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Non-transitive points and porosity

T. K. Subrahmonian Moothathu (2013)

Colloquium Mathematicae

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show...

Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

Mochammad Idris, Hendra Gunawan, A. Eridani (2018)

Mathematica Bohemica

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent 1 , and obtain a new estimate for their norm.

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

For X ⊆ [0,1], let D X denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of D X . For α ≥ 3, X is properly D ξ ( Π α 0 ) iff D X is properly D ξ ( Π 1 + α 0 ) . We also show that for every nonempty set X ⊆[0,1], D X is Π 3 0 -hard. For each nonempty Π 2 0 set X ⊆ [0,1], in particular for X = x, D X is Π 3 0 -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is Π 3 0 -complete. Moreover, D ℚ , the...

Normal spaces and the Lusin-Menchoff property

Pavel Pyrih (1997)

Mathematica Bohemica

We study the relation between the Lusin-Menchoff property and the F σ -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the F σ -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.

Note on a discretization of a linear fractional differential equation

Jan Čermák, Tomáš Kisela (2010)

Mathematica Bohemica

The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.

Note on functions satisfying the integral Hölder condition

Josef, Jr. Král (1996)

Mathematica Bohemica

Given a modulus of continuity ω and q ∈ [ 1 , ∞ [ then H q ω denotes the space of all functions f with the period 1 on ℝ that are locally integrable in power q and whose integral modulus of continuity of power q (see(1)) is majorized by a multiple of ω . The moduli of continuity ω are characterized for which H q ω contains “many” functions with infinite “essential” variation on an interval of length 1 .

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