Geometric properties and algorithms for rational q-Bézier curves and surfaces

In this paper, properties and algorithms of q-Bézier curves and surfaces are analyzed. It is proven that the only q-Bézier and rational q-Bézier curves satisfying the boundary tangent property are the Bézier and rational Bézier curves, respectively. Evaluation algorithms formed by steps in barycentric form for rational q-Bézier curves and surfaces are provided

Quasi-Splines and their moduli

We study what we call quasi-spline sheaves over locally Noetherian schemes. This is done with the intention of considering splines from the point of view of moduli theory. In other words, we study the way in which certain objects that arise in the theory of splines can be made to depend on parameters. In addition to quasi-spline sheaves, we treat ideal difference-conditions, and individual quasi- splines. Under certain hypotheses each of these types of objects admits a fine moduli scheme. The moduli of quasi-spline sheaves is proper, and there is a natural compactification of the moduli of ideal difference-conditions. We include some speculation on the uses of these moduli in the theory of splines and topology, and an appendix with a treatment of the Billera-Rose homogenization in scheme theoretic language

The Effect of Malaysia General Election on Financial Network: An Evidence from Shariah-Compliant Stocks on Bursa Malaysia

Instead of focusing the volatility of the market, the market participants should consider on how the general election affects the correlation between the stocks during 14th general election Malaysia. The 14th general election of Malaysia was held on 9th May 2018. This event has a great impact towards the stocks listed on Bursa Malaysia. Thus, this study investigates the effect of 14th general election Malaysia towards the correlation between stock in Bursa Malaysia specifically the shariah-compliant stock. In addition, this paper examines the changes in terms of network topology for the duration, sixth months before and after the general election. The minimum spanning tree was used to visualize the correlation between the stocks. Also, the centrality measure, namely degree, closeness and betweenness were computed to identify if any changes of stocks that plays a crucial role in the network for the duration of before and after 14th general election Malaysia

Polynomials

Polynomial and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on polynomial and its applications has been done in mathematics, applied mathematics, and sciences. This book is based on recent results in all areas related to polynomial and its applications. This book provides an overview of the current research in the field of polynomials and its applications. The following papers have been published in this volume: ‘A Parametric Kind of the Degenerate Fubini Numbers and Polynomials’; ‘On 2-Variables Konhauser Matrix Polynomials and Their Fractional Integrals’; ‘Fractional Supersymmetric Hermite Polynomials’; ‘Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation’; ‘Iterating the Sum of Möbius Divisor Function and Euler Totient Function’; ‘Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations’; ‘Truncated Fubini Polynomials’; ‘On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights’; ‘Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity’; ‘Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials’; ‘Some Identities Involving Hermite Kampé de Fériet Polynomials Arising from Differential Equations and Location of Their Zeros.

BEST : Bézier-Enhanced Shell Triangle : a new rotation-free thin shell finite element

A new thin shell finite element is presented. This new element doesn’ t have rotational degrees of freedom. Instead, in order to overcome the C1 continuity requirement across elements, the author resorts to enhance the geometric description of the flat triangles of a mesh made out of linear triangles, by means of Bernstein polynomials and triangular Bernstein-Bézier patches. The author estimates the surface normals at the nodes of a mesh of triangles, in order to use them to define the Bernstein-Bézier patches. Ubach, Estruch and García-Espinosa performed a comprehensive statistical comparison of different weighting factors. The conclusion of that work is that the inverse of the area of the circumscribed circle to the triangle and the internal angle of the triangle at the node considered, should be used as weighting factor. Using this new weighting factor, we reduce by about 10% the root mean square error in the estimation of normals of randomly generated surfaces with respect to the previous best weighting factor found in the literature. The author uses the information of the normal vectors at the nodes and the triangular Bernstein-Bézier patches to build cubic Bézier triangles. These cubic Bézier triangles are surface interpolants; C1 continuous at the nodes and C0 continuous across the edges. Owing to this approach, the new element is called Bézier-enhanced shell triangle (BEST). The BEST element takes advantage of all the nodes’ connectivities in each triangle of the mesh. The computation of the normal vectors at the nodes doesn’ t depend on the number of triangles surrounding each node of the mesh. The BEST element is independent from the mesh topology. A new paradigm is presented consisting on the reconstruction of the geometry of a cubic triangular element. This geometric reconstruction exploits the properties of cubic B-spline functions (cubic Bézier triangle). This way, the author builds a conforming continuum-based shell finite element. A cubic Bézier triangle has 30 parameters (3 coordinates for each of the 10 control points). Therefore it needs to apply 30 independent conditions. 15 of these conditions are given directly by the positions of the 3 vertices of the triangle and the orientations of the normal vectors at the 3 vertices. 8 of the remaining conditions are imposed introducing energy minimization considerations. These energy minimization considerations serve also to define a well-posed element. The author defines 3 different reduced problems for the 3 different shell deformation modes: bending deformation, membrane (in-plane extension) deformation and in-plane shear (drilling rotation) deformation. The only degrees of freedom of the BEST element are the vertices’ coordinates (9 variables). The remaining 21 parameters are solved internally. In order to fix the values of these 21 internal parameters, each BEST element solves 9 systems of linear equations of rank 3. The BEST element is successfully applied to the analysis of thin shells in linear and geometrically non-linear regimes using an implicit method. The non-linearity is solved using a Total Lagrangian formulation. The author succeeds at pre-integrating through-the-thickness efficiently and accurately. The through-the-thickness integrals are evaluated just once: at the reference configuration. There are just 14 through-the-thickness scalar integrals to perform for each Gauss point. The numerical examples results show that the BEST element has the potential to achieve cubic convergence. Although they also cast doubts on the possibility of reproducing this result for a wide range of problems. For in-plane shear dominated problems, the formulation used in this thesis only achieves linear convergence. For membrane oriented tests with curvature, the convergence is quadratic. The BEST element exhibits membrane locking behavior. The author suggests exploiting further the drilling rotations kinematics in order to solve membrane locking.Se presenta un nuevo elemento finito de lámina delgada. Este nuevo elemento no usa rotaciones como grados de libertad. En su lugar, para sortear el requisito de mantener continuidad C1 entre elementos, el autor mejora la descripción geométrica de los triángulos planos de una malla de triángulos lineales, por medio de polinomios de Bernstein y particiones triangulares de Bernstein-Bézier. Para definir las particiones de Bernstein-Bézier, el autor estima las normales a la superficie en los nodos de una malla de triángulos. Ubach, Estruch y García-Espinosa hicieron una comparación estadística exhaustiva entre distintos factores de ponderación. La conclusión de dicho trabajo conduce a usar como factor de ponderación: el inverso del área de la circunferencia circunscrita al triángulo y el ángulo interno del triángulo en el nodo considerado. Con este nuevo factor de ponderación, se reduce en aproximadamente un 10% el error medio cuadrático cometido en la estimación de las normales de superficies generadas aleatoriamente, respecto del mejor factor usado previamente en la literatura. Con la información de los vectores normales en los nodos, el autor construye triángulos cúbicos de Bézier. Estos triángulos cúbicos de Bézier interpolan la superficie; con continuidad C1 en los nodos y C0 en las aristas. En virtud a este planteamiento, el nuevo elemento recibe el nombre de BEST. El elemento BEST aprovecha todas las conectividades nodales de cada triángulo de la malla. El número de triángulos que rodean cada nodo de la malla no afecta al cálculo de los vectores normales. El elemento BEST es independiente de la topología de la malla. Se propone un nuevo paradigma que consiste en reconstruir la geometría de un elemento triangular cúbico. Esta reconstrucción geométrica aprovecha las propiedades de las funciones cúbicas B-spline (triángulo cúbico de Bézier). Así, el autor crea un elemento de lámina conforme basado en el continuo. Un triángulo cúbico de Bézier tiene 30 parámetros (3 coordenadas para cada uno de los 10 puntos de control). Es necesario aplicar 30 condiciones independientes. 15 de estas condiciones se deducen de la posición de los 3 vértices del triángulo y de los vectores normales en los 3 vértices. De las otras 15 condiciones, 8 se obtienen a partir de criterios de minimización de la energía. Estos criterios de minimización de la energía sirven para definir un elemento bien planteado. El autor desarrolla 3 problemas reducidos para los 3 modos de deformación de la lámina: deformación de flexión, de membrana (extensión en el plano) y de cortante en el plano (rotación de taladro). Los únicos grados de libertad del elemento BEST son las posiciones de los vértices (9 variables). Los otros 21 parámetros se resuelven internamente. Para obtener estos 21 parámetros internos, hay que resolver 9 sistemas de ecuaciones lineales de rango 3 para cada elemento BEST. Se ha aplicado el elemento BEST con éxito al cálculo de láminas delgadas en régimen lineal y geométricamente no-lineal con un método implícito. La no-linealidad se plantea con una formulación Lagrangiana total. Se demuestra cómo pre-integrar en el espesor de manera eficiente y precisa. Solo es preciso evaluar las integrales en el espesor una vez: en la configuración de referencia. Solo hay 14 integrales escalares en el espesor para cada punto de Gauss. Los ejemplos numéricos muestran que el elemento BEST tiene potencial para converger cúbicamente. Pero también existen dudas sobre la capacidad de reproducir de manera consistente este resultado en un amplio rango de problemas. En problemas dominados por la deformación de cortante en el plano, la formulación utilizada en esta tesis solo alcanza convergencia lineal. En ejemplos orientados a la deformación de membrana que incluyen curvatura, la convergencia es cuadrática. El elemento BEST sufre de bloqueo por membrana. El autor sugiere desarrollar más profundamente la cinemática de las rotaciones de taladro para resolver el bloqueo por membrana.Es presenta un nou element finit de làmina prima. Aquest nou element no fa servir rotacions com a graus de llibertat. Enlloc d'això, per esquivar el requisit de mantenir continuïtat C1 entre els elements, l'autor millora la descripció geomètrica dels triangles plans d'una malla de triangles lineals, mitjançant polinomis de Bernstein i particions triangulars de Bernstein-Bézier.Per definir les particions de Bernstein-Bézier, l'autor estima les normals a la superfície en els nodes d'una malla de triangles. Ubach, Estruch i García-Espinosa varen fer una comparació estadística exhaustiva entre diferents factors de ponderació. La conclusió d'aquest treball condueix a fer servir com a factor de ponderació: l'invers de l'àrea de la circumferència circumscrita al triangle i l'angle intern del triangle en el node considerat. Amb aquest nou factor de ponderació, es redueix aproximadament en un 10% l'error quadràtic mig comès en l'estimació de les normals de superfícies generades aleatòriament, respecte del millor factor usat prèviament a la literatura.Amb la informació dels vectors normals en els nodes, l'autor construeix triangles cúbics de Bézier. Aquests triangles cúbics de Bézier interpolen la superfície; amb continuïtat C1 als nodes i C0 a les arestes. En virtut d'aquest plantejament, el nou element rep el nom de BEST (Bézier-enhanced shell triangle).L'element BEST aprofita totes les connectivitats nodals de cada triangle de la malla. El nombre de triangles que envolten cada node de la malla no afecta al càlcul dels vectors normals. L'element BEST és independent de la topologia de la malla.Es proposa un nou paradigma que consisteix en reconstruir la geometria d'un element triangular cúbic. Aquesta reconstrucció geomètrica aprofita les propietats de les funcions cúbiques B-spline (triangle cúbic de Bézier). D'aquesta manera l'autor crea un element de làmina que és conforme i basat en el continu.Un triangle cúbic de Bézier té 30 paràmetres (3 coordenades per cadascun dels 10 punts de control). Cal aplicar 30 condicions independents. 15 d'aquestes condicions es dedueixen de la posició dels 3 vèrtexs del triangle i dels vectors normals en els 3 vèrtexs.De les 15 condicions restants, 8 s'obtenen a partir de criteris de minimització de l'energia. Aquests criteris de minimització de l'energia serveixen per definir un element ben plantejat. L'autor desenvolupa 3 problemes reduïts per als 3 modes de deformació de la làmina: deformació de flexió, de membrana (extensió en el pla) i de tallant en el pla (rotació de barrina).Els únics graus de llibertat de l'element BEST són les posicions dels vèrtexs (9 variables). Els altres 21 paràmetres es resolen internament. Per obtenir aquests 21 paràmetres interns, cal resoldre 9 sistemes d'equacions lineals de rang 3 per cada element BEST.S'ha aplicat l'element BEST amb èxit al càlcul de làmines primes en règim lineal i geomètricament no-lineal fent servir un mètode implícit. La no-linealitat es planteja amb una formulació Lagrangiana total. Es demostra com es pot pre-integrar a través del gruix de manera eficient i precisa. Només cal avaluar les integrals a través del gruix un cop: a la configuració de referència. Només hi ha 14 integrals escalars a través del gruix per a cada punt de Gauss. Els exemples numèrics mostren que l'element BEST té potencial per convergir cúbicament. Però també hi ha dubtes de que aquest resultat es pugui reproduir de manera consistent per un ventall ampli de problemes. En problemes dominats per la deformació de tallant en el pla, la formulació emprada en aquesta tesi només assoleix convergència lineal. En exemples orientats a la deformació de membrana que incloguin curvatura, la convergència és quadràtica. L'element BEST pateix de bloqueig per membrana. L'autor suggereix desenvolupar en més profunditat la cinemàtica de les rotacions de barrina per resoldre el bloqueig per membrana

Integral Transformation, Operational Calculus and Their Applications

The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects

Mudança digital no desenho arquitetónico: uma outra visão para a arquitetura paisagista

Mestrado em Arquitetura Paisagista - Instituto Superior de Agronomia - ULA presente dissertação procura traçar uma perspetiva histórica do desenho digital, em particular, nos processos arquitetónicos. O uso do computador e de métodos computacionais no desenho arquitetónico, por meio de programas específicos, tem sido estudado por vários autores no campo da Arquitetura, levando em consideração as mudanças no processo criativo e noutras formas de projetar. Contudo, não tem sido dado relevo à origem dessa mudança, tornando premente resgatar a memória histórica sobre o contexto e os protagonistas dessa transformação que tanto tem marcado a arquitetura do século XXI. Neste sentido, pretende-se evidenciar as origens do desenho digital e olhar para as alterações que o desenho arquitetónico sofreu com a utilização de meios digitais, primeiro nos círculos académicos, nas décadas de sessenta e setenta e depois com uma massificação dos programas CAD (desenho assistido por computador) nas décadas de oitenta e noventa. A elaboração da tese de doutoramento em 1963, Sketchpad: A man-machine graphical communication system de Ivan Sutherland marca começo do digital no desenho arquitetónico. É a capacidade visionária de alguns académicos, e em particular de Sutherland, que possibilita a criação do primeiro programa de CAD interativo que permite desenhar sem papel, num monitor, num tempo em que os computadores eram dispendiosos, enormes, mas com monitores de dimensão inferior a alguns smartphones. A dissertação tem como principal objetivo esboçar uma perspetiva histórica do desenho digital a nível internacional, através de pesquisa bibliográfica do trabalho de académicos e escolas relevantes para a revolução digital do século XX. A dissertação estrutura-se em três partes. Na primeira, o primeiro capítulo destaca os antecedentes e a ligação da arquitetura à ciência e da computação à indústria. A segunda parte, descreve o começo da mudança digital através do desenvolvimento dos primeiros sistemas CAD interativos e dos contributos de cinco pioneiros do desenho digital arquitetónico. A terceira parte consiste na descrição de um caso prático de aplicação dos meios digitais à renovação do espaço público urbanoN/

Birefringent properties of the human cornea in vivo : towards a new model of corneal structure

The fundamental corneal properties of mechanical rigidity, maintenance of curvature and optical transparency result from the specific organisation of collagen fibrils in the corneal stroma. The exact arrangement of stromal collagen is currently unknown but several structural models have been proposed. The purpose of the present study is to investigate inconsistencies between current x‐ray derived structural models of the cornea and optically derived birefringence data. Firstly, the thesis reviews the current understanding of corneal structure, particularly in relation to corneal birefringence. It also reviews and develops the different analytical approaches used to model optical biaxial behaviour, particularly as applied to predict corneal optical phase retardation. The second part develops a novel technique of elliptic polarization biomicroscopy (EPB), enabling study of corneal birefringence in vivo. Using EPB, the pattern of corneal retardation is recorded for a range of human subjects. This dataset is then used to investigate both central and peripheral corneal birefringence as well as the corneal microstructure. A key finding is that the central parts of the cornea exhibit a retardation pattern compatible with a negative biaxial crystal, whereas the peripheral corneal regions do not. Furthermore, within the central regions of the cornea, orthogonal confocal conic fibrillar structures are identified which resemble the analytically derived contours of equal refractive index of an ideal negative biaxial crystal. The third part of this work presents a synthesis of previous published experimental, anatomical and theoretical findings and the experimental results presented in this thesis. Based on these findings, a novel corneal structural model is proposed that comprises overlapping spherical elliptic structural units. Finally, ensuing biomechanical and clinical consequences of the spherical elliptic structural model and of the EPB technique are discussed including their potential diagnostic and surgical applications