This dissertation explores bridge n-sections of knotted surfaces in 4-manifolds by defining invariants that measure the complexity of their topology and by giving geometric constructions that determine Lagrangian surfaces in 4-manifolds under certain conditions. First, we present an elegant geometric construction that generates all triple grid…

Graph Signal Processing (GSP) plays a crucial role in addressing the growing need for information processing across networks, especially in tasks like supervised classification. However, the success of GSP in such tasks hinges on accurately identifying the underlying relational structures, which are often not readily available and must be inferred…

Online networks, as digital images reflecting every aspects of real world especially under the current context of information technology, have become the most massive and opulent data source these days, where vast amount of information keeps being produced and transmitted. Their rapid expansion in scale and data volume draws wide attention and…

Topological data analysis (TDA) is a collection of powerful tools that can quantify shape and structure in data in order to answer questions from the data's domain. This is done by representing some aspect of the structure of the data in a simplified topological signature. In this article, we introduce two of the most commonly used topological…

It is known that active exploration of a new environment leads to better spatial learning than does passive visual exposure. We ask whether specific components of active learning differentially contribute to particular forms of spatial knowledge--the "exploration-specific learning hypothesis". Previously, we found that idiothetic…

This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real…

Patients with traumatic brain injury show clear impairments in behavioural flexibility and inhibition that often persist beyond the time of injury, affecting independent living and psychosocial functioning. Functional magnetic resonance imaging studies have shown that patients with traumatic brain injury typically show increased and more broadly…

This tricentennial tribute commemorates Euler's major contributions to mathematical and physical sciences. A brief biographical sketch is presented with his major contributions to certain selected areas of number theory, differential and integral calculus, differential equations, solid and fluid mechanics, topology and graph theory, infinite…

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

Although the topological index does not uniquely correspond to the individual structure of a graph, it roughly represents the topological nature of the graph. Examples are given for using the topological index as a first sorting device for coding and retrieving structures, especially of fused polycyclic systems. (14 references) (Author/NH)

The material in this booklet is concerned with a discussion and examination of geometric puzzles and the ideas which result from their study. The general idea of graphs is introduced as a tool which can be used to solve geometric puzzles. The fact that working with puzzles can lead to unexpected mathematical discoveries is stressed. Such topics as…

This bibliography presents a series of studies that focus on the topic of networks and on the manner in which they have been considered in the literature. Each section of the bibliography provides a working set of references for one of the five major areas of research in which network analysis has been used. The first section deals primarily with…

Presents a model for representing and storing binary topological and directional relationships between 2-dimensional objects that is used to provide a basis for fuzzy querying capabilities. A data structure called an abstract spatial graph (ASG) is defined for the binary relationships that maintains all necessary information regarding topology and…