A self‐adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application

The self‐adaptive intelligence gray predictive model (SAIGM) has an alterable‐flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

THE COMPACT COMPOSITION OPERATOR ON THE μ-BERGMAN SPACE IN THE UNIT BALL

nAbstract Let p 0 and μ be a normal function on [0, 1), ν(r) =(1-r~2)~(1+(n/p)) μ(r) for r ∈ [0, 1). In this article, the bounded or compact weighted composition operator T_φ,ψfrom the μ-Bergman space A~p(μ) to the normal weight Bloch type space β_ν in the unit ball is characterized. The briefly sufficient and necessary condition that the composition operator C_φis compact from A~p(μ) to β_ν is given. At the same time, the authors give the briefly sufficient and necessary condition that C_φis compact on β_μ for a 1.     

由Campanato型函数和与薛定谔算子相关的Riesz变换生成的Toeplitz算子的有界性

本文研究了由Campanato型函数及与Schrödinger算子相关的Riesz变换生成的Toeplitz算子的有界性.利用Sharp极大函数估计得到了Toeplitz算子Θ^b在Lebesgue空间的有界性,拓广了已有交换子的结果.     

On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

The purpose of this paper is to study the mixed Dirichlet‐Neumann boundary value problem for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces on a bounded Lipschitz domain in ${\mathbb{R}}^3$, with p in a neighborhood of 2. This system is obtained by adding the semilinear term | u | u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well‐posedness results for the Dirichlet and Neumann problems in L _p ‐based Besov spaces on bounded Lipschitz domains in ${\mathbb{R}}^n$(n ≥3) are also presented. Then, using integral potential operators, we show the well‐posedness in L ₂‐based Sobolev spaces for the mixed problem of Dirichlet‐Neumann type for the linear Brinkman system on a bounded Lipschitz domain in ${\mathbb{R}}^n$(n ≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann‐to‐Dirichlet operator, we extend the well‐posedness property to some L _p ‐based Sobolev spaces. Next, we use the well‐posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces, with p ∈(2?ε ,2+ε ) and some parameter ε >0.     

SL(n+1,R)/S(GL(1,R)×GL(n,R))上线丛的一个超几何方程

本文研究了伪黎曼对称空间SL（n+1,R）/S（GL（1,R）×GL（n,R））线丛上的微分方程.利用李代数方法,即Casimir算子得到这个微分算子.这个微分算子是一个超几何方程,这个结论推广了文献[1,3,5]中的微分方程.     

不同权的Bloch型空间之间的加权复合算子再刻画

本文研究了单位圆中从空间β_L到β_α的加权复合算子uC_φ为有界算子和紧算子的条件.利用阶估计等方法,获得了有界性和紧性的简捷充要条件,推广了叶善力的相应结果.     

一类双曲型方程的Hamilton算子半群方法

将一类双曲型方程混合问题转换成一阶抽象Cauchy问题,证明所得Hamilton算子矩阵H在相应空间中生成压缩半群,并借助Fourier变换,采用一致连续半群做逼近的方法,得到H所生成的压缩半群,进而给出了问题的古典解.     

可修复人机系统的指数稳定性

研究了两相同部件温储备可修的人机系统,运用C_0半群的相关理论,对系统主算子的谱界进行估值.估算系统的算子产生的半群的增长界,然后运用了共尾的概念及相关的理论,得到了系统算子A+B的谱界与系统算子产生的半群的增长界相同.进而运用相关代数知识证得,0为系统算子的简单本征值,并分析了系统算子的谱分布,得到系统的指数稳定性.并研究了系统算子预解式的特性.对任意给定的δ0,γ=a+bi,-μ+δa_1≤a≤a_2,得到lim|b|→∞‖R(γ;A+B)‖=0.进而得到在~sRγ≥a_1的右半平面内相应于系统算子A+B的谱点由有限个本征值组成.