|
Showing 1 - 5 of
5 matches in All Departments
This book is an introduction to the modern theory of Markov chains,
whose goal is to determine the rate of convergence to the
stationary distribution, as a function of state space size and
geometry. This topic has important connections to combinatorics,
statistical physics, and theoretical computer science. Many of the
techniques presented originate in these disciplines. The central
tools for estimating convergence times, including coupling, strong
stationary times, and spectral methods, are developed. The authors
discuss many examples, including card shuffling and the Ising
model, from statistical mechanics, and present the connection of
random walks to electrical networks and apply it to estimate
hitting and cover times. The first edition has been used in courses
in mathematics and computer science departments of numerous
universities. The second edition features three new chapters (on
monotone chains, the exclusion process, and stationary times) and
also includes smaller additions and corrections throughout. Updated
notes at the end of each chapter inform the reader of recent
research developments.
Starting around the late 1950s, several research communities began
relating the geometry of graphs to stochastic processes on these
graphs. This book, twenty years in the making, ties together
research in the field, encompassing work on percolation,
isoperimetric inequalities, eigenvalues, transition probabilities,
and random walks. Written by two leading researchers, the text
emphasizes intuition, while giving complete proofs and more than
850 exercises. Many recent developments, in which the authors have
played a leading role, are discussed, including percolation on
trees and Cayley graphs, uniform spanning forests, the
mass-transport technique, and connections on random walks on graphs
to embedding in Hilbert space. This state-of-the-art account of
probability on networks will be indispensable for graduate students
and researchers alike.
This is a mathematically rigorous introduction to fractals which
emphasizes examples and fundamental ideas. Building up from basic
techniques of geometric measure theory and probability, central
topics such as Hausdorff dimension, self-similar sets and Brownian
motion are introduced, as are more specialized topics, including
Kakeya sets, capacity, percolation on trees and the traveling
salesman theorem. The broad range of techniques presented enables
key ideas to be highlighted, without the distraction of excessive
technicalities. The authors incorporate some novel proofs which are
simpler than those available elsewhere. Where possible, chapters
are designed to be read independently so the book can be used to
teach a variety of courses, with the clear structure offering
students an accessible route into the topic.
We live in a highly connected world with multiple self-interested
agents interacting and myriad opportunities for conflict and
cooperation. The goal of game theory is to understand these
opportunities. This book presents a rigorous introduction to the
mathematics of game theory without losing sight of the joy of the
subject. This is done by focusing on theoretical highlights (e.g.,
at least six Nobel Prize winning results are developed from
scratch) and by presenting exciting connections of game theory to
other fields such as computer science (algorithmic game theory),
economics (auctions and matching markets), social choice (voting
theory), biology (signaling and evolutionary stability), and
learning theory. Both classical topics, such as zero-sum games, and
modern topics, such as sponsored search auctions, are covered.
Along the way, beautiful mathematical tools used in game theory are
introduced, including convexity, fixed-point theorems, and
probabilistic arguments. The book is appropriate for a first course
in game theory at either the undergraduate or graduate level,
whether in mathematics, economics, computer science, engineering,
or statistics. The importance of game-theoretic thinking transcends
the academic setting-for every action we take, we must consider not
only its direct effects, but also how it influences the incentives
of others.
This eagerly awaited textbook covers everything the graduate
student in probability wants to know about Brownian motion, as well
as the latest research in the area. Starting with the construction
of Brownian motion, the book then proceeds to sample path
properties like continuity and nowhere differentiability. Notions
of fractal dimension are introduced early and are used throughout
the book to describe fine properties of Brownian paths. The
relation of Brownian motion and random walk is explored from
several viewpoints, including a development of the theory of
Brownian local times from random walk embeddings. Stochastic
integration is introduced as a tool and an accessible treatment of
the potential theory of Brownian motion clears the path for an
extensive treatment of intersections of Brownian paths. An
investigation of exceptional points on the Brownian path and an
appendix on SLE processes, by Oded Schramm and Wendelin Werner,
lead directly to recent research themes.
|
You may like...
Sermons
Frederick William Robertson
Paperback
R534
Discovery Miles 5 340
|