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The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.

The analysis of body composition (fat, bone and muscle) is an important process throughout the biomedical sciences. This is the first book to offer a clear and detailed introduction to the key methods and techniques in body composition analysis and to explain the importance of body composition data in the context of sport, exercise and health. With contributions from some of the world's leading body composition specialists, the book goes further than any other in demonstrating the practical and applied value of body composition analysis in areas such as performance sport and weight control in clinical populations. The book pays particular attention to the important concept of change in body composition, and includes discussion of ethical issues in the collection, interpretation and presentation of data, and considerations when working with special populations. Bridging the gap between research methods and practical application, this book is important reading for advanced students and practitioners working in sport and exercise science, health science, anatomy, nutrition, physical therapy or ergonomics.

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for

We analyze a class of nonlinear partial differential equations (PDEs) defined on

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than

We prove that for Bernoulli percolation on

Dietary Reference Intakes for Water, Potassium, Sodium, Chloride, and Sulfate The Dietary Reference Intakes (DRIs) are quantitative estimates of nutrient intakes to be used for planning and assessing diets for healthy people. This new report, the sixth in a series of reports presenting dietary reference values for the intakes of nutrients by Americans and Canadians, establishes nutrient recommendations on water, potassium, and salt for health maintenance and the reduction of chronic disease risk. Dietary Reference Intakes for Water, Potassium, Sodium, Chloride, and Sulfate discusses in detail the role of water, potassium, salt, chloride, and sulfate in human physiology and health. The major findings in this book include the establishment of Adequate Intakes for total water (drinking water, beverages, and food), potassium, sodium, and chloride and the establishment of Tolerable Upper Intake levels for sodium and chloride. The book makes research recommendations for information needed to advance the understanding of human requirements for water and electrolytes, as well as adverse effects associated with the intake of excessive amounts of water, sodium, chloride, potassium, and sulfate. This book will be an invaluable reference for nutritionists, nutrition researchers, and food manufacturers.

The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure-- so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.

This volume contains the proceedings of the AMS Special Session on Celebrating M. M. Rao's Many Mathematical Contributions as he Turns 90 Years Old, held from November 9-10, 2019, at the University of California, Riverside, California.The articles show the effectiveness of abstract analysis for solving fundamental problems of stochastic theory, specifically the use of functional analytic methods for elucidating stochastic processes and their applications. The volume also includes a biography of M. M. Rao and the list of his publications.