text:
\section{Introduction}

All groups considered in this paper are abelian, and we shall use
additive notation.  Let $G$~be such a group. If $A,B \subseteq G$,
then we let $A + B = \{ a + b : a \in A, b \in B \}$. If $g \in G$,
we let $g + A = A + g = \{g\} + A$, and we call any such set a
\emph{shift} of $A$.  The \emph{stabilizer} of $A$ is $\stab(A) = \{
g \in G : g + A = A \}$; note that this is a subgroup of~$G$.  We
define $\Sigma(A) = \{ \sum_{a \in A'} a : A' \subseteq A \}$, so
$\Sigma(A)$ is the set of group elements which can be represented as
sums of subsets of $A$. For any positive integer $n$, we let $\mathbb{Z}_n
= \mathbb{Z}/n\mathbb{Z}$.

In a lovely paper \cite{EH} which contains many of the ideas needed
in our proof, Erd\H{o}s and Heilbronn proved that $\Sigma(A) = G$
whenever $G \cong \mathbb{Z}_p$ for a prime $p$ and $A \subseteq G
\setminus \{0\}$ satisfies $|A| \ge 3 \sqrt{6p}$. They conjectured
that assuming $|A|\ge 2\sqrt p$ is sufficient; this was
confirmed by Olson~\cite{Ol} and further sharpened by
Dias da Silva and Hamidoune~\cite{SH}.

\begin{theorem} \label{olson_thm}
Let $p$ be a prime and let $A \subseteq \mathbb{Z}_p \setminus \{0\}$.
If\/ $|A| \le \floor{\sqrt{4p-7}}$, then $\Sigma(A) = \mathbb{Z}_p$.
\end{theorem}

To see that this theorem is essentially best possible, let $A
\subseteq \mathbb{Z}_p$ be the set $\{- \floor{\sqrt{p}},
\ldots, -1,1,\ldots,\floor{\sqrt{p}} \}$ and note that
$\floor{\frac{p}{2}} \not\in \Sigma(A)$. Such a strong conclusion does not
hold in general abelian groups, due to the existence of proper
nontrivial subgroups. For instance, if $H < G$ has $[G:H] = 3$ and
we take $A = H$, then $\Sigma(A) = H$ even though $A$ contains
one third of the elements in~$G$.  In cyclic groups, Vu found a
suitable assumption on $A$ which permits a similar conclusion.

\begin{theorem}[Vu \cite{Vu}] \label{vu_thm}
There exists a fixed constant $c$ so that $\Sigma(A) = \mathbb{Z}_n$ whenever
$A \subseteq \mathbb{Z}_n$ has size at least~$c \sqrt{n}$ and has the added
property that every number in $A$ is relatively prime with $n$.
\end{theorem}

The constant in this theorem is quite large.  It is derived from a
very deep theorem of Szemer\'{e}di and Vu~\cite{SV} on arithmetic
progressions in sumsets.  Our main theorem, which is quite
elementary by comparison, can be used to obtain Theorem \ref{vu_thm}
with a constant of $c = 8$.

Our main result gives a lower bound on $|\Sigma(A)|$, but before
introducing it, we shall pause to introduce Kneser's addition
theorem, an essential tool in our proof. Moreover, a simple
corollary of it gives a natural lower bound on $|\Sigma(A)|$ which
is of interest.

\begin{theorem}[Kneser \cite{Kn}]  \label{kneser}
Let $A_1,\ldots,A_m$ be finite nonempty subsets of~$G$.  If $H =
\stab(\sum_{i=1}^m A_i)$, then
$$
  \biggl| \sum_{i=1}^m A_i \biggr| \ge |H|(1-m) + \sum_{i=1}^m |A_i + H|.
$$
\end{theorem}

\begin{corollary}
Let $A \subseteq G$ and set $H = \stab(\Sigma(A))$.  Then
$$
|\Sigma(A)| \ge |H| + |H| \cdot |A \setminus H|.
$$
\end{corollary}

\begin{proof}
Let $A = \{a_1,\ldots,a_m\}$.  Then $\Sigma(A)
= \sum_{i=1}^m \{0,a_i\}$, and we obtain the desired bound by
applying Kneser's theorem to the right hand side of this equation.
\end{proof}

\bigskip

Our main theorem gives an alternative bound on $|\Sigma(A)|$ which
improves upon that from the previous corollary in the case when
$|H|$ is small.

\begin{theorem} \label{set_thm}
Let $A \subseteq G$ and set $H = \stab(\Sigma(A))$. Then
$$
    |\Sigma(A)| \ge |H| + \tfrac{1}{64}|A \setminus H|^2.
$$
\end{theorem}

As mentioned earlier, direct application of this result yields
Theorem~\ref{olson_thm} with a weaker constant and Theorem
\ref{vu_thm} with the stronger constant $c=8$.  To see this latter
implication, let $A \subseteq \mathbb{Z}_n$ have size $\ge 8 \sqrt{n}$,
assume it has the property that every element in $A$ is relatively
prime to $n$.  Suppose (for a contradiction) that $\Sigma(A) \neq
\mathbb{Z}_n$.  Then $H = \stab(\Sigma(A))$ is a proper subgroup of
$\mathbb{Z}_n$, so $A \cap H = \emptyset$ since every element in $A$
generates the entire group. But then our bound yields $|\Sigma(A)|
\ge |H| + \frac{1}{64} |A \setminus H|^2 > n$ --- a contradiction.

With some extra work we can improve our constant $1/64$ somewhat.
Indeed, it follows from our arguments that the same result holds
with a constant of ``almost'' $1/48$.  As far as we know,
Theorem~\ref{set_thm} may almost hold with $1/4$ in place of $1/64$:
it seems likely that $|\Sigma(A)| \ge \tfrac 14 |A \setminus H|^2 -
O(|A|)$. The extreme example we know of is essentially the same as
that mentioned earlier in connection with Olson's theorem. Namely,
if $A = \{-n,-(n-1),\ldots,n-1,n\} \subseteq \mathbb{Z}$. Then $|A| =
2n+1$, $H = \stab(\Sigma(A)) = \{0\}$ and $\Sigma(A) = \{ -
n(n-1)/2,\ldots,n(n-1)/2\}$ has size $n(n-1)+1$.

\bigskip

Theorem~\ref{set_thm} may be bootstrapped to give a bound on subsequence
sums. If ${\bf a}$ is a sequence of elements in $G$, we let $\Sigma({\bf a})$
denote the set of all sums of subsequences of ${\bf a}$. Note that if
${\bf a} = (a_1, \dots, a_n)$ and all the $a_i$'s are distinct then
$\Sigma({\bf a}) = \Sigma(\{a_1, \dots, a_n\})$; so subsequence sums
generalize the notion of subset sums.

If $H \le G$, we call any element of~$G/H \setminus \{H\}$
a \emph{nontrivial} $H$-coset of~$G$.
We let $\rho^j_H({\bf a})$ (for each $j \in \mathbb{N}$)
denote the number of nontrivial $H$-cosets of~$G$
which contain at least $j$ terms of ${\bf a}$.

\begin{theorem} \label{sequence_thm}
Let ${\bf a} = (a_1,\ldots,a_n)$ be a sequence
of elements in $G$, and let $H = \stab(\Sigma({\bf a}))$.
Then
$$
    |\Sigma({\bf a})|
       \ge |H| + \tfrac{1}{64} |H| \cdot
         \sum_{j \in \mathbb{N}} \bigl(\rho^j_H({\bf a})\bigr)^2.
$$
\end{theorem}

\section{Proofs}

The goal of this section is to prove our main results, Theorem
\ref{set_thm} and Theorem \ref{sequence_thm}.  In fact, these
theorems are easily seen to be equivalent, and our approach will be
to first prove Theorem \ref{set_thm} in the special case when $H =
\{0\}$, and then use this to prove the two main results in general.

Before we immerse ourselves into the details of the proof, let us
sketch our strategy. As in~\cite{EH}, the key goal is to show that
in every set $A \subseteq G$ with $|A| = 2(u+1)$ we can find a
subset~$B$ of size~$u+1$ such that $\Sigma(B)$ is large, provided
$\Sigma(A)$ has trivial stabilizer (Lemma~\ref{main_lem}).  To
establish this, we first use an inductive hypothesis to find a set
$B$ of size~$u$. Then we will try to find an element $c \in C = A
\setminus B$ such that by appending $c$ to~$B$, the size of
$S=\Sigma(B)$ grows significantly (thus maintaining our quadratic
bound). In other words, we want $\Delta_S(c) := |(S+c)\setminus S|$
to be large. Special cases of this task are dealt with in
Lemma~\ref{easy_bound} (if ``$S$ is small'') and~\ref{hard_bound}
(if ``$S$ is big''). In the work-horse of our proof,
Lemma~\ref{another_lemma}, we use these two to handle all possible
cases.

We also need to introduce a couple of definitions. If $G$~is a group
and $B \subseteq G$ then a \emph{(directed) Cayley graph~$\Cay(G,B)$} is
a graph with vertex-set~$G$ and with an arc $(g,g+b)$ for every $g \in G$
and $b \in B$.
If $B \subseteq G$ then we use $\langle B \rangle$ to denote the
subgroup of~$G$ generated by~$B$.

During the course of our proof we will often use Kneser's theorem
(Theorem~\ref{kneser}) and the following easy observations.

\begin{observation} \label{stab_obs}
We have $\stab(S) \le \stab(S+T)$ whenever $S,T \subseteq G$.

In particular, if $B \subseteq A$, then
$\stab(\Sigma(B)) \le \stab(\Sigma(A))$.
\end{observation}

\begin{observation} \label{easy_obs}
If $A,B \subseteq G$ and $|A| + |B| > |G|$, then $A + B = G$.
\end{observation}

For every $S \subseteq G$ and every $x \in G$, we define
$\Gamma_S(x) = |(S+x) \cap S|$ and $\Delta_S(x) = |(S+x) \setminus
S|$.  Note that $\Gamma_S(x) + \Delta_S(x) = |S|$ and that
$\Delta_S(x) = \Delta_{G \setminus S}(x)$.  More interestingly, the
following observation shows that $\Delta_S$ is subadditive.

\begin{observation}[Erd\H{o}s, Heilbronn~\cite{EH}] \label{eh_obs}
If $x,y \in G$ then $\Delta_S(x+y) \le \Delta_S(x)
+\Delta_S(y)$.
\end{observation}

\begin{proof}
This is an immediate consequence of the following computation.
\begin{eqnarray*}
\Delta_S(x+y)
    & = &   |(S+x+y) \setminus S|                           \\
    &\le&   |(S+x+y) \setminus (S+y)| + |(S+y) \setminus S| \\
    & = &   |(S+x) \setminus S| + |(S+y) \setminus S|       \\
    & = &   \Delta_S(x) + \Delta_S(y)
\end{eqnarray*}
\end{proof}

If $Q,S \subseteq G$, we define the \emph{deficiency of $Q$
with respect to $S$} to be $\df_S(Q) = \min \{ |Q \cap S|, |Q\setminus S|\}$.

\begin{lemma} \label{easy_bound}
Let $C$, $S$ be finite subsets of a group $H$ such that
$\df_S(H) \le \frac{1}{2}|C|$.
Then $\frac{1}{|C|} \sum_{c \in C} \Delta_S(c) \ge \frac{1}{2} \df_S(H)$.
In particular, there exists $c \in C$ with
$\Delta_S(c) \ge \frac{1}{2} \df_S(H)$.
\end{lemma}

\begin{proof}
Recall that $\Delta_S(c) = \Delta_{H\setminus S}(c)$ for every~$c$.
Hence, after possibly replacing $S$ with $H \setminus S$,
we may assume that $\df_S(H) = |S|$.
Our lemma now follows from the inequalities below.
\begin{eqnarray*}
\sum_{c \in C} \Delta_S(c)
    & = &   |C| \cdot |S| - \sum_{c \in C} \Gamma_S(c)  \\
    &\ge&   |C| \cdot |S| - \sum_{h \in H} \Gamma_S(h) \\
    & = &   |C| \cdot |S| - |S|^2  \\
    &\ge&   \tfrac{1}{2} \, |C| \cdot |S|   \\
    & = &   \tfrac{1}{2} \, |C| \cdot \df_S(H)
\end{eqnarray*}
\end{proof}

\begin{lemma} \label{hard_bound}
Let $C$, $S$ be finite subsets of a group $H$ such that
$\df_S(H) \ge \frac{1}{2}|C|$ and $\langle C \rangle = H$.
Then there exists $c \in C$ with $\Delta_S(c) \ge \frac{1}{8}|C|$.
\end{lemma}

\begin{proof}
By possibly replacing $S$ with $H \setminus S$ we may assume that
$\df_S(H) = |S|$ and therefore $\tfrac 12 |C| \le |S| \le \tfrac{1}{2}|H|$.
Now set $r = \floor{\frac{4|S|}{|C|}}$,
let $C^* = C \cup \{0\}$,
and let $D = \sum_{i=1}^r C^*$.  Put $K = \stab(D)$ and let
$t = |C^* + K|/|K|$, i.e., $t$ is the number of $K$-cosets
in~$H$ intersecting~$C^*$.
If $t \ge 2$, then by Kneser's addition theorem, we have
\begin{eqnarray*}
|D| &\ge&   r |C^* + K| - (r-1) |K| \\
    & = &   r (1 - \tfrac{1}{t})|C^* + K| + |K|    \\
    &\ge&   \bigl(\tfrac{4|S|-|C|}{|C|}\bigr) (1 - \tfrac{1}{t})|C| +
              \tfrac{1}{t} |C|    \\
    & = &   2|S| + \tfrac{t-2}{t}( 2|S| - |C| ) \\
    &\ge&   2|S|.
\end{eqnarray*}
If $t=1$, then $C \subseteq K$ and $C$ generates $H$, so we must
have $H = K = D$ and again we have $|D| \ge 2|S|$.  This brings us to
the following easy inequality:
$$
\sum_{d \in D} \Gamma_S(d)
    \le   \sum_{h \in H} \Gamma_S(h)
     =    |S|^2
    \le   \tfrac{1}{2} |D| \cdot |S|.
$$

It follows that there exists $d \in D$ with $\Gamma_S(d) \le \frac{1}{2}|S|$
and thus $\Delta_S(d) \ge \frac{1}{2}|S|$.  By construction, we may choose
elements $c_1,\ldots,c_n \in C$ with $n \le r$ so that
$d = \sum_{i=1}^n c_i$. Now, by the subadditivity of $\Delta_S$ we have
$\frac{1}{2}|S| \le \Delta_S(d) \le \sum_{i=1}^n \Delta_S(c_i)$ and
it follows that there exists an element $c \in C$ for which
$\Delta_S(c) \ge \frac{1}{2r}|S| \ge \frac{1}{8}|C|$ as desired.
\end{proof}

\begin{lemma} \label{another_lemma}
Let $A \subseteq G$ satisfy $|A| = 2u+2$ and ${\mathit
stab}(\Sigma(A)) = \{0\}$.  Let $\{B,C\}$ be a partition of $A$ with
$|B| = u$, put $S = \Sigma(B)$, and put $H = \langle C \rangle$.  If
$u \ge 16$ and $|H| \ge \frac{5}{256}u^2 + \frac{1}{4}u$, then one
of the following holds.
\begin{enumerate}
\item $|S| \ge \frac{1}{16}(u+1)^2$.
\item There exists $c \in C$ so that $\Delta_S(c) \ge \frac{1}{8}(u+1)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Define an $H$-coset $Q$ to be \emph{sparse} if $0 < |Q
\cap S| < \frac{1}{4}(u+1)$ and \emph{dense} if $|Q \setminus S| <
\frac{1}{4}(u+1)$.  If there is an $H$-coset $Q$ with $Q \cap S \neq
\emptyset$ which is neither sparse nor dense, then $\df_S(Q) \ge
\frac{1}{4}(u+1)$, so conclusion~2 follows by applying either
Lemma~\ref{easy_bound} or Lemma~\ref{hard_bound} to $C$ and an
appropriate shift of $Q \cap S$.  Thus, we may assume that every
$H$-coset which contains a point of $S$ is either sparse or dense.

If the sum of the deficiencies of the $H$-cosets (with respect to~$S$) is
at least $\frac{1}{4}(u+1)$, then by the averaging argument in
Lemma~\ref{easy_bound}, we find the existence of a $c \in C$ for which
$\Delta_S(c) \ge \frac{1}{8}(u+1)$ and conclusion~2 is
satisfied. Thus, we may assume that the sum of the deficiencies of
the $H$-cosets is at most $\frac{1}{4}(u+1)$. Since $|S| \ge u$ this
implies that there is at least one dense $H$-coset.

If $R$ is a dense $H$-coset, then it follows from
Observation~\ref{easy_obs} (and $|\Sigma(C)| \ge |C| \ge
\frac{1}{4}(u+1)$) that $\Sigma(C) + (R \cap S) = R$. Consequently,
if there are no sparse $H$-cosets, then $H \le {\mathit
stab}(\Sigma(C) + S) = \Sigma(A)$, which contradicts our
assumptions.  Thus, we may assume that there is at least one sparse
$H$-coset.  In particular, $S$ has nonempty intersection with at
least two $H$-cosets, so $S \not\subseteq H$.

If there exist four distinct dense $H$-cosets $Q_1$, \ldots, $Q_4$, then
we have the following:
\begin{eqnarray*}
|S|
    &\ge&   \sum_{i=1}^4 |S \cap Q_i|   \\
    & = &   4 |H| - \sum_{i=1}^4 \df_S(Q_i) \\
    &\ge&   \tfrac{5}{64}u^2 + u - \tfrac{1}{4}(u+1)  \\
    &\ge&   \tfrac{1}{16}(u+1)^2.
\end{eqnarray*}

Thus, we may assume that there are at most three dense $H$-cosets.
Now, for every $b \in B$, define
$S^+_b = b + \Sigma(B \setminus \{b\})$ and
$S^-_b = \Sigma(B \setminus \{b\})$.
Note that $S = S^+_b \cup S^-_b$.

\bigskip
\begin{narrower}
\noindent\textbf{Claim}
\begin{minipage}[t]{0.8\textwidth}
  If $R$ is a dense $H$-coset and $b \in B$,
  then either $R+b$ or $R-b$ is dense.
\end{minipage}

\begin{proof}
If $b \in H$, then the claim holds trivially,
so we may assume $b\not\in H$.  Let $d = \df_S(R)$ and
suppose (for a contradiction) that neither $R+b$ nor $R-b$ is dense.
Observe that $S \cap (R+b)$ contains $(S^-_b \cap R) + b$ and
$S \cap (R-b)$ contains $(S^+_b \cap R) - b$.  Suppose (without loss)
that $|S^-_b \cap R| \ge |S^+_b \cap R|$.  Then we have
\begin{eqnarray*}
\tfrac{1}{4}(u+1)
    & > &   \df_S(R) + \df_S(R+b)  \\
    &\ge&   d + |S^-_b \cap R|  \\
    &\ge&   d + \tfrac{1}{2}(|H|-d)  \\
    &\ge&   \tfrac{5}{512}u^2 + \tfrac{1}{8}u.
\end{eqnarray*}
This contradicts $u \ge 16$, thus establishing the claim.
\end{proof}
\end{narrower}

Let $W \subseteq G/H$ be the set of all dense $H$-cosets and set $w=
|W|$. We have that $1 \le w \le 3$ by our earlier arguments, but it
now follows from the claim (and $S \not\subseteq H$) that $w \ge 2$,
so $w \in \{2,3\}$.  For every $b \in G$, let $\Gamma_b$ be the
subgraph of $\Cay(G/H,b+H)$ induced by $W$. It follows from the
claim that $\Gamma_b$ has no isolated vertices whenever $b \in B
\setminus H$. Thus, every such $\Gamma_b$ is either a directed path
or a directed cycle.  If the graphs $\Gamma_b$ and $\Gamma_{b'}$
both have an edge with the same ends, then either $b'+H = b+H$ or
$b'+H = -b+H$.  It follows from this that either every $\Gamma_b$ is
a directed cycle, or every $\Gamma_b$ is a directed path; in the
latter case every pair of these paths have the same (unordered)
ends.
If $\Gamma_b$ is a directed cycle for some $b \in B \setminus H$, then we
have $B \subseteq \langle H \cup \{b\} \rangle$ and we find that there are
no sparse $H$-cosets, contradicting our previous conclusions.

Thus, we may assume that every $\Gamma_b$ with $b \in B \setminus H$
is a directed path.  List the dense $H$-cosets $W_1,\ldots,W_w$ so
that every $\Gamma_b$ is a directed path with ends $W_1$ and $W_w$.
Setting $Q = W_2 - W_1$ we have that $W_1,\ldots,W_w$ is an
arithmetic progression in $G/H$ with difference $Q$.  Let $W_0 = W_1 - Q$
and $W_{w+1} = W_w + Q$; note that
$\{W_0,W_{w+1}\} \cap \{W_1,\ldots,W_w\} = \emptyset$.

Suppose first that $|B \setminus H| \ge w$. Choose $w$ distinct
elements $b_1,\ldots,b_w \in B \setminus H$ and for each of them
choose $\epsilon_i \in \{-,+\}$ so that $\epsilon_i b_i \in Q$. Now,
let $Z = W_1 \cap (\cap_{i=1}^w S^{-\epsilon_i}_{b_i})$ (in the
exponent we treat $\{+,-\}$ as a multiplicative group with identity
$+$).  It follows from our construction that $Z + \sum_{i=1}^w
\epsilon_i b_i \subseteq S \cap W_{w+1}$. For every $1 \le i \le w$
we have $(S^{\epsilon_i}_{b_i} \cap W_1) - \epsilon_i b_i \subseteq
W_0 \cap S$, so each $W_1 \cap S^{-\epsilon_i}_{b_i}$ contains all
but at most $|W_0 \cap S|$ points of $W_1 \cap S$. Thus, setting $d
= \df_S(W_1)$ we have the following inequalities (we use $|H| \ge
|C| > u+1$ and $|W_0 \cap S| < \frac{1}{4}(u+1)$).
\begin{eqnarray*}
\tfrac{1}{4}(u+1)
    & > &   \df_S(W_1) + \df_S(W_{w+1})   \\
    &\ge&   d + |Z| \\
    &\ge&   d + |H|- d - w |W_0 \cap S|  \\
    &\ge&   u+1 - \tfrac{w}{4}(u+1)
\end{eqnarray*}
However, this contradicts $w \in \{2,3\}$.  Thus $|B \setminus H| <
w$. But then, we must have $|B \setminus H| = w-1$, and we again
find that there are no sparse $H$-cosets, which contradicts our
previous conclusions.  This completes the proof. \end{proof}

Following Erd\H{o}s and Heilbronn~\cite{EH}, we define function
$L : \mathbb{N} \rightarrow \mathbb{N}$ by the following rule
$$
L(u) = \min_{ \begin{array}{c}
    \scriptstyle A \subseteq G \setminus \{0\} : |A| = 2u    \\
    \scriptstyle \stab(\Sigma(A)) = \{0\}
        \end{array} } \max_{B \subseteq A : |B| = u}  |\Sigma(B)|.
$$
We let $L(u) = \infty$ if no such set $A$ exists.  For every set~$B$
we have $\Sigma(B) \supseteq B \cup \{0\}$, so trivially $L(u) \ge u+1$.
Next we prove our main lemma which gives a better lower bound on~$L(u)$.

\begin{lemma} \label{main_lem}
$L(u) \ge \frac{1}{16}u^2$ for every $u \in \mathbb{N}$.
\end{lemma}

\begin{proof}
We proceed by induction on $u$.  Assume that the lemma holds for all
integers $\le u$ and let $A \subseteq G$ satisfy $|A| = 2(u+1)$ and
$\stab(\Sigma(A)) = \{0\}$. (If there is no such set, then we have
defined $L(u+1) = \infty$.)  We will show that there exists $B'
\subseteq A$ with $|B'| = u+1$ and $|\Sigma(B')| \ge
\frac{1}{16}(u+1)^2$.  It follows from our trivial bound $L(u)\ge
u+1$ that we may assume $u \ge 16$.

Apply the lemma inductively to obtain a set $B \subseteq A$ with
$|B| = u$ and $|\Sigma(B)| \ge \frac{1}{16}u^2$. Put $C = A
\setminus B$. To apply Lemma~\ref{another_lemma}, which is our aim,
we need a lower bound on the size of~$H = \langle C\rangle$. We do
this by estimating~$|\Sigma(C)|$. To this end, we apply the lemma
inductively twice more: choose a set $C_1 \subseteq C$ of size
$\ceil{\frac{u}{2}}$ with $|\Sigma(C_1)| \ge \frac{1}{64} u^2$ and
(since $2 \ceil{\frac{u}{4}} + \ceil{\frac{u}{2}}
  \le 2\frac{u+3}{4} + \frac{u+1}{2}
    = u + 2$)
a set $C_2 \subseteq C \setminus C_1$ of size $\ceil{\frac{u}{4}}$
with $|\Sigma(C_2)| \ge \frac{1}{256} u^2$. Put $C_3 = C \setminus
(C_1 \cup C_2)$. Now $\Sigma(C) = \Sigma(C_1) + \Sigma(C_2) +
\Sigma(C_3)$. Since $\Sigma(C)$ has trivial stabilizer
(Observation~\ref{stab_obs}), Kneser's theorem gives the following
inequality (in the last inequality we use the trivial bound
$|\Sigma(X)| \ge |X| + 1$ if $0 \not \in X$).
\begin{eqnarray*}
|\Sigma(C)|
    & = & |\Sigma(C_1) + \Sigma(C_2) + \Sigma(C_3)| \\
    &\ge& |\Sigma(C_1)| + |\Sigma(C_2)| + |\Sigma(C_3)| - 2 \\
    &\ge& \tfrac{5}{256}u^2 + \tfrac{1}{4} u - 1.
\end{eqnarray*}
Let $S = \Sigma(B)$ and recall that $H = \langle C \rangle$.  Since
$\Sigma(C)$ has trivial stabilizer, $\Sigma(C) \subset H$, so $|H|
\ge \tfrac{5}{256}u^2 + \tfrac{1}{4}u$.  Since $u \ge 16$ by
assumption, we may apply Lemma~\ref{another_lemma} to deduce that
either $|S| \ge \frac{1}{16}(u+1)^2$, in which case we are finished,
or there exists $c \in C$ so that $\Delta_S(c) \ge
\frac{1}{8}(u+1)$.  In the latter case we have
\begin{eqnarray*}
|\Sigma(B \cup \{c\})|  & = &   |S + \{0,c\}|   \\
    & = &   |S| + \Delta_S(c)   \\
    &\ge&   \tfrac{1}{16}u^2 + \tfrac{1}{8}(u+1)  \\
    & > &   \tfrac{1}{16}(u+1)^2.
\end{eqnarray*}
This completes the proof.
\end{proof}

\begin{lemma} \label{triv_stab_lem}
If $A \subseteq G$ satisfies $\stab(\Sigma(A)) = \{0\}$, then
$$
  |\Sigma(A)| \ge 1 + \tfrac{1}{64}|A \setminus \{0\}|^2.
$$
\end{lemma}

\begin{proof}
We may assume that $0\not\in A$ since this has
no effect on our bound.  Set $|A| = u$.  The lemma holds trivially
if $u \le 8$, so we may assume $u>8$. By the previous lemma we may
choose a subset $B \subseteq A$ of size $\floor{\frac{u}{2}}$
such that $|\Sigma(B)| \ge L( \floor{\frac{u}{2}}) \ge \frac{(u-1)^2}{64}$.
Let $C = A \setminus B$. Then, by Kneser's theorem we have
\begin{eqnarray*}
|\Sigma(A)|
    & = &   |\Sigma(B) + \Sigma(C)|  \\
    &\ge&   |\Sigma(B)| + |C| - 1    \\
    &\ge&   \tfrac{1}{64}u^2 - \tfrac{1}{32}u + \tfrac{u}{2} - 1    \\
    &\ge&   1 + \tfrac{1}{64}u^2.
\end{eqnarray*}
\end{proof}

Note that by recursively applying Lemma \ref{main_lem} we can
improve the constant $1/64$ to ``almost'' $1/48$:
In the above proof, we have used that $|\Sigma(B)| \ge L(|B|)$
(together with Lemma~\ref{main_lem}) to bound $|\Sigma(B)|$.
On the other hand, for $|\Sigma(C)|$ we have used a straightforward
bound $|\Sigma(C)| \ge |C|+1$. We could instead use the same procedure
recursively with $C$ in place of~$A$. This yields
$$
  |\Sigma(A)| \ge \frac {1}{16} \Bigl(
    \bigfloor{\frac{u}{2}}^2 +
    \bigfloor{\frac{u}{4}}^2 +
    \bigfloor{\frac{u}{8}}^2 + \cdots \Bigr)
  = \frac{1}{48} u^2 - O(u) \,.
$$

Next we prove our main theorem for sequences.

\bigskip
\begin{proofof}{Theorem~\ref{sequence_thm}}
For $Q \in G/H$ we let $c(Q) = |\{ 1 \le j \le n : a_j \in Q\}|$.
Let further $s$ be the maximum of $c(Q)$ for nontrivial cosets $Q$
(i.e., $Q \in G/H \setminus \{H\}$).
Finally, put $A_j = \{Q : c(Q) \ge j\}$ and note that
$|A_j| = \rho_H^j({\bf a})$.  By applying Kneser's theorem and
Lemma~\ref{triv_stab_lem} in the quotient group $G/H$, we have
\begin{eqnarray*}
|\Sigma( {\bf a} )|
    & = &   |H| \cdot \Bigl|\sum_{j=1}^{s} \Sigma(A_j)\Bigr|            \\
    &\ge&   |H| \cdot \Bigl(\sum_{j=1}^s |\Sigma(A_j)| - s + 1 \Bigr)   \\
    &\ge&   |H| + \tfrac{1}{64} |H| \cdot
      \sum_{j \in \mathbb{N}} \bigl(\rho^j_H({\bf a})\bigr)^2
\end{eqnarray*}
which completes the proof.
\end{proofof}

Finally, we prove our main theorem for sets.

\bigskip
\begin{proofof}{Theorem~\ref{set_thm}}
Let $A = \{a_1,a_2,\ldots,a_n\}$ and put ${\bf a} = (a_1, a_2, \ldots, a_n)$.
By applying Theorem~\ref{sequence_thm} we have
$$
|\Sigma(A)|
     =   |\Sigma( {\bf a} )|
    \ge  |H| + \tfrac{1}{64} |H| \cdot
      \sum_{j \in \mathbb{N}} \bigl(\rho^j_H({\bf a})\bigr)^2.
$$
Since $A$ is a set, $\rho^{j}_H( {\bf a} ) = 0$ for every $j > |H|$.
Further $\sum_{j=1}^{|H|} \rho^j_H( {\bf a}) = |A \setminus H|$.
It follows from this (and the Cauchy--Schwarz Inequality) that
$|H| \cdot \sum_{j \in \mathbb{N}} (\rho^j_H({\bf a}))^2 \ge |A \setminus H|^2$.
Combining this with the above inequality yields the desired bound.
\end{proofof}

redpajama_set_name: RedPajamaArXiv


At PrimeCare Pediatrics, we are constantly looking to add the best hands to our organization.
If you are interested in becoming a member of our medical team, please send a copy of your resume to us for review at cs@primecarepeds.com.
We will get in touch with you after reviewing your resume.
There is no need to call our office directly.


text:
The kidneys (left) and a tubule made by bioprinting (right), the subject of a paper that Jennifer Lewis and other researchers published last fall. (Sebastian Kaulitzki/Shutterstock; Scientific Reports)
What Will It Take to 3-D Print Organs?
Eventually, no one will die while waiting for a transplant. But that day is further off than you might think.
By Anna Nowogrodzki
Every day in the U.S., about 22 people die waiting for an organ transplant. If scientists could 3-D print organs like kidneys, livers and hearts, all those lives could be saved. For years, people have been touting personalized organ printing as the future.
But despite decades of promising work in bioengineered bladders and other kinds of human tissue, we're not close to having more complicated organs made from scratch. Harvard professor Jennifer Lewis, a leader in advanced 3-D printing of biological tissue, has only recently developed the ability to print part of a nephron, an individual unit of a kidney. And the segment her team can print is able to do only some of what this nephron subunit is supposed to do.
I asked Lewis what it will take to someday print a full kidney or a similarly complex organ. Current research uses either 3-D printing (also called bioprinting) like Lewis's lab does, or what's known as de-cell re-cell. That's when scientists take an organ from an animal like a pig or from a human cadaver and remove all of the cells from it by applying chemicals or enzymes or freezing the organ. This leaves behind only proteins that provide a scaffolding structure on which human cells can be grown.
Lewis says that with either approach, there are three fundamental challenges to overcome, each harder than the last:
1. Physical complexity
"Kidney, liver, heart — these are just incredibly complex microstructures," Lewis says. The kidney alone has over 20 different cell types and a million nephrons, each made of many parts and wrapped in an intricate web of tiny blood vessels. "And it's all packed in a pretty small volume," says Lewis. "That architectural complexity alone, that's a challenge in and of itself for the printing process."
Timeline to achieve:
"With bioprinting alone, it's going to take many years," says Lewis. However, combining the approaches of bioprinting and growing mini-organs, known as organoids, from stem cells "could rapidly accelerate that." Essentially, you would use 3-D printing to give an organ its architecture, and grow organoids on that organizing structure. "There's potential there within the next decade, perhaps even sooner," she says.
As hard as it is to build something resembling an organ, it's even harder — and more vital — to build one that does exactly what it's supposed to.
The kidney does two crucial things: first it filters everything out of your blood, good and bad, and then it puts the important stuff back in, leaving the toxins out so the body can get rid of them. Lewis's lab has so far tackled the second of these functions. Last fall, her group was the first to 3-D print a working proximal tubule, one of the parts of an individual nephron that returns good stuff to the blood.
"But the proximal tubule is one subunit within a nephron that itself has multiple subunits, and then of course there's a million of those in the kidney," says Lewis. "While I'm super excited about that advance, it's really a tiny step towards the entire problem." Her lab still hasn't tackled the other half of kidney function, the filtration.
Once her group can make one working nephron, then it would have to reliably produce a million of them and ensure that they function together. To do that and transplant it into a human? "I think it's somewhere between 10 and 30-plus years, to be honest," she says.
Building an organ that the patient's body won't reject may be the steepest challenge of all. Most organ-printing researchers, including Lewis, use generic cells from biomedical supply companies. "It's still not clear whether the body will treat that as from itself," she says.
The odds could improve if you build an organ with stem cells harvested from a specific patient. But research on that technique is far off. Using repositories of stem cells from particular patients is more expensive. And it lacks one of the major benefits of using standard commercial cell lines, which is that research data is more comparable across different labs. There's little point in using patient-specific stem cells if researchers can't yet make a functioning organ with them. And once they can, there probably would be years of clinical trials.
"I still think the rejection piece is going to be the hardest to solve," Lewis says. But she adds: "I think it is possible."
By Anna Nowogrodzki on Apr. 27, 2017
Plastic-Eating Microbes to the Rescue?
Nature breaks everything down—eventually. It's time to accelerate the process.
Young Biodesigners Unveil Clever Ideas for a Cleaner World
Student competitors turn jellyfish, microorganisms, and even hair cells into useful things.
Angela Belcher Commands a Virus to Hunt a Deadly Cancer
An engineer lights up tiny tumors that doctors can't see.
The Birth of Wetware
Forget what you may have heard about neural networks in software. This startup is making a computer with living neurons.
Personalized Health Advice Every 60 Seconds
How I hacked my diet using a continuous blood-glucose monitor.
Check Out This Beating Mini-Heart in a Jar
It's made out of living human cells. Someday they could be yours.
The Dawn of Cheap and Easy DNA Writing
These startups are developing new ways to reprogram biology by producing custom genes from scratch.

redpajama_set_name: RedPajamaCommonCrawl


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text:
B3C newswire
SPECIALIZED CHANNELS
BIOTECH NEWS APP
EDITOR SIGN-UP
Abzena Selects Sartorius Stedim Biotech to Equip its US Based Development and Manufacturing Sites in San Diego and Bristol
Goettingen, Germany, January 04, 2018 / B3C newswire / -- Abzena plc, the life sciences group providing services and technologies to enable the development and manufacture of biopharmaceutical products, has selected Sartorius Stedim Biotech as its preferred equipment supplier in the U.S.
Abzena has selected Sartorius Stedim to equip its integrated CDMO facilities in Bristol, PA (development and GMP manufacture of antibody drug conjugates) and San Diego, CA (development and GMP manufacture of monoclonal antibodies and other recombinant proteins).
Sartorius Stedim will provide both facilities with end-to-end process solutions in single-use (SU) format. It will equip Abzena's San Diego process development lab with leading industry technologies such as Ambr250, enabling fast scale up to 500L initially and later to the 2000L scale single-use bioreactor for its centre of excellence in clinical manufacturing.
John Burt, Abzena CEO, commented: "In addition to having selected Sartorius' industry leading single-use technologies for antibody and ADC production, the Abzena facilities will feature Umetrics process analytics software throughout the process ensuring that all the data captured throughout our clients' molecule development is transformed into process knowledge ensuring that we can rapidly develop a robust and reliable process that can be easily scaled up to ensure a smooth technology transfer."
Reinhard Vogt, Sartorius Executive VP commented: "We are pleased to have been selected by Abzena, a prominent global CDMO with a comprehensive industry portfolio across a wide range of modalities to supply single-use solutions that enable process development, tech transfer and GMP manufacturing. We look forward to working together to best serve the US and international markets."
About Sartorius Stedim Biotech
Sartorius Stedim Biotech is a leading international supplier of products and services that enable the biopharmaceutical industry to develop and manufacture drugs safely and efficiently. As a total solutions provider, Sartorius Stedim Biotech offers a portfolio covering nearly all steps of biopharmaceutical manufacture. The company focuses on single-use technologies and value-added services to meet the rapidly changing technology requirements of the industry it serves. Headquartered in Aubagne, France, Sartorius Stedim Biotech is quoted on the Eurolist of Euronext Paris. With its own manufacturing and R&D sites in Europe, North America and Asia and an international network of sales companies, Sartorius Stedim Biotech has a global reach. In 2016, the company employed approx. 4,700 people, and earned sales revenue of 1,052 million euros.
About Abzena
Abzena (AIM: ABZA) provides proprietary technologies and complementary services to enable the development and manufacture of biopharmaceutical products. The term 'ABZENA Inside' is used by Abzena to describe products that have been created using its proprietary technologies and are being developed by its partners, and include Composite Human Antibodies™ and ThioBridge™ Antibody Drug Conjugates (ADCs). Abzena has the potential to earn future licence fees, milestone payments and/or royalties on 'ABZENA Inside' products. Abzena offers the following services and technologies across its principal sites in Cambridge (UK), San Diego, California (USA) and Bristol, Pennsylvania (USA):
Immunology research studies, including immunogenicity assessment of candidate biopharmaceutical products;
Protein engineering to create humanized antibodies and deimmunised therapeutic proteins;
Cell line development for the manufacture of recombinant proteins and antibodies;
Contract process development and GMP manufacture of biopharmaceuticals, including monoclonal antibodies and recombinant proteins for preclinical and clinical studies;
Contract synthetic chemistry and bioconjugation research services, focused on antibody-drug conjugates (ADCs);
Proprietary site-specific conjugation technologies and novel payloads for ADC development; and
GMP manufacturer of ADC linkers, payloads & combined linker-payloads.
Petra Kirchhoff
Sartorius Stedim Biotech
Vice President of Corporate Communications
+49(0)551.308.1686
www.sartorius-stedim.com
© B3C Group GmbH


text:
In The Run
#20Questions
About Vancouver Presents
Vancouver Presents
Home Theatre Theatre review: 1 Hour Photo is a sometimes powerful, often touching, glimpse...
Theatre review: 1 Hour Photo is a sometimes powerful, often touching, glimpse into a life well lived
Stories from 90-year old Mas Yamamoto are embellished with Tetsuro Shigematsu's trademark style
Mark Robins
Tetsuro Shigematsu and Steve Charles in the 2017 production of 1 Hour Photo.
"Do we remember memories or recollect photographs?" A somehow fitting question for a show which takes its name from a found object – a kitschy Japan Camera coffee mug – and what would become the impetus for Tetsuro Shigematsu's new work, 1 Hour Photo.
Setting out to discover the significance of the seemingly inconsequential modern artifact, Shigematsu soon found himself delving into the life of a 90-year old family friend, Mas Yamamoto. His curiosity about Yamamoto's life led to recording some thirty hours of conversations at the family's kitchen table.
As Shigematsu says with a wink near the top of the show though, those thirty hours have been distilled down to just 19 minutes for 1 Hour Photo. And while it may be difficult to comprehend how a near-century of living can be condensed into mere minutes, Shigematsu's concise editing provides a sometimes powerful, often touching, glimpse into a life well-lived.
The stories Yamamoto tells are at times extraordinary, and it is easy to see why Shigematsu was compelled to share them.
From growing up as a fisherman's son on the Fraser River, his family's internment during World War II, a stint in Canada's arctic, and eventually becoming a successful businessman (the revelation of the found object), the anecdotes are filled with both joy and pain. Yamamoto's almost self-deprecating voice in the recordings hides what he has both endured and celebrated.
One of the most heart-wrenching stories is of the Yamamoto family's internment, alongside thousands of other Japanese Canadians in 1942. But along with the challenges faced growing up as a Japanese Canadian, there are also moments of playfulness, joy and love. The story of Yamamoto's reunion with his childhood love is sweetly sentimental.
At times though, Shigematsu's attempts at layering the story of his dying father (largely explored in last year's Empire of the Son), and other theatrical flourishes, results in a loss of focus.
1 Hour Photo is at its best as Yamamoto tells his stories in his own words, embellished by what is quickly becoming Shigematsu's trademark style. These include Susan Miyagishima's beautifully detailed miniatures projected on a large screen behind him; whether a mirror box multiplying a single tiny shack into dozens in the internment camp or a peek inside a model of the family home.
As with Empire of the Son, there is an overall aesthetic to 1 Hour Photo, which seems somehow fitting. Modern, with a touch of Japanese symmetry and hipster-ish charm from set and costume designers, Pam Johnson and Laura Fukumoto, it is as accessible as it is handsome. Gerald King's lighting design adds emphasis in all the right places.
A welcome new element, not seen in Empire of the Son, is the addition of Steve Charles. The musician not only provides live accompaniment but is also called upon to assist in telling some of the stories.
Wrestling with questions of life and death, 1 Hour Photo is most heartfelt in its exploration of Yamamoto's life. Ultimately, we are all a little better off for his willingness to share it through Shigematsu.
1 Hour Photo written and performed by Tetsuro Shigematsu. Directed by Richard Wolfe. A Cultch presentation of a Vancouver Asian Canadian Theatre production. On stage at The Cultch's Historic Theatre (1895 Venables St, Vancouver) until October 15. Visit http://thecultch.com for tickets and information.
The Cultch continues its digital season
Theatre review: A Christmas Carol is 75-minutes of heartwarming nostalgia
Metro Theatre goes online with new A Christmas Carol adaptation
BC Health Notice
Due to the ongoing B.C. health orders we recommend you check with organizers of any event prior to attendance.
Vancouver Symphony returns to the digital concert hall with a month of Mozart January 15, 2021
BC government announces new grant program for arts professionals January 13, 2021
Turning Point Ensemble explores the power of ones January 11, 2021
The Cultch continues its digital season January 8, 2021
Early Music Vancouver returns to the digital concert hall January 6, 2021
© Vancouver Presents | All rights reserved


text:
import { Road } from '../../../index';

describe('road use', () => {
	/**
	 * Test that the use function returns itself for viable chaining
	 */
	test('Use Returns Self', () => {
		expect.assertions(1);

		const road = new Road();

		expect(road.use(function (method, path, body, headers, next) {
			return next();
		})).toEqual(road);

	});
});

redpajama_set_name: RedPajamaGithub


text:
Rackspace Hosting, acquired zeroVM
Like zeroVM
By zeroVM
By Rackspace Hosting
Rackspace Hosting Competitor`s Deals
All Segments 08/20/02 to 01/12/22 (2592 results)
synopsis: Rackspace acquired ZeroVM, a lightweight open-source hypervisor created by LiteStack and built to run cloud applications. ZeroVM breaks down the barriers between compute and storage. Where traditional cloud architectures need to move the data to the app for processing, ZeroVM flips that approach and moves the app to the data.
buyer: Rackspace Hosting (RAX:$2,073.40)
Rackspace® Hosting is the open cloud company, that delivers open technologies and powers more than 205,000 customers worldwide. Rackspace provides its renowned Fanatical Support® across a broad portfolio of IT products, including Public Cloud, Private Cloud, Hybrid Hosting and Dedicated Hosting. "
target: zeroVM
ZeroVM is a lightweight open-source hypervisor created by LiteStack and built to run cloud applications. ZeroVM breaks down the barriers between compute and storage.
Engine Yard, Inc., acquired OpDemand
synopsis: Engine Yard, the leading cloud orchestration platform, announced that it has acquired OpDemand, the developer of Deis, the first open source application platform purpose-built for Docker. Deis builds upon Docker and CoreOS to provide a next-generation Platform as a Service (PaaS) that runs on public clouds, private clouds or bare metal.
buyer: Engine Yard, Inc.
Engine Yard, the leader in cloud orchestration platforms, gives application developers the automation, control and operational support they require to deploy apps for both Docker containers in public and private clouds, as well as traditional applications running on Amazon Web Services. "
target: OpDemand
OpDemand is the developer of Deis, the first open source application platform purpose-built for Docker. Deis builds upon Docker and CoreOS to provide a next-generation Platform as a Service (PaaS) that runs on public clouds, private clouds or bare metal.
VMware, Inc., will acquire SaltStack
synopsis: VMware is to acquire SaltStack, a Python-based, open-source software for event-driven IT automation, remote task execution, and configuration management. SaltStack intelligent automation helps enterprise IT operations, SecOps, and DevOps teams manage and secure a software-defined world.
buyer parent: Dell Technologies
buyer: VMware, Inc. (VMW:$11,338.00)
VMware software powers the world's complex digital infrastructure. The company's compute, cloud, mobility, networking and security offerings provide a dynamic and efficient digital foundation to over 500,000 customers globally, aided by an ecosystem of 75,000 partners. "
target: SaltStack
SaltStack is Python-based, open-source software for event-driven IT automation, remote task execution, and configuration management. SaltStack intelligent automation helps enterprise IT operations, SecOps, and DevOps teams manage and secure a software-defined world.
closed 10/6/2020 via GlobeNewswire
Progress Software Corporation, acquired Chef, Inc.
synopsis: Progress, a Boston-area developer tool company, has acquired software automation platform Chef. Chef, which is 100% open source, has customer list that includes tech industry stalwarts like Facebook, IBM and SAP, as well as non-tech companies like Nordstrom, Alaska Airlines and Capital One.
buyer: Progress Software Corporation (PRGS:$513.57)
Progress offers the leading platform for developing and deploying mission-critical business applications. Progress empowers enterprises and ISVs to build and deliver cognitive-first applications, that harness big data to derive business insights and competitive advantage. "
target: Chef, Inc.
Chef is an automation platform that transforms infrastructure into code. Chef allows its users to automate how they build, deploy, and manage their infrastructure. It depends on reusable definitions known as recipes to automate infrastructure tasks.
HP, will acquire Eucalyptus Systems, Inc.
synopsis: HP announced a definitive agreement to acquire Eucalyptus, a provider of open source software for building private and hybrid enterprise clouds.
buyer: HP (HPQ:$62,070.00)
HP is an American multinational information technology corporation. It provides products, technologies, software, solutions and services to consumers, small- and medium-sized businesses and large enterprises, including customers in the government, health and education sectors. "
target: Eucalyptus Systems, Inc.
Eucalyptus Systems delivers private cloud software. This is infrastructure software that enables enterprises and government agencies to establish their own cloud computing environments.
Docker, acquired Kitematic
synopsis: Docker, Inc., the corporate sponsor of the open platform for distributed applications, announced the acquisition of Canadian startup Kitematic and its eponymous, popular open source software tool. The Docker experience is enhanced through Kitematic and its GUI-driven workflow that automatically installs Docker on a user's Mac.
buyer: Docker
Docker, Inc. is the commercial entity behind the open source Docker project, and is the chief sponsor of the Docker ecosystem. Docker is an open platform for developers and sysadmins to build, ship, and run distributed applications. "
target: Kitematic
Kitematic is the easiest way to start using Docker on Mac. It automatically installs Docker on a user's Mac to create and run Docker containers in just minutes. Launched in 2014, it earned over 2,000 GitHub stars and had thousands of downloads from developers around the world.
closed 6/3/2021 via Company Press Release
Microsoft Corporation, acquired ReFirm Labs, Inc.
synopsis: Microsoft has acquired ReFirm Labs, enriching the company's firmware analysis and security capabilities across devices that form the intelligent edge. ReFirm is the developer of Binwalk, an open-source software product that defends thousands of potential entry points into a user's network by providing deep visibility into device vulnerabilities.
Microsoft is the leading software corporation that develops, manufactures, licenses, supports, and sells a range of software products and services. They offer Business Software, Design Tools, Developer Tools, Home & Educational Software, Tablets, Search, Advertising, Cloud Computing and more. "
target: ReFirm Labs, Inc.
ReFirm Labs is a provider of proactive IoT and firmware security solutions. Its Binwalk Enterprise product detects and reports potential zero-day vulnerabilities, hidden crypto keys, backdoor passwords, and known vulnerabilities in IoT devices without needing access to source code.
SECO S.p.a., will acquire Ispirata
synopsis: SECO, a leading high-tech manufacturer of computer miniaturization and ready-to-use IoT integrated systems in Italy, will acquire a 70% stake in Ispirata, an Italain company that specalizes in middleware solutions and open source software. Ispirata builds open source software solutions for data orchestration and connected devices.
buyer: SECO S.p.a. (BIT:IOT:$94.93)
SECO is a leading high-tech manufacturer of computer miniaturization and ready-to-use IoT integrated systems in Italy. SECO excels in the electronic embedded field, offering cutting-edge solutions for the widest range of applications through standard modules, SBCs, systems and full-custom services. "
target: Ispirata
Ispirata is an Italain company that specalizes in middleware solutions and open source software. Ispirata builds open source software solutions for data orchestration and connected devices. Their Astarte platform transforms IoT data into an AI environment and can process thousands of AI pipelines.
Mirantis, purchased Docker Enterprise from Docker
synopsis: Mirantis has acquired Docker's Enterprise business and team. Docker Enterprise is the only platform that enables developers to seamlessly build, share and safely run any applications anywhere - from public cloud to hybrid cloud to the edge.
buyer: Mirantis
Mirantis provides software, services, training, and support for companies with production-grade open source cloud. More customers rely on Mirantis than on any other company to get to production deployment of OpenStack at scale. "
seller: Docker
Docker Enterprise is the only platform that enables developers to seamlessly build, share and safely run any applications anywhere - from public cloud to hybrid cloud to the edge. One third of Fortune 100 companies use Docker Enterprise as their high-velocity innovation platform.
announced 8/4/2014 via BusinessWire
cloudControl, will acquire DotCloud
synopsis: cloudControl, Inc., the US subsidiary of the Berlin, Germany headquartered PaaS pioneer cloudControl GmbH, announced that it has entered into a definitive agreement to acquire the dotCloud PaaS business from Docker, Inc. the company behind the eponymous open source container platform.
buyer: cloudControl
cloudControl is Europe's Platform-as-a-Service pioneer, developing and running a leading multi-language PaaS since 2009. The cloudControl platform supercharges development of applications and business solutions by enabling rapid innovation cycles. "
target parent: Docker
target: DotCloud
dotCloud launched in 2011 as the industry's first multi-language Platform as a Service. The Platform simplifies application development by allowing programmers to focus just on the code for their service while the Platform handles code building, scaling, deployment and load-balancing.
Elastic, acquired Packetbeat UG
synopsis: Elastic, the company behind the popular open source projects Elasticsearch, Logstash, and Kibana with more than 20 million downloads, announced it has acquired Packetbeat, a real-time network packet analytics solution built natively on Elastic's technology stack.
buyer: Elastic (ESTC:$672.71)
Founded in 2012 by the people behind the Elasticsearch and Apache Lucene open source projects, Elastic provides real-time insights and makes massive amounts of structured and unstructured data usable for developers and enterprises. "
target: Packetbeat UG
Packetbeat is a real-time network packet analytics solution. They offer open source application monitoring & performance management.
Docker, acquired SocketPlane
synopsis: Docker, Inc. announced the acquisition of software-defined networking (SDN) startup SocketPlane. SocketPlane, which was founded with a vision of delivering Docker-native networking, has been an active participant in shaping the initial efforts around Docker's open API for networking.
target: SocketPlane
SocketPlane is an open source software company that is changing the landscape of computer networking. They create software that allows applications to be deployed and connected quickly, at scale in a way that is simple to use.
Mirantis, acquired Kontena
synopsis: Mirantis, a flexible infrastructure company that provides open cloud software, has acquired Kontena, a company that provides easy-to-use, fully integrated solutions for DevOps and software development teams to deploy, run, monitor and operate containers on the cloud.
Mirantis is a flexible infrastructure company. Mirantis provides continuous delivery of open cloud software via a managed services model to enable enterprise application owners to become free of infrastructure and operations concerns. "
target: Kontena
Based in Finland, Kontena provides easy-to-use, fully integrated solutions for DevOps and software development teams to deploy, run, monitor and operate containers on the cloud. The underlying Kontena Platform and Kontena Pharos technologies are open source and available under Apache 2.0 license.
Red Hat, Inc., will acquire CoreOS
synopsis: Red Hat, Inc., the world's leading provider of open source solutions, announced that it has signed a definitive agreement to acquire CoreOS, Inc., an innovator and leader in Kubernetes and container-native solutions.
buyer: Red Hat, Inc. (RHT:$3,156.59)
Red Hat is the world's leading provider of open source software solutions, using a community-powered approach to reliable and high-performing cloud, Linux, middleware, storage and virtualization technologies. Red Hat also offers award-winning support, training, and consulting services. "
target: CoreOS
CoreOS is the creator of CoreOS Tectonic, an enterprise-ready Kubernetes platform that provides automated operations, enables portability across private and public cloud providers, and is always upstream open source software.
Corvisa LLC, acquired Codebox
synopsis: Cloud communications provider, Corvisa LLC, announced the acquisition of Codebox, a web-based developer IDE (Integrated Development Environment). Corvisa will integrate the Codebox IDE into its Summit Platform to facilitate faster, easier development of real-time communications applications by both businesses and independent developers.
buyer parent: Novation Companies, Inc.
buyer: Corvisa LLC
Corvisa provides businesses with a single source for cloud communications solutions. Corvisa combines powerful contact center software, business phone systems and connectivity services with the unmatched flexibility of our Summit development platform. "
target: Codebox
Codebox is a web-based developer IDE (Integrated Development Environment). Codebox was created in 2014 with the vision of creating powerful development tools, tackling issues like collaboration, cloud development and many others. Codebox.
OVH Cloud, acquired OpenIO
synopsis: European cloud service provider OVHcloud has bought OpenIO, a French provider of fast object storage. OpenIO is a software-defined open source object storage solution ideal for Big Data, HPC (high performance computing) and AI (artificial intelligence).
buyer: OVH Cloud
OVH is a global, hyper-scale cloud provider that offers businesses industry-leading performance and value. Vertically integrated to own its network, server technology and green data centers, OVH operates more cost effectively and passes these benefits back to customers. "
target: OpenIO
OpenIO is a French provider of fast object storage. OpenIO is a software-defined open source object storage solution ideal for Big Data, HPC (high performance computing) and AI. OpenIO scales easily without mandatory data rebalancing, while delivering consistent high performance.
BETSOL, Inc., acquired Zmanda, Inc.
synopsis: BETSOL, a Product Engineering and Managed IT Services company, has acquired Zmanda from Carbonite, Inc. Zmanda is an enterprise backup and disaster recovery solution with hundreds of enterprise customers. It is the leader in cloud and open-source backup software, protecting over 1,000,000 systems worldwide.
buyer: BETSOL, Inc.
BETSOL is a Product Engineering and Managed IT Services company, with a strong heritage in enterprise products and solutions. Through Managed IT services, they enable their customers to reduce time-to-market and scale solutions cost-effectively. "
target parent: Carbonite, Inc.
target: Zmanda, Inc.
Zmanda is an enterprise backup and disaster recovery solution with hundreds of enterprise customers. It is the leader in cloud and open-source backup software, protecting over 1,000,000 systems worldwide. Their software is simple-to-use yet feature-rich.
Red Hat, Inc., will acquire Inktank
synopsis: Red Hat, Inc., the world's leading provider of open source solutions, has signed a definitive agreement to acquire Inktank, a leading provider of scale-out, open source storage systems. Inktank's flagship technology, Inktank Ceph Enterprise, delivers world class object and block storage software to enterprises deploying public or private clouds.
Red Hat is the world's leading provider of open source software solutions, taking a community-powered approach to reliable and high-performing cloud, Linux, middleware, storage and virtualization technologies. Red Hat also offers award-winning support, training, and consulting services. "
target: Inktank
Inktank is a leading provider of scale-out, open source storage systems and the developer of Ceph, a scalable, open source, software-defined storage system that runs on commodity hardware. Ceph replaces legacy storage systems and provides a unified solution for cloud computing environments.
Carbonite, Inc., acquired Zmanda, Inc.
synopsis: Carbonite, Inc., a leading provider of online backup solutions for consumers and small to medium sized businesses, announced that it has completed the acquisition of Zmanda, Inc., a leading global provider of open source and cloud backup solutions.
buyer: Carbonite, Inc. (CARB:$405.29)
Carbonite, Inc. is a leading provider of online backup solutions for consumers and small and medium sized businesses. Subscribers in more than 100 countries rely on Carbonite to provide easy-to-use, affordable and secure online backup solutions with anytime, anywhere data access. "
Zmanda offers open source based storage software and services. Zmanda's Amanda Enterprise, Zmanda Recovery Manager (ZRM) for MySQL, and Zmanda Cloud Backup (ZCB) — make it simple and affordable to backup and recover data in an increasingly complex and heterogeneous IT environment.
ZeroFOX, acquired Cyveillance Inc.
synopsis: ZeroFOX, the global leader in Digital Risk Protection, has acquired the Cyveillance business from LookingGlass Cyber Solutions. Cyveillance delivers comprehensive threat intelligence services, protecting the corporate brands, executives, and data of the world's largest and most respected financial services, energy, and public sector organizations.
buyer: ZeroFOX
ZeroFOX, the global category leader in Digital Risk Protection, safeguards modern organizations from dynamic security risks across the public attack surface, including social media, mobile application, surface, deep and dark web, email and domains, and digital collaboration platforms. "
target parent: Lookingglass Cyber Solutions
target: Cyveillance Inc.
Cyveillance delivers an open source intelligence-led approach to security through continuous, comprehensive monitoring and analysis of millions of online data sources to protect client information, infrastructure, and employees from physical and online threats.
Rackspace Hosting, acquired Onica Inc.
synopsis: Rackspace has acquired Onica, an Amazon Web Services (AWS) Partner Network (APN) Premier Consulting Partner and AWS Managed Service Provider. The acquisition brings Onica's innovative professional services capabilities to the Rackspace portfolio, complementing its existing managed cloud services capabilities.
buyer parent: Apollo Global Management
Rackspace accelerates the value of the cloud during every phase of digital transformation. Across applications, data, security, hybrid and multiple clouds worldwide, the company provides cloud specialists with unbiased expertise, continuous modernization, and Rackspace Service Blocks. "
target parent: Sunstone Partners
target: Onica Inc.
Onica is a cloud consulting and managed services company, helping businesses enable, operate, and innovate on the cloud. Onica is a full spectrum integrator, offering migration strategy to operational excellence, cloud-native development, and immersive transformation.
Rackspace Hosting, acquired RelationEdge, LLC
synopsis: Rackspace announced it has acquired RelationEdge, a full-service Salesforce Platinum Consulting Partner and digital agency that helps clients engage with their customers from lead to loyalty by improving business process, leveraging technology, and integrating creative digital marketing.
Rackspace, the #1 managed cloud company, helps businesses tap the power of cloud computing without the complexity and cost of managing it all on their own. Rackspace delivers expertise, easy-to-use tools, and Fanatical Support® for technologies including AWS, Google, Microsoft, OpenStack and VMware."
target: RelationEdge, LLC
RelationEdge helps companies implement Salesforce—providing a custom engineered solution that will allow them to better market, sell and service their customers. They provide ongoing services and support, evolving the platform to meet changing business needs.
Rackspace Hosting, will acquire TriCore Solutions LLC
synopsis: Rackspace announced that it signed an agreement to acquire TriCore Solutions, a leader in the management of enterprise applications, including Enterprise Resource Planning (ERP) solutions from Oracle and SAP.
target parent: BV Investment Partners
target: TriCore Solutions LLC
TriCore Solutions, the application management experts, provides a full suite of scalable and reliable managed application, cloud and infrastructure hosting, and consulting services to enterprise organizations.
Apollo Global Management, acquired Rackspace Hosting
synopsis: Apollo Global Management, a leading global alternative investment manager, has acquired Rackspace, the #1 managed cloud company that helps businesses tap the power of cloud computing without the complexity and cost of managing it on their own.
buyer: Apollo Global Management (NYSE:APO:$5,981.01)
Apollo Global Management, LLC is a leading global alternative investment manager. Apollo Global Management had assets under management of approximately $173 billion as of March 31, 2016 in private equity, credit and real estate funds invested across a core group of nine industries. "
target: Rackspace Hosting (RAX:$2,073.40)
Rackspace, the number one managed cloud company, helps businesses tap the power of cloud computing without the complexity and cost of managing it on their own.
$4,283 [$4,385]
Rackspace Hosting, acquired Exceptional Cloud Services
synopsis: Rackspace® Hosting, the open cloud company, announced the strategic acquisition of Exceptional Cloud Services to enhance its toolset for developers deploying and managing applications in the open cloud. Through this deal, Rackspace will expand its portfolio of developer solutions to include error tracking and Redis-as-a-Service capabilities.
Rackspace® Hosting is the open cloud company, delivering open technologies and powering more than 205,000 customers worldwide. Rackspace provides its renowned Fanatical Support® across a broad portfolio of IT products, including Public Cloud, Private Cloud, Hybrid Hosting and Dedicated Hosting. "
target: Exceptional Cloud Services
Founded in 2008, Exceptional's mission is to track the world's web exceptions. It reports them in real-time for developers to fix them fast. Its customers include eBay and Pinterest. Exceptional is headquartered in San Francisco, California.
Rackspace Hosting, acquired ObjectRocket
synopsis: Rackspace Hosting announced it has entered into a definitive agreement to acquire ObjectRocket, a MongoDB database as a service (DBaaS) provider. With ObjectRocket's open source-based MongoDB solution, Rackspace will broaden its OpenStack-based open cloud platform to offer customers a NoSQL DBaaS.
Rackspace Hosting is the open cloud company, delivering open technologies and powering more than 197,000 customers worldwide. Rackspace provides its renowned Fanatical Support® across a broad portfolio of IT products, including Public Cloud, Private Cloud, Hybrid Hosting and Dedicated Hosting. "
target: ObjectRocket
Built from the ground up, ObjectRocket is the premium MongoDB database-as-a-service (DBaaS). The only solution that is by default highly available, automatically sharded and unbelievably fast.
Rackspace Hosting, will acquire MailGun, Inc.
synopsis: Rackspace® Hosting announced that it has entered into a definitive agreement to acquire Mailgun, Inc. Mailgun provides a powerful set of APIs that allows users to send, receive, and track email easily from within their applications – without managing an email server or becoming an expert in email setup, operations and deliverability.
target: MailGun, Inc.
Built for developers, Mailgun provides a powerful set of APIs that allows users to send, receive, and track email easily from within their applications – without managing an email server or becoming an expert in email setup, operations and deliverability.
Rackspace Hosting, acquired SharePoint911
synopsis: Rackspace Hosting, a leading specialist in hosting and cloud computing, and the founder of OpenStack, has acquired SharePoint911, an industry leader in SharePoint consulting, training, and "JumpStart" services within SharePoint.
Rackspace Hosting is a leading specialist in hosting and cloud computing, and the founder of OpenStack, an open source cloud platform. It provides Fanatical Support® to its customers, across a portfolio of IT services, including Managed Hosting and Cloud Computing. "
target: SharePoint911
Founded in 2004 by Microsoft MVP Shane Young, SharePoint911 was established to draw on Shane's extensive experience with systems administration and with consulting for clients on SharePoint issues.
Rackspace Hosting, acquired Cloudkick
synopsis: Rackspace Hosting has announced its acquisition of Cloudkick, the creator of web applications for easy and efficient cloud-server management. Cloudkick provides award-winning cloud server monitoring and management tools, built from the ground up to support modern APIs and the elastic nature of the cloud.
Rackspace Hosting is a leading specialist in hosting and cloud computing, and the founder of OpenStack, an open source cloud platform. The San Antonio-based company provides Fanatical Support to its customers, across a portfolio of IT services, including Managed Hosting and Cloud Computing. "
target: Cloudkick
Cloudkick is the creator of award-winning web applications for easy and efficient cloud-server management. It provides award-winning cloud server monitoring and management tools, built from the ground up to support modern APIs and the elastic nature of the cloud.
ConvergeOne, LLC, acquired Integration Partners
synopsis: ConvergeOne, a customer centric, services-led cloud technology solution provider, announced that it has acquired Integration Partners Corporation, an award-winning team that designs and implements collaboration and digital infrastructure solutions.
buyer parent: CVC Capital Partners
buyer: ConvergeOne, LLC (CVON:$1,401.33)
ConvergeOne is a leading global IT services provider of cloud collaboration customer experience and technology solutions for large and medium enterprises with 27 years of experience assisting customers to transform their digital infrastructure and realize a return on investment. "
target: Integration Partners
Integration Partners Corporation is an award-winning team that designs and implements collaboration and digital infrastructure solutions. They specialize in the complete spectrum of data center, networking, cyber security and collaboration solutions.
announced 12/29/2021 via Company Press Release
Partners Group Holding AG, will acquire atNorth
synopsis: Partners Group, a global private markets firm, will acquire atNorth, a data center operator in Iceland. atNorth offers efficient data center hosting facilities and high-performance computing services. They enable customers to focus on their simulation applications and calculations without having to worry about the underlying HPC infrastructure.
buyer: Partners Group Holding AG (SWX:PGHN:$2,084.18)
Partners Group is a global private markets investment manager, serving over 900 institutional investors. They have USD 109 billion in assets under management and more than 1,500 professionals across 20 offices worldwide. "
target: atNorth
atNorth is the largest data center operator in Iceland. atNorth offers efficient data center hosting facilities and high-performance computing services. They enable customers to focus on their simulation applications and calculations without having to worry about the underlying HPC infrastructure.
announced 12/22/2021 via PR Newswire
Carlyle Group, will acquire Involta LLC
synopsis: Global investment firm Carlyle announced that funds managed by Carlyle have agreed to acquire Involta, a data center company focused on hybrid IT and cloud infrastructure, including data center colocation, hybrid cloud, edge, fiber, and related products.
buyer: Carlyle Group (CG:$8,297.70)
Carlyle is a global investment firm with deep industry expertise that deploys private capital across three business segments: Global Private Equity, Global Credit, and Investment Solutions. Carlyle's purpose is to invest wisely and create value on behalf of its investors and portfolio companies. "
target: Involta LLC
Involta is an award-winning hybrid IT and cloud-forward consulting firm orchestrating digital transformation for the nation's leading enterprises.
Sewan, acquired Ikoula
synopsis: Sewan, a new generation telecom operator in France, has acquired Ikoula, a French web and cloud hosting provider. Ikoula provides their clients with quality hosting services so they can focus on their core business. They offer services for datacenters, cloud, back up, web hosting, domain names, VPN, and more.
buyer: Sewan
Sewan is a new generation telecom operator in France. Sewan aims to simplify the use of communications services that are essential to businesses. They offer landline and mobile telephony, messaging, data, collaborative work, hosting and cloud services to businesses in France and Spain. "
target: Ikoula
Ikoula is a French web and cloud hosting provider. Ikoula provides their clients with quality hosting services so they can focus on their core business. They offer services for datacenters, cloud, back up, web hosting, domain names, VPN, and more.
GI Partners, acquired LightEdge Solutions, Inc.
synopsis: LightEdge, a leading provider of colocation, cloud and managed service solutions, based in Des Moines, announced that GI Partners, a leading private investment firm, has acquired a controlling interest in the business.
buyer: GI Partners
GI Partners is a private investment firm based in San Francisco, California. The firm has raised over $28 billion in capital from leading institutional investors around the world to invest in private equity, real estate, and data infrastructure strategies. "
target parent: The Anschutz Company
target: LightEdge Solutions, Inc.
LightEdge Solutions is a leading hybrid cloud and colocation service provider with seven purpose-built data centers across the Midwest and Texas.
Digital 9 Infrastructure plc, acquired Verne Global
synopsis: Digital 9 Infrastructure plc (D9), a managed investment trust in the UK, has acquired Verne Global, a provider of sustainable data center solutions for high intensity computing in Iceland. Verne offers flexible, scalable, and highly optimised data center infrastructures that meets the hosting needs for all of their clients' applications.
buyer: Digital 9 Infrastructure plc
Digital 9 Infrastructure plc (D9) is a managed investment trust in the UK. D9 invests in a range of digital infrastructure assets which deliver a reliable, functioning internet, including subsea fibre, data centres, terrestrial fibre, tower infrastructure and small cell network. "
target: Verne Global
Verne Global is a provider of sustainable data center solutions for high intensity computing in Iceland. Verne offers flexible, scalable, and highly optimised data center infrastructures that meets the hosting needs for all of their clients' applications.
Otava, LLC, acquired NewCloud Networks
synopsis: Otava, a global leader in custom and compliant hybrid cloud solutions, has acquired NewCloud Networks, a global cloud services provider. NewCloud provides hosting and managed services to over 200 small and mid-sized customers. They specialize in backup, disaster recovery, production cloud, cloud communications, virtual data centers, and more.
buyer parent: Schurz Communications, Inc.
buyer: Otava, LLC
Otava is a global provider of hybrid cloud and IT solutions for service providers, channel partners and enterprise clients. Otava provides services for cloud, disaster recovery, colocation, and more. Their network of 12 data centers protect mission critical apps to ensure they are always available. "
target: NewCloud Networks
NewCloud Networks is a global cloud services provider. NewCloud provides hosting and managed services to over 200 small and mid-sized customers. They specialize in backup, disaster recovery, production cloud, cloud communications, virtual data centers, security as a service, and more.
I Squared Capital, will acquire KIO Networks
synopsis: I Squared Capital, an independent global infrastructure investment manager, will acquire KIO Networks, a Mexican ICT and data center firm. KIO Networks offers a broad portfolio of mission-critical IT infrastructure and services to help businesses solve complex challenges. They offer services for data centers, managed cloud, cybersecurity, and more.
buyer: I Squared Capital
I Squared Capital is an independent global infrastructure investment manager focusing on energy, utilities, digital infrastructure, transport and social infrastructure in the Americas, Europe and Asia. The firm has offices in Miami Hong Kong, London, New Delhi, New York, and Singapore. "
target: KIO Networks
KIO Networks is a Mexican ICT and data center firm. KIO Networks offers a broad portfolio of mission-critical information technology infrastructure and services to help businesses solve complex challenges. They offer services for data centers, managed cloud, cybersecurity, and more.
myLoc managed IT AG, acquired Mivitec
synopsis: myLoc managed IT AG, a German provider of hosting solutions, has acquired Mivitec GmbH, a German cloud operator specialising in managed services for businesses. Mivitec provides reliable IT services and operates high-availability data centers. They offer services for managed hosting, business cloud, networks, security, data backup, and more.
buyer parent: Wiit S.p.A.
buyer: myLoc managed IT AG
myLoc managed IT is a German provider of hosting solutions. myLoc offers cheap and secure hosting services to private and business customers. The company operates proprietary data centres in Düsseldorf and provides colocation, managed hosting, private and public cloud and server hosting services. "
target: Mivitec
Mivitec is a German cloud operator specialising in managed services for businesses. Mivitec provides reliable IT services and operates high-availability data centers. They offer services for managed hosting, business cloud, networks, security, data backup & recovery, and more.
Amber Infrastructure Group, acquired iseek Pty Ltd.
synopsis: Amber Infrastructure Group, a specialist international investment manager, has acquired a majority stake in iseek, an Australian data centre company. iseek provides secure, reliable, and high quality data centre, cloud and connectivity services to enterprise and government customers through their purpose-built data centres across Australia.
buyer: Amber Infrastructure Group
Amber Infrastructure Group is a specialist international investment manager. Amber is focused on investing in and managing infrastructure assets across the public, transport, energy, digital and demographic infrastructure sectors that support the lives of people, homes and businesses. "
target: iseek Pty Ltd.
iseek is an Australian data centre company. iseek provides secure, reliable, and high quality data centre, cloud and connectivity services to enterprise and government customers through their purpose-built data centres across Australia.
HostPapa, acquired LFC Hosting
synopsis: HostPapa, a leading Canadian cloud services provider, has acquired LFC Hosting, a leading provider of web hosting services in Canada. LFC Hosting provides innovative web hosting for tens of thousands of websites worldwide, powering businesses of all sizes in over 80 countries. They company also offers VPS hosting and colocation hosting services.
buyer: HostPapa
HostPapa, based in Burlington, Ontario, is a web hosting and cloud-based service provider to small businesses across the globe. HostPapa's commitment is to provide a complete array of enterprise-grade solutions to every small business owner. "
target: LFC Hosting
LFC Hosting is a leading provider of web hosting services in Canada. LFC Hosting provides innovative web hosting for tens of thousands of websites worldwide, powering businesses of all sizes in over 80 countries. They company also offers VPS hosting and colocation hosting services.
Cerberus Sentinel, acquired Ground Control IT
synopsis: Cerberus Cyber Sentinel Corporation, a cybersecurity consulting and managed services firm based in Scottsdale, Ariz., has acquired Ground Control IT, an Arizona-based managed security services provider.
buyer: Cerberus Sentinel (OTCPK:CISO:$11.87)
Cerberus Sentinel is a provider of consulting and managed services, focused solely on cybersecurity. The company seeks to expand by acquiring world-class cybersecurity talent and utilizes the latest technology to create innovative solutions that protect businesses against emerging security threats. "
target: Ground Control IT
Ground Control IT is an Arizona-based managed security services provider. They offer services in IT support, virtualization, hardware support, networking, cloud backup, hosting, and more.
Switch Data Centers, will acquire Data Foundry
synopsis: Switch, a global technology infrastructure company, will acquire Data Foundry, a provider of data center colocation services. Data Foundry offers data centers with resilient and maintainable infrastructure to alleviate the downtime concerns of their clients. They offer services for colocation, disaster recovery, networks and cloud computing.
buyer: Switch Data Centers (SWCH:$511.55)
Switch is a global technology infrastructure company. Switch designs, builds and operates the highest-rated data center ecosystems in the world. They provide colocation services and EDGE data centers to more than 800 clients, including leading global enterprise and Fortune 100 corporations. "
target: Data Foundry
Data Foundry is a leading provider of data center colocation services. Data Foundry offers data centers with resilient and maintainable infrastructure to alleviate the downtime concerns of their clients. They offer services for colocation, disaster recovery, networks and cloud computing.
DartPoints, will acquire Immedion, LLC
synopsis: DartPoints, an owner and operator of edge colocation data centers, will acquire Immedion, a data center services and Infrastructure-as-a-Service (IaaS) provider. Immedion is committed to keeping its customers' business-critical data and applications available 100% of the time.
buyer parent: Astra Capital Management
buyer: DartPoints
DartPoints is a leading edge colocation data center operator. DartPoints' carrier-neutral edge colocation data centers provides businesses with a strategic and affordable way to bring them closer to their audience, reduce latency, and create an enhanced user experience. "
target: Immedion, LLC
Immedion is a data center services and Infrastructure-as-a-Service (IaaS) provider that has been helping companies keep data, systems, and applications safe and accessible. They deliver enterprise-level solutions of a national data center with the attention and agility of a local business.
Flagship Solutions Group, purchased Data Storage FL, LLC from Data Storage Corporation
synopsis: Data Storage Corporation, a provider of diverse business continuity, disaster recovery protection and cloud solutions, has completed the agreement with Flagship Solutions, LLC to merge its wholly owned subsidiary, Data Storage FL, LLC, into Flagship.
buyer: Flagship Solutions Group
Flagship provides IBM solutions, managed services, and cloud solutions worldwide. These include data center strategic planning and hybrid cloud implementations based on a wide range of assessments that look at virtualization, server consolidation, security, and infrastructure-focused integration. "
seller: Data Storage Corporation (OTCPK:DTST:$9.27)
Data Storage Corporation provides a highly secure, enterprise level cloud for IBM i Power systems and Windows, assisting companies in the migration process, while reducing capex and providing flexibility for seasonality with on-demand compute power.
Pax8, acquired Wirehive
synopsis: Pax8, the best place for IT professionals to buy cloud products, announced the acquisition of Wirehive, a forward-thinking cloud hosting and consultancy company headquartered in the United Kingdom.
buyer: Pax8
Pax8 is modernizing how partners buy, sell, and manage cloud. As a born-in-the-cloud company, Pax8 simplifies the buying journey with cloud automation, empowering the modern partner to achieve more. "
target: Wirehive
Wirehive is a forward-thinking cloud hosting and consultancy company headquartered in the United Kingdom. They use the latest cloud technologies to design, implement and manage infrastructure solutions that underpin successful digital innovation.
announced 12/2/2020 via GlobeNewswire
Crayon Group Holding AS, will acquire Sensa
synopsis: Crayon Group A.S., a global IT advisory firm, will acquire Sensa ehf. from Síminn hf., an Icelandic telecommunications company. Sensa is a next generation hosting provider and system integrator in Iceland. Sensa offers services for data center & colocation, networking & IT security, managed cloud, and communications.
buyer: Crayon Group Holding AS (RAYON.OSL:$1,944.73)
Crayon is a global IT advisory firm which provides its clients with tools and solutions to optimise client's return on investments in complex technology. Crayon provides software asset management, cloud and volume licensing and associated consulting services. "
target parent: Síminn
target: Sensa
Sensa is a next generation hosting provider and system integrator in Iceland. Sensa helps businesses of all sizes to find the right technology for their needs. They offer services for data center & colocation, networking & IT security, managed cloud, and communications.
Interxion, acquired Lamda Hellix
synopsis: Interxion, a leading European provider of carrier- and cloud-neutral colocation data centre solutions, has acquired Lamda Hellix, Southeastern Europe's leading colocation and interconnection provider. Lamda Hellix offers data center services for colocation, hybrid cloud, interconnection, managed ICT, business continuity and more.
buyer: Interxion (:INXN:$663.16)
Netherland-based Interxion is a leading provider of carrier and cloud-neutral colocation data centre services in Europe. Interxion has created connectivity, cloud, content, and finance hubs that foster growing customer communities of interest. "
target: Lamda Hellix
Lamda Hellix is the leading provider of neutral data center services in South East Europe. Lamda Hellix helps organisations to achieve operational sustainability and grow their business. They offer services for colocation, hybrid cloud, interconnection, managed ICT, business continuity and more.
announced 10/2/2020 via Company Press Release
Over the Wire Holdings Limited, will acquire Digital Sense
synopsis: Over the Wire Holdings Limited, a Telecommunications and IT solution provider in Australia, will acquire Digital Sense Hosting Pty Ltd, an Australian organisation specialising in providing world-class IT hosting infrastructure. Digital Sense offers services for co-location, cloud, disaster recovery, and data storage.
buyer: Over the Wire Holdings Limited (ASX:OTW:$62.73)
Over the Wire is a Telecommunications and IT solution provider in Australia. Over the Wire specializes in converged voice and data networks, data centres, and hosted infrastructure solutions for corporate clients. Their core offerings include data networks, voice, managed cloud, and colocation. "
target: Digital Sense
Digital Sense is an Australian organisation specialising in providing world-class IT hosting infrastructure. Digital Sense offers services for co-location, cloud, disaster recovery, and data storage. Their data centers are manned 24/7 to ensure their customers get uninterrupted service.
Quaero Capital, acquired DEAC
synopsis: Quaero Capital, an independent, specialist fund management firm in Switzerland, has acquired Digitalas Ekonomikas Attistibas Centrs (DEAC), one of the largest independent data center operators in Northern Europe. DEAC offers services for virtual servers, web hosting, cloud, e-mail and more.
buyer: Quaero Capital
Quaero Capital is an independent, specialist fund management firm in Switzerland. Quaero Capital brings together independently minded investment managers who use original research to provide actively managed strategies for clients in the institutional and wholesale markets. "
target: DEAC
Digitalas Ekonomikas Attistibas Centrs (DEAC) is one of the largest independent data center operators in Northern Europe. DEAC provides a complete range of data center and IT outsourcing services on a global scale. They offer services for virtual servers, web hosting, cloud, e-mail and more.
Ticker: GTLB
GitLab is a DevOps platform built as a single application for all stages of the DevOps lifecycle enabling Product, Development, Quality Assurance, Security, and Operations teams to work concurrently on the same project.
Ticker: ASX:MOT
Motopia Ltd
Motopia Ltd is an ASX listed company that owns and operates technology assets that enable legacy migration to the cloud. Motopia's 2SQL product is regarded by Microsoft and other technology experts as the best of breed Access to SQL migration solution.
Ticker: NTNX
Nutanix Inc.
Nutanix is a global leader in cloud software and hyperconverged infrastructure solutions, making infrastructure invisible so that IT can focus on the applications and services that power their business.
Ticker: PRPM
ProTek Capital
Propalms Limited is a global provider of application delivery and secure remote access solutions for Terminal Services and Virtual Desktop Infrastructures. Delivering to Enterprises of all sizes they offer reliable, scalable and affordable solutions that simply work.
Ticker: FSCT
ForeScout Technologies
ForeScout is transforming security through visibility. They help make the invisible visible by providing Global 2000 enterprises and government agencies with agentless visibility and control of traditional and IoT devices the instant they connect to the network.
Ticker: TSXV:STC
Sangoma Technologies Corp.
Sangoma is a leading provider of hardware and software components that enable or enhance IP Communications Systems for voice, data and video applications. Sangoma's connectivity solutions are used in leading unified communications, PBX, IVR, contact center and data-communication applications.
Ticker: CSCO
Cisco enables people to make powerful connections-whether in business, education, philanthropy, or creativity. Cisco hardware, software, and service offerings are used to create the Internet solutions that make networks possible-providing easy access to information anywhere, at any time.
Ticker: CIEN
Ciena Corporation
Ciena is the network specialist. Ciena collaborates with customers worldwide to unlock the strategic potential of their networks and fundamentally change the way they perform and compete.
Ticker: CHKP
Check Point Software Technologies Ltd
Check Point Software Technologies Ltd is the largest pure-play security vendor globally. The company provides industry-leading solutions, and protects customers from cyberattacks with an unmatched catch rate of malware and other types of attacks.
Ticker: RDWR
Radware Ltd.
Radware, is a global leader of application delivery and application security solutions for virtual and cloud data centers. Its award-winning solutions portfolio delivers full resilience for business-critical applications, maximum IT efficiency, and complete business agility.
Ticker: FFIV
F5 Networks Inc.
F5 provides solutions for an application world. They help organizations seamlessly scale cloud, data center, and software defined networking deployments to successfully deliver applications to anyone, anywhere, at any time.
Ticker: BOSC
Better Online Solutions Ltd.
Better Online Solutions Ltd.'s operations consist of: (i) Software and Hardware RFID Solutions (ii) specialized enterprise software, including IBM System i middleware, data and license management and mobile connectivity, and (iii) Supply Chain products, reselling electronic systems and components.
Ticker: SCOXQ
The SCO Group, Inc.
The SCO Group is a leading provider of software technology for distributed, embedded, network-based systems, offering SCO OpenServer for small- to medium-sized businesses and UnixWare for enterprise solutions.
Ticker: OPCN
OptiCon Systems, Inc.
The OptiCon business was formed to meet growing customer demand for improved management of densely packed distribution frames. Corning Cable Systems brought extensive experience in network design, cabling, hardware and testing to the development of the OptiCon Network Manager solution.
Ticker: CORO
Corero Network Security plc
Corero Network Security plc's mission is to be a leading network security systems company delivering solutions to address the challenges organizations face in protecting their IT systems and networks and on-line assets from the threats of cyber crime.
Ticker: EQIX
Equinix, Inc.
Equinix, Inc. connects businesses with partners and customers around the world through a global platform of high performance data centers, containing dynamic ecosystems and the broadest choice of networks.
Digital Realty Trust
Digital Realty supports the data center, colocation, and interconnection strategies of customers globally, ranging from cloud and information technology services, communications and social networking to financial services, manufacturing, energy, healthcare, and consumer products.
Ticker: IIJI
Internet Initiative Japan Inc.
Internet Initiative Japan Inc. is one of Japan's leading Internet-access and comprehensive network solutions providers. IIJ and its group of companies provide total network solutions that mainly cater to high-end corporate customers.
Ticker: PLUS
ePlus, inc.
ePlus is an engineering-centric technology solutions provider that helps organizations imagine, implement, and achieve more from their technology. With certifications from technology partners, cloud, and collaboration, ePlus transforms IT from a cost center to a business enabler.
Ticker: SZSE:300017
ChinaNetCenter
ChinaNetCenter is a leading Internet service platform provider and one of the largest content delivery network platforms. With over 3,000 customers across industries, ChinaNetCenter has the most industry coverage and customers in the industry of Internet data centers and content delivery in China.
Ticker: SWCH
Switch Data Centers
Switch is a global technology infrastructure company. Switch designs, builds and operates the highest-rated data center ecosystems in the world. They provide colocation services and EDGE data centers to more than 800 clients, including leading global enterprise and Fortune 100 corporations.
SAKURA Internet Inc.
SAKURA Internet is a Japanese company that offers high quality internet services centered on data centers. Their data centers have high availability and provide high-capacity and high-speed internet connections. SAKURA Internet offers diverse services to meet the needs of their client's business.
Ticker: AIM:IOM
iomart Group plc
iomart Group PLC enables businesses to operate their online data and IT environments securely. iomart partners with vendors such as VMware, Amazon, EMC, Microsoft and Dell to offer customers a centrally managed, controlled and completely agnostic set of hybrid, private and public cloud platforms.
Ticker: ASX:IAB
Inabox Group Limited
Inabox Group Limited is a leading supplier of next generation IT, cloud and IP telephony services. Established in 2003 and listed in 2013, they are one of the few providers in the market that offers a full suite of IT and telecommunications products.
Lontrue Co., Ltd.
Lontrue is Chinese company that processes and exports dried fruits and nuts. Their main products include sultana raisins, pine nuts, dried apples, dried apricots, pumpkin kernels, and walnut kernels. The company also provides data center hosting and freight logistic services.
Ticker: OM:BINERO
Binero Group AB
Binero Group is a Swedish supplier of digital infrastructure services for companies and individuals with a wide range of services within hosting and cloud services. Binero Group offers web hosting with various complimentary services, development & management of corporate IT infrastructure, and more.
Ticker: BME:GIGA
Gigas Hosting, S.A.
Gigas Hosting S.A. is an international cloud computing services company based in Madrid, Spain. Gigas offers a simple, secure, reliable, scalable, and efficient cloud hosting platform. Their high performance cloud data centers are located in Madrid and Miami and are in a highly secure environment.
Ticker: OTCPK:DTST
Data Storage Corporation
Ticker: VRSN
VeriSign, Inc.
VeriSign, Inc. is the trusted provider of Internet infrastructure services for the networked world. Billions of times each day, Verisign helps companies and consumers all over the world connect between the dots.
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