It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted attention many scientists. At this paper, we provide Euler’s De Moivre’s formulas for 4×4 matrices associated with by using trigonometric identities. Also, give roots Moreover, some illustrative examples to support main formulas.

The function g (which contains the singular part of the free energy) must have a special symmetry that relates its values for dual arguments. (For example the function f(x) = x/(1 + x) equals f(x), establishing a duality between the arguments at x and x.) Eq.(VI.23) has the following properties: 1. Low temperatures are mapped to high temperatures, and vice versa. 2. The mapping connects pairs o...

An overview is given of effective gravitational field theories with fixed background fields that break spacetime symmetry. The behavior of the background fields and the types of excitations that can occur depend on whether the symmetry breaking is explicit or spontaneous. For example, when the breaking is spontaneous, the background field is dynamical and massless Nambu–Goldstone and massive Hi...

Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can ...

by repeated partial integration. The topic is usually reserved for more advanced calculus courses. The purpose of this note is to show that (1) can be derived using only the mathematics taught in elementary school, that is, basic algebra, the Pythagorean theorem, and the formula π · r 2 for the area of a circle of radius r . Viggo Brun gives an account of Wallis’s method in [1] (in Norwegian). ...

We study some properties of the Askey-Wilson polynomials (AWP) when q is a primitive Nth root of unity. For general four-parameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval i...