Abstract:
The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor. The restrictions arising from the associated conservation equations are investigated. It is shown that there are, for general $d$, three linearly independent conformal invariant forms for the three point function of the energy momentum tensor, although for $d=3$ there are two and for $d=2$ only one. The form of the three point function is also demonstrated to simplify considerably when all three points lie on a straight line. Using this the coefficients of the conformal invariant point functions are calculated for free scalar and fermion theories in general dimensions and for abelian vector fields when $d=4$. Ward identities relating three and two point functions are also discussed. This requires careful analysis of the singularities in the short distance expansion and the method of differential regularisation is found convenient. For $d=4$ the coefficients appearing in the energy momentum tensor three point function are related to the coefficients of the two possible terms in the trace anomaly for a conformal theory on a curved space background.

Abstract:
The quality of a soil is often viewed in relation to its ability to suppress plant disease and enhance agricultural productivity. A soil is considered suppressive when, in spite of favourable conditions for disease incidence and development, a pathogen cannot become established, or establishes but produces no disease, or establishes and produces disease for a short time and then declines. The interplay of biotic and abiotic factors has long been known to assert disease suppressive capabilities or otherwise. However, the multi-functionality of soil makes the identification of a single property as a general indicator of soil health an uphill task. In this paper, therefore, some indicators of soil health important to agriculture are reviewed with emphasis on pea footrot disease suppression potentials. Findings show that footrot disease due to Nectria haematococca (anamorph Fusarium solani f.sp pisi) is a globally, economically important disease of peas, and an initial inoculum density of ? 100 pathogenic forms of N. haematococca cells would produce an appreciable level of pea footrot disease depending on the relative amount of phosphorus, carbon and nitrogen present in soil. It would be desirable to confirm pea footrot disease models obtained from pot experiments with results from field experiments.

Abstract:
Footrot disease due to Nectria haematococca is an economically important disease of peas all over the world where peas are grown. The combined effect of pathogenicity genes on disease severity has not been adequately addressed. Hence in this research, molecular PCR-based assays have been developed and/or used to detect all six (PDA, PEP1, PEP2, PEP3, PEP4 and PEP5) pea pathogenicity genes in fifteen fungal isolates previously isolated from fields with footrot disease. The pathogenicity of these isolates on pea was also assessed. Results showed that all six pathogenicity genes (PDA, PEP1, PEP2, PEP3, PEP4 and PEP5) are required for high virulence (DI ≥ 3.75). Isolates that possessed only the PEP1 or a combination of the PEP1 and PEP4 genes generally caused a relatively low degree of disease (DI≤2.75). Similarly, reference isolate 156-30-6 which possesses only the PDA gene was non pathogenic on peas (DI = 1.50±0.87). However, without exception, isolates that possessed the PDA gene alongside other gene(s), especially PEP2, PEP3 and PEP5 genes caused high degrees of disease (DI≥3.50). The PEP2, PEP3 and PEP5 genes were observed only in isolates with high pathogenicity (DI≥3.75).

Abstract:
PCR based assays were developed in this study to quantitatively predict pea footrot infections in agricultural soils prior to cultivation. Pea footrot disease due to Nectria haematococca (anamorph Fusarium solani f.sp. pisi) is linked to the presence of six pea pathogenicity (PEP) genes (PDA1, PEP1, PEP2, PEP3, PEP4 and PEP5). Whilst molecular assays have been used recently to selectively detect these genes in soil- DNA, quantitative molecular assay has been extended to only the PEP3 gene whose role in pea pathogenicity is yet unknown. In this research, PCR-based quantification assays were developed to quantify the two pea pathogenicity genes (PDA and PEP5) with identified roles in pea pathogenicity from soil-DNA obtained from fields with pea footrot histories. Results showed that the quantitative molecular assays developed herein were both efficient. Amplification efficiency of the Q-PCR assay for the PDA and PEP5 gene were 97 and 89%, respectively. PDA and PEP5 gene copy numbers were shown to vary significantly (p = 0.01) between fields. However, the PDA gene copy numbers were relatively higher than those of the PEP5 gene in agricultural fields. The genes, especially PEP5 gene, were comparable to and positively correlated to the number of spores of pathogenic N. haematococca, and footrot disease. The PDA gene alone in soil could not cause footrot disease in peas after 8 weeks of planting; assays directed at it alone may therefore be insufficient to predict pea footrot disease. However, the molecular assay targeting the PDA alongside the PEP5 gene offers the opportunity for quantitative prediction of pea footrot infections in agricultural soils prior to cultivation.

Abstract:
Footrot disease due to N. haematococca (anamorph Fusarium solani f. sp. pisi) is a globally, economically important disease of peas. The disease has been linked to the presence of six pea pathogenicity (PEP) genes (PDA1, PEP1, PEP2, PEP3, PEP4 and PEP5) inherent in pathogenic forms of the causal fungus N. haematococca MPIV. The disease is prevented only through avoidance of fields with high disease potential. Identifying agricultural fields with a high disease potential prior to pea cultivation has been paramount in the implementation of preventive measures. Although molecular techniques have been successfully used to quantify pathogenic strains of N. haematococca in agricultural soils, targeting all six pathogenicity genes in these assays would not be cost effective. This study therefore attempts to review the functions and roles of the different genes linked with pea pathogenicity with the aim of identifying gene(s) that would serve as a logical target in a quantitative molecular assay. Findings suggest that, whilst the PDA gene may be targeted in a preliminary diagnostic measure, a conclusive assay, targeting the PEP3 gene may be required to affirm pea footrot disease potential of agricultural fields. Agricultural fields with the PEP3 gene copy numbers of up to 100 per g soil prior to cultivation may be deemed unsafe for peas.

Abstract:
Possible short and semi-short representations for $\N=2$ and $\N=4$ superconformal symmetry in four dimensions are discussed. For $\N=4$ the well known short supermultiplets whose lowest dimension conformal primary operators correspond to $\half$-BPS or ${1\over 4}$-BPS states and are scalar fields belonging to the $SU(4)_r$ symmetry representations $[0,p,0]$ and $[q,p,q]$ and having scale dimension $\Delta =p$ and $\Delta = 2q+p$ respectively are recovered. The representation content of semi-short multiplets, which arise at the unitarity threshold for long multiplets, is discussed. It is shown how, at the unitarity threshold, a long multiplet can be decomposed into four semi-short multiplets. If the conformal primary state is spinless one of these becomes a short multiplet. For $\N=4$ a ${1\over 4}$-BPS multiplet need not have a protected dimension unless the primary state belongs to a $[1,p,1]$ representation.

Abstract:
Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.

Abstract:
Superconformal transformations are derived for the $\N=2,4 supermultiplets corresponding to the simplest chiral primary operators. These are applied to two, three and four point correlation functions. When $\N=4$, results are obtained for the three point function of various descendant operators, including the energy momentum tensor and SU(4) current. For both $\N=2$ or 4 superconformal identities are derived for the functions of the two conformal invariants appearing in the four point function for the chiral primary operator. These are solved in terms of a single arbitrary function of the two conformal invariants and one or three single variable functions. The results are applied to the operator product expansion using the exact formula for the contribution of an operator in the operator product expansion in four dimensions to a scalar four point function. Explicit expressions representing exactly the contribution of both long and possible short supermultiplets to the chiral primary four point function are obtained. These are applied to give the leading perturbative and large N corrections to the scale dimensions of long supermultiplets.

Abstract:
Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for a complete set of observables formed from the photon creation and annihilation operators. These Moyal solutions incorporate the asymptotics of the classical limit in a simple explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.

Abstract:
The Moyal--Weyl description of quantum mechanics provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in $\hbar$. Its semiclassical expansion `coefficients,' acting on symbols that represent observables, are simple, globally defined differential operators constructed in terms of the classical flow. Two methods of constructing this expansion are discussed. The first introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold's formula for the Weyl product of symbols. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of `quantum trajectories.' Their Green function solutions construct the regular $\hbar\downarrow0$ asymptotic series for the Heisenberg--Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the $\hbar$ coefficients recursively. The Heisenberg--Weyl description of evolution involves no essential singularity in $\hbar$, no Hamilton--Jacobi equation to solve for the action, and no multiple trajectories, caustics or Maslov indices.