Spherical 7-Designs in 2 n -Dimensional Euclidean Space

We consider a finite subgroup _n of the group O(N) of orthogonal matrices, where N = 2 ⁿ , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 ⁿ -dimensional Euclidean space R ^N . We prove that representations of the group _n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit _n,2 x ^t of any initial point x on the sphere S _{N – 1} is a 7-design in the Euclidean space of dimension 2 ⁿ .     

同轴高功率超宽带功分器研究

 运用同轴结构，研制出高功率超宽带功分器，采取先阻抗变换，再功分的形式，以简化机械加工难度。运用高频仿真软件进行仿真分析，研制出一分为二和一分为四的功分器，再组合成为一分为八的功分器，高压实验表明该功分器能耐高压，驻波测试表明阻抗匹配较好，从测量的传输参数可以看出输出各端口幅度一致性较好。证明了该高功率超宽带功分器可以用作实际的高功率天线阵列。     

Economical Covers with Geometric Applications

A cover of a hypergraph is a collection of edges whose unioncontains all vertices. Let H = (V, E) be a k-uniform, D-regularhypergraph on n vertices, in which no two vertices are containedin more than o(D/e^2k log D) edges as D tends to infinity. Ourresults include the fact that if k = o(log D), then there isa cover of (1 + o(1))n/k edges, extending the known result thatthis holds for fixed k. On the other hand, if k 4 log D thenthere are k-uniform, D-regular hypergraphs on n vertices inwhich no two vertices are contained in more than one edge, andyet the smallest cover has at least (nk) log (k log D)) edges.Several extensions and variants are also obtained, as well asthe following geometric application. The minimum number of linesrequired to separate n random points in the unit square is,almost surely, (n^2/3 / (log n)^1/3). 2000 Mathematical SubjectClassification: 05C65, 05D15, 60D05.     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

Characterizing matchings as the intersection of matroids

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function (G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, (n), for graphs with n vertices. We describe an integer programming formulation for deciding whether (G)k. Using combinatorial arguments, we prove that (n)(log logn). On the other hand, we establish that (n)O(logn/ log logn). Finally, we prove that (n)=4 for n=5,,12, and sketch a proof of (n)=5 for n=13,14,15.An earlier version appears as an extended abstract in the Proceedings of COMB01 [5]. Supported by the Gerhard-Hess-Forschungs-Förderpreis (WE 1462) of the German Science Foundation (DFG) awarded to R. Weismantel.     

Backward Stability of Clenshaw's Algorithm

We study numerical properties of Clenshaw's algorithm for summing the series w = _{n = 0} ^N b _n p _n where p _n satisfy the linear three-term recurrence relation. We prove that under natural assumptions Clenshaw's algorithm is backward stable with respect to the data b _n, n = 0,N.     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Designs in Grassmannian Spaces and Lattices

We introduce the notion of a t-design on the Grassmann manifold of the m-subspaces of the Euclidean space ⁿ . It generalizes the notion of antipodal spherical design which was introduced by P. Delsarte, J.-M. Goethals and J.-J. Seidel. We characterize the finite subgroups of the orthogonal group which have the property that all its orbits are t-designs. Generalizing a result due to B. Venkov, we prove that, if the minimal m-sections of a lattice L form a 4-design, then L is a local maximum for the Rankin function _n,m .