Bounded mean oscillation and bandlimited interpolation in the presence of noise

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.     

Reconstruction from local discrete averages on the plane

The aim of the paper is to construct a solution for the equation f?μ=g, where f?μ is the convolution of f and μ given by , g is an arbitrary continuous function and μ is a finitely supported measure on the plane.     

On the well-posedness of a two-phase minimization problem

We prove a series of results concerning the emptiness and non-emptiness of a certain set of Sobolev functions related to the well-posedness of a two-phase minimization problem, involving both the p(x)-norm and the infinity norm. The results, although interesting in their own right, hold the promise of a wider applicability since they can be relevant in the context of other problems where minimization of the p-energy in a part of the domain is coupled with the more local minimization of the L^∞-norm on another region.     

Large degree asymptotics of generalized Bessel polynomials

Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.     

Hyperbolically convex functions and the generalized Fekete-Szegö functional

We describe extremal functions for the generalized Fekete-Szegö functional over the class of hyperbolically convex functions. We apply the Julia variational formula to reduce the problem to mappings onto hyperbolic polygons having no more than two proper sides. In general, this is the best result possible. We show that both one-sided and two-sided maps can be extremal for different sub-classes.     

Faster computation of the Tate pairing

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This paper proposes new explicit formulas for the doubling and addition steps in Miller's algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller's algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nideQo-K9ME/.     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A hybrid shuffled complex evolution approach based on differential evolution for unconstrained optimization

Numerous optimization methods have been proposed for the solution of the unconstrained optimization problems, such as mathematical programming methods, stochastic global optimization approaches, and metaheuristics. In this paper, a metaheuristic algorithm called Modified Shuffled Complex Evolution (MSCE) is proposed, where an adaptation of the Downhill Simplex search strategy combined with the differential evolution method is proposed. The efficiency of the new method is analyzed in terms of the mean performance and computational time, in comparison with the genetic algorithm using floating-point representation (GAF) and the classical shuffled complex evolution (SCE-UA) algorithm using six benchmark optimization functions. Simulation results and the comparisons with SCE-UA and GAF indicate that the MSCE improves the search performance on the five benchmark functions of six tested functions.     

Finite-time boundedness and L2-gain analysis for switched delay systems with norm-bounded disturbance

Finite-time boundedness and finite-time weighted L₂-gain for a class of switched delay systems with time-varying exogenous disturbances are studied. Based on the average dwell-time technique, sufficient conditions which guarantee the switched linear system with time-delay is finite-time bounded and has finite-time weighted L₂-gain are given. These conditions are delay-dependent and are given in terms of linear matrix inequalities. Detail proofs are given by using multiple Lyapunov-like functions. An example is employed to verify the efficiency of the proposed method.