C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

Über eine Verallgemeinerung des Plateauschen Problems

Let T be the domain in ^N defined by the inequalities O < ₁ < ... < _N < +. Put _N+k = /2(1+k) (k=1,2,3), _N+4=₁+2, and denote byF() the set of functions x=x(u,v)=(x¹(u,v),...,x^p(u,v)), (p2) of class , where B is the unit disk u²+v²<1, which maps the circular arcs _k={w=eⁱ:_k<<_K+1} (k=1,..., N+3) into the straight lines containing the edges a_k, a_k+1 (a_N+4=a₁) of a polygon IR^p. Then we show that the function ()= _inf ^xF() D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

---

---

Hans Lewy und Charles B. Morrey gewidmet     

Hypercubic 4 and 5-Designs from Double-Error-Correcting BCH Codes

An ordered orthogonal array OOA(, k, n) is a binary 2 ^k × n matrix with the property that for each complete -set of columns, each possible -tuple occurs in exactly 2 ^k– rows of those columns (for definition of a complete -set, see below). Constructions of OOA(, k, n) for = 4 and = 5 are given.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Asymptotic Distribution of Extremes of Randomly Indexed Random Variables

Let {X _n} be a sequence of i.i.d. random variables and let {_k} be a sequence of random indexes. We study the problem of the existence of non-degenerated asymptotic distribution for min{X ₁,..., X _n}.     

Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

C r -finite elements of Powell-Sabin type on the three direction mesh

Let be the triangulation generated by a uniform three direction mesh of the plane. Let ₆ be the Powell-Sabin subtriangulation obtained by subdividing each triangleT by connecting each vertex to the midpoint of the opposite side.Given a smooth functionu, we construct a piecewise polynomial function C ^r (²) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle of ₆, interpolating derivatives ofu up to orderr at the vertices of.     

Opérateurs différentiels associés à certaines représentations unitaires d'un groupe de Lie résoluble exponentiel

Soient G = exp g un groupe de Lie résoluble exponentiel et H = exp h un sous-groupe connexe de G. Soient un caractère unitaire de H et = Ind_H ^G. Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalente à la commutativité de D (G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre D (G) ^H , des opérateurs différentiels qui laissent l'espace des vecteurs C de invariant et qui commuttent avec l'action de H sur cet espace.     

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday     

On exact D-optimal designs for regression models with correlated observations

Let ^* be an exact D-optimal design for a given regression model Y = X + Z . In this paper sufficient conditions are given for sesigning how the covariance matrix of Z may be changed so that not only ^* remains D-optimal but also that the best linear unbiased estimator (BLUE) of stays fixed for the design ^*, although the covariance matrix of Z * is changed. Hence under these conditions a best, according to D-optimality, BLUE of is known for the model with the changed covariance matrix. The results may also be considered as determination of exact D-optimal designs for regression models with special correlated observations where the covariance matrices are not fully known. Various examples are given, especially for regression with intercept term, polynomial regression, and straight-line regression. A real example in electrocardiography is treated shortly.     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

A homology transfer for a class of simplicial maps

We construct a homology transfer _*H_*(B) H_*(E) for a certain class of proper simplicial maps pE B. Roughly, the important hypothesis is that there is a compact space F, so that all fibres F_x>=p^–1(x), xB, are quotient spaces of F, in a certain locally controlled manner. The composition p_** H_*(B)H_*(E)H_*(B) is multiplication by the Euler characteristic of F.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

Upper bound for transversals of tripartite hypergraphs

We prove 2 7/9v for 3-partite hypergraphs. (This is an improvement of the trivial bound 3v.)     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).