Binyamin Schwarz

Some results on the frequencies of nonhomogeneous rods

In this paper we consider the transverse vibrations of a thin cylindrical rod of variable density and constant f lexural r igidity which is clamped at both ends. In Section I we prove two results on the principal frequency of these rods. These theorems are analogous to theorems on the principal frequency of nonhomogeneous strings. Theorem 1 corresponds to a result of Beesack [1] and Theorem 2 to a result of Krein [2]. In Section I I we consider nonhomogeneous clamped rods with equimeasurable density and we prove an inequality for the min imum of the nth frequency (n =2~). This Theorem 3 should be compared with a much stronger statement on the extrema of the frequencies of equimeasurable strings. We rely heavily upon results of Beesack [1, 3]. Notation and methods are similar to those used in our paper on nonhomogeneous strings [4].

Geometries of the projective matrix space

In our paper on matrix Mobius transformations (Comm. Algebra 9 (19) (1981), 1913–1968) we introduced the one-dimensional left-projective space over the complex n×n matrices P = P1(Mn(ℂ)). For n = 1 this space is the projective complex line P1(ℂ) and so is homeomorphic to the Riemann sphere. We studied the topology of P and the projective mappings of P onto itself. Here we present some generalizations of certain parts of the Euclidean, the spherical and the non-Euclidean geometry from the scalar to the multidimensional case.

Disconjugacy of complex differential systems and equations

where W(z) = (wik(z))n. In [12], (1) was called disconjugate in D if, for every choice of n (not necessarily distinct) points z1, . . *, Znin D, the only solution of (1) which satisfies wi(zi) = 0, i = 1,..., n, is the trivial one w(z) 0_ . Disconjugacy of (1) in D is equivalent to the fact that for any fundamental solution W(z) = (wik(z))l of (2) (i.e., for any solution W(z) for which the determinant W(z) = wik(z)l#O for all z of D), the determinant Iwik(z?) In#0 for every choice of n (not necessarily distinct) points z1,..., Zn of D [12, Theorem 3]. (If this holds for one fundamental solution of (2), then it holds for all of them.) This property may thus serve as the definition of disconjugacy of the systems (2), and (1), in D. We define now a more restrictive property than this above defined (ordinary) disconjugacy in D. DEFINITION 1. The differential systems (1) and (2) are called zo-absolute disconjugate in D if there exists a point zo E D such that the solution W(z) = (wik(z)) of (2), determined by

On the product of the distances of a point from the vertices of a polytope

AbstractLetx1,...,xm be points in the solid unit sphere ofEn and letx belong to the convex hull ofx1,...,xm. Then

Geometry of matrix differential systems

\prod\limits_{i = 1}^m {\left| {x - x_i } \right.\left\| \leqq \right.(1 - \left\| x \right\|)(1 + \left\| x \right\|)m^{ - 1} }

Contraction of the Matrix Unit Disk

. This implies that all such products are bounded by (2/m)m(m −1)m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z0)| where

Bounds for the carathéodory distance of the unit ball

p(z) = \prod\limits_{i = 1}^m {(z - z_i ),\left| {z_i } \right| \leqq 1,i = 1,...m,}