相关论文: Distributed Order Calculus: an Operator-Theoretic …

This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained and purely Hilbert space operator explanation of their…

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, that contains all operators of Helffer-Sj\"ostrand type and is closed under the action of smooth proper mappings.…

The fractional q-calculus is the q-extension of the ordinary fractional calculus and dates back to early 20-th century. The theory of q-calculus operators are used in various areas of science such as ordinary fractional calculus, optimal…

In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational…

Approximate solutions of the Fisher equation obtained by different splitting methods are investigated. The error of this nonlinear problem is analyzed. The order of different splitting methods coupled with numerical methods of different…

Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…

In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of…

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…

We associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them.

In this article, we impose a new class of fractional analytic functions in the open unit disk. By considering this class, we define a fractional operator, which is generalized Salagean and Ruscheweyh differential operators. Moreover, by…

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to…

This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of…

We collect and organise known results and add some new ones of the following nature: if A is a bounded operator in a Hilbert or Banach space, does there exist a nonconstant polynomial p(z) such that p(A) is "simpler", "nicer" than A. The…

We determine the boundedness and compactness of a large class of operators, mapping from general Banach spaces of holomorphic functions into a particular type of spaces of functions determined by the growth of the functions, or the growth…

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS…

We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the…

In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted…

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalizing and sharpening estimates, and…