positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

an integrable deformation of an ellipse of small eccentricity is an ellipse the classical birkho conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse or a circle as a special case in this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

a generalized multi point boundary value problem for second order ordinary differential equations let f 0 1 x r 2 r be a function satisfying caratheodory s conditions and e t l 1 0 1 let x i t j 0 1 a i b j r i 1 2 m 2 j 1 2 n 2 0 x 1 x 2 x m 2 1 0 t 1 t 2 t n 2 1 be given this paper is concerned with the problem of existence of a solution for the generalized multi point boundary value problems x t f t x t x t e t 0

conditional entropy and data processing an axiomatic approach based on core concavity this work presents an axiomatization for entropy based on an extension of concavity called core concavity we show that core concavity characterizes the largest class of functions for which the data processing inequality holds under the assumption that conditional entropy is defined as a generalized average also under the same assumption we show that data processing and conditioning reduces entropy properties are equivalent we prove several properties of core concave functions including generalization of perfect secrecy and of fano s inequality we also show that definitions of conditional entropy based on worst case can be retrieved as limit cases of generalized averages a connection between statistical decision making and this axiomatic approach is also presented

on the marked length spectrum of generic strictly convex billiard tables in this paper we show that for a generic strictly convex domain one can recover the eigendata corresponding to aubry mather periodic orbits of the induced billiard map from the maximal marked length spectrum of the domain

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

positive solutions for second order m point boundary value problems by using the schauder fixed point theorem and analysis method we establish the existence of solutions for the m point boundary value problem u t a t f u 0 u 0 0 u 1 i 1m 2kiu i b where b ki 0 i 1 2 m 2 0 b

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

multiple positive solutions for second order m point boundary value problems abstract in this paper the second order m point boundary value problem u t f t u 0 0 t 1 u 0 0 u 1 i 1 m 2 k i u i 0 is studied where k i 0 i 1 2 m 2 0 1 2 m 2 1 we give at first the associated green s function and its properties for the m point boundary value problem and growth conditions are imposed on f which yield the existence of at least three positive solutions by using the five functionals fixed point theorem and the properties

existence of at least three solutions of nonlinear three point boundary value problems with super quadratic growth abstract we establish the existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x 0 t 1 x 0 0 x 1 x 0 1 0 1 we employ a condition weaker than the well known nagumo condition which allows the nonlinearity f t x x to grow faster than quadratically with respect to x in some cases our method uses some degree theory arguments

conditional renyi entropies there is no generally accepted definition of conditional renyi entropy the unconditional renyi entropy depends on a parameter which for the case of min entropy takes the value even for this particular case there are several proposals for the definition of conditional entropy this paper describes three general definitions of conditional renyi entropy that were found or suggested in the literature their properties are studied and their values as a function of are compared the particular case of min entropy is widely used in cryptography as a security parameter this case is studied in some detail

moduli space heights and isospectral sets of plane domains let q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type then there exists a flat metric g on 2 such that q is isometric with 29g in what follows we do not distinguish between isometric domains by the spectrum of 29 we mean the spectrum of the laplace beltrami operator ag on 29 with dirichlet boundary conditions the height h eg log det ag is a spectral invariant and plays a central role in this paper among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum this metric is characterized by the fact that d 2 has constant geodesic curvature we call such a metric uniform and denote it by u the set of all such metrics is denoted by u 2 we can therefore identify with the moduli space 2 of conformal structures on e for n 3 we introduce a special parametrization for mu 2 by means of which we show that h u oo as u approaches the boundary of fu 2 using this along with the heat invariants for the laplacian we then show that any isospectral set of plane domains is compact in the c topology similar results hold for n 1 and 2

inverse spectral problem for analytic domains ii ℤ2 symmetric domains this paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit the algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary and on a feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral the algorithm is particularly effective for euclidean plane domains possessing a 2 symmetry which reverses the orientation of a bouncing ball orbit it is also very effective for domains with dihedral symmetries for simply connected analytic euclidean plane domains in either symmetry class we prove that the domain is determined within the class by either its dirichlet or neumann spectrum this improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

the existence of positive solutions for second order multi point bvps with the first derivative by using the fixed point index theory several theorems of existence of positive solutions for second order m point boundary value problem with the first derivative are obtained where the nonlinear term may be noncontinuous an example illustrating the main results is given

positive solutions for nonhomogeneous boundary value problems in banach spaces abstract by using darbo fixed point theorem the existence and nonexistence of fixed point are obtained for the following integral operator t y t 0 1 g t s f s y s ds tv using these results we investigate the existence and nonexistence of positive solutions for second order nonhomogeneous local and nonlocal boundary value problems in banach spaces

exponential decreasing rate of leaked information in universal random privacy amplification we derive a new upper bound for eve s information in secret key generation from a common random number without communication this bound improves on bennett s bound based on the renyi entropy of order 2 because the bound obtained here uses the renyi entropy of order 1 s for s 0 1 this bound is applied to a wire tap channel then we derive an exponential upper bound for eve s information our exponent is compared with hayashi s exponent for the additive case the bound obtained here is better the result is applied to secret key agreement by public discussion