非线性抛物组非均匀网格差分解的唯一性和稳定性

１．引言 １．对一维非线性抛物组,在文献山中已构造一般非均匀网格差分格式,其中差分逼近的组合系数对不同的网格点和不同的网格层可以不同,并且运用不动点原理证明了差分解的存在性和收敛性．在非均匀网格差分格式中差分逼近的组合系数为常数的情形,文献［２］证明了具有有界二阶差商的离散向量解的存在性、唯一性和稳定性．本文将对文献［１］中构造的一般非均匀网格差分格式,证明所得到的差分解的唯一性和稳定性． 考虑如下非线性抛物组其中是未知的ｍ－维向量函数是给定的矩阵函数,ｊ（ｘ,ｔ,ｕ,ｐ）。是给定的ｍ－维向量函数…     

Generalized Transvectants-Rankin–Cohen Brackets

We introduce o(p+1q+1)-invariant bilinear differential operators on the space of tensor densities on Rⁿ generalizing the well-known bilinear sl₂-invariant differential operators in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear o(p+1q+1)-invariant differential operators given by powers of the Laplacian.     

Spin and Pseudospin Symmetries of Dirac Equation and the Yukawa Potential as the Tensor Interaction

Considering of a tensor interaction in Dirac equation removes the degeneracy in spin and pseudospin doublets and consequently leads to results consistent with the experimental data. Here, instead of the commonly used Coulomb or linear terms, we investigate a tensor interaction of Yukawa form. We obtain arbitrary state solutions of Dirac equation under vector, scalar and tensor Yukawa potentials via a physical approximation and the Nikiforov-Uvarov methodology. The solutions are discussed in detail.     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Existence and multiplicity of positive solutions for a system of difference equations with coupled boundary conditions

We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to coupled multi-point boundary conditions.     

Dynamics of a higher-order rational difference equation

We consider a higher order rational difference equation. Firstly, we skillfully give a sufficient and necessary condition for the existence and uniqueness of the initial value problem. And then we investigate the local stability, asymptotic behavior, periodicity and oscillation of solutions for the difference equation. Finally, we give some numerical simulations to illustrate our results.     

荧光褪色效应与拉曼光谱基线干扰消除

荧光是直接测定的拉曼光谱中背景的最主要来源,需要采用真实、准确的方法消除,以得到纯净的拉曼响应。基线拟合消除和查找荧光贡献扣除是解决背景问题的两条思路,目前多采用基线拟合方法,其优点是满足用户“视觉”要求,无需额外硬件,但并非机理或实质上的解释,因而难以保证数据的真实性与合理性;查找荧光的方法,更为真实,但是目前提出的方法,需要增加光源等额外设计和成本。另外,在实验方法上,也有采用消荧光剂和长时间照射漂白的,存在操作繁琐、效率低等不足。利用稳定体系中拉曼和荧光的时间差异解决体系中荧光问题。在微小的时间段内,例如几个毫秒,激发光不会导致体系性质发生显著变化,荧光具有寿命周期,会随激发时间延长强度下降的“褪色”,“褪色”的强度差异可以被认为是整体荧光的一个微元;与此同时,由于体系组成未发生显著变化,拉曼光对于短时间照射可以保持稳定。利用此差异可以区分出混合信号中的荧光和拉曼光。根据该原理,提出了荧光褪色差分法(FBDA),实现拉曼光谱的背景校正。方法的主要步骤：测量微小时刻内的多张直接拉曼光谱,求取系列光谱的差分,对差分值作高频滤波降噪,可获得荧光强度微元;然后,多个荧光微元平均归一化后,得到荧光强度单元。以拉曼光谱2 000～2 500 cm-1的静默区,即通常不会出现拉曼信号的频段为基准,对荧光单元作逆差分,逆差分累计值与原始光谱在此频段一致时,得到整体荧光响应;最终,从原始光谱中扣除荧光成分,完成背景扣除和基线校正。以盐酸二甲双胍片的拉曼测量为例,说明和讲解了所提出的原理和方法,验证方法的有效性。与目前效果较好的基线校正方法(不对称最小二乘和自适应迭代再加权惩罚偏最小二乘)进行了对比,表明FBDA方法更为客观真实,FBDA的另一个优势是不需要额外的设计和成本,所有数据都是在现有设备直接采集和完成。需要说明的是,微小时刻光谱差异的要求,可以确保FBDA光谱实时性,长时间的光谱差异,将会影响结果的准确性;另外,对于光化学反应体系和其他非荧光引起的复杂背景,FBDA的适用性有待改善。     

The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra of dimension attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra and a primitive root of unity q of order n² is equivalent to Lusztig's small quantum group (under some conditions on n). We also give a conceptual construction of using the notion of de-equivariantization of tensor categories.     

-Difference sequence spaces of fuzzy numbers

In this paper, using the difference operator of order m and an Orlicz function, we introduce and examine some classes of sequences of fuzzy numbers. We give the relations between the strongly Cesàro type convergence and statistical convergence in these spaces. Furthermore, we study some of their properties like completeness, solidity, symmetricity, etc. We also give some inclusion relations related to these classes.