scholarly journals Generic combinatorial rigidity of periodic frameworks

2013 ◽  
Vol 233 (1) ◽  
pp. 291-331 ◽  
Author(s):  
Justin Malestein ◽  
Louis Theran
Keyword(s):  
Periodic Frameworks ◽  
Combinatorial Rigidity

scholarly journals Geometric auxetics

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150033 ◽  
Author(s):  
Ciprian Borcea ◽  
Ileana Streinu
Keyword(s):  
Computer Simulations ◽  
Mathematical Theory ◽  
Three Dimensional ◽  
Numerical Approximations ◽  
Periodic Frameworks ◽  
Infinite Supply

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.

scholarly journals Parabolic limits of renormalization

2000 ◽  
Vol 20 (1) ◽  
pp. 173-229 ◽  
Author(s):  
BENJAMIN HINKLE
Keyword(s):  
Fixed Point ◽  
Period Doubling ◽  
Unit Interval ◽  
Local Analysis ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Combinatorial Description ◽  
Combinatorial Rigidity

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

scholarly journals A DESIGN METHOD FOR OPTIMAL TRUSS STRUCTURES WITH REDUNDANCY BASED ON COMBINATORIAL RIGIDITY THEORY

2014 ◽  
Vol 79 (699) ◽  
pp. 583-592
Author(s):  
Rie KOHTA ◽  
Makoto YAMAKAWA ◽  
Naoki KATOH ◽  
Yoshikazu ARAKI ◽  
Makoto OHSAKI
Keyword(s):  
Design Method ◽  
Truss Structures ◽  
Rigidity Theory ◽  
Combinatorial Rigidity

scholarly journals Liftings and Stresses for Planar Periodic Frameworks

2015 ◽  
Vol 53 (4) ◽  
pp. 747-782 ◽  
Author(s):  
Ciprian Borcea ◽  
Ileana Streinu
Keyword(s):  
Periodic Frameworks

ChemInform Abstract: Topological Motifs in Cyanometallates: From Building Units to Three-Periodic Frameworks

ChemInform ◽  
2016 ◽  
Vol 47 (3) ◽  
Author(s):  
Eugeny V. Alexandrov ◽  
Alexander V. Virovets ◽  
Vladislav A. Blatov ◽  
Eugenia V. Peresypkina
Keyword(s):  
Periodic Frameworks
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