On Matrix-Valued Herglotz Functions

We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schr\"odinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanlinna-Riesz-Herglotz representation. In particular, we study a class of linear fractional transformations M_A(z) of a given n \times n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuos part of the measure associated to M_A(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self-adjoint finite-rank perturbations of self-adjoint operators, self-adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.Comment: LaTe

Cellulosic materials as natural fillers in starch-containing matrix-based films: a review

In this work, the different cellulosic materials, namely cellulose and lignin are analyzed. In addition, the starch-containing matrices (isolated starch and flour) reinforced with cellulosic materials to be used in packaging applications are described. Many efforts have been exerted to develop biopackaging based on renewable polymers, since these could reduce the environmental impact caused by petrochemical resources. Special attention has had the starch as macromolecule for forming biodegradable packaging. For these reasons, shall also be subject of this review the effect of each type of cellulosic material on the starch-containing matrix-based thermoplastic materials. In this manner, this review contains a description of films based on starch-containing matrices and biocomposites, and then has a review of cellulosic material-based fillers. In the same way, this review contains an analysis of the works carried out on starch-containing matrices reinforced with cellulose and lignin. Finally, the manufacturing processes of starch/cellulose composites are provided as well as the conclusions and the outlook for future works.Fil: Gutiérrez Carmona, Tomy José. Universidad Central de Venezuela; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Alvarez, Vera Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; Argentin