Class Of Hyperbolic Functions Research Articles

Nanoscale excitations of solids in the form of axially symmetric solitons

Soliton excitations in solids are a development of the traditional concept of exciton-type electronic excitations. Their main feature is the space and, generally speaking, time distributed amplitude. The spatial distribution of the amplitude is of several nanometers (up to 10 nm). It occurs due to the nonlinearity caused by the interaction of excitation with the lattice. New aspects of the cubic nonlinearity are considered. It is known that the nonlinear Schrodinger equation (NSE) with cubic nonlinearity has analytical solutions in the class of hyperbolic functions and only for the spatial-one-dimensional case. It turned out that there is a physically consistent case where it is possible to evenly divide variables in a spatial-three-dimensional NSE in terms of generalized functions, as well as to find and analyze an analytical solution.

Refinements on Some Classes of Complex Function Spaces

Some weighted classes of hyperbolic function spaces are defined and studied in this paper. Finally, by using the chordal metric concept, some investigations for a class of general hyperbolic functions are also given.

A cryptography method based on hyperbolic balancing and Lucas-balancing functions

The goal is to study a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring balancing and Lucas-balancing numbers. These functions are indeed the extension of Binet formulas for both balancing and Lucas-balancing numbers in continuous domain. Some identities concerning hyperbolic balancing and Lucas-balancing functions are also established. Further, a new class of square matrices, a generalization of balancing QB-matrices for continuous domain, is considered. These matrices indeed enable us to develop a cryptography method for secrecy purpose.

The q-pell Hyperbolic Functions

In 2005 Stakhov and Rozin introduced a new class of hyperbolic functions which is called Fibonacci hyperbolic functions. The aim of this study to give q-analogue of the Pell hyperbolic functions. These functions can be regarded as q extensions of classical hyperbolic functions. We introduce the q-analogue of classical Silver ratio as follow δq = 1+qn−12+4qn−2+(1+qn−1)22, n ≥ 2. Making use of this q-analogue of the Silver ratio, we defined sin Pqh(x) and cos Pqh(x) functions. We investigated some properties and gave some relationships between these functions.

The q-Fibonacci Hyperbolic Functions

In 2005 Stakhov and Rozin introduced a new class of hyperbolic functions which is called Fibonacci hyperbolic functions. In this paper, we study q-analogue of Fibonacci hyperbolic functions. These functions can be regarded as q extensions of classical hyperbolic functions. We introduce the q-analogue of classical Golden ratio as follow φq = 1+1+4qn−22, n ≥ 2. Making use of this q-analogue of the Golden ratio, we defined sin Fqh(x) and cos Fqh(x) functions. We investigated some properties and gave some relationships between these functions.

On a new class of hyperbolic functions

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet’s formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into ‘‘continuous’’ theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, (‘‘Lobatchevski’s hyperbolic geometry’’, ‘‘Four-dimensional Minkowski’s world’’, etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631]. 2004 Elsevier Ltd. All rights reserved.