非对角二元二次函数逼近的存在性和局部性质

§ 1　IntroductionThis paper is concerned with the properties of the simple off-diagonal bivariatequadratic Hermite-Padé approximation.Thisapproximation may be defined asfollows(see,for example,[1 ] ) .Let f(x,y) be a bivariate function,analytic in some neighbourhood of the origin(0 ,0 ) ,whose series expansion about the origin is known.Let a0 (x,y) ,a1 (x,y) ,a2 (x,y) bebivariate polynomials,a0 (x,y) = ki,j=0 a(0 )ij xiyj,a1 (x,y) = ni,j=0 a(1 )ij xiyj,a2 (x,y) = mi,j=0 a(2 )ij xiyj,such th…     

Optimal objective function approximation for separable convex quadratic programming

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.Corresponding author.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.     

An algorithm for the quadratic approximation

The quadratic approximation is a three dimensional analogue of the two dimensional Padé approximation. A determinantal expression for the polynomial coefficients of the quadratic approximation is given. A recursive algorithm for the construction of these coefficients is derived. The algorithm constructs a table of quadratic approximations analogous to the Padé table of rational approximations.     

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.     

On the asymptotic behaviour of the ideal counting function in quadratic number fields

LetK be a quadratic number field with discriminantD and denote byF(n) the number of integral ideals with norm equal ton. Forr≥1 the following formula is proved $$\sum\limits_{n \leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$ HereM _k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by \(|E_K (x,r)|<< |D|^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2} + \varepsilon } x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6} + \varepsilon } \) uniformly for \(r<< |D|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-0em} 6}} \) Moreover,E _k (x,r) is small on average, i.e \(\int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + \varepsilon } X^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2} + \varepsilon } \) uniformly for \(r<< |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0em} 4}} \) .     

The symmetric quadratic knapsack problem: approximation and scheduling applications

This paper reviews two problems of Boolean non-linear programming: the Symmetric Quadratic Knapsack Problem and the Half-Product Problem. The problems are related since they have a similar quadratic non-separable objective function. For these problems, we focus on the development of fully polynomial-time approximation schemes, especially of those with strongly polynomial time, and on their applications to various scheduling problems.     

A counterexample to the existence of a local plurisubharmonic peak function at infinity

We give an example of an unbounded pseudoconvex Reinhardt domain in , which is Kobayashi complete but admits no local plurisubharmonic peak function at infinity.

     

The quadratic schur subgroup over local and global fields

Support by N.S.E.R.C. grant #A8778 gratefully acknowledged     

On the best approximation and polynomials of the least quadratic deviation

Analysis Mathematica - ПустьС 2π — пространс тво...     

The asymptotic probabilistic behaviour of quadratic sum assignment problems

In this paper a surprising probabilistic behaviour of quadratic sum assignment problems is shown. The relative difference between worst and optimal solution value tends to zero with probability tending to one as the size of the problem goes to infinity. This result suggests that for high dimensional quadratic assignment problems even very simple approximation algorithms can in practice yield good suboptimal solutions.     

On the complexity of finding a local minimizer of a quadratic function over a polytope

Mathematical Programming - We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $$c^n$$ (for any constant $$c ge 0$$ ) of a local...     

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.     

On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations

A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action.     

Strong approximation of quadrics and representations of quadratic forms

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V. Suppose that M is a lattice on V, and that N is a lattice on U which is represented by M locally everywhere. The main result of this paper is a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.