B-splines, Pólya curves, and duality

AbstractLocal duality between B-splines and Pólya curves is examined, mostly from the viewpoint of computer-aided geometric design. Certain known results for the two curve types are shown to be related. A few new results for Pólya curves and a curve scheme related to B-splines also follow from these investigations

Fastest mixing Markov chain on graphs with symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.Comment: 39 pages, 15 figure

Matrices de subdivisión para curvas beta-spline cúbicas

En este trabajo se emplea una técnica de subdivisión para calcular los puntos de control que subdividen a las curvas polinómicas. Sea P = (P0, ..., Pn) el polígono de control y B[P] la curva polinómica de grado n para la cual se construye el algoritmo de subdivisión. Mediante operaciones matriciales se obtienen L y R, polígonos a izquierda y a derecha, respectivamente, que aproximan a la curva B[P]. Cada uno de los polígonos P, L, y R representan un conjunto de puntos en el plano. Consideramos el caso de las curvas Beta-spline cúbicas, con parámetros β1 y β2, y realizamos la subdivisión para distintos valores de estos parámetros. Detallamos explícitamente las matrices de subdivisión utilizadas para cada caso, así como también la representación gráfica de los subpolígonos obtenidos en los distintos pasos de la subdivisión.In this paper we use a subdivision technique to calculate the control points that subdivide polinomial curves. If P = (P0, ..., Pn) is the control polygon and B[P] is the polinomial curve of degree n for which a subdivision algorithm is to be constructed, we use matrix operations to obtain the left polygon L and the right polygon R that aproximate the curve B[P]. Each one of the polygons P, L, and R represent a set of points in the plane. In this work we have considered the case of cubic Beta-spline curves, with parameters β1 and β2, and we obtained the subdivision curves for different values of these parameters. We explicitely detail the subdivision matrices that we have used for each case, and present the graphic representation of the subpolygons obtained in the different steps of the subdivision.V Workshop de Computación Gráfica, Imágenes Y VisualizaciónRed de Universidades con Carreras en Informática (RedUNCI

Matrices de subdivisión para curvas beta-spline cúbicas

En este trabajo se emplea una técnica de subdivisión para calcular los puntos de control que subdividen a las curvas polinómicas. Sea P = (P0, ..., Pn) el polígono de control y B[P] la curva polinómica de grado n para la cual se construye el algoritmo de subdivisión. Mediante operaciones matriciales se obtienen L y R, polígonos a izquierda y a derecha, respectivamente, que aproximan a la curva B[P]. Cada uno de los polígonos P, L, y R representan un conjunto de puntos en el plano. Consideramos el caso de las curvas Beta-spline cúbicas, con parámetros β1 y β2, y realizamos la subdivisión para distintos valores de estos parámetros. Detallamos explícitamente las matrices de subdivisión utilizadas para cada caso, así como también la representación gráfica de los subpolígonos obtenidos en los distintos pasos de la subdivisión.In this paper we use a subdivision technique to calculate the control points that subdivide polinomial curves. If P = (P0, ..., Pn) is the control polygon and B[P] is the polinomial curve of degree n for which a subdivision algorithm is to be constructed, we use matrix operations to obtain the left polygon L and the right polygon R that aproximate the curve B[P]. Each one of the polygons P, L, and R represent a set of points in the plane. In this work we have considered the case of cubic Beta-spline curves, with parameters β1 and β2, and we obtained the subdivision curves for different values of these parameters. We explicitely detail the subdivision matrices that we have used for each case, and present the graphic representation of the subpolygons obtained in the different steps of the subdivision.V Workshop de Computación Gráfica, Imágenes Y VisualizaciónRed de Universidades con Carreras en Informática (RedUNCI

PHYSICS-AWARE MODEL SIMPLIFICATION FOR INTERACTIVE VIRTUAL ENVIRONMENTS

Rigid body simulation is an integral part of Virtual Environments (VE) for autonomous planning, training, and design tasks. The underlying physics-based simulation of VE must be accurate and computationally fast enough for the intended application, which unfortunately are conflicting requirements. Two ways to perform fast and high fidelity physics-based simulation are: (1) model simplification, and (2) parallel computation. Model simplification can be used to allow simulation at an interactive rate while introducing an acceptable level of error. Currently, manual model simplification is the most common way of performing simulation speedup but it is time consuming. Hence, in order to reduce the development time of VEs, automated model simplification is needed. The dissertation presents an automated model simplification approach based on geometric reasoning, spatial decomposition, and temporal coherence. Geometric reasoning is used to develop an accessibility based algorithm for removing portions of geometric models that do not play any role in rigid body to rigid body interaction simulation. Removing such inaccessible portions of the interacting rigid body models has no influence on the simulation accuracy but reduces computation time significantly. Spatial decomposition is used to develop a clustering algorithm that reduces the number of fluid pressure computations resulting in significant speedup of rigid body and fluid interaction simulation. Temporal coherence algorithm reuses the computed force values from rigid body to fluid interaction based on the coherence of fluid surrounding the rigid body. The simulations are further sped up by performing computing on graphics processing unit (GPU). The dissertation also presents the issues pertaining to the development of parallel algorithms for rigid body simulations both on multi-core processors and GPU. The developed algorithms have enabled real-time, high fidelity, six degrees of freedom, and time domain simulation of unmanned sea surface vehicles (USSV) and can be used for autonomous motion planning, tele-operation, and learning from demonstration applications

SIA matrices and non-negative stationary subdivision

This dissertation is concerned with SIA matrices and non-negative stationary subdivision, and is organized as follows: After an introducing chapter where some basic notation is given we describe, in Chapter 3, how non-negative subdivision is connected to a corresponding non-homogenous Markov process. The family of matrices A, built from the mask of the subdivision scheme, is introduced. Among other results, Lemma 3.1 and Lemma 3.2 relate the coefficients of the iterated masks to matrix products from the family A, and in the limiting case the values of the basic limit function are found from the entries in an infinite product of matrices. Chapter 4 and Chapter 5 are the core of this dissertation. In Chapter 4, we first review some spectral and graph properties of row-stochastic matrices and, in particular, of SIA matrices. We point to the notion of scrambling power, introduced by Hajnal [16], and of the related coefficient of ergodicity. We also consider the directed graph of such matrices, and we improve upon a condition given by Ren and Beard in [30]. Then we study finite families of SIA matrices, the properties of their indicator matrices and the connectivity of their directed graphs. We consider this chapter to be an important contribution to the theory of non-negative subdivision, since it explains the background in order to apply the convergence result of Anthonisse and Tijms [2], which we reprove in Section 4.6, to rank one convergence of infinite products of row stochastic matrices. It does not use the notion of joint spectral radius but the (equivalent) coefficient of ergodicity. Properties equivalent to SIA are listed in Lemma 4.7 and in the subsequent Lemma 4.8; they connect the SIA property to equivalent conditions (scrambling property, positive column property) as they appear in the existing literature dealing with convergence of non-negative subdivision. The fifth chapter of the dissertation contains the full proof of the characterization of uniform convergence for non-negative subdivision, for the univariate and bivariate case, the latter one being a representative for multivariate aspects. It uses the pointwise definition of the limit function at dyadic points - refering to the dyadic expansion of real vectors from the unit cube - using the Anthonisse-Tijms pointwise convergence result, and employs the proper extension of the Micchelli-Prautzsch compatibility condition to the multivariate case, taking care of the ambiguity of representation of dyadic points. As a consequence, the Hölder exponent of the basic limit function can be expressed in terms of the coefficient of ergodicity of the family A. Our convergence theorems, in Theorem 5.1 and Theorem 5.8, include the existing characterizations of uniform convergence for non-negative univariate and bivariate subdivision from the literature, except for the GCD condition, which seems to be a condition applicable to univariate subdivision only. Chapter 5 also reports on some further attempts where we have tried to extend conditions from univariate subdivision, which are sufficient for convergence, to the bivariate case. We could find a bivariate analogue of Melkman's univariate string condition, which we call - in the bivariate case - a rectangular string condition. The chapter concludes with stating the fact that uniform convergence of non-negative stationary subdivision is a property of the support of the mask alone, modulo some apparent necessary conditions such as the sum rules. A typical application of this support property characterizes uniform convergence in the case where the mask is a convex combination of other masks. The dissertation ends with two short chapters on tensor product and box spline subdivision, and an appendix where some definitions and useful lemmas and theorems about matrix and graph theory are stated without proofs.Die Dissertation behandelt den Zusammenhang zwischen SIA-Matrizen und nicht-negativer Subdivision. Sie ist folgendermaßen aufgebaut: Nach einem einleitenden Kapitel wird in Kapitel 2 die grundlegende Notation bereit gestellt. Anschließend beschreiben wir in Kapitel 3 zunächst den formalen Zusammenhang zwischen nicht-negativer Subdivision und einem hierzu gehörenden Markov-Prozess. Wir führen dazu eine Familie A von Matrizen ein, die aus der Maske des Subdivisionsschemas aufgebaut werden. Unter anderem beschreiben Lemma 3.1 and Lemma 3.2, wie die Koeffizienten der iterierten Masken sich durch Matrix-Produkte aus der Familie A deuten lassen. Im Grenzfall ergeben sich so die Funktionswerte der Fundamentalfunktion des Subdivisionsprozesses aus den Einträgen eines unendlichen Matrix-Produktes. Die Kapitel 4 und 5 stellen den zentralen Beitrag dieser Dissertation dar. Zunächst geben wir dort einen Überblick über Spektraleigenschaften von zeilenstochastischen Matrizen und Eigenschaften ihrer gerichteten Graphen, wobei die SIA-Eigenschaft wieder im Vordergrund steht. Wir verweisen auf den Begriff der 'scrambling power', eingeführt von Hajnal [16], und den zugehörigen ergodischen Koeffizienten. Hinsichtlich der Eigenschaften gerichteter Graphen von SIA-Matrizen verbessern wir eine Aussage von Ren und Beard [30]. Anschließend studieren wir Familien von SIA-Matrizen, deren Indikator-Matrizen und die Zusammenhangseigenschaften der betreffenden gerichteten Graphen. Wir glauben, dass dies einen wichtigen Beitrag zur Theorie nicht-negativer Subdivision darstellt, weil dieser Hintergrund nunmehr eine Anwendung des Konvergenzsatzes von Anthonisse und Tijms [2] zulässt. Diesen Konvergenzsatz greifen wir in Abschnitt 4.6 auf. Er beschreibt die Rang-Eins-Konvergenz ohne Bezug auf den 'joint spectral radius', sondern verwendet hierzu den (äquivalenten) Begriff des ergodischen Koeffizienten. Eine Reihe von Eigenschaften, die zur SIA-Eigenschaft äquivalent sind, werden in Lemma 4.7 und dem anschlieÿenden Lemma 4.8 aufgelistet; diese nehmen Bezug auf Eigenschaften (scrambling property, positive column property), wie sie in der bisherigen Literatur zur Konvergenz nicht-negativer Subdivision auftauchen. Kapitel 5 enthält einen vollständigen Beweis der Charakterisierung gleichmäßiger Konvergenz für nicht-negative Subdivision, im Fall einer und zweier Veränderlichen, wobei letzterer Fall repräsentativ ist für den Fall mehrerer Variablen. Er benutzt die punktweise Definition der Grenzfunktion in dyadischen Punkten - wobei auf die Binärentwicklung reeller Vektoren aus dem Einheitswürfel Bezug genommen wird - unter Bezug auf den Konvergenzsatz von Anthonisse-Tijms. Eine geeignete Verallgemeinerung der Kompatibilitätsbedingung von Micchelli und Prautzsch berücksichtigt hierbei die Mehrdeutigkeit der Binärentwicklung in dyadischen Punkten. Als Folge hiervon lässt sich der Hölder Exponent der Fundamentalfunktion durch den ergodischen Koeffizienten der Familie A ausdrücken. Unsere Ergebnisse zur Konvergenz, in Theorem 5.1 und Theorem 5.8, fassen die existierenden Ergebnisse zur nicht-negativen Subdivision zusammen. Ausgenommen ist hiervon die GCD-Bedingung, die offensichtlich einen Spezialfall darstellt, der sich auf den Fall einer einzigen Variablen bezieht. Kapitel 5 enthält auch einige Ansätze zu unserem Versuch, hinreichende Bedingungen zur gleichmäßigen Konvergenz, die im Fall einer Variablen bekannt sind, auf den Fall zweier oder mehrerer Variablen zu verallgemeinern. Ein Analogon zu Melkmans [27] univariater 'string condition' ist unsere 'rectangular string condition' für den bivariaten Fall. Das Kapitel schließt mit einem Hinweis auf die Tatsache, dass die Eigenschaft der gleichmäßigen Konvergenz tatsächlich allein eine Trägereigenschaft der Maske ist, modulo offensichtlicher notwendiger Zusatzeigenschaften wie z. B. die 'sum rule'. Eine typische Anwendung dieser Trägereigenschaft liefert die Charakterisierung der gleichmäßigen Konvergenz bei Masken, die sich als Konvexkombinationen einfacherer Masken deuten lassen. Die Dissertation schließt mit zwei kurzen Kapiteln zur Tensorprodukt- und zur Box-Spline-Subdivision, sowie einem Anhang, in dem Definitionen und nützliche Hilfsergebnisse und Theoreme zur Theorie von Matrizen und deren Graphen ohne Beweise aufgeführt sind

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.