a unifying variational perspective on some fundamental information theoretic inequalities this paper proposes a unifying variational approach for proving and extending some fundamental information theoretic inequalities fundamental information theory results such as maximization of differential entropy minimization of fisher information cramer rao inequality worst additive noise lemma entropy power inequality and extremal entropy inequality are interpreted as functional problems and proved within the framework of calculus of variations several applications and possible extensions of the proposed results are briefly mentioned

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

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

beyond the entropy power inequality via rearrangements a lower bound on the renyi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements for the special case of boltzmann shannon entropy this lower bound is better than that given by the entropy power inequality several applications are discussed including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of levy processes

dynamics of convection thermal convection in a fluid layer is an example of a dynamical system governed by partial differential equations as the relevant control parameter the rayleigh number is increased successive bifurcations may lead to chaos and the nature of the transition depends on the spatial structure of the flow numerical experiments with idealized symmetry and boundary conditions make it possible to explore nonlinear behaviour in some detail and to relate bifurcation structures to those found in appropriate low order systems two examples are used to illustrate transitions to chaos in two dimensional thermosolutal convection where the spatial structure is essentially trivial chaos is caused by a heteroclinic bifurcation involving a symmetric pair of saddle foci when convection is driven by internal heating several competing spatial structures are involved and the transition to chaos is more complicated in both two and three dimensional configurations although the first few bifurcations can be isolated a statistical treatment is needed for behaviour at high rayleigh numbers

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 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

on the nature of the torus in the complex lorenz equations the complex lorenz equations are a nonlinear fifth order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus in this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter and between zero and nonzero r2 complexity parameter in so doing we are able to gain insight on the effect of the rotational invariance of the system and on how extra weak dispersion r2 0 affects the chaotic behavior of the real lorenz system which describes a weakly dissipative dispersive instability

crame spl acute r rao and moment entropy inequalities for renyi entropy and generalized fisher information the moment entropy inequality shows that a continuous random variable with given second moment and maximal shannon entropy must be gaussian stam s inequality shows that a continuous random variable with given fisher information and minimal shannon entropy must also be gaussian the crame spl acute r rao inequality is a direct consequence of these two inequalities in this paper the inequalities above are extended to renyi entropy p sup th moment and generalized fisher information generalized gaussian random densities are introduced and shown to be the extremal densities for the new inequalities an extension of the crame spl acute r rao inequality is derived as a consequence of these moment and fisher information inequalities

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

optimal m estimation in high dimensional regression we consider in the modern setting of high dimensional statistics the classic problem of optimizing the objective function in regression using m estimates when the error distribution is assumed to be known we propose an algorithm to compute this optimal objective function that takes into account the dimensionality of the problem although optimality is achieved under assumptions on the design matrix that will not always be satisfied our analysis reveals generally interesting families of dimension dependent objective functions

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

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

hardy littlewood sobolev inequalities via fast diffusion flows we give a simple proof of the d 2 cases of the sharp hardy littlewood sobolev inequality for d 3 and the sharp logarithmic hardy littlewood sobolev inequality for d 2 via a monotone flow governed by the fast diffusion equation

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

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

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity ii this article continues the analysis of the preceding article showing that the lyapunov functional introduced by newman can be used to prove the stability of the barenblatt pattle solutions of the porous medium equation

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

nonlinear dynamos a complex generalization of the lorenz equations abstract plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the lorenz system in the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations ending in chaos as a stability parameter d the dynamo number is increased we show that solutions undergo three successive hopf bifurcations followed by a transition to chaos the system possesses a symmetry and can therefore be reduced to a fifth order system with trajectories that lie on a 2 torus after the third bifurcation as d is then increased frequency locking occurs followed by a sequence of period doubling bifurcations that leads to chaos this behaviour is probably caused by the shil nikov mechanism with a conjectured homoclinic orbit when d is infinite

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

logarithmic sobolev and multilinear holder s inequalities via heat flow monotonicity formulas abstract heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriving functional and geometric inequalities which are in turn useful in mathematical analysis and applications this paper aims mainly at proving logarithmic sobolev and multilinear holder s inequalities through the heat flow method precisely two entropy monotonicity formulas are constructed via the heat flow it is shown that the first entropy monotonicity formula is intimately related to the concavity of the power of shannon entropy and fisher information from which the associated logarithmic sobolev inequality for probability measure in euclidean setting is recovered the second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to establish the proof of the multilinear holder inequalities

counterexample to the vector generalization of costa s entropy power inequality and partial resolution we give a counterexample to the vector generalization of costa s entropy power inequality due to liu et al in particular the claimed inequality can fail if the matrix valued parameter in the convex combination does not commute with the covariance of the additive gaussian noise conversely the inequality holds if these two matrices commute

heteroclinic cycles and periodic orbits for the o 2 equivariant 0 1 2 mode interaction we study the quadratic normal form describing the generic interaction of fourier modes of wavenumbers 0 1 and 2 under the symmetry group o 2 of rotations and reflections in the case that the homogeneous quadratic terms preserve energy the sum of the squares of absolute values of the complex variables this system is a generalization of the 1 2 mode interaction studied by dangelmayr g dangelmayr steady state mode interactions in the presence of o 2 symmetry dyn stab syst 1 2 1986 159 185 armbruster et al d armbruster j guckenheimer p holmes heteroclinic cycles and modulated travelling waves in systems with o 2 symmetry physica d 29 1988 257 282 and others and its restriction to the 1 2 subspace is a degenerate case of that system it displays all the classes of fixed points periodic orbits standing and travelling waves invariant tori modulated travelling waves and heteroclinic cycles found in the 1 2 interaction as well as new heteroclinic cycles connecting pure and mixed modes chaotic cycles and strange periodic orbits we describe the key dynamical features show that the degenerate 1 2 case possesses a second organizing center at which bifurcation curves coalesce provide representative bifurcation sets and diagrams for the 1 2 and 0 1 2 systems and use a conservative limit to understand the periodic orbits in the latter system 2005 elsevier b v all rights reserved

possible new strange attractors with spiral structure we define a class of three dimensional differential equations which might present strange attractors of a new kind this is illustrated by numerical observations on an explicit example

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

on the similarity of the entropy power inequality and the brunn minkowski inequality corresp the entropy power inequality states that the effective variance entropy power of the sum of two independent random variables is greater than the sum of their effective variances the brunn minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii both these inequalities are recast in a form that enhances their similarity in spite of this similarity there is as yet no common proof of the inequalities nevertheless their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found two applications of this reasoning are presented first an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the fisher information matrix given an entropy constraint just as a sphere minimizes the surface area given a volume constraint second a theorem involving the effective radii of growing convex sets is proved

best matching barenblatt profiles are delayed the growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self similar solutions however at next order there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self similar barenblatt solutions this distance can be measured in terms of a relative entropy to the best matching barenblatt profile this best matching barenblatt function determines a scale in new variables based on this scale which are given by a self similar change of variables if and only if the initial datum is one of the barenblatt profiles the typical scale is monotone and has a limit coming back to original variables the best matching barenblatt profile is delayed compared to the self similar solution with same initial second moment as the initial datum such a delay is a new phenomenon which has to be taken into account for instance when fitting experimental data

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

one parameter class of uncertainty relations based on entropy power we use the concept of entropy power to derive a one parameter class of information theoretic uncertainty relations for pairs of conjugate observables in an infinite dimensional hilbert space this class constitutes an infinite tower of higher order statistics uncertainty relations which allows one in principle to determine the shape of the underlying information distribution function by measuring the relevant entropy powers we illustrate the capability of this class by discussing two examples superpositions of vacuum and squeezed states and the cauchy type heavy tailed wave function

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

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 lyapunov functional for the evolution of solutions to the porous medium equation to self similarity i we employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a lyapunov functional which characterizes the evolution of the profile to its self similar solution

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

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

entropy power inequality for the renyi entropy the classical entropy power inequality is extended to the renyi entropy we also discuss the question of the existence of the entropy for sums of independent random variables

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

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

convexity and regularity properties for entropic interpolations abstract in this paper we prove a convexity property of the relative entropy along entropic interpolations solutions of the schrodinger problem and a regularity property of the entropic cost along the heat flow then we derive a dimensional evi inequality and a contraction property for the entropic cost along the heat flow as a consequence we recover the equivalent results in the wasserstein space proved by erbar kuwada and sturm

equilibria of lower semicontinuous games abstract in this paper we first introduce the notion of majorized i majorized correspondences which generalize majorized correspondences and differs from l majorized correspondences and then we give a new existence theorem of a maximal element as its applications we obtain some new existence theorems of an equilibrium for a qualitative game and an abstract economy

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

monotonicity thinning and discrete versions of the entropy power inequality we consider the entropy of sums of independent discrete random variables in analogy with shannon s entropy power inequality where equality holds for normals in our case infinite divisibility suggests that equality should hold for poisson variables we show that some natural analogues of the epi do not in fact hold but propose an alternative formulation which does always hold the key to many proofs of shannon s epi is the behavior of entropy on scaling of continuous random variables we believe that renyi s operation of thinning discrete random variables plays a similar role to scaling and give a sharp bound on how the entropy of ultra log concave random variables behaves on thinning in the spirit of the monotonicity results established by artstein ball barthe and naor we prove a stronger version of concavity of entropy which implies a strengthened form of our discrete epi

capacity and rate regions of a class of broadcast interference channels in this paper a class of broadcast interference channels bic is investigated where one of the two broadcast receivers is subject to interference coming from a point to point transmission for a general discrete memoryless broadcast interference channel dm bic an achievable scheme based on message splitting superposition and binning is proposed and a concise representation of the corresponding achievable rate region mathcal r is obtained two partial order broadcast conditions interference oblivious less noisy and interference cognizant less noisy are defined thereby extending the usual less noisy condition for a regular broadcast channel by taking interference into account under these conditions a reduced form of mathcal r is shown to be equivalent to a rate region based on a simpler scheme where the broadcast transmitter uses only superposition furthermore if interference is strong for the interference oblivious less noisy dm bic the capacity region is given by the aforementioned two equivalent rate regions for the interference cognizant less noisy dm bic it is argued that the strong but not very strong interference condition does not exist and in this case the capacity region for the very strong interference is obtained for a gaussian broadcast interference channel channel parameters are categorized into three regimes for the first two regimes which are closely related to the two partial order broadcast conditions achievable rate regions are derived by specializing the corresponding achievable schemes of dm bics with gaussian input distributions the entropy power inequality based outer bounds are obtained by combining bounding techniques for a gaussian broadcast channel and a gaussian interference channel these inner and outer bounds lead to either exact or approximate characterizations of capacity regions and sum capacity under various conditions for the remaining complementing regime inner and outer bounds are also provided

chaos in a three dimensional single loop feedback system with a piecewice linear feedbackk function abstract this paper describes chaotic behaviour in a single loop feedback system of low order the system is particularly simple and because it is piecewise linear and hence can be undersdtood globally a description of the flow much more complete than usual can be given an orbit structure is suggested for the system after a minor approximation is considered the approximation gives two equivalent discrete one dimensional maps of the interval to itself and these maps are discussed in the light of recent work the validity of the approximation is confirmed by computer simulation

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

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

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

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

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

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

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

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

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

existence and stability of multiple impulse solutions of a nerve equation a piecewise linear partial differential equation which models the propagation of a nerve impulse along an axon is studied for the equation parameters used there are two solutions which represent solitary impulses traveling with fixed form one with a slower velocity c s1 than the velocity c f1 of the other we show that near these velocities there exist multiple impulse solutions and exhibit four such solutions each of which has two pulses the local stability of these solutions is analyzed the two solutions whose velocities are near c s1 demonstrate instabilities which indicate growth or decay of the separate pulses for one of the multiple impulse solutions with velocity near c s1 and for one of those with velocity near c f1 the second pulse deomonstrates an instability which indicates a change in the spacing between the two pulses the other solution with velocity near c f1 is stable

on the equivalence between stein and de bruijn identities this paper focuses on illustrating 1 the equivalence between stein s identity and de bruijn s identity and 2 two extensions of de bruijn s identity first it is shown that stein s identity is equivalent to de bruijn s identity under additive noise channels with specific conditions second for arbitrary but fixed input and noise distributions under additive noise channels the first derivative of the differential entropy is expressed by a function of the posterior mean and the second derivative of the differential entropy is expressed in terms of a function of fisher information several applications over a number of fields such as signal processing and information theory are presented to support the usefulness of the developed results in this paper

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

remote source coding under gaussian noise dueling roles of power and entropy power the distributed remote source coding the so called ceo problem is studied in the case where the underlying source not necessarily gaussian has finite differential entropy and the observation noise is gaussian the main result is a new lower bound for the sum rate distortion function under arbitrary distortion measures when specialized to the case of mean squared error it is shown that the bound exactly mirrors a corresponding upper bound except that the upper bound has the source power variance whereas the lower bound has the source entropy power bounds exhibiting this pleasing duality of power and entropy power have been well known for direct and centralized source coding since shannon s work while the bounds hold generally their value is most pronounced when interpreted as a function of the number of agents in the ceo problem

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

oscillators with chaotic behavior an illustration of a theorem by shil nikov using an explicit one parameter family of differential equations describing oscillators with feedback effects we prove the existence of values of the parameters such that there exist infinitely many unstable periodic orbits of saddle type the proof relies on a theorem by shil nikov which we propose as an explanation for the origin and structure of the chaotic behavior displayed by many well known third order differential systems

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

fixed points theorems for mappings with non compact and non convex domains this note gives some fixed point theorems for lower and upper semi continuous mappings and mappings with open lower sections defined on non compact and non convex sets it will be noted that the conditions of our theorems are not only sufficient but also necessary also our theorems generalize some well known fixed point theorems such as the kakutani fixed point theorem and the brouwer schauder fixed point theorem by relaxing the compactness and convexity conditions

information measures inequalities and performance bounds for parameter estimation in impulsive noise environments recent studies found that many channels are affected by additive noise that is impulsive in nature and is best explained by heavy tailed symmetric alpha stable distributions dealing with impulsive noise environments comes with an added complexity with respect to the standard gaussian environment the alpha stable probability density functions do not possess closed form expressions except in few special cases furthermore they have an infinite second moment and the nice hilbert space structure of the space of random variables having a finite second moment is lost along with its tools and methodologies this is indeed the case in estimation theory where classical tools to quantify the performance of an estimator are tightly related to the assumption of having finite variance variables in alpha stable environments expressions such as the mean square error and the cramer rao bound are hence problematic in this paper we tackle the parameter estimation problem in the impulsive noise environments and develop novel tools that are tailored to the alpha stable and heavy tailed noise environments tools that coincide with the standard ones adopted in the gaussian setup namely a generalized power measure and a generalized fisher information we generalize known information inequalities commonly used in the gaussian context the de bruijn identity the fisher information inequality the isoperimetric inequality for entropies and the cramer rao bound additionally we derive upper bounds on the differential entropy of independent sums having a stable component intermediately the new power measure is used to shed some light on the additive alpha stable noise channel capacity in a setup that generalizes the linear average power constrained additive white gaussian noise channel our theoretical findings are paralleled with numerical evaluations of various quantities and bounds using developed matlab packages

single and multiple pulse waves for the fitzhugh nagumo the fitzhugh nagumo equations describe the propagation of impulses along a nerve axon we study traveling wave solutions which satisfy a set of ordinary differential equations with a cubic nonlinearity f v v 1 v v a our emphasis is on solutions which tend to zero at pm infty an improved proof of a previous result that there are at least two such solutions is presented further by allowing the middle root v a of the nonlinearity to be a small parameter better estimates of the parameter range where such solutions exist are given in view of recent work of evans fenichel feroe and sil nikov our results imply that there are a great many more bounded traveling waves than was known previously in particular there is an infinite family of periodic waves all traveling with the same speed a reasonable transversality hypothesis introduced by evans fenichel and feroe implies also that there are multiple pulse waves at speeds close to that of a single pulse

discrete versions of the transport equation and the shepp olkin conjecture we introduce a framework to consider transport problems for integer valued random variables we introduce weighting coecients which allow us to characterise transport problems in a gradient ow setting and form the basis of our introduction of a discrete version of the benamou brenier formula further we use these coecients to state a new form of weighted log concavity these results are applied to prove the monotone case of the shepp olkin entropy concavity conjecture msc2000 primary 60e15 secondary 94a17 60d99

wasserstein continuity of entropy and outer bounds for interference channels it is shown that under suitable regularity conditions differential entropy is o sqrt n lipschitz as a function of probability distributions on mathbb r n with respect to the quadratic wasserstein distance under similar conditions discrete shannon entropy is shown to be o n lipschitz in distributions over the product space with respect to ornstein s bar d distance wasserstein distance corresponding to the hamming distance these results together with talagrand s and marton s transportation information inequalities allow one to replace the unknown multi user interference with its independent identically distributed approximations as an application a new outer bound for the two user gaussian interference channel is proved which in particular settles the missing corner point problem of costa 1985

on renyi entropy power inequalities this paper gives improved renyi entropy power inequalities r epis consider a sum s n sum k 1 n x k of n independent continuous random vectors taking values on mathbb r d and let alpha in 1 infty an r epi provides a lower bound on the order alpha renyi entropy power of s n that up to a multiplicative constant which may depend in general on n alpha d is equal to the sum of the order alpha renyi entropy powers of the n random vectors x k k 1 n for alpha 1 the r epi coincides with the well known entropy power inequality by shannon the first improved r epi is obtained by tightening the recent r epi by bobkov and chistyakov which relies on the sharpened young s inequality a further improvement of the r epi also relies on convex optimization and results on rank one modification of a real valued diagonal matrix

the secrecy capacity region of the gaussian mimo multi receiver wiretap channel in this paper we consider the gaussian multiple input multiple output mimo multi receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper we derive the secrecy capacity region of this channel for the most general case we first show that even for the single input single output siso case existing converse techniques for the gaussian scalar broadcast channel cannot be extended to this secrecy context to emphasize the need for a new proof technique our new proof technique makes use of the relationships between the minimum mean square error and the mutual information and equivalently the relationships between the fisher information and the differential entropy using the intuition gained from the converse proof of the siso channel we first prove the secrecy capacity region of the degraded mimo channel in which all receivers have the same number of antennas and the noise covariance matrices can be arranged according to a positive semi definite order we then generalize this result to the aligned case in which all receivers have the same number of antennas however there is no order among the noise covariance matrices we accomplish this task by using the channel enhancement technique finally we find the secrecy capacity region of the general mimo channel by using some limiting arguments on the secrecy capacity region of the aligned mimo channel we show that the capacity achieving coding scheme is a variant of dirty paper coding with gaussian signals

rate of convergence to barenblatt profiles for the fast diffusion equation we study the asymptotic behaviour of positive solutions of the cauchy problem for the fast diffusion equation as t approaches the extinction time we find a continuum of rates of convergence to a self similar profile these rates depend explicitly on the spatial decay rates of the initial data

rate region of gaussian multiple description coding with individual and central distortion constraints the rate region of gaussian multiple description coding with individual and central distortion constraints is completely characterized specifically a lower bound and an upper bound are derived for each supporting hyperplane of the rate region where the lower bound is associated with a max min game while the upper bound is associated with a min max game furthermore it is shown that these two bounds coincide due to the existence of a saddle point