361 research outputs found
A congruence property of Ramanujan's function τ(n)
This article does not have an abstract
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
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 , \lsb Q_+ , Q_- \rsb ,
where are constants or central elements of the algebra, are
constructed using a generalization of the well known two-mode bosonic
realizations of and . 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 in
detail. We also examine the quadrature and amplitude-squared squeezing effects.
Finally, we derive expression for the -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
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