Let (V, M) be a valuation domain which is distinct from its quotient field K , and let π: V → V/M be the canonical surjection. Let D be a subring of V/M . It is proved that the pullback R := π^{− 1}(D) has the property that V (and hence each integral overring of R ) is a small R -submodule of K . Applications include all classical D + M constructions and locally pseudo-valuation domains.

It is proved that there exist no simple finite-dimensional Filippov superalgebras of type B(0, n) over an algebraically closed field of characteristic 0.

We prove the existence of a positive solution for singular elliptic problems of the type − Δu = α(x)u^{−β} + λf(x, u) in Ω , u_{| ∂Ω} = 0 , where β is any positive real number.

We introduce a set of quadratic forms for the solutions of the Liouville equation w_{tt}+ λ^2a(t)w = 0 . From these forms we derive estimates for the wave equation u_{tt} − a(t)Δu = 0 and then prove the global solvability for the Kirchhoff equation in suitable classes of not necessarily smooth or small initial data.

We describe a general method, based on a Lyapunov–Schmidt reduction and perturbative techniques, recently used by the authors to find periodic and quasi-periodic solutions both in finite and in infinite dimensional hamiltonian systems. We also illustrate some concrete applications to celestial mechanics and nonlinear wave equation.

À partir de la décomposition en base b(X) (un polynôme non constant), nous donnons un critère d'irréductibilité des polynômes en deux variables. Puis nous étudions un problème de spécialisation qui a un lien avec le théorème d'irréductibilité de Hilbert. Starting from the decomposition in base b(X) (a nonconstant polynomial), we give an irreducibility criterion of the bivariate polynomials. Moreover we study a problem of specialization which has a link with Hilbert’s irreducibility theorem.

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.

We consider the Cauchy problem for nonlinear Schrödinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the H1-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly – no matter how large it is, unless the initial datum decays sufficiently fast at infinity.

Let A be a reduced abelian p -group with a nice subgroup G . It is proved that if A/G is simply presented, then A is simply presented precisely when G is strongly simply presented in A . Moreover, the same type theorem for the class of ℵ_1 - Σ -cyclic p -groups is also established without the niceness of G in A . Some analogous assertions for other exotic sorts of abelian groups are also considered. The results obtained strengthen previous results due to J. A. Dieudonné (Portugal. Math., 1952), P. D. Hill (Trends in Math., 1999), and some other authors.

The paper deals with the problem of existence and calculation of solutions to Lax equations that define finite-dimensional integrable systems. The method presented in the paper is based on Wiener–Hopf factorization and related Riemann–Hilbert problems on Riemann surfaces. The idea behind the method was first proposed by Semenov–Tian–Shansky but, to the authors' knowledge, is here applied, for the first time, in an infinite dimensional setting. The method dealt with in the paper enables one to analyse the global existence of solutions which seems more difficult by other methods. An example of a dynamical system associated with an elliptic curve is completely worked out in the paper.