scholarly journals Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

2003 ◽  
Vol 64 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Lee-Peng Teo>
Keyword(s):  
Analytic Functions ◽  
Integrable Hierarchies ◽  
Tau Functions

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

scholarly journals Properties of the algebra Psd related to integrable hierarchies

2020 ◽  
pp. 183-195
Author(s):  
Gerard F. Helminck ◽  
Elena A. Panasenko
Keyword(s):  
Integrable Hierarchies ◽  
Pseudodifferential Operators ◽  
Darboux Transformations ◽  
Fredholm Determinants

In this paper we discuss and prove various properties of the algebra of pseudodifferential operators related to integrable hierarchies in this algebra, in particular the KPhierarchy and its strict version. Some explain the form of the equations involved or giveinsight in why certain equations in these systems are combined, others lead to additionalproperties of these systems like a characterization of the eigenfunctions of the linearizations ofthe mentioned hierarchies, the description of elementary Darboux transformations of bothhierarchies and the search for expressions in Fredholm determinants for the constructedeigenfunctions and their duals.

scholarly journals The Ultrametric Corona Problem and spherically complete fields

2010 ◽  
Vol 53 (2) ◽  
pp. 353-371 ◽  
Author(s):  
Alain Escassut
Keyword(s):  
Maximal Ideal ◽  
Analytic Functions ◽  
Unit Disc ◽  
Unique Continuation ◽  
Open Unit ◽  
Closed Field ◽  
Corona Problem ◽  
Open Unit Disc ◽  
Bounded Analytic Functions

AbstractLet K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ‘open’ unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).

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