Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).Comment: 77 pages, LaTeX version 2.09. This is the text of the revised version which will appear in Annals of Mathematics. An error in Section 2 of the previous version has been correcte

Localization for nonabelian group actions

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction \xred = \mu^{-1}(0)/K. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X)$ of $X$ to the cohomology H^*(\xred). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of \xred of any \eta_0 \in H^*(\xred) whose degree is the dimension of \xred, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ isComment: 42 pages, LaTex version no. 2.09, Introduction and Section 8 have been rewritten in revised versio

Recovering low molecular weight extractives from degraded straw by oyster mushroom at the farm scale for high value use

The cultivation of mushrooms on wheat straw can be considered a solid state fermentation, yet following harvest the residual, partially degraded straw is discarded. During cultivation, the degradation of lignocellulose in the straw takes place by the fungus under the action of enzymes releasing degradation products with small molecular weight, some of which are potentially valuable. These compounds may be extracted from straw after mushroom cultivation in two stages: an aqueous extraction followed by a solvent extraction. The present work is focused on the first stage of the process. The aqueous extraction releases water soluble compounds, such as sugars and phenolics with lower molecular weight, which are easily obtained. The partially degraded straw may then be treated with organic solvents to release water insoluble lignin breakdown products, such as fatty acids, phenolics and other aromatics. It is important to conduct scale-up experiments at a scale that would reflect the amount of waste straw generated by a mushroom farm. A study was performed using a vessel of 300 L capacity with mixing impeller, by observing the influence of the temperature (20oC, 25oC, 40oC, 60oC and 80oC) and water-to-dry straw ratio (from 40:1 to 90:1) on the total extracted matter and especially on sugar and phenolic compounds yields. A microbial study of the aqueous extract was also performed at 20oC and 25oC to explain the high concentration of organic carbon in the extract under certain circumstances. The optimum extraction conditions were determined by taking into account the yield and the energy consumption of the process. The conclusion was that the extraction temperature can be conducted between 20oC and 25oC with good results for obtaining liquor which can be used in a biogas installation. The extraction should be conducted at 80oC to obtain greater yields of sugars and phenolics

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e. by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.Comment: 22 page

The impact of study support : a report of a longitudinal study into the impact of participation in out-of-school-hours learning on the academic attainment, attitudes and school attendance of secondary school students

Study support makes a difference. It has an impact on three key aspects of students’ school careers: • attainment at GCSE and KS3 SATs; • attitudes to school; • attendance at school. These findings were consistent for all groups of students in all schools in the study. - Study support can help to improve schools and can influence the attitudes to learning of teachers and parents as well as students

Critical sets of the total variance of state detect all SLOCC entanglement classes

We present a general algorithm for finding all classes of pure multiparticle states equivalent under Stochastic Local Operations and Classsical Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also discuss the Morse indices of critical points which have the interpretation of the number of independent non-local perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.Comment: 4 page

Optimal hurricane overwash thickness for maximizing marsh resilience to sea level rise

The interplay between storms and sea level rise, and between ecology and sediment transport governs the behavior of rapidly evolving coastal ecosystems such as marshes and barrier islands. Sediment deposition during hurricanes is thought to increase the resilience of salt marshes to sea level rise by increasing soil elevation and vegetation productivity. We use mesocosms to simulate burial of Spartina alterniflora during hurricane-induced overwash events of various thickness (0-60 cm), and find that adventitious root growth within the overwash sediment layer increases total biomass by up to 120%. In contrast to most previous work illustrating a simple positive relationship between burial depth and vegetation productivity, our work reveals an optimum burial depth (510 cm) beyond which burial leads to plant mortality. The optimum burial depth increases with flooding frequency, indicating that storm deposition ameliorates flooding stress, and that its impact on productivity will become more important under accelerated sea level rise. Our results suggest that frequent, low magnitude storm events associated with naturally migrating islands may increase the resilience of marshes to sea level rise, and in turn, slow island migration rates. Synthesis: We find that burial deeper than the optimum results in reduced growth or mortality of marsh vegetation, which suggests that future increases in overwash thickness associated with more intense storms and artificial heightening of dunes could lead to less resilient marshes