1. CMB Online first
 Kurdyka, Krzysztof; Paunescu, Laurentiu

Nuij type pencils of hyperbolic polynomials
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.
Keywords:hyperbolic polynomial, stable polynomial, determinantal representa tion, symmetric Toeplitz matrix Categories:15A15, 30C10, 47A56 

2. CMB Online first
 Liu, Zhongyun; Qin, Xiaorong; Wu, Nianci; Zhang, Yulin

The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices
It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=CS$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the GaussSeidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
GaussSeidel iteration, no matter whether the CSCS itself is
convergent or not.
Keywords:Hermitian positive definite, CSCS splitting, GaussSeidel splitting, iterative method, Toeplitz matrix Categories:15A23, 65F10, 65F15 

3. CMB Online first
 Reichstein, Zinovy; Vistoli, Angelo

On the dimension of the locus of determinantal hypersurfaces
The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$tuple $A := (A_1, \dots, A_r)$ of $n \times n$matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$tuple of $n \times n$matrices in general position,
then up to conjugacy, there are only finitely many $r$tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r1)n^2 + 1$.
Keywords:determinantal hypersurface, matrix invariant, $q$binomial coefficient Categories:14M12, 15A22, 05A10 

4. CMB 2016 (vol 59 pp. 585)
 Lin, Minghua

A Determinantal Inequality Involving Partial Traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$norms:
\[
1+\\mathbf{A}\_p\ge \\tr_1 \mathbf{A}\_p+\\tr_2 \mathbf{A}\_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]
Keywords:determinantal inequality, partial trace, block matrix Categories:47B65, 15A45, 15A60 

5. CMB 2014 (vol 58 pp. 196)
6. CMB 2012 (vol 57 pp. 25)
 Bourin, JeanChristophe; Harada, Tetsuo; Lee, EunYoung

Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.
Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities Categories:47A63, 15A45 

7. CMB 2011 (vol 56 pp. 92)
 Jacob, Benoît

On Perturbations of Continuous Maps
We give sufficient conditions for the following problem: given a
topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and
a continuous map $f$ from $X$ to $Y$, is it possible, by applying to
$f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not
meet $Z$? We also give a relative variant: if $f(X')$ does not meet
$Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged
on $X'$. We also develop a variant for continuous sections of
fibrations and discuss some applications to matrix perturbation
theory.
Keywords:perturbation theory, general topology, applications to operator algebras / matrix perturbation theory Category:54F45 

8. CMB 2009 (vol 52 pp. 145)
 Wang, Z.; Chen, J. L.

$2$Clean Rings
A ring $R$ is said to be $n$clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$module of rank at least 2 is $2$clean and that the ring $B(R)$
of all $\omega\times \omega$ row and columnfinite matrices over
any ring $R$ is $2$clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$clean rings, $2$good rings, free modules, row and columnfinite matrix rings, group rings Categories:16D70, 16D40, 16S50 

9. CMB 2009 (vol 52 pp. 95)
 Miranian, L.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory
In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
ChristoffelDarboux formulas are presented for the first time.
Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, ChristoffelDarboux Category:42C99 

10. CMB 2006 (vol 49 pp. 281)
 Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.

Correction to a Theorem on Total Positivity
A wellknown theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(z)$ generates a PF sequence.
Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur function Categories:15A48, 15A45, 15A57, 05E05 

11. CMB 2000 (vol 43 pp. 145)
 Chang, JaeHo; Lee, Sang Youl; Park, ChanYoung

On the 2Parallel Versions of Links
In this paper, we show that the absolute value of the signature of
the $2$parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2parallel version Category:57M25 
