On approximation of definite integrals by compound quadrature formulas using derivatives

Рассмотрена задача вычисления определенного интеграла функции, для которой известны значения ее самой и набора производных до заданного порядка в точках отрезка интегрирования. Построены составные квадратурные формулы, которые используют значения функции и ее производных до m-го порядка включительно. Получено представление остаточного члена, выраженное через производную соответствующего порядка и число узловых точек. Приведены примеры интегрирования заданных функций с исследованием погрешности и ее оценки. Дано сравнение с известными численными методами и формулой Эйлера-Маклорена, которое показало повышенную точность и лучшую сходимость метода двухточечного интегрирования The problem of computing a definite integral of a function for which the values of itself and the set of derivatives up to a given order at the points of the interval of integration are known is considered. Composite quadrature formulas are constructed that use the values of the function and its derivatives up to the m-th order inclusive. A representation of the remainder is obtained, expressed in terms of the derivative of the corresponding order and the number of nodal points. Examples of integration of the given functions with the study of the error and its estimation are given. A comparison is made with the known numerical methods and the Euler-Maclaurin formula, which showed increased accuracy and better convergence of the two-point integration method.

Exploring the Reliability of DFT Calculations of the Infrared and Terahertz Spectra of Sodium Peroxodisulfate

A number of DFT programs with various combinations of pseudo-potentials and van der Waals’ dispersive corrections have been used to optimize the structure of sodium peroxodisulfate, Na2(SO4)2, and to calculate the infrared, attenuated total reflectance and terahertz absorption spectra of the powdered crystal. Comparison of the results from the different methods highlights the problems of calculating the absorption spectrum reliably. In particular the low frequency phonon modes are especially sensitive to the choice of grids to represent the wavefunction or the charge distribution, k-point integration grid and the energy cutoff. A comparison is made between the Maxwell-Garnett (MG) and Bruggeman effective medium methods used to account for the effect of crystal shape on the predicted spectrum. Possible scattering of light by air inclusions in the sample and by larger particles of Na2(SO4)2 is also considered using the Mie method. The results of the calculations are compared with experimental measurements of the transmission and attenuated total reflection spectra.

PTO-QuickStep: A Fast and Efficient Method for Cloning Random Mutagenesis Libraries

QuickStep is a cloning method that allows seamless point integration of a DNA sequence at any position within a target plasmid using only Q5 High-Fidelity DNA Polymerase and DpnI endonuclease. This efficient and cost-effective method consists of two steps: two parallel asymmetric PCRs, followed by a megaprimer-based whole-plasmid amplification. To further simplify the workflow, enhance the efficiency, and increase the uptake of QuickStep, we replaced the asymmetric PCRs with a conventional PCR that uses phosphorothioate (PTO) oligos to generate megaprimers with 3′ overhangs. The ease and speed of PTO-QuickStep were demonstrated through (1) right-first-time cloning of a 1.8 kb gene fragment into a pET vector and (2) creating a random mutagenesis library for directed evolution. Unlike most ligation-free random mutagenesis library creation methods (e.g., megaprimer PCR of whole plasmid [MEGAWHOP]), PTO-QuickStep does not require the gene of interest to be precloned into an expression vector to prepare a random mutagenesis library. Therefore, PTO-QuickStep is a simple, reliable, and robust technique, adding to the ever-expanding molecular toolbox of synthetic biology and expediting protein engineering via directed evolution.

The Calculation of the Infrared and Terahertz Spectrum of Sodium Peroxodisulfate.

A number of DFT programs with various combinations of pseudo-potentials and van der Waals’ dispersive corrections have been used to optimize the structure of sodium peroxodisulfate, Na₂(SO₄)₂, and to calculate the infra-red and terahertz absorption spectrum of the powdered crystal. Comparison of the results from the different methods highlights the problems of calculating the absorption spectrum reliably. In particular the low frequency phonon modes are especially sensitive to the choice of grids to represent the wavefunction or the charge distribution, k-point integration grid and the energy cutoff. A comparison is made between the Maxwell-Garnett (MG) and Bruggeman effective medium methods used to account for the effect of crystal shape on the predicted spectrum. Possible scattering of light by air inclusions in the sample and by larger particles of Na₂(SO₄)₂ is also considered using the Mie method. The results of the calculations are compared with experimental measurements of the transmission and attenuated total reflection spectra <br>

The Calculation of the Infrared and Terahertz Spectrum of Sodium Peroxodisulfate.

A number of DFT programs with various combinations of pseudo-potentials and van der Waals’ dispersive corrections have been used to optimize the structure of sodium peroxodisulfate, Na₂(SO₄)₂, and to calculate the infra-red and terahertz absorption spectrum of the powdered crystal. Comparison of the results from the different methods highlights the problems of calculating the absorption spectrum reliably. In particular the low frequency phonon modes are especially sensitive to the choice of grids to represent the wavefunction or the charge distribution, k-point integration grid and the energy cutoff. A comparison is made between the Maxwell-Garnett (MG) and Bruggeman effective medium methods used to account for the effect of crystal shape on the predicted spectrum. Possible scattering of light by air inclusions in the sample and by larger particles of Na₂(SO₄)₂ is also considered using the Mie method. The results of the calculations are compared with experimental measurements of the transmission and attenuated total reflection spectra <br>

Bending and vibration analyses of plate and shell using an element decomposition method

Purpose The purpose of this paper is to apply the element decomposition method (EDM) in the study of the bending and vibration properties of plate and shell. Design/methodology/approach In the present method, each quadrilateral element is first divided into four sub-triangular cells, and the local strains are obtained in those sub-triangles based on linear interpolation. The whole strain filed is formulated through a weighted averaging operation of local strains, implying that only one integration point is adopted to construct the stiffness matrix. To reduce the instability of one-point integration and increase the accuracy of the present method, a stabilization item of the stiffness matrix is formulated by variance of the local strains. A mixed interpolated tensorial components (MITC) method is used in eliminating the shear locking phenomenon. Findings The novel EDM based on linear interpolation is effective in bending and vibration analyses of plate and shell, and the present method used in practical problems is reliable for static and free vibration analysis. Originality/value This method eliminated the instability of one-point integration and increased the accuracy by a stabilization item and performed stably in engineering analysis including large-scale problems of vehicle components.