weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

measuring time frequency information content using the renyi entropies the generalized entropies of renyi inspire new measures for estimating signal information and complexity in the time frequency plane when applied to a time frequency representation tfr from cohen s class or the affine class the renyi entropies conform closely to the notion of complexity that we use when visually inspecting time frequency images these measures possess several additional interesting and useful properties such as accounting and cross component and transformation invariances that make them natural for time frequency analysis this paper comprises a detailed study of the properties and several potential applications of the renyi entropies with emphasis on the mathematical foundations for quadratic tfrs in particular for the wigner distribution we establish that there exist signals for which the measures are not well defined

comparison of maximum entropy and higher order entropy estimators abstract we show that generalized maximum entropy gme is the only estimation method that is consistent with a set of five axioms the gme estimator can be nested using a single parameter into two more general classes of estimators gme estimators each of these gme estimators violates one of the five basic axioms however small sample simulations demonstrate that certain members of these gme classes of estimators may outperform the gme estimation rule

weighted norm inequalities for oscillatory integrals with finite type phases on the line abstract we obtain two weighted l 2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type the conditions imposed on the weights involve geometrically defined maximal functions and the inequalities are best possible in the sense that they imply the full l p r l q r mapping properties of the oscillatory integrals our results build on work of carbery soria vargas and the first author

robust jump linear quadratic control a mode stabilizing solution this correspondence is concerned with optimal feedback control of a class of linear stochastic systems for a quadratic performance index the solution to this problem is known but the resulting optimal regulator does not ensure stability in every mode of operation to improve robustness a new mode stabilizing solution is defined an example illustrates the advantages of the proposed controller

expressions for renyi and shannon entropies for bivariate distributions exact forms of renyi and shannon entropies are determined for 27 continuous bivariate distributions including the kotz type distribution truncated normal distribution distributions with normal and centered normal conditionals natural exponential distribution freund s exponential distribution marshall and olkin s exponential distribution exponential mixture distribution arnold and strauss s exponential distribution mckay s gamma distribution distribution with gamma conditionals gamma exponential distribution dirichlet distribution inverted beta distribution distribution with beta conditionals beta stacy distribution cuadras and auge s distribution farlie gumbel morgenstern distribution logistic distribution pearson type vii distribution pearson type ii distribution distribution with cauchy conditionals bilateral pareto distribution muliere and scarsini s pareto distribution distribution with pareto conditionals and the distribution with gumbel conditionals we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

comments on robust jump linear quadratic control a mode stabilizing solution in the above paper an optimal control problem with discounting was proposed to achieve closed loop mode stabilization for linear systems with jump parameters the paper used the unstated hypothesis that the diagonal entries of the generator of the jump process are equal this correspondence discusses the situation in the general case

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

two weight extrapolation via the maximal operator abstract we give several extrapolation theorems for pairs of weights of the form w m k w and w mw w r w where w is any non negative function r 1 and m k is the k th iterate of the hardy littlewood maximal operator as an application we show that our results can be used to extend and sharpen results for square functions and singular integral operators by chang et al 1985 comment math helv 60 217 246 chanillo and wheeden 1987 indiana univ math j 36 277 294 wilson 1987 duke math j 55 879 887 1989 trans amer math soc 314 661 692 1989 illinois j math 33 361 366 and uchiyama 1995 studia math 115 135 149 in the process we prove a conjecture due to wilson

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

weighted estimates for conic fourier multipliers we prove a weighted inequality which controls conic fourier multiplier operators in terms of lacunary directional maximal operators by bounding the maximal operators this enables us to conclude that the multiplier operators are bounded on l p mathbb r 3 with 1 p infty

on maximum entropy principle superstatistics power law distribution and renyi parameter abstract the equilibrium distributions of probabilities are derived on the basis of the maximum entropy principle mep for the renyi and tsallis entropies new s forms for the renyi and tsallis distribution functions are found which are normalised with corresponding entropies in contrast to the usual z forms normalised with partition functions z the superstatistics based on the gibbs distribution of energy fluctuations gives rise to a distribution function of the same structure that the renyi and tsallis distributions have the long range tail of the renyi distribution is the power law distribution with the exponent s expressed in terms of the free renyi parameter q as s 1 1 q the condition s 0 gives rise to the requirement q q can be uniquely determined with the use of a further extension of mep as the condition for maximum of the difference between the renyi and boltzmann entropies for the same power law distribution dependent on q it is found that the maximum is realized for q within the range from 0 25 to 0 5 and the exponent s varies from 1 3 to 2 in dependence on parameters of stochastic systems

about closedness by convolution of the tsallis maximizers in this paper we study the stability under convolution of the maximizing distributions of the tsallis entropy under energy constraint called hereafter tsallis distributions these distributions are shown to obey three important properties a stochastic representation property an orthogonal invariance property and a duality property as a consequence of these properties the behaviour of tsallis distributions under convolution is characterized at last a special random convolution called kingman convolution is shown to ensure the stability of tsallis distributions

solution and asymptotic behavior of coupled riccati equations in jump linear systems a new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled riccati equations occurring in jump linear systems is derived by verifying a riccati inequality it is shown that such a solution exists in addition two numerical algorithms are given to compute it an example is given to illustrate the proposed method

divergence statistics based on entropy functions and stratified sampling abstract r h v 1 2 divergence statistics are obtained by either replacing both distributions involved in the argument by their samples estimates or replacing one distribution and considering the other as given asymptotic properties of r h v 1 2 divergence statistics are investigated in a stratified random sampling with independence among strata test for goodness of fit for homogeneity of two parallel samples and for independence can be constructed on the basis of these results to finish an example is given

formulas for renyi information and related measures for univariate distributions in a recent paper song j stat plan infer 93 2001 51 considered renyi information of order and established its connection to the loglikelihood from this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and renyi information were derived for some standard continuous distributions in this paper we derive analytical formulas for renyi and shannon entropies as well as for song s measure for 26 flexible families of univariate continuous distributions we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

a bilinear extension inequality in two dimensions abstract we provide sharp decay estimates for circular averages of a certain bilinear extension operator on l 2 s 1 l 2 s 1

subdyadic square functions and applications to weighted harmonic analysis abstract through the study of novel variants of the classical littlewood paley stein g functions we obtain pointwise estimates for broad classes of highly singular fourier multipliers on r d satisfying regularity hypotheses adapted to fine subdyadic scales in particular this allows us to efficiently bound such multipliers by geometrically defined maximal operators via general weighted l 2 inequalities in the spirit of a well known conjecture of stein our framework applies to solution operators for dispersive pde such as the time dependent free schrodinger equation and other highly oscillatory convolution operators that fall well beyond the scope of the calderon zygmund theory

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

a stein conjecture for the circle we discuss the manner in which one might expect directional maximal functions to control the fourier extension operator via l 2 weighted inequalities we prove a general inequality of this type for the extension operator restricted to circles in the plane

on weighting clustering recent papers and patents in iterative unsupervised learning have emphasized a new trend in clustering it basically consists of penalizing solutions via weights on the instance points somehow making clustering move toward the hardest points to cluster the motivations come principally from an analogy with powerful supervised classification methods known as boosting algorithms however interest in this analogy has so far been mainly borne out from experimental studies only this paper is to the best of our knowledge the first attempt at its formalization more precisely we handle clustering as a constrained minimization of a bregman divergence weight modifications rely on the local variations of the expected complete log likelihoods theoretical results show benefits resembling those of boosting algorithms and bring modified weighted versions of clustering algorithms such as k means fuzzy c means expectation maximization em and k harmonic means experiments are provided for all these algorithms with a readily available code they display the advantages that subtle data reweighting may bring to clustering

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

renyi information measure for a used item recently in the literature a measure of tail heaviness has been proposed based on renyi entropy this measure is very useful in the sense that it can be used to measure tail heaviness even for the distributions for which the kurtosis measure b 2 m 4 m 2 2 does not exist nadarajah and zografos nadarajah zografos information sciences 153 2003 119 138 have derived the expression for this measure for different univariate continuous distributions but this measure can only be used for the lifetime of a new item in case of used item this measure needs some modification in this paper we have modified the measure accordingly so that it can be used in the case of used item and also for the new item we have also derived expressions for this measure for sixteen different univariate distributions and ten other standard distributions derived from the general distributions used in reliability and survival analysis the most of the results obtained in the literature in this direction can be obtained as particular cases of our general results

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

renyi s entropy as an index of diversity in simple stage cluster sampling abstract shannon s measure of entropy and gini simpson s diversity index are usually used by social scientists for measuring diversity of a multinomial population in this paper entropy of order r renyi a 1961 in proceedings of fourth berkeley symposium on mathematics statistics and probability 1 547 561 renyi 1961 is presented as a generalization of shannon s entropy to measure and analyze the diversity asymptotic properties of r order index in a simple stage cluster sampling are investigated also applications to testing hypotheses are discussed finite sample results are computed and to finish a comparison of family members via moments is given

restriction estimates for space curves with respect to general measures in this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition the argument here is new in that it doesn t rely on the offspring curve method which has been extensively used in the previous works our work was inspired by the recent argument due to bourgain and guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces

some new information measures for fuzzy sets abstract after reviewing some existing measures for fuzzy sets we introduce a new informative measure for discrimination between two fuzzy sets this discriminating measure reduces to the nonprobabilistic entropy of deluca and termini 7 under a special condition the divergence measure between two sets has been defined along with a large set of properties it has also been used to define an ambiguity fuzziness measure renyi s 17 probabilistic entropy of order has been extended to define nonprobabilistic entropy of a fuzzy set various properties of this definition have also been proved applications of these measures to clustering image processing vision etc are highlighted

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators