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The shape of spherical quartics

Bert Jüttler, Wenping Wang
2003 Computer Aided Geometric Design  
With the help of the generalized stereographic projection (Dietz et al., 1993), we construct a two-parameter family of spherical quartics solving this problem.  ...  We study the shape of these solutions and derive criteria which guarantee solutions without cusps or self-intersections.  ...  In particular, we are indebted to one of them for providing the proof of Proposition 2. Wenping Wang's work is supported by a grant (HKU 7032/99E) of Hong Kong Research Grant Council.  ... 
doi:10.1016/j.cagd.2003.07.003 fatcat:7hqs457g4jh2jlkhezhmop7byy

Jamming in Systems Composed of Frictionless Ellipse-Shaped Particles

Mitch Mailman, Carl F. Schreck, Corey S. O'Hern, Bulbul Chakraborty
2009 Physical Review Letters  
Finally, we find that the degree of hypostaticity is determined by the number of quartic modes of the packing.  ...  We study the structural and mechanical properties of jammed ellipse packings, and find that the nature of the jamming transition in these systems is fundamentally different from that for spherical particles  ...  We find significant differences between the behavior near jamming of ellipse-shaped and spherical particles: (i) The lowest-energy vibrational excitations are composed of predominantly rotational particle  ... 
doi:10.1103/physrevlett.102.255501 pmid:19659093 fatcat:tbiyzhzrareotkbxqxudwv6dai

The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions [article]

Dong Wang, John D. Treado, Arman Boromand, Blake Norwick, Michael P. Murrell, Mark D. Shattuck, Corey S. O'Hern
2021 arXiv   pre-print
In contrast, β≈ 0.5 for packings of deformable particles with non-zero bending energy, which matches the value for jammed packings of soft, spherical particles with fixed shape.  ...  We find that the contributions to the eigenmodes of the dynamical matrix from the shape degrees of freedom are significant over the full range of frequency and shape parameters for particles with zero  ...  The average quartic eigenmode frequency scales as ω 0 ∼ ( A − 1) −1/3 , which is different than the scaling behavior of the quartic modes in jammed packings of rigid, frictionless non-spherical particles  ... 
arXiv:2108.09616v1 fatcat:l264eyayzfc7vfshfi3o4tlvhq

Jammed packings of 3D superellipsoids with tunable packing fraction, contact number, and ordering

Ye Yuan, Kyle VanderWerf, Mark D. Shattuck, Corey S/ O'Hern
2019 Soft Matter  
We consider more than 200 different particle shapes by varying the three shape parameters...  ...  We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions.  ...  Quartic modes always correspond to the lowest frequency modes, and the translational content of these eigenmodes is small for nearly spherical particle shapes, and increases with asphericity.  ... 
doi:10.1039/c9sm01932d pmid:31742301 pmcid:PMC6902436 fatcat:s5tnewwngzb2dgmyvpxovas5qi

Jammed packings of 3D superellipsoids with tunable packing fraction, contact number, and ordering [article]

Ye Yuan, Kyle VanderWerf, Mark D. Shattuck, Corey S. O'Hern
2019 arXiv   pre-print
Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value  ...  We consider more than 200 different particle shapes by varying the three shape parameters that define superellipsoids.  ...  In addition, we acknowledge the High-performance Computing Platform of Peking University. We also thank S. Li at Peking University for helpful discussions.  ... 
arXiv:1909.12191v1 fatcat:j6j65dhl2bb4pnfppn42rxuh6m

On Spherical Quartics, with a Quadruple Cyclic Arc, and a Triple Focus

Henry M. Jeffery
1880 Proceedings of the London Mathematical Society  
If cP < 1, there may be two cusped quartics in the group, pear-shaped ( Fig. 1) , and cardioidal (Fig. 2) ; these are of the fifth class.  ...  Such a group of spherical quartics may be thus defined: * = (l+dx)(l+x*+y')+\(l+x'+y*)* : (A), if x, y denote by Gudermann's system of spherical coordinates the tangents of arcs, intercepted on the arcs  ... 
doi:10.1112/plms/s1-12.1.168 fatcat:j5l7wsjafzdpnl2uktm4e6buz4

Second shape transition temperature: prolate noncollective to oblate noncollective

Alan L. Goodman, Taihao Jin
1997 Zeitschrift für Physik A Hadrons and Nuclei  
For temperatures between T c1 (I) and T c2 (I), rotation of the hot spherical shape produces a prolate shape rotating about its symmetry axis. This is the opposite of the classical response.  ...  Many nuclei have a temperature interval where the rotation of a hot spherical equilibrium shape generates a prolate equilibrium shape rotating around its symmetry axis. PACS: 21.60.  ...  For temperatures between T c1 (I) and T c2 (I), rotation of the hot spherical shape produces a prolate shape rotating about its symmetry axis. This is the opposite of the classical response.  ... 
doi:10.1007/s002180050287 fatcat:xkfaat4jwzg3vbbo445e6khrta

Bridging particle deformability and collective response in soft solids [article]

John D. Treado, Dong Wang, Arman Boromand, Michael P. Murrell, Mark D. Shattuck, Corey S. O'Hern
2021 arXiv   pre-print
We find that jammed packings of deformable particles with excess shape degrees of freedom possess low-frequency quartic vibrational modes that stabilize the packings even though they possess fewer interparticle  ...  We find that the mechanical response of jammed packings of deformable particles with shape degrees of freedom differs significantly from that of jammed packings of rigid-shape particles, which emphasizes  ...  As shown in Fig. 2 (a), we find that the number of missing contacts in jammed DP packings always matches the number of quartic modes N q across a wide range of shape parameters.  ... 
arXiv:2012.02915v2 fatcat:trvawy265fgblmduha2ifem5mi

Effect of anharmonicities in the critical number of trapped condensed atoms with an attractive two-body interaction

Victo S. Filho, Arnaldo Gammal, Lauro Tomio
2002 Physical Review A. Atomic, Molecular, and Optical Physics  
As an example of how a cubic or quartic anharmonic term can affect the maximum number of particles, we consider the trap parameters and the results given by Roberts et al. [Phys. Rev.  ...  We determine the quantitative effect, in the maximum number of particles and other static observables, due to small anharmonic terms added to the confining potential of an atomic condensed system with  ...  L.T. also thanks the Conselho Nacional de Desenvolvimento Científico e Tecnológico ͑CNPq͒ of Brazil for support.  ... 
doi:10.1103/physreva.66.043605 fatcat:f2psqy36ovh6jptyfrbw2y5n7a

Optical cross-connect system based on the White cell and three-state microelectromechanical system: experimental demonstration of the quartic cell

Víctor Argueta-Díaz, Betty Lise Anderson
2006 Applied Optics  
We describe in detail the implementation of an underpopulated quartic White cell configuration. We discuss the aberrations associated with the output of the system.  ...  We present a proof of concept (design, simulations, and experimental results) for an optical crossconnection device based on the optical White cell and a three-state microelectromechanical system tilting  ...  We thank William Thalgott and Feras Abou Galala for their time and expertise to construct the pseudo-MEMS and cut the silicon wafers, respectively.  ... 
doi:10.1364/ao.45.004658 pmid:16799680 fatcat:6djaz7ru5zg3bpeq7mjbbvdxti

The Schrödinger equation on a Lagrange mesh

Gaotsiwe J Rampho
2017 Journal of Physics, Conference Series  
The accuracy of the matrix elements is tested with the quartic anharmonic oscillator and a double ring-shaped harmonic oscillator potentials.  ...  generalized spherical coordinates Gaotsiwe J Rampho -Recent citations Lagrange-mesh solution of the Schrödinger equation in generalized spherical coordinates Gaotsiwe J Rampho -This content was downloaded  ...  discussed for the quartic anharmonic oscillator in two dimensions and the double ring-shaped harmonic oscillator in three dimensions.  ... 
doi:10.1088/1742-6596/905/1/012037 fatcat:dknj2zyucnagxjftetsmon6o5a

Page 9943 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
disk.” 2004m:41003 41A10 65D05 65D18 Jiittler, Bert (A-LINZ-AG; Linz); Wang, Wenping [Wang, Wen Ping'| (PRC-HK-CIS; Hong Kong) The shape of spherical quartics.  ...  Given two points and associated tangent vectors (first deriva- tives) on a sphere, the focus is on the interpolation of this data with a spherical quartic.  ... 

Method of solving triplets consisting of a singlet and airspaced doublet with given primary aberrations

Chao-Hsien Chen, Shin-Gwo Shiue, Mao-Hong Lu
1998 Journal of Modern Optics  
The roots give the element powers. Second, the lens shapes are obtained by solving the quartic polynomial equation which is obtained by combining the equations of spherical aberration and coma.  ...  The triplet is required to yield specified amounts of lens power and four primary aberrations: spherical aberration, coma, longitudinal chromatic aberration and secondary spectrum.  ...  Second, the lens shapes are obtained b y solving the quartic polynomial equation which is obtained by combining the equations of S 1 and S2.  ... 
doi:10.1080/095003498150547 fatcat:um6k7t2urbej5dtdjpwh46qws4

Method of solving triplets consisting of a singlet and air-spaced doublet with given primary aberrations

Chao-Hsien Chen, Shin-Gwo Shiue, Mao-Hong Lu
1998 Journal of Modern Optics  
The roots give the element powers. Second, the lens shapes are obtained by solving the quartic polynomial equation which is obtained by combining the equations of spherical aberration and coma.  ...  The triplet is required to yield specified amounts of lens power and four primary aberrations: spherical aberration, coma, longitudinal chromatic aberration and secondary spectrum.  ...  Second, the lens shapes are obtained b y solving the quartic polynomial equation which is obtained by combining the equations of S 1 and S2.  ... 
doi:10.1080/09500349808231743 fatcat:qvvq7aeebbcktncdpxsphaym7i

Stationary states in two-dimensional systems driven by bivariate Lévy noises

Krzysztof Szczepaniec, Bartłomiej Dybiec
2014 Physical Review E  
Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e. its spectral measure.  ...  In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of  ...  The studied model is presented in Section II. Section III discusses obtained, mainly numerically, results. The paper is closed with concluding remarks (Section IV).  ... 
doi:10.1103/physreve.90.032128 pmid:25314416 fatcat:ezuw7xpx7fbo5lspvorbrbakcq
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