In this thesis, we develops a Pocket PC-based 3D object design system using implicit functions, which allows users to design and view 3D objects easily at anytime and anywhere. With the growing wireless techniques, Pocket PC can easily connect to the Internet via WLAN, GPRS, WCDMA, etc. The problems of insufficient computing power caused by the limited hardware on Pocket PCs can be solved by the client-server model, which performs complex computation on the workstation, and transmit polygon mesh data to the Pocket PC. Via the proposed system, one can design and visualize his any innovative ideas instantly wherever he is.

In this thesis we develop a 3D soft object modeling system with implicit functions. Our system provides implicit functions including the primitive functions and blending functions to modeling soft object. The system satisfies some requirements, such as intuitive operating environment, the Model-View-Controller (MVC) system architecture, and extensibility, etc. The system consists of four major parts: (1) the graphical user interface (GUI); (2) the constructive tree viewing Subsystem; (3) the implicit function operating Subsystem; (4) the 3D scene graph rendering Subsystem. The GUI initializes the window frame and responds to the event messages dispatched from the OS or applications. The constructive tree viewing Subsystem presents the composition of the object modal with the tree structure. The implicit function operating system accomplishes the polygon tessellation, that is, it translates the blending tree with the implicit functions to polygon meshes. The 3D scene graph rendering system renders the polygon mesh and provides 3D scene graph manipulations.

Consider the D-optimal designs for a combined polynomial of degree d and trigonometric of order m regression on a partial circle [see Graybill (1976), p. 324]. It is shown that the structure of the optimal design depends only on the length of the design interval and that the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure.

A novel idea for perception of object surfaces is presented by so called "shape descriptors". Such idea is as an abstract level to represent the object surface by some real numbers. It has the similar idea like as the Fourier coefficients of mapping a function f(x) to frequency domain by Fourier transform. The main goal of this thesis is to define some of the key issues in understanding of an object shape and also to find a modeling methodology to create the "shape descriptors". The modeling methodology is designed based on a variational interpolation technique. Such technique is used to generate a group of variational implicit functions with help of radial basis functions. In our modeling methodology, we randomly choose some reference points on a set of related concentric spheres around a 3D point cloud data as known values in variational implicit functions. The "shape descriptors" are found from these implicit functions implementing LU decomposition. We show that the "shape descriptors" are invariant to size and positioning (rotation and translation) changes of a shape and they are also effective tools for matching of two similar objects surfaces.

Many chemical engineering applications, such as safety verification and parameter estimation, require global optimization of dynamic models. Global optimization algorithms typically require obtaining global bounding information of the dynamic system, to aid in locating and verifying the global optimum. The typical approach for providing these bounds is to generate convex relaxations of the dynamic system and minimize them using a local optimization solver. Tighter convex relaxations typically lead to tighter lower bounds, so that the number of iterations in global optimization algorithms can be reduced. To carry out this local optimization efficiently, subgradient-based solvers require gradients or subgradients to be furnished. Smooth convex relaxations would aid local optimization even more. To address these issues and improve the computational performance of global dynamic optimization, this thesis proposes several novel formulations for constructing tight convex relaxations of dynamic systems. In some cases, these relaxations are smooth. Firstly, a new strategy is developed to generate convex relaxations of implicit functions, under minimal assumptions. These convex relaxations are described by parametric programs whose constraints are convex relaxations of the residual function. Compared with established methods for relaxing implicit functions, this new approach does not assume uniqueness of the implicit function and does not require the original residual function to be factorable. This new strategy was demonstrated to construct tighter convex relaxations in multiple numerical examples. Moreover, this new convex relaxation strategy extends to inverse functions, feasible-set mappings in constraint satisfaction problems, as well as parametric ordinary differential equations (ODEs). Using a proof-of-concept implementation in Julia, numerical examples are presented to illustrate the convex relaxations produced for various implicit functions and optimal-value functions. In certain cases, these convex relaxations are tighter than those generated with existing methods. Secondly, a novel optimization-based framework is introduced for computing time-varying interval bounds for ODEs. Such interval bounds are useful for constructing convex relaxations of ODEs, and tighter interval bounds typically translate into tighter convex relaxations. This framework includes several established bounding approaches, but also includes many new approaches. Some of these new methods can generate tighter interval bounds than established methods, which are potentially helpful for constructing tighter convex relaxations of ODEs. Several of these approaches have been implemented in Julia. Thirdly, a new approach is developed to improve a state-of-the-art ODE relaxation method and generate tighter and smooth convex relaxations. Unlike state-of-the-art methods, the auxiliary ODEs used in these new methods for computing convex relaxations have continuous right-hand side functions. Such continuity not only makes the new methods easier to implement, but also permits the evaluation of the subgradients of convex relaxations. Under some additional assumptions, differentiable convex relaxations can be constructed. Besides that, it is demonstrated that the new convex relaxations are at least as tight as state-of-the-art methods, which benefits global dynamic optimization. This approach has been implemented in Julia, and numerical examples are presented. Lastly, a new approach is proposed for generating a guaranteed lower bound for the optimal solution value of a nonconvex optimal control problem (OCP). This lower bound is obtained by constructing a relaxed convex OCP that satisfies the sufficient optimality conditions of Pontryagin's Minimum Principle. Such lower bounding information is useful for optimizing the original nonconvex OCP to a global minimum using deterministic global optimization algorithms. Compared with established methods for underestimating nonconvex OCPs, this new approach constructs tighter lower bounds. Moreover, since it does not involve any numerical approximation of the control and state trajectories, it provides lower bounds that are reliable and consistent. This approach has been implemented for control-affine systems, and numerical examples are presented. / Thesis / Doctor of Philosophy (PhD)

This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w'(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

This paper is concerned with the problem of constructing A-optimal design for polynomial regression with analytic weight function on the interval [m-a,m+a]. It is shown that the structure of the optimal design depends on a and weight function only, as a close to 0. Moreover, if the weight function is an analytic function a, then a scaled version of optimal support points and weights is analytic functions of a at $a=0$. We make use of a Taylor expansion which coefficients can be determined recursively, for calculating the A-optimal designs.

This paper is devoted to studying the problem of constructing Ds-optimal design for d-th degree polynomial regression with analytic weight function on the interval [m-a,m+a],m,a in R. It is demonstrated that the structure of the optimal design depends on d, a and weight function only, as a close to 0. Moreover, the Taylor polynomials of the scaled versions of the optimal support points and weights can be computed via a recursive formula.

Orientador: Sandra Augusta Santos / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T21:01:17Z (GMT). No. of bitstreams: 1 SilvaJunior_EpitacioPedroda_M.pdf: 1234663 bytes, checksum: 67cac0299f5435bbce39b930a12c49c4 (MD5) Previous issue date: 2008 / Resumo: Este trabalho tem dois objetivos principais. O primeiro é apresentar um contexto aplicado para o uso do Teorema da Função Implícita. Este resultado permite analisar a influência da precisão dos relógios envolvidos no funcionamento do GPS (Global Positioning System), cujo receptor é usado para determinar as coordenadas de um ponto da Terra, O segundo objetivo é estabelecer algumas conexões entre conceitos da Geometria Analítica do Ensino Médio com a,do Ensino Superior, bem como com o Cálculo de Várias Variáveis, aparentemente desconectados para o aluno do Ensino Superior. Para tanto, a idéia foi partir do cálculo da área de regiões simples, como triângulos e polígonos, e chegar à computação de áreas de regiões mais sofisticadas, por meio do Teorema de 'Green. Este resultado permite justificar o funcionamento do aparelho mecânico denominado planímetro / Abstract: The objective of this work is twofold. First, it presents an applied context for using the Implicit Function Theorem. This result allows to analyze the influence of the accuracy of the clocks involved in the working of the GPS (Global Positioning System), the receiver of which is a device used to locate the position of a point on the surface of the earth. Second, it points some connections among concepts of Analytic Geometry, together with Calculus of Several Variables, apparently not linked for the university student. To achieve such goal, the idea was to start with the calculus of the area of simple regions, like triangles and polygons, and reach the computation of more sophisticated areas by using Green's theorem. This result allows to justify the working of the mechanical device called planimeter / Mestrado / Matematica / Mestre em Matemática

The thesis follows on Petr Vopìnka's alternative theory of sets and semisets by extending notions of in nite closeness and monad for real spaces of several variables. It speci es and explains on examples the basic terminology of this theory, namely notions of sets, semisets and domains. It brings up two worlds | an ancient and a classical one | by which it shows a dual way of looking at real functions of several variables. That is used for examining local properties like continuity, limit or derivative of a function at a point. The peak of the thesis is an alternative formulation of the implicit function theorem and the inverse function theorem. The thesis also contains translation rules, which allow us to reformulate all these results from an alternative into a traditional formulation used in mathematical analysis.