25 research outputs found

    Generalized Branching in Circle Packing

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    Circle packings are configurations of circle with prescribed patterns of tangency. They relate to a surprisingly diverse array of topics. Connections to Riemann surfaces, Apollonian packings, random walks, Brownian motion, and many other topics have been discovered. Of these none has garnered more interest than circle packings\u27 relationship to analytical functions. With a high degree of faithfulness, maps between circle packings exhibit essentially the same geometric properties as seen in classical analytical functions. With this as motivation, an entire theory of discrete analytic function theory has been developed. However limitations in this theory due to the discreteness of circle packings are shown to be unavoidable. This thesis explores methods to introduce continuous parameters for the purpose of overcoming these difficulties. Our topics include, packings with deep overlaps, fractional branching, and shift-points. Using the software package CirclePack, examples of some previously non-realizable discrete functions in circle packing are shown to computational exist using these techniques. Some necessary theory is developed including a generalization for overlapping packings and some results for expressing singularities associated with faces

    An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros

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    A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula extends to nn-dimensional simplices for all n>3n > 3. Remarkably, for n=3n = 3 the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to Z24\mathbb Z_2^4, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 33 chirotope, aka affine oriented matroid.Comment: 51 pages, 6 sections, 5 appendices, 7 figures, 2 tables, 81 references; v7 clarifies the definitions made in the text leading up to Theorem 5.4, along with the usual miscellaneous minor corrections and improvement

    3D Composer: A Software for Micro-composition

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    The aim of this compositional research project is to find new paradigms of expression and representation of musical information, supported by technology. This may further our understanding of how artistic intention materialises during the production of a musical work. A further aim is to create a software device, which will allow the user to generate, analyse and manipulate abstract musical information within a multi-dimensional environment. The main intent of this software and composition portfolio is to examine the process involved during the development of a compositional tool to verify how transformations applied to the conceptualisation of musical abstraction will affect musical outcome, and demonstrate how this transformational process would be useful in a creative context. This thesis suggests a reflection upon various technological and conceptual aspects within a dynamic multimedia framework. The discussion situates the artistic work of a composer within the technological sphere, and investigates the role of technology and its influences during the creative process. Notions of space are relocated in the scope of a personal compositional direction in order to develop a new framework for musical creation. The author establishes theoretical ramifications and suggests a definition for micro-composition. The main aspect focuses on the ability to establish a direct conceptual link between visual elements and their correlated musical output, ultimately leading to the design of a software called 3D-Composer, a tool for the visualisation of musical information as a means to assist composers to create works within a new methodological and conceptual realm. Of particular importance is the ability to transform musical structures in three-dimensional space, based on the geometric properties of micro-composition. The compositions Six Electroacoustic Studies and Dada 2009 display the use of the software. The formalisation process was derived from a transposition of influences of the early twentieth century avant-garde period, to a contemporary digital studio environment utilising new media and computer technologies for musical expression

    Natural Communication

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    In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science

    Characterising substructures of finite projective spaces

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    UNOmaha Problem of the Week (2021-2022 Edition)

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    The University of Omaha math department\u27s Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester\u27s end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes. Now there are three difficulty tiers to POW problems, roughly corresponding to easy/medium/hard difficulties, with each tier getting twelve problems per semester, and three problems (one of each tier) per week posted online and around campus. The tiers are named after the EPH classification of conic sections (which is connected to many other classifications in math), and in the present compilation they abide by the following color-coding: Cyan, Green, and Magenta. In practice, when creating the problem sets, we begin with a large enough pool of problem drafts and separate out the ones which are most obviously elliptic or hyperbolic, and then the remaining ones fall into parabolic. The tiers don\u27t necessarily reflect workload, though, only prerequisite mathematical background. Ideally, the solutions to elliptic problems, and any parts of solutions to parabolic and hyperbolic problems not covered in standard undergraduate courses, are meant to test participants\u27 creativity. Beware, though, many solutions also include additional commentary which varies wildly in the reader\u27s assumed mathematical maturity
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