Smooth points on semi-algebraic sets

Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n = 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research.

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Numerical irreducible decomposition over a number field

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501955 ◽

2018 ◽

Vol 17 (10) ◽

pp. 1850195

Author(s):

Timothy M. McCoy ◽

Chris Peterson ◽

Andrew J. Sommese

Keyword(s):

Algebraic Geometry ◽

Number Field ◽

Polynomial System ◽

Field Extension ◽

General Point ◽

Numerical Algebraic Geometry ◽

Irreducible Decomposition ◽

Algebraic Set ◽

Field Of Definition ◽

Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

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Emulating the local Kuramoto model with an injection-locked photonic crystal laser array

Scientific Reports ◽

10.1038/s41598-021-86982-w ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Naotomo Takemura ◽

Kenta Takata ◽

Masato Takiguchi ◽

Masaya Notomi

Keyword(s):

Photonic Crystal ◽

Nearest Neighbor ◽

Multiple Quantum Well ◽

Kuramoto Model ◽

Multiple Quantum ◽

Laser Array ◽

Novel Strategy ◽

A Chain ◽

The One ◽

The Kuramoto Model

AbstractThe Kuramoto model is a mathematical model for describing the collective synchronization phenomena of coupled oscillators. We theoretically demonstrate that an array of coupled photonic crystal lasers emulates the Kuramoto model with non-delayed nearest-neighbor coupling (the local Kuramoto model). Our novel strategy employs indirect coupling between lasers via additional cold cavities. By installing cold cavities between laser cavities, we avoid the strong coupling of lasers and realize ideal mutual injection-locking with effective non-delayed dissipative coupling. First, after discussing the limit cycle interpretation of laser oscillation, we demonstrate the synchronization of two indirectly coupled lasers by numerically simulating coupled-mode equations. Second, by performing a phase reduction analysis, we show that laser dynamics in the proposed device can be mapped to the local Kuramoto model. Finally, we briefly demonstrate that a chain of indirectly coupled photonic crystal lasers actually emulates the one-dimensional local Kuramoto chain. We also argue that our proposed structure, which consists of periodically aligned cold cavities and laser cavities, will best be realized by using state-of-the-art buried multiple quantum well photonic crystals.

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Partially Phase-Locked Solutions to the Kuramoto Model

Journal of Statistical Physics ◽

10.1007/s10955-021-02783-5 ◽

2021 ◽

Vol 183 (3) ◽

Author(s):

Jared C. Bronski ◽

Lan Wang

Keyword(s):

Kuramoto Model ◽

The Kuramoto Model

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Emergent synchronous beating behavior in spontaneous beating cardiomyocyte clusters

Scientific Reports ◽

10.1038/s41598-021-91466-y ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Kazufumi Sakamoto ◽

Yoshitsune Hondo ◽

Naoki Takahashi ◽

Yuhei Tanaka ◽

Rikuto Sekine ◽

...

Keyword(s):

Stem Cell ◽

Embryonic Stem Cell ◽

Human Embryonic Stem Cell ◽

Single Cells ◽

Embryonic Stem ◽

Kuramoto Model ◽

Distribution Analysis ◽

Cell Clusters ◽

Overdrive Suppression ◽

The Kuramoto Model

AbstractWe investigated the dominant rule determining synchronization of beating intervals of cardiomyocytes after the clustering of mouse primary and human embryonic-stem-cell (hES)-derived cardiomyocytes. Cardiomyocyte clusters were formed in concave agarose cultivation chambers and their beating intervals were compared with those of dispersed isolated single cells. Distribution analysis revealed that the clusters’ synchronized interbeat intervals (IBIs) were longer than the majority of those of isolated single cells, which is against the conventional faster firing regulation or “overdrive suppression.” IBI distribution of the isolated individual cardiomyocytes acquired from the beating clusters also confirmed that the clusters’ IBI was longer than those of the majority of constituent cardiomyocytes. In the complementary experiment in which cell clusters were connected together and then separated again, two cardiomyocyte clusters having different IBIs were attached and synchronized to the longer IBIs than those of the two clusters’ original IBIs, and recovered to shorter IBIs after their separation. This is not only against overdrive suppression but also mathematical synchronization models, such as the Kuramoto model, in which synchronized beating becomes intermediate between the two clusters’ IBIs. These results suggest that emergent slower synchronous beating occurred in homogeneous cardiomyocyte clusters as a community effect of spontaneously beating cells.

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Investigation of Preferred Orientations in Planar Polycrystals

Journal of Applied Mechanics ◽

10.1115/1.2912930 ◽

2008 ◽

Vol 75 (5) ◽

Cited By ~ 3

Author(s):

M. R. Tonks ◽

A. J. Beaudoin ◽

F. Schilder ◽

D. A. Tortorelli

Keyword(s):

Texture Evolution ◽

Mean Field ◽

Homogenization Method ◽

Process Models ◽

Kuramoto Model ◽

Taylor Model ◽

Crystal Plasticity Finite Element ◽

Polycrystal Plasticity ◽

Taylor Hypothesis ◽

The Kuramoto Model

More accurate manufacturing process models come from better understanding of texture evolution and preferred orientations. We investigate the texture evolution in the simplified physical framework of a planar polycrystal with two slip systems used by Prantil et al. (1993, “An Analysis of Texture and Plastic Spin for Planar Polycrystal,” J. Mech. Phys. Solids, 41(8), pp. 1357–1382). In the planar polycrystal, the crystal orientations behave in a manner similar to that of a system of coupled oscillators represented by the Kuramoto model. The crystal plasticity finite element method and the stochastic Taylor model (STM), a stochastic method for mean-field polycrystal plasticity, predict the development of a steady-state texture not shown when employing the Taylor hypothesis. From this analysis, the STM appears to be a useful homogenization method when using representative standard deviations.

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