Generalized Analytic Functions Research Articles

A Regularization Process for Electrical Impedance Equation Employing Pseudoanalytic Function Theory

The electrical impedance equation is considered an ill-posed problem where the solution to the forward problem is more easy to achieve than the inverse problem. This work tries to improve convergence in the forward problem method, where the Pseudoanalytic Function Theory by means of the Taylor series in formal powers is used, incorporating a regularization method to make a solution more stable and to obtain better convergence. In addition, we include a comparison between the designed algorithms that perform proposed method with and without a regularization process and the autoadjustment parameter for this regularization process.

On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators

Pseudoanalytic function theory is considered to study a two-dimensional supersymmetric quantum mechanics system. Hamiltonian components of the superHamiltonian are factorized in terms of one Vekua and one Bers derivative operators. We show that imaginary and real solutions of a Vekua equation and its Bers derivative are ground state solutions for the superHamiltonian. The two-dimensional Darboux and pseudo-Darboux transformations correspond to Bers derivatives in the complex plane. Results on the completeness of the ground states are obtained. Finally, the superpotential is studied in the separable case in terms of transmutation operators. We show how Hamiltonian components of the superHamiltonian are related to the Laplacian operator using these transmutation operators.

On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies withf(R)dark energy

The problem of matching different regions of spacetime in order to construct inhomogeneous cosmological models is investigated in the context of Lagrangian theories of gravity constructed from general analytic functions f(R), and from non-analytic theories with f(R)=R^n. In all of the cases studied, we find that it is impossible to satisfy the required junction conditions without the large-scale behaviour reducing to that expected from Einstein's equations with a cosmological constant. For theories with analytic f(R) this suggests that the usual treatment of weak-field systems as perturbations about Minkowski space may not be compatible with late-time acceleration driven by anything other than a constant term of the form f(0), which acts like a cosmological constant. In the absence of Minkowski space as a suitable background for weak-field systems, one must then choose and justify some other solution to perform perturbative analyses around. For theories with f(R)=R^n we find that no known spherically symmetric vacuum solutions can be matched to an expanding FLRW background. This includes the absence of any Einstein-Straus-like embeddings of the Schwarzschild exterior solution in FLRW spacetimes.

First Characterization of a New Method for Numerically Solving the Dirichlet Problem of the Two-Dimensional Electrical Impedance Equation

Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary conditions are exact solutions of the electrical impedance equation, performing a brief comparison with the finite element method. Finally, we discuss the possible contributions of these results to the field of the electrical impedance tomography.

Fundamentals of Bicomplex Pseudoanalytic Function Theory: Cauchy Integral Formulas, Negative Formal Powers and Schrödinger Equations with Complex Coefficients

The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane naturally leads to bicomplex Vekua-type equations (Campos et al. in Adv Appl Clifford Algebras, 2012; Castaneda et al. in J Phys A Math Gen 38:9207–9219, 2005; Kravchenko in J Phys A Math Gen 39:12407–12425, 2006). To the difference of complex pseudoanalytic (or generalized analytic) functions (Bers in Theory of pseudo-analytic functions. New York University, New York, 1952; Vekua in Generalized analytic functions. Nauka, Moscow (in Russian); English translation Oxford, 1962. Pergamon Press, Oxford, 1959) the theory of bicomplex pseudoanalytic functions has not been developed. Such basic facts as, e.g., the similarity principle or the Liouville theorem in general are no longer available due to the presence of zero divisors in the algebra of bicomplex numbers. In the present work we develop a theory of bicomplex pseudoanalytic formal powers analogous to the developed by Bers (Theory of pseudo-analytic functions. New York University, 1952) and especially that of negative formal powers. Combining the approaches of Bers and Vekua with some additional ideas we obtain the Cauchy integral formula in the bicomplex setting. In the classical complex situation this formula was obtained under the assumption that the involved Cauchy kernel is global, a very restrictive condition taking into account possible practical applications, especially when the equation itself is not defined on the whole plane. We show that the Cauchy integral formula remains valid with the Cauchy kernel from a wider class called here the reproducing Cauchy kernels. We give a complete characterization of this class. To our best knowledge these results are new even for complex Vekua equations. We establish that reproducing Cauchy kernels can be used to obtain a full set of negative formal powers for the corresponding bicomplex Vekua equation and present an algorithm which allows one their construction. Bicomplex Vekua equations of a special form called main Vekua equations are closely related to stationary Schrodinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrodinger operators which allow one to construct the Cauchy kernel when the fundamental solution is known and vice versa. Moreover, using these results we construct the fundamental solutions for the Darboux transformed Schrodinger operators.

Construction and application of Bergman-type reproducing kernels for boundary and eigenvalue problems in the plane

We show how the Bergman-type reproducing kernels for the elliptic operator D = div p grad + q with variable coefficients defined in a bounded domain in the plane can be constructed using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q and with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be obtained following a simple algorithm consisting in recursive integration. Then the complete system of solutions is used for constructing the corresponding reproducing kernel. We study theoretical and numerical aspects of the method and apply it to solve boundary value and eigenvalue problems for the stationary Schrödinger operator in bounded domains.

Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane

An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {φk}k=0∞ such that Lφk=0 for k=0,1, Lφk=k(k−1)φk−2, for k=2,3,… and all φk satisfy certain prescribed initial conditions. We study the transmutation operators related to L in terms of the transformation of powers of the independent variable {(x−x0)k}k=0∞ to the elements of the L-basis and establish a precise form of the transmutation operator realizing this transformation. We use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary Schrödinger equation from a certain class. The system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result. Its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification. The obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory.

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua-type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered. Copyright © 2011 John Wiley & Sons, Ltd.

A Method for the Recovery of Gaps in General Analytic Signals

We propose a generalized Papoulis-Gerchberg algorithm for the recovery of gaps in general analytic functions. The continuous-time algorithm is based on signal expansions in terms of the Chebyshev polynomials of the first kind, and the discrete-time implementation is based on a suitable nonuniform sampling scheme and the discrete cosine transform.

Zeros of analytic functions, with or without multiplicities

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.

ALGEBRAIC ASPECTS OF SPECTRAL THEORY

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of n×n matrices with entries that are polynomials or more general analytic functions.

On Bers Generating Functions for First Order Systems of Mathematical Physics

Considering one of the fundamental notions of Bers’ theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.

An abc theorem on the disk

We extend the classical abc theorem for polynomials (also known as Mason's, or Mason–Stothers', theorem) to general analytic functions on the disk.

On the two-dimensional stationary Schrödinger equation with a singular potential

We study the equation−△u(x,y)+ν(x,y)u(x,y)=0 when the potential ν has the following expressionν(r,θ)=A(θ)r2 where (r,θ) are the polar coordinates in R2 and A is a 2π-periodic function of class Ck with k⩾1. We obtain series representations for solutions in a full neighborhood of the singular point and we also give representations in terms of pseudoanalytic formal powers in sectors having the singular point as a vertex. The results are obtained with the aid of the reduction of the stationary Schrödinger equation to a Vekua equation of a special form and by using recent developments in pseudoanalytic function theory.

Solution of boundary and eigenvalue problems for second-order elliptic operators in the plane using pseudoanalytic formal powers

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D = div p grad + qin the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method. Copyright © 2010 John Wiley & Sons, Ltd.

Explicit solutions of generalized Cauchy–Riemann systems using the transplant operator

In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a “complicated” Vekua equation from positive formal powers and a generating sequence of a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.

Zakharov–Shabat system and hyperbolic pseudoanalytic function theory

AbstractIn J. Phys. A: Math. Theor. 2008; 41:65205 a hyperbolic analogue of pseudoanalytic function theory was developed. In the present contribution we show that one of the central objects of the inverse problem method, the Zakharov–Shabat system is closely related to a hyperbolic Vekua equation for which among other results a generating sequence and hence a complete system of formal powers can be constructed explicitly. Copyright © 2009 John Wiley & Sons, Ltd.

A representation for solutions of the Sturm–Liouville equation

A representation for the general solution of the equation (pu′)′ + qu = ω2 u in terms of a non-trivial solution of is given. This result is obtained with the aid of the theory of pseudoanalytic functions and their relationship to solutions of the stationary two-dimensional Schrödinger equation. The representation has a simple and easily verifiable form and lends itself to numerical computation. Its applications to spectral problems are discussed.

On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation

Using three different representations of the bicomplex numbers , which is a commutative ring with zero divisors defined by where i_1² = − 1, i_2² = − 1, j 2 = 1 and i1 i2 = j = i2 i1 , we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two complex variables and we established some connections between one of these systems and the classical Vekua equations. We also consider the complexification of the real stationary two-dimensional Schrödinger equation. With the aid of any of its particular solutions, we construct a specific bicomplex Vekua equation possessing the following special property. The scalar parts of its solutions are solutions of the original complexified Schrödinger equation and the vectorial parts are solutions of another complexified Schrödinger equation.

Prediction of volcanic eruptions by pseudoanalytic functions

The imminent Oshima large volcanic eruption in Oshima Island near metropolitan Tokyo, Japan was successfully predicted by a practical method of modeling and numerical simulation. The event showed the same tendency against earth tides with a possible 2023-June-6-Mid-Europe-Gaigantic earthquake and 2042-August-6-Biscay-Alps-Black-Sea-Trech-Gigantic earthquake. Huge and large earthquakes and volcanic eruptions were triggered by their peculiar effective earth tides. The event scrutinized here fully acted as plate tectonics and a mathematical model, as viscoelastic fluid phenomena. At the first stage of the simulation, an elastic-viscoplastic porous medium replaced the actual geologic structure, and then conventional theories for random signals and noises determined the most possible vent of eruption. At the next stage, the temporal three-dimensional pseudoanalytic function solver gave the dynamical simulation of the event in Oshima Island. An integrated procedure of direct and indirect solvers minimized variety types of inevitable ambiguities in the modeling and numerical simulation. The event was given in minute order and in Japan Standard Time.