On a Problem of Fenchel

We give a necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n -simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles of a hyperbolic tetrahedron. We also present a simple proof of the known necessary and sufficient condition for the dihedral angles of Euclidean n-simplexes.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

On a Problem of Daubechies

Abstract. We solve a problem posed by Daubechies [12] by showing the nonexistence of orthonormal wavelet bases with good time-frequency localization associated with irrational dilation factors.     

On a Problem of Daubechies

   Abstract. We solve a problem posed by Daubechies [12] by showing the nonexistence of orthonormal wavelet bases with good time-frequency localization associated with irrational dilation factors.     

On a Problem of Gonska

This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.     

On a Problem of Solynin

A number of authors have solved the problem of determining the minimal harmonic measure at the origin of a continuum in the closed unit disc which meets every radius. Solynin has given an extension of this problem by requiring that the competing continua have a certain specific index about the origin and has provided an analytically implicit solution. In this paper is given a simpler treatment which leads to a geometrically explicit solution.     

On a Problem of Ulam

We prove stability of solutions to one minimization problem for an integral functional.     

On a Hypergraph Matching Problem

Let H = (V, E) be an r-uniform hypergraph and let A matching M of H is (α, )-perfect if for each F ∈ , at least α|F| vertices of F are covered by M. Our main result is a theorem giving sufficient conditions for an r-uniform hypergraph to have a -perfect matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote the complete r-uniform hypergraph with n vertices. Let t and r be fixed positive integers where t≥r≥2. Then, K(n, r) can be packed with edge-disjoint copies of K(t, r) such that each vertex is incident with only o(n ^{r ?1}) unpacked edges. This extends a result of Rödl [9].     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

On a Generalized Anti-Ramsey Problem

For positive integers , a coloring of is called a -coloring if the edges of every receive at least and at most colors. Let denote the maximum number of colors in a -coloring of . Given we determine the largest such that all -colorings of have at most O(n) colors and we determine asymptotically when it is of order equal to . We give several bounds and constructions. Received May 3, 1999     

On a Fourth-order Eigenvalue Problem

WeconsidertheexistenceofpositivesolutionsfortheequationwithoneofthefollowingsetsofboundaryvalueconditionsBVP(1)-(2)describesthedeformationsofallelasticbeambothofwhoseendssimplysupported,andBVP(1)-(3)describesthedeformationsofanelasticbeamwithoneendsimplysupportedandtheotherendclampedbyslidingclamps[1].SeveralresultsontheexistenceofsolutionsfornonlinearbeamequationshavebeenestablishedbyseveralauthorsbyusingtheLeray-SchauderContinuationTheorem,seetilereferencesof[1,2].Recently,MaRuyunandWan…     

关于一类几何极值问题

设平面上边长为１和２的闭矩形区域为Ｒ，Ｓ是Ｒ上一个有限点集，ｆ（Ｓ）是Ｓ中任意两点之间的最小距离，ｆＲ（ｎ）＝ｍａｘｆ（Ｓ），本文给出了当２≤ｎ≤６时，ｆＲ（ｎ）的精确值以及相应的图形．     

On a Problem Proposed by Kartsatos

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＸｂｅａｒｅａｌｒｅｆｌｅｘｉｖｅＢａｎａｃｈｓｐａｃｅｗｉｔｈｎｏｒｍａｎｄｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇＪ．Ａｓｕｓｕａｌ,ｗｅｓｕｐｐｏｓｅｔｈａｔｔｈｅｓｐ...     

On a Minimum Linear Classification Problem

We study the following linear classification problem in signal processing: Given a set Bof n black point and a set W of m white points in the plane (m = O(n)), compute a minimum number of lines L such that in the arrangement of L each face contain points with the same color (i.e., either all black points or all white points). We call this the Minimum Linear Classification (MLC) problem. We prove that MLC is NP-complete by a reduction from the Minimum Line Fitting (MLF) problem; moreover, a C-approximation to MLC implies a C-approximation to the MLF problem. Nevertheless, we obtain an O(log n)-factor algorithm for MLC and we also obtain an O(log Z)-factor algorithm for MLC where Z is the minimum number of disjoint axis-parallel black/white rectangles covering B and W.