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We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set {π(λ)g : λ ∈ Λ} for a discrete subset Λ ⊆ N/Z(N) forms a frame for Hπ, then its density satisfies the strict inequality D − (Λ) > dπ, where D − (Λ) is the lower Beurling density. An analogous density condition D+(Λ) < dπ holds for a Riesz sequence in Hπ contained in the orbit of (π,Hπ). The proof is based on a deformation theorem for coherent systems, a universality result for p-frames and p-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

We show that a real-valued function $f$ in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $\{|f(\lambda)|: \lambda \in \Lambda \}$ on any set $\Lambda \subseteq \mathbb{R}$ with lower Beurling density $D^{-}(\Lambda)>2$., Comment: 7 pages

A cooperative breeding case was found in Lesser Spotted Woodpecker (LSW) in which a male and three different females were involved. The contributions of each member of the cooperative breeding team were quantified during nest excavation and incubation. This is the first such report on LSW. During daytime, the nest was mainly occupied by the male during the final excavation phase, nest guarding, and egg laying. Aggressive interactions were recorded between two of the females during nest excavation, egg laying and incubation. The male was recorded destroying an egg presumably laid by the dominant female. The highest contribution during daytime incubation was made by the dominant female (39%), followed by the male (34%) and then by a second female (27%). The third female contributed very little at any stage. A picture of the male removing the egg was analyzed to estimate the egg maximum width, ruling out a possible runt egg. Motivations behind the behaviour of the male destroying the egg remain unclear.

We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.

During a study of a colour-marked population of Lesser Spotted Woodpecker (LSW) in NE Spain we found two cases of cooperative breeding, in which two females and a male were rearing the brood. One of the females was involved in both cooperative breeding cases, in two consecutive years. We analysed the feeding contribution of each individual to nestling care as well as feeding rates to nestlings. In both cases the male showed a (compensatory) reduction in nestling feeding due to the involvement of the second female. Total feeding rate increased slightly compared to regular (noncooperative) monogamous pairs. The high breeding success found in the second case (six eggs/six nestlings) is thought to be due to the cooperation of the three adults in nestling care. The two cooperative breeding cases were found in territories that suffered from significant selective logging of dead trees. This logging is thought to be a possible anthropogenic example of the ecological constraints hypothesis and a significant factor that could force atypical cooperative breeding attempts. A locally biased sex-ratio might result in the cooperation of two females and a male in both cases. No evidence of kinship was found among members of the cooperative breeding teams. We could not determine whether the cooperative breeding cases were monogamous or polygamous.

This paper explains the process of producing healthy trees and sound wood for Gmelina arborea Roxb. (gmelina) in Costa Rica. Ston Forestal S. A. began planting gmelina in the early 1990s and currently has 14,000 ha of plantations. The company is involved in vertical production, from the development of improved seeds in its own seed orchard to the finished solid wood product. This article describes the entire process from site selection to the production of logs for specific products. The solid wood products that can be produced from gmelina are highlighted.