Applications of topology in computer algorithms

The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorithms. The applications of general topology in other branches of mathematics are not discussed, since they are not applications of mathematics outside of mathematics.Comment: This paper is based on the invited lecture at International Conference on Topology and Applications held in August 23--27, 1999, at Kanagawa University in Yokohama, Japa

Derivation of graph and pointer algorithms

We introduce operators and laws of an algebra of formal languages, a subalgebra of which corresponds to the algebra of (multiary) relations. This algebra is then used in the formal specification and derivation of some graph and pointer algorithms

Tracing evolutionary links between species

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM

Functional programming and graph algorithms

This thesis is an investigation of graph algorithms in the non-strict purely functional language Haskell. Emphasis is placed on the importance of achieving an asymptotic complexity as good as with conventional languages. This is achieved by using the monadic model for including actions on the state. Work on the monadic model was carried out at Glasgow University by Wadler, Peyton Jones, and Launchbury in the early nineties and has opened up many diverse application areas. One area is the ability to express data structures that require sharing. Although graphs are not presented in this style, data structures that graph algorithms use are expressed in this style. Several examples of stateful algorithms are given including union/find for disjoint sets, and the linear time sort binsort. The graph algorithms presented are not new, but are traditional algorithms recast in a functional setting. Examples include strongly connected components, biconnected components, Kruskal's minimum cost spanning tree, and Dijkstra's shortest paths. The presentation is lucid giving more insight than usual. The functional setting allows for complete calculational style correctness proofs - which is demonstrated with many examples. The benefits of using a functional language for expressing graph algorithms are quantified by looking at the issues of execution times, asymptotic complexity, correctness, and clarity, in comparison with traditional approaches. The intention is to be as objective as possible, pointing out both the weaknesses and the strengths of using a functional language

Modelling and Design of Resilient Networks under Challenges

Communication networks, in particular the Internet, face a variety of challenges that can disrupt our daily lives resulting in the loss of human lives and significant financial costs in the worst cases. We define challenges as external events that trigger faults that eventually result in service failures. Understanding these challenges accordingly is essential for improvement of the current networks and for designing Future Internet architectures. This dissertation presents a taxonomy of challenges that can help evaluate design choices for the current and Future Internet. Graph models to analyse critical infrastructures are examined and a multilevel graph model is developed to study interdependencies between different networks. Furthermore, graph-theoretic heuristic optimisation algorithms are developed. These heuristic algorithms add links to increase the resilience of networks in the least costly manner and they are computationally less expensive than an exhaustive search algorithm. The performance of networks under random failures, targeted attacks, and correlated area-based challenges are evaluated by the challenge simulation module that we developed. The GpENI Future Internet testbed is used to conduct experiments to evaluate the performance of the heuristic algorithms developed

Transforming data by calculation

Thispaperaddressesthefoundationsofdata-modeltransformation.A catalog of data mappings is presented which includes abstraction and representa- tion relations and associated constraints. These are justified in an algebraic style via the pointfree-transform, a technique whereby predicates are lifted to binary relation terms (of the algebra of programming) in a two-level style encompassing both data and operations. This approach to data calculation, which also includes transformation of recursive data models into “flat” database schemes, is offered as alternative to standard database design from abstract models. The calculus is also used to establish a link between the proposed transformational style and bidi- rectional lenses developed in the context of the classical view-update problem.Fundação para a Ciência e a Tecnologia (FCT

THE DISTANCE CENTRALITY: MEASURING STRUCTURAL DISRUPTION OF A NETWORK

This research provides an innovative approach to identifying the influence of vertices on the topology of a graph by introducing and exploring the neighbor matrix and distance centrality. The neighbor matrix depicts the “distance profile” of each vertex, identifying the number of vertices at each shortest path length from the given vertex. From the neighbor matrix, we can derive 11 oft-used graph invariants. Distance centrality uses the neighbor matrix to identify how much influence a given vertex has over graph structure by calculating the amount of neighbor matrix change resulting from vertex removal. We explore the distance centrality in the context of three synthetic graphs and three graphs representing actual social networks. Regression analysis enables the determination that the distance centrality contains different information than four current centrality measures (betweenness, closeness, degree, and eigenvector). The distance centrality proved to be more robust against small changes in graphs through analysis of graphs under edge swapping, deletion, and addition paradigms than betweenness and eigenvector centrality, though less so than degree and closeness centralities. We find that the neighbor matrix and the distance centrality reliably enable the identification of vertices that are significant in different and important contexts than current measures.http://archive.org/details/thedistancecentr1094559576Lieutenant Colonel, United States ArmyApproved for public release; distribution is unlimited

Network Resilience Improvement and Evaluation Using Link Additions

Computer networks are getting more involved in providing services for most of our daily life activities related to education, business, health care, social life, and government. Publicly available computer networks are prone to targeted attacks and natural disasters that could disrupt normal operation and services. Building highly resilient networks is an important aspect of their design and implementation. For existing networks, resilience against such challenges can be improved by adding more links. In fact, adding links to form a full mesh yields the most resilient network but it incurs an unfeasibly high cost. In this research, we investigate the resilience improvement of real-world networks via adding a cost-efficient set of links. Adding a set of links to an obtain optimal solution using an exhaustive search is impracticable for large networks. Using a greedy algorithm, a feasible solution is obtained by adding a set of links to improve network connectivity by increasing a graph robustness metric such as algebraic connectivity or total graph diversity. We use a graph metric called flow robustness as a measure for network resilience. To evaluate the improved networks, we apply three centrality-based attacks and study their resilience. The flow robustness results of the attacks show that the improved networks are more resilient than the non-improved networks

AN EXPLORATORY MIXED METHODS STUDY OF PROSPECTIVE MIDDLE GRADES TEACHERS\u27 MATHEMATICAL CONNECTIONS WHILE COMPLETING INVESTIGATIVE TASKS IN GEOMETRY

With the implementation of No Child Left Behind legislation and a push for reform curricula, prospective teachers must be prepared to facilitate learning at a conceptual level. To address these concerns, an exploratory mixed methods investigation of twenty-eight prospective middle grades teachers’ mathematics knowledge for teaching geometry and mathematical connection-making was conducted at a large public southeastern university. Participants completed a diagnostic assessment in mathematics with a focus on geometry and measurement (CRMSTD, 2007), a mathematical connections evaluation, and a card sort activity. Mixed methods data analysis revealed prospective middle grades teachers’ mathematics knowledge for teaching geometry was underdeveloped and the mathematical connections made by prospective middle grades teachers were more procedural than conceptual in nature

New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations

This Special Issue puts forward for discussion state-of-the-art papers on new topics related to neutrosophic theories, such as neutrosophic algebraic structures, neutrosophic triplet algebraic structures, neutrosophic extended triplet algebraic structures, neutrosophic algebraic hyperstructures, neutrosophic triplet algebraic hyperstructures, neutrosophic n-ary algebraic structures, neutrosophic n-ary algebraic hyperstructures, refined neutrosophic algebraic structures, refined neutrosophic algebraic hyperstructures, quadruple neutrosophic algebraic structures, refined quadruple neutrosophic algebraic structures, neutrosophic image processing, neutrosophic image classification, neutrosophic computer vision, neutrosophic machine learning, neutrosophic artificial intelligence, neutrosophic data analytics, neutrosophic deep learning, neutrosophic symmetry, and their applications in the real world. This book leads to the further advancement of the neutrosophic and plithogenic theories of NeutroAlgebra and AntiAlgebra, NeutroGeometry and AntiGeometry, Neutrosophic n-SuperHyperGraph (the most general form of graph of today), Neutrosophic Statistics, Plithogenic Logic as a generalization of MultiVariate Logic, Plithogenic Probability and Plithogenic Statistics as a generalization of MultiVariate Probability and Statistics, respectively, and presents their countless applications in our every-day world