A new dimension-reduction graphical method for testing highdimensional normality is developed by using the theory of spherical distributions and the idea of principal component analysis. The dimension reduction is realized by projecting high-dimensional data onto some selected eigenvector directions. The asymptotic statistical independence of the plotting functions on the selected eigenvector directions provides the principle for the new plot. A departure from multivariate normality of the raw data could be captured by at least one plot on the selected eigenvector direction. Acceptance regions associated with the plots are provided to enhance interpretability of the plots. Monte Carlo studies and an illustrative example show that the proposed graphical method has competitive power performance and improves the existing graphical method significantly in testing high-dimensional normality.

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.

Copula method has been widely applied to model the correlation among underlying assets in financial market. In this paper, we propose to use the multivariate Fréchet copula family presented in J. P. Yang et al. [Insurance Math. Econom., 2009, 45: 139–147] to price multivariate financial instruments whose payoffs depend on the k^th realization of the underlying assets and collateralized debt obligation (CDO). The advantage of the multivariate Fréchet copula is discussed. Empirical study shows that such copula family gives a better fitting to CDO’s market price than Gaussian copula for some derivatives.

We construct fermionic-bosonic representations for a class of generalized B(m, n), C(n), D(m, n)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions.

Given α ∈[0, 1], let h_α(z) := z/(1 − αz), z ∈ D := {z ∈ C: |z| <1}. An analytic standardly normalized function f in D is called close-to-convex with respect to hα if there exists δ ∈ (−π/2, π/2) such that Re{e^iδzf′(z)/h_α(z)} >0, z ∈ D. For the class C(h_α) of all close-to-convex functions with respect to h_α, the Fekete-Szegö problem is studied.

A discrete three-dimensional three wave interaction equation with self-consistent sources is constructed using the source generation procedure. The algebraic structure of the resulting fully discrete system is clarified by presenting its discrete Gram-type determinant solution. It is shown that the discrete three-dimensional three wave interaction equation with self-consistent sources has a continuum limit into the three-dimensional three wave interaction equation with self-consistent sources.

We obtain the global smooth solution of a nonlinear Schrödinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let ℱ denote the set of all simple 2-edge-connected graphs on n≥4 vertices such that G ∈ ℱ if and only if d(e) + d(e')≥2n for every pair of independent edges e, e' of G. We prove in this paper that for each G ∈ ℱ, G is not Z₃-connected if and only if G is one of K₂,_n−2, K₃,_n−3, K⁺₂,_n−2, K⁺₃,_n−3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].

This article studies the Floer theory of Landau-Ginzburg (LG) model on ℂⁿ: We perturb the Kähler form within a xed Kähler class to guarantee the transversal intersection of Lefschetz thimbles. The C⁰ estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/AG of G is called a G-boundary factor of A. For any Gboundary factor H/AG of A, the subgroup (A ∩ H)/AG of G/AG is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p?-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.

We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p>2, −Δpu=f(u) in ?2m for all dimensions satisfying 2m≥p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m≥2p.

This paper is concerned with numerical simulations for the GBrownian motion (defined by S. Peng in Stochastic Analysis and Applications, 2007, 541–567). By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman (HJB) equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations (SDEs)/stochastic partial differential equations (SPDEs) driven by the G-Brownian motions.