The great storm of October 2013?
For the past few days, all I've been hearing about is the storm that is about to hit this evening/tomorrow. Multiple reports from good sources predicting hurricane force gusts, localised flooding, etc.
The American part of me quietly rejoices – of course I don't want people to get hurt, buildings to get blown down – but I miss storms. Some of them, anyway. At the same time, I find myself skeptical.
In what now feels like another lifetime, I lived and worked in the woods. As a camp director at a large summer camp in the middle of nowhere, I found myself addicted to checking hourly weather predictions. This one website would say a storm would start at 3 pm, and at 2:58 black clouds would roll in. I took to printing out a little forecast for every day so we could adjust activities accordingly – though life at camp went on pretty much as normal unless it was a prolonged thunderstorm, severe thunderstorm, or tornado.
You see, I grew up in a part of the world where we were hit with severe thunderstorms multiple times a summer…often more than one per week. The winds were strong enough to uproot massive, old trees, leaving six foot craters behind. Hail was heavy and large enough to break windows and cause bruises. I personally knew one child (though this did not happen at camp, thankfully!) who was struck by lightning and had both kneecaps blown out at the age of five.
Thunderstorms, even severe ones, were my secret delight – especially at camp. Elsewhere, you had a chance to just enjoy booming thunder so loud your teeth would vibrate in your head…for hours on end. At camp, as a part of the supervisory staff, we rode golf carts around, wildly careening to make sure each and every child was safe in a building or basement. We tied down tent flaps to try to save possessions. One on memorable occasion, I went into a platform tent that had a downed tree on top in order to get the kids' stuff out.
Tornados were another matter – they hit the area I live a good few times each summer, leaving more uprooted trees, damaged buildings, power outages for days. It was just part of my life to notice when the skies went green, when the air went deadly still in the middle of a howling thunderstorm, when the leaves on the trees flipped upside down to reveal the silvery undersides.
I spent hours of my childhood in the basement of our family home. Where I'm from, you see, every house has a basement. While they have evolved into awesome rec rooms, extra lounges, and storage spaces – they exist because of the severity of the weather. And it should be noted I'm not from a part of America that gets totally destroyed by tornadoes.
One of my earliest memories is being shoved in a cardboard box under the stairs with my baby sister, my mother throwing a mattress over us and lying on it to hold it in place.
One year while I attended camp as a child, a tornado hit my neighbourhood. My sister and birth father were locked out of the house, holding onto the handle of the door in the garage that led into the house, and they were literally lifted off their feet and almost sucked into the tornado. The large RV parked a foot from our house was lifted, and twisted like a wet cloth being wrung out, before being thrown hundreds of metres away in a neighbour's garden. A huge tree was thrown through our roof and my parents' bedroom was full of leaves and branches….a real live treehouse.
So you can see why I am a bit skeptical about the UK. This started my first year here, with snowfall. I lived in South London, and we had less than a centimetre of snow AND THE WHOLE CITY SHUT DOWN. My father in law was scared to drive, workplaces were closed, and I was just amazed. A few years later a 'bad' snowstorm caused people to be trapped on a motorway for hours, and food and water had to be airlifted in.
The storm predicted for tonight/tomorrow seems to consist of gusts of gale force and heavy rain. Like many in the UK, we have cleared the garden of (most) toys in preparation. Unlike other people, we have not bought extra food, batteries, candles, etc. I don't doubt severe winds could cause absolute upheaval in the UK – and I also imagine a power outage would take a considerably long time to repair, compared to America, where we had storm related power outages around the year. But I can't imagine a storm so bad that we'd be trapped at home for days, unable to get supplies if needed.
I don't want anyone to get hurt, to lose their roof, for a flood to damage a business. But I wouldn't mind some real storm action.
We've planned to stay in tomorrow, to wear pyjamas and watch films and play. To basically hang out and watch the rain and wind.
If the storm is as bad as predicted, much of the country will be facing what the UK terms as severe weather. I hope those of you living in this country remain safe, your homes and loved ones secure. Suzy will be going to work tomorrow in all likelihood, a half hour drive on narrow roads along massive drops, up and down hills. I secretly hope the weather is obviously bad enough for her to stay in and play with us, as I will be genuinely worried during her drive to and from work.
But assuming we all stay safe….well….the truth is, I wouldn't mind a big storm. They are one of the things I get homesick for.


text:
\section{Introduction
}\label{sgenp}

It is well known that even for a nonsingular and well conditioned input matrix,
Gaussian elimination
fails in numerical
computations with rounding errors
as soon as it encounters
a vanishing or nearly vanishing
leading (that is northwestern) entry.
In practice the
users avoid such encounters by applying
GEPP, which stands for Gaussian elimination
with partial pivoting and has some limited
formal but ample empirical support.
Partial pivoting, that is an appropriate row interchange,
 however, takes its toll:
"pivoting usually degrades the performance"
\cite[page 119]{GL96}.
It
interrupts the stream of arithmetic operations
with foreign operations of comparison,
involves book-keeping, compromises data locality,
increases communication overhead and data dependence,
 and
tends to destroy matrix structure.
The users apply {\em Gaussian elimination with no pivoting} (hereafter
we refer to it as {\em GENP}) or its variants
wherever this application  is numerically safe,
in particular where
a well  conditioned input matrix
 is
positive definite,
 diagonally
dominant,  or totally positive.
By following the idea in \cite[Section 12.2]{PGMQ},
briefly revisited in \cite[Section 6]{PQZ13},
we generalize
this class to all
nonsingular and well
conditioned matrices
by {\em preconditioning} GENP with
standard Gaussian random multipliers (hereafter we call
them just {\em Gaussian}).
We provide formal support for
this approach, which
 can be extended
to block Gaussian elimination
(see Section \ref{sbgegenpn}).
Our tests are in good accordance with our formal study.
Moreover multiplication by circulant
(rather than Gaussian) random multipliers
has the same empirical power, although it
uses by a factor of $n$ fewer random parameters and
by a factor of $n/\log (n)$ fewer flops
in the case of $n\times n$ inputs.
The latter saving is particularly dramatic in the important case where
the input matrix has Toeplitz structure.
Our study is
 a new demonstration of the power of randomized matrix algorithms
(cf. \cite{M11}, \cite{HMT11}, and the bibliography therein).

\section{Some definitions and basic results}\label{sdef}

We assume computations in the field $\mathbb R$ of real numbers,
but the extension to the case of the complex field $\mathbb C$
is quite straightforward.
Hereafter ``flop" stands for ``arithmetic operation",
and ``Gaussian matrix" stands for
``standard Gaussian random matrix".
 The concepts ``large", ``small", ``near", ``closely approximate",
``ill  conditioned" and ``well conditioned" are
quantified in the context. By saying ``expect" and ``likely" we
mean ``with probability $1$ or close to $1$"
(we do not use the concept of the expected value).
Next we recall and extend some customary definitions of matrix computations
\cite{GL96}, \cite{S98}.

\subsection{Some basic definitions of matrix computations}\label{smat}

$\mathbb  R^{m\times n}$ is the class of real $m\times n$ matrices $A=(a_{i,j})_{i,j}^{m,n}$.
$(B_1~|~\dots~|~B_k)$ is a $1\times k$ block matrix with the blocks $B_1,\dots,B_k$.
$\diag (B_1,\dots,B_k)=\diag(B_j)_{j=1}^k$ is a $k\times k$ block diagonal matrix
with the diagonal blocks $B_1,\dots,B_k$. In both cases the blocks $B_j$ can be rectangular.
${\bf e}_i$ is the $i$th coordinate vector of dimension $n$ for
$i=1,\dots,n$. These vectors define
the  $n\times n$ identity
matrix $I_n=({\bf e}_1~|~\dots~|~{\bf e}_n)$.
$O_{k,l}$ is the $k\times l$ matrix filled with zeros.
We write $I$ and $O$ where the  matrix size
 is defined by context.
$\rank (A)$ denotes the rank of a
matrix $A$. $A^T$ is its transpose.
 $A_{k,l}$  is its leading,
that is northwestern  $k\times l$ block
submatrix, and
in Section \ref{sbgegenpn}
we also write  $A^{(k)}=A_{k,k}$.
 $A_s^T$ denotes the transpose $(A_s)^T$
of a matrix $A_s$, e.g., $A_{k,l}^T$ stands for $(A_{k,l})^T$.
A matrix of a rank $\rho$ has {\em generic rank profile}
if all its leading $i\times i$ blocks are nonsingular for $i=1,\dots,\rho$.
If such a matrix is nonsingular itself, then  it is called {\em strongly nonsingular}.
{\em Preprocessing} $A\rightarrow FA$, $A\rightarrow AH$, and $A\rightarrow FAH$,
for nonsingular matrices $F$ and $H$,
reduces
the inversion of a matrix $A$ to
the inversion of the products $FA$, $AH$, or $FAH$,
and similarly for the solution of a linear system of equations.
\begin{fact}\label{faprepr}
Assume three nonsingular matrices $F$, $A$, and $H$
 and a vector ${\bf b}$.
Then
$A^{-1}=H(FAH)^{-1}F$,
$FAH{\bf y}=F{\bf b}$, ${\bf x}=H{\bf y}$ if $A{\bf x}={\bf b}$.
\end{fact}

\subsection{Matrix norms, orthogonality, SVD, and pseudo-inverse}\label{sosvdi}

$||A||=||A||_2=\sup_{{\bf v}^T{\bf v}=1}||A{\bf v}||$
is the spectral norm of a matrix $A=(a_{i,j})_{i,j=1}^{m,n}$,
$||AB||\le ||A||~||B||$.
A real matrix $Q$ is called
{\em orthogonal} if $Q^TQ=I$
 or $QQ^T=I$.

An {\em SVD} or {\em full SVD} of an $m\times n$ matrix $A$ of a rank
 $\rho$ is a factorization
\begin{equation}\label{eqsvd}
A=S_A\Sigma_AT_A^T
\end{equation}
where $S_A=({\bf s}_i)_{i=1}^m$ and $T_A=({\bf t}_j)_{j=1}^n$ are square orthogonal matrices,
$\Sigma_A=\diag(\widehat \Sigma_A,O_{m-\rho,n-\rho})$,
$\widehat \Sigma_A=\diag(\sigma_j(A))_{j=1}^{\rho}$,
$\sigma_j=\sigma_j(A)=\sigma_j(A^T)$
is the $j$th largest singular value of a matrix $A$, and
$\sigma_j=0$ for $j>\rho$,
$\sigma_{\rho}>0$,
 $\sigma_1=\max_{||{\bf x}||=1}||A{\bf x}||=||A||$,
and
\begin{equation}\label{eqmins}
\min_{\rank(B)\le s-1}~~~||A-B||=\sigma_{s}(A),~s=1,2,\dots,
\end{equation}

\begin{fact}\label{faccondsub}
If $A_0$ is a
submatrix of a
matrix $A$,
then
$\sigma_{j} (A)\ge \sigma_{j} (A_0)$ for all $j$.
\end{fact}

\begin{proof}
Extend \cite[Corollary 8.6.3]{GL96}.
\end{proof}

\begin{fact}\label{faintpr} (Cf. \cite[Corollary 8.6.3]{GL96}.)
Suppose $r+l\le n\le m$, $l\ge 0$,
 $1\le k\le r$,
 $A\in \mathbb R^{m\times n}$,
and
$A_{m,r}$ is the leftmost $m\times r$ block of the matrix $A$.
Then $\sigma_{k} (A_{m,r})\ge \sigma_{k+l} (A_{m,r+l})$.
\end{fact}

$A^+=T_A\diag(\widehat \Sigma_A^{-1},O_{n-\rho,m-\rho})S_A^T$ is the Moore--Penrose
pseudo-inverse of the matrix $A$ of (\ref{eqsvd}).
  $A^{+T}$ stands for $(A^+)^T=(A^T)^+$,
and $A_s^+$ stands for $(A_s)^{+}$, e.g., $A_{k,l}^+$ denotes $(A_{k,l})^+$.

If a matrix $A$ has full column rank $\rho$, then
 $A^+=(A^TA)^{-1}A^T$ and
\begin{equation}\label{eqnrm+}
||A^+||=1/\sigma_{\rho}(A).
\end{equation}

\begin{corollary}\label{cointpr}
Assume that $\rank(A_{m,r})=r$ and $\rank(A_{m,r+l})=r+l$
for the matrices $A_{m,r}$  and $A_{m,r+l}$ of Fact \ref{faintpr}.
Then $||A_{m,r}^+||\le ||A_{m,r+l}^+||$.
\end{corollary}
\begin{proof}
Combine Fact  \ref{faintpr} for $k=r$ with equation (\ref{eqnrm+}).
\end{proof}

\begin{theorem}\label{thpert}  \cite[Corollary 1.4.19]{S98}.
Assume a pair of square matrices $A$ (nonsingular) and $E$ such that
$||A^{-1}E||\le 1$. Then $||(A+E)^{-1}||\le\frac{||A^{-1}||}{1-||A^{-1}E||}$ and
moreover $\frac{||(A+E)^{-1}-A^{-1}||}{||A^{-1}||}\le \frac{||A^{-1}||}{1-||A^{-1}E||}$.
\end{theorem}

\subsection{Condition number, numerical rank and  generic conditioning profile
}\label{scnpn}

$\kappa (A)=\frac{\sigma_1(A)}{\sigma_{\rho}(A)}=||A||~||A^+||$ is the condition
number of an $m\times n$ matrix $A$ of a rank $\rho$. Such matrix is
{\em ill conditioned}
if the ratio $\sigma_1(A)/\sigma_{\rho}(A)$
is large. If the ratio is reasonably bounded,
then the matrix is {\em well conditioned}.
An $m\times n$ matrix $A$ has a {\em numerical rank}
$r=\nrank(A)\le \rho=\rank (A)$
if the ratios $\sigma_{j}(A)/||A||$
are small for $j>r$ but not for $j\le r$.

\begin{remark}\label{renul}
One can specify the adjective ``small"
above as
``smaller than  a fixed positive tolerance"
and similarly specify ``closely" and ``well conditioned".
The specification can be a challenge,
e.g., for the matrix $\diag(1.1^{-j})_{j=0}^{999}$.
\end{remark}

If a well conditioned  $m\times n$ matrix $A$ has a
rank $\rho<l=\min\{m,n\}$,
 then all its close neighbors
have
numerical rank $\rho$ and  almost
all of them have rank $l$.
Conversely, if a matrix $A$
has a positive
numerical rank $r$,
then the $r$-truncation $A_{r}$,
obtained by setting to 0
all singular values $\sigma_{j}(A)$ for $j>r$, is
a  well conditioned  rank-$r$ approximation to the matrix $A$
within the error norm bound $\sigma_{r+1}(A)$ (cf. (\ref{eqmins})).
It follows that a matrix is ill conditioned if and only if
it is close to a matrix having a smaller rank
and
a matrix has a numerical rank $r$
if and only if it can be closely approximated by a well conditioned
matrix having full rank $r$.
Rank-revealing factorizations  of a matrix $A$ that  has a small numerical rank $r$,
but possibly has  a large rank $\rho$,
 produce its rank-$r$ approximations
at a lower computational cost
 \cite{P00}.
The
 randomized algorithms
of \cite{HMT11} decrease the computational cost
further.
An $m\times n$ matrix
has {\em generic conditioning profile}
if it
has a numerical rank $r$ and if
its leading $i\times i$ blocks are nonsingular and well conditioned for $i=1,\dots,r$.
Such matrix is {\em strongly well conditioned}
if it has full numerical rank  $r=\min\{m,n\}$.

\subsection{Block Gaussian elimination and GENP
}\label{sbgegenpn}

For a
nonsingular
$2\times 2$ block matrix $A=\begin{pmatrix}
B  &  C  \\
D  &  E
\end{pmatrix}$
with a nonsingular {\em pivot block} $B=A^{(k)}$,
 define
$S=S(A^{(k)},A)=E-DB^{-1}C$,
the {\em Schur complement} of $A^{(k)}$ in $A$,
and
the block factorization,

\begin{equation}\label{eqgenp}
\begin{aligned}
A=\begin{pmatrix}
I_k  &  O_{k,r}  \\
DB^{-1}  & I_r
\end{pmatrix}
\begin{pmatrix}
B  &  O_{k,r} \\
O_{r,k}  &  S
\end{pmatrix}
\begin{pmatrix}
I_k  &  B^{-1}C  \\
O_{k,r}  & I_r
\end{pmatrix}
\end{aligned}.
\end{equation}

Apply this factorization recursively
to  the pivot block
$B$ and its Schur complement $S$
 and arrive at
 the block Gaussian elimination process,
completely defined
by the sizes of the pivot blocks.
The recursive process
 either fails, where its pivot block turns out to be singular,
in particular where it is a vanishing pivot entry of GENP,
or can continue until all pivot blocks become
nonzero scalars. When this occurs  we arrive at GENP.
Factorization (\ref{eqgenp}) defines the block elimination of the
first $k$
columns of the matrix $A$,
whereas $S=S(A^{(k)},A)$
is the matrix produced at this elimination step.
Now assume that the pivot dimensions
$d_1,\dots,d_r$ and $\bar d_1,\dots,\bar d_{\bar r}$ of
two block elimination processes
 sum to the same integer $k$, that is
$k=d_1+\dots+d_r=\bar d_1+\dots+\bar d_{\bar r}$.
Then verify that
both processes produce the same Schur complement $S=S(A^{(k)},A)$.

\begin{theorem}\label{thsch}
In every step of the recursive block factorization process based on (\ref{eqgenp})
 every diagonal block of a block diagonal factor is either
a leading block of the input matrix $A$ or the Schur complement $S(A^{(h)},A^{(k)})$
 for some integers $h$ and $k$ such that $0<h<k\le n$ and
$S(A^{(h)},A^{(k)})=(S(A^{(h)},A))^{(h)}$.
\end{theorem}

\begin{corollary}\label{corec}
The recursive block factorization process based on equation (\ref{eqgenp})
can be completed by involving no vanishing pivot elements and no singular
pivot blocks
if and only if the input matrix $A$ has
generic rank profile.
\end{corollary}
\begin{proof}
Combine Theorem \ref{thsch} with equation
$\det A=(\det B)\det S$,
implied by (\ref{eqgenp}).
\end{proof}

The following result shows that
the  pivot blocks are not close to singular matrices
where block Gaussian elimination (e.g., GENP) is applied
to a strongly
well conditioned input matrix.

\begin{theorem}\label{thnorms} (Cf. \cite[Theorem 5.1]{PQZ13}.)
Assume GENP
or block Gaussian elimination
applied to
 an
$n\times n$
matrix $A$ and
write $N=||A||$ and $N_-=\max_{j=1}^n ||(A^{(j)})^{-1}||$,
and so $N_-N\ge ||A||~||A^{-1}||\ge 1$.
Then the absolute values of all pivot elements of GENP
and the norms of all pivot blocks of
block Gaussian elimination
do not exceed $N_+=N+N_-N^2$,
whereas the absolute values of the reciprocals of these
elements and the norms of the inverses of the blocks do not
exceed $N_-$.
\end{theorem}
\begin{proof}
Observe that the inverse $S^{-1}$ of the Schur complement $S$
in (\ref{eqgenp}) is the southeastern block of the inverse $A^{-1}$
and obtain
$||B||\le N$, $||B^{-1}||\le N_-$, and
$||S^{-1}||\le ||A^{-1}||\le N_-$. Moreover
$||S||\le N+N_-N^2$, due to (\ref{eqgenp}). Now the claimed bound follows
from Theorem \ref{thsch}.
\end{proof}
\noindent Invert (\ref{eqgenp}) to obtain
$\begin{aligned}
A^{-1}=\begin{pmatrix}
I_k  &  -B^{-1}C  \\
O_{k,r}  & I_r
\end{pmatrix}
\begin{pmatrix}
B^{-1}  &  O_{k,r} \\
O_{r,k}  &  S^{-1}
\end{pmatrix}
\begin{pmatrix}
I_k  &  O_{k,r}  \\
-DB^{-1}  & I_r
\end{pmatrix}
\end{aligned}$,
 extend this factorization  recursively,
apply it to the inversion of the matrix $A$ and the solution of a linear system $A{\bf x}={\bf b}$,
and extend the above analysis.

\begin{remark}\label{reir}
For a strongly nonsingular input matrix $A$
 block factorization (\ref{eqgenp}) can be extended
to computing the complete recursive factorization,
which  defines GENP.  By virtue of Theorem \ref{thnorms}
the norms of the inverses of
all pivot blocks involved in this computation
are at most $N_-$.
If the matrix $A$ is also strongly well conditioned, then
 we have a reasonable upper bound on $N_-$,
and  so
in view of Theorem \ref{thpert}
the inversion of all pivot blocks is numerically
safe. In this case we say that {\em GENP and block Gaussian elimination are locally safe} for the matrix $A$.
In a sense locally safe GENP and
 block Gaussian elimination are at least as safe numerically as
GEPP because a locally safe recursive factorization involves
neither divisions by absolutely small pivot entries
(recall that  pivoting has been introduced
precisely in order to avoid such divisions)
 nor inversions of ill conditioned pivot blocks.
 Let us also compare the
 magnification of
the perturbation norm bounds of  Theorem \ref{thpert} in GEPP and
in  the process of recursive factorization, which defines GENP
and block Gaussian elimination.
We observe immediately that in the recursive factorization
only the factors of the leading blocks
and the Schur complements
can contribute to this magnification, namely at most $\log_2(n)$
such factors can contribute to the norm of each of the output triangular or block triangular
factors $L$ and $U$. This implies the upper bound $(N_+N_-)^{\log_2(n)}$
on their norms, which can be compared favorably to the sharp upper bound on the growth factor
$2^{n-1}$ for GEPP \cite[page 119]{GL96} and \cite[Theorem 3.4.12]{S98}.
\end{remark}

\section{Singular values of the  matrix products (deterministic estimates)
and GENP with preprocessing}\label{smrc}

Fact \ref{faprepr} reduces the tasks of inverting a nonsingular and well
conditioned matrix $A$
and solving a linear system  $A{\bf x}={\bf b}$ to similar tasks for the
matrix $FAH$ and multipliers $F$ and $H$ of our choice.
 Remark \ref{reir} motivates the choice for which the
matrix $FAH$ is strongly nonsingular
and strongly well conditioned. In Section \ref{ssgnp}
we
prove that this is likely to occur already where one of the multipliers
$F$ and $H$
 is the identity matrix $I$
 and another one is Gaussian matrix,
and therefore also
where both $F$ an $H$ are independent Gaussian matrices.
In this section we prepare  background for that proof by
estimating the norms of the matrices $(FA)_{k,k}=F_{k,m}A_{m,k}$
and $(AH)_{k,k}=A_{k,n}H_{n,k}$ and of their inverses
for general (possibly
nonrandom) multipliers
$F$ and $H$.
We will keep writing $M_{s}^T$ and $M_{s}^+$  for $(M_{s})^T$ and $(M_{s})^+$,
respectively,
where $M$ can stand for  $A$, $F$, or $H$, say,
and where $s$ can be any subscript such as a pair $(k,l)$ or $(A,r)$.
We begin with  two simple lemmas.

\begin{lemma}\label{lepr2}
Suppose $S$ and $T$
are square orthogonal matrices. Then
$\sigma_{j}(SA)=\sigma_j(AT)=\sigma_j(A)$ for all $j$.
\end{lemma}

\begin{lemma}\label{lepr1}
Suppose $\Sigma=\diag(\sigma_i)_{i=1}^{n}$, $\sigma_1\ge \sigma_2\ge \cdots \ge \sigma_n$,
$F\in \mathbb R^{r\times n}$, $H\in \mathbb R^{n\times r}$.
Then
$\sigma_{j}(F\Sigma)\ge\sigma_{j}(F)\sigma_n$ and
$\sigma_{j}(\Sigma H)\ge\sigma_{j}(H)\sigma_n$ for all $j$.
If also $\sigma_n>0$, then
 $\rank (F\Sigma)=\rank (F)$ and $\rank (\Sigma H)=\rank (H)$.
\end{lemma}

\noindent The following theorem bounds the norms $||(FA)^+||$ and $||(AH)^+||$
for three matrices $A$, $F$ and $H$.

\begin{theorem}\label{1}
Suppose $A\in \mathbb R^{m\times n}$,
$F\in\mathbb R^{r\times m}$, $H\in\mathbb R^{n\times r}$,
 $r\le \rho$
for
$\rho=\rank (A)$,
 $A=S_A\Sigma_AT^T_A$ (cf. (\ref{eqsvd})),
 $\widehat F=FS_A$, and
$\widehat H=T_A^TH$.

 Then
\begin{equation}\label{eqfa}
\sigma_j(FA)\ge \sigma_k(A) ~\sigma_j(\widehat F_{r,k})~{\rm for~all}~k\le m~{\rm and~all}~j,
\end{equation}
\begin{equation}\label{eqah}
\sigma_j(AH)\ge \sigma_l(A) ~\sigma_j(\widehat H_{l,r})~{\rm for~all}~l\le n~{\rm and~all}~j.
\end{equation}
\end{theorem}

\begin{proof}
Note that
 $AH=S_A\Sigma_AT_A^TH$, and so
$\sigma_j(AH)=\sigma_j(\Sigma_AT_A^TH)=
\sigma_j(\Sigma_A\widehat H)$ for all $j$
by virtue of Lemma \ref{lepr2}, because
$S_A$ is a square orthogonal matrix.
Furthermore it follows from Fact \ref{faccondsub} that
$\sigma_j(\Sigma_A\widehat H)\ge
\sigma_j(\Sigma_{l,A}\widehat H_{l,r})$
for all $l\le n$. Combine this bound with the latter equations,
then apply Lemma \ref{lepr1}, and obtain bound (\ref{eqah}).
Similarly deduce bound (\ref{eqfa}).
\end{proof}

\begin{corollary}\label{cor1}
Keep the assumptions and definitions of Theorem \ref{1}. Then

(i)  $\sigma_{r}(AH)\ge
\sigma_{\rho}(A)\sigma_{r}(\widehat H_{\rho,r})=
\sigma_{r}(\widehat H_{\rho,r})/||A^+||$,

(ii) $||(AH)^+||\le ||A^+||~||\widehat H_{\rho,r}^+||$
if $\rank(AH)=\rank(\widehat H_{\rho,r})=r$,

(iii) $\sigma_{r}(FA)\ge
\sigma_{\rho}(A)\sigma_{r}(\widehat F_{r,\rho})=
\sigma_{r}(\widehat F_{r,\rho})/||A^+||$, and

(iv) $||(FA)^+||\le ||A^+||~||\widehat F_{r,\rho}^+||$
if $\rank(FA)=\rank(\widehat F_{r,\rho})=r$.
\end{corollary}

\begin{proof}
Substitute $j=r$ and $l=\rho$
into bound (\ref{eqah}), recall
(\ref{eqnrm+}), and
obtain part (i).
If $\rank(AH)=\rank(\widehat H_{l,r})=r$,
then apply (\ref{eqnrm+})
to obtain that
$\sigma_{r}(AH)=1/||(AH)^+||$ and
$\sigma_{r}(\widehat H_{l,r})=1/||\widehat H_{l,r}^+||$.
Substitute these equations
into part (i) and obtain part (ii).
Similarly prove parts (iii) and (iv).
\end{proof}

Let us  extends Theorem \ref{1}  to the leading
 blocks of the matrix products.

\begin{corollary}\label{cogh}
Keep the assumptions and definitions of Theorem \ref{1} and fix
two positive integers $k$ and $l$
such that
$k\le m$,  $l\le n$.

Then
(i) $||(FA)_{k,l}^+||\le ||\widehat F_{k,m}^+||~||A_{m,l}^+||\le ||\widehat F_{k,m}^+||~||A^+||$
 if $m\ge n=\rho$ and if the matrices $(FA)_{k,l}$
and  $\widehat F_{k,m}$ have  full rank, whereas
(ii) $||(AH)_{k,l}^+||\le ||\widehat H_{n,l}^+||~||A_{k,n}^+||\le
||\widehat H_{n,l}^+||~||A^+||$ if $m=\rho\le n$
and if the matrices  $(AH)_{k,l}$ and $\widehat H_{n,l}$
have full rank.
\end{corollary}

\begin{proof}
Recall that
$(FA)_{k,l}=F_{k,m}A_{m,l}$
and the matrix $A_{m,l}$ has full rank if $m\ge n=\rho$.
Apply Corollary \ref{cor1}  for $A$ and $F$ replaced by
$A_{m,l}$ and $F_{k,m}$, respectively,
and obtain that $||(FA)_{k,l}^+||\le ||\widehat F_{k,m}^+||~||A_{m,l}^+||$.
Combine (\ref{eqnrm+}) and Corollary \ref{cointpr} and deduce that
$||A_{m,l}^+||\le||A^+||$. Combine  the two latter
inequalities
to complete the proof of part (i).
Similarly prove part (ii).
\end{proof}

The following definition formalizes the assumptions of
Corollaries \ref{cor1} and  \ref{cogh}.

\begin{definition}\label{def1}
Assume the matrices $A$, $F$, $\widehat F$, $H$, and  $\widehat H$ of Theorem \ref{1}.
Then the matrix pair $(A,H)$ (resp. $(F,A)$) has {\em full rank}
if the matrices $AH$, $\widehat H_{\rho,r}$ (resp. $FA$, $\widehat F_{r,\rho})$)
and $A$ have full rank.
This pair has full rank and is {\em well conditioned}
if in addition the matrices  $\widehat H_{\rho,r}$ (resp. $\widehat F_{r,\rho})$)
and $A$ are well conditioned,
whereas it has {\em generic rank profile} if  $\rank(A)=\rho$ and
$\rank((AH)_{k,k})=\rank(\widehat H_{\rho,k})=k$
(resp. $\rank((FA)_{k,k})=\rank(\widehat F_{k,\rho})=k$) for $k=1,\dots,r$.
 The pair has generic rank profile
and  is {\em strongly well conditioned} if
in addition
the matrices $\widehat H_{\rho,k}$
(resp. $\widehat F_{k,\rho}$) for $k=1,\dots,r$

 are well conditioned.
\end{definition}

\begin{remark}\label{regenppr}
Fact \ref{faprepr},
Corollary \ref{corec} and Theorem \ref{thnorms} together imply the
following guiding rule.
 Suppose
$A\in \mathbb R^{m\times n}$,
$F\in\mathbb R^{r\times m}$, $H\in\mathbb R^{n\times r}$,
 $r\le \rho=\rank (A)$, and
 the matrix pair $(A,H)$ for $m\le n$ or $(F,A)$
 for $m\ge n$
has generic rank profile and
 is  strongly well conditioned. Then
 GENP is locally safe for the
matrix products $AH$ or $FA$, respectively
 (see Remark \ref{reir} on the concept ``locally safe").
\end{remark}

\section{Benefits of using random matrix multipliers
}\label{sapsr1}

Next we
define Gaussian matrices and recall their basic properties.
In Section \ref{ssgnp}  we show
that these multipliers are expected to support
locally safe GENP, and
in Section \ref{srnd} comment on using non-Gaussian random
multipliers.

\subsection{Gaussian
matrix, its rank, norm and condition estimates}\label{srrm}

\begin{definition}\label{defrndm}
A matrix is  {\em standard Gaussian random} or just
{\em Gaussian} if it is filled with i.i.d.
Gaussian random
variables  having mean $0$ and variance $1$.
\end{definition}

\begin{fact}\label{facdgr}
A Gaussian matrix  is rank deficient with probability 0.
\end{fact}
\begin{proof}
Assume a rank deficient $m\times n$ matrix where $m\ge n$, say.
Then the determinants
of all its $n\times n$ submatrices vanish. This implies
$\begin{pmatrix} m \\n \end{pmatrix}$
polynomial
equations on the entries, that is the rank deficient matrices form
an algebraic variety of a lower dimension in the linear space
$\mathbb R^{m\times n}$.
($V$ is an algebraic variety of a dimension $d\le N$
in the space $\mathbb  R^{N}$ if it is defined by $N-d$
polynomial equations and cannot be defined by fewer equations.)
Clearly such a variety has Lebesgue (uniform) and
Gaussian measure 0, both being absolutely continuous
with respect to one another.
\end{proof}

\begin{corollary}\label{codgr}
 A Gaussian
matrix has generic rank profile
with probability 1.
\end{corollary}
Hereafter $\nu_{j,m,n}$ denote the random variables
$\sigma_j(G)$ for Gaussian $m\times n$ matrix $G$
and all $j$, whereas
$\nu_{m,n}$, $\nu_{m,n}^+$, and $\kappa_{m,n}$ denote the
 random variables
 $||G||$,
 $||G^+||$, and $\kappa(G)=||G||~||G^+||$, respectively.
Note that $\nu_{j,n,m}=\nu_{j,m,n}$, $\nu_{n,m}=\nu_{m,n}$,
$\nu_{n,m}^+=\nu_{m,n}^+$, and $\kappa_{n,m}=\kappa_{m,n}$.

\begin{theorem}\label{thsignorm}
(Cf. \cite[Theorem II.7]{DS01}.)
Suppose
$h=\max\{m,n\}$, $t\ge 0$,  and
$z\ge 2\sqrt {h}$.
Then

Probability$\{\nu_{m,n}>z\}\le
\exp(-(z-2\sqrt {h})^2/2\}$ and

Probability$\{\nu_{m,n}>t+\sqrt m+\sqrt n\}\le
\exp(-t^2/2)$.
\end{theorem}

\begin{theorem}\label{thsiguna}
Suppose
$m\ge n$, and $x>0$ and write $\Gamma(x)=
\int_0^{\infty}\exp(-t)t^{x-1}dt$
and $\zeta(t)=
t^{m-1}m^{m/2}2^{(2-m)/2}\exp(-mt^2/2)/\Gamma(m/2)$.
Then

(i) Probability $\{\nu_{m,n}^+\ge m/x^2\}<\frac{x^{m-n+1}}{\Gamma(m-n+2)}$
for $n\ge 2$ and

(ii)  Probability $\{\nu_{m,1}^+\ge x\}\le
(m/2)^{(m-2)/2}/(\Gamma(m/2)x^m)$.
\end{theorem}
\begin{proof}
(i) See \cite[Proof of Lemma 4.1]{CD05}.
(ii) $G\in \mathbb R^{m\times 1}$ is a vector of length $m$.
 So, with probability 1,
$G\neq 0$,
$\rank (G)=1$, $||G^+||=1/||G||$,
 and consequently
Probability $\{||G^+||\ge x\}=$
 Probability $\{||G||\le 1/x\}=\int_{0}^{1/x} \zeta(t) dt$.
Note that $\exp(-mt^2/2)\le 1$, and so
$\int_{0}^{1/x} \zeta(t) dt< c_m\int_{0}^{1/x}t^{m-1}dt=c_m/(mx^m)$
where $c_m=m^{m/2}2^{(2-m)/2}/\Gamma(m/2)$.

\end{proof}

The following condition estimates
 from \cite[Theorem 4.5]{CD05}
are quite tight for large values $x$, but for $n\ge 2$ even tighter
estimates (although more involved)
 can be found in \cite{ES05}. (See \cite{D88} and  \cite{E88}
on the early study.)

\begin{theorem}\label{thmsiguna}
 With probability 1, we have
$\kappa_{m,1}=1$. If $m\ge n\ge 2$, then

Probability$\{\kappa_{m,n}m/(m-n+1)>x\}\le \frac{1}{2\pi}(6.414/x)^{m-n+1}$
for $x\ge m-n+1$.
\end{theorem}

\begin{corollary}\label{cogfrwc}
 A Gaussian
matrix has generic rank profile
with probability 1
and is expected to be well conditioned.
\end{corollary}
\begin{proof}
Combine Corollary \ref{codgr} and Theorem \ref{thmsiguna}.
\end{proof}

\subsection{Supporting  GENP with Gaussian multipliers
}\label{ssgnp}

\begin{lemma}\label{lepr3}
Suppose $H$ is Gaussian matrix,
$S$ and $T$ are orthogonal matrices, $H\in \mathbb R^{m\times n}$,
$S\in \mathbb R^{k\times m}$, and $T\in \mathbb R^{n\times k}$
for some $k$, $m$, and $n$.
Then $SH$ and $HT$ are Gaussian matrices.
\end{lemma}

\begin{theorem}\label{thrkpg}
Suppose $A\in \mathbb R^{m\times n}$, $F\in \mathbb R^{r\times m}$,
$H\in \mathbb R^{m\times r}$, $F$ and $H$ are Gaussian matrices,
and $\rank (A)=\rho$.
Then $\rank (FA)=\rank(AH)=\min \{r,\rho\}$ with probability 1.
\end{theorem}

\begin{proof}
Suppose $A=S_{A}\Sigma_{A}T_{A}^T$ is SVD of (\ref{eqsvd}).
Then $FA=FS_{A}\Sigma_{A}T_{A}^T=G\Sigma_{A}T_{A}^T$
where $G=FS_A$ is Gaussian $r\times m$ matrix by virtue of
Lemma \ref{lepr3}.
Clearly $\rank(FA)=\rank(G\Sigma_{A}T_{A}^T)=\rank(G\Sigma_{A})$
because $T_{A}$ is a square orthogonal matrix.
Moreover $\rank(G\Sigma_{A})=\rank(GD_{\rho})$
where $D_{\rho}=\diag(I_{\rho},O_{m-\rho,n-\rho}))$, and so
$GD_{\rho}$ is Gaussian  $r\times \rho$ matrix because it is a submatrix of
the Gaussian matrix $G$.
Therefore $\rank(FA)=\rank(GD_{\rho})$ is equal to $\min\{r,\rho\}$
  with probability 1  by virtue of
Fact \ref{facdgr}. Similarly obtain that $\rank(AH)=\min\{r,\rho\}$
with probability 1.
\end{proof}

\begin{corollary}\label{cor10}
Keep the assumptions and definitions of Theorem \ref{1}.
Suppose the matrix $A$ has full rank $\rho=\min\{m,n\}$,
 $k\le r\le\rho$,
and $F=FS_A$
and $H=T_A^TH$ are Gaussian matrices.
Then (i) so
are the matrices $\widehat F$,
$\widehat H$ and
 all their submatrices, in particular
$\widehat F_{k,\rho}$ and
 $\widehat H_{\rho,k}$, and
 (ii) with probability $1$,
$\rank((AH)_{k,k})=k$ if $m\le n$, $\rank((FA)_{k,k})=k$ if $m\le n$, and
$\rank(\widehat H_{\rho,k})=\rank(\widehat F_{k,\rho})=k$.
\end{corollary}

\begin{proof}
If $H$ and $F$ are Gaussian matrices, then so are the matrices
$\widehat H$ and $\widehat F$ by virtue of Lemma \ref{lepr3}.
Consequently so are all their submatrices. This proves
 parts (i) and  by virtue of Fact \ref{facdgr} also implies the
 equations $\rank(\widehat H_{\rho,k})=\rank(\widehat F_{k,\rho})=k$ of part (ii).
Now recall that $(AH)_{k,k}=A_{k,n}H_{n,k}$,
and hence $\rank ((AH)_{k,k})=\rank(A_{k,n}H_{n,k})$.
This is equal to $\rank(A_{k,n})$ with probability 1
by virtue of Theorem \ref{thrkpg} because
$H_{n,k}$ is a Gaussian matrix and because $k\le \rho\le n$.
Finally obtain that
 $\rank(A_{k,n})=k$ for $k\le \rho=m$, and so
$\rank((AH)_{k,k})=k$.
Similarly prove that $\rank ((FA)_{k,k})=k$ for $k\le \rho=n$.
\end{proof}

\begin{corollary}\label{cogmforalg}
The  choice of
Gaussian multipliers
$F$ where $m\le n$ or $H$ where $m\ge n$ is expected
to satisfy the assumptions of Remark \ref{regenppr}
(thus supporting application of
GENP
to the matrix $FA$ where $m\le n$ or $AH$
where $m\ge n$)
provided that the $m\times n$ matrix $A$
is nonsingular and well conditioned.
\end{corollary}
\begin{proof}
Combine Corollaries \ref{cogfrwc}
and \ref{cor10}.
\end{proof}

\subsection{Structured random multipliers}\label{srnd}

Given $n\times n$ matrices $A$,
$F$
and  $H$, we need $2n^3-n^2$ flops
to compute each of the products $FA$ and $AH$,
but we only need order of $n^2\log (n)$ flops
 to compute such products where
 $F$ and $H$ are circulant matrices (cf. \cite{p01}).
Furthermore we need just $n$ random parameters to define
a Gaussian circulant  $n\times n$ matrix $C=(c_{i-j\mod n})_{i,j=0}^{n-1}$,
expected to be very well conditioned \cite{PQa}.
Can we extend our results to these random  multipliers?
The proof and consequently the claim of Fact \ref{facdgr}
can be immediately extended,
and so we can still
satisfy
 with probability 1
the generic rank profile assumption
of  Remark \ref{regenppr}.
Moreover we can satisfy that assumption
with probability close to 1 even where we
fill the multipliers $F$ and $H$
with i.i.d. random variables defined
under the discrete uniform probability distribution
over a fixed sufficiently large finite set
(see Appendix \ref{srsnrm}).
We cannot extend our proof that the pair
$(T_A,H)$ is strongly well conditioned, however,
because we cannot extend Lemma \ref{lepr3} to this case.
Nevertheless empirically circulant random multipliers
turn out to support GENP
 quite strongly
(see our next section and also compare \cite{HMT11}, and \cite{M11}
on randomized low-rank approximation of a matrix).
By engaging simultaneously
 two independent structured random multipliers $F$ and $H$
we may enhance their power.

\section{Numerical Experiments}\label{sexp}

Our numerical experiments with general, Hankel, Toeplitz and circulant
random  matrices
have been performed in the Graduate Center of the City University of New York
on a Dell server with a dual core 1.86 GHz
Xeon processor and 2G memory running Windows Server 2003 R2. The test
Fortran code has been compiled with the GNU gfortran compiler within the Cygwin
environment.  Random numbers have been generated
assuming the
standard Gaussian
probability distribution.
 The tests have been designed by the first author
and performed by his coauthors.

Tables \ref{tab44g} and  \ref{tab44} show the results of our tests of the solution
of  nonsingular well conditioned linear systems $A{\bf y}={\bf b}$ of
$n$ equations with random normalized vectors ${\bf b}$
whose coefficient matrices had
singular
$n/2\times n/2$   leading principal blocks for $n=64, 256,1024$.
Namely by following
\cite[Section 28.3]{H02}
we generated the  $n\times n$
input matrices
$A=\begin{pmatrix}
A_k  &  B  \\
C    &  D
\end{pmatrix}$
where
 $A_k$ was a $k\times k$  matrix,
$B$, $C$ and $D$ were Toeplitz random  matrices such that
 $||B||\approx ||C||\approx ||D||\approx ||A_k||\approx 1$,
$n=2^s$, $s=6,7,8, 10$,
and $A_k=U\Sigma V^T$ for $k=n/2$, $\Sigma=\diag (\sigma_i)_{i=1}^k$,
$\sigma_i=1$ for $i=1,\dots,k-h$, $\sigma_i=0$ for $i=k-h+1,\dots,k$, $h=4$,
and $U$ and $V$ were $k\times k$ orthonormal random  matrices, computed as
the unique $k\times k$ factors $Q(V)$ in the QR factorization of $k\times k$ random matrices $V$.
We have performed 100 numerical tests
 for each dimension $n$ and have computed
the maximum, minimum and average relative residual norms
$||A{\bf y}-{\bf b}||/||{\bf b}||$ as well as the standard deviation.

In our tests the norms $||A^{-1}||$ ranged from
70 to $4\times 10^6$ (see Table \ref{tabinorm}),
and so   GEPP was expected to output accurate solutions
to the linear systems $A{\bf y}={\bf b}$, and indeed we always observed this
 in our tests
 (see Table \ref{tab44gpp}).
GENP, however, was expected to fail for these systems,
because the leading block $A_k$ of the matrix $A$  was singular, having nullity
$k-\rank (A_k)=4$.
Indeed  this has caused poor performance of GENP in our tests,
which
has consistently
output corrupted solutions,   with the residual norms
ranging from 10 to $10^8$.
In view of Remark \ref{cogmforalg}
we expected to fix this deficiency by means of multiplication
by Gaussian random matrices, and indeed  in our tests
 we observed consistently
the residual norms below
 $4\times 10^{-9}$ for all inputs
(see Table \ref{tab44g}). Furthermore the tests
showed the same power of preconditioning
where we used the Gaussian circulant multipliers (see Table \ref{tab44}).
As could be expected
the output accuracy of GENP
preprocessed with nonunitary  random multipliers
tended to deteriorate a little versus GEPP
 in our tests,
but the output residual norms were small enough to support
application of the inexpensive iterative refinement,
whose single step decreased the output residual norm by  factors of $10^h$ for $h>1$
in the case of Gaussian multipliers and $h\ge 2$ in the case of Gaussian circulant multipliers
(see Tables \ref{tab44g} and  \ref{tab44} and Remark \ref{reir}).

\begin{table}[ht]
  \caption{the norms $||A||^{-1}$ for the input matrices $A$}
  \label{tabinorm}
  \begin{center}
    \begin{tabular}{| c | c | c | c | c | c |}
      \hline
      \bf{dimension} &  \bf{min} & \bf{max} & \bf{mean} & \bf{std} \\ \hline
 $64$ &  $6.9\times 10^1$ & $2.3\times 10^3$ & $4.6\times 10^4$ & $6.4\times 10^3$ \\ \hline
 $256$ &  $8.4\times 10^2$ & $1.1\times 10^4$ & $5.8\times 10^5$ & $ 5.8\times 10^4$ \\ \hline
 $1024$ & $3.5\times 10^3$ & $9.9\times 10^4$ & $4.0\times 10^6$ & $4.3\times 10^5$ \\ \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[ht]
  \caption{Relative residual norms  of GEPP}
  \label{tab44gpp}
  \begin{center}
    \begin{tabular}{| c | c | c | c | c | c |}
      \hline
      \bf{dimension}  & \bf{min} & \bf{max} & \bf{mean} & \bf{std} \\ \hline

 $64$ &  $2.0\times 10^{-15}$ & $6.9\times 10^{-13}$ & $3.2\times 10^{-14}$ & $8.9\times 10^{-14}$ \\ \hline
 $256$ &  $ 1.4\times 10^{-14}$ & $1.3\times 10^{-12}$ & $1.2\times 10^{-13}$ & $1.9\times 10^{-13}$ \\ \hline
 $512$ & $5.2\times 10^{-14}$ & $4.6\times 10^{-11}$ & $1.0\times 10^{-12}$ & $4.9\times 10^{-12}$ \\ \hline
 $1024$  & $1.2\times 10^{-13}$ & $1.0\times 10^{-09}$ & $1.2\times 10^{-11}$ & $1.0\times 10^{-10}$ \\ \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[ht]
  \caption{Relative residual norms:
 GENP with Gaussian random multipliers}
  \label{tab44g}
  \begin{center}
    \begin{tabular}{| c | c | c | c | c | c | c |}
      \hline
      \bf{dimension} & \bf{iterations} & \bf{min} & \bf{max} & \bf{mean} & \bf{std} \\ \hline
 $64$ & $0$ & $3.6\times 10^{-13}$ & $1.4\times 10^{-11}$ & $5.3\times 10^{-12}$ & $8.4\times 10^{-12}$ \\ \hline
 $64$ & $1$ & $7.3\times 10^{-15}$ & $2.8\times 10^{-13}$ & $3.2\times 10^{-14}$ & $6.4\times 10^{-14}$ \\ \hline
 $256$ & $0$ & $5.1\times 10^{-12}$ & $3.8\times 10^{-9}$ & $9.2\times 10^{-10}$ & $4.7\times 10^{-11}$ \\ \hline
 $256$ & $1$ & $4.8\times 10^{-15}$ & $9.2\times 10^{-10}$ & $8.6\times 10^{-11}$ & $2.1\times 10^{-11}$ \\ \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}[ht]
  \caption{Relative residual norms:
 GENP with Gaussian circulant random  multipliers
}
  \label{tab44}
  \begin{center}
    \begin{tabular}{| c | c | c | c | c | c | c |}
      \hline
      \bf{dimension} & \bf{iterations} & \bf{min} & \bf{max} & \bf{mean} & \bf{std} \\ \hline
 $64$ & $0$ & $4.7\times 10^{-14}$ & $8.0\times 10^{-11}$ & $4.0\times 10^{-12}$ & $1.1\times 10^{-11}$ \\ \hline
 $64$ & $1$ & $1.9\times 10^{-15}$ & $5.3\times 10^{-13}$ & $2.3\times 10^{-14}$ & $5.4\times 10^{-14}$ \\ \hline
 $256$ & $0$ & $1.7\times 10^{-12}$ & $1.4\times 10^{-7}$ & $2.0\times 10^{-9}$ & $1.5\times 10^{-8}$ \\ \hline
 $256$ & $1$ & $8.3\times 10^{-15}$ & $4.3\times 10^{-10}$ & $4.5\times 10^{-12}$ & $4.3\times 10^{-11}$ \\ \hline
 $1024$ & $0$ & $1.7\times 10^{-10}$ & $4.4\times 10^{-9}$ & $1.4\times 10^{-9}$ & $2.1\times 10^{-9}$ \\ \hline
 $1024$ & $1$ & $3.4\times 10^{-14}$ & $9.9\times 10^{-14}$ & $6.8\times 10^{-14}$ & $2.7\times 10^{-14}$ \\ \hline
    \end{tabular}
  \end{center}
\end{table}

\section{Conclusions}\label{sconcl}

It is well known that Gaussian
 (that is standard Gaussian random) matrices
tend to be well conditioned,
and
this property motivates our preprocessing of
 well conditioned nonsingular  input matrices
with Gaussian multipliers to support the application of
GENP and block
 Gaussian elimination.
 Both of these algorithms can fail in
practical numerical computations without
preprocessing, but we prove that  with Gaussian multipliers
the algorithms are expected to be locally safe,
which would achieve the purpose of pivoting. Namely the
absolute values of the reciprocals of
all pivot elements of GENP  and
the norms of the inverses of
all pivot blocks of block Gaussian elimination
 are likely to be reasonably bounded.
Our tests were in good accordance with that formal study.
We generated matrices that
were hard for GENP, but the problems were consistently
avoided where we preprocessed the inputs with Gaussian multipliers.
Moreover empirically we obtained essentially the same results even
where we used
 circulant random  multipliers.
Their choice accelerates multiplication significantly, and
particularly dramatically in the important case where
the input matrix has Toeplitz structure. That choice also limits
randomization  to $n$
random parameters for an $n\times n$
input.
 Formal support for the
empirical power of these multipliers is a natural research challenge.
\bigskip

{\bf Acknowledgements:}
Our research has been supported by NSF Grant CCF--1116736 and
PSC CUNY Awards 64512--0042 and 65792--0043.

redpajama_set_name: RedPajamaArXiv


text:
RE/MAX begins major push for new brokerages, layoffs to come with third-party tech adoption
RE/MAX Holdings Inc. is launching an array of strategic initiatives to grow its business by attracting new brokerages and agents, building on the agent-teams model, investing in its mortgage line and introducing a third-party technology platform that will replace existing in-house systems and result in the layoff of 17% of the company's workforce, or about 120 tech-focused employees.
"RE/MAX Holdings continues to focus on increasing our near- and long-term profitable growth," CEO Steve Joyce said in a press release. "After a thoughtful and thorough review, we are moving forward with these strategic endeavors, each of which is expected to contribute to our growth objectives."
Brokerage mergers and conversions
To bring in new brokerages, the company formally launched a mergers-and-conversions initiative under which franchise sales personnel have been given the go-ahead to offer financial support to entice brokerages into converting to the RE/MAX network, either through upfront capital to assist with conversion costs or a temporary waiver of recurring franchise fees.
"Our data shows that U.S. and Canadian agents who join RE/MAX and stick with the brand typically increase their sales over time," Bailey said. "These statistics address a chief concern about agent productivity that tends to hold many of these potential mergers or conversions back. And that's an important message for us to continue to reiterate across the industry, especially now as the market is fluctuating and we see many experienced, successful brokers looking to transition their businesses."
Pilot program to boost teams
RE/MAX is also launching a pilot program in California, Florida, Maryland, New Jersey and Texas that will offer an alternative fee structure for medium to large teams. The program will reduce the continuing-franchise and marketing fees for team members, not including team leaders, and it will cap the broker fee for participating teams at $1,000 per team member, including leaders.
"Teams have scale and capabilities that give them a clear competitive edge," RE/MAX President and CEO Nick Bailey said. "We realize not everyone wants or needs to be on a team — and in our network we have many individual top producers — but more and more teams are seeing RE/MAX as the network to join if they want to grow and thrive."
Motto Mortgage growth
RE/MAX will make additional investments in its Motto Mortgage and wemlo lines with the goal of boosting its annual mortgage-related revenue to $100 million by 2028. Future Motto Mortgage franchises in most cases will be required to use wemlo's processing services, the company said.
"Motto has sold over 300 franchises to date and we have 200 open offices," said Motto Franchising LLC and wemlo President and CEO Ward Morrison. "We believe we can eventually grow that number to more than 1,000 open franchises. Motto has a unique and attractive value proposition, and we would like to get to 1,000 open offices much sooner than our current growth trajectory suggests."
Beginning in the fourth quarter, RE/MAX will begin rolling out Inside Real Estate's kvCORE technology platform for brokers and agents, replacing the booj products developed by the company. These will sunset mid-2023, and as a result of the phase-out, it will reduce staff, most of whom work in the tech department.
The new kvCORE platform will provide an integrated customer relationship-management (CRM) system with customizable websites, enhanced lead-generation services, "robust" listing tools and presentations, a marketing autopilot feature and business analytics, the company said in an SEC filing.
Other in-house tech systems, like RE/MAX's First app and its Seller Star ratings that use predictive analytics to help identify potential sellers, as well as certain functions of the Megaphone marketing tool, will remain available, while other systems, like the RE/MAX Hustle custom-video tool and Photofy, which allows agents to send RE/MAX-exclusive graphics and videos via social media or text, will be integrated into the new kvCORE platform.
"In today's highly competitive market, much of the agent-facing technology — CRM, websites, etc. — has become table stakes," Bailey said. "Providing our network with RE/MAX technology powered by kvCORE, including our proprietary tech and branding integrations, alongside an ongoing commitment to invest in and enhance the www.remax.com website with internal resources will enable us to focus on what we do best: Help small business owners build their businesses."
Posted In: Making Moves, National News
Tags: brokerage technology, franchise, Inside Real Estate, kcCORE, Nick Bailey, RE/MAX, Steve Joyce
'Greenest States to Move to in 2023' study praises Arizona for solar power efforts

redpajama_set_name: RedPajamaCommonCrawl


Hiring a person with a disability may be easier than you think. Many individuals with disabilities have a college education, long work histories, and a variety of marketable skills. In addition, often individuals with disabilities looking for employment have a variety of support available to them. Find out more below.
Colleges and universities often have specialized supports available to individuals with disabilities. As a result, many qualified candidates may be accessible through college placement or Work One offices. In addition, other external resources like the State Vocational Rehabilitation office may have already helped the person purchase equipment and other services that will help them meet the expectation of the job. The need for additional accommodation may not be great.
Some individuals with disabilities already have a long work history and good work skills. A person may have been injured in some type of accident that has required them to seek alternatives to their previous type of work. A person suffering from a back injury may have had the opportunity to be retrained for work that does not require lifting. Again, resources like the State Vocational Rehabilitation office, workman's compensation or insurance settlements from accidents may have supplied the person with the retraining and services needed to return to work in an alternative setting.
Individuals with developmental disabilities or mental health concerns are often working with a specialized placement agency. The placement agency works with the employer to find appropriate candidates to fill the job, to restructure jobs if needed, help determine any accommodations that may be needed, assist with training the individual and assist with training the other employees about the disability if needed. Most of the time the placement agency will work with the employer to provide support beyond the employer's normal training period.
Most communities, including Bloomington, have many organizations assisting employers and potential employees with disabilities to make the connection and develop win-win outcomes for both parties.


Magnesium is a mineral that plays a role in hundreds of reactions throughout your body. It helps your body make fuel for energy and helps relax your nerves and arteries. Recent studies are beginning to show that magnesium may have a much larger impact on your body than we previously thought. Like vitamin D, magnesium may play a role in several different systems throughout your body from making proteins and enzymes, to regulating your central nervous system.
Magnesium is a calcium channel blocker. This means that it counters the effects of calcium in your body. It may sound counterintuitive but your body needs calcium and magnesium to balance each other out. Too much calcium, and your nerves and body functions become too relaxed. Likewise, too much magnesium and your body functions become too energetic. Having the proper amounts of calcium and magnesium in your body promotes a healthy heart, nerves, and digestive tract.
Magnesium is very important for your heart because it can help keep your heart pumping efficiently, which reduces your risk of having a heart attack or stroke. On top of that, magnesium also acts as a natural blood thinner, allowing your blood to flow more smoothly through your arteries and veins. This also helps reduce your risks of heart attacks and strokes. Magnesium causes your arteries to dilate, which reduces your blood pressure. Magnesium is also usually given to people who have just had a heart attack in order to help restore their normal heart rhythm.
Magnesium is also important for your bones. It helps regulate the rate at which your bones and teeth are made. It prevents too much calcium from going into your bones and teeth so they do not become overgrown. It also helps power your body's detoxification process, helping it flush out toxins and heavy metals. Glutathione, the most powerful antioxidant in your body, needs magnesium in order to be made.
Some signs of magnesium deficiency include loss of appetite, nausea and vomiting and fatigue. There is no accurate blood test for magnesium deficiency because most of the magnesium in your body is in your tissues and not your blood. There are urine and tissue tests your doctor can perform to check for magnesium deficiency. Along with any symptoms you're experiencing, your doctor can diagnose whether you have a magnesium deficiency or not.
People who are most prone to magnesium deficiencies include, those with kidney problems, digestive disorders like IBS or Crohn's disease, diabetics, alcoholics, those taking diuretic medications, cancer medication or antibiotics, and elderly adults. Those who fall into any of these categories should keep a close eye on their magnesium intake. People who suffer with migraines may not have enough magnesium. About half of all people who suffer from migraines are deficient in magnesium.
With long-term deficiencies you may begin to experience numbness or tingling in your extremities, muscle cramps, muscle spasms, heart palpitations, abnormal heart rhythms or seizures.
Magnesium can help keep type 2 diabetes at bay but those who have type-2 diabetes may be urinating out magnesium faster than those who do not have type 2 diabetes. Therefore, it's important to make sure you get 100% your daily value of magnesium.
It's very important that you balance your magnesium and calcium intake together because they counter balance each other. It's also important to make sure you are getting enough vitamin D because our bodies need this vitamin present in order to absorb calcium. Here are the National Institute of Health (NIH) recommendations for our daily intake of magnesium.
Seaweed has about 216 mg of magnesium in 1 ounce (28 grams), which is over 50% of your daily value.
½ cup of pumpkin seeds has over 100% of your daily value of magnesium, which is about 370 mg. You'll also get over 100% of your DV of manganese, 80% of your DV of phosphorus and about 50% of your DV of copper. Read more about the health benefits of pumpkin seeds here.
1 ounce of unsweetened cocoa powder has about 35% of your daily value of magnesium.
These seeds have almost 82 mg of magnesium in a few tablespoons of ground up flaxseeds, which is about 20% of your daily value.
These butters include Peanut butter, Sunflower seed butter, Almond butter, and Tahini (sesame butter). All of these nut butters have about 25-30% of your daily value of magnesium in two tablespoons, which is about 100 -120 mg of magnesium.
1 ounce or about 22 almonds has about 20% of your daily value of magnesium.
Red and black caviars can have just over 20% of your daily value of magnesium in one ounce, which is about two tablespoons worth of caviar. What makes caviar special in this case is that it's one of the few natural foods that's also a good source of calcium (about 10% of your daily value in one ounce) and vitamin D (over 15% of your daily value in one ounce).
It's important to note that most of these foods that are rich in magnesium are poor sources of calcium. Many of the foods rich in calcium tend to also be poor sources of magnesium. You should try to include foods that are good sources of magnesium in your diet as well as foods that are good sources of calcium. Click here for more information about calcium.
Foods should always be your #1 source for magnesium. However, if you decide to take supplements there are several forms of magnesium on the market. The main forms of magnesium on the market are magnesium carbonate, magnesium citrate, and chelated magnesium (magnesium chelate or magnesium glycinate). Of these three, chelated magnesium is usually absorbed best by the body. Magnesium citrate and Magnesium carbonate are forms that will also be absorbed by your body but they may also have a stool softening or mild laxative effect on some people as well. However, these other two forms are safe to take.
Magnesium hydroxide or magnesium sulfate are forms of magnesium that are used in laxatives like Milk of Magnesia. These two forms of magnesium should not be used as a supplement at all. Take them as directed by a physician.
Note: It's especially important that the magnesium supplement you take also has calcium and vitamin D as well. There is also some evidence that vitamin K2 is important to have as well because it keeps calcium and vitamin D from depositing in the wrong places like you arteries, increasing he risk for heart attacks or strokes. How much vitamin D and vitamin K2 a supplement should have is still being debated. However, if you are eating a diet healthy in vitamins D and K2 you should be in good shape. Since your body makes vitamin D in the summer months using sunlight, our vitamin D levels tend to be sufficient in the summer/spring months (March/April – September). To keep you vitamin K2 levels up, eat dark leafy greens like Kale and Swiss chard. Foods like Parsley, dried Basil are rich in vitamin K2 as well.


Recent news suggests that the Home Office have increased the number of unannounced visits to Tier 2 sponsors, to ensure that Tier 2 Sponsors are complying with their legal duties. The ability to Sponsor skilled workers from outside the European Economic Area (EEA) to come and work in the UK has previously been described by the courts as a privilege. UKVI officials are therefore increasingly vigilant to ensure that businesses maintain compliance with the UK immigration rules and the process which allows businesses to employ skilled non- EEA workers continues to be used appropriately.
The Tier 2 visa route allows businesses to recruit skilled workers from abroad and employers wishing to employ non-EEA migrants must obtain a sponsorship licence before they can sponsor a migrant.
Sponsors should be aware that UKVI officials can visit a sponsor at any time during the duration of the Tier 2 sponsor licence. Most sponsors can expect to be visited at least once every four years however if issues have been discovered during a previous visit, a sponsor can expect more frequent visits from the Home Office.
Failure to comply with UKVI rules could result in downgrading, suspension or revocation of a Sponsor's Licence which means that the employer will be banned from recruiting migrant workers in the future. Worst of all the loss of sponsorship due to one employee will have a knock on effect on all existing Tier 2 sponsored employees who will also lose their jobs. All employees sponsored under the revoked Tier 2 licence would be forced to leave the UK unless they can find a new sponsor to employ them almost immediately. If you are concerned about compliance issues following a visit from the UKVI or at any other time we would recommend that legal advice is sought immediately.
To contact one of our Immigration Solicitors or Immigration Barristers please complete our legal case assessment form and we will get in touch or call us now on 02071830570 for a telephone case assessment.


Start by washing then drying your arugula. Set aside to make the Lemon Vinaigrette (this is my favorite go-to salad dressing that my Mom taught me years ago).
Now squeeze the lemon half into a small bowl. Using a whisk, or a fork if you don't have one, gently pour the olive oil into the lemon juice, stirring as you go. You want to ensure that you have equal parts lemon juice and olive oil. Now season the dressing with a few pinches of salt and pepper. Mix to distribute evenly.
With a potato peeler, carefully run the blade alongside the block of Parmigiano-Reggiano to create thin, delicate shards of cheese.
Toss the salad with the dressing and then carefully lay your cheese on top.
Fieldhouse Fun Fact: This Simple Arugula Salad is a Fieldhouse Kitchen staple — I love putting it on top of a breaded chicken cutlet, alongside a steak or having it on its own. Hope you do, too!


Austin Pullar (SS), Logan Cline (2B), Steve Tiersma (DH), Cam Pfafman (C), Andrew Pullar (3B), Jason Holwerda (RF), David Laurenti (LF), Chris Schrieber (1B), Parker Hamann (CF), Lucas Nelson (P).
Ben DE BOER (3B), Drew SCHUTT (SS), Austin ZYLSTRA (1B), Colton HAROLD (RF), Josh FAKKEMA (DH), Britton YODER (LF), A.J. NITZSCHKE (CF), Zachary ROSSON (C), Mason O'DONNELL (2B), Ben ZEUTENHORST (P).
Top of 1st- Austin Pullar struck out swinging. Logan Cline grounded out(6-3). Steve Tiersma singled. Cam Pfafman lined out(8).
Bottom of 1st- Ben DE BOER flied out(9). Drew SCHUTT homered(1) over the fence on a 1-0 count. Austin ZYLSTRA reached on a walk. Passed Ball by Cam Pfafman, Austin ZYLSTRA advanced to second. Colton HAROLD flied out(9). Josh FAKKEMA flied out(8).
Top of 2nd- Andrew Pullar flied out(7). Jason Holwerda flied out(8). David Laurenti grounded out(3).
Bottom of 2nd- Britton YODER grounded out(6-3). A.J. NITZSCHKE homered(4) over the fence on a 0-0 count. Zachary ROSSON hit by pitch. FUHS pinch running for ROSSON, Alex FUHS was caught stealing (2-6). Mason O'DONNELL singled. Ben DE BOER singled, Mason O'DONNELL advanced to second. Drew SCHUTT reached first on the fielder's choice, Mason O'DONNELL advanced to third, Ben DE BOER was forced out(6).
Top of 3rd- ROSSON came in to play C, FUHS goes out. Chris Schrieber singled. Parker Hamann struck out swinging. Austin Pullar popped out(5). Logan Cline lined out(4).
Bottom of 3rd- Austin ZYLSTRA homered(4) over the fence on a 0-0 count. Colton HAROLD grounded out(6-3). Josh FAKKEMA reached on a walk. Britton YODER singled, Josh FAKKEMA advanced to second. A.J. NITZSCHKE singled, Josh FAKKEMA advanced to third, Britton YODER advanced to second. Zachary ROSSON grounded into a (1-2-3) double play.
Top of 4th- Steve Tiersma singled. Cam Pfafman popped out(4). Andrew Pullar singled, Steve Tiersma advanced to second. Jason Holwerda singled, Steve Tiersma advanced to third, Andrew Pullar advanced to second. David Laurenti hit by pitch, Steve Tiersma Scored, Andrew Pullar advanced to third, Jason Holwerda advanced to second. Chris Schrieber hit a grand slam(1) over the fence on a 1-0 count, Andrew Pullar Scored, Jason Holwerda Scored, David Laurenti Scored. Parker Hamann singled. Austin Pullar doubled, Parker Hamann was thrown out (8-4-2). OLESEN now in to pitch, replacing ZEUTENHORST. Logan Cline lined out(8).
Bottom of 4th- Mason O'DONNELL grounded out(6-3). Ben DE BOER grounded out(4-3). Drew SCHUTT reached on a walk. Austin ZYLSTRA reached on a walk, Drew SCHUTT advanced to second. Colton HAROLD hit a 3 run home run(1) over the fence on a 3-2 count, Drew SCHUTT Scored, Austin ZYLSTRA Scored. Josh FAKKEMA struck out looking.
Top of 5th- Steve Tiersma struck out swinging. Cam Pfafman reached on a walk. Branderhorst pinch running for Pfafman. Andrew Pullar grounded out(6-3), Dillon Branderhorst advanced to second. Jason Holwerda flied out(7).
Bottom of 5th- Pfafman came in to play C, Branderhorst goes out. Horne now in to pitch, replacing Nelson. Britton YODER reached base on Andrew Pullar's wild throw(E5). A.J. NITZSCHKE hit a 2 run home run(5) over the fence on a 3-2 count, Britton YODER Scored. Zachary ROSSON grounded out(4-3). Mason O'DONNELL lined out(8). Ben DE BOER flied out(6).
Top of 6th- David Laurenti grounded out(3). Chris Schrieber reached base on Evan OLESEN's dropped popup(E1). Branderhorst pinch running for Schrieber. Parker Hamann lined into a (4-3) double play.
Bottom of 6th- Schrieber came in to play 1B, Branderhorst goes out. Drew SCHUTT flied out(7). Austin ZYLSTRA reached base on Austin Pullar's wild throw(E6). FUHS pinch running for ZYLSTRA. Colton HAROLD hit a 2 run home run(2) over the fence on a 1-2 count, Alex FUHS Scored. Josh FAKKEMA struck out looking. Britton YODER flied out(8).
Top of 7th- ZYLSTRA came in to play 1B, FUHS goes out. Austin Pullar struck out swinging. Logan Cline reached on a walk. Steve Tiersma struck out swinging. Bond pinch hitting for Pfafman. Justin Bond grounded out(5-3).


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DC Water to Assess Fairness of Skyrocketing Fees After News4 I-Team Investigation of Surging Bills
By Jodie Fleischer, Rick Yarborough and Steve Jones	• Published at 1:57 pm on October 27, 2017
DC Water says it will assess the fairness of skyrocketing fees after a News4 I-Team investigation revealed bills that have jumped by thousands of dollars and customers who are unable to pay.
This does not necessarily mean DC Water is planning to change the fees, but the agency has vowed to take a look at who pays the most and whether the right people are paying the right amount.
The fees are part of DC Water's Clean Rivers program which started in 2009; the fees have grown exponentially every year since then.
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DC Water also launched a new web page Thursday night to explain the program's history and address growing concerns from the community.
The web page also promises to assess the overall fairness of the Clean Rivers fees, which fund a $2.6 billion project to build underground tunnels designed to stop sewage and stormwater from overflowing into our rivers. The federal government mandated DC Water do something to fix that problem.
Earlier this week, the News4 I-Team revealed how leaders in Washington's faith community said they're having trouble paying their bills and could have to cut community programs. That prompted a meeting from various religious institutions to discuss the problem.
Historic cemeteries also told the I-Team they could have to shut down as a result of the high fees.
In one case, the water bill has jumped from $3,500 a year to $200,000 a year and, the fee makes up 80 percent of that bill.
DC Water's general manager said the problem with lowering fees for the faith community and cemeteries is that if they get a break, all of the other customers will have to pay more to make up the difference.
DC Water customers are the main funding source for the Clean Rivers project, which is the largest public works project in Washington, D.C., since Metro was built.
Skyrocketing DC Water Bills Leave Some Customers Unable to Pay
Water bills are skyrocketing across the District by hundreds, even thousands, of dollars a month, and the News4 I-Team found it has nothing to do with how much water customers are using.
The fees are funding a multi-billion dollar project, and DC Water is charging each of its customers to pay for it.
A group of church leaders reached out to the News4 I-Team saying they could no longer afford to pay their growing bills, so we crunched the numbers and found Washington, D.C.'s faith community, which survives largely on donations, has already shelled out millions for these fees.
"It's dire, because we're looking at our budget now for 2018 and we can't balance it," Velma Wyman told the I-Team.
On Sunday mornings at the First Baptist Church on Minnesota Avenue, the members celebrate a higher power.
But it's the church's higher water bills that have Wyman on edge.
The church members pass the plate, but now, many of their donations can't go to the ministry or the long list of needed projects for the 144-year-old congregation.
"Our people are very generous, and they will do that. but at a certain point, they don't have the money," said Wyman. "They're getting hit the same way we're getting hit."
Down the road, at Bethesda New Life Gospel Church, Pastor Jesse Richardson, Jr. is also getting hit.
"We started to notice this great big ol' water bill all of a sudden," recalls Richardson, saying at first DC Water said the church must have a leak.
"They came by and they checked all the plumbing and these things, and one of the deacons happened to look at the itemized bill and say, 'Hey, what is this?'" Richardson said.
"We're Being Robbed Without a Gun!"
It's called the Clean Rivers Impervious Area Charge, or IAC, and it's on every DC water bill.
The fee pays for the building of large underground tunnels to keep sewage and stormwater from flooding our rivers. The project stems from a federal mandate to fix the environmental problem.
But when the News4 I-Team was flooded with complaints from across the District, we took a look at the money DC Water is charging customers for the project.
"It's clearly made up, they gave absolutely no thought to the process or how much they would charge," said Richardson. "And I think it's totally unfair."
DC Water does have a formula based on aerial photos of every D.C. property. Those photos are used to calculate the square footage of the buildings and any areas of concrete, like sidewalks or parking lots.
The thinking is the properties with the most concrete would contribute the most stormwater runoff and should pay more.
"Some churches in our least affluent communities are paying the most, like close to $50,000 at a Ward 8 church. How does that even make sense?" said Craig Muckle, with the Archdiocese of Washington, many of whose member churches are also at a breaking point.
The same is true for synagogues and mosques, where the main source of income is donations. And congregants who live in the District are also paying the fee on their own water bills at home.
"It's not fair," said Muckle. "Some people are paying it twice, or three times if they happen to own a business."
For months the News4 I-Team filed records requests and crunched the numbers. The District's faith-based community has already paid more than $11 million just in Clean Rivers fees, largely because of their parking lots, which city law requires.
"When you build a church in District of Columbia, you have to have one off-street parking for every 10 members," said Pastor George Gilbert, Sr. of Holy Trinity United Baptist Church.
When the charge first started in 2009, it was so small, no one really noticed. Now some churches pay $6,000, $9,000, even $12,000 a month, just for the IAC fee.
"We're being robbed without a gun," said Pastor Richardson. "You have to remember, we cannot charge, we cannot go up on our rates to allow people to come into the church."
"Unfortunately Someone's Got to Pay the Bill."
The News4 I-Team took the pastors' concerns to the head of DC Water, George Hawkins.
"It pains me to explain this to our ratepayers but it is the hard reality. We have a giant project. It's the biggest public works project in Washington, D.C., since Metro was built," he said.
Hawkins pointed out that one benefit of the current fee system is it allows DC Water to collect IAC fees from properties that are only parking lots, with plenty of stormwater runoff and no need for regular water service.
But Hawkins says he's aware the faith community and other non-profits are hurting, and it's something of great concern.
"One of the challenges we have in the District, perhaps more than most cities, is an enormous number of our customers are nonprofit," said Hawkins. "Government is nonprofit. We have every nonprofit known to humankind located here."
And some of those non-profits have not expressed any trouble paying the higher bills. So the board would have to consider how to help the ones who can't pay while making sure the ones who can don't get an unnecessary break.
"Every discount we give to one customer is money we have to raise from everybody else," said Hawkins, pointing out that regular homeowners, local business owners, even the government-owned buildings whose bills are paid with tax dollars will all have to make up the difference.
"We look at every option. And there aren't that many. Unfortunately someone's got to pay the bill," said Hawkins.
A $2 Credit on a $420 Fee
First Baptist Church of Minnesota Avenue has tried everything church leaders can think of to lower the fees.
In 2013, D.C.'s Department of Energy and Environment and the Anacostia Watershed Society helped build the church two rain gardens and funnel water from their roof. They also replaced a large section of concrete with porous pavers to limit the stormwater runoff.
"If we get a heavy rain, it looks like a little river coming through the channels here and there," said Wyman.
But the credit on the church's bill only amounts to $2.45 off of the $420 fee.
"Oh yeah, you feel good, but your purse doesn't feel good because it's costly," said Wyman.
And the I-Team found DC Water actually increased the church's fee, instead of lowering it.
"I think it's words, they say they're going to look at that. But looking at it and doing something about it are two different things," said Wyman, adding that churches could be forced to cut things like feeding programs, bookbag giveaways and holiday help for the needy -- things that really impact the community.
The I-Team found the average church is paying about $400 per month just for the IAC fee, which will continue to rise each year for the foreseeable future.
"It's a it's a painful reality for what is necessary to improve our system," said Hawkins. "My comment to any of those customers is we are aware of the issue and we know it's a challenge."
He committed that the board will thoroughly evaluate the hardship for churches and other non-profits, but that would likely not happen until the beginning of 2018 at the earliest.
"We understand their dilemma but it's hard, it's really hard on us," said Wyman.
Historic D.C. Cemeteries in Jeopardy of Closing Due to Growing Water Bills
When you look out across the rolling hills of Rock Creek Cemetery, you find a resting place filled with beauty, sculptures and a significant piece of Washington DC history. Over the years, the grounds have attracted people of all faiths, races, and backgrounds.
"We have four Supreme Court justices here. We have a signer of the Constitution here," said Jim Jones, who sits on the finance committee for St. Paul's Rock Creek Church, which runs the cemetery. "This is the most historic cemetery in the city. And we want to be here. We've been here since 1712."
But the cemetery's future is in jeopardy because it can't pay its $200,000-a-year water bill, Jones said. That bill used to be about $3,500 per year.
"Our actual water usage is declining, incidentally. But the actual water bill is still climbing," said Jones. "It makes absolutely no sense whatsoever."
It's because of the Clean Rivers IAC, which stands for Impervious Area Charge. It's a fee DC Water charges based on estimated stormwater runoff for each property.
Roadways, parking spaces and rooftops of buildings all generate increased fees. In the cemetery's case, DC Water is also counting the surface area of the mausoleums on the property.
"It was a real shock and a surprise," said Jones. "Most of these mausoleums were constructed as places for people's loved ones 150, 200 years ago. There's no way we go back and recapture that. And to, in effect, be punished for that, it's a bit much."
Rock Creek Cemetery is paying nearly $15,000 a month for the IAC fee, which accounts for about 80 percent of the cemetery's total water bill.
The News4 I-Team spent months requesting records from DC Water, matching up properties and calculating how much each has to pay, and found D.C.'s cemeteries have already paid millions toward funding the District's multi-billion dollar Clean Rivers project, construction of underground tunnels to keep sewage and stormwater from flooding our rivers.
The federal government mandated D.C. do something to solve its overflow issues, and DC Water opted to embark on the Clean Rivers project in 2009. It's the most expensive public works project in Washington, D.C., since Metro was built.
"It just seems patently unfair that the heaviest burden appears to be placed in a place that's actually supporting the city in its work to serve the community," said Craig Muckle, public policy manager for the Archdiocese of Washington.
Cemeteries and churches are especially hard hit, Muckle added, because they have to have parking areas.
Some larger cemeteries also have roads to get to their gravesites. It all factors into that IAC fee.
But the I-Team found DC Water does not make the District pay for its public roadways; those are exempt.
"If it's good for the goose, it's good for the gander," said Muckle.
D.C. government does pay millions each year for its buildings and parking lots, and so does the federal government.
The fee even allows DC Water to charge customers who don't get a regular water bill but do cause stormwater runoff, like commercial parking lots.
Cemeteries are different, Jones said. "We're not a commercial parking lot or the evil chemical company incorporated; we're a green space!"
Rock Creek Cemetery even spent $100,000 ripping out some of its roadways and planting grass instead, Jones said.
But when it appealed to DC Water, an inspector came out and noticed more impervious area, so the monthly IAC fees went up.
"We were absolutely dumbfounded," said Jones. "We have the largest variety of trees outside of the arboretum here in Washington, D.C."
The National Arboretum is a 400-acre sanctuary of public green space which pays nearly $400,000 a year in IAC fees, even though it isn't hooked to a storm drain and re-uses its stormwater run-off to water the plants.
The News4 I-Team asked the head of DC Water, George Hawkins, whether the cemeteries and arboretum have a valid argument to avoid the growing fees.
"It's an argument that has a point to it," replied Hawkins, adding that if bills are lowered for high-dollar customers like cemeteries and the arboretum, the rest of the city's customers would have to make up the difference, because that massive tunnel project isn't getting any cheaper.
"We're going to evaluate various options of what we could or should do just on this basis. What is the cost of it?" said Hawkins, vowing that the water board will take a look at the issue, but probably not until sometime in 2018.
"I think there needs to be a whole rethinking of how revenue for this project is derived," said Jones. "And there ought to be at least a cap or an exemption for historic places."
Other cities, like Baltimore, have already exempted cemetery streets from stormwater fees and offer a hardship exemption for non-profits who can't pay.
Jones said the cemeteries won't be able to pay anything if they're all out of business.
"I think over time many operators are going to be simply left with no other choice than to close their gates and walk away," said Jones. "And I think for an institution like Rock Creek Cemetery ... that is just unthinkable."
Reported by Jodie Fleischer, produced by Rick Yarborough, and shot and edited by Steve Jones.


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text:
She Decides South Africa
SheDecides South Africa, still very much in its inception stages, has shown great potential for growth and success.
The Movement in the country boasts footprints in four of South Africa's nine provinces: Eastern Cape, Gauteng, Kwa-Zulu Natal and Western Cape.
In reference to the liberation struggle against colonialism and apartheid, members of the emergent SheDecides South Africa call themselves "Comrades".
The Comrades in Eastern Cape are hard at work with the Department of Education, trying to create a world (or rather an Eastern Cape) where young girls can decide their own future. The programme planned will be a hackathon where children can learn how to code and develop their software skills, giving them opportunities for the future.
The Western Cape Comrades have occupied the fashion space, taking visible action at the Khayelitsha Fashion Week. On Freedom Day courageous women gathered as part of the "Do Not Rape" campaign and the Comrades spoke about the relevance of SheDecides for women in South Africa.
The movement in South Africa promises to retain the enthusiasm with which it started off. A working group has been created to formulate comprehensive strategies and plans on how to develop the movement and see it thrive in South Africa's ever volatile political, civil and economic society. Check out the shedecides_sa Instagram to follow the action of the Comrades on the ground.
Main SheDecides Facebook Page
UNSAFE ABORTIONS DURING COVID-19 LOCKDOWN
Written by Tshiamo Wilson, Founder of SheSpoke Organisation and Saidy Brown, SheDecides 25x25 Young Leader
SheDecides Kenya announces its Local Champions
SheDecides Kenya has unveiled the global Movement's first ever SheDecides Local Champions!
Global Commitments, Local Actions: Reflections on ICFP
This past month the government of Rwanda successfully hosted the 5th International Conference on Family Planning (ICFP), with many SheDecides Champions and local movements playing central roles in the vibrant programme.


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Escanaba Trough
Hydrothermal Systems in Escanaba Trough
Escanaba Trough: Exploring the Seafloor and Oceanic Footprints
May 26 - June 14, 2022
Expedition Home
Hydrothermal systems occur when seawater percolates down through fractures in the oceanic crust, heating up as it nears the earth's hot interior. These systems are often found near mid-ocean ridges such as Escanaba Trough, where tectonic plates diverge, and new seafloor is created.
In hydrothermal systems, descending seawater interacts with oceanic crust, removing chemicals from the rocks as it heats to 350-400 degrees Celsius (650-750 degrees Fahrenheit), about four times hotter than boiling water. (Extreme pressure in the ocean's depths prevents fluids from boiling). This interaction of seawater and crust produces hydrothermal fluid, a chemically modified slurry of gases and dissolved elements, including metals.
The superheated fluid is then ejected back up to the seafloor and promptly chilled by near-freezing ocean bottom waters. Chemicals dissolved in the fluid precipitate at the vent, forming chimney-like deposits and supporting deep-sea chemosynthetic communities—organisms that rely on chemicals rather than photosynthesis to fuel their metabolism.
The discovery of modern hydrothermal systems dates to a 1977 research expedition to the Galapagos Rift and East Pacific Rise, which encountered chemosynthetic fauna that the expedition's geology team famously preserved in vodka. Less attention is paid to the fact that this team of geologists, geochemists, and geophysicists found exactly what they had predicted: A heat-driven region of seafloor spreading and mineral precipitation, based on studying ancient analogues on land and studying heat flow on the seafloor.
Now, after 45 years of excitement and discovery around active hydrothermal systems and the ecosystems they support, interest has turned towards the more subtle ways in which hydrothermal systems evolve over time, eventually becoming severed from the heat and fluid-flow pathways responsible for their formation.
Photo from expedition to Escanaba Trough in late 1980s of older, oxidized sulfide deposit colonized by sponges at SESCA site. Image in public domain. Download largest version (jpg, 814 KB).
At Escanaba Trough, for example, there is little current hydrothermal activity, making it a good place to study how hydrothermal systems transform and persist after heat sources become less active, as well as in regions further from high-temperature fluid flow.
Central to this research is the concept of chemical equilibrium. Minerals and chemicals formed under one set of circumstances are said to be in equilibrium—their composition and concentration are balanced, given the circumstances. Picture a sealed bottle of soda: The liquid inside the bottle contains dissolved carbon dioxide, and there is carbon dioxide gas in the space between the liquid and the cap. Carbon dioxide is moving constantly between the liquid and gas phases within the bottle, but the contents don't appear to change. The system is in equilibrium.
When minerals and chemicals shift to a different set of circumstances, they go out of equilibrium, and must seek balance with their new surroundings. These transformations from equilibrium to disequilibrium may occur rapidly, in less than a second—imagine opening that sealed soda bottle with an effervescent hiss—or they may occur slowly, over millions of years.
The atmosphere of Earth is an example of how chemical equilibrium changes over geologic time: The concentrations of oxygen, nitrogen, methane, and carbon dioxide on Earth today are different than they were when the planet formed 4.5 billion years ago. This is largely because Earth supports life, and organisms produce gases that constantly modify the composition of Earth's atmosphere.
The disequilibrium that occurs in hydrothermal systems is both very rapid—when hydrothermal fluid vents to cold seawater—and slowly unfolding, as minerals transform through their interaction with seawater. The chemosynthetic organisms making use of this phenomenon remain poorly understood; samples taken during the Escanaba Trough expedition will help to characterize the chemical and biological conditions that accompany the evolution of hydrothermal systems.
By Amy Gartman, Principal Investigator; Research Oceanographer, U.S. Geological Survey Pacific Coastal and Marine Science Center
Re-posted with permission from original article.


On Friday 15th September, Milverton Kindergarten and ABC Marne came to College Street to visit our Kapa Haka group. It is always so special performing to and with our preschool tamariki who may one day come to our kura and join our ropu!
Nga mihi nui to the kaiako from both centres who helped make this possible and Pa Smiler as always!


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Purdue receives 400 doses of H1N1 vaccine | Purdue News | Lafayette Online
News | Purdue News
Purdue receives 400 doses of H1N1 vaccine
By Staff Reports, Purdue University News Service
WEST LAFAYETTE, Ind. — The Purdue Student Health Center has received 400 doses of H1N1 vaccine through the Tippecanoe County Health Department.
The Health Center also reported that the supply of seasonal flu vaccine has been exhausted and that no additional supplies are expected to be available this year.
H1N1 vaccines will be available from 8:30 a.m. to 4:30 p.m. Monday through Friday in Room 43 at the center. Because of the limited supply, only those in established priority groups will be eligible for vaccines. The priority guidelines have been set by the Tippecanoe County Health Department based on guidance from the Centers for Disease Control and Prevention.
Those considered top priority include the following:
Pregnant women.
Caregivers for children younger than 6 months.
University health-care and emergency medical personnel.
Individuals with underlying health conditions that put them at risk for complications of flu-related illness.
Individuals considered to be at risk are those who have: cancer, blood disorders, chronic lung disease (asthma, chronic obstructive pulmonary disease, etc.), diabetes, heart disease, kidney or liver disorders, neurological or neuromuscular disorders, or otherwise weakened immune systems.
Individuals may call the Student Health Center's Urgent Care Unit at 765-494-1724 to determine if they have a qualifying health condition.
Purdue continues to offer H1N1 vaccine
Purdue reports 1 case of H1N1 since Dec 23
Purdue opens H1N1 vaccinations to all employees, retirees, students
Purdue to vaccinate 8,900 for H1N1
H1N1 vaccine offered at 4 locations for Purdue Wellness Week


I agree with you about AA philosophy. It puts me off as well. I never lasted more than 1-2 meetings even when I was majorly in need of something. Those little slogans... It's not something I choose to be involved with, although I know it helps alot of people.
Please let me know of your experience/progress with your group and what their philosophy is. I'm looking for something that will understand that an occasional Margarita (Cadillac, of course) with friends is not something to flagellate myself over. So far, the meditation/spiritual practice is keeping me on track and is probably all I need as long as I stick with it. Since I'm not in such desperate need of immediate relief I don't feel the need to get so fu**ed up like I used to. But I can't kid myself - booze combined with hypomania is a potent brew. Just wish it didn't cost so much in every way imaginable.
>AA's philosophy puts me off. I know it helps people, but I've been repelled to it, not drawn. I know of another group in the area that I'll start going to.


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- A civilian was killed and 3 others wounded in a parked car bomb explosion. The car exploded in a park near Al Sakhra restaurant in Palestine street east Baghdad at 1,00 pm.
- A parked car explosion in Al attar street in Karrada Dakhil (Karrada in) neighborhood downtown Baghdad at 1,00 claiming the life of a civilian and injuring an other 4.
- 2 Iraqi army members were killed and 4 others were injured when a suicide car bomber attacked a check point of the Iraqi army in Al Liqa'a square in Al Mansour neighborhood south west Baghdad around 3,30 pm.
- A policeman was killed and his colleague was injured in a random fire by insurgents in Slekh neighborhood north east Baghdad around 5 ,00 pm.
- A security source in the combined operation room in Diyala province said that 3 policemen were killed by Qaida insurgents. The source said that the Qaida insurgents stopped the car of the three policemen in Al Mustafa intersection west Baquba and shot them dead.
- A security source said that gunmen killed 3 civilians and injured a fourth one in Khirnabat area north of Baquba today morning.
- Police sources in Muqdadiyah town north eat of Baquba city said that 3 men including a policeman were injured when insurgent opened fire targeting them in Al Mu'alimeen neighborhood north Muqdadiyah town today afternoon. The source said that another civilian was kidnapped todayby another gunmen group in the same neighborhood.
- Iraqi police said that 2 civilians were killed and 2 others were injured by unknown gunmen in tow incidents on Tikrit- Kirkuk motorway north of Baghdad early morning today.
- Iraqi police said that a child was killed by American soldiers when they opened fire on a house in Al sajla neighborhood close to Dujail town north of Baghdad yesterday evening. The source added that the shooting caused the death of some cows and sheep.
- Iraqi police said that gunmen killed a man and burnt his car in Yathrib neighborhood east Balad town north of Baghdad.
- Eyewitnesses said that 2 students were killed and 2 others were injured in clashes between the British forces and gunmen in Al Qurna town north Basra today afternoon. The eyewitnesses said that the students were leaving home from school which is in the area of the clashes.