A note on homogeneous locally symmetric spaces

Let denote an orthogonal symmetric Lie algbra and let (G, K) be an associated pair, i.e., Lie(G = and Lie(K°) = . In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.     

Volumes of symmetric random polytopes

We consider the moments of the volume of the symmetric convex hull of independent random points in an n-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for n points when the given body is (and all of the moments for the case q = 2), and derive from these the asymptotic behavior, as , of the expected volume of a random simplex in those bodies. Received: 5 February 2003     

A Correction Theorem for Functions with Integral Smoothness

A theorem similar to the correction theorem of Oskolkov is proved. Namely, for a function with a given kth continuity modulus calculated in a symmetric space X and for every , there exists a set of measure at least and such that we can give a sharp quantitative estimate of the uniform kth continuity modulus of the considered function. It is shown that this estimate depends only on and on the fundamental function of the symmetric space. Bibliography: 9 titles.     

A homotopy algorithm for a symmetric generalized eigenproblem

In this paper, a homotopy algorithm for finding all eigenpairs of a real symmetric matrix pencil (A, B) is presented, whereA andB are realn×n symmetric matrices andB is a positive semidefinite matrix. In the algorithm, pencil (A, B) is first reduced to a pencil , where is a symmetric tridiagonal matrix and is a positive definite and diagonal matrix. Then, the Divide and Conquer strategy with homotopy continuation approach is used to find all eigenpairs of pencil . One can easily form the eigenpair (x,) of pencil (A, B) from the eigenpair (y, ) of pencil with a few computations. Numerical comparisons of our algorithm with the QZ algorithm in the widely used EISPACK library are presented. Numerical results show that our algorithm is strongly competitive in terms of speed, accuracy and orthogonality. The performance of the parallel version of our algorithm is also presented.Research supported in part by NSF under Grant CCR-9024840.     

Projectively flat affine surfaces

We study non-degenerate affine surfaces in A³ with a projectively flat induced connection. The curvature of the affine metric , the affine mean curvature H, and the Pick invariant J are related by . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.     

Compression semigroups of open orbits on real flag manifolds

Let (G, H) be an irreducible semisimple symmetric pair,P G a parabolic subgroup. Suppose that theL-orbit of the base point in the flag manifoldG/P is open and writeS(L,P)={gG:gL LP} for the compression semigroup of this orbit. We show that ifP is minimal andS(L, P)=G, then (G, H) is Riemannian and we give a geometric characterization of those cases whereS(L, P) has non-empty interior different fromG. IfG/H is a symmetric space of regular type, then we show under certain additional assumptions thatS(L, Q) is an Ol'shanskiî semigroup.Supported by a DFG Heisenberg-grant.     

Finite splitting fields of normal subgroups

Let G be a finite group, a normal subgroup, p a prime, a finite splitting field of characteristic p for G and We prove that is a splitting field for N, using the action of the Galois group of the field extension on the irreducible representations of N. As is a splitting field for the symmetric group S_n we get as a corollary that is a splitting field for the alternating group A_n. Received: 31 July 2003     

OnQ-matrices

Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ Q , is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive.     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Two Splittings of a Square Matrix

In this note it is shown that any square matrix AC ^n×n can be represented as the sum A= , where is complex symmetric and rank . The corresponding persymmetric result can be used in finding the terms of a small rank perturbed Toeplitz matrix via an O(n ²) computation. This allows one to perform fast matrix–vector products in case n is large.     

On the mutual noncompatibility of homogeneous analytic non-power means

Summary Homogeneous symmetric meansµ and , defined on ₊ ⁿ and ₊ ⁿ⁺¹ , respectively, are calledcompatible if the value of remains unchanged upon replacing n of its arguments by theirµ-mean. Power means (of a common exponent) are a model example, which turns out to be unique, given analyticity of at least one of the two means considered. This is proved by fixing all but one argument in both and , which leads to a functional equation with two unknown functions, involving their mutual superpositions. The equation is solved in the class of analytic functions by comparing the power series coefficients.     

Extremal Distances between Sections of Convex Bodies

Let K, D be centrally symmetric convex bodies in Let k < n and let d_k(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define

Canonical Heights on Elliptic Curves in Characteristic p

Let be the rational function field with finite constant field and characteristic , and let K/k be a finite separable extension. For a fixed place v of k and an elliptic curve E/K which has ordinary reduction at all places of K extending v, we consider a canonical height pairing which is symmetric, bilinear and Galois equivariant. The pairing for the infinite place of k is a natural extension of the classical Néron–Tate height. For v finite, the pairing plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.     

Spin Models on Finite Cyclic Groups

The concept of spin model is due to V. F. R. Jones. The concept of nonsymmetric spin model, which generalizes that of the original (symmetric) spin model, is defined naturally. In this paper, we first determine the diagonal matrices T satisfying the modular invariance or the quasi modular invariance property, i.e., or (respectively), for the character table P of the group association scheme of a cyclic group G of order m. Then we show that a (symmetric or nonsymmetric) spin model on G is constructed from each of the matrices T satisfying the modular or quasi modular invariance property.     

The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL _n (), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL _n () for n 3 cannot act on a visibility space X without fixing a point in . Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.     

Almost sure invariance principles when EX 1 2 =∞

Summary Let {Y _i} be iid with EY ₁=0, EY ₁ ² =1. Let {X_i} be iid normal mean zero, variance one random variables. According to Strassen's first almost sure invariance principle {X _i} and {Y _i} can be reconstructed on a new probability space without changing the distribution of each sequence such that a.s., thus improving on the trivial bound obtainable from the law of the iterated logarithm: a.s. In this work we establish analogous improvements for symmetric {Y _i} in the domain of normal attraction to a symmetric stable law with index 0<<2. (We make this assumption of symmetry in order to avoid messy details concerning centering constants.) Let {X _i} be iid symmetric stable random variables with index 0<<2. Then, for example, hypotheses are stated which imply for a given satisfying 2> that a.s., thus improving on the trivial bound: a.s., >0.This research was supported in part by a National Science Foundation grant, USA