scholarly journals Iterated Functions Systems Composed of Generalized θ-Contractions

2021 ◽  
Vol 5 (3) ◽  
pp. 69
Author(s):  
Pasupathi Rajan ◽  
María A. Navascués ◽  
Arya Kumar Bedabrata Chand
Keyword(s):  
Product Space ◽  
Complete Metric Space ◽  
Cartesian Product ◽  
Iterated Function Systems ◽  
Theoretical Work ◽  
Finite Collection ◽  
Self Similarity ◽  
Iterated Function ◽  
Finite Cartesian Product ◽  
Cartesian Product Space

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

2018 ◽  
Vol 7 (3.31) ◽  
pp. 126
Author(s):  
Minirani S ◽  
. .
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Finite Collection ◽  
Integral Type ◽  
Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor. 

BIPROJECTIVITY AND BIFLATNESS OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2014 ◽  
Vol 91 (1) ◽  
pp. 134-144 ◽  
Author(s):  
F. ABTAHI ◽  
A. GHAFARPANAH ◽  
A. REJALI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractLet ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

Pascal's triangle, dynamical systems and attractors

1992 ◽  
Vol 12 (3) ◽  
pp. 479-486 ◽  
Author(s):  
Fritz V. Haeseler ◽  
Heinz-Otto Peitgen ◽  
Gencho Skordev
Keyword(s):  
Dynamical Systems ◽  
General Framework ◽  
Prime Power ◽  
Systems Approach ◽  
Iterated Function Systems ◽  
Binomial Coefficients ◽  
Self Similarity ◽  
Iterated Function ◽  
Fractal Patterns ◽  
Pascal's Triangle

AbstractThis paper establishes a global dynamical systems approach for the fractal patterns which are obtained when analysing the divisibility of binomial coefficients modulo a prime power. The general framework is within the class of hierarchical iterated function systems. As a consequence we obtain a complete deciphering of the hierarchical self-similarity features.

scholarly journals Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes

2011 ◽  
Vol 21 (4) ◽  
pp. 757-767 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Diana Domańska
Keyword(s):  
Pattern Recognition ◽  
Iterated Function System ◽  
Global Analysis ◽  
Recognition Rate ◽  
Iterated Function Systems ◽  
Dependence Graph ◽  
Local Analysis ◽  
Self Similarity ◽  
Pattern Recognition Methods ◽  
Iterated Function

Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapesOne of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system. The proposed method uses local analysis of the shape, which improves the recognition rate. The effectiveness of our method is shown on two test databases. The first one was created by the authors and the second one is the MPEG7 CE-Shape-1 PartB database. The obtained results show that the proposed methodology has led to a significant improvement in the recognition rate.

scholarly journals Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

2021 ◽  
Vol 9 (1) ◽  
pp. 90-119
Author(s):  
Matthew Badger ◽  
Vyron Vellis
Keyword(s):  
Metric Spaces ◽  
Optimal Parameter ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Interesting Phenomenon ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Path Connected

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

scholarly journals ARENS REGULARITY AND AMENABILITY OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2012 ◽  
Vol 87 (2) ◽  
pp. 195-206 ◽  
Author(s):  
S. J. BHATT ◽  
P. A. DABHI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Arens Regularity ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractGiven a morphism T from a Banach algebra ℬ to a commutative Banach algebra 𝒜, a multiplication is defined on the Cartesian product space 𝒜×ℬ perturbing the coordinatewise product resulting in a new Banach algebra 𝒜×Tℬ. The Arens regularity as well as amenability (together with its various avatars) of 𝒜×Tℬ are shown to be stable with respect to T.

Self-Similar Functions Generated by Cellular Automata

Fractals ◽  
1998 ◽  
Vol 06 (04) ◽  
pp. 371-394 ◽  
Author(s):  
Heinz-Otto Peitgen ◽  
Anna Rodenhausen ◽  
Gencho Skordev
Keyword(s):  
Cellular Automata ◽  
Iterated Function Systems ◽  
The Self ◽  
Self Similarity ◽  
Hausdorff Dimensions ◽  
Fractal Sets ◽  
Sequential Machines ◽  
Iterated Function ◽  
Self Similar

The self-similarity properties of the functions (closed relations) associated with one- and two-sided cellular automata are studied. It turns out that these functions are generated by sequential machines, and their graphs are fractal sets generated by hierarchical iterated function systems. The Hausdorff dimensions of the graphs is one for one-sided cellular automata and two for two-sided automata.

On the uniform ergodicity of Markov processes of order 2

2003 ◽  
Vol 40 (2) ◽  
pp. 455-472 ◽  
Author(s):  
Ulrich Herkenrath
Keyword(s):  
Time Series ◽  
Markov Process ◽  
Markov Processes ◽  
Product Space ◽  
Transition Probability ◽  
Cartesian Product ◽  
Time Series Models ◽  
Uniform Ergodicity ◽  
General State Space ◽  
Cartesian Product Space

We study the uniform ergodicity of Markov processes (Zn, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z2n–1 and Z2n into one variable Yn with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Yn, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Zn, n ≥ 1) is defined in terms of the same property for (Yn, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Zn, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

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