scholarly journals On classification problems in the theory of differential equations: Algebra + geometry

2018 ◽  
Vol 103 (117) ◽  
pp. 33-52
Author(s):  
Pavel Bibikov ◽  
Alexander Malakhov
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Geometric Theory ◽  
Differential Invariants ◽  
Classification Problems ◽  
Cremona Group ◽  
Right Hand ◽  
Global Classification ◽  
Birational Automorphisms

We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs y?? = F(x, y). In the first part of the paper we consider the case of smooth right-hand side F. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs y?? = F(x, y). In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations y?? = F(x, y) with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C2, which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others.

scholarly journals Projective structures of elliptic differential operators

1985 ◽  
Vol 99 ◽  
pp. 111-130 ◽  
Author(s):  
Kazushige Ueno
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Operators ◽  
Point Of View ◽  
Differential Invariants ◽  
Projective Structures ◽  
General Study ◽  
Elliptic Differential Operators ◽  
The Given

In the study of differential equations from the standpoint of the automorphism pseudogroups, the differential invariants of the pseudogroups play an important role.A general study of pseudogroups and their differential invariants originated with Sophus Lie. He applied his study to the classification of ordinary and partial differential equations. So as to study differential equations from his point of view, it is very important to write the given differential equation by the differential invariants of the automorphism pseudogroup. That is to say, the geometric structure of a differential equation is contained in the expression of the equation by its differential invariants.

scholarly journals A TOPOLOGICAL CLASSIFICATION OF DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECT

1994 ◽  
Vol 25 (1) ◽  
pp. 15-27
Author(s):  
D. D. BAINOV ◽  
S. I. KOSTADINOV ◽  
NGUYEN VAN MINH ◽  
P. P. ZABREIKO
Keyword(s):  
Differential Equations ◽  
Exponential Dichotomy ◽  
Linear Part ◽  
Nonlinear Perturbation ◽  
Topological Classification ◽  
Impulse Effect ◽  
Right Hand ◽  
The Right

The present paper is concerned with the problem of topological clas- sification of differential equations with impulse effect. Under the assumption that the linear part of the right-hand side of the equation considered has an exponential dichotomy and the nonlinear perturbation is small enough it is proved that for the underlying equations there exist $N + 1$ types topologicall'y,,different from one another.

scholarly journals Limit Behavior of Solutions of Ordinary Linear Differential Equations

1994 ◽  
Vol 1 (3) ◽  
pp. 315-323
Author(s):  
František Neuman
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Linear Differential Equations ◽  
Limit Behavior ◽  
Limit Sets ◽  
Equivalent Linear ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Abstract A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

scholarly journals Improving Land Cover Classification Using Genetic Programming for Feature Construction

2021 ◽  
Vol 13 (9) ◽  
pp. 1623
Author(s):  
João E. Batista ◽  
Ana I. R. Cabral ◽  
Maria J. P. Vasconcelos ◽  
Leonardo Vanneschi ◽  
Sara Silva
Keyword(s):  
Land Cover ◽  
Genetic Programming ◽  
Satellite Images ◽  
State Of The Art ◽  
Binary Classification ◽  
Feature Construction ◽  
Classification Problems ◽  
Construction Methods ◽  
Box Models

Genetic programming (GP) is a powerful machine learning (ML) algorithm that can produce readable white-box models. Although successfully used for solving an array of problems in different scientific areas, GP is still not well known in the field of remote sensing. The M3GP algorithm, a variant of the standard GP algorithm, performs feature construction by evolving hyperfeatures from the original ones. In this work, we use the M3GP algorithm on several sets of satellite images over different countries to create hyperfeatures from satellite bands to improve the classification of land cover types. We add the evolved hyperfeatures to the reference datasets and observe a significant improvement of the performance of three state-of-the-art ML algorithms (decision trees, random forests, and XGBoost) on multiclass classifications and no significant effect on the binary classifications. We show that adding the M3GP hyperfeatures to the reference datasets brings better results than adding the well-known spectral indices NDVI, NDWI, and NBR. We also compare the performance of the M3GP hyperfeatures in the binary classification problems with those created by other feature construction methods such as FFX and EFS.

scholarly journals Global classification of NPGCs

2014 ◽  
Vol 13 (12) ◽  
pp. 888-888
Author(s):  
Sarah Crunkhorn
Keyword(s):  
Global Classification
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