‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.     

关于平面19-点集的空凸分划问题

主要研究了平面上处于一般位置的19-点集,根据其凸包边数的不同,分别讨论了其所含空凸多边形的个数,得出G(19)≤5.在此基础上,对平面上处于一般位置的n-点集得出G(n)≤[11n/42],从而改进了G(n)的上界.     

超声提取-高效液相色谱法同时测定新疆罗布麻茶中绿原酸和芦丁的含量

以绿原酸和芦丁的提取率为指标,选用L9(34)正交试验对新疆罗布麻茶中绿原酸和芦丁的超声工艺进行优化,并建立同时测定绿原酸和芦丁含量的反相高效液相色谱(RP-HPLC)法。结果表明,甲醇浓度是影响绿原酸和芦丁提取的显著因素;最佳提取工艺条件是:60%甲醇溶液,超声时间为40 min,超声功率为400 W,料液比为1∶25,在此条件下罗布麻茶中绿原酸和芦丁的提取率分别为0.71%和0.49%。采用Agilent TC-C18色谱柱(150×4.6mm,5μm),流动相为甲醇-0.5%冰乙酸溶液(体积比40∶60),流速为1.0 mL/min,检测波长326nm,柱温30℃。绿原酸在0.356~2.495μg(r=0.99993)和芦丁在0.317~2.218μg(r=0.99992)间呈良好线性关系;绿原酸和芦丁的平均加标回收率分别为103.41%和99.68%,其相对标准偏差(RSD,n=6)分别为0.79%和0.69%;精密度实验RSD(n=5)分别为0.27%和0.73%。     

Bivariate version of the Hahn-Sonine theorem

We consider orthogonal polynomials in two variables whose derivatives with respect to are orthogonal. We show that they satisfy a system of partial differential equations of the form where , , is a vector of polynomials in and for , and is an eigenvalue matrix of order for . Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.

     

Projection orthogonale sur le graphe d'une relation linéaire fermé

Let denote the set of all closed linear relations on a Hilbert space (which contains all closed linear operators on ). In this paper, for every we define and study two associated linear operators on , and , which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of , a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the and operators to operator theory will be given in a forthcoming paper.

     

Koassoziierte Primideale in 2-Dimensionalen Regulären Ringen

Suppose that R is a 2-dimensional regular local ring, pa prime ideal of height 1, and u = r/s an element of the quotient field of R. We give criteria for pbeing coassociated to the R-module B = R[u], i.e., that there is an Artinian factor module B/X such that p= Ann _R (B/X). These criteria are formulated using the ideal a= (r, s) and the intersection numbers ( _i , r), ( _i , s), where ₁, ... _h are the branches of pin the completion R.     

On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.     

正交表L_8的几个性质

张建方．正交表Ｌ８的几个性质．本文介绍了正交表Ｌ８的几个性质。在二位级正交表当中,这些性质很基本,内容比较丰富,对于应用和了解其它正交表有一定帮助     

Biosynthesis optimization of silver nanoparticles (AgNPs) using Trichoderma longibranchiatum and biosafety assessment with silkworm (Bombyx mori)

Silver nanoparticles (AgNPs) have raised public concern due to their widespread application in the field of agriculture, medicine, and environment and their potential toxic effects on humans and the environments. In this study, biosynthesis of nanosilver particles mediated by Trichoderma longibranchiatum using orthogonal experimental design (OED) was optimized. Silkworm larvae were exposed via the mulberry leaves to AgNPs to evaluate their toxic effects. The results showed that 2 mmol/L silver nitrate and 55 °C of reaction temperature at pH 7.0 for 24 h were the optimum values for AgNPs biosynthesis with the synthesis amount and antifungal activity of AgNPs as the indices. The characterization of the biosynthesized AgNPs was conducted using electron microscopy, energy dispersive X-ray analysis (EDS), UV/visible spectrophotometry, Fourier transform infrared spectroscopy (FTIR), and X-ray diffraction (XRD). The crystalline structured nanoparticles were spherical or polyhedral with a mean size ranging about 5–50 nm. FTIR showed that many functional group moieties (–OH, –CH₃, –C–O, etc.) involved as a capping and reducing agent in AgNPs biosynthesis. After the larvae were fed with 50 mg/mL AgNPs, there were no obvious adverse effects on the growth of larvae and cocoon quality. Further supplement of AgNPs-B could promote the weight of larvae and the cocoon shell ratio. The data presented herein provided valuable information on a reliable eco-friendly, simple, low-cost biosynthesis of AgNPs and its biosafety evaluation which may contribute to its increased application in the future.     

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form

W(t)=t^αe^-te^Att^Bt^B*e^A*t,