1. CMB Online first
 Koşan, Tamer; Sahinkaya, Serap; Zhou, Yiqiang

Additive maps on units of rings
Let $R$ be a ring. A map $f: R\rightarrow R$
is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$
of $R$.
Here a map $f: R\rightarrow R$ is called unitadditive if $f(u+v)=f(u)+f(v)$
for all units $u$ and $v$ of $R$. Motivated by a recent result
of Xu, Pei and Yi
showing that, for any field $F$, every
unitadditive map of ${\mathbb M}_n(F)$ is additive for all $n\ge
2$, this paper is about the question when every unitadditive
map of a ring is additive. It is proved that every unitadditive
map of a semilocal ring $R$ is additive if and only if either
$R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong
\mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal
ring $R$, every unitadditive map of ${\mathbb M}_n(R)$ is additive
for all $n\ge 2$. These results are further extended to rings
$R$ such that $R/J(R)$ is a direct product of exchange rings
with primitive factors Artinian. A unitadditive map $f$ of a
ring $R$ is called unithomomorphic if $f(uv)=f(u)f(v)$ for all
units $u,v$ of $R$. As an application, the question of when every
unithomomorphic map of a ring is an endomorphism is addressed.
Keywords:additive map, unit, 2sum property, semilocal ring, exchange ring with primitive factors Artinian Categories:15A99, 16U60, 16L30 

2. CMB 2014 (vol 57 pp. 780)
 Erzakova, Nina A.

Measures of Noncompactness in Regular Spaces
Previous results by the author on the connection
between three of measures
of noncompactness obtained for $L_p$, are extended
to regular spaces of measurable
functions.
An example of advantage
in some cases one of them in comparison with another is given.
Geometric characteristics of regular spaces are determined.
New theorems for $(k,\beta)$boundedness of partially additive
operators are proved.
Keywords:measure of noncompactness, condensing map, partially additive operator, regular space, ideal space Categories:47H08, 46E30, 47H99, 47G10 

3. CMB 2013 (vol 57 pp. 877)
 Schoen, Tomasz

On Convolutions of Convex Sets and Related Problems
We prove some results concerning covolutions, the
additive energy and sumsets of convex sets and its generalizations. In
particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq
\mathbb R$ has
the property that for every fixed
$1\leqslant d\lt n,$ all differences $a_ia_{id}$, $d\lt i\lt n,$ are distinct, then
$A+A\gg A^{3/2+c}$ for a constant $c\gt 0.$
Keywords:convex sets, additive energy, sumsets Category:11B99 

4. CMB 2010 (vol 54 pp. 180)
 Spurný, J.; Zelený, M.

Additive Families of Low Borel Classes and Borel Measurable Selectors
An important conjecture in the theory of Borel sets in nonseparable
metric spaces is whether any pointcountable Boreladditive family in
a complete metric space has a $\sigma$discrete refinement. We confirm the conjecture for
pointcountable $\mathbf\Pi_3^0$additive families, thus generalizing results of
R. W. Hansell and the first author. We apply this result to the
existence of Borel measurable selectors for multivalued mappings of
low Borel complexity, thus answering in the affirmative a particular
version of a question of J. Kaniewski and R. Pol.
Keywords:$\sigma$discrete refinement, Boreladditive family, measurable selection Categories:54H05, 54E35 

5. CMB 2009 (vol 52 pp. 295)
 P{\l}otka, Krzysztof

On Functions Whose Graph is a Hamel Basis, II
We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, \emph{i.e.,} for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.
Keywords:Hamel basis, additive, Hamel functions Categories:26A21, 54C40, 15A03, 54C30 

6. CMB 2008 (vol 51 pp. 399)
 Meng, Xianmeng

Linear Equations with Small Prime and Almost Prime Solutions
Let $b_1, b_2$ be any integers such that
$\gcd(b_1, b_2)=1$ and $c_1b_1<b_2\leq c_2b_1$, where
$c_1, c_2$ are any given positive constants. Let $n$ be any
integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote
any integer with no more than $k$ prime factors, counted according
to multiplicity. In this paper, for almost all $b_2$, we prove (i)
a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$
is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$
whenever both $b_i$ are positive, and (ii) a sharp upper bound for the
least solutions $p, m$ of the above equation whenever $b_i$ are
not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.
Keywords:sieve method, additive problem Categories:11P32, 11N36 
