prof. dr hab. Grzegorz Graff
Zatrudnienie
- Kierownik Studiów Doktoranckich w Wydział Fizyki Technicznej i Matematyki Stosowanej
- Kierownik zakładu w Zakład Równań Różniczkowych i Zastosowań Matematyki
- Profesor w Instytut Matematyki Stosowanej
Obszary badawcze
Publikacje
Filtry
wszystkich: 64
Katalog Publikacji
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Ordinal pattern statistics for the assessment of heart rate variability
PublikacjaThe recognition of all main features of a healthy heart rhythm (the so-called sinus rhythm) is still one of the biggest challenges in contemporary cardiology. Recently the interesting physiological phenomenon of heart rate asymmetry has been observed. This phenomenon is related to unbalanced contributions of heart rate decelerations and accelerations to heart rate variability. In this paper we apply methods based on the concept...
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Relationship between heart rate variability, blood pressure and arterial wall properties during air and oxygen breathing in healthy subjects
PublikacjaPrevious studies reported that normobaric hyperoxia influences heart rate, arterial pressure, cardiac output and systemic vascular resistance, but the mechanisms underlying these changes are still not fully understood. Several factors are considered including degeneration of endothelium-derived nitric oxide by reactive oxygen species, the impact of oxygen-free radicals on tissues and alterations of autonomic nervous system function....
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Entropy Measures in the Assessment of Heart Rate Variability in Patients with Cardiodepressive Vasovagal Syncope
PublikacjaSample entropy (SampEn) was reported to be useful in the assessment of the complexity of heart rate dynamics. Permutation entropy (PermEn) is a new measure based on the concept of order and was previously shown to be accurate for short, non-stationary datasets. The aim of the present study is to assess if SampEn and PermEn obtained from baseline recordings might differentiate patients with various outcomes of the head-up tilt test...
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Fixed point indices of iterated smooth maps in arbitrary dimension
PublikacjaWe give a complete description of possible sequences ofindices of iterations of f at an isolated fixed point, answering inaffirmative the Chow, Mallet-Paret and Yorke conjecture posed in[S.N. Chow, J. Mallet-Parret, J.A. Yorke, A periodic point index whichis a bifurcation invariant, in: Geometric Dynamics, Rio de Janeiro,1981, in: Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983,pp. 109-131].
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Minimizing the number of periodic points for smooth maps. Non-simply connected case
PublikacjaNiech f będzie gładkim odwzorowaniem zamkniętej rozmaitości o wymiarze wiekszym niż 2, a r ustaloną liczbą naturalną. W artykule zdefiniowany został niezmiennik topologiczny równy minimalnej liczbie punktów r-periodycznych w gładkiej klasie homotopii f.
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Persistent homology as a new method of the assessment of heart rate variability
PublikacjaHeart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent...
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Computing algebraic transfer entropy and coupling directions via transcripts
PublikacjaMost random processes studied in nonlinear time series analysis take values on sets endowed with a group structure, e.g., the real and rational numbers, and the integers. This fact allows to associate with each pair of group elements a third element, called their transcript, which is defined as the product of the second element in the pair times the first one. The transfer entropy of two such processes is called algebraic transfer...
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Electrophysiological features in patients with sinus node dysfunction and vasovagal syncope
PublikacjaThe aim of the study was to identify electrophysiological criteria that can be used for identification of patients with sinus node dysfunction and concurrent vasovagal syncope.
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Lefschetz periodic point free self-maps of compact manifolds
PublikacjaLet f be a self-map of a compact connected manifold M. We characterize Lefschetz periodic point free continuous self-maps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous self-maps on connected compact manifolds, Topology Appl. 158 (16) (2011) 2165-2169].
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Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds
PublikacjaLet M be a smooth compact and simply-connected manifold with simply-connected boundary ∂M, r be a fixed odd natural number. We consider f, a C1 self-map of M, preserving ∂M . Under the assumption that the dimension of M is at least 4, we define an invariant Dr(f;M,∂M) that is equal to the minimal number of r-periodic points for all maps preserving ∂M and C1-homotopic to f. As an application, we give necessary and sufficient...
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Lefschetz periodic point free self-maps of compact manifolds
PublikacjaLet f be a self-map of a compact connected manifold M. We characterize Lefschetz periodic point free continuous self-maps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous self-maps on connected compact manifolds,
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Generating sequences of Lefschetz numbers of iterates
PublikacjaDu, Huang and Li showed in 2003 that the class of Dold–Fermat sequences coincides with the class of Newton sequences, which are defined in terms of socalled generating sequences. The sequences of Lefschetz numbers of iterates form an important subclass of Dold–Fermat (thus also Newton) sequences. In this paper we characterize generating sequences of Lefschetz numbers of iterates.
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Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds
PublikacjaLet M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a fixed natural number. The topological invariant D^m_r [f], defined by the authors in [Forum Math. 21 (2009), 491-509], is equal to the minimal number of r-periodic points in the smooth homotopy class of f, a given self-map of M. In this paper, we present a general combinatorial scheme of computing D^m_r [f] for arbitrary dimension...
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Dold sequences, periodic points, and dynamics
PublikacjaIn this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.
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Topological-numerical analysis of a two-dimensional discrete neuron model
PublikacjaWe conduct computer-assisted analysis of a two-dimensional model of a neuron introduced by Chialvo in 1995 [Chaos, Solitons Fractals 5, 461–479]. We apply the method of rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 [SIAM J. Appl. Dyn. Syst. 8, 757–789] and improved and expanded afterward. Additionally, we introduce a new algorithm to analyze the return times...
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Periodic points of latitudinal maps of the $m$-dimensional sphere
PublikacjaLet f be a smooth self-map of the m-dimensional sphere Sm. Under the assumption that f preserves latitudinal foliations with the fibres S1, we estimate from below the number of fixed points of the iterates of f. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz.
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Shub’s conjecture for smooth longitudinal maps of S^m
PublikacjaLet f be a smooth map of the m-dimensional sphere Sm to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f , reduce Shub’s conjecture for longitudinal maps to a lower dimensional classical version, and prove the conjecture in case m = 2 and in a weak form for m = 3.
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Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms
PublikacjaWe apply the representation of Lefschetz numbers of iterates in the form of periodic expansion to determine the minimal sets of Lefschetz periods of Morse–Smale diffeomorphisms. Applying this approach we present an algorithmic method of finding the family of minimal sets of Lefschetz periods for Ng, a non-orientable compact surfaces without boundary of genus g. We also partially confirm the conjecture of Llibre and Sirvent (J Diff...
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Local fixed point indices of iterations of planar maps
PublikacjaW artykule podana zostaje postać indeksów iteracji dla pewnej klasy odwzorowań planarnych. Podstawowymi narzędziami stosowanym w pracy są liczba Nielsena i indeks Conleya.
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A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions
PublikacjaIn this paper we discuss a topological treatment for the planar system z' = f (t, z) + g(t, z) where f and g are T -periodic in time and g(t, z) is bounded. Namely, we study the effect of g(t, z) in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding...
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Fixed point indices of iterates of a low-dimensional diffeomorphism at a fixed point which is an isolated invariant set
PublikacjaLet f be an R^n-diffeomorphism, where n = 2, 3, for which {0} is an isolated invariant set. We determine all possible forms of the sequences of fixed point indices of iterates of f at 0, {ind(f n, 0)}_n, confirming in R3 the conjecture of Ruiz del Portal and Salazar (J Differ Equ 249, 989–1013, 2010).
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Detecting coupling directions with transcript mutual information: A comparative study
PublikacjaCausal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between...
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Dynamics of Field Line Mappings in Magnetic Flux Tubes
PublikacjaWe study the topological constraints on the dynamics of magnetic field lines in flux tubes. Our approach is based on the application of the topological invariant: fixed point index. We consider periodic flux tubes and find various restrictions on the field lines that come from the sequence of fixed point indices of iterations. We also analyze the case of a tube with a cylindrical obstacle, deducing some special dynamical properties...
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Periodic Points for Sphere Maps Preserving MonopoleFoliations
PublikacjaLet S^2 be a two-dimensional sphere. We consider two types of its foliations with one singularity and maps f:S^2→S^2 preserving these foliations, more and less regular. We prove that in both cases f has at least |deg(f)| fixed points, where deg(f) is a topological degree of f. In particular, the lower growth rate of the number of fixed points of the iterations of f is at least log|deg(f)|. This confirms the Shub’s conjecture in...
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The Efficiency of Polish Stock Market: Ordinal Patterns Approach
PublikacjaZunino et al. analyzed the problem of discrimination of developed and emergent markets by the use of ordinal patterns methods: number of forbidden patterns and ordinal pattern probability distribution as a basis for entropy and statistical measure of complexity. In this paper we apply the same methodology for the analysis of Polish stock market (index WIG). The results indicate that Polish market belongs neither to developed, nor...
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General form of fixed point indices of an iterated C^1 map andinfiniteness of minimal periods
PublikacjaDla zwartego podzbioru punktów periodycznych gładkiego odwzorowania podana zostaje formuła na indeksy iteracji. Wynik stanowi uogólnienie rezultatu Chowa, Malleta-Pareta i Yorke'a.
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Entropia w badaniach zaburzeń rytmu serca
PublikacjaArtykuł prezentuje zastosowanie ''Approximate Entropy'', będącej miarą stopnia złożoności szeregów czasowych, do analizy zmiennosci rytmu serca.
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Minimal number of periodic points for smooth self-maps of two-holed 3-dimensional closed ball
PublikacjaDla ciągłego odwzorowania f przestrzeni określonej w tytule w siebie, które posiada rzeczywiste wartości własne na drugiej grupie homologii, wyznaczona została minimalna liczba punktów r-periodycznych w klasie wszystkich gładkich odwzorowań homotopijnych z f.
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Minimal number of periodic points for C^1 self-maps of compact simply-connected manifolds
PublikacjaNiech f będzie odwzorowaniem gładkiej zwartej i jednospójnej rozmaitości o wymiarze większym lub równym 3. W pracy zdefiniowany został topologiczny niezmiennik będący najlepszym dolnym oszacowaniem liczby punktów periodycznych w klasie gładkich odwzorowań homotopijnych z f.
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Minimal number of periodic points for smooth self-maps of S^3
PublikacjaW pracy wyznaczona została najmniejsza liczba punktów periodycznych w gładkiej klasie homotopii odwzorowania sfery trójwymiarowej w siebie.
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On the growth of the number of periodic points for smooth self maps of a compact manifold
PublikacjaDla ciągłego przekształcenia jednospójnej rozmaitości wymiaru co najmniej 3 w siebie, wykazujemy, że wzrost liczby punktów r-periodycznych w klasie homotopii może być nie szybszy niż liniowy, dla dowolnego, ustalonego r.
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Existence of periodic orbits for a perturbed vector field
PublikacjaPrzy nałożeniu pewnego warunku na odwzorowanie Poincarego, wyrażonego w języku indeksów iteracji, dowodzi się istnienia orbit periodycznych dla zaburzonego pola wektorowego.
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Fixed points of planar homeomorphisms of the form Identity + Contraction
PublikacjaW pracy dowodzi się, przy użyciu indeksu, istnienia punktów stałych dla planarnych homeomorfizmów, których orbity spełniają pewien geometryczny warunek.
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Periodicity of a sequence of local fixed point indices of iterations
PublikacjaPraca uogólnia klasyczne twierdzenie Shuba i Sullivana o periodyczności ciągu indeksów punktu stałego iteracji odwzorowań gładkich na szerszą klasę przekształeń.
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Fixed point indices of iterated planar maps
PublikacjaW artykule dokonuje się przeglądu wyników dotyczących indeksów punktu stałego iteracji odwzorowań planarnych, sformułowane zostają otwarte pytania i podane nowe dowody w przypadku gładkim.
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Indices of iterations of planar maps
PublikacjaArtykuł omawia najnowsze rezultaty badań dotyczące postaci indeksów punktu stałego iteracji odwzorowań planarnych.
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Algebraic periods of self-maps of a rational exterior space of rank 2
PublikacjaArtykuł stanowi kompletny opis okresów algebraicznych dla odwzorowań wymiernej przestrzeni zewnętrznej rangi 2 w siebie.
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Fixed point indices of iterations of C^1 maps in R^3
PublikacjaW przypadku gładkiego odwzorowania w R^3 dowiedziona została hipoteza Chowa, Malleta-Pareta i Yorka dotycząca postaci ciągów indeksow iteracji oraz podano kompletny opis możliwych ciągów indeksow.
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Minimal number of periodic points for smooth self-maps of RP^3
PublikacjaNiech f będzie gładkim odwzorowaniem 3-wymiarowej rzeczywistej przestrzeni rzutowej w siebie, r będzie ustaloną liczbą naturalną. W artykule wyznaczona została minimalna liczba punktów r-periodycznych w gładkiej klasie homotopii odwzorowania f.
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Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds
PublikacjaLet M be a smooth closed simply-connected 4-dimensional manifold, f be a smooth self-map of M with fast grow of Lefschetz numbers and r be a product of different primes. The authors calculate the invariant equal to the minimal number of r-periodic points in the smooth homotopy class of f.
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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
PublikacjaLet 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....
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Entropy measures of heart rate variability for short ECG datasets in patients with congestive heart failure
PublikacjaWe investigated the usefulness of entropy measures calculated for short ECG series in distinguishing healthy subjects from patients with congestive heart failure (CHF). Four entropy measures were tested: Approximate Entropy (ApEn), Sample Entropy (SampEn), Fuzzy Entropy (FuzzyEn) and Permutation Entropy (PE), each computed for ECG series of 1000, 500, 250 and 100 RR intervals. We found that with a reduction of the data set length...
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Jak gładkość generuje punkty periodyczne
PublikacjaJednym z ważnych problemów teorii układów dynamicznych i topologii jest pytanie, jaka jest najmniejsza liczba punktów stałych lub periodycznych w danej klasie odwzorowań. Na przykład klasyczne twierdzenie Brouwera stwierdza, że każde ciągłe odwzorowanie kuli domkniętej w siebie ma przynajmniej jeden punkt stały. Szczególnie interesujące staje się powyższe pytanie w odniesieniu do klasy homotopii danego odwzorowania f. Artykuł poświęcony...
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Computations of the least number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds
PublikacjaLet $r$ be an odd natural number, $M$ a compact simply-connected smooth manifold, $\dim M\geq 4$, such that its boundary $\partial M$ is also simply-connected. We consider $f$, a $C^1$ self-maps of $M$, preserving $\partial M$. In [G. Graff and J. Jezierski, Geom. Dedicata 187 (2017), 241-258] the smooth Nielsen type periodic number $D_r(f;M,\partial M)$ was defined and proved to be equal to the minimal number of $r$-periodic points...
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Visualization of short-term heart period variability with network tools as a method for quantifying autonomic drive
PublikacjaWe argue that network methods are successful in detecting nonlinear properties in the dynamics of autonomic nocturnal regulation in short-term variability. Two modes of visualization of networks constructed from RR-increments are proposed. The first is based on the handling of a state space. The state space of RR-increments can be modified by a bin size used to code a signal and by the role of a given vertex as the representation...
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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
PublikacjaLet f be a smooth self-map of an m-dimensional (m >3) closed connected and simply-connected manifold such that the sequence of the Lefschetz num- bers 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 defned in combinatorial terms and is...
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Estimation of the minimal number of periodic points for smooth self-maps of odd dimensional real projective spaces
PublikacjaLet f be a smooth self-map of a closed connected manifold of dimension m⩾3. The authors introduced in [G. Graff, J. Jezierski, Minimizing the number of periodic points for smooth maps. Non-simply connected case, Topology Appl. 158 (3) (2011) 276-290] the topological invariant NJD_r[f], where r is a fixed natural number, which is equal to the minimal number of r-periodic points in the smooth homotopy class of f. In this paper smooth...
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O Centrum Zastosowań Matematyki
PublikacjaCentrum Zastosowań Matematyki to projekt realizowany w ramach Programu Operacyjnego Kapitał Ludzki wyłoniony w drodze konkursu zorganizowanego przez Narodowe Centrum Badań i Rozwoju.
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Entropy Measures of heart rate variability for short ECG datasets in patients with congestive heart failure
PublikacjaWe investigated the usefulness of entropy measures calculated for short ECG series in distinguishing healthy subjects from patients with congestive heart failure (CHF). Four entropy measures were tested: Approximate Entropy (ApEn), Sample Entropy (SampEn), Fuzzy Entropy (Fuzzy En) and Permutation Entropy (PE), each computed for ECG series of 1000, 500, 250 and 100 RR intervals. We found that with a reduction of the data set length...
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Network Approach to Increments of RR-intervals for Visualization of Dynamics of Cardiac Regulation
PublikacjaThe transition network for RR -increments is pre- sented in a directed and weighted graph, with vertices represent- ing RR -increments and edges corresponding to the order in a sequence of increments. The adjacency matrix and the transition matrix of this network provide a graphical tool which could be useful in the assessment of cardiac regulation. As an example, the method is applied in detecting differences between diurnal activity...
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