Single-Vertex Origami and Spherical Expansive Motions

2005 ◽  
pp. 161-173 ◽  
Author(s):  
Ileana Streinu ◽  
Walter Whiteley
Keyword(s):  
Single Vertex

scholarly journals Jumping over baselines with new methods to predict activation maps from resting-state fMRI

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Eric Lacosse ◽  
Klaus Scheffler ◽  
Gabriele Lohmann ◽  
Georg Martius
Keyword(s):  
Resting State ◽  
Brain Activity ◽  
Resting State Fmri ◽  
Behavioral Differences ◽  
Large Variability ◽  
Single Vertex ◽  
Spontaneous Brain Activity ◽  
New Methods ◽  
Group Averaging ◽  
Proposed Modification

AbstractCognitive fMRI research primarily relies on task-averaged responses over many subjects to describe general principles of brain function. Nonetheless, there exists a large variability between subjects that is also reflected in spontaneous brain activity as measured by resting state fMRI (rsfMRI). Leveraging this fact, several recent studies have therefore aimed at predicting task activation from rsfMRI using various machine learning methods within a growing literature on ‘connectome fingerprinting’. In reviewing these results, we found lack of an evaluation against robust baselines that reliably supports a novelty of predictions for this task. On closer examination to reported methods, we found most underperform against trivial baseline model performances based on massive group averaging when whole-cortex prediction is considered. Here we present a modification to published methods that remedies this problem to large extent. Our proposed modification is based on a single-vertex approach that replaces commonly used brain parcellations. We further provide a summary of this model evaluation by characterizing empirical properties of where prediction for this task appears possible, explaining why some predictions largely fail for certain targets. Finally, with these empirical observations we investigate whether individual prediction scores explain individual behavioral differences in a task.

scholarly journals Boundary Quotient -algebras of Products of Odometers

2019 ◽  
Vol 71 (1) ◽  
pp. 183-212 ◽  
Author(s):  
Hui Li ◽  
Dilian Yang
Keyword(s):  
Single Vertex ◽  
Standard Product

AbstractIn this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.

scholarly journals Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

2021 ◽  
Vol 182 (3) ◽  
pp. 219-242
Author(s):  
Mostafa Haghir Chehreghani ◽  
Albert Bifet ◽  
Talel Abdessalem
Keyword(s):  
Betweenness Centrality ◽  
Directed Graphs ◽  
Randomized Algorithm ◽  
Exact Algorithm ◽  
Directed Network ◽  
Weighted Graphs ◽  
Single Vertex ◽  
Small Constant ◽  
Positive Weights ◽  
Real World Datasets

Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network G and a vertex r ∈ V (G), we propose an exact algorithm to compute betweenness score of r. Our algorithm pre-computes a set ℛ𝒱(r), which is used to prune a huge amount of computations that do not contribute to the betweenness score of r. Time complexity of our algorithm depends on |ℛ𝒱(r)| and it is respectively Θ(|ℛ𝒱(r)| · |E(G)|) and Θ(|ℛ𝒱(r)| · |E(G)| + |ℛ𝒱(r)| · |V(G)| log |V(G)|) for unweighted graphs and weighted graphs with positive weights. |ℛ𝒱(r)| is bounded from above by |V(G)| – 1 and in most cases, it is a small constant. Then, for the cases where ℛ𝒱(r) is large, we present a simple randomized algorithm that samples from ℛ𝒱(r) and performs computations for only the sampled elements. We show that this algorithm provides an (ɛ, δ)-approximation to the betweenness score of r. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

Waterbomb base: a symmetric single-vertex bistable origami mechanism

2014 ◽  
Vol 23 (9) ◽  
pp. 094009 ◽  
Author(s):  
Brandon H Hanna ◽  
Jason M Lund ◽  
Robert J Lang ◽  
Spencer P Magleby ◽  
Larry L Howell
Keyword(s):  
Single Vertex

Kinematics and Dynamics of Nanostructured Origami™

2005 ◽  
Author(s):  
P. Stellman ◽  
W. Arora ◽  
S. Takahashi ◽  
E. D. Demaine ◽  
G. Barbastathis
Keyword(s):  
Dynamic Modeling ◽  
Periodic Structures ◽  
Metal Layer ◽  
Three Dimensional ◽  
Spatial Dimension ◽  
Index Of Refraction ◽  
Coulomb Energy ◽  
Single Vertex ◽  
Revolute Joints ◽  
Final Design

Two-dimensional (2D) nanofabrication processes such as lithography are the primary tools for building functional nanostructures. The third spatial dimension enables completely new devices to be realized, such as photonic crystals with arbitrary defect structures and materials with negative index of refraction [1]. Presently, available methods for three-dimensional (3D) nanopatterning tend to be either cost inefficient or limited to periodic structures. The Nanostructured Origami method fabricates 3D devices by first patterning nanostructures (electronic, optical, mechanical, etc) onto a 2D substrate and subsequently folding segments along predefined creases until the final design is obtained [2]. This approach allows almost arbitrary 3D nanostructured systems to be fabricated using exclusively 2D nanopatterning tools. In this paper, we present two approaches to the kinematic and dynamic modeling of folding origami structures. The first approach deals with the kinematics of unfolding single-vertex origami. This work is based on research conducted in the origami mathematics community, which is making rapid progress in understanding the geometry of origami and folding in general [3]. First, a unit positive “charge” is assigned to the creases of the structure in its folded state. Thus, each configuration of the structure as it unfolds can be assigned a value of electrostatic (Coulomb) energy. Because of repulsion between the positive charges, the structure will unfold if allowed to decrease its energy. If the energy minimization can be carried out all the way to the completely unfolded state, we are simultaneously guaranteed of the absence of collisions for the determined path. The second method deals with dynamic modeling of folding multi-segment (accordion style) origamis. The actuation method for folding the segments uses a thin, stressed metal layer that is deposited as a hinge on a relatively stress free structural layer. Through the use of robotics routines, the hinges are modeled as revolute joints, and the system dynamics are calculated.

MATCHING PRECLUSION AND CONDITIONAL MATCHING PRECLUSION FOR CROSSED CUBES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250005 ◽  
Author(s):  
EDDIE CHENG ◽  
SACHIN PADMANABHAN
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Single Vertex ◽  
Minimum Number ◽  
Important Variant

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find the matching preclusion number and the conditional matching preclusion number with the classification of the optimal sets for the class of crossed cubes, an important variant of the class of hypercubes. Indeed, we will establish more general results on the matching preclusion and the conditional matching preclusion problems for a larger class of interconnection networks.

scholarly journals Fixation probabilities for simple digraphs

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120676 ◽  
Author(s):  
Burton Voorhees ◽  
Alex Murray
Keyword(s):  
Complete Bipartite Graph ◽  
Directed Graphs ◽  
Closed Form Solution ◽  
Form Solution ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Fixation Probability ◽  
Single Vertex ◽  
Fixation Probabilities ◽  
Birth Death

The problem of finding birth–death fixation probabilities for configurations of normal and mutants on an N -vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the single-vertex fixation probabilities. This also yields a closed-form solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r >1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.

MATCHING PRECLUSION AND CONDITIONAL MATCHING PRECLUSION FOR AUGMENTED CUBES

2010 ◽  
Vol 11 (01n02) ◽  
pp. 35-60 ◽  
Author(s):  
EDDIE CHENG ◽  
RANDY JIA ◽  
DAVID LU
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Single Vertex ◽  
Minimum Number

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented cubes, a class of networks designed as an improvement of the hypercubes.

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