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1
Content available remote

On time reparametrizations and isomorphisms of impulsive dynamical systems

100%
Ciesielski K.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 1
1-25
EN
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).
2
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Sum and difference free partitions of vector spaces

100%
Ciesielski K.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 2
263-271
3
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On Semicontinuity in Impulsive Dynamical Systems

100%
Ciesielski K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
tom 52
|
nr 1
71-80
EN
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. We characterize the function ϕ from the point of view of its semicontinuity. Also, we show the analogous properties for impulsive systems given by semidynamical systems. In the last section we investigate the continuity properties of the escape time function in impulsive systems.
4
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On Stability in Impulsive Dynamical Systems

100%
Ciesielski K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
tom 52
|
nr 1
81-91
EN
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.
5
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The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

100%
Ciesielski K.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2110-2128
EN
The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.
6
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Strong Fubini properties for measure and category

64%
Ciesielski K., Laczkovich M.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 2
171-188
EN
Let (FP) abbreviate the statement that $∫_{0}^{1} (∫_{0}^{1} fdy)dx = ∫_{0}^{1} (∫_{0}^{1} fdx)dy$ 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 of the ideal of null sets. In particular, we establish the equivalence of (SFP) to the nonexistence of certain sets with paradoxical properties, a phenomenon that was already known for (FP). We also give the category analogues of these statements and, whenever possible, we try to put the statements in a setting of general ideals as initiated by Recław and Zakrzewski.
7
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A big symmetric planar set with small category projections

64%
Ciesielski K., Natkaniec T.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 3
237-253
EN
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 holds in the iterated perfect set model. We also prove that the existence of a set A as in (i) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of ℝ. On the other hand, for the class of real-analytic functions a Bernstein set A satisfying (ii) exists in ZFC.
8
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Covering Property Axiom $CPA_{cube}$ and its consequences

64%
Ciesielski K., Pawlikowski J.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 1
63-75
EN
We formulate a Covering Property Axiom $CPA_{cube}$, 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. (c) cof(𝓝) = ω₁ < 𝔠, i.e., the cofinality of the measure ideal 𝓝 is ω₁. (d) For every uniformly bounded sequence $⟨fₙ ∈ ℝ^{ℝ}⟩_{n<ω}$ of Borel functions there are sequences: $⟨P_ξ ⊂ ℝ: ξ < ω₁⟩$ of compact sets and $⟨W_ξ ∈ [ω]^ω: ξ < ω₁⟩$ such that $ℝ = ⋃_{ξ<ω₁}P_ξ$ and for every ξ < ω₁, $⟨fₙ ↾ P_ξ⟩_{n∈W_ξ}$ is a monotone uniformly convergent sequence of uniformly continuous functions. (e) Total failure of Martin's Axiom: 𝔠 > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them
9
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The characterization of the cut of funnel in a planar semidynamical system

63%
Ciesielski K.
Annales Polonici Mathematici
|
1990
|
tom 51
|
nr 1
99-121
10
Content available remote

On nowhere weakly symmetric functions and functions with two-element range

52%
Ciesielski K., Muthuvel K., Nowik A.
Fundamenta Mathematicae
|
2001
|
tom 168
|
nr 2
119-130
EN
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 ignore some countable set from which we can choose the sequence hₙ, then there exists a function f: ℝ → {0,1} which is nowhere weakly symmetric in this weaker sense if and only if the continuum hypothesis holds.
11
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Functions characterized by images of sets

52%
Ciesielski K., Dikrajan D., Watson S.
Colloquium Mathematicum
|
1998
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tom 77
|
nr 2
211-232
EN
For non-empty topological spaces X and Y and arbitrary families $\cal A$ ⊆ $\cal P(X)$ and $\cal B ⊆ \cal P(Y)$ we put $\cal C_{\cal A,\cal B}$={f ∈ $Y^X$ : (∀ A ∈ $\cal A$)(f[A] ∈ $\cal B)$}. We examine which classes of functions $\cal F$ ⊆ $Y^X$ can be represented as $\cal C_{\cal A,\cal B}$. We are mainly interested in the case when $\cal F=\cal C(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\cal F=\cal C$(X,ℝ) is not equal to $\cal C_{\cal A,\cal B}$ for any $\cal A$ ⊆ $\cal P(X)$ and $\cal B$ ⊆ $\cal P$(ℝ). Thus, $\cal C$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as $\cal C_{\cal A,\cal B}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.
12
Content available remote

Uncountable γ-sets under axiom $CPA_{cube}^{game}$

52%
Ciesielski K., Millán A., Pawlikowski J.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 2
143-155
EN
We formulate a Covering Property Axiom $CPA_{cube}^{game}$, 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 < 𝔠, since every γ-set is universally null, while $CPA_{cube}^{game}$ implies that every universally null has cardinality less than 𝔠 = ω₂. We also show that $CPA_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets.
13
Content available remote

Semantic information within the BEATCA framework

46%
Ciesielski K., Czerski D., Dramiński M., Kłopotek M. A., Wierzchoń S. T.
Control and Cybernetics
|
2010
|
tom 39
|
nr 2
377-399
14
Content available remote

Extensions of invariant measures on Euclidean spaces

45%
Ciesielski K., Pelc A.
Fundamenta Mathematicae
|
1985
|
tom 125
|
nr 1
1-10
15
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Continuity in semidynamical systems

44%
Ciesielski K.
Annales Polonici Mathematici
|
1985
|
tom 46
|
nr 1
61-70
16
Content available remote

On the netweight of subspaces

32%
Ciesielski K.
Fundamenta Mathematicae
|
1983
|
tom 117
|
nr 1
37-46
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