Riemann sums based on $\delta$-fine partitions are illustrated with a Maple procedure.

Let $f\in V_r\left(\right){\cup }_r\left(\right)$, where, for 1 ≤ r < ∞, $V_r\left(\right)$ (resp., ${}_r\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

ℒ denotes the Lebesgue measurable subsets of ℝ and ${ℒ}_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0$hasaperfectsubsetQ\in ℒ\${ℒ}_0$ which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes ${ℒ}_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal $\left(s^0\right)$ which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure ${\lambda }_{f*}:=\lambda ◦{(f*)}^{-1}$, where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional...

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_w^n:{\lambda }_p\left(A\right)\to {\lambda }_p\left(A\right)$ defined on the Köthe sequence space ${\lambda }_p\left(A\right)$ exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$ and any $n\in \mathbb{N}$ is obtained. Under this assumption, the principal measure of $B_w^n$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$.

This paper is a continuation of [1], where a explicit description of the scrambled sets of weakly unimodal functions of type 2∞ was given. Its aim is to show that, for an appropriate non-trivial subset of the above family of functions, this description can be made in a much more effective and informative way.

The McShane integral of functions $fI\to ℝ$ defined on an $m$-dimensional interval $I$ is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.

Several mean value theorems for higher order divided differences and approximate Peano derivatives are proved.

For a differentiable function $𝐟:I\to {ℝ}^k,$ where $I$ is a real interval and $k\in \mathbb{N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M:I^2\to I$ such that$$𝐟\left(x\right)-𝐟\left(y\right)=(x-y){𝐟}^{\text{'}}\left(M(x,y)\right),x,y\in I,$$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.