Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

A simple example is considered of Hill's equation x" + (a^2 + bp(t))x = 0, where the forcing term p, instead of periodic, is quasi periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitrary b, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

Effect of management spontaneous cover crop on rosy apple aphid, green apple aphid and their natural enemies in an apple organic orchard

Cover crops have been reported as shelters for pest but also for natural enemies. Nevertheless, there is not agreement about their influence on pest presence on apple trees. An experiment was conducted in 2004-2006 in an IRTA-Estació Experimental de Lleida organic apple orchard located in Les Borges Blanques (Lleida, Spain) in order to evaluate the influence of cover crops on the presence of pest and natural enemies on apple trees. Three cover crops were tested and compared to a bare soil: (1) spontaneous cover crop where Plantago lanceolata L. was the most abundant specie, (2) no-mowed spontaneous cover crop, (3) spontaneous cover crop mowed when weeds were 30 cm tall. Cover crops tested affected neither the presence the rosy apple aphid nor aphid natural enemies. Nevertheless, the green apple aphid was more abundant in bare soil trees than in cover crop trees

Choreographic Three Bodies on the Lemniscate

We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn^2(t)) parameterized by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/4, conserve the center of mass and the angular momentum, where x-hat and y-hat are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of square of the curvature, the product of distance and the sum of square of distance between bodies. We find that they satisfy the equation of motion under the potential energy sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij} -sum_{i}sqrt{3}/8 r_{i}^2, where r_{ij} the distance between the body i and j, and r_{i} the distance from the origin. The first term of the potential energies is the Newton's gravity in two dimensions but the second term is the mutual repulsive force or a repulsive force from the origin, respectively. Then, geometric construction methods for the positions of the choreographic three bodies are given

The degree/diameter problem in maximal planar bipartite graphs

The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D) and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version

The degree/diameter problem in maximal planar bipartite graphs

The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft

Toward a collective intelligence recommender system for education

The development of Information and Communication Technology (ICT), have revolutionized the world and have moved us into the information age, however the access and handling of this large amount of information is causing valuable time losses. Teachers in Higher Education especially use the Internet as a tool to consult materials and content for the development of the subjects. The internet has very broad services, and sometimes it is difficult for users to find the contents in an easy and fast way. This problem is increasing at the time, causing that students spend a lot of time in search information rather than in synthesis, analysis and construction of new knowledge. In this context, several questions have emerged: Is it possible to design learning activities that allow us to value the information search and to encourage collective participation?. What are the conditions that an ICT tool that supports a process of information search has to have to optimize the student's time and learning? This article presents the use and application of a Recommender System (RS) designed on paradigms of Collective Intelligence (CI). The RS designed encourages the collective learning and the authentic participation of the students. The research combines the literature study with the analysis of the ICT tools that have emerged in the field of the CI and RS. Also, Design-Based Research (DBR) was used to compile and summarize collective intelligence approaches and filtering techniques reported in the literature in Higher Education as well as to incrementally improving the tool. Several are the benefits that have been evidenced as a result of the exploratory study carried out. Among them the following stand out: • It improves student motivation, as it helps you discover new content of interest in an easy way. • It saves time in the search and classification of teaching material of interest. • It fosters specialized reading, inspires competence as a means of learning. • It gives the teacher the ability to generate reports of trends and behaviors of their students, real-time assessment of the quality of learning material. The authors consider that the use of ICT tools that combine the paradigms of the CI and RS presented in this work, are a tool that improves the construction of student knowledge and motivates their collective development in cyberspace, in addition, the model of Filltering Contents used supports the design of models and strategies of collective intelligence in Higher Education.Postprint (author's final draft

Study of the derivative expansions for the nuclear structure functions

We study the convergence of the series expansions sometimes used in the analysis of the nuclear effects in Deep Inelastic Scattering (DIS) proccesses induced by leptons. The recent advances in statistics and quality of the data, in particular for neutrinos calls for a good control of the theoretical uncertainties of the models used in the analysis. Using realistic nuclear spectral functions which include nucleon correlations, we find that the convergence of the derivative expansions to the full results is poor except at very low values of $x$