ન્યાયમજરીમાં રજૂ થયેલ પ્રત્યક્ષ પ્રમાણનું તુલનાત્મક અને સમીક્ષાત્મક અધ્યયન

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Zariski Closures and Subgroup Separability

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.Comment: Final version. To appear in Selecta Mat

Observation of a Degenerate Fermi Gas Trapped by a Bose-Einstein Condensate

We report on the formation of a stable quantum degenerate mixture of fermionic $^6$Li and bosonic $^{133}$Cs in an optical trap by sympathetic cooling near an interspecies Feshbach resonance. New regimes of the quantum degenerate mixtures are identified. With moderate attractive interspecies interactions, we show that a degenerate Fermi gas of Li can be fully confined in the Cs condensate without external potentials. For stronger attraction where mean-field collapse is expected, no such instability is observed. In this case, we suggest the stability is a result of dynamic equilibrium, where the interspecies three-body loss prevents the collapse. Our picture is supported by a rate equation model, and the crossover between the thermalization rate and the observed inelastic loss rate in the regime where the mean-field collapse is expected to occur.Comment: 6 Pages, 4 Figure

Charged analogue of Finch-Skea stars

We present solutions to the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes with a specified form of the electric field intensity. The condition of pressure isotropy yields three category of solutions. The first category is expressible in terms of elementary functions and does not have an uncharged limit. The second category is given in terms of Bessel functions of half-integer order. These charged solutions satisfy a barotropic equation of state and contain Finch-Skea uncharged stars. The third category is obtained in terms of modified Bessel functions of half-integer order and does not have an uncharged limit. The physical features of the charged analogue of the Finch-Skea stars are studied in detail. In particular the condition of causality is satisfied and the speed of sound does not exceed the speed of light. The physical analysis indicates that this analogue is a realistic model for static charged relativistic perfect fluid spheres.Comment: 17 pages, To appear in Int. J. Mod. Phys.

Quark distributions and gluon contents of eta and etaprime

Using QCD sum rule, we calculate valance quark distribution functions of {\eta} and of {\eta}' in chiral limit. Extrapolating the calculated quark distribution functions at both end regions, their first two moments have been calculated and compared with those of other hadrons.Comment: 14 pages, 7 figure