1. CMB Online first
 ShahrokhiDehkordi, M. S.

Traceless maps as the singular minimizers in the multidimensional calculus of variations
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and
consider
the energy functional
\begin{equation*}
{\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x},
\end{equation*}
over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand
${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly
convex function
with bounded second derivatives. In this paper we address the
question of
regularity for solutions of the corresponding system of
EulerLagrange equations.
In particular we introduce a class of singular maps referred
to as traceless and
examine them as a new counterexample to the regularity of minimizers
of the energy
functional $\mathcal F[\cdot,\Omega]$ using a method based on
null Lagrangians.
Keywords:traceless map, singular minimizer, nullLagrangian Categories:49K27, 49N60, 49J30, 49K20 

2. CMB Online first
 Le Fourn, Samuel

Nonvanishing of central values of $L$functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic
(or rational) field of discriminant $D$ and Dirichlet character
$\chi$, if a prime $p$ is large enough compared to $D$, there
is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with
respect to the AtkinLehner involution $w_{p^2}$ such that $L(f
\otimes \chi,1) \neq 0$. This result is obtained through an estimate
of a weighted sum of twists of $L$functions which generalises
a result of Ellenberg. It relies on the approximate functional
equation for the $L$functions $L(f \otimes \chi, \cdot)$ and
a Petersson trace formula restricted to AtkinLehner eigenspaces.
An application of this nonvanishing theorem will be given in
terms of existence of rank zero quotients of some twisted jacobians,
which generalises a result of Darmon and Merel.
Keywords:nonvanishing of $L$functions of modular forms, Petersson trace formula, rank zero quotients of jacobians Categories:14J15, 11F67 

3. CMB 2016 (vol 60 pp. 111)
 Ghaani Farashahi, Arash

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the
abstract Plancherel (trace) formulas over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be
the normalized $G$invariant measure on $G/H$ associated to the
Weil's formula.
Then, we present a generalized abstract notion of Plancherel
(trace) formula for the Hilbert space $L^2(G/H,\mu)$.
Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula Categories:20G05, 43A85, 43A32, 43A40 

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 2009 (vol 52 pp. 416)
 Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor

Hamiltonian Properties of Generalized Halin Graphs
A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no
vertex of degree two, and $C$ is a cycle connecting the endvertices
of $T$ in the cyclic order determined by a plane embedding of $T$.
In this paper, we define classes of generalized Halin graphs, called
$k$Halin graphs, and investigate their Hamiltonian properties.
Keywords:$k$Halin graph, Hamiltonian, Hamiltonian connected, traceable Categories:05C45, 05C38 

6. CMB 2008 (vol 51 pp. 140)
 Rossi, Julio D.

First Variations of the Best Sobolev Trace Constant with Respect to the Domain
In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 

7. CMB 2000 (vol 43 pp. 60)
 Farkas, Daniel R.; Linnell, Peter A.

Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the
normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$
as a subgroup of finite index, then $U = G$. This result can be used
to give an alternative proof of a recent result of Marciniak and
Sehgal on units in the integral group ring of a crystallographic group.
Keywords:units, trace, finite conjugate subgroup Categories:16S34, 16U60 
