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Author: Eduardo M.R.A. Engel
Publisher: Springer Science & Business Media
ISBN: 1441986847
Category : Mathematics
Languages : en
Pages : 166
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Book Description
There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as "the method of arbitrary funetionJ. " It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented.
Author: Eduardo M.R.A. Engel
Publisher: Springer Science & Business Media
ISBN: 1441986847
Category : Mathematics
Languages : en
Pages : 166
Get Book Here
Book Description
There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as "the method of arbitrary funetionJ. " It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented.
Author: Eduardo Engel
Publisher:
ISBN: 9783540977407
Category : Convergence
Languages : en
Pages : 0
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Book Description
There are many ways of introducing the concept of probability in classical, deterministic physics. This volume is concerned with one approach, known as 'the method of arbitrary functions', which was first considered by Poincare. Essentially, the method proceeds by associating some uncertainty to our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. By modeling this uncertainty by a probability density distribution, it is then possible to analyze how the state of the system evolves through time. This approach may be applied to a wide variety of classical problems and the author considers here examples as diverse as bouncing balls, simple and coupled harmonic oscillators, integrable systems (such as spinning tops), planetary motion, and billiards. An important aspect of this account is to study the speed of convergence for solutions in order to determine the practical relevance of the method of arbitrary functions for specific examples. Consequently, both new results on convergence, and tractable upper bounds are derived and applied.
Author: Karl Svozil
Publisher: World Scientific
ISBN: 9789810208097
Category : Science
Languages : en
Pages : 314
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Book Description
Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua.Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the physical properties of its hardware. Computers are a medium to virtual realities. The foreseeable importance of such virtual realities stimulates the investigation of an ?inner description?, a ?virtual physics? of these universes of computation. Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery.One motive of this book is the recognition that what is often referred to as ?randomness? in physics might actually be a signature of undecidability for systems whose evolution is computable on a step-by-step basis. To give a flavour of the type of questions envisaged: Consider an arbitrary algorithmic system which is computable on a step-by-step basis. Then it is in general impossible to specify a second algorithmic procedure, including itself, which, by experimental input-output analysis, is capable of finding the deterministic law of the first system. But even if such a law is specified beforehand, it is in general impossible to predict the system behaviour in the ?distant future?. In other words: no ?speedup? or ?computational shortcut? is available. In this approach, classical paradoxes can be formally translated into no-go theorems concerning intrinsic physical perception.It is suggested that complementarity can be modelled by experiments on finite automata, where measurements of one observable of the automaton destroys the possibility to measure another observable of the same automaton and it vice versa.Besides undecidability, a great part of the book is dedicated to a formal definition of randomness and entropy measures based on algorithmic information theory.
Author: Peter Hellekalek
Publisher: Springer Science & Business Media
ISBN: 9780387985541
Category : Mathematics
Languages : en
Pages : 358
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Book Description
This book sumarizes recent theoretical and practical developments. The generation and the assessment of pseudo- and quasi-random point sets is one of the basic tasks of applied mathematics and statistics, with implications for Monte Carlo methods, stochastic simulation, and applied statistics. They are also of strong theoretical interest, with applications to algebraic geometry, metric number theory, probability theory, and cryptology.
Author: Jesper Moller
Publisher: Springer Science & Business Media
ISBN: 146122652X
Category : Mathematics
Languages : en
Pages : 144
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Book Description
Tessellations are subdivisions of d-dimensional space into non-overlapping "cells". Voronoi tessellations are produced by first considering a set of points (known as nuclei) in d-space, and then defining cells as the set of points which are closest to each nuclei. A random Voronoi tessellation is produced by supposing that the location of each nuclei is determined by some random process. They provide models for many natural phenomena as diverse as the growth of crystals, the territories of animals, the development of regional market areas, and in subjects such as computational geometry and astrophysics. This volume provides an introduction to random Voronoi tessellations by presenting a survey of the main known results and the directions in which research is proceeding. Throughout the volume, mathematical and rigorous proofs are given making this essentially a self-contained account in which no background knowledge of the subject is assumed.
Author: Ibrahim Rahimov
Publisher: Springer Science & Business Media
ISBN: 1461242169
Category : Mathematics
Languages : en
Pages : 207
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Book Description
The aim of this monograph is to show how random sums (that is, the summation of a random number of dependent random variables) may be used to analyse the behaviour of branching stochastic processes. The author shows how these techniques may yield insight and new results when applied to a wide range of branching processes. In particular, processes with reproduction-dependent and non-stationary immigration may be analysed quite simply from this perspective. On the other hand some new characterizations of the branching process without immigration dealing with its genealogical tree can be studied. Readers are assumed to have a firm grounding in probability and stochastic processes, but otherwise this account is self-contained. As a result, researchers and graduate students tackling problems in this area will find this makes a useful contribution to their work.
Author: Shelemyahu Zacks
Publisher: Springer
ISBN: 1461226902
Category : Mathematics
Languages : en
Pages : 188
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Book Description
The present monograph is a comprehensive summary of the research on visibility in random fields, which I have conducted with the late Professor Micha Yadin for over ten years. This research, which resulted in several published papers and technical reports (see bibliography), was motivated by some military problems, which were brought to our attention by Mr. Pete Shugart of the US Army TRADOC Systems Analysis Activity, presently called US Army TRADOC Analysis Command. The Director ofTRASANA at the time, the late Dr. Wilbur Payne, identified the problems and encouraged the support and funding of this research by the US Army. Research contracts were first administered through the Office of Naval Research, and subsequently by the Army Research Office. We are most grateful to all involved for this support and encouragement. In 1986 I administered a three-day workshop on problem solving in the area of sto chastic visibility. This workshop was held at the White Sands Missile Range facility. A set of notes with some software were written for this workshop. This workshop led to the incorporation of some of the methods discussed in the present book into the Army simulation package CASTFOREM. Several people encouraged me to extend those notes and write the present monograph on the level of those notes, so that the material will be more widely available for applications.
Author: Persi Diaconis
Publisher: Princeton University Press
ISBN: 0691196397
Category : Mathematics
Languages : en
Pages : 272
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Book Description
In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact.
Author: Michael Strevens
Publisher: Harvard University Press
ISBN: 0674076028
Category : Science
Languages : en
Pages : 260
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Book Description
Tychomancy—meaning “the divination of chances”—presents a set of rules for inferring the physical probabilities of outcomes from the causal or dynamic properties of the systems that produce them. Probabilities revealed by the rules are wide-ranging: they include the probability of getting a 5 on a die roll, the probability distributions found in statistical physics, and the probabilities that underlie many prima facie judgments about fitness in evolutionary biology. Michael Strevens makes three claims about the rules. First, they are reliable. Second, they are known, though not fully consciously, to all human beings: they constitute a key part of the physical intuition that allows us to navigate around the world safely in the absence of formal scientific knowledge. Third, they have played a crucial but unrecognized role in several major scientific innovations. A large part of Tychomancy is devoted to this historical role for probability inference rules. Strevens first analyzes James Clerk Maxwell’s extraordinary, apparently a priori, deduction of the molecular velocity distribution in gases, which launched statistical physics. Maxwell did not derive his distribution from logic alone, Strevens proposes, but rather from probabilistic knowledge common to all human beings, even infants as young as six months old. Strevens then turns to Darwin’s theory of natural selection, the statistics of measurement, and the creation of models of complex systems, contending in each case that these elements of science could not have emerged when or how they did without the ability to “eyeball” the values of physical probabilities.
Author: Doug Fisher
Publisher: Springer Science & Business Media
ISBN: 1461224047
Category : Mathematics
Languages : en
Pages : 444
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Book Description
Ten years ago Bill Gale of AT&T Bell Laboratories was primary organizer of the first Workshop on Artificial Intelligence and Statistics. In the early days of the Workshop series it seemed clear that researchers in AI and statistics had common interests, though with different emphases, goals, and vocabularies. In learning and model selection, for example, a historical goal of AI to build autonomous agents probably contributed to a focus on parameter-free learning systems, which relied little on an external analyst's assumptions about the data. This seemed at odds with statistical strategy, which stemmed from a view that model selection methods were tools to augment, not replace, the abilities of a human analyst. Thus, statisticians have traditionally spent considerably more time exploiting prior information of the environment to model data and exploratory data analysis methods tailored to their assumptions. In statistics, special emphasis is placed on model checking, making extensive use of residual analysis, because all models are 'wrong', but some are better than others. It is increasingly recognized that AI researchers and/or AI programs can exploit the same kind of statistical strategies to good effect. Often AI researchers and statisticians emphasized different aspects of what in retrospect we might now regard as the same overriding tasks.
