Algorithms for Sparse Linear Systems

Large sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are used in sparse direct methods and discusses the computation of approximate direct and inverse factorizations that are key to constructing general-…

Algebra in Cuneiform: Introduction to an Old Babylonian Geometrical Technique

This textbook analyzes a number of texts in "conformal translation," that is, a translation in which the same Babylonian term is always translated in the same way and, more importantly, in which different terms are always translated differently. Appendixes are provided for readers who are familiar with basic Assyriology but otherwise philological details are avoided. All of these texts are from…

A Game-Theoretic Perspective on Coalition Formation

The formation of coalitions to achieve both collaborative and competitive goals is a phenomenon we see all around us. The list of examples of this phenomenon is long and varied: production cartels, political lobbies, customs unions, environmental coalitions, and ethnic alliances are just a few everyday instances. This book looks at coalition formation from the perspective of game theory. How ar…

Metric Algebraic Geometry

Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances. After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equatio…

Branes and DAHA Representations

In recent years, there has been an increased interest in exploring the connections between various disciplines of mathematics and theoretical physics such as representation theory, algebraic geometry, quantum field theory, and string theory. One of the challenges of modern mathematical physics is to understand rigorously the idea of quantization. The program of quantization by branes, which com…

Notes on algebraic potentials

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Flexibility Principles in Boolean Semantics: The Interpretation of Coordinati…

An investigation of the logical flexibility principles needed for a formal semantic account of coordination, plurality, and scope in natural language.Since the early work of Montague, Boolean semantics and its subfield of generalized quantifier theory have become the model-theoretic foundation for the study of meaning in natural languages. This book uses this framework to develop a new semantic…

Communication Complexity: A New Approach to Circuit Depth

Communication Complexity describes a new intuitive model for studying circuit networks that captures the essence of circuit depth. Although the complexity of boolean functions has been studied for almost 4 decades, the main problems the inability to show a separation of any two classes, or to obtain nontrivial lower bounds remain unsolved. The communication complexity approach provides clues as…

What is the Genus?

Exploring several of the evolutionary branches of the mathematical notion of genus, this book traces the idea from its prehistory in problems of integration, through algebraic curves and their associated Riemann surfaces, into algebraic surfaces, and finally into higher dimensions. Its importance in analysis, algebraic geometry, number theory and topology is emphasized through many theorems. Al…

Algebraic semantics of imperative programs

Algebraic Semantics of Imperative Programs presents a self-contained and novel "executable" introduction to formal reasoning about imperative programs. The authors' primary goal is to improve programming ability by improving intuition about what programs mean and how they run. The semantics of imperative programs is specified in a formal, implemented notation, the language OBJ; this makes the s…