Analytical solutions of a class of inverse coefficient problems

This paper is devoted to some class of inverse coefficient problems. By using a well-known transformation, the inverse problem is transformed to a new problem without the unknown time dependent coefficient. Therefore, the new inverse problem can be solved easily. To show the efficiency of the present method, some examples are presented.     

Determination of an unknown function in a parabolic equation with an overspecified condition

In this paper we consider the inverse problems of identifying some space-dependent unknown coefficients in parabolic equations subject to initial boundary value conditions along with an overspecified condition at the final time t = T. We use the overspecified information to transform the problems into non-linear parabolic equations involving a functional of the solution with respect to the time variable. This transformation allows us to establish existence theorems for these inverse problems by employing the Schauder fixed-point theorem.     

Uniqueness of the Solution of Inverse Initial Value Problems for the Burgers Equation with an Unknown Source

Differential Equations - We consider inverse problems of determining the initial conditions and a time-invariant inhomogeneity in boundary value problems for the Burgers equation. A transformation...     

Inverse Scattering Problem For The Schrödinger Equation With An Additional Quadratic Potential On The Entire Axis

We consider the Schrödinger equation with an additional quadratic potential on the entire axis and use the transformation operator method to study the direct and inverse problems of the scattering theory. We obtain the main integral equations of the inverse problem and prove that the basic equations are uniquely solvable.     

矩阵的分块初等变换与分块初等阵及其应用

讨论了矩阵分块初等变换和分块初等阵的定义和性质,利用这一工具研究了行列式的分块运算,分块矩阵的求逆和对称阵的分块合同变换等问题.     

Application of the differential transformation method to the free vibrations of strongly non-linear oscillators

This paper adopts the differential transformation method to obtain the free vibration behavior of an oscillator with fifth-order non-linearities. The principle of differential transformation is briefly introduced, and is then applied in the derivation of a set of difference equations for the free vibration oscillator problem. The solutions are subsequently solved by a process of inverse transformation. The time responses of the oscillator are presented under different parameter conditions, and the current results are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the proposed method. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the proposed differential transformation method is a powerful and efficient tool for solving non-linear problems.     

Canonical transformations associated with second order problems in the calculus of variations

Summary Our object is a systematic investigation of some of the properties of canonical transformations associated with second order problems in the calculus of variations. After the introduction of such transformations, together with the related concepts of Lagrange and Poisson brackets, the bracket relationships are found which characterize canonical transformations. This characterization is also achieved by means of so-called reciprocity relations between the original transformation and its inverse (which always exists). The effect of the canonical transformation on the underlying variational problem is discussed. It is also shown that the Jacobian of such a transformation always has the value unity. The special case when the canonical transformation is independent of the parameter (a generalization of the so-called time-independent canonical transformation of mechanics) is treated in some detail. Finally it is indicated how the present theory can be extended to problems of higher order. Some of the results of this paper are contained in a doctoral thesis ([2]) which was presented to the University of South Africa. The writer wishes to express his gratitude to his supervisor, ProfessorH. Rund, for his interest, encouragement and advice concerning this work.     

Stable barrier-projection and barrier-Newton methods in linear programming

The present paper is devoted to the application of the space transformation techniques for solving linear programming problems. By using a surjective mapping the original constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem the stable version of the gradient-projection and Newton's methods are used. After an inverse transformation to the original space a family of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of the starting and current points, but they preserve feasibility. As a result of a space transformation the vector fields of differential equations are changed and additional terms are introduced which serve as a barrier preventing the trajectories from leaving the feasible set. A proof of a convergence is given.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Research was supported by the grant N93-012-450 from Russian Scientific Fund.     

基于Arnold变换的图像逆置乱算法

针对Arnold变换的周期依赖于图像的阶数这一特性,提出了一种反变换算法.该算法通过分析加密图像任一点处两坐标分量间关系,得到原图像相应点的坐标,从而实现图像的解密.该反变换也可作为图像置乱的正变换,相应的反变换就是Arnold变换.在此基础上,把二维反变换算法推广到m维的情形.实验结果表明,对于已应用Arnold变换进行预处理的置乱图像,在无须计算原图像变换周期的前提下可快速实现图像的逆置乱,该过程具有确定性,其迭代次数与预处理置乱次数相等.     

Method of approximations for solving problems of the linear theory of thermoviscoelasticity

A general approximate method of solving problems of the linear theory of thermoviscoelasticity is proposed. Use is made of the Laplace transformation and certain properties of the dependence of solutions to problems of the theory of elasticity on Poisson's ratio that make possible a simple approximation. As a result, the inverse transforms become elementary and the general solution of the problem is expressed by creep and relaxation functions.Mekhanika Polimerov, Vol. 4, No. 2, pp. 210–221, 1968     

Generalized Cole–Hopf Transformation

On the basis of generalization of the Cole–Hopf transformation for parabolic equations with a source, we obtain some new representations of solutions and coefficients of nonlinear parabolic equations of mathematical physics which in fact are differential-algebraic identities. These representations can be used in studying the multidimensional direct and inverse problems.     

Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa–Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.     

Algorithms for the Split Variational Inequality Problem

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.     

The Eshelby tensor

The problem of the existence of a tensor that is inverse to the well-known Eshelby tensor, which connects the free homogeneous and hindered strains of an ellipsoidal elastic inclusion undergoing transformation, is investigated. It is shown that this tensor exists for inclusions in the form of oblate and prolate spheroids in isotropic elastic space. Certain applications are considered, in particular problems of determining the stresses in ellipsoidal rigid and rigid plastic inclusions.     

The formulae and algorithms for Lagrange-power basis transformation and Lagrange-Newton transformation

The interpolation polynomials based on Lagrange, Newton and power basis play important roles in applied mathematics, computing method and many other emerging applications. In this paper, we present some coordinate transformation formulae and algorithms as demonstrated below. Firstly, we put forward the formulae of the Lagrange-power basis transformation and its inverse transformation, and as a byproduct, we provide a new method to arrive at the inversion of the Vandermonde matrix. Secondly, we give the formulae of Lagrange-Newton transformation and its inverse transformation. Moreover, we construct related algorithms of Lagrange-power basis transformation, Lagrange-Newton transformation and their inverse transformations.     

High‐Order Soliton Solution of Landau–Lifshitz Equation

The Landau–Lifshitz equation is analyzed via the inverse scattering method. First, we give the well‐posedness theory for Landau–Lifshitz equation with the frame of inverse scattering method. The generalized Darboux transformation is rigorous considered in the frame of inverse scattering transformation. Finally, we give the high‐order soliton solution formula of Landau–Lifshitz equation and vortex filament equation.     

Generating low-discrepancy sequences from the normal distribution: Box–Muller or inverse transform?

Quasi-Monte Carlo simulation is a popular numerical method in applications, in particular, economics and finance. Since the normal distribution occurs frequently in economic and financial modeling, one often needs a method to transform low-discrepancy sequences from the uniform distribution to the normal distribution. Two well known methods used with pseudorandom numbers are the Box–Muller and the inverse transformation methods. Some researchers and financial engineers have claimed that it is incorrect to use the Box–Muller method with low-discrepancy sequences, and instead, the inverse transformation method should be used. In this paper we prove that the Box–Muller method can be used with low-discrepancy sequences, and discuss when its use could actually be advantageous. We also present numerical results that compare Box–Muller and inverse transformation methods.     

Variational methods of constructing monotone approximations for atmospheric chemistry models

A new method of constructing efficient monotone numerical schemes for solving direct, adjoint, and inverse atmospheric chemistry problems is presented. It is a synthesis of variational principles combined with splitting and decomposition methods and a constructive implementation of Euler integrating multipliers (EIM) bymeans of a local adjoint problem technique. To increase the efficiency of calculations, a method of decomposing the multicomponent substance transformation operators in terms of the mechanisms of reactions is also proposed. With analytical EIMs, the systems of stiff ODEs are decomposed and reduced to equivalent systems of integral equations solved by noniterative multistage algorithms of a given order of accuracy. An unconventional variational method of constructing mutually consistent algorithms for direct and adjoint problems and sensitivity studies for complex models with constraints is described.     

哈明距离下的网络逆问题研究综述

逆优化问题研究的是如何改变原问题中的权参数,使得某些给定的解是问题在新的权参数下的最优解,且使总的改造费用尽可能少.作为逆优化问题中相对较新的一个分支,哈明距离下的网络逆问题具有较大的理论研究及实际应用价值.此文首先介绍了逆优化问题和哈明距离下的网络逆问题以及它们的应用,然后详细介绍了哈明距离下的网络逆问题的研究动态及使用的研究方法.最后给出了该领域中的一些值得研究的问题.     

Some properties of Umar semigroups: isomorphism theorems,ranks and maximal inverse subsemigroups

In a paper published in 1994, Umar defined an interesting class of transformation semigroups which naturally generalizes the Vagner one-point completion of the symmetric inverse semigroup. In this paper we prove some isomorphism theorems for finite such semigroups and compute their ranks. Moreover, we determine all maximal inverse subsemigroups of an arbitrary transformation semigroup of this type which is not inverse.