Related papers: A Commutative Alternative to Fractional Calculus o…
In 1931 Koopman and von Neumann extended previous work of Liouville and provided an operatorial version of Classical Mechanics (CM). In this talk we will review a path-integral formulation of this operatorial version of CM. In particular we…
We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville's theorem…
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
This is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. Functional analysis takes us beyond finite matrices, allowing us to work with infinite sets of…
We show that every operator in $L^{2}$ has an associated measure on a space of functions and prove that it can be used to find solutions to abstract Cauchy problems, including partial differential equations. We find explicit formulas to…
In this paper, we further investigate the problem of commutativity up to a factor (or $\lambda$-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
In this paper, a new notion, named Riemann-Liouville fractional cosine function is presented. It is proved that a Riemann-Liouville $\alpha$-order fractional cosine function is equivalent to Riemann-Liouville $\alpha$-order fractional…
In this paper, we consider the locally convex spaces of entire functions with growth given by proximate orders, and study the representation as a differential operator of a continuous homomorphism from such a space to another one. As a…
We introduce an $\mathcal{M}$-operator approach to establish the uniqueness of continuous or bounded solutions for a broad class of Landau-type nonlinear kinetic equations. The specific $\mathcal{M}$-operator, originally developed in [3],…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional…
The conformable derivative has been promoted in numerous publications as a new fractional derivative operator. This article provides a critical reassessment of this claim. We demonstrate that the conformable derivative is not a fractional…
We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of…
A new model of quantum computing has recently been proposed which, in analogy with a classical lambda-calculus, exploits quantum processes which operate on other quantum processes. One such quantum meta-operator takes N unitary…
Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of…
The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed…
In the present paper, we investigate the fractional analog of the Sturm-Liouville problem on a metric graph using a combination of left Riemann-Liouville and right Caputo fractional derivatives. This combination creates a symmetric and…
This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space…
In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus…