scholarly journals Satellite constructions and geometric classification of Brunnian links

2021 ◽  
Author(s):  
Sheng Bai ◽  
Jiming Ma
Keyword(s):  
Building Blocks ◽  
Classification Theorem ◽  
Jsj Decomposition ◽  
Link Complements ◽  
Hopf Formula ◽  
Geometric Decomposition ◽  
Jsj Decompositions ◽  
Brunnian Links

We study satellite operations on Brunnian links. First, we find two special satellite operations, both of which can construct infinitely many distinct Brunnian links from almost every Brunnian link. Second, we give a geometric classification theorem for Brunnian links, characterize the companionship graph defined by Budney in [JSJ-decompositions of knot and link complements in [Formula: see text], Enseign. Math. 3 (2005) 319–359], and develop a canonical geometric decomposition, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links then turn out to be Hopf [Formula: see text]-links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements. Third, we define an operation to reduce a Brunnian link in an unlink-complement into a new Brunnian link in [Formula: see text] and point out some phenomena concerning this operation.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

scholarly journals A classification of unsplittable-link complements.

1976 ◽  
Vol 23 (3) ◽  
pp. 261-266 ◽  
Author(s):  
Wilbur Whitten
Keyword(s):  
Link Complements

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

A Sequential Two-Step Design Framework for Deformable Mechanical Metamaterials

2020 ◽  
Author(s):  
Sree Kalyan Patiballa ◽  
Girish Krishnan
Keyword(s):  
Shape Change ◽  
Building Blocks ◽  
Load Flow ◽  
Stretchable Electronics ◽  
Design Framework ◽  
Shape Morphing ◽  
Mechanical Metamaterials ◽  
Global Deformation ◽  
Deformation Patterns

Abstract Deformable metamaterials are materials that are made up of several repeating elastic building blocks whose geometries can be tailored to obtain a specified global shape change or stiffness behavior. They are deemed useful in soft robotics, shape morphing mechanisms, stretchable electronics, wearable devices, and devices that adapt according to their environment. This paper presents a two-step sequential design framework for the synthesis of deformable mechanical metamaterials where (a) topology optimization is used to map global deformation requirement to local elasticity matrix, followed by (b) a selection of building block microstructure geometry from a database and refining it to match the elasticity requirement. The first step is accomplished through a unique parameterization scheme that enables the classification of the planar orthotropic elasticity matrix into four distinct classes. The second step uses a kinetostatic framework known as load flow visualization to populate candidate microstructure geometries within these four classes. Finally, the framework is validated for the design of a cantilever beam with a specified lateral stiffness requirement and the design of planar sheets that exhibit sinusoidal deformation patterns.

scholarly journals QUASI-ISOMETRIES, BOUNDARIES AND JSJ-DECOMPOSITIONS OF RELATIVELY HYPERBOLIC GROUPS

2013 ◽  
Vol 05 (04) ◽  
pp. 451-475 ◽  
Author(s):  
BRADLEY W. GROFF
Keyword(s):  
Hyperbolic Groups ◽  
Isometry Groups ◽  
Geometric Structures ◽  
Relatively Hyperbolic Groups ◽  
Jsj Decomposition ◽  
Outer Automorphisms ◽  
Jsj Decompositions ◽  
Relatively Hyperbolic

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.

scholarly journals UHF-slicing and classification of nuclear C*-algebras

2014 ◽  
Vol 06 (04) ◽  
pp. 465-540 ◽  
Author(s):  
Karen R. Strung ◽  
Wilhelm Winter
Keyword(s):  
Dynamical Systems ◽  
Classification Theorem ◽  
Discrete Version ◽  
C Algebras ◽  
Minimal Dynamical Systems ◽  
Tracial States

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

scholarly journals Classification of Roberts actions of strongly amenable C∗-tensor categories on the injective factor of type III1

2017 ◽  
Vol 28 (07) ◽  
pp. 1750052
Author(s):  
Toshihiko Masuda
Keyword(s):  
Classification Theorem ◽  
Finitely Generated ◽  
Tensor Categories ◽  
Type Iii ◽  
Type Iii1 ◽  
Injective Factor

In this paper, we generalize Izumi’s result on uniqueness of realization of finite C[Formula: see text]-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C[Formula: see text]-tensor categories by applying Popa’s classification theorem of strongly amenable subfactors of type III1.

scholarly journals Classification of self-assembling protein nanoparticle architectures for applications in vaccine design

2017 ◽  
Vol 4 (4) ◽  
pp. 161092 ◽  
Author(s):  
G. Indelicato ◽  
P. Burkhard ◽  
R. Twarock
Keyword(s):  
Self Assembly ◽  
Vaccine Design ◽  
Building Blocks ◽  
Coiled Coils ◽  
Specific Class ◽  
Regular Graphs ◽  
Humoral Immune ◽  
Self Assembling ◽  
Display Systems

We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coiled coils that are designed to assemble into trimeric and pentameric clusters. This allows a mathematical description of particle architectures in terms of bipartite (3,5)-regular graphs. Exploiting the relation with fullerene graphs, we provide a complete atlas of SAPN morphologies. The classification enables a detailed understanding of the spectrum of possible particle geometries that can arise in the self-assembly process. Moreover, it provides a toolkit for a systematic exploitation of SAPNs in bioengineering in the context of vaccine design, predicting the density of B-cell epitopes on the SAPN surface, which is critical for a strong humoral immune response.

Designing an activity-based costing model for a non-admitted prisoner healthcare setting

2013 ◽  
Vol 37 (4) ◽  
pp. 418
Author(s):  
Xiao Cai ◽  
Elizabeth Moore ◽  
Martin McNamara
Keyword(s):  
Healthcare Setting ◽  
Building Blocks ◽  
Design Phase ◽  
Activity Based Costing ◽  
Implementation Phase ◽  
Design And Implementation ◽  
Model Based ◽  
Solution Design

Aim. To design and deliver an activity-based costing model within a non-admitted prisoner healthcare setting. Method. Key phases from the NSW Health clinical redesign methodology were utilised: diagnostic, solution design and implementation. Results. The diagnostic phase utilised a range of strategies to identify issues requiring attention in the development of the costing model. The solution design phase conceptualised distinct ‘building blocks’ of activity and cost based on the speciality of clinicians providing care. These building blocks enabled the classification of activity and comparisons of costs between similar facilities. The implementation phase validated the model. Conclusions. The project generated an activity-based costing model based on actual activity performed, gained acceptability among clinicians and managers, and provided the basis for ongoing efficiency and benchmarking efforts.

scholarly journals A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes

2019 ◽  
Vol 11 (01) ◽  
pp. 181-204
Author(s):  
Zhichao Liu
Keyword(s):  
Inductive Limit ◽  
Decomposition Theorem ◽  
Building Blocks ◽  
Classification Theorem ◽  
Real Rank ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Rank Zero ◽  
The Real ◽  
Cw Complexes

In this paper, we consider the real rank zero [Formula: see text]-algebras which can be written as an inductive limit of the Elliott–Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.

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