the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

universality of non extensive tsallis statistics and time series analysis theory and applications abstract we investigate tsallis q statistics for various complex physical systems in particular the tsallis q triplet is estimated for space plasmas atmospheric dynamics seismicity and brain and cardiac activity remarkable agreement between theoretical tsallis predictions and experimental estimations was observed in all cases we also present theoretical extensions of tsallis q statistics for intermittent turbulence theory non equilibrium statistics and fractal extension of dynamics

approximate independence of distributions on spheres and their stability properties let zeta be chosen at random on the surface of the p sphere in mathbb r n 0 p n x in mathbb r n sum n i 1 x i p n if p 2 then the first k components zeta 1 ldots zeta k are for k fixed in the limit as n rightarrow infty independent standard normal considering the general case p 0 the same phenomenon appears with a distribution f p in an exponential class mathscr e f p can be characterized by the distribution of quotients of sums by conditional distributions and by a maximum entropy condition these characterizations have some interesting stability properties some discrete versions of this problem and some applications to de finetti type theorems are discussed

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

l p nested symmetric distributions in this paper we introduce a new family of probability densities called lp nested symmetric distributions the common property shared by all members of the new class is the same functional form x f x where f is a nested cascade of lp norms x p xi p 1 p lp nested symmetric distributions thereby are a special case of spherical distributions for which f is only required to be positively homogeneous of degree one while both spherical and lp nested symmetric distributions contain many widely used families of probability models such as the gaussian spherically and elliptically symmetric distributions lp spherically symmetric distributions and certain types of independent component analysis ica and independent subspace analysis isa models spherical distributions are usually computationally intractable here we demonstrate that lp nested symmetric distributions are still computationally feasible by deriving an analytic expression for its normalization constant gradients for maximum likelihood estimation analytic expressions for certain types of marginals as well as an exact and efficient sampling algorithm we discuss the tight links of lp nested symmetric distributions to well known machine learning methods such as ica isa and mixed norm regularizers and introduce the nested radial factorization algorithm nrf which is a form of non linear ica that transforms any linearly mixed non factorial lp nested symmetric source into statistically independent signals as a corollary we also introduce the uniform distribution on the lp nested unit sphere

distributions for the eigenvalues of large random matrices generated from four manifolds in this paper we will first study the spectrum of certain large euclidean random matrices the entries of these matrices are functions of n random points lp norm uniformly sampled in n dimensional lp ball or lp sphere under the setting n n 0 as n and n tend to it is shown that the empirical distributions of eigenvalues of normalized euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law we also investigate certain sample covariance type matrices whose entries are determined by n random positions from an n dimensional lp ellipsoid or its surface when n and n with n n 0 we prove that their empirical spectral distribution converge to the same fixed distribution

cone volume measures of polytopes abstract the cone volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition as a consequence a conjectured a dozen years ago fundamental sharp affine isoperimetric inequality for the u functional is completely established along with its equality conditions

a brunn minkowski theory for coconvex sets of finite volume abstract let c be a closed convex cone in r n pointed and with interior points we consider sets of the form a c k where k c is a closed convex set if a has finite volume lebesgue measure then a is called a c coconvex set and k is called c close the family of c coconvex sets is closed under the addition defined by c a 1 a 2 c a 1 c a 2 we develop first steps of a brunn minkowski theory for c coconvex sets which relates this addition to the notion of volume in particular we establish the equality condition for a brunn minkowski type inequality with reversed inequality sign and introduce mixed volumes and their integral representations for c close sets surface area measures and cone volume measures can be defined in analogy to similar notions for convex bodies they are borel measures on the intersection of the unit sphere with the interior of the polar cone of c we prove a minkowski type uniqueness theorem for c close sets with equal surface area measures concerning minkowski type existence problems we give conditions for a borel measure to be either the surface area measure or the cone volume measure of a c close set these conditions are sufficient in the first case and necessary and sufficient in the second case

portevin le chatelier effect and tsallis nonextensive statistics abstract in this paper we estimate tsallis entropic index q q s t a t stat denotes stationary states for various stress serration time series of cu 15 al alloys corresponding to different deformation temperatures and types a b c of portevin le chatelier plc bands tsallis statistical analysis is also conducted on experimental data derived from correlation diagrams focusing on plc type c bands spatial organization at the surface of various specimens the results concerning the stress serrations analysis reveal the non gaussian tsallis q gaussian nonextensive sub additive character of the underlying dynamics of the plc type a bands q 1 indicating dynamics at the edge of chaos with global long range correlations and power law scaling similarly for plc type b bands a tsallis q gaussian non extensive super additive statistical profile was found with q 1 while for the plc type c bands a near boltzmann gibbs bg statistical character was verified indicating gaussian dynamics with short weak range correlations finally tsallis q index for the spatial data of plc type c bands was found above unity q 1 indicating that type c bands are not randomly nucleated in the specimen e g yielding a gaussian distribution but there exist spatial coupling and long range deformation correlations in the plc type c bands spatial localization

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

spectral properties of distance matrices distance matrices are matrices whose elements are the relative distances between points located on a certain manifold in all cases considered here all their eigenvalues except one are non positive when the points are uncorrelated and randomly distributed we investigate the average density of their eigenvalues and the structure of their eigenfunctions the spectrum exhibits delocalized and strongly localized states that possess different power law average behaviour the exponents depend only on the dimensionality of the manifold

statistical analysis of the extreme values of stress time series from the portevin le châtelier effect in an effort to understand the deterministic vs stochastic character of the portevin char21 le ch atelier plc phenomenon we investigate the structure of the underlying mechanism that generates the stick slip patterns of stress over time the stress time series is reduced to a series of successive pairs of minimum and maximum values representing the stick slip patterns and a statistical analysis by means of hypothesis testing is applied to it the null hypothesis of least deterministic structure is that the time series of extreme values is a bounded random walk of alternating direction brwad that is besides the constraint of succession of minima to maxima bounded at a predefined range there are no other correlations in the data to implement the test we use surrogate data generated by a model consistent with a brwad type process which also uses the statistics of the original data to best mimic them the proposed hypothesis testing is found to perform properly on simulated data from stochastic and deterministic systems for the plc time series the null hypothesis is rejected at a high level of confidence giving evidence for some deterministic structure in the succession of the extreme stress values this result allows for further statistical analysis including also the time aspect of the stick slip patterns

multifractality and randomness in the unstable plastic flow near the lower strain rate boundary of instability the unstable plastic flow of an almg alloy associated with the portevin le chatelier effect was studied near the lower strain rate boundary of instability using multifractal analysis self similarity of deformation curves indicating long range time correlations of stress serrations was detected within the strain rate range where serrations are commonly ascribed to the occurrence of uncorrelated deformation bands the deformation curves display a wide range of shapes that are characterized by different groupings of serrations multifractal analysis provides a method to quantify the observed complexity and compare it to known portevin le chatelier effect regimes the measurement noise effect on the multifractal spectra determined from experimental data was mimicked by superposing multifractal cantor sets with random noise such tests using standard multifractal data sets justify the separation of self similar and random components of the serrated deformation curves furthermore these results shed light on the general problem of the effect of experimental noise on the apparent multifractal properties of physical fractals

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

distributions of eigenvalues of large euclidean matrices generated from lp balls and spheres abstract let x 1 x n be points randomly chosen from a set g r n and f x be a function the euclidean random matrix is given by m n f x i x j 2 n n where is the euclidean distance when n is fixed and n we prove that m n the empirical distribution of the eigenvalues of m n converges to 0 for a big class of functions of f x assuming both n and n go to infinity proportionally we obtain the explicit limit of m n when g is the l p unit ball or sphere with p 1 as corollaries we obtain the limit of a n with a n d x i x j n n and d being the geodesic distance on the ordinary unit sphere in r n we also obtain the limit of a n for the euclidean distance matrix a n x i x j n n the limits are a b v where a and b are constants and v follows the marcenko pastur law the same are also obtained for other examples appeared in physics literature including exp x i x j n n and exp d x i x j n n our results partially confirm a conjecture by do and vu 14

multifractal analysis of evolving noise associated with unstable plastic flow plastic flow instability attracts increasing interest as a self organization phenomenon showing various dynamical regimes including deterministic chaos and self organized criticality the analysis of the associated nonrandom noise drastic jumps of the mechanical stress however confronts the variation of the noise average parameters due to the evolution of the dislocation microstructure the present paper examines some limitations of the multifractal approach to the study of the evolving noise the applicability of the multifractal analysis to practical situations is proven using the example of discontinuous deformation curves observed under conditions of the portevin le ch telier effect in an a1mg alloy as well as model signals generated by stretching multifractal cantor sets it is found that the smooth trends in the stress serration parameters may narrow the range of the scale invariant behavior associated with the multifractal structure but do not essentially mask it

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

superconcentration and randomized dvoretzky s theorem for spaces with 1 unconditional bases abstract let n be a sufficiently large natural number and let b be an origin symmetric convex body in r n in the l position and such that the space r n b admits a 1 unconditional basis then for any e 0 1 2 and for random c e log n log 1 e dimensional subspace e distributed according to the rotation invariant haar measure the section b e is 1 e euclidean with probability close to one this shows that the worst case dependence on e in the randomized dvoretzky theorem in the l position is significantly better than in john s position it is a previously unexplored feature which has strong connections with the concept of superconcentration introduced by s chatterjee in fact our main result follows from the next theorem let b be as before and assume additionally that b has a smooth boundary and e n b n c e n grad b 2 for a small universal constant c 0 where grad b is the gradient of b and n is the standard gaussian measure in r n then for any p 1 c log n the p th power of the norm b p is c log n superconcentrated in the gauss space

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

the spectrum of kernel random matrices we place ourselves in the setting of high dimensional statistical inference where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n we consider the spectrum of certain kernel random matrices in particular n x n matrices whose i j th entry is f x i x j p or f x i x j 2 p where p is the dimension of the data and x i are independent data vectors here f is assumed to be a locally smooth function the study is motivated by questions arising in statistics and computer science where these matrices are used to perform among other things nonlinear versions of principal component analysis surprisingly we show that in high dimensions and for the models we analyze the problem becomes essentially linear which is at odds with heuristics sometimes used to justify the usage of these methods the analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high dimensional data encountered in practice

eigenvalue distributions of large euclidean random matrices for waves in random media we study probability distributions of eigenvalues of hermitian and non hermitian euclidean random matrices that are typically encountered in the problems of wave propagation in random media

tsallis non extensive statistical mechanics in the ionospheric detrended total electron content during quiet and storm periods abstract in this paper the tsallis non extensive q statistics in ionospheric dynamics was investigated using the total electron content tec obtained from two global positioning system gps receiver stations this investigation was carried out considering the geomagnetically quiet and storm periods the micro density variation of the ionospheric total electron content was extracted from the tec data by method of detrending the detrended total electron content which represent the variation in the internal dynamics of the system was further analyzed using for non extensive statistical mechanics using the q gaussian methods our results reveals that for all the analyzed data sets the tsallis gaussian probability distribution q gaussian with value q 1 were obtained it was observed that there is no distinct difference in pattern between the values of q q u i e t and q s t o r m however the values of q varies with geophysical conditions and possibly with local dynamics for the two stations also observed are the asymmetric pattern of the q gaussian and a highly significant level of correlation for the q index values obtained for the storm periods compared to the quiet periods between the two gps receiver stations where the tec was measured the factors responsible for this variation can be mostly attributed to the varying mechanisms resulting in the self reorganization of the system dynamics during the storm periods the result shows the existence of long range correlation for both quiet and storm periods for the two stations

cone volume measure and stability abstract we prove stability results for two central inequalities involving the cone volume measure of a centered convex body the subspace concentration conditions and the u functional volume inequality

an investigation of the hidden structure of states in a mean field spin glass model we study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses the p spin spherical model this structure cannot be revealed by the standard methods mainly due to the presence of an exponentially high number of states each one having a vanishing weight in the thermodynamic limit performing a purely entropic computation based on the tap equations for this model we define a constrained complexity which gives the overlap distribution of the states we find that this distribution is continuous non random and highly dependent on the energy range of the considered states furthermore we show which is the geometrical shape of the threshold landscape giving some insight into the role played by threshold states in the dynamical behaviour of the system

the surface measure and cone measure on the sphere of ell_p n we prove a concentration inequality for the l n p norm on the l n p sphere for p q 0 this inequality which generalizes results of schechtman and zinn 2000 is used to study the distance between the cone measure and surface measure on the sphere of l n p in particular we obtain a significant strengthening of the inequality derived by naor and romik 2003 and calculate the precise dependence of the constants that appeared there on p

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1