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### Absolute continuity and hyponormal operators.

International Journal of Mathematics and Mathematical Sciences

### Absolute continuity with respect to a subset of an interval

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset $M$ of an interval $I$. This generalization is based on adding more requirements to disjoint systems ${\left\{\left({a}_{k},{b}_{k}\right)\right\}}_{K}$ from the classical definition of absolute continuity – these systems should be not too far from $M$ and should be small relative to some covers of $M$. We discuss basic properties of relative absolutely continuous functions and compare this class with other classes of...

### Addendum to the paper "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅ ²₊" (Studia Math. 201 (2010), 287-304)

Studia Mathematica

### Cauchy's means of Levinson type.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### Changes of variable which preserve almost everywhere approximate differentiability

Colloquium Mathematicae

### Darboux property of the Wronski determinant

Mathematica Slovaca

### Fields of sets, set functions, set function integrals, and finite additivity.

International Journal of Mathematics and Mathematical Sciences

### Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Studia Mathematica

We investigate the convergence behavior of the family of double sine integrals of the form ${\int }_{0}^{\infty }{\int }_{0}^{\infty }f\left(x,y\right)sinuxsinvydxdy$, where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals ${\int }_{a₁}^{b₁}{\int }_{a₂}^{b₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and ${b}_{j}>{a}_{j}\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

### Idempotents in quotients and restrictions of Banach algebras of functions

Annales de l'institut Fourier

Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $\left(1+|n|{\right)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let ${J}_{{𝒜}_{\beta }}\left(E\right)=\left\{f\in {𝒜}_{\beta }:f=0\phantom{\rule{0.166667em}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{neighbourhood}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{0.166667em}{0ex}}E\right\}$. We prove that, for $\beta >\frac{1}{2}$, there exists a closed set $E\subseteq 𝕋$ of measure zero such that the quotient algebra ${𝒜}_{\beta }/\stackrel{‾}{{J}_{{𝒜}_{\beta }}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda }_{\gamma }$ and the algebra $𝒜𝒞$ of absolutely continuous functions on $𝕋$, we characterize the closed sets $E\subseteq 𝕋$ for which the restriction algebras ${\lambda }_{\gamma }\left(E\right)$ and $𝒜𝒞\left(E\right)$ are generated by their idempotents.

### Increasing functions.

Aequationes mathematicae

### Lipschitzian superposition operators on metric semigroups and abstract convex cones of mappings of finite $\text{Λ}$-variation.

Sibirskij Matematicheskij Zhurnal

### On a generalization of the Hermite-Hadamard inequality. II.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### On absolutely kth continuous functions

Fundamenta Mathematicae

### On certain inequalities improving the Hermite-Hadamard inequality.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### On functions of bounded (p,2)-variation.

Collectanea Mathematica

### On some integral inequalities of block type

Colloquium Mathematicae

### On some integral inequalities of Weyl type

Colloquium Mathematicae

### Ordinary differential equations the solution of which are $AC{G}_{*}$-functions

Archivum Mathematicum

### Some remarks on descriptive characterizations of the strong McShane integral

Mathematica Bohemica

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f:W\to X$ defined on a non-degenerate closed subinterval $W$ of ${ℝ}^{m}$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure ${V}_{ℳ}F$ generated by the primitive $F:{ℐ}_{W}\to X$ of $f$, where ${ℐ}_{W}$ is the family of all closed non-degenerate subintervals of $W$.

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