arXiv subjects

J. Vaz

Publications and source records attributed to J. Vaz.

4 recordsLinked to original sources

arXiv2026

Beta-weighted non-local differential operators and related stochastic processes

In this work we introduce a class of non-local differential operators defined through a beta-weighted averaging of the ordinary derivative. We investigate their analytical properties and establish connections with the Caputo and Erd\'elyi-Kober operators. Differential equations involving the beta-weighted derivative are studied by Mellin transform methods, leading to solutions represented in terms of Barnes G-functions and a new class of G-hypergeometric functions. We also analyze asymptotic properties, Laplace transforms, and the second-order equation involving the sequential beta-weighted derivative. Finally, we present stochastic applications of these results, showing that continuous-time random walks, with waiting times characterized by the beta-weighted derivative, converge to Brownian motions time-changed by a scaled inverse stable subordinator. We compare this anomalous-diffusion model with a time-changed Brownian motion whose one-dimensional distribution solve a heat-type equation with beta-weighted derivative.

math.PR↗

arXiv2025

On fractional differential equations, dimensional analysis, and the double gamma function

In this paper we discuss some issues that arise in the process of writing a fractional differential equation (FDE) by replacing an integer order derivative by a fractional order derivative in a given differential equation. To address these issues, we propose a dimensional regularization of the Caputo fractional derivative, ensuring consistency in physical dimensions. Then we solve some FDEs using this proposed dimensional regularization. We show that the solutions of these FDEs are most conveniently written using the double gamma function. We also compare these solutions with those from equations involving the standard Caputo fractional derivative.

math.GM↗

arXiv1999

The Dirac-Hestenes Lagrangian

We discuss the variational principle within Quantum Mechanics in terms of the noncommutative even Space Time sub-Algebra, the Clifford $\Ra$-algebra $Cl_{1,3}^+$. A fundamental ingredient, in our multivectorial algebraic formulation, is the adoption of a $\D $-complex geometry, $\D \equiv span_{\RR} \{1,γ_{21} \}$, $γ_{21} \in Cl_{1,3}^+$. We derive the Lagrangian for the Dirac-Hestenes equation and show that such Lagrangian must be mapped on $\D \otimes {\cal F}$, where $\cal F$ denotes an $\Ra$-algebra of functions.

hep-ph↗

arXiv1999

Complex Geometry and Dirac Equation

Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate complex geometries is strictly related to the geometric interpretation of the complex imaginary unit $i=\sqrt{-1}$. We discuss {\em two} possibilities which appear in the multivector algebra approach: the $σ_{123}$ and $σ_{21}$ complex geometries. Our formalism permits to perform a set of rules which allows an immediate translation between the complex standard Dirac theory and its version within geometric algebra. The problem concerning a double geometric interpretation for the complex imaginary unit $i=\sqrt{-1}$ is also discussed.

hep-th↗