Hadamard Problem Research Articles

NONLOCAL BOUNDARY VALUE PROBLEM IN SPACES OF EXPONENTIAL TYPE OF DIRICHLET-TAYLOR SERIES FOR THE EQUATION WITH COMPLEX DIFFERENTIATION OPERATOR

Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.

Quantum computation with coherent spin states and the close Hadamard problem

We study a model of quantum computation based on the continuously-parameterized yet finite-dimensional Hilbert space of a spin system. We explore the computational powers of this model by analyzing a pilot problem we refer to as the close Hadamard problem. We prove that the close Hadamard problem can be solved in the spin system model with arbitrarily small error probability in a constant number of oracle queries. We conclude that this model of quantum computation is suitable for solving certain types of problems. The model is effective for problems where symmetries between the structure of the information associated with the problem and the structure of the unitary operators employed in the quantum algorithm can be exploited.

On the existence of flat orthogonal matrices

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $${\frac{1}{\sqrt{n}}}$$ . Entries of $${\pm \frac{1}{\sqrt{n}}}$$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem.

Нелокальна крайова задача для рівняння з частинними похідними у комплексній області

Досліджено нелокальну крайову задачу для рівняння з частинними похідними з оператором узагальненого диференціювання $B=z\frac{\partial}{\partial z}$, який діє на функції скалярної комплексної змінної $z$. Доведено теорему єдиності та теореми існування розв'язку задачі у просторі $\mathbf{H}_{q}^n(\mathcal{D})$. Встановлено умови бієктивності оператора нелокальних умов задачі. Показано коректність за Адамаром задачі, що відрізняє її від некоректної за Адамаром задачі з багатьма просторовими комплексними змінними, розв'язність якої пов'язана з проблемою малих знаменників.

Features of gravitational waves in higher dimensions

There are several fundamental differences between four-dimensional and higher-dimensional gravitational waves, namely in the so called braneworld set-up. One of them is their asymptotic behaviour within the Cauchy problem. This study is connected with the so called Hadamard problem, which aims at the question of Huygens principle validity. We investigate the effect of braneworld scenarios on the character of propagation of gravitational waves on FRW background.

Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients

Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients

Decomposition of Binary Matrices and Fast Hadamard Transforms

Binary matrices or (± 1)-matrices have numerous applications in coding, signal processing, and communications. In this paper, a general and efficient algorithm for decomposition of binary matrices and the corresponding fast transform is developed. As a special case, Hadamard matrices are considered. The difficulties of the construction of 4n-point Hadamard transforms are related to the Hadamard problem: the question of the existence of Hadamard matrices. (It is not known whether for every integer n, there is an orthogonal 4n × 4n matrix with elements ± 1.) In the derived fast algorithms, the number of real operations is reduced from O(N2) to O(N log N) compared to direct computation. The proposed scheme requires no zero padding of the input data. Comparisions revealing the efficiency of the proposed algorithms with respect to the known ones are given. In particular, it is demonstrated that, in typical applications, the proposed algorithm is significantly more efficient than the conventional Walsh-Hadamard transform. Note that for Hadamard matrices of orders ≥ 96 the general algorithm is more efficient than the classical Walsh-Hadamard transform whose order is a power of 2. The algorithm has a simple and symmetric structure. The results of numerical examples are presented.

Multidimensional Baker–Akhiezer Functions and Huygens' Principle

A notion of the rational Baker–Akhiezer (BA) function related to a configuration of hyperplanes in Cn is introduced. It is proved that the BA function exists only for very special configurations (locus configurations), which satisfy a certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrodinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.

Multidimensional Baker–Akhiezer Functions and Huygens' Principle

A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C^n is introduced. It is proved that BA function exists only for very special configurations (locus configurations), which satisfy certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrodinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard's problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.

Solution of a restricted Hadamard problem on Minkowski spaces

Solution of a restricted Hadamard problem on Minkowski spaces

Calogero quantum problem, Knizhnik-Zamolodchikov equation, and Huygens principle

The interrelations between Calogero quantum problem and Knizhnik-Zamolodchikov equation are described following Matsuo, Cherednik, Felder, and the author. As the basic tool of the considerations the Dunkl operator is used. The generalizations related to an arbitrary Coxeter group and the applications to the Hadamard problem about the hyperbolic equations with the Huygens principle are discussed.

Hadamard's problem and coxeter groups: New examples of Huygens' equations

Hadamard's problem and coxeter groups: New examples of Huygens' equations

Solution of the Hadamard problem by a Tikhonov-regularizing algorithm

Solution of the Hadamard problem by a Tikhonov-regularizing algorithm

Reverse problem of penetration of electromagnetic field into conducting region

The following paper presents the use of the method of regularisation for solving the ill-posed, in the sense of Hadamard problem, reverse problem of penetration of the time-dependant electromagnetic field into the conducting region. By applying the method of regularisation it was possible to obtain the boundary condition for the electromagnetic field, if the time dependence of the magnetic field intensity in a point of the investigated region is given. By use of a computer, theoretical considerations were illustrated with numerical calculations.

On Hadamard's problem for higher dimensions.

On Hadamard's problem for higher dimensions.

A Solution of Hadamard's Problem for a Restricted Class of Operators

A Solution of Hadamard's Problem for a Restricted Class of Operators

The algebraic character of a class of harmonic functions in three variables

It is obvious that the coefficients An, of the series development (1) determine completely the function F(r, cos 0, 4). Thus the sequence of coefficients determines the analytic character of F, the locations and nature of singular points, and so on. It seems therefore natural to look for and extract those properties of a sequence of coefficients Any which most readily yield relevant information. In the case of harmonic functions of two variables, the problem reduces immediately to that of detecting the singularities of an analytic function given by its Taylor development. This last mentioned problem, also known as the Hadamard problem, has occupied the attention of mathematicians for a considerable period and a great many results have been obtained. When dealing, however, with harmonic functions of three variables, the problem becomes much more difficult and it seems impossible to apply directly the methods developed in the theory of functions of one complex variable. We have, however, at our disposal the method of integral operators introduced by S. Bergman, which enables us to represent harmonic functions in three variables by means of an integral operator on a function of a complex variable. This representation makes it possible to carry over certain results of the well known theory of functions of one complex variable into the relatively little developed theory of harmonic functions of three variables. The present paper is to be considered as a part of an extended program of study of harmonic functions given by the de-