On convex perturbations with a bounded isotropic constant

Let be a convex body and ɛ > 0. We prove the existence of another convex body , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain. The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590. Received: November 2005; Accepted: February 2006     

A Translative Integral Formula for Absolute Curvature Measures

Absolute curvature measures are certain positive variants of curvature measures which are defined by means of the kinematic measure of flats of a given dimension touching the examined body. A kinematic flat section formula for absolute curvature measures was proved by Baddeley and Rother and Zähle. We present here a translative version of this formula.     

The unitary connections on the complex Grassmann manifold

In the complex Grassmann manifold ℱ(m,n), the space of complexn-planes passes through the origin of C^m+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G2,5

In this paper, we classify all nonsingular holomorphic spheres in complex Grassmann manifolds G_2,5 with the induced constant curvatures K=4, 2, 4/3,1 and 4/5 into some classes, up to unitary equivalence, in which none of the spheres are congruent; At the same time we also prove that there does not exist the nonsingular holomorphic sphere in G_2,5 with constant curvature 2/3.     

Recursions on the marginals and exact computation of the normalizing constant for Gibbs processes

This paper presents different recursive formulas for computing the marginals and the normalizing constant of a Gibbs distribution \(\pi \) . The common thread is the use of the underlying Markov properties of such processes. The procedures are illustrated with several examples, particularly the Ising model.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Grassmann流形上Pontrjagin示性式的积分式公式

命Ｇ_n+m,n是一个Ｇｒａｓｓｍａｎｎ流形，Ｚ_2k＝是Ｇ_n+m,n的一个Ｓｃｈｕｂｅｒｔ流形。命Ｅ是Ｇ_n+m,n上的规范矢丛，ＥＣ是Ｅ的变化，Ｅ是ＥＣ的相配酉（ｎ－２ｋ）－标架丛，并且Ｅ′是ＥＣ的一个矢丛，本文证明了下列积分公式：其中ｃ_4k是Ｇ_n+m,n的４ｋ链，Ｐ_k（Ω）是Ｇ_n+m,n的第ｋ个Ｐｏｎｔｒｊａｇｉｎ示性式，是定义在Ｅ和Ｅ′上的（４ｋ－１）－形式。     

Iteration-complexity of the subgradient method on Riemannian manifolds with lower bounded curvature

Abstract

The subgradient method for convex optimization problems on complete Riemannian manifolds with lower bounded sectional curvature is analysed in this paper. Iteration-complexity bounds of the subgradient method with exogenous step-size and Polyak's step-size are stablished, completing and improving recent results on the subject.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

A cellation of the Grassmann manifold

Motivated by the computation of equilibria in economic models with incomplete asset markets, a cellation of the Grassmann manifold is constructed by restricting a common atlas. The Grassmann manifold ofm-planes inn-dimensional space is shown to be a union ofn choosem congruentm(n−m)-dimensional topological disks whose interiors are disjoint.     

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Let be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector and the point . Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces (). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. . For known examples of 4-dimensional Osserman manifolds of signature we check also that . By the presentation of a class of examples we show that curvature homogeneity and do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.

     

The double exponential map on spaces of constant curvature

In the article, we derive an explicit formula for the double exponential map on spaces of constant curvature. In addition, we consider some applications of the resulting formula to computing the principal symbol of the product of two pseudodifferential operators on a manifold with connection.     

An infinite family of manifolds with bounded total curvature

The negative answer to the following problem of V. I. Arnold is given: Is the number of topologically different -manifolds of bounded total curvature finite?

     

Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below

We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric_∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric_∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.     

A generalization of imaginary parts of eigenvalues for matrices: Chain rotation numbers

Rotation numbers and chain rotation numbers may be interpreted as a generalization of the imaginary parts for matrices. In dimension two they measure how the solutions of a linear autonomous differential equation rotate in the phase space, and they reduce to the imaginary parts of the eigenvalues of the system’s matrix. In higher dimensions they measure how a two-frame of vectors rotate under the induced flow in the plane which is spanned by the frame. For their calculation, only special sets in the oriented Grassmann manifold of planes are relevant, and to each of these sets corresponds a compact interval of chain rotation numbers. In this paper we will determine these relevant sets and calculate the corresponding sets of chain rotation numbers.     

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R ⁿ . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.