Carleman Shift Research Articles

Обобщённая математическая модель основных задач теории упругости для анизотропных тел

Introduction. The assessment of the strength of underground structures is a complex task that requires a comprehensive approach. In many cases empirical methods based on experience are not sufficient. For effective application of analytical methods, it is necessary to use mathematical modelling. This allows to simplify the problem considerably, to consider different variants, to identify cases to obtain a solution in a closed form. Materials and methods. The model is based on a system of singular integral equations with a shift and complex conjugate terms. The solution of the system is the complex elastic potential. Systems of singular equations with a Carleman shift are considered on the space of random functions converging in the mean square to expand the possibility of practical use of the work results. Results. The results of the study are: a mathematical model formulation, the development of an algorithm for reducing a system of singular equations to a system of known Fredholm integral equations of the second kind, a method for determining the solvability conditions and the number of solutions. Discussion. As a result of the study: (1) a general algorithm for finding a solution to a mathematical model is obtained; (2) the methodology for determining the number of solutions and the type of solvability conditions is given; (3) The solution algorithm is extended for the stochastic case; (4) The possibility of wider application of the mathematical apparatus of functions of complex variable for research of strength properties of rocks is proved. Conclusion. The proposed comprehensive approach to assessing the strength characteristics of underground structures, combining a set of analytical methods in conjunction with mathematical modelling, expands the variability of calculation and allows to simplify the problem significantly and obtain a solution in closed form. Resume. The paper develops a mathematical model of the stress state of an elastic body, which allows solving the basic problems of elasticity theory for a variety of media. Suggestions for practical applications and directions for future research. The results of the study make it possible to improve the efficiency of analytical methods in the study of the strength of underground structures. The obtained results are applicable for predicting the strength and safety of underground structures.

The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift

The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift

Noetherian solvability and explicit solution of a singular integral equation with weighted Carleman shift in Besov spaces

In this paper, we prove Noetherian solvability and obtain the index formula for a singular integral equation with a Carleman shift in Besov spaces. We also obtain the existence of the solution of the singular integral equation as well as their explicit representations in Besov spaces.

Noetherian solvability of an operator singular integral equation with a Carleman shift in fractional spaces

In this paper, we obtain conditions of Noetherian solvability and the Index formula for a singular integral equation with a Cauchy kernel and a Carleman shift in Besov space, which is embedded into the space of continuous functions on a closed Lyapunov contour but not into the class of functions satisfying Hölder condition.

Riemann–Hilbert problems with Carleman shift and conjugation for polynomial Beltrami equations

ABSTRACT In this paper, we study Riemann–Hilbert problems with Carleman shift and conjugation on the Lyapunov curve for null solutions of some certain polynomial Beltrami equations. Firstly, by constructing and verifying an explicit weakly singular kernel via a shift transformation between Lyapunov curves, and using the Fredholm integral equations method based on Cauchy's formula associated with the Beltrami equations, new integral representations of these generalized analytic functions with order one are obtained. Next, we introduce the notion of the canonical matrix in the context of some Beltrami equations and develop its theory, especially the method of constructing the explicit canonical matrix for triangular matrix functions. Finally, by using the decomposition theorem, the conversion method as well as constructing the canonical matrix, we obtain explicit formulae of solutions and conditions of solvability for the problems mentioned above.

Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.

Generalized Toeplitz Plus Hankel Operators: Kernel Structure and Defect Numbers

Generalized Toeplitz plus Hankel operators $$T(a)+H_{\alpha }(b)$$ generated by functions a, b and a linear fractional Carleman shift $$\alpha $$ changing the orientation of the unit circle $${\mathbb {T}}$$ are considered on the Hardy spaces $$H^p({\mathbb {T}})$$ , $$1<p<\infty $$ . If the functions $$a,b\in L^\infty ({\mathbb {T}})$$ and satisfy the condition $$\begin{aligned} a(t) a(\alpha (t))=b(t) b(\alpha (t)),\quad t\in {\mathbb {T}}, \end{aligned}$$ the defect numbers of the operators $$T(a)+H_{\alpha }(b)$$ are established and an explicit description of the structure of the kernels and cokernels of the operators mentioned is given.

Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted H&amp;#246;lder Spaces

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Hölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Hölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

A Collocation Method for a Class of Nonlinear Singular Integral Equations with a Carleman Shift

The paper is concerned with the applicability of the collocation method to a class of nonlinear singular integral equations with a Carleman shift preserving orientation on simple closed smooth Jordan curve in the generalized Holder space Hφ(L).

Defect numbers of singular integral operators with Carleman shift and almost periodic coefficients

This paper is devoted to the study of properties of the kernel and the cokernel of singular integral operators with almost periodic coefficients and a Carleman shift. In particular, the dimensions of their kernels and cokernels are obtained. This is done by considering appropriate properties of the related almost periodic elements and, in special, the partial indices of some of their relevant factorizations.

Solvability of some singular integral equations on the circle with the shift

The paper is devoted to investigation of the solvability of some characteristic singular integral equations on the circle with the Carleman shift ω(t) = −t. Under some additional requirements on the coefficients, an explicit description of the kernels and the cokernels of the corresponding operators is given. The investigation method is based on the reduction of the solvability problem to the factorization of some matrix function.

Explicit solutions of Cauchy singular integral equations with weighted Carleman shift

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.

Reduction of singular integral operators with flip and their Fredholm property

This paper deals with singular integral operators with a reverting orientation Carleman shift defined on the classic Lebesgue space and having essentially bounded functions as coefficients. We will use similarity relations to show that the mentioned operators are equivalent tomatrix Toeplitz plus Hankel operators acting on the corresponding Hardy space. The main purpose is to extract Fredholm characteristics of the initial operators (in the form of necessary and sufficient conditions). Namely, Fredholm criteria are obtained for some of the operators under study when they have coefficients in the classes of continuous, piecewise continuous, and semi-almost-periodic functions. In addition, Fredholm index formulas are also provided in some of these cases.

On a Boundary Value Problem for Pairs of Polyanalytic Functions

The necessary and sufficient conditions of Fredholmness and the index formula for a boundary value problems for a pair of polyanalytic functions, containing differential operators as well as operators with Carleman shift and conjugations, are obtained.

On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel

This article deals with the solvability, the explicit solutions of a class of singular integral equations with a linear-fractional Carleman shift and the degenerate kernel on the unit circle by means of the Riemann boundary value problem and of a system of linear algebraic equations. All cases about index of the coefficients in the equations are considered in detail.

Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case

AbstractThis paper deals with what we call modified singular integral operators. When dealing with (pure) singular integral operators on the unit circle with coefficients belonging to a decomposing algebra of continuous functions it is known that a factorization of the symbol induces a factorization of the original operator, which is a representation of the operator as a product of three singular integral operators where the outer operators in that representation are invertible.The main purpose of this paper is to obtain a similar operator factorization for the case of singular integral operators with a backward shift and to extract from there some consequences for their Fredholm characteristics. At the end of the paper it is shown that the operator factorization is also possible for other classes of singular integral operators, namely those including either a conjugation operator or a composition of a conjugation with a forward shift operator. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Calculation of the Defect Numbers of the Generalized Hilbert and Carleman Boundary Value Problems with Linear Fractional Carleman Shift

The defect numbers of the generalized Hilbert and Carleman boundary value problems with a direct or an inverse linear fractional Carleman shift of order 2 (α (α (t)) ≡ t) on the unit circle Open image in new window are computed. The approach followed consists of the reduction of the mentioned problems to singular integral equations with linear fractional Carleman shift and of the factorization of Hermitian matrix functions with negative determinant.

Singular Integral Operators with Carleman Shift and Discontinuous Coefficients in the Spaces $H^{\omega}_0(\Gamma,\rho)$

For the singular integral operators with Carleman shift, preserving or changing orientation, and piecewise continuous coefficients we prove the theorem on Fredholmness and obtain the formula for index in the generalized Holder spaces \(H^{\omega}_0(\Gamma,\rho)\) defined by an arbitrary continuity modulus ω from the Bari-Stechkin class and some general weights on a closed or open finite Lyapunov curve Γ.

On a Class of Nonlinear Singular Integral Equations with Carleman Shift in Generalized Holder Spaces

The paper concerns with the investigation of a class of nonlinear singular integral equations with Carlernan shift. Existence results are given by application of Schauder's fixed-point theorem to this type of equations in generalized Holder space.

On a class of nonlinear singular integral equations with shift on a closed contour

In this article we discuss the applicability of an approximation method to a class of nonlinear singular integral equations with Carleman shift, forming the finite group of iterations, preserving or changing orientation.