A new Euclidean tight 6-design

We give a new example of Euclidean tight 6-design in $\mathbb R^{22}$.Comment: 9 page

An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

NcPred for accurate nuclear protein prediction using n-mer statistics with various classification algorithms

Prediction of nuclear proteins is one of the major challenges in genome annotation. A method, NcPred is described, for predicting nuclear proteins with higher accuracy exploiting n-mer statistics with different classification algorithms namely Alternating Decision (AD) Tree, Best First (BF) Tree, Random Tree and Adaptive (Ada) Boost. On BaCello dataset [1], NcPred improves about 20% accuracy with Random Tree and about 10% sensitivity with Ada Boost for Animal proteins compared to existing techniques. It also increases the accuracy of Fungal protein prediction by 20% and recall by 4% with AD Tree. In case of Human protein, the accuracy is improved by about 25% and sensitivity about 10% with BF Tree. Performance analysis of NcPred clearly demonstrates its suitability over the contemporary in-silico nuclear protein classification research

Amici Curiae Brief on Behalf of the Fred T. Korematsu Center for Law and Equality, the Asian Bar Association of Washington, the Pacific Northwest District of the Japanese American Citizens League, and the Vietnamese Bar Association of Washington, in Support of Petitioner

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Convergence to equilibrium under a random Hamiltonian

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.Comment: 11 pages, 1 figure, v1-v3: some minor errors and typos corrected and new references added; v4: results for the degenerated spectrum added; v5: reorganized and rewritten version; to appear in PR

Amicus Curiae Brief on Behalf of the Fred T. Korematsu Center for Law and Equality, in Support of Neither Party

Civil Rights Amicus Brief Projec