On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

L 2(R n )-extremal problems on the basis of incomplete information

. ( ), R ⁿ L ₂(R ²).

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The author is supported by the National Natural Science Found of China.     

Spectral properties of Neumann Laplacian of horns

We study the Neumann Laplacian of unbounded regions in ⁿ with cusps at infinity so that the corresponding Dirichlet Laplacian has compact resolvent. Typical of our results is that of the region {(x, y)²x, y|<1} the Neumann Laplacian has absolutely continuous spectrum [0, ) of uniform multiplicity four and an infinity of eigenvaluesE _o<E ₁... and that for the region {(x, y)²y|1}, it has absolutely continuous spectrum [1/4, ) of uniform multiplicity 2 and an infinity of eigenvaluesE ₀=0<E ₁.... We use the Enss theory with a suitable asymptotic dynamics.The second author's research is partially funded under NSF grand number DMS-8801918     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Occupation time large deviations of the voter model

This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model _t (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are classical in high dimensions (d5 for _t) but fat in low dimensions (d4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.Dedicated to Frank Spitzer on his 60th birthdayPartially supported by the National Science Foundation under Grant DMS-831080Partially supported by the National Science Foundation under Grant DMS-841317Partially supported by the National Science Foundation under Grant DMS-830549     

Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

A Weakly Nonideal Bose Gas Model Leading to the Fountain Effect

The ordinary thermodynamics corresponds to the equilibrium ground state of the system. We present a model making it possible to construct a second thermodynamics, an analogue of an equilibrium state of a superfluid, based on which we explain the effect of the appearance of a bottleneck in a thin capillary for a moving superfluid if it is heated to the critical point in the second thermodynamics.     

Asymptotic behavior for t→∞ of the solutions of the equation ψxx+u(x,t)ψ+(λ/4)ψ=0 with a potential u,satisfying the Korteweg-de Vries equation. II

This is the third paper in a series of papers of the authors, devoted to a rigorous investigation of the asymptotic behavior of the solutions of the KdV equation as t. The immediate purpose of the paper is the investigation of the solution of the Schrödinger equation in the neighborhood of the singular point x=3t for a special class of potentials, introduced in the previous papers. As it will be proved, in the final analysis this class of potentials describes the asymptotic behavior of the solutions, decreasing for x, of the KdV equation as t. The solution , far from the singular point, has been investigated earlier. In the paper we investigate a series which gives the solution of the Schrödinger equation and we consider the asymptotic properties of this series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 8–32, 1984.     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Optimal selection of the four best of a sequence

We consider the situation in which the decision-maker is allowed to have four choices with purpose to choose exactly the four absolute best candidates fromN applicants. The optimal stopping rule and the maximum probability of making the right choice are given for largeNN, the maximum asymptotic value of the best choice being lim _N P(win)0.12706.     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

A n 4 -polyhedra with free homology

An A _n ^k -polyhedron is a CW-complex which is (n–1)-connected and of dimension (n+k). We compute an algebraic category which is equivalent to the homotopy category of A _n ² -polyhedra with free homology (n3). This computation and a calculation of the -group _n+3(X) (n3) is used to obtain a complete algebraic homotopy invariant for A _n ⁴ -polyhedra with free homology (n3).As an application we compute the group of self-homotopy equivalences Aut(X) for a bouquet X=P².     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

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Model reduction techniques for sampled-data systems

An approximately balanced realization of linear finite-dimensional sampled-data systems is proposed. The theoretical support of the approximately balancing algorithm is represented by a result on the asymptotic expansions with respect to the sampling step of the sampled controllability and observability graminas. Reduced order models obtained as singular perturbational approximations of approximately balanced realizations of sampled-data systems are shown to be acceptable solutions to the sampled-data system model reduction problem in the sense that, enjoying some asymptotic properties, they come close to the exact solutions as the sampling step decreases. An example illustrates the results.     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.