On the 2- and 4-dissections of the Rogers–Ramanujan functions

Basil Gordon showed that if the two Rogers–Ramanujan functions are expanded as series, in each case half the coefficients are almost always even. In order to do this, he gave formulae modulo 2 for the 2-dissections of each of the Rogers–Ramanujan functions. In this paper we give the exact 2- and 4-dissections. We then show how our 2-dissections lead to Gordon’s formulae, we go on to give examples of what he calls “linear zero congruences modulo 2” , and finally we give a simple proof of Gordon’s main result that \(g(2n+1)\) and \(h(2n)\) are almost always even.     

Graded Limits of Minimal Affinizations over the Quantum Loop Algebra of Type <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

The aim of this paper is to study the graded limits of minimal affinizations over the quantum loop algebra of type G ₂. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and obtain defining relations of them. As an application, we obtain a polyhedral multiplicity formula for the decomposition of minimal affinizations of type G ₂ as a \(U_{q}(\mathfrak {g})\)-module, by showing the corresponding formula for the graded limits. As another application, we prove a character formula of the least affinizations of generic parabolic Verma modules of type G ₂ conjectured by Mukhin and Young.     

Generalized Skew Derivations Cocentralizing Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are generalized skew derivations of R and \({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over C with n non-commuting variables. If F and G satisfy the following condition:

$$F(f(r_1,\ldots, r_n))f(r_1, \ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))\in C$$

for all \({r_1, \ldots, r_n \in R}\), then we describe all possible forms of F and G.

     

Pricing <Emphasis Type="Italic">k</Emphasis><Superscript>th</Superscript> realization derivatives and collateralized debt obligation with multivariate Fréchet copula

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.     

Exponential family mixed membership models for soft clustering of multivariate data

For several years, model-based clustering methods have successfully tackled many of the challenges presented by data-analysts. However, as the scope of data analysis has evolved, some problems may be beyond the standard mixture model framework. One such problem is when observations in a dataset come from overlapping clusters, whereby different clusters will possess similar parameters for multiple variables. In this setting, mixed membership models, a soft clustering approach whereby observations are not restricted to single cluster membership, have proved to be an effective tool. In this paper, a method for fitting mixed membership models to data generated by a member of an exponential family is outlined. The method is applied to count data obtained from an ultra running competition, and compared with a standard mixture model approach.     

悬链线模型在系泊系统设计中的应用--2016年全国大学生数学建模竞赛A题解答评述

通过力和力矩的平衡、悬链线方程等给出2016年全国大学生数学建模竞赛A题的一种解答,并对学生答卷的不同方法和结果进行简单的点评。     

一类拟线性椭圆方程的非平凡解

运用环绕理论和对称型山路理论对一类具有次临界多项式增长和次临界指数增长的$p$-Laplacian方程建立一个非平凡解(无穷多个非平凡解)的存在性结果.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

围长至少是7的1-平面图是$(1,1,1,0)$-可染的

一个图称为是1-平面图的, 如果它可以画在一个平面上使得它的每条边最多交叉另外一条边.本文证明了围长大于等于7的1-平面图是$(1,1,1,0)$-可染的.     

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This paper considers a dividend strategy with investment in Omega model. If at a potential dividend-payment time the surplus is above, part of the excess are paid as dividends directly, the other part are used as dynamic investment capital, at a particular time, the sum of profits and investment capital will be paid as another dividend. Under this dividend policy, we get the optimal dividend strategy and the optimal portfolio policy.