Envision Geometry

In Pi (π) in Nature, Art, and Culture Marcel Danesi investigates the manifestations of π in science, nature, symbolism, and culture, arguing that these are intrinsically intertwined.

Publisher Description

Enabling Technologies for Computational Science assesses future application computing needs, identifies research directions in problem-solving environments (PSEs), addresses multi-disciplinary environments operating on the Web, proposes methodologies and software architectures for building adaptive and human-centered PSEs, and describes the role of symbolic computing in scientific and engineering PSEs. The book also includes an extensive bibliography of over 400 references. Enabling Technologies for Computational Science illustrates the extremely broad and interdisciplinary nature of the creation and application of PSEs. Authors represent academia, government laboratories and industry, and come from eight distinct disciplines (chemical engineering, computer science, ecology, electrical engineering, mathematics, mechanical engineering, psychology and wood sciences). This breadth and diversity extends into the computer science aspects of PSEs. These papers deal with topics such as artificial intelligence, computer-human interaction, control, data mining, graphics, language design and implementation, networking, numerical analysis, performance evaluation, and symbolic computing. Enabling Technologies for Computational Science provides an assessment of the state of the art and a road map to the future in the area of problem-solving environments for scientific computing. This book is suitable as a reference for scientists from a variety of disciplines interested in using PSEs for their research.

Sixteen-year-old Blanche forms a band that wins a spot at Coachella, a southern California music festival, where she hopes to reconnect with her father, a famous but reclusive musician who left when she was six years old.

The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.