Collectanea Mathematica

Let (Ω, Σ, μ) be a σ-finite measure space and let $$\mathcal{L}(X,Y)$$ stand for the space of all bounded linear operators between Banach spaces (X; ‖ • ‖ X ) and (Y; ‖ • ‖ Y ). We study the problem of integral representation of linear operators from an Orlicz-Bochner spaceL ϕ(μ,X) toY with respect to operator measures $$m : \sum \to \mathcal{L}(X,Y) $$ . It is shown that a linear operatorT:L ϕ (μ,X) →Y has the integral representationT(f = ∫Ω f(ω)dm with respect to a ϕ*-variationally μ-continuous operator measurem if and only ifT is (γϕ ‖ • ‖ Y )-continuous, where γϕ stands for a natural mixed topology onL ϕ (μ,X). As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.

The purpose of this paper is to prove a new topological fact about the Poincaré and related groups. IfG is a group, say that G is an algebraically determined Polish (i.e., complete separable metric topological) group if, wheneverH is a Polish group and ϕ:H → G is an algebraic isomorphism, then ϕ is a topological isomorphism. The proper Lorentz group, the proper orthochronous Lorentz group and the Heisenberg group are examples of Polish groups that are not algebraically determined. On the other hand it will be shown that the Lorentz group, the orthochronous Lorentz group and the Poincaré group and the other closely associated semi-direct products are algebraically determined Polish groups.

In the present paper, we study the existence of a mild solution of a fractional order nonlocal differential inclusion with impulsive condition in a Banach space E. We obtain the sufficient condition for the existence of the mild solution by using a fixed point theorem for multi-valued operators due to Dhage and resolvent semigroup theory with approximate techniques.

LetX be the base locus of a linear system L of hypersurfaces in ℙ r (ℂ). In this paper it is showed that the existence of linear syzygies for the ideal ofX has strong consequences on the fibres of the rational map associated toL. The case of hyperquadrics is particularly addressed. The results are applied to the study of rational maps and to the Perazzo’s map for cubic hypersurfaces.

Let $${\mathcal {M}}$$ be a smooth complex projective variety and let $${\mathcal {L}}$$ be a line bundle on it. Rays-positive manifolds, namely pairs $${(\mathcal {M},\mathcal {L})}$$ such that $${\mathcal {L}}$$ is numerically effective and $${\mathcal {L}\cdot R >0 }$$ for all extremal rays R on $${\mathcal {M}}$$ , are studied. Several illustrative examples and some applications are provided. In particular, projective varieties with crepant singularities and of small degree with respect to the codimension are classified, and the non-negativity of the sectional genus $${g(\mathcal {M},\mathcal {L})}$$ is proven, describing as well the pairs with $${g(\mathcal {M},\mathcal {L})=0,1}$$ .

The pseudo-parabolic regularization technique is applied to establish the well-posedness of local solutions for a generalized Novikov equation in the Sobolev space $$H^s(R)$$ with $$s>\frac{3}{2}$$ . The existence of local weak solutions for the equation in the lower order Sobolev space $$H^s$$ with $$1\le s\le \frac{3}{2}$$ is also investigated.

Let X be a Banach function space over a nonatomic probability space. We associate with X the weak space $${\mathrm {w}}\text {-}{X}$$ , which is a quasi-Banach space defined in a natural way. We give necessary and sufficient conditions for Doob’s inequality in $${\mathrm {w}}\text {-}{X}$$ to be valid, and necessary and sufficient conditions for Burkholder’s inequality in $${\mathrm {w}}\text {-}{X}$$ to be valid.