Volume 8, Issue 3

- Considering some integral geometry problems it is used a so-called Fourier - Gelfand - Graev transformation. This mapping transforms an arbitrary function defined on the Lobachevsky space into a homogeneous function on a cone. For a function to be the Fourier -Gelfand -Graev image it is necessary and sufficient for some condition (symmetry condition) to be valid. In the present paper different forms of the symmetry condition are studied. A relation between the symmetry condition and a theorem of the expansion in continuous spectrum eigenfunctions of the Beltrami - Laplace operator on the Lobachevsky space is established.

- A class of inverse problems in viscous flow is studied. A subdifferential condition is considered as an additional information about flow to determination not only the flow field but also the external conditions that bring on a flow. The abstract evolution inverse problem is reduced to evolution variational inequality. The applications to special inverse problems of hydrodynamics are considered.

- We study a noncharacteristic Cauchy problem for a linear parabolic equation in one space variable. This problem is ill-posed. We prove that solutions that are bounded in the supremum norm depend continuously upon Cauchy data and the same is true for regular level lines of such solutions.

- Consider an anisotropic conductor of electric current which occupies a domain Ω ⊂ℝ 2 with conductivity coefficient A = A 0 +BχD, where D is a subdomain of Ω, A 0 and A 0 +B are real, symmetric, uniformly positive definite and 2×2-matrix-valued functions. Denote by ΛA(f) the flux across ∂Ω induced by an electric potential f on ∂Ω. We prove that D can be uniquely determined by Λ A (f) for infinitely many f provided det A is discontinuous on ∂D and B is small on ∂D.

- The heat equation and wave equation are considered. Then class of their exact solutions is constructed both in the one-dimensional and multidimensional case. Multidimensional solutions have functional arbitrariness which can be used for solution of inverse problems.

- Assume that the potential q(r), r > 0, is known for r ≥ a > 0 and the phase shifts δ l (k) are given at a fixed energy, i. e., at a fixed k > 0, for l = 0, 1, 2, . . .. The inverse scattering problem is: find q(r) on the interval 0 ≤ r ≤ a using the above data. The proposed numerical method consists of a reduction of this problem to a moment problem for q(r) on the interval r ∈ [0, a]. The moment problem is solved numerically, the results are presented.

- We define a notion of graphical convergence of a net of functions between topological spaces X and Y . This convergence, which differs from that of Kuratowsky by being basically function theoretic rather than set theoretic, is used to define a space of multifunctions Multi(X, Y ) as a natural extension of function spaces, in as far as functional relations are expected to be dense in the space of all relations between two sets. This convergence admits a new definition of continuity of multifunctions that is defined in Section 3. Multi(X, Y ) appears to be a natural space for the evolution of discrete dynamical systems towards chaos, and Sections 4 and 5 explore this possibility based on the fundamental ill-posedness of the problem.

- The paper is devoted to the reconstruction problems of a compactly supported symmetric tensor field in ℝ n , if its ray transform is known along the lines, intersecting a given curve. Two versions of the reconstruction are considered.