1. CMB 2015 (vol 58 pp. 673)
 Achter, Jeffrey; Williams, Cassandra

Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields
Consider a quartic $q$Weil polynomial $f$. Motivated by equidistribution
considerations, we define, for each prime $\ell$, a local factor
that
measures the relative frequency with which $f\bmod \ell$ occurs
as the
characteristic polynomial of a symplectic similitude over $\mathbb{F}_\ell$.
For a certain
class of polynomials, we show that the resulting infinite product
calculates the number of principally polarized abelian surfaces
over $\mathbb{F}_q$
with Weil polynomial $f$.
Keywords:abelian surfaces, finite fields, random matrices Category:14K02 

2. CMB 2015 (vol 59 pp. 211)
 Totik, Vilmos

Universality Under SzegÅ's Condition
This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesguepoints both
for the weight function $w$ and for $\log w$.
Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory Categories:42C05, 60B20, 30C85, 31A15 

3. CMB 2011 (vol 55 pp. 339)
 Loring, Terry A.

From Matrix to Operator Inequalities
We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$relations
being residually finite dimensional.
Our main result is a metatheorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$polynomials.
Keywords:$C*$algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional Categories:46L05, 47B99 

4. CMB 2011 (vol 55 pp. 172)
 Rhoades, B. E.

Hausdorff Prime Matrices
In this paper we give the form of every multiplicative Hausdorff
prime matrix, thus answering a longstanding open question.
Keywords:Hausdorff prime matrices Category:40G05 

5. CMB 2011 (vol 55 pp. 208)
6. CMB 2011 (vol 54 pp. 255)
 Dehaye, PaulOlivier

On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to
compute asymptotics for the determinants of its reductions to finite
rank. One can also consider how those asymptotics are affected when
shifting an initial set of rows and columns (or, equivalently,
asymptotics of their minors). Bump and Diaconis
obtained a formula for such shifts involving Laguerre polynomials and
sums over symmetric groups. They also showed how the Heine identity
extends for such minors, which makes this question relevant to Random
Matrix Theory. Independently, Tracy and Widom
used the WienerHopf factorization to
express those shifts in terms of products of infinite matrices. We
show directly why those two expressions are equal and uncover some
structure in both formulas that was unknown to their authors. We
introduce a mysterious differential operator on symmetric functions
that is very similar to vertex operators. We show that the
BumpDiaconisTracyWidom identity is a differentiated version of the
classical JacobiTrudi identity.
Keywords:Toeplitz matrices, JacobiTrudi identity, SzegÅ limit theorem, Heine identity, WienerHopf factorization Categories:47B35, 05E05, 20G05 

7. CMB 2008 (vol 51 pp. 57)
 Dobrowolski, Edward

A Note on Integer Symmetric Matrices and Mahler's Measure
We find a lower bound on the absolute value of the discriminant of
the minimal polynomial of an integral symmetric matrix and apply
this result to find a lower bound on Mahler's measure of related
polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure Categories:11C20, 11R06 

8. CMB 2006 (vol 49 pp. 560)
 Luijk, Ronald van

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

9. CMB 2005 (vol 48 pp. 267)
 Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.

Continuous Adjacency Preserving Maps on Real Matrices
It is proved that every adjacency preserving continuous map
on the vector space of real matrices of fixed size, is either a
bijective affine tranformation
of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if
the matrices are of square size, or its range is contained
in a linear subspace consisting of matrices of rank at most one
translated by some matrix $R$. The result
extends previously known
theorems where the map was assumed to be also injective.
Keywords:adjacency of matrices, continuous preservers, affine transformations Categories:15A03, 15A04. 

10. CMB 2001 (vol 44 pp. 270)
 Cheung, WaiShun; Li, ChiKwong

Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$.
The $c$numerical range of an $n \times n$ matrix $A$ is defined by
$$
W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\}
\text{ an orthonormal basis for } \IC^n \Bigr\},
$$
and the $c$numerical radius of $A$ is defined by $r_c (A) = \max
\{z : z \in W_c (A)\}$. We determine the structure of those linear
operators $\phi$ on algebras of block triangular matrices, satisfying
$$
W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or}
\quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.
$$
Keywords:linear operator, numerical range (radius), block triangular matrices Categories:15A04, 15A60, 47B49 

11. CMB 2000 (vol 43 pp. 448)
 Li, ChiKwong; Zaharia, Alexandru

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a
subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider
the {\it generalized matrix function} on $m\times m$ matrices $B =
(b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m
b_{j\sigma(j)}$ and the {\it generalized numerical range} of an
$n\times n$ complex matrix $A$ associated with $d^H$ defined by
$$
\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that }
X^*X = I_m\}.
$$
It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$.
We show that there exist normal matrices $A$ for which $\wmp(A)$ is
not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove
that a normal matrix $A $ with eigenvalues lying on a straight line
has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some
nonzero $\nu \in \IC$. These results extend those of Hu, Hurley
and Tam, who studied the special case when $2 \le m \le 3 \le n$
and $H = S_m$.
Keywords:convexity, generalized numerical range, matrices Category:15A60 

12. CMB 1998 (vol 41 pp. 10)
 Borwein, David

Simple conditions for matrices to be bounded operators on $l_p$
The two theorems proved yield simple yet reasonably
general conditions for triangular matrices to be bounded
operators on $l_p$. The theorems are applied to N\"orlund and
weighted mean matrices.
Keywords:Triangular matrices, NÃ¶rlund matrices, weighted means, operators, on $l_p$. Categories:47B37, 47A30, 40G05 
