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Author: Azriel Rosenfeld Publisher: ISBN: Category : Convex geometry Languages : en Pages : 6
Book Description
Abstract: "A pebbled graph is called '(geodesically) convex' if at least one shortest path between any two unpebbled nodes has no pebbles on any of its nodes. There exist conditions on the node neighborhoods in a pebbled graph that imply convexity; but no such conditions can be necessary for convexity. The convex pebblings can be characterized for various special types of graphs, such as cycles, trees, and cliques. For a graph L whose nodes are the lattice points in the plane under the relation of row or column adjacency, we show that a pebbling of L is convex iff the set of unpebbled nodes is conected and orthoconvex."
Author: Azriel Rosenfeld Publisher: ISBN: Category : Convex geometry Languages : en Pages : 6
Book Description
Abstract: "A pebbled graph is called '(geodesically) convex' if at least one shortest path between any two unpebbled nodes has no pebbles on any of its nodes. There exist conditions on the node neighborhoods in a pebbled graph that imply convexity; but no such conditions can be necessary for convexity. The convex pebblings can be characterized for various special types of graphs, such as cycles, trees, and cliques. For a graph L whose nodes are the lattice points in the plane under the relation of row or column adjacency, we show that a pebbling of L is convex iff the set of unpebbled nodes is conected and orthoconvex."
Author: Ignacio M. Pelayo Publisher: Springer Science & Business Media ISBN: 1461486998 Category : Mathematics Languages : en Pages : 117
Book Description
Geodesic Convexity in Graphs is devoted to the study of the geodesic convexity on finite, simple, connected graphs. The first chapter includes the main definitions and results on graph theory, metric graph theory and graph path convexities. The following chapters focus exclusively on the geodesic convexity, including motivation and background, specific definitions, discussion and examples, results, proofs, exercises and open problems. The main and most studied parameters involving geodesic convexity in graphs are both the geodetic and the hull number which are defined as the cardinality of minimum geodetic and hull set, respectively. This text reviews various results, obtained during the last one and a half decade, relating these two invariants and some others such as convexity number, Steiner number, geodetic iteration number, Helly number, and Caratheodory number to a wide range a contexts, including products, boundary-type vertex sets, and perfect graph families. This monograph can serve as a supplement to a half-semester graduate course in geodesic convexity but is primarily a guide for postgraduates and researchers interested in topics related to metric graph theory and graph convexity theory.
Author: Laurent Najman Publisher: Springer ISBN: 3319580027 Category : Mathematics Languages : en Pages : 378
Book Description
This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics, differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected. Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging field.
Author: M.L.J. van de Vel Publisher: Elsevier ISBN: 0080933106 Category : Mathematics Languages : en Pages : 556
Book Description
Presented in this monograph is the current state-of-the-art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one; specifically, it is not restricted to the context of vector spaces. Classical concepts of order-convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly inspired by intuition; they go back to the first half of this century. An axiomatic approach started to develop in the early Fifties. The author became attracted to it in the mid-Seventies, resulting in the present volume, in which graphs appear side-by-side with Banach spaces, classical geometry with matroids, and ordered sets with metric spaces. A wide variety of results has been included (ranging for instance from the area of partition calculus to that of continuous selection). The tools involved are borrowed from areas ranging from discrete mathematics to infinite-dimensional topology. Although addressed primarily to the researcher, parts of this monograph can be used as a basis for a well-balanced, one-semester graduate course.
Author: Kenʼichi Ōshika Publisher: American Mathematical Soc. ISBN: 9780821820803 Category : Mathematics Languages : en Pages : 212
Book Description
This book deals with geometric and topological aspects of discrete groups. The main topics are hyperbolic groups due to Gromov, automatic group theory, invented and developed by Epstein, whose subjects are groups that can be manipulated by computers, and Kleinian group theory, which enjoys the longest tradition and the richest contents within the theory of discrete subgroups of Lie groups. What is common among these three classes of groups is that when seen as geometric objects, they have the properties of a negatively curved space rather than a positively curved space. As Kleinian groups are groups acting on a hyperbolic space of constant negative curvature, the technique employed to study them is that of hyperbolic manifolds, typical examples of negatively curved manifolds. Although hyperbolic groups in the sense of Gromov are much more general objects than Kleinian groups, one can apply for them arguments and techniques that are quite similar to those used for Kleinian groups. Automatic groups are further general objects, including groups having properties of spaces of curvature 0. Still, relationships between automatic groups and hyperbolic groups are examined here using ideas inspired by the study of hyperbolic manifolds. In all of these three topics, there is a ``soul'' of negative curvature upholding the theory. The volume would make a fine textbook for a graduate-level course
Author: Jiri Matousek Publisher: Springer Science & Business Media ISBN: 1461300398 Category : Mathematics Languages : en Pages : 491
Book Description
The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.
Author: Larry S. Davis Publisher: Springer Science & Business Media ISBN: 1461515297 Category : Computers Languages : en Pages : 496
Book Description
Computer systems that analyze images are critical to a wide variety of applications such as visual inspections systems for various manufacturing processes, remote sensing of the environment from space-borne imaging platforms, and automatic diagnosis from X-rays and other medical imaging sources. Professor Azriel Rosenfeld, the founder of the field of digital image analysis, made fundamental contributions to a wide variety of problems in image processing, pattern recognition and computer vision. Professor Rosenfeld's previous students, postdoctoral scientists, and colleagues illustrate in Foundations of Image Understanding how current research has been influenced by his work as the leading researcher in the area of image analysis for over two decades. Each chapter of Foundations of Image Understanding is written by one of the world's leading experts in his area of specialization, examining digital geometry and topology (early research which laid the foundations for many industrial machine vision systems), edge detection and segmentation (fundamental to systems that analyze complex images of our three-dimensional world), multi-resolution and variable resolution representations for images and maps, parallel algorithms and systems for image analysis, and the importance of human psychophysical studies of vision to the design of computer vision systems. Professor Rosenfeld's chapter briefly discusses topics not covered in the contributed chapters, providing a personal, historical perspective on the development of the field of image understanding. Foundations of Image Understanding is an excellent source of basic material for both graduate students entering the field and established researchers who require a compact source for many of the foundational topics in image analysis.
Author: Gabor T. Herman Publisher: Springer Science & Business Media ISBN: 1461241367 Category : Mathematics Languages : en Pages : 221
Book Description
"La narraci6n literaria es la evocaci6n de las nostalgias. " ("Literary narration is the evocation of nostalgia. ") G. G. Marquez, interview in Puerta del Sol, VII, 4, 1996. A Personal Prehistory In 1972 I started cooperating with members of the Biodynamics Research Unit at the Mayo Clinic in Rochester, Minnesota, which was under the direction of Earl H. Wood. At that time, their ambitious (and eventually realized) dream was to build the Dynamic Spatial Reconstructor (DSR), a device capable of collecting data regarding the attenuation of X-rays through the human body fast enough for stop-action imaging the full extent of the beating heart inside the thorax. Such a device can be applied to study the dynamic processes of cardiopulmonary physiology, in a manner similar to the application of an ordinary cr (computerized tomography) scanner to observing stationary anatomy. The standard method of displaying the information produced by a cr scanner consists of showing two-dimensional images, corresponding to maps of the X-ray attenuation coefficient in slices through the body. (Since different tissue types attenuate X-rays differently, such maps provide a good visualization of what is in the body in those slices; bone - which attenuates X-rays a lot - appears white, air appears black, tumors typically appear less dark than the surrounding healthy tissue, etc. ) However, it seemed to me that this display mode would not be appropriate for the DSR.
Author: Azriel Rosenfeld
Author: Azriel Rosenfeld
Author: Athanase Papadopoulos
Author: Ignacio M. Pelayo
Author: Laurent Najman
Author: M.L.J. van de Vel
Author: Kenʼichi Ōshika
Author: 
Author: Jiri Matousek
Author: Larry S. Davis
Author: Gabor T. Herman