Origami Folding and Bar Frameworks

2014 ◽  
Author(s):  
Alejandro R. Diaz
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Optimization Problems ◽  
Undirected Graph ◽  
Geometric Optimization ◽  
Order Effects ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Higher Order Effects

One of the more computationally demanding tasks in a process of synthesizing “from scratch” origami crease patterns designed for a given purpose involve a simulation capability to track the progression of the folding process as the pattern folds. This work presents an approach to simulate origami folding based on bar frameworks. The work is related to joint frameworks and projected polyhedral, as they apply to folding. The analysis starts from a representation of a crease pattern as an undirected graph G(E,V) formed by edges E and vertices V. A framework G(p) is an instance of G where the vertex locations are assigned positions according to a vector valued function p(t), where t marks the folding progression and t=0 represents the initial, flat configuration. The strategy presented is based on finding a sequence of instances {p(1), p(2), …} corresponding to an analytic flex p, i.e., functions such that edges in all G(p(t)) have the same length. The method is based on using a finite element description of a bar framework corresponding to a truss-like structure congruent with G(p). Solutions to an eigenvalue problem associated with this structure provide the means to update from p(t) to p(t+1). Two simple (purely geometric) optimization problems adjust the update to compensate for higher order effects, guaranteeing that the length of the edges remain constant. The methodology can be used to achieve configurations close to “flat folding”, provided that no interference of the faces occurs along the way. We expected that physically-motivated constraints (stresses, deformations, etc.) and sensitivity analysis computations will be more easily represented in this framework and therefore this formulation will have an advantage over more standard “origami mathematics” approaches. The approach is illustrated with an example of folding a simple 10-crease pattern.

Time after time: support for memory accounts of temporal preparation.

2019 ◽  
Author(s):  
Joe Butler ◽  
Samuel Ngabo ◽  
Marcus Missal
Keyword(s):  
Response Times ◽  
Reaction Times ◽  
Evidence Base ◽  
Higher Order ◽  
Order Effects ◽  
Previous Trial ◽  
Adaptive Responses ◽  
Temporal Preparation ◽  
Subsequent Trial ◽  
Higher Order Effects

Complex biological systems build up temporal expectations to facilitate adaptive responses to environmental events, in order to minimise costs associated with incorrect responses, and maximise the benefits of correct responses. In the lab, this is clearly demonstrated in tasks which show faster response times when the period between warning (S1) and target stimulus (S2) on the previous trial was short and slower when the previous trial foreperiod was long. The mechanisms driving such higher order effects in temporal preparation paradigms are still under debate, with key theories proposing that either i) the foreperiod leads to automatic modulation of the arousal system which influences responses on the subsequent trial, or ii) that exposure to a foreperiod results in the creation of a memory trace which is used to guide responses on the subsequent trial. Here we provide data which extends the evidence base for the memory accounts, by showing that previous foreperiod exposures are cumulative with reaction times shortening after repeated exposures; whilst also demonstrate that the higher order effects associated with a foreperiod remain active for several trials.

scholarly journals Responsible innovation, anticipation and responsiveness: case studies of algorithms in decision support in justice and security, and an exploration of potential, unintended, undesirable, higher-order effects

AI and Ethics ◽  
2021 ◽  
Author(s):  
Marc Steen ◽  
Tjerk Timan ◽  
Ibo van de Poel
Keyword(s):  
Decision Support ◽  
Case Studies ◽  
Personal Data ◽  
Higher Order ◽  
Popular Media ◽  
Responsible Innovation ◽  
Order Effects ◽  
Key Dimensions ◽  
Higher Order Effects

AbstractThe collection and use of personal data on citizens in the design and deployment of algorithms in the domain of justice and security is a sensitive topic. Values like fairness, autonomy, privacy, accuracy, transparency and property are at stake. Negative examples of algorithms that propagate or exacerbate biases, inequalities or injustices have received ample attention, both in academia and in popular media. To supplement this view, we will discuss two positive examples of Responsible Innovation (RI): the design and deployment of algorithms in decision support, with good intentions and careful approaches. We then explore potential, unintended, undesirable, higher-order effects of algorithms—effects that may occur despite good intentions and careful approaches. We do that by engaging with anticipation and responsiveness, two key dimensions of Responsible Innovation. We close the paper with proposing a framework and a series of tentative recommendations to promote anticipation and responsiveness in the design and deployment of algorithms in decision support in the domain of justice and security.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

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