We discuss automorphisms of infinite metacyclic groups and present their automorphism groups, and then we prove the following result: All automorphisms of an infinite metacyclic group G are inner automorphisms if and only if G is isomorphism to one of following groups: (i) Z∝_ξ Z_pl. where p is an odd prime, l is an integer, ξ maps the generator 1 of Z to ξ(1) = m, p m, and Z_pl=Z_pl^*=Z_p(l-1)(p-1)=<m>; (ii) The infinite dihedral group D∞.

In this article the author considers the contact process on the product graph of a homogeneous tree with degree d≥q3 and an arbitrary finite connected simple graph, showing that an intermediate phase for weak survival exists.

We will study the condition of blow-up for the generalized hyperelastic-rod equation. The solution blowing up in finite time lies in the scope of minimal initial velocity and the maximal variation of intial date and generalized functional G(μ).

Let (M,T) be a closed and smooth manifold with an involution T. The fixed point set of T is F. In this paper, we determine bordism classes of the involutions with fixed point set F=∪^m_i=1 RP_i(3)×HP_i(k), where HP(k) is the k-dimensional quaternionic projective space, and k=2^λ-2 or k=2^λ-1 (λ>1).

In this note, using the spectral theory of linear operators, a spectral representation of infimum of two self-adjoint operators with respect to the logic order is established.

We classify a class of Randers metrics of isotropic S-curvature by navigation problem. This class includes all Randers metrics of isotropic S-curvature and scalar flag curvature. In particular, all 2-dimensional Randers metrics of isotropic S-curvature are classified. Besides, we construct for the first time many Randers metrics of constant S-curvature with other special curvature properties.

Sequential quadratic programming (SQP) type method is one of the most effective methods for solving nonlinear programming. Recently, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems. In this paper, a new modified approach combined the filter technique and SQP trust region method is proposed to tackle the original problem, which ensures that every trail point will not be far away from the feasible region. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.

The present paper give an estimation of the upper and lower bounds of the (lower) q-order and the (lower) q-type of a new product function defined by two Dirichlet series, named Dirichlet--Hadamard product function. It is of ρ_[q]-regular growth or perfectly ρ_[q]-regular growth, when two constituent Dirichlet series satisfy some special conditions. Finally we give an example and generalize a result concerning the subject of linear substitution.

We introduced the iterated maximal operators on martingale space and established some weighted Kolmogorov inequality and weighted double Φ-inequality for this kind of operators.

In this paper, the theorem of cone and the techniques of partial order theory are used to discuss the existence of coupled quasi-solutions and solutions for a mixed monotone operator equation Lx = N(x,y) in complete metric space and Banach space, where neither L nor N need to be continuous. We also construct some new non-symmetric iterative sequences and study their approximation, then we get some new theorems. Finally, we apply the new results presented in this paper to Hammerstain integral equations.

We study some twist second order differential equations. We prove that in some cases, the geometric property of angle function implies the existence of periodic solutions. As the applications, we obtain some new results for the existence of periodic solutions for some second order equations or reversible systems without a priori bounds, such as superlinear second order equations or reversible systems with time-dependent potential.

Using the tools of Hausdorff dimension and Hausdorff measure, we give quantitative version for the set of admission kneading sequences with product metric. It is proved for the set that the Hausdorff dimension is s and the s-dimension Hausdorff measure is zero in Σ₂, which are more profound than those obtained by traditional qualitative analysis.

In the paper stated here, Hausdorff convergence is introduced to discuss the differentiable sphere theorem with positive Ricci curvature. Finally, a type of rigidity phenomenon on Riemannian manifolds will be derived.

Let (F(S),ρ₃) be the three-valued logic metric space. The present paper characterizes consistency, full divergency and logical closedness of theories in propositional logic system L^*₃ by means of topological concepts in the three-valued logic metric space. It is proved that a closed theory Γ is consistent iff it contains no interior points, iff it possesses the truth-forget property, iff it contains no non-empty regular sphere. It is also proved that a theory Γ is fully divergent iff its logic closure is dense in (F(S),ρ₃). Finally, it is proved that the logic closure of finite theory is closed in (F(S),ρ₃).

In this paper, the definition of path algebras of finite automata is given, the relations of characteristics of finite automata and algebraic properties of their path algebra are studied, and some results are given.