Recently some extensions of the classical Darboux integrability theory to autonomous and non-autonomous polynomial vector fields have been done. The classical Darboux integrability theory and its recent extensions are based on the existence of algebraic invariant hypersurfaces. However the algebraicity of the invariant hypersurfaces is not necessary and the unique necessary condition is the alg...

Abstract. We consider the integrability of φ(f∗) for various maximal functions f∗ and various increasing functions φ. We show that for some of the standard maximal functions arising in harmonic analysis and ergodic theory, there is never integrability of φ(f∗) for all Lebesgue integrable functions f except in cases where the growth of φ is slow enough so that the integrability follows from the ...

In this paper we shall primarily study polynomial integrability of the differential system ẋ = −y + Pn(x, y), ẏ = x + Qn(x, y), n = 2, 3, where Pn and Qn are homogeneous polynomials of degree n. By taking various yet very elementary ways, we not only straightforwardly find the necessary and sufficient integrability conditions but also explicitly present the corresponding polynomial first integr...

We answer a question posed in [12] on exponential integrability of functions of restricted n-energy. We use geometric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.

We give a definition of uniform PU-integrability for a sequence of μ-measurable real functions defined on an abstract metric space and prove that it is not equivalent to the uniform μ-integrability.

We have succeeded in applying Ziglin’s test on non-integrability to nonhomogeneous nonlinear lattices (Fermi-Pasta-Ulam lattices). By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lamé type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the nonexistence of additional analyt...

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and coLegendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the exis...

Two important notions of integrability for discrete mappings are algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities whereas singularity confinement is associated with the local analysis of singularities. In this paper, the relationship between these two notions i...