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Search results for: FIXED POINT INDEX
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Fixed point index for $G$-equivariant multivalued maps
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Fixed orbit index for equivariant maps
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Fixed point indices of iterates of a low-dimensional diffeomorphism at a fixed point which is an isolated invariant set
PublicationLet 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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Fixed point indices of iterated planar maps
PublicationW 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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On the Nielsen fixed point theory for multivalued mappings
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Fixed point indices of iterations of planar homeomorphisms.
PublicationW pracy bada się postać indeksów iteracji lokalnych homeomorfizmów płaszczyzny.
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Periodicity of a sequence of local fixed point indices of iterations
PublicationPraca uogólnia klasyczne twierdzenie Shuba i Sullivana o periodyczności ciągu indeksów punktu stałego iteracji odwzorowań gładkich na szerszą klasę przekształeń.
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Local fixed point indices of iterations of planar maps
PublicationW 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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Sequences of fixed point indices of iterations in dimension 2.
PublicationW pracy dowodzi się, że każdy ciąg liczb całkowitych spełniający relacje Dolda może zostać zrealizowany jako ciąg indeksów punktu stałego iteracji ciągłego odwzorowania dwuwymiarowego dysku w siebie.
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Fixed point indices of iterated smooth maps in arbitrary dimension
PublicationWe 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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Fixed point indices of iterations of C^1 maps in R^3
PublicationW 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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Fixed point theorems for weakly commuting and compatible multi-valued mappings.
PublicationW pracy podano twierdzenia o wspólnych punktach stałych czwórki odwzorowań: dwóch jednowartościowych T i S oraz dwóch wielowartościowych F i G, spełniających nieliniowy uogólniony warunek kontrakcyjny, przy pewnych założeniach dotyczących uogólnionej komutatywności T,S i F,G. W pracy zamieszczono przykłady ilustrujące udowodnione twierdzenia.
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Solving boundary value problems for delay differential equations by a fixed-point method
PublicationOgólne liniowe zagadnienie brzegowe dla nieliniowego układu równań różniczkowych z opóźnieniem jest redukowane do zagadnienia o punkcie stałym odpowiedniego operatora a następnie poszukiwany punkt stały tego operatora jest przybliżany funkcją kawałkami liniową zdefiniowaną poprzez jej wartości w węzłach. Przy odpowiednich założeniach istnienie tego punktu stałego jest równoważne istnieniu tzw. epsilon przybliżonych punktów stałych...
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General form of fixed point indices of an iterated C^1 map andinfiniteness of minimal periods
PublicationDla 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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A new approach to numerical solution of fixed-point problems and its application to delay differential equations
PublicationW pracy rozpatruje się pewne aproksymacje punktu stałego ciągłego operatora A odwzorowującego przestrzeń metryczną w siebie. Wspomniany punkt stały przybliża się tzw. epsilon przybliżonym punktem stałym z przestrzeni skończenie wymiarowej. Udowodnione zostało twierdzenie dające warunki konieczne i dostateczne istnienia punktu stałego w ogólnej przestrzeni metrycznej. Warunki te wyrażone są w terminach epsilon przybliżonego punktu...
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Solving Boundary Value Problems for Second Order Singularly Perturbed Delay Differential Equations by ε-Approximate Fixed-Point Method
PublicationIn this paper, the boundary value problem for second order singularly perturbed delay differential equation is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by cubic spline vh defined by its values vi = vh(ti) at grid points ti, i = 0, 1, ... ,N. The necessary for construction the cubic spline and missing the first derivatives at the boundary...
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The Maslov index and the spectral flow—revisited
PublicationWe give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of self-adjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell,...
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Some remarks on the Euler ring U(G)
PublicationNiech G będzie zwartą grupą Liego i niech U(G) oznacza pierściń Eulera G skonstruoawany przez tom Diecka w [5,6]. Główny wynikpracy (Twierdzenie 4.1) opisuje homomorfizm pierścienia U(SO(3)) w pierścień U(SO(2))indukowany przez włożenie grupy SO(2) w grupę SO(3).
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Bernstein-type theorem for ϕ-Laplacian
PublicationIn this paper we obtain a solution to the second-order boundary value problem of the form \frac{d}{dt}\varPhi'(\dot{u})=f(t,u,\dot{u}), t\in [0,1], u\colon \mathbb {R}\to \mathbb {R} with Sturm–Liouville boundary conditions, where \varPhi\colon \mathbb {R}\to \mathbb {R} is a strictly convex, differentiable function and f\colon[0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R} is continuous and satisfies a suitable growth...
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Morse cohomology in a Hilbert space via the Conley index
PublicationThe main theorem of this paper states that Morse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed.
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Spectral splittings in the Conley index theory
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The Conley index, cup-length and bifurcation
PublicationZastosowano strukturę modułu w indeksie kohomologicznym Conleya do dowodu twierdzenia o minimalnej ilości rozwiązań okresowych dla układów Hamiltonowskich. Wykazano też ogólne twierdzenia dotyczące nietrywialności struktury mudułu.
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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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Algebraic periods and minimal number of periodic points for smooth self-maps of 1-connected 4-manifolds with definite intersection forms
PublicationLet M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map f: M-->M. This task is related to determining a topological invariant D^4_r[f], defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of...
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Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds
PublicationLet 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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Two families of infinitely many homoclinics for singular strong force Hamiltonian systems
PublicationWe are concerned with a planar autonomous Hamiltonian system with a potential possessing a single well of infinite depth at a point X and a unique strict global maximum 0 at a point A. Under a strong force condition around the singularity X, via minimization of an action integral and using a shadowing chain lemma together with simple geometrical arguments, we prove the existence of infinitely many geometrically distinct homoclinic...
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Otopy classes of equivariant maps
PublicationW artykule definiuje się stopień topologiczny niezmienniczych odwzorowań lokalnych w przypadku gradientowym i niegradientowym. Wyniki dotyczą relacji pomiędzy tymi dwoma niezmiennikami topologicznymi.
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On relations between gradient and classical equivariant homotopy groups of spheres
PublicationWe investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted...
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Connecting orbits for a periodically forced singular planar Newtonian system
PublicationW niniejszym artykule badamy problem istnienia i krotności rozwiązań homoklinicznych i heteroklinicznych dla nieautonomicznych układów Newtonowskich na płaszczyźnie z potencjałem okresowym ze względu na zmienną czasową, mającym maksimum globalne właściwe przyjmowane w dwóch punktach płaszczyzny i punkt osobliwy (studnię nieskończonej głębokości), w otoczeniu którego potencjał spełnia warunek Gordona (gradient potencjału ze względu...
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Connected components of the space of proper gradient vector fields
PublicationWe show that there exist two proper gradient vector fields on Rn which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.
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Periodic expansion in determining minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms
PublicationWe 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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The Hopf type theorem for equivariant gradient local maps
PublicationWe construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite-dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of Z.
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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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The Arnold conjecture in $ \mathbb C\mathbb P^n $ and the Conley index
Publicationn this paper we give an alternative, purely Conley index based proof of the Arnold conjecture in CP^n asserting that a Hamiltonian diffeomorphism of CP^n endowed with the Fubini-Study metric has at least (n+1) fixed points.
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Production of fuel briquette from solid waste biomass using natural resin as a binder
PublicationAbstract Background: This research aimed to evaluate the use of natural resin as a binder for the production of fuel briquette from solid waste biomass. Methods: Proximate analyses were made for fuel briquette prepared from solid waste biomass using natural resin as a binder in comparison with using starch as a binder. Results: The results for density, percent content of moisture, percent content of volatile matter, percent content...
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Production of fuel briquette from solid waste biomass using natural resin as a binder
PublicationBackground: This research aimed to evaluate the use of natural resin as a binder for the production of fuel briquette from solid waste biomass.Methods: Proximate analyses were made for fuel briquette prepared from solid waste biomass using natural resin as a binder in comparison with using starch as a binder.Results: The results for density, percent content of moisture, percent content of volatile matter, percent...
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Wpływ termoplastycznych kompozycji napełnianych miałem gumowym na wybrane właściwości asfaltu drogowego
PublicationDynamiczny rozwój transportu samochodowego zmusza do wykonywania nawierzchni drogowych z materiałów o wysokiej wytrzymałości oraz odporności na odkształcenia trwałe, co znacznie wydłuża żywotność nawierzchni. Polskie przedsiębiorstwo BISEK Sp. z o.o. z Wrocławia we współpracy z Politechniką Wrocławską, jako pierwsze wykorzystało miał gumowy do modyfikacji asfaltów metodą mokrą. W chwili obecnej firma Lotos Asfalt Sp. z o.o. wprowadziła...
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Thermal visualization of Ostwald-de Waele liquid in wavy trapezoidal cavity: Effect of undulation and amplitude
PublicationThe present study is concerned with the numerical simulations of Ostwald-de Waele fluid flow in a wavy trapezoidal cavity in the presence of a heated cylinder situated at the center of the cavity. The work consists in characterizing the mixed convection as a function of the intensity of heat flow. The flow behaviour and temperature distribution in a cavity are the main focus of this study. The lower wall of the cavity is fixed...
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Digital Gaps and Economic Inequalities in MENA Countries: An Empirical Investigation
PublicationThis research contributes to the present state of the art by examining changes in cross-country inequalities in digital technologies deployment and economic performance. Our empirical target builds on the hypothesis that rapid global diffusion of digital technologies unequivocally leads to gradual eradication of cross-country digital gaps and cross-country inequalities in terms of deployment of ICT; the technology convergence occurs....
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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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Spatial Calibration of a Dual PTZ-Fixed Camera System for Tracking Moving Objects in Video
PublicationA dual camera setup is proposed, consisting of a fixed (stationary) camera and a pan-tilt-zoom (PTZ) camera, employed in an automatic video surveillance system. The PTZ camera is zoomed in on a selected point in the fixed camera view and it may automatically track a moving object. For this purpose, two camera spatial calibration procedures are proposed. The PTZ camera is calibrated in relation to the fixed camera image, using interpolated...
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E-cohomological Conley index
PublicationIn this thesis we continue with developing the E-cohomological Conley index which was introduced by A.Abbondandolo. In particular, we generalize the index to non-gradient flows, we show that it an possesses additional multiplicative structure and we prove the continuation principle. Then, using continuation principle, we show how the computation of the E-cohomological Conley index can be reduced to the computation of the classical...
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Wieloobszarowa rozmyta regulacja PID mocy reaktora jądrowego
PublicationW artykule przedstawiono wieloobszarowy regulator rozmyty z lokalnymi regulatorami PID dla sterowania mocą reaktora jądrowego typu PWR. Wykorzystano model matematyczny o parametrach skupionych reaktora PWR obejmujący procesy generacji i wymiany ciepła oraz efektów reaktywnościowych. Nastawy lokalnych regulatorów PID zostały dobrane w sposób optymalny, minimalizując całkowy wskaźnik jakości ISE. Na przykładzie pokazano że zastosowane...
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Numerical Modelling for Prediction of Compression Index from Soil Index Properties in Jimma town, Ethiopia
PublicationIn this study, correlations are developed to predict compression index (Cc) from index parameters so that one can be able to model Jimma soils with compression index using simple laboratory tests. Undisturbed and disturbed soil samples from twelve different locations in Jimma town were collected. Laboratory tests like specific gravity, grain size analysis, Atterberg limit, and one-dimensional consolidation test for a total of twenty-four...
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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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Propagation analysis of the point-to-point radio links operating in the band
PublicationThe theoretical possibilities of utilizing the EHF-band frequencies for point-to-point short range communications have been discussed. The most important factors that affect propagation attenuation in this band have been briefly described, as well as relevant mathematical formulas that allow for numerical analysis of these phenomena. After that, a research have been carried out in which the authors have analyzed propagation losses...
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Performance evaluation and model-based optimization of the mainstream deammonification in an integrated fixed-film activated sludge reactor
PublicationThis study aimed to model and optimize mainstream deammonification in an integrated fixed-film activated sludge (IFAS) pilot plant under natural seasonal temperature variations. The effect of gradually decreasing temperature on the performance was evaluated during a winter season and a transition period to summer conditions, and the correlation of the performance parameters was investigated using principal component analysis (PCA)....
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Morse inequalities via Conley index theory
PublicationThe relation known as the Morse inequalities can be extended to a more general setting of flows on a locally compact metric spaces (Conley index) as well as dynamical systems on Hilbert spaces (LS-index). This paper is a discourse around this extension. Except the part concerning the LS-index the material is self-contained and has a character of a survey.
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Visual Object Tracking System Employing Fixed and PTZ Cameras
PublicationThe paper presents a video monitoring system utilizing fixed and PTZ cameras for tracking of moving objects. First type of camera provides image for background modelling, being employed for foreground objects localization. Estimated objects locations are then utilised for steering of PTZ cameras when observing targeted objects with high close-ups. Objects are classified into several classes, then basic event detection is being...
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Opitmalising Human Development Index with sensitivity analysis
PublicationResearch background: Composite indicators are commonly used not only to measure economic development, the standard of livin g, competitiveness, fairness, effectiveness but are also willingly implemented in to many different fields. How- ever, it seems that in most cases the variable weig hting procedure is avoided or erroneous since in most cases so-called “wights by belief” are...