The Boden-Hu conjecture holds precisely up to rank eight

Consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point. Boden and Hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme, and they conjectured that one can always obtain a small resolution this way. The present text proves this conjecture in some cases (including all bundles of rank up to eight) and gives counterexamples in all other cases (in particular in every rank beyond eight). The main tool is a generalisation of Ext-groups involving more than two quasiparabolic bundles.Comment: 17 page

Decision blocks: A tool for automating decision making in CLIPS

The human capability of making complex decision is one of the most fascinating facets of human intelligence, especially if vague, judgemental, default or uncertain knowledge is involved. Unfortunately, most existing rule based forward chaining languages are not very suitable to simulate this aspect of human intelligence, because of their lack of support for approximate reasoning techniques needed for this task, and due to the lack of specific constructs to facilitate the coding of frequently reoccurring decision block to provide better support for the design and implementation of rule based decision support systems. A language called BIRBAL, which is defined on the top of CLIPS, for the specification of decision blocks, is introduced. Empirical experiments involving the comparison of the length of CLIPS program with the corresponding BIRBAL program for three different applications are surveyed. The results of these experiments suggest that for decision making intensive applications, a CLIPS program tends to be about three times longer than the corresponding BIRBAL program

Dispersion theoretic perturbation methods

The manuscript is organized as follows. In Chapter 1 the Chew-Mandelstam equations are derived and there is a general discussion of the partial wave disperison relations and the ODD ambiguity. The dispersion theoretic method of Dashen and Frautschi is presented in Chapter 2 both for single as well as multi channel case. PATCH's investigation of the Dashen-Frautschi method is reviewed in Chapter 5.One of the criticisms concerned the poor convergence of the equations in the presence of short range forces, while the other dealt with the problem of including contributions coming from infra-red divergent terms in the input to the DF expressions. In order to handle the first difficulty a method of modified perturbed dispersion relations is presented and applied to a model calculation in potential theory with good results. A modified Pagels-type procedure to solve the resulting equations for N and D functions is employed. This procedure is then applied to investigate the modified perturbed dispersion relations technique in the presence of long range forces. All this is done in Chapter 4.The modified Pagels-type procedure is employed in Chapter 5 to generate Regge trajectories, the object being to see whether reasonable it is possible to Reggeize the direct channel while using unreggeized input in the crossed channels' is shown that this is possible provided the cut-off is chosen suitably. In Chapter 6 the problem of infra-red divergent contributions to the input in the Dashen-Frautschi method is again treated along the lines of a suggestion due to SQUIRES. The procedure is carried out within the context of potential theory where it is shown to give satisfactory results. The full details of the method are exposed in an Appendix to this Chapter

Ultrafast optical control using the Kerr nonlinearity in hydrogenated amorphous silicon microcylindrical resonators

Microresonators are ideal systems for probing nonlinear phenomena at low thresholds due to their small mode volumes and high quality (Q) factors. As such, they have found use both for fundamental studies of light-matter interactions as well as for applications in areas ranging from telecommunications to medicine. In particular, semiconductor-based resonators with large Kerr nonlinearities have great potential for high speed, low power all-optical processing. Here we present experiments to characterize the size of the Kerr induced resonance wavelength shifting in a hydrogenated amorphous silicon resonator and demonstrate its potential for ultrafast all-optical modulation and switching. Large wavelength shifts are observed for low pump powers due to the high nonlinearity of the amorphous silicon material and the strong mode confinement in the microcylindrical resonator. The threshold energy for switching is less than a picojoule, representing a significant step towards advantageous low power silicon-based photonic technologies

Green's Functions and the Adiabatic Hyperspherical Method

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$-dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three-particles with s-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semi-analytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom-loss due to three-body recombination for a three-component fermi-gas of $^{6}$Li atoms is presented.Comment: 14 pages, 8 figures, 2 table