Self-organized critical behavior: the evolution of frozen spin networks model in quantum gravity

In quantum gravity, we study the evolution of a two-dimensional planar open frozen spin network, in which the color (i.e. the twice spin of an edge) labeling edge changes but the underlying graph remains fixed. The mainly considered evolution rule, the random edge model, is depending on choosing an edge randomly and changing the color of it by an even integer. Since the change of color generally violate the gauge invariance conditions imposed on the system, detailed propagation rule is needed and it can be defined in many ways. Here, we provided one new propagation rule, in which the involved even integer is not a constant one as in previous works, but changeable with certain probability. In random edge model, we do find the evolution of the system under the propagation rule exhibits power-law behavior, which is suggestive of the self-organized criticality (SOC), and it is the first time to verify the SOC behavior in such evolution model for the frozen spin network. Furthermore, the increase of the average color of the spin network in time can show the nature of inflation for the universe.Comment: 5 pages, 5 figure

Efficient Real Space Solution of the Kohn-Sham Equations with Multiscale Techniques

We present a multigrid algorithm for self consistent solution of the Kohn-Sham equations in real space. The entire problem is discretized on a real space mesh with a high order finite difference representation. The resulting self consistent equations are solved on a heirarchy of grids of increasing resolution with a nonlinear Full Approximation Scheme, Full Multigrid algorithm. The self consistency is effected by updates of the Poisson equation and the exchange correlation potential at the end of each eigenfunction correction cycle. The algorithm leads to highly efficient solution of the equations, whereby the ground state electron distribution is obtained in only two or three self consistency iterations on the finest scale.Comment: 13 pages, 2 figure

Functional Inequalities and Subordination: Stability of Nash and Poincar\'e inequalities

We show that certain functional inequalities, e.g.\ Nash-type and Poincar\'e-type inequalities, for infinitesimal generators of $C_0$ semigroups are preserved under subordination in the sense of Bochner. Our result improves \cite[Theorem 1.3]{BM} by A.\ Bendikov and P.\ Maheux for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.Comment: 15 page

On the Coupling Property of L\'{e}vy Processes

We give necessary and sufficient conditions guaranteeing that the coupling for L\'evy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a L\'{e}vy process admits a successful coupling, if it is a strong Feller process or if the L\'evy (jump) measure has an absolutely continuous component.Comment: 14 page

The vortex dynamics of a Ginzburg-Landau system under pinning effect

It is proved that the vortices are attracted by impurities or inhomogeities in the superconducting materials. The strong H^1-convergence for the corresponding Ginzburg-Landau system is also proved.Comment: 23page

Spin Sum Rules at Low Q2Q^2

Recent precision spin-structure data from Jefferson Lab have significantly advanced our knowledge of nucleon structure at low $Q^2$. Results on the neutron spin sum rules and polarizabilities in the low to intermediate $Q^2$ region are presented. The Burkhardt-Cuttingham Sum Rule was verified within experimental uncertainties. When comparing with theoretical calculations, results on spin polarizability show surprising disagreements with Chiral Perturbation Theory predictions. Preliminary results on first moments at very low $Q^2$ are also presented.Comment: 4 pages, to be published in the Proceedings of the 10th Conference on Intersections of Nuclear and Particle Physics (CIPANP

On generating functions of Hausdorff moment sequences

The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$. Also we provide a simple analytic proof that for any real $p$ and $r$ with $p>0$, the Fuss-Catalan or Raney numbers $\frac{r}{pn+r}\binom{pn+r}{n}$, $n=0,1,\ldots$ are the moments of a probability distribution on some interval $[0,\tau]$ {if and only if} $p\ge1$ and $p\ge r\ge 0$. The same statement holds for the binomial coefficients $\binom{pn+r-1}n$, $n=0,1,\ldots$.Comment: 23 pages, LaTeX; Minor corrections and explanations added, literature update. To appear in Transactions Amer. Math. So