Endpoint resolvent estimates for compact Riemannian manifolds

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.Comment: 14 page

Concerning LpL^p resolvent estimates for simply connected manifolds of constant curvature

We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas restriction theorem.Comment: 25 pages, 2 figure

Directed Random Market: the equilibrium distribution

We find the explicit expression for the equilibrium wealth distribution of the Directed Random Market process, recently introduced by Mart\'inez-Mart\'inez and L\'opez-Ruiz, which turns out to be a Gamma distribution with shape parameter $\frac{1}{2}$. We also prove the convergence of the discrete-time process describing the evolution of the distribution of wealth to the equilibrium distribution

[Review of] Ellen Carol DuBois and Vicki L. Ruiz, eds. Unequal Sisters: A Multicultural Reader in U. S. Women\u27s History

Edited by Ellen C. DuBois and Vicki L. Ruiz, two respected historians, Unequal Sisters: A Multicultural Reader in U.S. Women\u27s History is a welcome response to the call for a more complex approach to women\u27s history. Central to this approach are the integration of women of color into women\u27s history and a definition of community that reflects both conflict and concord

Conjugated Polymers for Organic Electronics: Structural and Electronic Characteristics

The use of organic materials to design electronic devices has actually presented a broad interest for because they constitute an ecological and suitable resource for our current "electronic world". These materials provide several advantages (low cost, light weight, good flexibility and solubility to be easily printed) that cannot be afforded with silicium. They can also potentially interact with biological systems, something impossible with inorganic devices. Between these materials we can include small molecules, polymers, fullerenes, nanotubes, graphene, other carbon-based molecular structures and hybrid materials. Actually these materials are being used to build electronic structures into electronic devices, like organic light-emitting diodes (OLEDs), organic solar cells (OSCs), and organic field-effect transistors (OFETs), constituting and already commercial reality. Some of them are used on a widespread basis1, and are the focus of some recent researches in molecules2,3 and polymers4-6 suitable for these purposes. In this study we analyze the electronic and molecular characteristics of some different π-conjugated structures in order to evaluate their potential as semiconducting materials for organic electronics. For this purpose we focus on the study of conjugated polymers with different backbones configurations: (i) donor-acceptor configuration, (ii) 1D lineal or 2D branched conjugated backbones, and (iii) encapsulated polymers. To achieve this goal, we use a combined experimental and theoretical approach that includes electronic spectroscopies (i.e., absorption, emission and microsecond transient absorption), vibrational Raman spectroscopy and DFT calculations. These structural modifications are found to provoke a strong impact on the HOMO and LUMO levels and the molecular morphology, and, consequently, on their suitability as semiconductors in organic electronic applications.References 1. S. R. Forrest, M. E. Thompson. Chem. Rev., 2007, 107, 923 2. R. C. González-Cano, G. Saini, J. Jacob, J. T. López Navarrete, J. Casado and M. C. Ruiz Delgado. Chem. Eur. J. 2013, 19, 17165 3. J. L. Zafra, R. C. González-Cano, M. C. Ruiz Delgado, Z. Sun, Y. Li, J. T. López Navarrete, J. Wu and J. Casado. J. Chem. Phys. , 2014, 140, 054706 4. M. Goll, A. Ruff, E. Muks, F. Goerigk, B. Omiecienski, I. Ruff, R. C. González-Cano, J. T. López Navarrete, M. C. Ruiz Delgado, S. Ludwigs. Beilstein J. Org. Chem., 2015, 11, 335. 5. D. Herrero-Carvajal, A. de la Peña, R. C. González-Cano, C. Seoane, J. T. López Navarrete, J. L. Segura, J. Casado, M. C. Ruiz Delgado, J. Phys. Chem. C, 2014, 118, 9899. 6. M. Scheuble, Y. M. Gross, D. Trefz, M. Brinkmann, J. T. López Navarrete, M. C. Ruiz Delgado, and S. Ludwigs, Macromolecules, 2015, 48, 7049.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech