Generalization of Cardinality Estimates for Plane Circuits Realizing Partial Boolean Operators

The order of the Shannon function of plane circuit activity is considered for a class of partial Boolean operators with restrictions on the number of different values. It is proved that for the class of partial operators with m outputs, domain of cardinality d, and the number of different values not exceeding r the mean and maximal orders of activity are equal to \((\sqrt {\text{a}} + m\sqrt r /\log r)\sqrt {\log r} \) by the order.     

Triply-Logarithmic Parallel Upper and Lower Bounds for Minimum and Range Minima over Small Domains

We consider the problem of computing the minimum ofnvalues, and several well-known generalizations [prefix minima, range minima, and all nearest smaller values (ANSV)] for input elements drawn from the integer domain [1···s], wheres ≥ n. In this article we give simple and efficient algorithms for all of the preceding problems. These algorithms all takeO(log log log s) time using an optimal number of processors andO(ns^ε) space (for constant ε < 1) on the COMMON CRCW PRAM. The best known upper bounds for the range minima and ANSV problems were previouslyO(log log n) (using algorithms for unbounded domains). For the prefix minima and for the minimum problems, the improvement is with regard to the model of computation. We also prove a lower bound of Ω(log log n) for domain sizes = 2^{Ω(log n log log n)}. Since, forsat the lower end of this range, log log n = Ω(log log log s), this demonstrates that any algorithm running ino(log log log s) time must restrict the range ofson which it works.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

An Adaptive Local (AL) Basis for Elliptic Problems with Complicated Discontinuous Coefficients

We develop a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. In [5] a semidiscrete method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size H which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires O(log(1/H)^d+1) basis functions per mesh point where d denotes the spatial dimension of the computational domain. We present a fully discrete version of this method, where the AL basis is constructed by solving finite-dimensional localized problems, and which preserves the optimal convergence rates. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error estimate in a finite volume approximation of the partial asymptotic domain decomposition

The method of asymptotic partial domain decomposition has been proposed for partial differential equations set in rod structures, depending on a small parameter. It reduces the dimension of the problem (or simplifies it in another way) in the main part of the domain keeping the initial formulation in the remaining part and prescribing the asymptotically precise conditions on the interface. This paper is devoted to the finite volume implementation of the method of asymptotic partial domain decomposition. We consider a model problem in a thin domain (its thickness is a small parameter). We obtain an error estimate, expressed in terms of the small parameter and the step of the mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

A Brunn-Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R³, as an application we get the Brunn-Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R³.     

Cremer fixed points and small cycles

Let , , and let denote the sequence of convergents to the regular continued fraction of . Let be a function holomorphic at the origin, with a power series of the form . We assume that for infinitely many we simultaneously have (i) , (ii) the coefficients stay outside two small disks, and (iii) the series is lacunary, with for . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

     

On the Solution Sets of Constrained Differential Inclusions with Applications

We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion in a closed domain is an R -set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given.     

The $$\bar \partial $$ -problem with support conditions on some weakly pseudoconvex domainswith support conditions on some weakly pseudoconvex domains

We consider a domain Ω with Lipschitz boundary, which is relatively compact in ann-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the\(\bar \partial \)-equation with exact support in ω admits a solution in bidegrees (p, q), 1≤q≤n?1. Moreover, the range of\(\bar \partial \) acting on smooth (p, n?1)-forms with support in\(\bar \Omega \) is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flatCR manifolds of arbitrary codimension.     

Secret sharing schemes with partial broadcast channels

In a conventional secret sharing scheme a dealer uses secure point-to-point channels to distribute the shares of a secret to a number of participants. At a later stage an authorised group of participants send their shares through secure point-to-point channels to a combiner who will reconstruct the secret. In this paper, we assume no point-to-point channel exists and communication is only through partial broadcast channels. A partial broadcast channel is a point-to-multipoint channel that enables a sender to send the same message simultaneously and privately to a fixed subset of receivers. We study secret sharing schemes with partial broadcast channels, called partial broadcast secret sharing schemes. We show that a necessary and sufficient condition for the partial broadcast channel allocation of a (t, n)-threshold partial secret sharing scheme is equivalent to a combinatorial object called a cover-free family. We use this property to construct a (t, n)-threshold partial broadcast secret sharing scheme with O(log n) partial broadcast channels. This is a significant reduction compared to n point-to-point channels required in a conventional secret sharing scheme. Next, we consider communication rate of a partial broadcast secret sharing scheme defined as the ratio of the secret size to the total size of messages sent by the dealer. We show that the communication rate of a partial broadcast secret sharing scheme can approach 1/O(log n) which is a significant increase over the corresponding value, 1/n, in the conventional secret sharing schemes. We derive a lower bound on the communication rate and show that for a (t,n)-threshold partial broadcast secret sharing scheme the rate is at least 1/t and then we propose constructions with high communication rates. We also present the case of partial broadcast secret sharing schemes for general access structures, discuss possible extensions of this work and propose a number of open problems.      

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.     

Analysis and Geometry on Worm Domains

In this primarily expository article, we study the analysis of the Diederich-Fornæss worm domain in complex Euclidean space. We review its importance as a domain with nontrivial Nebenhülle, and as a counterexample to a number of basic questions in complex geometric analysis. Then we discuss its more recent significance in the theory of partial differential equations: the worm is the first smoothly bounded, pseudoconvex domain to exhibit global non-regularity for the \(\overline{\partial}\)-Neumann problem. We take this opportunity to prove a few new facts. Next, we turn to specific properties of the Bergman kernel for the worm domain. An asymptotic expansion for this kernel is considered, and applications to function theory and analysis on the worm are provided.     

Large-Amplitude Periodic Oscillations in Simple and Complex Mechanical Systems: Outgrowths from Nonlinear Analysis

In this paper, we survey developments in nonlinear partial differential equations and applications since the original paper of Ambrosetti and Prodi on nonlinearities that cross eigenvalues. We show how this work has important developments in nonlinear analysis, and survey the implications for periodically forced mechanical systems, from simple one-particle vibrating systems to suspension bridges and ships. Lecture held in the Seminario Matematico e Fisico on March 9, 2005 Received: December 2005     

Arithmetical properties of some series with logarithmic coefficients

We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler’s constant γ, log and log σ, where σ is Somos’s quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.     

On the curvature conjecture of Hua Loo-keng

It is proved that both the holomorphic sectional and the bisectional curvatures of the conformal Bergman metric ds21 = K2(z,)2log K(z, )/zαβdzαdβ are always negative, where K(z,) is the Bergman kernel of a bounded domain Din Cn . As a subsequent result, the Weyl tensor for a Hermitian manifold is obtained.     

A quasi-wavelet algorithm for second kind boundary integral equations

In solving integral equations with a logarithmic kernel, we combine the Galerkin approximation with periodic quasi-wavelet (PQW) [4]. We develop an algorithm for solving the integral equations with only O(N log N) arithmetic operations, where N is the number of knots. We also prove that the Galerkin approximation has a polynomial rate of convergence.     

Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f L ₁ such that the partial sums S _n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly bad functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S _n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense optimal, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L ₁ functions.