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Search results for: HOMOTOPY, MORPHISMS, COINCIDENCE INDEX, ADMISSIBLE MAPPING.
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On homotopies of morphisms and admissible mappings
PublicationThe notion of homotopy in the category of morphisms introduced by G´orniewicz and Granas is proved to be equivalence relation which was not clear for years. Some simple properties are proved and a coincidence point index is described.
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On homotopy Conley index for multivalued flows in Hilbert spaces
PublicationPodano aproksymacyjną definicję indeksu homotopijnego, otrzymując naturalne związki z podobnymi niezmiennikami. Zbadano własności tego niezmiennika i zastosowano do badania gradientowych potoków wykazując pewne geometryczne własności zbiorów niezmienniczych
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Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces
PublicationIn this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain...
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Topological invariants for equivariant flows: Conley index and degree
PublicationAbout forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical...
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The cohomological span of LS-Conley index
PublicationIn this paper we introduce a new homotopy invariant – the cohomological span of LS-Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form Lx+K(x), where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula...
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Index filtrations and Morse decomposition for discrete dynamical systems
PublicationOn a Morse decomposition of an isolated invariant set of a homeomorphism(discrete dynamical system) there are partial orderings defined by the homeomorphism.These are called admissible orderings of the...
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On a comparison principle and the uniqueness of spectral flow
PublicationThe spectral flow is a well-known quantity in spectral theory that measures the variation of spectra about 0 along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly, we consider homotopy invariance properties of the spectral flow and establish a simple formula which comprises its classical homotopy invariance and yields a comparison theorem for the spectral flow under compact perturbations. We apply...
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Offshore benthic habitat mapping based on object-based image analysis and geomorphometric approach. A case study from the Slupsk Bank, Southern Baltic Sea
PublicationBenthic habitat mapping is a rapidly growing field of underwater remote sensing studies. This study provides the first insight for high-resolution hydroacoustic surveys in the Slupsk Bank Natura 2000 site, one of the most valuable sites in the Polish Exclusive Zone of the Southern Baltic. This study developed a quick and transparent, automatic classification workflow based on multibeam echosounder and side-scan sonar surveys to...
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Dynamics of Field Line Mappings in Magnetic Flux Tubes
PublicationWe 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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A new index for statistical analyses and prediction of travelling ionospheric disturbances
PublicationTravelling Ionospheric Disturbances (TIDs) are signatures of atmospheric gravity waves (AGWs) observed in changes in the electron density. The analysis of TIDs is relevant for studying coupling processes in the thermosphere–ionosphere system. A new TID index is introduced, which is based on an easy extension of the commonly used approach for TID detection. This TID activity index, which can be applied for individual Global Navigation...
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Automatic classification and mapping of the seabed using airborne LiDAR bathymetry
PublicationShallow coastal areas are among the most inhabited areas and are valuable for biodiversity, recreation and the economy. Due to climate change and sea level rise, sustainable management of coastal areas involves extensive exploration, monitoring, and protection. Current high-resolution remote sensing methods for monitoring these areas include bathymetric LiDAR. Therefore, this study presents a novel methodological approach to assess...
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Dietary behaviour and physical activity policies in Europe: learnings from the Policy Evaluation Network (PEN)
PublicationThe European Policy Evaluation Network (PEN), initiated in autumn 2018, aimed at advancing the evidence base for public policies impacting dietary behaviour, physical activity and sedentary behaviours in Europe. This is needed because non-communicable diseases—the leading cause of global mortality—are substantially caused by physical inactivity and unhealthy dietary behaviours, which in turn are driven by upstream factors that...
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Multicultural coastal cities: what are the differences in culture of urban planning management. Comparison analysis of Gdańsk and Gdynia
PublicationThe goal of this book chapter is the analysis of the multiculturalism of the seaside cities (Gdansk and Gdynia). We define multiculturalism by ethnic aspects, as concentration of multiple nationalities and ethnic groups over the given area and non ethnic aspects, as occurrence of multiple symbols, signs or cultures. We sense multiculturalism of the cities through the cultural and social properties of the citizens and urban architecture....