‎Let $M^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de Sitter space or an anti-de‎ ‎Sitter space‎, ‎$S$ and $K$ be the squared norm of the second‎ ‎fundamental form and Gauss-Kronecker curvature of $M^n$‎. ‎If $S$ or‎ ‎$K$ is constant‎, ‎nonzero and $M^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

We consider the problem of selecting the “best” subset of exactly k columns from an m× n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2,m2n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-...

We study Thurston’s norm on the second homology of a 3-manifold. The novelty of our approach consists in the use of methods of the C∗algebra theory. Namely, for Stallings’ fibration M with pseudo-Anosov monodromy, we associate a C∗-algebra whose K-theory gives rise to an algebraic number field K. We show that the trace function on the ring of integers of K induces a norm on the second homology ...

My research in graph theory focuses on asymptotic maximal bounds for Cartesian powers of finite graphs. This line of inquiry was opened up by [4], in which the authors proved that the maximal spherical averaging operator on the N -dimensional hypercube K 2 satisfies a bound in the L 2 norm independent of the dimension N . That is, if Sk(x) := {y : d(x, y) = k} in the graph distance metric on K ...

This paper treats the problem of approximating an nth order continuous system by a system of order k < n. Optimal solutions minimizing the H1 norm exist for k = 0; n ? 1. This paper presents an iterative two-step LMI method for improving the H1 model error compared to Hankel norm reduction. As an intermediate step, the algorithm nds a generalized balanced realization such that the n ?k Hankel s...

Let K be a global field. The aim of this paper is to study the basic properties of the global non-abelian norm-residue symbol NR φWeil K of K, which is defined following Chevalley-Miyake philosophy of idèles by “glueing” the local non-abelian norm-residue symbol {•,Kν}φν of Kν in the sense of Koch, for each ν ∈ hK . Mathematics Subject Classification (2010). Primary 11R37,11R39; Secondary 11S15...

We consider the problem of selecting the “best” subset of exactly k columns from an m× n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2,m2n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-...

For P 2 Z[x], let kPk denote the Euclidean norm of the coe cient vector of P . For an algebraic number , with minimal polynomial A, de ne the Euclidean norm of by k k = kkAk ; where k is the smallest positive integer for which kA 2 Z[x]. De ne the minimal Euclidean norm of by k k min = min kPk : P 2Z[x]; P ( ) = 0; P 6 0 : Given an algebraic number , we show there exists a P 2 Z[x] with P ( ) =...

The linear spaces generated by the eigenfunctions of a diierential operator are well-known in applied mathematics. In this paper we examine their interpolation properties, connection with Sobolev spaces and apply these results to the solving of hyperbolic equation in Sobolev spaces of fractional order. 1. Interpolation and function spaces Let A 1 and A 2 be two Banach spaces, linearly and conti...

Max-convolution is an important problem closely resembling standard convolution; as such, maxconvolution occurs frequently across many fields. Here we extend the method with fastest known worst-case runtime, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm ‖ · ‖∞, and use this approach to derive two numerically stable methods based on the idea of compu...