Generalized Trigonometric Functions and Matrix Parameterization

The generalized trigonometric functions (GTF) have been introduced using an appropriate redefinition of Euler type identities involving non-standard forms of imaginary numbers, realized by different types of matrices. In this paper we use the GTF to get parameterization of practical interest for non-singular matrices. The possibility of using this procedure to deal with applications in electron transport is also touched on

Theory of generalized trigonometric functions: From Laguerre to Airy forms

We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired by the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transforms allowing the framing of the previous concepts within the context of generalized Borel transform

Inverse derivative operator and umbral methods for the harmonic numbers and telescopic series study

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism

Dual numbers and operational umbral methods

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus

About the use of generalized forms of derivatives in the study of electromagnetic problems

The use of non-local operators, defining Riemann–Liouville or Caputo derivatives, is a very useful tool to study problems involving non-conventional diffusion problems. The case of electric circuits, ruled by non-integer derivatives or capacitors with fractional dielectric permittivity, is a fairly natural frame of relevant applications. We use techniques, involving generalized exponential operators, to obtain suitable solutions for this type of problems and eventually discuss specific problems in applications

Las representaciones pictóricas como recursos semióticos. Caso equilibrio químico

De acuerdo con Orlandi (1996) tres variables intervienen en el proceso de producción de sentidos: la intencionalidad del autor, la materialidad del texto y las posibilidades de resignificación del lector. Las dos primeras no pueden ser modificadas por el lector. Las posibilidades del lector para atribuir significados pueden ser modificadas conociendo los diferentes tipos de Representaciones Externas Pictóricas (REP) y los recursos semióticos que se utilizan en su construcción. Se diseña una intervención dirigida a dar a conocer este contenido para dos tipos REP y así facilitar operar con éstas. Los resultados indican que las REP ejercen una doble influencia: a) cuando la tarea se realiza a partir de una REP o b) cuando la tarea es construir la REP. En ambos casos se encuentra que cada tipo de representación permite la recuperación de una información específica y diferente

Fractional derivatives, memory kernels and solution of a free electron laser volterra type equation

The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed

Quasi Exact Solution of the Fisher Equation

We propose an accurate non numerical solution of the Fisher Equation (FE), capable of reproducing the known analy- tical solutions and those obtained from a numerical analysis. The form we propose is based on educated guesses con- cerning the possibility of merging diffusive and logistic behavior into a single formul