On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Interpolation of spectra

By the M.Riesz Convexity Theorem, an operator T on the space of simple integrable functions into the measurable functions (on some measure space) which has continuous extensions to L^p() and L^q() , where 1 p q , also has continuous exten — sions to all L^r () , p r q . It is shown that, whenever (T_p) and (T_q) are o-dimensional (in particular, countable) then the spectra (T_r) (p r q) are pairwise identical. For q = , only w^*-continuous extensions are considered. An example due to Dayanithy shows that the conclusion fails in general.     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

The “Full Müntz Theorem” Revisited

We give an elementary proof of the convergent sum part of the full Müntz Theorem in L_p(A) and in C(A), together with the Clarkson--Erdös–Schwartz phenomenon for all p (0,), and for all compact A [0,) with positive lower density at 0. This extends earlier results of Müntz, Szász, Clarkson and Erdös, L. Schwartz, P. Borwein and Erdélyi, and Operstein, and offers an arguably shorter and more elementary approach to reprove a large part of the result W. B. Johnson achieved with the author. This approach does not require the usage of Basteros extension of the Krivine–Maurey stable theory. It requires only a standard undergraduate level familiarity with real and complex analysis.     

On a partition analog of the Cauchy-Davenport theorem

Summary Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A_1,A_2,\ldots,A_n of G satisfies \bigg|\sum_{i=1}^n A_i\bigg| \ge \min \bigg\{|G|,\,\sum_{i=1}^n |A_i|-n+1\bigg\}. M.~Kneser generalized the Cauchy--Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy--Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J.~E. Olson in the context of the Erdős--Ginzburg--Ziv Theorem.     

Algorithm for a sum of reciprocals

An algorithm is described for the approximate calculation of a collection of sums of the form _k= _j–1 ⁿ c_j/(_j+_k), 1kn, where 0<_j. The working time of the algorithm is 0(n(t+ log n)(t+log n)) if _k calculated to within 2^–t; here the function (l) denotes the time of multiplication of twoZ-bit numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 3–6, 1984.In conclusion, the author expresses thanks to A. O. Slisenko and Yu. A. Kuznetsov, who drew the attention of the author to the problem considered.     

On very weak entire solutions of nonlinear partial differential equations

Let us consider the variational equation in R ⁿ

Multipliers with singularities along a curve in ℝ n

We consider in ⁿ ,n2, the curve = (t,t ² ,...,t ⁿ ), ₀t₀,₀>0 a small number. We study the boundedness of operatorsT ,>0, defined by multipliers which present singularities along . Our results are derived from a sharp estimate on a suitable maximal function. In the casen=2 theT 's are Bochner-Riesz operators and our results coincide with the known ones.     

On regular embedding integrally in R3 of metrics of class C4 of negative curvature

On the x₀y plane let there be specified a complete metric of negative curvature K by means of the line element ds²=dx²+B²(x, y) dy², and, in the strip _a={0xa, -4-bounded function B>0,K-²<0 ( and are constants). Then, the metric in strip _a is embedded in R³ by means of a surface of class C³.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 261–266, August, 1973.     

Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

Variations of random processes with independent increments

In the paper one considers random processes _{s ost} with independent increments, continuous in the mean (_P<). One establishes relations among multiple integrals, variations, i.e., the limits of sums of the form , and the Itô stochastic integrals.Translated from Zapiski Nauchnykh Seminarov Leningradskago Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 25–35, 1983.     

Complemented subspaces of sums and products of banach spaces

Summary We prove that, when X is one of the Banach spaces l^p (1p ) or c₀, then every infinite-dimensional complemented subspace of X^N (resp. X^(N)) is isomorphic to one of the following spaces: (, X, × X, X^N (resp. , X, X, X^(N)). Therefore, X^N and X^(N) are primary. We also give some consequences and related results.The second author acknowledges partial support from the Italian Ministero della Pubblica Istruzione.     

Maximum principle and second-order conditions for minimax problems of optimal control

In this paper, we study the optimal control problem of minimizing the functionalJ(x, u)=max_t1tt2(x(t),t). We formulate and prove necessary optimality conditions for this problem. We establish the equivalence between the initial minimax problem and a problem involving a terminal functional and phase constraints.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

On the support of midconvex operators

Summary LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D Y is midconvex,g: D Y {– } is midconcave andg f onD, then there exists a Jensen mappingh: D Y {– } such thatg h f onD. Using this theorem we show that a mappingf: D Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Asymptotic behaviour of minimum problems with bilateral obstacles

Summary In this paper we study the asymptotic behaviour (as h) of the solutions of minimum problems for the functional [¦Du¦²+g(x, u)]dx with bilateral obstacles of the type _hu_h, where _h and _h are sequences of arbitrary functions fromR ⁿ into ¯R.