scholarly journals Invariants of germs of analytic differential equations in the complex plane

2005 ◽  
Vol 77 (1) ◽  
pp. 1-11
Author(s):  
Leonardo M. Câmara
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Straightforward Manner ◽  
Dicritical Singularities ◽  
Complete Set

We study the classification of germs of differential equations in the complex plane giving a complete set of analytic invariants determining the analytic type of the underlying foliation. In particular we answer in affirmative a conjecture of S. Voronin, and generalize some previous results about dicritical singularities in a straightforward manner. Such problem has its origins in a conjecture proposed by R. Thom in the mid-1970s.

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 LOCAL MONODROMY OF p-ADIC DIFFERENTIAL EQUATIONS: AN OVERVIEW

2005 ◽  
Vol 01 (01) ◽  
pp. 109-154 ◽  
Author(s):  
KIRAN S. KEDLAYA
Keyword(s):  
Differential Equations ◽  
Local Monodromy ◽  
Expository Article

This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent ∇-modules, the Christol–Mebkhout construction of the ramification filtration, and the Christol–Dwork Frobenius antecedent theorem. We also briefly discuss the p-adic local monodromy theorem without proof.

scholarly journals Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions

2015 ◽  
Vol 28 (1) ◽  
pp. 51-76 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Short Introduction ◽  
The Core ◽  
Boolean Equation ◽  
General Difference ◽  
Derivatives Of

The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

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