Cauchy Operator Research Articles

Spectrum of the Kerzman-Stein Operator for the Ellipse

The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.

Evolutionary programming using a mixed mutation strategy

Different mutation operators have been proposed in evolutionary programming, but for each operator there are some types of optimization problems that cannot be solved efficiently. A mixed strategy, integrating several mutation operators into a single algorithm, can overcome this problem. Inspired by evolutionary game theory, this paper presents a mixed strategy evolutionary programming algorithm that employs the Gaussian, Cauchy, Lévy, and single-point mutation operators. The novel algorithm is tested on a set of 22 benchmark problems. The results show that the mixed strategy performs equally well or better than the best of the four pure strategies does, for all of the benchmark problems.

Optimal Power Flow by Improved Evolutionary Programming

This article proposes an improved evolutionary programming (IEP) for solving optimal power flow (OPF) with nonsmooth and nonconvex generator fuel cost curves. Initially, the whole population is divided into multiple subpopulations, which are used to perform the parallel search in divided solution space. IEP includes Gaussian and Cauchy mutation operators in different subpopulations to enhance the search diversity, selection operators with probabilistic updating strategy to avoid entrapping in local optimum, and reassignment operator for every subpopulation to exchange search information. The proposed IEP was tested on the IEEE 30 bus system with three different types of generator fuel cost curves. It is shown that IEP total generator fuel cost is less expensive than those of evolutionary programming, tabu search, hybrid tabu search and simulated annealing, and improved tabu search, leading to substantial generator fuel cost savings. Moreover, IEP can easily facilitate parallel implementation to reduce the computing time without sacrificing the quality of solution.

Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge--Dirac operator on compact manifolds depend analytically on $L_\infty$ changes in the metric. We also recover a unified proof of many results in the Calder\'on program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.

Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative

A diagonalization algorithm for a matrix pencil depending on a variable and a parameter in the cases when the limiting matrix has a simple spectrum or a multiple eigenvalue for all values of the variable is put forward. The algorithm uses an exhaustive superposition of special similarity transformations. Formulae for the Cauchy operator of a linear non-stationary equation with a small parameter at the derivative and with a matrix pencil are obtained for various degeneracy orders of the structure matrix.

On the existence of mild solutions of semilinear evolution differential inclusions

In this paper we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion: x ′ ( t ) ∈ A ( t ) x ( t ) + F ( t , x ( t ) ) and with initial data x ( 0 ) = x 0 ∈ E , where { A ( t ) } t ∈ [ 0 , d ] is a family of linear operators in the Banach space E generating an evolution operator and F is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii, Obukhovskii and Zecca [Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 7, de Gruyter, Berlin, 2001] for inclusions with constant operator.

Spectrum of the Kerzman?Stein Operator for Model Domains

For a domain Ω ⊂ C, the Kerzman-Stein operator is the skew- hermitian part of the Cauchy operator acting on L 2 (bΩ), which is defined with respect to Euclidean measure. In this paper we compute the spectrum of the Kerzman-Stein operator for three domains whose boundaries consist of two circular arcs: a strip, a wedge, and an annulus. We also treat the case of a domain bounded by two logarithmic spirals.

On finite difference potentials and their applications in a discrete function theory

AbstractWe present a potential theoretical method which is based on the approximation of the boundary value problem by a finite difference problem on a uniform lattice. At first the discrete fundamental solution of the Laplace equation is studied and the theory of difference potentials is described. In the second part we define a discrete Cauchy integral operator and a Teodorescu transform. In addition a Borel–Pompeiu formula can be formulated. Copyright © 2002 John Wiley & Sons, Ltd.

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderon-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szego projections and the corresponding Lp-decompositions. Our approach relies on an extension of the classical Calderon-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

Modeling of aerodynamic and wave diffraction problems and extensions of Cauchy type integral operators on closed and open curves

For a long time, numerous problems in various applied fields [1-3] have been investigated with the use of singular integral equations. For example, some problems of aerodynamics [1], diffraction of electromagnetic and acoustic waves [2], elasticity [3], etc. can be reduced to the solution of such equations. The theory of singular integral equations was developed long ago [3], but related applied problems required the solution of these equations in the class of absolutely integrable functions. It was mentioned in [4] that, under specific conditions imposed on the coefficients of a singular integral equation, one can find solutions in the class of fimctions with nonintegrable singularity of the form 1/x as x -* 0. But the absence of applied problems leading to such solutions, numerical methods for their construction, and a theory of distributions developed for singular integral equations hindered the investigation of such singular solutions. However, in the framework of aerodynamics [5, 6], the mechanics of wings has recently been analyzed with the use of ejection of an external flow. It was shown in [6] that, in the case of a thin profile, this problem can be reduced to solving a singular integral equation of the first kind on a closed interval in the class of fimctions that have a nonintegrable singularity of the above-mentioned type at the ejection point. By [7], the investigation of the diffraction of an electromagnetic wave on an ideal conductor with a current thread on the surface leads to a similar mathematical problem. Such problems were numerically solved in [1, 7, 8] in sorne special cases with the use of the discrete vortex and discrete current methods. However, the integral operators constructed in these papers are continuously invertible in related pairs of spaces but have the property that their squares map the entire subspace of singular solutions to zero. In the present paper, in view of [9, 10], we give another mathematical treatment of the same applied problems, which permits one to consider them in the framework of the theory of pseudodifferential operators on a closed curve. Then we describe the construction of Sobolev type weighted spaces and develop the theory of linear operators (similar to pseudodifferential ones) in these spaces for open curves. We present examples illustrating the difference between the notions of integral in the sense of the Cauchy principal value and pseudodifferential operators on distributions that are singular solutions. Similar linear operators are constructed for singular integral equations of the second kind with Hilbert kernel and constant coefficients on a closed interval. We indicate the possibility of the generalization of the suggested construction of linear operators to some spatial problems.

Fast evolving multi-layer perceptrons for noisy chaotic time series modeling and predictions

A fast evolutionary programming (FEP) is proposed to train multi-layer perceptrons (MLP) for noisy chaotic time series modeling and predictions. This FEP, which uses a Cauchy mutation operator that results in a significantly faster convergence to the optimal solution, can help MLP to escape from local minima. A comparison against back-propagation-trained networks was performed. Numerical experimental results show that the FEP can help MLP better capturing dynamics from noisy chaotic time series than the back-propagation algorithm and produce a more consistently modeling and prediction.

Effects of crack surface convection for rapid crack growth in a thermoelastic solid

As a Green’s function for rapid steady-state crack growth with crack surface convection, semi-infinite Mode I crack growth at subcritical speeds in an unbounded solid under the action of compressive line loads moving on the crack surfaces is considered. A standard convection law that relates heat flux to change in temperature is employed, and the solid obeys the fully-coupled (dynamic) equations of thermoelasticity. The use of robust asymptotic forms reduces the problem to the solution of coupled integral equations. These exhibit both Cauchy and Abel operators, but an exact solution is possible. The solution indicates that convection can give rise to temperature changes in the crack plane that are both more prominent and extensive than those that occur for an insulated crack surface. Exact expressions for the thermoelastic Rayleigh speed, which is the critical crack speed, and for speeds that arise for a particular value of an important characteristic parameter are also presented.

PROCESS-ALTERED SURFACE CONVECTION EFFECTS IN A COUPLED THERMOELASTIC ANALYSIS OF RAPID SLIDING INDENTATION WITH FRICTION

Rapid sliding rigid indentation with friction of a half-space is analyzed as a two-dimensional steady-state process governed by fully coupled thermoelasticity. While rapid sliding justifies an insulated contact zone, the half-space surface outside the zone is allowed to (i) exchange heat with the atmosphere via convection and (ii) have its convective properties altered by the sliding process itself. The problem boundary conditions are mixed in both mechanical and thermal fields, yet robust asymptotic forms are used to reduce analysis to a simply coupled integral equation set that exhibits both Abel and Cauchy operators. Exact solutions show that convection has no influence on contact zone extent, although other thermoelastic parameters do. The surface temperature, however, is both controlled by friction and influenced by convection; effects of the latter are even more pronounced far away from the indentor.

Effects of Mixed-Mode and Crack Surface Convection in Rapid Crack Growth in Coupled Thermoelastic Solids

Two Green’s function problems for rapid two-dimensional steady-state crack growth governed by fully coupled (dynamic) linear thermoelasticity are analyzed. In Problem A, normal and in-plane shear line loads move on the insulated surfaces of a semi-infinite crack growing at a subcritical speed. Problem B involves only normal line loads, but crack surface convection is allowed. Problem A involves, therefore, mixed traction/displacement boundary conditions, while Problem B also exhibits mixed thermal boundary conditions. Robust asymptotic forms based on exact solutions for related problems reduce Problems A and B to coupled sets of integral equations. Both sets exhibit both Cauchy and Abel operators, but are solved exactly. The solutions show that Mode II loading couples the tangential crack face separation and discontinuity in crack-face temperature changes, while crack surface convection enhances thermal response, especially at large distances. [S0021-8936(00)03101-9]

Spectral Properties of the Cauchy Operator and Its Product with Bergman's Projection on a Bounded Domain

Spectral Properties of the Cauchy Operator and Its Product with Bergman's Projection on a Bounded Domain

Reflected double layer potentials and Cauchy's operators

Necessary and sufficient conditions are given for the reflected Cauchy's operator (the reflected double layer potential operator) to be continuous as an operator from the space of all continuous functions on the boundary of the investigated domain to the space of all holomorphic functions on this domain (to the space of all harmonic functions on this domain) equipped with the topology of locally uniform convergence.

Maxwell's square roots from the metric tensors of wave surfaces and branches of solutions of the photon and phonon wave equations

For the wave equations of optics and acoustics of isotropic media the infinite families of three-dimensional plane wave Cauchy operators are found by a direct tensor method. These families form involutive Lie groups. Their generators N can be found by taking square roots from the unit tensors of the wavefront subspaces with outer normal n. In optics the basic structural elements of N are complex involutive operators (reflection isometries) described by pairs of complex vectors S and C, which satisfy the metric condition , and also by a pair of projective operators of the two-dimensional space of a plane orthogonal to n. In the acoustics of isotropic media, in view of the inequality of the longitudinal and transverse wave velocities, the generators N are represented as a linear combination of an involutive operator and a diad . It is shown that the projection of the average energy flux of the wave is conserved in the general case . The families of vectors , , , , , , being a part of N, are indicated. For these families the global operators acting on initial-field vectors give states described by the right-hand and left-hand elliptical helices. The wave normal n characterizes the direction of the angular momentum of the field and for the case turns out to be equivalent to the Darboux vector known in geometry.

On a Generalized Integro-Differential Equation of Barbashin Type

We study a class of integro-differential equations containing multiplication opera-tors, partial integral operators, and ordinary integral operators. Building on the usual identification of real functions of several variables and abstract functions, such integro- differential equations may be reformulated as ordinary differential equations in suitable Banach spaces. We give a representation theorem for the corresponding Cauchy operator and study the (unique) solvability of a general boundary value problem.

On the Boundedness of Cauchy Singular Operator from the Space Lp to Lq , p > q ≥ 1

Abstract It is proved that for a Cauchy type singular operator, given by equality (Khvedelidze, Current problems of mathematics 7: 5-162, 1975), to be bounded from the Lebesgue space Lp (Γ) to Lq (Γ), as , it is necessary and sufficient that either condition (Calderon, Proc. Nat. Acad. Sci. USA 4: 1324-1327, 1977) or (David, L'integrale de Cauchy sur le courbes rectifiables, Prepublications Univ. Paris–Sud; Dept. Math., 1982) be fulfilled.

On the boundedness of Cauchy singular operator from the spaceL p toL q ,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue space Lp(€) to Lq(€), as € = ( 1=1 €n, €n = fz : jzj = rng, it is necessary and sucient that either condition (4) or (5) be fulÞlled.