A concentration of mass property for isotropic convex bodies in high dimensions

It is shown that, for a certain subclass of istropic convex sets in ℝⁿ, the mass concentrates in a spherical shell, asymptotically for largen. This in turn shows that the inequality is close to an equality for the mentioned class of isotropic convex sets, asymptotically for largen. It also implies a ‘central limit property’ for this class. Partly supported by the DFG.     

Hessian measures of semi-convex functions and applications to support measures of convex bodies

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. Received: 15 July 1999     

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L ₁. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations, we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models (parallel case) in three dimensions.     

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALáT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.     

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

Convex integration with constraints and applications to phase transitions and partial differential equations

We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ ⁿ →ℝ ^m is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite. Received April 23, 1999 / final version received September 11, 1999     

Entropy and the combinatorial dimension

We solve Talagrand’s entropy problem: the L ₂-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley’s theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton’s Theorem and estimates on the uniform central limit theorem in the real valued case. Oblatum 10-XII-2001 & 4-IX-2002?Published online: 8 November 2002     

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

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.     

Cotorsion pairs associated with Auslander categories

Let N ≥ n + 1, and denote by K the convex hull of N independent standard gaussian random vectors in ℝⁿ. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes. Supported by the Clay Mathematics Institute and by NSF grant #DMS-0456590     

Congruences between derivatives of abelian <Emphasis Type="Italic">L</Emphasis>-functions at <Emphasis Type="Italic">s</Emphasis>=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field. Mathematics Subject Classification (1991)  Primary 11G40; Secondary 11R65; 19A31; 19B28     

Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model

We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang, and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91–100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev, Probability for sums of independent random variables, Theory Probab. Appl., 16:600–675, 1971].     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/∞ system revisited: finiteness,summability, long range dependence,and reverse engineering

We explore M/G/∞ systems ‘fed’ by Poissonian inflows with infinite arrival rates. Three processes – corresponding to the system's state, workload, and queue-size – are studied and analyzed. Closed form formulae characterizing the system's stationary structure and correlation structure are derived. And, the issues of queue finiteness, workload summability, and Long Range Dependence are investigated. We then turn to devise a ‘reverse engineering’ scheme for the design of the system's correlation structure. Namely: how to construct an M/G/∞ system with a pre-desired ‘target’ workload/queue auto-covariance function. The ‘reverse engineering’ scheme is applied to various examples, including ones with infinite queues and non-summable workloads. AMS Subject Classifications Primary: 60K25; Secondary: 60G55, 60G10     

Ploucquet’s “Refutation” of the Traditional Square of Opposition

In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”. My thanks are due to Hanno von Wulfen for helpful discussions and for transforming the word-document into a Latex-file.     

On an inequality of Edgar

LetE′ be the separable dual of a Banach spaceE. LetK be the class of all non-empty convex, weak^*-compact subsets ofE′. In this paper we prove that Edgar’s inequality, given in [2], extends to adapted sequences ofK-valued random variables.     

Smoothing algorithms for state–space models

Two-filter smoothing is a principled approach for performing optimal smoothing in non-linear non-Gaussian state–space models where the smoothing distributions are computed through the combination of ‘forward’ and ‘backward’ time filters. The ‘forward’ filter is the standard Bayesian filter but the ‘backward’ filter, generally referred to as the backward information filter, is not a probability measure on the space of the hidden Markov process. In cases where the backward information filter can be computed in closed form, this technical point is not important. However, for general state–space models where there is no closed form expression, this prohibits the use of flexible numerical techniques such as Sequential Monte Carlo (SMC) to approximate the two-filter smoothing formula. We propose here a generalised two-filter smoothing formula which only requires approximating probability distributions and applies to any state–space model, removing the need to make restrictive assumptions used in previous approaches to this problem. SMC algorithms are developed to implement this generalised recursion and we illustrate their performance on various problems.     

Isotropic bodies and Bourgain''''s problem

Let K (?) Rn be a convex body of volume 1 whose barycenter is at the origin, LK be the isotropic constant of K. Finding the least upper bound of LK , being called Bourgain's problem, is a well known open problem in the local theory of Banach space. The best estimate known today is LK < cn1/4 log n, recently shown by Bourgain, for an arbitrary convex body in any finite dimension. Utilizing the method of spherical section function, it is proven that if K is a convex body with volume 1 and r1Bn2 (?) K (?) r2Bn2,(r1≥1/2, r2≤(?)/2), then (?) ≤(?) and find the conditions with equality. Further, the geometric characteristic of isotropic bodies is shown.     

Lipschitz continuity of quantile functions on spaces of random variables

We prove that quantile functions on spaces of random variables satisfy the Lipschitz condition with constant 1 with respect to any norm on a subspace of a space of random variables that majorizes L^∞-norm. The considered random variables not necessarily belong to this Banach space. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351–358, 2008, pp. 253–258.     

On a Modified Conjecture of De Giorgi

We study the Γ-convergence of functionals arising in the Van der Waals–Cahn–Hilliard theory of phase transitions. Their limit is given as the sum of the area and the Willmore functional. The problem under investigation was proposed as modification of a conjecture of De Giorgi and partial results were obtained by several authors. We prove here the modified conjecture in space dimensions n = 2,3.This work was supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00274, FRONTS-SINGULARITIES.