Grzegorz Graff - Publikacje - MOST Wiedzy

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Rok 2024
  • An absorbing set for the Chialvo map

    The classical Chialvo model, introduced in 1995, is one of the most important models that describe single neuron dynamics. In order to conduct effective numerical analysis of this model, it is necessary to obtain a rigorous estimate for the maximal bounded invariant set. We discuss this problem, and we correct and improve the results obtained by Courbage and Nekorkin (2010). In particular, we provide an explicit formula for an...

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Rok 2023
Rok 2022
  • Generalized Dold sequences on partially-ordered sets
    Publikacja

    - ELECTRONIC JOURNAL OF COMBINATORICS - Rok 2022

    Dold sequences constitute an important class of integer sequences that play an important role in combinatorics, number theory, topology and dynamical systems. We generalize the notion of Dold sequence for the case of partially ordered sets and describe their properties. In particular we give two alternative descriptions of generalized Dold sequences: by some class of elementary sequences as well as by different...

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  • Minimal Sets of Lefschetz Periods for Morse-Smale Diffeomorphisms of a Connected Sum of g Real Projective Planes
    Publikacja

    - Rok 2022

    The dataset titled Database of the minimal sets of Lefschetz periods for Morse-Smale diffeomorphisms of a connected sum of g real projective planes contains all of the values of the topological invariant called the minimal set of Lefschetz periods, computed for Morse-Smale diffeomorphisms of a non-orientable compact surface without boundary of genus g (i.e. a connected sum of g real projective planes), where g varies from 1 to...

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Rok 2021
Rok 2020
Rok 2019
  • Detecting coupling directions with transcript mutual information: A comparative study
    Publikacja

    - DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B - Rok 2019

    Causal 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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  • Generating sequences of Lefschetz numbers of iterates
    Publikacja
    • G. Graff
    • M. Lebiedź
    • P. Nowak-Przygodzki

    - MONATSHEFTE FUR MATHEMATIK - Rok 2019

    Du, 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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  • Jak gładkość generuje punkty periodyczne
    Publikacja

    Jednym 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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  • Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms

    We 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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  • Periodic Points for Sphere Maps Preserving MonopoleFoliations
    Publikacja

    Let 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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Rok 2018
Rok 2017
  • Visualization of short-term heart period variability with network tools as a method for quantifying autonomic drive
    Publikacja
    • D. Makowiec
    • B. Graff
    • A. Kaczkowska
    • G. Graff
    • D. Wejer
    • J. Wdowczyk-Szulc
    • M. Żarczyńska-Buchowiecka
    • M. Gruchała
    • Z. R. Struzik

    - Rok 2017

    We 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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Rok 2016
  • Computing algebraic transfer entropy and coupling directions via transcripts
    Publikacja
    • J. Amigó
    • R. Monetti
    • B. Graff
    • G. Graff

    - CHAOS - Rok 2016

    Most 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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  • Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds
    Publikacja

    Let 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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