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Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein’s relation). The main aim of this book is to reveal similar connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies the heat diffusion at this general level and discusses the following topics: The multiplicative Einstein relation, Isoperimetric inequalities, Heat kernel estimates Elliptic and parabolic Harnack inequality.
The main aim of this book is to reveal connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies heat diffusion at this general level and discusses the multiplicative Einstein relation; Isoperimetric inequalities; and Heat kernel estimates; Elliptic and parabolic Harnack inequality.
Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein's relation). The main aim of this book is to reveal similar connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies the heat diffusion at this general level and discusses the following topics: The multiplicative Einstein relation; Isoperimetric inequalities; Heat kernel estimates and Elliptic and parabolic Harnack inequality.
First Steps in Random Walks by J. Klafter,I. M. Sokolov Pdf
The name "random walk" for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of "Nature". The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier such a problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays the theory of random walks has proved useful in physics and chemistry (diffusion, reactions, mixing flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub- and super-diffusive transport processes as well. This book discusses the main variants of random walks and gives the most important mathematical tools for their theoretical description.
Random Walk in Random and Non-random Environments by P l Rvsz Pdf
The simplest mathematical model of the Brownian motion of physics is the simple, symmetric random walk. This book collects and compares current results OCo mostly strong theorems which describe the properties of a random walk. The modern problems of the limit theorems of probability theory are treated in the simple case of coin tossing. Taking advantage of this simplicity, the reader is familiarized with limit theorems (especially strong ones) without the burden of technical tools and difficulties. An easy way of considering the Wiener process is also given, through the study of the random walk.Since the first and second editions were published in 1990 and 2005, a number of new results have appeared in the literature. The first two editions contained many unsolved problems and conjectures which have since been settled; this third, revised and enlarged edition includes those new results. In this edition, a completely new part is included concerning Simple Random Walks on Graphs. Properties of random walks on several concrete graphs have been studied in the last decade. Some of the obtained results are also presented.
Providing an elementary introduction to branching random walks, the main focus of these lecture notes is on the asymptotic properties of one-dimensional discrete-time supercritical branching random walks, and in particular, on extreme positions in each generation, as well as the evolution of these positions over time. Starting with the simple case of Galton-Watson trees, the text primarily concentrates on exploiting, in various contexts, the spinal structure of branching random walks. The notes end with some applications to biased random walks on trees.
Computer Vision -- ECCV 2010 by Kostas Daniilidis,Petros Maragos,Nikos Paragios Pdf
The 2010 edition of the European Conference on Computer Vision was held in Heraklion, Crete. The call for papers attracted an absolute record of 1,174 submissions. We describe here the selection of the accepted papers: Thirty-eight area chairs were selected coming from Europe (18), USA and Canada (16), and Asia (4). Their selection was based on the following criteria: (1) Researchers who had served at least two times as Area Chairs within the past two years at major vision conferences were excluded; (2) Researchers who served as Area Chairs at the 2010 Computer Vision and Pattern Recognition were also excluded (exception: ECCV 2012 Program Chairs); (3) Minimization of overlap introduced by Area Chairs being former student and advisors; (4) 20% of the Area Chairs had never served before in a major conference; (5) The Area Chair selection process made all possible efforts to achieve a reasonable geographic distribution between countries, thematic areas and trends in computer vision. Each Area Chair was assigned by the Program Chairs between 28–32 papers. Based on paper content, the Area Chair recommended up to seven potential reviewers per paper. Such assignment was made using all reviewers in the database including the conflicting ones. The Program Chairs manually entered the missing conflict domains of approximately 300 reviewers. Based on the recommendation of the Area Chairs, three reviewers were selected per paper (with at least one being of the top three suggestions), with 99.
A Non-Random Walk Down Wall Street by Andrew W. Lo,A. Craig MacKinlay Pdf
For over half a century, financial experts have regarded the movements of markets as a random walk--unpredictable meanderings akin to a drunkard's unsteady gait--and this hypothesis has become a cornerstone of modern financial economics and many investment strategies. Here Andrew W. Lo and A. Craig MacKinlay put the Random Walk Hypothesis to the test. In this volume, which elegantly integrates their most important articles, Lo and MacKinlay find that markets are not completely random after all, and that predictable components do exist in recent stock and bond returns. Their book provides a state-of-the-art account of the techniques for detecting predictabilities and evaluating their statistical and economic significance, and offers a tantalizing glimpse into the financial technologies of the future. The articles track the exciting course of Lo and MacKinlay's research on the predictability of stock prices from their early work on rejecting random walks in short-horizon returns to their analysis of long-term memory in stock market prices. A particular highlight is their now-famous inquiry into the pitfalls of "data-snooping biases" that have arisen from the widespread use of the same historical databases for discovering anomalies and developing seemingly profitable investment strategies. This book invites scholars to reconsider the Random Walk Hypothesis, and, by carefully documenting the presence of predictable components in the stock market, also directs investment professionals toward superior long-term investment returns through disciplined active investment management.
Harness the power of randomness (and Python code) to solve real-world problems in fun, hands-on experiments—from simulating evolution to encrypting messages to making machine-learning algorithms! The Art of Randomness is a hands-on guide to mastering the many ways you can use randomized algorithms to solve real programming and scientific problems. You’ll learn how to use randomness to run simulations, hide information, design experiments, and even create art and music. All you need is some Python, basic high school math, and a roll of the dice. Author Ronald T. Kneusel focuses on helping you build your intuition so that you’ll know when and how to use random processes to get things done. You’ll develop a randomness engine (a Python class that supplies random values from your chosen source), then explore how to leverage randomness to: Simulate Darwinian evolution and optimize with swarm-based search algorithms Design scientific experiments to produce more meaningful results by making them truly random Implement machine learning algorithms like neural networks and random forests Use Markov Chain Monte Carlo methods to sample from complex distributions Hide information in audio files and images, generate art, and create music Reconstruct original signals and images from only randomly sampled data Scientific anecdotes and code examples throughout illustrate how randomness plays into areas like optimization, machine learning, and audio signals. End-of-chapter exercises encourage further exploration. Whether you’re a programmer, scientist, engineer, mathematician, or artist, you’ll find The Art of Randomness to be your ticket to discovering the hidden power of applied randomness and the ways it can transform your approach to solving problems, from the technical to the artistic.
Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices by Göran Högnäs,Arunava Mukherjea Pdf
A Scientific American article on chaos, see Crutchfield et al. (1986), illus trates a very persuasive example of recurrence. A painting of Henri Poincare, or rather a digitized version of it, is stretched and cut to produce a mildly distorted image of Poincare. The same procedure is applied to the distorted image and the process is repeated over and over again on the successively more and more blurred images. After a dozen repetitions nothing seems to be left of the original portrait. Miraculously, structured images appear briefly as we continue to apply the distortion procedure to successive images. After 241 iterations the original picture reappears, unchanged! Apparently the pixels of the Poincare portrait were moving about in accor dance with a strictly deterministic rule. More importantly, the set of all pixels, the whole portrait, was transformed by the distortion mechanism. In this exam ple the transformation seems to have been a reversible one since the original was faithfully recreated. It is not very farfetched to introduce a certain amount of randomness and irreversibility in the above example. Think of a random miscoloring of some pixels or of inadvertently giving a pixel the color of its neighbor. The methods in this book are geared towards being applicable to the asymp totics of such transformation processes. The transformations form a semigroup in a natural way; we want to investigate the long-term behavior of random elements of this semigroup.
A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing (Ninth Edition) by Burton G. Malkiel Pdf
Updated with a new chapter that draws on behavioral finance, the field that studies the psychology of investment decisions, the bestselling guide to investing evaluates the full range of financial opportunities.
Learning for a Better Future by Suzanne Hattingh Pdf
Various international scholars and associates of the PASCAL (Place, Social Capital and Learning Regions) International Observatory (Africa hub), under the auspices of the Centre for Local Economic Development (CENLED) based at the University of Johannesburg (UJ), have contributed chapters in this scholarly book. The book aims to demonstrate how a combination of globalisation, pandemics and the impact of innovation and technologies are driving towards a world in which traditional ideas are being challenged. The book carries forward a dual context and relevance: to South African social, educational, economic and cultural development, and the broader international context and action directed at how lifelong learning for all can be fostered in communities as a foundation for a just, human-centred, sustainable world. The distinctive contribution of this book to the production of a local body of knowledge lies in the symbiotic relationships between these objectives, so that South Africa could serve as a test case in working towards approaches that have a wider international significance.
