A note on Gabor frames in finite dimensions

The purpose of this note is to present a proof of the existence of Gabor frames in general linear position in all finite dimensions. The tools developed in this note are also helpful towards an explicit construction of such a frame, which is carried out in the last section. This result has applications in signal recovery through erasure channels, operator identification, and time-frequency analysis.Comment: 10 page

Discrete squeezed states for finite-dimensional spaces

We show how discrete squeezed states in an $N^{2}$-dimensional phase space can be properly constructed out of the finite-dimensional context. Such discrete extensions are then applied to the framework of quantum tomography and quantum information theory with the aim of establishing an initial study on the interference effects between discrete variables in a finite phase-space. Moreover, the interpretation of the squeezing effects is seen to be direct in the present approach, and has some potential applications in different branches of physics.Comment: 16 pages; 3 figure

Soil and crop responses following application of biosolids-derived organomineral fertilisers to ryegrass (Lolium perenne L.) grown in pots

Biosolids-derived organomineral fertilisers (OMF) were produced using a novel technique reported in earlier studies. This technique enables addition of N and potash to biosolids granules to form a balanced NPK fertiliser. Two fertiliser products; OMF10 (10:4:4) and OMF15 (15:4:4), were formulated and tested in a glasshouse facility on pot-grown ryegrass in comparison with urea and biosolids granules at N application rates ranging from 0 to 300 kg ha-1. The aim of this research was to contribute to the understanding of nutrients management and dynamics in grass crops fertilised with OMF. The study focused upon dry matter yield (DMY) and crop responses to applied fertiliser, nitrogen use efficiency (NUE) and fertilisers’ effect on soil fertility. Results indicated that ryegrass responds linearly to application of OMF increasing DMY by about 2% to 27% compared with biosolids but to a lesser extent than urea (range: 17% to 55%). NUE was related to the concentration of readily available N in the fertiliser; urea and OMF showed significantly greater (P<0.05) N recoveries than biosolids (26% to 75%, and 19% to 29%, respectively). Total nitrogen in soil and SOM increased (P<0.05) depending on the concentration of organic-N in the fertiliser applied. DMY was lower but more sustained overtime in biosolids-treated pots. OMF application did not result in significant changes in soil extractable-P levels whereas for urea, it decreased significantly while it showed a significant increase in biosolids-treated pots, where soil-P Index changed from 5 to 6. In OMF-treated soil, soil P Index remained close to constant overtime thereby supporting the purpose of the formulations tested

Libertad de enseñanza o la perpetuación del poder

Abstract not availabl

Polyhedral Gauss Sums, and polytopes with symmetry

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $\mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $\mathcal G$ be the group generated by $\mathcal{W}$ as well as all integer translations in $\mathbb Z^d$. We prove that if $P$ multi-tiles $\mathbb R^d$ under the action of $\mathcal G$, then we have the closed form $G_P(n) = \text{vol}(P) G(n)^d$. Conversely, we also prove that if $P$ is a lattice tetrahedron in $\mathbb R^3$, of volume $1/6$, such that $G_P(n) = \text{vol}(P) G(n)^d$, for $n \in \{ 1,2,3,4 \}$, then there is an element $g$ in $\mathcal G$ such that $g(P)$ is the fundamental tetrahedron with vertices $(0,0,0)$, $(1, 0, 0)$, $(1,1,0)$, $(1,1,1)$.Comment: 18 pages, 2 figure

Phosphorus Release Characteristics from Biosolids-Derived Organomineral Fertilizers

This study investigated the availability of phosphorus (P) following soil application of a novel biosolids-derived organomineral fertilizer (OMF15; 15:4:4) in comparison with single superphosphate (0:18:0). Two soil types of contrasting characteristics were incubated over a period of 90 days at 25 °C and maintained near field capacity. Phosphorus was applied at rates equivalent to 0 (control), 150, and 300 kg ha−1 of P2O5, respectively. Availability of P from OMF15 was low throughout the experiment accounting for less than 6.5% of total OMF15-P applied. It was shown that after the 90 days incubation period, the overall increase in soil extractable P in OMF15-treated soil was marginal in both soil types. For single superphosphate (SSP), P availability ranged from 16% to 46% of total SSP-P applied. Application of SSP increased soil extractable P levels significantly (P < 0.001) compared with unfertilized control soils. The results of this study aided the development of fertilization strategies for the best use of OMF produced from nutrient-enriched biosolids granules for applications in winter cereal and grass crops in England