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In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincaré map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.

By the new spectrum originated from the single-valued extension property, we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property (ω) holds. Meanwhile, the relationship between hypercyclic property (or supercyclic property) and property (ω) is discussed.

Let Λ ⊂ ℝn be a uniformly discrete set and let CΛ be the vector space consisting of all mean periodic functions whose spectrum is simple and contained in Λ. If Λ is a gentle set then for every f ∈ CΛ we have f(x) = O(ωΛ(x)) as |x| → ∞ and ωΛ(x) can be estimated (Theorem 4.1). This line of research was proposed by Jean-Pierre Kahane in 1957.

The growth of meromorphic solutions of linear difference equations containing Askey—Wilson divided difference operators is estimated. The ϕ-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order ρlog (f) and the classical order ρ(f) of a meromorphic function f.

We prove several existence and uniqueness results for L p (p > 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.

It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero k flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet’s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.

In author’s one previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birth-death processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the corresponding constants in the inequalities are presented. As typical applications, the Nash inequalities and logarithmic Sobolev inequalities are examined.

This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs.