Bounds on the sizes of constant weight covering codes

Motivated by applications in universal data compression algorithms we study the problem of bounds on the sizes of constant weight covering codes. We are concerned with the minimal sizes of codes of lengthn and constant weightu such that every word of lengthn and weightv is within Hamming distanced from a codeword. In addition to a brief summary of part of the relevant literature, we also give new results on upper and lower bounds to these sizes. We pay particular attention to the asymptotic covering density of these codes. We include tables of the bounds on the sizes of these codes both for small values ofn and for the asymptotic behavior. A comparison with techniques for attaining bounds for packing codes is also given. Some new combinatorial questions are also arising from the techniques.Part of this work was done while the first and third authors were visiting Bellcore. The third author was supported in part by National Science Foundation under grant NCR-8905052. Part of this work was presented in the Coding and Quantization Workshop at Rutgers University, NJ, October 1992.     

Quantum Reduction for Affine Superalgebras

We extend the homological method of quantization of generalized Drinfeld–Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.     

A connection between a convex programming problem and the LYM property on perfect graphs

We derive three equivalent conditions on a perfect graph concerning the optimal solution of a convex programming problem, the length-width inequality, and the simultaneous vertex covering by cliques and anticliques. By combining proof techniques including Lagrangian dual, Dilworth's Theorem, and Kuhn-Tucker Theorem, we establish a strong connection between the three topics. This provides new insights into the structure of perfect graphs. The famous Lubell-Yamamoto-Meschalkin (LYM) Property or Sperner Property for partially ordered sets is a specialization of our results to a subclass of perfect graphs.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Notes on Banach Ideals

The aim of the present article is to study two constructions of new space ideals from given ones. Properties of some classes of BANACH spaces scattered in the literature are obtained as particular cases of the above constructions.     

Radiation-induced processes and defects in alkali and alkaline-earth borate crystals

The paper presents the results of a study of the radiation-induced processes and defects in nonlinear optical crystals Li₂B₄O₇ (LTB), LiB₃O₅ (LBO), CsLiB₆O₁₀, KB₅O₈·4H₂O, β-BaB₂O₄. It was revealed that a pulsed electron beam irradiation at 290 K forms the radiation-induced pairs of the ‘vacancy—interstitial atom’ defects in the cation sublattice of these crystals. This gives rise to a creation of metastable electronic (interstitial atom) and hole (small-radius polaron near the cation vacancy) centers in high concentrations. Optical hole-transitions from the local level of the trapped hole centers to the valence band states are responsible for the transient optical absorptions (TOA) of borates in the visible and UV spectral ranges. A sublattice of the weakly bound mobile lithium cations in LTB and LBO favors a spatial separation of the radiation-induced pair defects ‘hole polaron near Li-vacancy—mobile interstitial Li⁰ atom’. Their decay rated by the electron–hole nonradiative tunnel recombination determines a peculiar feature of the TOA decay kinetics in LTB and LBO.