Bicomplex Numbers Research Articles

A bicomplex finite element method for wave propagation in homogeneous media

PurposeThe purpose of this paper is to present the advantageous applicability of the bicomplex analysis in the context of the Finite Element Method (FEM). This method can be applied for wave propagation problems in various environments.Design/methodology/approachIn this paper, the bicomplex number system is introduced and accordingly the differential equation for time-harmonic Maxwell’s equations in homogeneous media is derived in detail. Besides that, numerical simulations of wave propagation are performed and compared to the traditional approach based on classical FEM related to the Helmholtz equation. The appropriate error norm is investigated for different discretizations.FindingsThe results show that the use of bicomplex analysis in FEM leads to the higher accuracy of the electromagnetic field determination compared to the traditional Helmholtz approach. By using the bicomplex-valued formulation, the complex-valued electric and magnetic fields can be found directly and no additional FEM calculations are necessary to get the whole field.Originality/valueThe direct bicomplex formulation overcomes the use of the second order derivatives, which leads to the higher accuracy. In general, accurate calculations of the wave propagation in FEM is still an open problem and the approach described in this paper is a contribution to this class of problems.

Fractal-type sets in the four-dimensional space using bicomplex and hyperbolic numbers

In the first part of this work we study the set of bicomplex numbers from the point of view of a hyperbolic module. We make use of the partial order defined on the set of hyperbolic numbers. We recall some properties of the hyperbolic geometrical objects that were defined in previous papers. With the help of these notions some fractal-type sets in the four dimensional space are constructed.

On bicomplex Fourier–Wigner transforms

We consider the [Formula: see text]d and [Formula: see text]d bicomplex analogues of the classical Fourier–Wigner transform. Their basic properties, including Moyal’s identity and characterization of their ranges giving rise to new bicomplex–polyanalytic functional spaces are discussed. Details concerning a special window function are developed explicitly. An orthogonal basis for the space of bicomplex-valued square integrable functions on the bicomplex numbers is constructed by means of a specific class of bicomplex Hermite functions.

On bicomplex numbers with coefficients from the complex Fibonacci sequence

On bicomplex numbers with coefficients from the complex Fibonacci sequence

Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus

Abstract Human being live in three-dimensional space; they can accurately visualize processes taking place in one, two and three dimensions. Although the set of bi-complex numbers and quaternion have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located. We suggested a new set of complex number called “the Trinition”. The new set is comprised between complex number with one imaginary part and complex number with three imaginary parts called quaternion/bi-complex numbers. We established a bijection between the new set and the three-dimensional space. We presented some important properties of the new set. We showed that all chaotic attractors in three dimension are simply three-dimensional mapping in the new set. Fewer examples of mapping in such set were presented. A new methodology that can be used to obtain more strange attractors are equally suggested. The methodology combines fractional chaotic models and some fractal mapping within the new set. Some illustrative figures are presented.

Bicomplex extensions of zero mean curvature surfaces in [formula omitted] and [formula omitted

In this paper, we construct zero mean curvature complex surfaces in CN with a various type of standard metric, each of which changes its type from positive definite to neutral, by means of bicomplex numbers. By applying them as bicomplex extensions, we describe the correspondence between fold singularities and type-changing of zero mean curvature real surfaces in R2,1 and R2,2. In particular, we show that any fold singularity consists of branch points of the bicomplex extension. We also show that type-changing across a lightlike line segment occurs on an incomplete end on a fold singularity.

Comparison of Neural Network–Based Adaptive Controllers Using Hypercomplex Numbers for Controlling Robot Manipulator

This study investigates an adaptive controller by applying a neural network, in which all the network parameters, states, signals and functions are expressed using hypercomplex numbers and algebras; its application to dynamics control of a robot manipulator. To design hypercomplex–valued neural networks where each neural network is a multilayer feedforward network with a split–type activation function of neurons using a tapped–delay–line input, we consider the following four types of hypercomplex numbers: complex, hyperbolic, bicomplex and quaternion numbers. In the control system, we utilise a feedback error–learning scheme to conduct the training of the network through a back–propagation algorithm. In the computational experiments, we explore a hypercomplex–valued neural network–based controller as a trajectory control problem of a three–link robot manipulator, in which the position of the end–effector follows to the desired trajectory in a 3–dimensional space. The simulation results validate the feasibility and effectiveness of the quaternion neural network–based controller for this task.

On Bicomplex Pell and Pell-Lucas Numbers

In this paper, bicomplex Pell and bicomplex Pell-Lucas numbers are defined. Also, negabicomplex Pell and negabicomplex Pell-Lucas numbers are given. Some algebraic properties of bicomplex Pell and bicomplex Pell-Lucas numbers which are connected between bicomplex numbers and Pell and Pell-Lucas numbers are investigated. Furthermore, d'Ocagne's identity, Binet's formula, Cassini's identity and Catalan's identity for these numbers are given.

Note on Bicomplex Matrices

In this paper, we consider bicomplex numbers and bicomplex matrices. Firstly, we give some propertiesof bicomplex numbers.After that we investigate bicomplex matrices using properties of complex matrices.Then we define the complex adjoint matrix of bicomplex matrices and we describe some of theirproperties. Furthermore, we give the definition of q-determinant of bicomplex matrices.

Fibonacci, Lucas Numbers with Daul Bicomplex Numbers

Recently, the authors give some results about Fibonacci and Lucas numbers. In this present paper, our object introduce a detailed study of a new generation of dual bicomplex Fibonacci numbers. We define new dual vector which is called dual Fibonacci vector. We give properties of dual Fibonacci vector to expert in geometry and then we introduce some formulas, facts and properties about dual bicomplex Fibonacci numbers and variety of geometric and algebraic properties which are not generally known.

Complex Symmetric Formulation of Maxwell Equations for Fields and Potentials

Maxwell equations have two types of asymmetries between the electric and magnetic fields. The first asymmetry is the inhomogeneity induced by the absence of magnetic charge sources. The second asymmetry is due to parity. We show how both asymmetries are naturally resolved under an alternative formulation of Maxwell equations for fields or potentials that uses a compact complex vector operator representation. The developed complex symmetric operator formalism can be easily applied to performing the continuity equation, the field wave equations, the Maxwell equations for potentials, the gauge transformations, and the 4-momentum representation; in general, the developed formalism constitutes a simple way of unfolding the Maxwell theory. Finally, we provide insights for extending the presented analysis within the context of (i) bicomplex numbers and tessarine algebra; and (ii) Lp-spaces in nonlinear Maxwell equations.

Holographic coordinates

The Laplace equation in the two-dimensional Euclidean plane is considered in the context of the inverse stereographic projection. The Lie algebra of the conformal group as the symmetry group of the Laplace equation can be represented solely in terms of the solutions and derivatives of the solutions of the Laplace equation. It is then possible to put contents from differential geometry and quantum systems, like the Hopf bundle, relativistic spin, bicomplex numbers, and the Fock space into a common context. The basis elements of the complex numbers, considered as a Clifford paravector algebra, are reinterpreted as differential tangent vectors referring to dilations and rotations. In relation to this a homogeneous space is defined with the Lie algebra of the conformal group, where dilations and rotations are the coset representatives. Potential applications in physics are discussed.

On Integrated and Differentiated C_2-Sequence Spaces

The integrated and differentiated C_2-sequence spaces are defined and studied by using the norm on the bicomplex space C_2, infinite matrices of the bicomplex number and the Orlicz functions. We also studied some topological properties of the C_2-sequence spaces We define the α-duals of the integrated and differentiated C_2-sequence spaces.

Bicomplex Fibonacci quaternions

In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.

From the theory of “congeneric surd equations” to “Segre's bicomplex numbers”

We will study the historical emergence of Tessarines or Bicomplex numbers, from their origin as “imaginary” solutions of irrational equations, to their insertion in the context of study of the algebras of hypercomplex numbers.

Bicomplex Harmonic and Isotonic Oscillators: The Excited States

We investigate the complete analytical solutions of quantum mechanical harmonic and isotonic oscillators formulated in the commutative ring of bicomplex numbers. We obtain the explicit closed form expressions for the excited eigenstates and corresponding energy eigenvalues associated with the harmonic and isotonic oscillator potentials by extending the formalism adopted in Banerjee (Ann Phys 377:493, 2017) recently to find the analytical closed form solutions for ground states.

On the quantum mechanics of bicomplex Hamiltonian system

We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponding to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian PT symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving x2 and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken PT symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.

Generalization of the Gauss--Lucas theorem for bicomplex polynomials

The aim of this paper is to extend the domain of the Gauss—Lucas theorem from the set of complex numbers to the set of bicomplex numbers. We also discuss a bicomplex version of another compact generalization of the Gauss—Lucas theorem.

Bifurcations and exceptional points in a -symmetric dipolar Bose–Einstein condensate

We investigate the bifurcation structure of stationary states in a dipolar Bose–Einstein condensate located in an external -symmetric potential. The imaginary part of this external potential allows for the effective description of in- and outcoupling of particles. To unveil the complete bifurcation structure and the properties of the exceptional points we perform an analytical continuation of the Gross–Pitaevskii equation, which is used to describe the system. We use an elegant and numerically efficient method for the analytical continuation of the Gross–Pitaevskii equation with dipolar interactions containing bicomplex numbers. The Bose–Einstein condensate with dipole interaction shows a much richer bifurcation scenario than a condensate without long-range interactions. The inclusion of analytically continued states can also explain property changes of bifurcation points which were hidden before, and allows for the examination of the properties of the exceptional points associated with the branch points. With the new analytically continued states we are able to prove the existence of an exceptional point of fifth order.

Inverse Fourier transform for bi-complex variables

In this paper we examine the existence of bicomplexified inverse Fourier transform as an extension of its complexified inverse version within the region of convergence of bicomplex Fourier transform. In this paper we use the idempotent representation of bicomplex-valued functions as projections on the auxiliary complex spaces of the components of bicomplex numbers along two orthogonal,idempotent hyperbolic directions.