Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2012 (vol 65 pp. 1020)
Monotone Hurwitz Numbers in Genus Zero Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.
Keywords:Hurwitz numbers, matrix models, enumerative geometry Categories:05A15, 14E20, 15B52 |
2. CJM 2010 (vol 63 pp. 3)
Free Bessel Laws
We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.
Keywords:Poisson law, Bessel function, Wishart matrix, quantum group Categories:46L54, 15A52, 16W30 |
3. CJM 2010 (vol 62 pp. 758)
General Preservers of Quasi-Commutativity Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.
Keywords:general preservers, matrix algebra, quasi-commutativity Categories:15A04, 15A27, 06A99 |
4. CJM 2008 (vol 60 pp. 1050)
Adjacency Preserving Maps on Hermitian Matrices Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.
Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices Categories:15A03, 15A04, 15A57, 15A99 |
5. CJM 2007 (vol 59 pp. 638)
Distance from Idempotents to Nilpotents We give bounds on the distance from a non-zero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.
Keywords:operator, matrix, nilpotent, idempotent, projection Categories:47A15, 47D03, 15A30 |
6. CJM 2004 (vol 56 pp. 776)
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 |
7. CJM 2003 (vol 55 pp. 91)
Some Convexity Features Associated with Unitary Orbits Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex
Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}
(C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices
unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$
such that every matrix in the convex hull of $\mathcal{U}(C)$ can
be written as the average of two matrices in $\mathcal{U}(C)$. The
result is used to study spectral properties of submatrices of
matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U}
(C)$ under linear transformations, and some related questions
concerning the joint $C$-numerical range of Hermitian matrices.
Analogous results on real symmetric matrices are also discussed.
Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical range Categories:15A60, 15A42 |