The Cauchy‐Kowalewski product for bicomplex holomorphic functions

In this paper we study the Cauchy‐Kowalewski extension of real analytic functions satisfying a system of differential equations connected to bicomplex analysis, and we use this extension to study the product in the space of bicomplex holomorphic functions. We also show how these ideas can be used to define a Fourier transform for bicomplex holomorphic functions.     

Bicomplex holomorphic functional calculus

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus.     

Topological Bicomplex Modules

In this paper, we develop topological modules over the ring of bicomplex numbers. We discuss bicomplex convexity, hyperbolic-valued seminorms and hyperbolic-valued Minkowski functionals in bicomplex modules. We also study the conditions under which topological bicomplex modules and locally bicomplex convex modules become hyperbolic normable and hyperbolic metrizable respectively.     

Bicomplex hyperfunctions

In this paper, we consider bicomplex holomorphic functions of several variables in _boxclose C^n{{\mathbb B}{\mathbb C}^n} .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space \mathbb Rⁿ{{\mathbb R}^n} within the bicomplex space \mathbb B\mathbb Cⁿ{{\mathbb B}{\mathbb C}^n}, and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.     

Bicomplex Weighted Hardy Spaces and Bicomplex <Emphasis Type="Italic">C</Emphasis>*-algebras

In this paper we study the bicomplex version of weighted Hardy spaces. Further, we describe reproducing kernels for the bicomplex weighted Hardy spaces. In particular, we generalize some results which holds for the classical weighted Hardy spaces. We also introduce the notion of bicomplex C*-algebra and discuss some of its properties.     

Bicomplex signals with sparsity constraints

In this paper, we aim to prove the possibility to reconstruct a bicomplex sparse signal, with high probability, from a reduced number of bicomplex random samples. Due to the idempotent representation of the bicomplex algebra, this case is similar to the case of the standard Fourier basis, thus allowing us to adapt in a rather easy way the arguments from the recent works of Rauhut and Candés et al.     

SUMUDU TRANSFORM IN BICOMPLEX SPACE AND ITS APPLICATIONS

In this paper, we define Sumudu transform with convergence conditions in bicomplex space. Also, we derive some of its basic properties and its inverse. Applications of bicomplex Sumudu transform are illustrated to find the solution of differential equation of bicomplex-valued functions and find the solution for Cartesian transverse electric magnetic (TEM) waves in homogeneous space.     

On the Bicomplex Gleason–Kahane–Żelazko Theorem

We prove that a bicomplex linear functional acting on a bicomplex Banach algebra (with a hyperbolic-valued norm) in such a way that invertible elements are transformed into invertible bicomplex numbers is, in fact, a multiplicative functional and thus, an algebra homomorphism. We give two proofs of this. The first of them is based on the theory of bicomplex holomorphic functions and we present here a number of previously not published facts; the second uses its complex antecedent (classic Gleason–Kahane–?elazko theorem).     

Singularities of functions of one and several bicomplex variables

In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.     

Infinite Dimensional Bicomplex Spectral Decomposition Theorem

This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space M′ imbedded in any bicomplex Hilbert space M.     

Complete Families of Solutions for the Dirac Equation Using Bicomplex Function Theory and Transmutations

The Dirac equation with a scalar and an electromagnetic potential is considered. In the time-harmonic case and when all the involved functions depend only on two spatial variables it reduces to a pair of decoupled bicomplex Vekua-type equations [8]. Using the technique developed for complex Vekua equations a system of exact solutions for the bicomplex equation is constructed under additional conditions, in particular when the electromagnetic potential is absent and the scalar potential is a function of one Cartesian variable. Introducing a transmutation operator relating the involved bicomplex Vekua equation with the Cauchy-Riemann equation we prove the expansion and the Runge approximation theorems corresponding to the constructed family of solutions.     

Conservation laws and the variational bicomplex for second-order scalar hyperbolic equations in the plane

In this paper, we announce several new results concerning the cohomology of the variational bicomplex for a second-order scalar hyperbolic equation in the plane. These cohomology groups are represented by the conservation laws, and certain form-valued generalizations, for the equation. Our methods are based upon the introduction of an adapted coframe for the the variational bicomplex which is constructed by generalizing the classical Laplace transformation used to integrate certain linear hyperbolic equations in the plane.     

Existentially closed fields with holomorphy rings

In this paper we show that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally -adic fields with a -adic holomorphy ring have no model companions in the language of fields augmented by a unary predicate for the corresponding ring. Received February 6, 1993     

Non-Commutative Ricci and Calabi Flows

Starting from an improved version of the bicomplex structure associated the continual Lie algebra with non-commutative base algebra, we obtain dynamical systems resulting from the bicomplex conditions. General expressions for conserved currents associated to a continual Lie algebra bicomplex are found explicitly in two first orders. The Moyal-product counterparts for two-dimensional Ricci and Calabi flow equations depending on non-commutative variables are introduced. Communicated by Petr Kulish Dedicated to the memory of Daniel Arnaudon Submitted: January 30, 2006; Accepted: March 8, 2006     

复Banach空间中C-R方程的全纯解

二重复数是复数的一种推广，在其上的全纯映照族对应于Ｃ２上满足复Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯映照族．可以证明，这样的映照族本质上是由二个单复变数的全纯函数的直乘积所组成的族．本文证明：即使在Ｂａｎａｃｈ空间中，Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯解，具有同样的性质．     

Bicomplex Analogue of Zalcman Lemma

In this paper a bicomplex analogue of Zalcman’s heuristic principle on normal families of meromorphic functions is obtained and as an application a bicomplex version of Lappan’s five-point theorem is proved.     

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; Castañeda 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 Schrödinger equations with complex-valued potentials. We use this relation to establish useful connections between the reproducing Cauchy kernels and the fundamental solutions for the Schrödinger 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 Schrödinger operators.     

On different geometric formulations of Lagrangian formalism

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the -spectral sequence. On one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's -spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences modulo the space of Euler-Lagrange morphisms.     

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.     

Cyclic modules with ∞-simplicial faces and the cyclic homology of A ∞-algebras

Mathematical Notes - A chain bicomplex for A ∞-algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the...