Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton

The question of the possibility of holomorphic continuation into some domain of functions defined on the entire boundary of this domain has been well studied. The problem of describing functions defined on a part of the boundary that can be extended holomorphically into a fixed domain is attracting more interest. In this article, we reformulate the problem under consideration: Under what conditions can we extend holomorphically to a matrix ball the functions given on a part of its skeleton? We describe the domains into which the integral of the Bochner—Hua Luogeng type for a matrix ball can be extended holomorphically. As the main result, we present the criterion of holomorphic continuation into a matrix ball of functions defined on a part of the skeleton of this matrix ball. The proofs of several results are briefly presented. Some recent advances are highlighted. The results obtained in this article generalize the results of L.A. Aizenberg, A.M. Kytmanov and G. Khudayberganov.

Определение типа отражателя по амплитудам бликов изображений, восстановленных по разным акустическим схемам

In ultrasonic flaw detection, methods for recording and analyzing echo signals have been developed to determine the type of reflector and its size. The method of digital antenna focusing (DFA) allows you to restore the image of the entire boundary of the discontinuity, using echo signals reflected from the bottom of the control object, taking into account the transformation of the wave type. However, this approach is not always applicable in practice, since the shape of the bottom of the object of control may be unknown. Using the features of the behavior of the reflection coefficient for different types of waves, it is possible to make a conclusion about the type of reflector from the images only on the direct beam. Numerical and model experiments confirmed the efficiency of the proposed approach.

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.

Petz map and Python’s lunch

We look at the interior operator reconstruction from the point of view of Petz map and study its complexity. We show that Petz maps can be written as precursors under the condition of perfect recovery. When we have the entire boundary system its complexity is related to the volume/action of the wormhole from the bulk operator to the boundary. When we only have access to part of the system, Python’s lunch appears and its restricted complexity depends exponentially on the size of the subsystem one loses access to.

Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary

The finite element method (FEM) with a special graded mesh is constructed for the Dirichlet boundary value problem with degeneration of the solution on the entire boundary of the two-dimensional domain. A comparative numerical analysis is performed for the proposed method and the classical finite element method for a set of model problems in symmetric domain. Experimental confirmation of theoretical estimates of accuracy is obtained and conclusions are made.

IDEA OF ELECTRICAL TOMOGRAPHY SYSTEM FOR MONITORING LUNG VENTILATION

In many applications of electrical tomography, such as monitoring the lungs of unconscious intensive care patients, data acquisition on the entire boundary of the body is impractical. The boundary area available for electrical tomography measurements is restricted. Physiological processes that produce changes in the electrical conductivity of the body can be monitored by hybrid algorithms. This paper presents the architecture of the system based on electrical tomography.

The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions

It is shown that the Dirichlet problem for the slab (a, b) × ℝd with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function g, the inhomogeneous difference equation h(t + 1, y) − h(t, y) = g(t, y) has an entire harmonic solution h.

Shape Optimization of Acoustic Enclosures Based on a Wavelet–Galerkin Formulation

The problem of the shape optimization of acoustic enclosures is investigated in this paper. A general procedure, comprising a Wavelet–Garlerkin formulation and a so-called vertex-driven shape optimization is proposed to deal with the general problem of internal sound field prediction and the optimization of the boundary shape. It is shown that, owing to the compactly supported orthogonal property and the remarkable fitting ability, Daubechies Wavelet can be used as a global basis to approximate the unknown sound field on a relatively large interval globally instead of piecewise approximation like most of element based methods do. This feature avoids meshing the boundary of the enclosure, although vertex points are needed to define the boundary shape, whose positions keep updating during the shape optimization process. A rectangular enclosure is used as benchmark to assess and validate the proposed formulation, by investigating the influence of some key parameters involved in the formulation. It was shown that the sound pressure along the entire boundary of the rectangular enclosure can be accurately approximated without meshing. The same enclosure with an inner rigid acoustic screen is then used to reduce the sound pressure level within a chosen area through optimizing the shape of the screen, which shows the remarkable potentials of the proposed approach as a shape optimal tool for inner sound field problems.