[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

Topological Structure and Analysis of Interconnection Networks

This is a cumulative status report on the 1100 problems listed in the volume Open Problems in Topology (North-Holland, 1990), edited by J. van Mill and G.M. Reed.

Open Problems in Topology

AN ILLUSTRATED GUIDE TO THEORETICAL ECOLOGY by Ted J. case Oxford University Press, 2000, 449 pp. An Illustrated Guide to Theoretical Ecology may seem an unusual book review subject for this journal. After all, our readers are mathematics and computer science teachers, not ecologists or biologists. Nevertheless, I believe that this book warrants examination by mathematics teachers for several reasons. First, An Illustrated Guide to Theoretical Ecology is well written. The ecology is clear, even for a novice, and the mathematics is presented thoroughly and meticulously. Second, I am convinced that we best connect with our non-mathematics majors when we show some understanding of how the mathematics we are teaching plays a role in other disciplines. Ecology is a particularly well-suited field for showing mathematics at work. Finally, mathematical ideas often are best presented in context. It has been my experience that consumers of mathematics (scientists, economists, etc.) frequently have unique points of view on mathematical concepts that may help mathematics students learn their subject. An Illustrated Guide to Theoretical Ecology has two main foci: population ecology and community ecology. The author, Ted Case, a professor of biology at the University of California-San Diego, is a respected population and community ecologist. Case has used the mathematical methods he teaches in this book in his own work with lizards, desert ants, and habitat fragmentation. He has a wonderful grasp of mathematics and the ability to make the formalism come alive. Population ecology is concerned with the growth and decay of populations of species. Case takes the student from simple exponential growth through the whole panoply of ecological modeling techniques that address spatial, temporal, and individual variations in birth and death rates, age structures, density-dependent population growth, life history trade-offs, reproductive value, natural selection, and aging. The mathematics involved in studying these topics is equally broad. The student will learn about difference and differential equations, log scales, Gaussian distributions, geometric and arithmetic means, variance, central limit theorem, matrices and vectors, eigenvalues and eigenvectors, limits, Ricker diagrams, stability, combinatorics, and probability. Supporters of the reform movement in algebra and calculus will be pleased to see that An Illustrated Guide to Theoretical Ecology takes the Rule of Three quite seriously and uses symbolic, numeric, and graphic views of functions to describe and explain. …

An Illustrated Guide to Theoretical Ecology

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati. The original concept is that of “contracted triangulation”, which was introduced with the main goal of finding a “minimal” atlas for topological manifolds ([P1 1968], [P2 1974], [P3 1974], [FG2 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P4 1975], [F1 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject. For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.

A graph-theoretical representation of PL-manifolds — A survey on crystallizations

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

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Very Colorful Theorems

Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.

The combinatorics of colored triangulations of manifolds

Characterization of n-path graphs and of graphs having nth root

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

https://paginas.matem.unam.mx/luis/phocadownloadpap/articulos/48lm.pdf

Carousels, zindler curves and the floating body problem

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies.

/pdf/order-types-of-convex-bodies-4k9sntgsss.pdf

Luis Montejano

Papers

Very Colorful Theorems

The combinatorics of colored triangulations of manifolds

Characterization of n-path graphs and of graphs having nth root

Carousels, zindler curves and the floating body problem

Order Types of Convex Bodies