, black and white; X, 365 p. 69 illus., 61 illus. in color.
Series: RSME Springer Series 10
Pub. Date: 08-Nov-2023
ISBN-13: 9783031399152
The aim of this book is to provide an overview of some of the progress made by the Spanish Network of Geometric Analysis (REAG, by its Spanish acronym) since its born in 2007. REAG was created with the objective of enabling the interchange of ideas and the knowledge transfer between several Spanish groups having Geometric Analysis as a common research line. This includes nine groups at Universidad Autonoma de Barcelona, Universidad Autonoma de Madrid, Universidad de Granada, Universidad Jaume I de Castellon, Universidad de Murcia, Universidad de Santiago de Compostela and Universidad de Valencia. The success of REAG has been substantiated with regular meetings and the publication of research papers obtained in collaboration between the members of different nodes. On the occasion of the 15th anniversary of REAG this book aims to collect some old and new contributions of this network to Geometric Analysis. The book consists of thirteen independent chapters, all of them authored by current members of REAG. The topics under study cover geometric flows, constant mean curvature surfaces in Riemannian and sub-Riemannian spaces, integral geometry, potential theory and Riemannian geometry, among others. Some of these chapters have been written in collaboration between members of different nodes of the network, and show the fruitfulness of the common research atmosphere provided by REAG. The rest of the chapters survey a research line or present recent progresses within a group of those forming REAG. Surveying several research lines and offering new directions in the field, the volume is addressed to researchers (including postdocs and PhD students) in Geometric Analysis in the large.
Snapsots of non-local constrained mean curvature type flows.- Spherical
curves whose curvature depends on distance to a great circle.- Conjugate
Plateau constructions in product spaces.- Integral Geometry of pairs of lines
and planes.- Homogeneous hypersurfaces in symmetric spaces.- First Dirichlet
Eigenvalue and Exit Time Moments: A survey.- Area-minimizing horizontal
graphs with low-regularity in the sub-Finsler Heisenberg group H1.- On the
double soul conjecture.- Consequences and extensions of the Brunn-Minkowski
theorem.- An account on links between Finsler and Lorentz Geometries for
Riemannian Geometers.- Geometric and architectural aspects of the singular
minimal surface equation.- Geometry of [ , e3]-minimal surfaces in R3.-
Uniqueness of constant mean curvature spheres.
Format: Hardback, 260 pages, height x width: 240x168 mm, 24 Illustrations, color; 1 Illustrations, black and white; XIV, 260 p. 25 illus., 24 illus. in color.
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 08-Oct-2023
ISBN-13: 9783031389849
This new edition presents the essential topics in probability and statistics from a rigorous standpoint. Any discipline involving randomness, including medicine, engineering, and any area of scientific research, must have a way of analyzing or even predicting the outcomes of an experiment. The authors focus on the tools for doing so in a thorough, yet introductory way. After providing an overview of the basics of probability, the authors cover essential topics such as confidence intervals, hypothesis testing, and linear regression. These subjects are presented in a one semester format, suitable for engineers, scientists, and STEM students with a solid understanding of calculus. There are problems and exercises included in each chapter allowing readers to practice the applications of the concepts.
Probability.- Random Variables.- Distributions of Sample Mean and Sample
SD.- Confidence and Prediction Intervals.- Hypothesis Testing.- Linear
Regression.- Answers to Problems.
Format: Hardback, 362 pages, height x width: 235x155 mm, 75 Illustrations,
color; 17 Illustrations, black and white; XVIII, 362 p. 92 illus., 75 illus. in color.,
Series: Springer Proceedings in Complexity
Pub. Date: 10-Oct-2023
ISBN-13: 9783031374036
This volume of proceedings contains research results within the framework of the fields of Chaos, Fractals and Complexity, written by experienced professors, young researchers, and applied scientists. It includes reviews of the fields, which are presented in an educational way for the widest possible audience, analytical results, computer simulations and experimental evidence, focusing on mathematical modelling.
The papers presented here are selected from lectures given at the 28th Summer School "Dynamical Systems and Complexity", July 18 - 27, 2022. Topics cover applications of complex systems in Neuroscience, Biology, Photonics, Seismology, Meteorology, and more broadly Physical and Engineering systems. The summer school has a long history, which began at the University of Patras in 1987 and continues with great success to this day. The original main purpose was to introduce young students and researchers of Greece to a new science that emerged several decades ago and continues to grow internationally at an ever increasing rate around the world.
Introduction by the Editorial Committee
Chapter
1. Nonlinear Dynamics and Chaos
(a) Review of advances in Hamiltonian Dynamics
(b) Progress in Conservative and Dissipative Systems
(c) Integrable systems, Solitons and Wave Propagation
(d) Applications in Physics and Engineering
Chapter
2. Fractal Geometry and Applications
(a) Review of advances in Fractal Analysis
(b) Progress in the study of fractal basins of attraction
(c) Applications of fractal geometry in Biology
(d) Fractal analysis of paintings
Chapter
3. Complexity Science and Applications
(a) Understanding Complex Systems
(b) Methodologies on the analysis of Complex Systems
(c) Progress in Complex Networks
(d) Multiscale Evolutionary Processes
(e) Complexity in Geosciences
Chapter
4. Neuroscience and Photonics
(a) Neuronal synchronization and brain dynamics
(b) Chimera states in biological networks
(c) Electrophysiology, imaging and machine learning
(d) Laser synchronization in non-Hermitian photonics
(e) Phase locking and waveform generation in coupled lasers
Chapter 5: Nonlinear Time Series Analysis
(a) Correlation and causality networks from time series
(b) Data analysis, dynamics and prediction in seismic time series
(c) Dynamics and prediction in Economics and Finance
(d) Data analysis and prediction in Meteorology