An Efficient Hierarchical Optical Path Network Design Algorithm based on a Traffic Demand Expression in a Cartesian Product Space

2006 ◽  
Author(s):  
Isao Yagyu ◽  
Hiroshi Hasegawa ◽  
Ken-ichi Sato
Keyword(s):  
Network Design ◽  
Product Space ◽  
Cartesian Product ◽  
Optical Path ◽  
Design Algorithm ◽  
Traffic Demand ◽  
Cartesian Product Space

Hierarchical Optical Path Network Design Algorithm Considering Waveband Protection

2009 ◽  
Vol 27 (24) ◽  
pp. 5736-5748 ◽  
Author(s):  
Yoshiyuki Yamada ◽  
Hiroshi Hasegawa ◽  
Ken-Ichi Sato
Keyword(s):  
Network Design ◽  
Optical Path ◽  
Design Algorithm

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 ${\it\varphi}$.

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 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.

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.

On the uniform ergodicity of Markov processes of order 2

2003 ◽  
Vol 40 (02) ◽  
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 (Z n , 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 Z 2n–1 and Z 2n into one variable Y n with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Y n , n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Z n , n ≥ 1) is defined in terms of the same property for (Y n , n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Z n , 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.

Sign in / Sign up

Export Citation Format

Share Document