Оптимальний вибір площин, на яких розміщені томограми, в комп’ютерній томографії

The solution of the problem of reconstructing the internal structure of a three-dimensional body by the known tomograms produced by a computer to-mograph using interflatation of functions and blending approximation is proposed. The known methods ofapproximating functions of one and two variables by interpolation type piecewise constant splines using means and medians are also considered. The paper presents an algorithm for optimizing the choice of the planes in which the tomogramsproduced by a computer tomograph are placed. The case is considered when all the tomograms are parallel to each other. The algorithm developeduses approximations of objects by classical piecewise constant splines. The internal structure of a three-dimensional body (density or absorption coefficient) is assumed to be given by a function of three variables of the form h(x, y, z ) = f (x)g( y, z), where g is an arbitrary function, provided that f is a monotone function on a closed segment. The method of optimal choice of the planes for placing the tomograms is implemented using MathCad computer software.Представлено розв’язок задачі відновлення внутрішньої структури тривимірного тіла за відомими томограмами, що поступають з комп’ютерного томографу, за допомогою інтерфлетації функцій та мішаної апроксимації. Розглянуто також відомі методи наближення функцій однієї та двоx змінних кусково-сталими сплайнами інтерполяційного типу, з використанням середніх та медіан. В статті пропонується алгоритм оптимізації вибору площин, на яких розміщені томограми, що поступають з комп’ютерного томографу. Розглядається випадок, коли всі томограми паралельні одна одній. Запропонований алгоритм використовує наближення об’єктів класичними кусково-сталими сплайнами. При побудові алгоритму істотно використовується припущення про те, що внутрішня структура тривимірного тіла (щільність або коефіцієнт поглинання) є функцією від трьох змінних вигляду h(x, y, z ) = f (x)g( y, z), де g – довільна функція, при умові, що f – монотонна функція на замкненому відрізку. Представлена чисельна реалізація методу оптимального вибору площин, на яких лежать томограми, в системі компʼютерної математики MathCad

Classical Functional Bethe Ansatz for SL(N)SL(N): separation of variables for the magnetic chain

The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which gives separation variables for systems for which an $R$-matrix is known. Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated) $R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing rational functions \A(u) and \B(u) of the matrix elements of $T(u)$, so that, in the generic case, the zeros $x_i$ of \B(u) are the separation coordinates and the P_i=\A(x_i) provide their conjugate momenta. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed.Comment: 14pp LaTex,DAMTP 94-1

On moduli of rings and quadrilaterals: algorithms and experiments

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds

Explicit local time-stepping methods for time-dependent wave propagation

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leap-frog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.Comment: overview paper, typos added, references updated. arXiv admin note: substantial text overlap with arXiv:1109.448

Asymptotic theory of two-dimensional trailing-edge flows

Problems of laminar and turbulent viscous interaction near trailing edges of streamlined bodies are considered. Asymptotic expansions of the Navier-Stokes equations in the limit of large Reynolds numbers are used to describe the local solution near the trailing edge of cusped or nearly cusped airfoils at small angles of attack in compressible flow. A complicated inverse iterative procedure, involving finite-difference solutions of the triple-deck equations coupled with asymptotic solutions of the boundary values, is used to accurately solve the viscous interaction problem. Results are given for the correction to the boundary-layer solution for drag of a finite flat plate at zero angle of attack and for the viscous correction to the lift of an airfoil at incidence. A rational asymptotic theory is developed for treating turbulent interactions near trailing edges and is shown to lead to a multilayer structure of turbulent boundary layers. The flow over most of the boundary layer is described by a Lighthill model of inviscid rotational flow. The main features of the model are discussed and a sample solution for the skin friction is obtained and compared with the data of Schubauer and Klebanoff for a turbulent flow in a moderately large adverse pressure gradient

Curve network interpolation by C1C^1 quadratic B-spline surfaces

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate $C^1$ quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.Comment: With respect to the previous version, this version of the paper is improved. The results have been reorganized and it is more general since it deals with non uniform knot partitions. Accepted for publication in Computer Aided Geometric Design, October 201

Classes of confining gauge field configurations

We present a numerical method to compute path integrals in effective SU(2) Yang-Mills theories. The basic idea is to approximate the Yang-Mills path integral by summing over all gauge field configurations, which can be represented as a linear superposition of a small number of localized building blocks. With a suitable choice of building blocks many essential features of SU(2) Yang-Mills theory can be reproduced, particularly confinement. The analysis of our results leads to the conclusion that topological charge as well as extended structures are essential elements of confining gauge field configurations.Comment: 18 pages, 16 figures, several sections adde