25 research outputs found
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
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
Monte Carlo and hydrodynamic simulation of a one dimensional n+ – n – n+ silicon diode
An improved closure relation - based on the entropy principle - is implemented in a Hydrodynamic
model for electron transport. Steady-state electron transport in the "benchmark" n+ - n - n+ submicron silicon diode is simulated and the quality of the model is assessed by comparison
with Monte Carlo results
Solar differential rotation and meridional flow: The role of a subadiabatic tachocline for the Taylor-Proudman balance
We present a simple model for the solar differential rotation and meridional
circulation based on a mean field parameterization of the Reynolds stresses
that drive the differential rotation. We include the subadiabatic part of the
tachocline and show that this, in conjunction with turbulent heat conductivity
within the convection zone and overshoot region, provides the key physics to
break the Taylor-Proudman constraint, which dictates differential rotation with
contour lines parallel to the axis of rotation in case of an isentropic
stratification. We show that differential rotation with contour lines inclined
by 10 - 30 degrees with respect to the axis of rotation is a robust result of
the model, which does not depend on the details of the Reynolds stress and the
assumed viscosity, as long as the Reynolds stress transports angular momentum
toward the equator. The meridional flow is more sensitive with respect to the
details of the assumed Reynolds stress, but a flow cell, equatorward at the
base of the convection zone and poleward in the upper half of the convection
zone, is the preferred flow pattern.Comment: 15 pages, 7 figure
Repeated derivatives of hyperbolic trigonometric functions and associated polynomials
Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functionscan usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec(.), tan(.) and for their hyperbolic counterparts
Hybrid Complex Numbers: The Matrix Version
In this paper we review the notion of hybrid complex numbers, recently introduced to provide a comprehensive conceptual and formal framework to deal with circular, hyperbolic and dual complex. We exploit the established isomorphism between complex numbers as abstract entities and as two dimensional matrices in order to derive the associated algebraic properties. Within such a respect we derive generalized forms of Euler exponential formula and explore the usefulness and relevance of operator ordering procedure of the Wei-Norman type. We also discuss the properties of dual numbers in terms of Pauli matrices. Finally we explore generalized forms of Dirac-like factorization, emerging from the properties of these numbers