The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.

Based on a concept of asymptotic exponential arbitrage proposed by Föllmer-Schachermayer, the author introduces a new formulation of asymptotic arbitrage with two main differences from the previous one: Firstly, the realising strategy does not depend on the maturity time while the previous one does, and secondly, the probable maximum loss is allowed to be small constant instead of a decreasing function of time. The main result gives a sufficient condition on stock prices for the existence of such asymptotic arbitrage. As a consequence, she gives a new proof of a conjecture of Föllmer and Schachermayer.

In this paper, the author studies the existence and uniqueness of discrete pseudo asymptotically periodic solutions for nonlinear Volterra difference equations of convolution type, where the nonlinear perturbation is considered as Lipschitz condition or non-Lipschitz case, respectively. The results are a consequence of application of different fixed point theorems, namely, the contraction mapping principle, the Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.

In this paper the forward asymptotical behavior of non-autonomous dynamical systems and their attractors are investigated. Under general conditions, the authors show that every neighborhood of pullback attractor has forward attracting property.

Let t ≥ 2 be an integer, and let p ₁, ⋯, p _t be distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring ${Z_{{p_1}{p_2} \cdots {p_t}}}$ and the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over ${Z_{{p_1}{p_2} \cdots {p_t}}}$.

The aim of this paper is to study the operator (dd ^c▪) ^q ∧ T on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where T is a positive closed current of bidimension (q, q) on an open set Ω of ℂ ⁿ. The author introduces two classes ${\cal F}_p^T\left( {\rm{\Omega }} \right)$ and ${\cal E}_p^T\left( {\rm{\Omega }} \right)$ and shows first that they belong to the domain of definition of the operator (dd ^c▪) ^q ∧ T. Then the author proves that all functions that belong to these classes are C _T-quasi-continuous and that the comparison principle is valid for them.

Let (X, d, μ) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of Hytönen. In this paper, the authors obtain the boundedness of the commutators of θ-type Calderón-Zygmund operators with RBMO functions from L ^∞ (μ) into RBMO(μ) and from $H_{{\rm{at}}}^{1,\;\infty }\left( \mu \right)$ into L ¹ (μ), respectively. As a consequence of these results, they establish the L ^p (μ) boundedness of the commutators on the non-homogeneous metric spaces.

If V is an irreducible quasi-Kähler complex variety and E is a vector bundle over reg(V), the author proves that W ₀ ^1,2(reg(V), E) = W^1,2(reg(V), E), and that for dim_ℂ reg(V) > 1, the natural inclusion W^1,2(reg(V), E) ↪ L ²(reg(V), E) is compact, the natural inclusion ${W^{1,2}}\left( {{\rm{reg}}\left( V \right),\;E} \right)\hookrightarrow {L^{{{2v} \over {v - 1}}}}\left( {{\rm{reg}}\left( V \right),\;E} \right)$ is continuous.

Let G be a finite p-group with a cyclic Frattini subgroup. In this paper, the automorphism group of G is determined.