On hyperbolic curves in dimension four using bicomplex numbers

2019 ◽  
Author(s):  
M. Elena Luna-Elizarrarás
Keyword(s):  
Bicomplex Numbers

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

scholarly journals The bicomplex quantum Coulomb potential problem

2013 ◽  
Vol 91 (12) ◽  
pp. 1093-1100 ◽  
Author(s):  
J. Mathieu ◽  
L. Marchildon ◽  
D. Rochon
Keyword(s):  
Differential Equation ◽  
Coulomb Potential ◽  
Potential Problem ◽  
Orthonormal System ◽  
Mathematical Formulation ◽  
Algebraic Method ◽  
Hamiltonian Operator ◽  
Number System ◽  
Bicomplex Numbers ◽  
Bicomplex Number

Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [Formula: see text], where ξ is a bicomplex number. Following Pauli’s algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrödinger’s time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.

scholarly journals Generalized Bicomplex Numbers and Lie Groups

2015 ◽  
Vol 25 (4) ◽  
pp. 943-963 ◽  
Author(s):  
Sıddıka Özkaldı Karakuş ◽  
Ferdag Kahraman Aksoyak
Keyword(s):  
Lie Groups ◽  
Bicomplex Numbers

scholarly journals Fibonacci, Lucas Numbers with Daul Bicomplex Numbers

2018 ◽  
Vol 10 (1-2) ◽  
pp. 161-172
Author(s):  
Faik Babadag
Keyword(s):  
Lucas Numbers ◽  
Bicomplex Numbers

scholarly journals Some inequalities in quasi-Banach algebra of non-Newtonian bicomplex numbers

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2231-2243
Author(s):  
Nilay Sager ◽  
Birsen Sağır
Keyword(s):  
Banach Algebra ◽  
Bicomplex Numbers

In this paper, we construct the quasi-Banach algebra BC(N) of non-Newtonian bicomplex numbers and we generalize some topological concepts and inequalities as Schwarz?s, H?lder?s and Minkowski?s in the set of bicomplex numbers in the sense of non-Newtonian calculus.

Elements of probability theory with values in bihyperbolic algebra

2021 ◽  
Vol 34 ◽  
pp. 36-49
Author(s):  
Tamila Kolomiiets
Keyword(s):  
Statistical Physics ◽  
Random Variable ◽  
Measurable Space ◽  
Probabilistic Measure ◽  
Basic Properties ◽  
Bicomplex Numbers ◽  
Zero Divisors ◽  
Special Cases ◽  
Testing Complex ◽  
Significant Probability

In this paper we expand the concept of a really significant probabilistic measure in the case when the measure takes values in the algebra of bihyperbolic numbers. The basic properties of bihyperbolic numbers are given, in particular idempotents, main ideals generated by idempotents, Pierce's decompo\-sition and the set of zero divisors of the algebra of bihyperbolic numbers are determined. We entered the relation of partial order on the set of bihyperbolic numbers, by means of which the bihyperbolic significant modulus is defined and its basic properties are proved. In addition, some bihyperbolic modules can be endowed with a bihyperbolic significant norms that take values in a set of non-negative bihyperbolic numbers. We define $\sigma$-additive functions of sets in a measurable space that take appropriately normalized bihyperbolic values, which we call a bihyperbolic significant probability. It is proved that such a bihyperbolic probability satisfies the basic properties of the classical probability. A representation of the bihyperbolic probability measure is given and its main properties are proved. A bihyperbolically significant random variable is defined on a bihyperbolic probability space, and this variable is a bihyperbolic measurable function in the same space. We proved the criterion of measurability of a function with values in the algebra of bihyperbolic numbers, and the basic properties of bihyperbolic random variables are formulated and proved. Special cases have been studied in which the bihyperbolic probability and the bihyperbolic random variable take values that are zero divisors of bihyperbolic algebra. Although bihyperbolic numbers are less popular than hyperbolic numbers, bicomplex numbers, or quaternions, they have a number of important properties that can be useful, particularly in the study of partial differential equations also in mathematical statistics for testing complex hypotheses, in thermodynamics and statistical physics.

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

2017 ◽  
Vol 44 (3) ◽  
pp. 232-251 ◽  
Author(s):  
Cinzia Cerroni
Keyword(s):  
Bicomplex Numbers

Geometry and Trigonometric Representations of Bicomplex Numbers

2015 ◽  
pp. 51-72
Author(s):  
M. Elena Luna-Elizarrarás ◽  
Michael Shapiro ◽  
Daniele C. Struppa ◽  
Adrian Vajiac
Keyword(s):  
Bicomplex Numbers

Homothetic Motions at E 4 with Bicomplex Numbers

2010 ◽  
Vol 21 (3) ◽  
pp. 541-546 ◽  
Author(s):  
Hesna Kabadayi ◽  
Yusuf Yayli
Keyword(s):  
Bicomplex Numbers

On F(p, n)-Fibonacci bicomplex numbers

2018 ◽  
Vol 23 ◽  
pp. 35-44 ◽  
Author(s):  
Mirosław Liana ◽  
Anetta Szynal-Liana ◽  
Iwona Włoch
Keyword(s):  
Bicomplex Numbers
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