A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

Cationic gold carbonyl complex on a phosphine support

A cationic gold carbonyl complex has been synthesized and characterized using several techniques including X-ray crystallography. [(Mes(3)P)Au(CO)][SbF(6)] (Mes = 2,4,6-Me(3)C(6)H(2)) has a linear, two-coordinate gold atom. This compound displays the CO stretching frequency at 2185 cm(-1). The (13)C NMR signal of the gold-bound (13)CO appears as a doublet centered at δ 182.6 ((2)J(C,P) = 115 Hz). A computational study shows that the Au-CO bond consists of electrostatic attraction, Au ← CO donation, and significant Au → CO π-back-bonding components. Polarization of the CO bond caused by the electrostatic effect of the cationic gold center is mainly responsible for the large blue shift in the CO stretching frequency.     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

Commutativity of first‐order discrete‐time linear time‐varying systems

After introducing the concept of commutativity for continuous‐time linear time‐varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete‐time linear time‐varying systems, the problem is formulated for first‐order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first‐order discrete‐time linear time‐varying systems are derived, and their advantageous use in digital system design is illustrated, which are the main objectives of the paper. The results are verified by examples which include an application in amplitude modulation for digital telecommunication.     

Change-point inference on volatility in noisy Itô semimartingales

This work is concerned with tests on structural breaks in the spot volatility process of a general Itô semimartingale based on discrete observations contaminated with i.i.d. microstructure noise. We construct a consistent test building up on infill asymptotic results for certain functionals of spectral spot volatility estimates. A weak limit theorem is established under the null hypothesis relying on extreme value theory. We prove consistency of the test and of an associated estimator for the change point. A simulation study illustrates the finite-sample performance of the method and efficiency gains compared to a skip-sampling approach.     

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q ‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.     

Comments on The Co-emergence of Machine Techniques, Paper-and-Pencil Techniques, and Theoretical Reflection

We comment on the paper The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra by Carolyn Kieran and Paul Drijvers. We look at aspects of Kieran and Drijvers' analysis with regard to ‚task-technique-theory’ in the light of a model of abstraction in context and suggest that this re-viewing brings mutual analytic benefits.     

Crystal structure of 3,3-dichloro-N-p-methoxyphenyl-4-(2-phenylstryl)-2-azetidinone.

3,3-Dichloro-N-p-methoxyphenyl-4-(2-phenylstryl)-2-azetidinone (C22H15Cl2NO2) was studied by X-Ray analysis, which indicated a monoclinic space group, P2(1)/c, with a = 9.619(5), b = 13.879(4), c = 14.161(5)A, beta = 100.16(3)degrees, V = 1860.8(13)A3, Z = 4, Dc = 1.414 g cm(-3), micro(Mo Kalpha) = 0.366 mm(-1) and F000 = 816. The structure was solved by direct methods and refined to R = 0.041 for 4026 reflections [I > 2sigma(I). The beta-lactam ring (2-azetidinone) has antimicrobial affects. The substituents of the methoxyphenyl and phenyl substituents do not change the activity property of the beta-lactam ring, and the activity properties depend on the planarity of the beta-lactam ring.     

Non-linear vibrations of a simple-simple beam with a non-ideal support in between

A simply supported Euler-Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.