Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

The new forms of Voronovskaya’s theorem in weighted spaces

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.     

关于Taylor公式中Lagrange余项的再研究

在Azpeitja对Taylor公式中Lagrange余项的"中间点"渐近性的研究基础上,又建立几个易于验证和推广的结果.     

Basics of Riemann Delta and Nabla Integration on Time Scales

In this paper we introduce and investigate the concepts of Riemann's delta and nabla integrals on time scales. Main theorems of the integral calculus on time scales are proved.     

Almost periodic solutions for neutral‐type neural networks with the delays in the leakage term on time scales

In this paper, a class of neutral‐type neural networks with delays in the leakage term on time scales are considered. By using the Banach fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. The results of this paper are new and complementary to the previously known results. Finally, an example is presented to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.     

Taylor type formula with Fréchet polynomials

We extend Taylor’s formula for functions defined on uniquely 2-divisible Abelian semigroups with values in Banach spaces. As corollaries, we give sufficient conditions for the representation of such functions as sums of generalized power series, and also a stability criterion for Fréchet’s polynomial equation which extends the known results.     

Hermite’s equations on time scales

We present the exact formula for multiplication by a polynomial of degree one. The obtained results are applied in solving a second-order linear equation of Hermite’s type on time scales.     

The First Attempt on the Stochastic Calculus on Time Scale

The aim of this article is to study the Doob–Meyer decomposition theorem, ?-stochastic integration and Ito's formula for stochastic processes defined on time scale. The obtained results can be considered as a first attempt on the stochastic calculus on time scale.     

Teachers’ beliefs towards teaching calculus

This paper focuses on the belief systems towards teaching calculus of 29 upper secondary mathematics teachers. Firstly, we discuss different educational trends in teaching calculus, and the theoretical approach based on the construct of belief systems. Afterwards we describe the method of our study used to analyse the belief systems of the calculus teachers. Referring to findings of our research, we firstly focus on central and peripheral beliefs and thus discuss the structure of the teachers’ belief systems. Further, we compare the teachers’ belief systems to established educational trends of teaching calculus. Finally, we conclude the paper by reflecting on our main findings and by discussing possible directions of further research.     

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Grönwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.     

Environmental dispersion in wetland flow

For the typical case of a pulsed contaminant emission into a shallow wetland channel, a theoretical analysis is presented in this paper for the decay of the width-averaged mean concentration under environmental dispersion. The velocity profile of a fully developed steady flow through the wetland channel is obtained with that for the well-known plane Poiseuille flow as a special case. An environmental dispersion model for the mean concentration is devised as an extension of Taylor’s classic analysis on dispersion, and corresponding environmental dispersivity is obtained by Aris’s method of moments and illustrated with an asymptotic time variation with stem dominated, transitional, and width-stem dominated stages. Analytical solution for the longitudinal decay of mean concentration due to environmental dispersion is rigorously derived and characterized with multiple time scales.     

On the equivalence of Rolle's and generalized mean value theorems on time scales

In this note, by using elementary calculus on time scales, we recall the direct proof of the equivalence between Rolle's and generalized (Cauchy's) mean value theorems on time scales.     

Weighted Hardy-Type Inequalities on Time Scales with Applications

In this paper, we will prove some new dynamic Hardy-type inequalities on time scales with two different weighted functions. The study is to determine conditions on which the generalized inequalities hold using some known hypothesis. The main results will be proved by employing Hölder’s inequality, Minkowski’s inequality and a chain rule on time scales. As special cases of our results, when the time scale is the real numbers, we will derive some well-known results due to Copson, Bliss, Flett and Bennett by a suitable choice of the weighted functions. We will apply the results to investigate the oscillation and nonoscillation of a half-linear second order dynamic equation on time scales.     

Oscillation criteria for second-order dynamic equations on time scales

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.     

Some Integral Inequalities on Time Scales

In the paper, some new integral inequalities on time scales are presented by using elementarily analytic methods in calculus of time scales.     

Stochastic Integral Representations of the Extrema of Time-homogeneous Diffusion Processes

The stochastic integral representations (martingale representations) of square integrable processes are well-studied problems in applied probability with broad applications in finance. Yet finding explicit expression is not easy and typically done through the Clack-Ocone formula with the advanced machinery of Malliavin calculus. To find an alternative, Shiryaev and Yor (Teor Veroyatnost i Primenen 48(2):375–385, 2003) introduced a relatively simple method using Itô’s formula to develop representations for extrema of Brownian motion. In this paper, we extend their work to provide representations of functionals of time-homogeneous diffusion processes based on the Itô’s formula.     

Duality principle of time scales and inequalities

Here we adopt, develop further and use the principle of duality in time scales Caputo, [Time scales: from nabla calculus to delta calculus and vice versa via duality, arxiv: 0910.0085v1 [math.OC] 1 Oct. 2009]. Using this principle and based on a variety of important delta inequalities we produce the corresponding nabla ones. We give several applications.     

Estimates for the density of functionals of SDEs with irregular drift

We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable. The argument consists of using Girsanov’s theorem together with an Itô–Taylor expansion of the change of measure. One then applies Malliavin calculus techniques in a non-trivial manner so as to avoid the irregularity of the drift. An integration by parts formula for this set-up is obtained.