Two results concerning symmetric bi-derivations on prime rings

Summary LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R R is called a symmetric bi-derivation if, for any fixedy R, the mappingx D(x, y) is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD ₁ andD ₂ are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD ₁(x, x)D ₂(x,x) = 0 holds for allx R, then eitherD ₁ = 0 orD ₂ = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] Z(R) holds for allx R, whereZ(R) denotes the center ofR, forcesR to be commutative.     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Приближение функции ограниченнойp-флукту адии полиномами по мульти пликативным система м

A new discrete modulus of continuity is introduced for functions of boundedp-fluctuation, and direct and converse theorems are proved on the approximation of these functions by polynomials with respect to multiplicative systems. Sufficient conditions for the convergence of Fourier series with respect to multiplicative systems are also obtained and these are the best possible in a certain sense.

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Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Proof of two conjectures of H. Brezis and L.A. Peletier

In this paper, we consider the problem: –u=N(N–2)u ^p– , u>0 on ; u=0 on , where is a smooth and bounded domain inR ^N, N3, p= , and >0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as goes to zero. We give similar results concerning the related problem: –u=N(N–2)u^p+u, u>0 on ; u=0 on , for N is larger than 4.     

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

Skew Hopf fibrations

We introduce class F_R(S²⁺¹) of analytic fibrations of sphere S²ⁿ⁺¹,n1, by great circles for which there exists a tensor R, with the algebraic properties of a curvature tensor, such that 1) for almost everyx (R ²ⁿ +2 there exists a unique plane )x, Ofith the condition R (x, u, x)=x ² u, (u x, u (); 2) for planes spanned by fibers condition R(x, u x)=x ² u, (u x, u, x () is fulfiled. We show that F_R(S²ⁿ⁺¹) consists of skew Hopf fibrations (for n=1 see Rzh. Mat. 1987, 11A822). This implies a negative answer to the conjecture expressed in Rzh. Mat. 1972, 11A559 that this class consists of Hopf fibrations. The proof is based on the following result: skew Hopf fibrations are characterized, in the class of all analytic fibrations of a sphere by great circles, by the property that for any pair of orthogonal fibers the great three-dimensional sphere containing them inherits a skew Hopf fibration.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 101–104, 1990.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

Extremal problems in classes of univalent functions,omitting prescribed values

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f()=c₁+c₂²+... regular and univalent in the circleU={||<1} and such that f(₁)f(₂)=1 for ₁₂U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f(₀)| in the class of functions f()R with a fixed value f(₀, where ₀ is an arbitrary point U, and of the maximum of |f(₀)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.     

Symmetric bi-derivations on prime and semi-prime rings

Summary LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R R is called a symmetric bi-derivation if, for any fixedy R, a mappingx D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD ₁,D ₂ are nonzero derivations onR, then the mappingx D ₁(D ₂ (x)) cannot be a derivation, is also presented.     

Gibbs states of the Hopfield model in the regime of perfect memory

Summary We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N 0 asN , then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N , asN for small enough, we show that for temperaturesT smaller than someT()<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

Skew Brownian motion-type of extensions

We consider a class of possible extensions of a given symmetric Feller processZ _t fromR{0} to the entire real lineR, depending on a parameter [0, 1]. It is proved that the proposed extension exists if =1/2; for 1/2, exists if and only ifZ _t does not jump over 0 (e.g., ifZ _t is a diffusion).     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

Hyperaffine planes over hyperfields

Summary In this paper we solve the problem of finding suitable conditions for an hyperaffine plane (,) under which a planar hyperfield K exists such that (,) is isomorphic to the hyperaffine plane (K²,').Research partially supported by C.N.R.-G.N.S.A.G.A.