bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

rate of convergence and edgeworth type expansion in the entropic central limit theorem an edgeworth type expansion is established for the entropy distance to the class of normal distributions of sums of i i d random variables or vectors satisfying minimal moment conditions

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 cubic siegel polynomials we study the one dimensional parameter space of cubic polynomials in the complex plane which have a fixed siegel disk of rotation number where is a given irrational number of brjuno type the main result of this work is that when is of bounded type the boundary of the siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial we also show that these boundaries vary continuously as one moves in the parameter space this is most nontrivial near the set of cubics with both critical points on the boundary of their siegel disk we prove that this locus is a jordan curve in the parameter space most of the techniques and results can be generalized to polynomials of higher degrees

shannon s monotonicity problem for free and classical entropy we give a short unified proof of the following theorem valid in the context of both classical probability theory and voiculescu s free probability theory let x j 1 x j n be independent resp freely independent n tuples of random variables let z n p n 1 2 x 1 p x n p be their central limit sums then the entropy resp free entropy of the n tuple z n 1 z n n is a monotone function of n the classical case for n 1 is a celebrated result of artstein ball barthe and naor and our proof is an adaptation and simplification of their argument

herman s condition and siegel disks of bi critical polynomials we extend a theorem of herman from the case of unicritical polynomials to the case of polynomials with two finite critical values this theorem states that siegel disks of such polynomials under a diophantine condition called herman s condition on the rotation number must have a critical point on their boundaries

on the rate of convergence in the entropic central limit theorem we study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition

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

entropy inequalities for stable densities and strengthened central limit theorems we consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function by showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a levy centered density of parameter lambda 1 this introduces a new information theoretic approach to the central limit theorem for stable laws in which the main argument is shown to be the relative fractional fisher information recently introduced in toscani ricerche mat 65 1 71 91 2016 in particular it is proven that with respect to the relative fractional fisher information the levy density satisfies an analogous of the logarithmic sobolev inequality which allows to pass from the monotonicity and decay to zero of the relative fractional fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate

non computable impressions of computable external rays of quadratic polynomials we discuss computability of impressions of prime ends of compact sets in particular we construct quadratic julia sets which possess explicitly described non computable impressions

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

estimation in gaussian noise properties of the minimum mean square error consider the minimum mean square error mmse of estimating an arbitrary random variable from its observation contaminated by gaussian noise the mmse can be regarded as a function of the signal to noise ratio snr as well as a functional of the input distribution of the random variable to be estimated it is shown that the mmse is concave in the input distribution at any given snr for a given input distribution the mmse is found to be infinitely differentiable at all positive snr and in fact a real analytic function in snr under mild conditions the key to these regularity results is that the posterior distribution conditioned on the observation through gaussian channels always decays at least as quickly as some gaussian density furthermore simple expressions for the first three derivatives of the mmse with respect to the snr are obtained it is also shown that as functions of the snr the curves for the mmse of a gaussian input and that of a non gaussian input cross at most once over all snrs these properties lead to simple proofs of the facts that gaussian inputs achieve both the secrecy capacity of scalar gaussian wiretap channels and the capacity of scalar gaussian broadcast channels as well as a simple proof of the entropy power inequality in the special case where one of the variables is gaussian

a local proof of the dimensional prekopa s theorem abstract the aim of this paper is to find an expression for second derivative of the function t defined by t v t x d x 1 n n where u r and v r n are open bounded subsets and u v r is a c 2 smooth function as a consequence this result gives us a direct proof of the dimensional prekopa s theorem based on a local approach

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

information inequalities for joint distributions with interpretations and applications upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies these inequalities generalize shannon s chain rule for entropy as well as inequalities of han fujishige and shearer a duality between the upper and lower bounds for joint entropy is developed all of these results are shown to be special cases of general new results for submodular functions thus the inequalities presented constitute a richly structured class of shannon type inequalities the new inequalities are applied to obtain new results in combinatorics such as bounds on the number of independent sets in an arbitrary graph and the number of zero error source channel codes as well as determinantal inequalities in matrix theory a general inequality for relative entropies is also developed finally revealing connections of the results to literature in economics computer science and physics are explored

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

asymptotic linear bounds for the castelnuovo mumford regularity we prove asymptotic linear bounds for the castelnuovo mumford regularity of certain filtrations of homogeneous ideals whose rees algebras need not be noetherian

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

weighted covering numbers of convex sets abstract in this paper we define the notions of weighted covering number and weighted separation number for convex sets and compare them to the classical covering and separation numbers this sheds new light on the equivalence of classical covering and separation we also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions

non linear behaviour of castelnuovo mumford regularity abstract it is shown that the castelnuovo mumford regularity of lex segment ideals of ordinary powers as well as of frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree

the b conjecture for the gaussian measure of dilates of symmetric convex sets and related problems we show that given a symmetric convex set k rd the function t etk is log concave on r where denotes the standard d dimensional gaussian measure we also comment on the extension of this property to unconditional log concave measures and sets and on the complex case

maximal correlation and the rate of fisher information convergence in the central limit theorem we consider the behaviour of the fisher information of scaled sums of independent and identically distributed random variables in the central limit theorem regime we show how this behaviour can be related to the second largest non trivial eigenvalue of the operator associated with the hirschfeld gebelein renyi maximal correlation we prove that assuming this eigenvalue satisfies a strict inequality an o 1 n rate of convergence and a strengthened form of monotonicity hold

moments of the scores upper bounds on absolute moments of the scores are derived for sums of independent random variables in terms of the moments of the scores as well as in terms of the total variation norm of densities of summands

fisher information inequalities and the central limit theorem we give conditions for an o 1 n rate of convergence of fisher information and relative entropy in the central limit theorem we use the theory of projections in l 2 spaces and poincare inequalities to provide a better understanding of the decrease in fisher information implied by results of barron and brown we show that if the standardized fisher information ever becomes finite then it converges to zero

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

on the entropy of compound distributions on nonnegative integers some entropy comparison results are presented concerning compound distributions on nonnegative integers the main result shows that under a log concavity assumption two compound distributions are ordered in terms of shannon entropy if both the ldquonumbers of claimsrdquo and the ldquoclaim sizesrdquo are ordered accordingly in the convex order several maximum minimum entropy theorems follow as a consequence most importantly two recent results of johnson 2008 on maximum entropy characterizations of compound poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

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

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

monotonic decrease of the non gaussianness of the sum of independent random variables a simple proof artstein ball barthe and naor have recently shown that the non gaussianness divergence with respect to a gaussian random variable with identical first and second moments of the sum of independent and identically distributed i i d random variables is monotonically nonincreasing we give a simplified proof using the relationship between non gaussianness and minimum mean square error mmse in gaussian channels as artstein we also deal with the more general setting of nonidentically distributed random variables

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

fisher information and the central limit theorem an edgeworth type expansion is established for the relative fisher information distance to the class of normal distributions of sums of i i d random variables satisfying moment conditions the validity of the central limit theorem is studied via properties of the fisher information along convolutions

entropy and the fourth moment phenomenon we develop a new method for bounding the relative entropy of a random vector in terms of its stein factors our approach is based on a novel representation for the score function of smoothly perturbed random variables as well as on the de bruijn s identity of information theory when applied to sequences of functionals of a general gaussian field our results can be combined with the carbery wright inequality in order to yield multidimensional entropic rates of convergence that coincide up to a logarithmic factor with those achievable in smooth distances such as the 1 wasserstein distance in particular our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components with explicit rates of convergence in total variation that are independent of the order of the associated wiener chaoses the results proved in the present paper are outside the scope of other existing techniques such as for instance the multidimensional stein s method for normal approximations

constructing locally connected non computable julia sets a locally connected quadratic siegel julia set has a simple explicit topological model such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution we constructively produce parameter values for siegel quadratics for which the julia sets are non computable yet locally connected

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