Stable Determination of X-Ray Transforms of Time Dependent Potentials from Partial Boundary Data

We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term q(t, x), we can recover the X-ray transform of time dependent potentials q(t, x) from the dynamical Dirichlet-to-Neumann map in a stable way. We derive conditional Hölder stability estimates for the X-ray transform of q(t, x). The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.     

Quasijets in anisotropic media, Finsler geometry, and Fermi coordinates

The Hamilton-Jacobi equations for the phase function of quasijet solutions in the case of Finsler geometry are considered. This case corresponds to the physical problem of wave propagation in anisotropic media. The wave field corresponding to a quasijet solution propagates along a geodesic. For this reason, all computations are performed in Fermi coordinates near a geodesic. Upon extracting the frequency factor, the quadratic term of the phase function satisfies the covariant Riccati equation. A notably simple form for the equation is obtained in the case of Riemannian geometry. The nontrivial coefficients of the Riccati equation coincide with the elements of the curvature tensor. In the case of Finsler geometry, all considerations are more complicated. Nevertheless, of cricial importance in the Riccati equation are the elements of the third Cartan curvature tensor computed at tangential elements to the geodesic. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 48–69.     

Painlevé transcendents and the Hankel determinants generated by a discontinuous Gaussian weight

This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension in a previous study, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlevé IV. In addition, we consider the large n behavior of the corresponding orthogonal polynomials and prove that they satisfy the biconfluent Heun equation. We also consider the jump at the edge under a double scaling, from which a Painlevé XXXIV appeared. Furthermore, we study the Gaussian weight with two jumps and show that a quantity related to the Hankel determinant satisfies a two variables' generalization of the Jimbo‐Miwa‐Okamoto σ‐form of the Painlevé IV.     

Power expansions for the self-similar solutions of the modified Sawada-Kotera equation

The fourth-order ordinary differential equation that defines the self-similar solutions of the Kaup—Kupershmidt and Sawada—Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points z = 0, z = ∞ and near an arbitrary point z = z ₀ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.      

Nonstationary Electromagnetic Gaussian Beams in Inhomogeneous Anisotropic Media

Nonstationary Gaussian beams of quasiphoton type for the Maxwell equation with an arbitrary anisotropy are constructed. The solutions of the Maxwell equations can be described as ray-type solutions with complex phases and amplitudes. Owing to a large parameter p, they are concentrated in small neighborhoods of space-time rays corresponding to different types of electromagnetic waves in an anisotropic medium. Bibliography: 6 titles.     

Riesz transform,Gaussian bounds and the method of wave equation

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL^– on L^p for some > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.As an application of the obtained results we prove boundedness of the Riesz transform on L^p for all p (1,2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L^p of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.Mathematics Subject Classification (1991): 42B20The author was partially supported by Summer Research Award from New Mexico State University.in final form: 8 June 2003     

Decaying modes for the wave equation in the exterior of an obstacle

In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ_k} of Δ on the geometry of the obstacle ??. In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle ??, both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λ_k} can be determined as roots of special functions—upper and lower bounds for the density of the real {λ_k}, and upper and lower bounds for λ₁, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on ??. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.     

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

A second-order Monte Carlo method for the solution of the Ito stochastic differential equation

A difference approximation that is second-order accurate in the time step his derived for the general Ito stochastic differential equation. The difference equation has the form of a second-order random walk in which the random terms are non-linear combinations of Gaussian random variables. For a wide class of problems, the transition pdf is joint-normal to second order in h; the technique then reduces to a Gaussian random walk, but its application is not limited to problems having a Gaussian solution. A large number of independent sample paths are generated in a Monte Carlo solution algorithm; any statistical function of the solution (e.g., moments or pdf's) can be estimated by ensemble averaging over these paths     

Image analysis using p -Laplacian and geometrical PDEs

Minimizing the integral ∫_Ω1/p |∇L |^p d Ω for an image L under suitable boundary conditions gives PDEs that are well-known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p -Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L ( x , t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

On the inner curvature of the second fundamental form of helicoidal surfaces

Let M be a helicoidal surface in E ³, free of points of vanishing Gaussian curvature. Let H be the mean curvature and K _II the curvature of the second fundamental form. In this note it is shown that the helicoidal surfaces satisfying K _II =H are locally characterized by constancy of the ratio of the principal curvatures. Moreover it is proved that these helicoidal surfaces are determined by a first order differential equation. Research supported by E.E.C. contract CHRX-CT92-0050.     

Matrix Measures, Random Moments, and Gaussian Ensembles

We consider the moment space M_n\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S_1,n, ... , S_n,n)^* ~ U (M_n)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S _1,n ,…,S _k,n )^∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space M_n\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.     

Steady Ricci flows

Steady solutions for Ricci flows are given. A class of Riemannian 3-manifolds related to the geometry of a surface is considered. The components of the metric tensor, which reproduce the Riemannian space and a triorthogonal coordinate system, are determined by a system of partial differential equations. In the stationary case, the curvature tensor of the space satisfies six equations determining the metric of the space. The exact analytic solutions corresponding to surfaces of constant Gaussian and mean curvature (n = 3) are written. Arbitrary curvilinear coordinate systems are constructed, on which the construction of structured grids is based.     

SL(2)-Solution of the Pentagon Equation and Invariants of Three-Dimensional Manifolds

We construct an algebraic complex corresponding to a triangulation of a three-manifold starting with a classical solution of the pentagon equation, constructed earlier by the author and Martyushev and related to the flat geometry, which is invariant under the group SL(2). If this complex is acyclic (which is confirmed by examples), we can use it to construct an invariant of the manifold.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Type constants and (q, 2)-summing norms defined byn vectors

Forq≧2, the (q, 2)-summing norm of an operator of rankn can be computed, up to a constantc _q, by an appropriate choice of at mostn vectors. A corresponding statement is true for the Gaussian type and cotype constants ofn-dimensional spaces.     

Influence of helicity on the turbulent Prandtl number: Two-loop approximation

Using the field theory renormalization group technique in the two-loop approximation, we study the influence of helicity (spatial parity violation) on the turbulent Prandtl number in the model of a scalar field passively advected by the helical turbulent environment given by the stochastic Navier-Stokes equation with a self-similar Gaussian random stirring force δ-correlated in time with the correlator proportional to k ^4−d−2ɛ. We briefly discuss the influence of helicity on the stability of the corresponding scaling regime. We show that the presence of helicity increases the value of the turbulent Prandtl number up to 50% of its nonhelical value.