Companions and an Essential Motion of a Reaction System

2020 ◽  
Vol 175 (1-4) ◽  
pp. 187-199
Author(s):  
Daniela Genova ◽  
Hendrik Jan Hoogeboom ◽  
Nataša Jonoska
Keyword(s):  
Directed Graph ◽  
Reaction System ◽  
Global Dynamics ◽  
Transition Graph ◽  
The Family ◽  
Transition Graphs

For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices.

Companions and an Essential Motion of a Reaction System

2020 ◽  
Author(s):  
Daniela Genova ◽  
Hendrik Jan Hoogeboom ◽  
Nataša Jonoska
Keyword(s):  
Directed Graph ◽  
Reaction System ◽  
Global Dynamics ◽  
Transition Graph ◽  
The Family ◽  
Transition Graphs

For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices.

scholarly journals On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

A New Hazard Identification Method-State Transition Graph

2011 ◽  
Vol 48-49 ◽  
pp. 71-78 ◽  
Author(s):  
Min Hu ◽  
Fang Fang Wu ◽  
Bo Zhu ◽  
Bo Lu ◽  
Jing Lei Pu
Keyword(s):  
State Transition ◽  
Hazard Identification ◽  
New Method ◽  
System State ◽  
Tunnel Engineering ◽  
Transition Graph ◽  
State Transition Graph ◽  
Engineering Data ◽  
Stimulation Experiments ◽  
Transition Graphs

It is important and difficult to identify the Hazard before a disaster happen because disaster often happens suddenly. This paper proposes a new method – State Transition Graph, which based on visual data space reconstruction, to identify hazard. The change process of the system state movement from one state to another in a certain period is described by some state transition graphs. The system state, which is safe or hazard, could be distinguished by its state transition graphs. This paper conducted experiments on single-dimension and multi-dimension benchmark data to prove the new method is effectiveness. Especially the result of stimulation experiments, based on the Yangtze River tunnel engineering data, showed that state transition graph identifies hazard easily and has better performances than other method. The State transition graph method is worth further researching.

scholarly journals Finite-state discrete-time Markov chain models of gene regulatory networks

2014 ◽  
Author(s):  
V.P. Skornyakov ◽  
M.V. Skornyakova ◽  
A.V. Shurygina ◽  
P.V. Skornyakov
Keyword(s):  
Markov Chain ◽  
Gene Regulatory Networks ◽  
Regulatory Networks ◽  
Finite Graph ◽  
Transition Graph ◽  
Markov Chain Models ◽  
Finite State ◽  
Gene Regulatory ◽  
Maximal Independent Sets ◽  
Transition Graphs

AbstractIn this study Markov chain models of gene regulatory networks (GRN) are developed. These models gives the ability to apply the well known theory and tools of Markov chains to GRN analysis. We introduce a new kind of the finite graph of the interactions called the combinatorial net that formally represent a GRN and the transition graphs constructed from interaction graphs. System dynamics are defined as a random walk on the transition graph that is some Markovian chain. A novel concurrent updating scheme (evolution rule) is developed to determine transitions in a transition graph. Our scheme is based on the firing of a random set of non-steady state vertices of a combinatorial net. We demonstrate that this novel scheme gives an advance in the modeling of the asynchronicity. Also we proof the theorem that the combinatorial nets with this updating scheme can asynchronously compute a maximal independent sets of graphs. As proof of concept, we present here a number of simple combinatorial models: a discrete model of auto-repression, a bi-stable switch, the Elowitz repressilator, a self-activation and show that this models exhibit well known properties.

scholarly journals Complete transition diagrams of generic Hamiltonian flows with a few heteroclinic orbits

2020 ◽  
pp. 2150023
Author(s):  
Tetsuo Yokoyama ◽  
Tomoo Yokoyama
Keyword(s):  
Integer Programming ◽  
Equivalence Classes ◽  
Heteroclinic Orbits ◽  
Topological Equivalence ◽  
Integer Programming Problem ◽  
Transition Graph ◽  
Surface Flows ◽  
Hamiltonian Flows ◽  
Vector Representations ◽  
Transition Graphs

We study the transition graph of generic Hamiltonian surface flows, whose vertices are the topological equivalence classes of generic Hamiltonian surface flows and whose edges are the generic transitions. Using the transition graph, we can describe time evaluations of generic Hamiltonian surface flows (e.g., fluid phenomena) as walks on the graph. We propose a method for constructing the complete transition graph of all generic Hamiltonian flows. In fact, we construct two complete transition graphs of Hamiltonian surface flows having three and four genus elements. Moreover, we demonstrate that a lower bound on the transition distance between two Hamiltonian surface flows with any number of genus elements can be calculated by solving an integer programming problem using vector representations of Hamiltonian surface flows.

scholarly journals Finite-state discrete-time Markov chain models of gene regulatory networks

F1000Research ◽  
2014 ◽  
Vol 3 ◽  
pp. 220
Author(s):  
Vladimir Skornyakov ◽  
Maria Skornyakova ◽  
Antonina Shurygina ◽  
Pavel Skornyakov
Keyword(s):  
Markov Chain ◽  
Gene Regulatory Networks ◽  
Regulatory Networks ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Transition Graph ◽  
Markov Chain Models ◽  
Finite State ◽  
Gene Regulatory ◽  
Transition Graphs

In this study, Markov chain models of gene regulatory networks (GRN) are developed. These models make it possible to apply the well-known theory and tools of Markov chains to GRN analysis. A new kind of finite interaction graph called a combinatorial net is introduced to represent formally a GRN and its transition graphs constructed from interaction graphs. The system dynamics are defined as a random walk on the transition graph, which is a Markov chain. A novel concurrent updating scheme (evolution rule) is developed to determine transitions in a transition graph. The proposed scheme is based on the firing of a random set of non-steady-state vertices in a combinatorial net. It is demonstrated that this novel scheme represents an advance in asynchronicity modeling. The theorem that combinatorial nets with this updating scheme can asynchronously compute a maximal independent set of graphs is also proved. As proof of concept, a number of simple combinatorial models are presented here: a discrete auto-regression model, a bistableswitch, an Elowitz repressilator, and a self-activation model, and it is shown that these models exhibit well-known properties.

scholarly journals Spider Covers and Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Filomena De Santis ◽  
Luisa Gargano ◽  
Mikael Hammar ◽  
Alberto Negro ◽  
Ugo Vaccaro
Keyword(s):  
Combinatorial Optimization ◽  
Approximation Algorithm ◽  
Greedy Algorithm ◽  
Directed Graph ◽  
Optimization Problems ◽  
Disjoint Paths ◽  
Combinatorial Optimization Problems ◽  
The Family ◽  
Minimum Number ◽  
Cover Problem

We introduce two new combinatorial optimization problems: the Maximum Spider Problem and the Spider Cover Problem; we study their approximability and illustrate their applications. In these problems we are given a directed graph , a distinguished vertex , and a family D of subsets of vertices. A spider centered at vertex s is a collection of arc-disjoint paths all starting at s but ending into pairwise distinct vertices. We say that a spider covers a subset of vertices X if at least one of the endpoints of the paths constituting the spider other than s belongs to X. In the Maximum Spider Problem the goal is to find a spider centered at s that covers the maximum number of elements of the family D. Conversely, the Spider Cover Problem consists of finding the minimum number of spiders centered at s that covers all subsets in D. We motivate the study of the Maximum Spider and Spider Cover Problems by pointing out a variety of applications. We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a -approximation algorithm for the Spider Cover Problem.

scholarly journals Dynamics of the Picking transformation on integer partitions

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Thi Ha Duong Phan ◽  
Eric Thierry
Keyword(s):  
Point Of View ◽  
Dynamical Process ◽  
Integer Partitions ◽  
Finite Sets ◽  
Transition Graph ◽  
Heterogeneous Distribution ◽  
Fixed Integer ◽  
The Family ◽  
International Audience ◽  
Almost All

International audience This paper studies a conservative transformation defined on families of finite sets. It consists in removing one element from each set and adding a new set composed of the removed elements. This transformation is conservative in the sense that the union of all sets of the family always remains the same.We study the dynamical process obtained when iterating this deterministic transformation on a family of sets and we focus on the evolution of the cardinalities of the sets of the family. This point of view allows to consider the transformation as an application defined on the set of all partitions of a fixed integer (which is the total number of elements in the sets).We show that iterating this particular transformation always leads to a heterogeneous distribution of the cardinalities, where almost all integers within an interval are represented.We also tackle some issues concerning the structure of the transition graph which sums up the whole dynamics of this process for all partitions of a fixed integer.

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