The characterization of the cut of funnel in a planar semidynamical system
Continuity in semidynamical systems
On the netweight of subspaces
Sum and difference free partitions of vector spaces
The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century
The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.
On Semicontinuity in Impulsive Dynamical Systems
In the important paper on impulsive systems [K1] several notions are introduced and several properties of these systems are shown. In particular, the function ϕ which describes "the time of reaching impulse points" is considered; this function has many important applications. In [K1] the continuity of this function is investigated. However, contrary to the theorem stated there, the function ϕ need not be continuous under the assumptions given in the theorem. Suitable examples are shown in this paper....
On time reparametrizations and isomorphisms of impulsive dynamical systems
We prove that for a given impulsive dynamical system there exists an isomorphism of the basic dynamical system such that in the new system equipped with the same impulse function each impulsive trajectory is global, i.e. the resulting dynamics is defined for all positive times. We also prove that for a given impulsive system it is possible to change the topology in the phase space so that we may consider the system as a semidynamical system (without impulses).
On Stability in Impulsive Dynamical Systems
Several results on stability in impulsive dynamical systems are proved. The first main result gives equivalent conditions for stability of a compact set. In particular, a generalization of Ura's theorem to the case of impulsive systems is shown. The second main theorem says that under some additional assumptions every component of a stable set is stable. Also, several examples indicating possible complicated phenomena in impulsive systems are presented.
Extensions of invariant measures on Euclidean spaces
A big symmetric planar set with small category projections
We show that under appropriate set-theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set A ⊂ ℝ such that (i) the set {c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set {c ∈ ℝ: (f+c) ∩ (A×A) = ∅} is nowhere meager for each continuous function f: ℝ → ℝ. The existence of such a set also follows from the principle CPA, which...
Covering Property Axiom and its consequences
We formulate a Covering Property Axiom , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence of Borel functions there are sequences:...
Strong Fubini properties for measure and category
Let (FP) abbreviate the statement that holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties...
Analytic functions are -density continuous
A real function is -density continuous if it is continuous with the -density topology on both the domain and the range. If is analytic, then is -density continuous. There exists a function which is both and convex which is not -density continuous.
Functions characterized by images of sets
For non-empty topological spaces X and Y and arbitrary families ⊆ and we put =f ∈ : (∀ A ∈ )(f[A] ∈ . We examine which classes of functions ⊆ can be represented as . We are mainly interested in the case when is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class (X,ℝ) is not equal to for any ⊆ and ⊆ (ℝ). Thus, (X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...
On nowhere weakly symmetric functions and functions with two-element range
A function f: ℝ → {0,1} is weakly symmetric (resp. weakly symmetrically continuous) at x ∈ ℝ provided there is a sequence hₙ → 0 such that f(x+hₙ) = f(x-hₙ) = f(x) (resp. f(x+hₙ) = f(x-hₙ)) for every n. We characterize the sets S(f) of all points at which f fails to be weakly symmetrically continuous and show that f must be weakly symmetric at some x ∈ ℝ∖S(f). In particular, there is no f: ℝ → {0,1} which is nowhere weakly symmetric. It is also shown that if at each point x we...
Uncountable γ-sets under axiom
We formulate a Covering Property Axiom , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while implies that every universally null has cardinality less than = ω₂. We also show that implies the existence of a partition of ℝ into ω₁ null compact sets....
On negative escape time in semidynamical systems.
On Stefan Banach and some of his results.
Semantic information within the BEATCA framework
Restricted continuity and a theorem of Luzin
Let P(X,ℱ) denote the property: For every function f: X × ℝ → ℝ, if f(x,h(x)) is continuous for every h: X → ℝ from ℱ, then f is continuous. We investigate the assumptions of a theorem of Luzin, which states that P(ℝ,ℱ) holds for X = ℝ and ℱ being the class C(X) of all continuous functions from X to ℝ. The question for which topological spaces P(X,C(X)) holds was investigated by Dalbec. Here, we examine P(ℝⁿ,ℱ) for different families ℱ. In particular, we notice that P(ℝⁿ,"C¹") holds, where...
Page 1 Next