Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

Explicit Bézier control net of a PDE surface

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface. In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.The work was partially supported by Spanish Ministry of Economy and Competitiveness DGICYT grant MTM2015-64013

Bézier surfaces with prescribed diagonals

The affine space of all tensor product Bézier patches of degree n × n with prescribed main diagonal curves is determined. First, the pair of Bézier curves which can be diagonals of a Bézier patch is characterized. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves or tangent planes along boundary curves are also prescribed.Funding for open access charge: CRUE-Universitat Jaume

The infrared and molecular environment surrounding the Wolf-Rayet star WR130

We present a study of the molecular CO gas and mid/far infrared radiation arising from the environment surrounding the Wolf-Rayet (W-R) star 130. We use the multi-wavelength data to analyze the properties of the dense gas and dust, and its possible spatial correlation with that of Young Stellar Objects (YSOs). We use CO J=1-0 data from the FCRAO survey as tracer of the molecular gas, and mid/far infrared data from the recent WISE and Herschel space surveys to study the dust continuum radiation and to identify a population of associated candidate YSOs. The spatial distribution of the molecular gas shows a ring-like structure very similar to that observed in the HI gas, and over the same velocity interval. The relative spatial distribution of the HI and CO components is consistent with a photo-dissociation region. We have identified and characterized four main and distinct molecular clouds that create this structure. Cold dust is coincident with the dense gas shown in the CO measurements. We have found several cYSOs that lie along the regions with the highest gas column density, and suggest that they are spatially correlated with the shell. These are indicative of regions of star formation induced by the strong wind and ionization of the WR star.Comment: 15 pages, 12 figures, 6 Tables. Accepted for publication in MNRA

Understanding proof practices of pre-service mathematics teachers in geometry

In this work, we show the results of a research with pre-service mathematics secondary teachers about their Van Hiele level regarding the proof in Geometry. We observe three different profiles whose characteristics are described. These descriptions allow us to foresee certain differences when carrying out proof teaching in secondary school. The presence of a profile with a lower level than that assumed for some high school students stands out. The other two profiles show differences regarding the presence of some advanced proof strategies

Triggered star formation in a molecular shell created by a SNR?

We present a study of a new molecular shell, G126.1-0.8-14, using available multiwavelegth Galactic plane surveys and optical Gemini observations. A well defined shell-like structure is observed in the CO(1--0) line emission at (l,b) = (126.1, -0.8), in the velocity range --10.5 to --15.5 km/s. The HI, emission shows a region of low emissivity inside G126.1-0.8-14, while radio continuum observations reveal faint non-thermal emission possibly related to this shell. Optical spectra obtained with Gemini South show the existence of B-type stars likely to be associated with G126.1-0.8-14. An estimate of the stellar wind energy injected by these stars show that they alone can not be able to create such a structure. On the other hand, one supernova explosion would provide enough energy to generate the shell. Using the MSX, IRAS, and WISE Point Source Catalogues we have found about 30 young stellar objects candidates, whose birth could have been triggered by the expansion of G126.1-0.8-14. In this context, Sh2-187 could be a consequence of the action on its surroundings of the most massive (and thus most evolve) of the stars formed by the expanding molecular shell.Comment: 13 pages, 10 figures, accepted in MNRA

Exponential Perturbative Expansions and Coordinate Transformations

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation

Trabajando la demostración con profesorado de secundaria en formación

Trabajar la demostración en el aula es interesante no sólo por su importancia desde el punto de vista del quehacer matemático, sino por su contribución a la comprensión de los conceptos matemáticos involucrados (Hanna, 1995). Esta necesidad ha sido recogida de forma desigual en los distintos planes de estudios españoles desde 1934 (Ibañes & Ortega, 2002). La LOMCE establece el aprendizaje de la demostración como obligatorio y transversal a todos los contenidos. Sin embargo, los contenidos relativos a la demostración se presentan en forma de lista sin un orden lógico que ayude al profesor a planificar su enseñanza. Así, surge el interés de llevar a cabo actividades de formación de profesorado de secundaria relacionadas con la demostración, que involucren tanto aspectos relativos a la práctica de la demostración, como a su enseñanza y aprendizaje