HOMOTHETIC MOTIONS VIA GENERALIZED BICOMPLEX NUMBERS

In this paper, by using the matrix representation of generalized bicomplexnumbers, we dene the homothetic motions on some hypersurfaces infour dimensional generalized linear space R4 alpha-beta. Also, for some special cases we give some examples of homothetic motions in R4 and R42and obtainsome rotational matrices, too. So, we investigate some applications about kinematics of generalized bicomplex numbers

Elements of probability theory with values in bihyperbolic algebra

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.

Stability Conditions of Bicomplex-Valued Hopfield Neural Networks

Hopfield neural networks have been extended using hypercomplex numbers. The algebra of bicomplex numbers, also referred to as commutative quaternions, is a number system of dimension 4. Since the multiplication is commutative, many notions and theories of linear algebra, such as determinant, are available, unlike quaternions. A bicomplex-valued Hopfield neural network (BHNN) has been proposed as a multistate neural associative memory. However, the stability conditions have been insufficient for the projection rule. In this work, the stability conditions are extended and applied to improvement of the projection rule. The computer simulations suggest improved noise tolerance.

Some inequalities in quasi-Banach algebra of non-Newtonian 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.

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.