Abstract:
A graph is said to be vertex-transitive, if its automorphism group is transitive on its vertices. In this paper, it is proven that a connected cubic vertex-transitive graph of order 4p (p a prime) is either a Cayley graph or isomorphic to one of the following: the generalized Petersen graph P(10,2), the Dodecahedron, the Coxeter graph, or the generalized Petersen graph P(2p,k) where k^2\equiv -1(\mod 2p).

Abstract:
In this paper we develop the continuous averaging method of Treschev to work on the simultaneous Diophantine approximation and apply the result to give a new proof of the Nekhoroshev theorem. We obtain a sharp normal form theorem and an explicit estimate of the stability constants appearing in the Nekhoroshev theorem.

Abstract:
This paper constructs a certain planar four-body problem which exhibits fast energy growth. The system considered is a quasi-periodic perturbation of the Restricted Planar Circular three-body Problem (RPC3BP). Gelfreich-Turaev's and de la Llave's mechanism is employed to obtain the fast energy growth. The diffusion is created by a heteroclinic cycle formed by two Lyapunov periodic orbits surrounding $L_1$ and $L_2$ Lagrangian points and their heteroclinic intersections. Our model is the first known example in celestial mechanics of the a priori chaotic case of Arnold diffusion.

Abstract:
In this paper, we show that there is a Cantor set of initial conditions in a planar four-body problem such that all the four bodies escape to infinity in finite time avoiding collisions. This proves the Painlev\'e conjecture for the four-body case, thus settles the conjecture completely.

Abstract:
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.

Abstract:
A direced Cayley graph $X=\cay(G, S)$ is called normal for $G$ if the right representation $R(G)$ of $G$ is normal in the full automorphism group $\Aut (X)$. In this paper, we determine all non-normal directed Cayley graphs of finite abelian groups with valencies 2 and 3. Using the result, we give a complete classification of connected directed arc-transitive graphs of order $p^n$ ($n\leq 2, p$ an odd prime) with valency at most 3.

This paper proposes a kind of optimization software for
substation operation mode, which can not only read data on-line from EMS, but also calculate total loss of substations in
parallel operation, split operation or individual operation mode. It can also
select the most optimized way and feed the conclusion back to EMS
to make substations operate in the most optimized way. The software is suitable
for optimization of substation in rural power grid.

Abstract:
A new converter with spherical cap for energy scavenging is proposed. Based on the method of separated variables within the torrid coordinate system, a corresponding analytical model for spherical cap converter is further established so as to obtain the analytic expressions of the topology capacitance and the output voltage. The concept of energy increment factor is specifically defined to denote the improvement of energy storage efficiency. With regard to spherical cap converters of different dimensions, the measured values of energy increment factor coincide well with the theoretical equivalents, indicating an effective verification of the proposed analytical model for the spherical cap converter topology.

Abstract:
In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers $Q_1=(-\chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $\mu\ll 1$. They interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlev\'{e} conjecture.