Tensor Rank, Invariants, Inequalities, and Applications

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$.Comment: 31 pages, 1 figur

Skunk River Fall 1998

https://openspace.dmacc.edu/skunkriver/1019/thumbnail.jp

Trends in postpartum hemorrhage in high resource countries: a review and recommendations from the International Postpartum Hemorrhage Collaborative Group

<p>Abstract</p> <p>Background</p> <p>Postpartum hemorrhage (PPH) is a major cause of maternal mortality and morbidity worldwide. Several recent publications have noted an increasing trend in incidence over time. The international PPH collaboration was convened to explore the observed trends and to set out actions to address the factors identified.</p> <p>Methods</p> <p>We reviewed available data sources on the incidence of PPH over time in Australia, Belgium, Canada, France, the United Kingdom and the USA. Where information was available, the incidence of PPH was stratified by cause.</p> <p>Results</p> <p>We observed an increasing trend in PPH, using heterogeneous definitions, in Australia, Canada, the UK and the USA. The observed increase in PPH in Australia, Canada and the USA was limited solely to immediate/atonic PPH. We noted increasing rates of severe adverse outcomes due to hemorrhage in Australia, Canada, the UK and the USA.</p> <p>Conclusion</p> <p><it>Key Recommendations</it></p> <p indent="1">1. Future revisions of the International Classification of Diseases should include separate codes for atonic PPH and PPH immediately following childbirth that is due to other causes. Also, additional codes are required for placenta accreta/percreta/increta.</p> <p indent="1">2. Definitions of PPH should be unified; further research is required to investigate how definitions are applied in practice to the coding of data.</p> <p indent="1">3. Additional improvement in the collection of data concerning PPH is required, specifically including a measure of severity.</p> <p indent="1">4. Further research is required to determine whether an increased rate of reported PPH is also observed in other countries, and to further investigate potential risk factors including increased duration of labor, obesity and changes in second and third stage management practice.</p> <p indent="1">5. Training should be provided to all staff involved in maternity care concerning assessment of blood loss and the monitoring of women after childbirth. This is key to reducing the severity of PPH and preventing any adverse outcomes.</p> <p indent="1">6. Clinicians should be more vigilant given the possibility that the frequency and severity of PPH has in fact increased. This applies particularly to small hospitals with relatively few deliveries where management protocols may not be defined adequately and drugs or equipment may not be on hand to deal with unexpected severe PPH.</p

B-cell survival and development controlled by the coordination of NF-κB family members RelB and cRel.

Targeted deletion of BAFF causes severe deficiency of splenic B cells. BAFF-R is commonly thought to signal to nuclear factor κ-light-chain-enhancer of activated B cells (NF-κB)-inducing kinase dependent noncanonical NF-κB RelB. However, RelB-deficient mice have normal B-cell numbers. Recent studies showed that BAFF also signals to the canonical NF-κB pathway, and we found that both RelB and cRel are persistently activated, suggesting BAFF signaling coordinates both pathways to ensure robust B-cell development. Indeed, we report now that combined loss of these 2 NF-κB family members leads to impaired BAFF-mediated survival and development in vitro. Although single deletion of RelB and cRel was dispensable for normal B-cell development, double knockout mice displayed an early B-cell developmental blockade and decreased mature B cells. Despite disorganized splenic architecture in Relb(-/-)cRel(-/-) mice, generation of mixed-mouse chimeras established the developmental phenotype to be B-cell intrinsic. Together, our results indicate that BAFF signals coordinate both RelB and cRel activities to ensure survival during peripheral B-cell maturation

Reporter

Subtitled: "Tuition Hikes