PCMO: Partial Classification from CNN-Based Model Outputs

2021 ◽  
pp. 150-163
Author(s):  
Jiarui Xie ◽  
Violaine Antoine ◽  
Thierry Chateau
Keyword(s):  
Partial Classification

scholarly journals Second-order PDEs in four dimensions with half-flat conformal structure

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190642
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. Kruglikov ◽  
V. Novikov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Partial Differential ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Partial Classification ◽  
Monge Ampere

We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.

C-Nodal Surfaces of Order Three

1983 ◽  
Vol 35 (1) ◽  
pp. 68-100
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Surface ◽  
Partial Classification ◽  
Differentiability Condition ◽  
Nodal Surfaces

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of synthetic surfaces of order three in [9]. While his theory resulted in a partial classification of a now larger class of surfaces, it was too general to permit a global description. In [1], we added a differentiability condition to Marchaud's definition. This resulted in a partial classification and description of surfaces of order three with exactly one singular point in [2]-[5]. In the present paper, we examine C-nodal surfaces and thus complete this survey.

scholarly journals The algebraic structure of relativistic wave equations

1978 ◽  
Vol 20 (4) ◽  
pp. 446-486 ◽  
Author(s):  
A. Cant ◽  
C. A. Hurst
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Mass Spectra ◽  
Wave Equations ◽  
Relativistic Wave Equations ◽  
Relativistic Wave ◽  
Partial Classification ◽  
The Family ◽  
Image Position

The algebraic structure of relativistic wave equations of the formis considered. This leads to the problem of finding all Lie algebrasLwhich contain the Lorentz Lie algebraso(3, 1) and also contain a “four-vector” αμa such anLgives rise to a family of wave equations. The simplest possibility is the Bhabha equations whereL≅so(5). Some authors have claimed that this is theonlyone, but it is shown that there are many other possibilities still in accord with physical requirements. Known facts about representations, along with Dynkin's theory of the embeddings of Lie algebras, are used to obtain a partial classification of wave equations. The discrete transformationsC, P, Tare also discussed, along with reality properties. Finally, a simple example of a family of wave equations based onL=sp(12) is considered in detail. Theso(3, 1) content and mass spectra are given for the low order members of the family, and the problem of causality is briefly discussed.

scholarly journals Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

2008 ◽  
Vol 23 (2) ◽  
pp. 137-162 ◽  
Author(s):  
M.J. Parker ◽  
I.N. Stewart ◽  
M.G.M. Gomes
Keyword(s):  
Partial Classification

scholarly journals Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 294
Author(s):  
Daniel López-Aguayo ◽  
Servando López Aguayo
Keyword(s):  
Lower Bound ◽  
Abelian Groups ◽  
Additive Group ◽  
Exact Formula ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Partial Classification ◽  
Open Questions ◽  
Odd Order

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

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