Frames of subspaces and operators

We study the relationship between operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space $\mathcal{H}$. We get sufficient conditions on an orthonormal basis of subspaces $\mathcal{E} = \{E_i \}_{i\in I}$ of a Hilbert space $\mathcal{K}$ and a surjective $T\in L(\mathcal{K}, \mathcal{H})$ in order that $\{T(E_i)\}_{i\in I}$ is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J. A. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames. Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinament of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.Comment: 21 pages, LaTeX; added references and comments about fusion frame

Multiplicative Lidskii's inequalities and optimal perturbations of frames

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame \cF for \hil\cong\C^d we compute those dual frames \cG of \cF that are optimal perturbations of the canonical dual frame for \cF under certain restrictions on the norms of the elements of \cG. On the other hand, for a fixed finite frame \cF=\{f_j\}_{j\in\In} for \hil we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame V\cdot \cF=\{V\,f_j\}_{j\in\In} - which is equivalent to \cF - is optimal among such perturbations of \cF. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.Comment: 22 page

Optimal dual frames and frame completions for majorization

In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $\mathcal F$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F$ by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus' frame potential. On the other hand, given a fixed frame $\mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $\mathcal G$ that are in duality with $\mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.Comment: 29 pages, with modifications related with the exposition of the materia

Some properties of frames of subspaces obtained by operator theory methods

AbstractWe study the relationship among operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space H. We get sufficient conditions on an orthonormal basis of subspaces E={Ei}i∈I of a Hilbert space K and a surjective T∈L(K,H) in order that {T(Ei)}i∈I is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J.A. Antezana, G. Corach, M. Ruiz, D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006) 1031–1037], which relate frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinement of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given

The ecology of larger microzooplankton in the Weddell-Scotia Confluence Area: Horizontal and vertical distribution patterns

The distribution of microzooplankton \u3e 15 μm (large dinoflagellates, foraminifers, radiolarians, tintinnids, microcrustaceans and various invertebrate larvae) was studied in samples retrieved from 10 to 400 m in two overlapping transects along 49W, between 57S and 61°30′S (27 Nov.–12 Dec. 1988, and 27 Dec. 1988–4 Jan. 1989). Dinoflagellates and tintinnids concentrated at 50–90 m (10–400 m weighted averages, dinoflagellates: 103 ind./I, 131 mg C/m2; tintinnids: 9.7 ind./I, 53 mg C/m2). Copepod nauplii had a more variable vertical pattern with maximum numbers at 100–200 m (10–400 m av.: 2.6 ind./I, 27 mg C/m2). Foraminifers and radiolarians were most abundant in noticeably deeper waters peaking below 150 m (10–400 m av., foraminifers: 0.2 ind./I, 11 mg C/m2; radiolarians: 2.7 ind./I, 12 mg C/m2). Large dinoflagellates accounted, on the average, for 55% of the biomass of the heterotrophs considered in the 10–400 m layer, followed by the tintinnids (23%), copepod nauplii (11%), foraminifers (5%), and radiolarians (5%). The 100–400 m layer hosted up to 87% (mean: 49%) of total 10–400 m integrated microzooplanktonic biomass. The distribution of loricate ciliates was strongly correlated with those of chlorophyll a, and especially dinoflagellates (r = 0.832, for log-transformed data), suggesting close trophic relationships between these two groups. The northern sites were generally richer in microzooplankton than the area closer to the ice-edge, and the southernmost ice-covered zone yielded the lowest microplanktonic values. This biological pattern, which was but loosely coupled with the Weddell-Scotia Confluence, with the vertical stability of the water column, and with near-surface concentrations of chlorophyll a, can at least partly be explained by differential grazing pressure by crustacean mesozooplankton. The time elapsed between the two transects did not affect the microzooplanktonic assemblages noticeably. Comparisons with previous abundance estimates carried out earlier and later in the growth season suggest that microzooplanktonic abundances increase toward the late summer-fall, probably in response to enhanced availability of nano- and pico-sized producers, characteristic of Antarctic post-bloom conditions

Oblique projections and abstract splines

Given a closed subspace L of a Hilbert space ℋ and a bounded linear operator A ∈ L(ℋ) which is positive, consider the set of all A-self-adjoint projections onto Y: ℘(A,Y) = {Q ∈ L(ℋ): Q2 = Q, Q(ℋ) = Y, AQ = Q*A}. In addition, if ℋ1 is another Hilbert space, T : ℋ → ℋ1 is a bounded linear operator such that T*T = A and ξ ∈ ℋ, consider the set of (T, Y) spline interpolants to ξ: sp(T, Y, ξ) = { η ε ξ + Y : ∥Tη∥ = min ∥T(ξ + σ)∥}. A strong relationship exists between ℘(A, Y) and s p(T, Y, ξ). In fact, ∥(A, Y) is not empty if and only if s p(T, Y, ξ) is not empty for every ξ ∈ ℋ. In this case, for any ξ ∈ ℋ\Y it holds s p(T, Y, ξ) = {(1 - Q)ξ:Q ∈ ℘(A, Y)} and for any ξ ∈ ℋ, the unique vector of s p(T, Y, ξ) with minimal norm is (1 - PA,Y)ξ, where PA,L is a distinguished element of ℘(A, Y). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.Facultad de Ciencias Exacta

Integración de procesos de negocio basados en servicios Web: Coreografía y satisfacción de restricciones

Los procesos de negocio en ocasiones necesitan funcionalidades adicionales que pueden alcanzarse empleando servicios del mismo dominio o de uno externo. Durante este procedimiento, los servicios web involucrados deben integrarse, de tal forma que sus funcionalidades individuales generen un comportamiento global. La coreografía de servicios web controla la secuencia de las interacciones entre los servicios durante el proceso de integración. Este artículo presenta una aproximación al modelado basado en restricciones de la coreografía de servicios web, a fin de lograr su automatización

Modelamiento dinámico y estático del proceso de coreografía de servicios web

La coreografía de servicios web se encarga de la definición de la vista global de un proceso de composición de servicios; en esta definición se identifica los componentes y sus distintas interacciones. La coreografía puede ser definida teniendo en cuenta sus aspectos estáticos y dinámicos. La vista estática está definida por el orden funcional requerido por el servicio compuesto. La vista dinámica se enfoca en las interacciones entre los servicios, ya que a través de estas se intercambia información. En este artículo se presenta un modelamiento a la coreografía de servicios web, con el fin de ser solucionada por medio de definición de restricciones

Epilithic diatoms in the Upper Laja River Basin, Guanajuato, Mexico

This study investigates the epilithic diatoms from the Dolores and Laja Rivers, located within the Upper Laja River Basin, Guanajuato, Mexico. The samples were collected during the cold-dry (January) and rainy (October) seasons of 2011. One-hundred seventy three infrageneric taxa were identified, 7 of which are new records for Mexico: Fallacia monoculata, Hantzschia abundans, Navigiolum uruguayense, Neidium bisulcatum, Nitzschia solita, Planothidium incuriatum and Sellaphora bacilloides; it is worth mentioning that Navigiolum is a new record at the genus level. Furthermore, 3 species are new records for the state of Guanajuato: Gomphonema laticollum, G. subclavatum and Gomphosphenia lingulatiformis. Teratological forms of 6 species were also found in the sampling sites with higher nutrient concentrations, with relative abundances varying from 0.8 to 8.3%