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On algebraic sets invariant by one-dimensional foliations on 𝐂 P ( 3 )

Marcio G. Soares (1993)

Annales de l'institut Fourier

We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on C P ( 2 ) without algebraic solutions to the case of foliations by curves on C P ( 3 ) . We give an example of a foliation on C P ( 3 ) with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.

On algebraic solutions of algebraic Pfaff equations

Henryk Żołądek (1995)

Studia Mathematica

We give a new proof of Jouanolou’s theorem about non-existence of algebraic solutions to the system ẋ = z s , ẏ = x s , ż = y s . We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.

On aliases of differential equations

Rutherford Aris, Gianni Astarita (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Theory of chemical reactions in complex mixtures exhibits the following problem. Single reacting species follow an intrinsic kinetic law. However, the observable quantity, which is a mean of individual concentrations, follows a different law. This one is called «alias» of intrinsic kinetics. In this paper the phenomenon of alias of uniform families of differential equations is discussed in general terms.

On almost periodicity defined via non-absolutely convergent integrals

Dariusz Bugajewski, Adam Nawrocki (2025)

Czechoslovak Mathematical Journal

We investigate some properties of the normed space of almost periodic functions which are defined via the Denjoy-Perron (or equivalently, Henstock-Kurzweil) integral. In particular, we prove that this space is barrelled while it is not complete. We also prove that a linear differential equation with the non-homogenous term being an almost periodic function of such type, possesses a solution in the class under consideration.

On almost specification and average shadowing properties

Marcin Kulczycki, Dominik Kwietniak, Piotr Oprocha (2014)

Fundamenta Mathematicae

We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...

On almost-Riemannian surfaces

Roberta Ghezzi (2010/2011)

Séminaire de théorie spectrale et géométrie

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its...

On an antiperiodic type boundary value problem for first order linear functional differential equations

Robert Hakl, Alexander Lomtatidze, Jiří Šremr (2002)

Archivum Mathematicum

Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem u ' ( t ) = ℓ ( u ) ( t ) + q ( t ) , u ( a ) + λ u ( b ) = c are established, where ℓ : C ( [ a , b ] ; R ) → L ( [ a , b ] ; R ) is a linear bounded operator, q ∈ L ( [ a , b ] ; R ) , λ ∈ R + , and c ∈ R . The question on the dimension of the solution space of the homogeneous problem u ' ( t ) = ℓ ( u ) ( t ) , u ( a ) + λ u ( b ) = 0 is discussed as well.

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