CONSTRUCTION OF LATTICES OF BALANCED EQUIVALENCE RELATIONS FOR REGULAR HOMOGENEOUS NETWORKS USING LATTICE GENERATORS AND LATTICE INDICES

2009 ◽  
Vol 19 (11) ◽  
pp. 3691-3705 ◽  
Author(s):  
HIROKO KAMEI
Keyword(s):  
Special Kind ◽  
Building Blocks ◽  
Equivalence Relations ◽  
Lattice Structures ◽  
Internal Dynamics ◽  
One Dimensional ◽  
Cell Network ◽  
Coupled Cell Networks ◽  
Cell Networks ◽  
Number Of Cells

Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.

THE EXISTENCE AND CLASSIFICATION OF SYNCHRONY-BREAKING BIFURCATIONS IN REGULAR HOMOGENEOUS NETWORKS USING LATTICE STRUCTURES

2009 ◽  
Vol 19 (11) ◽  
pp. 3707-3732 ◽  
Author(s):  
HIROKO KAMEI
Keyword(s):  
Steady State ◽  
Equivalence Relations ◽  
Lattice Structures ◽  
Bifurcation Structure ◽  
Internal Dynamics ◽  
One Dimensional ◽  
Synchronized Cells ◽  
Codimension One ◽  
Generic Bifurcation

For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.

AN OPTIMAL LIFTING THEOREM FOR COUPLED CELL NETWORKS

2011 ◽  
Vol 21 (09) ◽  
pp. 2481-2487 ◽  
Author(s):  
IAN STEWART
Keyword(s):  
Minimal Size ◽  
Coupled Cell Networks ◽  
Cell Networks ◽  
Number Of Cells

The multiarrow formalism for coupled cell networks permits multiple arrows and self-loops. The Lifting Theorem states that any such network is a quotient of a network in which all arrows are single and self-loops do not occur. Previous proofs are inductive, and give no useful estimate of the minimal size of the lift. We give a noninductive proof of the Lifting Theorem, and identify the number of cells in the smallest possible lift. We interpret this construction in terms of the type matrix of the network, which encodes its topology and labeling.

ENUMERATION OF HOMOGENEOUS COUPLED CELL NETWORKS

2005 ◽  
Vol 15 (08) ◽  
pp. 2361-2373 ◽  
Author(s):  
FALIH ALDOSRAY ◽  
IAN STEWART
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Directed Graph ◽  
Schematic Diagram ◽  
Natural Manner ◽  
Cell Network ◽  
Different Types ◽  
Coupled Cell Networks ◽  
Cell Networks

A coupled cell network is a schematic diagram employed to define a class of differential equations, and can be thought of as a directed graph whose nodes (cells) represent dynamical systems and whose edges (arrows) represent couplings. Often the nodes and edges are labeled to distinguish different types of system and coupling. The associated differential equations reflect this structure in a natural manner. The network is homogeneous if there is one type of cell and one type of arrow, and moreover, every cell lies at the head end of the same number r of arrows. This number is the valency of the network. We use a group-theoretic formula usually but incorrectly attributed to William Burnside to enumerate homogeneous coupled cell networks with N cells and valency r, in both the disconnected and connected cases. We compute these numbers explicitly when N, r ≤ 6.

ELIMINATION OF MULTIPLE ARROWS AND SELF-CONNECTIONS IN COUPLED CELL NETWORKS

2007 ◽  
Vol 17 (01) ◽  
pp. 99-106 ◽  
Author(s):  
IAN STEWART
Keyword(s):  
Directed Graphs ◽  
Formal Theory ◽  
Coupled Systems ◽  
Inductive Step ◽  
Combinatorial Objects ◽  
Cell Network ◽  
Coupled Cell Networks ◽  
Multiple Edges ◽  
Cell Networks ◽  
Identical Edge

A coupled cell network is a finite directed graph in which nodes and edges are classified into equivalence classes. Such networks arise in a formal theory of coupled systems of differential equations, as a schematic indication of the topology of the coupling, but they can be studied independently as combinatorial objects. The edges of a coupled cell network are "identical" if they are all equivalent, and the network is "homogeneous" if all nodes have isomorphic sets of input edges. Golubitsky et al. [2005] proved that every homogeneous identical-edge coupled cell network is a quotient of a network that has no multiple edges and no self-connections. We generalize this theorem to any coupled cell network by removing the conditions of homogeneity and identical edges. The problem is a purely combinatorial assertion about labeled directed graphs, and we give two combinatorial proofs. Both proofs eliminate self-connections inductively. The first proof also eliminates multiple edges inductively, the main feature being the specification of the inductive step in terms of a complexity measure. The second proof obtains a more efficient result by eliminating all multiple edges in a single construction.

scholarly journals Coupled cell networks: minimality

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1030501-1030502
Author(s):  
Manuela A. D. Aguiar ◽  
Ana Paula S. Dias
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

Shape memory alloy–actuated prestressed composites with application to morphing automotive fender skirts

2018 ◽  
Vol 30 (3) ◽  
pp. 479-494 ◽  
Author(s):  
Venkata Siva C Chillara ◽  
Leon M Headings ◽  
Ryohei Tsuruta ◽  
Eiji Itakura ◽  
Umesh Gandhi ◽  
...  
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory Alloys ◽  
Shape Memory ◽  
Smart Materials ◽  
Design Procedure ◽  
Building Blocks ◽  
Smart Composite ◽  
Thermally Induced ◽  
One Dimensional ◽  
Flat Shape

This work presents smart laminated composites that enable morphing vehicle structures. Morphing panels can be effective for drag reduction, for example, adaptive fender skirts. Mechanical prestress provides tailored curvature in composites without the drawbacks of thermally induced residual stress. When driven by smart materials such as shape memory alloys, mechanically-prestressed composites can serve as building blocks for morphing structures. An analytical energy-based model is presented to calculate the curved shape of a composite as a function of force applied by an embedded actuator. Shape transition is modeled by providing the actuation force as an input to a one-dimensional thermomechanical constitutive model of a shape memory alloy wire. A design procedure, based on the analytical model, is presented for morphing fender skirts comprising radially configured smart composite elements. A half-scale fender skirt for a compact passenger car is designed, fabricated, and tested. The demonstrator has a domed unactuated shape and morphs to a flat shape when actuated using shape memory alloys. Rapid actuation is demonstrated by coupling shape memory alloys with integrated quick-release latches; the latches reduce actuation time by 95%. The demonstrator is 62% lighter than an equivalent dome-shaped steel fender skirt.

scholarly journals Is the Responsibility to Protect an Accepted Norm of International Law in the post-Libya Era?

2018 ◽  
pp. 11
Author(s):  
Jeremy Sarkin
Keyword(s):  
Human Rights ◽  
International Law ◽  
Transitional Justice ◽  
Building Blocks ◽  
Responsibility To Protect ◽  
Human Rights Violations ◽  
One Dimensional

This article explores the Responsibility to Protect (RtoP) in the post-Libya era to determinewhether it is now an accepted norm of international law. It examines what RtoP means intoday`s world and whether the norm now means that steps will be taken against states thatare committing serious human rights violations. The building blocks of RtoP are examined tosee how to make the doctrine more relevant and more applicable. It is contended that theresponsibility to react should be viewed through a much wider lens and that it needs to bemore widely interpreted to allow it to gain greater support. It is argued that there is a need tofocus far more on the responsibility to rebuild and that it ought to focus on the transitionallegal architecture as well as transitional justice. It is contended that these processes ought notto be one-dimensional, but ought to have a variety of constituent parts. It is further arguedthat the international and donor community ought to be far more engaged and far moredirective in these projects.

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