Functional equations on finite groups of substitutions

Motivated by some investigations of Babbage, we study a class of single variable functional equations. These are functional equations involving one unknown function and a finite set of known functions that form a group under the operation of composition. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. In the proof of the main result, the Implicit Function Theorem and, when the stabilizer is nontrivial, the Global Existence and Uniqueness Theorem play a key role.     

關於區域的映照半徑

<正> §1.設α_1,α_2,…,α_n是z平面上n個相異的點,G_1,G_2,…,G_n是z平面上n個不互相重叠的有限區域,α_k屬於G_k,記G_k對於α_k的映照半徑為     

An Extremal Relation of the Theory of Approximation of Functions by Algebraic Polynomials

In this paper we find an extremal relation of the theory of approximation of functions on a finite interval by algebraic polynomials. In the Legendre case we find some estimations and representation in the form of a series of the constant from the extremal relation.     

Equilibrium of vector potentials and uniformization of the algebraic curves of genus 0

We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the Hermite-Padé approximants. Solutions of such problems can be expressed by means of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the Hermite-Padé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system.     

A numerical-analytical solution of multi-term fractional-order differential equations

In this paper, a solution to initial value problems for fractional-order linear commensurate multi-term differential equations with Caputo derivatives is presented. The solution is obtained in the form of a finite sum of the Mittag-Leffler–type functions and the meta-trigonometric cosine function by using a numerical-analytical method. The results of presented numerical experiments show that for high accuracy calculations of these functions, the multi-precision arithmetic must be applied. The approach for solving of the initial value problems for generalized Basset equation, generalized Bagley-Torvik equation, and multi-term fractional equation is demonstrated.     

Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.     

Algebraic solutions of the Lamé equation, revisited

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S₄, then the degree parameter of the equation may differ by ±1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.     

Solution of the advective-dispersive transport equation using a least squares collocation,Eulerian-Lagrangian method

Numerical solution of the advective-dispersive transport equation is difficult when advection dominates. Difficulties arise because of the first-order spatial derivatives which can be elminated by a local coordinate transformation to the characteristic lines of the first order hyperbolic portion of the equation. The resulting differential equation is discretized using a finite difference in time and finite elements in space employing cubic Hermite basis functions. The residuals at individual collocation points are then computed. The sum of the squares of the residuals is minimized to form the necessary set of algebraic equations. The method has performed well in one-dimensional test problems.     

Error analysis of collocation method based on reproducing kernel approximation

Solving partial differential equations (PDE) with strong form collocation and nonlocal approximation functions such as orthogonal polynomials, trigonometric functions, and radial basis functions exhibits exponential convergence rates; however, it yields a full matrix and suffers from ill conditioning. In this work, we discuss a reproducing kernel collocation method, where the reproducing kernel (RK) shape functions with compact support are used as approximation functions. This approach offers algebraic convergence rate, but the method is stable like the finite element method. We provide mathematical results consisting of the optimal error estimation, upper bound of condition number, and the desirable relationship between the number of nodal points and the number of collocation points. We show that using RK shape function for collocation of strong form, the degree of polynomial basis functions has to be larger than one for convergence, which is different from the condition for weak formulation. Numerical results are also presented to validate the theoretical analysis. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 554–580, 2011     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

Extremal problems for Fredholm eigenvalues

The study of extremal problems for Fredholm eigenvalues was initiated by Schiffer in the context of the existence of conformal maps onto canonical domains. We present a different approach to solving rather general extremal problems for Fredholm eigenvalues related to appropriate univalent functions with quasiconformal extensions. It involves the complex geometry of the universal Teichmüller space.     

Toral algebraic sets and function theory on polydisks

A toral algebraic set A is an algebraic set in ℂ ⁿ whose intersection with T ⁿ is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.     

与一族单叶函数相关的平均直径的下界问题

<正> §1.区域1<|ζ|<∞上单叶的正则函数 w=F(ζ)=ζ+α_0+α_1/ζ+α_2/ζ~2+…的全体聚成一族,记它做∑.∑中的函数,在区域1<|ζ|<∞上无零点的,其全体是Σ的一子族∑~0.设 p 是一正整数,∑~0中函数 w=F(ζ)经过变换 F_1(ζ)=(?)而得的函数 w=F_1(ζ)仍属于∑~0,其全体是Σ~0的一子族∑_p.设函数 w=F(ζ)映     

On the homogeneous algebraic graphs of large girth and their applications

Families of finite graphs of large girth were introduced in classical extremal graph theory. One important theoretical result here is the upper bound on the maximal size of the graph with girth ?2d established in Even Circuit Theorem by P. Erdös. We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth ?2d is established. Getting the lower bound, we use the family of bipartite graphs D(n,K) with n?2 over a field K, whose partition sets are two copies of the vector space Kⁿ. We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory. Finally, we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography.     

Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems

Many applied problems are described by differential algebraic systems. Complex Rosenbrock schemes are proposed for the numerical integration of differential algebraic systems by the ?-embedding method. The method is proved to converge quadratically. The scheme is shown to be applicable even to superstiff systems. The method makes it possible to perform computations with a guaranteed accuracy. An equation is derived that describes the leading term of the error in the method as a function of time. An algorithm extending the method to systems of differential equations for complex-valued functions is proposed. Examples of numerical computations are given.     

Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.     

不相混流体在多孔介质中置换不稳定现象的三维数值模拟

本文采用不连续势函数在待定界面上变化的方法模拟不相混流体在多孔介质中置换．描述流体在多孔介质中的流动的连续方程在三维直角坐标系下被转化为七点式代数方程．采用强隐式法（Stronglyimplictprocedure）求解，再确定变动的流体间界面．计算中只模拟从稳定向不稳定置换转化这一过程中界面的变动．计算中考虑了多孔介质的浸润特性、毛细管压力、及多孔介质渗透不均匀性．     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

关于Borg定理

Borg定理是判定周期系数二阶线性微分方程稳定的一个重要定理,这个定理在一定意义下是不可改进的.本文利用判别式的一个新形式,在弱的限制下,得到判定周期系数二阶线性微分方程稳定的定理,所得结论改进了Borg定理的判定结果.