Collapse of a Bose gas: Kinetic approach

We have analytically explored the temperature dependence of critical number of particles for the collapse of a harmonically trapped attractively interacting Bose gas below the condensation point by introducing a kinetic approach within the Hartree?CFock approximation. The temperature dependence obtained by this easy approach is consistent with that obtained from the scaling theory.     

Unitary representations of the fundamental group of orbifolds

Let X be a smooth complex projective variety of dimension n and \(\mathcal {L}\) an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E on X with \(c_{1}(E) = 0 = c_{2} (E) \cdot c_{1} (\mathcal {L})^{n-2}\) and the equivalence classes of unitary representations of π₁(X). We show that this bijective correspondence extends to smooth orbifolds.     

On rationality of moduli spaces of vector bundles on real Hirzebruch surfaces

Let X be a real form of a Hirzebruch surface. Let M _H (r,c ₁, c ₂) be the moduli space of vector bundles on X. Under some numerical conditions on r, c ₁ and c ₂, we identify those M _H (r,c ₁,c ₂) that are rational.     

Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.     

Spherical aberration from trajectories in real and hard-edge solenoid fields

For analytical, real and hard-edge solenoidal axial magnetic fields, the low-energy electron trajectories are obtained using the third-order paraxial ray equation. Using the particle trajectories, it is shown that the spherical aberration in the hard-edge model is high and it increases monotonously with hard edginess, although the focal length converges, in agreement with a recent field and spherical aberration model. The model paved the way for a hard-edge approximation that gives correct focal length and spherical aberration, which is verified here by the trajectory method. In essence, we show that exact hard-edge fields give infinite spherical aberrations.     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.