scholarly journals Counting equilibria of large complex systems by instability index

2021 ◽  
Vol 118 (34) ◽  
pp. e2023719118 ◽  
Author(s):  
Gérard Ben Arous ◽  
Yan V. Fyodorov ◽  
Boris A. Khoruzhenko
Keyword(s):  
Autonomous System ◽  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Interaction Strength ◽  
Absolute Instability ◽  
Instability Index ◽  
Random Interactions ◽  
Stable Equilibria ◽  
The Mean ◽  
Nonlinear Autonomous System

We consider a nonlinear autonomous system of N≫1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

scholarly journals Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Phan Thành Nam ◽  
Marcin Napiórkowski
Keyword(s):  
Density Matrix ◽  
Ground State ◽  
Mean Field ◽  
Interaction Strength ◽  
Bose Gas ◽  
Body Density ◽  
The Mean ◽  
The One ◽  
Mean Field Regime ◽  
Number Of Particles

AbstractWe consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy and the ground state.

Maximum Total Point Motion of Five Points Versus All Points in Assessing Tibial Baseplate Stability

2021 ◽  
Author(s):  
Abigail Niesen ◽  
Anna L Garverick ◽  
Maury Hull
Keyword(s):  
Degrees Of Freedom ◽  
Stability Limit ◽  
3D Models ◽  
Six Degrees Of Freedom ◽  
Percent Error ◽  
Model Based ◽  
Point Motion ◽  
The Mean ◽  
The Difference ◽  
Total Point

Abstract Maximum total point motion (MTPM), the point on a baseplate that migrates the most, has been used to assess the risk of tibial baseplate loosening using radiostereometric analysis (RSA). Two methods for determining MTPM for model-based RSA are to use either 5 points distributed around the perimeter of the baseplate or to use all points on the 3D model. The objectives were to quantify the mean difference in MTPM using 5 points vs. all points, compute the percent error relative to the 6-month stability limit for groups of patients, and to determine the dependency of differences in MTPM on baseplate size and shape. A dataset of 10,000 migration values was generated using the mean and standard deviation of migration in six degrees of freedom at 6 months from an RSA study. The dataset was used to simulate migration of 3D models (two baseplate shapes and two baseplate sizes) and calculate the difference in MTPM using 5 virtual points vs. all points and the percent error (i.e. difference in MTPM/stability limit) relative to the 6-month stability limit. The difference in MTPM was about 0.02 mm, or 4% percent relative to the 6-month stability limit, which is not clinically important. Furthermore, results were not affected by baseplate shape or size. Researchers can decide whether to use 5 points or all points when computing MTPM for model-based RSA. The authors recommend using 5 points to maintain consistency with marker-based RSA.

Jacobi analysis for an unusual 3D autonomous system

2020 ◽  
Vol 17 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Chunsheng Feng ◽  
Qiujian Huang ◽  
Yongjian Liu
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Chen System ◽  
Parameter Region ◽  
Stable Equilibria ◽  
The Lorenz System ◽  
Deviation Vector

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

AC0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem

2018 ◽  
Vol 52 (5) ◽  
pp. 1981-2001 ◽  
Author(s):  
Yuan Bao ◽  
Zhaoliang Meng ◽  
Zhongxuan Luo
Keyword(s):  
Singular Perturbation ◽  
Fourth Order ◽  
Degrees Of Freedom ◽  
Perturbation Parameter ◽  
Singular Perturbation Problem ◽  
Mean Values ◽  
Perturbation Problem ◽  
Convex Quadrilateral ◽  
The Mean ◽  
Quadrilateral Finite Element

In this paper, aC0nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateralQ, the shape function space is the union ofS21(Q*) and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

AN AUTONOMOUS SYSTEM FOR GROUND-BASED CLOUD TRACKING USING THE MEAN-SHIFT ALGORITHM

2011 ◽  
Author(s):  
M.D. O’Toole ◽  
S.A. Wormald ◽  
D. Kerr ◽  
J. Coupland ◽  
A.P. Sandford
Keyword(s):  
Autonomous System ◽  
Mean Shift ◽  
Mean Shift Algorithm ◽  
The Mean ◽  
Cloud Tracking

Pentachlorophenol Residues in Porcine Tissue Following Preslaughter Exposure to Treated Wood Shavings

1991 ◽  
Vol 54 (6) ◽  
pp. 448-450 ◽  
Author(s):  
KELLY M. BUTLER ◽  
RICHARD FRANK
Keyword(s):  
Degrees Of Freedom ◽  
Treated Wood ◽  
Test Group ◽  
Liver Fat ◽  
Control Group ◽  
Porcine Tissue ◽  
Wood Shavings ◽  
Significant Difference ◽  
The Mean ◽  
Level Of Significance

Sixty market hogs originating from one producer and finished in a concrete and steel facility were divided into two groups of 30 and housed for approximately 60 h on either straw (control group) or pentachlorophenol (PCP) treated wood shavings (test group). Feed, straw, and shavings were analyzed for PCP residues. Both feed and straw yielded nondetectable levels of PCP residues, while shavings ranged from 0.03 to 12.0 ppm. The hogs were shipped to slaughter without bedding, and liver, fat and muscle (muscle from the test group only) samples were collected postmortem. The mean level of PCP residue in control (straw) hog livers was 0.037 ppm, while that of livers of hogs bedded with contaminated shavings was 0.342 ppm, a highly significant difference. The t-value using Welch's approximation equalled 9.77 using 28.5 degrees of freedom, indicating the mean PCP residue level was higher for the treated than the control group at a 0.01% level of significance.

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