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

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

stability in the energy space of the sum of n peakons for the degasperis procesi equation the degasperis procesi equation possesses well known peaked solitary waves that are called peakons their stability has been established by lin and liu in 5 in this paper we localize the proof in some suitable sense detailed in section 3 of the stability of a single peakon thanks to this we extend the result of stability to the sum of n peakons traveling to the right with respective speeds c1 cn such that the difference between consecutive locations of peakons is large enough

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

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

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

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

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

a remark on the stability of peakons for the degasperis procesi equation in this paper we present a new argument see lemma 3 4 that allows us to simplify the proof of stability of peakons established in lin and liu 2009 theorem 1 1

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

elimination of cusps in dimension 4 and its applications we study a class of homotopies between maps from 4 manifolds to surfaces which we call cusp merges these homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4 manifolds derived from maps to the 2 sphere for example broken lefschetz fibrations wrinkled fibrations or morse 2 functions our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy the latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges we apply our methods to the uniqueness problem of surface diagrams of 4 manifolds and describe algorithms to obtain surface diagrams for total spaces of achiral lefschetz fibrations and 4 manifolds of the form m s1 where m is a 3 manifold along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in morse theory

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 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

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

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

stability of peakons for the degasperis procesi equation the degasperis procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the euler equations with the same asymptotic accuracy as that of the camassaholm equation in this paper we study the orbital stability problem of the peaked solitons to the degasperis procesi equation on the line by constructing a liapunov function we prove that the shapes of these peakon solitons are stable under small perturbations

topology of surface diagrams of smooth 4 manifolds surface diagrams are a new way to specify any smooth closed orientable 4 manifold by an orientable surface decorated with simple closed curves these curves are cyclically indexed and each curve has a unique transverse intersection with the next the aim of this paper is to announce a uniqueness theorem for these objects within a fixed homotopy class that turns out to be similar to the reidemeister singer theorem for heegaard splittings of 3 manifolds

on the orbital stability of the degasperis procesi antipeakon peakon profile in this paper we prove an orbital stability result for the degasperis procesi peakon with respect to perturbations having a momentum density that is first negative and then positive this leads to the orbital stability of the antipeakon peakon profile with respect to such perturbations

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

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

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

transversality in families of mappings by a classical result of thom 20 if n and p are smooth manifolds and q is a smooth submanifold of p the set of maps f n p transverse to q is residual in c n p if q is a closed subset of p the set of transverse maps is also open however if u is a further smooth manifold and fu a family of smooth mappings we cannot expect to be able to deform the family to make each fu transverse to q this situation was considered by bruce 2 who showed that for a residual set of maps f n p u we have a for each u u the set of points in n at which fu is not transverse to q is discrete and b at each such point the class of the contact of fu n with q has finite codimension this is the result he established the formal statement given was more explicit but weaker this result is not entirely satisfactory we may hope also for example that each contact class presented is universally unfolded in the family we address this problem and show that this hope is justified under a hypothesis of nice dimensions after recalling mather s theory of topological stability we establish a result valid in all dimensions with topologically versal unfoldings since the proofs of these results involve numerous translations between equivalent versality conditions we begin with a section developing a notation and approach to facilitate such proofs we then consider the corresponding question with the submanifold q replaced by a stratified subset of p this requires an extensive study of theories of contact equivalence relative to a subset of the target good algebraic conditions and criteria are only known under an analyticity hypothesis for some results this needs to be strengthened to a holonomic condition with these preliminaries we obtain a parallel to the theory of the earlier section however a full account of a topological analogue remains beyond our reach

fiber connected indefinite morse 2 functions on connected n manifolds we discuss generic smooth maps from smooth manifolds to smooth surfaces which we call morse 2 functions and homotopies between such maps the two central issues are to keep the fibers connected in which case the morse 2 function is fiber connected and to avoid local extrema over one dimensional submanifolds of the range in which case the morse 2 function is indefinite this is foundational work for the long range goal of defining smooth invariants from morse 2 functions using tools analogous to classical morse homology and cerf theory

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

equation of motion for interacting pulses we develop a systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension the theory is applicable to both dissipative and dispersive systems in the case of the time dependent ginzburg landau equation which is a typical example of a dissipative system the front equation obtained is the same as has been obtained previously the pulse interaction is also derived for the korteweg char21 de vries equation emphasizing the difference between the cases with and without dissipative terms

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

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

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

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

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

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