scholarly journals Strictifying and taming directed paths in Higher Dimensional Automata

2021 ◽  
Vol 31 (2) ◽  
pp. 193-213
Author(s):  
Martin Raussen
Keyword(s):  
State Space ◽  
Algebraic Topology ◽  
Computational Topology ◽  
Topological Properties ◽  
Engineering Communication ◽  
Higher Dimensional ◽  
Directed Paths

AbstractDirected paths have been used by several authors to describe concurrent executions of a program. Spaces of directed paths in an appropriate state space contain executions with all possible legal schedulings. It is interesting to investigate whether one obtains different topological properties of such a space of executions if one restricts attention to schedulings with “nice” properties, e.g. involving synchronisations. This note shows that this is not the case, i.e. that one may operate with nice schedulings without inflicting any harm. Several of the results in this note had previously been obtained by Ziemiański in Ziemiański (2017. Applicable Algebra in Engineering, Communication and Computing28 497–525; 2020a. Journal of Applied and Computational Topology4 (1) 45–78). We attempt to make them accessible for a wider audience by giving an easier proof for these findings by an application of quite elementary results from algebraic topology; notably the nerve lemma.

scholarly journals Analytical Consideration of Growth in Population via Homological Invariant in Algebraic Topology

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Lewis Brew ◽  
William Obeng-Denteh ◽  
Fred Asante-Mensa
Keyword(s):  
Topological Space ◽  
Population Growth ◽  
Algebraic Topology ◽  
Homology Theory ◽  
Significant Feature ◽  
Topological Properties ◽  
Analytical Consideration ◽  
Abstract Approach ◽  
Dimensional Surface

This paper presents an abstract approach of analysing population growth in the field of algebraic topology using the tools of homology theory. For a topological space X and any point vn∈X, where vn is the n-dimensional surface, the group η=X,vn is called population of the space X. The increasing sequence from vin∈X to vjn∈X for i<j provides the bases for the population growth. A growth in population η=X,vn occurs if vin<vjn for all vin∈X and vjn∈X. This is described by the homological invariant Hηk=1. The aim of this paper is to construct the homological invariant Hηk and use Hηk=1 to analyse the growth of the population. This approach is based on topological properties such as connectivity and continuity. The paper made extensive use of homological invariant in presenting important information about the population growth. The most significant feature of this method is its simplicity in analysing population growth using only algebraic category and transformations.

scholarly journals Computational Topology Techniques Help to Solve a Long-Lasting Forensic Dilemma: Aldo Moro’s Death

2018 ◽  
Author(s):  
James F. Peters ◽  
Arturo Tozzi
Keyword(s):  
Algebraic Topology ◽  
Computational Topology ◽  
Forensic Analysis ◽  
Recent History ◽  
The Novel ◽  
Analysis Method ◽  
Spatial Clusters ◽  
Red Brigades ◽  
Aldo Moro

Here we show how a recently-introduced method from algebraic topology, namely proximal planar vortex 1-cycles, might be helpful in detecting hidden features of the shapes and holes in images, therefore contributing to the solution of both cold and fresh forensic cases. In particular, we test the efficacy of this technique by assessing one of the most puzzling cases of recent history, i.e., Aldo Moro&rsquo;s death. Terrorists of the Red Brigades claimed that they killed Moro when he was placed inside the trunk of a car,shooting him with a barrage of bullets. We demonstrate, based on the analysis of the photographs taken during the autoptic procedure, that the terrorist&rsquo;s account does not hold true. Our results, showing different series of shots, point towards a three-step execution, with the first phasestaking place outside the car. In conclusion, the novel forensic analysis method introduced in this paper permits the evaluation of a collection of vortex cycles/nerves equipped with a connectedness proximity, which makes it possible to assess unexpected spatial clusters in photographs.

On the Nonlinear Observability of Polynomial Dynamical Systems

2021 ◽  
Vol 1 (1) ◽  
pp. 88-94
Author(s):  
Daniel Gerbet ◽  
Klaus Röbenack
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
State Space ◽  
Polynomial Dynamical Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
System Properties ◽  
Controllability And Observability ◽  
Important System ◽  
Nonlinear Observability

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

An Unsupervised Ensemble Learning Method for Nonlinear Dynamic State-Space Models

2002 ◽  
Vol 14 (11) ◽  
pp. 2647-2692 ◽  
Author(s):  
Harri Valpola ◽  
Juha Karhunen
Keyword(s):  
State Space ◽  
Ensemble Learning ◽  
Latent Variable ◽  
State Space Model ◽  
Dynamic State ◽  
Learning Method ◽  
Nonlinear Prediction ◽  
Higher Dimensional ◽  
Prediction Techniques ◽  
Multilayer Perceptron Networks

A Bayesian ensemble learning method is introduced for unsupervised extraction of dynamic processes from noisy data. The data are assumed to be generated by an unknown nonlinear mapping from unknown factors. The dynamics of the factors are modeled using a nonlinear state-space model. The nonlinear mappings in the model are represented using multilayer perceptron networks. The proposed method is computationally demanding, but it allows the use of higher-dimensional nonlinear latent variable models than other existing approaches. Experiments with chaotic data show that the new method is able to blindly estimate the factors and the dynamic process that generated the data. It clearly outperforms currently available nonlinear prediction techniques in this very difficult test problem.

Topological properties of flat electroencephalography’s state space

2016 ◽  
Author(s):  
Tan Lit Ken ◽  
Tahir bin Ahmad ◽  
Mohd Sham bin Mohd ◽  
Su Kong Ngien ◽  
Tohru Suwa ◽  
...  
Keyword(s):  
State Space ◽  
Topological Properties

Topological Properties of a Class of Higher-dimensional Self-affine Tiles

2019 ◽  
Vol 62 (4) ◽  
pp. 727-740
Author(s):  
Guotai Deng ◽  
Chuntai Liu ◽  
Sze-Man Ngai
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Higher Dimensional ◽  
Necessary And Sufficient

AbstractWe construct a family of self-affine tiles in $\mathbb{R}^{d}$ ($d\geqslant 2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in $\mathbb{R}^{2}$, and its extension to $\mathbb{R}^{3}$ by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

scholarly journals CARTESIAN PRODUCT PARTITIONING OF MULTI-DIMENSIONAL REACHABLE STATE SPACES

2016 ◽  
Vol 30 (3) ◽  
pp. 413-430 ◽  
Author(s):  
Tuǧrul Dayar ◽  
M. Can Orhan
Keyword(s):  
State Space ◽  
Cartesian Product ◽  
Model Systems ◽  
Dimensional System ◽  
Reachable State ◽  
State Spaces ◽  
Cartesian Products ◽  
Higher Dimensional ◽  
The Matrix ◽  
Minimum Number

Markov chains (MCs) are widely used to model systems which evolve by visiting the states in their state spaces following the available transitions. When such systems are composed of interacting subsystems, they can be mapped to a multi-dimensional MC in which each subsystem normally corresponds to a different dimension. Usually the reachable state space of the multi-dimensional MC is a proper subset of its product state space, that is, Cartesian product of its subsystem state spaces. Compact storage of the matrix underlying such a MC and efficient implementation of analysis methods using Kronecker operations require the set of reachable states to be represented as a union of Cartesian products of subsets of subsystem state spaces. The problem of partitioning the reachable state space of a three or higher dimensional system with a minimum number of partitions into Cartesian products of subsets of subsystem state spaces is shown to be NP-complete. Two algorithms, one merge based the other refinement based, that yield possibly non-optimal partitionings are presented. Results of experiments on a set of problems from the literature and those that are randomly generated indicate that, although it may be more time and memory consuming, the refinement based algorithm almost always computes partitionings with a smaller number of partitions than the merge-based algorithm. The refinement based algorithm is insensitive to the order in which the states in the reachable state space are processed, and in many cases it computes partitionings that are optimal.

scholarly journals A Review of and Some Results for Ollivier–Ricci Network Curvature

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1416
Author(s):  
Nazanin Azarhooshang ◽  
Prithviraj Sengupta ◽  
Bhaskar DasGupta
Keyword(s):  
Ricci Curvature ◽  
Network Science ◽  
Topological Spaces ◽  
Topological Properties ◽  
Approximate Computation ◽  
Curvature Measure ◽  
Higher Dimensional ◽  
Science Domain ◽  
And Mathematics

Characterizing topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures is by now quite common in mainstream physics and mathematics, and it is therefore natural to try to extend these notions from the non-network domains in a suitable way to the network science domain. In this article we discuss one such extension, namely Ollivier’s discretization of Ricci curvature. We first motivate, define and illustrate the Ollivier–Ricci Curvature. In the next section we provide some “not-previously-published” bounds on the exact and approximate computation of the curvature measure. In the penultimate section we review a method based on the linear sketching technique for efficient approximate computation of the Ollivier–Ricci network curvature. Finally in the last section we provide concluding remarks with pointers for further reading.

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