Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

A generalization of the Subbarao identity

Arithmetical functionsf andh are said to satisfy the Subbarao identity if

On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation

We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x – t,y) – 2f(x,y)/t ² =f(x,y + s) + f(x,y – s) – 2f(x,y)/s ² (N) analogous to the well-known wave equation ( ²/x ² – ²/y ²)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows. Theorem.Let R be the set of all real numbers. A function f: R × R R satisfies the functional equation (N)for all x, y R, s, t R\{0}, and s t if and only if there exist

On a question of H. Kraljević

In this paper, assuming a certain set-theoretic hypothesis, a positive answer is given to a question of H. Kraljevi, namely it is shown that there exists a Lebesgue measurable subsetA of the real line such that the set {c R: A + cA contains an interval} is nonmeasurable. Here the setA + cA = {a + ca: a, a A}. Two other results about sets of the formA + cA are presented.     

Solutions rationnelles de certaines équations fonctionnelles

Dans cet article, nous démontrons essentiellement les deux résultats suivants, qui montrent que les solutions séries formelles à coefficients dans de certaines équations fonctionnelles sont rationnelles. Soient tout d'abords un entier naturel non nul, eta _i ,b _i ,(i = 1, , s), 2s nombres complexes, lesa _i étant non nuls. On définit l'ensembleA comme étant l'intersection des parties de , contenant l'origine et stables par toutes les applicationsg _i (x) = a _i x + b _i . On a alors le résultat suivant: Théorème 1.Soient f, R ₁, ,R _s s + 1 fractions rationnelles de (x), régulières à l'origine, et a_i, b_i (i = 1,, s), 2s éléments de . On suppose que les a_i sont non nuls et de module strictement inférieur à un pour tout i = 1,, s. Soit y(x) un élément de [[x]], vérifiant l'équation fonctionnelle

Initial value problem of a higher order parabolic equation

In the present work it is studied the initial value problem for an equation in the form

Survival time of a random graph

LetV _n ={1, 2, ...,n} ande ₁,e ₂, ...,e _N ,N= be a random permutation ofV _n ⁽²⁾. LetE _t={e ₁,e ₂, ...,e _t} andG _t=(V _n ,E _t ). If is a monotone graph property then the hitting time() for is defined by=()=min {t:G _t }. Suppose now thatG starts to deteriorate i.e. loses edges in order ofage, e ₁,e ₂, .... We introduce the idea of thesurvival time =() defined by t = max {u:(V _n, {e _u,e _u+1, ...,e _T }) }. We study in particular the case where isk-connectivity. We show that

The Binet-Pexider functional equation for rectangular matrices

IfK is a field of characteristic 0 then the following is shown. Iff, g, h: M _n (K) K are non-constant solutions of the Binet—Pexider functional equation

Oscillating solutions of integral equations and linear recursions

Answering a question of Z. Daróczy we show that there are positive sequences _n satisfying the recursion

On the error of compound quadrature formulas forr-convex functions

LetC _m be a compound quadrature formula, i.e.C _m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ _n to every subinterval. LetR _m be the corresponding error functional. Iff ^(r) > 0 impliesR _m [f] > 0 (orR _m [f] < 0),=" then=" we=" say=">C _m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf ^(r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC _m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR _m [f], where , denotes the modulus of continuity of orderr:

On a joint generalization of Cauchy's and d'Alembert's functional equations

LetX be a locally compact abelian group and let a compact groupG act onX. We find solutionsf: X inL(X) of the following functional equation:

Jacobi elliptic functions and change of variable in a convolution

We solve the functional equation

New closure conditions and some problems in loop theory

A loopQ(·) is said to be anA _l-loop (A _r-loop) if x, y Q, l _x,y AutQ (r _x,y AutQ) hold, where

A new Pythagorean functional equation

The purpose of this paper is to solve the following Pythagorean functional equation:(e ^p(x,y) ) ² ) = q(x,y) ² + r(x, y) ², where each ofp(x,y), q(x, y) andr(x, y) is a real-valued unknown harmonic function of the real variablesx, y on the wholexy-planeR ².The result is as follows.     

On the quasiarithmetic mean in a mean value property and the associated functional equation

The functional equation

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

Some notes about affine diameters of convex figures

Given a pointx in a convex figureM, let(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.

The existence of non-trivial hyperfactorizations ofK 2n

A -hyperfactorization ofK _2n is a collection of 1-factors ofK _2n for which each pair of disjoint edges appears in precisely of the 1-factors. We call a -hyperfactorizationtrivial if it contains each 1-factor ofK _2n with the same multiplicity (then =(2n–5)!!). A -hyperfactorization is calledsimple if each 1-factor ofK _2n appears at most once. Prior to this paper, the only known non-trivial -hyperfactorizations had one of the following parameters (or were multipliers of such an example)

A note on induced cycles in Kneser graphs

Letg(n,r) be the maximal order of an induced cycle in the Knesser graph Kn([n] r), whose vertices are ther-sets of [n]={1, ...,n} and whose adjacency relation is disjointness. Thusg(n, r) is the largestm for which there is a sequenceA ₁,A ₂,...,A _m [n] ofr-sets withA _i A _j= if and only if |i-j|=1 orm–1. We prove that there is an absolute constantc>0 for which

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits