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1
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Minimal periods of maps of rational exterior spaces

100%
Graff G.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 2
99-115
EN
The problem of description of the set Per(f) of all minimal periods of a self-map f:X → X is studied. If X is a rational exterior space (e.g. a compact Lie group) then there exists a description of the set of minimal periods analogous to that for a torus map given by Jiang and Llibre. Our approach is based on the Haibao formula for the Lefschetz number of a self-map of a rational exterior space.
2
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Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers

64%
Graff G., Kaczkowska A.
Annales Polonici Mathematici
|
2013
|
tom 107
|
nr 1
29-48
EN
Let f be a smooth self-map of an m-dimensional (m ≥ 4) closed connected and simply-connected manifold such that the sequence ${L(fⁿ)}_{n=1}^{∞}$ of the Lefschetz numbers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defined in combinatorial terms and is constant for all sufficiently large r. We compute J[f] for self-maps of some manifolds with simple structure of homology groups.
3
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Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

64%
Graff G., Kaczkowska A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2160-2172
EN
Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.
4
Content available remote

Minimal number of periodic points for smooth self-maps of S³

64%
Graff G., Jezierski J.
Fundamenta Mathematicae
|
2009
|
tom 204
|
nr 2
127-144
EN
Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant $D^m_r[f]$, introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate $D³_r[f]$ for all self-maps of S³.
5
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The Project Center for Applications of Mathematics

64%
Bartłomiejczyk A., Graff G.
Mathematica Applicanda
|
2013
|
tom 41
|
nr 2
PL
Centrum Zastosowań Matematyki to projekt realizowany w ramach Programu Operacyjnego Kapitał Ludzki, wyłoniony w drodze konkursu zorganizowanegoprzez Narodowe Centrum Badan i Rozwoju. Trzyletni okres funkcjonowania projektu obejmujacy lata 2012–2015 umożliwia prowadzenie szeroko zakrojonej działalności,której celem jest rozwój matematyki stosowanej w Polsce. Centrum umiejscowione jest na Wydziale Fizyki Technicznej i Matematyki Stosowanej Politechniki Gdanskiej, alejego działalność ma charakter ogólnopolski. W Radzie Naukowej Centrum zasiadaja przedstawiciele trzech głównych trójmiejskich uczelni, reprezentujacy matematyke, fizyke i medycyne:dr hab. Grzegorz Graff, prof.nadzw. PG (kierownik projektu),prof. dr hab. Sergey Leble z Politechniki Gdanskiej,dr hab. Henryk Leszczynski, prof. nadzw. UG,prof. dr hab. Danuta Makowiec z Uniwersytetu Gdańskiegoorazprof. dr hab. Krzysztof Narkiewicz z Gdanskiego Uniwersytetu Medycznego.
EN
Abstract. Center of Applied Mathematics is the project co-financed by EuropeanUnion within the Human Capital Operational Programme. Its main aim is to promoteinterdisciplinary cooperation between mathematicians and the representativesof other disciplines as well as the development of the mathematical methods whichcould be useful in the sphere of applications. The project is realized at Faculty ofApplied Physics and Mathematics of Gdansk University o Technology. In the articlethe main tasks realized within the project are described.
6
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Entropy analysis in cardiac arrythmias

51%
Graff B., Graff G., Kolesiak A.
Mathematica Applicanda
|
2008
|
tom 36
|
nr 50/09
PL
Artykuł prezentuje zastosowanie Approximate Entropy, będącej miarą stopnia złożoności szeregów czasowych, do analizy zmienności rytmu serca.
EN
Healthy human heart rate is characterized by oscillations observed in intervals between consecutive heartbeats (RR intervals). Conventional methods of heart rate variability analysis measure the overall magnitude of RR interval fluctuations around its mean value or the magnitude of fluctuations in predetermined frequencies. The new methods of chaos theory and nonlinear dynamics provide powerful tools, which allow to predict clinical outcome in patients with cardiovascular diseases. The main aim of our article is to present Approximate Entropy (ApEn), a measure of system regularity and complexity, introduced by Pincus in 1991. ApEn estimation used for clinical purposes is applied for finite number of records, divided in vectors, and depends on two fixed parameters m and r. Then Approximate Entropy may be interpreted as the average of negative natural logarithms of conditional probability, that two vectors of length m + 1 are similar (we define here r-similarity), if two vectors of the length m are similar. The article provides a formal mathematical description of ApEn and presents a simple algorithm for its assessment. The choice of input parameters m and r is also discussed. In vast majority of publications r depends on standard deviation (SD) of average of all records, when individual features of heart rhythm are taken into account. The fraction of r, equal to 0, 2SD, and m = 2 are usually chosen on the basis of previous findings of good statistical validity. With the above set of parameters we can avoid the influence of outliers and do not loose too much information. ApEn has also some disadvantages - the main is counting self similarities. To reduce this kind of bias some improvements of the methods based on Pincus’ algorithm were developed. For example Sample Entropy (SampEn), which has similar algorithm but does not count self-matches, was proposed and easily applied to clinical time-series. In the article we present also an application of ApEn in predicting atrial fibrillation (AF), a type of arrhythmia which is the most common sustained heart rhythm disturbance. Both ApEn and SampEn decrease before the spontaneous onset of AF. What is more, ApEn is not sensitive to ectopy beats and therefore can be assessed fully automatically. The potential application of ApEn is the possibility to detect an increased vulnerability to AF before the onset of arrhythmia during continuous heart rate recording, for example for patients with implantable pacemakers. The recognition of the higher risk of AF would be followed by immediate pacemaker reprogramming to prevent an episode of arrhythmia. It would result not only in better quality of life of the patient but also in decreased number of hospitalization and cost of treatment.
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