Finite-Dimensional Theory—Ground Zero: The Homogenous Case

2014 ◽  
pp. 123-158
Keyword(s):  
Dimensional Theory ◽  
Ground Zero ◽  
Finite Dimensional ◽  
Homogenous Case

Minimax differential game with delay

2020 ◽  
pp. 359-369
Author(s):  
Arkadii V. Kim ◽  
Gennady A. Bocharov
Keyword(s):  
Dynamic Programming ◽  
Differential Games ◽  
Differential Game ◽  
Programming Method ◽  
Dynamic Programming Method ◽  
Mixed Strategies ◽  
Dimensional Theory ◽  
Analysis Methodology ◽  
Finite Dimensional ◽  
Finite Dimensional Case

The paper considers a minimax positional differential game with aftereffect based on the i-smooth analysis methodology. In the finite-dimensional (ODE) case for a minimax differential game, resolving mixed strategies can be constructed using the dynamic programming method. The report shows that the i-smooth analysis methodology allows one to construct counterstrategies in a completely similar way to the finite-dimensional case. Moreover as it is typical for the use of i-smooth analysis, in the absence of an aftereffect, all the results of the article pass to the corresponding results of the finite-dimensional theory of positional differential games.

Regular types in nonmultidimensional ω-stable theories

1984 ◽  
Vol 49 (3) ◽  
pp. 880-891
Author(s):  
Anand Pillay
Keyword(s):  
Dimensional Theory ◽  
Finite Dimensional

AbstractWe define a hierarchy on the regular types of an ω-stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that any resplendent model of an ω-stable finite-dimensional theory is saturated.

Stability of universal unfoldings: a correction

1981 ◽  
Vol 90 (1) ◽  
pp. 195-196 ◽  
Author(s):  
R. J. Magnus
Keyword(s):  
Banach Spaces ◽  
Bifurcation Theory ◽  
Singularity Theory ◽  
Stability Theorem ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Potential Case ◽  
The Stability ◽  
Imperfect Bifurcation

The author is grateful to Les Lander for pointing out an error in the stability section of (1). In fact Theorems 5 and 7 are incorrect. Recently Arkeryd proved a stability theorem for the infinite-dimensional case in the context of the imperfect bifurcation theory of Golubitsky and Schaeffer(3). In his result finitely many derivatives are controlled, the number depending on the codimension of the singularity unfolded. In this note we shall present a stability theorem involving the determinacy of the singularity. The context is the parameter-free potential case, that is, catastrophe theory. The proof is without recourse to the finite-dimensional results, and the theorem concludes an account of a part of singularity theory in Banach spaces, in which the author has tried to use as little as possible of the finite-dimensional theory (1, 2).

scholarly journals Differential Galois theory of infinite dimension

1996 ◽  
Vol 144 ◽  
pp. 59-135 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
19Th Century ◽  
Galois Theory ◽  
Infinite Dimension ◽  
Differential Galois Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The 19Th Century

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

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