Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.     

The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings

We give some sufficient conditions for the Domínguez-Lorenzo condition in terms of the James constant, the Jordan-von Neumann constant, and the coefficient of weak orthogonality. As a consequence, we obtain fixed point theorems for multivalued nonexpansive mappings.     

Vector-Orthogonality and Lanczos-Type Methods

A method for solving a linear system is defined. It is a Lanczos-type method, but it uses formal vector orthogonality instead of scalar orthogonality. Moreover, the dimension of vector orthogonality may vary which gives a large freedom in leading the algorithm, and controlling the numerical problems. The ideas of truncated and restarted methods are revisited. The obtained residuals are exactly orthogonal to a space of increasing dimension. Some experiments are done, the problem of finding automaticaly good directions of projection remains partly open.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Tuning the Reactivity of a Substrate for SNAP-Tag Expands Its Application for Recognition-Driven DNA-Protein Conjugation

Recognition-driven modification has been emerging as a novel approach to modifying biomolecular targets of interest site-specifically and efficiently. To this end, protein modular adaptors (MAs) are the ideal reaction model for recognition-driven modification of DNA as they consist of both a sequence-specific DNA-binding domain (DBD) and a self-ligating protein-tag. Coupling DNA recognition by DBD and the chemoselective reaction of the protein tag could provide a highly efficient sequence-specific reaction. However, combining an MA consisting of a reactive protein-tag and its substrate, for example, SNAP-tag and benzyl guanine (BG), revealed rather nonselective reaction with DNA. Therefore new substrates of SNAP-tag have been designed to realize sequence-selective rapid crosslinking reactions of MAs with SNAP-tag. The reactions of substrates with SNAP-tag were verified by kinetic analyses to enable the sequence-selective crosslinking reaction of MA. The new substrate enables the distinctive orthogonality of SNAP-tag against CLIP-tag to achieve orthogonal DNA-protein crosslinking by six unique MAs.     

一类正交空间填充设计的构造

空间填充设计在计算机试验中应用十分广泛,当拟合回归模型时,正交的空间填充设计保证了因子效应估计的独立性.基于广义正交设计,文章给出了构造二阶正交拉丁超立方体设计和列正交设计的方法,新构造的设计不仅满足任意两列之间相互正交,还能保证每一列与任一列元素平方组成的列以及任两列元素相乘组成的列都正交.当某些正交的空间填充设计不存在时,具有较小相关系数的近似正交设计可作为替代设计使用.设计构造的灵活性为计算机试验在实践中的广泛应用提供了必要的支持.     

A 3D mesh deformation technique for irregular in‐flight ice accretion

Aircraft holding around busy airports may be requested to sustain as much as 45 min of icing before landing or being diverted to another airport. In this paper, a three‐dimensional mesh deformation scheme, based on a structural frame analogy, is proposed for the numerical simulation of ice accretion during extended exposure to adverse weather conditions. The goal is to provide an approach that is robust and efficient enough to delay or altogether avoid re‐meshing while preserving (enforcing) nearly orthogonal elements at the highly distorted ice surface. Robustness is achieved by suitably modifying the axial and torsional stiffness components of the frame elements in order to handle large and irregular grid displacements typical of in‐flight icing. Computational efficiency is obtained by applying the mesh displacement to an automatically selected small subset of the entire computational domain. The methodology is validated first in the case of deformations typical of fluid‐structure interaction problems, including wing bending, a helicopter rotor in forward flight, and the twisting of a high‐lift wing configuration. The approach is then assessed for aero‐icing on two swept wings and compared against experimental measurements where available. Copyright © 2015 John Wiley & Sons, Ltd.     

Uncorrelatedness and orthogonality for vector-valued processes

For a square integrable vector-valued process on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for are presented. The process is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for involving the biorthogonal representation for the conditional expectation of with respect to the usual product -algebra is presented.

     

Matrix Continued Fraction for the Resolvent Function of the Band Operator

The aim of this paper is the expansion of a matrix function in terms of a matrix-continued fraction as defined by Sorokin and Van Iseghem. The function under study is the Weyl function or resolvent function of an operator, given in the standard basis by a bi-infinite band matrix, with p subdiagonals and q superdiagonals, where the p + q – 1 intermediate diagonals are zero. The main goal of this paper is to find, for the moments, an explicit form in terms of the coefficients of the continued fraction, called genetic sums, which lead to a generalization of the notion of a Stieltjes continued fraction. These results are extension of some results already found for the vector case (p = 1) and are a step in the direction towards the solution of the direct and inverse spectral problem. The actual computation of the approximants of the given function as the convergents of the continued fraction is shown to be effective. Some possible extensions are considered.     

一种用于光机热集成分析的新方法——干涉图插值法

Zernike多项式是光机热集成分析中实现光、机、热各分析软件之间数据传递的重要工具。许多光学元件表面形状并不满足Zernike多项式的正交性条件，使有限元分析不能精确拟合光学元件表面变形及变形产生的各项像差。讨论影响Zernike正交性的几个因素，详细介绍将有限元分析得到的光学表面变形数据导入光学分析软件的一种新方法——干涉图插值法。该方法避开Zernike多项式拟合，在去除镜面刚体位移后将镜面畸变精确地表示成干涉图数据，直接生成数据接口。