A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite operators A_j. In part I we only considered the case q=1 and proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2. We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also able to include the parameter q\geq 1 and still retain the convexity. Among other things this leads to a definition of an L^q(L^p) norm for operators when 1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces -- which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.Comment: Proof of a conjecture in math/0701352. Revised version replaces earlier draft. 18 pages, late

Hypercontractivity on the qq-Araki-Woods algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.Comment: 17 page

The distribution of an invasive species, clidemia hirta along roads and trails in Endau Rompin National Park, Malaysia

Invasive species pose a grave threat to many national parks. Construction of roads and trails for tourism may facilitate invasion of alien species. To understand the effect of road and trail construction on invasive species, we established six transects in three land-use types (forest interior and road or trail edges) in Endau Rompin National Park, Johor, Malaysia, where we measured the number of an invasive shrub, Clidemia hirta (Melastomataceae), canopy openness, and soil properties; compared the density of C. hirta between the three land-use types; and finally, identified soil and canopy variables affecting its abundance using generalized linear mixed models. C. hirta was found along the road and trail with density ranging from 0.0 m2 to 33 m2 (average: 3.8 m2), but was not found in the forest interior. Generalized linear mixed models suggested that canopy openness and soil pH negatively affected the density of C. hirta along the road, as did total soil nitrogen along the trail. This suggests that C. hirta was more abundant along dark and nutrient-poor road and trail edges. The construction of narrow roads (2.0–3.8 m) and trails (0.5–2.0 m wide) at our site would be considered a relatively minor disturbance without intensive clear cuts, and C. hirta seemed to prefer habitats with such minor disturbances. In the tropical rainforests, the managers or conservationists of the national park should include consideration of the effects such as minor disturbances have on invasive species