2,443 research outputs found
Admissible orders on quotients of the free associative algebra
An admissible order on a multiplicative basis of a noncommutative algebra A is a term order satisfying additional conditions that allow for the construction of Grobner bases for A -modules. When A is commutative, a finite reduced Grobner basis for an A -module can always be obtained, but when A is not commutative this is not the case; in fact in many cases a Grobner basis theory for A may not even exist.
E. Hinson has used position-dependent weights, encoded in so-called admissible arrays, to partially order words in the free associative algebra in a way which produces a length-dominant admissible order on a particular quotient of the free algebra, where the ideal by which the quotient is taken is an ideal generated by pure homogeneous binomial differences and is determined by the array A.
This dissertation investigates the properties of two large classes of admissible arrays A. We prove that weight ideals associated to arrays in the first class are finitely generated and we describe the generating sets. We exhibit instances of trivial and nontrivial finitely generated weight ideals associated to arrays in the second class and we partially characterize the corresponding arrays. We also exhibit instances of weight ideals associated to arrays in the second class which do not admit a finite generating set. We identify an algebro-combinatorial property on weight ideals, which we call saturation, that is connected to finite generation. In addition, we look at actions of the multiplicative monoid generated by the set of transvections and diagonal matrices with non-negative entries on the set of equivalence classes of admissible arrays under order-isomorphism and we analyze the stabilizers and orbits of these actions
Adventures in Invariant Theory
We provide an introduction to enumerating and constructing invariants of
group representations via character methods. The problem is contextualised via
two case studies arising from our recent work: entanglement measures, for
characterising the structure of state spaces for composite quantum systems; and
Markov invariants, a robust alternative to parameter-estimation intensive
methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
The Dirichlet Markov Ensemble
We equip the polytope of Markov matrices with the normalized
trace of the Lebesgue measure of . This probability space
provides random Markov matrices, with i.i.d. rows following the Dirichlet
distribution of mean . We show that if \bM is such a random
matrix, then the empirical distribution built from the singular values
of\sqrt{n} \bM tends as to a Wigner quarter--circle
distribution. Some computer simulations reveal striking asymptotic spectral
properties of such random matrices, still waiting for a rigorous mathematical
analysis. In particular, we believe that with probability one, the empirical
distribution of the complex spectrum of \sqrt{n} \bM tends as to
the uniform distribution on the unit disc of the complex plane, and that
moreover, the spectral gap of \bM is of order when is
large.Comment: Improved version. Accepted for publication in JMV
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
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