Basic Calculus on Time Scales and some of its Applications

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.     

Metric Inequality, Subdifferential Calculus and Applications

In this paper we establish characterizations of Asplund spaces in terms of conditions ensuring the metric inequality and intersection formulae. Then we establish chain rules for the limiting Fréchet subdifferentials. Necessary conditions for constrained optimization problems with non-Lipschitz data are derived.     

在微积分中Dirichlet核的性质及应用

在微积分中,Dirichlet核的内容虽然较简单,但它的应用却非常广泛,而且效果独特.将进一步研究它的具体内容、性质和应用.     

Hausdorff Calculus and Some of Its Applications

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排序原理在微积分中的一些应用

利用排序原理证明了数列{(1 1/n)~n}收敛性,级数∑(n!e~n)/(n~n p)在p≤3/2时是发散的和几个不等式。     

Differential Calculus in Riesz Spaces and Applications to g-Calculus

We apply the theory of Differential and Integral Calculus in Riesz Spaces introduced in [1] and [4] to investigate some properties of the g-calculus and to solve some types of differential, functional and stochastic equations.     

Nonconvex Separation Theorem for Multifunctions,Subdifferential Calculus and Applications

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multi-objective optimization problem.     

Fractional Calculus and Its Applications Involving Certain Classes of Functional Relations

An interesting functional relation between Fox's H-function and the Digamma function ψ(z) was derived recently by applying the Riemann-Liouville fractional differintegral operator of (real or complex) order µ The object of this paper is to present much simpler alternative derivations of substantially more general classes of functional relations without using fractional calculus. Some relevant historical details are also provided.     

Calculus and its teaching: an accumulation effect

The teaching of elementary calculus during the last twenty‐five years in the United States is comparable to the development of the calculus. The development of the calculus and the teaching of calculus share a similar background in that both are characterized by periods of time in which ideas slowly accumulate, awaiting the arrival of some person, who will, with a new method of discovery, synthesize the fragmentary pieces and lift the subject to a higher plane.     

Continuous-Time Stochastic Calculus: A Survey of Applications in Finance and Accountancy

In recent years, the methods of stochastic calculus have beenincreasingly applied in both the finance and accountancy literatures.To the uninitiated the techniques of stochastic calculus areconfusing and counterintuitive, since few of the rules of ordinarycalculus appear to be satisfied. The present paper providesa simple and somewhat heuristic introduction to the basic concepts,with special attention being paid to applications in the financeand accountancy disciplines. After introducing the concept ofwhite noise, the standard Brownian, Ehrenfest, and tied-downor bridge processes are explained in a discrete-time binomialcontext. Limits are then taken to obtain the continuous-timeanalogues. The paper also provides a brief introduction to theimportant topic of Ito calculus and its applications in accountancyand finance.     

Malliavin Calculus for Functionals with Generalized Derivatives and Some Applications to Stable Processes

We introduce the notion of a generalized derivative of a functional on a probability space with respect to some formal differentiation. We establish a sufficient condition for the existence of the distribution density of a functional in terms of its generalized derivative. This result is used for the proof of the smoothness of the distribution of the local time of a stable process.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Continuity Properties for Modulation Spaces,with Applications to Pseudo-Differential Calculus,II

We discuss continuity for weighted modulation spaces, andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces. We also discussthe trace operator aa(0, ·) acting on modulationspaces. The results are used to get inclusions betweenmodulation spaces and Besov spaces, and proving continuityfor pseudo-differential operators and Toeplitz operators.     

Correlation Wavelet and its Applications

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广义次微分及其应用

通过应用广义次微分来研究不可微规划的最优解,得到了适当函数类在强意义下的最优性条件,并给出了广义次微分在稳定性理论和极小化方法中的应用