The planar approximation: Part II

1993 ◽  
pp. 605-615
Author(s):  
C. Itzykson ◽  
J.-B. Zuber
Keyword(s):  
Planar Approximation

Two-stage planar approximation of non-planar crack growth

2012 ◽  
Vol 96 ◽  
pp. 147-164 ◽  
Author(s):  
V.K. Hombal ◽  
Y. Ling ◽  
K.A. Wolfe ◽  
S. Mahadevan
Keyword(s):  
Crack Growth ◽  
Two Stage ◽  
Planar Approximation ◽  
Planar Crack

SPONTANEOUS CHIRAL-SYMMETRY BREAKING AND SCALING LAW IN THREE-DIMENSIONAL QED

1990 ◽  
Vol 05 (06) ◽  
pp. 407-416 ◽  
Author(s):  
KEI-ICHI KONDO ◽  
HAJIME NAKATANI
Keyword(s):  
Chiral Symmetry ◽  
Order Parameter ◽  
Chiral Symmetry Breaking ◽  
Scaling Law ◽  
Three Dimensional ◽  
Dyson Equation ◽  
Dirac Fermion ◽  
Dynamical Mass ◽  
Planar Approximation ◽  
Spontaneous Breakdown

We analyze the critical behavior associated with spontaneous breakdown of chiral symmetry in QED3 (three-dimensional QED with four-component Dirac fermion using the SD (Schwinger-Dyson) equation. In the quenched planar approximation, we find an approximate solution such that QED3 resides in only one phase where the chiral symmetry is broken. Moreover, we predict the scaling law for the dynamical mass and chiral order parameter by an analytic study of the SD equation, which is then confirmed by solving the SD equation numerically. This scaling law is consistent with the Monte Carlo result in the quenched approximation.

DIMENSIONAL DEPENDENCE OF CHIRAL-SYMMETRY- BREAKING PHASE TRANSITION IN QUENCHED PLANAR QED

1989 ◽  
Vol 04 (22) ◽  
pp. 2155-2166 ◽  
Author(s):  
KEI-ICHI KONDO ◽  
HAJIME NAKATANI
Keyword(s):  
Phase Transition ◽  
Chiral Symmetry ◽  
Chiral Symmetry Breaking ◽  
Scaling Law ◽  
Mean Field ◽  
Dyson Equation ◽  
Essential Singularity ◽  
Spontaneous Breaking ◽  
Planar Approximation ◽  
Singularity Type

We consider the critical behavior of the phase transition associated with the spontaneous breaking of chiral-symmetry in (QED) D, in the framework of the Schwinger-Dyson equation. Special attention is paid on the scaling law. While it is well known that quenched planar QED 4 obeys the Miransky scaling of the essential singularity type, our numerical calculations show that QED 5 and QED 6 do obey the mean-field type scaling, even in the quenched planar approximation. Thus the essential singularity type scaling in the cutoff QED is considered to be possible only when D=4 under the quenched planar approximation.

scholarly journals PIECEWISE-PLANAR APPROXIMATION OF LARGE 3D DATA AS GRAPH-STRUCTURED OPTIMIZATION

2019 ◽  
Vol IV-2/W5 ◽  
pp. 365-372 ◽  
Author(s):  
S. Guinard ◽  
L. Landrieu ◽  
L. Caraffa ◽  
B. Vallet
Keyword(s):  
Approximation Error ◽  
Region Growing ◽  
Point Clouds ◽  
Approximation Problem ◽  
Planar Approximation ◽  
Computation Efficiency ◽  
Planar Structures ◽  
3D Data ◽  
Working Set ◽  
Structured Optimization

<p><strong>Abstract.</strong> We introduce a new method for the piecewise-planar approximation of 3D data, including point clouds and meshes. Our method is designed to operate on large datasets (e.g. millions of vertices) containing planar structures, which are very frequent in anthropic scenes. Our approach is also adaptive to the local geometric complexity of the input data. Our main contribution is the formulation of the piecewise-planar approximation problem as a non-convex optimization problem. In turn, this problem can be efficiently solved with a graph-structured working set approach. We compare our results with a state-of-the-art region-growing-based segmentation method and show a significant improvement both in terms of approximation error and computation efficiency.</p>

Gravité quantique et modèles de matrices

1991 ◽  
Vol 69 (7) ◽  
pp. 837-854 ◽  
Author(s):  
David Sénéchal
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Gravity ◽  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Continuum Limit ◽  
Two Dimensional ◽  
Planar Approximation ◽  
Conformal Field ◽  
The Matrix

A review of the main results recently obtained in the study of two-dimensional quantum gravity is offered. The analysis of two-dimensional quantum gravity by the methods of conformal field theory is briefly described. Then the treatment of quantum gravity in terms of matrix models is explained, including the notions of continuum limit, planar approximation, and orthogonal polynomials. Correlation fonctions are also treated, as well as phases of the matrix models.

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