We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi- Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.

We first propose a new class of smoothing functions for the nonlinear complementarity function which contains the well-known Chen-Harker- Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P₀ nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.

Weak log-Sobolev and L^p weak Poincaré inequalities for general symmetric forms are investigated by using newly defined Cheeger’s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.

This paper is a further contribution to the classification of linetransitive finite linear spaces. We prove that if φ is a non-trivial finite linear space such that the Fang-Li parameter gcd(k, r) is 9 or 10, and the group G≤Aut(φ) is line-transitive and point-imprimitive, then φ is the Desarguesian projective plane PG(2, 9).

We consider the ?n-Galois covering Λn of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872-1893]. We calculate the dimensions of all Hochschild cohomology groups of Λn and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg’s conjecture.

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CPⁿ. Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CPⁿ.

This work is mainly concerned with the rotating Newtonian stars with prescribed angular velocity law. For general compressible fluids, the existence of rotating star solutions was proved by using concentrationcompactness principle. In this paper, we establish the asymptotic estimates on the diameters of the stars with small rotation. The novelty of this paper is that a direct and concise definition of slowly rotating stars is given, different from the case with given angular momentum law, and the most general fluids are considered.

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.

Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω?. Let Ω denote the even part of the Lie superalgebra Ω?.We first give the generator sets of the Lie algebra Ω. Then, we reduce the derivation of Ω to a certain form. With the reduced derivation and a torus of Ω, we determine the derivation algebra of Ω.

Let w∈A∞. In this paper, we introduce weighted-(p, q) atomic Hardy spaces Hwp,q(?n×?m) for 0?p≤1, q?qw and show that the weighted Hardy space Hwp,q(?n×?m) defined via Littlewood-Paley square functions coincides with Hwp,q(?n×?m) for 0?p≤1, q?qw. As applications, we get a general principle on the Hwp,q(?n×?m) to Lwp,q(?n×?m) boundedness and a boundedness criterion for two parameter singular integrals on the weighted Hardy spaces.

In this paper, we consider the lattice Schr¨odinger equations iq ˙n(t)=tan?π(nα+x)qn(t)+?(qn+1(t)+qn-1(t))+δυn(t)|qn(t)|2τ-2qn(t), with α satisfying a certain Diophantine condition, x∈?/?, and τ = 1 or 2, where υn(t) is a spatial localized real bounded potential satisfying |υn(t)|≤Ce-ρ|n|. We prove that the growth of H¹ norm of the solution {qn(t)}n∈? is at most logarithmic if the initial data {qn(0)}n∈?∈H1 for ? sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e., iq ˙n(t)=tan?π(nα+x)qn(t)+?(qn+1(t)+qn-1(t))+δυn(θ0+tw)qn(t). Then the linear equation can be reduced to an autonomous equation for a.e. x and most values of the frequency vectors ω if ? and δ are sufficiently small.