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Alexandru Mihail - One of the best experts on this subject based on the ideXlab platform.
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Iterated function systems consisting of phi-max-contractions have attractor
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Flavian Georgescu, Radu Miculescu, Alexandru MihailAbstract:We associate to each iterated function system consisting of phi-max-contractions an operator (on the space of continuous functions from the shift space on the metric space corresponding to the system) having a unique fixed point whose image turns out to be the attractor of the system. Moreover, we prove that the unique fixed point of the operator associated to an iterated function system consisting of convex contractions is the Canonical Projection from the shift space on the attractor of the system.
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A generalization of Istratescu's fixed point theorem for convex contractions
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Radu Miculescu, Alexandru MihailAbstract:In this paper we prove a generalization of Istr\u{a}\c{t}escu's theorem for convex contractions. More precisely, we introduce the concept of iterated function system consisting of convex contractions and prove the existence and uniqueness of the attractor of such a system. In addition we study the properties of the Canonical Projection from the code space into the attractor of an iterated function system consisting of convex contractions.
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Reich-type iterated function systems
Journal of Fixed Point Theory and Applications, 2015Co-Authors: Radu Miculescu, Alexandru MihailAbstract:In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the Canonical Projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.
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The Canonical Projection between the shift space of an IIFS and its attractor as a fixed point
Fixed Point Theory and Applications, 2015Co-Authors: Alexandru MihailAbstract:An important class of fractal sets is given by the attractors of iterated function systems which are defined as the fixed points of the associated fractal operators. In the study of such an attractor, an important place is taken by the Canonical Projection between the shift space associated with the system and the attractor. In this paper, by using different fixed point theorems, we present the Canonical Projection as the fixed point of a certain operator defined on the space of continuous functions from the shift space on the metric space associated with the system.
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THE SHIFT SPACE FOR AN INFINITE ITERATED FUNCTION SYSTEM
2009Co-Authors: Alexandru Mihail, Radu MiculescuAbstract:The aim of the paper is to define the shift space for an infinite iterated function systems (IIFS) and to describe the relation between this space and the attractor of the IIFS. We construct a Canonical Projection (which turns out to be continuous) from the shift space of an IIFS on its attractor and provide sucient conditions The shift (or code) space of an iterated function system (IFS for short) and the address of the points lying on the attractor of the IFS are very good tools to get a more precise description of the invariant dynamics of the IFS. The theory of fractal tops provides a useful mapping from an IFS attractor into the associated code space that may be applied to assign colors to the IFS attractor via a method introduced by M.F. Barnsley and J. Hutchinson (which they refer as colour-stealing) and to construct homeomorphisms between attractors (roughly speaking, if the symbolic dynamical systems associated with the tops of two IFSs are topologically conjugate, then the attractors of the IFSs are homeomorphic). Moreover, Barnsley (2) proved that if two hyperbolic IFS attractors are homeomorphic, then they have the same entropy. In this paper we present a generalization of the notion of shift space associated with an IFS. More precisely, we define the shift space of an infinite iterated function systems (IIFS) and describe the relation between this space and the attractor of the IIFS. We construct a Canonical Projection (which turns out to be continuous) from the shift space of an IIFS on its attractor and provide sucient conditions for this function to be onto.
J Cadzow - One of the best experts on this subject based on the ideXlab platform.
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ICASSP - Signal enhancement using Canonical Projection operators
ICASSP '87. IEEE International Conference on Acoustics Speech and Signal Processing, 1Co-Authors: J CadzowAbstract:A commonly occurring signal processing application is concerned with the task of resurrecting a signal from a noise and distorted measurement of that signal. It is often known that the underlying signal possesses well-defined properties which are obscured through the measurement process. A signal enhancement algorithm is herein developed which slightly modifies the measured signal so that the modified (or enhanced) signal takes on these prescribed properties. As such, the modified signal often provides a more accurate characterization of the underlying signal.
Radu Miculescu - One of the best experts on this subject based on the ideXlab platform.
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The Canonical Projection associated with certain possibly infinite generalized iterated function systems as a fixed point
Journal of Fixed Point Theory and Applications, 2018Co-Authors: Radu Miculescu, Silviu UrziceanuAbstract:In this paper, influenced by the ideas from Mihail (Fixed Point Theory Appl 2015:15, 2015), we associate to every generalized iterated function system $$\mathcal {F}$$ (of order m) an operator $$H_{\mathcal {F}}:\mathcal {C} ^{m}\rightarrow \mathcal {C}$$ , where $$\mathcal {C}$$ stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of $$\mathcal {F}$$ ) for the operator $$H_{\mathcal {F}}$$ to be continuous, contraction, $$\varphi $$ -contraction, Meir–Keeler or contractive. We also give sufficient condition under which $$H_{\mathcal {F}}$$ has a unique fixed point $$\pi _{0}$$ . Moreover, we prove that, under these circumstances, the closure of the imagine of $$\pi _{0}$$ is the attractor of $$\mathcal {F}$$ and that $$\pi _{0}$$ is the Canonical Projection associated with $$\mathcal {F}$$ . In this way we give a partial answer to the open problem raised on the last paragraph of the above-mentioned Mihail’s paper.
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The Canonical Projection associated to certain possibly infinite generalized iterated function system as a fixed point
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Radu Miculescu, Silviu UrziceanuAbstract:In this paper, influenced by the ideas from A. Mihail, The Canonical Projection between the shift space of an IIFS and its attractor as a fixed point, Fixed Point Theory Appl., 2015, Paper No. 75, 15 p., we associate to every generalized iterated function system F (of order m) an operator H defined on C^m and taking values on C, where C stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of F) for the operator H to be continuous, contraction, phi-contraction, Meir-Keeler or contractive. We also give sufficient condition under which H has a unique fixed point. Moreover, we prove that, under these circumstances, the closer of the imagine of the fixed point is the attractor of F and that the fixed point is the Canonical Projection associated to F. In this way we give a partial answer to the open problem raised on the last paragraph of the above mentioned Mihail's paper.
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Iterated function systems consisting of phi-max-contractions have attractor
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Flavian Georgescu, Radu Miculescu, Alexandru MihailAbstract:We associate to each iterated function system consisting of phi-max-contractions an operator (on the space of continuous functions from the shift space on the metric space corresponding to the system) having a unique fixed point whose image turns out to be the attractor of the system. Moreover, we prove that the unique fixed point of the operator associated to an iterated function system consisting of convex contractions is the Canonical Projection from the shift space on the attractor of the system.
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A generalization of Istratescu's fixed point theorem for convex contractions
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Radu Miculescu, Alexandru MihailAbstract:In this paper we prove a generalization of Istr\u{a}\c{t}escu's theorem for convex contractions. More precisely, we introduce the concept of iterated function system consisting of convex contractions and prove the existence and uniqueness of the attractor of such a system. In addition we study the properties of the Canonical Projection from the code space into the attractor of an iterated function system consisting of convex contractions.
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Reich-type iterated function systems
Journal of Fixed Point Theory and Applications, 2015Co-Authors: Radu Miculescu, Alexandru MihailAbstract:In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the Canonical Projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.
Johannes Kellendonk - One of the best experts on this subject based on the ideXlab platform.
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Integer Cech Cohomology of Icosahedral Projection Tilings
Zeitschrift für Kristallographie, 2008Co-Authors: Franz Gähler, John Hunton, Johannes KellendonkAbstract:The integer Cech cohomology of Canonical Projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional Canonical Projection tilings, where many examples without torsion are known.
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Torsion in Tiling Homology and Cohomology
arXiv: Mathematical Physics, 2005Co-Authors: Franz Gähler, John Hunton, Johannes KellendonkAbstract:The first author's recent unexpected discovery of torsion in the integral cohomology of the T\"ubingen Triangle Tiling has led to a re-evaluation of current descriptions of and calculational methods for the topological invariants associated with aperiodic tilings. The existence of torsion calls into question the previously assumed equivalence of cohomological and K-theoretic invariants as well as the supposed lack of torsion in the latter. In this paper we examine in detail the topological invariants of Canonical Projection tilings; we extend results of Forrest, Hunton and Kellendonk to give a full treatment of the torsion in the cohomology of such tilings in codimension at most 3, and present the additions and amendments needed to previous results and calculations in the literature. It is straightforward to give a complete treatment of the torsion components for tilings of codimension 1 and 2, but the case of codimension 3 is a good deal more complicated, and we illustrate our methods with the calculations of all four icosahedral tilings previously considered. Turning to the K-theoretic invariants, we show that cohomology and K-theory agree for all Canonical Projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the K-theory of the T\"ubingen Triangle Tiling. The question of the equivalence of cohomology and K-theory for tilings of higher dimensional euclidean space remains open.
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Cohomology of Canonical Projection Tilings
Communications in Mathematical Physics, 2002Co-Authors: Alan Forrest, John Hunton, Johannes KellendonkAbstract:We define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the Canonical Projection method in Schlottmann's formulation. We prove the cohomology to be equivalent to a certain cohomology of the lattice. We discuss one of its qualitative features, namely that it provides a topological obstruction for a generic tiling to be substitutional. We develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2, presenting explicit formulae. These in turn give computations for the $K$-theory of certain associated non-commutative C * algebras.
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Topological invariants for Projection method patterns
arXiv: Algebraic Topology, 2000Co-Authors: Alan Forrest, John Hunton, Johannes KellendonkAbstract:We analyze and compare different dynamical systems and groupoids which can be obtained from Projection point patterns. We define the cohomology of a point pattern as the cocycle cohomology of the pattern groupoid. We describe this cohomology qualitatively and calculate it for Canonical Projection method tilings with codimension smaller or equal than 3.
Yunyun Yang - One of the best experts on this subject based on the ideXlab platform.
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The Fourier transform of thick distributions
Analysis and Applications, 2020Co-Authors: Ricardo Estrada, Jasson Vindas, Yunyun YangAbstract:We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a Canonical Projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the Canonical Projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.