Optimal interleaving schemes for correcting two-dimensional cluster errors

We present an elementary theory of optimal interleaving schemes for correcting cluster errors in two-dimensional digital data. It is assumed that each data page contains a fixed number of, say n, codewords with each codeword consisting of m code symbols and capable of correcting a single random error (or erasure). The goal is to interleave the codewords in the m×n array such that different symbols from each codeword are separated as much as possible, and consequently, an arbitrary error burst with size up to t can be corrected for the largest possible value of t. We show that, for any given m, n, the maximum possible interleaving distance, or equivalently, the largest size of correctable error bursts in an m×n array, is given by if n?⌈m²/2⌉, and t=m+⌊(n-⌈m²/2⌉)/m⌋ if n?⌈m²/2⌉. Furthermore, we develop a simple cyclic shifting algorithm that can provide a systematic construction of an m×n optimal interleaving array for arbitrary m and n. This extends important earlier work on the complementary problem of constructing interleaving arrays that, given the burst size t, minimize the interleaving degree, that is, the number of different codewords in a 2-D (or 3-D) array such that any error burst with given size t can be corrected. Our interleaving scheme thus provides the maximum burst error correcting power without requiring prior knowledge of the size or shape of an error burst.     

Exhaustive determination of (1023, 511, 255)-cyclic difference sets

An exhaustive search for (1023, 511, 255)-cyclic difference sets has been conducted. A total of 10 non-equivalent (1023, 511, 255)-cyclic difference sets have been found, all of which are members of previously known or conjectured infinite families. A fast and effective autocorrelation test method was utilized that can also facilitate the testing of longer sequences.

     

A new arithmetic function of combinatorial significance

The function γ(n) is defined for n > 1 as the number of representations n = a^b with positive integers a and b. From the basic identity

$\text{∑}\text{n=2}\text{∫}\text{γ(n)f(n) =}\text{∑}\text{m=2}\text{∫}\text{∑}\text{k=1}\text{∫}\text{f (m}{}^{\mathrm{k}}\text{) =}\text{∑}\text{k=1}\text{∫}\text{∑}\text{m=2}\text{∫}\text{f(m}{}^{\mathrm{k}}\text{)}$

, valid for any function f(n) for which Σ_n=2^∞γ(n) |f(n)| < ∞, several zeta-function identities are derived, and certain sums involving γ(n) are evaluated while others are proved irrational.     

Optically induced birefringence and holography in silk

In this study, we demonstrated that thin films of azobenzene‐modified Bombyx mori silk showed optically induced effects similar to those found in other polymers incorporating azobenzene moieties. Optically induced anisotropy in refractive index reached 7 × 10^?3 with 18 J cm^?2 required to reach 50% of steady state. Holograms were formed with diffraction efficiencies reaching 2.5%. The holograms were accompanied by surface relief gratings and measurements of the mechanical properties of films with surface relief gratings showed that the raised areas were more adhesive than the lower areas. We also demonstrated the fidelity of holographic images. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

The metric gradient in normed linear spaces

Summary In this paper we give a definition for the gradient of a functional defined on a normed linear space which in the case of Hilbert space reduces to the usual definition. We also establish some interesting properties of the gradient which allow us to extend the well-known theorem of Curry to a large class of normed linear spaces.A part of this research was sponsored by the U.S. Army under contract No. DA-31-124-ARO-D-462.     

Zur Theorie der nichtlinearen Integralgleichungen,Integralgleichungssysteme und allgemeinen Funktionalgleichungen

Ohne ZusammenfassungDissertation der Philosophischen Fakultät Berlin.