There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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Table Of Integrals Series And Products

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

Exactly solvable models for many-body systems far from equilibrium

We consider an exclusion process with particles injected with rate α at the origin and removed with rate β at the right boundary of a one-dimensional chain of sites. The particles are allowed to hop onto unoccupied sites, to the right only. For the special case of α =β = 1 the model was solved previously by Derridaet al. Here we extend the solution to general α, β. The phase diagram obtained from our exact solution differs from the one predicted by the mean-field approximation.

Phase transitions in an exactly soluble one-dimensional exclusion process

Fluctuation theorems make use of time reversal to make predictions about entropy production in many-body systems far from thermal equilibrium. Here we review the wide variety of distinct, but interconnected, relations that have been derived and investigated theoretically and experimentally. Significantly, we demonstrate, in the context of Markovian stochastic dynamics, how these different fluctuation theorems arise from a simple fundamental time-reversal symmetry of a certain class of observables. Appealing to the notion of Gibbs entropy allows for a microscopic definition of entropy production in terms of these observables. We work with the master equation approach, which leads to a mathematically straightforward proof and provides direct insight into the probabilistic meaning of the quantities involved. Finally, we point to some experiments that elucidate the practical significance of fluctuation relations.

https://iopscience.iop.org/article/10.1088/1742-5468/2007/07/P07020/pdf

Fluctuation theorems for stochastic dynamics

We consider the asymmetric simple exclusion process (ASEP) with open boundaries and other driven stochastic lattice gases of particles entering, hopping and leaving a one- dimensional lattice. The long-term system dynamics, stationary states, and the nature of phase transitions between steady states can be understood in terms of the interplay of two characteristic velocities, the collective velocity and the shock (domain wall) velocity. This interplay results in two distinct types of domain walls whose dynamics is computed. We conclude that the phase diagram of the ASEP is generic for one-component driven lattice gases with a single maximum in the current-density relation.

/pdf/phase-diagram-of-one-dimensional-driven-lattice-gases-with-r1gm5wg52z.pdf

Phase diagram of one-dimensional driven lattice gases with open boundaries

Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilitiesP(x1,...,xN;t/y
 1,...,yN; 0) of findingN particles on lattices sitesx
 1,...,xN at timet with initial occupationy
 1,...,yN at timet=0.

Gunter M. Schütz

Papers

Exactly solvable models for many-body systems far from equilibrium

Phase transitions in an exactly soluble one-dimensional exclusion process

Fluctuation theorems for stochastic dynamics

Phase diagram of one-dimensional driven lattice gases with open boundaries

Exact solution of the master equation for the asymmetric exclusion process