scholarly journals Schreier graphs of spinal groups

2021 ◽  
pp. 1-26
Author(s):  
Tatiana Nagnibeda ◽  
Aitor Pérez
Keyword(s):  
Dynamical Systems ◽  
Rooted Trees ◽  
Topological Spaces ◽  
Geometric Properties ◽  
Intermediate Growth ◽  
Isomorphism Classes ◽  
Schreier Graphs ◽  
Number Of Ends

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on [Formula: see text]-regular rooted trees and on their boundaries, viewed as topological spaces or as spaces with measure. They form interesting families of finitely ramified graphs, and we study their combinatorics, their isomorphism classes and their geometric properties, such as growth and the number of ends.

scholarly journals Simulating and Predicting Dynamical Systems with Spatial Semantic Pointers

2021 ◽  
pp. 1-35
Author(s):  
Aaron R. Voelker ◽  
Peter Blouw ◽  
Xuan Choo ◽  
Nicole Sandra-Yaffa Dumont ◽  
Terrence C. Stewart ◽  
...  
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Cognitive Processes ◽  
Deep Neural Networks ◽  
Learning Task ◽  
Topological Spaces ◽  
Multiple Objects ◽  
Symbolic Structure ◽  
High Level ◽  
Vector Representations

Abstract While neural networks are highly effective at learning task-relevant representations from data, they typically do not learn representations with the kind of symbolic structure that is hypothesized to support high-level cognitive processes, nor do they naturally model such structures within problem domains that are continuous in space and time. To fill these gaps, this work exploits a method for defining vector representations that bind discrete (symbol-like) entities to points in continuous topological spaces in order to simulate and predict the behavior of a range of dynamical systems. These vector representations are spatial semantic pointers (SSPs), and we demonstrate that they can (1) be used to model dynamical systems involving multiple objects represented in a symbol-like manner and (2) be integrated with deep neural networks to predict the future of physical trajectories. These results help unify what have traditionally appeared to be disparate approaches in machine learning.

scholarly journals On a family of Schreier graphs of intermediate growth associated with a self-similar group

2012 ◽  
Vol 33 (7) ◽  
pp. 1408-1421 ◽  
Author(s):  
Ievgen Bondarenko ◽  
Tullio Ceccherini-Silberstein ◽  
Alfredo Donno ◽  
Volodymyr Nekrashevych
Keyword(s):  
Similar Group ◽  
Intermediate Growth ◽  
Schreier Graphs ◽  
Self Similar

scholarly journals On the bialgebra of functional graphs and differential algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Maurice Ginocchio
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Discrete Dynamical Systems ◽  
Rooted Trees ◽  
Connected Components ◽  
Differential Algebras ◽  
Noncommutative Polynomials ◽  
International Audience

International audience We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.

scholarly journals Neutrosophic Orbit Topological Spaces

2021 ◽  
Vol 18 (24) ◽  
pp. 1443
Author(s):  
T Madhumathi ◽  
F NirmalaIrudayam
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Space ◽  
Necessary Conditions ◽  
Neutrosophic Set ◽  
Topological Spaces ◽  
Open Set ◽  
Evolution Function

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

scholarly journals Proper shape invariants: Tameness and movability

1996 ◽  
Vol 19 (2) ◽  
pp. 291-298
Author(s):  
Zvonko Čerin
Keyword(s):  
Topological Spaces ◽  
Shape Theory ◽  
Geometric Properties ◽  
Basic Properties ◽  
Proper Maps ◽  
Dual Forms

We study geometric properties of topological spaces called properNCℬ-tameness, properPCℬ-tameness, and properNℬ-movability, whereℬandCdenote classes of spaces. They are related to properMCℬ-tameness and properMℬ-movability from [5] and could be regarded as their dual forms. All three are invariants of a recently invented author's proper shape theory and are described by the use of proper multi-valued functions. We explore their basic properties and prove several results on their preservation under proper maps.

scholarly journals Entropy dimension for deterministic walks in random sceneries

2021 ◽  
pp. 1-18
Author(s):  
DOU DOU ◽  
KYEWON KOH PARK
Keyword(s):  
Growth Rate ◽  
Dynamical Systems ◽  
Base System ◽  
Irrational Rotations ◽  
Intermediate Growth ◽  
Skew Products ◽  
Entropy Dimension ◽  
Deterministic Walks ◽  
Different Levels ◽  
Bernoulli Systems

Abstract Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Discrete dynamical systems in L-topological spaces

2005 ◽  
Vol 156 (1) ◽  
pp. 25-42 ◽  
Author(s):  
Li Chen ◽  
Hui Kou ◽  
Mao-Kang Luo ◽  
Weinian Zhang
Keyword(s):  
Dynamical Systems ◽  
Discrete Dynamical Systems ◽  
Topological Spaces

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 365-427 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Positive Cone ◽  
Point Of View ◽  
Order Structure ◽  
Topological Spaces ◽  
Projective Modules ◽  
Finitely Generated ◽  
C Algebras ◽  
Isomorphism Classes

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

scholarly journals Attractors for dynamical systems in topological spaces

2002 ◽  
Vol 8 (3) ◽  
pp. 585-597 ◽  
Author(s):  
Alfredo Marzocchi ◽  
◽  
Sara Zandonella Necca
Keyword(s):  
Dynamical Systems ◽  
Topological Spaces
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