Mechanical Theorem Proving in Geometries

Mechanical Theorem Proving in Geometries
Author: Wen-tsün Wu
Publisher: Springer Science & Business Media
Total Pages: 301
Release: 2012-12-06
Genre: Computers
ISBN: 370916639X

There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations.

Machine Proofs in Geometry

Machine Proofs in Geometry
Author: Shang-Ching Chou
Publisher: World Scientific
Total Pages: 490
Release: 1994
Genre: Mathematics
ISBN: 9789810215842

This book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems.The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method.This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers. By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education.

An Introduction to Riemannian Geometry

An Introduction to Riemannian Geometry
Author: Leonor Godinho
Publisher: Springer
Total Pages: 476
Release: 2014-07-26
Genre: Mathematics
ISBN: 3319086669

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

Computer Mathematics - Proceedings Of The Special Program At Nankai Institute Of Mathematics

Computer Mathematics - Proceedings Of The Special Program At Nankai Institute Of Mathematics
Author: Wen-tsun Wu
Publisher: World Scientific
Total Pages: 173
Release: 1993-09-09
Genre:
ISBN: 9814552518

The area of molecular imaging has matured over the past decade and is still growing rapidly. Many concepts developed for molecular biology and cellular imaging have been successfully translated to in vivo imaging of intact organisms. Molecular imaging enables the study of processes at a molecular level in their full biological context. Due to the high specificity of the molecular readouts the approach bears a high potential for diagnostics. It is fair to say that molecular imaging has become an indispensable tool for biomedical research and drug discovery and development today.This volume familiarizes the reader with the concepts of imaging and molecular imaging in particular. Basic principles of imaging technologies, reporter moieties for the various imaging modalities, and the design of targeted probes are described in the first part. The second part illustrates how these tools can be used to visualize relevant molecular events in the living organism. Topics covered include the studies of the biodistribution of reporter probes and drugs, visualization of the expression of biomolecules such as receptors and enzymes, and how imaging can be used for analyzing consequences of the interaction of a ligand or a drug with its molecular target by visualizing signal transduction, or assessing the metabolic, physiological, or structural response of the organism studied. The final chapter deals with visualization of cell migration, for example in the context of cell therapies.The second edition covers novel developments over recent years, in particular regarding imaging technologies (hybrid techniques) and novel reporter concepts. Novel biomedical applications have been included, where appropriate. All the chapters have been thoroughly reworked and the artwork updated.

Learning and Geometry: Computational Approaches

Learning and Geometry: Computational Approaches
Author: David Kueker
Publisher: Springer Science & Business Media
Total Pages: 217
Release: 2012-12-06
Genre: Computers
ISBN: 1461240883

The field of computational learning theory arose out of the desire to for mally understand the process of learning. As potential applications to artificial intelligence became apparent, the new field grew rapidly. The learning of geo metric objects became a natural area of study. The possibility of using learning techniques to compensate for unsolvability provided an attraction for individ uals with an immediate need to solve such difficult problems. Researchers at the Center for Night Vision were interested in solving the problem of interpreting data produced by a variety of sensors. Current vision techniques, which have a strong geometric component, can be used to extract features. However, these techniques fall short of useful recognition of the sensed objects. One potential solution is to incorporate learning techniques into the geometric manipulation of sensor data. As a first step toward realizing such a solution, the Systems Research Center at the University of Maryland, in conjunction with the Center for Night Vision, hosted a Workshop on Learning and Geometry in January of 1991. Scholars in both fields came together to learn about each others' field and to look for common ground, with the ultimate goal of providing a new model of learning from geometrical examples that would be useful in computer vision. The papers in the volume are a partial record of that meeting.

Automated Deduction in Geometry

Automated Deduction in Geometry
Author: Dongming Wang
Publisher: Springer Science & Business Media
Total Pages: 252
Release: 1998-03-18
Genre: Computers
ISBN: 9783540642978

This book constitutes the thoroughly refereed and revised post-workshop proceedings of the International Workshop on Automated Deduction in Geometry, held in Toulouse, France, in September 1996. The revised extended papers accepted for inclusion in the volume were selected on the basis of double reviewing. Among the topics covered are automated geometric reasoning and the deduction applied to Dixon resultants, Gröbner bases, characteristic sets, computational geometry, algebraic geometry, and planet motion; furthermore the system REDLOG is demonstrated and the verification of geometric statements as well as the automated production of proof in Euclidean Geometry are present.

Mathematics Mechanization and Applications

Mathematics Mechanization and Applications
Author: Xiao-Shan Gao
Publisher: Academic Press
Total Pages: 584
Release: 2000-07-25
Genre: Computers
ISBN:

Mathematics Mechanization and Applications provides surveys for major research developments on mechanizing algebraic equations-solving and geometric theorem proving with diverse applications accomplished in Wu's extended Chinese group. The book: * addresses the frontiers of research, with new and original ideas and results * includes sophisticated and successful applications to scientific and engineering problems * covers polynomial system solving; geometric reasoning; computer algebra; and mathematical software * is comprehensive and focused, and easy to read with a uniform presentation * contains an extensive bibliography, of high value for reference to western readers. This book is of interest to researchers, software developers and graduate students in symbolic and algebraic computation, automated theorem-proving, algorithmic mathematics, and computer-aided mathematical problem solving. It is relevant for researchers and university teachers in computer-aided instruction and education; and for engineers and practitioners in mechanics, computer-aided geometric design, geometric modelling and robotics. People in many other related areas, from pure mathematics to computer-aided design, particularly those who know of the Wu method, but have little knowledge of it or the work that has arisen around it, will also find the book good reading.

A Panoramic View of Riemannian Geometry

A Panoramic View of Riemannian Geometry
Author: Marcel Berger
Publisher: Springer Science & Business Media
Total Pages: 835
Release: 2012-12-06
Genre: Mathematics
ISBN: 3642182453

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

Riemannian Geometry

Riemannian Geometry
Author: Manfredo P. do Carmo
Publisher: Copernicus
Total Pages: 328
Release: 1992
Genre: Mathematics
ISBN:

Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text. A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.

Geometric Mechanics on Riemannian Manifolds

Geometric Mechanics on Riemannian Manifolds
Author: Ovidiu Calin
Publisher: Springer Science & Business Media
Total Pages: 285
Release: 2006-03-15
Genre: Mathematics
ISBN: 0817644210

* A geometric approach to problems in physics, many of which cannot be solved by any other methods * Text is enriched with good examples and exercises at the end of every chapter * Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics