We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDXT of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(εκ(X)), where ε is the machine precision and κ(X) ≡ ‖X‖2 · ‖X−1‖2 is the spectral condition number of X . The eigenvectors are a...

We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset. It follows from [B1] (see also [BP1, Theorem 1]) that for each partition of the n-dimensional space R into n pieces one of the pieces contains an unbounded centrally symmetric subset. On the other hand, R admits a partition...

X iv :m at h/ 03 05 42 3v 3 [ m at h. R T ] 1 1 N ov 2 00 3 Stein’s Method and Plancherel Measure of the Symmetric Group Running head: Stein’s Method and Plancherel Measure By Jason Fulman University of Pittsburgh Department of Mathematics 301 Thackeray Hall Pittsburgh, PA 15260 Email: [email protected] Abstract: We initiate a Stein’s method approach to the study of the Plancherel measure of...

We propose a class of synthetic optical materials in which the refractive index satisfies n(−x) = −n∗(x). We term such systems antisymmetric parity-time (APT ) structures. Unlike PT -symmetric systems, which require balanced gain and loss, i.e., n(−x) = n∗(x), APT systems consist of balanced positiveand negative-index materials. Despite the seemingly PT -symmetric optical potential V (x) ≡ n(x)...

We study a network of non-collaborating sensors that make noisy measurements of some physical process X and communicate their readings to a central processing unit. Limited power resources of the sensors severely restrict communication rates. Sensors and their communication links are both subject to failure, however, the central unit is guaranteed to receive data from a minimum fraction of the ...

Let U be a real n x n matrix whose entries uij are random variables, and let A and B be fixed n x n real symmetric matrices. Statisticians (e.g., see [OU]) have been interested in the distribution of the eigenvalues d l , . . . , O n of the matrix AUBUt, where denotes transpose. Of particular interest are the quantities tr((AU = C 0; for k = 1,2,. . . , since these determine the eigenvalues. Mo...

When considering σr(X), the variety of r-secant P r−1 to a projective variety X, one question which arises is what are the possible values of the X-rank of points on σr(X), apart from the generic value r? This geometric problem is of particular relevance (also for Applied Math) when X is a variety parameterizing some kind of tensors. We study here the case when X is a Veronese variety (i.e. the...

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators for Pic(X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit...

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that spanK = X, there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee’s result for separable spaces. The well-known Mazur’s theorem says that a continuous conv...

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matri...