Physical Sciences and Mathematics Commons^™

Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola

Dartmouth Scholarship

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We …

Invariant And Coinvariant Spaces For The Algebra Of Symmetric Polynomials In Non-Commuting Variables, Francois Bergeron, Aaron Lauve

Mathematics and Statistics: Faculty Publications and Other Works

We analyze the structure of the algebra K⟨x⟩Sn of symmetric polynomials in non-commuting variables in so far as it relates to K[x]Sn, its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of K⟨x⟩Sn analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.

Résumé. Nous analysons la structure de l'algèbre K⟨x⟩Sn des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau K[x]Sn des polynômes symétriques en des variables …

Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky

CMC Faculty Publications and Research

Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of K^N where N≥ 2. Let Z_K be a union of varieties defined over K such that V ⊈ Z_K. We prove the existence of a point of small height in V \ Z_K, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing Z_K, where dependence on …

An Investigation Of Kurosh's Theorem, Keith Anthony Earl

Theses Digitization Project

The purpose of this project will be an exposition of the Kurosh Theorem and the necessary and suffcient condition that A must be algebraic and satisfy a P.I. to be locally finite.