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Today's news release reports that the recently completed geophysical surveys over the Iceberg gold deposit indicate the possibility of a deeper horizon of gold mineralization. It is located beneath the current near-surface deposits of the North and Central zones of the Iceberg!
Forty years ago, when the famed Canadian geologist Dr. N.B. Keevil Sr., (founder of Teck Corporation, whom I was fortunate to know) first championed the beginnings of the science of geo-physics, many of his colleagues scoffed and slurred it as 'geo-phantasy'.
As the good Dr. Keevil said then, "it's an infant science that will depend on the integrity of the users to grow." Grown it has indeed and contributed to numerous discoveries over the years.
NuLegacy's Chief Geoscience Officer, Dr. Roger Steininger, and his associate Mr. Derick Unger (our Chief Geologist) are both strong proponents of geo-physics.
They successfully used it to locate the Avocado 'anomaly' which was recently drilled and confirmed as the second gold deposit on our property. Now they have identified a 'deep' IP anomaly under the near-surface Iceberg gold deposit which we will drill in the spring 2017.


Most natural signals (i.e cosmic sources) are almost always non-polarised (which is the same as "random polarised"), so the use of any single polarisation method either linear or circular will achieve the same result. The slight polarisation present in such signals do not bring any significant "power advantage" so in practice linear polarised antennas are preferred more in Radio Astronomy as they are more practical to construct for a specific gain over a circular polarised antenna.
Polarisation can however carry interesting information about the source, so radio astronomers sometimes want to measure this. However it is quite difficult to do, because the signal characteristics are so weak, and below a few 100 MHz, the polarisation information is usually too mixed up by the ionosphere to be of any practical use.


text:
Home News Scottish News Venison expert calls for multi-million pound investment into deer farms
Venison expert calls for multi-million pound investment into deer farms
By Xantha Leatham -
A TOP venison supplier is calling for a multi-million pound investment into large-scale deer farms – as demand for the red meat skyrockets.
Plans for Britain's first "mega" deer farm in Scotland are well underway, but an expert believes that £10m is needed to build more.
Christian Nissen, who provides meat for many of the country's supermarkets and top restaurants, says that the government needs to "think big" amid expectation of soaring global demand.
Plans for a "mega" deer farm are underway
Nissen currently imports about 600 tons of venison from New Zealand per year, and believes Scotland should step up to provide its own meat.
The entrepreneur said: "Why are we importing venison when we have the most fantastic pastures here in Scotland to produce top-quality venison?"
"The fact is that deer farming is not getting anywhere near the level of government funding that is needed.
"We need to think big and more land needs to be turned over to farming deer. A fund of £10m, which is peanuts compared with the overall funding for farmers in Scotland, would encourage more to diversify into deer farming."
A 1,000-acre site near Edinburgh has been chosen as the location for the country's flagship deer farm, set to be five times larger than anything that currently exists.
1,000 hinds and 40 stags will rear up to 1,000 calves for slaughter every year – equalling about 80 tons of meat.
But this only represents a fraction of the 1,200 tons needed to satisfy demand in Britain.
John Goffin, who is setting up the farm, said: "The problem is that we can't keep up with our demand, let alone develop an export market."
Earlier this year, a report by Mintel, the market research firm, highlighted British game as one of the world's top 50 consumer growth markets.
The success of venison as an alternative to mainstream red meats has made it a favourite among chefs such as Jamie Oliver, Andrew Farlie and Albert Roux.
According to a recent study, 81 farms were involved in raising deer in 2013 but the number of deer farmed throughout Scotland has dropped from 16,000 in 1990 to 6,000 in 1996 and has remained static ever since.
Ministers have pledged to develop Scotland's venison sector and are encouraging farmers to produce the meat.
A government spokesman said: "Scotland's venison sector is playing an increasing role in our food and drink success.
"We have worked to expand support in the new common agricultural policy so that new entrants and others who were wholly or partly frozen out of the old system – such as deer farmers – will receive basic payments at the regional average."
Previous articleStella McCartney shoots new collection in top secret Scottish location
Next articleScotland's most delayed trains revealed – and they all pass through Glasgow
Xantha Leatham
Aldi shopper claims to have discovered world's biggest chip after finding 9″ monster in bag of sweet potato fries
Stormzy praises man who bought stolen bike for £80 – so he could give it back to owner
Baffled bank customers struggle to use cash machine placed for people who are 8ft tall
http://www.deadlinenews.co.uk/2015/08/16/venison-expert-calls-for-multi-million-pound-investment-into-deer-farms/

redpajama_set_name: RedPajamaCommonCrawl


Many new options included for adding the technique used when performing an adjustment.
Pick any 4 technique buttons as shortcuts in user settings. All other techniques can be quickly accessed in behind the 5th button on the screen.
The overview screen for plan schedule data now includes a count of future appointments booked for the patient against the plan.
Quickly see if the correct patient appointments have been booked for the next six weeks for this patient.
Note that Front Desk appointments are only available for the upcoming week, and week's 2-6 will show zero appointments booked even though they are available in Front Desk.


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text:
The Round-Up: Some hope remains for Harry Dunn's family
8 December 2020 by William Craig Cohen
Credit: Getty Images/Tetra Images
On Thursday, Harry Dunn's family were granted permission to appeal against the High Court ruling handed down on 24 November, which held in no uncertain terms that Mrs Sacoolas did enjoy diplomatic immunity at the time she killed 19 year-old Harry Dunn while driving on the wrong side of the road in August of last year. The US state department has refused to waive her immunity under Article 32 of the Vienna Convention on Diplomatic Relations, stating that to allow the waiver, and thereby the extradition request that would inevitably follow would set an "extraordinarily troubling precedent". The arrests of diplomats Michael Kovrig in China and Rob Macaire in Iran over the last year highlight the continued importance of the inviolability of diplomatic agents serving abroad. However, where there has been an unlawful killing by a family member of an agent, natural inclinations of justice are upset by the failure of a longstanding diplomatic ally to simply do the right thing.


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Charlotte also has experience in dealing with family immigration matters including entry clearance, leave to remain and settlement for fiancés, spouses, unmarried partners and children as well as EEA and British citizenship matters.
Charlotte read Law at the University of Surrey, during which she undertook a placement year at a top boutique immigration firm in London dealing with various corporate and high net worth private clients.
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Outside of work, Charlotte is an avid skier and qualified scuba diver. She regularly blogs about vegan and cruelty-free cooking.


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

The first part of the Weil conjectures, the rationality of the zeta-function of algebraic varieties over finite fields, was proved by Dwork in a 1960 paper \cite{Dw1} using $p$-adic analysis. Subsequently, Dwork further developed his $p$-adic techniques and studied the special case of zeta functions of non-singular projective hypersurfaces. In particular, he examined how his $p$-adic constructions varied within a family, his so-called \emph{deformation theory} \cite{Dw2, Dw3}. The family
\begin{equation}\label{Dwork_HypS}
 x_1^n  + x_2^n \dots + x_n^n - n \lambda \, x_1 x_2 \dots x_n=0
 \end{equation}
appears often in his work and is now known as the Dwork family of hypersurfaces.
In the early 1990's, the $n=5$ case appeared in the celebrated work of Candelas et al \cite{COGP} on mirror symmetry, thus reviving interest in the Dwork family.

More recently, formulas for counting the number of points over finite fields on the Dwork hypersurface using hypergeometric functions have been of special interest. Let $\mathbb{F}_{q}$ denote the finite field with $q$ elements, where $q$ is a power of a prime $p$. Koblitz \cite{Ko4} provides a formula for the number of $\mathbb{F}_{q}$-points on monomial deformations of a diagonal hypersurface, of which the Dwork family is a special case, in terms of Gauss sums, when $q$ is in a particular congruence class. He then highlights the analogy between this formula and the Barnes integral for classical hypergeometric series. In \cite{McC5}, this author provided a simple formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface, in the case ${n=5}$ and ${\lambda=1}$, in terms of a finite field hypergeometric function, when $p \equiv 1 \pmod 5$.
We then extended this result to all odd primes using a hypergeometric type function defined in terms of the $p$-adic gamma function (see Definition \ref{def_Gp} below).
In \cite{Go}, Goodson considers the ${n=4}$ case and gives formulas for the number of $\mathbb{F}_{q}$-points in terms of finite field hypergeometric functions when $q \equiv 1 \pmod 4$, and extends to all odd primes using this author's $p$-adic hypergeometric function. She also conjectures a formula in the special case that $n$ is prime and that $p \not\equiv 1 \pmod n$, which was proven by Barman et al in \cite{BRS}.

The purpose of this paper is to provide a formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface in terms of the $p$-adic hypergeometric function in the general case, i.e for any $n$ and $\lambda$ in $\mathbb{F}_p^{*}$, and which holds for all odd primes $p$.

 \section{Statement of Results}\label{sec_Results}

We first define the $p$-adic hypergeometric function. Let $\gfp{\cdot}$ denote Morita's $p$-adic gamma function and let $\omega$ denote the Teichm\"{u}ller character of $\mathbb{F}_p$ with $\bar{\omega}$ denoting its character inverse.
For $x \in \mathbb{Q}$ we let  $\left\lfloor x \right\rfloor$ denote the greatest integer less than or equal to $x$ and
$\langle x \rangle$ denote the fractional part of $x$, i.e. $x- \left\lfloor x \right\rfloor$.
\begin{defi}\cite[Definition 1.1]{McC7}\label{def_Gp}
Let $p$ be an odd prime and let $x \in \mathbb{F}_p$. For $m \in \mathbb{Z}^{+}$ and $1 \leq i \leq m$, let $a_i, b_i \in \mathbb{Q} \cap \mathbb{Z}_p$.
Then we define
\begin{multline*
{_{m}G_{m}}
\biggl[ \begin{array}{cccc} a_1, & a_2, & \dotsc, & a_m \\
 b_1, & b_2, & \dotsc, & b_m \end{array}
\Big| \; x \; \biggr]_p
: = \frac{-1}{p-1}  \sum_{j=0}^{p-2}
(-1)^{jm}\;
\bar{\omega}^j(x)\\
\times \prod_{i=1}^{m}
\frac{\biggfp{\langle a_i -\frac{j}{p-1}\rangle}}{\biggfp{\langle a_i \rangle}}
\frac{\biggfp{\langle - b_i +\frac{j}{p-1}\rangle}}{\biggfp{\langle -b_i \rangle}}
(-p)^{-\lfloor{\langle a_i \rangle -\frac{j}{p-1}}\rfloor -\lfloor{\langle -b_i \rangle +\frac{j}{p-1}}\rfloor}.
\end{multline*}
\end{defi}
\noindent Throughout the paper we will refer to this function as ${_{m}G_{m}}[\cdots]$.
We note that the value of ${_{m}G_{m}}[\cdots]$ depends only on the fractional part of the $a$ and $b$ parameters, and is invariant if we change the order of the parameters.

We now describe our main result. We consider the Dwork hypersurface as described in (\ref{Dwork_HypS}).
Let $d:=\gcd({p-1},n)$
and
\begin{equation}\label{Def_W}
W:=\{w=(w_1, w_2, \ldots, w_n) \in \mathbb{Z}^n : 0 \leq w_i < d, \sum_{i=1}^n w_i \equiv 0 \pmod d\}.
\end{equation}
Define an equivalence relation $\sim$ on $W$ by
\begin{equation}\label{Def_tilde}
w \sim w \prime \textup{ if } w- w\prime \textup{ is a multiple modulo $d$ of $(1,1, \ldots, 1)$}.
\end{equation}
We note $|W|=d^{n-1}$ and $|{W/\sim}|=d^{n-1}$ as every equivalence class has $d$ elements. We will denote the class containing $w$ by $[w]$. We note also that each class contains a representative $w$ where some $w_i = 0$, for $1 \leq i \leq n$. We will write $[w^{\ast}]$ to indicate that we have chosen such a representative for a particular class.

For a given $w=(w_1, w_2, \ldots, w_n) \in W$, define $n_k$ to be the number of $k$'s appearing in $w$, i.e.,
$n_k = |\{w_i \mid  1 \leq i \leq n, w_i=k \} |.$
We then let $S_w := \{ k \mid 0 \leq k \leq {d-1}, n_k=0 \}$ and $S_w^{c}$ denote its complement in $\{0,1 \cdots, {d-1}\}$. So the elements of $S_w$ are the numbers from $0$ to ${d-1}$, inclusive, which do not appear in $w$.
We define the following lists
\begin{equation}\label{def_Aw}
A_w: \left[ \tfrac{d-k}{d} \mid k \in S_w \right] \cup \left[ \tfrac{h}{n} \mid 0 \leq h \leq n-1, h \not\equiv 0 \imod{\tfrac{n}{d}} \right];
\end{equation}
\begin{equation}\label{def_Bw}
B_w: \left[  \tfrac{d-k}{d} \, \textup{repeated $n_k$-1 times} \mid k \in S_w^c \right] .
\end{equation}
We note both lists contain
\begin{equation}\label{def_s}
s:=n-|S_w^c|
\end{equation}
numbers.

\begin{theorem}\label{thm_Main}
For a prime $p$, let $N_p(\lambda)$ be the number of points in $\mathbb{P}^{n-1}(\mathbb{F}_p)$ on
$$ x_1^n  + x_2^n \dots + x_n^n - n \lambda \, x_1 x_2 \dots x_n=0,$$
for some $n, \lambda \in \mathbb{F}_p^{*}$.
Define $d:=\gcd({p-1},n)$ and let $W$, $\sim$, $A_w$, $B_w$ and $s$ be defined by (\ref{Def_W})
- (\ref{def_s})
respectively.
 Then for $p$ odd,
\begin{equation*}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} + (-1)^n \sum_{[w^{\ast}] \in W/\sim} (-p)^{\frac{1}{d} \sum_{i=1}^{n} w_i} \;
\prod_{i=1}^{n} \gfp{\tfrac{w_i}{d}} \;
{_{s}G_{s}}
\biggl[ \begin{array}{c} A_w \\
 B_w \end{array}
\Big| \; \lambda^n \; \biggr]_p.
\end{equation*}
\end{theorem}
\noindent If $p \mid n$ then the problem reduces to the $\lambda=0$ case, formulas for which are well known and can be found in \cite{IR, W}.
We now look at a couple of special cases.

\begin{cor}\label{cor_RelPrime}
If $d=\gcd({p-1},n) = 1$ then
\begin{equation*}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} + (-1)^n  \;
{_{n-1}G_{n-1}}
\biggl[ \begin{array}{cccc} \frac{1}{n} & \frac{2}{n} & \dots & \frac{n-1}{n} \\
 1 & 1 & \dots & 1 \end{array}
\Big| \; \lambda^n \; \biggr]_p.
\end{equation*}
\end{cor}
\noindent Corollary \ref{cor_RelPrime} is a slight generalization of the result we mentioned in Section \ref{sec_Intro} for the case $n$ is prime and $p \not\equiv 1 \pmod n$, which was conjectured by Goodson \cite{Go} and proven by Barman et al in \cite{BRS}.

As noted in Section \ref{sec_Intro} some of the previous results counting the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface have been in terms of finite field hypergeometric functions and were valid only for primes in certain congruence classes. This restriction to primes in certain congruence classes is a common theme in results involving finite field hypergeometric functions. Establishing results for all primes is the main reason we developed the $p$-adic function defined above. (See \cite{McC7} for a more complete discussion.) Conversely, if we choose to restrict results involving ${_{m}G_{m}}[\cdots]$ to primes in certain congruence classes then it is always possible to reduce these results to expressions in terms of $_mF_m(\cdots)$. Our next corollary does exactly that for Theorem \ref{thm_Main} in the case $p \equiv 1 \pmod n$.

Let $\widehat{\mathbb{F}^{*}_{q}}$ denote the group of multiplicative characters of $\mathbb{F}^{*}_{q}$.
We extend the domain of $\chi \in \widehat{\mathbb{F}^{*}_{q}}$ to $\mathbb{F}_{q}$ by defining $\chi(0):=0$ (including for the trivial character $\varepsilon$) and denote $\bar{\chi}$ as the inverse of $\chi$.
Let $\theta$ be a fixed non-trivial additive character of $\mathbb{F}_q$ and for $\chi \in \widehat{\mathbb{F}^{*}_{q}}$ we define the Gauss sum $g(\chi):= \sum_{x \in \mathbb{F}_p} \chi(x) \theta(x)$.
We define the finite field hypergeometric function as follows. (See \cite{McC6} for the relationship to the functions of Greene and Katz.)
\begin{defi}\cite[Definition 1.4]{McC6}\label{def_F}
For $A_1, A_2, \dotsc, A_m, B_1, B_2 \dotsc, B_m \in \widehat{\mathbb{F}_q^{*}}$ and $x \in \mathbb{F}_q$,
\begin{equation*}\label{def_HypFnFF}
{_{m}F_{m}} {\biggl( \begin{array}{cccc} A_1, & A_2, & \dotsc, & A_m \\
 B_1, & B_2, & \dotsc, & B_m \end{array}
\Big| \; x \biggr)}_{q}\\
:= \frac{-1}{p-1}  \sum_{\chi \in \widehat{\mathbb{F}_p^{*}}}
\prod_{i=1}^{m} \frac{g(A_i \chi)}{g(A_i)} \frac{g(\bar{B_i \chi})}{g(\bar{B_i})}

 \chi(-1)^{m}
 \chi(x).
\end{equation*}
\end{defi}
\noindent We note Definition \ref{def_HypFnFF} is stated slightly differently than the original in \cite{McC6}, which has an implied $B_1 = \varepsilon$, as is often the custom in hypergeometric functions. The relationship between $_mF_m(\dots)$ and ${_{m}G_{m}}[\cdots]$ is outlined in \cite{McC7} and reproduced below in a slightly altered form to take account of the altered definition of $_mF_m(\dots)$ above.
\begin{lemma}[\cite{McC7} Lemma 3.3]\label{lem_G_to_F}
For a fixed odd prime $p$, let $A_i, B_k \in \widehat{\mathbb{F}_p^{*}}$ be given by $\bar{\omega}^{a_i(p-1)}$ and $\bar{\omega}^{b_k(p-1)}$ respectively, where $\omega$ is the Teichm\"{u}ller character . Then
\begin{equation*}
{_{m}F_{m}} {\biggl( \begin{array}{cccc} A_1, & A_2, & \dotsc, & A_m \\
 B_1, & B_2, & \dotsc, & B_m \end{array}
\Big| \; t \biggr)}_{p}
=
{_{m}G_{m}}
\biggl[ \begin{array}{cccc} a_1, & a_2, & \dotsc, & a_m \\
 b_1, & b_2, & \dotsc, & b_m \end{array}
\Big| \; t^{-1} \; \biggr]_p.
\end{equation*}
\end{lemma}
Let $p \equiv 1 \pmod{n}$ and so $d:=\gcd(p-1,n) = n$ and $t:=\frac{p-1}{d}=\frac{p-1}{n}$. Let $T$ be a fixed generator for $\widehat{\mathbb{F}_p^{*}}$ and define the following lists
\begin{equation*}\label{def_ATw}
A_{T,w}^{\prime}: \left[ T^{(n-k)t} \mid k \in S_w \right] ;
\end{equation*}
\begin{equation*}\label{def_BTw}
B_{T,w}^{\prime}: \left[  T^{(n-k)t} \, \textup{repeated $n_k$-1 times} \mid k \in S_w^c \right] .
\end{equation*}

\begin{cor}\label{cor_FF}
If $d=\gcd(p-1,n) = n$, i.e., $p \equiv 1 \pmod{n}$, and $t:=\frac{p-1}{d}$, then
\begin{equation*}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} +  \sum_{[w^{\ast}]  \in W/\sim}
\prod_{i=1}^{n} g(T^{w_i t})\;
{_{|S_w|}F_{|S_w|}} {\biggl( \begin{array}{c} A_{T,w}^{\prime} \\
 B_{T,w}^{\prime} \end{array}
\Big| \; \lambda^{-n} \; \biggr)}_{p}.\\
\end{equation*}
\end{cor}

 \section{An Example}\label{sec_Examples}
In this section we give an example of how Theorem \ref{thm_Main} works in practice, when $n=4$.
Let $p$ be an odd prime and let $N_p(\lambda)$ be the number of points in $\mathbb{P}^{3}(\mathbb{F}_p)$ on
$$ x_1^4  + x_2^4 +x_3^4 + x_4^4 - 4 \lambda \, x_1 x_2 x_3 x_4=0,$$
for some $\lambda \in \mathbb{F}_p^{*}$.

If $p \equiv 3 \pmod{4}$ then $d=\gcd({p-1},4)= 2$. We now evaluate the sets $W$ and $W/\sim$. We first note that the contribution of any $w=(w_1, w_2, w_3, w_4)$ to the sum in Theorem \ref{thm_Main} is the same as that for any permutation of $w$. We therefore list the elements of these sets up to permutation. We will however indicate, using a superscript, the total number of distinct permutations. So
$$W = \{ (0,0,0,0), (0,0,1,1)^6, (1,1,1,1) \}$$
and
$${W/\sim} = \{[0,0,0,0], [0,0,1,1]^3 \}.$$
When $w= (0,0,0,0)$ we see that $n_0=4, n_1=0$, and so $S_w=\{1\}$ and $S_w^c=\{0\}$ with $s:=n - |S_w^c|=3$.
Thus we get $A_w: \frac{1}{2}, \frac{1}{4}, \frac{3}{4}$ and $B_w: 1,1,1$.
Now when $w=(0,0,1,1)$ we get that $n_0=2, n_1=2$, and so $S_w=\emptyset$ and $S_w^c=\{0, 1\}$ with $s=2$.
Thus we get $A_w: \frac{1}{4}, \frac{3}{4}$ and $B_w: 1,\frac{1}{2}$.
So when $p \equiv 3 \pmod 4$ we get
\begin{equation*}
N_p(\lambda) = \frac{p^{3}-1}{p-1} +
{_{3}G_{3}}
\biggl[ \begin{array}{ccc} \frac{1}{4} & \frac{1}{2} & \frac{3}{4} \\
 1 & 1 & 1 \end{array}
\Big| \; \lambda^4 \; \biggr]_p
-3p \cdot
{_{2}G_{2}}
\biggl[ \begin{array}{cc} \frac{1}{4} & \frac{3}{4} \\
 1 & \frac{1}{2} \end{array}
\Big| \; \lambda^4 \; \biggr]_p.
\end{equation*}
This corresponds to Theorem 1.2 in \cite{Go}.

If  $p \equiv 1 \pmod{4}$ then $d=4$. Here we have
$${W/\sim} = \{[0,0,0,0], [0,0,2,2]^3, [0,0,1,3]^{12} \}.$$
When $w= (0,0,0,0)$ we have $A_w: \frac{1}{2}, \frac{1}{4}, \frac{3}{4}$ and $B_w: 1,1,1$.
When $w= (0,0,2,2)$ we have $A_w: \frac{1}{4}, \frac{3}{4}$ and $B_w: 1,\frac{1}{2}$.
Finally, when $w= [0,0,1,3]$ we get $A_w: \frac{1}{2}$ and $B_w: 1$.
So when $p \equiv 1 \pmod 4$ we get
\begin{equation*}
N_p(\lambda) = \frac{p^{3}-1}{p-1} +
{_{3}G_{3}}
\biggl[ \begin{array}{ccc} \frac{1}{4} & \frac{1}{2} & \frac{3}{4} \\
 1 & 1 & 1 \end{array}
\Big| \; \lambda^4 \; \biggr]_p
-3p \cdot
{_{2}G_{2}}
\biggl[ \begin{array}{cc} \frac{1}{4} & \frac{3}{4} \\
 1 & \frac{1}{2} \end{array}
\Big| \; \lambda^4 \; \biggr]_p
-12p \cdot
{_{1}G_{1}}
\biggl[ \begin{array}{c} \frac{1}{2}  \\
 1 \end{array}
\Big| \; \lambda^4 \; \biggr]_p.
\end{equation*}
This corresponds to Theorem 1.3 in \cite{Go}, after simplification of the final term.

\section{Preliminaries}\label{sec_Prelim}
Let $\mathbb{Z}_p$ denote the ring of $p$-adic integers, $\mathbb{Q}_p$ the field of $p$-adic numbers, $\bar{\mathbb{Q}_p}$ the algebraic closure of $\mathbb{Q}_p$, and $\mathbb{C}_p$ the completion of $\bar{\mathbb{Q}_p}$.

Let $\zeta_p$ be a fixed primitive $p$-th root of unity in $\bar{\mathbb{Q}_p}$. We define the additive character $\theta : \mathbb{F}_p \rightarrow \mathbb{Q}_p(\zeta_p)$ by $\theta(x):=\zeta_p^{x}$.
We note that $\mathbb{Z}^{*}_p$ contains all $({p-1})$-st roots of unity.
Thus we can consider multiplicative characters of $\mathbb{F}_p^{*}$ to be maps $\chi: \mathbb{F}_p^{*} \to \mathbb{Z}_{p}^{*}$.
Recall that for $\chi \in \widehat{\mathbb{F}_p^{*}}$, the Gauss sum $g(\chi)$ is defined by
$g(\chi):= \sum_{x \in \mathbb{F}_p} \chi(x) \theta(x).$

The following useful result gives a simple expression for the product of two Gauss sums. For $\chi \in \widehat{\mathbb{F}_p^{*}}$ we have
\begin{equation}\label{for_GaussConj}
g(\chi)g(\bar{\chi})=
\begin{cases}
\chi(-1) p & \text{if } \chi \neq \varepsilon,\\
1 & \text{if } \chi= \varepsilon.
\end{cases}
\end{equation}

In \cite{Ko4}, Koblitz provides a formula for the number of points in $\mathbb{P}^{n-1}(\mathbb{F}_p)$ on the hypersurface
$x_1^a + x_2^a + \dots + x_n^a - a \lambda x_1 x_2 \dots x_n=0$,
for some $a, \lambda \in \mathbb{F}_p$, where $p \equiv 1 \pmod a$.
Koblitz's result builds on the work of Weil \cite{W} which provides a formula in the $\lambda=0$ case. Weil's results hold for all primes $p$ and it is a relatively straightforward exercise to extend Koblitz's result to all primes in the case $a=n$, as follows.

\begin{theorem}[cf Koblitz \cite{Ko4} {Thm.~2}, Weil \cite{W}]\label{thm_Koblitz}
Let $N_p(\lambda)$ be the number of points in $\mathbb{P}^{n-1}(\mathbb{F}_p)$ on $\sum_{i=1}^n x_i^n - n \lambda \prod_{j=1}^{n} x_i=0$, for some $n \in \mathbb{F}_p^{*}$, $\lambda \in \mathbb{F}_p$. Let $T$ be a fixed generator for $\widehat{\mathbb{F}_p^{*}}$, $d:=\gcd(p-1,n)$ and $t:=\frac{p-1}{d}$.
Let $W$ be defined by (\ref{Def_W}).
Then
\begin{equation*}
N_p(\lambda) = \sum_{w \in W} N_p(0,w) + \frac{1}{p-1} \sum_{w \in W} \; \sum_{j=0}^{t-1}
\frac{\prod_{i=1}^{n} g(T^{w_i t + j})}{g(T^{nj})}\; T^{nj}(n \lambda).\\
\end{equation*}
where
\begin{equation*}\label{Def_Np0}
N_p(0,w):=
\begin{cases}
0 & \text{if some but not all } w_i=0,\\[6pt]
\frac{p^{n-1}-1}{p-1} & \text{if all } w_i=0,\\[6pt]
\frac{1}{p} \prod_{i=1}^{n} g(T^{w_i t}) & \text{if all } w_i \neq 0.
\end{cases}
\end{equation*}
\end{theorem}
Theorem \ref{thm_Koblitz} can also be proved directly using the point counting technique in \cite{W}. This technique is also often used to establish results involving finite field hypergeometric functions \cite{BRS, McC8, F2, Go, L2, McC5}.

We define the Teichm\"{u}ller character to be the primitive character $\omega: \mathbb{F}_p \rightarrow\mathbb{Z}^{*}_p$ satisfying $\omega(x) \equiv x \pmod p$ for all $x \in \{0,1, \ldots, p-1\}$.
We now recall the $p$-adic gamma function. For further details, see \cite{Ko}.
Let $p$ be an odd prime.  For $n \in \mathbb{Z}^{+}$ we define the $p$-adic gamma function as
\begin{align*}
\gfp{n} &:= {(-1)}^n \prod_{\substack{0<j<n\\p \nmid j}} j \\
\intertext{and extend it to all $x \in\mathbb{Z}_p$ by setting $\gfp{0}:=1$ and}
\gfp{x} &:= \lim_{n \rightarrow x} \gfp{n}
\end{align*}
for $x\neq 0$, where $n$ runs through any sequence of positive integers $p$-adically approaching $x$.
This limit exists, is independent of how $n$ approaches $x$, and determines a continuous function
on $\mathbb{Z}_p$ with values in $\mathbb{Z}^{*}_p$.
We now state a product formula for the $p$-adic gamma function.
If $m\in\mathbb{Z}^{+}$, $p \nmid m$ and $x=\frac{r}{p-1}$ with $0\leq r \leq p-1$, then
\begin{equation}\label{for_pGammaMult}
\prod_{h=0}^{m-1} \gfp{\tfrac{x+h}{m}}=\omega\left(m^{(1-x)(1-p)}\right)
\gfp{x} \prod_{h=1}^{m-1} \gfp{\tfrac{h}{m}}.
\end{equation}
We note also that
\begin{equation}\label{for_pGammaOneMinus}
\gfp{x}\gfp{1-x} = {(-1)}^{x_0},
\end{equation}
where $x_0 \in \{1,2, \dotsc, {p}\}$ satisfies $x_0 \equiv x \pmod {p}$. The Gross-Koblitz formula \cite{GK} allows us to relate Gauss sums and the $p$-adic gamma function. Let $\pi \in \mathbb{C}_p$ be the fixed root of $x^{p-1}+p=0$ that satisfies ${\pi \equiv \zeta_p-1 \pmod{{(\zeta_p-1)}^2}}$. Then we have the following result.
\begin{theorem}[Gross, Koblitz \cite{GK}]\label{thm_GrossKoblitz}
For $ j \in \mathbb{Z}$,
\begin{equation*}
g(\bar{\omega}^j)=-\pi^{(p-1) \langle{\frac{j}{p-1}}\rangle} \: \gfp{\langle{\tfrac{j}{p-1}}\rangle}.
\end{equation*}
\end{theorem}
\noindent We recall also the following result which can be derived from (\ref{for_pGammaMult}).
\begin{lemma}[\cite{McC7} Lemma 4.1] \label{lem_pGamma}
Let $p$ be prime. For $0 \leq j \leq p-2$ and $n \in \mathbb{Z}^{+}$ with $p \nmid n$,
\begin{equation*}\label{lem_gammamtj}
\gfp{\Big\langle {\tfrac{-nj}{p-1}} \Big\rangle}\; {\omega(n^{-nj}) \displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}} = \displaystyle\prod_{h=0}^{n-1} \gfp{\Big\langle \tfrac{1+h}{n} - \tfrac{j}{p-1} \Big\rangle}.
\end{equation*}
\end{lemma}

\section{Proofs}\label{sec_Proofs}

\begin{proof}[Proof of Theorem \ref{thm_Main}]
By Theorem \ref{thm_Koblitz} we have
\begin{equation*}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} + \frac{1}{p} \sum_{\substack{w \in W \\ w_i \neq 0}}  \prod_{i=1}^{n} g(T^{w_i t})+ \frac{1}{p-1} \sum_{w \in W} \; \sum_{j=0}^{t-1}
\frac{\prod_{i=1}^{n} g(T^{w_i t + j})}{g(T^{nj})}\; T^{nj}(n \lambda).\\
\end{equation*}
From (\ref{for_GaussConj}) we get that
\begin{equation*}
g(T^{nj})g(T^{-nj})=
\begin{cases}
T^{nj}(-1) \; p & \text{if } T^{nj} \neq \varepsilon,\\
1 & \text{if } T^{nj}= \varepsilon.
\end{cases}
\end{equation*}
Now $T^{nj}= \varepsilon$ if and only if $j=0$, as $0 \leq j < \frac{p-1}{d}$. Therefore
\begin{align*}
\notag N_p(\lambda)
&= \frac{p^{n-1}-1}{p-1} + \frac{1}{p} \sum_{\substack{w \in W \\ \text{all } w_i \neq 0}}  \prod_{i=1}^{n} g(T^{w_i t})
-\frac{1}{p-1} \sum_{w \in W}  \prod_{i=1}^{n} g(T^{w_i t}) \\
\notag & \qquad \qquad  \qquad \qquad +\frac{1}{p(p-1)} \sum_{w \in W} \; \sum_{j=1}^{t-1}
\prod_{i=1}^{n} g(T^{w_i t + j}) \; g(T^{-nj})\; T^{nj}(-n \lambda)\\
\notag &= \frac{p^{n-1}-1}{p-1} + \frac{1}{p} \sum_{\substack{w \in W \\ \text{all } w_i \neq 0}}  \prod_{i=1}^{n} g(T^{w_i t})
-\frac{1}{p-1} \sum_{w \in W}  \prod_{i=1}^{n} g(T^{w_i t}) \left[ 1 - \frac{1}{p} \right]\\
\notag & \qquad \qquad  \qquad \qquad +\frac{1}{p(p-1)} \sum_{w \in W} \; \sum_{j=0}^{t-1}
\prod_{i=1}^{n} g(T^{w_i t + j}) \; g(T^{-nj})\; T^{nj}(-n \lambda)\\
\notag &= \frac{p^{n-1}-1}{p-1} - \frac{1}{p} \sum_{\substack{w \in W \\ \text{some } w_i=0}}  \prod_{i=1}^{n} g(T^{w_i t}) \\
\notag & \qquad \qquad  \qquad \qquad +\frac{1}{p(p-1)} \sum_{w \in W} \; \sum_{j=0}^{t-1}
\prod_{i=1}^{n} g(T^{w_i t + j}) \; g(T^{-nj})\; T^{nj}(-n \lambda).\\
\notag &= \frac{p^{n-1}-1}{p-1} - \frac{1}{p} \sum_{\substack{w \in W \\ \text{some } w_i=0}}  \prod_{i=1}^{n} g(T^{w_i t}) \\
& \qquad \qquad  \qquad \qquad +\frac{1}{p(p-1)} \sum_{[w] \in W/\sim} \; \sum_{j=0}^{p-2}
\prod_{i=1}^{n} g(T^{w_i t + j}) \; g(T^{-nj})\; T^{nj}(-n \lambda).
\end{align*}
We now examine the inner sum in the last term above, which we will denote $R_{[w]}$, i.e.,
\begin{equation*}
R_{[w]} = \sum_{j=0}^{p-2}
\prod_{i=1}^{n} g(T^{w_i t + j}) \; g(T^{-nj})\; T^{nj}(-n \lambda),
\end{equation*}
and
\begin{equation}\label{Np_for1}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} - \frac{1}{p} \sum_{\substack{w \in W \\ \text{some } w_i=0}}  \prod_{i=1}^{n} g(T^{w_i t}) +\frac{1}{p(p-1)} \sum_{[w] \in W/\sim} \; R_{[w]}.
\end{equation}
We note that $R_{[w]}$ is independent of choice of representative for the equivalence class.
Recalling the notation from Section \ref{sec_Results} we see that
\begin{align*}
R_{[w]}  = \sum_{j=0}^{p-2} \; \prod_{k \in S_w^c} g(T^{k t + j})^{n_k} \; g(T^{-nj})\; T^{nj}(-n \lambda).
\end{align*}
Again using (\ref{for_GaussConj}) we get that
\begin{equation*}
g(T^{k t + j}) \, g(T^{-k t - j})=
\begin{cases}
T^{k t + j}(-1) \; p & \text{if } T^{k t + j} \neq \varepsilon,\\
1 & \text{if } T^{k t + j}= \varepsilon.
\end{cases}
\end{equation*}
Now $T^{k t + j}= \varepsilon$ if and only if $k=j=0$ or $k>0, j=(d-k)t$. So, as $n_k\geq1$ when $k \in S_w^c$,
\begin{align*}
R_{[w]}  &= \sum_{\substack{j=0\\j \not\equiv 0 \imod{t}}}^{p-2} \;
\prod_{k \in S_w^c} \frac{g(T^{k t + j})^{n_k-1} \; T^{k t + j}(-1) \; p}{g(T^{-k t - j})}
 \; g(T^{-nj})\; T^{nj}(-n \lambda)\\
 &  \qquad \qquad \qquad \qquad -\sum_{a=0}^{d-1} \, \prod_{k \in S_w^c} \frac{g(T^{(k+a) t })^{n_k-1} }{g(T^{-(k+a)t})}
 \prod_{\substack{k \in S_w^c\\k \not\equiv -a \imod{d}}} \left(T^{(k+a) t}(-1) \; p \right)\\
&= \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \frac{g(T^{k t + j})^{n_k-1} \; T^{k t + j}(-1) \; p}{g(T^{-k t - j})}
 \; g(T^{-nj})\; T^{nj}(-n \lambda)\\
  &  \qquad -\sum_{a=0}^{d-1} \, \prod_{k \in S_w^c} \frac{g(T^{(k+a) t })^{n_k-1} }{g(T^{-(k+a)t})}
 \left[ \prod_{\substack{k \in S_w^c\\k \not\equiv -a (d)}} \left(T^{(k+a) t}(-1) \; p \right)
 -  \prod_{k \in S_w^c} \left(T^{(k+a) t}(-1) \; p \right) \right].\\
\end{align*}
For a given $0 \leq a \leq d-1$ define
$$v_a:=
\begin{cases}
0 & \text{if } a=0\\
d-a & \text{if } a>0.
\end{cases}$$
Then $0 \leq v_a \leq d-1$ and $v_a \equiv - a \pmod{d}$.
So
\begin{align*}
R_{[w]}
&= \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \frac{g(T^{k t + j})^{n_k-1} \; T^{k t + j}(-1) \; p}{g(T^{-k t - j})}
 \; g(T^{-nj})\; T^{nj}(-n \lambda)\\
  &  \qquad + (p-1) \sum_{\substack{a=0\\v_a \in S_w^c}}^{d-1} \, \prod_{k \in S_w^c} \frac{g(T^{(k+a) t })^{n_k-1} }{g(T^{-(k+a)t})}
 \prod_{\substack{k \in S_w^c\\k \neq v_a}} \left(T^{(k+a) t}(-1) \; p \right).
\end{align*}
We will analyze the two terms appearing on the right-hand side of the above equation separately, and refer to them as $R_{[w]}^{\prime}$ and $R_{[w]}^{\prime\prime}$ respectively. It is easy to see that $R_{[w]}^{\prime}$ is independent of choice of equivalence class representative thus so is $R_{[w]}^{\prime\prime}$.
We note first that for a given $0 \leq a \leq d-1$, if $v_a \in S_w^c$ then
\begin{align*}
\prod_{k \in S_w^c} {g(T^{-(k+a)t})}  \prod_{\substack{k \in S_w^c\\k \neq v_a}} {g(T^{(k+a)t})}
&= - \prod_{\substack{k \in S_w^c\\k \neq v_a}} {g(T^{-(k+a)t})}  \prod_{\substack{k \in S_w^c\\k \neq v_a}} {g(T^{(k+a)t})}\\
&= - \prod_{\substack{k \in S_w^c\\k \neq v_a}} \left[ T^{(k+a)t}(-1) \, p \right]
\end{align*}
using (\ref{for_GaussConj}) and the fact that $g(\varepsilon)=-1$. Thus
\begin{align*}
R_{[w]}^{\prime\prime}
&=-(p-1) \sum_{\substack{a=0\\v_a \in S_w^c}}^{d-1} \, \prod_{k \in S_w^c} g(T^{(k+a) t })^{n_k-1}
 \prod_{\substack{k \in S_w^c\\k \neq v_a}} {g(T^{(k+a)t})}\\
&=(p-1) \sum_{\substack{a=0\\v_a \in S_w^c}}^{d-1} \, \prod_{k \in S_w^c} g(T^{(k+a) t })^{n_k}\\
&=(p-1) \sum_{\substack{a=0\\v_a \in S_w^c}}^{d-1} \, \prod_{i=1}^{n} g(T^{(w_i+a) t }).
\end{align*}
For a given $0 \leq a \leq d-1$ let $\bar{a}$ be the $n$-tuple $(a,a, \cdots, a)$. Note then that
$$v_a \in S_w^c \Longleftrightarrow 0 \in S_{w+\bar{a}}^{c}$$
where the addition $w+\bar{a}$ is considered modulo $d$ so $w+\bar{a} \in W$.
Therefore
\begin{align*}
\sum_{[w] \in W/\sim} R_{[w]}^{\prime\prime}
&= (p-1) \sum_{[w] \in W/\sim} \sum_{\substack{a=0\\0 \in S_{w+\bar{a}}^{c}}}^{d-1} \, \prod_{i=1}^{n} g(T^{(w_i+a) t })\\
&=  (p-1) \sum_{\substack{w \in W \\ \text{some } w_i=0}}  \prod_{i=1}^{n} g(T^{w_i t }).
\end{align*}
So now (\ref{Np_for1}) becomes
\begin{align}\label{Np_for2}
\notag N_p(\lambda)
\notag &= \frac{p^{n-1}-1}{p-1} - \frac{1}{p} \sum_{\substack{w \in W \\ \text{some } w_i=0}}  \prod_{i=1}^{n} g(T^{w_i t})
+\frac{1}{p(p-1)} \sum_{[w] \in W/\sim} \; \left(R_{[w]}^{\prime} +R_{[w]}^{\prime\prime} \right)\\
&= \frac{p^{n-1}-1}{p-1} +\frac{1}{p(p-1)} \sum_{[w] \in W/\sim} \; R_{[w]}^{\prime},
\end{align}
where
\begin{equation*}
R_{[w]}^{\prime} = \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \frac{g(T^{k t + j})^{n_k-1} \; T^{k t + j}(-1) \; p}{g(T^{-k t - j})}
 \; g(T^{-nj})\; T^{nj}(-n \lambda).
\end{equation*}
We now switch to the $p$-adic setting to analyze $R_{[w]}^{\prime}$. We let $T=\bar{\omega}$ and use the Gross-Koblitz formula, Theorem \ref{thm_GrossKoblitz}, to get
\begin{equation*}
R_{[w]}^{\prime} = \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \frac{\biggfp{\langle \frac{k}{d} +\frac{j}{p-1} \rangle}^{n_k-1} \; \bar{\omega}^{k t + j}(-1) \; p}{\biggfp{\langle -\frac{k}{d} -\frac{j}{p-1} \rangle}}
 \; \biggfp{\langle -\tfrac{nj}{p-1} \rangle} \; \bar{\omega}^{nj}(-n \lambda) \cdot \pi^{(p-1)x} \cdot (-1)^y,
\end{equation*}
where
\begin{align*}
x&=\sum_{k \in S_w^c} (n_k-1) \langle \tfrac{k}{d} +\tfrac{j}{p-1} \rangle - \sum_{k \in S_w^c}  \langle -\tfrac{k}{d} -\tfrac{j}{p-1} \rangle + \langle -\tfrac{nj}{p-1} \rangle,
\intertext{and}
y&=\sum_{k \in S_w^c} (n_k-1)- \sum_{k \in S_w^c}1 + 1.
\end{align*}
Using the facts that $\langle x \rangle=x- \left\lfloor x \right\rfloor$,
\begin{equation*}
 \sum_{k \in S_w^c} n_k = \sum_{i=0}^{d-1} n_k = n,
\end{equation*}
and
\begin{equation*}
 \sum_{k \in S_w^c} k \, n_k = \sum_{i=0}^{n-1} w_i \equiv 0 \pmod{d},
\end{equation*}
it is easy to see that
\begin{align}\label{for_x}
x&=  \sum_{k \in S_w^c} \frac{k \, n_k}{d} - \sum_{k \in S_w^c} (n_k-1) \left\lfloor \tfrac{k}{d} +\tfrac{j}{p-1} \right\rfloor
+ \sum_{k \in S_w^c} \left\lfloor  -\tfrac{k}{d} -\tfrac{j}{p-1} \right\rfloor - \left\lfloor -\tfrac{nj}{p-1} \right\rfloor \in \mathbb{Z},
\intertext{and}
\notag y&= n+1.
\end{align}
So
\begin{equation}\label{Rw1_for1}
R_{[w]}^{\prime} = \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \frac{\biggfp{\langle \frac{k}{d} +\frac{j}{p-1} \rangle}^{n_k-1} \; \bar{\omega}^{k t + j}(-1) \; p}{\biggfp{\langle -\frac{k}{d} -\frac{j}{p-1} \rangle}}
 \; \biggfp{\langle -\tfrac{nj}{p-1} \rangle} \; \bar{\omega}^{nj}(-n \lambda) \cdot (-p)^{x} \cdot (-1)^{n+1}.
\end{equation}
From Lemma \ref{lem_gammamtj} we see that as $p\nmid n$,
\begin{equation}\label{lem_gammamtj_1}
\gfp{\Big\langle {\tfrac{-nj}{p-1}} \Big\rangle}
=
\frac{\displaystyle\prod_{h=0}^{n-1} \gfp{\Big\langle \tfrac{1+h}{n} - \tfrac{j}{p-1} \Big\rangle}}
{{\omega(n^{-nj}) \displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}}}
=
\frac{\displaystyle\prod_{h=0}^{n-1} \gfp{\Big\langle \tfrac{h}{n} - \tfrac{j}{p-1} \Big\rangle}}
{{\omega(n^{-nj}) \displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}}}.
\end{equation}
Now
\begin{align}\label{lem_gammamtj_2}
\notag \displaystyle\prod_{h=0}^{n-1} \gfp{\Big\langle \tfrac{h}{n} - \tfrac{j}{p-1} \Big\rangle}
&=
\displaystyle\prod_{k=0}^{d-1} \gfp{\Big\langle \tfrac{k}{d} - \tfrac{j}{p-1} \Big\rangle}
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\Big\langle \tfrac{h}{n} - \tfrac{j}{p-1} \Big\rangle}\\
\notag &=
\displaystyle\prod_{k=1}^{d} \gfp{\Big\langle \tfrac{d-k}{d} - \tfrac{j}{p-1} \Big\rangle}
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\Big\langle \tfrac{h}{n} - \tfrac{j}{p-1} \Big\rangle}\\
&=
\displaystyle\prod_{k=0}^{d-1} \gfp{\Big\langle -\tfrac{k}{d} - \tfrac{j}{p-1} \Big\rangle}
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\Big\langle \tfrac{h}{n} - \tfrac{j}{p-1} \Big\rangle}
\end{align}
Accounting for (\ref{lem_gammamtj_1}) and (\ref{lem_gammamtj_2}) in (\ref{Rw1_for1}) we get
\begin{multline}\label{Rw1_for2}
R_{[w]}^{\prime} = \frac{(-1)^{n-1}}{\displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}} \;
 \sum_{j=0}^{p-2} \;
\prod_{k \in S_w^c} \left(\biggfp{\langle \tfrac{k}{d} +\tfrac{j}{p-1} \rangle}^{n_k-1} \; \bar{\omega}^{k t + j}(-1) \; p \right)
\prod_{k \in S_w} \gfp{\langle -\tfrac{k}{d} - \tfrac{j}{p-1} \rangle}
\\
\cdot \displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\langle \tfrac{h}{n} - \tfrac{j}{p-1} \rangle}
\cdot \bar{\omega}^{nj}(- \lambda) \cdot (-p)^{x} .
\end{multline}
Our aim now is to convert (\ref{Rw1_for2}) in to the appropriate $_mG_m[\cdots]$.
We start with a few preliminary observations.
Similar to (\ref{lem_gammamtj_2}) we have
\begin{equation}\label{for_obs1}
\displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}
= \displaystyle\prod_{k=1}^{d-1} \gfp{\tfrac{k}{d}} \displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\tfrac{h}{n}}.
\end{equation}
Furthermore, and noting that $\gfp{0}:=1$, we see that
\begin{equation}\label{for_obs2}
\displaystyle\prod_{k=1}^{d-1} \gfp{\tfrac{k}{d}}
= \displaystyle\prod_{k=1}^{d-1} \gfp{\langle \tfrac{d-k}{d} \rangle}
= \displaystyle\prod_{k=0}^{d-1} \gfp{\langle -\tfrac{k}{d} \rangle}
= \displaystyle\prod_{k \in S_w} \gfp{\langle -\tfrac{k}{d} \rangle}
\displaystyle\prod_{k \in S_w^c} \gfp{\langle -\tfrac{k}{d} \rangle}.
\end{equation}
Using the Gross-Koblitz formula (Theorem  \ref{thm_GrossKoblitz}) and (\ref{for_GaussConj}) we also observe that
\begin{align}\label{for_obs3}
\notag \displaystyle\prod_{k \in S_w^c}  \gfp{\langle -\tfrac{k}{d} \rangle} \; \gfp{\langle \tfrac{k}{d} \rangle}
&=
\displaystyle\prod_{k \in S_w^c} g(\bar{\omega}^{-k t}) g(\bar{\omega}^{k t})
\; \pi^{-(p-1) \left[ \langle\frac{-k}{d}\rangle + \langle\frac{k}{d}\rangle \right] }\\
\notag &=
(-p)^{-|S_w^c\setminus \{0\}|} \displaystyle\prod_{k \in S_w^c\setminus \{0\}} \left( \bar{\omega}^{k t}(-1) \, p \right) \\
&=
(-1)^{|S_w^c\setminus \{0\}|} \displaystyle\prod_{k \in S_w^c} \bar{\omega}^{k t}(-1),
\end{align}
as
$$\langle\tfrac{-k}{d}\rangle + \langle\tfrac{k}{d}\rangle = - \lfloor \tfrac{-k}{d}\rfloor - \lfloor\tfrac{k}{d}\rfloor
=
-
\begin{cases}
0 & k=0\\
-1 & 1 \leq k \leq {d-1}.
\end{cases}
$$
Combining (\ref{for_obs1}), (\ref{for_obs2}) and (\ref{for_obs3}) we get that
\begin{equation*}
\frac
{
\displaystyle\prod_{k \in S_w^c} \biggfp{\langle \tfrac{k}{d} \rangle}^{n_k-1}
\displaystyle\prod_{k \in S_w} \gfp{\langle -\tfrac{k}{d}  \rangle}
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \gfp{\langle \tfrac{h}{n}  \rangle}
}
{\displaystyle\prod_{h=1}^{n-1} \gfp{\tfrac{h}{n}}}
=
\frac
{\displaystyle\prod_{k \in S_w^c} \biggfp{\langle \tfrac{k}{d} \rangle}^{n_k}}
{(-1)^{|S_w^c\setminus \{0\}|} \displaystyle\prod_{k \in S_w^c} \bar{\omega}^{k t}(-1)},
\end{equation*}
and so (\ref{Rw1_for2}) becomes
\begin{multline}\label{Rw1_for3}
R_{[w]}^{\prime} = (-1)^{n-1}
 \sum_{j=0}^{p-2} \;
 (-1)^{|S_w^c\setminus \{0\}|}
\displaystyle\prod_{k \in S_w^c} \frac{\biggfp{\langle \tfrac{k}{d} +\tfrac{j}{p-1} \rangle}^{n_k-1}}{\biggfp{\langle \tfrac{k}{d} \rangle}^{n_k-1} }
\displaystyle\prod_{k \in S_w} \frac{\gfp{\langle -\tfrac{k}{d} - \tfrac{j}{p-1} \rangle}}{\gfp{\langle -\tfrac{k}{d}  \rangle}}
\\ \cdot
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \frac{\gfp{\langle \tfrac{h}{n} - \tfrac{j}{p-1} \rangle}}
{\gfp{\langle \tfrac{h}{n}  \rangle}}
\displaystyle\prod_{k \in S_w^c} \biggfp{\langle \tfrac{k}{d} \rangle}^{n_k}
\cdot  \left(\bar{\omega}^{ j}(-1) \; p \right)^{|S_w^c|}
\cdot \bar{\omega}^{nj}(- \lambda) \cdot (-p)^{x}
\end{multline}
We now turn our attention to the power of $(-p)$. Comparing (\ref{Rw1_for3}) to Definition \ref{def_Gp} for $_mG_m[\cdots]$ we see that for our particular arguments of the $p$-adic gamma function we would like the power of $(-p)$ to be
$$z := - \left[\sum_{k \in S_w^c} (n_k-1) \lfloor \langle \tfrac{k}{d} \rangle +\tfrac{j}{p-1} \rfloor
+\sum_{k \in S_w} \lfloor \langle -\tfrac{k}{d} \rangle - \tfrac{j}{p-1} \rfloor
+\sum_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \lfloor \langle \tfrac{h}{n} \rangle - \tfrac{j}{p-1} \rfloor
\right]. $$
Comparing to (\ref{for_x}) we see that
\begin{multline*}
x-z= \sum_{k \in S_w^c} \frac{k \, n_k}{d}
+ \sum_{k \in S_w^c} \left\lfloor  -\tfrac{k}{d} -\tfrac{j}{p-1} \right\rfloor
- \left\lfloor -\tfrac{nj}{p-1} \right\rfloor
+\sum_{k \in S_w} \lfloor \langle -\tfrac{k}{d} \rangle - \tfrac{j}{p-1} \rfloor
+\sum_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \lfloor \langle \tfrac{h}{n} \rangle - \tfrac{j}{p-1} \rfloor.
\end{multline*}
A straightforward calculation yields
\begin{equation*}
\left\lfloor -\tfrac{nj}{p-1} \right\rfloor = \sum_{h=0}^{n-1} \lfloor \tfrac{h}{n} - \tfrac{j}{p-1} \rfloor
= \sum_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \lfloor \langle \tfrac{h}{n} \rangle - \tfrac{j}{p-1} \rfloor
+ \sum_{k=0}^{d-1} \lfloor \langle -\tfrac{k}{d} \rangle - \tfrac{j}{p-1} \rfloor.
\end{equation*}
So
\begin{align*}
x-z
&=\sum_{k \in S_w^c} \frac{k \, n_k}{d}
+ \sum_{k \in S_w^c} \left\lfloor  -\tfrac{k}{d} -\tfrac{j}{p-1} \right\rfloor
-\sum_{k \in S_w^c} \lfloor \langle -\tfrac{k}{d} \rangle - \tfrac{j}{p-1} \rfloor\\
&=\sum_{k \in S_w^c} \frac{k \, n_k}{d}
+ \sum_{k \in S_w^c\setminus \{0\}} \left\lfloor  -\tfrac{k}{d} -\tfrac{j}{p-1} \right\rfloor
-\sum_{k \in S_w^c\setminus \{0\}} \lfloor  \tfrac{d-k}{d} - \tfrac{j}{p-1} \rfloor\\
&=\sum_{k \in S_w^c} \frac{k \, n_k}{d} - {|S_w^c\setminus \{0\}|}.
\end{align*}
Recall $s:= n-|S_w^c|$, $\bar{\omega}(-1)=-1$ and $\sum_{k \in S_w^c} k \, n_k = \sum_{i=1}^{n} w_i$. Therefore
\begin{multline*}
R_{[w]}^{\prime} =
\displaystyle\prod_{i=1}^{n} \biggfp{\tfrac{w_i}{d}}
\cdot (-p)^{\frac{1}{d} \sum_{i=1}^{n} w_i}
\cdot p^{\delta_w}
\cdot (-1)^{n-1} \\
 \times \sum_{j=0}^{p-2} \;
 (-1)^{js} \; \bar{\omega}^j(\lambda^n)
\displaystyle\prod_{k \in S_w} \frac{\gfp{\langle \tfrac{d-k}{d} - \tfrac{j}{p-1} \rangle}}{\gfp{\langle \tfrac{d-k}{d}  \rangle}}
\displaystyle\prod_{\substack{h=0\\h \not\equiv 0 \, (\frac{n}{d})}}^{n-1} \frac{\gfp{\langle \tfrac{h}{n} - \tfrac{j}{p-1} \rangle}}
{\gfp{\langle \tfrac{h}{n}  \rangle}}
\\ \cdot
\displaystyle\prod_{k \in S_w^c} \frac{\biggfp{\langle -\tfrac{d-k}{d} +\tfrac{j}{p-1} \rangle}^{n_k-1}}{\biggfp{\langle -\tfrac{d-k}{d} \rangle}^{n_k-1} }
\cdot (-p)^z
\end{multline*}
where
\begin{equation*}
\delta_w :=
\begin{cases}
1 & \text{if } 0 \in S_w^c\\
0 & \text{if } 0 \in S_w.
\end{cases}
\end{equation*}
So
\begin{equation*}
R_{[w]}^{\prime} =
\displaystyle\prod_{i=1}^{n} \biggfp{\tfrac{w_i}{d}}
\cdot (-p)^{\frac{1}{d} \sum_{i=1}^{n} w_i}
\cdot p^{\delta_w}
\cdot (-1)^{n}
\cdot (p-1) \cdot
{_{s}G_{s}}
\biggl[ \begin{array}{c} A_w \\
 B_w \end{array}
\Big| \; \lambda^n \; \biggr]_p,
\end{equation*}
where $A_w$ and $B_w$ are the parameter lists defined in (\ref{def_Aw}) and (\ref{def_Bw}). Therefore, from (\ref{Np_for2}) we get that
\begin{equation}
\notag N_p(\lambda)
= \frac{p^{n-1}-1}{p-1} +
 \sum_{[w] \in W/\sim} \;
\displaystyle\prod_{i=1}^{n} \biggfp{\tfrac{w_i}{d}}
\cdot (-p)^{\frac{1}{d} \sum_{i=1}^{n} w_i}
\cdot p^{\delta_w-1}
\cdot (-1)^{n} \cdot
{_{s}G_{s}}
\biggl[ \begin{array}{c} A_w \\
 B_w \end{array}
\Big| \; \lambda^n \; \biggr]_p.
\end{equation}
If the representative $w$ we chose in each equivalence class is such that $w_i=0$ for some $1 \leq i \leq n$ then $\delta_w=1$. Therefore, choosing only representatives of this form yields
\begin{equation}
\notag N_p(\lambda)
= \frac{p^{n-1}-1}{p-1} +
 \sum_{[w^{\ast}] \in W/\sim} \;
\displaystyle\prod_{i=1}^{n} \biggfp{\tfrac{w_i}{d}}
\cdot (-p)^{\frac{1}{d} \sum_{i=1}^{n} w_i}
\cdot (-1)^{n} \cdot
{_{s}G_{s}}
\biggl[ \begin{array}{c} A_w \\
 B_w \end{array}
\Big| \; \lambda^n \; \biggr]_p.
\end{equation}
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor_RelPrime}]
If $d=1$ then $w=(0,0,\cdots, 0)$ is the only element  in $W$, and
$S_w= \emptyset$ and  $S_w^c=\{0\}.$
Thus
$A_w$ is  $\tfrac{1}{n}, \tfrac{2}{n}, \cdots , \tfrac{n-1}{n}$ and $B_w$ is $1, 1, \cdots, 1$ (${n-1}$ times).
Therefore in this case the result in Theorem \ref{thm_Main} reduces to
\begin{equation}
\notag N_p(\lambda)
= \frac{p^{n-1}-1}{p-1} +
(-1)^{n} \cdot
{_{n-1}G_{n-1}}
\biggl[ \begin{array}{cccc} \frac{1}{n} & \frac{2}{n} & \dots & \frac{n-1}{n} \\
 1 & 1 & \dots & 1 \end{array}
\Big| \; \lambda^n \; \biggr]_p.
\end{equation}
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor_FF}]
Corollary \ref{cor_FF} can be proved as a stand alone result, in a similar manner to Theorem \ref{thm_Main} but without having to transfer to the $p$-adic setting. But having proved Theorem \ref{thm_Main} above, we now derive Corollary \ref{cor_FF} from that result.

When $d=n$ then the list
$\left[ \tfrac{h}{n} \mid 0 \leq h \leq n-1, h \not\equiv 0 \imod{\tfrac{n}{d}} \right]$
is empty and so we get the lists
\begin{equation*}
A_w: \left[ \tfrac{n-k}{n} \mid k \in S_w \right] ; \textup{ and } B_w: \left[  \tfrac{n-k}{n} \, \textup{repeated $n_k$-1 times} \mid k \in S_w^c \right] .
\end{equation*}
We note that $|A_w|=|S_w| = d- |S_w^c| = n- |S_w^c| = \sum_{k \in S_w^c} (n_k - 1) = |B_w|$.
Then using Lemma \ref{lem_G_to_F} and Theorem \ref{thm_GrossKoblitz} we see that in this case Theorem \ref{thm_Main} reduces to
\begin{equation}\label{cor_FF_for1}
N_p(\lambda) = \frac{p^{n-1}-1}{p-1} +  \sum_{[w^{\ast}] \in W/\sim}
\prod_{i=1}^{n} g(\bar{\omega}^{w_i t})\;
{_{|S_w|}F_{|S_w|}} {\biggl( \begin{array}{c} A_{\bar{\omega},w}^{\prime} \\
 B_{\bar{\omega},w}^{\prime} \end{array}
\Big| \; \lambda^{-n} \; \biggr)}_{p}.\\
\end{equation}
This equation holds if we replace $\bar{\omega}$ by any generator $T$ for $\widehat{\mathbb{F}^{*}_{p}}$. To see this, let $T = \bar{\omega}^{\alpha}$ for some $0 \leq \alpha \leq {p-2}$ with $\gcd(\alpha, {p-1})=1$. Define a map
$f_{\alpha}: \; {W/\sim} \, \to \, {W/\sim}$
given by
$$ \quad f_{\alpha}[w] = [\alpha w \imod{n}],$$
where if $w=(w_1, w_2, \cdots w_n)$ then $\alpha w \imod{n} = (\alpha w_1 \imod{n}, \alpha w_2 \imod{n}, \cdots, \alpha w_n \imod{n})$. Then, as $\gcd(\alpha, n)=1$, $f_{\alpha}$ is a well-defined isomorphism on ${W/\sim}$. Now replacing $[w]$ by $[\alpha w \imod{n}]$ in (\ref{cor_FF_for1}) yields the result.
\end{proof}

\vspace{9pt}

redpajama_set_name: RedPajamaArXiv


Deborah King | Who goes out in the sun anymore? - Duration: 64 seconds.
Deborah King | My Story - Duration: 78 seconds.
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Deborah King | Self Forgiveness - Duration: 69 seconds.
Deborah King on Holistic Treatment | "Over The Hump" on Veria.com - Duration: 17 minutes.
Deborah King is a spiritual teacher and master healer who can pinpoint physical and nonverbal body language messages and signals of liars that indicates that someone is lying. Watch the videos in the playlist for more details.
Deborah King Explains How To Know If Your Relationship Might Be In Trouble - Duration: 3 minutes, 56 seconds.
Healing Courses | Deborah King Discusses Physical Signs of Lying - Duration: 3 minutes, 39 seconds.
Deborah King Discusses How to Use Body Language to Spot Lies - Duration: 5 minutes, 32 seconds.
Deborah King on Why People Lie - Duration: 4 minutes, 28 seconds.
Deborah King Talks About Nonverbal Communication In Relationships - Duration: 5 minutes, 11 seconds.
Deborah King on Lying - Duration: 5 minutes, 2 seconds.
Deborah King on How to Use Body Language as a Lie Detector - Duration: 4 minutes, 13 seconds.
Spiritual Teacher | Deborah King Tells Us How To Spot a Lie - Duration: 5 minutes, 14 seconds.
Deborah King on Tennessee Mornings | How to Tell When Someone is Lying - Duration: 6 minutes, 52 seconds.
Deborah King | Explains Why People Lie - Duration: 5 minutes, 16 seconds.
Spiritual Teacher Deborah King | How to Spot a Liar - Duration: 2 minutes, 17 seconds.
Deborah King | How To Spot A Lie - Duration: 6 minutes, 57 seconds.
Deborah King discusses why celebrities lie and what lessons we can learn from celebrity liars and celebrity relationships.
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Join Deborah Tuesdays at 1pm ET/4pm PT for her live show. You can join her on camera at https://www.crowdcast.io/e/the-deborah-king-show.
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This monograph is the second volume of a graduate text book on the modern theory of linear one-dimensional singular integral equations. Both volumes may be regarded as unique graduate text books. Singular integral equations attract more and more attention since this class of equations appears in numerous applications, and also because they form one of the few classes of equations which can be solved explicitly. The present book is to a great extent based upon material contained in the second part of the authors' monograph  which appeared in 1973 in Russian, and in 1979 in German translation. The present text includes a large number of additions and complementary material, essentially changing the character, structure and contents of the book, and making it accessible to a wider audience. Our main subject in the first volume was the case of closed curves and continuous coeffi cients. Here, in the second volume, we turn to general curves and discontinuous coefficients. We are deeply grateful to the editor Professor G. Heinig, to the translator Dr. S. Roeh, and to the typist Mr. G. Lillack, for their patient work. The authors Ramat-Aviv, Ramat-Gan, May 26, 1991 11 Introduction This book is the second volume of an introduction to the theory of linear one-dimensional singular integral operators. The main topics of both parts of the book are the invertibility and Fredholmness of these operators. Special attention is paid to inversion methods.


text:
Pit People, Heroes of Hammerwatch, & More New Indie Releases
Pictured above: Worshippers
As I mentioned in my previous release highlights article, last week was full of new games worthy of your attention. Here's five additional games that didn't make the original article.
The Behemoth
After around a year in Early Access, the latest game by the developers of Castle Crashers and BattleBlock Theater has been released. Pit People takes The Behemoth's signature aesthetic and quirky humor to a post-apocalyptic world after a giant magical space bear has collided with the planet. This turn-based strategy game streamlines and simplifies the experience by not allowing you to directly control your units' actions. That might sound infuriating, but the game's mechanics are based around manipulating your units in ways that influence their actions. Multiplayer plays a fairly large role in the game, including the ability to play the entire campaign cooperatively and 2v2 arena matches. You can also find Pit People on Xbox One.
Battlezone: Combat Commander
Big Boat Interactive
The original Battlezone was remastered back in 2016, and now its Battlezone 2's turn with Battlezone: Combat Commander. This rerelease keeps the core gameplay and story of this classic FPS/RTS hybrid intact, but features updated visuals, improved AI, restored online multiplayer with competitive and cooperative modes, and full Steam Workshop support. You can also find both remasters on the Humble Store and GOG.
Mulaka
Mulaka is a 3D adventure game inspired by the folklore of the Tarahumara tribe of northern Mexico. Cleanse your native land of a mysterious corruption as a Sukurúame, a traditional Tarahumara shaman and spiritual leader. Explore a colorful, low-poly world based on a mixture of real Mexican landscapes and the tribe's ancient mythology. Use shamanistic magic to transform into animal deities to fight enemies, solve puzzles, and traverse the environment. Mulaka is also available on the Humble Store, PS4, Switch, and Xbox One.
Vagante
Nuke Nine
Vagante is a 2D action roguelite platformer that exited Early Access a few weeks back. Explore a procedurally generated cave network packed with monsters, traps, and magical artifacts waiting to be recovered. You can play the entire game alone or with up to three friends in both local and online co-op. Vagante is also on the Humble Store.
Heroes of Hammerwatch
Another recent roguelite action-adventure to hit Steam is Heroes of Hammerwatch, the sequel to 2013's Hammerwatch. Like Vagante, Heroes of Hammerwatch allows you and up to three friends to battle your way through procedurally generated dungeons in search of enchanted items and other loot. Spend your newly-acquired wealth on improvements to the town center to give your party various buffs for future expeditions. Heroes of Hammerwatch is also available on the Humble Store and GOG.
Worshippers
Brainwashing Games
Finally we come to Worshippers, a 4X-lite inspired by popular strategy board games. Control a small tribe of warriors in a low-magic fantasy world based on Slavic folklore as you attempt to expand your god's influence and convert new worshippers to your religion. Your units and upgrades come in the form of cards that you'll collect during each session, adding an element of deckbuilding to every scenario. Worshippers is designed with quick sessions in mind, with the average scenario only taking about half an hour to complete.
Which of these games caught your attention?


text:
[Docs] [txt|pdf] [draft-ietf-dmar...] [Tracker]
PROPOSED STANDARD
Internet Engineering Task Force (IETF)                         J. Levine
Request for Comments: 8616                          Taughannock Networks
Updates: 6376, 7208, 7489                                      June 2019
Category: Standards Track
Email Authentication for Internationalized Mail
Sender Policy Framework (SPF) (RFC 7208), DomainKeys Identified Mail
(DKIM) (RFC 6376), and Domain-based Message Authentication,
Reporting, and Conformance (DMARC) (RFC 7489) enable a domain owner
to publish email authentication and policy information in the DNS.
In internationalized email, domain names can occur both as U-labels
and A-labels.  This specification updates the SPF, DKIM, and DMARC
specifications to clarify which form of internationalized domain
names to use in those specifications.
Internet Standards is available in Section 2 of RFC 7841.
https://www.rfc-editor.org/info/rfc8616.
Levine                       Standards Track                    [Page 1]
RFC 8616                   EAI Authentication                  June 2019
3.  General Principles  . . . . . . . . . . . . . . . . . . . . .   3
4.  SPF and Internationalized Mail  . . . . . . . . . . . . . . .   3
5.  DKIM and Internationalized Mail . . . . . . . . . . . . . . .   3
6.  DMARC and Internationalized Mail  . . . . . . . . . . . . . .   4
7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   5
8.  Security Considerations . . . . . . . . . . . . . . . . . . .   5
9.  Normative References  . . . . . . . . . . . . . . . . . . . .   5
Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .   6
SPF [RFC7208], DKIM [RFC6376], and DMARC [RFC7489] enable a domain
owner to publish email authentication and policy information in the
DNS.  SPF primarily publishes information about what host addresses
are authorized to send mail for a domain.  DKIM places cryptographic
signatures on email messages, with the validation keys published in
the DNS.  DMARC publishes policy information related to the domain in
the From: header field of email messages.
In conventional email, all domain names are ASCII in all contexts, so
there is no question about the representation of the domain names.
All internationalized domain names are represented as A-labels
[RFC5890] in message header fields, SMTP sessions, and the DNS.
Internationalized mail [RFC6530] (generally called "EAI" for Email
Address Internationalization) allows U-labels in SMTP sessions
[RFC6531] and message header fields [RFC6532].
Every U-label is equivalent to an A-label, so in principle, the
choice of label format will not cause ambiguities.  But in practice,
consistent use of label formats will make it more likely that code
for mail senders and receivers interoperates.
Internationalized mail also allows UTF-8-encoded Unicode characters
in the local parts of mailbox names, which were historically only
ASCII.
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
The term "IDN", for Internationalized Domain Name, refers to a domain
name containing either U-labels or A-labels.
Since DMARC is not currently a Standards Track protocol, this
specification offers advice rather than requirements for DMARC.
3.  General Principles
In headers in EAI mail messages, domain names that were restricted to
ASCII can be U-labels, and mailbox local parts can be UTF-8.  Header
field names and other text intended primarily to be interpreted by
computers rather than read by people remains ASCII.
Strings stored in DNS records remain ASCII since there is no way to
tell whether a client retrieving a DNS record expects an EAI or an
ASCII result.  When a domain name found in a mail header field
includes U-labels, those labels are translated to A-labels before
being looked up in the DNS, as described in [RFC5891].
4.  SPF and Internationalized Mail
SPF [RFC7208] uses two identities from the SMTP session: the host
name in the EHLO command and the domain in the address in the MAIL
FROM command.  Since the EHLO command precedes the server response
that tells whether the server supports the SMTPUTF8 extension, an IDN
host name MUST be represented as A-labels.  An IDN in MAIL FROM can
be either U-labels or A-labels.
All U-labels MUST be converted to A-labels before being used for an
SPF validation.  This includes both the labels in the name used for
the original DNS lookup, described in Section 3 of [RFC7208], and
those used in the macro expansion of domain-spec, described in
Section 7.  Section 4.3 of [RFC7208] states that all IDNs in an SPF
DNS record MUST be A-labels; this rule is unchanged since any SPF
record can be used to authorize either EAI or conventional mail.
SPF macros %{s} and %{l} expand the local part of the sender's
mailbox.  If the local part contains non-ASCII characters, terms that
include %{s} or %{l} do not match anything, because non-ASCII local
parts cannot be used as the DNS labels the macros are intended to
match.  Since these macros are rarely used, this is unlikely to be an
issue in practice.
5.  DKIM and Internationalized Mail
DKIM [RFC6376] specifies a mail header field that contains a
cryptographic message signature and a DNS record that contains the
validation key.
Section 2.11 of [RFC6376] defines dkim-quoted-printable.  Its
definition is modified in messages with internationalized header
fields so that non-ASCII UTF-8 characters need not be quoted.  The
ABNF [RFC5234] for dkim-safe-char in those messages is replaced by
the following, adding non-ASCII UTF-8 characters from [RFC3629]:
dkim-safe-char        =  %x21-3A / %x3C / %x3E-7E /
UTF8-2 / UTF8-3 / UTF8-4
; '!' - ':', '<', '>' - '~', non-ASCII
UTF8-2                = <Defined in Section 4 of RFC 3629>
Section 3.5 of [RFC6376] states that IDNs in the d=, i=, and s= tags
of a DKIM-Signature header field MUST be encoded as A-labels.  This
rule is relaxed only for internationalized message header fields
[RFC6532], so IDNs SHOULD be represented as U-labels.  This provides
improved consistency with other header fields.  (A-labels remain
valid to allow a transition from older software.)  The set of
allowable characters in the local part of an i= tag is extended in
the same fashion as local parts of email addresses as described in
Section 3.2 of [RFC6532].  When computing or verifying the hash in a
DKIM signature as described in Section 3.7 of [RFC6376], the hash
MUST use the domain name in the format it occurs in the header field.
Section 3.4.2 of [RFC6376] describes relaxed header canonicalization.
Its first step converts all header field names from uppercase to
lowercase.  Field names are restricted to printable ASCII (see
[RFC5322], Section 3.6.8), so this case conversion remains ASCII case
DKIM key records, described in Section 3.6.1 of [RFC6376], do not
contain domain names, so there is no change to their specification.
6.  DMARC and Internationalized Mail
DMARC [RFC7489] defines a policy language that domain owners can
specify for the domain of the address in an RFC5322.From header
Section 6.6.1 of [RFC7489] specifies, somewhat imprecisely, how IDNs
in the RFC5322.From address domain are to be handled.  That section
is updated to say that all U-labels in the domain are converted to
A-labels before further processing.  Section 7.1 of [RFC7489] is
similarly updated to say that all U-labels in domains being handled
are converted to A-labels before further processing.
DMARC policy records, described in Sections 6.3 and 7.1 of [RFC7489],
can contain email addresses in the "rua" and "ruf" tags.  Since a
policy record can be used for both internationalized and conventional
mail, those addresses still have to be conventional addresses, not
internationalized addresses.
This document has no IANA actions.
Email is subject to a vast range of threats and abuses.  This
document attempts to slightly mitigate some of them but does not, as
far as the author knows, add any new ones.  The updates to SPF, DKIM,
and DMARC are intended to allow the respective specifications to work
as reliably on internationalized mail as they do on ASCII mail, so
that applications that use them, such as some kinds of mail filters
that catch spam and phish, can work more reliably on
internationalized mail.
9.  Normative References
[RFC3629]  Yergeau, F., "UTF-8, a transformation format of ISO
10646", STD 63, RFC 3629, DOI 10.17487/RFC3629, November
2003, <https://www.rfc-editor.org/info/rfc3629>.
[RFC5234]  Crocker, D., Ed. and P. Overell, "Augmented BNF for Syntax
Specifications: ABNF", STD 68, RFC 5234,
DOI 10.17487/RFC5234, January 2008,
[RFC5322]  Resnick, P., Ed., "Internet Message Format", RFC 5322,
DOI 10.17487/RFC5322, October 2008,
[RFC5890]  Klensin, J., "Internationalized Domain Names for
Applications (IDNA): Definitions and Document Framework",
RFC 5890, DOI 10.17487/RFC5890, August 2010,
[RFC5891]  Klensin, J., "Internationalized Domain Names in
Applications (IDNA): Protocol", RFC 5891,
DOI 10.17487/RFC5891, August 2010,
[RFC6376]  Crocker, D., Ed., Hansen, T., Ed., and M. Kucherawy, Ed.,
"DomainKeys Identified Mail (DKIM) Signatures", STD 76,
RFC 6376, DOI 10.17487/RFC6376, September 2011,
[RFC6530]  Klensin, J. and Y. Ko, "Overview and Framework for
Internationalized Email", RFC 6530, DOI 10.17487/RFC6530,
February 2012, <https://www.rfc-editor.org/info/rfc6530>.
[RFC6531]  Yao, J. and W. Mao, "SMTP Extension for Internationalized
Email", RFC 6531, DOI 10.17487/RFC6531, February 2012,
[RFC6532]  Yang, A., Steele, S., and N. Freed, "Internationalized
Email Headers", RFC 6532, DOI 10.17487/RFC6532, February
[RFC7208]  Kitterman, S., "Sender Policy Framework (SPF) for
Authorizing Use of Domains in Email, Version 1", RFC 7208,
DOI 10.17487/RFC7208, April 2014,
[RFC7489]  Kucherawy, M., Ed. and E. Zwicky, Ed., "Domain-based
Message Authentication, Reporting, and Conformance
(DMARC)", RFC 7489, DOI 10.17487/RFC7489, March 2015,
[RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>.
John Levine
Taughannock Networks
Trumansburg, NY  14886
Email: standards@taugh.com
URI:   http://jl.ly


text:
'use strict';
if (require('./_descriptors')) {
    var LIBRARY = require('./_library')
        , global = require('./_global')
        , fails = require('./_fails')
        , $export = require('./_export')
        , $typed = require('./_typed')
        , $buffer = require('./_typed-buffer')
        , ctx = require('./_ctx')
        , anInstance = require('./_an-instance')
        , propertyDesc = require('./_property-desc')
        , hide = require('./_hide')
        , redefineAll = require('./_redefine-all')
        , toInteger = require('./_to-integer')
        , toLength = require('./_to-length')
        , toIndex = require('./_to-index')
        , toPrimitive = require('./_to-primitive')
        , has = require('./_has')
        , same = require('./_same-value')
        , classof = require('./_classof')
        , isObject = require('./_is-object')
        , toObject = require('./_to-object')
        , isArrayIter = require('./_is-array-iter')
        , create = require('./_object-create')
        , getPrototypeOf = require('./_object-gpo')
        , gOPN = require('./_object-gopn').f
        , getIterFn = require('./core.get-iterator-method')
        , uid = require('./_uid')
        , wks = require('./_wks')
        , createArrayMethod = require('./_array-methods')
        , createArrayIncludes = require('./_array-includes')
        , speciesConstructor = require('./_species-constructor')
        , ArrayIterators = require('./es6.array.iterator')
        , Iterators = require('./_iterators')
        , $iterDetect = require('./_iter-detect')
        , setSpecies = require('./_set-species')
        , arrayFill = require('./_array-fill')
        , arrayCopyWithin = require('./_array-copy-within')
        , $DP = require('./_object-dp')
        , $GOPD = require('./_object-gopd')
        , dP = $DP.f
        , gOPD = $GOPD.f
        , RangeError = global.RangeError
        , TypeError = global.TypeError
        , Uint8Array = global.Uint8Array
        , ARRAY_BUFFER = 'ArrayBuffer'
        , SHARED_BUFFER = 'Shared' + ARRAY_BUFFER
        , BYTES_PER_ELEMENT = 'BYTES_PER_ELEMENT'
        , PROTOTYPE = 'prototype'
        , ArrayProto = Array[PROTOTYPE]
        , $ArrayBuffer = $buffer.ArrayBuffer
        , $DataView = $buffer.DataView
        , arrayForEach = createArrayMethod(0)
        , arrayFilter = createArrayMethod(2)
        , arraySome = createArrayMethod(3)
        , arrayEvery = createArrayMethod(4)
        , arrayFind = createArrayMethod(5)
        , arrayFindIndex = createArrayMethod(6)
        , arrayIncludes = createArrayIncludes(true)
        , arrayIndexOf = createArrayIncludes(false)
        , arrayValues = ArrayIterators.values
        , arrayKeys = ArrayIterators.keys
        , arrayEntries = ArrayIterators.entries
        , arrayLastIndexOf = ArrayProto.lastIndexOf
        , arrayReduce = ArrayProto.reduce
        , arrayReduceRight = ArrayProto.reduceRight
        , arrayJoin = ArrayProto.join
        , arraySort = ArrayProto.sort
        , arraySlice = ArrayProto.slice
        , arrayToString = ArrayProto.toString
        , arrayToLocaleString = ArrayProto.toLocaleString
        , ITERATOR = wks('iterator')
        , TAG = wks('toStringTag')
        , TYPED_CONSTRUCTOR = uid('typed_constructor')
        , DEF_CONSTRUCTOR = uid('def_constructor')
        , ALL_CONSTRUCTORS = $typed.CONSTR
        , TYPED_ARRAY = $typed.TYPED
        , VIEW = $typed.VIEW
        , WRONG_LENGTH = 'Wrong length!';

    var $map = createArrayMethod(1, function (O, length) {
        return allocate(speciesConstructor(O, O[DEF_CONSTRUCTOR]), length);
    });

    var LITTLE_ENDIAN = fails(function () {
        return new Uint8Array(new Uint16Array([1]).buffer)[0] === 1;
    });

    var FORCED_SET = !!Uint8Array && !!Uint8Array[PROTOTYPE].set && fails(function () {
            new Uint8Array(1).set({});
        });

    var strictToLength = function (it, SAME) {
        if (it === undefined)throw TypeError(WRONG_LENGTH);
        var number = +it
            , length = toLength(it);
        if (SAME && !same(number, length))throw RangeError(WRONG_LENGTH);
        return length;
    };

    var toOffset = function (it, BYTES) {
        var offset = toInteger(it);
        if (offset < 0 || offset % BYTES)throw RangeError('Wrong offset!');
        return offset;
    };

    var validate = function (it) {
        if (isObject(it) && TYPED_ARRAY in it)return it;
        throw TypeError(it + ' is not a typed array!');
    };

    var allocate = function (C, length) {
        if (!(isObject(C) && TYPED_CONSTRUCTOR in C)) {
            throw TypeError('It is not a typed array constructor!');
        }
        return new C(length);
    };

    var speciesFromList = function (O, list) {
        return fromList(speciesConstructor(O, O[DEF_CONSTRUCTOR]), list);
    };

    var fromList = function (C, list) {
        var index = 0
            , length = list.length
            , result = allocate(C, length);
        while (length > index)result[index] = list[index++];
        return result;
    };

    var addGetter = function (it, key, internal) {
        dP(it, key, {
            get: function () {
                return this._d[internal];
            }
        });
    };

    var $from = function from(source /*, mapfn, thisArg */) {
        var O = toObject(source)
            , aLen = arguments.length
            , mapfn = aLen > 1 ? arguments[1] : undefined
            , mapping = mapfn !== undefined
            , iterFn = getIterFn(O)
            , i, length, values, result, step, iterator;
        if (iterFn != undefined && !isArrayIter(iterFn)) {
            for (iterator = iterFn.call(O), values = [], i = 0; !(step = iterator.next()).done; i++) {
                values.push(step.value);
            }
            O = values;
        }
        if (mapping && aLen > 2)mapfn = ctx(mapfn, arguments[2], 2);
        for (i = 0, length = toLength(O.length), result = allocate(this, length); length > i; i++) {
            result[i] = mapping ? mapfn(O[i], i) : O[i];
        }
        return result;
    };

    var $of = function of(/*...items*/) {
        var index = 0
            , length = arguments.length
            , result = allocate(this, length);
        while (length > index)result[index] = arguments[index++];
        return result;
    };

    // iOS Safari 6.x fails here
    var TO_LOCALE_BUG = !!Uint8Array && fails(function () {
            arrayToLocaleString.call(new Uint8Array(1));
        });

    var $toLocaleString = function toLocaleString() {
        return arrayToLocaleString.apply(TO_LOCALE_BUG ? arraySlice.call(validate(this)) : validate(this), arguments);
    };

    var proto = {
        copyWithin: function copyWithin(target, start /*, end */) {
            return arrayCopyWithin.call(validate(this), target, start, arguments.length > 2 ? arguments[2] : undefined);
        },
        every: function every(callbackfn /*, thisArg */) {
            return arrayEvery(validate(this), callbackfn, arguments.length > 1 ? arguments[1] : undefined);
        },
        fill: function fill(value /*, start, end */) { // eslint-disable-line no-unused-vars
            return arrayFill.apply(validate(this), arguments);
        },
        filter: function filter(callbackfn /*, thisArg */) {
            return speciesFromList(this, arrayFilter(validate(this), callbackfn,
                arguments.length > 1 ? arguments[1] : undefined));
        },
        find: function find(predicate /*, thisArg */) {
            return arrayFind(validate(this), predicate, arguments.length > 1 ? arguments[1] : undefined);
        },
        findIndex: function findIndex(predicate /*, thisArg */) {
            return arrayFindIndex(validate(this), predicate, arguments.length > 1 ? arguments[1] : undefined);
        },
        forEach: function forEach(callbackfn /*, thisArg */) {
            arrayForEach(validate(this), callbackfn, arguments.length > 1 ? arguments[1] : undefined);
        },
        indexOf: function indexOf(searchElement /*, fromIndex */) {
            return arrayIndexOf(validate(this), searchElement, arguments.length > 1 ? arguments[1] : undefined);
        },
        includes: function includes(searchElement /*, fromIndex */) {
            return arrayIncludes(validate(this), searchElement, arguments.length > 1 ? arguments[1] : undefined);
        },
        join: function join(separator) { // eslint-disable-line no-unused-vars
            return arrayJoin.apply(validate(this), arguments);
        },
        lastIndexOf: function lastIndexOf(searchElement /*, fromIndex */) { // eslint-disable-line no-unused-vars
            return arrayLastIndexOf.apply(validate(this), arguments);
        },
        map: function map(mapfn /*, thisArg */) {
            return $map(validate(this), mapfn, arguments.length > 1 ? arguments[1] : undefined);
        },
        reduce: function reduce(callbackfn /*, initialValue */) { // eslint-disable-line no-unused-vars
            return arrayReduce.apply(validate(this), arguments);
        },
        reduceRight: function reduceRight(callbackfn /*, initialValue */) { // eslint-disable-line no-unused-vars
            return arrayReduceRight.apply(validate(this), arguments);
        },
        reverse: function reverse() {
            var that = this
                , length = validate(that).length
                , middle = Math.floor(length / 2)
                , index = 0
                , value;
            while (index < middle) {
                value = that[index];
                that[index++] = that[--length];
                that[length] = value;
            }
            return that;
        },
        some: function some(callbackfn /*, thisArg */) {
            return arraySome(validate(this), callbackfn, arguments.length > 1 ? arguments[1] : undefined);
        },
        sort: function sort(comparefn) {
            return arraySort.call(validate(this), comparefn);
        },
        subarray: function subarray(begin, end) {
            var O = validate(this)
                , length = O.length
                , $begin = toIndex(begin, length);
            return new (speciesConstructor(O, O[DEF_CONSTRUCTOR]))(
                O.buffer,
                O.byteOffset + $begin * O.BYTES_PER_ELEMENT,
                toLength((end === undefined ? length : toIndex(end, length)) - $begin)
            );
        }
    };

    var $slice = function slice(start, end) {
        return speciesFromList(this, arraySlice.call(validate(this), start, end));
    };

    var $set = function set(arrayLike /*, offset */) {
        validate(this);
        var offset = toOffset(arguments[1], 1)
            , length = this.length
            , src = toObject(arrayLike)
            , len = toLength(src.length)
            , index = 0;
        if (len + offset > length)throw RangeError(WRONG_LENGTH);
        while (index < len)this[offset + index] = src[index++];
    };

    var $iterators = {
        entries: function entries() {
            return arrayEntries.call(validate(this));
        },
        keys: function keys() {
            return arrayKeys.call(validate(this));
        },
        values: function values() {
            return arrayValues.call(validate(this));
        }
    };

    var isTAIndex = function (target, key) {
        return isObject(target)
            && target[TYPED_ARRAY]
            && typeof key != 'symbol'
            && key in target
            && String(+key) == String(key);
    };
    var $getDesc = function getOwnPropertyDescriptor(target, key) {
        return isTAIndex(target, key = toPrimitive(key, true))
            ? propertyDesc(2, target[key])
            : gOPD(target, key);
    };
    var $setDesc = function defineProperty(target, key, desc) {
        if (isTAIndex(target, key = toPrimitive(key, true))
            && isObject(desc)
            && has(desc, 'value')
            && !has(desc, 'get')
            && !has(desc, 'set')
            // TODO: add validation descriptor w/o calling accessors
            && !desc.configurable
            && (!has(desc, 'writable') || desc.writable)
            && (!has(desc, 'enumerable') || desc.enumerable)
        ) {
            target[key] = desc.value;
            return target;
        } else return dP(target, key, desc);
    };

    if (!ALL_CONSTRUCTORS) {
        $GOPD.f = $getDesc;
        $DP.f = $setDesc;
    }

    $export($export.S + $export.F * !ALL_CONSTRUCTORS, 'Object', {
        getOwnPropertyDescriptor: $getDesc,
        defineProperty: $setDesc
    });

    if (fails(function () {
            arrayToString.call({});
        })) {
        arrayToString = arrayToLocaleString = function toString() {
            return arrayJoin.call(this);
        }
    }

    var $TypedArrayPrototype$ = redefineAll({}, proto);
    redefineAll($TypedArrayPrototype$, $iterators);
    hide($TypedArrayPrototype$, ITERATOR, $iterators.values);
    redefineAll($TypedArrayPrototype$, {
        slice: $slice,
        set: $set,
        constructor: function () { /* noop */
        },
        toString: arrayToString,
        toLocaleString: $toLocaleString
    });
    addGetter($TypedArrayPrototype$, 'buffer', 'b');
    addGetter($TypedArrayPrototype$, 'byteOffset', 'o');
    addGetter($TypedArrayPrototype$, 'byteLength', 'l');
    addGetter($TypedArrayPrototype$, 'length', 'e');
    dP($TypedArrayPrototype$, TAG, {
        get: function () {
            return this[TYPED_ARRAY];
        }
    });

    module.exports = function (KEY, BYTES, wrapper, CLAMPED) {
        CLAMPED = !!CLAMPED;
        var NAME = KEY + (CLAMPED ? 'Clamped' : '') + 'Array'
            , ISNT_UINT8 = NAME != 'Uint8Array'
            , GETTER = 'get' + KEY
            , SETTER = 'set' + KEY
            , TypedArray = global[NAME]
            , Base = TypedArray || {}
            , TAC = TypedArray && getPrototypeOf(TypedArray)
            , FORCED = !TypedArray || !$typed.ABV
            , O = {}
            , TypedArrayPrototype = TypedArray && TypedArray[PROTOTYPE];
        var getter = function (that, index) {
            var data = that._d;
            return data.v[GETTER](index * BYTES + data.o, LITTLE_ENDIAN);
        };
        var setter = function (that, index, value) {
            var data = that._d;
            if (CLAMPED)value = (value = Math.round(value)) < 0 ? 0 : value > 0xff ? 0xff : value & 0xff;
            data.v[SETTER](index * BYTES + data.o, value, LITTLE_ENDIAN);
        };
        var addElement = function (that, index) {
            dP(that, index, {
                get: function () {
                    return getter(this, index);
                },
                set: function (value) {
                    return setter(this, index, value);
                },
                enumerable: true
            });
        };
        if (FORCED) {
            TypedArray = wrapper(function (that, data, $offset, $length) {
                anInstance(that, TypedArray, NAME, '_d');
                var index = 0
                    , offset = 0
                    , buffer, byteLength, length, klass;
                if (!isObject(data)) {
                    length = strictToLength(data, true)
                    byteLength = length * BYTES;
                    buffer = new $ArrayBuffer(byteLength);
                } else if (data instanceof $ArrayBuffer || (klass = classof(data)) == ARRAY_BUFFER || klass == SHARED_BUFFER) {
                    buffer = data;
                    offset = toOffset($offset, BYTES);
                    var $len = data.byteLength;
                    if ($length === undefined) {
                        if ($len % BYTES)throw RangeError(WRONG_LENGTH);
                        byteLength = $len - offset;
                        if (byteLength < 0)throw RangeError(WRONG_LENGTH);
                    } else {
                        byteLength = toLength($length) * BYTES;
                        if (byteLength + offset > $len)throw RangeError(WRONG_LENGTH);
                    }
                    length = byteLength / BYTES;
                } else if (TYPED_ARRAY in data) {
                    return fromList(TypedArray, data);
                } else {
                    return $from.call(TypedArray, data);
                }
                hide(that, '_d', {
                    b: buffer,
                    o: offset,
                    l: byteLength,
                    e: length,
                    v: new $DataView(buffer)
                });
                while (index < length)addElement(that, index++);
            });
            TypedArrayPrototype = TypedArray[PROTOTYPE] = create($TypedArrayPrototype$);
            hide(TypedArrayPrototype, 'constructor', TypedArray);
        } else if (!$iterDetect(function (iter) {
                // V8 works with iterators, but fails in many other cases
                // https://code.google.com/p/v8/issues/detail?id=4552
                new TypedArray(null); // eslint-disable-line no-new
                new TypedArray(iter); // eslint-disable-line no-new
            }, true)) {
            TypedArray = wrapper(function (that, data, $offset, $length) {
                anInstance(that, TypedArray, NAME);
                var klass;
                // `ws` module bug, temporarily remove validation length for Uint8Array
                // https://github.com/websockets/ws/pull/645
                if (!isObject(data))return new Base(strictToLength(data, ISNT_UINT8));
                if (data instanceof $ArrayBuffer || (klass = classof(data)) == ARRAY_BUFFER || klass == SHARED_BUFFER) {
                    return $length !== undefined
                        ? new Base(data, toOffset($offset, BYTES), $length)
                        : $offset !== undefined
                        ? new Base(data, toOffset($offset, BYTES))
                        : new Base(data);
                }
                if (TYPED_ARRAY in data)return fromList(TypedArray, data);
                return $from.call(TypedArray, data);
            });
            arrayForEach(TAC !== Function.prototype ? gOPN(Base).concat(gOPN(TAC)) : gOPN(Base), function (key) {
                if (!(key in TypedArray))hide(TypedArray, key, Base[key]);
            });
            TypedArray[PROTOTYPE] = TypedArrayPrototype;
            if (!LIBRARY)TypedArrayPrototype.constructor = TypedArray;
        }
        var $nativeIterator = TypedArrayPrototype[ITERATOR]
            , CORRECT_ITER_NAME = !!$nativeIterator && ($nativeIterator.name == 'values' || $nativeIterator.name == undefined)
            , $iterator = $iterators.values;
        hide(TypedArray, TYPED_CONSTRUCTOR, true);
        hide(TypedArrayPrototype, TYPED_ARRAY, NAME);
        hide(TypedArrayPrototype, VIEW, true);
        hide(TypedArrayPrototype, DEF_CONSTRUCTOR, TypedArray);

        if (CLAMPED ? new TypedArray(1)[TAG] != NAME : !(TAG in TypedArrayPrototype)) {
            dP(TypedArrayPrototype, TAG, {
                get: function () {
                    return NAME;
                }
            });
        }

        O[NAME] = TypedArray;

        $export($export.G + $export.W + $export.F * (TypedArray != Base), O);

        $export($export.S, NAME, {
            BYTES_PER_ELEMENT: BYTES,
            from: $from,
            of: $of
        });

        if (!(BYTES_PER_ELEMENT in TypedArrayPrototype))hide(TypedArrayPrototype, BYTES_PER_ELEMENT, BYTES);

        $export($export.P, NAME, proto);

        setSpecies(NAME);

        $export($export.P + $export.F * FORCED_SET, NAME, {set: $set});

        $export($export.P + $export.F * !CORRECT_ITER_NAME, NAME, $iterators);

        $export($export.P + $export.F * (TypedArrayPrototype.toString != arrayToString), NAME, {toString: arrayToString});

        $export($export.P + $export.F * fails(function () {
                new TypedArray(1).slice();
            }), NAME, {slice: $slice});

        $export($export.P + $export.F * (fails(function () {
                return [1, 2].toLocaleString() != new TypedArray([1, 2]).toLocaleString()
            }) || !fails(function () {
                TypedArrayPrototype.toLocaleString.call([1, 2]);
            })), NAME, {toLocaleString: $toLocaleString});

        Iterators[NAME] = CORRECT_ITER_NAME ? $nativeIterator : $iterator;
        if (!LIBRARY && !CORRECT_ITER_NAME)hide(TypedArrayPrototype, ITERATOR, $iterator);
    };
} else module.exports = function () { /* empty */
};

redpajama_set_name: RedPajamaGithub


Why look for FDC3 enabled applications?
Why should I FDC3-enable my applications?
There is a trend towards breaking up monolithic desktop applications, replacing them with adaptible wokflows which involve teh collaboration of multiple best-of-breed applications. Still much of the integration on the desktop is done by the actual end-user; copy/paste between applications, exporting/importing CSV files etc. Every application that has manual user input is a candidate for being FDC3-enabled, being able to demonstrate that your application can effectively take part in a worklow (without manual dual-entry or other tedious operations) is a easy route to happier users. Allowing your application to reach out to others is another way of extending the power of your offering; your app might not offer charting, but can let the end-user chart in an FDC3 enabled companion application based on context passed from your own app.
Why should my development team look at adopting FDC3?
Deploying effective end-user workflows with as little development effort as possible, should be the goal for all internal/platform integration development teams. Implementing or developing on a platform that is FDC3 enabled, if done right, results in more bang for the buck. FDC3 is all about (re)usability and low-touch integration, with an App directory in place and a platform to develop on, each new enabled app broadens the value of the workflow offering.


text:
World War II Veterans Oral History Collection
Home Oral History Collection John Ellis oral history interview audio
John Ellis oral history interview
John Ellis oral history interview audio
Title John Ellis oral history interview
Subject United States. Marine Corps. Marine Division, 2nd. Marines, 8th.
World War, 1939-1945--Communications.
World War, 1939-1945--Campaigns--Guam
United States. Marine Corps. Division, 3rd.
Iwo Jima, Battle of, Japan, 1945.
Description The National Museum of the Pacific War presents an oral interview with John Ellis. Ellis was born 17 December 1917 in Chillicothe, Ohio. After graduating from high school in 1935, he attended Miami University in Oxford, Ohio. While there he joined the Marine Corps Reserve and entered into platoon leader classes. Upon graduating from college in 1939, he was commissioned a second lieutenant and went to Philadelphia for advanced courses. Upon completion, he reported to San Diego where he was assigned as battalion communications officer for the 2nd Battalion, 8th Marines. During October 1941, he attended the battalion regimental communications officer course at Fort Benning, Georgia. In 1942 he went to American Samoa aboard the USS Zeilin (APA-3). In January 1944 he was ordered to return to Camp Pendleton to establish and command the 3rd Joint Assault Signal Company (JASCO). On 10 April 1944 the unit boarded the USS Algorab (AKA-8) and sailed for Guadalcanal, where they began training for the invasion of Guam. The day after going ashore during the invasion of Guam, Ellis was wounded by shell fragments. He returned after being treated aboard an LST offshore. After Guam was secured the unit began preparing for the invasion of Iwo Jima. Ellis comments on the difficulty in moving over the volcanic ash on Iwo Jima. After Iwo Jima was secured, Ellis returned to Guam and remained there until he returned to the United States in December 1945. Ellis stayed in the Marine Corps until 1967 when he retired as a colonel.
Creator Ellis, John H.
Publisher National Museum of the Pacific War
Contributors Misenhimer, Richard
Cook, Wanda
Type Sound
Format audio/mp3
Identifier OH00794
Rights CC-BY-SA (http://creativecommons.org/licenses/by-sa/4.0/)
Title John Ellis oral history interview audio
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OH00794aud Ellis, John
OH00794trs Ellis, John
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