Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.47 (2015)

Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in.

We present some existence and multiplicity results for positive solutions to the Dirichlet problem associated with; under suitable conditions on the nonlinearity g(u)and thew eight function a(x): The assumptions considered are related to classical theorems about positive solutions to a sublinear elliptic equation due to Brezis-Oswald and Brown-Hess.

Some discrete inequalities of Jensen type for X-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

We use Fink’s identity to obtain new identities related to generalizations of Steffensen’s inequality. Ostrowski-type inequalities related to these generalizations are also given. Using inequalities for the Cebysev functional we obtain bounds for these identities. Further, we use these identities to obtain new generalizations of Steffensen’s inequality for n-convex functions. Finally, we use these generalizations to construct a linear functional that aenerates exvonentiallv convex functions.

In this note we present a notion of fundamental scheme for Cohen-Macaulay, order I, irreducible congruences of lines. We show that such a congruence is formed by the k-secant lines to its fundamental scheme for a number k that we call the secant, index of the congruence. if the fundamental scheme X is a smooth connected variety in FN, then k = (N — l)/(c — 1) (where c is the codimension of X) and X comes equipped with a special tangency divisor cut out by a virtual hypersurface of degree k — 2 (to be precise, linearly equivalent to a section by an hypersurface of degree (k — 2) without being cut by one). This is explained in the main theorem of this paper. This theorem is followed by a complete classification of known locally Cohen-Macaulay order 1 congruences of lines with smooth fundamental scheme. To conclude we remark that according to Zak’s classification of Severi Varieties and Hartshome conjecture for low codimension varieties, this classification is complete.