Pure Point spectrum for measure dynamical systems on locally compact Abelian groups

We show equivalence of pure point diffraction and pure point dynamical spectrum for measurable dynamical systems build from locally finite measures on locally compact Abelian groups. This generalizes all earlier results of this type. Our approach is based on a study of almost periodicity in a Hilbert space. It allows us to set up a perturbation theory for arbitrary equivariant measurable perturbations.Comment: 22 page

Completability and optimal factorization norms in tensor products of Banach function spaces

[EN] Given s-finite measure spaces ( 1, 1, mu 1) and ( 2, 2, mu 2), we consider Banach spaces X1(mu 1) and X2(mu 2), consisting of L0(mu 1) and L0(mu 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X1(mu 1). p X2(mu 2) is continuously included in the metric space of measurable functions L0(mu 1. mu 2). In particular, we prove that the elements of the completion of the projective tensor product of L p-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally showthat given a bounded linear operator T : X1(mu 1). p X2(mu 2). E (where E is a Banach space), a norm can be found for T to be bounded, which is ` minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.J. M. Calabuig and M. Fernandez-Unzueta were supported by Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernandez-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovacion (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sanchez Perez was supported by Ministerio de Economia, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.Calabuig, JM.; Fernández-Unzueta, M.; Galaz-Fontes, F.; Sánchez Pérez, EA. (2019). Completability and optimal factorization norms in tensor products of Banach function spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3513-3530. https://doi.org/10.1007/s13398-019-00711-7S351335301134Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, Vol 50, AMS (2002)Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)Bu, Q., Buskes, G., Kusraev, A.G.: Bilinear maps on products of vector lattices: a survey. In: Boulabiar, K., Buskes, G., Triki, A. (eds.) Positivity-Trends in Mathematics. Birkhäser Verlag AG, Basel, pp. 97–26 (2007)Buskes, G., Van Rooij, A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47–59 (2003)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108(2), 353–367 (2014)Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F., Sánchez-Pérez, E.A.: Equivalent norms in a Banach function space and the subsequence property. J. Korean Math. Soc. https://doi.org/10.4134/JKMS.j180682Curbera, G.P., Ricker, W.J.: Optimal domains for kernel operators via interpolation. Math. Nachr. 244, 47–63 (2002)Curbera, G.P., Ricker, W.J.: Vector measures, integration and applications. In: Positivity. Birkhäuser Basel, pp. 127–160 (2007)Gil de Lamadrid, J.: Uniform cross norms and tensor products. J. Duke Math. 32, 797–803 (1965)Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience Publishers Inc., New York (1958)Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94(3), 777–798 (1972)Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211(2), 87–106 (1974)Yew, K.L.: Completely $$p$$-summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362–1402 (2008)Kwapien, S., Pelczynski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34(1), 43–68 (1970)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)Milman, M.: Some new function spaces and their tensor products. Depto. de Matemática, Facultad de Ciencias, U. de los Andes, Mérida, Venezuela (1978)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Oper. Theory Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 579–591 (1984)Zaanen, A.C.: Integration. North-Holland Publishing Company, Amsterdam-New York (1967)Zaanen, A.C.: Riesz Spaces II. North-Holland Publishing Company, Amsterdam (1983

Close-packed dimers on the line: diffraction versus dynamical spectrum

The translation action of \RR^{d} on a translation bounded measure $\omega$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $\omega$, which is the carrier of the diffraction measure, live on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) in \cite{EM}. Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of `atomic' versus `molecular' spectrum suggests a way to come to a more general correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement

Equicontinuous delone dynamical systems

We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with nite local complexity the only equicontinuous systems are then shown to be the crystalline ones. On the other hand, within the class without nite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems which are not crystalline. Our results solve the problem posed by Lagarias whether a Delone set whose Dirac comb is strongly almost periodic must be crystalline

Mathematical diffraction of aperiodic structures

Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details

Topology Of Mappings In Locally Convex Topological Vector Spaces, Their Differentiation And Integration, And Application To Gradient Mappings.

PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/181634/2/0018604.pd

Contribuci?n de diferentes arreglos silvopastoriles a la conservaci?n de la biodiversidad, mediante la provisi?n de h?bitat y conectividad en el paisaje de la sub-cuenca del R?o Cop?n, Honduras

Tesis (M. Sc) -- CATIE, Turrialba (Costa Rica), 2008however, when based on field data, they serve as an important tool for making landscape scale management decisions, and for conservation planning in agricultural landscapes. Este trabajo tuvo como objetivos caacterizar el paisaje de la subcuenca del R?o Cop?n, Honduras en t?rminos de usos del suelo y m?tricas de composici?n estructural del paisaje. La elaboraci?n de modelos de calidad de h?bitat y conectividad funcional para los g?neros de aves Dendroica, Trogon e Icterus y evaluar la contribuci?n actual de los sistemas silvopastoriles a la provisi?n de h?bitat y conectividad. Iniciativas de conservaci?n dentro del paisaje deben incluir estrategias que busquen modificar la matriz productiva como la incorporaci?n de sistemas silvopastoriles en combinaci?n con estrategias que busquen conservar y expandir las ?reas naturales dentro de este paisaje. This objective of this project is to characterize land use and landscape metrics of the Copan River watershed and also model functional habitat and connectivity. We develop habitat quality and landscape connectivity models for three genera of avifauna, Trogon, Icterus and Dendroica, and we evaluate the contribution of current landuse, particularly silvopastoral systeMON to habitat provision and landscape connectivity. Species specific landscape models need to be validated in the fiel

Contribuci?n de diferentes arreglos silvopastoriles a la conservaci?n de la biodiversidad, mediante la provisi?n de h?bitat y conectividad en el paisaje de la sub-cuenca del R?o Cop?n, Honduras

Tesis (M. Sc) -- CATIE, Turrialba (Costa Rica), 2008however, when based on field data, they serve as an important tool for making landscape scale management decisions, and for conservation planning in agricultural landscapes. Este trabajo tuvo como objetivos caacterizar el paisaje de la subcuenca del R?o Cop?n, Honduras en t?rminos de usos del suelo y m?tricas de composici?n estructural del paisaje. La elaboraci?n de modelos de calidad de h?bitat y conectividad funcional para los g?neros de aves Dendroica, Trogon e Icterus y evaluar la contribuci?n actual de los sistemas silvopastoriles a la provisi?n de h?bitat y conectividad. Iniciativas de conservaci?n dentro del paisaje deben incluir estrategias que busquen modificar la matriz productiva como la incorporaci?n de sistemas silvopastoriles en combinaci?n con estrategias que busquen conservar y expandir las ?reas naturales dentro de este paisaje. This objective of this project is to characterize land use and landscape metrics of the Copan River watershed and also model functional habitat and connectivity. We develop habitat quality and landscape connectivity models for three genera of avifauna, Trogon, Icterus and Dendroica, and we evaluate the contribution of current landuse, particularly silvopastoral systeMON to habitat provision and landscape connectivity. Species specific landscape models need to be validated in the fiel