Multiple orthogonal polynomials applied to matrix function evaluation

Multiple orthogonal polynomials generalize standard orthogonal polynomials by requiring orthogonality with respect to several inner products. This paper discusses an application to the approximation of matrix functions and presents quadrature rules that generalize the anti-Gauss rules proposed by Laurie.     

A Review with Updated Perspectives on Nutritional and Therapeutic Benefits of Apricot and the Industrial Application of Its Underutilized Parts

Fruits maintain the image as the richest sources of vitamins. Focusing on apricots, utilization of apricot species for many applications is possible due to its various benefits. Many research studies demonstrated different perspectives of apricot, especially in medical used as it can act as antioxidant, anti-inflammatory, and antimicrobial agents. Moreover, in the industrial sectors, apricots can be used in the production of biofuels and batteries. All components of the apricot fruit, including seeds and kernels have been found to possess significant interest. This review is to breach the knowledge gap regarding the key nutrients and chemicals of apricot fruit, contributing to its health-promoting properties to emphasize the noble importance of this fruit in the diet and in the management of several diseases. We also cover the application of apricots in the industry that could be developed as a promising and sustainable source.     

Voltammetric determination of itopride using carbon paste electrode modified with Gd doped TiO2 nanotubes

In the present work TiO₂ nanotubes (TNT) have been synthesized by alkaline hydrothermal transformation. Then they have been doped with Gd element. Characterizations of doped and undoped TNT have been done with TEM and SEM. The chemical composition was analyzed by EDX, Raman and FTIR spectroscopy. The crystal structure was characterized by XRD. Carbon paste electrode has been fabricated and mixed with Gd doped and undoped TNT to form a nanocomposite working electrode. Comparison of bare carbon paste electrode and Gd doped and undoped TNT carbon paste electrode for 1.0 ×10⁻³ M K₄ [Fe(CN)₆] voltammetric analysis; it was observed that Gd doped TNT modified electrode has advantage of high sensitivity. Gd doped TNT modified electrode has been used as working electrode for itopride assay in a pharmaceutical formulation. Cyclic voltammetry analysis showed high correlation coefficient of 0.9973 for itopride (0.04–0.2 mg/mL) with a limit of detection (LOD) and limit of quantitation values (LOQ) of 2.9 and 23.0 μg.mL⁻¹ respectively.     

Simplified anti-Gauss quadrature rules with applications in linear algebra

The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w ^T f(A)v with \(A\in {\mathbb {R}}^{n\times n}\) a large matrix, f a function, and \(v,w\in {\mathbb {R}}^{n}\) arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v ^T f(A)v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A ^T . The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A ^T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.