一类深度1的可迁模Lie超代数

建立了满足如下条件的可迁$\mathbb{Z}$-分次模Lie超代数$\frak{g}=\oplus_{-1\leq i\leq r}\frak{g}_{i}$的嵌入定理:(i) $\frak{g}_{0}\simeq \widetilde{\mathrm{p}}(\frak{g}_{-1}) $ 并且$\frak{g}_{0}$-模 $\frak{g}_{-1}$ 同构于$\widetilde{\mathrm{p}}(\frak{g}_{-1})$的自然模;(ii) $\dim \frak{g}_1=\frac 23 n(2n^2+1),$ 其中 $n=\frac{1}{2} \dim \frak{g}_{-1}.$特别地, 证明了满足上述条件的有限维单模Lie超代数同构于奇Hamilton模Lie超代数.对局限Lie超代数也做了相应的讨论.     

非齐性广义Morrey空间上双线性强奇异Calder\''{o}n-Zygmund算子及交换子

本文的主要建立非齐性度量测度空间上双线性强奇异积分算子$\widetilde{T}$及交换子$\widetilde{T}_{b_{1},b_{2}}$在广义Morrey空间$M^{u}_{p}(\mu)$上的有界性. 在假设Lebesgue可测函数$u, u_{1}, u_{2}\in\mathbb{W}_{\tau}$, $u_{1}u_{2}=u$,且$\tau\in(0,2)$. 证明了算子$\widetilde{T}$是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到空间$M^{u}_{p}(\mu)$有界的, 也是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到广义弱Morrey空间$WM^{u}_{p}(\mu)$有界的,其中$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$及$1相似文献   

正则函数的修正的Cauchy型积分算子的不动点定理及迭代构造

本文第一部分讨论了正则函数的{\small Cauchy}型积分算子$T[f]$的{\small H\"{o}lder}连续性及此积分算子$T[f]$的范数与$f$的范数之间的关系.第二部分引入了修正的Cauchy型积分算子$\small \widetilde{T}$,首先利用压缩映射原理证明了$\small \widetilde{T}$算子具有不动点,然后给出了其不动点的迭代序列并证明了此序列强收敛于$\small \widetilde{T}$算子的不动点.     

与群PSL2(\mathcal{R})相关的交叉积{\mathcal{R}(\mathcal{A}},α)的一点注记

在上半复平面$\mathbb{H}$上给定双曲测度$dxdy/y^{2}$, 群$G={\rm PSL}_{2}(\mathbb{R})$ 在$\mathbb{H}$上的分式线性作用导出了$G$在Hilbert空间$L^{2}(\mathbb{H}, dxdy/y^{2})$上的酉表示$\alpha$. 证明了交叉积 $\mathcal{R}(\mathcal{A}, \alpha)$是$\mathrm{I}$型von Neumann代数, 其中$\mathcal{A}= \{M_{f}:f\in L^{\infty}(\mathbb{H},dxdy/y^{2} )\}$. 具体地, 交叉积代数$\mathcal{R}(\mathcal{A}, \alpha)$与von Neumann代数$\mathcal{B}(L^{2}(P, \nu))\overline{\otimes}\mathcal{L}_{K}$是*-同构的, 其中$\mathcal{L}_{K}$是$G$中子群 $K$的左正则表示生成的群von Neumann代数.     

在区间截断的情况下,两点分布估计及其收敛速度

假设总体$X$服从两点均匀分布, 即$\pr(X=x_1)=\pr(X=x_2)=1/2$, 但是随机变量$X$的取值$x_1$和$x_2$是未知的\bd 在区间截断的情况下, 利用样本获得了$x_1$和$x_2$估计量$\wh{x}_1$和$\wh{x}_2$, 并给出了估计量$\wh{x}_1$和$\wh{x}_2$的收敛速度$o(n^{-1/3+\xs})$.     

算子AB和BA的Drazin可逆性

给定Hilbert空间${\cal H}$上的有界线性算子$A$和$B$, 本文证明了$AB$和$BA$的Drazin可逆性是等价的. 作为应用, 我们证明了$\sigma_D(AB)=\sigma_D(BA)$和$\sigma_D(A)=\sigma_D(\widetilde{A})$,这里$\sigma_D(M)$和$\widetilde{M}$分别表示算子$M$的Drazin谱和Aluthge变换.     

带有数值条件的丰富向量丛和高维射影流形的结构

$X$是复数域上的$n$维光滑射影簇$(n \ge 3)$, $K_X $是$X$的典范丛, $E$是$X$上秩为$n - k$的丰富向量丛$(k \ge 0)$．$c_1 (E)$表示$E$的第1陈类, $\Omega $表示$X$的满足$(K_X + c_1 (E)) \cdot R \le 0$的极端半线$R ={\R_+} [C]$的集合, $\R_+$是正实数集．$\ell (R)$表示$R$的长度．定义 $ \Lambda (E,K_X ) = \max \{( - K_X - c_1 (E)) \cdot C|R ={\R_+} [C] \in \Omega ,\,\mbox{且}\,\ell (R) = - K_X \cdot C\}.$ 如果 $\Lambda (E,K_X ) \ge k$, 那么 $(X,E)$是以下五者之一: (i) $(X,E) \cong (P^n,O_{P^n} (1)^{ \oplus (n - k)}), $ (ii) $(X,E) \cong (P^n,O_{P^n} (2) \oplus O_{P^n} (1)^{ \oplus (n - k - 1)}),$ (iii) $(X,E) \cong (P^n,T_{P^n} ),$ (iv) $(X,E) \cong (Q^n,O_{Q^n} (1)^{ \oplus (n - k)}), $ (v) $(X,E)$是一条光滑曲线$Y$上的涡卷, 即$X$是$Y$上的线性$P^{n - 1}$丛, $g:X \to Y,$且对$g$的每个纤维$F$有$(F,\left. E \right|_F ) \cong (P^{n - 1},O_{P^{n - 1}} (1)^{ \oplus (n - k)})$． 这里$Q^n$是$n + 1$维射影空间$P^{n + 1}$中的超二次曲面．     

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

连续函数之集在上半连续函数之集中的一种拓扑位置

设$(X,\rho)$是一个度量空间. 用$\dd {\rm USCC}(X)$和$\dd {\rm CC}(X)$ 分别表示从$X$ 到 $\I=[0,1]$的紧支撑的上半连续函数和紧支撑的连续函数下方图形全体. 赋予 Hausdorff 度量后, 它们是拓扑空间. 文中证明了, 如果 $X$ 是一个无限的且孤立点集稠密的紧度量空间, 则 $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx(Q,c_0\cup (Q\setminus \Sigma))$, 即存在一个同胚 $h:~\dd {\rm USCC}(X)\to Q$, 使得 $h(\dd {\rm CC}(X))=c_0\cup (Q\setminus \Sigma)$, 这里 $Q=[-1,1]^{\omega},\,\Sigma=\{(x_n)_{n}\in Q: {\rm sup}|x_n|<1\},\, c_0=\Big\{(x_n)_{n}\in \Sigma: \lim\limits_{n\to +\infty}x_n=0\Big\}.$ 结合这个论断和另一篇文章的结果, 可以得到: 如果 $X$ 是一个无限的紧度量空间, 则 $(\uscc(X), \cc(X))\approx \left\{ \begin{array}{ll} (Q,c_0\cup (Q\setminus \Sigma)), &;\quad \text{如 果 孤 立 点 集 在} X \text{中稠密},\\ (Q, c_0), &;\quad \text{ 其他}. \end{array} \right.$ 还证明了, 对一个度量空间$X$, $(\dd {\rm USCC}(X),\dd {\rm CC}(X))\approx (\Sigma,c_0)$ 当且仅当 $X$是一个非紧的、局部紧的、非离散的可分空间.     

球覆盖性质不是同胚不变的

Banach空间$X$中的一个开球族$\ss$是$X$的球覆盖, 如果$B$中的任一元素不包含原点,且$B$中元素之并覆盖了$X$的单位球面$S_{X}$. Banach空间$X$称为具有球覆盖性质,如果$X$有一个由可数多个球组成的球覆盖. 通过在 $l^\infty$上构造等价范数证明了Banach空间$X$的球覆盖性质既不是线性同胚不变的, 也不是在商映射下不变的, 同时, 它也不具有子空间的可继承性.     

A new discrete integrable system and its discrete integrable coupling system

In terms of the properties of the known loop algebra and difference operators, a new algebraic system X is constructed, which is devote to working out the well-known generalized cubic Volterra lattice equations hierarchy. Then an extended algebraic system of X is presented, from which the integrable coupling system of generalized cubic Volterra lattice equations are obtained.     

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.     

微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

Collocation method using sinc and Rational Legendre functions for solving Volterra’s population model

This paper proposes two approximate methods to solve Volterra’s population model for population growth of a species in a closed system. Volterra’s model is a nonlinear integro-differential equation on a semi-infinite interval, where the integral term represents the effect of toxin. The proposed methods have been established based on collocation approach using Sinc functions and Rational Legendre functions. They are utilized to reduce the computation of this problem to some algebraic equations. These solutions are also compared with some well-known results which show that they are accurate.     

Integrable nonholonomic deformation of modified Volterra lattice equation

The modified Volterra lattice equation with nonholonomic constrain has been considered in this paper. The integrability of the deformed model has been demonstrated by providing a Lax pair. Applying the gauge transformation to the Lax pair, we establish Darboux transformation theorem for the nonholonomic deformation equation. Some analytic solutions of the system are obtained via the one-fold and two-fold Darboux transformations. The deformation on explicit solutions exhibits different curvy profiles and propagation trajectories that were not found in modified Volterra lattice equation.     

A new method for solving nonlinear differential-difference equation

A new approach is presented which enables one to construct the exact solutions of nonlinear differential difference equations. As its application, the soliton solutions and periodic solutions of Hybrid lattice, discretized mKdV lattice and modified Volterra lattice are conveniently obtained by computing the solutions for a lattice equation introduced by Wadati.     

Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets

A computational method for numerical solution of a nonlinear Volterra integro-differential equation of fractional (arbitrary) order which is based on CAS wavelets and BPFs is introduced. The CAS wavelet operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.     

An approach to directly construct exact solutions of nonlinear differential-difference equations

In this paper, a constructive method for exactly solving nonlinear differential-difference equations (NDDEs) is presented. The NDDE which includes Hybrid lattice, discretized mKdV lattice and modified Volterra lattice is chosen to illustrate this approach. Some new solutions of these lattices are obtained.