On direct and inverse problems for fuzzy equation systems with tolerances

We investigate the resolution of fuzzy (relational) equation systems with tolerances which are a certain extension of fuzzy equations considered f.i. in [3–5]. The extension of the concept of Higashi and Klir [3] enables us to describe the set of solutions to our problem (for given tolerances) by means of posets. In a second part we investigate an inverse problem: Given upper (lower) tolerances how to determine lower (upper) tolerances such that the arising problem becomes consistent? Numerical examples are given.     

On inverse crack problems in elastostatics

In solid mechanics, nondestructive testing has been an important technique in gathering information about unknown cracks, or defects in material. From a mathematical point of view, this is described as an inverse problem of partial differential equations, that is, the problem is to extract information about the location and shape of an unknown crack from the surface displacement field and traction on the boundary of the elastic material. By using the enclosure method introduced by Prof. Ikehata we can derive the extraction formula of an unknown linear crack from a single set of measured boundary data. Then, we need to have precise properties of a solution of the corresponding boundary value problem; for instance, an expansion formula around the crack tip. In this paper we consider the inverse problem concentrating on this point. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On inverse traveling salesman problems

The inverse traveling salesman problem belongs to the class of ??inverse combinatorial optimization?? problems. In an inverse combinatorial optimization problem, we are given a feasible solution for an instance of a particular combinatorial optimization problem, and the task is to adjust the instance parameters as little as possible so that the given solution becomes optimal in the new instance. In this paper, we consider a variant of the inverse traveling salesman problem, denoted by ITSP _W,A , by taking into account a set W of admissible weight systems and a specific algorithm. We are given an edge-weighted complete graph (an instance of TSP), a Hamiltonian tour (a feasible solution of TSP) and a specific algorithm solving TSP. Then, ITSP _W,A , is the problem to find a new weight system in W which minimizes the difference from the original weight system so that the given tour can be selected by the algorithm as a solution. We consider the cases ${W \in \{\mathbb{R}^{+m}, \{1, 2\}^m , \Delta\}}$ where ?? denotes the set of edge weight systems satisfying the triangular inequality and m is the number of edges. As for algorithms, we consider a local search algorithm 2-opt, a greedy algorithm closest neighbor and any optimal algorithm. We devise both complexity and approximation results. We also deal with the inverse traveling salesman problem on a line for which we modify the positions of vertices instead of edge weights. We handle the cases ${W \in \{\mathbb{R}^{+n}, \mathbb{N}^n\}}$ where n is the number of vertices.     

On the dual of linear inverse problems

In linear inverse problems considered in this paper a vector with positive components is to be selected from a feasible set defined by linear constraints. The selection rule involves minimization of a certain function which is a measure of distance from a priori guess. Csiszar made an axiomatic approach towards defining a family of functions, we call it α-divergence, that can serve as logically consistent selection rules. In this paper we present an explicit and perfect dual of the resulting convex programming problem, prove the corresponding duality theorem and optimality criteria, and make some suggestions on an algorithmic solution.     

On the one-dimensional two-phase inverse Stefan problems

New formulations of the inverse nonstationary Stefan problems are considered: (a) forx [0,1] (the inverse problem IP₁; (b) forx [0, (t)] with a degenerate initial condition (the inverse problem IP). Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase {x [0, y(t)]{, the solution of the inverse problem is found in the form of a series; on the second phase {x [y(t), 1] orx [y(t), (t)]{, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IP is found for the self similar motion of the boundariesx=y(t) andx=(t).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1058–1065, August, 1993.     

On direct and inverse images ofD-modules in prime characteristic

Letf:XY be a morphism of smooth varieties over an algebraically closed fieldK. IfK=C (or more generallyChar(K)=0) there are well defined and well known functors of direct and inverse images on the category of left resp. rightD-modules as described e. g. in the first chapter of Hotta's book [8]. We generalize these constructions to the caseChar(K)=p>0 roughly following the concept of [8, Chap. 1] using characteristic-p-methods. Finally we prove Kashiwara's equivalence in characteristicp.     

An algorithm for direct and inverse three-dimensional problems in vertical seismic profiling

The article discusses the propagation of plane waves in a three-dimensional homogeneous medium with anisotropy. The zeroth approximation solution (the ray solution) is considered. Algorithms are proposed for solving the direct dynamic VSP problem and for projecting the wave field on the model medium by multiple solution of the direct problem. Simulation results and calculations with real data are reported. __________ Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 106–124, 2005.     

On some structured inverse eigenvalue problems

This work deals with various finite algorithms that solve two special Structured Inverse Eigenvalue Problems (SIEP). The first problem we consider is the Jacobi Inverse Eigenvalue Problem (JIEP): given some constraints on two sets of reals, find a Jacobi matrix J (real, symmetric, tridiagonal, with positive off-diagonal entries) that admits as spectrum and principal subspectrum the two given sets. Two classes of finite algorithms are considered. The polynomial algorithm which is based on a special Euclid–Sturm algorithm (Householder's terminology) and has been rediscovered several times. The matrix algorithm which is a symmetric Lanczos algorithm with a special initial vector. Some characterization of the matrix ensures the equivalence of the two algorithms in exact arithmetic. The results of the symmetric situation are extended to the nonsymmetric case. This is the second SIEP to be considered: the Tridiagonal Inverse Eigenvalue Problem (TIEP). Possible breakdowns may occur in the polynomial algorithm as it may happen with the nonsymmetric Lanczos algorithm. The connection between the two algorithms exhibits a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix. This result is used to illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow convergence, even for a symmetric matrix, since an outer eigenvalue of the tridiagonal matrix of order n − 1 can be arbitrarily far from the spectrum of the original matrix. This revised version was published online in August 2006 with corrections to the Cover Date.     

On inverse problems for the cycle graph operator

The cycle graph of a graph is the intersection graph of the edge set of all the induced cycles ofH. The main result of this paper is: A (K ₄−e)-free graph is a cycle graph if and only if it is a block graph where each vertex lies in a finite number of blocks. Some additional results are also given.     

On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems

We present a principle of approximate solutions of constrained inverse Lipschitz function problems. As corollaries and applications of the principle, we obtain a result of convergence of an approximate solutions sequence for the constrained problems, a conclusion relating direct and inverse images of upper and lower limits of a sequence of subsets, and several versions of inverse Lipschitz function theorems. Finally we give local uniqueness criteria for solutions to constrained nonlinear problems in finite dimension spaces.     

On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

On the unsolvability of inverse eigenvalues problems almost everywhere

Using Sard's theorem we show that the inverse eigenvalue problems are unsolvable almost everywhere if “too many” of the eigenvalues are equal.     

On the realizability of open nonnegative inverse eigenvalue problems

This work is concerned with answering three open nonnegative inverse eigenvalue problems (NIEPs) which have been around for 70 years. Our approach is quite straightforward; it offers effective ways to judge whether a given NIEP is realizable.     

On some inverse problems for elliptic equations and systems

Under study are the inverse problems of determining the right-hand side of a particular form and the solution for elliptic systems, including a series of elasticity systems. (On the boundary of the domain the solution satisfies either the Dirichlet conditions or mixed Dirichlet-Neumann conditions.) We assume that on a system of planes the normal derivatives of the solution can have discontinuities of the first kind. The conjugating boundary conditions on the discontinuity surface are analogous to the continuity conditions for the fields of displacements and stresses for a horizontally laminated medium. The overdetermination conditions are integral (the average of the solution over some domain is specified) or local (the values of the solution on some lines are specified). We study the solvability conditions for these problems and their Fredholm property.