The Cayley-Hilbert metric and positive operators

AbstractThe Cayley-Hilbert metric is defined for a real Banach space containing a closed cone. By restricting the domain of a particular type of positive nonlinear operator, the Banach contraction-mapping theorem is used to prove the existence of a unique fixed point of the operator with explicit upper and lower bounds. Applications to quasilinear elliptic partial differential equations and to matrix theory are considered

Trace inequalities for positive definite matrix power products

AbstractAn elementary proof is given for the best possible upper and lower bonds of tr(AB)n for Hermitian positive semidefinite N × N matrices A and B

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

The mammalian gene function resource: The International Knockout Mouse Consortium

In 2007, the International Knockout Mouse Consortium (IKMC) made the ambitious promise to generate mutations in virtually every protein-coding gene of the mouse genome in a concerted worldwide action. Now, 5 years later, the IKMC members have developed highthroughput gene trapping and, in particular, gene-targeting pipelines and generated more than 17,400 mutant murine embryonic stem (ES) cell clones and more than 1,700 mutant mouse strains, most of them conditional. A common IKMC web portal (www.knockoutmouse.org) has been established, allowing easy access to this unparalleled biological resource. The IKMC materials considerably enhance functional gene annotation of the mammalian genome and will have a major impact on future biomedical research

Nutrient loss and water quality under extensive grazing in the upper Burdekin river catchment, North Queensland

Increased sediment and nutrient losses resulting from unsustainable grazing management in the Burdekin River catchment are major threats to water quality in the Great Barrier Reef Lagoon. To test the effects of grazing management on soil and nutrient loss, five 1 ha mini-catchments were established in 1999 under different grazing strategies on a sedimentary landscape near Charters Towers. Reference samples were also collected from watercourses in the Burdekin catchment during major flow events. Soil and nutrient loss were relatively low across all grazing strategies due to a combination of good cover, low slope and low rainfall intensities. Total soil loss varied from 3 to 20 kg ha−1 per event while losses of N and P ranged from 10 to 1900 g ha−1 and from 1 to 71 g ha−1 per event respectively. Water quality of runoff was considered moderate across all strategies with relatively low levels of total suspended sediment (range: 8–1409 mg l−1), total N (range: 101–4000 μg l−1) and total P (range: 14–609 μg l−1). However, treatment differences are likely to emerge with time as the impacts of the different grazing strategies on land condition become more apparent. \ud Samples collected opportunistically from rivers and creeks during flow events displayed significantly higher levels of total suspended sediment (range: 10–6010 mg l−1), total N (range: 650–6350 μg l−1) and total P (range: 50–1500 μg l−1) than those collected at the grazing trial. These differences can largely be attributed to variation in slope, geology and cover between the grazing trial and different catchments. In particular, watercourses draining hillier, grano-diorite landscapes with low cover had markedly higher sediment and nutrient loads compared to those draining flatter, sedimentary landscapes. These preliminary data suggest that on relatively flat, sedimentary landscapes, extensive cattle grazing is compatible with achieving water quality targets, provided high levels of ground cover are maintained. In contrast, sediment and nutrient loss under grazing on more erodable land types is cause for serious concern. Long-term empirical research and monitoring will be essential to quantify the impacts of changed land management on water quality in the spatially and temporally variable Burdekin River catchment

Increased seizure susceptibility in mice lacking metabotropic glutamate receptor 7

To study the role of mGlu7 receptors (mGluR7), we used homologous recombination to generate mice lacking this metabotropic receptor subtype (mGluR7-/-). After the serendipitous discovery of a sensory stimulus-evoked epileptic phenotype, we tested two convulsant drugs, pentylenetetrazole (PTZ) and bicuculline. In animals aged 12 weeks and older, subthreshold doses of these drugs induced seizures in mGluR7-/-, but not in mGluR7+/-, mice. PTZ-induced seizures were inhibited by three standard anticonvulsant drugs, but not by the group III selective mGluR agonist (R,S)-4-phosphonophenylglycine (PPG). Consistent with the lack of signs of epileptic activity in the absence of specific stimuli, mGluR7-/- mice showed no major changes in synaptic properties in two slice preparations. However, slightly increased excitability was evident in hippocampal slices. In addition, there was slower recovery from frequency facilitation in cortical slices, suggesting a role for mGluR7 as a frequency-dependent regulator in presynaptic terminals. Our findings suggest that mGluR7 receptors have a unique role in regulating neuronal excitability and that these receptors may be a novel target for the development of anticonvulsant drugs