Controllability Degree of Directed Line Networks: Nodal Energy and Asymptotic Bounds

2018 ◽  
Author(s):  
Shiyu Zhao ◽  
Fabio Pasqualetti
Keyword(s):  
Directed Line ◽  
Asymptotic Bounds

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

An Efficient Improved Algorithm of Convex Hull

2014 ◽  
Vol 602-605 ◽  
pp. 3104-3106
Author(s):  
Shao Hua Liu ◽  
Jia Hua Zhang
Keyword(s):  
Convex Hull ◽  
Line Segment ◽  
Main Idea ◽  
Discrete Point ◽  
Directed Line ◽  
Generation Algorithm ◽  
Simple Program ◽  
Point Set ◽  
High Computational Efficiency ◽  
Improved Algorithm

This paper introduced points and directed line segment relation judgment method, the characteristics of generation and Graham method using the original convex hull generation algorithm of convex hull discrete points of the convex hull, an improved algorithm for planar discrete point set is proposed. The main idea is to use quadrilateral to divide planar discrete point set into five blocks, and then by judgment in addition to the four district quadrilateral internally within the point is in a convex edge. The result shows that the method is relatively simple program, high computational efficiency.

scholarly journals Entropy of systolically extremal surfaces and asymptotic bounds

2005 ◽  
Vol 25 (4) ◽  
pp. 1209-1220 ◽  
Author(s):  
MIKHAIL G. KATZ ◽  
STÉPHANE SABOURAU
Keyword(s):  
Asymptotic Bounds

Asymptotic bounds on overall moduli of cracked bodies

2000 ◽  
Vol 37 (43) ◽  
pp. 6221-6237 ◽  
Author(s):  
J. Wang ◽  
J. Fang ◽  
B.L. Karihaloo
Keyword(s):  
Asymptotic Bounds

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

Characterization and Coordination of Point-line Trajectories

2004 ◽  
Vol 127 (3) ◽  
pp. 502-505 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Yi Zhang ◽  
Ruj Bunduwongse
Keyword(s):  
Orientation Relationship ◽  
Complete Characterization ◽  
Free Form ◽  
Directed Line ◽  
Free Form Surface ◽  
Point Line ◽  
Line Trajectory ◽  
The Relationship ◽  
Line Direction

A point-line refers to a rigid combination of a directed line and an endpoint along the line. To trace a point-line trajectory, one must control not only the trajectory of the endpoint (the directrix) but also the direction of the point-line (the indicatrix). This paper addresses three issues on point-line trajectories. First of all, by considering the relationship between the point trajectory and the corresponding point-line direction, it offers the complete characterization of point-line trajectories. It presents the coordination of the point-line axis with a free point trajectory and also with a point trajectory constrained on a free form surface. The issue of maintaining an invariant orientation relationship between the point-line axis and the point trajectory is also addressed.

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