Numerical solution of conservative finite-dimensional stochastic Schrodinger equations

The paper deals with the numerical solution of the nonlinear Ito stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrodinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.Comment: Published at http://dx.doi.org/10.1214/105051605000000403 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schroedinger equations

A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schr\"odinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, weaker than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups.Comment: 16 page

Regional and farm specialisation in Spanish agriculture before and after integration in the European Union

In this paper, we study the evolution of agricultural product specialisation at farm and county level from 1979 to 1997 in Spain, thus covering all the stages of the gradual implementation of the Common Agricultural Policy. We use a multiproduct version of Theil and Finizza's index of segregation that allows us to decompose farm product specialisation into county specialisation with respect to the national level, i.e., the usual measure of regional specialisation, and farm specialisation within counties. Our results confirm the importance of increasing regional specialisation but also highlight that trends of farm specialisation within counties have varied across large agricultural areas. In particular, regions more specialised in export-oriented products seem to have speeded regional specialisation

REGIONAL AND FARM SPECIALISATION IN SPANISH AGRICULTURE BEFORE AND AFTER INTEGRATION IN THE EUROPEAN UNION

In this paper, we study the evolution of agricultural product specialisation at farm and county level from 1979 to 1997 in Spain, thus covering all the stages of the gradual implementation of the Common Agricultural Policy. We use a multiproduct version of Theil and Finizza's index of segregation that allows us to decompose farm product specialisation into county specialisation with respect to the national level, i.e., the usual measure of regional specialisation, and farm specialisation within counties. Our results confirm the importance of increasing regional specialisation but also highlight that trends of farm specialisation within counties have varied across large agricultural areas. In particular, regions more specialised in export-oriented products seem to have speeded regional specialisation.

Antimagic Labelings of Caterpillars

A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic.Comment: 13 pages, 4 figure

Experimentación y conjetura en el aula

El poder de la tecnología informática permite, entre otras cosas, potenciar la exploración dirigida al descubrimiento y formulación de conjeturas en un ambiente dinámico en el aula, que supera la praxis tradicional centrada en la algorítmica y la manipulación simbólica