On a problem of danzer

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A congruence property of Ramanujan's function τ(n)

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On polar reciprocal convex domains

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On a problem of G Fejes Toth

A solution is given for the following Problem of G Fejes Toth: In 3-space find the thinnest lattice of balls such that every straight line meets one of the balls

Fluctuations of Spatial Patterns as a Measure of Classical Chaos

In problems where the temporal evolution of a nonlinear system cannot be followed, a method for studying the fluctuations of spatial patterns has been developed. That method is applied to well-known problems in deterministic chaos (the logistic map and the Lorenz model) to check its effectiveness in characterizing the dynamical behaviors. It is found that the indices $\mu _q$ are as useful as the Lyapunov exponents in providing a quantitative measure of chaos.Comment: 10 pages + 7 figures (in ps file), LaTex, Submitted to Phys. Rev.

Qubit-Qutrit Separability-Probability Ratios

Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric -- and also their analogous formulas (quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same seven billion well-distributed (``low-discrepancy'') sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically rank-six (rank-five) density matrices. The (rank-six) separability probabilities obtained based on the 35-dimensional volumes appear to be -- independently of the metric (each of the seven inducing Haar measure) employed -- twice as large as those (rank-five ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last paragraph presenting qubit-qutrit probability ratios disaggregated by the two distinct forms of partial transpositio

Three dimensional quadratic algebras: Some realizations and representations

Four classes of three dimensional quadratic algebras of the type \lsb Q_0 , Q_\pm \rsb $=$ $\pm Q_\pm$, \lsb Q_+ , Q_- \rsb $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of $su(2)$ and $su(1,1)$. The resulting matrix representations and single variable differential operator realizations are obtained. Some remarks on the mathematical and physical relevance of such algebras are given.Comment: LaTeX2e, 23 pages, to appear in J. Phys. A: Math. Ge

A report on the nonlinear squeezed states and their non-classical properties of a generalized isotonic oscillator

We construct nonlinear squeezed states of a generalized isotonic oscillator potential. We demonstrate the non-existence of dual counterpart of nonlinear squeezed states in this system. We investigate statistical properties exhibited by the squeezed states, in particular Mandel's parameter, second-order correlation function, photon number distributions and parameter $A_3$ in detail. We also examine the quadrature and amplitude-squared squeezing effects. Finally, we derive expression for the $s$-parameterized quasi-probability distribution function of these states. All these information about the system are new to the literature.Comment: Accepted for publication in J. Phys. A: Math. Theo