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Full-Text Articles in Physical Sciences and Mathematics
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang
CMC Senior Theses
In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative …
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
HMC Senior Theses
Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this …