Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

In this paper we consider interpolatory quadrature formulae with multiple nodes that have the maximal trigonometric degree of exactness. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, Academie-Verlag, Berlin, 1970). We introduce and consider the so-called σ-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Also, some numerical examples are included. The authors were supported in part by the Serbian Ministry of Science and Technological Development (Project: Orthogonal Systems and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”).     

On the degree distribution of the nodes in increasing trees

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set {1,…,n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity degree of node j in a random tree of size n and give closed formulae for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via a tree evolution process. Furthermore limiting distribution results of this parameter are given, which completely characterize the phase change behavior depending on the growth of j compared to n.     

Approximating the integral of a multifunction

Given a weakly converging sequence of measures, we study the convergence of the corresponding integrals of a continuous unbounded multifunction. We also study the implication of these results to variational problems, and provide further approximating results for the integral of a multifunction, involving both truncation of the multifunction and measure approximation.     

一个四分子饱和可逆生化反应模型的定性分析

应用微分方程定性理论的相关知识研究了生化反应中一类具有二重饱和度的四分子可逆生化反应模型,得到了该系统极限环存在,不存在和唯一的充分条件.     

A Formula Approximating the Root of a Function

A formula is presented to approximate the root of a single variablefunction given two x-values, one on either side of the root,and the corresponding function values (of opposite sign) andderivatives. The method is to interpolate a quadratic splinewith a knot on the x-axis, between the two x-values; the positionof the knot is determined, and is taken as the estimate of theroot.     

Dynamic reaction of a pseudo-ceramic half-space with a longitudinal shear crack to concentrated forces

The dynamic problem of electroelasticity for a piezoelectric half-space with tunnel cavities-cuts is examined. The problem is reduced to the solution of an integrodifferential singular equation in terms of the amplitude of the jump in displacements at the cut. Equations are obtained for the dynamic mechanial-stress intensity factor. The influence exerted by the curvature of the crack, its orientation, and the normalized wave number on the intensity factor is studied numerically.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 50–55, 1989.     

Approximating and learning by Lipschitz kernel on the sphere

This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.     

Approximating the Stance Map of a 2-DOF Monoped Runner

Summary. We report in this paper a relatively simple means of generating closed-form approximants to the return map associated with a family of nonintegrable Hamiltonian systems. These systems arise in consideration of legged locomotion by animals and robots. The approximations proceed through the iterated application of the mean value theorem for integral operators applied to a nonintegrable perturbation of the system of interest. Both the accuracy of these approximants and their algebraic intractability grow in a relatively controlled manner. Received May 26, 1999; accepted January 6, 2000 Online publication July 19, 2000     

Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid

By means of a result due to Fiedler, we obtain a relation between the lowest degree of the Jones polynomial and the unknotting number for any link which has a closed positive braid diagram. Furthermore, we obtain relations between the lowest degree and the slice euler characteristic or the four-dimensional clasp number.     

Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

This article is concerned with an initial-boundary value problem for an elliptic-parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed in Alber and Zhu (2007) [4], and investigated in Alber and Zhu (2006) [3] the existence of weak solutions which are defined in a standard way, however the key technique used in Alber and Zhu (2006) [3] is not applicable to multi-dimensional problem. One of the motivations of this study is to solve this multi-dimensional problem, and another is to investigate the sharp interface limits. Thus we define weak solutions in a way, which is different from Alber and Zhu (2006) [3], by using the notion of viscosity solution. We do prove successfully the existence of weak solutions in this sense for one-dimensional problem, yet the multi-dimensional problem is still open.     

On the number of nodes in self-contained integration formulas of degree two for compact planar regions

Approximate integration formulas of precision two for compact planar regions are studied. We restrict our attention to formulas for which the nodes lie in the region and consider the problem of determining the minimum number of nodes attainable. While Stroud has shown that this minimum number is always at least 3, we find a two-parameter family of symmetric star-shaped regions for which it is equal to 4. That is, the conditions of symmetry and star-shapedness are not sufficient for the existence of such formulas with the minimal number of nodes.Work performed under the auspices of the U.S. Atomic Energy Commission.     

Variational sensitivity analysis of physical reaction forces and application to multiscale problems

Nowadays profitability and efficiency of practical and industrial applications and therefore of material design is becoming a more important issue. Due to different well-known and established approaches for analysis and simulation of complex heterogeneous materials on multiple scales based on numerical homogenisation techniques, development and production of high performance materials (using 3d printers for example) and as a consequence optimisation and design of materials is reality. The objective is to find optimal structures with optimal material distribution under given constraints related to posed problems and tailoring applications to their special requirements. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximating eigenvectors and eigenvalues across a wide range of design

This paper concerns optimization of the design of large structures in which the solution to a finite-element-based vibration problem is part of the objective function. A method is presented where a spline-fit is used as an approximation of the eigenvalues and eigenvectors across a given design range. subroutines facilitate nonlinear sensitivity calculations which provide the eigenvalue and eigenvector derivatives used in the approximation. This paper presents an example that illustrates good approximations to frequency and mode shapes valid across a wide range of design changes. The method provides the potential for changing mode shapes interactively, and using that information as part of the optimization process.     

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients.

A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities.

In addition, we study the accuracy of the approximations and present some numerical examples.

     

On the stability of the stationary front of an exothermic reaction in a condensed phase

A system in the theory of combustion is considered in which the Arrhenius dependence of the rate of reaction on temperature is modified at temperatures close to the initial temperature in order to ensure the existence of a stationary front. Investigation of the stability of the stationary wave leads to the determination of the neutral curves and the amplitude equation. A coefficient, which defines the nature of the loss of stability, is calculated for the case when there is loss of stability with the occurrence of selfexcited oscillations of the plane front. This coefficient is always negative, which corresponds to soft excitation of selfexcited oscillations. The results obtained are compared with the data from numerical experiments /1–5/.     

赋范线性空间中一类算子的不动点逼近问题

使用新的分析技巧研究了一类φ-强伪压缩影像的不动点和φ-强增生影像的零点的迭代逼近问题,推广并改进了许多相应的结果.