Non-Beiter ternary cyclotomic polynomials with an optimally large set of coefficients

Let l>=1 be an arbitrary odd integer and p,q and r primes. We show that there exist infinitely many ternary cyclotomic polynomials \Phi_{pqr}(x) with l^2+3l+5<= p<q<r such that the set of coefficients of each of them consists of the p integers in the interval [-(p-l-2)/2,(p+l+2)/2]. It is known that no larger coefficient range is possible. The Beiter conjecture states that the cyclotomic coefficients a_{pqr}(k) of \Phi_{pqr} satisfy |a_{pqr}(k)|<= (p+1)/2 and thus the above family contradicts the Beiter conjecture. The two already known families of ternary cyclotomic polynomials with an optimally large set of coefficients (found by G. Bachman) satisfy the Beiter conjecture.Comment: 20 pages, 7 Table

Ternary cyclotomic polynomials having a large coefficient

Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\ge 11$. We also prove that given any $\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<...$ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\epsilon)p_j$ for $j\ge 1$.Comment: 19 pages, 6 tables, to appear in Crelle's Journal. Revised version with many small change

Supersymmetric Drinfeld-Sokolov reduction

The Drinfeld-Sokolov construction of integrable hierarchies, as well as its generalizations, may be extended to the case of loop superalgebras. A sufficient condition on the algebraic data for the resulting hierarchy to be invariant under supersymmetry transformation is given. The method used is a construction of the hierarchies in superspace, where supersymmetry is manifest. Several examples are discussed.Comment: 25 pages, LaTeX fil

N=2 KP and KdV hierarchies in extended superspace

We give the formulation in extended superspace of an $N=2$ supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP hierarchy. In particular we find two different hierarchies with the $N=2$ classical super-${\cal W}_n$ algebra as a hamiltonian structure. The relation with the formulation in $N=1$ superspace is carried out.Comment: 18 pages, LaTeX file, important reference adde

Extra energy coupling through subwavelength hole arrays via stochastic resonance

Interaction between metal surface waves and periodic geometry of subwavelength structures is at the core of the recent but crucial renewal of interest in plasmonics. One of the most intriguing points is the observation of abnormal strong transmission through these periodic structures, which can exceed by orders of magnitude the classical transmission given by the filling factor of the plate. The actual paradigm is that this abnormal transmission arises from the periodicity, and then that such high transmission should disappear in random geometries. Here, we show that extra energy can be coupled through the subwavelength structure by adding a controlled quantity of noise to the position of the apertures. This result can be modelled in the statistical framework of stochastic resonance. The evolution of the coupled energy with respect to noise gives access to the extra energy coupled at the surface of the subwavelength array.Comment: 12 page

Experimental evidence of percolation phase transition in surface plasmons generation

Carrying digital information in traditional copper wires is becoming a major issue in electronic circuits. Optical connections such as fiber optics offers unprecedented transfer capacity, but the mismatch between the optical wavelength and the transistors size drastically reduces the coupling efficiency. By merging the abilities of photonics and electronics, surface plasmon photonics, or 'plasmonics' exhibits strong potential. Here, we propose an original approach to fully understand the nature of surface electrons in plasmonic systems, by experimentally demonstrating that surface plasmons can be modeled as a phase of surface waves. First and second order phase transitions, associated with percolation transitions, have been experimentally observed in the building process of surface plasmons in lattice of subwavelength apertures. Percolation theory provides a unified framework for surface plasmons description

The Erd\H{o}s--Moser equation 1k+2k+...+(m−1)k=mk1^k+2^k+...+(m-1)^k=m^k revisited using continued fractions

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.Comment: 17 page