3,854 research outputs found
Decomposing numerical ranges along with spectral sets
This note is to indicate the new sphere of applicability of the method
developed by Mlak as well as by the author. Restoring those ideas is summoned
by current developments concerning -spectral sets on numerical ranges
Optimal General Matchings
Given a graph and for each vertex a subset of the
set , where denotes the degree of vertex
in the graph , a -factor of is any set such that
for each vertex , where denotes the number of
edges of incident to . The general factor problem asks the existence of
a -factor in a given graph. A set is said to have a {\em gap of
length} if there exists a natural number such that and . Without any restrictions the
general factor problem is NP-complete. However, if no set contains a gap
of length greater than , then the problem can be solved in polynomial time
and Cornuejols \cite{Cor} presented an algorithm for finding a -factor, if
it exists. In this paper we consider a weighted version of the general factor
problem, in which each edge has a nonnegative weight and we are interested in
finding a -factor of maximum (or minimum) weight. In particular, this
version comprises the minimum/maximum cardinality variant of the general factor
problem, where we want to find a -factor having a minimum/maximum number of
edges.
We present an algorithm for the maximum/minimum weight -factor for the
case when no set contains a gap of length greater than . This also
yields the first polynomial time algorithm for the maximum/minimum cardinality
-factor for this case
Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers
This is the first of three papers that develop structures which are counted
by a "parabolic" generalization of Catalan numbers. Fix a subset R of
{1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are
determined by R. These are the "inverses" of (parabolic) multipermutations
whose multiplicities are determined by R. The standard forms of the ordered
partitions are refered to as "R-permutations". The notion of 312-avoidance is
extended from permutations to R-permutations. Let lambda be a partition of N
such that the set of column lengths in its shape is R or R union {n}. Fix an
R-permutation pi. The type A Demazure character (key polynomial) in x_1, ..,
x_n that is indexed by lambda and pi can be described as the sum of the weight
monomials for some of the semistandard Young tableau of shape lambda that are
used to describe the Schur function indexed by lambda. Descriptions of these
"Demazure" tableaux developed by the authors in earlier papers are used to
prove that the set of these tableaux is convex in Z^N if and only if pi is
R-312-avoiding if and only if the tableau set is the entire principal ideal
generated by the key of pi. These papers were inspired by results of Reiner and
Shimozono and by Postnikov and Stanley concerning coincidences between Demazure
characters and flagged Schur functions. This convexity result is used in the
next paper to deepen those results from the level of polynomials to the level
of tableau sets. The R-parabolic Catalan number is defined to be the number of
R-312-avoiding permutations. These special R-permutations are reformulated as
"R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless
R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the
publication data for the DMTCS journal (dates and volume/issue/number). This
is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v
UNITARILY INVARIANT NORMS ON FINITE VON NEUMANN ALGEBRAS
John von Neumann’s 1937 characterization of unitarily invariant norms on the n × n matrices in terms of symmetric gauge norms on Cn had a huge impact on linear algebra. In 2008 his results were extended to Ifactor von Neumann algebras by J. Fang, D. Hadwin, E. Nordgren and J. Shen. There already have been many important applications. The factor von Neumann algebras are the atomic building blocks from which every von Neumann algebra can be built. My work, which includes a new proof of the Ifactor case, extends von Neumann’s results to an arbitrary finite von Neumann algebra on a separable Hilberts space. A major tool is the theory of direct integrals.
The main idea is to associate to a von Neumann algebra R a measure space (Λ, λ) and a group G (R) of invertible measure-preserving transformations on L∞ (Λ, λ). Then we show that there is a one-to-one correspondence between the unitarily invariant norms on R and the normalized G (R)-symmetric gauge norms on (Λ, λ)
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