Studies on the determination of multiple numbers of experimental parameters. 2. Simultaneous multiple relaxation equations

Simultaneous multiple relaxation equations are solved. Possible use will be extraction of such parameters from a number of discrete data without resort to graphical techniques. Error tolerances of parameters due to the data errors are formulated.     

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d > 0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k -fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.     

Euler sums of multiple hyperharmonic numbers

Lithuanian Mathematical Journal - For k ≔ (k1, …, kr) ∈ ℕr and n, m ∈ ℕ, we extend the definition of classical hyperharmonic...     

Quasi-socle ideals and Goto numbers of parameters

Goto numbers for certain parameter ideals Q in a Noetherian local ring (A,m) with Gorenstein associated graded ring are explored. As an application, the structure of quasi-socle ideals I=Q:m^q (q≥1) in a one-dimensional local complete intersection and the question of when the graded rings are Cohen-Macaulay are studied in the case where the ideals I are integral over Q.     

Design of experimental and measurement strategy for the determination of cross sections of multiple nuclear processes

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 38–46, 1988.     

Bounds on chromatic numbers of multiple factors of a complete graph

Bounds on the sum and product of the chromatic numbers of n factors of a complete graph of order p are shown to exist. The well-known theorem of Nordhaus and Gaddum solves the problem for n = 2. Strict lower and some upper bounds for any n and strict upper bounds for n = 3 are given. In particular, the sum of the chromatic numbers of three factors is between 3p^1/3 and p + 3 and the product is between p and [(p + 3)/3]³.     

On condition numbers of a nondefective multiple eigenvalue

Summary This paper concerns with measures of the sensitivity of a nondefective multiple eigenvalue of a matrix. Different condition numbers are introduced starting from directional derivatives of the multiple eigenvalue. Properties of the condition numbers defined by Stewart and Zhang [4] are studied; especially, the Wilkinson's theorem on matrices with a very ill-conditioned eigenproblem is extended.This research was supported by the Institute for Advanced Computer Studies of the University of Maryland and the Swedish Natural Science Research Council     

Combinatorial formulas for certain sequences of multiple numbers

Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half-integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.     

Ramsey numbers for multiple copies of complete graphs

The Ramsey numbers r(m₁K_p, …, m_cK) are calculated to within bounds which are independent of m₁, …, m_c when c > 2 and p₁, …, p_c > 2.     

The geometry of multiple integrals depending on parameters

We consider the geometry of an integral in the situation where the multiple integral depends on a finite number of parameters; the problems of exhibiting properties invariant with respect to certain classes of transformations (involving the underlying manifold and the parameter manifold) and the corresponding classification of integrals are solved by the methods of modern differential geometry.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 37–58.     

An exact determination of serial correlations of pseudo-random numbers

Summary Exact expressions for serial correlations of sequences of pseudo-random numbers are derived. The reduction to generalized Dedekind sums is of optimum simplicity and covers all cases of the linear congruential method. The subsequent evaluation of the generalized Dedekind sums is based on a modified Euclidean algorithm whose quotients are recognized as the main contributors to the size of the serial correlations. This leads to the establishment of bounds as well as of fast computer programs. Moreover, some light is thrown upon the general question of quality in random number generation.Dedicated to Prof. H. Schubert, Kiel on the occasion of his 50th birthdayResearch supported by the DFG (Deutsche Forschungsgemeinschaft) and Nova Scotia Technical College, Halifax, N.S. Canada.     

Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

The biplanar crossing number cr₂(G) of a graph G is min{cr(G₁) + cr(G₂)}, where cr is the planar crossing number. We show that cr₂(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) ‐ 2 ≤ Kcr₂(G)^0.4057 log₂n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr₂(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On worst-case condition numbers of a nondefective multiple eigenvalue

Summary. This paper is a continuation of the author [6] in Numerische Mathematik. Let be a nondefective multiple eigenvalue of multiplicity of an complex matrix , and let be the secants of the canonical angles between the left and right invariant subspaces of corresponding to the multiple eigenvalue . The analysis of this paper shows that the quantities are the worst-case condition numbers of the multiple eigenvalue . Received September 28, 1992 / Revised version received January 18, 1994     

Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue

Summary This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of the canonical angles between the left and right invariant of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed.This work was supported in part by the Air Force Office of Sponsored Research under Contract AFOSR-87-0188