Une généralisation d'un résultat de J. Aczél et M. Hosszú sur l'équation de translation

Résumé En généralisant un résultat de J. Aczél et M. Hosszú on donne des conditions nécessaires et suffisantes pour qu'une solution de l'équation de translationF(F(, x), y) = F(, xy), oùF: × G , est un ensemble arbitraire,G forme un groupe, soit de la formeF(, x) = f ^–1(f()·1(x)), oùf est une bijection de au groupeG ₁ isomorphe avecG et 1 est un homomorphisme deG àG ₁. On considère aussi le cas oùG forme un espace vectoriel sur le corps des nombres rationels.Si est un intervalle ayant plus qu'un point etG = R ^m avec l'addition comme l'opération on trouve des conditions pour que la fonction continueF soit de la formeF(, x ₁,, x _m ) =f ^–1(f() + c ₁ x ₁ + +c _m x _m ), oùf est une homéomorphie de àR et (c ₁,,c _m ) R ^m .

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On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

Monomial gorenstein curves in A4 as set-theoretic complete intersections

It is shown that if the prime ideal ,, x₄], k an arbitrary field, has generic zero x_i=tⁿ _i, n_i positive integers with g.c.d. equal l, l i 4, then P(S) is a set-theoretic complete intersection if the numerical semigroup S=1,, n₄> is symmetric (i.e. if the extension of P(S) in k[[x₁,, x₄]] is a Gorenstein ideal).     

A basis for the type 〈n〉 hyperidentities of the medial variety of semigroups

Summary The medical varietyMV of semigroups is the variety defined by the medial identityxyzw = xzyw. This variety is known to satisfy the medial hyperidentitiesF(G(x ₁₁ ,, x _1n ),, G(x _n1 ,, x _nn )) = G(F(x ₁₁ ,, x _n1 ),, F(x _1n ,, x _nn )), forn 1. Taylor has observed in [2] thatMV also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of then = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for theMV hyperidentities with one binary operation symbol.In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arityn 2. We show that theMV hyperidentities with onen-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.Research supported by NSERC of Canada.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

Asymptotically minimax tests of composite hypotheses

Summary X ₁,,X> _n are independent, identically distributed random variables with common density function f(x¦ ₁ ,, _k , _k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that _k+1 =b against the alternative that _k+1 =b+cn ^–1/2 . ₁ ,, _k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦ ₁ -a _i ¦n ^–1/2 M(n) for i=1,,k, where a ₁ , ,a _k are known, and M(n) approaches infinity as n increases but n ^–1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to _k+1 =b, and weight 1-A to _k+1 =b+cn ^–1/2 , in each case the weight being spread uniformly over the possible values of ₁ ,, _k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of ₁ ,..., _k are substituted for a ₁ ..., a _k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a ₁ ..., a _k are.Research supported by the U.S. Air Force under Grant AF-AFOSR-68-1472.     

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.     

Characterization of the projections of geodesics of the Sasakian metric ofTCP n andT 1 CP n

Curves that are projections of geodesics of the Sasakian metric of the tangent and tangent sphere bundles of a complex projective space are considered. The main result is: THEOREM. If is a geodesic of TCPⁿ (T₁ CPⁿ) then ₀is a curve inCP ⁿ for which curvatures k₁, , k₅ are constant and k₆ = = k_2n = 0.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 121–126, 1991.     

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence f _n in LSC(X) and a functionf LSC(X), we show that convergence of f _n tof in several variational convergence modes implies that for each , the sublevel set at height off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef _n, at height _n approaching . The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.     

On the Ranked Excursion Heights of a Kiefer Process

Let (K(s,t), 0s1, t1) be a Kiefer process, i.e., a continuous two-parameter centered Gaussian process indexed by [0,1]×₊ whose covariance function is given by (K(s₁,t₁) K(s₂,t₂))=(s₁s₂-s₁s₂)t₁t₂, 0s₁, s₂1, t₁, t₂ 0. For each t>0, the process K(·,t) is a Brownian bridge on the scale of . Let M ₁ ^* (t) M ₂ ^* (t) M _j ^* (t) 0 be the ranked excursion heights of K(,t). In this paper, we study the path properties of the process tM _j ^* (t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M _j ^* (t) as t goes to infinity.     

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

Regular (2, 2)-systems

Let (E,I) be an independence system over the finite setE = {e ₁, ,e _n }, whose elements are orderede ₁ e _n . (E,I) is called regular, if the independence of {e _l , ,e _{l k} },l ₁ < <l _k , implies that of {e _{m l} , ,e _{m k} }, wherem _l < ··· <m _k andl ₁ m ₁, ,l _k m _k . (E,I) is called a 2-system, if for anyI I,e E I the setI {e } contains at most 2 distinct circuitsC, C I and the number 2 is minimal with respect to this property. If, in addition, for any two independent setsI andJ the family (C J, C C (J, I)), whereC(J, I) denotes {C C:e J I C {e}}, can be partitioned into 2 subfamilies each of which possesses a transversal, then (E,I) is called a (2, 2)-system. In this paper we characterize regular 2-systems and we show that the classes of regular 2-systems resp. regular (2, 2)-systems are identical.     

On products of additive functions (a third approach)

Summary Leta ₁, , a_s : G K be additive functions from an abelian groupG into a fieldK such thata ₁(g)··a_s(g) = 0 for allg G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.     

Convergence des solutions formelles de certaines équations fonctionnelles

Résumé Soitq un nombre algébrique de module 1, qui ne soit pas une racine de l'unité, etP [X, Y ₀,Y ₁] un polynôme non nul. Dans cet article, nous montrons que toute solution de l'équation fonctionnelleP(z, (z), (qz))=0, qui est une série formelle (z) dansQ[[z]], a un rayon de convergence non nul.

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Summary Letq Q be an algebraic number of modulus one that is not a root of unity. LetP Q[X, Y ₀,Y ₁] be a non zero polynomial. In this paper, we show that every formal power series,(z) Q[[z]], solution of the functional equationP(z), (z), (qz)) = 0 has a non zero radius of convergence.

     

The complete figure for second order multiple integral problems in the calculus of variations

Notation Throughout this paper Greek indices, , , and Latin indicesi, j, h, k, assume the values 1, ,m, and 1, ,n respectively. The summation convention is operative in respect of both sets of indices.This work was supported by the South African Council for Scientific and Industrial Research.At time of writing Professor Grässer was Visiting Scholar at the University of Arizona, Tucson, Arizona.     

On the convexity of the sum of the first eigenvalues of operators depending on a real parameter

Let ₁ (k) ₂ (k) be the eigenvalues of an operator of a certain type depending on a real parameterk. The paper shows that under certain requirements on the operator and on the nature of its dependence onk, the sum ₁ (k)++ _N (k) is a concave function ofk, for any positive integerN.

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Zusammenfassung Seien ₁ (k) ₂ (k) die Eigenwerte eines von einem reellen Parameterk abhängigen Operators. Man zeigt, daß unter gewissen Voraussetzungen über den Operator und seine Abhängigkeit vonk die Summe ₁ (k)++ _N (k) für jedesN eine konkave Funktion vonk ist.

     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.