Related papers: Complete Reduction for Derivatives in a Primitive …

Let $R$ be a commutative unital ring, $\textsf{X}$ a subshift, and $\widetilde{\mathcal{A}}_R(\textsf{X})$ the corresponding unital subshift algebra. We establish the reduction theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$. As a…

A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…

In this paper, we propose new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of…

The form factor bootstrap in integrable quantum field theory allows one to capture local fields in terms of infinite sequences of Laurent polynomials called `towers'. For the sine-Gordon model, towers are systematically described by…

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is…

The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…

Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where…

We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal…

The phase retrieval problem in the classical setting is to reconstruct real/complex functions from the magnitudes of their Fourier/frame measurements. In this paper, we consider a new phase retrieval paradigm in the…

Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient,…

Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh--Abe…

A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…

In this article we study the basic theoretical properties of Mellin-type fractional integrals, known as generalizations of the Hadamard-type fractional integrals. We give a new approach and version, specifying their semigroup property,…

In this paper we prove that the fractional derivative or the fractional primitive of a $T$-periodic function cannot be a $\tilde{T}$-periodic function, for any period $\tilde{T}$, with the exception of the zero function.

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner…

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of $\Pi\Sigma^*$-fields. More…

In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory…

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first…

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we…