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October 20, 2010
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The inverse nodal problem of recovering integral-differential operators with the Sturm–Liouville differential part and the integral part of Volterra type is studied. We reconstruct the potential and the boundary conditions provided the kernel of integral perturbation is known. We prove a uniqueness theorem and provide reconstruction formulae for the potential and parameters of the boundary conditions.
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October 20, 2010
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The algorithm of reconstruction of pressure velocities in a thinly stratified layer and of the location of the gap points of medium is suggested. The analytic expressions for derivatives of residual functional with respect to the pressure velocities in thin layers and by the coordinate of gap points of medium are obtained. The suggested algorithm is tested by numerical examples.
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October 20, 2010
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In this work the possibility of restoration of real symmetrical five diagonal final matrices using four numerically sequences is studied. Three from these four numerically sequences are interpreted as sets of eigenvalues of the considered matrix and else of two matrices, obtained from considered matrix deleting some diagonal elements. The concrete formulas of construction of matrix elements using four sets of eigenvalues are obtained.
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October 20, 2010
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We generalize the inversion formulas obtained by Pestov–Uhlmann for the geodesic ray transform of functions and vector fields on 2-dimensional manifolds with boundary of constant curvature. Our formulas hold for simple 2-dimensional manifolds whose curvatures are close to a constant.
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October 20, 2010
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The regularized inversion of real-valued Laplace transforms computable at any point on the real axis is discussed from the point of view of practical calculations. New criterion for selection of free parameters is suggested. Selection of optimal values of free parameters allows to improve the numerical results significantly. The effectiveness of the proposed criterion is demonstrated with examples. Method can be used in conjunction with other numerical methods for problems where the inverse Laplace transform is expected to tend to a monotonic function.
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October 20, 2010
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In this paper we investigate the generalized Gauss–Newton method in the following form x n +1 = ξ n – θ ( F ′*( x n ) F ′( x n ), τ n ) F ′*( x n ){( F ( x n ) – ƒ δ ) – F ′( x n )( x n – ξ n )}, x 0 , ξ n ∈ 𝒟 ⊂ H 1 . The modified source condition which depends on the current iteration point x n , is used. We call this inclusion the undetermined reverse connection . The new source condition leads to a much larger set of admissible control elements ξ n as compared to the previously studied versions, where ξ n = ξ . The process is combined with a novel a posteriori stopping rule, where is the number of the first transition of ‖ F ( x n ) – ƒ δ ‖ through the given level δ ω , 0, < ω < 1, i.e., The convergence analysis of the proposed algorithm is given.
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October 20, 2010
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In this paper we study two problems concerned with recovering memory kernels related to two sub-bodies Ω 1 and Ω 2 of an open thermal body under the assumptions that and is not accessible for the measurements. Additional measurements of temperature gradient or flux type are provided on ∂ Ω. In the first problem the memory kernel related to Ω 1 is unknown and a single measurement is given. In the second problem both kernels are to be determined from two measurements on ∂ Ω. Making use of Laplace transforms, we prove the uniqueness for these identification problems in the infinite time interval (0, ∞).
