INEQUALITIES FOR CHORD POWER INTEGRALS

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.

A convex body C in d-dimensional Euclidean space E ~ is a compact convex subset of lE a with non-empty interior. We say that C is of differentiability class ~k if its boundary considered as a manifold is of class ~k. The symmetric difference metric 6 s, also called Nikodym metric, on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference. Given a convex body C, let ~ = ~ ( C ) (n = d + l, d + 2,...) denote the family of convex polytopes P having at most n vertices and being inscribed into C, that is, all vertices are on the boundary bdC of C. It is well known that for each n there is a polytope P, ~ ~ such that ~s(C, P.) = ~s(C, ~,~)( = inf {~s(C, P): P ~ ~.~}).

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

The goal of this paper is to study some properties about the isoperimetric profile of convex bodies in the plane, find some conditions that make the conjecture true and give some types of curves that the convex body isoperimetric conjecture holds on them. We have verified any ellipses satisfies the conjecture. Moreover, we discover a kind of important properties from ellipse, called completely 4-symmetric, and we also check that the convex body isoperimetric conjecture remains true on any smooth completely 4-symmetric curves. We verify that the isoperimetric profile of a unit ball B has a local maximum for all available area under the area-preserving variation on B. Finally, we try to relate curve-shortening flow and convex body isoperimetric conjecture, that is, we discover some properties about smooth convex curves that breaks the conjecture by using relate curve-shortening flow.

The systematic study of various mean-values of nonnegative valued functions is due in large part to the work of J. L. W. V. Jensen [12], A. Kolmogoroff [11], and B. Jessen [13]. In this paper we present a similar theory of certain means of functions whose values are taken from a collection of convex bodies in a finite dimensional Euclidean space. The range of nonnegative valued functions is totally ordered while the collection of convex bodies to be considered is partially ordered by set inclusion; corresponding to inequalities, we shall have inclusions, here to be called inequalities, between mean-values. Thus the discussion will furnish an example of a partially ordered system whose algebraic and topological structure is sufficient to admit a fairly elaborate theory of inequalities analogous to that of the real numbers. The first section is a resume of those parts of the work of Jensen, Kolmogoroff, and Jessen for which analogues will be developed. ?2 treats of pertinent material about star and convex bodies. Certain families of star bodies and of convex bodies are defined in the third section; these latter play the role of nonnegative valued functions. Two systems of power means of such families are described. Each system is, in its way, anialogous to the power means of nonnegative valued functions. In ?4 we discuss some special cases of power means including elementary means and the Riemaiin-Minkowski integrals of A. Dinghas [2], [3]. Also some rotation invariants of a convex body are described as power means of special families determined by the convex body. The fifth section begins with crucial properties of the means and contains an extremal characterization for the two systems of means defined in the third section. Analogues of Jensen's and Jessen's inequalities make up ?6. We also mentioni the limiting cases of power means as the power index becomes infinite, positively and negatively. As an applicationl of Jessen's inequality, a Brunn-Minkowski type theorem is proved. In the final section we discuss some further systems of meanls of convex bodies. 1. The definition of the elementary power means extends in a natural way to include power means of certain functions f which have finite, nonnegative values and whose domain T is a compact topological group equipped with its Haar measure. Iff is bounded and measurable over its domain, its pth- power mean is defined by

Let \mathcal{G}_{n} denote the space of n -generated marked groups. We prove that, for every n\ge 2 , there exist 2^{\aleph_0} non-atomic, \operatorname{Out}(F_{n}) -invariant, mixing probability measures on \mathcal{G}_{n} . On the other hand, there are non-empty closed subsets of \mathcal{G}_{n} that admit no \operatorname{Out}(F_{n}) -invariant probability measure. Acylindrical hyperbolicity of the group \operatorname{Aut}(F_{n}) plays a crucial role in the proof of both results. We also discuss model-theoretic implications of the existence of \operatorname{Out}(F_{n}) -invariant, ergodic probability measures on \mathcal{G}_{n} .

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.

As usual, a convex body of the Euclidean space \(\mathbb{E}^d\) is a compact convex set with non-empty interior. Let C ⊂ \(\mathbb{E}^d\) be a convex body, and let H ⊂ \(\mathbb{E}^d\) be a hyperplane. Then the distance w(C;H) between the two supporting hyperplanes of C parallel to H is called the width of C parallel to H. Moreover, the smallest width of C parallel to hyperplanes of \(\mathbb{E}^d\) is called the minimal width of C and is denoted by w(C).

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.

Duality has been one of the most powerful methods in Mathematics, with a wide range of applications. In the Calculus of Variations we have probably the most important use of this method in modern mathematics. Hamilton, when applying the duality of poles and polars to the Calculus of Variations, the much in use by geometers, was responsible for a decisive step towards the introduction of a separate dual space. Although poles and polars were of substantive importance in Apollonius work on the conies (III century B.C.) and reappeared in the “Brouillon project” of Gerard Desar-gues (1591–1661). Although analysed by Fermat and by Pascal, this “project” was forgotten for about two centuries and was revived by Chasles, making its way to Hamilton. Together with Legendre’s transformation, Hamiltonian theory is related to Minkowski’s duality of convex figures. This is not surprising, since Minkowski’s dual convex figures is the key for Hahn-Banach Theorem. In fact, the existence of supporting hyperplane was important in Minkowski’s work on convex bodies. But even before, the proof of Euler’s condition relies on a lemma which says that if ƒ is continuous in an interval (0,1) and ∫ 0 1 ƒ g dt = 0 for every sufficiently smooth g, then ƒ is identically zero.

The problem of quantization in an Euclidean space with unitary constraints can be formulated as an unconstrained problem on a Grassmann manifold. Such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver. Due to the constraints, the distortion rate analysis developed for Euclidean spaces cannot be applied directly. This paper extends Gersho's asymptotic (large rate, small distortion) distortion bounds to the case when the source is distributed on the complex Grassmann manifold. The special structure of the Grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in Euclidean spaces.

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.

In [2], we have shown that the extended diameter family of a convex body C in n-space has the property of outward simplicity if and only if there is no pair of parallel line segments in the boundary of C respectively in an opposite pair of parallel contact planes of C. In [3] Professor Sobczyk and I show that every outwardly simple family of lines in the plane is the extended diameter family of a constant breadth convex body and incidentally give a simple construction for all constant breadth planar convex bodies. The purpose of this note is to show that not every continuous outwardly simple line family in Euclidean 3-space is the extended diameter family of a convex body. A diameter of a convex body in En (n > 2) is a longest chord in its direction. In [2] it is proved that the diameters of a convex body cover the body.

In this paper, classification via sparse representation of monogenic signal on Grassmann manifolds is presented for target recognition in SAR image. To capture the broad spectral information with maximal spatial localization of SAR image, a recently proposed vector-valued analytic signal, namely monogenic signal is exploited. Different from the conventional methods, where a single feature descriptor is generated using the monogenic signal in an Euclidean space, the multiple components of monogenic signal at various scale spaces are viewed as points on a special type of Riemannian manifolds, Grassmann manifolds. The similarity between a pair of patterns (points) is measured by Grassmann distance metric. To exploit the nonlinear geometry structure further, we embed the sets of monogenic components into an implicit reproducing kernel Hilbert space (RKHS), where the kernel-based sparse signal modeling can be learnt to reach the inference. Specifically, the sets of monogenic components resulting from the training samples are concatenated first to build a redundant dictionary. Then, the counterpart of the query is efficiently approximated by superposition of atoms of the dictionary. Notably, the representation coefficients of superposition are very parsimonious. The inference is drawn by evaluating which class of training patterns could recover the query as accurately as possible. The novelty of this paper comes from 1) the development of Grassmann manifolds formed by the multiresolution monogenic signal; 2) the definition of similarity between the sets of monogenic components on Grassmann manifolds for target recognition; 3) the generalization of sparse signal modeling on Grassmann manifold; and 4) multiple comparative experiments for performance assessment.

Let IRd be the Euclidean space with the usual norm |ċ|2, Pnd be the set of all polynomials over IRd of degree n, and K ⊂ IRd be a convex body. An algorithm for calculation of the Bernstein-Szego factor: BS(K) := supx∈int(K) P∈ Pnd, n∈IN {|gradP(x)|2w(K) √1 - α2(K, x)/n√||P||C(K)2 - P2(x) is considered, where w(K) is the width of K and α(K, x) is the generalized Minkowsky functional. It is known that BS(K) ∈ [2, 2√2]. On the basis of computer experiments, we show that the existing in the literature hypothesis, that BS(K)= 2 for any convex body K ⊂ IRd, fails to hold.