Connectivity calculus of fractal polyhedrons

The paper analyzes the connectivity information (more precisely, numbers of tunnels and their homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combinatorial counterparts, called homological spanning forest (HSF), are presented here as an useful topological tool, which codifies such information and provides an hierarchical directed graph-based representation of the initial polyhedra. The Menger sponge and the Sierpiński pyramid are presented as examples of these computational algebraic topological techniques and results focussing on the number of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self-similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, the underlying concepts apply to general cell complexes and digital images and are suitable for progressing in the computation of advanced algebraic topological information of 3-dimensional objects

Fractal Topological Analysis for 2D Binary Digital Images

Fractal dimension is a powerful tool employed as a measurement of geometric aspects. In this work we propose a method of topological fractal analysis for 2D binary digital images by using a graph-based topological model of them, called Homological Spanning Forest (HSF, for short). Defined at interpixel level, this set of two trees allows to topologically describe the (black and white) connected component distribution within the image with regards to the relationship “to be surrounded by”. This distribution is condensed into a rooted tree, such that its nodes are connected components determined by some special sub-trees of the previous HSF and the levels of the tree specify the degree of nesting of each connected component. We ask for topological auto-similarity by comparing this topological description of the whole image with a regular rooted tree pattern. Such an analysis can be used to directly quantify some characteristics of biomedical images (e.g. cells samples or clinical images) that are not so noticeable when using geometrical approaches.Ministerio de Economía y Competitividad TEC2016-77785-PMinisterio de Economía y Competitividad MTM2016-81030-

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

Interplay of Geometry and Mechanics: Disordered Spring Networks and Shape-changing Cerebella

This thesis reports work in three topics - I) isotropic strain-induced rigidity transitions in under-constrained spring networks II) uniaxial strain-induced stiffening transitions in semiflexible networks with area-conserving inclusions and III) non-linearities in the buckling without bending morphogenesis model for growing cerebella. All models are two dimensional and we employ either discrete and continuum models to study these systems.In I), motivated by the rigidity transitions in isotropically strained disordered spring networks, we study rigidity transitions in isotropically strained area-conserving random polygonal loops. We find a crossover transition in these loops. We also provide arguments towards showing convexity as a necessary condition for the transition and cyclic polygonal configurations, a sufficient condition. In II), towards modeling the uniaxial compression stiffening experiments in mEF cells and composite systems of fibrin and dextran beads, we construct area-conserving regular polygonal loops. These loops demonstrate compression stiffening. We also report the compression softening of on-lattice semiflexible polymer networks. The softening mechanism is independent of Euler-buckling instabilities. Introduction of area-conserving regular polygons as inclusions in the semiflexible network introduces non-affinities in the elastic response of the system. The non-affine bending of filaments leads to compression stiffening of the semiflexible network. In III), we find that by adding non-linearities to the buckling without bending morphogenesis model, we obtain cusped folds which visually resemble the cusped folds of the cerebellum. Introduction of non-linearities in the energy functional of the model robustly develops a quadratic non-linearity in the Euler-Lagrange equations. We study the effect of such a non-linear force for a simple harmonic oscillator like system and see that there too we obtain cusped `folds\u27. We also discuss steric confinements on the growing cerebellum and a paradigmatic demonstration of hierarchical folds in the cerebellum

Variational principles and optimality in biological systems

The aim of this thesis is to investigate the signatures of evolutionary optimization in biological systems, such as in proteins, human behaviours and transport tissues in vascular plants (xylems), by means of the Pareto optimality analysis and the calculus of variations. In the first part of this thesis, we address multi-objective optimization problems with tradeoffs through the Pareto optimality analysis ( [132],[69]), according which the best tradeoff solutions correspond to the optimal species, enclosed onto low-dimensional geometrical polytopes, defined as Pareto optimal fronts, in the space of physical traits, called morphospace. Chapter 3 is devoted to the Pareto optimality analysis in the Escherichia coli proteome by projecting proteins onto the space of solubility and hydrophobicity. In chapter 4 we analyze the HCP dataset of cognitive and behavioral scores in 1206 humans, in order to identify any signature of Pareto optimization in the space of Delay Discounting Task (DDT), which measures the tendency for people to prefer smaller, immediate monetary rewards over larger, delayed rewards. The second part of this thesis is devoted to solving an optimization problem regarding xylems, which are the internal conduits in angiosperms that deliver water and other nutrients from roots to petioles in plants. Based on the optimization criteria of minimizing the energy dissipated in a fluid flow, we propose in chapter 5 a biophysical model with the goal of explaining the underlying physical mechanism that affects the structure of xylem conduits in vascular plants, which results in tapered xylem profiles [104, 105, 117, 164]. We address this optimization problem by formulating the model in the context of the calculus of variations. The results of these investigations, besides providing quantitative support to previous theories of natural selection, demonstrate how processes of optimization can be identified in different biological systems by applying statistical methods such as the Pareto optimality and the variational one, showing the relevance of employing these statistical approaches to various biological systems

High-Quality Simplification and Repair of Polygonal Models

Because of the rapid evolution of 3D acquisition and modelling methods, highly complex and detailed polygonal models with constantly increasing polygon count are used as three-dimensional geometric representations of objects in computer graphics and engineering applications. The fact that this particular representation is arguably the most widespread one is due to its simplicity, flexibility and rendering support by 3D graphics hardware. Polygonal models are used for rendering of objects in a broad range of disciplines like medical imaging, scientific visualization, computer aided design, film industry, etc. The handling of huge scenes composed of these high-resolution models rapidly approaches the computational capabilities of any graphics accelerator. In order to be able to cope with the complexity and to build level-of-detail representations, concentrated efforts were dedicated in the recent years to the development of new mesh simplification methods that produce high-quality approximations of complex models by reducing the number of polygons used in the surface while keeping the overall shape, volume and boundaries preserved as much as possible. Many well-established methods and applications require "well-behaved" models as input. Degenerate or incorectly oriented faces, T-joints, cracks and holes are just a few of the possible degenaracies that are often disallowed by various algorithms. Unfortunately, it is all too common to find polygonal models that contain, due to incorrect modelling or acquisition, such artefacts. Applications that may require "clean" models include finite element analysis, surface smoothing, model simplification, stereo lithography. Mesh repair is the task of removing artefacts from a polygonal model in order to produce an output model that is suitable for further processing by methods and applications that have certain quality requirements on their input. This thesis introduces a set of new algorithms that address several particular aspects of mesh repair and mesh simplification. One of the two mesh repair methods is dealing with the inconsistency of normal orientation, while another one, removes the inconsistency of vertex connectivity. Of the three mesh simplification approaches presented here, the first one attempts to simplify polygonal models with the highest possible quality, the second, applies the developed technique to out-of-core simplification, and the third, prevents self-intersections of the model surface that can occur during mesh simplification

Integrating Parametric Structural Analysis and Optimization in the Architectural Schematic Design Phase

The implementation of computational form-finding structural optimization methods has recently mushroomed in the architectural research area. There has been a few emerging architectural parametric Computer-Aided Design (CAD) systems that enable architects to perform early schematic form-finding structural optimization such as the coupling of Grasshopper (a visual programming language), Karamba (a structural analysis plugin) and Galapagos (an optimization plugin). However, the application of the method is very rare in both educational and design practice environments. Also, the architectural schematic design phase is commonly characterized by free-form shapes without the embedded considerations of the material and structural system. On the other hand, the considerations of materiality and structural system are often more properly imposed by structural engineers, who usually prefer to be involved as early as possible in the project. Seen from this perspectives, this research examines the implementation of structural optimization in the architectural schematic design phase; investigate the accessibility and usability of existing architectural structural optimization tools; and study the interoperability and integration of architectural parametric CAD tools and engineering analysis and optimization tools as well as the usability of these tools. This research uses Grounded Theory for data collection and analysis procedure to investigate those research concerns. A comparative study of software is also used to examine the second research concern. Semi-structured interviews are used to acquire in-depth understanding of the participants\u27 responses towards incorporating architectural structural optimization procedure in the context of the collaboration between architects and engineers. Students and faculty, with years of design practice experience, in Clemson University are used as the target population for the interview process. Five architectural, form-finding structural optimization methods are developed to facilitate the interview process. Improvements of the tools are made based on the participants\u27 responses towards the usefulness of the tools. Finally, guidelines concerning the implementation of the developed architectural structural optimization for the educational and design practice purposes were developed. The design guidelines are developed with the aim to better the communication between architects and engineers during the collaboration process. This research believes that participants\u27 in-depth responses toward the contemporary architectural design issues and the developed methods are the essential driving forces that help this research in finding ways to improve the collaboration between architects and structural engineers

Automorphs

Thesis (M. Arch.)--Massachusetts Institute of Technology, Dept. of Architecture, 1984.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ROTCH.Includes bibliographical references (p. 164-166).The purpose of this thesis is the delineation and investigation of a general pattern or mechanism which I have termed 'automorphic,' a word derived from the Latin roots, 'autos' meaning self, and 'morphe' referring to form. The thesis further characterizes the mechanism as 'recursive,' 'self-similar,' 'scaling' -- adjectives referring to form (and) phenomenon in or from which the same configuration is repeated many times at many different scales. The attempt is to maintain the generality of the term in order to establish it as a fundamental attribute of form or persistent structure, (or as a necessary component to a conception of form, as inherent to "order" as "modular coordination" or symmetry). I will explore briefly several disciplinary fragments of contemporary physical theory where this mechanism can be said to be operational including an analogy to basic life processes -- the most elegant of the physical automorphisms. Its analytic and thus generative power in fields as diverse as astrophysics, geomorphology, biology and particle physics, carry important implications for the understanding our own human physical and cognitive processes and subsequently important bearings on the artifacts we generate. The study will then focus on those topics specific to built form particularly that of this type of pattern's inherent structural and energetic stability. Central to this section is a note on spatial perception ( and therefore 'space' itself) as a function of automorphism, or inherent recursive perceptual thresholds. A graphic presentation of two built projects is meant as an attempt at unified synthesis and application.by David E. Nelson.M.Arch

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition