text:
\section*{Introduction}

Let $(R,\m)$ be a $d$-dimensional Noetherian local ring of positive characteristic $p$. For
every $\m$-primary ideal $I$ in $R$,
one can define an asymptotic invariant that measures the  singularity of
$R$ via the Frobenius powers of $I$. For $q$ a power of $p$, we let $I^{[q]}$ be the ideal generated by
the $q$th powers of all elements of $I$.  In 1983, Monsky \cite{M} proved that there is a
real number $e_{HK}(I) \geq 1$ such that for $q = p^e$,
$$\ell(R/I^{[q]}) = e_{HK}(I)q^d + O(q^{d-1}).$$
When $I = \m$, we set $e_{HK}(\m) = e_{HK}(R)$. We use $\ell(\quad)$ to denote the length of a module,
and $\mu(\quad)$ to denote the minimal number of generators of a module throughout this paper.

Later Huneke, McDermott and Monsky \cite{HMM} extended this result in the case $R$ is integrally closed and
excellent with perfect residue field
to show there is a constant $\beta$ such that
$$\ell(R/I^{[q]}) = e_{HK}(I)q^d + \beta q^{d-1}+ O(q^{d-2}).$$
See \cite{HoY} for further refinements.

The number $e_{HK}(R)$ is called the Hilbert-Kunz multiplicity of $R$.
If $R$ is regular it is easy to see that $e_{HK}(R) = 1$.
The converse is true if $R$ is unmixed, but this is highly non-obvious. In fact, it was not until
2000 that Watanabe and Yoshida \cite{WY1} proved this fact. A somewhat simpler proof can be found in \cite{HY}.

In this paper we are interested in giving lower bounds on the Hilbert-Kunz multiplicity.
Let $e$ denote the usual Hilbert-Samuel multiplicity of
$R$. It is not hard to prove the following bounds:

$$\text{max}\{\frac {e}{d!},\, 1\}\leq e_{HK}(R)\leq e.$$
The last inequality, for example, follows at once from Lech's Lemma \cite[11.2.10]{SH}.

A natural way to think about the Hilbert-Kunz multiplicity is that as it nears $1$,
the singularities of the ring should be better.
Blickle and Enescu were able to prove an explicit version of this principle:

\medskip
\begin{theorem}[Blickle-Enescu, 2004 \cite{BE}]\label{b-e} Let $(R,\m,k)$ be an unmixed local ring of
prime characteristic $p > 0$.
\newline 1) If $e_{HK}(R) < 1 + \frac {1}{d!}$, then $R$ is Cohen-Macaulay and F-rational.
\newline 2) If $e_{HK}(R) < 1 + \frac {1}{p^dd!}$, then $R$ is regular.
\end{theorem}

A Noetherian local ring $R$ of prime characteristic $p$ is said to be \it F-rational \rm if ideals generated by
systems of parameters are tightly closed. For details concerning the theory of tight
closure and its relationship to the Hilbert-Kunz multiplicity, we refer to \cite{HH} and \cite{H}.
We will not be directly using tight closure in this paper, except at the very end of the paper.

The second result in Theorem \ref{b-e} is unsatisfactory due to its dependence on $p$.
Blickle and Enescu \cite{BE} raised the question of whether
or not the Hilbert-Kunz multiplicity is uniformly bounded away from $1$ for unmixed local rings
which are not regular. Specifically, the following was asked:

\smallskip

\begin{question}\label{question} Is there a constant $\epsilon(d)$, depending only on the dimension $d$ of $R$, such that if
$R$ is an unmixed local ring which is not regular, then $e_{HK}(R) -1 \geq \epsilon(d)$?
\end{question}
\smallskip

As a further question, Watanabe and Yoshida asked that if one can determine the infinmum of
Hilbert-Kunz multiplicities over all unmixed local Noetherian rings of characteristic $p$ and
dimension $d$ which are not regular, and if that infinmum is attained, to classify those rings which have the lowest value.

Aberbach and Enescu gave a positive answer to Question \ref{question} in \cite{AE}. Our main result, Theorem~\ref{mainthm},
improves upon their bound.

One could extend these questions to ask whether or not the Hilbert-Kunz multiplicities of
rings of fixed characteristic and dimension are discrete within any bounded region, or are
there limit points?

Question \ref{question} was made even more explicit in \cite{WY3}, where the following conjecture is posed:

\begin{conj} Let $d \geq 1$ be an integer and $p > 2$ be a prime number. Set
$A_{p,d} :=K[[X_0,X_1,...,X_d]]/(X_0^2 +\ldots +X_d^2)$, where $K$ is the algebraic closure of the field with $p$
elements.  Let $(R, \m, K)$ be a $d$-dimensional
unmixed local ring. Then the
following two statements hold.
(1) If $R$ is not regular, then $e_{HK}(R) \geq  e_{HK}(A_{p,d}) \geq  1+ \frac{c_d}{d!}$.
(2) If $e_{HK}(R) = e_{HK}(A_{p,d})$, then $\widehat{R}\cong A_{p,d}$.
\end{conj}

In this conjecture, $c_d$ is defined by the equation,
$$\mathrm{sec}(x) + \mathrm{tan}(x) = \sum_{d=0}^{\infty} \frac {c_d}{d!}x^d.$$
Monsky proved that $\mathrm{lim}_{p\rightarrow \infty}  e_{HK}(A_{p,d}) = 1 + \frac{c_d}{d!}$.
To get a feeling for what the conjecture says for small dimension, the coefficients $\frac{c_d}{d!}$
are as follows: $\frac{c_2}{2!} = \frac{1}{2}$, $\frac{c_3}{3!} = \frac{1}{3}$,
$\frac{c_4}{4!} = \frac{5}{24}$, and $\frac{c_5}{5!} = \frac{2}{15}$.
Enescu and Shimomoto prove the conjecture for complete intersections in \cite{ES}.

In \cite{WY3} and \cite{WY4} both the problem of lower bounds and classification are solved in dimensions at most four. Moreover
in low dimension they showed that there is a minimal value for the Hilbert-Kunz multiplicity for non-regular rings,
and rings which take this value are classified.
For example, they proved
that if  $(R,\m,k)$ is a two-dimensional Cohen-Macaulay local ring of prime positive characteristic $p$, algebraically closed
residue field,  and multiplicity $e$
then $e_{HK}(R)\geq \frac {e+1}{2}$ with equality if and only if
the associated graded ring of $R$ is isomorphic
to the $e$th Veronese subring of $k[x,y]$.
They also proved that
if  $(R,\m,k)$ is a three-dimensional unmixed local ring of positive prime characteristic $p$ which is not regular,
then $e_{HK}(R) \geq \frac {4}{3}.$ Furthermore,
if $k$ is algebraically closed of characteristic not $2$, then equality occurs
if and only if  $R$ is analytically isomorphic with $k[[x,y,z,w]]/(x^2+y^2+z^2+w^2)$.

Using an intricate and beautiful argument, Aberbach and Enescu solved the basic problem of bounds independent of the
characteristic in all dimensions:

\begin{theorem}[Aberbach-Enescu]\label{a-ebound} Let $(R,\m,k)$ be an unmixed ring of characteristic
$p$ and dimension $d\geq 2$.
If $R$ is not regular, then
$$e_{HK}(R)\geq 1 + \frac{1}{d(d!(d-1)+1)^d}.$$
\end{theorem}

Our main results in this paper build from the work of Aberbach and Enescu.\footnote{As this paper was being completed,
Aberbach and Enescu posted on the ArXiv a new paper with better bounds that their previous work. We will compare these bounds to ours in the last
section; our bounds given in Theorem \ref{mainthmintro} are better in general. See Remark \ref{remarka-e}.}
In the next section  we give a bound in the Gorenstein case which
is often better than bounds in \cite{AE}. Our improvement comes  by introducing the
F-signature into the proofs, and consideration of a certain dual sequence.  However, our main theorem, in Section 3,
is an improvement of Theorem~\ref{a-ebound}:

\begin{theorem}\label{mainthmintro} Let $(R,\m,k)$ be a Noetherian local unmixed ring of prime characteristic $p$ with infinite perfect residue field and dimension $d\geq 2$.
Let $\boldsymbol{x}$ be a minimal reduction of $\m$, and let $\mu$ be the maximal number of minimal generators of any ideal in
$R/(\boldsymbol{x})$. Let $t$ be the largest integer such that $\overline{\m^t}$ is not contained in $(\boldsymbol{x})$.
If $R$ is not regular, then
$$e_{HK}(R)\geq 1 +  (\text{min}\{\frac{1}{d!}, (\frac{\mu}{e-\mu}) \cdot \frac{1}{(\lceil\frac{d}{t}\rceil)^d}\}).$$
\end{theorem}

Here and throughout the paper, by $\overline{J}$ we denote the integral closure of an ideal $J$.
Though the bound in Theorem \ref{mainthmintro}  is a substantial improvement on Theorem \ref{a-ebound}, it needs to be pointed out that our bound is still probably
far from best, as suggested by the conjecture of Watanabe and Yoshida above.
However, the method we use has some interest in its own right, and may be useful in other contexts.
For instance in Proposition \ref{Wat}, we use these ideas to give an affirmative answer to an old conjecture
of Watanabe in many cases.

\vskip.3truein

\section{New Estimates}

\bigskip

In this section we give new bounds  on the Hilbert-Kunz multiplicity which depend upon the F-signature.
We first recall that a Noetherian ring $R$ of prime characteristic $p$ is said to be \it F-finite \rm if
$R$ is a finitely generated $R$-module via the Frobenius.

\begin{definition} Let $(R,\m,k)$ be a Noetherian local ring of prime charactersitic $p$ and dimension $d$. Assume that
$R$ is F-finite, and reduced. For $q = p^e$, write $R^{1/q} \cong R^{a_q}\oplus M$, where $M$ has no free summands.
Set $\alpha(R) = \text{log}_p [k : k^p]$. Consider the sequence $\{\frac{a_q}{q^{d+\alpha(R)}}\}$.
We denote by $s_{-}(R)$ and $s^+(R)$ the liminf and
limsup respectively of the above sequence as $q \rightarrow \infty$. If $s_{-}(R) = s^+(R)$, then
the limit, denoted by $s(R)$, is called the F-signature of R.
\end{definition}

Throughout the rest of the paper we will  write $s$ for $s(R)$ if there is no ambiguity about the ring $R$.

In very recent and striking work, Kevin Tucker \cite{T} has shown that the F-signature exists in general. In our main technical results the ring will be Gorenstein, in such case the existence of $F$-signature is known and easy to prove.
For the statement of our first result, Theorem \ref{T:ee}, it is worth noting that the F-signature is always at
most $1$, and is equal to $1$ if and only if the ring is regular \cite[Corollary 16]{HL}.
Thus the fraction appearing in this theorem is always at least $1$, and is only exactly $1$
when the ring is regular.

\begin{theorem} \label{T:ee}
Suppose that  $(R,\m,k)$ is a reduced F-finite Gorenstein local ring of prime characteristic $p$ with
infinite perfect residue field.  Let
$e$ denote the multiplicity of $R$, and let $s$ denote the F-signature of $R$. We choose a minimal reduction $\boldsymbol{x}$ of the maximal ideal.
For any ideal $I$ of $R$ such that $I\supseteq (\boldsymbol{x}),$ let
$\mu=\mu(I/(\boldsymbol{x})),$ then $$e_{HK}(R)\geq \frac{e-s\mu}{e-\mu} = 1 + \frac{\mu}{e-\mu}(1-s).$$
\end{theorem}

\begin{proof}
We write $e_{HK}$ for $e_{HK}(R)$. Let $F=R^{\mu(R^{1/q})}$. In the proof of this theorem
we use the notation $M^* = \text{Hom}_R(M,R)$. Note that $(R^{1/q})^*\cong R^{1/q}$ since $R$ is Gorenstein: the dual
of $R^{1/q}$ is isomorphic to the canonical module of $R^{1/q}$, and since $R^{1/q}$ is also
Gorenstein, this canonical module is isomorphic to the ring. Hence there is a short exact sequence
$0\to N\to F\to (R^{1/q})^*\to0.$
Its dual is exact since $(R^{1/q})^*$ is maximal Cohen-Macaulay and $R$ is Gorenstein:
$$0\to R^{1/q}\to F^*\to N^*\to 0$$
(see  \cite[Theorem 3.3.10(c)]{BH}).
We now write $R^{1/q}\cong R^{a_q}\oplus M$, where $M$ has no free summand. Note that
this means $a_q$ copies of $R$ split from the above short exact sequence, giving rise to
an exact sequence,
$$0\to M\to G^*\to N^*\to 0$$
where $G = R^{\mu(R^{1/q})-a_q}$.
Observe that the image of $M$ is contained in $\m G^*$ since $M$ has no free summand
(if not, then dualizing would give a free summand of $M^*$, and hence of $M^{**}\cong M$).

Since $\mu(F)=\ell(R/\m^{[q]}) = e_{HK}q^d+O(q^{d-1}),$
we can write $G^* = R^{e_{HK}q^d-a_q+d_q}$, where $d_q = O(q^{d-1})$.  Since $N$ is a
maximal Cohen-Macaulay module (MCM), so is $N^*$ by \cite[Theorem 3.3.10(d)]{BH}.
Then $\text{Tor}_1^R(N^*,R/(\boldsymbol{x}))=0.$ Therefore, we have an injection
$$\frac{M}{\boldsymbol{x}M} \hookrightarrow \frac{G^*}{\boldsymbol{x}G^*},$$
Then $$\frac{IR^{1/q}}{\boldsymbol{x}R^{1/q}} = \frac{IR^{a_q}}{\boldsymbol{x}R^{a_q}}\oplus \frac{IM}{\boldsymbol{x}M}
\hookrightarrow \frac{IR^{a_q}}{\boldsymbol{x}R^{a_q}} \oplus \frac{(\m I+\boldsymbol{x})G}{\boldsymbol{x}G}.$$
Computing the length, we see that
\begin{align*}
&\ell(IR^{1/q}/\boldsymbol{x}R^{1/q})=\ell(I^{[q]}R/\boldsymbol{x}^{[q]}R)\\ =&\ell(R/\boldsymbol{x}^{[q]}R)- \ell(R/I^{[q]})\\
\leqslant& a_q\ell(I/(\boldsymbol{x})) +\ell((\m I+(\boldsymbol{x}))/(\boldsymbol{x}))(e_{HK}q^d-a_q+d_q).
\end{align*}
Since $R$ is Cohen-Macaulay, we obtain that
$$\ell(R/\boldsymbol{x}^{[q]}R) = eq^d\leq \ell(R/I^{[q]})+ a_q\ell(I/(\boldsymbol{x})) +\ell((\m I+(\boldsymbol{x}))/(\boldsymbol{x}))(e_{HK}q^d-a_q+d_q).$$

Let $I\subset J_1\subset J_2\subset \cdots \subset R$ be a cyclic filtration of ideals of $R$ such that $J_{i+1}=J_i+(s_i)$ and
such that $J_{i+1}/J_i\cong k$.
Since there are surjections $R/\m^{[q]}\twoheadrightarrow J^{[q]}_{i+1}/J^{[q]}_i,$ we see that
$\ell(R/I^{[q]})\leqslant \ell(R/\m^{[q]})\cdot \ell(R/I).$
Therefore, taking the $q$th Frobenius of the filtration of $I\subset R,$ we get
\begin{align*}
eq^d&\leqslant \ell(R/I^{[q]})+a_q\ell(I/(\boldsymbol{x}))+\ell(\frac{\m I+(\boldsymbol{x})}{(\boldsymbol{x})})(e_{HK}q^d-a_q+d_q)\\
&\leqslant \ell(R/\m^{[q]})\ell(R/I)+a_q\mu+e_{HK}q^d\ell(\frac{mI+(\boldsymbol{x})}{(\boldsymbol{x})})+d_q\ell(\frac{mI+(\boldsymbol{x})}{(\boldsymbol{x})}).\\
\end{align*}
(Note that $a_q\mu = a_q\ell(I/(\m I + (\boldsymbol{x}))$.)
Dividing by $q^d$ and taking limits as $q\rightarrow \infty$, and observing that $\text{lim}\frac{a_q}{q^d} = s$ and
$\ell(R/(\boldsymbol{x})) = e$,  we see that
$e\leqslant e_{HK}(e-\mu)+s\mu,$ hence \[\frac{e-s\mu}{e-\mu}\leqslant e_{HK}.\qedhere\]
\end{proof}

This result can be used to recover and sometimes improve estimates in \cite{AE}. For example,
compare the following to \cite[Cor. 3.7]{AE}.

\begin{corollary}\label{bound} Let  $(R,\m,k)$ be a reduced F-finite Gorenstein local ring of prime characteristic $p$ with
infinite perfect residue field.
Then $$e_{HK}(R)\geq \frac{e-s(v-d)}{e-v+d},$$ where $v$ is the embedding dimension of $R$, i.e., the number of
minimal generators of $\m$.  If $R$ is not F-regular, then $$e_{HK}(R)\geq \frac{e}{e-v+d},$$
\end{corollary}

\begin{proof}  The proof of the first statement follows immediately from Theorem~\ref{T:ee} by applying this theorem to
the ideal $I = \m$. The second statement holds since if $R$ is not F-regular, then $s= 0$ \cite{HL}
(the converse is also true, see \cite{AL}).
\end{proof}

\begin{remark}\label{F-sing}{\rm If $R$ is not F-rational  (or even not strongly F-regular), then $s(R) = 0$, and Corollary \ref{bound}, already in \cite{AE}
in this case, gives a very good uniform bound away from $1$ for the Hilbert-Kunz multiplicity. In fact, when $R$ is Gorenstein, $s(R)>0$ if and only if $R$ is $F$-rational (equivalently in this case, strongly $F$-regular, weakly $F$-regular or $F$-regular).

In general, even in the F-rational case,
we would have a very good estimate bounding the Hilbert-Kunz multiplicity away from $1$, if we could bound $s$ below $1$
uniformly. In fact the problem of bounding the F-signature below $1$ and that of bounding the Hilbert-Kunz multiplicity above $1$ are
in some sense the same problem. In \cite[Proposition 14]{HL}, the following upper bound is given:
$$(e-1)(1-s) \geq e_{HK}(R) -1.$$ Combining this with Theorem \ref{T:ee} and rewriting gives:
$$\frac {\mu}{e-\mu}\leq \frac {e_{HK}(R)-1}{1-s}\leq e-1.$$
This means that the ratio in the middle of the two terms we are interested in are trapped between numbers depending on the usual
multiplicity of the ring and the number $\mu$. Note that equality holds between the top and bottom terms in the inequality if
and only if $\mu = e-1$ if and only if $R$ has minimal multiplicity.}
\end{remark}

\vskip.3truein

\section{Chains of Integrally Closed Ideals}
\bigskip
We begin this section with a seemingly unrelated result. The idea behind this is found in the
thesis of Choi \cite{Ch}. Note that we do not assume $R$ to be of prime characteristic.

\begin{proposition} \label{ranklemma} Let $(R,\m)$ be a local Noetherian domain, and
let $I=(J,u)$ where $J$  is an integrally closed $\mathfrak{m}$-primary ideal of $R$ and $u\in J:\m$.
If $M$ is a finitely generated torsion-free $R$-module, then $$\ell(IM/JM)\geq {\mathrm{rank\;}} M. $$
\end{proposition}

\begin{proof}
Set $N=(JM:_Mu)$. Since $\displaystyle{\frac{M}{N}\cong \frac{(J,u)M}{JM}}$ and $\mathfrak{m}M\subseteq N$, we can write $M=N+N'$ with $\displaystyle{\mu(N')=\ell\left(\frac{IM}{JM}\right)}$.
Thus it suffices to prove $\mu(N') \geq {\mathrm{rank\;}}(M)$. Since $u(M/N')\subseteq J(M/N')$, it follows from the
determinantal trick \cite[2.1.8]{SH} that there is an element $ r = u^n+j_{1} \cdot u^{n-1}+\cdots + j_{n}$
with $j_{i}\in J^{i}$ for all $i$ such that
$rM \subseteq N'$.
Observe that $r \neq 0$ since $J$ is integrally closed and $u\notin J$. Since
$M_r = N'_r$, this implies that   $\mu(N')\geq {\mathrm{rank\;}}(N')={\mathrm{rank\;}}(M)$.
\end{proof}

Given two ideals $I$ and $J$ with $J \subseteq I$, $\overline{\ell}(I/J)$ will denote the length of the longest chain of integrally closed ideals between $J$ and $I$.

\begin{corollary} \label{rankcor} Let $(R,\m)$ be a Noetherian local domain.  Let $J$ be an integrally closed $\mathfrak{m}$-primary ideal of $R$ and let $I$ be an ideal containing $J$.
If $M$ is a finitely generated torsion-free $R$-module, then $$\ell(IM/JM)\geq \overline{\ell}(I/J) \cdot {\mathrm{rank\;}}(M).$$
\end{corollary}

\begin{proof} Set $n = \overline{\ell}(I/J)$. Then  there is a chain of ideals $$J=K_0\subset K_1\subset \ldots \subset K_{n-1}\subset K_{n}\subset I$$ with $\overline{K_{i}}=K_{i}$ for all $i$. Then
$$\ell\left(IM/JM\right) \geq  \sum_{j=0}^{n} \ell(K_{j+1}M/K_{j}M) \geq \sum_{j=0}^{n} \ell((K_{j},u_{j})M/K_{j}M)$$
for some $u_{j}\in K_{j+1} \cap \Soc(K_{j})$. Thus the result follows from Proposition \ref{ranklemma}.
\end{proof}

In \cite[2.17]{WY1}, the following conjectures were raised:

\smallskip

{\it  Let $A$ be a Cohen-Macaulay local ring of characteristic
$p > 0$. Then for any $\m$-primary ideal $I$, we have
   (1) $e_{HK}(I) \geq \ell(A/I).$
   (2) If pd$_A(A/I) < \infty$, then $e_{HK}(I) = \ell(A/I)$.}

\smallskip

Although this conjecture has turned out to not be true (for example, see the paper of Kurano \cite{Ku}), our next result shows that
(1) is true for many $\m$-primary ideals:

\begin{proposition}\label{Wat} Assume $(R,\m)$ is an excellent normal ring of prime characteristic $p$ with an algebraically closed residue field. If $I$ is an integrally closed $\mathfrak{m}$-primary ideal of $R$, then
$$e_{HK}(I) \geq \ell(R/I)+e_{HK}(R)-1.$$
If $I$ is an $\m$-primary ideal such that there is an integrally closed ideal $K\subset I$ with
$\ell(I/K) = 1$, then
$$e_{HK}(I) \geq \ell(R/I).$$
\end{proposition}

\begin{proof} We prove the first statement. From \cite[2.1]{W}, we obtain that there is a composition series $\displaystyle{I=I_0\subset I_1\subset \cdots \subset I_{l-1}=\mathfrak{m}\subset R}$
such that $\overline{I}_i=I_i$ for all $i$.
It follows from Corollary \ref{rankcor} that:
$$\ell \left(\frac{\m R^{1/q}}{IR^{1/q}}\right)\geq \ell(\m/I) \cdot {\mathrm{rank\;}}(R^{1/q}). $$
Therefore,
$$\ell \left(\frac{R^{1/q}}{IR^{1/q}}\right)\geq \ell(\m/I) \cdot {\mathrm{rank\;}}(R^{1/q})+ \ell \left(\frac{R^{1/q}}{\m R^{1/q}}\right).$$
Dividing by $q^d,$ we get \[e_{HK}(I)\geq \ell(R/I)+e_{HK}(R)-1.\]

We prove the second statement. Note that $\ell(R/K^{[q]}) = \ell(R/I^{[q]}) + \ell(I^{[q]}/K^{[q]}).$
On the other hand, $\ell(I^{[q]}/K^{[q]})\leq \ell(R/\m^{[q]})$. Hence $\ell(R/I^{[q]}) + \ell(R/\m^{[q]})\geq \ell(R/K^{[q]})$.
Dividing by $q^d$ and taking limits, we obtain that
$$e_{HK}(I)+e_{HK}(R)\geq e_{HK}(K) \geq \ell(R/K) + e_{HK}(R) - 1$$ by the first
part of the theorem. The result follows.
\end{proof}

In \cite{WY1} (see Theorem 2.7), Watanabe and Yoshida raised the following question:
If  $(R,\m)$ is an unmixed local ring of prime characteristic $p$, then is it true that for every $\m$-primary integrally closed
ideal $I$, that $e_{HK}(I)\geq \ell(R/I)$? Furthermore, if equality holds for some $I$, does it force $R$ to be regular.

Obviously, the first part of this question is answered in the positive by Proposition~\ref{Wat} above provided $R$ is in addition
excellent normal with algebraically closed residue field. We are grateful to the referee for providing the following extension, which
also answers positively the second question:

\begin{corollary}\label{ref} Let $(R,\m)$ be an excellent local analytically irreducible domain of prime characteristic $p$ with an algebraically closed residue field. If $I$ is an integrally closed
$\m$-primary ideal then $e_{HK}(I)\geq \ell(R/I)$, with equality if and only if $R$ is regular.
\end{corollary}

\begin{proof} (due to the anonymous referee) If $R$ is normal, the inequality is given in Proposition~\ref{Wat}. The normalization
$S$ of $R$ is an excellent normal local domain, and it is a finitely generated $R$-module. Let $\n$ be the maximal ideal of $S$. Then
$S/\n = R/\m$. In particular, the length of an $S$-module is the same as the length of the same module viewed as an $R$-module, so
in computing length we don't need to specify which ring we are working over. Set $L = \overline{IS}$. Then $L$ is an integrally closed ideal of $S$ and $L\cap R = I$. Thus we have that
$e_{HK}(I)  = e_{HK}(IS) \geq e_{HK}(L) \geq \ell(S/L)$ by Proposition~\ref{Wat}, and $\ell(S/L)\geq \ell(R/I)$, since $R/I\subset S/L$.

For the last claim, suppose first that $R$ is normal but not regular. Then $e_{HK}(I) > \ell(R/I)$ by Proposition~\ref{Wat}. If $R$ is not normal
and we have equality, then equality must occur in all the inequalities in the paragraph above. Then $S$ is regular, and $e_{HK}(IS) = e_{HK}(L)$.
By basic facts of tight closure theory, see \cite{HH}, it follows that $IS = L$. Moreover, from the same set of inequalities, we must have that $R/I = S/L$. Thus $S = R +IS$ as an
$R$-module and Nakayama's lemma gives that $R = S$ is regular.
\end{proof}
\vskip.3truein

\section{Main Result}

\bigskip

Aberbach and Enescu improved the Blickle-Enescu Theorem~\ref{b-e} (2) by proving:

\begin{theorem}[Aberbach-Enescu, 2008, \cite{AE}]\label{a-e2} Let $(R,\m,k)$ be an unmixed ring of prime characteristic $p$ and dimension $d\geq 2$. If
$e_{HK}(R)\leq 1 + \text{max}\{\frac {1}{d!},\, \frac {1}{e}\}$, then $R$ is Gorenstein
and F-regular.
\end{theorem}
\medskip

In particular, for the purposes of bounding the Hilbert-Kunz multiplicity away from $1$, we may assume that $R$ is Gorenstein and F-regular.
Recall that F-regular means that every ideal in every localization of $R$ is tightly closed (see \cite{H} for the definition
of tight closure and basic properties. We will use that F-regular rings are Cohen-Macaulay and normal.)

We will use the following fact, which appears in \cite[Theorem 2.7]{WY1}, for example:

\begin{theorem}\label{finiteext} Let $(R,\m)\rightarrow (S,\n)$ be a module-finite extension of Noetherian local domains.
Then for every $\m$-primary ideal $I$ of $R$,
$$e_{HK}(I)\cdot [Q(S) : Q(R)] =e_{HK}(IS)\cdot [S/\n : R/\m]$$
where $Q(\quad)$ denotes the field of fractions of a domain.
\end{theorem}

We also need the following discussion, which is found in \cite[Remark 4.3]{AE}.

\begin{discussion}\label{disc}
{\rm Let $(R,\m)$ be a local domain. Let $z \in \m$, and let $n$ be a positive integer.
Let $y\in R^+$ be any root of $f(X)=X^n-z$. We call $S=R[y]$ a radical extension for the pair $R$ and $z$.

Whenever $S$ is radical for $R$ and $z$, then $b := [Q(S) : Q(R)] \leq n$. Assume
also that $R$ is normal and $z$ is a minimal generator of $\m$. Then in fact, $b = n$. Moreover,
$S=R[y]\cong R[X]/(X^n-z)$ so that if $R$ is Cohen-Macaulay (respectively Gorenstein) then so is $S$. Also, $S$ is local with maximal ideal $(\mathfrak{m}, y)$,
because $\m$ is certainly in the Jacobson radical of $S$ as $S$ is a finite extension of $R$, and
$S/\m S\cong k[X]/(X^n)$ where $k$ is the residue field of $R$.}
\end{discussion}

\begin{proposition} \label{intprop} Assume $(R,\m)$ is Cohen-Macaulay and normal, and let $x\in \mathfrak{m}-\mathfrak{m}^2$ be part of a minimal reduction
of $\mathfrak{m}$. Let $S=R[y]$ with $y^n=x$. Then $\m S+(y^i)$ is integrally closed for any nonnegative integer $i$.
\end{proposition}
\begin{proof} Assume the claim is wrong. We may assume that $i\leq n$. Since $S/\m S\cong k[X]/(X^n)$, the ideals in $S/\m S$ are linearly ordered, and thus  $y^{i-1}\in \overline{\mathfrak{m}S+(y^i)}$. By the valuative criterion \cite[6.8.3]{SH},  $\overline{\mathfrak{m}S+(y^i)}=\bigcap (\mathfrak{m}S+(y^i))V\cap R$ where $(V,v)$ runs over all discrete valuation domains of the field of fractions of $R$
centered on the maximal ideal of $R$.
Therefore $$v(y^{i-1})\geq v(\mathfrak{m}S+(y^i))\geq \min \{v(\mathfrak{m}S),v(y^i)\},$$ which is equivalent to $$(i-1)v(y)\geq \min\{v(\mathfrak{m}S),iv(y)\}.$$
Hence $(i-1)v(y)\geq v(\mathfrak{m}S),$ that is, $\displaystyle{y^{i-1}\in \overline{\mathfrak{m}S}}$. By \cite[6.8.12]{SH},
this is equivalent to $$c\cdot (y^{i-1})^{nl}=c\cdot (x^{i-1})^l \in \mathfrak{m}^{nl}S,$$ for some $c\in R$ and for all $l\gg 0$.
Since $R$ is normal, Discussion \ref{disc}  shows that $S$ is a free $R$-module. Therefore
$$c\cdot (x^{i-1})^l \in \mathfrak{m}^{nl}.$$
Thus $\displaystyle{x^{i-1} \in \overline{\mathfrak{m}^{n}}}$.
Now set $j=i-1$ and choose a minimal reduction $(x,x_2,\ldots,x_d)$ of $\mathfrak{m}$.
Then $\displaystyle{x^j\in \overline{(x,x_2,\ldots, x_d)^n}}$ and hence $\displaystyle{x^{jl}\in (x,x_2,\ldots,x_d)^{n(l-t)}}$
for some $t$ and for all $l\gg 0$. Since $x,x_2,\ldots,x_d$ is a system of parameters, this is not possible.
\end{proof}

\begin{corollary}\label{RS}
Assume that $(R,\m,k)$ is a Cohen-Macaulay normal local ring of prime characteristic $p$ which is F-finite with
infinite perfect residue field. Let $x\in \mathfrak{m}-\mathfrak{m}^2$
be part of a minimal reduction of $\mathfrak{m}$ and let $S=R[y]$ with $y^n=x.$
Then $$e_{HK}(R)-1\geq \frac{e_{HK}(S)-1}{n}.$$
\end{corollary}

\begin{proof}
We use Proposition \ref{intprop} and Corollary \ref{rankcor}. By  Proposition \ref{intprop}, there is a chain of integrally closed
ideals, $\m S\subset (\m S + (y^{n-1}))\subset \ldots \subset (\m S + (y)),$ and so using the torsion-free $S$-module $S^{1/q}$ and applying  Corollary \ref{rankcor} to the pair of integrally closed ideals $\m S\subset (\m S, y)$,
yields that $\ell(S^{1/q}/\m S^{1/q})\geq (n-1)\text{rank}(S^{1/q}) + \ell(S^{1/q}/(\m S, y)S^{1/q})$. Dividing by $q^d$ and taking limits gives  that $$e_{HK}(\m S)\geq n +e_{HK}(S)-1$$
Moreover, $e_{HK}(\mathfrak{m}S)=
n\cdot e_{HK}(R)$ by Theorem \ref{finiteext}.
Therefore, $$n\cdot e_{HK}(R)\geq n+e_{HK}(S)-1$$ and hence the result follows.
\end{proof}

The last corollary is in some sense our main improvement upon the methods used by Aberbach and Enescu. We use their
strategy of adjoining roots of a minimal reduction until we obtain a non F-regular ring; at this point previous
estimates are good. The new work of Aberbach and Enescu \cite{AE1} which appeared in ArXiv as we were finishing this
paper gives the following result, among other estimates in lower dimension. Their method of comparing the Hilbert-Kunz
multiplicities of radical extensions seem much different from the one we developed above.
The new result of Aberbach and Enescu \cite{AE1} is:

\begin{theorem}\label{newa-e} Let $(R,\m,k)$ be a local Gorenstein F-regular ring of dimension $d \geq 2$ and Hilbert-Samuel multiplicity $e \geq 6$.
Let $k = \mu(\m) -\dim(R)$. Assume further that R is not a complete intersection.
Then if $e\geq d!+1$, then $e_{HK}(R)\geq 1+\frac{1}{d!}$. Otherwise, if $k = e-2$, then
$$e_{HK}(R)\geq 1 + 3(\frac {4}{6(\lceil\frac{d}{2}\rceil)-2})^d$$
while if $k\ne e-2$, then
$$e_{HK}(R)\geq 1 + (\frac {4}{(\lceil\frac{d}{3}\rceil)d!+4})^d(\frac{4}{(6d-16)}).$$
\end{theorem}

Our main result is the following:

\begin{theorem}\label{mainthm}
Let $(R,\m,k)$ be a Noetherian local unmixed ring of prime characteristic $p$ which is F-finite  with infinite perfect residue field and dimension $d\geq 2$.
Let $(\boldsymbol{x})$ be a minimal reduction of $\m$, and let $\mu$ be the maximal number of minimal generators of any ideal in
$R/(\boldsymbol{x})$. Let $t$ be the largest integer such that $\overline{\m^t}$ is not contained in $(\boldsymbol{x})$.
If $R$ is not regular, then
$$e_{HK}(R)\geq 1 +  (\text{min}\{\frac{1}{d!}, (\frac{\mu}{e-\mu}) \cdot \frac{1}{(\lceil\frac{d}{t}\rceil)^d}\}).$$
\end{theorem}

\begin{proof}
If $e_{HK}(R)\geq 1+ 1/d!,$ there is nothing to prove. Hence we may assume that
$e_{HK}(R)< 1+ 1/d!,$ and  then $R$ is $F$-regular and Gorenstein by \cite[3.6]{AE}. Thus we may assume that $R$ is $F$-regular and Gorenstein.

Let $(\boldsymbol{x})=(x_1,\cdots,x_d)$ be a minimal reduction of $\m,$
let $\mu=\mu(\m/(\boldsymbol{x})).$ Consider the set of overrings $S=R[x^{1/n}_1, \dots, x_{i}^{1/n}]=R_{i,n}$ which are
not $F$-regular. Choose $n$ and $i$ such that we attain
$\text{min } \{n^i: R_{i,n} \text{ is not F-regular} \}$. Set $S = R_{i,n}$.
Then by Theorem \ref{T:ee} applied to $S$ and the minimal reduction $x_1^{1/n},...,x_i^{1/n}, x_{i+1},...,x_d$ and the fact that $s(S)=0$ (see Remark \ref{F-sing}),
$$e_{HK}(S)\geq \frac{e(S)-s(S)\cdot \mu(S)}{e(S)-\mu(S)}= \frac{e(S)}{e(S)-\mu(S)}.$$
Here, we define  $\mu(S)$ to be the maximal number of minimal generators of any ideal in
$S/(x_1^{1/n},...,x_i^{1/n}, x_{i+1},..,x_d)$.

However, since $S/(x_1^{1/n},...,x_i^{1/n}, x_{i+1},..,x_d)\cong R/(\boldsymbol{x}),$ we have $e(S)=e(R)$, and $\mu(S) = \mu$. Therefore,
$$e_{HK}(S)\geq 1 + \frac{\mu}{e-\mu}.$$

Let $R_0=R,$ and for each $i\geq j\geq 1,$ let $R_j=R_{j-1}[x^{1/n}_j],$ then by Corollary \ref{RS} $$e_{HK}(R_j)-1\geq \frac{e_{HK}(R_{j-1})-1}{n}.$$

Therefore, we get $$e_{HK}(R)-1=e_{HK}(R_0)-1\geq \frac{e_{HK}(S)-1}{n^i}\geq \frac{\mu}{e-\mu}(\frac{1}{n^i}). $$

Thus it remains to prove that
$$\text{min } \{n^i: R_{i,n} \text{ is not F-regular} \}\leq (\lceil\frac{d}{t}\rceil)^d.$$

To do this we note that it suffices to prove that $A= R[x_1^{1/\lceil\frac{d}{t}\rceil},...,x_d^{1/\lceil\frac{d}{t}\rceil}]$
is not F-regular.
Set $y_i=x_{i}^{1/\lceil\frac{d}{t}\rceil}, i=1,\ldots, d$. Then a socle representative  of $A/(\boldsymbol{x})$ is:
$$u \cdot y_{1}^{\lceil\frac{d}{t}\rceil-1} \ldots y_{d}^{\lceil\frac{d}{t}\rceil-1},$$
where $u$ generates the socle of $(\boldsymbol{x}R)$. By our assumption on $t$, we may in addition assume that $u\in\overline{\m^t}$.
Let $v$ be any discrete valuation centered on the maximal ideal of $S$.
Then we claim that
$$v(u) + (((\lceil\frac{d}{t}\rceil-1)d)/\lceil\frac{d}{t}\rceil)v(\m) \geq dv(\m).$$
Since $v(u)\geq tv(\m)$, it suffices to prove that
$$t + (((\lceil\frac{d}{t}\rceil-1)d)/\lceil\frac{d}{t}\rceil)\geq d.$$
This simplifies to $t(\lceil\frac{d}{t}\rceil)\geq d$, which is true.

It follows that $u \cdot y_{1}^{\lceil\frac{d}{t}\rceil-1} \ldots y_{d}^{\lceil\frac{d}{t}\rceil-1}\in
\overline{(\m S)^d}$.
By the tight closure Brian\c con-Skoda theorem \cite[Section 5]{HH} this implies that $(x_1^{1/\lceil\frac{d}{t}\rceil},...,x_d^{1/\lceil\frac{d}{t}\rceil})A$
is not tightly closed, which gives the desired conclusion.
\end{proof}

\begin{remark}{\rm Aberbach and Enescu also state \cite[Page 15]{AE} an inequality in their
text which is closer to the one we give. Let $r$ be the maximum $i$
such that $\m^i$ is not contained in $(\boldsymbol{x})$. (Note that $r\leq t$.)
Let $i_0$ be the least integer such that $R_i = R[x_1^{1/m},...,x_i^{1/m}]$ is
not F-regular, where $m = \lceil\frac{d}{r}\rceil$. (Observe that $m\geq \lceil\frac{d}{t}\rceil$.) Then
$$e_{HK}(R) \geq 1 + (\frac{1}{e(m-1)+1)})^{i_0}(\frac{1}{d}).$$
In this situation the estimate is closely related to our estimate above which we give  with
a multiple of  $\frac{1}{n^i}$; the main difference now is that the
estimate of Aberbach and Enescu has an extra $e^{i_0}$ in the denominator.}
\end{remark}

\smallskip

\begin{remark}\label{remarka-e}{\rm To compare this theorem to the new result of Aberbach and Enescu, note that since we
may assume that $R$ is Gorenstein, $t\geq 2$ unless $R$ is a hypersurface of degree $2$, in which case
good bounds are known by \cite{ES}. Moreover, in the notation of Theorem \ref{newa-e}, if $k\ne e-2$, then
$t\geq 3$, and in this case we have essentially removed a factor of $(d!)^d$ from the denominator of the
estimate given in Theorem \ref{newa-e}.}
\end{remark}

\begin{remark}{\rm We can also improve the statements by bringing in the idea of the \it core \rm of the
maximal ideal, which is the intersection of all reductions of the maximal ideal. The $t$ we choose in the
statement of Theorem \ref{mainthm} can actually be chosen maximal so that $\overline{\m^t}$ is not
contained in the core of $\m$; then there will be some minimal reduction which does not contain $\overline{\m^t}$.
} \end{remark}

\begin{acknowledgement} The authors thank the referee for a careful reading of the paper and pointing out the statement
and proof of Corollary~\ref{ref}. The second and third authors gratefully acknowledge support of
the NSF during this research.
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


Portland-by-way-of-Portugal producer and multi-instrumentalist André Allen Anjos, a.k.a. RAC, is a remix artist extraordinaire. In fact, RAC originally stood for "remix artist collective," and the name referred to an international group of musicians known for transforming pre-existing songs. According to RAC.fm, the project began "with an ambitious goal to change the way things were remixed: Taking the focus away from danceability and attempting different and interesting arrangements." RAC's portfolio of such arrangements is vast — numbering in the hundreds — and there's an emphasis on indie beats, from renditions of Tegan and Sara, St. Lucia, Penguin Prison, to The Shins and beyond. Nowadays, Anjos, still under the name RAC, produces original music in addition to working his remix magic. Anjos has seen a lot of success with his breed of music; he released full-length Strangers in April, he just wrapped a fall tour peppered with sold-out dates and his work has been widely streamed — RAC has almost 4 million followers on SoundCloud.
When you write music, it's a melding of your own influences — whether it's intentional or not, it comes through.
Twenty-seven-year-old Matthew Koma has worn many hats in the music world. Originally a rocker, he snagged a publishing deal at age 16 and eventually pivoted. He began making waves in the EDM world in 2012, co-writing hit "Calling (Lose My Mind)" with Alesso and Swedish House Mafia's Sebastian Ingrosso, working again with Alesso on "Years," in which Koma was featured. The EDM achievements continue to rack up: Koma's since joined forces with artists including Nicky Romero, Hardwell and Zedd— with whom he co-wrote the Grammy-winning track "Clarity" and was featured in single "Find You." With seven Beatport #1s under his belt, Koma is known for adding hook and melody everywhere he goes and has toured alongside top artists, including Charli XCX, Ellie Goulding and LMFAO. Beyond the plethora of writing credits and collaborations, there's a lot to look forward to on the Koma solo front: He'll be releasing his own single in 2015 and is wrapping a forthcoming album.
André Allen Anjos (RAC) and Matthew Koma are no strangers. In fact, their work in the different facets of the music industry has brought them together for September-released single, "Cheap Sunglasses," of which there are seven versions — one of which (the track that lives on RAC's 2014 album, Strangers — listen to it here), has been streamed over 2 million times on Spotify.
The two met up on a rainy morning in New York City at NoHo coffee shop La Colombe. There was a lot to be done: Later that night, RAC would play a sold-out headlining show at Terminal 5, which Koma would attend — and surprise fans with an appearance, chiming in for "Cheap Sunglasses" during the encore.
Koma and Anjos take off toward their first stop, Bleecker Street Records — a classic neighborhood spot in the charming West Village. En route, they talk about what went into making "Cheap Sunglasses," a process that began three years ago when Koma wrote the song with producer Tim Pagnotta. "It's funny how ideas like that float around for a while before you find a home for them," says Koma.
Koma and Anjos arrive at Bleecker Street Records. As they browse through the racks, they talk influences. Anjos draws inspiration from Paul Simon, and, fittingly, he leaves with a copy of Simon's iconic album Graceland.
From there, the two head to Terminal 5, where Anjos — and Koma — would be taking the stage in a just few hours' time. As they hang backstage, energy and anticipation is thick in the air. "When you're inspired by something and you feel like there's something you could serve that's different from what the other person brought to the table, but it works, the puzzle clicks. It's a rare thing, but a cool experience," says Koma. As the cheers from the packed venue would later indicate, it's clear this is one of those coveted, extraordinary circumstances.


Estany Blau on Wikimapia. GPS coordinates: 42.49677, 1.62225.
The nearest object on Wikimapia: . You can also see Estany Blau on GoogleMaps and OpenStreetMaps. If you are looking for hotel near Estany Blau try this HotelsMap.


INDIANAPOLIS—The Ruth Lilly Philanthropic Foundation has awarded $7 million to strengthen program development and establish two endowed chairs at the Indiana University Center for Global Health for its AMPATH initiative in Kenya. The purpose of the gift is to address health care disparities and advance global health research, education, and health care delivery to underserved populations.
AMPATH (Academic Model Providing Access to Healthcare) is the cornerstone of the IU Center for Global Health. AMPATH was created in 2001 in response to the HIV crisis in western Kenya. It is built on a partnership between Moi Teaching and Referral Hospital and the Moi University School of Medicine in Eldoret, Kenya, and a consortium of North American academic health centers led by Indiana University School of Medicine. Working with the Kenyan Ministry of Health, AMPATH aims to address the challenges of access to health care in resource-limited communities. Serving a population of 4 million people in western Kenya, AMPATH aspires to restore patients' lives, not just their health.
The director of the IU Center for Global Health, Robert Einterz, M.D., said the Ruth Lilly Philanthropic Foundation's continuous support demonstrates their belief in AMPATH's mission to strengthen sustainable health care services in Kenya where access to health care is a daily challenge.
"This new commitment will help us to better serve Kenyan families and continue to expand our initiatives to address the critical needs for primary health care, chronic disease care, and specialty care for the entire population, leaving no one behind," he said.
The Ruth Lilly Philanthropic Foundation has been a long-time supporter of the AMPATH program, including the donation of $4 million in 2013 to help construct the four-story state-of-the-art Chandaria Cancer and Chronic Diseases Centre on the campus of Moi Teaching and Referral Hospital in Eldoret. The center will be a hub for training, research, and chronic disease care and is the largest public facility of its kind in sub-Saharan Africa. Space for training and telemedicine is still being developed; however, last year the doors opened for patient care and research.
Since 1989, Indiana University School of Medicine and Moi University School of Medicine and Moi Teaching and Referral Hospital in Eldoret, Kenya, have worked together to deliver health services, conduct health research, and develop leaders in health care for both North America and Africa. From this partnership, AMPATH was formed in 2001 creating one of Africa's largest, most comprehensive and effective HIV/AIDS management and treatment models. AMPATH is a formal partner with the United States government through a $75 million grant from USAID and has continually transitioned its successful HIV approach into a more comprehensive primary health care system. Since, other North American academic health centers joined the AMPATH consortium. Those universities include Brown University, Duke University, Icahn School of Medicine at Mount Sinai, Lehigh Valley Health Network, Providence Portland Medical Center, Purdue University, University of California-San Francisco, University of Massachusetts, University of Notre Dame, and University of Toronto.


We would like to say a BIG CONGRATULATIONS to all the winners and highly commended finalists at the eleventh Suffolk Chamber of Commerce in Lowestoft & Waveney Business Awards.
We would also like to thank all of the 2018 sponsors and everyone who has helped us to put on this event. Without them the Awards wouldn't be the success that it is.
Photos from the Gala evening coming soon!
Josh experienced various setbacks in his working life, he has shown great perseverance in overcoming these setbacks and recovered well from them and has since worked very hard in his ambition to be an engineer and in pursuing his career in this industry.
A well-deserved winner showing great initiative and sheer determination and a role model for future apprentices.
A really close final 2, but judges were impressed with the way that having changed role Julian was able to transfer those skills to his new business and is working towards tackling the scourge of the planets plastic waste.
Well done to both finalists and keep up the good work.
A very close run this year, but Access Community Trusts unique approach to their staff development regardless of previous knowledge and work experience makes them a worthy winner.
The judges were really impressed with the innovative Pakefield Postcard Auction and took particular note of how it has raised significant funds for local organisations in the community – some of which were also entrants into this category.
Access Community Trust stood out for its commitment to working with vulnerable, disadvantaged and excluded young people and for the success of its projects that it has developed and delivered locally.
It is the care and treatment they provide to their customers, which was displayed by several testimonials. They show excellent examples of going above and beyond for their customers.
ASAP has clearly built not only a successful brand but a very successful service and reputation for being one of the best providers in its field. An impressive, successful, but friendly and engaging locally-born business that has been an exemplar in providing years of commendable high quality customer service and product and after-sales support. Very well done and many congratulations to all of those involved in this remarkable achievement – and for an impressive second year running.
The judges applaud the efforts that the team are making to take their business out into the community reaching out to groups that are often hard to reach whilst continuing to grow and look after their own staff.
They wish Sara, Paul and the Total Balance family team all the best for a successful future, and hope that they can expand their current endeavors to other locations in the coming years.
Adapting to change, see's their new factory accommodating international trade and forging key alliances with leading automation brands, alongside this PCE have not only seen growth in staff but now also run an apprenticeship programme.
Great facilities giving local people the chance to get out of their houses away from television and social media and to get exercise and fresh air while learning, having fun and meeting new people.
Donnatella's has quickly established itself as one of the leading boutiques in Lowestoft and the surrounding area. Donna makes extensive use of social media to promote her business and is always happy to "go the extra mile" for her customers.
Sarah shows dedication to not only her business, clients and staff but also to her family, her aspiration to achieve her dream, build and invest in her business and with plans to grow whilst bringing up her young family is truly impressive.
PCE Automation has shown a considerable commitment to supporting the local business community of Beccles and the wider Waveney area as well as continued investment in young people through their apprenticeship scheme and connection to the New Anglia Youth Pledge. As such we are pleased to select them as our winner of the Business of the Year.


Why do you workout? Everyone has a different reason and they can all be right for each individual. A lot of us work out to stay fit. What does that mean exactly? What is your definition of fitness and are your current workout and nutrition routines getting you there? In October of 2002, Greg Glassman wrote the CrossFit Journal article "What is Fitness?" This discusses CrossFit's definition of fitness and gives standards for measuring it. Read the article and think about the three models for measuring fitness and decide if your current routine is getting you fit.
Also, note that fitness can be put on a continuum with sickness and wellness and every measure of good health will be realized by the fit people.
Fitness is even more clearly defined as in this article as "increased work capacity across broad time and modal domains." And when that is expanded over your lifetime, you have a clear definition of health.
A lot of people I know workout with no clear objective and never see any results. No measurable data that can be analyzed to see improvement over time. If you know people like that, or you are that person, let's talk about getting real results and really building a broad fitness over many physical domains.
Please read these articles and decide if you are getting fit and reaching your goals. Also, take a look at your current regimen and decide if you are getting a well rounded workout that will improve your fitness in a lot of areas. So I ask you again, what is your definition of fitness and is your current routine getting you fit?


Asus ZenFone Max M1 ZB555KL has 3 GB RAM and comes up with Android 8.0 Oreo OS.
Asus ZenFone Max M1 ZB555KL has a 5.5 inch Screen Size and with 720 x 1280 pixels.
Asus ZenFone Max M1 ZB555KL has WCDMA 2100 3G and LTE 4G connectivity.


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Metagenomi closes USD 65 million Series A financing led by Leaps by Bayer and Humboldt Fund
Metagenomi, a next generation gene editing company launched by UC Berkeley scientists in 2018, successfully completed a USD 65 million Series A financing round and emerged from stealth mode today. The Series A round was led by Leaps by Bayer and Humboldt Fund. Other investors included Sozo Ventures, Agent Capital, InCube Ventures and HOF Capital.
CRISPR-based gene editing is one of the most important discoveries in recent years, and a critical technology for next generation therapeutic development. With the ability to edit DNA, scientists can address a whole range of diseases ? but there are still hurdles to overcome such as delivery, immunogenicity, selectivity, and genomic accessibility with current technologies. To enable efficient technological solutions, Metagenomi is building a proprietary suite of CRISPR-based gene editing systems by applying computational algorithms to screen thousands of genomes from microorganisms around the world. The approach leverages big data and the science of metagenomics to quickly identify novel gene editing systems with properties that leapfrog first generation technologies. These include the capability to precisely edit genomes via single base changes, knockouts, or integrations, with lower potential for off-target effects. While in stealth mode, Metagenomi has developed a vast portfolio of these innovative gene editing systems, and it is Metagenomi's goal to improve patient access to gene editing technologies by providing safe, accurate modalities for use in therapeutic development.
"Gene editing is a paradigm shifting technology which will continue to move the needle from treatment to cure in many genetically inherited diseases and significantly improve the world around us for the better," said Dr. Juergen Eckhardt, Head of Leaps by Bayer. "Leaps by Bayer is committed to advancing new breakthroughs in gene editing and is thrilled to partner with Metagenomi in its mission to discover tomorrow's gene editing technologies."
"Metagenomi is accelerating innovation in gene editing technology to help patients with genetic diseases that are incurable today. This means developing a vast database of gene editing capabilities to enable unprecedented therapeutic approaches," said Brian C. Thomas, CEO and co-founder of Metagenomi. "Working with visionary investors, such as Leaps by Bayer and Humboldt Fund, will allow us to deliver on our promise to partners and fuel the development of our own pipeline of innovative curative medicines."
Capital raised during this round will enable Metagenomi to accelerate the expansion of its gene editing systems for potential therapeutic use. In addition, Metagenomi will advance research and preclinical validation for its own pipeline. New hires will help sustain the growth Metagenomi has seen in 2020 and carry the company through further scientific advancements. Joining Metagenomi as Chief Business Officer is Jak Knowles who will oversee exploration of partnerships with industry leaders in the cell and gene therapy space.
"With the Nobel Prize in Chemistry this year honoring CRISPR, the technology will continue to sweep through laboratories around the world and be examined for its countless potential applications," said Dr. Sebastian Bernales, General Partner of Humboldt Fund.
About Bayer and Leaps by Bayer
Bayer is a global enterprise with core competencies in the life science fields of health care and nutrition. Its products and services are designed to benefit people by supporting efforts to overcome the major challenges presented by a growing and aging global population. At the same time, the Group aims to increase its earning power and create value through innovation and growth. Bayer is committed to the principles of sustainable development, and the Bayer brand stands for trust, reliability and quality throughout the world. In fiscal 2019, the Group employed around 104,000 people and had sales of 43.5 billion euros. Capital expenditures amounted to 2.9 billion euros, R&D expenses to 5.3 billion euros. For more information, go to www.bayer.com.
Leaps by Bayer, a unit of Bayer AG, leads impact investments into solutions to some of today's biggest challenges in health and agriculture. The investment portfolio includes more than 30 companies. They are all working on potentially breakthrough technologies to overcome some specific challenges such as, e.g. regenerating lost tissue function, reducing the environmental impact of agriculture, preventing or curing cancer, and others. For more information, go to www.leaps.bayer.com
About Humboldt Fund
Humboldt Fund is a venture capital fund that supports innovative, early-stage companies by investing in the biotech revolution. At Humboldt Fund, we identify and invest in pioneers of engineering, physical sciences and life sciences to bring emerging biotech enterprises to a global market. Together we explore, discover and create new technologies that disrupt global conventions and solve the critical issues of our world, while creating value and prosperity. For a more information, please go to www.humboldtfund.com
About Metagenomi
At Metagenomi, we are accelerating innovation in cell and gene therapy with a wave of proprietary CRISPR-based systems to accurately edit DNA where current technologies cannot. Our metagenomics-powered discovery platform and analytical expertise reveal novel cellular machinery sourced from otherwise unknown organisms. We adapt and forge these naturally evolved systems into powerful therapeutic tools that can be leveraged by partners and fuel our own pipeline of potentially curative medicines. Our goal is to revolutionize gene editing and unlock its power for the benefit of patients around the world. For more information, please visit https://metagenomi.co/
scw (2020-0245)
This release may contain forward-looking statements based on current assumptions and forecasts made by Bayer management. Various known and unknown risks, uncertainties and other factors could lead to material differences between the actual future results, financial situation, development or performance of the company and the estimates given here. These factors include those discussed in Bayer's public reports which are available on the Bayer website at www.bayer.com. The company assumes no liability whatsoever to update these forward-looking statements or to conform them to future events or developments.
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text:
Deadly 'Dr Opiate' retired on bumper NHS pension which funds second home in Spain and regular trips to Australia
Dr Jane Barton is at the heart of a scandal which saw 656 patients needlessly having their lives shortened at Gosport War Memorial Hospital
Alex Matthews
21 Jun 2018, 8:45
Updated: 21 Jun 2018, 9:51
A DEADLY doctor whose blunders saw her dubbed "Dr Opiate" retired on a bumper NHS pension which funded a holiday home in Spain.
Dr Jane Barton is at the heart of a scandal which saw 656 patients needlessly having their lives shortened at Gosport War Memorial Hospital in Hampshire.
Dr Jane Barton is at the heart of a scandal which saw 656 patients needlessly having their lives shortened at Gosport War Memorial Hospital in HampshireCredit: Solent News
But while devastated families campaigned for justice during a 20 year cover up she retired to her £700,000 home with her Royal Navy commodore husband.
As the tragic failings at Gosport were finally brought to light yesterday, Barton was sunning herself in the Mediterranean.
A neighbour told reporters: "She's gone to Menorca."
While another said: "She doesn't have many friends around here.
Dr Jane Barton's £700,000 home in Gosport, HampshireCredit: Solent News
"I only know her from seeing her across the street but she's friendly.
"Both their cars have gone, so I presume they're not there. I think they will have gone away."
When she isn't visiting her holiday home in Spain, Barton is known to enjoy trips to Australia
While Down Under she goes on birdwatching trips with her husband Tim and even wrote about them for the Royal Naval Birdwatching Society journal, the Daily Mail reports.
But as Barton enjoys the comfortable trappings of life as a upper-middle class retiree, she is accused of leaving a trail of destruction behind.
While devastated families campaigned for justice during a 20 year cover up – Barton retired to her £700,000 home with her Royal Navy commodore husband.Credit: PA:Press Association
She is alleged to have prescribed deadly doses of medical heroin diamorphine to patients at Gosport.
Diamorphine was the painkiller used by Harold Shipman to murder his estimated 260 victims.
There were so many deaths on her watch that the two wards she ran became known as "the end of the line".
She was found guilty of failings in her care of 12 patients at the hospital between 1996 and 1999.
But she wasn't sacked over her conduct and instead retired from medicine after the findings.
Barton was found to be responsible for 12 deaths of "multiple instances of serious professional misconduct" by the General Medical Council in 2010.
The GMC had been told of her "brusque and indifferent" manner, "intransigence and worrying lack of insight" and "failure to recognise the limits of her professional competence".
A shocking report has revealed up to 656 people died prematurely because they were given opioid drugs at Gosport War Memorial HospitalCredit: PA:Press Association
Barton is currently the only medical professional to face disciplinary action – and public condemnation for the blunders at Gosport – but the findings of the Gosport report prove she won't be the last.
The Gosport Independent Panel investigation, first launched in 2014, examined more than one million documents.
It found hospital management, Hampshire Police, the Crown Prosecution Service, General Medical Council, and Nursing and Midwifery Council "all failed to act in ways that would have better protected patients and relatives".
The report also highlighted failings by healthcare organisations, local politicians and the coronial system.


text:
LOT SOLD.	302,500 GBP
18 October 2013 | 10:30 AM BST
Alex Branczik
European Head of Department
alex.branczik@sothebys.com
Isabelle Paagman
European Head of Private Sales
isabelle.paagman@sothebys.com
Head of Evening Sale
emma.baker@sothebys.com
alexander.williams@sothebys.com
Andrea Usai
Post Sale Services and Business Admin
andrea.usai@sothebys.com
signed and dated '91 on the overturn edge; signed, titled and dated 1991 on the overlap; signed, titled and dated 1991 on the stretcher
165 by 227.5cm.; 65 by 89 3/8 in.
PaceWildenstein, New York
Acquired directly from the above by the present owner
Nuremberg, Kunsthalle Nuremberg; Quadrat Bottrop, Josef Albers Museum; London, Hayward Gallery; Birmingham, Ikon Gallery, Bridget Riley: Paintings 1982-1992, 1992-93, p. 55, illustrated in colour
New York, PaceWildenstein, Bridget Riley: Paintings 1982 - 2000 and Early Works on Paper, 2000
New York, PaceWildenstein, Bridget Riley: Recent Paintings, 2004, p. 9, illustrated in colour
Dark Light summates over thirty years of Bridget Riley's dedicated engagement to the language of painting and the possibilities of colour. Deceptively composed from a schema of vertical stripes, the interspersion of multi-coloured truncated diagonals contained within the width of each strip imparts a visual staccato that causes the eye to ceaselessly hover over a glorious chromatic field. Belonging to the series of diagonally characterised Lozenge Paintings initiated in 1986, the present work signifies a dramatic change in artistic direction. Following the series of vertiginous striped works known as the Egyptian Paintings, the ensuing Lozenge works inverted the effect of Riley's previous output. Rather than stimulating optical effects upon the spectator, with these works Riley looked to imitate our visual experience of nature itself. Indeed, dominated by a field of dusk-blue tones and kaleidoscopically punctuated with red, yellow and emerald green, Dark Light magnificently defies the strict geometry of its composition to deliver an immersive experience akin to the twilight hour at which nature's colours are freed from the dominion of the sun.
In response to this change of direction, which has informed the entirety of her later production, Riley explains: "colours are organised on the canvas so that the eye can travel over the surface in a way parallel to the way it moves over nature. It should feel caressed and soothed, experience frictions and ruptures, glide and drift. Vision can be arrested, tripped up or pulled back in order to float free again… one moment there will be nothing to look at and the next second the canvas suddenly seems to refill, to be crowded with visual events" (Bridget Riley, 'The Pleasures of Sight (1984)' in: Exhibition Catalogue, London, Tate Britain, Bridget Riley, 2003, p. 213). As in the present work, we are immersed in the nuanced and undulating colour juxtapositions that mediate chromatic passages of differing intensities. One detects echoes, repetitions and inversions. However, any search for cogent patternation is utterly thwarted and our sight is disarmed by sheer immersive splendour. Unable to settle, our vision constantly traverses the latticed geometry of Riley's composition to impart an unstable viewing encounter.
This radical departure and inversion of effect in Riley's work can be directly linked to a renewed interest in her earliest artistic beginnings. During the very onset of her professional career Riley had studied closely the sophisticated colourist methods of Georges Seurat alongside the scrutiny of appearance versus perception articulated in the work of Paul Cézanne and Claude Monet. As a young artist, Riley made a number of studies in the Neo-Impressionist style and it was out of a clear understanding of the pioneering masters of Classical Modernism that her progression into pure optical abstraction was facilitated. In returning to her earliest preoccupation with colour and an experience of nature outside of ordered representation, Riley tackles the question laid down on the cusp of Modernity at the Fin de Siècle: how do we negotiate the disparity between the way our eye sees things and the actuality of appearance? In contrast to her previous series which looked to impact optical effects on the spectator, these works sought to emulate more closely the effect of external stimulus on the eye, to replicate the experience of seeing and absorbing the ever-changing perception of the physical world. Nonetheless, the reality of appearance is not the central inspiration or point of departure here; rather, these paintings originate from a will to weave a relationship with nature from within the language of abstraction itself. Speaking in 1992, Robert Kudielka described how this very notion was crystallised in the preparatory stages for the present work: "When she had completed the full-size cartoon for Dark Light last winter she remarked: 'Isn't it curious that nature immediately looks so much better once I've got a painting together?' That is roughly how the balance between the making of abstract art and nature should be seen" (Robert Kudielka quoting the artist in: Robert Kudielka on Bridget Riley: Essays and Interviews 1972-2003, London 2005, p. 134).
Colour: The colours in the catalogue illustration are fairly accurate, although the overall tonality is slightly more vibrant in the original. Condition: This work is in very good condition. No restoration is apparent when examined under ultraviolet light.


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text:
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	<Description>should be added by deploy script</Description>
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redpajama_set_name: RedPajamaGithub


The power of legacy lies in the power to change lives. That was the unofficial theme of this year's Legacy Award Dinner benefiting The Cooper Institute. Held at the beautiful Belo Mansion in downtown Dallas, the event was a veritable who's who of health and wellness champions.


text:
St. Herman of Alaska Orthodox Church, a parish of the Russian Orthodox Church Outside Russia (ROCOR), was originally granted mission status by the Orthodox Church in America (OCA) on May 1, 2004. That particular day's icons on the revised Julian calendar were, providentially, the "Unexpected Joy" of the Theotokos and the "Myrrh-bearing Women." In the evening of May 9, our founding members gathered together at the historic Aquia Episcopal Church's fellowship hall for a Prayer of Thanksgiving for our new mission community. At the time we did not realize that this was just the first of God's many blessings upon us through our Episcopalian friends at Aquia.
Our founding priest, Archpriest Michael Roshak, celebrated our first Divine Liturgy as a mission community at the Stafford Courthouse Community Center on May 30, 2004 (Pentecost Sunday). Since then, God the Holy Spirit has clearly engendered a true spirit of Christian fellowship and love that is still quite evident today.
On August 15, 2004, we celebrated our twelfth Divine Liturgy together at our new location, Fredericksburg Square, in downtown Fredericksburg, Virginia. We witnessed our first chrismation of a convert to Orthodoxy only two months later on October 24. On November 6, Metropolitan Herman of the OCA provided his blessing to dedicate the mission to "Christ the Savior." A mere 22 days later, we witnessed our first two baptisms.
We have worshipped at our current location, the historic Clifton Chapel in Stafford, Virginia, since January 1, 2005. At first, we had access to the property only for Saturday Vespers, but one month later Aquia church granted us full-time use of the chapel and grounds. We celebrated our first Divine Liturgy in our long-standing home on February 13, 2005 (Zacchaeus Sunday).
The historic Clifton Chapel was built in the 1850's and provided a convenient place to worship for Wide Water, Virginia, residents when poor weather rendered the roads impassible. Both sides of the Civil War ("War Between the States") occupied this chapel on a hill overlooking the Potomac River. At that time there were few trees in the vicinity, and the chapel offered an excellent observation post and a high ground for artillery to guard the Potomac River's shoreline. After the war, local families repaired the damaged church, and it continued to serve the community. In 1896 Clifton Chapel was officially dedicated as a property of Aquia Episcopal Church. Now, as the home of St. Herman of Alaska Orthodox Church, its quiet, wooded location offers a peaceful worship environment.
On November 26, 2006, our founding priest served his final Divine Liturgy with us. We were then without an assigned parish priest until September 9, 2007, when we welcomed Archpriest Michael Sekela as our permanent rector. Father Michael came out of retirement as a priest to serve the Lord yet again, relocating from Chicago, Illinois, to Stafford, Virginia, and bringing us our first matushka (or "Little Mother" in Russian), his wife Theresa. Father Michael and Matushka Theresa continued together to expand and enhance the family we had started, especially by introducing the concept of congregational singing. Under Father Michael's visionary leadership, our chapel conformed more closely to the image of a traditional Orthodox church, when we moved all the pews to the perimeter of the chapel and installed a beautiful, rustic iconostas handcrafted by Father Joseph Toroney. Through Matushka Theresa's guidance, we officially established the Sisterhood of the Myrrh-bearing Women, inspired by one of the icons on the day of our founding as an Orthodox mission (May 1, 2004).
God the Holy Trinity has continued to bless us through the friendship and generosity of Aquia Episcopal Church, which granted us occupancy of a newly-built parish hall in 2009 and, in 2010, a ten-year lease for the entire property, including Clifton Chapel, the parish hall, and children's playground.
On September 1, 2011, upon Father Michael Sekela's second and "final" retirement, we welcomed Archpriest Alexander F.C. Webster, PhD, and his wife Matushka Kathleen. Fr. Alexander retired as a U.S. Army chaplain (in the rank of colonel) in 2010, but he continues to teach as an adjunct professor at Northern Virginia Community College (comparative religion) and George Mason University (emergency management). While revitalizing our community's liturgical, educational, evangelistic, and humanitarian missions, Father Alexander has steadfastly maintained our congregational singing, now led by the angelic voices of our "liturgical singers" directed by Areti Stylianopoulos. On April 1, 2012, our very own William Laymon was tonsured a reader, set aside as a subdeacon, and ordained into the diaconate as "Alexander" by Metropolitan Jonah, former primate of the OCA. Previously, in January 2012, Matushka Joanna Laymon formed the Iconography Guild of St. Theodosia.
On December 19, 2012, the congregation met in Special Assembly and voted unanimously to petition Metopolitan Hilarion to receive the mission as a parish of the Russian Orthodox Church Outside Russia (ROCOR). On January 31, 2013, we cleared the last hurdle for Metropolitan Tikhon of the OCA to dissolve the OCA mission and free us to enter ROCOR. His Eminence Metropolitan Hilarion inspired us to begin our new ecclesial life in ROCOR as a parish under the heavenly protection of St. Herman of Alaska, Russian missionary monk sent to evangelize Alaska in the eighteenth century and the first canonized American saint of the Church. Meanwhile, we have dedicated our parish hall to Christ the Savior, the original name of the mission community. Today our growing congregation numbers approximately 40 adult members and 20 children. Prospective new members, visitors, and other seekers of the truth of the Gospel of Our Lord God and Savior Jesus Christ will always find a warm southern welcome among us.
Classes, Sermons and Special Events by +Metropolitan Jonah
The Sisterhood of Myrrhbearing Women
About St. Herman of Alaska
Jordanville Bookstore
Iconography Guild
Iconography Workshops
About Our Iconostasis
January February March April May June July August September October November December 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 (Printing Instructions)
Event and Services - This Week
Saturday, 10 / 23 January
3:00 Catechumen's Class
4:45 Confessions
5:00 Vigil and Confessions
Sunday, 11 / 24 January
9:30 Reading of the Hours
10:00 Divine Liturgy followed by Fellowship Hour
Wednesday, 14 / 27 January
6:30 Vespers
7:00 Adult Religious Education
Monthly Calendar >
For weather related cancellations, please stay tuned to the following channels for updates:
TV - WJLA channel 7, News Channel 8
Radio - WTOP (FM 103.5, 103.9, 107.7, WTOP.com)


text:
fertility tracking device
Digital fertility news: Ava launches new wearable, Glow hits security hitch
By Heather Mack 12:27 pm July 28, 2016
San Francisco, California-based Ava, a medical technology company focused on women's reproductive health, announced the launch of its fertility tracking bracelet and companion app. Named simply "Ava," the direct-to-consumer bracelet uses sensor-enabled technology to detect a woman's fertility window in a given month. The company, which was founded in Switzerland in 2014 and has raised $2.6...
Swiss startup Ava raises $2.6M for fertility tracking device, app
By Aditi Pai 07:22 am November 16, 2015
Switzerland-based Ava, which is developing a fertility tracking system, raised $2.6 million from ZKB and Swisscom as well as angel investors. The company, which has offices in Zurich and San Francisco, is working on creating a wearable device, worn only at night, that it claims can measure nine physiological parameters. The company lists seven such parameters on its website: neurological,...
Connected fertility tracking device, spirometer and 7 other digital health crowdfunding campaigns
By Aditi Pai 10:18 am October 20, 2015
There have been a number of digital health products that have launched on Indiegogo and Kickstarter in the last few weeks. These products, which all include a hardware element this time around, range from a mole -scanning smartphone peripheral, from a company that's been active in digital health for a couple years now, to a smartphone-connected spirometer. Notably, there are two women's health...


text:
Аблати́в (від  < ablātus < aufero — «відношу», «відкладаю»), відкладний відмінок, іноді орудно-місцевий відмінок — відмінок, який вказує на вихідний пункт траєкторії руху одного з учасників ситуації. Характерний для багатьох мов, слов'янським мовам не властивий.

Слова у відкладному відмінку відповідають на такі запитання: «звідки?», «від кого?», «від чого?», «чому?».

Українськими відповідниками відкладного відмінка є прийменники від, з у поєднанні з родовим відмінком.

Індоєвропейські мови

Латинська мова

Аблатив у латинській мові (ablativus) зазвичай використовується як прислівник і є ознакою дії, вираженої дієсловом. Має багато застосувань та походить від трьох відмінків праіндоєвропейської мови: аблативу, орудного та місцевого.

Інколи має назву прислівникового через своє значення: magnā (cum) celeritāte («дуже швидко», «з величезною швидкістю»).

Латинський супін в аблативі (так званий «супін II», «супін на-U») служить додатком при прикметнику в деяких стійких виразах: jūcundus cōgnitū («приємний для пізнання»), horribile dictū («страшно сказати»).

Відкладний відмінок вживається для позначення агента, знаряддя або способу дії: oppidum vallo cingitur («поселення оточене валом»), libri a discipulis leguntur («книжки читаються учнями»), docendo et cogitando («вченням і міркуванням»), igni et ferro («вогнем і залізом», «вогнем і мечем»). У цьому значенні аблатив поширений у девізах, українською він перекладається зазвичай орудним відмінком (virtute et educatione — «доблестю і просвітою»). Окрім того, його вживають для позначення місцезнаходження (in Romā — «у Римі», in horto — «у саду», in domu — «у домі», in mari — «у морі/на морі»), на українську в цьому разі він перекладається місцевим відмінком.

Слово «ребус» () за походженням являє собою форму відкладного відмінка множини іменника res («річ»), тобто «від речей», «речами».

Албанська мова
В албанській мові аблатив є п'ятим відмінком та має назву rasa rrjedhore.

Санскрит
У граматиці санскриту аблатив є п'ятим відмінком (panchami) та має схожі граматичні функції з латинським аналогом.

Іменники у цьому відмінку зазвичай відповідають предмету, від чи з якого щось з'явилось чи виникло — наприклад, patram vṛkṣāt patati («листок падає з дерева»).

Відмінок має кілька інших призначень, зокрема, коли дія відбувається з причини чи без певного суб'єкта; або з іменником, що позначає відстань чи напрям.

Вірменська мова
У західновірменській мові аблатив утворюється суфіксом -e (невизначений) чи -en (визначений).

 Mart — чоловік
 Marten — від чоловіка (визн.)
 Marte — від чоловіка (невизн.)

 Doon — будинок
 D'nen — з будинку (визн.)
 D'ne — з будинку (невизн.)

У східновірменській мові суфікс -its вживається для усіх іменників.

Mard — чоловік
Mardits — від чоловіка

Toon — будинок
T'nits — з будинку

Обидва суфікси мають походження з класичної вірменської. Західний суфікс -e походить з форми однини у класичній, східний -its відповідно з множинної, проте в обох сучасних діалектах вони не залежать від числа.

Приклади застосування (західний варіант):

 Рух від вказаного предмета.
 KAGHAKEN katsi. — Я прийшов з міста.
 ASTEGHEN heroo g'abrei. — Я жив далеко звідси.

 Вказує на агента при використанні пасиву.
 INE misht g' sirveis. — Ти завжди була любима мною.
 AZAD'CHNEREN azadetsank. — Ми були звільнені визволителями.

 У розмовній вірменській — для конструкції порівняння:
 Inch MEGHREN anoosh eh? — Що є солодшим за мед? (приказка)
 Mariam EKHPEREN b'zdig eh. — Марія менша (молодша) за свого брата.

 Керування відмінками у випадку післяйменників ().
 INE var — піді мною.
 KEZME ver — над тобою.
 ANONTSME verch — після них.
 MEZME arach — перед нами.

Слов'янські та балтійські мови
У балто-слов'янських мовах індоєвропейський аблатив злився з родовим відмінком. До праіндоєвропейських значень відкладного відмінка сходять такі значення слов'янського родового відмінка: 1) вираження проявлення якихось почуттів; 2) вираження відділення, позбуття від чогось; 3) вираз додатка при вищому ступені порівняння прикметників.

Тюркські мови

Азербайджанська мова

Аблатив в азербайджанській мові () виражений суфіксами -dan або -dən. Приклади: Ev («будинок») — evdən («від/з будинку»).

Турецька мова
Аблатив у турецькій мові () виражається суфіксами -den, -dan, -ten або -tan, де вибір зумовлюється гармонією голосних та приголосних. Приклади:
 Ev («будинок») — evden («від/з будинку»)
 At («кінь») — attan («від/з коня»)
 Ses («звук», «гучність») — sesten («від звуку», «гучності»)

У деяких випадках аблатив відповідає значенню причини, у цих ситуаціях після аблативу може йти необов'язкове слово dolayı («через», «з причини»).

 ondan dolayı — «тому, з цієї причини»
 Yüksek sesten (dolayı) rahatsız oldum — «Мені було некомфортно через гучний звук».

Деякі мови, що мають аблатив
 З індоєвропейських мов:
 Давньогрецька мова
 Вірменська мова
 Хеттська мова
 Італьські мови
 Санскрит
 Палі
 З тюркських мов:
 Азербайджанська мова
 Башкирська мова
 Казахська мова
 Кримськотатарська мова
 Татарська мова
 Турецька мова
 Чуваська мова
 З угро-фінських:
 Карельська мова
 Фінська мова
 Естонська мова
 Угорська мова
 З картвельських:
 Лазька мова
 З палеоазійських:
 Чукотська мова
 Ітельменська мова
 З індіанських мов:
 Кечуа
 Японська мова
 Квенья — вигадана мова ельфів Толкіна

Примітки

Література
 Oda Buchholz, Wilfried Fiedler: Albanische Grammatik. Verlag Enzyklopädie, Leipzig 1987, ISBN 3-324-00025-4.
 Margarete I. Ersen-Rasch: Baschkirisch. Lehrbuch für Anfänger und Fortgeschrittene. Harrassowitz, Wiesbaden 2009, ISBN 978-3-447-05730-1, S. 43.
 Irja Grönholm: Estnisch Wort für Wort (= Kauderwelsch. Bd. 55). 3. Auflage. Reise-Know-How-Verlag Rump, Bielefeld 2002, ISBN 3-89416-245-7.
 József Tompa: Kleine Ungarische Grammatik. Verlag Enzyklopädie, Leipzig 1972.
 Hans-Peter Vietze: Lehrbuch der Mongolischen Sprache. 5. durchgesehene Auflage. Verlag Enzyklopädie, Leipzig 1988, ISBN 3-324-00242-7.

Посилання
 Аблатив  // ВУЕ

 What is the ablative case?
 Ablative case in Hungarian

Латинська мова
Відмінки

redpajama_set_name: RedPajamaWikipedia


Who is who in the Slovak music business?
International professionals evaluate Slovak music.
Face to Face Meeting Artists, bands and music professionals are welcome to attend this networking event where they can meet music professionals, agents and bookers from festivals in order to establish a first contact, present their project and develop their professional network.
These short (max 5 minutes) personal meetings are the tool how to offer your bands to festivals, break into international media or ask for advice from experts.
To book your Face to Face meeting please use the Face2Face REGISTRATION FORM and highlight the requested professionals.
Stephen Budd (UK) – Is a British music industry executive based in London. He is a director of artist and producer management company Stephen Budd Music Ltd, the OneFest Festival, the Africa Express project and is the co-founder of the NH7 Weekender festivals in India. He is a TV commentator on music industry issues. In June 2017 he completed his 3-year term as co- chairman of the MMF (Music Managers Forum).
Tarmo Noormaa (EST) – is the director of Estonian Traditional Music Center and program manager of Viljandi Folk Music Festival has been organized since 1993. The purpose of the festival is to advocate and teach traditional music amongst Estonians of all ages.
Minna Huuskonen (FIN) – has over 20 years of experience on various fields of music covering all genres. 10 years passed at Finnish Music Information Centre (Fimic) working at promotional and export projects. In 2008-2018 Minna worked at Helsinki Festival programming and producing Huvila Festival Tent. She remains at the festival as an artistic advisor. Recently Minna has done mentoring work for e.g. Music Finland, Music Estonia, Sibelius Academy and Arcada university of applied sciences. She has a long membership in the marketing and export committee of The Finnish Music Foundation. She has been a jury member for WOMEX, jazzahead!, Young Nordic Jazz Comets, and Israeli World & Jazz showcase.
Anna Mašátová (CZ) – Music publicists, agent and PR manager. Anna regularly contributes to Full Moon Magazine, Harmony, cultural magazine UNI and music portals such as FullMoonZine.cz and Frontaman.cz. You can also find her articles in Lidové noviny, magazine Spark etc. She closely cooperates with the music agency, Jazz Meets World, booking agency Earth Music, festivals such as Na jednom břehu and Colour Meeting. She's a member of the organizing team of the music conference and showcase festival, Czech Music Crossroads, pro-export office, SoundCzech and is also a member of the grand committee of the Ministry of Culture for Alternative Music.
The 2-day music conference offers a meeting point for Slovakian and international professionals. Part of the conference will be devoted to introduction of the local music scene. The conference events will include: panels, presentations by music professionals and networking sessions. The presence of festival promoters and many invited international experts will be an oportunity for networking with the local and regional community.
The festival is an unique opportunity to meet the protagonists of the music scene and establish valuable bussiness contacts.


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Home Entertainment More than 22,000 cases worth R1.5 trillion are under investigation by the...
More than 22,000 cases worth R1.5 trillion are under investigation by the Hawks
Despite the police's soft approach to crime, Hawkes's Directorate for Major Crime Investigation (DPCI) on Tuesday held 45 high-profile cases under investigation or prosecution.
At least 22,477 cases worth R1.5 trillion are being investigated and 4,447 convictions have been handed down in the past four years, said lawyer Godfrey Lebeya, a DPCI lieutenant-general.
Hawkmom on Phala Phala
Notably absent from the presentation was any mention of the theft of the $4 million (about 68 million rand) the president claimed – or just $600,000 (just over 10 million rand), according to News24. Cyril Ramaphosathe Phala Phala game farm, and the apparently irregular means used to track down the alleged thieves and the money.
READ ALSO: Hawks: Eight more suspects arrested in July riots
There was also no mention of what progress had been made in ending illegal mining activities.
Lebeya explained these lacunae with brevity and the fact that these cases are really being investigated.
"We have more than 22,000 cases, so I didn't list them all," he said.
"The complainant [in the Phala Phala case] made a statement, and we entered 41 statements into the case. I will not delve into the minutiae of the courts."
Asked why no known person had been arrested in connection with the July 2021 riots in KwaZulu-Natal and Gauteng, Lebeya said 43 arrests had been made, some this week alone, and more were expected a lot
Good luck to the Hawks
Some of the high-profile successes noted by DPCI include:
Interception of 706 bricks of cocaine worth R706 million in the Eastern Cape. In another incident, 805kg of cocaine, worth R400 million, was found in a boat being towed along the N1 in Gauteng.
27 arrests in connection with 188 charges related to VBS Bank, a failed bank that lost approximately 1.6 billion rand, owned by approximately 20 municipalities. The investigation collected more than 2,391 statements, reports DCPI. The case will be heard later this year.
12 suspects, including former Member of Parliament lawyer Bongani Bong, were arrested in October 2020 on fraud and corruption charges in connection with fraudulent land deals worth almost 124 million rand. In Mpumalanga, farms are believed to have been bought and sold at great profit. This case will be transferred to a pre-trial hearing later this year.
Arrest of former JB Marks executive mayor Kgotso Moses Khumalo and director Cyril Hendry in 2020 on charges of fraud involving R5 million that was meant for the municipality but ended up in private pockets.
22 suspects have been charged with fraud, corruption and racketeering in connection with the eThekwini (Durban) solid waste scandal, which is estimated to be worth R208 million. Former mayor Zandile Gumede is among the accused.
373 suspects have been arrested since 2018 for crimes involving endangered species, including abalone, rhino horn, elephant tusks, lion and lizard bones.
26 arrests have been made in connection with the July 2021 riots in KwaZulu-Natal and Gauteng, which caused billions of rand in damage to infrastructure and businesses. The case will be heard in the Durban Magistrates Court later this week.
France Hlakudi, Abram Masanga, Hudson Kgamoeswana and seven others were arrested in 2019 on charges of fraud and corruption at Eskom's Kusile power station project. It is alleged that the tender was fraudulently awarded to Tubular Construction for services originally awarded to Alstom S&E Africa.
Former Group Five CEO Mike Lomas is expected to be extradited from the UK in connection with the above case, which returns to court on November 4.
Seven suspects, including Angelo Agrisi and a former correctional services commissioner, were arrested in 2019 on charges of corruption, money laundering and fraud in connection with four tenders allegedly irregularly awarded to Bosasa facilities management company.
ALSO READ: Hawks already processing 41 affidavits linked to Phala Phala
DPCI has a staff strength of 23,672 and is currently advertising for another 200 vacancies.
Since 2018, the unit has arrested 12,157 suspects and issued 4,447 convictions, although the number of allegations investigated was much higher.
In one case alone, two people were convicted of 487 VAT fraud cases totaling 110 million rand, potentially reaching 440 million rand.
They were found guilty and sentenced to 17 and 16 years in prison, respectively
"The DPCI noted that corruption is one of the biggest threats in South Africa," Lebeya said.
More muscles
"It also allows the flow of illicit funds, which is the reason that informs the Directorate to monitor the implementation of money laundering allegations initiated as part of criminal investigations.
In order to strengthen the fight against corruption, the work of the Anti-Corruption Task Force (ACTT) and the Task Force have been intensified to speed up criminal investigations, prosecutions and asset recovery."
The task force has established a system of prioritization of crimes in the Top-100, Top-33 and Top-10 categories.
It has the power to confiscate and recover assets derived from illegal activities. The results of this new system have been "overwhelming," Lebeya said, with 554 arrests and 142 convictions.
One of these is an arms dealing case which will be heard in the Pietermaritzburg High Court in October.
Several of the cases covered by Lebeya involved fraud in tenders for personal protective equipment, misuse of funds and fraud in municipalities, and the killing of several police officers.
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Associate Membership is available to companies that supply services to the consumer packaged goods industry and fall under one of the following categories: consulting firm; technology provider; audit firm; logistics or other service provider.
Supplier membership is available to companies that manufacture or process agribusiness products, or ingredients used in products for human or pet consumption or sold to other manufacturers at any point along the CPG industry value chain as an ingredient.
Apply for GMA membership now! Click the link to to download General Membership and Associate Membership Application.


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Chelsea Manning Opens Up in Her First Television Interview: 'I Had to Be Who I Am'
On Friday morning, whistleblower Chelsea Manning made her first television appearance since her release from prison in mid-May.
Speaking with Good Morning America's Juju Chang, Manning touched on a number of sensitive topics, including why she chose to leak what became known as the Iraq War Logs to Wikileaks, as well as her decision to come out as transgender shortly after entering prison.
"I had to be who I am," Manning explained.
At one point during the interview, Manning grew visibly emotional, when asked about President Obama's decision to commute her prison sentence—the largest ever handed down to a leaker in U.S. history.
"Thank you," Manning said, holding back tears. "I've been given a chance. It's all I wanted."
On Twitter, the night before her segment aired, Manning was decidedly more upbeat, gently mocking the tone of the GMA interview:
Welcome back Chelsea. It's great to hear your voice.


India, a vast and diverse country, will not fail to captivate and enthral, with it's rich history and breathtaking beauty. A unique blend of cultures, politics and religions seen through it's monuments, beautiful palaces and also in the faces of its people.
India has traditionally been a popular choice for the adventurous traveller, drawn to the kaleidoscope of experiences, however now as word spreads of the first class hospitality, improved transportation links and luxury hotel chains developing, more families are inspired to enjoy a tailored itinerary that affords a taste of India which invariably whets the appetite for a lifelong attraction.
India is a country that has much to discover, from its exotic wildlife, spiritual cities to its dramatic landscapes of lush jungles, arid deserts, snow-capped mountains and excellent beaches.


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Fire danger increasing in Colorado as hot, dry weather persists
Posted: 7:18 AM, Jun 20, 2016
By: Mark Boyle

Camp site at Bear Creek Lake Park in Lakewood (Photo by: Mark Boyle)

Small wildfires are starting to pop up around the state of Colorado and fire crews say the big fire could spark at any time.
Captain Jay Jackson, with West Metro Fire Department said the hot, dry weather has increased the fire danger around much of the state.
The winds are also drying out the grasses quicker than normal, Jackson said.
Those camping at Bear Creek Lake Park say even close to the city, some people aren't responsible with their camp fires.
"This morning, people left their fire. They packed up and left and it smoked until 11 o'clock, they just didn't put the fire out," said Brian Smith, who's camping with his family. "We've always been taught, if you're in something like this you try to keep the fire maybe four or five inches above your ring."
Fire crews continue to work to contain a fire near Rifle and last Friday, a fire charred portions of grassland around Horsetooth Reservoir in Fort Collins.
Jackson said it's especially important now to be careful when starting and putting out your camp fire.
"Make sure that you've got a way to put it out. When you put it out, have water to put it out so you can stir around the coals," said Jackson.
Jackson advises if you're in a situation where water is scarce, bury your fire until it's out.


When you're a fancy restaurant, it's not enough to be just posh these days.
You have to claim a certain level of creativity.
Who cannot be bewitched, therefore, by a dish being served at very upscale San Francisco restaurant Quince?
It's called "A Dog In Search Of Gold" and I fear the less polite might wonder what the person who invented this delicacy was in search of at the time.
My own guess is either art or publicity. Or, perhaps, both.
For this dish involves white truffle croquettes with a slight twist. They're reportedly being served on iPads that feature a video of water dogs on the hunt for truffles.
How very imaginative, you might think. Or how very San Francisco. Or even how very, thoroughly nauseating.
An image of the dish was tweeted by local chef Richie Nakano on Tuesday. Although I'm sure it tastes delicious -- Quince has three Michelin stars -- some might think it has all the visual edification of expectorating in church.
It's tempting to see serving such a thing as an example of precisely the excess for which San Francisco is currently renowned -- excess of hubris, excess of sanctimoniousness and excess of tastelessness.
Michael Tusk, chef and owner of Quince, told me he never set out to capture a younger audience, but "really to work on fun collaborations and have a humorous or whimsical start to the meal."
But what about the hygienic aspect? Tusk told me: "The iPad slips in and out of the box and we designed a sheath that is cleaned after each order. The food does not directly sit on top of the iPad at all."
"The iPads are removable and sit on top of cleanable screen that is sanitized after every order," he added.
Tusk said he first experimented with the idea two years ago. He wasn't alone.
Last year, the Daily Mail reported that British chefs were suffering a burst of creative plating, that included serving food on garbage-can lids and, yes, iPads.
It may be that, in person, these truffles offer a divinity of taste coupled with a sublime sense of place, as we consider precisely how the truffles were found and by whom. After all, it's part of a $220 tasting menu, so it should have a good story attached.
Tusk's story? "Living in San Francisco for over 20 years I have witnessed the tech boom and I wanted to combine a little bit of gastronomy with technology and a little bit of education," he told me.
I fear, though, that it will only inspire other elevated chefs to start putting caviar on Amazon Echos, delicate amuse-bouches on iPhones and Belgian cheese plates on the new MacBook Pros.
Although, in the last case, I know that some think that's as good a use for the new MacBooks Pro as you can find.
Updated: 2:15 p.m. PT: adds comment from Quince Chef Michael Tusk.
Tusk's own image of the dish.