OnBMO regularity for linear elliptic systems

Summary We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO()», a class neither containing, nor contained in . We also prove local and global L^p regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.Work partially supported by M.P.I.Project 40% «Equazioni di evoluzione e applicazioni fisico-matematiche».     

Lightlike periodic trajectories in space-times

Let ., _z be a Lorentz metric on a manifold such that isnot compact. We prove the existence of infinitely many lightlike periodic trajectories in by using variational methods and Ljusternik-Schnirelman theory.Sponsored by M.U.R.S.T. (fondi 60% «Problemi differenziali nonlineari e teoria dei punti critici»; fondi 40% «Equazioni differenziali e calcolo delle variazioni»).     

Lubin-Hensel factorization for Laurent series

Summary Let K be a complete ultrametric algebraically closed field. Let D be a bounded closed strongly infraconnected set in K with no T-filter, and let H(D) be the Banach algebra of the analytic elements in D. Let r, r be functions from D toR with bounds a, b such that 0 (D,r,r) be the Banach algebra of the Laurent series with coefficients a_s in H(D) such that , provided with a suitable norm. In (D, r, r) we give a kind of Hensel Factorization for series whose dominating coefficients at r(x) and at r(x) conserve the same rank. We take advantage of this method to correcting a mistake that happened in our previous article on the Hensel Factorization for Taylor series.And Erratum to «Maximum principle for analytic elements and Lubin-Hensel's Theorem inH(D)Y»,135, pp. 265–278 of this Journal.     

Lo spazio duale di un prodotto di algebre di Boole e le compattificazioni di Stone

Summary This paper is concerned with the Stone space X of a direct product of infinitely many Boolean algebras. In paragraph 2, after recalling that X is the Stone-ech compactification of the sum (disjoint union) of the Stone spaces of the algebras B_i, we exhibit a compactification of which is not a Stone space and we give a method to construct all the «Stone compactifications» of (the corresponding Boolean algebras are easily characterised). In paragraph 3, a set of ultrafilters of B (the «decomposable» ultrafilters) are introduced: this set properly contains , but, as is shown in paragraph 5, there are direct products that admit nondeeomposable ultrafilters (this is the case iff the set {Card B_i: i I } is not bounded by a natural number). In paragraph 4, among other things, we prove, for the set of decomposable ultrafilters, a weak form of countable compactness, in the sense that every countable clopen cover has a finite subcover; then, we deduce that the set of decomposable ultrafilters is pseudocompact, while obviously is not. Lastly, in paragraph 6, we give a second characterisation of the Stone space of B, showing that every ultrafilter of B can be obtained by iterating in a suitable way the procedure which leads to the construction of decomposable ultrafilters.

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Lavoro eseguito nell'ambito dell'attività del Comitato Nazionale per le Scienze Matematiche del C.N.R.     

Fermeture en probabilité de certains sous-espaces d'un espace L 2

Summary Let _i , iI be a family of equidistributed, independant random variables, defined on a probability space (, , P). Let {f _m , mN{ be a sequence of functions such that the f _m ( _i ) are, for every i, centered random variables in L ²(, , P) and in an L _p (, , P) (where p is an even integer at least equal to 4).In this paper the closed linear subspaces of L ² (, , P) generated by the variables of the form , where for fixed M, are studied. A uniform bound of the L ^p norm of elements of the unit ball of the above defined subspaces and also the closure in probability of these subspaces is thus obtained. These results are applied to Wiener's chaos of gaussian variables.

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Equipe de Recherche n⁰ 1 « Processus stochastiques et applications » dépendant de la Section n⁰ 2 «Théories Physiques et Probabilités », associée au C.N.R.S.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Natural operations with projectable tangent valued forms on a fibred manifold

Summary Let p: EB be a fibred manifold. Then, we consider the sheaf (E)=(B)P(E) of (local) projectable tangent valued forms on E, where (B) is the sheaf of (local) differential forms on B andP(E) is the sheaf of (local) projectable vector fields on E. The Frölicher-Nijenhuis bracket makes (E) to be a sheaf of graded Lie algebras [18]. In this paper we study all natural R -bilinear operations on (E) which are of Frölicher-Nijenhuis type. By using the analytical method of [16], we prove that there is a three-parameter family of such operators on (E). As a consequence, we obtain a result on the unicity of the covariant differential of tangent valued forms and of the curvature associated with a given connection on E. All manifolds and mappings are assumed to be infinitely differentiable.This paper has been written during the author's visit at the Institute of Applied Mathematics «Giovanni Sansone». Florence, Italy. The author would like to thank ProfessorMarco Modugno for his kind hospitality and for stimulating discussions.     

On a condition on the radical of a Banach algebra ensuring strong decomposability

The main result is the following theorem. Let be a commutative Banach algebra with radical R, where the factor algebra is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If rⁿ¹ /n 0 as n uniformly for r R, rl, then the algebra is strongly decomposable, i.e., there exists a closed subalgebra B isomorphic to such that =BR.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed . There are 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 589–592, December, 1967.     

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.     

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit

We consider the time evolved states of the free motion _t (q, v)=(q+tv,v),q,v ^d , starting in some non-equilibrium state and look at the associated processX _t of fluctuations of the actual number _t/ () of particles of the realization in with velocities inB at timet/ around its mean as 0 (i.e., in the hydrodynamic limit). It is shown that under natural conditions on the initial state , especially a mixing condition in the space variables, for eacht the laws of the fluctuations become Gaussian in the hydrodynamic limit in the following sense: as 0, where denotes weak convergence and is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution is also given by the free motion in the sense that On the other hand we shall see that ast, whereP _z× is the Poisson process with intensity measurez·×, i.e., the equilibrium state for the free motion with particle densityz and velocity distribution . In the hydrodynamic limit this behaviour corresponds to the ergodic theorem for the fluctuation process: ast. Here is a centered Gaussian state describing the equilibrium fluctuations, i.e., the fluctuations ofP _z× . Thus we prove the central limit theorem for the ideal gas: fluctuations are Gaussian even in non-equilibrium. The proofs rest on an adaption of the method of moments for sequences of generalized fields.     

Estimation of the Coefficient of Multiple Determination

Assume that we have iid observations on the random vector X = (X ,...,X ) following a multivariate normal distribution N (,) where both R and (p.d.) are unknown. Let denote the multiple correlation coefficient between X and (X ,...,X ). The parameter = , called the multiple coefficient of determination, indicates the proportion of variability in X explained by its best linear fit based on (X ,..., X ). In this paper we consider the point estimation of under the ordinary squared error loss function. The usual estimators (MLE, UMVUE) have complicated risk expressions and hence it is quite difficult to get exact decision-theoretic results. We therefore follow the asymptotic decision theoretic approach (as done by Ghosh and Sinha (1981, Ann. Statist., 9, 1334-1338)) and study Second Order Admissibility of various estimators including the usual ones.     

On Sign Changes in Weighted Polynomial L^1 -Approximation

For denote by a polynomial of best weighted -approximation to on of degree . In the present paper, we are dealing with the asymptotic distribution of sign changes of the error functions .     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

We introduce the notion of (, )-derivative of a function of one complex variable, and define on the basis of this the classes of (, )-differentiable analytic functions in a bounded domain G. The classes consist of the Cauchy-type integrals whose densities f() are such that the induced functions on the unit circle are periodic functions of classes . We consider approximation of functions by algebraic polynomials constructed from their series expansions in Faber polynomials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1556–1570, November, 1992.     

On τ-Closed Formations of n-Ary Group

We prove that if G is a nonsingle-element n-ary finite group that belongs to a -closed formation , then , where is the intersection of all maximal -closed subformations of the -closed formation of n-ary groups .     

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is 0 such that the essential spectral radius of is smaller than || (for example, if G R ³ is a domain with a piecewise smooth boundary and the restriction of the Newtonian potential on G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition for which (G) = 0.     

Capacitary Estimates for Solutions of the Dirichlet Problem for Second Order Elliptic Equations in Divergence Form

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equationdiv( in an n-dimensional domain , n 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain , which in turn lead to a sufficient condition for the Hölder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmén–Lindelöf type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div( in a domain and for the solutions of the nonhomogeneous equation -div with measure on the right-hand side.     

The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Let h(d) be the class number of the field and let be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value h(d) increases, roughly speaking, at the rate as grows. A similar result is obtained in the case where the value is close to , i.e., to the least possible value. In addition, it is shown that the interval contains no values of for prime p such that p 3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.