Statistical and Geometrical Way of Model Selection for a Family of Subdivision Schemes

The objective of this article is to introduce a generalized algorithm to produce the m-point n-ary approximating subdivision schemes(for any integer m, n ≥ 2). The proposed algorithm has been derived from uniform B-spline blending functions. In particular, we study statistical and geometrical/traditional methods for the model selection and assessment for selecting a subdivision curve from the proposed family of schemes to model noisy and noisy free data. Moreover, we also discuss the deviation of subdivision curves generated by proposed family of schemes from convex polygonal curve. Furthermore, visual performances of the schemes have been presented to compare numerically the Gibbs oscillations with the existing family of schemes.     

Flow of a mixture of a viscous fluid and a granular solid in an orthogonal rheometer

In this paper, we study the flow of a linearly viscous fluid and a granular solid, consisting of many particles, situated between two parallel plates rotating about different axes. Flow in orthogonal rheometers has been studied for many viscoelastic fluids so that their rheological properties can be measured. The mixture is modeled using the theory of interacting continua, and constitutive relations for the fluid phase, the granular phase, and the interaction forces are provided. For a very special case, an analytical solution to the equations of motion is also provided.     

Second-order differential equations in the Laguerre–Hahn class

Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.     

Orthogonal array design for the optimization of ionic liquid-based dispersive liquid-liquid microextraction of benzophenone-type UV filters

In the present study, dispersive liquid-liquid microextraction (DLLME) using an ionic liquid (IL) as the extractant was successfully developed to extract four benzophenone-type UV filters from the different water matrices. Orthogonal array experimental design (OAD), based on five factors and four levels (L(16)(4(5))), was employed to optimize IL-dispersive liquid-liquid microextraction procedure. The five factors included pH of sample solution, the volume of IL and methanol addition, extraction time and the amount of salt added. The optimal extraction condition was as follows. Sample solution was at a pH of 2.63 in the presence of 60 mg/mL sodium chloride; 30 μL IL and 15 μL methanol were used as extractant and disperser solvent, respectively; extraction was achieved by vortexing for 4 min. Using high-performance liquid chromatography-UV analysis, the limits of detection of the target analytes ranged between 1.9 and 6.4 ng/mL. The linear ranges were between 10 or 20 ng/mL and 1000 ng/mL. This procedure afforded a convenient, fast and cost-saving operation with high extraction efficiency for the model analytes. Spiked waters from two rivers and one lake were examined by the developed method. For the swimming pool water, the standard addition method was employed to determine the actual concentrations of the UV filters.     

Recursion relations for the extended Krylov subspace method

The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K^m,mi+1(A)=span{A^-m+1v,…,A^-1v,v,Av,…,A^miv}, m=1,2,3,…, with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.     

A study of Galerkin method for the heat convection equations

This article investigates the Galerkin method for an initial boundary value problem for heat convection equations. The new error estimates for the approximate solutions and their derivatives in strong norm are obtained.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.