Integrability and supersymmetry of the supersymmetric extension of the sine-Gordon theory on a half-line are examined and the boundary potential which preserves both the integrability and supersymmetry on the bulk is derived. It appears that unlike the boundary bosonic sine-Gordon theory, integrability and supersymmetry strongly restrict the form and parameters of the boundary potential, so tha...

Preface I would like to thank my supervisor, Dr. Andy Gordon for his patience and many helpful suggestions during the last few years. Secondly, I thank my parents for supporting me, both emotionally and nancially during my time as a student. Finally, I thank Silvano Dal-Zilio and SSren Lassen for providing helpful comments on earlier drafts of this thesis. Except where otherwise stated in the t...

Kathy Liu Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138 and Center for Excellence in Education, Research Science Institute, 8201 Greensboro Drive, Suite 215, McLean, Virginia 22102 Roy G. Gordon John A. Paulson School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, Massachusetts 02138 and ...

A row-complete Latin square (RCLS) of order n, L, is a Latin square in which for any two distinct symbols, X, y, there exist i, j, 1 <i < ~1, 1 <:j < n 1 such that L(i,j) k x and L(i,j I1) =y. Column complete Latin squares are defmed analogously, and a square is complete if it is both. row and column complete. RCLS are used in statistics for the design of sequential experiments. A finite group ...

We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricateg...

A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized.

We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald’s symmetric functions, and the “n!” and “(n + 1)n−1” conjectures relating Macdonald polynomials to the characters of doubly-graded Sn modules. To make the treatment self-contained, we include background ma...

In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsing...