scholarly journals Control of connectivity and rigidity in prismatic assemblies

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200485
Author(s):  
Gary P. T. Choi ◽  
Siheng Chen ◽  
Lakshminarayanan Mahadevan
Keyword(s):  
Number Theory ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Connected Components ◽  
Elementary Number Theory ◽  
Percolation Transition ◽  
Elementary Number ◽  
Topological Connectivity ◽  
Internal Degrees Of Freedom ◽  
Deterministic Setting

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

Elementary Number Theory

2016 ◽  
pp. 1-32
Author(s):  
Gary L. Mullen ◽  
James A. Sellers
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Result Section ◽  
Structural Result ◽  
Elementary Number

This chapter proves some number-theoretic results about the sequences defined in Chapter 23. It proceeds as follows. Section 24.2 proves Lemma 24.1, a multipart structural result. Section 24.3 takes care of several number-theoretic details left over from Section 23.6 and Section 23.7.

Infinite Orbits Revisited

2019 ◽  
pp. 227-238
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Model Section

This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.

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