Heisenberg equation for a nonrelativistic particle on a hypersurface: from the centripetal force to a curvature induced force

In classical mechanics, a nonrelativistic particle constrained on an $N-1$ curved hypersurface embedded in $N$ flat space experiences the centripetal force only. In quantum mechanics, the situation is totally different for the presence of the geometric potential. We demonstrate that the motion of the quantum particle is "driven" by not only the the centripetal force, but also a curvature induced force proportional to the Laplacian of the mean curvature, which is fundamental in the interface physics, causing curvature driven interface evolution.Comment: 4 page

Distributed Flow Scheduling in an Unknown Environment

Flow scheduling tends to be one of the oldest and most stubborn problems in networking. It becomes more crucial in the next generation network, due to fast changing link states and tremendous cost to explore the global structure. In such situation, distributed algorithms often dominate. In this paper, we design a distributed virtual game to solve the flow scheduling problem and then generalize it to situations of unknown environment, where online learning schemes are utilized. In the virtual game, we use incentives to stimulate selfish users to reach a Nash Equilibrium Point which is valid based on the analysis of the `Price of Anarchy'. In the unknown-environment generalization, our ultimate goal is the minimization of cost in the long run. In order to achieve balance between exploration of routing cost and exploitation based on limited information, we model this problem based on Multi-armed Bandit Scenario and combined newly proposed DSEE with the virtual game design. Armed with these powerful tools, we find a totally distributed algorithm to ensure the logarithmic growing of regret with time, which is optimum in classic Multi-armed Bandit Problem. Theoretical proof and simulation results both affirm this claim. To our knowledge, this is the first research to combine multi-armed bandit with distributed flow scheduling.Comment: 10 pages, 3 figures, conferenc

The centripetal force law and the equation of motion for a particle on a curved hypersurface

It is pointed out that the current form of extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version for it is established without regard to the fact that the particle can never depart from the geodesics on the surface. Once the fact be taken into consideration, the equation takes that same form as that for centripetal force law, provided that the symbols are re-interpreted so that the law is applicable for higher dimensions. The controversial issue of constructing operator forms of these equations is addressed, and our studies show the quantization of constrained system based on the extrinsic equation of motion is favorable.Comment: 5 pages, major revisio

General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$) curved surface, the Cartesian components of its momentum in $N$-dimensional flat space is believed to offer a proper form of momentum for the particle on the surface, which is called the geometric momentum as it depends on the mean curvature. Once the momentum is made general covariance, the spin connection part can be interpreted as a gauge potential. The present study consists in two parts, the first is a discussion of the general framework for the general covariant geometric momentum. The second is devoted to a study of a Dirac fermion on a two-dimensional sphere and we show that there is the generalized total angular momentum whose three cartesian components form the $su(2)$ algebra, obtained before by consideration of dynamics of the particle, and we demonstrate that there is no curvature-induced geometric potential for the fermion.Comment: 8 pages, no figure. Presentation improve

Charmed Baryon Weak Decays with SU(3) Flavor Symmetry

We study the semileptonic and non-leptonic charmed baryon decays with $SU(3)$ flavor symmetry, where the charmed baryons can be ${\bf B}_{c}=(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$, ${\bf B}'_{c}=(\Sigma_c^{(++,+,0)},\Xi_{c}^{\prime(+,0)},\Omega_c^0)$, ${\bf B}_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^+,\Omega_{cc}^+)$, or ${\bf B}_{ccc}=\Omega^{++}_{ccc}$. With ${\bf B}_n^{(\prime)}$ denoted as the baryon octet (decuplet), we find that the ${\bf B}_{c}\to {\bf B}'_n\ell^+\nu_\ell$ decays are forbidden, while the $\Omega_c^0\to \Omega^-\ell^+\nu_\ell$, $\Omega_{cc}^+\to\Omega_c^0\ell^+\nu_\ell$, and $\Omega_{ccc}^{++}\to \Omega_{cc}^+\ell^+\nu_\ell$ decays are the only existing Cabibbo-allowed modes for ${\bf B}'_{c}\to {\bf B}'_n\ell^+\nu_\ell$, ${\bf B}_{cc}\to {\bf B}'_c\ell^+\nu_\ell$, and ${\bf B}_{ccc}\to {\bf B}_{cc}^{(\prime)}\ell^+\nu_\ell$, respectively. We predict the rarely studied ${\bf B}_{c}\to {\bf B}_n^{(\prime)}M$ decays, such as ${\cal B}(\Xi_c^0\to\Lambda^0\bar K^0,\,\Xi_c^+\to\Xi^0\pi^+)=(8.3\pm 0.9,8.0\pm 4.1)\times 10^{-3}$ and ${\cal B}(\Lambda_c^+\to \Delta^{++}\pi^-,\,\Xi_c^0\to\Omega^- K^+)=(5.5\pm 1.3,4.8\pm 0.5)\times 10^{-3}$. For the observation, the doubly and triply charmed baryon decays of $\Omega_{cc}^{+}\to \Xi_c^+\bar K^0$, $\Xi_{cc}^{++}\to (\Xi_c^+\pi^+$, $\Sigma_c^{++}\bar K^0)$, and $\Omega_{ccc}^{++}\to (\Xi_{cc}^{++}\bar K^0,\Omega_{cc}^+\pi^+,\Xi_c^+ D^+)$ are the favored Cabibbo-allowed decays, which are accessible to the BESIII and LHCb experiments.Comment: 29 pages, no figure, a typo in the table correcte

A note on modular forms and generalized anomaly cancellation formulas

By studying modular invariance properties of some characteristic forms, we prove some new anomaly cancellation formulas which generalize the Han-Zhang and Han-Liu-Zhang anomaly cancellation formula