On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

An online identification algorithm of unknown time-varying delay and internal multimodel control for discrete non-linear systems

In this paper, an online algorithm is proposed for the identification of unknown time-varying input delay in the case of discrete non-linear systems described by decoupled multimodel. This method relies on the minimization of a performance index based on the error between the real system and the partial internal models outputs. In addition, a decoupled internal multimodel control is proposed for the compensation of discrete non-linear systems with time-varying delay. This control scheme incorporates partial internal model controls. Each partial controller is associated to a specified operating zone of the non-linear system. The switching between these controllers is ensured by a supervisor that contains a set of local predictors. A simulation example is carried out to illustrate the significance of the proposed time-varying delay identification algorithm and the proposed internal multimodel control scheme.     

Matrix Representation of Recursive Sequences of Order $3$ and Its Applications

Here presented is a matrix representation of recursive number sequences of order $3$ defined by $a_n=pa_{n-1}+qa_{n-2}+ra_{n-3}$ with arbitrary initial conditions $a_0,$ $a_1=0$, and $a_2$ and their special cases of Padovan number sequence and Perrin number sequence with initial conditions $a_0=a_1=0$ and $a_2=1$ and $a_0=3$, $a_1=0$, and $a_2=2$, respectively. The matrix representation is used to construct many well known and new identities of recursive number sequences as well as Pavodan and Perrin sequences.     

Several formulas for special values of the Bell polynomials of the second kind and applications

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.     

Improved bi-accelerator derivative free with memory family for solving nonlinear equations

The object of the present paper is to accelerate the $R$-order convergence of with memory derivative free family given by Lotfi et al. (2014) without adding any new evaluations. To achieve this goal one more iterative parameter is introduced, which is calculated with the help of Newton''s interpolatory polynomial. It is shown that the $R$-order convergence of the proposed scheme is increased from 12 to 14 without any extra evaluation. Smooth as well as non-smooth examples are presented to confirm theoretical result and significance of the new scheme.     

The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

On the number of \(n\)-dimensional invariant spheres in polynomial vector fields of \(\mathbb{C}^{n+1}\)

We study the polynomial vector fields \(\mathcal{X}= \displaystyle \sum_{i=1}^{n+1} P_i(x_1,\ldots,x_{n+1}) \frac{\partial}{\partial x_i}\) in \(\mathbb{C}^{n+1}\) with \(n\geq 1\) . Let \(m_i\) be the degree of the polynomial \(P_i\). We call \((m_1,\ldots,m_{n+1})\) the degree of \(\mathcal{X}\). For these polynomial vector fields \(\mathcal{X}\) and in function of their degree we provide upper bounds, first for the maximal number of invariant \(n\)-dimensional spheres, and second for the maximal number of \(n\)-dimensional concentric invariant spheres.     

Integrability conditions for complex systems with homogeneous

In this paper we obtain conditions for the existence  of a local analytic first integral for four eight-parameter families of quintic complex system. We also discuss computational difficulties arising in the study of the problem of integrability for these systems.     

光纤陀螺消偏结构与偏振度关系的研究

利用在光源出端加单段保偏光纤和在消偏器后端加Loyt消偏器,可以使进入光纤线圈的光的偏振度降到10－3数量级,降低了因外界因素对线圈中光偏振态的影响而产生的光纤陀螺偏振误差.并用琼斯矩阵计算的方法,分析了光路中四个胶合角与偏振度的关系及每个角度的取值范围.最后探讨了工艺上胶合角度测量与实现.