Locally Weyl invariant massless bosonic and fermionic spin-1/2 action in the (Wn(4),g)\bf (W_{n(4)},g) and (U4,g)\bf (U_{4},g) space-times

We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of non-metricity and contortion tensor. In the presence of the non-metricity tensor the investigation is extended to Weyl $(W_n, g)$ space-time while when the torsion is encountered we are restricted to the Riemann-Cartan $(U_4, g)$ space-time. Our results hold for a subgroup of the Weyl-Cartan $(Y_4, g)$ space-time and we also calculate extra contributions to the conformal gravity.Comment: 16 page

Topological restrictions for circle actions and harmonic morphisms

Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is nonnegative and even. In particular, $M^m$ has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example $CP^{n}$, $K3$ surfaces, $S^{2n}\times P_g$ ($n\geq2$) where $P_g$ is the closed surface of genus $g\geq2$ can never be the domain of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and $\phi:(M^4,g)\to(N^3,h)$ a non-constant harmonic morphism. Suppose that one of the following assertions holds: (1) $(M^4,g)$ is half-conformally flat and its scalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat, (3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemannian coverings, $\phi$ is the canonical projection $T^4\to T^3$ between flat tori.Comment: 18 pages; Minor corrections to Proposition 3.1 and small changes in Theorem 2.8, proof of Theorem 3.3 and Remark 3.

A Structure Theorem for Small Sumsets in Nonabelian Groups

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.Comment: 23 page

The potential to control Haemonchus contortus in indigenous South African goats with copper oxide wire particles

The high prevalence of resistance of Haemonchus contortus to all major anthelmintic groups has prompted investigations into alternative control methods in South Africa, including the use of copper oxide wire particle (COWP) boluses. To assess the efficacy of COWP against H. contortus in indigenous South African goats, 18 male faecal egg-count-negative goats were each given ca.1200 infective larvae of H. contortus three times per week during weeks 1 and 2 of the experiment. These animals made up an “established” infection group (ESTGRP). At the start of week 7, six goats were each given a 2-g COWP bolus orally; six goats received a 4-g COWP bolus each and six animals were not treated. A further 20 goats constituted a “developing” infection group (DEVGRP). At the beginning of week 1, seven of the DEVGRP goats were given a 2-g COWP bolus each; seven goats were treated with a 4-g COWP bolus each and no bolus was given to a further six animals. During weeks 1–6, each of these DEVGRP goats was given ca. 400 H. contortus larvae three times per week. All 38 goats were euthanized for worm recovery from the abomasa and small intestines in week 11. In the ESTGRP, the 2-g and 4-g COWP boluses reduced the worm burdens by 95% and 93%, respectively compared to controls (mean burden ± standard deviation, SD: 23 ± 33, 30 ± 56 and 442 ± 518 worms, P = 0.02). However, in the DEVGRP goats, both the 2-g and 4-g COWP treatments were ineffective in reducing the worm burdens relative to the controls (mean burdens ± SD: 1102 ± 841, 649 ± 855, 1051 ± 661 worms, P = 0.16). Mean liver copper levels did not differ between the ESTGRP goats treated with 2-g COWP, 4-g COWP or no COWP (mean ± standard error of the mean, SEM, in ppm: 93.7 ± 8.3; 101.5 ± 8.3; 71.8 ± 8.3, P = 0.07) nor did they differ between the DEVGRP goats (mean ± SEM, in ppm: 74.1 ± 9.1; 75.4 ± 9.1; 74.9 ± 10.0, P > 0.99). The copper values were considered adequate, but not high, for goats. The COWP boluses have the potential to be used in the place of conventional anthelmintics for the control of established H. contortus infections in indigenous South African goats, but their use as part of an integrated approach to control H. contortus in the field must be fully investigated

A new linear quotient of C⁴ admitting a symplectic resolution

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) &#60; Sp(4,C). Here Q_8 is the quaternionic group of order eight and D_8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C^4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V / G admitting symplectic resolutions