73,059 research outputs found
On Zudilin's q-question about Schmidt's problem
We propose an elemantary approach to Zudilin's q-question about Schmidt's
problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a
previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a
q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740]
derived from q-Pfaff-Saalschutz identity.Comment: 5 page
Some q-analogues of supercongruences of Rodriguez-Villegas
We study different q-analogues and generalizations of the ex-conjectures of
Rodriguez-Villegas. For example, for any odd prime p, we show that the known
congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv
(-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with
[p]=1+q+...+q^{p-1}:
\sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv
(-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where
(a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and
\varepsilon=\pm1. Several related conjectures are also proposed.Comment: 14 pages, to appear in J. Number Theor
A note on two identities arising from enumeration of convex polyominoes
Motivated by some binomial coefficients identities encountered in our
approach to the enumeration of convex polyominoes, we prove some more general
identities of the same type, one of which turns out to be related to a strange
evaluation of of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical
change
Neutrino propagation in Neutron Matter and the Nuclear Equation of State
We study the propagation of neutrinos inside dense matter under the
conditions prevailing in a proto-neutron star. Equations of state obtained with
different nuclear effective interactions (Skyrme type and Gogny type) are first
discussed. It is found that for many interactions, spin and/or isospin
instabilities occur at densities larger than the saturation density of nuclear
matter. From this study we select two representative interactions, SLy230b and
D1P. We calculate the response functions in pure neutron matter where nuclear
correlations are described at the Hartree-Fock plus RPA level. These response
functions allow us to evaluate neutrino mean free paths corresponding to
neutral current processes.Comment: 8 pages, 7 figures, to appear in "The Nuclear Many-Body Problem
2001", NATO Science Series II (Kluwer Academic Publishers
Relativistic calculations of quasi-one-electron atoms and ions using Laguerre and Slater spinors
A relativistic description of the structure of heavy alkali atoms and
alkali-like ions using S-spinors and L-spinors has been developed. The core
wavefunction is defined by a Dirac-Fock calculation using an S-spinors basis.
The S-spinor basis is then supplemented by a large set of L-spinors for the
calculation of the valence wavefunction in a frozen-core model. The numerical
stability of the L-spinor approach is demonstrated by computing the energies
and decay rates of several low-lying hydrogen eigenstates, along with the
polarizabilities of a hydrogenic ion. The approach is then applied to
calculate the dynamic polarizabilities of the , and states of
Sr. The magic wavelengths at which the Stark shifts between different pairs
of transitions are zero are computed. Determination of the magic wavelengths
for the and transitions near
~nm (near the wavelength for the transitions) would allow a
determination of the oscillator strength ratio for the
and transitions.Comment: 2 figures, 23 page
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