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Criminals are common to all societies. To fight against them the community takes different security measures as, for example, to bring about a police. Thus, crime causes a depletion of the common wealth not only by criminal acts but also because the cost of hiring a police force. In this paper, we present a mathematical model of a criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-of-criminals steady state is unstable under small perturbations of a certain socio-economical context. Finally, we compare two alternative strategies to control crime: (i) enhancing police efficiency, either by enlarging its size or by updating its technology, against (ii) either reducing criminal appealing or promoting social classes at risk.

We prove the existence of an appropriate function (very weak solution) u in the Lorentz space L(N') (,infinity)(Omega), N' = (N)(N - 1) satisfying Lu - Vu + g (x, u, del u) = mu in Omega an open bounded set of R(N), and u = 0 on partial derivative Omega in the sense that (u, L phi)(0) - (Vu, phi)(0) + (g(., u, del u),phi)(0) = mu(phi), for all phi is an element of C(c)(2)(Omega). The potential V

We show that the spectral radius ρ ( D ) of a digraph D with n vertices and c 2 closed walks of length 2 satisfies ρ ( D ) ⩾ c 2 n . Moreover, equality occurs if and only if D is the symmetric digraph associated to a c 2 n -regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.

We give mild conditions for the existence of optimal solutions for a Markov decision problem with average cost, under m constraints of the same kind, in Borel actions and states spaces. Moreover, there is an optimal policy that is a convex combination of at most m + 1 deterministic policies.

3er Premio de Investigación Fundación Ramiro Melendreras, La Confederación Española de Centros de Investigación Matemática y Estadística (CECIME) albergó hasta 1989 a los matemáticos del CSIC. Posteriomente se integraría en el Instituto de Matemáticas y Física Fundamental (IMAFF)., Estudiamos los principales tipos de conceptos de óptimo considerados en problemas de optimización multiobjetivo, cuando la ordenación de alternativas se regula mediante un cono K convexo: soluciones K-maximales, débilmente K-maximales, fuertemente K-maximales, propiamente K-maximales. Damos caracterizaciones en problemas generales de optimización vectorial y condiciones suficientes en problemas de maximización de funciones de valor vectoriales y escalares, particularizando después al caso de conos poliédricos y al de ortantes no negativos. Se concluye con algunas aplicaciones y cuestiones prácticas., We study the main kinds of optimum in multiobjective optimization problems, when alternatives are ordered by means of a convex cone K: K-maximal, weakly K-maximal, strongly K-maximal and properly K-maximal solutions. We give several characterisations and sufficient conditions, concluding with some applications.