Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna-Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

Blaschke product interpolation and its application to the design of digital filters

A new proof is given thatn distinct points on the unit circle can be mapped inton arbitrary points on the unit circle of the complex plane by a finite Blaschke product. A result of this proof is that the mapping can be done with at mostn?1 factors in the product. The problem is studied in the context of its application to frequency transformations used to design digital filters.     

Lozenge tilings of hexagons with arbitrary dents

Eisenkölbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.     

Counting matrices over finite fields having a given number of rows of unit weight

We present a formula enumerating matrices over a finite field of a given rank and a given number of rows of unit weight, i.e., each having a single nonzero entry. We also determine the number of subspaces of a given dimension containing a given number of vectors of unit weight.     

Unitary Geometric Algebra

The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40?years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.     

Synthesis of a control in the problem of the time-optimal transfer of a point mass to a specified position with zero velocity

The problem of the time-optimal control of the motion of a point mass by means of a force of bounded modulus is considered. It is required that the point be transferred from an arbitrary state of motion to the origin of the system of coordinates with zero velocity. By introducing self-similar conjugate variables, the solution of the two-point problem can be successfully reduced to a search for the optimal root of a certain function, specified analytically. A complete solution of the control problem in the form of a synthesis is obtained using mathematical modelling methods. The feedback coefficients along the unit vectors of the position and velocity vectors are found and a control algorithm and a Bellman function are constructed. Examples using practical initial data are presented.     

The permutahedron ofN-sparse posets

The permutahedron of a poset is the convex hull of all incidence vectors of linear extensions. For the case ofN-sparse posets in which any five elements induce at most oneN we give a characterization of the permutahedron in terms of linear inequalities. This yields an LP-solution for minimizing the weighted mean completion time for jobs with unit processing times andN-sparse precedence constraints. We close with an extension of our approach to arbitrary processing times.     

Low-degree minimal spanning trees in normed spaces

We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the strict Hadwiger number of the unit ball, i.e., the largest number of unit vectors such that the distance between any two is larger than 1.     

Factorization of transformations over a local ring

Let V be a finitely generated free module over a local ring R, and π an invertible linear transformation for V. Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R. The smallest number of factors needed is not greater than r+1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.     

On functional specification of latin squares

This paper studies functional methods for specification of Latin squares over the sets of n-dimensional Boolean vectors, n-dimensional vectors over an arbitrary finite prime field and over an arbitrary finite Abelian group. In conclusion, a method for constructing classes of nongroup Latin squares is presented.     

Units and class groups in cyclotomic function fields

A unit index-class number formula is proven for “cyclotomic function fields” in analogy with similar results in cyclotomic number fields. The main theorem generalizes to arbitrary modules work done earlier for prime power modules.     

Closed-form solution of a maximization problem

According to the characterization of eigenvalues of a real symmetric matrix A, the largest eigenvalue is given by the maximum of the quadratic form 〈xA, x〉 over the unit sphere; the second largest eigenvalue of A is given by the maximum of this same quadratic form over the subset of the unit sphere consisting of vectors orthogonal to an eigenvector associated with the largest eigenvalue, etc. In this study, we weaken the conditions of orthogonality by permitting the vectors to have a common inner product r where 0 ≤ r < 1. This leads to the formulation of what appears—from the mathematical programming standpoint—to be a challenging problem: the maximization of a convex objective function subject to nonlinear equality constraints. A key feature of this paper is that we obtain a closed-form solution of the problem, which may prove useful in testing global optimization software. Computational experiments were carried out with a number of solvers. We dedicate this paper to the memory of our great friend and colleague, Gene H. Golub.     

Cluster sets of interpolating Blaschke products

We construct interpolating Balschke products whose radial cluster sets at a given point of the unit circle can be prescribed to be one of the following: the closed unit disk; an arbitrary closed arc on the unit circle; an arbitrary interval of the form [x, y], wherexy ≠ 0 and −1≤1x≤y≤1. We also show that there does not exist an interpolating Blaschke product having [0,y] or [x, 0] as a radial cluster set. On the other hand, there do exist finite products of interpolating Blaschke products that have [0, 1] as a radial cluster set. Research supported by the RIP-program Oberwolfach, 2002/2003.     

Good lattice points,discrepancy, and numerical integration

Summary Methods based on Diophantine approximations lead to a simple and easy construction of rational vectors the multiple of which, reduced modulo 1, form finite sequences of points with certain properties of equipartition over the unit square. Given a function of bounded variation over this square, it is suggested that computing the average of its values at the points of such a sequence can be a practical method of numerical integration. Precise bounds for the error ore obtained. In the general case, these are of the order of the product of the reciprocal and of the logarithm of the number of points; in the case of a function satisfying stated conditions of regularity and periodicity, they are of the order of the logarithm of the number of points divided by an appropriately high power of this number. A slight sharpening of some well-known results on equipartiton is obtained incidentally. A part of this paper was read at a meeting of the Accademia Nazionale di Scienze Lettere e Arti in Modena on April 7th. 1965.     

Improved Hölder and reverse Hölder inequalities for Gaussian random vectors

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp–Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in the literature including the Prékopa–Leindler and Barthe inequalities.     

Minimal vectors in arbitrary Banach spaces

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

     

Compactness in fuzzy topological spaces

Since earlier approaches to compactness in fuzzy spaces have serious limitations, we propose a new definition of fuzzy space compactness. In doing so, we observe that it is possible to have degrees of compactness, which we call α-compactness (α a member of a designated lattice). We obtain a Tychonoff Theorem for an arbitrary product of α-compact fuzzy spaces and a 1-point compactification. We prove that the fuzzy unit interval is α-compact.     

Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand.An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with root-exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods.     

修理设备可更换且修理延迟的两相同部件冷储备可修系统(Ⅱ)

在文[1]的基础上,本文研究了修理有延迟和修理设备可更换的两单元冷储备可修系统.在假定单元的寿命服从指数分布、修理时间和延迟时间服从一般分布、修理设备的寿命和故障后的更换时间服从指数分布下,通过定义修理设备的"广义忙期",使用更新过程理论和全概率分解技术,提出一种新的分析技巧,讨论了修理设备的一些可靠性指标,获得了如修理设备的可用度和故障次数等可靠性结果.