Algebraic Concepts in the Study of Graphs and Simplicial Complexes

This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs

Survey of Graph Embeddings into Compact Surfaces

A prominent question of topological graph theory is what type of surface can a nonplanar graph be embedded into? This thesis has two main goals. First to provide a necessary background in topology and graph theory to understand the development of an embedding algorithm. The main purpose is developing and proving a direct constructive embedding algorithm that takes as input the graph with a particular order of edges about each vertex. The embedding algorithm will not only determine which compact surface the graph can be embedded into, but also determines the particular embedding of the graph on the surface. The embedding algorithm is then used to investigate surfaces into which trees and a class of the complete bipartite graphs can be embedded. Further, the embedding algorithm is used to investigate non-surface separating graph embeddings

The topological uniqueness of the deltahedra found in the boranes BnHn2− (6≤n≤12)

The deltahedra observed experimentally in the borane anions BnHn2− (6≤n≤12) are the only possible n-vertex deltahedra having only degree 4 and 5 vertices. The existence of an 11-vertex deltahedron having only degree 4 or 5 vertices is topologically impossible in accord with the presence of a degree 6 vertex in the observed structure for B11H112−

Mathematics at the eve of a historic transition in biology

A century ago physicists and mathematicians worked in tandem and established quantum mechanism. Indeed, algebras, partial differential equations, group theory, and functional analysis underpin the foundation of quantum mechanism. Currently, biology is undergoing a historic transition from qualitative, phenomenological and descriptive to quantitative, analytical and predictive. Mathematics, again, becomes a driving force behind this new transition in biology.Comment: 5 pages, 2 figure

Mathematical Aspects of Similarity and Quasi-analysis - Order, Topology, and Sheaves

The concept of similarity has had a rather mixed reputation in philosophy and the sciences. On the one hand, philosophers such as Goodman and Quine emphasized the „logically repugnant“ and „insidious“ character of the concept of similarity that allegedly renders it inaccessible for a proper logical analysis. On the other hand, a philosopher such as Carnap assigned a central role to similarity in his constitutional theory. Moreover, the importance and perhaps even indispensibility of the concept of similarity for many empirical sciences can hardly be denied. The aim of this paper is to show that Quine’s and Goodman’s harsh verdicts about this notion are mistaken. The concept of similarity is susceptible to a precise logico-mathematical analysis through which its place in the conceptual landscape of modern mathematical theories such as order theory, topology, and graph theory becomes visible. Thereby it can be shown that a quasi-analysis of a similarity structure S can be conceived of as a sheaf (etale space) over S

A containment-first search algorithm for higher-order analysis of urban topology

Research has revealed the importance of the concepts from the mathematical areas of both topology and graph theory for interpreting the spatial arrangement of spatial entities. Graph theory in particular has been used in different applications of a wide range of fields for that purpose, however not many graph-theoretic approaches to analyse entities within the urban environment are available in the literature. Some examples should be mentioned though such as, Bafna (2003), Barr and Barnsley (2004), Bunn et al. (2000), Krüger (1999), Nardinochi et al. (2003), and Steel et al. (2003). Very little work has been devoted in particular to the interpretation of initially unstructured geospatial datasets. In most of the applications developed up-to-date for the interpretation and analysis of spatial phenomena within the urban context, the starting point is to some extent a meaningful dataset in terms of the urban scene. Starting at a level further back, before meaningful data are obtained, the interpretation and analysis of spatial phenomena are more challenging tasks and require further investigation. The aim of retrieving structured information from initial unstructured spatial data, translated into more meaningful homogeneous regions, can be achieved by identifying meaningful structures within the initial random collection of objects and by understanding their spatial arrangement (Anders et al., 1999). It is believed that the task of understanding topological relationships between objects can be accomplished by both applying graph theory and carrying out graph analysis (de Almeida et al., 2007)

Chirality in Liquid Crystals: from Microscopic Origins to Macroscopic Structure

Molecular chirality leads to a wonderful variety of equilibrium structures, from the simple cholesteric phase to the twist-grain-boundary phases, and it is responsible for interesting and technologically important materials like ferroelectric liquid crystals. This paper will review some recent advances in our understanding of the connection between the chiral geometry of individual molecules and the important phenomenological parameters that determine macroscopic chiral structure. It will then consider chiral structure in columnar systems and propose a new equilibrium phase consisting of a regular lattice of twisted ropes.Comment: 20 pages with 6 epsf figure

Symmetries of spatial graphs and Simon invariants

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3+3 vertices in detail, and determine the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.Comment: 16 pages, 14 figure