Topological Complexity of Smooth Random Functions

Partition functions arise in combinatorics and related problems of statistical physics as they encode in a succinct way the combinatorial structure of complicated systems. The main focus of the

This book provides an overview of multiple aspects of spatial language understanding and explores recent trends of modeling based on very large foundational models and their applications. The authors address the following five main themes: spatial semantic representations in both symbolic and sub-symbolic spaces; spatial information extraction computational models; reasoning over spatial language; commonsense spatial understanding; and multiple modern and state-of-the-art downstream applications of spatial language understanding including dialogue systems, narrative discourse, and grounding language in the physical world with the multi-modal problem settings. One of the essential functions of natural language is to express spatial relationships between objects. Linguistic constructs can encode highly complex structures of objects, spatial relations between them, and patterns of motion through space relative to a reference point. The complexity of spatial language understanding and its

This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illu