Geometric Constructs Research Articles

Polynomiography and applications in art, education, and science

Polynomiography is the art and science of visualizing approximation of the zeros of complex polynomials. Informally speaking, polynomiography allows one to create colorful images of polynomials. These images can subsequently be re-colored in many ways, using one's own creativity and artistry. Polynomiography has tremendous applications in the visual arts, education, and science. The paper describes some of those applications. Artistically, polynomiography can be used to create quite a diverse set of images reminiscent of the intricate patterning of carpets and elegant fabrics, abstract expressionist and minimalist art, and even images that resemble cartoon characters. Educationally, polynomiography can be used to teach mathematical concepts, theorems, and algorithms, e.g., the algebra and geometry of complex numbers, the notions of convergence and continuity, geometric constructs such as Voronoi regions, and modern notions such as fractals. Scientifically, polynomiography provides not only a tool for viewing polynomials, present in virtually every branch of science, but also a tool to discover new theorems.

Proportional design in L. B. Alberti's Tempio Malatestiano, Rimini

Systems of proportion provoked much controversy in the 1950s. Recent research has moved away from the study of musical theory, to that of number. Applying Classical arithmetic techniques to analyze Alberti's Tempio Malatestiano, this paper demonstrates that its design corresponds to Alberti's system based on geometrical features of the cube depicting ‘natural relationships that cannot be defined as numbers’. For today's practitioners, the paper asserts that architectural works are geometric constructs and reminds them that CAD is based on the arithmeticization of geometry.

Time and Mind

Mind, it has recently been argued 1 , is a thoroughly temporal phenomenon: so temporal, indeed, as to defy description and analysis using the traditional computational tools of cognitive scientific understanding. The proper explanatory tools, so the suggestion goes, are instead the geometric constructs and differential equations of Dynamical Systems Theory. I consider various aspects of the putative temporal challenge to computational understanding, and show that the root problem turns on the presence of a certain kind of causal web: a web that involves multiple components (both inner and outer) linked by chains of continuous and reciprocal causal influence. There is, however, no compelling route from such facts about causal and temporal complexity to the radical anticomputationalist conclusion. This is because, interactive complexities notwithstanding, the computational approach provides a kind of explanatory understanding that cannot (I suggest) be recreated using the alternative resources of pure Dynamical Systems Theory. In particular, it provides a means of mapping information flow onto causal structure -- a mapping that is crucial to understanding the distinctive kinds of flexibility and control characteristic of truly mindful engagements with the world. Where we confront especially complex interactive causal webs, however, it does indeed become harder to isolate the syntactic vehicles required by the computational approach. Dynamical Systems Theory, I conclude, may play a vital role in recovering such vehicles from the burgeoning mass of real-time interactive complexity.

Linguistic Approach to 2D Geometric Modeling of Hierarchical Systems

Geometric modeling is the cornerstone of computer-aided design (CAD). Currently, geometric modeling is done either interactively, using the geometric constructs (i.e., line, circle, etc.) supported by the CAD system, or by writing a program that generates the geometry of interest automatically. While both interactive and automatic geometric modeling are more efficient than manual drafting, they have their limitations. This paper describes an approach to geometric modeling that is superior to the interactive and automated modeling approaches with respect to ease of use, flexibility, and cost-effectiveness. This approach involves defining a geometry linguistically and then generating the geometry from its linguistic definition automatically. The approach manifests itself in three activities: (1) linguistically representing the geometry using a suitable vocabulary; (2) translating the linguistic definition of the geometry into a logic program; and (3) generating the geometry via logical deduction in a logic-based CAD system. These activities are discussed and illustrated in the context of the geometry of a simple bridge structure.

Three-dimensional interplay of advective and diffusive processes in the generation of sequence boundaries

Quantitative stratigraphic models have been generated using either geometric constructs or diffusion algorithms to simulate the development and timing of sequence boundaries and how they vary in response to tectonics, eustasy, climate, and sediment supply. While diffusion models provide a physical basis for the generation of clinoforms, the resulting model predictions appear to do little more than track the transgressive and regressive movements of the shoreline. On the other hand, geometric models do a relatively good job in simulating the observed stratigraphy but offer little in actually ascertaining the physics controlling the processes of deposition and erosion. All these stratigraphic models fail to predict the most fundamental response of depositional systems, namely, the development of a downward shift in coastal onlap during a fall in relative sea level. This basinward shift and the development of an onlap surface are the critical observations that define chronostratigraphic sequences and are not simply a consequence of shoreline transgressions and regressions. A major assumption in these two-dimensional models is that the observed stacking patterns are solely the consequence of deposition processes and sediment transport within the modelling section, in marked contrast to the observation that the major sediment transport direction along margins is sub-parallel to the margin. We present a variation on the diffusion theme in which across-margin transport and deposition are dominated by advection-type processes and along-margin erosion and deposition processes are dominated by diffusion. With this combination of advection and diffusion it is possible to generate a lowstand system tract and sequence boundaries (i.e., onlap surfaces). In particular, the generation of the lowstand system tract is, in large part, the result of depositional processes operative along the margin (i.e., out of the section).

Spherical Maps: Their Construction, Properties, and Approximation

The Gaussian map and its allied visibility map on a unit sphere find wide applications for orientating the workpiece for machining by numerical control machines and for probing by coordinate measurement machines. They also provide useful aids in computerized scene analysis, computation of surface-surface intersection, component design for manufacturing and fabrication procedures. Spherical convex hulls and spherical circles are two geometric constructs used to approximate the Gaussian maps and the visibility maps. The duality and the complementarity of these spherical maps are examined so as to derive efficient algorithms.

Knowledge-based networks in classification problems

Abstract The paper proposes a distributed computational structure called knowledge-based network as a classification scheme in pattern recognition. Unlike the existing architectures and algorithms of pattern recognition the network allows for an explicit representation of domain classification knowledge while maintaining its learning capabilities. The knowledge-based network blends useful properties of knowledge-based systems (namely explicit knowledge representation) with those advantageous for neural networks (viz. learning). The network is composed of basic AND and OR neurons. Fuzzy clustering constitutes a preprocessing phase leading towards developing geometric constructs. They contribute to a conceptual level around which numerical processing of the classifier is centred.

Computer simulation of cortical polymaps: A proto-column algorithm

Neo-cortical sensory areas of the vertebrate brain are organized in terms of sensory (topographic) maps of peripheral sense organs. Known for over 40 years, cortical topography has been generally modeled in terms of a continuous map of a peripheral sensory surface onto a cortical surface. However, the details of cortical architecture do not conform to this concept: most, if not all cortical areas are represented by an interlacing of multiple featural or peripheral maps in the form of columns consisting of distinct classes of neural input to the cortex. Since such an architecture is at best piecewise continuous, it is useful to introduce a new term which focuses attention on the many-to-one interlaced nature of neo-cortex. We use the term “polymap” to refer to a cortical area which consists of more than one input system, interlaced in a globally topographic, but locally columnar fashion. The best known example of a cortical polymap is provided by the ocular dominance column system in layer IV of primate striate cortex, but the puff/extra-puff and orientation systems of surrounding layers also illustrate this concept, as do the thick-thin-interstripe columns of V-2, and the direction columns of MT. Since polymap architecture seems to be a common architectural pattern in neo-cortex, we have addressed the problem of computational modeling of polymap systems. In the present paper, we present an algorithm, based on the computational geometric constructs of Generalized Voronoi Polygon and Medial Axis, along with related image warps, which is capable of providing a general method for simulating polymap systems. We illustrate this idea with the ocular dominance column system of V-1, and show computer simulations of the structure of binocular stimuli, as they would appear at the level of layer IV in V-1. Comparison of these results to recent binocular 2DG experiments is presented, and methods of generalizing these techniques to other common polymap cortical areas are outlined. This work provides a demonstration of computational neuroscience. The basic anatomical structure of many neo-cortical areas cannot be understood even on a qualitative level, and certainly cannot be modeled, without the development of a detailed set of geometric structures and algorithms which make clear the relationship of an experimentally observed polymap to its afferent inputs.

Shape of concept: color percept. . .graphics, geometry, and gestalt

The relationship of shape and color, which are two of the parts of a composition, and their relation to the comprehension of written language are discussed. A composition is described as a gestalt that has shape and color within a geometric construct, affecting comprehension and meaning. Therefore, what is seen can be modified by changes in shape and color within a geometric construct modifying comprehension and meaning. Geometric constructs, shape concepts, and color percepts and their implementation in a composition are considered. >

Nonlinear Regression Analysis and Its Applications

Wiley-Interscience Paperback Series Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. The authors have put together an extraordinary presentation of concepts and methods concerning the use and analysis of nonlinear regression models ...highly recommend[ed] ...for anyone needing to use and/or understand issues concerning the analysis of nonlinear regression -Technometrics [This book] provides a good balance of relevant theory and application with many examples ...[and it] provides the most balanced approach to theory and application appropriate for a first course in nonlinear regression modeling for graduate statistics students. -Mathematical Reviews [This book] joins a distinguished list of publications with a reputation for balancing technical rigor with readability, and theory with application. [It] upholds tradition ...[and is] a worthwhile reference for the marketing researcher with a serious interest in linear models. -Journal of Marketing Research This book offers a balanced presentation of the theoretical, practical, and computational aspects of nonlinear regression and provides background material on linear regression, including the geometrical development for linear and nonlinear least squares. authors employ real data sets throughout, and their extensive use of geometric constructs and continuing examples makes the progression of ideas appear very natural. book also includes pseudocode for computing algorithms.

Estimates for the Bergman and Szego Kernels in C 2

The purpose of this paper (some of whose conclusions were announced in [NRSW]) is to study the Bergman and Szegd projection operators on pseudoconvex domains Q of finite type in C2. The results we obtain are of three kinds: (i) The size estimates of the Bergman and Szeg6 kernels, and their derivatives. (ii) The cancellation properties of those kernels, expressed in terms of the actions of these operators on suitable bump functions; these properties have an interest in their own right but are also crucial in (iii) below. (iii) The sharp mapping properties of these operators on functions spaces, such as LP, the nonisotropic Sobolev spaces, and the nonisotropic Lipschitz spaces, which are naturally attached to the geometry of the boundary. This also leads ultimately to sharp results in the regularity properties of solutions of the equation du = f in Q, and 9bU = f on the boundary. Let us describe these matters in more detail. Our starting points are certain basic geometric constructs associated to Q: a metric p defined in terms of the vector fields X1 and X2 which are the real and imaginary parts of the (tangential) Levi vector field on d 2; and a function A(p, 8), (a polynomial in 8 for p E d 2), which represents the higher Levi-invariant attached to d 9 . The significance of these to the geometry of d Q can be understood from the following facts: the ball B(p, 8) C d Q (centered at p, of radius 8 in the metric p) when viewed in the appropriate coordinates has width - 8 in the complex tangential directions, but its width is - A(p, 8) in the complementary direction. Thus the volume of the ball is - 82A(p, 8). Among other crucial properties

The versatility of automatic mesh generators based on tree structures and advanced geometric constructs

AbstractThe ability of the finite octree and finite quadtree automatic mesh generators to satisfy effectively a wide variety of meshing capabilities is demonstrated. Areas considered include the ability to integrate easily with geometric modelling systems, versatile interactive mesh control functions, the ability to reorder unknowns directly from the finite element data structure and the ability to support automated finite element modelling.

Theory of Dirac monopoles with a non-Abelian symmetry

From the viewpoint of a global formulation of Yang-Mills fields, a Lagrangian theory of non-Abelian classical Dirac monopoles is proposed. Dirac strings are used instead of coordinate patches; they are defined as purely geometric constructs. The formalism is free from pathologies such as Dirac's veto. While the analysis focuses on the gauge group $\mathrm{SU}\frac{(N)}{{Z}_{N}}$ allowing for only $N\ensuremath{-}1$ nontrivial and topologically distinct types of monopoles, it applies in general to any compact Lie group.