Variational measures related to local systems and the Ward property of $$\mathcal{P}$$ -adic path bases

Abstract

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a P-adic path system that defines a differentiation basis which does not possess Ward property.

A measure-theoretic characterization of the Henstock-Kurzweil integral revisited

“…This remarkable result has been generalized by several authors; see, for example, [1], [6], [7], [10], [11] and the references therein. In this paper we give a shorter proof of the corresponding result for the multiple Henstock-Kurzweil (equivalently, the Perron) integral; see Theorem 4.5.…”

Some Kurzweil-Henstock-type integrals and the wide Denjoy integral

“…Also, all of them, now except the dyadic local system, are neither left nor right strongly porous at each x ∈ . For ∆ d however (as for each P-adic local system [3] with the sequence P being bounded), one can construct a nonempty perfect set C with ∆ d being neither left nor right strongly porous at each x ∈ C. Recently, we have learnt (from Professor Valentin Skvortsov) a construction of a continuous ACG-function which is not an indefinite H ∆ -integral for ∆ being P-adic local system associated with any given sequence P. This construction will possibly appear elsewhere.…”

Power-Laws hereditariness of biomimetic ceramics for cranioplasty neurosurgery

“…Power-laws hereditariness in conjuction with Boltzmann superposition yields constitutive behavior in terms of the so-called fractional integrals and derivatives. Fractional calculus may be considered as generalization of the classical differential calculus to real-order integration and differentiation i.e.df dt → d β f dt β with β ∈ [0, 1] as reported in classical references 70 [11,12,13,14,15]. In such a context, uniaxial hereditariness [16,17,18,19] involving fractional order stress-strain relations have been reported since the beginning of the twentyth century [20,13] defining the so-called springpot element [21,22].…”