Abstract. Historically, the polylogarithm has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

No abstract

No abstract

The Kochen-Specker (KS) theorem is a central result in quantum theory and has applications in quantum information. Its proof requires several yes-no tests that can be grouped in contexts or subsets of jointly measurable tests. Arguably, the best measure of simplicity of a KS set is the number of contexts. The smaller this number is, the smaller the number of experiments needed to reveal the conflict between quantum theory and noncontextual theories and to get a quantum vs classical outperformance. The original KS set had 132 contexts. Here we introduce a KS set with seven contexts and prove that this is the simplest KS set that admits a symmetric parity proof.

A two-distance set in E d is a point set X in the d-dimensional Euclidean space such that the distances between distinct points in X assume only two different nonzero values. Based on results from classical distance geometry, we develop an algorithm to classify, for a given d, all maximal (largest possible) two-distance sets in E d . Using this algorithm we have completed the full classification for all d 7, and we have found one set in E 8 whose maximality follows from Blokhuis' upper bound on sizes of s-distance sets. While in the dimensions d 6 our classifications confirm the maximality of previously known sets, the results in E 7 and E 8 are new. Their counterpart in dimension d 10 is a set of unit vectors with only two values of inner products in the Lorentz space R d, 1 . The maximality of this set again follows from a bound due to Blokhuis. Academic Press

In this paper, we study embedding efficiency, which is an important attribute of steganographic schemes directly influencing their security. It is defined as the expected number of embedded random message bits per one embedding change. Constraining ourselves to embedding realized using linear covering codes (so called matrix embedding), we show that the quantity that determines embedding efficiency is not the covering radius but the average distance to code. We demonstrate that for linear codes of fixed block length and dimension, the highest embedding efficiency (the smallest average distance to code) is not necessarily achieved using codes with the smallest covering radius. Nevertheless, we prove that with increasing code length and fixed rate (i.e., fixed relative message length), the relative average distance to code and the relative covering radius coincide. Finally, we describe several specific examples of q-ary linear codes with q matched to the embedding operation and experimentally demonstrate the improvement in steganographic security when incorporating the coding methods to digital image steganography.

No abstract