bonnesenʼs inequality for john domains in rn abstract we prove sharp quantitative isoperimetric inequalities for john domains in r n we show that the bonnesen style inequalities hold true in r n under the john domain assumption which rules out cusps our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit we use the sharp quantitative inequalities proved in fusco et al 2008 7 and fuglede 1989 4 to reduce our problem to symmetric domains

rearrangements of functions let g be measurable non negative functions on r and let tf be their equimeasurable symmetric decreasing rearrangements let f r r r be continuous and suppose that the associated rectangle function defined by f r f a c f b d f a d f b c for r x y r2 a x b c y d is non negative then f g d f tf g d where is lebesgue measure the concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces and the inequality holds in the general case if f x y x y where is convex and 0 0 then we obtain tf g d g d in particular if x x p 1 p then we find that the operator s tf is a contraction on lp for 1 p

bispectral extensions of the askey wilson polynomials following the pioneering work of duistermaat and grunbaum we call a family pn x n 0 of polynomials bispectral if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators one consisting of difference operators acting on the degree index n and another one of operators acting on the variable x the goal of the present paper is to construct and parametrize bispectral extensions of the askey wilson polynomials where the second algebra consists of q difference operators in particular we describe explicitly measures on the real line for which the corresponding orthogonal polynomials satisfy higher order q difference equations extending all known families of orthogonal polynomials satisfying q difference difference or differential equations in x

sobolev inequalities the poisson semigroup and analysis on the sphere sn abstract hypercontractive estimates are obtained for the poisson semigroup on lp sn such estimates are determined by a sharp logarithmic sobolev inequality that relates the entropy of a function on sn to its smoothness

addendum to symmetrization symmetric stable processes and riesz capacities in 1 we proved the following theorem theorem 1 let 0 2 let k be a compact set in r with mn k v 0 let bv be a ball in r with mn bv v then c k c bv here mn k is the n dimensional lebesgue measure and c is the capacity defined by m riesz see e g 2 i as well as several other people who work on such symmetrization results in potential theory thought that this was an open problem t watanabe informed me that in his paper 3 from 1983 he proved an isoperimetric inequality theorem 1 for certain levy processes that include the symmetric stable processes moreover his theorem 2 contains an equality statement for the isoperimetric inequality in the introduction of 3 watanabe points out that his theorems 1 and 2 hold for riesz capacities i e theorem 1 above follows from theorem 1 in 3 the proof in 3 uses fukushima s theory of dirichlet forms and is completely different from the proof of theorem 1 in 1

some orthogonal very well poised functions recently ismail masson and suslov established a continuous orthogonality relation and some other properties of a bessel function on a q quadratic grid askey suggested that the bessel type orthogonality found in the above paper at the level really has a general character and can be extended up to the level very well poised fuctions are known as a nonterminating version of the classical askey wilson polynomials in this paper we prove askey s conjecture and discuss some properties of the orthogonal functions another type of the orthogonality relation for a very well poised function was recently found by askey rahman and suslov

symmetry of constrained minimizers of the cahn hilliard energy on the torus abstract we establish sufficient conditions for a function on the torus to be equal to its steiner symmetrization and apply the result to so called volume constrained minimizers of the cahn hilliard energy the resulting connectedness of superlevel sets is used in two dimensions together with the bonnesen inequality to quantitatively estimate the sphericity of minimizers we also show how two point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the cahn hilliard model

symmetrization symmetric stable processes and riesz capacities let x t be a symmetric stable process killed on exiting an open subset d of r n we prove a theorem that describes the behavior of its transition probabilities under polarization we show that this result implies that the probability of hitting a given set b in the complement of d in the first exit moment from d increases when d and b are polarized it can also lead to symmetrization theorems for hitting probabilities green functions and riesz capacities one such theorem is the following among all compact sets k in r n with given volume the balls have the least capacity 0 2

groundstates for a local nonlinear perturbation of the choquard equations with lower critical exponent abstract we prove the existence of ground state solutions by variational methods to the nonlinear choquard equations with a nonlinear perturbation u u i u n 1 u n 1 u f x u in r n where n 1 i is the riesz potential of order 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

nonlinear elliptic equations with subhomogeneous potentials abstract we prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation u v y 1 y k u g u in r n where x z y 1 y k r n 0 r n 1 r n k r n with n 3 k 1 n 0 0 and n i 2 for i 0 the nonlinearity g and the potential v are respectively a continuous function not necessarily superlinear at infinity and a positive measurable function not necessarily homogeneous but satisfying a subhomogeneity condition which implies vanishing at infinity and singularity at least at the origin this also yields the existence of nonrotating solitary waves and vortices with a critical frequency for nonlinear schrodinger and klein gordon equations with singular cylindrical potentials

a generalized hypergeometric function ii asymptotics and d 4 symmetry in previous work we introduced and studied a function r a a bf c v hat v that generalizes the hypergeometric function in this paper we focus on a similarity transformed function mathcal e a a gamma v hat v with parameters 4 related to the couplings c 4 by a shift depending on a a we show that the function is invariant under all maps w with w in the weyl group of type d 4 choosing a a positive and bf gamma hat v real we obtain detailed information on the re v asymptotics of the function in particular we explicitly determine the leading asymptotics in terms of plane waves and the c function that implements the similarity r

symmetric rearrangements around infinity with applications to levy processes we prove a new rearrangement inequality for multiple integrals which partly generalizes a result of friedberg and luttinger arch ration mech 61 35 44 1976 and can be interpreted as involving symmetric rearrangements of domains around infty as applications we prove two comparison results for general levy processes and their symmetric rearrangements the first application concerns the survival probability of a point particle in a poisson field of moving traps following independent levy motions we show that the survival probability can only increase if the point particle does not move and the traps and the levy motions are symmetrically rearranged this essentially generalizes an isoperimetric inequality of peres and sousi geom funct anal 22 4 1000 1014 2012 for the wiener sausage in the second application we show that the q capacity of a borel measurable set for a levy process can only decrease if the set and the levy process are symmetrically rearranged this result generalizes an inequality obtained by watanabe z wahrsch verw gebiete 63 487 499 1983 for symmetric levy processes

existence and stability for a non local isoperimetric model of charged liquid drops we consider a variational problem related to the shape of charged liquid drops at equilibrium we show that this problem never admits local minimizers with respect to l1 perturbations preserving the volume however we prove that the ball is stable under small c1 1 perturbations when the charge is small enough

the cauchy process and the steklov problem let xt be a cauchy process in rd d 1 we investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain d we establish a connection between the semigroup of this process and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem using this we derive a variational characterization for the eigenvalues of the cauchy process in d this characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the cauchy process inspired by those for brownian motion our results are new even in the simplest geometric setting of the interval 1 1 where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions such results although trivial for the laplacian take considerable work to prove for the cauchy processes and remain open for general symmetric stable processes along the way we present other general properties of the eigenfunctions such as real analyticity which even though well known in the case of the laplacian are not available for more general symmetric stable processes

on a general q fourier transformation with nonsymmetric kernels wiener used the poisson kernel for the hermite polynomials to deal with the classical fourier transform askey atakishiyev and suslov used this approach to obtain a q fourier transform by using the continuous q hermite polynomials rahman and suslov extended this result by taking the askey wilson polynomials considered to be the most general continuous classical orthogonal polynomials the theory of q fourier transformation is further extended here by considering a nonsymmetric version of the poisson kernel with askey wilson polynomials this approach enables us to obtain some new results for example the complex and real orthogonalities of these kernels

comparison and regularity results for the fractional laplacian via symmetrization methods abstract in this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional laplacian this allows to get sharp estimates for the solutions obtained by comparing them with solutions of suitable radial problems furthermore we use such result to prove a priori estimates for solutions in terms of the data providing several regularity results which extend the well known ones for the classical laplacian

two sided eigenvalue estimates for subordinate processes in domains let x xt t 0 be a symmetric markov process in a state space e and d an open set of e denote by x d the subprocess of x killed upon leaving d let s st t 0 be a subordinator with laplace exponent that is independent of x the processes x xst t 0 and x d x dt t 0 are called the subordinate processes of x and x d respectively under some mild conditions we show that if n n 1 and n n 1 denote the eigenvalues of the generators of the subprocess of x killed upon leaving d and of the process x d respectively then

rearrangement and polarization abstract the paper has two main goals the first is to take a new approach to rearrangements on certain classes of measurable real valued functions on r n rearrangements are maps that are monotonic up to sets of measure zero and equimeasurable i e they preserve the measure of super level sets of functions all the principal known symmetrization processes for functions such as steiner and schwarz symmetrization are rearrangements and these have a multitude of applications in diverse areas of the mathematical sciences the second goal is to understand which properties of rearrangements characterize polarization a special rearrangement that has proved particularly useful in a number of contexts in order to achieve this new results are obtained on the structure of measure preserving maps on convex bodies and of rearrangements generally

a comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity recently kreng and tan expert systems with applications 37 2010 5514 5522 developed an economic order quantity eoq model under two levels of trade credit policy in which the supplier offers to the wholesaler a permissible delay period m and the wholesaler also provides its retailers a permissible delay period n with m n in this paper we point out some inappropriate mathematical expressions in both interest charged and interest earned in kreng and tan for generality we then extend their model to allow the following facts 1 the interest rate i c charged by the supplier is not necessarily higher than the interest rate i e earned by the wholesaler and 2 the permissible delay period m offered by the supplier is independent of the permissible delay period n offered by the wholesaler furthermore we study the necessary and sufficient conditions for finding the optimal solution and thus establish several theoretical results to characterize the solution that provides the minimum annual total relevant cost finally numerical examples are given to illustrate the theoretical results and obtain some managerial insights

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

fuzzy inventory model for deteriorating items with permissible delay in payment in this paper we extend jamal et al a m m jamal b r sarker s wang an ordering policy for deteriorating items with allowable shortage and permissible delay in payment journal of operations research society 48 1997 826 833 model by fuzzifying the carrying cost rate interest paid rate and interest earned rate simultaneously based on the interval valued fuzzy numbers and triangular fuzzy number to fit the real world we then prove that the estimate of total variable cost per unit time in the fuzzy sense is a strictly pseudo convex function as a result there exists a unique optimal solution to our proposed model moreover we apply the jamal et al example to show the results and to compare with the jamal et al model

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

a generalized hypergeometric function iii associated hilbert space transform for generic parameters a a c 0 2 4 we associate a hilbert space transform to the relativistic hypergeometric function r a a bf c v hat v studied in previous papers restricting the couplings c to a certain polytope we show that the renormalized r function kernel gives rise to an isometry from the even subspace of l 2 mathbb r hat w hat v d hat v to the even subspace of l2 w v dv where hat w hat v and w v are positive and even weight functions we prove that the orthogonal complement of the range of this isometry is spanned by n pairwise orthogonal functions the latter are in essence askey wilson polynomials arising from the r function by choosing hat v i kappa n with kappa 0 ldots kappa n 1 distinct negative numbers the two commuting analytic difference operators acting on the variable v for which r is a joint eigenfunction give rise to two commuting self adjoint hamiltonians on the even subspace of l2 w v dv we explicitly determine the relation of the time dependent scattering theory for these dynamics to their joint spectral transform

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

some remarks on rearrangement for nonlocal functionals we prove that a nonlocal functional approximating the standard dirichlet p norm fails to decrease under two point rearrangement furthermore we get other properties related to this functional such as decay and compactness

symmetrization of levy processes and applications abstract it is shown that many of the classical generalized isoperimetric inequalities for the laplacian when viewed in terms of brownian motion extend to a wide class of levy processes the results are derived from the multiple integral inequalities of brascamp lieb and luttinger but the probabilistic structure of the processes plays a crucial role in the proofs

martin boundary and integral representation for harmonic functions of symmetric stable processes abstract martin boundaries and integral representations of positive functions which are harmonic in a bounded domain d with respect to brownian motion are well understood unlike the brownian case there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process one kind are functions harmonic in d with respect to the whole process x and the other are functions harmonic in d with respect to the process x d killed upon leaving d in this paper we show that for bounded lipschitz domains the martin boundary with respect to the killed stable process x d can be identified with the euclidean boundary we further give integral representations for both kinds of positive harmonic functions also given is the conditional gauge theorem conditioned according to martin kernels and the limiting behaviors of the h conditional stable process where h is a positive harmonic function of x d in the case when d is a bounded c 1 1 domain sharp estimate on the martin kernel of d is obtained

an eoq model with delay in payments and stock dependent demand in the presence of imperfect production abstract in real life situation supplier generally offers a delay period to the retailer to buy more in this connection a mathematical model is considered to investigate the retailer s optimal replenishment policy under permissible delay in payment with stock dependent demand within the eoq economic order quantity framework most of the eoq models are considered with infinite replenishment rate in this article the model is considered with finite replenishment rate under progressive payment scheme with the production of defective items within the cycle time the mathematical models have been derived for obtaining the optimal cycle time so that the annual total relevant cost is minimized numerical examples along with graphical representation and sensitivity analysis are provided to illustrate the model

on a comparison theorem via symmetrisation talenti s theorem allows us to estimate the norms of the solution of an elliptic boundary value problem posed on a bounded open set in terms of those of the symmetrised problem posed on the ball centred at 0 and such that we are interested in the limiting case when equality is achieved between the norms of the two solutions we show then that and that the solution is radially symmetric and decreasing the novelty of this exposition is that the method used is very elementary compared to the usual arguments

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

on optimization problems with prescribed rearrangements on considere certains resultats concernant des fonctions avec des distributions imposees et des prroblemes d optimisation qui y sont relies

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

an eoq model for a deteriorating item with time dependent quadratic demand under permissible delay in payment abstract in this paper an eoq economic order quantity model is developed for a deteriorating item having time dependent demand when delay in payment is permissible the deterioration rate is assumed to be constant and the time varying demand rate is taken to be a quadratic function of time mathematical models are also derived under two different circumstances i e case i the credit period is less than or equal to the cycle time for settling the account and case ii the credit period is greater than the cycle time for settling the account the results are illustrated with numerical examples justification for considering a time quadratic demand and permissible delay in payment are discussed

on the sharp effect of attaching a thin handle on the spectral rate of convergence abstract consider two domains connected by a thin tube it can be shown that the resolvent of the dirichlet laplacian is continuous with respect to the channel section parameter this in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation under an explicit nondegeneracy condition we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain as an application we prove that again under an explicit nondegeneracy condition the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions