Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

On the distribution of comparisons in sorting algorithms

We considered the following natural conjecture: For every sorting algorithm every key will be involved in(logn) comparisons for some input. We show that this is true for most of the keys and prove matching upper and lower bounds. Every sorting algorithm for some input will involven–n /2+1 keys in at leastlog₂ n comparisons,>0. Further, there exists a sorting algorithm that will for every input involve at mostn–n ^/c keys in greater thanlog₂ n comparisons, wherec is a constant and>0. The conjecture is shown to hold for natural algorithms from the literature.     

S-Continued Fraction Method for the Investigation of a Complex Dielectric Constant of Two-Components Composite

The aim of this contribution is to examine the S-continued fraction method of obtaining bounds on the effective dielectric constant _e of a two-phase composite for the case where the dielectric coefficients ₁and ₂ of both components are either complex or real. The starting point for our study is a power expansion of _e (z) at(z)=0 (z)=₂/₁-1. The obtained S-continued fraction bounds have an interesting mathematical structure convenient for theoretical and numerical investigations of _e. They also agree with the earlier estimations reported by Bergman and Milton. Specific examples of calculation of bounds on _e by theS-continued fraction method are also provided.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

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Support partill du CICYT, N^o PB86-0238     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Sharp Large Deviation Estimates for the Stochastic Heat Equation

We consider the family {X , 0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is _t X = X +b({X })+({X }) . Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({X }A) as 0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

On the Rate of Clustering to the Strassen Set for Increments of the Uniform Empirical Process

We obtain outer rates of clustering in the functional laws of the iterated logarithm of Deheuvels and Mason⁽¹¹⁾ and Deheuvels,⁽⁷⁾ which describe local oscillations of empirical processes. Considering increment sizes a _n 0 such that na _n and na _n(log n)^–7/3 we show that the sets of properly rescaled increment functions cluster with probability one to the _n-enlarged Strassen ball in B(0, 1) endowed with the uniform topology, where _n 0 may be chosen so small as (log (1/a _n) + log log n)^–2/3 for any sufficiently large . This speed of coverage is reduced for smaller a _n.     

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     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[ω]C

Suppose thatk, rz₊, W _o ^r H[]_C= {ff is a 2-periodic function,f C^r [–, ], (f^(r), ) ()}, T_k is the space of trigonometric polynomials of order k, p_k(f)T_k is the polynomial of best uniform approximation to f, and E_k(f) is the error of the best approximation. It is shown that for an arbitrary > 0 we have,where for 0<&#x2A7D;(1),k > 0.R () is the root of the equation , and for k = 0 or > (1) we have R()=.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 85–101, July, 1977.The author thanks S. B. Stechkin for posing the problem and for his attention to this work.     

On the sums of a family of Kapteyn series

The series ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) arise in studying the emission and absorption of radiation by a charged particle on a Kepler orbit. For the first few even,r, the sums are obtained in closed form, and for oddr they are given in terms of a certain definite integral. The integral is expressed as a power series in for ||<1, and, near =1, an asymptotic expansion in powers of (1–²)^1/2 may be obtained.

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Résumé La série ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) se trouve par l'émission et l'absorption du rayonnement d'une particule chargée sur l'orbite Keplerien. Pour les plus petites valuers paires der, les sommes s'obtienment en forme compacte, et pour les valuers impaires, elles se déterminent d'après une intégrale definie. Pour ||<1, cette intégrale se développe dans une série de puissances de , et dans le voisinage de =1, on obtient une série asymptotique et puissances de (1–²)^1/2.

     

Stability of Regularized Bilevel Programming Problems

A bilevel programming problem S is considered. First, sufficient conditions of minimal character are given on the data of the problem in order to guarantee the lower semicontinuity of the marginal function of the upper level problem. Then, for >0, a regularized problem S() is considered for which continuity of the regularized marginal function and convergence of the approximate value, as goes to zero, are obtained. Moreover, under perturbations on the data, convergence results for the perturbed marginal functions and the solutions to the problem S _n() are given for any >0.     

Decompositions of complete graphs into isomorphic bipartite subgraphs

Let |E(G)|= andf, a 1-1 mapping ofV(G) into {0,1,...,}. Thenf is called a -valuation ofG if the induced function given by , for alluvE(G) is 1-1. A -valuationf is called an -valuation ofG if there exists a nonnegative number such that for everyuvE(G) withf(u)<f(v),f(u)<f(v). Let denote the graph of then-dimensionalG-cube. ForG=K _{3, 3},K _{4, 4}, andP _k ,it is shown that for any positive integern, then-dimensionalG-cube has an -valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.     

Singularly perturbed functional-differential inclusions

Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0⁺ inC[0, 1]×(L ²(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L ¹(0, 1)–strong) is shown.