The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

The authors introduce an effective method to construct the rational function sheaf $\mathcal{K}$ on an elliptic curve $\mathbb{E}$, and further study the relationship between $\mathcal{K}$ and any coherent sheaf on $\mathbb{E}$. Finally, it is shown that the category of all coherent sheaves of finite length on $\mathbb{E}$ is completely characterized by $\mathcal{K}$.

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems $\dot z$(t) = J▽H(t, z(t)) with Lagrangian boundary conditions, where is a semipositive symmetric continuous matrix and satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.

The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.

The authors consider a differentiable manifold with Π-structure which is an isomorphic representation of an associative, commutative and unitial algebra. For Riemannian metric tensor fields, the Φ-operators associated with r-regular Π-structure are introduced. With the help of Φ-operators, the hyperholomorphity condition of B-manifolds is established.

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.

The authors explore a class of jump type Cahn-Hilliard equations with fractional noises. The jump component is described by a (pure jump) Lévy space-time white noise. A fixed point scheme is used to investigate the existence of a unique local mild solution under some appropriate assumptions on coefficients.

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kähler manifolds (e.g. Kähler hyperbolic manifolds, bounded domains of holomorphy) by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.

Let p denote a prime and P ₂ denote an almost prime with at most two prime factors. The author proves that for sufficiently large x, where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampére equations on compact Kähler manifolds.