The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.

The Rayleigh-Bénard convection is a classical problem in fluid dynamics. In this paper, we are concerned with the well-posedness for the compressible Rayleigh-Bénard convection in a bounded domain Ω ? ?². We prove the local well-posedness of the system with appropriate initial data. This is the result concerning compressible Rayleigh-Bénard convection, before only results about incompressible Rayleigh-Bénard convection were done.

We discuss continuity of the Poisson transform on Herz spaces B^p as well as its action on weighted versions of these sets. We also consider Banachvalued versions of Herz spaces and study some of their properties.

Determining whether a quantum state is separable or inseparable (entangled) is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47: 777] and Schr?odinger [Naturwissenschaften, 1935, 23: 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method (SAM) for this problem, which approximates a given quantum state by a so-called separable state: if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.

A finite group G is called an I N I-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-I N I-groups in which all proper subgroups are I N I-groups.

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang’s results.

The conditional independence structure of a common probability measure is a structural model. In this paper, we solve an open problem posed by Studeny [Probabilistic Conditional Independence Structures, Theme 9, p. 206]. A new approach is proposed to decompose a directed acyclic graph and its optimal properties are studied. We interpret this approach from the perspective of the decomposition of the corresponding conditional independence model and provide an algorithm for identifying the maximal prime subgraphs in a directed acyclic graph.

A total [k]-coloring of a graph G is a mapping ?: V (G) ∪E(G) → {1,2, ..., k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv ∈E(G),f(u)≠f(v). By χnsd″(G), we denote the smallest value k in such a coloring of G. Pil?niak and Wo?niak conjectured χnsd″(G)≤?(G)+3 for any simple graph with maximum degree Δ(G). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for K4-minor free graphs. Furthermore, we show that if G is a K4-minor free graph with ?(G)≥4, then χnsd″(G)≤?(G)+2 The bound Δ(G) + 2 is sharp.

The notion of a tilting pair over artin algebras was introduced by Miyashita in 2001. It is a useful tool in the tilting theory. Approximations and cotorsion pairs related to a fixed tilting pair were discussed. A contravariantly (covariantly) finite subcategory and a cotorsion pair associated with a fixed tilting pair were given in this paper.

We prove the H⁴-boundedness of the pullback attractor for a twodimensional non-autonomous non-Newtonian fluid in bounded domains.

The investigation of U-ample ω-semigroups is initiated. After obtaining some properties of such semigroups, a structure of U-ample ω-semigroups is established. It is proved that a semigroup is a U-ample ω-semigroup if and only if it can be expressed by WBR(T, θ), namely, the weakly Bruck-Reilly extensions of a monoid T. This result not only extends and amplifies the structure theorem of bisimple inverse ω-semigroups given by N. R. Reilly, but also generalizes the structure theorem of ?-bisimple type A ω-semigroups given by U. Asibong-Ibe in 1985.

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of almost equal fourth prime powers.

We introduce the class of strongly close-to-convex mappings of order α in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in ?n. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.

We investigate a model arising from biology, which is a hyperbolicparabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the H^s∩L¹-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L² decay rate of the solution and the almost optimal L² decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.

We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities.