1. CMB 2014 (vol 58 pp. 561)
 MartinezMaure, Yves

Plane Lorentzian and Fuchsian Hedgehogs
Parts of the BrunnMinkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of LorentzMinkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality Categories:52A40, 52A55, 53A04, 53B30 

2. CMB 2013 (vol 57 pp. 318)
 Huang, Zhaoyong

Duality of Preenvelopes and Pure Injective Modules
Let $R$ be an arbitrary ring and $()^+=\operatorname{Hom}_{\mathbb{Z}}(,
\mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers
and $\mathbb{Q}$ is the ring of rational numbers, and let
$\mathcal{C}$ be a subcategory of left $R$modules and $\mathcal{D}$
a subcategory of right $R$modules such that $X^+\in \mathcal{D}$
for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure
injective. Then a homomorphism $f: A\to C$ of left $R$modules with
$C\in \mathcal{C}$ is a $\mathcal{C}$(pre)envelope of $A$ provided
$f^+: C^+\to A^+$ is a $\mathcal{D}$(pre)cover of $A^+$. Some
applications of this result are given.
Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modules Categories:18G25, 16E30 

3. CMB 2011 (vol 55 pp. 783)
 Motallebi, M. R.; Saiflu, H.

Products and Direct Sums in Locally Convex Cones
In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$topologies and polars of the neighborhoods.
Keywords:product and direct sum, duality, locally convex cone Categories:20K25, 46A30, 46A20 

4. CMB 2011 (vol 56 pp. 424)
 Thom, Andreas

Convergent Sequences in Discrete Groups
We prove that a finitely generated group contains a
sequence of nontrivial elements that converge to the identity in
every compact homomorphic image if and only if the group is not
virtually abelian. As a consequence of the methods used, we show that a finitely generated
group satisfies Chu duality if and only if it is virtually abelian.
Keywords:Chu duality, Bohr topology Category:54H11 

5. CMB 2010 (vol 54 pp. 12)
 Bingham, N. H.; Ostaszewski, A. J.

Homotopy and the KestelmanBorweinDitor Theorem
The KestelmanBorweinDitor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.
Keywords:measure, category, measurecategory duality, differentiable homotopy Category:26A03 

6. CMB 2008 (vol 51 pp. 146)
 Zhou, Xiaowen

SteppingStone Model with Circular Brownian Migration
In this paper we consider the steppingstone model on a circle with
circular Brownian migration. We first point out a connection between
Arratia flow on the circle and the marginal distribution of this
model. We then give a new representation for the steppingstone
model using Arratia flow and circular coalescing Brownian motion.
Such a representation enables us to carry out some explicit
computations. In particular, we find the distribution for the first
time when there is only one type
left across the circle.
Keywords:steppingstone model, circular coalescing Brownian motion, Arratia flow, duality, entrance law Categories:60G57, 60J65 

7. CMB 2006 (vol 49 pp. 82)
 Gogatishvili, Amiran; Pick, Luboš

Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\f\_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$
Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality Categories:26D10, 46E20 
