The Abel-Zeilberger Algorithm

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.Comment: 18 page

Predictive protocol of flocks with small-world connection pattern

By introducing a predictive mechanism with small-world connections, we propose a new motion protocol for self-driven flocks. The small-world connections are implemented by randomly adding long-range interactions from the leader to a few distant agents, namely pseudo-leaders. The leader can directly affect the pseudo-leaders, thereby influencing all the other agents through them efficiently. Moreover, these pseudo-leaders are able to predict the leader's motion several steps ahead and use this information in decision making towards coherent flocking with more stable formation. It is shown that drastic improvement can be achieved in terms of both the consensus performance and the communication cost. From the industrial engineering point of view, the current protocol allows for a significant improvement in the cohesion and rigidity of the formation at a fairly low cost of adding a few long-range links embedded with predictive capabilities. Significantly, this work uncovers an important feature of flocks that predictive capability and long-range links can compensate for the insufficiency of each other. These conclusions are valid for both the attractive/repulsive swarm model and the Vicsek model.Comment: 10 pages, 12 figure

Study the Heavy Molecular States in Quark Model with Meson Exchange Interaction

Some charmonium-like resonances such as X(3872) can be interpreted as possible $D^{(*)}D^{(*)}$ molecular states. Within the quark model, we study the structure of such molecular states and the similar $B^{(*)}B^{(*)}$ molecular states by taking into account of the light meson exchange ($\pi$, $\eta$, $\rho$, $\omega$ and $\sigma$) between two light quarks from different mesons

Fluctuation and localization of the nonlinear Hall effect on a disordered lattice

The nonlinear Hall effect has recently attracted significant interest due to its potentials as a promising spectral tool and device applications. A theory of the nonlinear Hall effect on a disordered lattice is a crucial step towards explorations in realistic devices, but has not been addressed. We study the nonlinear Hall response on a lattice, which allows us to introduce disorder numerically and reveal a mechanism that was not discovered in the previous momentum-space theories. In the mechanism, disorder induces an increasing fluctuation of the nonlinear Hall conductance as the Fermi energy moves from the band edges to higher energies. This fluctuation is a surprise, because it is opposite to the disorder-free distribution of the Berry curvature. More importantly, the fluctuation may explain those unexpected observations in the recent experiments. We also discover an "Anderson localization" of the nonlinear Hall effect. This work shows an emergent territory of the nonlinear Hall effect yet to be explored

KN and KbarN Elastic Scattering in the Quark Potential Model

The KN and KbarN low-energy elastic scattering is consistently studied in the framework of the QCD-inspired quark potential model. The model is composed of the t-channel one-gluon exchange potential, the s-channel one-gluon exchange potential and the harmonic oscillator confinement potential. By means of the resonating group method, nonlocal effective interaction potentials for the KN and KbarN systems are derived and used to calculate the KN and KbarN elastic scattering phase shifts. By considering the effect of QCD renormalization, the contribution of the color octet of the clusters (qqbar) and (qqq) and the suppression of the spin-orbital coupling, the numerical results are in fairly good agreement with the experimental data.Comment: 20 pages, 8 figure

An SO(10) GUT Model with S4S4 Flavor Symmetry

We present a supersymmetric grand unification model based on SO(10) group with $S4$ flavor symmetry. In this model, the fermion masses are from Yukawa couplings involving $\bf{10}$ and $\bar{\bf{126}}$ Higgs multiplets and the flavor structures of mass matrices of both quarks and leptons are determined by spontaneously broken $S4$. This model fits all of the masses and mixing angles of the quarks and leptons. For the most general CP-violation scenario, this model gives $\sin\theta_{13}$ a wide range of values from zero to the current bound with the most probable values $0.02-0.09$. With certain assumptions where leptonic phases have same CP-violation source as CKM phase, one gets a narrower range $0.03-0.09$ for $\sin\theta_{13}$ with the most probable values $0.04-0.08$. This model gives leptonic Dirac CP phase the most probable values 2-4 radians in the general CP-violation case.Comment: 14 pages,2 figures. Version published in Physical Review