Topological Complexity of Smooth Random Functions

These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors’ 2007 Springer monograph “Random Fields and Geometry.” While not as exhaustive as the full monograp

Now in a paperback edition for the first time, this second edition of Some Random Series of Functions covers random series in Banach and Hilbert spaces, random Taylor or Fourier series, Brownian motion and other Gaussian processes, plus certain types of random sets and measures. The subject matter of this book is important and has wide application in mathematics, statistics, engineering, and physics. Professor Kahane's presentation is suitable even for beginning graduate students in probability and analysis (exercises are provided throughout), as well as non-specialists in the other disciplines to which this subject has application.

This book brings together into a general setting various techniques in the study of the topological properties of spaces of continuous functions. The two major classes of function space topologies stu

Partition functions arise in combinatorics and related problems of statistical physics as they encode in a succinct way the combinatorial structure of complicated systems. The main focus of the

This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.

Approximately 1,000 problems ? with answers and solutions included at the back of the book ? illustrate such topics as random events, random variables, limit theorems, Markov processes, and much more.

The book is based on the PhD thesis “Descriptive Set Theoretic Methods in Automata Theory,” awarded the E.W. Beth Prize in 2015 for outstanding dissertations in the fields of logic, language, and info

Topological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several

The earlier chapter of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavour and its linear algebra content. In this second edition Dr Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2. A number of corrections and improvements have also been made. There are many

Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.

This is an exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is pla

This text bridges the gap existing in the field of set theoretical topology between the introductory texts and the more specialised monographs. The authors review fit developments in general topology and discuss important new areas of research and the importance of defining a methodology applicable to this active field of mathematics. The concept of normal cover and related ideas is considered in detail, as are the characterisations of normal spaces, collectionwise normal spaces and their interrelationships with paracompact spaces (and other weaker forms of compactness). Various methods of embedding subspaces are studied, before considering newer concepts such as M-spaces and their relationships with established ideas. These ideas are applied to give new results pertaining to the extension of continuous vector-valued functions. Wallman–Frink compactifications and realcompactifications are also studied to assist in unifying the ideas through the use of the more general L-filter.

The main subject of this introductory book is simple random walk on the integer lattice, with special attention to the two-dimensional case. This fascinating mathematical object is the point of departure for an intuitive and richly illustrated tour of related topics at the active edge of research. It starts with three different proofs of the recurrence of the two-dimensional walk, via direct combinatorial arguments, electrical networks, and Lyapunov functions. After reviewing some relevant potential-theoretic tools, the reader is guided toward the relatively new topic of random interlacements - which can be viewed as a 'canonical soup' of nearest-neighbour loops through infinity - again with emphasis on two dimensions. On the way, readers will visit conditioned simple random walks - which are the 'noodles' in the soup - and also discover how Poisson processes of infinite objects are constructed and review the recently introduced method of soft local times. Each chapter ends with many e

This volume collects the proceedings of the conference 'Topological methods in group theory', held at Ohio State University in 2014 in honor of Ross Geoghegan's 70th birthday. It consists of eleven peer-reviewed papers on some of the most recent developments at the interface of topology and geometric group theory. The authors have given particular attention to clear exposition, making this volume especially useful for graduate students and for mathematicians in other areas interested in gaining a taste of this rich and active field. A wide cross-section of topics in geometric group theory and topology are represented, including left-orderable groups, groups defined by automata, connectivity properties and Σ-invariants of groups, amenability and non-amenability problems, and boundaries of certain groups. Also included are topics that are more geometric or topological in nature, such as the geometry of simplices, decomposition complexity of certain groups, and problems in shape theory.

This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science. Certain functions are highly sensitive to noise; this can be seen via Fourier analysis on the hypercube. The key model analyzed in depth is critical percolation on the hexagonal lattice. For this model, the critical exponents, previously determined using the now-famous Schramm–Loewner evolution, appear here in the study of sensitivity behavior. Even for this relatively simple model, beyond the Fourier-analytic set-up, there are three crucially important but distinct approaches: hypercontractivity of operators, connections to randomized algorithms, and viewing the spectrum as a random Cantor set. This book assumes a basic background in probability theory and integration theory. Each chapter ends with exercises, some straightforward, some challenging.

In four volumes, the "Principles" develop the theory of random functions, the main tool in statistical radiophysics and a variety of other branches of physics and engineering. The course addresses stu

Principles of Statistical Radiophysics is a four-volume series that introduces the newcomer to the theory of random functions. It aims at providing the background necessary to understand papers and mo

Principles of Statistical Radiophysics is a four-volume series that introduces the newcomer to the theory of random functions. It aims at providing the background necessary to understand papers and mo

Random matrix theory is an area of mathematics first developed by physicists interested in the energy levels of atomic nuclei, but it can also be used to describe some exotic phenomena in the number theory of elliptic curves. The purpose of this book is to illustrate this interplay of number theory and random matrices. It begins with an introduction to elliptic curves and the fundamentals of modelling by a family of random matrices, and moves on to highlight the latest research. There are expositions of current research on ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. Most of the material here had its origin in a Clay Mathematics Institute workshop on this topic at the Newton Institute in Cambridge and together these contributions provide a unique in-depth treatment of the subject.

This tract develops the purely mathematical side of the theory of probability, without reference to any applications. When originally published, it was one of the earliest works in the field built on the axiomatic foundations introduced by A. Kolmogoroff in his book Grundbegriffe der Wahrscheinlichkeitsrechnung, thus treating the subject as a branch of the theory of completely additive set functions. The author restricts himself to a consideration of probability distributions in spaces of a finite number of dimensions, and to problems connected with the Central Limit Theorem and some of its generalizations and modifications. In this edition the chapter on Liapounoff's theorem has been partly rewritten, and now includes a proof of the important inequality due to Berry and Esseen. The terminology has been modernized, and several minor changes have been made.

This is the second volume of "A Course in Analysis" and it is devoted to the study of mappings between subsets of Euclidean spaces. The metric, hence the topological structure is discussed as well as

The theory of random graphs is a vital part of the education of any researcher entering the fascinating world of combinatorics. However, due to their diverse nature, the geometric and structural aspects of the theory often remain an obscure part of the formative study of young combinatorialists and probabilists. Moreover, the theory itself, even in its most basic forms, is often considered too advanced to be part of undergraduate curricula, and those who are interested usually learn it mostly through self-study, covering a lot of its fundamentals but little of the more recent developments. This book provides a self-contained and concise introduction to recent developments and techniques for classical problems in the theory of random graphs. Moreover, it covers geometric and topological aspects of the theory and introduces the reader to the diversity and depth of the methods that have been devised in this context.

Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theo

In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negat

This book presents a model of computing and a measure of computational complexity which are intended to facilitate analysis of computations performed by people, machines, or a mixed system of people and machines. The model is designed to apply directly to models of economic theory, which typically involve continuous variables and smooth functions, without requiring analysis of approximations. The model permits analysis of the feasibility and complexity of the calculations required of economic agents in order for them to arrive at their decisions. The treatment contains applications of the model to game theory and economics, including comparison of the complexities of different solution concepts in certain bargaining games, and the trade-off between communication and computation in an example of an Edgeworth Box economy.

These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics. Apart from the theory of computational complexity concerned with enumeration problems, introductions are given to several of the topics treated, such as combinatorial knot theory, randomised approximation algorithms, percolation and random cluster models. To researchers in discrete mathematics, computer science and statistical physics, this book will be of great interest, but any non-expert should find it an appealing guide to a very active area of research.

一舉打入布克獎決選，斯特勞特再次藉主角露西巴頓，刻畫生命裡的幽微孤寂與失落。NEW YORK TIMES BESTSELLER - Pulitzer Prize winner Elizabeth Strout explores the mysteries of marriage and the secrets we keep, as a former couple reckons with where they've come from--and what they've left behind. ONE OF THE TEN BEST BOOKS OF THE YEAR: Maureen Corrigan, NPR's Fresh Air - ONE OF THE BEST BOOKS OF THE YEAR: The New York Times Book Review, The Washington Post, Time, Vulture, She Reads "Elizabeth Strout is one of my very favorite writers, so the fact that Oh William! may well be my favorite of her books is a mathematical equation for joy. The depth, complexity, and love contained in these pages is a miraculous achievement."--Ann Patchett, author of The Dutch House I would like to say a few things about my first husband, William. Lucy Barton is a writer, but her ex-husband, William, remains a hard man to read. William, she confesses, has always been a mystery to me. Another mystery is why the two have remained connected after all these

Tiến loves his family and his friends…but Tiến has a secret he’s been keeping from them, and it might change everything. An amazing YA graphic novel that deals with the complexity of family and how st

If you knew your future, would you try to fight fate?Aside from a delay, there will be no problems. The flight will be smooth, it will land safely. Everyone who gets on the plane will get off. But almost all of them will be forever changed.Because on this ordinary, short, domestic flight, something extraordinary happens. People learn how and when they are going to die. For some, their death is far in the future—age 103!—and they laugh. But for six passengers, their predicted deaths are not far away at all.How do they know this? There were ostensibly more interesting people on the flight (the bride and groom, the jittery, possibly famous woman, the giant Hemsworth-esque guy who looks like an off-duty superhero, the frazzled, gorgeous flight attendant) but none would become as famous as “The Death Lady.”Not a single passenger or crew member will later recall noticing her board the plane. She wasn’t exceptionally old or young, rude or polite. She wasn’t drunk or nervous or pregnant. Her a

Rediscover the Puffin Classics collection and bring the best-loved classics to a new generation - including this bold edition of The Adventures of Huckleberry Finn.Huckleberry Finn's childhood wasn't exactly smooth sailing but everything changed when he met Tom Sawyer. Bound by an oath to make trouble, the Gang embark on a daring adventure down the Mississippi river, every turn . . .until they meet Jim.Will the Gang work together to help him?

This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illu

A young Indian woman finds the false rumors that she killed her husband surprisingly useful―until other women in the village start asking for her help getting rid of their own husbands―in this razor-sharp debut.“Shroff captures the complexity of female friendship with acuity, wit, and a certain kind of magic irreverence…The Bandit Queens is tender, unpredictable, and brimming with laugh-out-loud moments.” –Téa Obreht, New York Times bestselling author of The Tiger’s WifeGeeta’s no-good husband disappeared five years ago. She didn’t kill him, but everyone thinks she did―no matter how much she protests. But she soon discovers that being known as a “self-made” widow has some surprising perks. No one messes with her, no one threatens her, and no one tries to control (ahem, marry) her. It’s even been good for her business; no one wants to risk getting on her bad side by not buying her jewelry. Freedom must look good on Geeta, because other women in the village have started asking for her he

An epic sci-fi graphic novel romance between two non-binary characters as they find one another through time, distance, and war. An amazing story that explores the complexity of human nature and what brings us together. When they were kids, Fassen's fighter spaceship crash-landed on a planet that Lu's survey force was exploring. It was a forbidden meeting between a kid from a war-focused resistance movement and a kid whose community and planet are dedicated to peace and secrecy.Lu and Fassen are from different worlds and separate solar systems. But their friendship keeps them in each other's orbit as they grow up. They stay in contact in secret as their communities are increasingly threatened by the omnipresent, ever-expanding empire.As the empire begins a new attack against Fassen's people--and discovers Lu's in the process--the two of them have the chance to reunite at last. They finally are able to be together...but at what cost? This beautifully illustrated graphic novel is an epic

Starting middle school is hard enough when you don’t know anyone; it’s even harder when you’re shy. A contemporary middle-grade graphic novel for fans of Guts and Real Friends about how dealing with anxiety and OCD can affect everyday life.As long as Maggie rolls the right number, nothing can go wrong…right?Maggie just wants to get through her first year of middle school. But between finding the best after-school clubs, trying to make friends, and avoiding the rumored monster on school grounds, she’s having a tough time…so she might need a little help from her twenty-sided dice. But what happens if Maggie rolls the wrong number?A touching middle-grade graphic novel that explores the complexity of anxiety, OCD, and learning to trust yourself and the world around you.“A charming, compassionate story that’s sure to resonate with anyone who’s ever stayed up worrying.” ―Gale Galligan, adaptor and illustrator of the Baby-Sitters Club graphic novel series

The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to

Exploring the origins and evolution of magnetic fields in planets, stars and galaxies, this book gives a basic introduction to magnetohydrodynamics and surveys the observational data, with particular focus on geomagnetism and solar magnetism. Pioneering laboratory experiments that seek to replicate particular aspects of fluid dynamo action are also described. The authors provide a complete treatment of laminar dynamo theory, and of the mean-field electrodynamics that incorporates the effects of random waves and turbulence. Both dynamo theory and its counterpart, the theory of magnetic relaxation, are covered. Topological constraints associated with conservation of magnetic helicity are thoroughly explored and major challenges are addressed in areas such as fast-dynamo theory, accretion-disc dynamo theory and the theory of magnetostrophic turbulence. The book is aimed at graduate-level students in mathematics, physics, Earth sciences and astrophysics, and will be a valuable resource for