A Noncommutative Algebraic Operational Calculus for Boundary Problems

We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the differential algebra underlying the given ring of boundary problems. Our methodology employs noncommutative localization in the theory of integro-differential algebras and operators. The resulting structure allows to build a symbolic calculus in the style of Heaviside and Mikusiński, but with the added benefit of incorporating boundary conditions where the traditional calculi allow only initial conditions. Admissible boundary conditions include multiple evaluation points and nonlocal conditions. The operator ring is noncommutative, containing all integrators initialized at any evaluation point.     

Banach空间半线性积微分方程的非局部Cauchy问题

运用预解算子理论和Schauder不动点定理,在Banach空间证明了一类非局部半线性积微分方程积分解的存在性.     

On the Cauchy Problems of Fractional Evolution Equations with Nonlocal Initial Conditions

In this paper, new criterions which allow us to relax the compactness and Lipschitz continuity on nonlocal item, ensuring the existence and uniqueness of mild solutions for the Cauchy problems of fractional evolution equations with nonlocal initial conditions, are established. The results obtained in this paper essentially extend some existing results in this area. Finally, we present two applications to the abstract results.     

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

The existence of at least one nontrivial solution to a class of semilinear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable re ection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity. Received July 1997; Revised February 1998     

Machine-Shop Problems: An Operational Research Approach

Production planning and scheduling in a batch production environment presents management with many problems, ranging from long-term capital investment in machine tools to short-term batch sequencing. This paper, which is aimed at the production manager, introduces the general concept of model building and then discusses the more specific technique of computer simulation modelling. A simple example of the latter is used to illustrate the basic properties of such a model and then the actual model currently being used is described. The possible applications of such models are discussed and developed. Finally, a case study is presented which shows how the model has been employed in an actual production environment.     

A White-Noise Approach to Stochastic Calculus

During the past 15 years a new technique, called the stochastic limit of quantum theory, has been applied to deduce new, unexpected results in a variety of traditional problems of quantum physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting commutation relations, new photon statistics in strong magnetic fields, etc. These achievements required the development of a new approach to classical and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and renormalization), some nontrivial extensions of known results in classical and quantum stochastic calculus can be obtained.     

Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach

We give a unified approach for studying the existence of multiplepositive solutions of nonlinear differential equations of theform where g, f are non-negativefunctions, subject to various nonlocal boundary conditions.We study these problems via new results for a perturbed integralequation, in the space C[0, 1], of the form where [u], ß[u] are linear functionalsgiven by Stieltjes integrals but are not assumed to be positivefor all positive u. This means we actually cover many more differentialequations than the simple equation written above. Previous resultshave studied positive functionals only, but even for positivefunctionals our methods give improvements on previous work.The well-known m-point boundary value problems are special casesand we obtain sharp conditions on the coefficients, which allowssome of them to have opposite signs. We also use some optimalassumptions on the nonlinear term.     

Feynman's Operational Calculus and Evolution Equations

We give an exposition and an extension of the ideas of Feynman's time-ordered operational calculus for noncommuting operators. Various directions for 'disentangling' functions of such operators are provided by measures on the time intervals in question. We concentrate especially on exponentials of sums of noncommuting operators and prove that the unique solution of a broad class of evolution equations is given by the time-dependent operators arrived at by disentangling such exponential expressions.     

Mikusinski算符演算在方程求解中的应用

就 Mikusinski算符演算在方程求解方面的研究进展情况和已获得的重要结果作一综述 ,其内容有常系数线性微分方程、差分方程的 M算符解法 ;变数算符概念及其相关结果 ;变系数线性常微分方程、差分方程、差分微分方程的 M算符解法以及 M算符演算在其他方程求解中的应用 .     

Approach to the Numerical Solution of Optimal Control Problems for Loaded Differential Equations with Nonlocal Conditions

Computational Mathematics and Mathematical Physics - An approach to the numerical solution of an optimal control problem described by systems of ordinary differential equations with point loading...     

A Language-Action Approach to Operational Research

This paper results from a collaboration between two academics, one a social psychologist who has explored creative work in engineering product design, the other engaged in the application of ‘soft’ methods in OR, but with a background in engineering. Much of the new thinking in OR is described in terms of a language-action perspective which is prominent in studies of computer supported work, negotiation, and other forms of discursive action. The perspective is used to examine some of the tasks needing to be undertaken if OR is to shift from a concern with social engineering to a concern with reflective social action. Particular attention is paid to the analysis of dilemmas, the growth of commitment, and the assessment of methods. It is concluded that, if OR makes the analysis of cognitive and political processes part of the core of the subject, it may become an epistemic subject with wide application to organisational decision-making.     

A General Stochastic Calculus Approach to Insider Trading

The purpose of this paper is to present a general stochastic calculus approach to insider trading. We consider a market driven by a standard Brownian motion $B(t)$ on a filtered probability space $\displaystyle (\Omega,\F,\left\{\F\right\}_{t\geq 0},P)$ where the coefficients are adapted to a filtration ${\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$, with $\F_t\subset\G_t$ for all $t\in [0,T]$, $T>0$ being a fixed terminal time. By an {\it insider} in this market we mean a person who has access to a filtration (information) $\displaystyle{\Bbb H}=\left\{\H_t\right\}_{0\leq t\leq T}$ which is strictly bigger than the filtration $\displaystyle{\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$. In this context an insider strategy is represented by an $\H_t$-adapted process $\phi(t)$ and we interpret all anticipating integrals as the forward integral defined in [23] and [25]. We consider an optimal portfolio problem with general utility for an insider with access to a general information $\H_t \supset\G_t$ and show that if an optimal insider portfolio $\pi^*(t)$ of this problem exists, then $B(t)$ is an $\H_t$-semimartingale, i.e. the enlargement of filtration property holds. This is a converse of previously known results in this field. Moreover, if $\pi^*$ exists we obtain an explicit expression in terms of $\pi^*$ for the semimartingale decomposition of $B(t)$ with respect to $\H_t$. This is a generalization of results in [16], [20] and [2].     

A Survey of the Development of Operational Calculus

Although the name of Oliver Heaviside is usually associated with the foundations of operational calculus, he was in fact preceded in introducing operational methods by a number of mathematicians. Heaviside's purely operational approach was often intuitive rather than rigorous and fell into disfavour in mathematical circles, being replaced by methods involving the concept of an integral transformation. In particular, the Laplace transformation came to be adopted by applied mathematicians and engineers despite the need for the Schwartz theory of distributions to account satisfactorily for the Dirac and other impulse functions. Jan Mikusinski,? ? It is intended that an expository article on Jan Mikusinski's operational calculus by H. G. Flegg will appear in a later issue ‐ Editor. however, has returned to the earlier operational approach and, by placing operational calculus upon a rigorous algebraic basis, has provided a calculus which is more general than those based upon integral transformations, and one which includes generalized functions without recourse to any specialized external theory.

     

Stability of Nonlocal Finite-Difference Problems

The article presents a brief review of the stability theory of finite-difference schemes for time-dependent problems of mathematical physics in which the spatial differential operator is constrained by boundary conditions joined on different pieces of the boundary. It is shown for the particular case of the heat-conduction equation that nonlocal boundary conditions may generate both simple finite-difference operators and finite-difference operators with an incomplete eigenfunction system. In both cases a rule is suggested for constructing the energy norm that ensures stability of the finite-difference scheme.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 72–83, 2004.     

Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

Nonlocal solvability of the Cauchy problem for second-order nonlinear parabolic equations

We prove the existence of smooth solutions of the Cauchy problem for some second-order nonlinear parabolic equations subject to natural smoothness conditions on the right side of the equation and on the initial function.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 581–586, April, 1974.     

Fourier Transform on the Lobachevsky Plane and Operational Calculus

Functional Analysis and Its Applications - The classical Fourier transform on the line sends the operator of multiplication by $$x$$ to $$ifrac{d}{dxi}$$ and the operator $$frac{d}{d x}$$ of...