Convex Optimal Designs for Compound Polynomial Extrapolation

The extrapolation design problem for polynomial regression model on the design space [–1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs | _m ^* (z), _2m ^* (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.     

Orlicz空间内正系数代数多项式倒数对非负连续函数的逼近

本文研究了Bernstein-Durrmeyer代数多项式倒数对非负连续函数在Orlicz空间中的逼近问题.利用光滑模和K-泛函等工具,获得了收敛速度的估计,所得的结果比Lp空间内的相应结果具有拓展的意义.     

Coboundaries, Flows, and Tutte Polynomials of Matrices

Using matroid duality and the critical problem, we show that certain evaluations of the Tutte polynomial of a matroid represented as a matrix over a finite field GF(q) can be interpreted as weighted sums over pairs f , g of functions defined from the ground set to GF(q) whose difference f – g is the restriction of a linear functional on the column space of the matrix. Similar interpretations are given for the characteristic polynomial evaluated at q. These interpretations extend and elaborate interpretations for Tutte and chromatic polynomials of graphs due to Goodall and Matiyasevich. Received July 14, 2006     

Roots and polynomials as Homeomorphic spaces

We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree n and the space of n roots with suitable topologies, we are able to formulate the classical theorems in the form of a homeomorphism. Related topological facts are also considered.     

Extensions of homogeneous polynomials on c 0(l 2 i )

We show that a 2-homogeneous polynomial on the complex Banach space c ₀ l ₂ ⁱ ) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c ₀(l ₂ ⁱ ). The second author was supported by FAPESP, Brazil, Research Grant 01/04220-8.     

The images of Lie polynomials evaluated on matrices

Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M₂(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial s_k is not a Lie polynomial, for k>2.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

Zero-sets of complex homogeneous polynomials

In this paper we study the zero-sets of continuous n-homogeneous polynomials on complex nonseparable Banach spaces. We prove that the zero-set of any complex n-homogeneous polynomial P is a subspace if, and only if, there is a functional ? such that P(x)=? (x)ⁿ for every x. We give sufficient conditions on the Banach space to ensure that every continuous 2-homogeneous polynomial is identically zero on a nonseparable subspace. Also, we prove that, in the 2-homogeneous case, one of the following three properties holds: P ^?1(0) is a subspace; P ^?1(0) is the union of two different subspaces; and P ^?1(0) is the union of infinitely many different subspaces.     

The Multivariate Integer Chebyshev Problem

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in ℂ ^d . We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete result.      

Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

A time-space harmonic polynomial for a stochastic process M=(M _t) is a polynomial P in two variables such that P(t, M _t) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, space, argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,⁽²⁾ where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a measure proposed by Hochberg⁽³⁾ on path space is obtained.     

Extension of vector-valued integral polynomials

We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X-valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they are extendible to X-valued polynomials. The Aron-Berner extension of an integral polynomial is also studied. A canonical integral representation is given for domains not containing ?₁.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X₁-Jacobi and X₁-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1,1] or the half-line [0,∞), respectively, and they are a basis of the corresponding L² Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions , then it must be either the X₁-Jacobi or the X₁-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X₁ polynomial sequences.     

An algorithm for the divisors of monic polynomials over a commutative ring

Gilmer and Heinzer proved that given a reduced ring R, a polynomial f divides a monic polynomial in R[X] if and only if there exists a direct sum decomposition of R = R₀ ⊕ … ⊕ R_m (m ≤ deg f), associated to a fundamental system of idempotents e₀, … , e_m, such that the component of f in each R_i[X] has degree coefficient which is a unit of R_i. We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.     

A branch-and-cut method for the obnoxious p-median problem

The obnoxious p-median (OpM) problem is the repulsive counterpart of the ore known attractive p-median problem. Given a set I of cities and a set J of possible locations for obnoxious plants, a p-cardinality subset Q of J is sought, such that the sum of the distances between each city of I and the nearest obnoxious site in Q is maximised. We formulate (OpM) as a {0,1} linear programming problem and propose three families of valid inequalities whose separation problem is polynomial. We describe a branch-and-cut approach based on these inequalities and apply it to a set of instances found in the location literature. The computational results presented show the effectiveness of these inequalities for (OpM). The work of the first author has been partially supported by the Coordinated Project C.A.M.P.O. and that of the third author by a short mobility grant, both of the Italian National Research Council.     

Asymptotic Conditions for Degree Diminution Along Prescribed Lines

One of the problems in bivariate polynomial interpolation is the choice of a space of polynomials suitable for interpolating a given set of data. Depending on the number of data, a usual space is that of polynomials in 2 variables of total degree not greater than k. However, these spaces are not enough to cover many interpolation problems. Here, we are interested in spaces of polynomials of total degree not greater than k whose degree diminishes along some prescribed directions. These spaces arise naturally in some interpolation problems and we describe them in terms of polynomials satisfying some asymptotic interpolation conditions. This provides a general frame to the interpolation problems studied in some of our recent papers.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).