the concavity of p renyi entropy power for doubly nonlinear diffusion equations and lp gagliardo nirenberg sobolev inequalities abstract we prove the concavity of p renyi entropy power for positive solutions to the doubly nonlinear diffusion equations on r n or compact riemannian manifolds with nonnegative ricci curvature as applications we give new proofs of the sharp l p sobolev inequality and l p gagliardo nirenberg inequalities on r n moreover two improvements of l p gagliardo nirenberg inequalities are derived

a hybrid molecular and continuum method for low reynolds number flows abstract a novel hybrid method is proposed and developed combining the continuum boundary singularity method bsm and the molecular direct simulation monte carlo method dsmc for low reynolds number flows in the transition regime the flow about a single cylinder is used to test the proposed hybrid method the dsmc is applied to an annular region enclosing the cylinder and the bsm is employed to solve the entire flow domain the parameters used in the dsmc and in the coupling procedure such as the number of simulated particles the cell size and the size of the coupling zone are determined by inspecting the convergence rate and accuracy of obtained solution it is observed that in the partial slip flow regime the results from the hybrid method match the ones from the bsm combined with the heuristic partial slip boundary conditions for higher knudsen numbers the differences in velocity and pressure drop between the hybrid and continuum bsm are significant that justify the use of hybrid method

topological degree theory and fixed point theorems in fuzzy normed space abstract in this paper the leray schauder topological degree theory is developed in a fuzzy normed space since the linear topology on this fuzzy normed space is not necessarily locally convex and since each menger probabilistic normed space can be considered as a special fuzzy normed space the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by nagumo amer j math 73 1951 497 511 and it also is an extension of the degree theory in menger probabilistic normed space studied by zhang and chen appl math mech 10 6 1989 477 486 applying this degree theory some fixed point theorems for operators are given in fuzzy normed spaces and some former corresponding results are extended and improved

extensions of the i mmse relationship to gaussian channels with feedback and memory unveiling a fundamental link between information theory and estimation theory the i mmse relationship by guo et al together with its numerous extensions has great theoretical significance and various practical applications on the other hand its influences to date have been restricted to channels without feedback or memory due to the absence of its extensions to such channels in this paper we propose the extensions of the i mmse relationship to discrete time and continuous time gaussian channels with feedback and or memory our approach is based on a very simple observation which can be applied to other scenarios such as a simple and direct proof of the classical de bruijn s identity

a new entropy power inequality for integer valued random variables the entropy power inequality epi yields lower bounds on the differential entropy of the sum of two independent real valued random variables in terms of the individual entropies versions of the epi for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy but no universal inequality is known beyond trivial ones more recently the sumset theory for the entropy function yields a sharp inequality h x x prime h x geq 1 over 2 o 1 when x x prime are independent identically distributed i i d with high entropy this paper provides the inequality h x x prime h x geq g h x where x x prime are arbitrary i i d integer valued random variables and where g is a universal strictly positive function on bbr satisfying g 0 0 extensions to nonidentically distributed random variables and to conditional entropies are also obtained

a new i vector topology generated by a fuzzy norm this paper is devoted to the research for a topological structure for i vector on the fuzzy normed linear space in the sense of felbin by using a natural method a new i vector topology on the fuzzy normed linear space is constructed we study some of its properties and prove that the fuzzy normed linear space is a locally convex i topological vector space with respect to the i topology under a proper condition we give a necessity and sufficienct condition for a fuzzy normed linear space to be hausdorff and weakly hausdorff with respect to the i vector topology respectively in addition we also discuss the relations between this i vector topology and other two known i vector topologies on the fuzzy normed linear space lastly we give some canonical examples of fuzzy normed linear spaces using this i vector topology

a strengthened entropy power inequality for log concave densities we show that shannon s entropy power inequality admits a strengthened version in the case in which the densities are log concave in such a case in fact one can extend the blachman stam argument to obtain a sharp inequality for the second derivative of shannon s entropy functional with respect to the heat semigroup

a generalization of the entropy power inequality with applications the authors prove the following generalization of the entropy power inequality h ax or h ax where h denotes joint differential entropy x x sub 1 x sub n is a random vector with independent components x x x sub n is a gaussian vector with independent components such that h x sub i h x sub i i 1 n and a is any matrix this generalization of the entropy power inequality is applied to show that a non gaussian vector with independent components becomes closer to gaussianity after a linear transformation where the distance to gaussianity is measured by the information divergence another application is a lower bound greater than zero for the mutual information between nonoverlapping spectral components of a non gaussian white process they also describe a dual generalization of the fisher information inequality

fuzzy bounded linear operators in felbin s type fuzzy normed linear spaces in this paper definitions of strongly fuzzy bounded linear operator and weakly fuzzy bounded linear operators are given and the idea of their fuzzy norms are introduced the concept of fuzzy dual spaces are introduced and the hahn banach theorem is established in fuzzy setting

on i topology generated by fuzzy norm abstract in this paper we point out that an i topology t on the fuzzy normed linear space x min max constructed by das and das fuzzy topology generated by fuzzy norm fuzzy sets and systems 107 1999 349 354 is incompatible with the linear structure on x that is x min max is not an i topological vector space with respect to the i topology t therefore we construct a new i topology t on the fuzzy normed linear space x l r by using fuzzy norm we study some of its properties and prove that if r max then x l r is a hausdorff locally convex i topological vector space with respect to the i topology t in addition we also study the relations among three i topologies t t and where is the induced i topology of the crisp vector topology determined by fuzzy norm

geometric inequalities from phase space translations we establish a quantum version of the classical isoperimetric inequality relating the fisher information and the entropy power of a quantum state the key tool is a fisher information inequality for a state which results from a certain convolution operation the latter maps a classical probability distribution on phase space and a quantum state to a quantum state we show that this inequality also gives rise to several related inequalities whose counterparts are well known in the classical setting in particular it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup as an application we derive a log sobolev inequality for the quantum ornstein uhlenbeck semigroup and argue that it implies fast convergence towards the fixed point for a large class of initial states

fixed point theorems for nonlinear contractions in kaleva seikkala s type fuzzy metric spaces in this paper the existence and unicity of fixed points for mappings in fuzzy metric spaces in the sense of kaleva and seikkala is discussed nonlinear contractions of the boyd wong s type alber guerre delabriere s type and kannan reich s type are considered and several new fixed point theorems for these contractions in complete fuzzy metric spaces are presented respectively also some error estimates are given for iterations to approximate fixed point previous work with respect to fixed point in fuzzy metric spaces is based on the t conorm max the presented work does away with this restriction by proposing weaker conditions defining a generic class of suitable binary operations as applications the corresponding fixed point theorems for menger probabilistic metric spaces are obtained

intrinsic entropies of log concave distributions the entropy of a random variable is well known to equal the exponential growth rate of the volumes of its typical sets in this paper we show that for any log concave random variable x the sequence of the lfloor n theta rfloor text th intrinsic volumes of the typical sets of x in dimensions n geq 1 grows exponentially with a well defined rate we denote this rate by h x theta and call it the theta text th intrinsic entropy of x we show that h x theta is a continuous function of theta over the range 0 1 thereby providing a smooth interpolation between the values 0 and h x at the endpoints 0 and 1 respectively

fast numerical method for the boltzmann equation on non uniform grids we introduce a new fast numerical method for computing discontinuous solutions to the boltzmann equation and illustrate it by numerical examples a combination of adaptive grids for approximation of the distribution function and an approximate fast fourier transform on non uniform grids for computing smooth terms in the boltzmann collision integral is used

a vector generalization of costa s entropy power inequality with applications this paper considers an entropy power inequality epi of costa and presents a natural vector generalization with a real positive semidefinite matrix parameter the new inequality is proved using a perturbation approach via a fundamental relationship between the derivative of mutual information and the minimum mean square error mmse estimate in linear vector gaussian channels as an application a new extremal entropy inequality is derived from the generalized costa epi and then used to establish the secrecy capacity regions of the degraded vector gaussian broadcast channel with layered confidential messages

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

a variational finite element method for the two dimensional stokes problem abstract this paper is devoted to the numerical solution of the steady stokes problem in stream function vorticity variables in two dimensions this is reduced to a dirichlet problem for the biharmonic operator a new formulation for the decomposition of the biharmonic operator using the potential theory is presented the problem comes down to the calculation of a function defined on the boundary which is the solution of a variational problem this kind of decomposition is used in electric potential theory to solve the laplacian problem outside a bounded domain for these problems the function defined on the boundary is called the electric charge of the conductor the first part of the paper is devoted to the analysis of the continuous problem and the variational formulation on the boundary is studied the method may be extended to the navier stokes problem by using a suitable finite element approximation similar properties for the discrete problem are found these properties imply that the discrete function is the solution of a linear system whose matrix is full but symmetric and positive definite among the iterative methods we use the conjugate gradient method which seems suitable in the context numerical results concerning the problem of steady flow in a square cavity are presented

fuzzy compact operators and topological degree theory in this paper we first prove that the definition of fuzzy norm continuity and fuzzy boundedness of linear operators are equivalent we show that the continuity assumption in the definition of compact linear operator is not necessary we also point out that there is a gap in the proof of a theorem of xiao and zhu and we give a corrected version of the theorem such that all results based on the revised theorem remain true furthermore we define a fuzzy product norm on the cartesian product of two fuzzy normed spaces and prove a multiplicative property for the leray schauder topological degree

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces in this paper we introduce the concept of f contractor couple and study the existence of solutions for a couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces we construct some new iterative algorithms for solving the couple of nonlinear equations with fuzzy mappings and prove the convergence of iterative sequences generated by the algorithms in fuzzy normed spaces as applications we obtain some new common fixed point theorems for fuzzy mappings in fuzzy normed spaces the results presented in this paper improve and generalize corresponding results of cho et al fuzzy sets and systems 110 2000 115 and fang and song fuzzy sets and systems 121 2001 267

the concavity of p entropy power and applications in functional inequalities abstract in this paper we prove the concavity of p entropy power of probability densities solving the p heat equation on closed riemannian manifold with nonnegative ricci curvature as applications we give new proofs of l p euclidean nash inequality and l p euclidean logarithmic sobolev inequality moreover an improvement of l p logarithmic sobolev inequality is derived

the concavity of renyi entropy power we associate to the pth renyi entropy a definition of entropy power which is the natural extension of shannon s entropy power and exhibits a nice behavior along solutions to the p nonlinear heat equation in rn we show that the renyi entropy power of general probability densities solving such equations is always a concave function of time whereas it has a linear behavior in correspondence to the barenblatt source type solutions this result extends costa s concavity inequality for shannon s entropy power to renyi entropies

a strong entropy power inequality when one of the random summands is gaussian we sharpen the entropy power inequality epi in terms of the strong data processing function for gaussian channels among other consequences this strong epi generalizes the vector extension of costa s epi to non gaussian channels in a precise sense this leads to a new reverse epi and as a corollary sharpens stam s uncertainty principle relating entropy power and fisher information or equivalently gross logarithmic sobolev inequality applications to network information theory are also given including a short self contained proof of the rate region for the two encoder quadratic gaussian source coding problem and a new outer bound for the one sided gaussian interference channel

on linearly topological structure and property of fuzzy normed linear space in this paper the simplified definition of fuzzy normed linear space is introduced the different structure of fuzzy normed linear space with variable right norm r is discussed in terms of topological vector space and its properties such as compactness completeness and density are studied under more general left norm l and right norm r as application the linearly topological structure of menger pn space is obtained

hessian and concavity of mutual information differential entropy and entropy power in linear vector gaussian channels within the framework of linear vector gaussian channels with arbitrary signaling the jacobian of the minimum mean square error and fisher information matrices with respect to arbitrary parameters of the system are calculated in this paper capitalizing on prior research where the minimum mean square error and fisher information matrices were linked to information theoretic quantities through differentiation the hessian of the mutual information and the entropy are derived these expressions are then used to assess the concavity properties of mutual information and entropy under different channel conditions and also to derive a multivariate version of an entropy power inequality due to costa

finite dimensional fuzzy normed linear space abstract the concept of a fuzzy normed linear space is introduced it is proved that in a finite dimensional fuzzy normed linear space fuzzy norms are the same upto fuzzy equivalence finite dimensional fuzzy subspaces of a fuzzy normed linear space are proved to be necessarily complete fuzzy normed linear spaces

a variational spectral method for the two dimensional stokes problem abstract a spectral galerkin scheme for solving the two dimensional stationary stokes problem is considered in vorticity stream function formulation the stokes problem is reduced to a biharmonic one a new formulation for the decomposition of the biharmonic operator is introduced main difficulty in this decomposition is to find the trace of the vorticity using the adjoint of the composition of trace operator and green s operator a variational form is obtained for computing the boundary value of the vorticity the discrete variational form preserves the properties such as coercivity and continuity of the continuous case which are important for our convergence analysis convergence of our scheme is proved and numerical experiments are performed for the steady driven cavity problem

statistical convergence in fuzzy normed linear spaces in this paper we introduce the notions of statistically convergent sequence and statistically cauchy sequence in a fuzzy normed linear space fns and establish some basic facts we also define statistical limit points and statistical cluster points of a sequence in an fns and investigate the relations between these concepts

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

simple proof of the concavity of the entropy power with respect to added gaussian noise a very simple proof of m h costa s result see ibid vol it 31 p 751 60 1985 that the entropy power of x sub t x n o ti is concave in t is derived as an immediate consequence of an inequality concerning fisher information this relationship between fisher information and entropy is found to be useful for proving the central limit theorem thus one who seeks new entropy inequalities should try first to find new equalities about fisher information or at least to exploit the existing ones in new ways

contractivity properties of a quantum diffusion semigroup we consider a quantum generalization of the classical heat equation and study contractivity properties of its associated semigroup we prove a nash inequality and a logarithmic sobolev inequality the former leads to an ultracontractivity result this in turn implies that the largest eigenvalue and the purity of a state with positive wigner function evolving under the action of the semigroup decrease at least inverse polynomially in time while its entropy increases at least logarithmically in time

a concavity property for the reciprocal of fisher information and its consequences on costa s epi abstract we prove that the reciprocal of fisher information of a log concave probability density x in r n is concave in t with respect to the addition of a gaussian noise z t n 0 t i n as a byproduct of this result we show that the third derivative of the entropy power of a log concave probability density x in r n is nonnegative in t with respect to the addition of a gaussian noise z t for log concave densities this improves the well known costa s concavity property of the entropy power costa 1985

on the entropy power inequality for the renyi entropy of order 0 1 using a sharp version of the reverse young inequality and a renyi entropy comparison result due to fradelizi madiman and wang 2016 the authors derive renyi entropy power inequalities for log concave random vectors when renyi parameters belong to 0 1 furthermore the estimates are shown to be sharp up to absolute constants

the entropy power inequality and mrs gerber s lemma for groups of order 2 n shannon s entropy power inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies the entropy power inequality has played a key role in resolving a number of problems in information theory it is therefore interesting to examine the existence of a similar inequality for discrete random variables in this paper we obtain an entropy power inequality for random variables taking values in a group of order 2 n i e for such a group g we explicitly characterize the function f g x y giving the minimum entropy of the sum of two independent g valued random variables with respective entropies x and y random variables achieving the extremum in this inequality are thus the analogs of gaussians in this case and these are also determined it turns out that f g x y is convex in x for fixed y and by symmetry convex in y for fixed x this is a generalization to groups of order 2 n of the result known as mrs gerber s lemma

resonance theorems for family of quasi homogeneous operators in fuzzy normed linear spaces in this paper we establish some resonance theorems for families of point wise bounded semi bounded and non unbounded quasi homogeneous and email protected subadditive operators in fuzzy normed linear spaces as their consequences we obtain the corresponding resonance theorems in classical normed spaces and menger probabilistic normed spaces

vorticity conditioning in the computation of two dimensional viscous flows abstract the problem of evaluating the boundary values of the vorticity in the calculation of two dimensional viscous flows is considered it is shown that the splitting of the fourth order equation for the stream function into two second order problems implies specific integral conditions which fix the abstract projection of the vorticity field with respect to the linear manifold of the harmonic functions these conditions are a direct consequence of the boundary conditions on the velocity and ensure satisfaction of physically essential conservation laws for the vorticity the discrete analogue of the projection conditions produces as many algebraic equations as the number of boundary points and requires the solution of an equal number of dirichlet problems in the particular case of stationary linearized equations stokes equations a direct i e noniterative method of solution is obtained steady and unsteady computational schemes relying on the projection conditions on the vorticity are introduced and extensive numerical results of finite difference calculations of the driven cavity model problem are reported and discussed

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

fuzzy normed space of operators and its completeness in this paper the new definition of the fuzzy norm of a linear operator from one fuzzy normed linear space into another is introduced and the boundedness of such an operator is described furthermore the space of all bounded linear operators endowed with this fuzzy norm is studied consequently its topological structure as well as completeness is given and that it can itself be made into a fuzzy normed linear space is also shown

condensing operators and topological degree theory in standard fuzzy normed spaces in this paper level cut measures of noncompactness are introduced in standard fuzzy normed spaces and condensing operators are studied using the extensions of nonlinear compact operators topological degree theory with respect to condensing operators is given in standard fuzzy normed spaces as an application of degree theory a fixed point theorem for condensing operators is presented

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power