Generalized KKM Type Theorems in FC-Spaces with Applications (I)

The class KKM(X,Y) (resp., s-KKM(X,Y,Z)) of set-valued mappings with KKM (resp., s-KKM) property is introduced in FC-spaces without any convexity structure. Some generalized KKM (resp., s-KKM) type theorems are proved in FC-spaces under much weak assumptions. As applications, some new section theorems and coincidence theorems are established in FC-spaces. These theorems generalize many known results in literature. The further applications of these results will be given in a follow-up paper.     

DBHF方法和热力学自洽性

采用Dirac Brueckner-Hartree-Fock理论方法, 计算了零温核物质中每核子的结合能、压强和单核子能量, 着重讨论了不同的T矩阵协变表示对核物质中Hugenholtz-Van Hove(HVH)定理满足程度的影响. 结果表明: 不同的协变表示对核子自能各分量的动量相关性和密度依赖性均有重要影响, 进而对核介质中HVH定理的满足程度产生重要影响. 在完全的膺矢量表示下, HVH定理遭到了相当大程度的破坏, 从而体现出基态关联效应对单核子性质的重要性, 并与非相对论BHF理论方法得到的结论一致, 因而完全的膺矢量表示要优于膺标量表示.     

Continuous spectral decompositions of Abelian group actions on -algebras

Let G be a locally compact Abelian group. Following Ruy Exel, we view Fell bundles over the Pontrjagin dual group of G as continuous spectral decompositions of G-actions on C^*-algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel's theory of square-integrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C^*-algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions.     

Further generalization of the Dubovicki-Milutin theorem

Combining the results of Walczak (Ref. 1) and Lasiecka (Refs. 2 and 3), we extend the well-known Dubovicki-Milutin theorem in two directions: (i) We allow for treatment of several equality constraints in optimization problems; and (ii) We assume less regularity of optimization problems by requiring the existence of external cones instead of tangent cones. An example of an optimization problem is also considered.     

Representations, convolution square roots and spherical vectors

We extend results on square integrable representations of a locally compact group to subrepresentations of a representations induced from a character of a closed subgroup. We do so working in the framework of quotient representations of ∗-algebras.     

Quadratic transformations and Guillera’s formulas for 1/π2

We prove two new series of Ramanujan type for 1/π².     

On a generalization of Pólya’s theorem

We consider generalizations of Pólya’s theorem to the case of entire functions taking algebraic values at natural or integer points.     

非自治时滞微分方程的正周期解

By utilizing a fixed point theorem on cone,some new results on the existence of positive periodic solutions for nonautonomous differential equations with delay are derived.     

Concentration inequalities for random fields via coupling

We present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields. Our method is based on coupling and does not use information inequalities. In case one has a uniform control on the coupling, one obtains exponential concentration inequalities. If such a uniform control is no more possible, then one obtains polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, both at high and low temperatures and in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.