451 research outputs found
A new Euclidean tight 6-design
We give a new example of Euclidean tight 6-design in .Comment: 9 page
Tight t-Designs and Squarefree Integers
The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5.Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F=ℝ,O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the author's result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1
An elementary approach to toy models for D. H. Lehmer's conjecture
In 1947, Lehmer conjectured that the Ramanujan's tau function
never vanishes for all positive integers , where is the -th
Fourier coefficient of the cusp form of weight 12. The theory of
spherical -design is closely related to Lehmer's conjecture because it is
shown, by Venkov, de la Harpe, and Pache, that is equivalent to
the fact that the shell of norm of the -lattice is a spherical
8-design. So, Lehmer's conjecture is reformulated in terms of spherical
-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 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
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise
in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are
the bispectral polynomials of a discrete argument obtained from a q-> -1 limit
of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators
including an involution, it is also a q-> -1 limit of the quantum algebra
sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator.
The algebra H, a two-parameter generalization of u(2) with an involution as
additional generator, is first derived from the recurrence relation of the -1
Hahn polynomials. It is then shown that H can be realized in terms of the
generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan
coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible
representation of H involving five-diagonal matrices and connected to the
difference equation of the dual -1 Hahn polynomials is constructed.Comment: 15 pages, Some minor changes from version #
Torsion divisors of plane curves and Zariski pairs
In this paper we study the embedded topology of reducible plane curves having
a smooth irreducible component. In previous studies, the relation between the
topology and certain torsion classes in the Picard group of degree zero of the
smooth component was implicitly considered. We formulate this relation clearly
and give a criterion for distinguishing the embedded topology in terms of
torsion classes. Furthermore, we give a method of systematically constructing
examples of curves where our criterion is applicable, and give new examples of
Zariski tuples.Comment: 19 page
Mixed Moore Cayley Graphs
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree.
There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph.
For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M(z,r,2)=(z+r)2+z+1 for the order of the graph. Apart from the case r=1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r,z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs
attaining the Moore bound for all orders up to 485
Exactly Solvable Birth and Death Processes
Many examples of exactly solvable birth and death processes, a typical
stationary Markov chain, are presented together with the explicit expressions
of the transition probabilities. They are derived by similarity transforming
exactly solvable `matrix' quantum mechanics, which is recently proposed by
Odake and the author. The (-)Askey-scheme of hypergeometric orthogonal
polynomials of a discrete variable and their dual polynomials play a central
role. The most generic solvable birth/death rates are rational functions of
( being the population) corresponding to the -Racah polynomial.Comment: LaTeX, amsmath, amssymb, 24 pages, no figure
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
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
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