Finite dimensional Hopf superalgebras u_q(sl(m|n)) corresponding to the Lie superalgebras sl(m|n) are constructed. The PBW type basis and the left and right integrals of u_q(sl(m|n)) are obtained. Furthermore, the group of Hopf superalgebra automorphisms is described.

In this paper, the Bayes estimator of the error variance is derived in a linear regression model, and the parametric empirical Bayes estimator (PEBE) is constructed. The superiority of the PEBE over the least squares estimator (LSE) is investigated under the mean square error (MSE) criterion. Finally, some simulation results for the PEBE are obtained.

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.

In this paper, we give a uniqueness theorem for meromorphic mappings from ?n into ?N(?) with rank≥μregardless of multiplicities.

We define an analogue of the Caldero-Chapoton map for the cluster category of finite-dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under this map). Moreover, we construct a ?-basis for the algebra generated by all generalized cluster variables.

In this paper, we use the constancy of certain subspace valued mappings on the components of the generalized Kato resolvent set and the equivalences of the single-valued extension property at a point 0 for operators which admit a generalized Kato decomposition to obtain a classification of the components of the generalized Kato resolvent set of operators. We also give some applications of these results.

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is called c^?-quasinormally embedded in G if there is a subgroup T of G such that G = HT and H∩T is s-quasinormally embedded in G.We investigate the influence of c^?-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.

In this paper, we study the evolution of hypersurfaces by powers of mean curvature minus an external force field. We prove that when the power is 2, the flow has a long-time smooth solution for all time under some conditions. Those conditions are that the second fundamental form on the initial submanifolds is not too large, the external force field, with its any order derivatives, is bounded, and the field is convex with its eigenvalues satisfying a pinch inequality.

A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.

Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign (or a root of unity) ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign (root of unity) ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established.

In this paper, we verify Vogan’s conjecture on quantization in the representation theory for G = GL(n, ?). Also we get some relationship between the induction of orbits and Howe’s θ-lifting of unitary representations.

In this paper, we consider the dividend optimization problem for a financial corporation with transaction costs. Besides the dividend control, the financial corporation takes proportional reinsurance to reduce risk and the surplus earns interest at the constant force ρ>0. Because of the presence of fixed transaction costs, the problem becomes a mixed classical-impulse stochastic control problem. We solve this problem explicitly and construct the value function together with the optimal policy.