text:
\section{Introduction}

Let $R$ be a finite ring equipped with a weight $w$.  Two linear codes
$C, D \le {_RR^n}$ are \emph{isometrically equivalent} if there is an
isometry between them, i.e., an $R$-linear bijection $\varphi: C
\longrightarrow D$ that satisfies $w(\varphi(c))=w(c)$ for all $c\in
C$. We say that $\varphi$ \emph{preserves} the weight $w$.

MacWilliams in her doctoral dissertation \cite{macw63} and later
Bogart, Goldberg, and Gordon~\cite{bogagoldgord78} proved that, in the
case where $R$ is a finite field and $w$ is the Hamming weight, every
isometry is the restriction of a {\em monomial transformation} $\Phi$
of the ambient space $_RR^n$.  A monomial transformation of $_RR^n$ is
simply a left linear mapping $\Phi: R^n\longrightarrow R^n$ the matrix
representation of which is a product of a permutation matrix and an
invertible diagonal matrix.  Said another way, every Hamming isometry
over a finite field extends to a monomial transformation.  This result
is often called the \emph{MacWilliams Extension Theorem} or the
\emph{MacWilliams Equivalence Theorem}.

With increased interest in linear codes over finite rings there arose
the natural question: could the Extension Theorem be proved in the
context of ring-linear coding theory? This question appeared
complicated, as two different weights were pertinent: the traditional
Hamming weight $w_{\rm H}$ and also a new weight $w_{\rm hom}$ called
the \emph{homogeneous} weight by its discoverers Constantinescu
and Heise~\cite{consheis97}.

In~\cite{wood99} Wood proved the MacWilliams Extension Theorem
for all linear codes over finite Frobenius rings equipped with the
Hamming weight. In the commutative case he showed in the same paper
that the Frobenius property was not only sufficient but also
necessary.  In the non-commutative case, the necessity of the
Frobenius property was proved in \cite{wood08a}.

Inspired by the paper of Constantinescu, Heise, and
Honold~\cite{consheishono96} which used combinatorial methods to prove
the Extension Theorem for homogeneous weights on $\mathbb Z_m$, Greferath
and Schmidt~\cite{grefschm00} showed that the Extension Theorem is
true for linear codes over finite Frobenius rings when using the
homogeneous weight.  Moreover, they showed that for all finite rings
every Hamming isometry between two linear codes is a homogeneous
isometry and vice versa.

The situation can be viewed as follows: for $R$ a finite ring, and
either the Hamming weight or the homogeneous weight, the Extension
Theorem holds for all linear codes in $R^n$ if and only if the ring is
Frobenius.  This is a special case of more general results by
Greferath, Nechaev, and Wisbauer~\cite{grefnechwisb04} who proved that
if the codes are submodules of a quasi-Frobenius bi-module $_RA_R$
over any finite ring $R$, then the Extension Theorem holds for the
Hamming and homogeneous weights.  The converse of this was proved by
Wood in \cite{wood09}. \medskip

Having understood all requirements on the algebraic side of the
problem, we now focus on the metrical aspect. This paper aims to
further develop a characterization of all weights on a finite
(Frobenius) ring, for which the corresponding isometries satisfy the
Extension Theorem.

In our discussion we will assume that the weights in question are
bi-invariant, which means that $w(ux) = w(x) =w(xu)$ for all $x\in R$
and $u\in R^\times$.  Our main results do not apply to weights with
smaller symmetry groups such as the Lee or Euclidean weight (on
$R=\ensuremath{\field{Z}}_m$, except for $m\in\{2,3,4,6\}$), despite their importance for
ring-linear coding theory.

The goal of this paper is to give a necessary and sufficient condition
that a bi-invariant weight $w$ must satisfy in order for the Extension
Theorem to hold for isometries preserving $w$.  We are not able to
characterize all such weights when the underlying ring is an arbitrary
Frobenius ring, but we do achieve a complete result for
\emph{principal ideal rings}. These are rings in which each left or
right ideal is principal, and they form a large subclass of the finite
Frobenius rings.

The present work is a continuation and generalization of earlier work
on this topic \cite{wood97, wood99a, grefhono05, grefhono06, wood09,
  grefmcfazumb13}.  As in \cite{grefhono06, grefmcfazumb13} the
M\"obius function on the partially ordered set of (principal, right)
ideals is crucial for the statement and proof of our main
characterization result; however, in contrast to these works we do not
need the values of the M\"obius function explicitly, but use its
defining properties instead to achieve a more general result.  Our
restriction to principal ideal rings stems from our method of proof,
which requires the annihilator of a principal ideal to be principal.
The main result was proved for the case of finite chain rings in
\cite[Theorem~3.2]{grefhono05} (and in a more general form in
\cite[Theorem~16]{wood97}), in the case $\ensuremath{\field{Z}}_m$ in
\cite[Theorem~8]{grefhono06}, for direct products of finite chain
rings in \cite[Theorem~22]{grefmcfazumb13}, and for matrix rings over
finite fields in \cite[Theorem~9.5]{wood09} (see
Example~\ref{ex:examples} below).  The main result gives a concrete
manifestation of \cite[Proposition~12]{wood97} and
\cite[Theorem~3.1]{wood99a}.  Further to \cite{grefmcfazumb13} we
prove that our condition on the weight is not only sufficient, but
also necessary for the Extension Theorem, using an argument similar to
that in \cite{grefhono06, wood08a}. \medskip

Here is a short summary of the contents of the paper.  In
Section~\ref{sec:prelims} we review the terminology of Frobenius
rings, M\"obius functions, and orthogonality matrices needed for the
statements and proofs of our main results.  In addition, we prove a
result (Corollary~\ref{cor_smult}) that says that a right-invariant
weight $w$ on $R$ satisfies the Extension Property if the Hamming
weight $w_{\rm H}$ is a correlation multiple of $w$.

In Section~\ref{sec:orthog-matrices} we show that the Extension
Property holds for a bi-invariant weight if and only if its
orthogonality matrix is invertible.  The main results are stated in
Section~\ref{sec:bi-inv-wts}.  By an appropriate unimodular change of
basis, the orthogonality matrix can be put into triangular form, with
a simple expression for the diagonal entries (Theorem~\ref{thm:WQ}).
The Main Result (Theorem~\ref{maintheorem}) then says that the
Extension Property holds if and only if all the diagonal entries of
the orthogonality matrix are nonzero.  A proof of Theorem~\ref{thm:WQ}
is given in Section~\ref{sec:proof}. \medskip

This paper is written in memory of our friend, teacher, and colleague
Werner Heise who, sadly, passed away in February 2013 after a long
illness. Werner has been very influential in ring-linear coding theory
through his discovery of the homogeneous weight on ${\mathbb Z}_m$
(``Heise weight'') and subsequent contributions.

\section{Notation and Background}%
\label{sec:prelims}

In all that follows, rings $R$ will be finite, associative and possess
an identity $1$. The group of invertible elements (units) will be
denoted by $R^\times$ or $\ensuremath{U}$.  Any module $_RM$ will be unital,
meaning $1m=m$ for all $m\in M$.

\subsection*{Frobenius Rings}

We describe properties of Frobenius rings needed in this paper, as in
\cite{honold01}.

The character group of the additive group of a ring $R$ is defined as
$\widehat{R}:={\rm Hom}_{\mathbb Z}(R,{\mathbb C}^\times)$.
This group has the structure of an $R,R$-bimodule by defining
$\chi^r(x):=\chi(rx)$ and $^r\chi(x):=\chi(xr)$ for all $r,x\in R$,
and for all $\chi\in \widehat{R}$.

The \emph{left socle} ${\rm soc}(_RR)$ is defined as the sum of all
minimal left ideals of $R$. It is a two-sided ideal.  A similar
definition leads to the \emph{right socle} ${\rm soc}(R_R)$ which is
also two-sided, but will not necessarily coincide with its left
counterpart.

A finite ring $R$ is
\emph{Frobenius} if one of the following four
equivalent statements holds:
\begin{itemize}\itemsep=1mm
\item $_RR \cong {_R\widehat{R}}$.
\item $R_R \cong {\widehat{R}_R}$.
\item ${\rm soc}(_RR)$ is left principal.
\item ${\rm soc}(R_R)$ is right principal.
\end{itemize}
For a finite Frobenius ring the left and right socles coincide.

Crucial for later use is the fact that finite Frobenius rings are
quasi-Frobenius and hence possess a perfect duality. This means the
following: Let $L(_RR)$ denote the lattice of all left ideals of $R$,
and let $L(R_R)$ denote the lattice of all right ideals of $R$. There
is a mapping $\perp: L(_RR) \longrightarrow L(R_R),\; I \mapsto
I^\perp$ where $I^\perp:= \{x\in R \mid Ix=0\}$ is the right
annihilator of $I$ in $R$. This mapping is an order anti-isomorphism
between the two lattices.  The inverse mapping associates to every
right ideal its left annihilator.

\subsection*{Principal Ideal Rings}

A ring $R$ is \emph{left principal} if every left ideal is left
principal, similarly a ring is \emph{right principal} if every right
ideal is right principal.  If a ring is both left principal and right
principal it is a {\em principal ideal ring}.  Nechaev in
\cite{nechaev73} proved that ``a finite ring with identity in which
every two-sided ideal is left principal is a principal ideal ring."
Hence every finite left principal ideal ring is a principal ideal
ring.  Further, as argued in \cite{nechaev73}, the finite principal
ideal rings are precisely the finite direct sums of matrix rings over
finite chain rings.  They form a subclass of the class of finite
Frobenius rings (since, for example, their one-sided socles are
principal).

\subsection*{M\"obius Function}

The reader who is interested in a more detailed survey of the
following is referred to \cite[Chapter~IV]{aigner}, \cite{rota64}, or
\cite[Chapter~3.6]{stanley}.

For a finite partially-ordered set (poset) $P$, we have the incidence
algebra
\[ {\mathbb A}(P) \;:=\; \{\,f: P\times P \longrightarrow {\mathbb Q}
\mid\, x \not\le y \;\; \mbox{implies} \;\; f(x,y)=0 \,\} \:. \]
Addition and scalar multiplication in ${\mathbb A}( P)$ are defined
point-wise; multiplication is convolution:
\[  (f*g)(a,b) = \sum_{a \le c \le b} f(a,c) \, g(c,b) \:. \]
The invertible elements are exactly the functions $f\in {\mathbb
  A}(P)$ satisfying $f(x,x) \ne 0$ for all $x\in P$.  In particular,
the characteristic function of the partial order of $P$ given by
\[ \zeta: P\times P \longrightarrow {\mathbb Q} \:,
\quad (x,y) \mapsto \left\{\begin{array}{lcl}
    1 & : & x\le y\\
    0 & : & \mbox{otherwise}
  \end{array}\right. \]
is an invertible element of ${\mathbb A}(P)$.  Its inverse is the {\em
  M\"obius function\/} $\mu: P\times P \longrightarrow {\mathbb Q}$
implicitly defined by $\mu(x,x) = 1$ and \[ \sum_{x\le t \le y}
\mu(x,t) \;=\; 0 \] if $x<y$, and $\mu(x,y) = 0$ if $x \not\le y$.

\subsection*{Weights and Code Isometries}

Let $R$ be any finite ring.  By a \emph{weight} $w$ we mean any
$\ensuremath{\field{Q}}$-valued function $w: R \longrightarrow {\mathbb Q}$ on $R$,
without presuming any particular properties.  As usual we extend $w$
additively to a weight on $R^n$ by setting
\[ w: R^n \longrightarrow {\mathbb Q} \:,\quad
x \mapsto \sum_{i=1}^n w(x_i) \:. \]
The \emph{left} and \emph{right symmetry groups} of $w$ are defined by
\[ G_{\mathrm{lt}}(w) := \{ u \in U: w(ux) = w(x), x \in R\} \:, \quad
G_{\mathrm{rt}}(w) := \{ v \in U: w(xv) = w(x), x \in R\} \:. \]
A weight $w$ is called \emph{left} (resp.\ \emph{right})
\emph{invariant} if $G_{\mathrm{lt}}(w) = U$ (resp.\ $G_{\mathrm{rt}}(w) = U$).

A (left) \emph{linear code} of length $n$ over $R$ is a submodule $C$
of $ {}_{R}R^{n}$. A {\em $w$-isometry\/} is a linear map $\varphi : C
\longrightarrow {}_{R}R^{n}$ with $w(\varphi(x)) = w(x)$ for all $x \in C$, i.e.,
a mapping that preserves the weight $w$.

A \emph{monomial transformation} is a bijective (left) $R$-linear
mapping $\Phi : R^{n} \longrightarrow R^{n}$ such that there is a
permutation $\pi\in S_n$ and units $u_1, \ldots, u_n\in \ensuremath{U}$ so that \[
\Phi(x_1, \ldots, x_n) \;=\; ( x_{\pi(1)} u_1 , \dots , x_{\pi(n)}
u_n ) \] for every $(x_1 , \dots , x_n ) \in R^{n}$. In other words,
the matrix that represents $\Phi$ with respect to the standard basis
of $_RR^n$ decomposes as a product of a permutation matrix and an
invertible diagonal matrix.  A \emph{$G_{\mathrm{rt}}(w)$-monomial
  transformation} is one where the units $u_i$ belong to the right
symmetry group $G_{\mathrm{rt}}(w)$.  A $G_{\mathrm{rt}}(w)$-monomial transformation is a
$w$-isometry of $R^n$.

We say that a finite ring $R$ and a weight $w$ on $R$ satisfy the
\emph{Extension Property} if the following holds: For every positive
length $n$ and for every linear code $C\le {_RR^n}$, every injective
$w$-isometry $\varphi : C \longrightarrow {_RR}^{n}$ is the
restriction of a $G_{\mathrm{rt}}(w)$-monomial transformation of $_RR^{n}$.  That
is, every injective $w$-isometry $\varphi$ extends to a monomial
transformation that is itself a $w$-isometry of $R^n$. \medskip

Let $w: R \longrightarrow {\mathbb Q}$ be a weight and let $f: R
\longrightarrow {\mathbb Q}$ be any function. We define a new weight
$wf$ as
\[ wf: R \longrightarrow {\mathbb Q} \:, \quad
x \mapsto \sum_{r\in R} w(rx)\,f(r) \:. \]
By the operation of \emph{right correlation} $(w,f)\mapsto wf$, the
vector space $V := \ensuremath{\field{Q}}^R$ of all weights on $R$ becomes a right module
$V_\ensuremath{A}$ over $\ensuremath{A} = \ensuremath{\field{Q}}[(R,\cdot)]$, the rational semigroup algebra of
the multiplicative semigroup $(R,\cdot)$ of the ring
(see~\cite{grefmcfazumb13}).  For $r\in R$ denote by $e_r$ the weight
where $e_r(r) = 1$ and $e_r(s) = 0$ for $s\ne r$.  Then $we_r$ is
simply given by $(we_r)(x) = w(rx)$.

Denote the natural additive extension of $wf$ to $R^n$ by $wf$ also.

\begin{lemma}\label{lem_smult}
  Let $C \le {_RR^n}$ be a linear code and let $\varphi: C
  \longrightarrow R^n$ be a $w$-isometry, then $\varphi$ is also a
  $wf$-isometry for any function $f: R \longrightarrow {\mathbb Q}$.
\end{lemma}

\begin{proof}
  For all $x\in C$ we compute
  \begin{align*}
    (wf) (\varphi(x))
    & \;=\; \sum_{r\in R} w(r\varphi(x)) \, f(r) \;=\;
    \sum_{r\in R} w(\varphi(rx)) \, f(r)  \\
    & \;=\;  \sum_{r\in R} w(rx) \, f(r)
    \;=\; (wf)(x) \:. \qedhere
  \end{align*}
\end{proof}

For a weight $w$ consider the $\ensuremath{\field{Q}}$-linear map $\tilde w: \ensuremath{A} \to V$,
$f\mapsto wf$.  By Lemma~\ref{lem_smult}, if $\varphi$ is a
$w$-isometry then $\varphi$ is a $w'$- isometry for all $w'
\in\im\tilde w$.  Note that $\im\tilde w = w\ensuremath{A} \le V_\ensuremath{A}$.

\subsection*{Weights on Frobenius Rings}

Now let $R$ be a finite Frobenius ring.  We describe two approaches
that ultimately lead to the same criterion for a weight $w$ to satisfy
the Extension Property.

\subsubsection*{Approach~1} From earlier work~\cite{wood99} we know
that the Hamming weight $w_{\rm H}$ satisfies the Extension Property.
Combining this fact with Lemma~\ref{lem_smult}, we immediately obtain
the following result.

\begin{cor}\label{cor_smult}
  Let $R$ be a finite Frobenius ring and let $w$ be a weight on $R$
  such that $G_{\mathrm{rt}}(w) = U$ and $wf = w_{\rm H}$ for some function
  $f:R\to\ensuremath{\field{Q}}$.  Then $w$ satisfies the Extension Property.
\end{cor}

In other words, if $w$ is right-invariant and $w_{\rm H}\in\im\tilde
w$ then $w$ satisfies the Extension Property.

How can we make sure that $w_{\rm H}\in\im\tilde w$?  One idea is to
show that the $\ensuremath{\field{Q}}$-linear map $\tilde w$ is bijective: Using the
natural basis $(e_r)_{r\in R}$ for $V$ and the property $(we_r)(s) =
w(rs)$ it is easy to see that $\tilde w$ is described by the transpose
of the matrix $(w(rs))_{r,s\in R}$.  However, if the weight function
$w$ is left- or right-invariant {\em or} satisfies $w(0) = 0$ then
this matrix is not invertible.  Therefore we work with a ``reduced''
version of the map $\tilde w$.

As before, let $V := \ensuremath{\field{Q}}^R$ be the vector space of all weights on $R$,
and let $V_0^\ensuremath{U}$ be the subspace of all weights $w$ satisfying $w(0) =
0$ that are right-invariant.  Similarly, we define the subspace $^\ensuremath{U}
V_0$ of all weights $w$ with $w(0) = 0$ that are left-invariant.  The
corresponding invariant subspaces of $\ensuremath{A} = \ensuremath{\field{Q}}[(R,\cdot)]$ are
$\ensuremath{A}_0^\ensuremath{U}$ and $^\ensuremath{U} \ensuremath{A}_0$, where $\ensuremath{A}_0 := \ensuremath{A} / \ensuremath{\field{Q}} e_0$.

If $w$ is a weight in $V_0^\ensuremath{U}$ then $wf\in V_0^\ensuremath{U}$ for {\em any}
function $f:R\to\ensuremath{\field{Q}}$, i.e., $\im\tilde w \le V_0^\ensuremath{U}$.  In this case we
could examine the bijectivity of the $\ensuremath{\field{Q}}$-linear map $\tilde w:
\ensuremath{A}_0^\ensuremath{U} \to V_0^\ensuremath{U}$ (the restriction of the above map $\tilde w$).
But this map does not have a nice matrix representation; setting
$e_{s\ensuremath{U}} = \sum_{r\in s\ensuremath{U}}e_r$ and letting $(e_{s\ensuremath{U}})_{s\ensuremath{U}\ne 0}$ be
the natural basis for $\ensuremath{A}_0^\ensuremath{U}$ and for $V_0^U$, the entries of the
matrix turn out to be sums of several values $w(rus)$.

However, if we work with the restriction $\tilde w: {}^\ensuremath{U} \ensuremath{A}_0 \to
V_0^\ensuremath{U}$ instead and if the weight $w$ is bi-invariant (i.e., both
left- and right-invariant), then, with respect to the natural bases,
this $\ensuremath{\field{Q}}$-linear map does have a nice matrix description, namely the
orthogonality matrix.  This will be explained below. If this map
$\tilde w$ is invertible, then $w$ satisfies the Extension Property by
Corollary~\ref{cor_smult}.

Note: Since $\im\tilde w$ is a submodule of $V_\ensuremath{A}$ it follows that
$w_{\rm H}\in\im\tilde w$ if and only if $\im\tilde w_{\rm H} \le \im\tilde w$.
Actually, $\im\tilde w_{\rm H} = V_0^\ensuremath{U}$ (see
Proposition~\ref{prop_om-reverse} below), so that $w_{\rm
  H}\in\im\tilde w$ if and only if $V_0^U \subseteq \im\tilde w$.
This is why it is a sensible approach to investigate the
surjectivity/bijectivity of the map $\tilde w$.

\subsubsection*{Approach~2} The same orthogonality matrix that appears
in Approach~1 also appears in \cite{wood97}.  By
\cite[Proposition~12]{wood97} (also, \cite[Theorem~3.1]{wood99a} and
\cite[Section~9.2]{wood09}), the invertibility of the orthogonality
matrix of $w$ implies that a $w$-isometry preserves the so-called
\emph{symmetrized weight composition} associated with $G_{\mathrm{rt}}(w)$.
Then, \cite[Theorem~10]{wood97} shows that any injective linear
homomorphism that preserves the symmetrized weight composition
associated with $G_{\mathrm{rt}}(w)$ extends to a $G_{\mathrm{rt}}(w)$-monomial
transformation.  Thus, if the orthogonality matrix is invertible, any
$w$-isometry extends to a $G_{\mathrm{rt}}(w)$-monomial transformation, and hence
$w$ satisfies the Extension Property.

\subsection*{Orthogonality Matrices}

Let $R$ be a finite Frobenius ring. There is a one-to-one
correspondence between left (resp., right) principal ideals and left
(resp., right) $\ensuremath{U}$-orbits.  Each $\ensuremath{U}$-orbit is identified with the
principal ideal of which its elements are the generators
(\cite[Proposition~5.1]{wood99}, based on work of Bass).  Define for
$r, s\in R\setminus \{0\}$ the functions $\varepsilon_{R r} (x) = \size{\ensuremath{U}
  r}^{-1} $ if $x \in \ensuremath{U} r$, i.e., if $Rr = Rx$, and zero otherwise;
similarly, let $e_{sR} (x) = e_{sU}(x) = 1$ if $xR = sR$ and zero
otherwise.  Then $(\varepsilon_{R r})$ and $(e_{sR})$ are bases for $^\ensuremath{U} \ensuremath{A}_0$
and $V_0^\ensuremath{U}$, as $R r$ and $sR$ vary over all left and right nonzero
principal ideals of $R$, respectively.

For a bi-invariant weight $w$, define the \emph{orthogonality matrix}
of $w$ by $W_0 = \big(w(rs) \big){}_{Rr\ne 0,\,sR\ne 0}$.  That is,
the entry in the $Rr, sR$-position is the value of the weight $w$ on
the product $rs \in R$.  The value $w(rs)$ is well-defined, because
$w$ is bi-invariant.  Note that $W_0$ is square; this follows from
work of Greferath \cite{gref02} that shows the equality of the number
of left and right principal ideals in a finite Frobenius ring.

\begin{prop}\label{prop_matrix}
  Suppose $w$ is bi-invariant with $w(0)=0$.  Then
  \[ w \, \varepsilon_{R r} = \sum_{sR\ne 0} w(rs) \, e_{s R} \]
  for nonzero $R r$, where the sum extends over all the nonzero right
  principal ideals $s R$.  In particular, the matrix representing the
  $\ensuremath{\field{Q}}$-linear map $\tilde w: {}^\ensuremath{U} \ensuremath{A}_0 \to V_0^\ensuremath{U}$, $f\mapsto wf$,
  with respect to the bases $(\varepsilon_{R r})$ and $(e_{sR})$, is the
  transpose of the matrix $W_0$.
\end{prop}

\begin{proof}
  Since $w\in V_0^\ensuremath{U}$ we have $w \, \varepsilon_{R r} \in V_0^\ensuremath{U}$, and therefore
  \[  w \, \varepsilon_{R r} = \sum_{sR\ne 0} (w \, \varepsilon_{R r})(s) \, e_{s R} \:.  \]
  Calculating, using that $w\in {}^\ensuremath{U} V_0$, we get:
  \[ (w \, \varepsilon_{R r})(s) = \sum_{t \in R} w(ts) \, \varepsilon_{R r}(t)
    = \sum_{t \in Ur} \size{\ensuremath{U} r}^{-1} w(ts)
    = w(rs) \:. \qedhere \]
\end{proof}

In the algebraic viewpoint of \cite{grefmcfazumb13}, $V_0^{\ensuremath{U}}$ is a
right module over $^\ensuremath{U} \!\ensuremath{A}_0$.  Then, $W_0$ is invertible if and only
if $w$ is a generator for $V_0^{\ensuremath{U}}$.

If $R$ is a finite field and $w = w_{\rm H}$, the Hamming weight on
$R$, then $W_0$ is exactly the orthogonality matrix considered by
Bogart, Goldberg, and Gordon~\cite[Section~2]{bogagoldgord78}.  More
general versions of the matrix $W_0$ have been utilized in
\cite{wood97, wood99a, wood09}.

\begin{example}
  For $R={\mathbb Z}_4$ the Lee weight $w_{\rm Lee}$ assigns $0
  \mapsto 0$, $1\mapsto 1$, $2\mapsto 2$ and $3\mapsto 1$. It is a
  bi-invariant weight function, as is the Hamming weight $w_{\rm H}$
  on $R$. Based on the natural ordering of the (nonzero) principal
  ideals of $R$ as $2R < R$ the orthogonality matrix for $w_{\rm Lee}$
  is
  \[ W_0^{\rm Lee} \, = \, \left[ \begin {array}{cc}
      0 & 2 \\
      2 & 1
    \end{array}\right], \]
  whereas the orthogonality matrix for $w_{\rm H}$ is given by
  \[ W_0^{\rm H} \, = \, \left[ \begin {array}{cc}
      0 & 1\\
      1 & 1
    \end{array}\right]. \]
  Both of these matrices are invertible over ${\mathbb Q}$ as observed
  in \cite{gref02}, where it was shown that the Extension Property is
  satisfied.
\end{example}

\section{Orthogonality Matrices and the Extension Theorem}%
\label{sec:orthog-matrices}

In the present section we will show that invertibility of the
orthogonality matrix of a bi-invariant weight is necessary and
sufficient for that weight to satisfy the Extension Property.  We
split this result into two statements.

\begin{prop}
  Let $R$ be a finite Frobenius ring and let $w$ be a bi-invariant
  weight on $R$.  If the orthogonality matrix $W_0$ of $w$ is
  invertible, then $w$ satisfies the Extension Property.
\end{prop}

\begin{proof}
  Approach~1: by Proposition~\ref{prop_matrix} the matrix $W_0$
  describes the $\ensuremath{\field{Q}}$-linear map $\tilde w: {}^\ensuremath{U} \ensuremath{A}_0 \to V_0^\ensuremath{U}$,
  $f\mapsto wf$.  Hence if $W_0$ is invertible the map $\tilde w$ is
  bijective, and in particular $w_{\rm H}\in\im\tilde w$.  Thus by
  Corollary~\ref{cor_smult} the weight $w$ satisfies the Extension
  Property.

  Approach~2: apply \cite[Proposition~12]{wood97} or
  \cite[Theorem~3.1]{wood99a}.
\end{proof}

We remark that in the foregoing discussion, $\ensuremath{\field{Q}}$ could be replaced
throughout by any field $K$ containing $\ensuremath{\field{Q}}$, for example $K = \ensuremath{\field{C}}$.

\begin{prop}\label{prop_om-reverse}
  Let $R$ be a finite Frobenius ring, and let $w$ be a bi-invariant
  rational weight on $R$ that satisfies the Extension Property.  Then
  the orthogonality matrix $W_0$ of $w$ is invertible.
\end{prop}

\begin{proof}
  The proof mimics that of \cite[Theorem~4.1]{wood08a} and
  \cite[Proposition~7]{grefhono06}.  Assume $W_0$ singular for the
  sake of contradiction. Then there exists a nonzero rational vector
  $v = (v_{cR})_{cR\ne 0}$ such that $W_0 v = 0$.  Without loss of
  generality, we may assume that $v$ has integer entries.  We proceed
  to build two linear codes $C_+, C_-$ over $R$.  Each of the codes
  will have only one generator.  The generator for $C_{\pm}$ is a
  vector $g_{\pm}$ with the following property: for each ideal $cR \le
  R_R$ with $v_{cR} > 0$ (for $g_+$), resp., $v_{cR} < 0$ (for $g_-$),
  the vector $g_{\pm}$ contains $\abs{v_{cR}}$ entries equal to
  $c$. To make these two generators annihilator-free, we append to
  both a trailing $1\in R$. The typical codeword in $C_{\pm}$ is hence
  of the form $a g_{\pm}$ for suitable $a \in R$.  We compare $w(a
  g_+)$ and $w(a g_-)$ for every $a \in R$ by calculating the
  difference $D( a ) = w(a g_+) - w(a g_-)$.  By our construction of
  the generators $g_{\pm}$, we have \[ D( a ) \;=\; \sum_{cR\ne 0}
  w(ac) \, v_{cR} \;=\; (W_0 v)_{Ra} \;=\; 0 \:, \] for all $a\in R$.
  Thus $a g_+ \mapsto a g_-$ forms a $w$-isometry from $C_+$ to
  $C_-$. The codes, however, are not monomially equivalent because
  their entries come from different right $\ensuremath{U}$-orbits.
\end{proof}

We summarize our findings in the following theorem.

\begin{theorem}
  A rational bi-invariant weight function on a finite Frobenius ring
  satisfies the Extension Property if and only if its orthogonality
  matrix is invertible.
\end{theorem}

The ultimate goal is to give necessary and sufficient conditions on a
bi-invariant weight $w$ on a finite Frobenius ring $R$ so that its
orthogonality matrix $W_0$ is invertible.  We are able to derive such
a result for finite principal ideal rings.

\subsection*{Extended Orthogonality Matrices}

Let $R$ be a finite Frobenius ring and let $w$ be a bi-invariant
weight function with $w(0) = 0$.  The orthogonality matrix for the
weight $w$ was defined as $W_0 = \big( w(rs) \big){}_{Rr\ne 0,\,sR\ne
  0}$.  Now define the {\em extended} orthogonality matrix for $w$ as
$W = \big( w(rs) \big){}_{Rr,\,sR}$.  In order to examine the
invertibility of $W_0$ we obtain a formula for $\det W$, the
determinant of the matrix $W$.  (Note that $\det W$ is well-defined up
to multiplication by $\pm1 $, the sign depending on the particular
orderings of the rows and columns of $W$.)  First we relate $\det W$
to $\det W_0$, viewing $w(0)$ as an indeterminate.

\begin{prop}\label{prop_W-W0}
  The determinant $\det W_0$ is obtained from $\det W$ by dividing
  $\det W$ by $w(0)$ and then setting $w(0) = 0$.
\end{prop}

\begin{proof}
  We treat $w(0)$ as an indeterminate $w_0$.  Up to a sign change in
  $\det(W)$, we may assume that the rows and columns of $W$ are
  arranged so that the first row is indexed by $R0$ and the first
  column is indexed by $0R$.  Then $W$ has the form
  \[ W = \left[ \begin{array}{c|c}
      w_0 & w_0 \,\cdots\, w_0 \\ \hline
      w_0 & \\
      \vdots & W' \\
      w_0 &
    \end{array} \right] . \]
  By subtracting the first row from every other row, we find that
  $\det W = w_0 \det(W'-w_0J)$, where $J$ is the all-one matrix.
  Finally the matrix $W_0$ equals the matrix $W'-w_0J$ evaluated at
  $w_0 = 0$, so that $\det W_0 = \det(W' - w_0J)|_{w_0 = 0}$.
\end{proof}

Note that the extended orthogonality matrix $W$ is not invertible for
weights $w$ satisfying $w(0) = 0$.

\section{Bi-invariant Weights with Invertible Orthogonality\\
  Matrix on Principal Ideal Rings}%
\label{sec:bi-inv-wts}

Let $R$ be a finite principal ideal ring, and let $w$ be a
bi-invariant weight on $R$.  Assume $W$ is the extended orthogonality
matrix of $w$.  We are interested in the determinant of $W$ and look
for a way to evaluate this determinant.

We will define an invertible matrix $(Q_{cR,Rx})_{cR,\, Rx}$ with
determinant $\pm 1$ and multiply $W$ by $Q$ from the right to arrive
at $WQ$; then $\det(WQ) = \pm\,\det(W)$.  The most significant
advantage of considering $WQ$, rather than $W$, is that $WQ$ will be a
lower triangular matrix for which we can easily calculate the
determinant.

Define for any finite ring the matrix $Q$ by
\[ Q_{cR, Rx} \;:=\; \mu( (Rx)^\perp, cR ) \: , \]
for $cR\le R_R$ and $Rx\le {_RR}$, where $\mu$ is the M\"obius
function of the lattice $L^*$ of all right ideals of $R$.

\begin{lemma}\label{lemQinvertible}
  For a finite principal ideal ring $R$, the matrix $Q$ is an
  invertible matrix with determinant $\pm1$.
\end{lemma}

\begin{proof}
  We claim that the inverse of $Q$ is given by $T_{Ra, bR} :=
  \zeta(bR, (Ra)^\perp)$, where $\zeta$ is the indicator function of
  the poset $L^*$, meaning
  \[ \zeta(xR, yR) \;=\; \left\{\begin{array}{ccl}
      1 & : & xR \le yR \:, \\
      0 & : & \mbox{otherwise} \:.
    \end{array}\right.\]
  We compute the product $TQ$,
  \[ (TQ)_{Ra,Rx} \;=\; \sum_{cR} {  \zeta(cR, (Ra)^\perp)
     \, \mu ( (Rx)^\perp , cR)} \:. \]
    By the definition of $\zeta$ and the fact that
  $\mu((Rx)^\perp,cR) =0$ unless $(Rx)^\perp \le cR$, the expression
  above simplifies to
  \[ (TQ)_{Ra,Rx} \;= \sum_{ (Rx)^\perp \le cR \le (Ra)^\perp} {\mu
    ((Rx)^\perp , cR)} \:, \] which is $1$ for $(Rx)^\perp =
  (Ra)^\perp$ and $0$ otherwise by the definition of the M\"obius
  function.

  The matrix $T$ is upper triangular with $1$s on the main diagonal.
  Thus $\det T$ and hence $\det Q$ equal $\pm 1$.  (The $\pm 1$ allows
  for different orders of rows and columns.)
\end{proof}

\begin{example}
  Let $R := \ensuremath{\field{F}}_q[x,y] / \langle x^2, y^2 \rangle$, which is a
  commutative local Frobenius ring.  (When $q=2^k$, $R$ is isomorphic
  to the group algebra over $\ensuremath{\field{F}}_{2^k}$ of the Klein $4$-group.)  Here,
  $(Rxy)^\perp = xR + yR$ is not principal and thus the above proof
  does not apply; in fact, the matrix $Q$ turns out to be singular in
  this case.

  On the other hand, the Frobenius ring $R$ is not a counter-example
  to the main result below.  In fact, $\det(W_0) = \pm q \,
  w(xy)^{q+3}$ satisfies the formula in
  \eqref{eq:det-W0-factorization} below (up to a nonzero
  multiplicative constant), so that the main result still holds over
  $R$.
\end{example}

We are now ready to state the main theorems.  The proof of the next
result is contained in the final section.

\begin{theorem}\label{thm:WQ}
  If $R$ is a finite principal ideal ring, then the matrix $WQ$ is
  lower triangular.  The diagonal entry at position $(Ra, Ra)$ is
  $\sum\limits_{dR \le aR} w( d ) \, \mu(0, dR)$.
\end{theorem}

We conclude that the determinant of $WQ$ and hence that of $W$ is
given by
\[ \det(W) \;=\; \pm\, \det(WQ) \;=\;
\pm\prod_{aR} \, \sum_{dR\le aR} w(d)\, \mu(0,dR) \:. \]
Applying Proposition~\ref{prop_W-W0} we find the determinant of $W_0$
to be
\begin{equation}
 \label{eq:det-W0-factorization}
 \det(W_0) \;=\; \pm\prod_{aR\ne 0} \, \sum_{0\ne dR\le
  aR} w(d)\, \mu(0,dR) \:,
\end{equation}
as in $\det(W)$ the term $aR = 0$ provides a factor of $w(0)$ which
gets divided away, and in each remaining term the contribution from
$dR =0R$ is $w(0)$ which is set equal to $0$.

This yields our main result: a characterization of all bi-invariant
weights on a principal ideal ring that satisfy the Extension Property.

\begin{theorem}[Main Result]\label{maintheorem}
  Let $R$ be a finite principal ideal ring and let $\mu$ be the
  M\"obius function of the lattice $L^*$ of all right ideals of $R$.
  Then a bi-invariant rational weight $w$ on $R$ satisfies the
  Extension Property if and only if
  \[ \sum_{0\ne dR\le aR} w(d)\, \mu(0,dR) \;\ne\, 0\quad \mbox{for all
    $aR \ne 0$} \:. \]
\end{theorem}

The condition in Theorem~\ref{maintheorem} needs to be checked only
for nonzero right ideals $aR\leq\soc(R_R)$, since we have
$\mu(0,dR)=0$ if $dR\not\leq\soc(R_R)$ (see
\cite[Proposition~2]{st:homippi}, for example) and since every right
ideal contained in $\soc(R_R)$ is principal.  As a consequence, the
Extension Property of $w$ depends only on the values of $w$ on the
socle of $R$.

\begin{example}
  For a chain ring $R$, the main result simply says that a
  bi-invariant weight function $w$ satisfies the Extension Property if
  and only if it does not vanish on the socle of $R$ (compare with
  \cite{grefhono05} and \cite[Theorem~9.4]{wood09}).  For $R =
  {\mathbb Z}_4$, it states that a bi-invariant weight will satisfy
  the Extension Property if and only if $w(2) \ne 0$.
\end{example}

\begin{example}
  Let $R := {\mathbb Z}_m$.  The nonzero ideals in $\soc(\ensuremath{\field{Z}}_m)$ are of
  the form $a\ensuremath{\field{Z}}_m$ with $a\mid m$ and $m/a>1$ square-free.  The
  M\"obius function of such an ideal is
  $\mu(0,a\ensuremath{\field{Z}}_m)=\mu(m/a)=(-1)^r$, where $\mu(\cdot)$ denotes the
  one-variable M\"obius function of elementary number theory and $r$
  is the number of different prime divisors of $m/a$.  According to
  Theorem~\ref{maintheorem}, an invariant weight $w$ on $\ensuremath{\field{Z}}_m$ has the
  Extension Property if and only if
  \[ \sum_{s\mid\frac{m}{a}} w(sa) \, \mu
  \Big( \frac{m}{sa} \Big) \;=\; (-1)^r \,
  \sum_{s\mid\frac{m}{a}} w(sa) \, \mu(s) \;\ne\; 0\]
  for all (positive) divisors $a$ of $m$ such that $m/a$ is
  square-free and $>1$.  We thus recover the main theorem of
  \cite{grefhono06}.
\end{example}

\begin{example}\label{ex:examples}
  Let $R := {\rm Mat}_n(\ensuremath{\field{F}}_q)$, $n\geq2$, the ring of
  $n\times n$ matrices over the finite field $\ensuremath{\field{F}}_q$ with $q$ elements,
  so that $\ensuremath{U} = {\rm GL}_n(\ensuremath{\field{F}}_q)$.  The ring $R$ is a finite principal
  ideal ring that is not a direct product of chain rings.  For each
  matrix $A\in R$, the left $\ensuremath{U}$-orbit $\ensuremath{U} A$ can be identified with
  the row space of $A$, and similarly, the right $\ensuremath{U}$-orbits
  correspond to the column spaces.

  Let $w$ be a bi-invariant weight on $R$.  Its value $w(A)$ depends
  only on the rank of the matrix $A$, and therefore we can write
  $w([{\rm rank}\,A]) := w(A)$.  Now for $n=2$, the main result says
  that $w$ satisfies the Extension Property if and only if $w([1])\ne
  0$ and $q\,w([2]) \ne (q+1)\,w([1])$.  For $n=3$, $w$ satisfies the
  Extension Property if and only if $w([1])\ne 0$, $\,q\,w([2]) \ne
  (q+1)\,w([1])$, and $q^3\,w([3]) + (q^2+q+1)\,w([1]) \ne
  (q^2+q+1)\,q\,w([2])$.

  It was shown in \cite[Theorem~9.5]{wood09} that the relevant
  non-vanishing sums are
  \begin{equation}\label{eq:wmatrixterm}
    \sum_{i=1}^{s}{(-1)^i q^{(\upontop{i}{2})}
      \left[\upontop{s}{i}\right]_q w([i]) }  \:,
  \end{equation}
  where $[\upontop{s}{i}]_q$ is the $q$-nomial (Cauchy binomial)
  defined as
  \[ \left[\upontop{k}{l}\right]_q \,:=\;
  \frac{(1 - q^k)(1-q^{k-1}) \dots (1-q^{k-l+1})}
  {(1-q^l)(1-q^{l-1}) \dots (1-q)}  \:. \]

  The {\em rank metric} $w([k]) := k$ satisfies these conditions.
  First we state the Cauchy binomial theorem:
  \begin{equation*}\label{eq:cauchythm}
    \prod_{i=0}^{k-1}{(1 + xq^i)} \;=\;
    \sum_{j=0}^{k}{\left[\upontop{k}{j}\right]_q  q^{(\upontop{j}{2})} x^j} \:.
  \end{equation*}
  Now we write the term in \eqref{eq:wmatrixterm} for the rank metric,
  changing the sign and including $i=0$ trivially in the sum. This can
  then be seen as the evaluation of a derivative.
  \[ \sum_{i=0}^{s}{i(-1)^{i-1}  \ q^{(\upontop{i}{2})}
    \left[\upontop{s}{i}\right]_q } \;=\;
  \left. \frac{d}{dx}\sum_{i=0}^{s}{x^i q^{(\upontop{i}{2})}
      \left[\upontop{s}{i}\right]_q  }\right|_{x=-1} \:. \]
  Applying the Cauchy binomial theorem and evaluating the derivative yields:
  \[ \left. \frac{d }{dx} \prod_{i=0}^{s-1}{(1+xq^i)} \right|_{x=-1}
  \;=\; \left( \prod_{i=0}^{s-1}{(1-q^i)} \right)
  \left( \sum_{i=0}^{s-1}{\frac{q^i}{1-q^i}} \right) \:. \]
  Both expressions on the right are nonzero provided $q$ is not $\pm
  1$, independent of $s$.  Hence the rank metric satisfies the
  Extension Property for all $q$ and $n$.
\end{example}

\begin{example}
  More generally, let $R = {\rm Mat}_n(S)$ be a matrix ring over a
  finite chain ring $S$.  Then $\soc(R_R) = \soc({}_RR) = {\rm
    Mat}_{n\times n}(\soc S) \cong {\rm Mat}_{n\times n}(\ensuremath{\field{F}}_q)$ as a
  (bi-)module over the residue class field $S/\rad S\cong\ensuremath{\field{F}}_q$.  Hence
  the previous example applies and characterizes all bi-invariant
  weights $w\colon R\to\ensuremath{\field{Q}}$ having the Extension Property.
\end{example}

\begin{example}
  Any finite semisimple ring is a direct product of matrix rings over
  finite fields and therefore a principal ideal ring.  Hence, the main
  result also applies to this case.
\end{example}

\section{A Proof of Theorem~\ref{thm:WQ}}%
\label{sec:proof}

We perform the matrix multiplication and see that the entry of $WQ$ in
position $(Ra,Rb)$ is given by the expression
\[ (WQ)_{Ra,Rb} \;=\; \sum_{cR} W_{Ra,cR} \, Q_{cR,Rb} \;=\;
\sum_{cR} w(ac)\, \mu((Rb)^\perp ,cR) \:. \]
According to the definition of the M\"obius function, $\mu((Rb)^\perp,
cR)$ can be nonzero only when $(Rb)^\perp \le cR$ (or: when $cR \in
[(Rb)^\perp, R]$, using interval notation on the lattice $L^*$ of all
right (necessarily principal) ideals of $R$).  With this in mind we
write
\begin{equation}\label{main}
  (WQ)_{Ra,Rb} \;= \sum_{cR \in [(Rb)^\perp, R]} w(ac) \, \mu((Rb)^\perp ,cR) \:.
\end{equation}

\subsection*{Diagonal Entries}

The diagonal terms of $WQ$ are given by
\[ (WQ)_{Ra,Ra} \;= \sum_{ cR \in [(Ra)^\perp,R]} w(ac)\, \mu((Ra)^\perp , cR) \:. \]

For an element $a\in R$ consider the left multiplication operator
$L_a: R \longrightarrow aR,\; t\mapsto at$.  The mapping $L_a$ is a
(right) $R$-linear mapping with kernel $(Ra)^\perp$, and the
isomorphism theorem yields an induced order isomorphism of intervals
\[ \nu_a:[(Ra)^\perp , R] \longrightarrow [0,aR] \:, \quad
J \mapsto aJ \:. \]
It follows that if $J_1, J_2\in [(Ra)^\perp,R]$, then $\mu(J_1,J_2) =
\mu ( \nu_a(J_1), \nu_a(J_2) ) = \mu (aJ_1, aJ_2)$.

The diagonal term simplifies to
\begin{align*}
  (WQ)_{Ra,Ra} &\;=\;
  \sum_{cR \in [(Ra)^\perp,R]} w(ac)\, \mu((Ra)^\perp , cR) \\
  &\;=\; \sum_{acR \in [0,a R]} w(ac)\, \mu(0, acR) \\
  &\;=\; \sum_{dR \in [0,a R]} w(d)\, \mu(0, dR) \:,
\end{align*}
where we have applied the above interval isomorphism with $J_1 =
(Ra)^\perp$ and $J_2 = cR$, followed by the relabeling $acR=dR$.

Finally, observe that the formula $(WQ)_{Ra,Ra} = \sum_{dR \in [0,a
  R]} w(d)\, \mu(0, dR)$ does not depend on the choice of generator
$a$ for the left ideal $Ra$.  Indeed, any other generator has the form
$ua$, where $u$ is a unit of $R$.  Left multiplication by $u$ induces
an order isomorphism of intervals $\nu_u: [0,aR] \longrightarrow
[0,uaR]$, so that $\mu(0,dR) = \mu(0,udR)$ for all $dR \in [0,aR]$.
Since $w$ is left-invariant, we have $w(ud) = w(d)$, and the right
side of the formula is well-defined.

\subsection*{Lower Triangularity}

Now let us return to the general form of the matrix $WQ$ given in
\eqref{main}.  We would like to prove that $WQ$ is lower triangular,
i.e., that $Rb \nleq Ra$ will imply that $(WQ)_{Ra,Rb}=0$. To that
end, assume
\begin{equation} \label{eq:lower-tri}
  Rb \nleq Ra \:.
\end{equation}
As above, the left multiplication operator $L_a$ induces a mapping
$\lambda_a : [0,R] \rightarrow [0,aR]$, which in turn induces a partition on
$[0,R]$ in a natural way. We first rewrite the general expression for
$(WQ)_{Ra,Rb}$ taking into account this partition.

\[ (WQ)_{Ra,Rb} \;= \sum_{dR \in [0, aR]} w(d)
\mathop{\sum_{cR \in [(Rb)^\perp,R] }}_{\lambda_a(cR)=dR} \mu((Rb)^\perp ,cR) \:. \]

Our goal is to examine the inner sum and show that it vanishes for
every $dR$ in question. In other words, we will show that \[
\mathop{\sum_{cR \in [(Rb)^\perp,R] }}_{\lambda_a(cR)=dR}
\mu((Rb)^\perp ,cR) \;=\; 0 \:,\quad \mbox{for all $dR \le aR$} \:. \]

We do this by induction on $dR$ in the partially ordered set $[0,aR]$.
Accordingly, we assume the existence of some $dR\in [0,aR]$ which is
minimal with respect to the property that
\[ \mathop{\sum_{cR \in [(Rb)^\perp,R] }}_{\lambda_a(cR)=dR} \mu((Rb)^\perp ,cR)
\;\ne\; 0 \:. \]

Consider the right ideal $K := L_a^{-1}(dR) = \sum\limits_{acR \le dR}
cR$. For this ideal we have $(Ra)^\perp \le K$, and moreover, $cR \le
K$ is equivalent to $acR \le dR$. For this reason
\[ \mathop{\sum_{cR \in [(Rb)^\perp,R] }}_{acR\le dR} \mu((Rb)^\perp ,cR)
\;=\; \sum_{ cR \in [(Rb)^\perp , K]} \mu((Rb)^\perp ,cR) \:. \]
By properties of $\mu$, the latter expression is nonzero if and only
if $K=(Rb)^\perp$. This would however imply $(Rb)^\perp \ge
(Ra)^\perp$ (because $(Ra)^\perp \le K$) and hence $Rb\le Ra$,
contrary to assumption \eqref{eq:lower-tri}.  Hence, we conclude
\[ 0 \;= \mathop{\sum_{cR \in [(Rb)^\perp,R]}}_{acR\le dR} \mu((Rb)^\perp ,cR)
\;= \mathop{\sum_{cR \in [(Rb)^\perp,R]}}_{acR=dR} \mu((Rb)^\perp ,cR) +
\mathop{\sum_{cR \in [(Rb)^\perp,R]}}_{acR<dR} \mu((Rb)^\perp ,cR) \:.  \]

In this equation the minimality property of $dR$ implies that the last
term vanishes.  This finally forces
\[ \mathop{\sum_{cR \in [(Rb)^\perp,R]}}_{acR=dR} \mu((Rb)^\perp ,cR) \;=\; 0 \:, \]
contradicting the minimality property of $dR$.  Lower triangularity
follows and this finishes the proof of Theorem~\ref{thm:WQ}.\qed

Note that this proof heavily relies on the hypothesis that $R$ is a
finite principal ideal ring. For a general finite Frobenius ring the
architecture of a proof will need to be vastly restructured.
Nonetheless, we conjecture that the main result, as stated, holds over
any finite Frobenius ring.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


I recently saw a presentation by Osprey Packs at a retail conference and was impressed with their approach to innovation. In fact, their mission statement: "to create innovative high-performance gear that reflects our love of adventure and our devotion to the outdoors" puts innovation at the forefront of all they've done for more than 40 years of making quality, highly functional products for outdoor enthusiasts.
Their head of product management, Erik Hamerschlag, shared with me that hiking trails and wilderness areas serve as their R&D lab where they see what could work better in real world user situations. A key breakthrough innovation, their AirSpeed™ Suspension backpack, is designed to make heavy loads feel lighter, making the user's long trail hikes a more pleasurable experience. Old pack designs, for decades, relied upon positioning the load primarily on the hips to leverage leg strength. Osprey found that by using a 'tensioned mesh panel' they could more evenly distribute the weight of the load across the entire back using ANTI-GRAVITY™ technology. Their 'breathable' custom mesh conforms to the back by pulling seamlines in different directions to naturally contour to the body while keeping the user cool.
A key to their innovation success and product quality is having a team in Asia near their factories who visit daily. Their own sampling and design center is nearby so they can do their own sewing and construction of new products prior to mass production; all of which saves time and increases quality. Their "All Mighty Guarantee" lifetime warranty states that Osprey will repair any damage or defect for any reason free of charge.
This summer, Osprey will launch their next innovation - the revolutionary new Super Ultra Light pack weighing in at under 2 lbs. I'm excited to see how Osprey will continue to keep innovation at the forefront of all they do and produce.


I'm not crying. There's just something in my eye.
Late night television appears to be developing a fondness for Tilt Brush. The popular VR painting app has already seen multiple appearances on The Tonight Show with Jimmy Fallon in the form of a Pictionary-esque party game with participants including stars such as Scarlett Johansson and SNL's Michael Che.
Now the immersive virtual art tool has returned to late night yet again in a segment that should leave any dedicated Disney fan on the verge of tears.
Keane was joined by NBA legend Kobe Bryant that evening in support of their previous collaboration, Dear Basketball, an animated short film which is currently on the shortlist for an Oscar for "Best Animated Short." The duo has a good shot at it too. Keane is an animation veteran who's responsible for the design of memorable characters such as Pocahontas, Aladdin, the Beast from Beauty and the Beast, Tarzan and of course the Little Mermaid. Combine this with Kobe Bryant's overall star power and you have a project that's as well-made as it is well-promoted.
Hopefully this kind of publicity from such a big name in animation shines a larger spotlight on Tilt Brush from within the professional community. We've already seen the kind of emotional rollercoasters capable of being produced in Tilt Brush with projects such as Dear Angelica. I can only imagine the level of production that would come from an official Disney Tilt Brush animation. Are you listening Disney!? I need a VR Tangled film!


Is it safe to leave a luggage in a car parking at JFK?
Hi, I will drive from Philadelphia to pick my friend at JFK. However, we need to go to Manhattan for 4-5 hours.
I afraid to drive to Manhattan, so we decided to take the subway there. Is it safe to leave my friend luggage in the car and park it at the JFK long term parking area?
Parking lot is monitored twenty four seven so it is safe to say that it is safe to leave luggage in a car.
I can not find my parking ticket for the short term lot at Terminal 8. What is the procedure for paying without a ticket? Thanks.
You will pay full amount. I know I have when I lost my ticket!
They can figure it out from your license plate when you leave the parking lot.
Does JFK have a cell phone waiting area?
Yes they do in each arrival hall at JFK.
Can you tell me if the online reserved parking at jfk in terminal 5 is legitimate. I saw it on a port authority web site.
It is on the PA site and is legit. Reserve space in short term lots, cost is $5 to reserve plus parking fees.
My car will be at the airport for about 18 hours. Is that considered short term or long term parking. Please advise so I know where to park.
It all depends on where you park. The lot across from the terminal is the short term lot and costs $33 per day, the lot out near Lefferts Blvd is the long term lot and costs $18 per day.
How much does it cost per day to park in JFK's Terminal 5 short term parking including tax? Is a reservation needed?
$33 per 24 hour period, no reservations.


text:
module Chaskiq
  module Manage::CampaignsHelper

    def metric_action_class(metric)
      case metric.action
      when "deliver"
        "plain"
      when "open"
        "info"
      when "click"
        "primary"
      when "bounce"
        "warning"
      when "spam"
        "danger"
      end
    end

    def metric_icon_action_class(metric)
      case metric.action
      when "deliver"
        "check"
      when "open"
        "check"
      when "click"
        "check"
      when "bounce"
        "exclamation"
      when "spam"
        "exclamation-triangle"
      end
    end

  end
end

redpajama_set_name: RedPajamaGithub


text:
/*
 * To change this license header, choose License Headers in Project Properties.
 * To change this template file, choose Tools | Templates
 * and open the template in the editor.
 */
package de.christofreichardt.diagnosis;

import de.christofreichardt.diagnosis.io.IndentablePrintStream;
import javax.xml.xpath.XPath;
import javax.xml.xpath.XPathConstants;
import javax.xml.xpath.XPathExpressionException;
import org.w3c.dom.Node;

/**
 * <div style="text-align: justify">
 * A special tracer intended for the use by a bounded blocking queue. This is an option to retrieve tracer within a multi-threaded environment
 * when we cannot control the creation of the threads. An example would be applications within a container that manages a thread pool. This tracer
 * honors the contract of an {@link AbstractTracer} but does so by wrapping and delegating to another {@link AbstractTracer}. In particular the
 * current tracing context will be automatically cleared when calling {@link #wayout()} and the method stack remains empty afterwards. Thus the tracer
 * can be reused for another thread's tracing context without any danger to create a memory leak.
 * </div>
 *
 * @author Christof Reichardt
 * @param <T> the wrapped tracer
 */
abstract public class QueueTracer<T extends AbstractTracer> extends AbstractTracer {

  private boolean online;
  private int level;

  /**
   * Some tracer.
   */
  final protected T tracer;

  /**
   * Constructor expects the tracer name and the to be wrapped tracer instance.
   *
   * @param name the name of the tracer
   * @param tracer the to be wrapped tracer
   */
  public QueueTracer(String name, T tracer) {
    super(name);
    this.tracer = tracer;
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @return the sync object of the wrapped tracer
   */
  @Override
  public Object getSyncObject() {
    return this.tracer.getSyncObject();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @return the buffer size of the wrapped tracer
   */
  @Override
  public int getBufferSize() {
    return this.tracer.getBufferSize();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @return indicates the autoflush mode of the wrapped tracer
   */
  @Override
  public boolean isAutoflush() {
    return this.tracer.isAutoflush();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @return indicates if the wrapped tracer is openend
   */
  @Override
  public boolean isOpened() {
    return this.tracer.isOpened();
  }

//  /**
//   *
//   * @return
//   */
//  @Override
//  public String getName() {
//    return this.tracer.getName();
//  }

  /**
   * Sets the buffer size of the wrapped tracer.
   *
   * @param bufferSize the bufferSize to set
   */
  @Override
  public void setBufferSize(int bufferSize) {
    this.tracer.setBufferSize(bufferSize);
  }

  /**
   * Sets the autoflush mode of the wrapped tracer
   * @param autoflush the autoflush to set
   */
  @Override
  protected void setAutoflush(boolean autoflush) {
    this.tracer.setAutoflush(autoflush);
  }

  @Override
  protected void readConfiguration(XPath xpath, Node node) throws XPathExpressionException, AbstractTracer.Exception {
    this.online = (boolean) xpath.evaluate("./dns:Online/text()", node, XPathConstants.BOOLEAN);
    this.level = Integer.parseInt((String) xpath.evaluate("./dns:DebugLevel/text()", node, XPathConstants.STRING));
    setAutoflush((boolean) xpath.evaluate("./dns:TraceLogger/dns:AutoFlush/text()", node, XPathConstants.BOOLEAN));
    setBufferSize(Integer.parseInt((String) xpath.evaluate("./dns:TraceLogger/dns:BufSize/text()", node, XPathConstants.STRING)));
  }

  /**
   * Opens the wrapped tracer.
   */
  @Override
  public void open() {
    this.tracer.open();
  }

  /**
   * Closes the wrapped tracer.
   */
  @Override
  public void close() {
    this.tracer.close();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer by using the config parameter global
   * to all queued tracer.
   */
  @Override
  public void initCurrentTracingContext() {
    this.tracer.initCurrentTracingContext(this.level, this.online);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @param debugLevel controls the extent of the output
   * @param online a value of false delivers no output of the current thread at all whereas a value of true delivers output controlled by debugLevel
   */
  @Override
  public void initCurrentTracingContext(int debugLevel, boolean online) {
    this.tracer.initCurrentTracingContext(debugLevel, online);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @param returnType the return type of the method as string representation
   * @param clazz the class to which that method belong
   * @param methodSignature the remaining method signature (without return type) inclusive parameter as string representation
   * @return the TraceMethod which has been put onto the stack - a mere data object for internal use primarily. May be null.
   */
  @Override
  public TraceMethod entry(String returnType, Class clazz, String methodSignature) {
    return this.tracer.entry(returnType, clazz, methodSignature);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @param returnType the return type of the method as string representation
   * @param object the object that owns the method
   * @param methodSignature the remaining method signature (without return type) inclusive parameter as string representation
   * @return the TraceMethod which has been put onto the stack - a mere data object for internal use primarily. May be null.
   */
  @Override
  public TraceMethod entry(String returnType, Object object, String methodSignature) {
    return this.tracer.entry(returnType, object, methodSignature);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @param logLevel one of the predefined levels INFO, WARNING, ERROR, FATAL and SEVERE
   * @param message the to be logged message
   * @param clazz the originating class
   */
  @Override
  public void logMessage(LogLevel logLevel, String message, Class clazz, String methodName) {
    this.tracer.logMessage(logLevel, message, clazz, methodName);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @param logLevel one of the predefined levels INFO, WARNING, ERROR, FATAL and SEVERE
   * @param throwable the to be logged throwable
   * @param clazz the originating class
   * @param methodName the name of the relevant method
   */
  @Override
  public void logException(LogLevel logLevel, Throwable throwable, Class clazz, String methodName) {
    this.tracer.logException(logLevel, throwable, clazz, methodName);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   * @return the thread map of the wrapped tracer
   */
  @Override
  protected AbstractThreadMap getThreadMap() {
    return this.tracer.getThreadMap();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer. Besides it checks if the
   * stack size of the current tracing context has decreased to zero. If so then the current tracing context
   * will be cleared.
   *
   * @return the TraceMethod which has been popped from the stack - a mere data object for internal use primarily. May be null.
   */
  @Override
  public TraceMethod wayout() {
    TraceMethod traceMethod = this.tracer.wayout();
    if (getThreadMap().getCurrentStackSize() == 0) {
      clearCurrentTracingContext();
      if (!TracerFactory.getInstance().offerTracer(this))
        System.err.printf("WARNING: Offer failed. Possible queue corruption.%n");
    }
    return traceMethod;
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   *
   * @return an {@link IndentablePrintStream}
   */
  @Override
  public IndentablePrintStream out() {
    return this.tracer.out();
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   * @param level the level of the to be printed data
   * @return an {@link IndentablePrintStream}
   */
  @Override
  protected IndentablePrintStream out(int level) {
    return this.tracer.out(level);
  }

  /**
   * Delegates to the corresponding method of the wrapped tracer.
   */
  @Override
  public void clearCurrentTracingContext() {
    this.tracer.clearCurrentTracingContext();
  }
}


text:
{ "item_title" : "The Tao of Bill Murray", "item_author" : [" Gavin Edwards", "R. Sikoryak "], "item_description" : "For the Bill Murray fan in all of us, this epic collection of Bill Murray stories--many reported for the first time here--distills a set of guiding principles out of his extraordinary ability to infuse the everyday with surprise, absurdity, and wonder. No one will ever believe you. New York Times bestselling author Gavin Edwards, like the rest of us, has always been fascinated with Bill Murray--in particular the beloved actor's adventures off-screen, which rival his filmography for sheer entertainment value. Edwards traveled to the places where Murray has lived, worked, and partied, in search of the most outrageous and hilarious Bill Murray stories from the past four decades, many of which have never before been reported. Bill once paid a child five dollars to ride his bike into a swimming pool. The star convinced Harvard's JV women's basketball team to play with him in a private game of hoops. Many of these surreal encounters ended with Bill whispering, No one will ever believe you into a stranger's ear. But The Tao of Bill Murray is more than just a collection of wacky anecdotes. A sideways mix of comedy and philosophy, full of photo bombs, late-night party crashes, and movie-set antics, this is the perfect book for anyone who calls themselves a Bill Murray fan--which is to say, everyone. Praise for The Tao of Bill Murray When confronted by life's challenges and opportunities, we should all be asking ourselves, 'What would Bill Murray do?' This book actually has the answers.--Jay Duplass, writer, director, actor, fan of Bill Murray Gavin Edwards] captures Murray's enigmatic, comic genius.--The Washington PostRib-tickling.--NewsdayIf you are among the multitudes who have a Bill Murray story (viz: 'I was in the airport bar, and who sat down next to me but Bill Murray?'), The Tao of Bill Murray will speak to you.--Elle Murray is an endless delight, and his knack for bons mots and non sequiturs will keep readers laughing before revealing an unexpectedly poignant vision for happiness. . . . A fun and revealing look behind the charm and mythos of Bill Murray that will only strengthen his legend.--Kirkus ReviewsThis book is bursting with anecdotes that underline Murray's unconventional and fun-loving life. . . . Murray's fans are sure to savor it] and walk away with a deeper appreciation of the actor and his work.--Publishers Weekly (starred review)I just can't fathom how anyone could not like the man. He lives as much as possible by his own rules; this is a guy who doesn't buy round-trip airline tickets because he prefers to make decisions on the fly. That old question of the three people you'd most like to have dinner with? I'd pick Bill Murray and leave the other two chairs empty just to see who he pulls over to join us. . . . You will love The Tao of Bill Murray. Period.--GeekDadA hilarious read--occasionally heartwarming, sometimes head-scratching. . . . Sure to please Murray fans.--Library Journal", "item_img_path" : "https://covers3.booksamillion.com/covers/bam/0/81/299/870/0812998707_b.jpg", "price_data" : { "retail_price" : "26.00", "our_price" : "6.97", "club_price" : "6.97", "savings_pct" : "73", "savings_amt" : "0.00", "club_savings_pct" : "73", "club_savings_amt" : "0.00", "discount_pct" : "10" } }
The Tao of Bill Murray
by Gavin Edwards and R. Sikoryak
Bargain info
Bargain items are publisher overstocks or other special purchases. While new, some books may have small markings or stickers showing their bargain price.
Our selection of bargain books changes hourly, and quantities are limited, so shop now and shop often. Due to unpredictable demand, it is possible for us to oversell the available quantity of a bargain item. In such cases, we will promptly cancel your order and your credit card will not be charged.
For the Bill Murray fan in all of us, this epic collection of "Bill Murray stories"--many reported for the first time here--distills a set of guiding principles out of his extraordinary ability to infuse the everyday with surprise, absurdity, and wonder. No one will ever believe you. New York Times bestselling author Gavin Edwards, like the rest of us, has always been fascinated with Bill Murray--in particular the beloved actor's adventures off-screen, which rival his filmography for sheer entertainment value. Edwards traveled to the places where Murray has lived, worked, and partied, in search of the most outrageous and hilarious Bill Murray stories from the past four decades, many of which have never before been reported. Bill once paid a child five dollars to ride his bike into a swimming pool. The star convinced Harvard's JV women's basketball team to play with him in a private game of hoops. Many of these surreal encounters ended with Bill whispering, "No one will ever believe you" into a stranger's ear. But The Tao of Bill Murray is more than just a collection of wacky anecdotes. A sideways mix of comedy and philosophy, full of photo bombs, late-night party crashes, and movie-set antics, this is the perfect book for anyone who calls themselves a Bill Murray fan--which is to say, everyone. Praise for The Tao of Bill Murray "When confronted by life's challenges and opportunities, we should all be asking ourselves, 'What would Bill Murray do?' This book actually has the answers."--Jay Duplass, writer, director, actor, fan of Bill Murray " Gavin Edwards] captures Murray's enigmatic, comic genius."--The Washington Post
"Rib-tickling."--Newsday
"If you are among the multitudes who have a Bill Murray story (viz: 'I was in the airport bar, and who sat down next to me but Bill Murray?'), The Tao of Bill Murray will speak to you."--Elle "Murray is an endless delight, and his knack for bons mots and non sequiturs will keep readers laughing before revealing an unexpectedly poignant vision for happiness. . . . A fun and revealing look behind the charm and mythos of Bill Murray that will only strengthen his legend."--Kirkus Reviews"This book is bursting with anecdotes that underline Murray's unconventional and fun-loving life. . . . Murray's fans are sure to savor it] and walk away with a deeper appreciation of the actor and his work."--Publishers Weekly (starred review)"I just can't fathom how anyone could not like the man. He lives as much as possible by his own rules; this is a guy who doesn't buy round-trip airline tickets because he prefers to make decisions on the fly. That old question of the three people you'd most like to have dinner with? I'd pick Bill Murray and leave the other two chairs empty just to see who he pulls over to join us. . . . You will love The Tao of Bill Murray. Period."--GeekDad"A hilarious read--occasionally heartwarming, sometimes head-scratching. . . . Sure to please Murray fans."--Library Journal
Publisher: American Book Company
Publish Date: June 2018

redpajama_set_name: RedPajamaCommonCrawl


7 Tips On How To Make Life Insurance Claims – Mortgage Finance Solutions – Finding your home loan for you!
Most people have no idea how to file any claim right up until something comes about. Then they will call his or her agent, that gives all of them some directions, and they're going from right now there. But a better idea would be to figure more or less everything out before something comes about, so you don't have to spend considerable time figuring that out while the bills are piling up. That means, it has to be lot speedier, and you'll not be likely to make virtually any mistakes.
Get the insurance plan service package deals and read through them. Enable the customer assist agent to debate exactly just what the policy requires and what exactly is expected involving you. Learn regarding the dos along with don'ts on the policy.
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Inspect intended for previous comments from customers and exactly what they claim. Examine online testimonials along with years involving experience ahead of signing whatever.
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It can be most effective to select 20 or maybe 30 year term plan to make certain you should be able to save income and see to it to scrutinize the alternate options that claim renewable.
Make certain that the firm will surely not supply the beneficiary a tough time throughout the truth. Ask exactly how they will process the actual claim in the event that something fails to you. Ask intended for sample cases who have occurred using previous buyers.
Will not buy your insurance policies collected from one of company. The financial status can become unpredictable and the most beneficial life insurance policy company can certainly go broke the subsequent year. It would certainly be most reliable to purchase various options from distinct providers as being a safety world wide web.
Ensure that the actual firm is associated with the most powerful ones throughout the world. For illustration, when you receive a medical care insurance, make sure you manage to deal conveniently using the medical services and team likely connected with. This will help you get the most beneficial life insurance. Get your ni card now!
If you are like most people, you've got a couple different insurance policies. When you think about it, insurance really is a great invention. It might not be the most exciting thing around, but it certainly protects us from a lot of financial misery and disaster. Imagine if you still had to pay your mortgage after your house burned down, or if you still had to make your car payments after it got stolen. With insurance, we don't need to worry about these items.


text:
/**
 * @author Igor Polevoy: 12/31/13 1:09 PM
 */

package org.javalite.db_migrator.mock;

import java.sql.*;
import java.util.ArrayList;
import java.util.List;

public class MockStatement implements Statement {

    protected List<String> executedStatements =  new ArrayList<String>();

    public List<String> getExecutedStatements() {
        return executedStatements;
    }

    public ResultSet executeQuery(String sql) throws SQLException {
        executedStatements.add(sql);
        return null;
    }

    public int executeUpdate(String sql) throws SQLException {
        executedStatements.add(sql);
        return 0;
    }

    public void close() throws SQLException {

    }

    public int getMaxFieldSize() throws SQLException {
        return 0;
    }

    public void setMaxFieldSize(int max) throws SQLException {

    }

    public int getMaxRows() throws SQLException {
        return 0;
    }

    public void setMaxRows(int max) throws SQLException {

    }

    public void setEscapeProcessing(boolean enable) throws SQLException {

    }

    public int getQueryTimeout() throws SQLException {
        return 0;
    }

    public void setQueryTimeout(int seconds) throws SQLException {

    }

    public void cancel() throws SQLException {

    }

    public SQLWarning getWarnings() throws SQLException {
        return null;
    }

    public void clearWarnings() throws SQLException {

    }

    public void setCursorName(String name) throws SQLException {

    }

    public boolean execute(String sql) throws SQLException {
        executedStatements.add(sql);
        return false;
    }

    public ResultSet getResultSet() throws SQLException {
        return null;
    }

    public int getUpdateCount() throws SQLException {
        return 0;
    }

    public boolean getMoreResults() throws SQLException {
        return false;
    }

    public void setFetchDirection(int direction) throws SQLException {

    }

    public int getFetchDirection() throws SQLException {
        return 0;
    }

    public void setFetchSize(int rows) throws SQLException {

    }

    public int getFetchSize() throws SQLException {
        return 0;
    }

    public int getResultSetConcurrency() throws SQLException {
        return 0;
    }

    public int getResultSetType() throws SQLException {
        return 0;
    }

    public void addBatch(String sql) throws SQLException {

    }

    public void clearBatch() throws SQLException {

    }

    public int[] executeBatch() throws SQLException {
        return new int[0];
    }

    public Connection getConnection() throws SQLException {
        return null;
    }

    public boolean getMoreResults(int current) throws SQLException {
        return false;
    }

    public ResultSet getGeneratedKeys() throws SQLException {
        return null;
    }

    public int executeUpdate(String sql, int autoGeneratedKeys) throws SQLException {
        executedStatements.add(sql);
        return 0;
    }

    public int executeUpdate(String sql, int[] columnIndexes) throws SQLException {
        executedStatements.add(sql);
        return 0;
    }

    public int executeUpdate(String sql, String[] columnNames) throws SQLException {
        executedStatements.add(sql);
        return 0;
    }

    public boolean execute(String sql, int autoGeneratedKeys) throws SQLException {
        executedStatements.add(sql);
        return false;
    }

    public boolean execute(String sql, int[] columnIndexes) throws SQLException {
        executedStatements.add(sql);
        return false;
    }

    public boolean execute(String sql, String[] columnNames) throws SQLException {
        executedStatements.add(sql);
        return false;
    }

    public int getResultSetHoldability() throws SQLException {
        return 0;
    }

    public boolean isClosed() throws SQLException {
        return false;
    }

    public void setPoolable(boolean poolable) throws SQLException {

    }

    public boolean isPoolable() throws SQLException {
        return false;
    }

    public void closeOnCompletion() throws SQLException {

    }

    public boolean isCloseOnCompletion() throws SQLException {
        return false;
    }

    public <T> T unwrap(Class<T> iface) throws SQLException {
        return null;
    }

    public boolean isWrapperFor(Class<?> iface) throws SQLException {
        return false;
    }
}


text:
IndiaVision Social and Lifestyle
IndiaVision Matrimonials, News and Information Sites site
Indians Abroard
Home » Feng shui
Feng Shui is the art of creating balance and harmony in our surroundings. Over 3,000 years ago, it was developed in China and attributable to its origin, it is also called Chinese Feng Shui. more..
Feng Shui Architecture
Feng Shui is the Chinese art, which teaches how to maintain balance and harmony around us and live happily. The concept of Feng Shui is quite ancient, but its principles still provide guidance on more..
Feng Shui Home
'Home is where the heart is', is a popular saying that reflects how important home is in our lives. It is the place where we spend the maximum time span of our lives. No more..
Feng Shui Bathroom
Feng Shui Bedroom
Feng Shui Kitchen
Feng Shui Living Room
Feng Shui Main Door
After home, an office holds the utmost importance in a man's life. It is the place where you work to realize your dreams. There is a saying 'Hard work is the key to success', more..
Feng Shui Office Desk
Feng Shui Office Interiors
Feng Shui Office Site
Feng Shui Products
Feng Shui is an ancient Chinese art, which transforms the lives of the people and is based on nature's principles. It lays down various guidelines to arrange things in our surroundings. It also emphasizes more..
Feng Shui Candles
Feng Shui Compass
Feng Shui Crystal
Feng Shui Dragon
Feng Shui Fountain
Feng Shui Mirror
Feng Shui Turtle
Feng Shui Types
Feng shui is the brain child of Chinese philosophy and originated thousands of years back. Though it is ancient, it still holds a great amount of significance in today's world. The reason behind this more..
Health Feng Shui
Love Feng Shui
Wealth Feng Shui
Feng Shui Colors
Colors play an important part in our lives. They fill our heart with vibrancy and beauty. The world is unperceivable without colors, as it would have made everything very dull. There would not have more..
Feng Shui Cures
People face a number of problems in different spheres of their lives, everyday. These problems can be due to various reasons like unbalanced energy flow, which is not realized by them. At such a more..
Feng Shui Decoration
Home decoration is one of the most loved activities, especially amongst women. Home is the place everyone is in love with and cherishes its comfort and warmth. An individual spends the maximum time of more..
Feng Shui Design
Design is basically how we live in our respective spaces. Feng Shui focuses on the flow of energy or Chi in the surroundings. At times, it is also considered the personal energy that should more..
Feng Shui Elements
Feng Shui follows the Five Element Theory. Infact, it is one of the main theories of Feng Shui. These five elements are considered the main manifestations of Chi, the vital energy. According to the more..
Feng Shui Garden
In Feng Shui, garden is considered a form of Yin, while the house is said to be Yang. Therefore, the garden and the house should present the balance of both yin and yang energies. more..
Feng Shui Gifts
Though Feng Shui is an ancient Chinese ideology, it is accepted throughout the world in the present scenario. It has become quite popular, as people are realizing the benefits of Feng Shui cure. The more..
Feng Shui Plants
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Feng Shui Principles
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Feng Shui Symbols
Feng Shui symbols serve as important tools for transforming negative energy into positive. These symbols can help you in improving the quality of your life. They serve the purpose of balancing and harmonizing energy more..
Feng Shui History
Feng Shui, the intuitive art of balance and harmony, originated thousands of years ago. Like any other form of art, its history & origin also comprises of many interesting facts. During ancient times, the more..
What is Feng Shui? It is a complex branch of knowledge, which was developed by the Chinese. It explains the phenomenon of balancing environmental energies to bring good health, wealth and fortune to individuals. more..
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text:
Rx Summit
Treatment Vouchers
Casey's Law
After-Care Programs
Help Lines
Treatment Referral Line
KY HELP Call Center
1-8338-KY-HELP
Audio / Video Library
Hale testifies before U.S. House Subcommittee
Nancy Hale, President and CEO of Operation UNITE, testified before the U.S. House Committee on Appropriations' Subcommittee on Labor, Health and Human Services, Education, and Related Agencies on Wednesday, April 5, 2017. Other witnesses were Barbara Cimaglio, Deputy Commissioner of the Vermont Department of Health; Bill Guy, Parents Helping Parents; and Rosalie Liccardo Pacula, Senior Economist and Co-Director of the Drug Policy Research Center, RAND Corporation. During the hearing, titled "Federal Response to the Opioid Abuse Crisis," Hale spoke about the strategies to effectively help those impacted by substance use and misuse, the importance of funding the fight against our country's deadly drug abuse epidemic, and cited UNITE initiatives as examples of what can be accomplished using an holistic approach.
Below is the complete copy of her written statement submitted for the record.
To view photos from the hearing, click here.
Click here to watch the full hearing.
Good morning. Chairman [Tom] Cole, Ranking Member [Rosa] DeLauro, and members of the subcommittee, thank you for giving me this opportunity to speak with you today. I am Nancy Hale, president and CEO of Operation UNITE.
UNITE is an acronym for Unlawful Narcotics Investigations, Treatment and Education. It is a three-pronged, comprehensive approach to create long-term success in combating substance abuse.
Operation UNITE was launched in April 2003 by Congressman Hal Rogers shortly after a special report, "Prescription for Pain," was published by the Lexington Herald-Leader. This series of articles exposed the addiction and corruption associated with drug abuse in southern and eastern Kentucky, which largely included Congressman Rogers' Fifth Congressional District.
Many of us were shocked to learn that, per capita, we were the top pain killer users in the entire world. Tragically, as a result, our commonwealth has been the epicenter for the explosion of opioid abuse: The drug overdose rate in Kentucky currently is more than 1.5 times higher than the national average. Rates in several counties are triple the national average.
Congressman Rogers and other local leaders feared that if we did not take swift and decisive action, an entire generation would have been wiped out. We held community meetings to find out the scope of the problem and what should be done. Teachers, preachers, parents, judges, and police officers. Everyone we spoke to had stories – personal stories. And they were ready for action.
Based on their feedback, Operation UNITE pioneered a holistic approach that has become a model for other states and the nation. This comprehensive method involves law enforcement, treatment, and education/prevention initiatives working together.
Through collaborative partnerships, UNITE's progress in our 32-county region is evident. Fourteen years later, more than 100,000 youth have participated in UNITE's programs, tens of thousands of community members have volunteered, and more than 4,000 people have entered treatment using a UNITE voucher.
Let me start with the first pillar: Investigations and Enforcement.
UNITE has long been a leader in the state, participating in or overseeing many of the largest drug busts in Kentucky history.
For example, UNITE had one-fifth of the cases in Operation Flamingo Road – a federal, state, and local law enforcement effort to arrest 518 people suspected of obtaining or distributing prescription pills from here to Florida. Over the last 14 years, UNITE detectives have:
Removed more than $12.3 million worth of drugs from the street,
Arrested more than 4,400 bad actors,
Achieved a conviction rate of more than 97 percent, and
Received and processed nearly 22,000 calls to our drug tip line.
But we have also long recognized that we cannot arrest our way out of this unique epidemic. As one law enforcement official so powerfully observed: Investigations will grab headlines. Treatment and education will result in long-term results.
That is why Treatment is our second pillar.
Getting justice is only part of the equation. Long-term recovery is what transforms substance users into healthy and productive members of their families and communities.
Many of the drug abusers who have their first experience with UNITE's law enforcement officers then benefit from our multi-faceted approach that includes treatment. We staff a treatment help line to connect people to resources and have supplied vouchers to help more than 4,000 low-income people enter long-term drug rehabilitation.
The UNITE treatment team responds to approximately 1,200 inquiries per month. Although the vast majority of these inquiries are seeking information about applying for a UNITE treatment voucher, a substantial number of inquiries are from individuals wanting information about Casey's Law (involuntary commitment), general information about the signs and symptoms of addiction, types of treatment available, or people who simply want to speak to someone about the addictive behavior of their loved one.
In addition, UNITE's assistance has helped increase the number of Drug Court programs in the region from five in 2003 to one in all 32 counties we serve. Participants obtain treatment and are more likely to return to productive lives, stay gainfully employed, pay child support, and meet other obligations.
Drug Courts in our service area have collected more than $1.4 million in fines, restitution, and court costs, along with more than $900,000 in child support. Participants also complete thousands of hours of community service each month.
UNITE has provided more than $4 million to create 30 new Drug Court programs in 24 counties in addition to programs operated by the Kentucky Administrative Office of the Courts. Kentucky drug courts currently operate in 113 of the state's 120 counties.
The final pillar is Education and Prevention.
To make progress, we must not only cut off the supply, but decrease the demand as well. Education and prevention are the keys to reducing the demand for abusing or misusing legal substances or using illegal drugs. When demand is high, users are willing to use what is most available and affordable, and suppliers are creative in meeting these needs, whether it is prescription pills, heroin, meth, or synthetic drugs.
Offering youth alternatives to drug use through programming and hands-on education makes a huge difference. We must give them the facts. Children should be taught the effects of drugs on their minds and bodies from K-12. Repetitive, consistent messaging is needed.
And our focus should not only be on presenting facts and providing information on the effects of drugs on their bodies and brains, but should be on helping our youth make that one decision to not use any addicting substances, including alcohol, tobacco, marijuana, and other drugs. That focus holds great promise of a stronger, clearer, and more effective goal for public education and prevention.
UNITE's education programs and activities introduce youth and adults to a life without drugs. Some programs are geared to help youth avoid the dangers of the streets, but, for many, the danger is much closer – it is at home. UNITE shows children a different path, and it also helps them teach their parents or caregivers. For example, one Leslie County parent sought help for an addicton after her 4th-grader told her about UNITE and what she was learning in her "Too Good for Drugs" class.
Thus far, we have reached more than 100,000 students through various drug education programs and summer activities.
Our anti-drug programming includes "On the Move," a mobile and interactive one-of-a-kind education initiative. It provides a hands-on experience to simulate distracted and impaired driving. "Life With A Record" is a prevention initiative that helps youth examine the criminal justice system and how seemingly harmless acts can impact their futures.
Camp UNITE is a free, weeklong leadership and adventure camp that provides middle school youth with an opportunity to engage in fun, constructive activities using a small group, peer mentorship format. Many participants have been directly impacted by substance abuse or are unable to afford a traditional summer camp program.
Other summer activities include "Shoot Hoops Not Drugs" and "Hooked On Fishing – Not On Drugs."
Federal funding has been critical. It has helped UNITE reach across jurisdictions and county lines – and across professional territories.
For example, ARC grants have enabled us to educate prescribers on addiction, pain management, and state monitoring systems for prescription drugs known as PDMP's.
SAMHSA has helped us provide treatment resources through UNITE's vouchers, which is vital in a region faced with high poverty and unemployment. It also funded substance abuse counselors in the middle school and high schools, which was extremely effective. The impact was large, not only in the schools but also in the community. Unfortunately, schools were not able to sustain that effort when the grant money ran out.
In addition, AmeriCorps has been an invaluable part of our education efforts. Our 54 UNITE ServiceCorps members serve 17 school systems in 14 counties. They provide math tutoring, teach anti-drug and wellness curricula, have recruited more than 8,200 volunteers, and sponsor anti-drug UNITE clubs that have impacted more than 4,000 students in the last year alone.
And the results are dramatic: Last school year alone, the more than 1,500 students they tutored showed an average 30 percent growth in math knowledge. And the 3,300 students who took the anti-drug and health information curricula showed an average of 35 percent growth in drug awareness and healthy decision-making knowledge.
I am pleased that the federal CARA legislation enacted by this Congress last year will enable regional organizations like UNITE to take advantage of these new federal funds focused on addressing the opioid epidemic, and I am grateful to each of you who supported that bill.
Congress' collaboration on CARA must be replicated elsewhere. In the anti-drug world, we certainly have to collaborate with stakeholders across a variety of professions, institutions, schools and faith-based organizations. Not just law enforcement. Not just treatment. Not just education. Everyone must work together.
We were founded on community input, and that involvement continues and grows. Our nonprofit UNITE Coalitions in each of our counties know what their communities need. These coalitions are the key to after-care. People in recovery will eventually come back to their communities. They need support when they come home. Our coalitions make that happen. UNITE provides guidance and small amounts of funding to create those strong, local partnerships.
As a result, tens of thousands of people have participated in UNITE events and coalition activities to educate and deter people from taking drugs.
When it became unfortunately clear that the challenges we had been experiencing in rural Kentucky had exploded across the country, we worked to share UNITE's holistic approach through the establishment of the National Rx Drug Abuse & Heroin Summit – now the largest gathering of medical professionals, advocates, law enforcement and policy makers in the United States.
Our next Summit is April 17th through 20th in Atlanta. Many of your colleagues have attended in the past, and I hope to see you there this year.
That is a quick overview of some of Operation UNITE's strategies. Now, I would like to touch on several of the lessons we have learned over the last 14 years that may benefit similar organizations in your home districts.
The first is that you must bring all stakeholders to the table at the beginning. For example, we did not engage the medical community early enough. It was not until a local physician was tragically murdered for refusing to give a patient pain medicine that we all rallied together at the same table.
The second lesson learned is that UNITE could have done a better job working with families and helping them understand that addiction is a chronic disease that their loved ones would deal with for the rest of their lives. We needed to do more to help the families understand the disease and how to support their loved one when in long-term recovery.
A third lesson learned is that you must have a champion to lead, to motivate, to encourage, and to fight alongside you. For us, that champion is Congressman Rogers. Today, there are bipartisan caucuses in both the House and Senate to facilitate bringing a unified national approach to this difficult effort.
A fourth lesson is that you cannot expect short-term treatment to yield long-term results. Models of recovery should be based on long-term goals.
The final, and most important, take-away is that education and prevention are the tools to achieve those long-term results. The longer I am involved in fighting this epidemic, the more I am convinced that education – particularly K-12 prevention education – is the key to saving our next generation.
Through private donations, we are able to provide $1,500 need-based scholarships to youth who have been actively involved in UNITE programs or have been impacted by substance abuse in their families.
It is only through collaboration and a holistic approach that we will succeed. And there is no better illustration of this than that of a young woman who was awarded an "I Am UNITE" college scholarship last year. I'll call her Sarah.
Sarah is a scholarship recipient who devoted 300 hours of service learning during high school. She also was one of only four students in the country selected by Jobs for America's Graduates for the honor of placing a wreath at the Tomb of the Unknown Soldier.
But before those successes, she had some stumbling blocks – like when she had to step over her father who was passed out on the floor from a drug overdose. Her father later was arrested as part of a UNITE drug investigation. But UNITE's efforts in her life did not end there. If they had, her story – and his – might have turned out differently.
Her father went to jail, but UNITE provided a voucher for him to enter long-term treatment. After he successfully completed treatment, he addressed an assembly at her high school. Sarah confessed that she was proud of him for the first time.
Sarah is now headed to college, and her father is making strides of his own with his recovery.
The day after she received a UNITE scholarship, a gentleman called us inquiring about funds to get a Celebrate Recovery group started in his county so he did not have to travel to a neighboring county. He explained he did not know anything about computers and was illiterate, so he would need help downloading and completing the forms. That man was Sarah's father.
Sarah and her family illustrate why the multi-pronged approach is the key to saving our families and communities. They also offer us hope, which is another important part of recovery.
That is why we created the Hope Wall, which features dozens of people who have been drug free for at least 18 months. When I look at those faces and think about these men and women returning to their families, my eyes are always drawn to one photo in particular – that of my own son. Knowing how each of these people, in long-term recovery, are giving back and helping others, is what should give us all hope.
Cadets learn connection between basketball and life
Sports balls donated for children of Eastern Kentucky
Gov. Beshear joins bipartisan group of state, community leaders to recognize Overdose Awareness Day
KY HELP Statewide Call Center adds evening, weekend hours
Death toll from overdoses rose 49% in 2020
Witt selected for BRIGHT Kentucky class
KY HELP featured on 'Best of the Bluegrass'
Freedom returns to delight of children
2 regional SHND camps offered
Miranda Baker
Olivia Broughton
Teresa Slaven
Earl Young
Michael Hicks
Mandy Bays
Dominic Carton
Billy Newsome
©2020 Operation UNITE. All rights reserved. | Hosted and maintained by The Center for Rural Development


First page of Playing the Game!
The first page of our new weekly webcomic is up at SlangShake. Check it out!
Playing The Game has a Teaser Trailer!.
A crudely put together motion comic of one of our older strips.


This is the reason nobody wants to drop the hammer on you when facing a big putt. Count it!
This is a classic trucker style cap with a cool fabric blend and Red lettering.
• 47% cotton/28% nylon/25% polyester.


« 2014 in review, Thank you All!
This entry was posted on January 1, 2015 at 1:26 pm and is filed under 1980s, Heavy Metal, Music, Uncategorized with tags Americans, Amon Amarth, Classic Rock, disco, hard rock, Heavy Metal, Mellow out rock, melodic rock, Ozzy Osbourne, progressive rock, Queen, The 1980s, The Sun, Toto, Wayne's World.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
Happy New Year. I hope the Ozzy disappointment is the last one you experience for the next 12 months. Thanks for giving Toto some attention. This is a great album that often gets overlooked (actually, the band themselves don't get a lot of respect from critics or rock fans). Obviously it was a slight disappointment for you but they went on to record some heavier material in later years. Have you explored their deep catalog or just the early years.
To be honest, I haven't explored much of their stuff after 1984 but I will be doing so as I progress through the years. It was a slight disappointment but I didn't think it was a bad album in anyway. Toto were talented musicians and should get a bit more respect.


text:
St. Louis, MO Highlighting St. Louis
Exhibit - LEAP Middle School Initiative: Lost Islands
Contemporary Art Museum
camstl.org
LEAP artists, in two bodies of work, consider pattern, form, place, time, and infinity. In the fall session, teaching artist Sukanya Mani introduced Tyvek, an artificial fabric that can be used as both a flat surface or three-dimensional object. The young artists cut, twist, fold, and paint the material, rendering repeated patterns and giving flat forms dimensionality with hints of the infinite. During the spring session, teaching artist Janie Stamm led LEAP to create the Lost Islands, complete worlds constructed from cardboard and papier-mâché. The islands are imagined and designed as idealized spaces with their own unique topographies, ecologies, and histories. The paintings of Derek Fordjour, also built from humble materials, serve as inspiration. The spring session was cut short due to COVID-19 and the exhibition includes various islands in different stages of completion, as projects remained unfinished as our community transitioned to shelter in place.
Designed for the young artist interested in an in-depth exploration of contemporary art practices and mediums, this free ten-week-long after-school program gives students the opportunity to work closely with professional St. Louis-based artists and educators in an atmosphere of focused attention and mentorship. Fifteen students from the greater St. Louis area are selected to focus on one particular contemporary practice or medium each semester.
LEAP Middle School Initiative is generously supported by The Strive Fund; Dana Brown Charitable Trust, US Bank, Trustee; and Crawford Taylor Foundation.
LEAP Middle School Initiative: Lost Islands is organized for the Contemporary Art Museum St. Louis by José Garza, Museum Educator. 2019/20 LEAP artists: Alexa Bishop, Shane Box, Emma Deaton, Saskia Dentman, Samiah Elmore, Jesse Heater, Laila Jaoko, Amora Jones, Abril Kaisin, Fern Kanak, Ivy Kanak, Cozzette Motherhead, Zenga Stroff, Sanai Todd, Oliver Vollman, and Yaa Williams.
Exhibit Date: Sep 11, 2020 – Feb 21, 2021.
This event listing provided for the St. Louis community events calendar. Community events are not associated with or sponsored by AARP, but may be of interest to you. If you have an event to share, let us know!
More from AARP in St. Louis
St. Louis Happenings | Upcoming AARP Events | About AARP
St. Louis Happenings
An ally on the issues that matter most to you in St. Louis


text:
Q: React Native - Strange error caught - {"line":25722,"column":33,"sourceURL":"http://10.0.2.2:8081/index.bundle?platform=android&dev=true&minify= My react native application is running on android virtual device (API 23)
Android Gradle Plugin Version - 7.2.0
Gradle Version - 7.4.2
build.gradle
buildscript {
    ext {
        kotlinVersion = "1.5.0"
        RNNKotlinVersion = kotlinVersion
        buildToolsVersion = "31.0.0"
        minSdkVersion = 23
        compileSdkVersion = 31
        targetSdkVersion = 28
...

}
An http request (http://10.0.2.2:8888/myapp/auth) is made to confirm login to my localhost server.
I get an http response code 500 (email/password is incorrect).
AccessService calls FetchTool to fetch http.
FetchTool throws a new Error('incorrect email/password').
But AccessService receives a ghost different error as described below:
LOG ERROR..:{"line":25722,"column":33,"sourceURL":"http://10.0.2.2:8081/index.bundle?platform=android&dev=true&minify=false&app=com.myapp&modulesOnly=false&runModule=true"}

Why don't I get the error 500 thrown in FetchTool in AccessService's catch?
export default class AccessService {

    static confirmLogon(email, password) {

        const login = {
            email: email,
            password: password
        };

        const headers = {
            'Content-Type': 'application/json',
        }

        const uri = properties.appurl + '/auth';

        return FetchTool.post(uri, headers, login)
            .then(credentials => {
                return credentials;
            })
            .catch(error => {
                // the ghost error is captured here
                console.log('ERROR..:' + JSON.stringify(error))
                throw error;
            });
    }

}

export default class FetchTool {

    static post(uri, headers, playload) {
        return this.executeRequest(uri, headers, playload, 'POST');
    }

    static executeRequest(uri, headers, playload, method) {

        if (headers['Content-Type'] && headers['Content-Type'] == 'application/json') {
            playload = JSON.stringify(playload);
        }

        const requestInfo = {
            method: method,
            body: playload,
            headers: headers
        }

        //uri = http://10.0.2.2:8888/myapp/auth

        return fetch(uri, requestInfo)
            .then(response => {

                if (response.ok) {
                    //some code return json ok
                } else {
                    return response.text()
                        .then(responseText => JSON.parse(responseText))
                        .then(responseObject => {
                            if(responseObject.statusCode === 500) {
                                //the error scenario takes place here
                                throw new Error(JSON.stringify(responseObject));
                            } else {
                                // some code
                            }
                        })
                }
            })

    }

}

A: Just remove JSON.stringify like so:
console.log('ERROR: ', error)

And you 'll see the error message as expected.

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