A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

Conjugacy and dynamics in almost automorphism groups of trees

We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson’s [Formula: see text] by Belk and Matucci. We also analyze the dynamics of a tree almost automorphism as a homeomorphism of the boundary of the tree.

Cost implications of cluster plot design choices for precise estimation of forest attributes in landscapes and forests of varying heterogeneity

Tradeoffs occur when deciding between improving forest inventory precision by increasing sample size or by augmenting cluster plot design factors like size or subplot separation distance. The nature of these tradeoffs changes with variation in type and scale of the spatial pattern of the attribute of interest. In order to understand the impacts of relationships between type and scale of spatial heterogeneity and cluster plot design efficiency, we constructed a factorial simulation experiment and analysed relationships between forest inventory cost, cluster plot design factors, and different spatial heterogeneity scenarios constructed via simulation. To calculate cost, we constructed a cost model that accounted for both on- and between-plot costs. We found that type and scale of heterogeneity have important implications for plot design choices. Homogeneous stands and landscapes are the least-costly to inventory. Subplot area and count have stronger impacts than subplot separation on cost efficiency, particularly in landscapes with aggregated forest patterns and in stands with homogeneous tree patterns. We discuss results in the context of the physical interaction between cluster plot geometry and spatial patterns at different scales, provide computer code for simulations, and suggest principles that forest inventory cluster plot design specialists should consider when designing inventories.

Riesz means on homogeneous trees

Let 𝕋 be a homogeneous tree. We prove that if f ∈ Lp (𝕋), 1 ≤ p ≤ 2, then the Riesz means Sz R (f) converge to f everywhere as R → ∞, whenever Re z > 0.

Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts

In the hyperbolic disc (or more generally in real hyperbolic spaces) we consider the horospherical Radon transform R and the geodesic Radon transform X. Composition with their respective dual operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree T, separately studied as acting on functions on the vertices or on the edges. This leads to a new theory of spherical functions and Radon inversion on the edges of a tree.

Experimental Analysis of Wind Interaction with Olive Grove and the Atmospheric Surface Boundary Layer

The emblematic olive grove covers 17% of the South Spain territory. Last years, intensive farming has been developed around its exploitation. This causes many environmental and social problems, ecological interactions and environmental variables are modified, accentuating the climate change effects. The aim of this work is optimize the layout to allow a sustainable agricultural production, adapting the exploitation to the climate change and minimizing alterations over environmental variables. Thus, interaction between the Surface Boundary Layer (SBL) and olive groves has been analyzed experimentally. Wind tunnel reduced-scale tests were carried out for different setups: grid with homogeneous tree models, over a flat surface and over a hill, and simulating different vegetation covers. The same tests were replicated with staggered pattern and heterogeneous tree models. Wind velocity and turbulence kinetic energy profiles were obtained using Hot Wire Anemometry, taking measurements between models and behind them. It has been proved that olive grove spatial layout and the presence or absence of vegetation cover modify the SBL dynamics. Consequently, not only fundamental environmental variables are altered, for example, the moisture flux and the evapotranspiration, but also important variables for human health are modified, as pollen flow dispersion to urban areas.Keywords: Olive grove, environmental management, sustainability, wind tunnel