Integro-differential Calculus Research Articles

Multidimensional scaling locus of memristor and fractional order elements

This paper combines the synergies of three mathematical and computational generalizations. The concepts of fractional calculus, memristor and information visualization extend the classical ideas of integro-differential calculus, electrical elements and data representation, respectively. The study embeds these notions in a common framework, with the objective of organizing and describing the "continuum" of fractional order elements (FOE). Each FOE is characterized by its behavior, either in the time or in the frequency domains, and the differences between the FOE are captured by a variety of distinct indices, such as the Arccosine, Canberra, Jaccard and Sørensen distances. The dissimilarity information is processed by the multidimensional scaling (MDS) computational algorithm to unravel possible clusters and to allow a direct pattern visualization. The MDS yields 3-dimensional loci organized according to the FOE characteristics both for linear and nonlinear elements. The new representation generalizes the standard Cartesian 2-dimensional periodic table of elements.

“Statistics 102” for Multisource-Multitarget Detection and Tracking

This tutorial paper summarizes the motivations, concepts and techniques of finite-set statistics (FISST), a system-level, “top-down,” direct generalization of ordinary single-sensor, single-target engineering statistics to the realm of multisensor, multitarget detection and tracking. Finite-set statistics provides powerful new conceptual and computational methods for dealing with multisensor-multitarget detection and tracking problems. The paper describes how “multitarget integro-differential calculus” is used to extend conventional single-sensor, single-target formal Bayesian motion and measurement modeling to general tracking problems. Given such models, the paper describes the Bayes-optimal approach to multisensor-multitarget detection and tracking: the multisensor-multitarget recursive Bayes filter. Finally, it describes how multitarget calculus is used to derive principled statistical approximations of this optimal filter, such as PHD filters, CPHD filters, and multi-Bernoulli filters.

Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation

The review was devoted to describing the dynamics of various systems and control processes in terms of the fractional integro-differential calculus. Consideration was given to particular types of the fractional differential equations and models of the fractional dynamic systems. Qualitative dynamics and the issues of stability and controllability of such systems were discussed. The analog and digital implementations of the fractional operations with the use of electrical and optical circuits were presented, as well as the models and methods of hardware implementation of the fractional controllers.

Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation

The review is devoted to using the fractional integro-differential calculus for description of the dynamics of various systems and control processes. Consideration was given to the basic notions of the fractional integro-differential calculus and the problem of interpretation of the fractional operators. Presented were examples of physical systems described in terms of the apparatus under consideration.

On the operad of associative algebras with derivation

Abstract We study the operad of associative algebras equipped with a derivation. We show that it is determined by polynomials in several variables and substitution. Replacing polynomials by rational functions gives an operad which is isomorphic to the operad of “moulds”. It provides an efficient environment for doing integro-differential calculus. Interesting variations are obtained by using formal group laws. The preceding case corresponds to the additive formal group law. We unravel the notion of homotopy associative algebra with derivation in the spirit of Kadeishvili's work.

A software package to manipulate space dependencies and geometry in magnetic confinement fusion

Improvement in the performance of magnetic confinement devices for nuclear fusion relies on the optimization of the geometry of the plasma: either the two-dimensional (2D) cross-section shape in tokamaks with toroidal symmetry or the 3D magnetic configuration in stellerators. The variation in time and space of the plasma parameters in these devices is measured using tomographic or imaging systems with a large number of detectors. To integrate the geometrical manipulations required for the analysis of experimental data, the description of the confining magnetic field configuration and the modeling and simulation of the physical processes within the plasma, an object oriented software package has been developed. Classes in this package are used to describe several coordinate systems, including magnetic flux coordinates, the geometry of the measurement systems, the configuration of the magnetic field and space, and time dependent functions representing plasma parameters. Methods applied on these classes can then easily implement coordinate system transformations, as well as interpolation of and integro-differential calculus on, space and time dependent functions. The geometrical description and characteristics of the magnetic flux surfaces have a natural representation in this environment, allowing the ready computation of the intersection of measurement viewing lines with a coordinate mesh and with flux surfaces, as well as the calculation of the corresponding transfer matrix used in tomographic inversion. The selected numerical methods used in these manipulations and their performances are also presented.

On some integral equations with the generalized Legendre function

We solve and investigate an integral equation with the generalized associated Legendre functionP k m,n (z) by using the fractional integro-differential calculus.