growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

uniform logarithmic sobolev inequalities for conservative spin systems with super quadratic single site potential we consider a noninteracting unbounded spin system with conservation of the mean spin we derive a uniform logarithmic sobolev inequality lsi provided the single site potential is a bounded perturbation of a strictly convex function the scaling of the lsi constant is optimal in the system size the argument adapts the two scale approach of grunewald villani westdickenberg and the second author from the quadratic to the general case using an asymmetric brascamp lieb type inequality for covariances we reduce the task of deriving a uniform lsi to the convexification of the coarse grained hamiltonian which follows from a general local cramer theorem

transference principles for log sobolev and spectral gap with applications to conservative spin systems we obtain new principles for transferring log sobolev and spectral gap inequalities from a source metric measure space to a target one when the curvature of the target space is bounded from below as our main application we obtain explicit estimates for the log sobolev and spectral gap constants of various conservative spin system models consisting of non interacting and weakly interacting particles constrained to conserve the mean spin when the self interaction is a perturbation of a strongly convex potential this partially recovers and partially extends previous results of caputo chafai grunewald landim lu menz otto panizo villani westdickenberg and yau when the self interaction is only assumed to be non strongly convex as in the case of the two sided exponential measure we obtain sharp estimates on the system s spectral gap as a function of the mean spin independently of the size of the system

rogers shephard and local loomis whitney type inequalities we provide functional analogues of the classical geometric inequality of rogers and shephard on products of volumes of sections and projections as a consequence we recover and obtain some new functional versions of rogers shephard type inequalities as well as some generalizations of the geometric rogers shephard inequality in the case where the subspaces intersect these generalizations can be regarded as sharp local reverse loomis whitney inequalities we also obtain a sharp local loomis whitney inequality

energy distribution of the quantum harmonic oscillator under random time dependent perturbations this paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time dependent forces the perturbations are small but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity we describe the effective evolution equation of the state vector which reads as a stochastic partial differential equation we exhibit a closed form equation for the transition probabilities which can be interpreted in terms of a jump process using standard probability tools we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle

a phase transition behavior for brownian motions interacting through their ranks consider a time varying collection of n points on the positive real axis modeled as exponentials of n brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time if at each time point we divide the points by their sum under suitable assumptions the rescaled point process converges to a stationary distribution depending on n and the vector of drifts as time goes to infinity this stationary distribution can be exactly computed using a recent result of pal and pitman the model and the rescaled point process are both central objects of study in models of equity markets introduced by banner fernholz and karatzas in this paper we look at the behavior of this point process under the stationary measure as n tends to infinity under a certain continuity at the edge condition on the drifts we show that one of the following must happen either i all points converge to 0 or ii the maximum goes to 1 and the rest go to 0 or iii the processes converge in law to a non trivial poisson dirichlet distribution the underlying idea of the proof is inspired by talagrand s analysis of the low temperature phase of derrida s random energy model of spin glasses the main result establishes a universality property for the bfk models and aids in explicit asymptotic computations using known results about the poisson dirichlet law

quantitative stability of the entropy power inequality we establish quantitative stability results for the entropy power inequality epi specifically we show that if uniformly log concave densities nearly saturate the epi then they must be close to gaussian densities in the quadratic kantorovich wasserstein distance furthermore if one of the densities is gaussian and the other is log concave or more generally has positive spectral gap then the deficit in the epi can be controlled in terms of the l 1 kantorovich wasserstein distance or relative entropy respectively as a counterpoint an example shows that the epi can be unstable with respect to the quadratic kantorovich wasserstein distance when densities are uniformly log concave on sets of measure arbitrarily close to one our stability results can be extended to non log concave densities provided certain regularity conditions are met the proofs are based on mass transportation

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

combined arrival time imaging and time reversal for scatterer identification abstract the computational inverse problem of identifying a scatterer in a time dependent wave field is considered the wave speed of the background medium and the wave source are assumed to be known wave measurements possibly noisy are given at chosen discrete points in space sensor locations and time the goal is to find scatterer parameters such as location size and shape the computational solution procedure consists of two steps in the first step a standard fast arrival time imaging ati algorithm is employed this results in a rough image which provides possible regions for the location of the scatterer in the second step an optimization scheme based on time reversal tr is used to determine the location size and shape of the scatterer the preliminary ati step has the effect of reducing considerably the search space for the tr optimization also proposed here is an improved definition of the objective function used for the optimization which tends to eliminate spurious solutions numerical experiments based on a finite element discretization in space and an explicit newmark time stepping show the identification capability of the proposed scheme for a model problem involving the linear scalar wave equation in a bounded domain two types of scatterers are considered a crack with a known orientation whose location and size are sought and a rectangular scatterer whose location aspect ratio and size are sought the performance of the scheme in the presence of measurement noise is also demonstrated

convergence of a non stiff boundary integral method for interfacial flows with surface tension boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities in addition surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics this makes the spatial discretization even more difficult and at the same time imposes a severe time step constraint for stable explicit time integration methods a proof of the convergence of a reformulated boundary integral method for two density fluid interfaces with surface tension is presented the method is based on a scheme introduced by hou lowengrub and shelley j comp phys 114 1994 pp 312 338 to remove the high order stability constraint or stiffness some numerical filtering is applied carefully at certain places in the discretization to guarantee stability the key of the proof is to identify the most singular terms of the method and to sho through energy estimates that these terms balance one another the analysis is at a time continuous space discrete level but a fully discrete case for a simple hele shaw interface is also studied the time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the cfl constraint depends on the curvature and regularity of the solution the robustness of the method is illustrated with several numerical examples a numerical simulation of an unstably stratified two density interfacial flow shows the roll up of the interface the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed the method remains stable even in the full nonlinear regime of motion another application of the method shows the process of drop formation in a falling single fluid

from concentration to logarithmic sobolev and poincare inequalities abstract we give a new proof of the fact that gaussian concentration implies the logarithmic sobolev inequality when the curvature is bounded from below and also that exponential concentration implies poincare inequality under null curvature condition our proof holds on non smooth structures such as length spaces and provides a universal control of the constants we also give a new proof of the equivalence between dimension free gaussian concentration and talagrand s transport inequality

on diffusion semigroups preserving the log concavity it is proved that if the semigroup with the generator akl xk xl m xm where akl and m are smooth functions sends log concave functions to log concave functions then a akl is constant and m is an affine mapping

isoperimetric and analytic inequalities for log concave probability measures we discuss an approach based on the brunn minkowski inequality to isoperimetric and analytic inequalities for probability measures on euclidean space with logarithmically concave densities in particular we show that such measures have positive isoperimetric constants in the sense of cheeger and thus always share poincare type inequalities we then describe those log concave measures which satisfy isoperimetric inequalities of gaussian type the results are precised in dimension 1

weighted poincare type inequalities for cauchy and other convex measures brascamp lieb type weighted poincare type and related analytic inequalities are studied for multidimensional cauchy distributions and more general concave probability measures in the hierarchy of convex measures in analogy with the limiting infinite dimensional log concave gaussian model the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures cheeger type isoperimetric inequalities are investigated similarly giving rise to a common weight in the class of concave probability measures under consideration

variational convergence over metric spaces we introduce a natural definition of l p convergence of maps p 1 in the case where the domain is a convergent sequence of measured metric space with respect to the measured gromov hausdorff topology and the target is a gromov hausdorff convergent sequence with the l p convergence we establish a theory of variational convergences we prove that the poincare inequality with some additional condition implies the asymptotic compactness the asymptotic compactness is equivalent to the gromov hausdorff compactness of the energy sublevel sets supposing that the targets are cat 0 spaces we study convergence of resolvents as applications we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of ricci curvature

torsional effects in high order viscoelastic thin filament models the authors present an approximation theory for thin filaments fibers or jets which yields families of transient 1 d models time dependent one dimensional closed systems of pdes the spatial reduction from three dimensions to one is achieved by axisymmetry together with a local expansion in the radial jet coordinate this reduction is in contrast to the predominant viscoelastic models in the literature which average out the radial dimension and thereby require moment equations for the computation of higher order corrections the authors also allow torsional flow effects which are usually ignored in a general johnson segalman constitutive law a formal perturbation theory based on a slenderness parameter and a compatible velocity pressure stress ansatz is then constructed for the full 3 d free surface boundary problem this formalism contains all 1 d transient models that govern slender axisymmetric flows of inviscid viscous or viscoelastic fluids specific models follow by positing the dominant

second order poincare inequalities and clts on wiener space we prove infinite dimensional second order poincare inequalities on wiener space thus closing a circle of ideas linking limit theorems for functionals of gaussian fields stein s method and malliavin calculus we provide two applications i to a new second order characterization of clts on a fixed wiener chaos and ii to linear functionals of gaussian subordinated fields

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

pulse propagation and time reversal in random waveguides mode coupling in a random waveguide can be analyzed with asymptotic analysis based on separation of scales when the propagation distance is large compared to the size of the random inhomogeneities which have small variance and when the wavelength is comparable to the scale of the inhomogeneities in this paper we study the asymptotic form of the joint distribution of the mode amplitudes at different frequencies we derive a deterministic system of transport equations that describe the evolution of mode powers this result is applied to the computations of pulse spreading in a random waveguide it is also applied to the analysis of time reversal in a random waveguide we show that randomness enhances spatial refocusing and that diffraction limited focal spots can be obtained even with small size time reversal mirrors the refocused field is statistically stable for broadband pulses in general we show here that it is also stable for narrowband pulses provided that the time reversal mirror is large enough

on log concave and log convex infinitely divisible sequences and densities we consider nonnegative infinitely divisible random variables whose levy measures are either absolutely continuous or supported by the integers necessary conditions are found ensuring that such distributions are log concave or log convex

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

entropy jumps in the presence of a spectral gap it is shown that if x is a random variable whose density satisfies a poincare inequality and y is an independent copy of x then the entropy of x y p 2 is greater than that of x by a fixed fraction of the entropy gap between x and the gaussian of the same variance the argument uses a new formula for the fisher information of a marginal which can be viewed as a local reverse form of the brunn minkowski inequality in its functional form due to prekopa and leindler

the first variation of the total mass of log concave functions and related inequalities abstract on the class of log concave functions on r n endowed with a suitable algebraic structure we study the first variation of the total mass functional which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions we prove some integral representation formulae for such a first variation which suggest to define in a natural way the notion of area measure for a log concave function in the same framework we obtain a functional counterpart of minkowski s first inequality for convex bodies as corollaries we derive a functional form of the isoperimetric inequality and a family of logarithmic type sobolev inequalities with respect to log concave probability measures finally we propose a suitable functional version of the classical minkowski s problem for convex bodies and prove some partial results towards its solution

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

extremal properties of half spaces for log concave distributions n w x n where m s m m is the product measure in r d s y1 1 is the n n dimensional cube in r n and the infimum is then over all borel measurable n n sets a r of measure m a g p 0 p 1 h 0 we are searching for necessary and sufficent conditions under which the n 4 infimum in 1 1 is attained at the half spaces a s x g r x f c of 1 measure p for all p and h the half spaces of this form will be called standard in these notes we give such conditions in the case where m is log concave there exists the following probabilistic consequence of the above property if z z are independent random variables on some probability space 1 n v i p with common distribution m consider a family of random variables s t indexed by a set t and formed by linear combinations n s t s a t z i i is1 where the coefficients a are arbitrary functions on t such that i n s s sup a t f 1 1 i t is1 set m s sup s t then the extremal property of the standard half spaces in t

entropy bounds on abelian groups and the ruzsa divergence over the past few years a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by ruzsa tao and others motivated by analogous results in additive combinatorics this paper extends these earlier results to the case of random variables taking values in mathbb r n or more generally in arbitrary locally compact and polish abelian groups we isolate and study a key quantity the ruzsa divergence between two probability distributions and we show that its properties can be used to extend the earlier inequalities to the present general setting the new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other although the setting is quite general the results are already of interest and new for random vectors in mathbb r n in that special case we discuss quantitative bounds for the stability of the equality conditions in the entropy power inequality a reverse entropy power inequality for log concave random vectors an information theoretic analog of the rogers shephard inequality for convex bodies and consequences of some of our results to determinant inequalities for sums of positive definite matrices moreover by considering various multiplicative subgroups of the complex plane one obtains new inequalities for the differential entropies of products and ratios of nonzero complex valued random variables

long time existence for a slightly perturbed vortex sheet consider a flat two dimensional vortex sheet perturbed initially by a small analytic disturbance by a formal perturbation analysis moore derived an approximate differential equation for the evolution of the vortex sheet we present a simplified derivation of moore s approximate equation and analyze errors in the approximation the result is used to prove existence of smooth solutions for long time if the initial perturbation is of size and is analytic in a strip m existence of a smooth solution of birkhoff s equation is shown for time t k2p if is sufficiently small with 1 as 0 for the particular case of sinusoidal data of wave length and amplitude e moore s analysis and independent numerical results show singularity development at time tc log o log log our results prove existence for t log if is sufficiently small with k 1 as 0 thus our existence results are nearly optimal

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

stable methods for vortex sheet motion in the presence of surface tension boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids several studies have revealed that standard numerical approximations tend to lead to unstable methods and various remedies have been introduced and tested in this paper we conduct a stability analysis of the linearized equations with a particular objective in mind the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes we discover two reasons for the typical failure in numerical discretizations one is the inability of the mesh to represent the vorticity created by surface tension effects on the finest scale and the other is the inaccuracies in the evaluation of the boundary integral for the velocity with the insight gained from our linear analysis we propose a new method that is spectrally accurate and linearly stable further the exact dispersion relation is obtained for all the available discrete modes numerical tests suggest that the method is also stable in the nonlinear regime however our method runs into difficulties generic to methods based on lagrangian motion the markers accumulate near a stagnation point on the interface forcing us to use an ever decreasing time step in our explicit method we introduce a redistribution of markers to overcome this difficulty when we redistribute according to equal arclength we find excellent agreement with a method based on preserving equal spacing in arclength

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

bounds on variance for unimodal distributions we show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power combining this bound with the well known entropy power lower bound on variance we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power as the differential entropy decreases the variance is sandwiched between two exponentially decreasing functions in the differential entropy this establishes that for the subclasses of unimodal distributions the differential entropy can be used as a surrogate for concentration of the distribution

concentration of the information in data with log concave distributions on which the distribution of xis supported in this case eh x is essentially the number of bits neededto represent x by a coding scheme that minimizes average code length sha48 in the continuous case with reference measure dx one may stillcall eh x the information content even though the coding interpretation nolonger holds in statistics one may think of the information content as thelog likelihood function the average value of the information content of xis known more com monly as the entropy indeed the entropy of xis de ned byh x zf x logf x dx elogf x

frequency content of randomly scattered signals the statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it the asymptotic analy

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

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

slender jets and thin sheets with surface tension simplified equations governing the potential flow and the shape of slender jets and thin sheets of liquid are derived taking into account surface tension families of similarity solutions of these equations are introduced for jets and for symmetrical sheets they satisfy ordinary differential equations the properties of these similarity solutions are examined analytically and numerically they can be used to describe the motion of a liquid sheet on a solid the thickening and flow following the breaking of jets the merging of two jets the formation or closing of holes or slits in sheets etc some applications to such problems are given

axisymmetric vortex sheet motion accurate evaluation of the principal value integral this paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion previous quadrature rules for this integral lose accuracy near the axis of symmetry an approximation by de bernadinis and moore dbm that converges pointwise at the rate of o h3 has maximal errors near the axis that are o h as a result the discretization error is not smooth it contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion this paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly o h3 the results are based on an analytic approximation of the integrand whose integral can be precomputed the modification is implemented at negligible additional cost per timestep as an example it is applied to compute the evolution of an initially spherical vortex sheet

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

analytic inequalities isoperimetric inequalities and logarithmic sobolev inequalities abstract there is a simple equivalence between isoperimetric inequalities in riemannian manifolds and certain analytic inequalities on the same manifold more extensive than the familiar equivalence of the classical isoperimetric inequality in euclidean space and the associated sobolev inequality by an isoperimetric inequality in this connection we mean any inequality involving the riemannian volume and riemannian surface measure of a subset and its boundary respectively we exploit the equivalence to give log sobolev inequalities for riemannian manifolds some applications to schrodinger equations are also given

distributed simulation of continuous random variables we establish the first known upper bound on the exact and wyner s common information of n continuous random variables in terms of the dual total correlation between them which is a generalization of mutual information in particular we show that when the pdf of the random variables is log concave there is a constant gap of n 2 log e 9n log n between this upper bound and the dual total correlation lower bound that does not depend on the distribution the upper bound is obtained using a computationally efficient dyadic decomposition scheme for constructing a discrete common randomness variable w from which the n random variables can be simulated in a distributed manner we then bound the entropy of w using a new measure which we refer to as the erosion entropy

droplet and bubble pinch off computations using level sets a two fluid potential flow model is employed to analyze the pinching characteristics of an inviscid fluid immersed in a second inviscid fluid of different density the system behavior is controlled by the relative density of the two fluids d r e r i where d 0 corresponds to droplets in air and d 100 to bubbles in water the numerical method employed combines the level set method for advancing the free surface position and boundary condition together with a 3d axisymmetric boundary integral formulation to obtain fluid velocities this approach provides a numerical methodology to analyze the pinch off behavior up to and beyond the initial break up of the inner fluid the combined algorithm is validated using the analytical solution for an oscillating sphere a series of numerical experiments up to and beyond the initial break up of the inner fluid have been carried out for the two extremes d 0 and d 100 the calculated scaling exponents match the theoretical values and the computed front profiles are in good agreement with recent experimental findings

moment entropy inequalities for a random vector the pth moment matrix is defined for a real random vector generalizing the classical covariance matrix sharp inequalities relating the pth moment and renyi entropy are established generalizing the classical inequality relating the second moment and the shannon entropy the extremal distributions for these inequalities are completely characterized

application of adaptive quadrature to axi symmetric vortex sheet motion studies of the formation of fine structures on free surfaces in liquids such as curvature singularities or interface pinching demand that the motion of the interface must be computed very accurately boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one on the other hand the boundary integrals are singular and their accurate evaluation can prove quite challenging in two dimensional motion the interface is just a curve when this curve is closed or periodic the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy unfortunately the nature of the singularity in the integrand for three dimensional motion is much more difficult to treat in this paper we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi symmetric flow the technique is based on a vector potential formulation which offers some computational advantages over other methods based on the biot savart integral direct numerical computations show that our technique is much more accurate and efficient than existing techniques we apply our technique to study the evolution of an initially spherical vortex sheet we present evidence of the formation of a 3 2 power singularity in the curvature of the vortex sheet

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

study of cylindrical jet breakup using one dimensional approximations of the euler equations we study the phenomenon of pinching of inviscid incompressible jets when surface tension is important this is done by analyzing one dimensional models derived by use of slender jet theory equations are derived for both single and two fluid arrangements higher order terms are retained for regularization purposes in the absence of a surrounding fluid these equations are the inviscid cosserat system it is shown that the leading order system terminates in infinite slope singularities before pinching happens this is achieved by studying a 2 x 2 system of conservation laws in the complex plane and by numerical solution of the evolution equations inclusion of higher order terms changes the order of the evolution equations and enables the one dimensional models to produce the phenomenon of pinching and drop formation in many cases the evolution equations are solved numerically and the numerical experiments suggest that the interface develops a cusp locally as the jet pinches local similarity solutions

multi valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems multi valued solutions are constructed for 2 2 first order systems using a generalization of the hodograph transformation the solution is found as a complex analytic function on a complex riemann surface for which the branch points move as part of the solution the branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds we perform an analysis of stability of these singularities with respect to perturbations of the initial data the generic singularity types are folds cusps and nondegenerate umbilic points with non zero 3 jet an isolated singularity is generically a square root branch point corresponding to a fold two types of collisions between singularities are generic at a tangential collision between two singularities moving at the same characteristic speed a cube root branch point is formed corresponding to a cusp a non tangential collision between two square root branch points moving at different characteristic speeds remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point these results are also valid for a diagonalizable n th order system for which there are exactly two speeds 1993 john wiley sons inc

the entropy per coordinate of a random vector is highly constrained under convexity conditions the entropy per coordinate in a log concave random vector of any dimension with given density at the mode is shown to have a range of just 1 uniform distributions on convex bodies are at the lower end of this range the distribution with i i d exponentially distributed coordinates is at the upper end and the normal is exactly in the middle thus in terms of the amount of randomness as measured by entropy per coordinate any log concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1 2 as applications we obtain an information theoretic formulation of the famous hyperplane conjecture in convex geometry entropy bounds for certain infinitely divisible distributions and quantitative estimates for the behavior of the density at the mode on convolution more generally one may consider so called convex or hyperbolic probability measures on euclidean spaces we give new constraints on entropy per coordinate for this class of measures which generalize our results under the log concavity assumption expose the extremal role of multivariate pareto type distributions and give some applications

a generalized localization theorem and geometric inequalities for convex bodies in this article we generalize a localization theorem of lovasz and simonovits random walks in a convex body and an improved volume algorithm random struct algorithms 4 4 1993 359 412 which is an important tool to prove dimension free functional inequalities for logconcave measures in a previous paper fradelizi and guedon the extreme points of subsets of s concave probabilities and a geometric localization theorem discrete comput geom 31 2004 327 335 we proved that the localization may be deduced from a suitable application of krein milman s theorem to a subset of log concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull here we generalize this result to more constraints give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem finally using this new localization theorem we solve an open question on the comparison of the volume of sections of non symmetric convex bodies in r n by hyperplanes a surprising feature of the result is

dimensional variance inequalities of brascamp lieb type and a local approach to dimensional prekopaʼs theorem abstract we give a new approach inspired by hormander s l 2 method to weighted variance inequalities which extend results obtained by bobkov and ledoux it provides in particular a local proof of the dimensional functional forms of the brunn minkowski inequalities we also present several applications of these variance inequalities including reverse holder inequalities for convex functions weighted brascamp lieb inequalities and sharp weighted poincare inequalities for generalized cauchy measures

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

on the maximum entropy of the sum of two dependent random variables investigates the maximization of the differential entropy h x y of arbitrary dependent random variables x and y under the constraints of fixed equal marginal densities for x and y we show that max h x y h 2x under the constraints that x and y have the same fixed marginal density f if and only if f is log concave the maximum is achieved when x y if f is not log concave the maximum is strictly greater than h 2x as an example identically distributed gaussian random variables have log concave densities and satisfy max h x y h 2x with x y more general inequalities in this direction should lead to capacity bounds for additive noise channels with feedback

a diffusion analysis approach to te mode propagation in randomly perturbed optical waveguides the aim of this work is to model the evolution of the modal distribution of the electromagnetic field as it propagates along a randomly deformed multimode optical waveguide when the number of guided modes becomes large we can regard the discrete set of modes as a quasi continuum in some cases nearest neighbor coupling predominates over other power transfer mechanisms and the coupling process can be ideally described in terms of a diffusion equation the theory is applied to the propagation of guided transverse electric te field waves in a slab waveguide with parabolic refractive index profile numerical simulations are in good agreement with theoretical results and the error is shown to behave as the inverse of the number of guided modes the technique allows the prediction of the long distance modal distribution for a very large number of guided modes within fixed computational resources

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

fractional smoothness of images of logarithmically concave measures under polynomials abstract we show that a measure on the real line that is the image of a log concave measure under a polynomial of degree d possesses a density from the nikolskii besov class of fractional order 1 d this result is used to prove an estimate for the total variation distance between such measures in terms of the fortet mourier distance

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

convex bodies and norms associated to convex measures isotropy like properties are considered for finite measures with heavy tails as a basic tool we extend k ball s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions satisfying convexity conditions of the brunn minkowski type

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

hyperplane sections of convex bodies we prove sharp inequalities for the volumes of hyperplane sections bisecting a convex body in rn this leads to a relative isoperimetric inequality for arbitrary hyperplane sections of a convex body

isoperimetric inequality for radial probability measures on euclidean spaces abstract we generalize the poincare limit which asserts that the n dimensional gaussian measure is approximated by the projections of the uniform probability measure on the euclidean sphere of appropriate radius to the first n coordinates as the dimension diverges to infinity the generalization is done by replacing the projections with certain maps using this generalization we derive a gaussian isoperimetric inequality for an absolutely continuous probability measure on euclidean spaces with respect to the lebesgue measure whose density is a radial function

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

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

a continuous model for mode mixing in graded index multimode fibres with random imperfections this article investigates the evolution of the distribution of propagating modes in a graded index multimode fibre with random imperfections these perturbations may be microscopic random bends and ellipticity or an index of refraction fluctuations introduced during manufacture for large diameter fibres with respect to a typical electromagnetic wavelength the number of guided modes becomes large and so we can regard the discrete set as a continuum in which the mode number is treated as a continuous variable the theory is applied to the propagation of guided optical modes in round fibres with a parabolic refractive index profile this continuous model offers the great advantage of elucidating the relevant physical parameters that participate in the mode mixing dynamics and leads to a numerically tractable problem numerical results are presented for the specific problem of random micro bends demonstrating that the theoretically predicted behaviour is achieved

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

dynamics of slender viscoelastic free jets previous studies of slender viscoelastic and newtonian free surface jets have focused mainly on steady state predictions guided primarily by industrial textile applications dynamical analyses and simulations of fiber flows even the linearized stability analysis of nontrivial steady states have lagged behind considerable experimental and textile processing observations and advances the foundational work of chang and lodge rheol acta 10 1971 pp 448 449 and petrie rheol acta 14 1975 pp 955 957 elongational flows pitman london 1979 has been extended by the axisymmetric modeling and simulation of beris and liu j non newtonian fluid mech 26 1988 pp 363 394 comput methods appl mech engrg 76 1989 pp 179 204 for maxwell fluids and markovich and renardy j non newtonian fluid mech 17 1985 pp 13 22 for johnson segalman fluids the model in markovich and renardy is a single nonlinear parabolic equation in which elastic retardation provides dominant smooth

numerical simulations of inviscid capillary pinchoff inviscid capillary pinchoff is studied numerically for an axisymmetric model problem in which a bubble pinches at two points on the symmetry plane breaking into two symmetric end bubbles and a satellite bubble in between results are presented for a range of density jumps across the bubble the numerical method uses a formulation in terms of arclength and tangent angle and incorporates a new procedure to redistribute the computational points dynamically in order to maintain resolution in regions of high curvature the results are compared with alternative computations by lepinnen and lister phys fluids 15 2003 568 where available new results include details about the cone crater structure near pinchoff and the dependence of satellite bubble volume and pinchoff time on the density ratio

high order quadratures for the evaluation of interfacial velocities in axi symmetric stokes flows we propose new high order accurate methods to compute the evolution of axi symmetric interfacial stokes flow the velocity at a point on the interface is given by an integral over the surface quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands both locally about the point of evaluation and about the poles where the interface crosses the axis of symmetry the local expansions yield methods that converge to the chosen order pointwise for fixed evaluation point the pole expansions yield corrections that remove maximal errors of low order introduced by singular behaviour of the integrands as the evaluation point approaches the poles an interesting example of roundoff error amplification due to cancellation is also addressed the result is a uniformly accurate fifth order method second order pointwise fifth order and uniform fifth order methods are applied to compute three sample flows each of which presents a different computational difficulty an initially bar belled drop that pinches in finite time a drop in a strain flow that approaches a steady state and a continuously extending drop in each case the fifth order methods significantly improve the ability to resolve the flow the examples furthermore give insight into the effect of the corrections needed for uniformity we determine conditions under which the pointwise method is sufficient to obtain resolved results and others under which the corrections significantly improve the results

affine moments of a random vector an affine invariant pth moment measure is defined for a random vector and used to prove sharp moment entropy inequalities that are more general and stronger than standard moment entropy inequalities

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

dielectric profile variations in high index contrast waveguides coupled mode theory and perturbation expansions perturbation theory formulation of maxwell s equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electromagnetic systems it is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fails when applied to the problem of shifted material boundaries in this paper we developed coupled mode and perturbation theory formulations for treating generic perturbations of a waveguide cross section based on hamiltonian formulation of maxwell equations in curvilinear coordinates we show that our formulation is accurate beyond the first order and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index contrast discontinuous dielectric profiles among others our formulation allows for an efficient numerical evaluation of such quantities as deterministic pmd and change in the gv and gvd of a mode due to generic profile distortion in a waveguide of arbitrary cross section to our knowledge this is the first time perturbation and coupled mode theories are developed to deal with arbitrary profile variations in high index contrast waveguides