Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ?γ? = 1. Namely, all such unitary perturbations are operators U_γ:= U + (γ ? 1)( ·, b₁)_Hb, where b ∈ H, ∥b∥ = 1, b₁ = U^?1b, ?γ? = 1. For ?γ? < 1, the operators U_γ are contractions with one-dimensional defects.     

An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure

We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ɛ in time polynomial in the problem size and in 1/ɛ. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.     

Embedding complete trees into the hypercube

We consider embeddings of the complete t-ary trees of depth k (denotation T^k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T^k,t), is known if max{k,t}2. First, we study the next open cases t=3 and k=3. We improve the known upper bound dim(T^k,3)2k+1 up to lim_k→∞dim(T^k,3)/k5/3 and show lim_t→∞dim(T^3,t)/t=227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T^k,t) for arbitrary k and t.     

Multi-parameter regularization and its numerical realization

In this paper we propose and analyse a choice of parameters in the multi-parameter regularization of Tikhonov type. A modified discrepancy principle is presented within the multi-parameter regularization framework. An order optimal error bound is obtained under the standard smoothness assumptions. We also propose a numerical realization of the multi-parameter discrepancy principle based on the model function approximation. Numerical experiments on a series of test problems support theoretical results. Finally we show how the proposed approach can be successfully implemented in Laplacian Regularized Least Squares for learning from labeled and unlabeled examples.     

Dynamical Systems Associated with Crossed Products

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C ^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A′. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.     

A hyperbolic surface with a square grid net

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by è. B. Vinberg. Supported by the Alexander von Humboldt Foundation. Supported by NSF grants DMS-0400636, DMS-0244421, and DMS-0244547.     

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.