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
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
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
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.
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
The investigation of
We prove that almost all integers
We introduce the class of strongly close-to-convex mappings of order
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
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.