Abstract

This lecture will be a continuation of the previous one with illustrative examples.

Abstract

It will be shown that from the symmetries or conservation law multipliers of a nonlinear PDE system one can determine systematically whether it can be mapped invertibly to some linear PDE system. We show to find systematically such a mapping when one exists. 

It will also be shown that from the symmetries of a linear PDE system with variable coefficients, one can determine sytematically whether it can be mapped invertibly to some linear PDE system with constant coefficients. We show how to find systematically such a mapping when one exists.  This leads to the solution of the problem posed by Kolmogorov on when can a diffusion process be mapped into a Wiener process.

These results will be extended systematically to noninvertible mappings.

Abstract

Continutation of seminar of July 10th.

Abstract

Continuation of July 10th seminar.

Details

ABSTRACT:
In this thesis, we study a natural noncommutative lift of the ubiquitous Schur functions, called noncommutative Schur functions. These functions were introduced by Bessenrodt, Luoto and van Willigenburg and resemble Schur functions in many regards. We prove some new results for noncommutative Schur functions that are analogues of classical results, and demonstrate that the resulting combinatorics in this setting is equally rich. First we prove a Murnaghan-Nakayama rule for noncommutative Schur functions. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box adding operators on compositions. We proceed to give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give new noncommutative Pieri rules for noncommutative Schur functions that use the jdt operators.

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ABSTRACT: Self-avoiding walks appear ubiquitously in the mathematical study of linear polymers as it naturally captures their volume exclusion property. However, self-avoiding walks are very difficult to analyse with few rigourous results available.

In 2008, Alvarez et al. determined numerical results for the forces induced by a self-avoiding walk in an interactive slit. These results resembled the exact results for a directed model in the same setting by Brak et al., suggesting the physical consistency of directed walks as polymer models. In the directed walk model, three phases were identified in the infinite slit limit as well as the regions of attractive and repulsive forces induced by the polymer on the walls.

Via the kernel method, we extend the model to include two directed walks as a way to find exact enumerative results for studying the behaviour of ring polymers near an interactive wall, or walls.

We first consider a ring polymer near an interactive surface via two friendly walks that begin and end together along a single wall. We find an exact solution and provide a full analysis of the phase diagram, which admits three phase transitions.

The model is extended to include a second wall so that two friendly walks are confined in an interactive slit. We find and analyse the exact solution of two friendly walks tethered to different walls where single interactions are permitted. That is, each walk interacts with the wall it is tethered to. This model exhibits repulsive force only in the parameter space. While these results differ from the single polymer models, they are consistent with Alvarez et al.

Finally, we consider the model with double interactions, where each walk interacts with both walls. We are unable to find exact solutions via the kernel method. Instead, we use transfer matrices to obtain numerical results to identify regions of attractive and repulsive forces. The results we obtain are qualitatively similar to those presented in Alvarez et al. Furthermore, we provide evidence that the zero force curve does not satisfy any simple polynomial equation.

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