text:
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Tag Archives: Spurlock Museum
"500 & 5" at the Spurlock Museum
Posted on December 16, 2014 by Katrina Spencer
Autorickshaws. Curry. Hijra. Tamil. Bindis. And rupees. Guess that country.
If you guessed India, you were right. This is the country that "500 & 5" highlighted at the International and Area Studies (IAS) Library's screening on Sunday, December 14th at the Spurlock Museum, organized by South Asian Studies Librarian Mara Thacker. Wanting to take advantage of one of the film creator's presence in Urbana-Champaign, 40 people gathered to view the Tamil-language piece, which was followed by a question and answer session with Kousalya Jeganathan.
"500 & 5" tells the tale of a 500 rupee note of Indian currency that travels through the hands of five different people in South India. While only valued at about $8 US, the bill's impact is immeasurable in deciding the fates of many, including those of a gangster, a chauffeur and a woman suffering from mental illness and drug addiction. Filmed in modern-day India, the piece evokes a variety of themes like organized crime, divorce and the illicit use of narcotics. The theme connecting the various stories is the overwhelming influence of money in a variety of social situations. While the rupees solve no problems in the film, they certainly increase the tensions in relationships based on authority and subservience.
The film has had some difficulty finding distribution, Jeganathan mentioned in the talkback afterwards. Some of the tropes reject the conventions of the traditional feature film. For example, the film is split into five shorts as opposed to films with one major story line. Women are featured as prominent characters and are not merely the object of amorous pursuits as is a pattern in many parts of the world, including the East and the West. And, more than anything else, the film is explicitly anti-consumerist and anti-capitalistic, frequently criticizing the power that money represents. Distributors have therefore been reluctant to risk supporting a film they are unsure will succeed at the box office.
The audience's response to the film was inquisitive and its questions revealed some unique details about the film making process. In order to dedicate themselves fully to the task, Jenganthan and other "500 & 5" creators quit their jobs, truly manifesting the idea that provoking thought was a higher priority than monetary gain. Certain scenes were filmed in Jeganathan's home. Many of the actors came from a theater background, and the role of the hijra character opened new discussions of a third gender that is widely accepted in Indian culture. Jeganathan shared that envisioning a moneyless culture was new and challenging for many audiences, but was perhaps still a worthy exercise, even if momentary and fleeting.
Calmly accepting the fate of the film, Jeganathan stated that "whenever the universe wants it, it will come out." "500 & 5" is recommended to audiences interested in the cinematic representation of South Asia. It offers a visually rich tapestry of several socioeconomic classes of Indian society, from the very poor and illiterate to the exaggeratedly rich figures of the entertainment industry. For more information on the film, visit accessiblehorizonfilms.com and be on the lookout for more events from the International and Area Studies Library by liking our Facebook page.
Posted in Campus, Culture, Events, Local	| Tagged 500 & 5, film, film screening, India, International and Area Studies Library, Mara Thacker, rupee, short films, South Asian Studies, Spurlock Museum	| Leave a reply
"Glocal" comes from the term "glocalization," which comes to us by way of Japan from the term "dochakuka" (土着化). In Japanese it means the process of indiginization but also the blending of global and local or cultural hybridity. The term focuses on the particularism of global phenomena and is thus a blend of international and area studies.
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text:
Chosen Survivors (1974)
The movie starts off with civilians being dropped via helicopter to a hidden location in he desert and put aboard an elevator which takes them nearly 2000 feet beneath the surface, to a self-contained habitat. As they recover from being drugged, they are informed that they have been chosen by the US government to carry on teh survival of the human race--and that they'll be able to go back to the surface in five years, when the radiation has tapered off.
However, they soon learn that they're not the only survivors. There's a huge colony of vampire bats--and as humans are now the only food source those bats want to get to them. But then about an hour into the movie there's a revelation by the psychologist in the group that changes everything--and then they plan to escape.
In the meantime they rig up a trap to electrocute the bats but only manage to destroy a portion of them.
This movie has the feel of OMEGA MAN and SOYLENT GREEN, though without Charlton Heston. But there's a solid cast with Jackie Cooper as the alcoholic rich guy, Alex Cord (GENESIS II) as a writer, Bradford Dillman (BUG!, ESCAPE FROM THE PLANET OF THE APES) as the psychologist, Diana Muldaur (PLANET EARTH) as a Congresswoman and Richard Jaeckel (THE GREEN SLIME) as the lone Army Major.
» Search Database for Chosen Survivors (1974)
Rating: 0.0 out of 10.0 - 0 votes cast total

redpajama_set_name: RedPajamaCommonCrawl


From film editing, to photographic editing at FreshCrewStudios, we can help. Below is an example of one of FreshCrewStudios photos taken in Manchester, and edited using Adobe Photoshop CC. The image of Sophie as female Gaston is shown from the first initial image, which was taken outside of Manchester Central through to the final edited version, where she is transformed into a wonderful forest setting.
Original un-edited image taken at Manchester Central.
Image as it progresses through the editing suite.
The final version of the image, with all layers completed.
Image editing can be extensive, as in the example above, or very subtle, as in the image to the right. The key is not to go overboard, and to ensure the final result is plausible, and not contrived. Image editing includes colouring and in some cases the addition, or removal of elements, to enhance the final outcome.
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Wedding image captured in RAW format and processed for natural look.


text:
'use strict';

let addons = require('./../../../lib/addons/addons');
let errors = require('./../../../lib/utils/errors');
let expect = require('chai').expect;
let Module = require('module');
let sinon = require('sinon');

describe('addons', function () {

  describe('loadAddons()', function () {
    it('should properly load configured addons', function () {
      let configuredAddons = [
        'basic',
        {name: 'withInit', settings: {port: 5000}}
      ];

      let initSpy = sinon.spy();
      let stubbedRequire = sinon.stub(Module, '_load');
      stubbedRequire.withArgs('basic').returns({});
      stubbedRequire.withArgs('withInit').returns({
        init: initSpy
      });
      stubbedRequire.throws(new Error('Unexpected call'));

      let loadedAddons;

      try {
        loadedAddons = addons.loadAddons(configuredAddons);
      } catch (e) {
        stubbedRequire.restore();
        throw e;
      }

      return Promise.all(loadedAddons).then(function (loaded) {
        expect(loaded).to.be.an('array').that.have.length(2);
        expect(initSpy.calledOnce).to.be.equal(true);
        expect(initSpy.calledWith(sinon.match({port: 5000}))).to.be.equal(true);
        stubbedRequire.restore();
      }).catch(function (e) {
        stubbedRequire.restore();
        throw e;
      });
    });

    it('should throw error when initialization fails', function () {
      let configuredAddons = [
        {name: 'failingAddon', settings: {foo: 'bar'}}
      ];

      let stubbedRequire = sinon.stub(Module, '_load');
      stubbedRequire.withArgs('failingAddon').returns({
        init: function () {
          throw new Error('Addon error');
        }
      });

      let loadedAddons;

      try {
        loadedAddons = addons.loadAddons(configuredAddons);
      } catch (e) {
        stubbedRequire.restore();
        throw e;
      }

      return Promise.all(loadedAddons).then(function () {
        throw new Error('Should be resolve due to failing addon.');
      }).catch(function (e) {
        stubbedRequire.restore();
        expect(e).to.be.instanceOf(errors.AddonError);
        expect(e.message).to.contains('during "init"');
      });
    });
  });

  describe('callHook()', function () {
    it('should call specified hook', function () {
      let firtsLoadedAddon = {
        name: 'first',
        hooks: {
          config: sinon.spy()
        }
      };
      let secondHook = {
        name: 'second',
        hooks: {
          config: sinon.spy(),
          testOptions: sinon.spy()
        }
      };

      return addons.prepareHooks([firtsLoadedAddon, secondHook]).then(function () {
        return addons.callHook(addons.hookTypes.config, 'data').then(function (results) {
          expect(results).to.be.an('array').that.have.length(2);
          expect(firtsLoadedAddon.hooks.config.calledTwice).to.be.equal(true);
          expect(secondHook.hooks.config.calledTwice).to.be.equal(true);
          expect(secondHook.hooks.testOptions.called).to.be.equal(false);
        });
      });
    });

    it('should handle failing hook gracefully', function () {
      let workingAddon = {
        name: 'working',
        hooks: {
          testOptions: sinon.stub().returns([{
            long: 'exampleParam',
            description: 'description of exampleParam'
          }]),
          addonError: sinon.spy()
        }
      };

      let failingAddon = {
        name: 'failing',
        hooks: {
          testOptions: sinon.stub().throws(),
          addonError: sinon.spy()
        }
      };

      return addons.prepareHooks([workingAddon, failingAddon]).then(function () {
        return addons.callHook(addons.hookTypes.testOptions).then(function (results) {
          expect(results).to.be.an('array').that.have.length(1);
          expect(results[0][0]).to.be.an('object').with.property('long', 'exampleParam');
          expect(failingAddon.hooks.addonError.calledOnce).to.be.equal(true);
          expect(workingAddon.hooks.addonError.called).to.be.equal(false);
        });
      });
    });
  });

});

redpajama_set_name: RedPajamaGithub


The growing demand for dynamic supply chains leads organizations to seek out robust warehouse management systems (WMS) that support increasingly sophisticated supply chain strategies. The trend is for deployment of a "smart" WMS that meets demanding process and efficiency requirements within the four walls of DC while integrating with upstream and downstream supply chain processes.
Labor costs, labor availability, and the need for more rapid order fulfillment in an omnichannel world are also driving increased adoption of automated systems in distribution, ushering in a new era of flexible, next-gen, near autonomous warehouses. As a result, the WMS must possess a "smart integration" capability to seamlessly work with various automated technologies deployed across the DC. Non-automated DCs are also looking for higher levels of synchronization and labor productivity, in some cases supported by technologies such as voice, collaborative mobile robots, smart carts, put walls, pick-to-light and more.
For both automated and non-automated DCs, Softeon delivers this smart WMS solution.
With deep roots in the WMS segment and a 20-year track record of innovation, Softeon is governed by a mission to solve supply chain problems with a holistic, end-to-end WMS solution that delivers real value. Across sectors, Softeon's primary focus is to enable customers to reduce costs and improve their own customers' experience through distribution excellence. With a powerful core WMS, Softeon also provides extended WMS solutions in areas such as yard and labor management, assembly/kitting, parcel shipping, 3PL Billing and more—all developed internally, on a single service-oriented architecture framework.
"Our components are like non-proprietary 'Lego blocks' that are combined to optimize WMS, reduce supply chain costs, and enhance customer experience for a competitive advantage," states Softeon's VP, Strategy, Dinesh Dongre.
Softeon offers several differentiated WMS capabilities. The includes a unique 'Pick route' capability that defines the type of order, channel, customer, strategy used, inventory, technology— RF, voice, pick-to-light, put walls, mobile robots—and services associated with the picking operation. The firm's WMS solution also includes a powerful configuration wizard to navigate clients around configuring the WMS with absolute precision and control right out-of-the-box, especially for multi-site deployments.
In addition to a robust WMS, Softeon also offers a Distributed Order Management (DOM) for automating omnichannel processes and determining optimal order fulfillment strategies. The DOM can be configured through a simple drag-and-drop process modeler. Now, Softeon is taking that process modeling further. "As a highly innovative company, we have recently introduced the concept of business process modeling to the WMS market as well, which is a game-changer," says Dan Gilmore, CMO of Softeon.
One of Softeon's retail clients recently moved from on-line only to a network of more than 40 stores. It also added a couple of third-party DC's to its distribution mix. The company implemented Softeon's WMS to power its own DC, and Softeon's DOM solution to enable processes such as buy on-line, pick-up in store, store fulfillment, store returns, and more. This allowed the retailer to reap a significant 30-40 percent reduction in the cost-to-serve along with benefits from both cost and customer affinity perspectives. "Our ability to squeeze out additional, meaningful percentages of savings is our differentiating factor," Gilmore adds.
With a highly successful track record, Softeon is committed to providing reliable support and visibility into emerging technologies and optimizing material f low in warehouses on par with the growing maturity and adoption levels in the WMS space.
Reston, Va - Softeon, a Global Supply Chain Software provider with the industry's best record of customer success, announced today that S3 Group, a premier management consulting firm providing solutions and services to the logistics and supply chain industries, will become a certified Implementation and Integration partner for Softeon solutions.
Softeon has a number of tools and programs to assist third-party integrators in working with its software solutions. That includes Softeon's innovative and unique Configuration Wizard, which enables customers, 3PLs and integrators to set-up its powerful WMS solution using an electronic dialog similar to how many income tax programs work.
The Wizard keeps track of what has been set-up and what hasn't, walks users through required configuration steps and much more.


text:
info@clubtennisdaro.com
Club Tennis d'Aro
The Tennis Club of Aro by history, tradition and facilities one of the best tennis clubs in the province of Girona. As a Board we have a duty to preserve this asset, constructed every day to many families and partners who have made the club their second home.
One of the commitments of the Board has been since the first day, the approach of managing partners. In this respect Mr Partners have a website, a channel open to suggestions that the club covers, as far as possible, their needs and concerns. With the contribution of everyone will be more effective management of the Club.
So we invite you to participate and encourage you to let us get historical photographs or topics of interest to this, our website the most inclusive and participatory as possible.
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text:
\section{Introduction}

Let $C$ be a projective algebraic curve defined over a finite field
$\mathbb{F}_{q}$ (with $q$ odd), assumed to be geometrically connected
and smooth. Let $K=\mathbb{F}_{q}(C)$ be its function field and let
$\Omega$ denote the set of all closed points of $C$. For any point
$\mathfrak{p}\in\Omega$ let $v_{\mathfrak{p}}$ be the induced discrete
valuation on $K$, $\hat{\mathcal{O}}_{\mathfrak{p}}$ the complete
discrete valuation ring with respect to $v_{\mathfrak{p}}$ and
$\hat{K}_{\mathfrak{p}}$ its fraction field. Any \emph{Hasse set} of
$K$, namely a non-empty finite set $S \subset\Omega$, gives rise to
an integral domain of $K$ called a \emph{Hasse domain}:
\begin{equation*}
\mathcal{O}_{S}:= \{x \in K: v_{\mathfrak{p}}(x) \geq0 \ \forall
\mathfrak{p}\notin S\}.
\end{equation*}
This is a Dedekind domain, regular and one dimensional. Schemes defined
over $\text{Spec} \,\mathcal{O}_{S}$ are denoted by an underline, being
omitted in the notation of their generic fibers.

As $2$ is invertible in $\mathcal{O}_{S}$, the $\mathcal{O}_{S}$-group
$\underline{\mu}_{2} := \text{Spec} \,\mathcal{O}_{S}[t]/(t^{2}-1)$ is
smooth, whence applying \'{e}tale cohomology to the Kummer sequence:
\begin{equation*}
1 \to\underline{\mu}_{2} \to\underline{\mathbb{G}}_{m}
\xrightarrow{x \mapsto x^{2}} \underline{\mathbb{G}}_{m} \to1
\end{equation*}
gives rise to the long exact sequence of abelian groups:
\begin{eqnarray*}
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mathbb{G}}_{m})
&\xrightarrow{[ \mathfrak{L}] \to[\mathfrak{L} \otimes_{\mathcal{O}_{S}}
\mathfrak{L}]}&
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mathbb{G}}_{m}) \to
H^{2}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{ \mu}_{2}) \to
H^{2}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{ \mathbb{G}}_{m})\\
&\xrightarrow{[A] \to[A \otimes_{\mathcal{O}_{S}} A]}& H^{2}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mathbb{G}}_{m}).
\end{eqnarray*}
Identifying
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mathbb{G}}_{m})$
with $\text{Pic~}(\mathcal{O}_{S})$ by Shapiro's lemma (cf.
\cite[XXIV,\break Prop.~8.4]{SGA3}), and $H^{2}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{\mathbb{G}}_{m})$ with the Brauer group
${\mathrm{Br}}(\mathcal{O}_{S})$, classifying Azumaya $\mathcal{O}
_{S}$-algebras (cf. \cite[\S2]{Mil}), we deduce the short exact
sequence:
\begin{equation}
\label{Kummermu2H1}
1 \to\text{Pic~}(\mathcal{O}_{S})/2 \xrightarrow{\partial} H^{2}
_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2}) \xrightarrow{i
_{*}^{2}} {_{2}}{\mathrm{Br}}(\mathcal{O}_{S}) \to1,
\end{equation}
in which the right non-trivial term is the $2$-torsion part in
${\mathrm{Br}}(\mathcal{O}_{S})$. We analyze some properties related to
this sequence in Section~\ref{Azumayasalgebras}, which will be used to
classify regular quadratic $\mathcal{O}_{S}$-spaces.

Let $(V,q)$ (not to be confused with $q=|\mathbb{F}_{q}|$) be a
quadratic $\mathcal{O}_{S}$-space of rank $n \geq3$, namely, $V$ is a
projective $\mathcal{O}_{S}$-module of rank $n$ and $q:V \to
\mathcal{O}_{S}$ is a $2$-order homogeneous $\mathcal{O}_{S}$-form.
Since $2$ is a unit, $q$ corresponds to the symmetric bilinear form
$B_{q}:V \times V \to\mathcal{O}_{S}$ such that:
\begin{equation*}
B_{q}(u,v) = q(u+v)-q(u)-q(v) \ \ \forall u,v \in V.
\end{equation*}
We assume it to be $\mathcal{O}_{S}$-\emph{regular}, namely, the induced
homomorphism $V \to V^{\vee}:= \text{Hom}(V,\mathcal{O}_{S})$ is an
isomorphism (\cite[I.~\S3, 3.2]{Knu}).
Two quadratic $\mathcal{O}_{S}$-spaces $(V,q)$ and $(V',q')$ are \emph{isomorphic} over an
extension $R$ of $\mathcal{O}_{S}$, if there exists an
$R$-\emph{isometry} between them, namely, an $R$-isomorphism
$T: V' \otimes_{\mathcal{O}_{S}} R \cong V \otimes_{\mathcal{O}_{S}} R$
such that $q \circ T = q'$. The notation of the
$\mathcal{O}_{S}$-isomorphism class $[(V,q)]$ is sometimes, when no
ambiguity arises, shortened to $[q]$.
The [proper] \emph{genus} of $(V,q)$ is the set of classes of all quadratic $\mathcal{O}_{S}$-spaces
that are [properly, i.e.,\vadjust{\goodbreak} with $\det=1$ isomorphisms] isomorphic to
$(V,q)$ over $K$ and over $\hat{\mathcal{O}}_{\mathfrak{p}}$ for any
prime $\mathfrak{p} \notin S$.
This [proper] genus bijects as a pointed-set with
the \emph{class set} $[\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})]$,
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$.

The results generalize the ones in \cite{Bit}, in which $S$ is assumed
to contain only one (arbitrary) point $\infty\in\Omega$, thus giving
rise to an affine curve $C^{\text{af}}= C-\{\infty\}$, for which
${\mathrm{Br}}(\mathcal{O}_{\{\infty\}})=1$ (cf.
\cite[Lemma~3.3]{Bit}). The quadratic $\mathcal{O}_{\{\infty\}}$-spaces
that are locally properly isomorphic to $(V,q)$ for the flat or the
\'{e}tale topology belong all to the same genus, and are classified by
the abelian group $H^{2}_{\text{\'{e}t}}(\mathcal{O}_{\{\infty\}
},\underline{\mu}_{2}) \cong\text{Pic~}(\mathcal{O}_{\{\infty\}})/2$.
Here we show more generally for any finite set $S$ that, as
$\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$ is the kernel of
what we call the \emph{relative} Witt-invariant
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})
\xrightarrow{w_{V}} {_{2} {\mathrm{Br}}( \mathcal{O}_{S})}$, the latter
abelian group bijects to the set of $2^{|S|-1}$ proper genera of
$(V,q)$ (Proposition~\ref{Wittexactness}).

Another consequence of passing to $|S|>1$ is that $\mathcal{O}_{S}
^{\times}\neq\mathbb{F}_{q}^{\times}$, whence
$\mathcal{O}_{S}$-regularity imposed on $(V,q)$, no longer guarantees
its isotropy. Requiring $(V,q)$ to be isotropic, we show that
$\text{Pic~}(\mathcal{O}_{S})/2$ still classifies the quadratic spaces
in the genus $\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$ being
equal to proper genus in this case (Lemma~\ref{samegenus}), for any $S$
(Theorem~\ref{isotropicclassification}). In particular in case $n \geq 5$,
in which all classes are isotropic, any proper genus in
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ --
corresponding as aforementioned to an element of $_{2} {\mathrm{Br}}(
\mathcal{O}_{S})$ -- is isomorphic to $\text{Pic~}(\mathcal{O}_{S})/2$,
whence their disjoint union $H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ is isomorphic to the
abelian group $H^{2}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2})$ (as for $|S|=1$
and $n \geq3$), fitting into the sequence \eqref{Kummermu2H1}
(Corollary~\ref{genera}).

In Section~\ref{hyperbolicplane}, we refer to the case in which $V$ is
split by a hyperbolic plane $H(L_{0})$, and provide an isomorphism
$\psi_{V}:\text{Pic~}(\mathcal{O}_{S})/2
\to\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$. In case $C$ is an
elliptic curve and $S=\{\infty \}$ where $\infty$ is
$\mathbb{F}_{q}$-rational, an algorithm, producing explicitly
representatives of classes in $H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$, is given (\ref{Generatoralgorithm}).

\section{A classification of Azumaya algebras}\label{Azumayasalgebras}

A faithfully flat projective (right) $\mathcal{O}_{S}$-module $A$ is an
\emph{Azumaya} $\mathcal{O}_{S}$-algebra if the map
\begin{equation*}
A \otimes A^{\text{op}} \to\text{End}_{\mathcal{O}_{S}}(A): \ a
\otimes b^{\text{op}} \mapsto(x \mapsto axb)
\end{equation*}
is an isomorphism. It is central, separable and finitely generated as
an $\mathcal{O}_{S}$-module. Two Azumaya $\mathcal{O}_{S}$-algebras
$A,B$ are \emph{Brauer equivalent} if there exist faithfully projective
modules $P,Q$ such that:
\begin{equation*}
A \otimes\text{End}_{\mathcal{O}_{S}}(P) \cong B \otimes\text{End}
_{\mathcal{O}_{S}}(Q).
\end{equation*}
The tensor product induces the structure of an abelian group
${\mathrm{Br}}(\mathcal{O}_{S})$ on the equivalence classes, in which
the neutral element is $[\mathcal{O}_{S}]$ and the inverse of $[A]$ is
$[A^{\text{op}}]$ (cf. \cite[III.~5.1 and 5.3]{Knu}).

Let $V \cong\mathcal{O}_{S}^{2}$. Consider the following exact and
commutative diagram of smooth $\mathcal{O}_{S}$-groups:
\begin{align}
\label{diagram}
\xymatrix{
& 1 \ar[d] & 1 \ar[d] \\
1 \ar[r] & \underline{\mu}_2 \ar[r] \ar[d] & \underline{\text{\textbf{SL}}}(V) \ar[r] \ar[d]
& \underline{\text{\textbf{PGL}}}(V) \ar[r] \ar@{=}[d] & 1 \\
1 \ar[r] & \underline{\mathbb{G}}_m \ar[r] \ar[d]^{x \mapsto x^2} & \underline{\textbf
{GL}}(V) \ar[r]^{\pi} \ar[d]^{\det} & \underline{\text{\textbf{PGL}}}(V) \ar[r] & 1 \\
& \underline{\mathbb{G}}_m \ar[r] \ar[d] & \underline{\mathbb{G}}_m \ar[d] \\
& 1 & 1.
}
\end{align}
The generalization of the Skolem--Noether Theorem to unital commutative
rings, applied to the Azumaya $\mathcal{O}_{S}$-algebra $A=\text{End}
_{\mathcal{O}_{S}}(V)$, is the exact sequence of groups (see
\cite[III.~5.2.1]{Knu}):
\begin{equation*}
1 \to\mathcal{O}_{S}^{\times}\to A^{\times}\to\text{Aut}_{
\mathcal{O}_{S}}(A) \to\text{Pic~}(\mathcal{O}_{S}).
\end{equation*}
This sequence induces by sheafification a short exact sequence of
sheaves in the \'{e}tale topology (cf. \cite[p.~145]{Knu}):
\begin{equation*}
1 \to\underline{\mathbb{G}}_{m} \to\underline{\text{\textbf{GL}}}(V)
\to\underline{\text{Aut}}(\text{End}_{\mathcal{O}_{S}}(V)) \to1
\end{equation*}
from which we see that: $\underline{\text{\textbf{PGL}}}(V) =
\underline{\text{Aut}}(\text{End}_{\mathcal{O}_{S}}(V))$. In this interpretation,
\'{e}tale cohomology applied to the diagram \eqref{diagram}, plus the
sequence \eqref{Kummermu2H1}, give rise to the exact and commutative diagram:
\begin{equation}
\label{PGLV(V)}
\xymatrix{
& & \text{Pic~}(\mathcal{O}_S) \ar[d]^\partial \\
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SL}}}(V)) \ar[r] \ar[d] & H^1_\text{\'et}(\mathcal{O}_S,\underline{\textbf
{PGL}}(V)) \ar[r]^{\partial^1} \ar@{=}[d] & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar[d] \\
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{Aut}}(V)) \ar[r]^{\pi_*} \ar[d]^{\Delta= \det_*} &
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{Aut}}(\text{End}_{\mathcal{O}_S}(V))) \ar[r] & {\mathrm{Br}}(\mathcal{O}_S) \\
\text{Pic~}(\mathcal{O}_S)
}
\end{equation}
in which $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},
\underline{\text{Aut}}(V))$ classifies twisted forms of $V$ in the
\'{e}tale topology, while its image in $H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{\text{Aut}}(\text{End}_{\mathcal{O}_{S}}(V))$
classifies these $\mathcal{O}_{S}$-modules up to scaling by an
$\mathcal{O}_{S}$-line, i.e., by an invertible $\mathcal{O}_{S}$-module.
Explicitly, $\pi_{*}: [P] \mapsto[\text{End}_{\mathcal{O}_{S}}(P)]$
(cf. \cite[III.~5.2.4]{Knu}).

\begin{cor}
\label{PictoH2}
In diagram \eqref{PGLV(V)}: $\partial([L]) = \partial^{1}([
\text{End}_{\mathcal{O}_{S}}(\mathcal{O}_{S}\oplus L)])$.
\end{cor}

\begin{proof}
By chasing diagram \eqref{PGLV(V)} we may deduce the following reduced one:
\begin{equation}
\label{reduceddiagram}
\xymatrix{
&& \text{Pic~}(\mathcal{O}_S)/2 \ar@{^{(}->}[d]^\partial \\
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{Aut}}(V)) \ar[rr]^{\partial^1 \circ\pi_*} \ar[urr]^{\Delta}
\ar[drr]^{0} && H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar@{->>}[d] \\
&& {_2 {\mathrm{Br}}(\mathcal{O}_S)}
}
\end{equation}
which shows that: $ \partial([L]) = \partial(\Delta([\mathcal{O}
_{S}\oplus L])) = \partial^{1}(\pi_{*}([\mathcal{O}_{S}\oplus L])) =
\partial^{1}([\text{End}_{\mathcal{O}_{S}}(\mathcal{O}_{S}\oplus L)])$.
\end{proof}

\begin{lem}
\label{no.genera}
$|{_{2} {\mathrm{Br}}(\mathcal{O}_{S})}| = 2^{|S|-1}$.
\end{lem}

\begin{proof}
Let $r_{\mathfrak{p}}: {\mathrm{Br}}(K) \to H^{1}(k_{\mathfrak{p}},
\mathbb{Q}/\mathbb{Z}) \cong\mathbb{Q}/\mathbb{Z}$ be the residue map
at a prime $\mathfrak{p}$. The ramification map $a:= \oplus_{
\mathfrak{p}}r_{\mathfrak{p}}$ yields the exact sequence from Class
Field Theory (see \cite[Theorem~6.5.1]{GS}):
\begin{equation}
\label{sumHassesinvariants}
1 \to{\mathrm{Br}}(K) \xrightarrow{a} \bigoplus_{
\mathfrak{p}}\mathbb{Q}/ \mathbb{Z}\xrightarrow{\sum_{\mathfrak{p}}
\text{Cor}_{\mathfrak{p}}} \mathbb{Q}/\mathbb{Z}\to1
\end{equation}
in which the corestriction map $\text{Cor}_{\mathfrak{p}}$ for any
$\mathfrak{p}$ is an isomorphism induced by the Hasse-invariant
${\mathrm{Br}}(\hat{K}_{\mathfrak{p}}) \cong\mathbb{Q}/\mathbb{Z}$ (cf.
\cite[Proposition~6.3.9]{GS}). On the other hand, as all residue fields
of $K$ are finite, thus perfect, and $\mathcal{O}_{S}$ is a
one-dimensional regular scheme, it admits due to Grothendieck the
following exact sequence (see \cite[Proposition~2.1]{Gro} and
\cite[Example~2.22, case (a)]{Mil}):
\begin{equation}
\label{Grothendieck}
1 \to{\mathrm{Br}}(\mathcal{O}_{S}) \to{\mathrm{Br}}(K) \xrightarrow{
\oplus_{\mathfrak{p}\notin S} r_{\mathfrak{p}}} \bigoplus
_{\mathfrak{p}\notin S} \mathbb{Q}/\mathbb{Z},
\end{equation}
which means that ${\mathrm{Br}}(\mathcal{O}_{S})$ is the subgroup of
${\mathrm{Br}}(K)$ of classes that vanish under $r_{\mathfrak{p}}$ at
any $\mathfrak{p}\notin S$. Thus omitting these $r_{\mathfrak{p}},
\mathfrak{p}\notin S$ in the sequence (\ref{sumHassesinvariants}),
results in ${\mathrm{Br}}(\mathcal{O}_{S}) = \ker\left[ \bigoplus_{
\mathfrak{p}\in S} \mathbb{Q}/\mathbb{Z}\xrightarrow{
\sum_{\mathfrak{p}\in S}\text{Cor}_{\mathfrak{p}}} \mathbb{Q}/
\mathbb{Z}\right] $, whence the cardinality of its $2$-torsion part is
$2^{|S|-1}$.
\end{proof}

\begin{lem}
\label{H1=1sc}
Let $\underline{G}$ be an affine, flat, connected and smooth
$\mathcal{O}_{S}$-group. Suppose that its generic fiber $G$ is almost
simple, simply connected and $\hat{K}_{\mathfrak{p}}$-isotropic for any
$\mathfrak{p}\in S$. Then $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},
\underline{G})=0$.
\end{lem}

\begin{proof}
The proof, basically relying on the strong approximation property
related to $\underline{G}$, is similar to that of Lemma~3.2 in \cite{Bit},
replacing $\{\infty\}$ by $S$.
\end{proof}

\section{Standard and relative invariants}\label{forms}

Let $\underline{\text{\textbf{O}}}_{V}$ be the \emph{orthogonal group} of
$(V,q)$ defined over $\text{Spec} \,\mathcal{O}_{S}$, namely, the
functor assigning to any $\mathcal{O}_{S}$-algebra $R$ the group of
self-isometries of $q$ over $R$:
\begin{equation*}
\underline{\text{\textbf{O}}}_{V}(R) = \{ A \in\underline{\text{\textbf{GL}}}_{n}(R):
q \circ A = q \}.
\end{equation*}
Since $2 \in\mathcal{O}_{S}^{\times}$ and $q$ is regular,
$\underline{\text{\textbf{O}}}_{V}$ is smooth as well as its connected
component, namely, the \emph{special orthogonal group}
$\underline{\text{\textbf{SO}}}_{V}:=
\ker[\underline{\text{\textbf{O}}}_{V}\xrightarrow{ \det}
\underline{\mu}_{2}]$ (see Definition~1.6, Theorem~1.7 and
Corollary~2.5 in \cite{Con}). Thus the pointed set $H^{1}_{
\text{fl}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ --
properly (i.e., with $\det=1$ isomorphisms) classifying
$\mathcal{O}_{S}$-forms that are locally everywhere isomorphic to $q$
in the flat topology -- coincides with the classification
$H^{1}_{\text{\'{e}t}}(\mathcal{O}
_{S},\underline{\text{\textbf{SO}}}_{V})$ for the \'{e}tale topology
(see \cite[VIII~Corollaire~2.3]{SGA4}).

Let $\text{\textbf{C}}(V) :=T(V)/(v \otimes v -q(v)\cdot1:v \in V)$ be the
\emph{Clifford algebra} associated to $(V,q)$ (see \cite[IV]{Knu}). The
linear map $v \mapsto-v$ on $V$ preserves $q$, thus extends to an
algebra automorphism $\alpha: \text{\textbf{C}}(V) \to\text{\textbf{C}}(V)$. As it
is an involution, the graded algebra $\text{\textbf{C}}(V)$ is decomposed into
positive and negative eigenspaces: $\text{\textbf{C}}_{0}(V) \oplus
\text{\textbf{C}}_{1}(V)$ where $\text{\textbf{C}}_{i}(V) = \{ x \in\text{\textbf{C}}(V):
\alpha(x) = (-1)^{i} x \}$ for $i=0,1$. Since $(V,q)$ is projective and
$\mathcal{O}_{S}$-regular, $\text{\textbf{C}}(V)$ is Azumaya over $
\mathcal{O}_{S}$ (cf. \cite[Theorem, p.~166]{Bas}).

The \emph{Witt-invariant} of $(V,q)$ is:
\begin{equation*}
w(q) = \left\{
\begin{array}{c@{\quad} c}
[\text{\textbf{C}}(V)] \in{\mathrm{Br}}(\mathcal{O}_{S}) & \ n \
\text{is even}
\\
\nonumber
[\text{\textbf{C}}_{0}(V)] \in{\mathrm{Br}}(\mathcal{O}_{S}) & \ n \ \text{is
odd}.
\end{array}
\right.
\end{equation*}
As $\text{\textbf{C}}(V)$ and $\text{\textbf{C}}_{0}(V)$ are algebras with involution,
$w(q)$ lies in ${_{2} {\mathrm{Br}}(\mathcal{O}_{S})}$
(\cite[IV.~8]{Knu}).

The \emph{Clifford group} associated to $(V,q)$ is
\begin{equation*}
\text{\textbf{CL}}(V) := \{ u \in\text{\textbf{C}}(V)^{\times}\ : \ \alpha(u)vu
^{-1} \in V \ \forall v \in V \}.
\end{equation*}
The group $\underline{\text{\textbf{Pin}}}_{V}(\mathcal{O}_{S}):=\ker[
\text{\textbf{CL}}(V) \xrightarrow{N} \mathcal{O}_{S}^{\times}]$ where
$N:v \mapsto v\alpha(v)$, admits an underlying $\mathcal{O}_{S}$-group
scheme, called the \emph{Pinor group} denoted by
$\underline{\text{\textbf{Pin}}}_{V}$. It is a double covering of $
\underline{\text{\textbf{O}}}_{V}$ and its center $\underline{\mu}_{2}$
is smooth. So applying \'{e}tale cohomology to the \emph{Pinor exact
sequence} of smooth $\mathcal{O}_{S}$-groups:
\begin{equation}
\label{Pinor}
1 \to\underline{\mu}_{2} \to\underline{\text{\textbf{Pin}}}_{V} \to
\underline{\text{\textbf{O}}}_{V}\to1
\end{equation}
gives rise to the coboundary map of pointed-sets
\begin{equation}
\label{H1Pinor} \delta_{V}:
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{V})
\to H^{2}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{ \mu}_{2}).
\end{equation}

Let $\underline{\text{\textbf{O}}}_{2n}$ and $\underline{\text{\textbf{O}}}_{2n+1}$
be the orthogonal groups of the hyperbolic spaces $H(\mathcal{O}_{S}
^{n})$ and $H(\mathcal{O}_{S}^{n}) \bot\langle1 \rangle$,
respectively, equipped with the \emph{standard split form} which we
denote by $q_{n}$ (see \cite[Definition~1.1]{Con}). The pointed set
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{n})$
classifies regular quadratic $\mathcal{O}_{S}$-modules of rank $n$
(\cite[IV.~5.3.1]{Knu}). It is identified with the pointed set
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{V})$
simply obtained by changing the base point to $(V,q)$ (cf.
\cite[IV, Prop.~8.2]{Knu}). We denote this identification by
$\theta$. One has the following commutative diagram of pointed sets
(cf. \cite[IV, Prop.~4.3.4]{Gir}):
\begin{equation}
\label{relativew}
\xymatrix{
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{O}}}_{n}) \ar[r]^{\theta}_{\cong} \ar[d]^{\delta} &
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{O}}}_{V}) \ar[d]_{\delta_V} \\
H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar[r]^{r_V}_{\cong} & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2)
}
\end{equation}
in which $\delta:= \delta_{q_{n}}$ and $r_{V}(x)=x-\delta([q])$.

\begin{definition}
\label{relativeWitt-invariant}
We call the composition of maps of pointed sets:
\begin{equation*}
w_{V}: H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}
_{V}) \xrightarrow{\delta_{V}} H^{2}_{\text{\'{e}t}}(\mathcal{O}
_{S},\underline{\mu}_{2}) \xrightarrow{i_{*}^{2}} {_{2}}{\mathrm{Br}}(
\mathcal{O}_{S})
\end{equation*}
(see sequence \eqref{Kummermu2H1} for $i_{*}^{2}$) the \emph{relative
Witt-invariant}. It is a ``shift'' of the Witt-invariant $w=i_{*}^{2}
\circ\delta$, such that the base-point $[(V,q)]$ is mapped to
$[0] \in{\mathrm{Br}}(\mathcal{O}_{S})$.
\end{definition}

\begin{remark}
\label{Stiefel-Whitney}
The $\delta$-image of a class represented by $(V',q')$, being a regular
$\mathcal{O}_{S}$-module, is its \emph{second Stiefel--Whitney class},
denoted $w_{2}(q')$ (cf.
\cite[Definition~1.6 and Corollary~1.19]{EKV}).
\end{remark}

The connected component $\underline{\text{\textbf{Spin}}}_{V}$ of
$\underline{\text{\textbf{Pin}}}_{V}$ is smooth, and it is the
universal covering of $\underline{\text{\textbf{SO}}}_{V}$:
\begin{equation}
\label{Spinshortsequence}
1 \to\underline{\mu}_{2} \to\underline{\text{\textbf{Spin}}}_{V} \stackrel{
\pi}{\rightarrow} \underline{\text{\textbf{SO}}}_{V}\to1.
\end{equation}
Then \'{e}tale cohomology gives rise to the exact sequence of pointed
sets:
\begin{equation}
\label{H1Spin}
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{Spin}}}_{V})
\to
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})
\xrightarrow{s\delta_{V}} H^{2}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\mu}_{2}) \to1
\end{equation}
in which the right exactness comes from the fact that $\mathcal{O}
_{S}$ is of Douai-type, thus $H^{2}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{Spin}}}_{V})=1$ (see Definition~5.2
and Example~5.4(iii) in \cite{Gon}). The inclusion
$i:\underline{\text{\textbf{SO}}}_{V}\subset
\underline{\text{\textbf{O}}}_{V}$ with the map $i_{*}^{2}$ from
sequence (\ref{Kummermu2H1}) induces the commutative diagram (cf.
\cite[IV, 8.3]{Knu})
\begin{equation}
\label{sdl}
\xymatrix{
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar[r]^{i_*} \ar[d]^{s\delta_V} & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{O}}}_{V}) \ar
[d]_{w_V} \ar[ld]^{\delta_V} \\
H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar[r]_{i_*^2} & {_2}{\mathrm{Br}}(\mathcal{O}_S).
}
\end{equation}

\begin{remark}
\label{restriction} The map $i_{*}$ does not have to be injective, yet
any form $q'$, properly isomorphic to $q$, represents a class in
$H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{n})$, so
the restriction of $w$ to
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{n})$
is well-defined. Similarly, we may write the restriction
$w_{V}|H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$
as $i_{*}^{2} \circ s \delta_{v}$, being surjective, as both
$i_{*}^{2}$ and $s \delta_{V}$ are such (see sequences \eqref{Kummermu2H1} and \eqref{H1Spin}).
\end{remark}

\begin{remark}
\label{projectiveplane1}
Unlike over fields, the Stiefel--Whitney class $w_{2}$ for quadratic
$\mathcal{O}_{S}$-spaces, referring to their Clifford algebras not only
as Azumaya algebras but as algebras with involution, is richer than the
Witt-invariant $w$ lying in ${_{2}}{\mathrm{Br}}(\mathcal{O}_{S})$. For
example, if $L$ is an invertible $\mathcal{O}_{S}$-module and
$H(L) = L \oplus L^{*}$ is the corresponding hyperbolic plane, then
$\text{\textbf{C}}(H(L))$ is isomorphic as a graded algebra to $\text{End}
_{\mathcal{O}_{S}}(\wedge L) = \text{End}_{\mathcal{O}_{S}}(
\mathcal{O}_{S}\oplus L)$ (\cite[IV, Prop.~2.1.1]{Knu}) being
Brauer-equivalent to $M_{2}(\mathcal{O}_{S})$, thus $w(H(L))=[0]
\in{\mathrm{Br}}(\mathcal{O}_{S})$, while:
\begin{equation*}
\delta([H(L)]) = \partial([L]) \stackrel{\text{Corollary}~\ref{PictoH2}}{=} \partial^{1}([\text{End}_{\mathcal{O}_{S}}(
\mathcal{O}_{S}\oplus L)]) \in H^{2}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\mu}_{2})
\end{equation*}
(see the left equality in the proof of the Proposition in
\cite[\S5.5]{EKV}), does not have to vanish as we shall see in Proposition~\ref{bijection}.
\end{remark}

\section{A classification of quadratic spaces via their genera}

Consider the ring of $S$-integral ad\`{e}les $\mathbb{A}_{S} :=
\prod_{\mathfrak{p}\in S} \hat{K}_{\mathfrak{p}}\times
\prod_{\mathfrak{p}\notin S} \hat{\mathcal{O}}_{\mathfrak{p}}$, being
a subring of the ad\`{e}les $\mathbb{A}$. Then the $S$-\emph{class set}
of an $\mathcal{O}_{S}$-group $\underline{G}$ is the set of double
cosets:
\begin{equation*}
\text{Cl}_{S}(\underline{G}) := \underline{G}(\mathbb{A}_{S}) \backslash
\underline{G}(\mathbb{A}) / G(K)
\end{equation*}
(where for any prime $\mathfrak{p}$ the geometric fiber $
\underline{G}_{\mathfrak{p}}$ of $\underline{G}$ is taken) and it is
finite (cf. \cite[Prop.~3.9]{BP}). If $\underline{G}$ is affine and
finitely generated over $\text{Spec} \,\mathcal{O}_{S}$, it admits
according to Nisnevich (\cite[Theorem~I.3.5]{Nis}) the following exact
sequence of pointed sets:
\begin{equation}
\label{Nissequence}
1 \to\text{Cl}_{S}(\underline{G}) \to H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{G}) \to H^{1}(K,G) \times
\prod_{\mathfrak{p}\notin S} H^{1}_{\text{\'{e}t}}(\hat{\mathcal{O}}_{\mathfrak{p}},(\underline{G})_{\mathfrak{p}}).
\end{equation}
If $\underline{G}$ admits, furthermore, the property:
\begin{equation}
\label{locallyembedded}
\forall\mathfrak{p} \notin S:
\ \
H^{1}_{\text{\'{e}t}}(\hat{\mathcal{O}}_{\mathfrak{p}},\underline{G}_{\mathfrak{p}}) \hookrightarrow H^{1}(\hat{K}_{\mathfrak{p}},G_{
\mathfrak{p}}),
\end{equation}
then Nisnevich's sequence for $\underline{G}$ reduces to (cf.
\cite[Corollary~I.3.6]{Nis}, \cite[Corollary~A.8]{GP}):
\begin{equation}
\label{Nissimple}
1 \to\text{Cl}_{S}(\underline{G}) \to H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{G}) \to H^{1}(K,G).
\end{equation}

\begin{remark}
\label{finiteetaleextensionisembeddedingenericfiber} Since $\text{Spec}
\,\mathcal{O}_{S}$ is normal, i.e., is integrally closed locally
everywhere (due to the smoothness of $C$), any finite \'{e}tale
covering of $\mathcal{O}_{S}$ arises by its normalization in some
separable unramified extension of $K$ (see \cite[Theorem~6.13]{Len}).
Consequently, if $\underline{G}$ is a finite $\mathcal{O}_{S}$-group,
then $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S}, \underline{G})$ is
embedded in $H^{1}(K,\underline{G})$. This is not true for infinite
groups like the multiplicative group $\underline{\mathbb{G}}_{m}$, for
which $H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\mathbb{G}}_{m}) \cong\text{Pic~}(\mathcal{O}_{S})$
clearly does not have to embed in $H^{1}(K,\mathbb{G}_{m})=1$.
\end{remark}

\begin{remark}
\label{genus}
In case $\underline{G}=\underline{\text{\textbf{O}}}_{V}$, the left exactness
of sequence \eqref{Nissequence} reflects the fact that $\text{Cl}
_{S}(\underline{\text{\textbf{O}}}_{V})$ is the \emph{genus} of the base point
$(V,q)$, namely, the set of classes of quadratic $\mathcal{S}$-forms that are $K$ and
$\hat{\mathcal{O}}_{\mathfrak{p}}$-isomorphic to it for all
$\mathfrak{p} \notin S$. Furthermore, being connected, $
\underline{\text{\textbf{SO}}}_{V}$ admits property \eqref{locallyembedded}
by Lang's Theorem (recall that all residue fields are finite), so the
\emph{proper genus} can be described as:
\begin{equation}
\label{propergenus}
\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) = \ker[H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V}) \to H^{1}(K,
\text{\textbf{SO}}_{V})].
\end{equation}
As $\underline{\text{\textbf{O}}}_{V}/\underline{\text{\textbf{SO}}}_{V}$ is the
finite representable $\mathcal{O}_{S}$-group $\underline{\mu}_{2}$ (cf.
\cite[Theorem~1.7]{Con}), $\underline{\text{\textbf{O}}}_{V}$ admits property \eqref{locallyembedded} as well (see in the proof of Proposition~3.4
in \cite{CGP}), so we may also write:
\begin{equation}
\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V}) = \ker[H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{V}) \to H^{1}(K,
\text{\textbf{O}}_{V})].
\end{equation}
As a pointed set, $\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$ is
bijective to the first Nisnevich cohomology set $H^{1}_{\text{Nis}}(
\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ (cf.
\cite[Theorem~I.2.8]{Nis} and \cite[4.1]{Mor}), classifying
$\underline{\text{\textbf{SO}}}_{V}$-torsors in the Nisnevich topology. But
Nisnevich covers are \'{e}tale, so it is a subset of $H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$. Similarly,
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V}) \subseteq H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{V})$.
\end{remark}

\begin{lem}
\label{isotropicreflection}
If $(V,q)$ is isotropic then $\underline{\text{\textbf{O}}}_{V}(\mathcal{O}
_{S}) \xrightarrow{\det} \underline{\mu}_{2}(\mathcal{O}_{S})$ is
surjective.
\end{lem}

\begin{proof}
Consider the following exact and commutative diagram that arises by
applying \'{e}tale cohomology to the short exact sequences related to
the smooth $\mathcal{O}_{S}$-groups $\underline{\text{\textbf{Pin}}}_{V}$ and
$\underline{\text{\textbf{O}}}_{V}$:
\begin{equation*}
\xymatrix{
& & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{Spin}}}_V) \ar[r] \ar[d]^{s\pi_*} & H^1_\text{\'et}
(\mathcal{O}_S,\underline{\text{\textbf{Pin}}}_V) \ar[d]^{\pi_*} \\
\underline{\text{\textbf{O}}}_{V}(\mathcal{O}_S) \ar[r]^{\det} &\underline{\mu}_2(\mathcal{O}_S) \ar[r]^{\partial_0} & H^1_\text{\'et}
(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar[r]^{h} \ar[d]^{s\delta_V} & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{O}}}_{V}) \ar[d]^{\delta_V}
\\
& & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar@{=}[r] & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2).
}
\end{equation*}
Denote $[\gamma] = \partial_{0}(-1)$. Then $s\delta_{V}([\gamma]) =
\delta_{V}(h([\gamma])=[0])=[0]$, hence $[\gamma] \in
\operatorname{Im}(s\pi_{*})$. But as $q$ is isotropic, $H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{Spin}}}_{V})$ vanishes
by strong approximation (cf. Lemma~\ref{H1=1sc}), so $[\gamma] = [0]$,
which means that $\partial_{0}$ is the trivial map and $\det(
\mathcal{O}_{S})$ surjects on $\underline{\mu}_{2}(\mathcal{O}_{S})$.
\end{proof}

\begin{lem}
\label{samegenus}
If $n$ is odd, or $(V,q)$ is isotropic, then $\text{Cl}_{S}(\underline{\text{\textbf{SO}}}
_{V}) = \text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$.
\end{lem}

\begin{proof}
Any representative $(V',q')$ of a class in
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$, being $K$
isomorphic to $q$, is regular and isotropic as well, whence
$\underline{\text{\textbf{O}}}_{V'}(\mathcal{O}_{S}) \to
\underline{\mu}_{2}(\mathcal{O}_{S})$ is surjective by Lemma~\ref{isotropicreflection}.
When $n$ is odd, this surjectivity is retrieved by the fact that $\underline{\text{\textbf{O}}}_{V'} \cong \underline{\text{\textbf{SO}}}_{V'} \times \underline{\mu}_2$ (cf. \cite[Thm.1.7]{Con}), and so applying \'{e}tale cohomology to the
exact sequence of smooth groups
\begin{equation*}
1 \to\underline{\text{\textbf{SO}}}_{V'} \to\underline{\text{\textbf{O}}}_{V'}
\to\underline{\mu}_{2} \to1
\end{equation*}
we get that
$\ker[H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V'})
\xrightarrow{\psi'} H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{\text{\textbf{O}}}_{V'})]=1$ for any
$[(V',q')] \in\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$, which
means that the restricted map
$\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) \xrightarrow{ \psi}
\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$ is injective.
Together with Remark~\ref{genus}, this amounts to the existence of the
following exact and commutative diagram:
\begin{equation*}
\xymatrix{
& \text{Cl}_S(\underline{\text{\textbf{SO}}}_{V}) \ar@{^{(}->}[r]^{\psi} \ar@{^{(}->}[d]^{i} & \text{Cl}_S(\underline{\text{\textbf{O}}}_{V}) \ar
@{^{(}->}[d]^{i'} \\
1 \ar[r] & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar[r]^{\psi'} \ar[d]^{m} & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{O}}}_{V}
) \ar[d]^{m'} \ar[r]^{d} & H^1_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar
@{^{(}->}[d]^{m''} \\
1 \ar[r] & H^1(K,\text{\textbf{SO}}_V) \ar@{^{(}->}[r]^{h} & H^1(K,\textbf
{O}_V) \ar[r]^{d'} & H^1(K,\mu_2)
}
\end{equation*}
in which as $m''$ is injective due to Remark~\ref{finiteetaleextensionisembeddedingenericfiber},
$\psi$ is also surjective, thus is the identity.
\end{proof}

\begin{prop}
\label{Wittexactness}
Let $(V,q)$ be a regular quadratic $\mathcal{O}_{S}$-space of rank
$n \geq3$ with proper genus $\text{Cl}_{S}(\underline{\text{\textbf{SO}}}
_{V})$. The relative Witt-invariant (cf. Definition~\ref{relativeWitt-invariant} and Remark~\ref{restriction}) induces an
exact sequence of pointed sets
\begin{equation*}
1 \to\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) \xrightarrow{h} H
^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})
\xrightarrow{w
_{V}} {_{2}{\mathrm{Br}}(\mathcal{O}_{S})} \to1
\end{equation*}
in which $h$ is injective and ${_{2}}{\mathrm{Br}}(\mathcal{O}_{S})$
bijects to the set of $2^{|S|-1}$ proper genera of $q$.
\end{prop}

\begin{proof}
Consider the short exact sequence induced by the double covering of the
generic fiber
\begin{equation*}
1 \to\mu_{2} \to\text{\textbf{Spin}}_{V} \to\text{\textbf{SO}}_{V} \to1.
\end{equation*}
As $\text{\textbf{Spin}}_{V}$ is simply connected, we know due to
Harder that $H^{1}(K,\text{\textbf{Spin}}_{V})=1$ (cf.
\cite[Satz~A]{Hard}). This is true for all twisted forms of $\text{\textbf{Spin}}_{V}$, whence Galois cohomology implies the embedding
$H^{1}(K,\text{\textbf{SO}}_{V}) \hookrightarrow H^{2}(K,\mu_{2})$. Due
to Hilbert's Theorem~90, applying Galois cohomology to the Kummer's
exact sequence related to $\mu_{2}$ over $K$ gives an isomorphism
$H^{2}(K,\mu_{2}) \cong{_{2} {\mathrm{Br}}(K)}$. Moreover, as shown in
the sequence \eqref{Grothendieck}, ${\mathrm{Br}}(\mathcal{O}_{S})$ is
a subgroup of ${\mathrm{Br}}(K)$. All together, the relative
Witt-invariant applied to classes in
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$
and on their generic fibers, yields the following exact and commutative
diagram:
\begin{equation}
\label{Wittdiagram}
\xymatrix{
H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar[r]^{w_V} \ar[d] & {_2 {\mathrm{Br}}(\mathcal{O}_S)} \ar@{^{(}->}[d] \\
H^1(K,\text{\textbf{SO}}_V) \ar@{^{(}->}[r]^{w_V} & {_2 {\mathrm{Br}}(K)}
}
\end{equation}
which justifies the left exactness in the asserted sequence:
\begin{eqnarray}
\label{genusinjection}
\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) \stackrel{\text{(\ref{propergenus})}}{=} \ker[H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{SO}}}_{V}) \to
H^{1}(K,\text{\textbf{SO}}_{V})] = \ker[H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})
\xrightarrow{w _{V}} {_{2} {\mathrm{Br}}(\mathcal{O}_{S})}].\nonumber\\
\end{eqnarray}
The surjectivity of $w_{V}:H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{SO}}}_{V})
\to{_{2}{\mathrm{Br}}(\mathcal{O}_{S})}$ (cf. Remark~\ref{restriction})
completes the proof.

The first equality in \eqref{genusinjection} suggests that for any
$[q'] \in
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$,
$q' \in\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$ if and only
if $q'$ is $K$-isomorphic to $q$ while the second equality claims that
this holds if and only if $w_{V}(q')=w_{V}(q)$. This is true for any
choice of base point, being regular as well. So as $H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ is
a disjoint union of the proper genera of $q$, together with the
surjectivity of $w_{V}$, we deduce that the set of proper genera
bijects with $_{2} {\mathrm{Br}}(\mathcal{O}_{S})$, whose cardinality
is computed in Lemma~\ref{no.genera}.
\end{proof}

\begin{theorem}
\label{isotropicclassification}
Let $(V,q)$ be a regular quadratic $\mathcal{O}_{S}$-space
of rank $n \geq3$. Then there exists a surjection of pointed-sets:
${Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) \twoheadrightarrow \text{Pic~}(\mathcal{O}_{S})/2$.
If $(V,q)$ is isotropic, then this is a bijection,
so the abelian group $\text{Pic~}(\mathcal{O}_{S})/2$ is isomorphic to ${Cl}_{S}(\underline{\text{\textbf{SO}}}_{V}) =
 {Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$.
\end{theorem}

\begin{proof}
Consider the following exact and
commutative diagram derived from sequences \eqref{Kummermu2H1} and \eqref{H1Spin}:
\begin{align}
\label{Snakediagram}
\xymatrix{
& 1 \ar[r] \ar[d] & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar@{=}[r] \ar@{->>}[d]^{s\delta_V}
& H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar@{->>}[d]^{w_V} \ar[r] & 1 \\
1 \ar[r] & \text{Pic~}(\mathcal{O}_S)/2 \ar[r]^{\partial} & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar
[r]^{i_*^2} & {_2 {\mathrm{Br}}(\mathcal{O}_S)} \ar[r] & 1.
}
\end{align}
We imitate the Snake Lemma argument (though the diagram terms are not
all groups): according to Proposition~\ref{Wittexactness} $\text{Cl}
_{S}(\underline{\text{\textbf{SO}}}_{V}) = \ker(w_{V})$, hence for any
$[q'] \in\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$ one has
$i_{*}^{2}(s\delta_{V}([q']))=[0]$, i.e., $[q']$ has a
$\partial$-preimage in $\text{Pic~}(\mathcal{O}_{S})/2$ which is unique
as $\partial$ is a monomorphism of groups. Moreover, any element in
$\text{Pic~}(\mathcal{O}_{S})/2$ arises in this way, since $s\delta
_{V}$ is surjective. As a result we have an exact sequence of pointed
sets:
\begin{equation*}
1 \to\mathfrak{K}_{1} \to\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})
\xrightarrow{s\delta_{V}} \text{Pic~}(\mathcal{O}_{S})/2 \to1.
\end{equation*}
If $(V,q)$ is isotropic, then $s\delta_{V}$ is also injective.
Indeed,
let $\underline{\text{\textbf{SO}}}_{V'}$ be a twisted form of $\underline{\text{\textbf{SO}}}_{V}$, properly
stabilizing a form $q'$, and let $\underline{\text{\textbf{Spin}}}_{V'}$ be its
Spin group. The lower row in the following exact diagram is the one
obtained when replacing the base point $q$ by $q'$,
as described in \cite[IV, Proposition~4.3.4]{Gir}:
\begin{equation} \label{Gir}
\xymatrix{
        \mathfrak{K}_1             \ar[r] & \text{Cl}_S(\underline{\textbf{SO}}_V)         \ar@{}[d]|-*[@]{\subset} \ar@{->>}[r]^{s\delta_V}   & \text{Pic~}(\mathcal{S}) / 2  \ar[d]  \\
H^1_\text{\'et}(\mathcal{S},\underline{\textbf{Spin}}_V)  \ar[r] & H^1_\text{\'et}(\mathcal{S},\underline{\textbf{SO}}_V)  \ar@{->>}[r]^{s\delta_V}                            & H^2_\text{\'et}(\mathcal{S},\underline{\mu}_2)    \\
H^1_\text{\'et}(\mathcal{S},\underline{\textbf{Spin}}_{V'}) \ar[r] & H^1_\text{\'et}(\mathcal{S},\underline{\textbf{SO}}_V') \ar@{->>}[r]^{\delta_{V'}} \ar[u]_{\cong}^{\theta}  & H^2_\text{\'et}(\mathcal{S},\underline{\mu}_2) \ar[u]^{r}_{\cong}
}
\end{equation}
If $[q'] \in\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$, then
$q'$, being $K$-isomorphic to $q$, is also $K$-isotropic, as well as
the generic fiber $\text{\textbf{Spin}}_{V'}$. Then by the
Hasse--Minkowsky Theorem (cf. \cite[VI.3.1]{Lam}), $q'$ is
$\hat{K}_{\mathfrak{p}}$-isotropic everywhere, in particular in $S$.
Hence $H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{\text{\textbf{Spin}}}_{V'})$ is trivial by Lemma~\ref{H1=1sc} for any class in
$\text{Cl}_{S}(\underline{\text{\textbf{SO}}} _{V})$
($\underline{\text{\textbf{SO}}}_{V}$ is not commutative for $n \geq3$
thus
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$
does not have to be a group, so the triviality of
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{Spin}}}_{V})$
does not imply the injectivity of $s\delta_{V}$, i.e., there still
might be distinct anisotropic classes in
$H^{1}_{\text{\'{e}t}}(\mathcal{O}
_{S},\underline{\text{\textbf{SO}}}_{V})$ whose images in $H^{2}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2})$ coincide).
So $\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$, being equal in the isotropic case to $\text{Cl} _{S}(\underline{\text{\textbf{O}}}_{V})$
by Lemma~\ref{samegenus}, embeds in $\text{Pic~}(\mathcal{O}_{S})/2$
and the assertion follows.
\end{proof}

\begin{definition}
We say that the \emph{local-global Hasse principle} holds for a
quadratic $\mathcal{O}_{S}$-space $(V,q)$ if
$|\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})|=1$.
\end{definition}

\begin{cor}
The Hasse principle holds for a regular isotropic quadratic
$\mathcal{O}_{S}$-form of rank $\geq3$ if and only if $|\text{Pic~}(
\mathcal{O}_{S})|$ is odd.
\end{cor}

\begin{cor}
\label{genera} Let $(V,q)$ be an $\mathcal{O}_{S}$-regular quadratic
space of rank $\geq5$. Then the pointed-set
$H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\textbf{SO}}_{V})$
is isomorphic to the abelian group $H^{2}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2})$,
i.e., any $\mathcal{O}_{S}$-isomorphism class in the proper classification corresponds to an Azumaya $\mathcal{O}_{S}$-algebra with involution. There are $2^{|S|-1}$ genera, each of them is isomorphic to $\text{Pic~}(\mathcal{O}_{S})/2$.
\end{cor}

\begin{proof}
In rank $\geq5$, any quadratic $\mathcal{O}_{S}$-space is isotropic;
indeed, for any such $(V',q')$, the generic fiber $q'_{K}:= q' \otimes
K$ is isotropic (cf. \cite[Theorem~66:2]{OMe}), i.e., there exists a
non-zero vector $v_{0} \in V' \otimes K$ such that $q'_{k}(v_{0})=0$.
Since $K$ is the fraction field of the Dedekind domain $\mathcal{O}
_{S}$, there exists a non-zero vector $\underline{v}_{0} \in Kv_{0}
\cap\mathcal{O}_{S}$ for which $q(\underline{v}_{0})=0$. Hence
according to Theorem~\ref{isotropicclassification}, we deduce that the
genus of any $[(V',q')] \in H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{SO}}}_{V})$ is isomorphic to the
abelian group $\text{Pic~}( \mathcal{O}_{S})/2$ and injects into
$H^{1}_{\text{\'{e}t}}(
\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$. Looking at the
obtained exact and commutative diagram:
\begin{equation*}
\xymatrix{
1 \ar[r] & \text{Cl}_S(\underline{\text{\textbf{SO}}}_{V}) \ar[r] \ar[d]^{\cong} & H^1_\text{\'et}(\mathcal{O}_S,\underline{\text{\textbf{SO}}}_{V}) \ar
[r]^{w_V} \ar@{->>}[d]^{s\delta_V} & {_2 {\mathrm{Br}}(\mathcal{O}_S)} \ar[r] \ar@{=}[d] & 1 \\
1 \ar[r] & \text{Pic~}(\mathcal{O}_S)/2 \ar[r]^{\partial} & H^2_\text{\'et}(\mathcal{O}_S,\underline{\mu}_2) \ar
[r]^{i_*^2} & {_2 {\mathrm{Br}}(\mathcal{O}_S)} \ar[r] & 1
}
\end{equation*}
we see that the cardinality of $H^{1}_{\text{\'{e}t}}(\mathcal
{O}_{S},\underline{\text{\textbf{SO}}}_{V})$, being the disjoint union
of its genera, equals the one of its $s\delta_{V}$-image
$H^{2}_{\text{\'{e}t}}(\mathcal {O}_{S},\underline{ \mu}_{2})$, hence
it is isomorphic to it. We have seen in Proposition~\ref{Wittexactness}
that ${_{2}}{\mathrm{Br}}(\mathcal{O}_{S})$ bijects with the set of
proper genera of $q$. Here, as any twisted form of $q$ is isotropic,
its proper genus is equal to its genus (Lemma~\ref{samegenus}), thus
there are $2^{|S|-1}$ such genera (Lemma~\ref{no.genera}).
\end{proof}

\begin{remark}
Since as we have seen any integral quadratic form of rank $\geq 5$ is isotropic, according to Lemma \ref{samegenus},
$\text{Cl}_S(\underline{\text{\textbf{SO}}}_{V})$ might not be equal to $\text{Cl}_S(\underline{\text{\textbf{O}}}_{V})$
(for $\text{rank}(V) \geq 3$) only when $(V,q)$ is anisotropic of rank $4$.
\end{remark}

\section{A splitting hyperbolic plane}\label{hyperbolicplane}

In this section, we refer to regular quadratic $\mathcal{O}_{S}$-spaces
being split by a hyperbolic plane $P$ (thus being isotropic and so
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V}) =
\text{Cl}_{S}(\underline{\text{\textbf{SO}}}_{V})$). Such $P$ contains
a hyperbolic pair $\{v_{0},v _{1}\}$ of $V$, namely, satisfying:
$q(v_{0})=q(v_{1})=0$ and $B_{q}(v_{0},v_{1})=1$, and it is of the form
$P_{\mathfrak{a}}= \mathfrak{a}v_{0} + \mathfrak{a}^{-1} v_{1}$ for
some fractional ideal $\mathfrak{a}$ of $\mathcal{O}_{S}$ (cf.
\cite[Proposition~2.1]{Ger}). L. J. Gerstein established in
\cite[Theorem~4.5]{Ger} for the ternary case the bijection:
\begin{equation*}
\psi: \text{Pic~}(\mathcal{O}_{S})/2 \xrightarrow{\sim} \text{Cl}
_{S}(\underline{\text{\textbf{O}}}_{V}): \ \ [\mathfrak{a}] \mapsto[V_{
\mathfrak{a}}] := [P_{\mathfrak{a}}\bot\mathfrak{b}\lambda]
\end{equation*}
where $\mathfrak{b}$ is again a fractional ideal of $\mathcal{O}_{S}$
and $\lambda\in\mathcal{O}_{S}^{\times}$, for which $V \cong V_{
\mathfrak{a}}$. The existence of $\psi$ can be viewed as a particular
case of our Theorem~\ref{isotropicclassification} (such a splitting
does not necessarily exist). The following Lemma suggests an alternative
bijection which resembles the one of Gerstein, but does not, however,
require finding and multiplying by a hyperbolic pair, and, more
important, is valid for any rank $n \geq3$.

\begin{prop}
\label{bijection} Suppose $V$ is split by a hyperbolic plane
$V=H(L_{0}) \bot V_{0}$ where $L_{0}$ is an $\mathcal{O}_{S}$-line.
Then $\psi_{V}: [L] \mapsto[V _{L} = H(L_{0} \otimes L^{*}) \bot
V_{0}]$ is an isomorphism of groups $\text{Pic~}(\mathcal{O}_{S})/2
\cong\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$.
\end{prop}

\begin{proof}
$L^{*}$ is locally free, thus $V_{L}$ obtained from $V$ by tensoring
$H(L_{0})$ with $L^{*}$ remains in
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$. Due to Theorem~\ref{isotropicclassification}, the groups
$\text{Pic~}(\mathcal{O}_{S})/2$ and
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$ are isomorphic
through diagram \eqref{Snakediagram}, so it is sufficient to show by
its commutativity that $s\delta_{V} \circ\psi_{V}$ coincides with the
groups embedding $\partial$. The $i$'th \emph{Stiefel--Whitney class}
$w_{i}(E)$ of a regular $\mathcal{O}_{S}$-module $E$, as defined in
\cite[\S1]{EKV}, gets values in
$H^{i}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{ \mu}_{2})$ for $i
\geq1$ and \mbox{$w_{0}(E)=1$}. Its basic axioms, namely, $w_{i}(E)=0$ for all
$i > \text{rank}(E)$ and for any direct sum of regular
$\mathcal{O}_{S}$-modules of finite rank
\begin{equation*}
w_{k}(E \oplus F) = \sum_{i+j=k} w_{i}(E) \cdot w_{j}(F),
\end{equation*}
imply that:
\begin{equation}
\label{w2}
w_{1}(E \oplus F) = w_{1}(E) + w_{1}(F)
\ \
\text{and:}
\ \
w_{2}(E \oplus F) = w_{2}(E) + w_{2}(F) + w_{1}(E) \cdot w_{1}(F).
\end{equation}
If $L$ is an $\mathcal{O}_{S}$-line and $H(L) = L \oplus L^{*}$ is the
corresponding hyperbolic plane, this reads:
\begin{equation*}
w_{1}(H(L)) = w_{1}(L) + w_{1}(L^{*})
\ \
\text{while}
\ \
w_{2}(H(L)) = w_{1}(L) \cdot w_{1}(L^{*}).
\end{equation*}
Moreover, $w_{1}$ furnishes an isomorphism of abelian groups
$\{ \mathcal{O}_{S}\text{-lines}, \otimes\}/\sim\cong H^{1}_{
\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mu}_{2})$ by
$$ w_1(L_1 \otimes L_2) = w_1(L_1) + w_1(L_2), \ \ \text{thus:} \ \ w_1(L) = - w_1(L^*),  $$
hence
\begin{align} \label{w1}
w_1(H(L_0 \otimes L^*)) - w_1(H(L_0)) &= w_1(L_0 \otimes L^*) + w_1(L_0^* \otimes L) - w_1(L_0) - w_1(L_0^*) \\ \nonumber
                                      &= w_1(L_0) + w_1(L^*) + w_1(L_0^*) + w_1(L) - w_1(L_0) - w_1(L_0^*) = 0
\end{align}
and:
\begin{align} \label{diffofw2}
w_2(H(L_0 \otimes L^*)) - w_2(H(L_0)) &= (w_1(L_0) + w_1(L^*)) \cdot (w_1(L_0^*) + w_1(L)) - w_1(L_0) \cdot w_1(L_0^*) \\ \nonumber
                                      &= w_1(L^*) \cdot w_1(L_0^*) + w_1(L_0) \cdot w_1(L) + w_1(L^*) \cdot w_1(L) \\ \nonumber
                                      &= w_1(L^*) \cdot w_1(L_0^*) - w_1(L_0^*) \cdot w_1(L) + w_1(L^*) \cdot w_1(L) \\ \nonumber
                                      &= w_1(L^*) \cdot w_1(L) = w_2(H(L)).
\end{align}
In our setting (recall that $\delta=w_{2}$, see Remark~\ref{Stiefel-Whitney}), we get:
\begin{align}
\label{diffofdl}
\delta([V_{L}]) - \delta([V])
&\stackrel{\text{(\ref{w2})}}{=} \delta([H(L
_{0} \otimes L^{*})]) + \delta([V_{0}]) + w_{1}(H(L_{0} \otimes L
^{*})) \cdot w_{1}(V_{0})
\\
\nonumber
& \ -\Big( \delta([H(L_{0})]) + \delta([V_{0}]) + w_{1}(H(L_{0}))
\cdot w_{1}(V_{0}) \Big)
\\
\nonumber
&\stackrel{\text{(\ref{w1})}}{=} \delta([H(L_{0} \otimes L^{*})]) - \delta
([H(L_{0})]) \stackrel{\text{(\ref{diffofw2})}}{=} \delta([H(L)]).
\end{align}
Altogether, we may finally conclude that for any $[L] \in\text{Pic~}(
\mathcal{O}_{S})/2$:
\begin{align*}
(s\delta_{V} \circ\psi_{V})([L])
&= s\delta_{V}([V_{L}]) \stackrel{\text{(\ref{relativew})}}{=} (r_{V} \circ\delta\circ\theta^{-1})([V_{L}])
= (r_{V} \circ\delta)([V_{L}])
\\
\nonumber
&= \delta([V_{L}])-\delta([V]) \stackrel{\text{(\ref{diffofdl})}}{=}
\delta([H(L)]) \stackrel{\text{Remark~\ref{projectiveplane1}}}{=}
\partial([L]). \qedhere
\end{align*}
\end{proof}

\subsection{$|S|=1$}

If $S=\{\infty\}$ where $\infty$ is an arbitrary closed point, then
$C^{\text{af}}:= C -\{\infty\}$ is an affine curve whence ${\mathrm{Br}}(
\mathcal{O}_{S})=1$ (cf. Lemma~\ref{no.genera}), i.e., there is a
single genus, and any $\mathcal{O}_{S}$-regular quadratic space
$(V,q)$ of rank $n \geq3$ is isotropic (cf. \cite{Bit}). Suppose
furthermore that $C$ is an elliptic curve and $\infty$ is $
\mathbb{F}_{q}$-rational. For any place $P$ on $C^{\text{af}}$ we define
the maximal ideal of $\mathcal{O}_{S}$
\begin{equation*}
\mathfrak{m}_{P} := \left\{ f \in\mathcal{O}_{S}\ : \ f(P) = 0
\right\} .
\end{equation*}
Then we have an isomorphism of abelian groups (see
\cite[II,~Proposition~6.5(c)]{Hart} for the first map and p.~393 in
\cite{Bau} for the second one):
\begin{equation*}
\varphi: C(\mathbb{F}_{q}) \cong\text{Pic~}^{0}(C) \cong
\text{Pic~}(\mathcal{O}_{S}): \ \ P \mapsto[P] - [\infty] \mapsto
\mathfrak{m}_{P}.
\end{equation*}
Since $\mathcal{O}_{S}$ is Dedekind, Steinitz's Theorem (cf. \cite[Corollary~6.1.9]{BK}
tells us that for any fractional ideal
$\mathfrak{a}$ of $\mathcal{O}_{S}$ there is an
$\mathcal{O}_{S}$-isomorphism of $\mathcal{O}_{S}$-modules
(though not of quadratic $\mathcal{O}_{S}$-spaces):
$$ \chi: \mathfrak{a} \oplus \mathfrak{a}^{-1} \to \mathcal{S} \oplus \mathcal{S}. $$
Towards our purpose of finding representatives of twisted $K$-forms of
$q$, we may choose such a $K$-isomorphism $\chi_{K}$ to be as in
\cite[Lemma~20.17]{Cla}:
\begin{equation*}
\chi_{K}:(\alpha,\beta) \mapsto(\alpha,\beta) \cdot A_{
\mathfrak{a}}, \ \ A_{\mathfrak{a}}=
\left(
\begin{array}{c@{\quad}c}
b_{1} & -a_{2}
\\
b_{2} & a_{1}
\end{array}
\right)
\end{equation*}
where $a_{1}b_{1} + a_{2}b_{2}=1, \ b_{1} \in\mathfrak{a}^{-1}, \ b
_{2} \in\mathfrak{a}$. According to Proposition~\ref{bijection}, the
matrix $A_{\mathfrak{m}_{P}}^{-t} B_{q} A_{\mathfrak{m}_{P}}^{-1}$
represents, for any coset $[P] \in C(\mathbb{F}_{q})/2$, a distinct
class of quadratic $\mathcal{O}_{S}$-forms in
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V}) =
H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\text{\textbf{SO}}}_{V})$.

We summarize this procedure by the following algorithm:

\begin{algorithm}
\caption{Generator of classes representatives isomorphic in the flat topology to a regular quadratic space split by a hyperbolic plane,
over the coordinate ring of an affine non-singular elliptic curve} \label{Generatoralgorithm}
\textbf{Input:} $C$ = elliptic projective $\mathbb{F}_q$-curve, $S=\{\infty \in C(\mathbb{F}_q) \}$,
$V = H(L_0) \bot V_0$ = quadratic regular $\mathcal{S}$-space of rank $n \geq 3$,
$H(L_0)$ is represented by $F_0 \in \textbf{GL}_2(\mathcal{S})$.
\begin{algorithmic}
\State Compute $C(\mathbb{F}_q)$ and
$\mathcal{S}:=C^\text{af}(\mathbb{F}_q)$ where $C^\text{af} := C-\{\infty\}$.
\For{each $[P] \in C(\mathbb{F}_q)/2$ }
\State $\mathfrak{m}_P = \{  f \in \mathcal{S} \ : \ f(P) = 0 \}$
for $P \neq \infty$ and $\mathfrak{m}_{\infty}=\mathcal{S}$.
\State Find $a_1,b_2 \in \mathfrak{m}_P$ and $a_2,b_1 \in \mathfrak{m}_P^{-1}$ such that $a_1 b_1 + a_2 b_2 =1$.
\State
$A_{\mathfrak{m}_P}^{-1} = \left( \begin{array}{cc}
    a_1  & a_2  \\
    -b_2 & b_1  \\
\end{array}\right)$
and: $F_P = A_{\mathfrak{m}_P}^{-t} F_0 A_{\mathfrak{m}_P}^{-1}$.
\EndFor
\State \textbf{Output:} $\text{Cl}_S(\underline{\textbf{O}}_V) = \{ [V_P \bot V_0] : [P] \in C(\mathbb{F}_q)/2 \}$, where $V_P$ is the quadratic $\mathcal{S}$-form represented by $F_P$.
\end{algorithmic}
\end{algorithm}

\begin{remark}
If $C$ is an elliptic curve and $\infty$ is $\mathbb{F}_{q}$-rational,
then for any $\mathcal{O}_{S}$-line $L_{0}$, the special orthogonal
group of $H(L_{0})$, being split, of rank $2$ and
$\mathcal{O}_{S}$-regular, is a one dimensional split $\mathcal{O}_{S}$-torus, i.e., isomorphic to $\underline{\mathbb{G}}_{m}$ (see in
the proof of \cite[Theorem~4.5]{Bit}), hence the proper classification
is via $H^{1}_{\text{\'{e}t}}(\mathcal{O}_{S},\underline{\mathbb{G}}
_{m}) \cong\text{Pic~}(\mathcal{O}_{S}) \cong C(\mathbb{F}_{q})$. Then
the class representatives are obtained by the above algorithm when replacing $C(\mathbb{F}_{q})/2$ by $C(\mathbb{F}_{q})$. This means that invertible fractional ideals, corresponding to non-trivial squares in
$C(\mathbb{F}_{q})$, i.e., $[L] \in2 \text{Pic~}(\mathcal{O}_{S})
\backslash\{0\}$, and only they, induce spaces $H(L_{0} \otimes L
^{*})$ that are \emph{stably isomorphic} to $H(L_{0})$ in the proper classification, namely, become
properly isomorphic after being extended by any non-trivial regular orthogonal
$\mathcal{O}_{S}$-space. In other words, the Witt Cancellation Theorem
fails over $\mathcal{O}_{S}$ in this case for the proper classification.
\end{remark}

\begin{example}\footnote{I take this opportunity to correct Example~4.12 in
\cite{Bit}.} Let $C = \{ Y^{2}Z = X^{3} + XZ^{2}\}$ defined over
$\mathbb{F}_{5}$. Then
\begin{equation*}
C(\mathbb{F}_{5}) = \{ (0:0:1), (1:0:2), (1:0:3), (0:1:0) \}.
\end{equation*}
Taking $S = \{\infty= (0:1:0)\}$ we get the affine elliptic curve
\begin{equation*}
C^{\text{af}}= \{y^{2} = x^{3} + x \} \ \ \text{with:} \ \
\mathcal{O}_{S}= \mathbb{F}_{5}[x,y]/(y^{2}-x^{3}-x).
\end{equation*}
The affine supports of the points in $C(\mathbb{F}_{5}) - \{ \infty
\}$ are: $ \{ (0,0), (1/2,0) = (3,0), (1/3,0) = (2,0) \}$. The
$y$-coordinate of these points vanishes which means that they are of
order $2$ according to the group law and $C(\mathbb{F}_{q}) \cong(
\mathbb{Z}/2)^{2}$. We get:
\begin{equation*}
\mathfrak{m}_{(0:0:1)} = \langle x,y \rangle, \ \mathfrak{m}_{(1:0:2)}
= \langle x-3,y \rangle, \ \mathfrak{m}_{(1:0:3)} = \langle x-2,y
\rangle, \ \mathfrak{m}_{(0:1:0)} = \mathcal{O}_{S}.
\end{equation*}
Now consider the standard ternary quadratic $\mathcal{O}_{S}$-space
$V = H(\mathcal{O}_{S}) \bot\langle1\rangle$, i.e., with
$B_{q}= \left(
\begin{array}{c@{\quad}c@{\quad}c}
0 & 1 & 0
\\
1 & 0 & 0
\\
0 & 0 & 1
\end{array}
\right) $. For any $[L] \in\text{Pic~}(\mathcal{O}_{S})/2=\text{Pic~}(
\mathcal{O}_{S})$, the quadratic space $V_{L} = H(L) \bot\langle1
\rangle$ belongs to $\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$
and there are four non-equivalent classes in
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$. For example,
$A_{\langle x,y \rangle}^{-1} = \left(
\begin{array}{c@{\quad}c}
x & -y/x
\\
y & -x
\end{array}
\right) $ induces the form represented by $\left(
\begin{array}{c@{\quad}c@{\quad}c}
2xy & -2x^{2}-1 & 0
\\
-2x^{2}-1 & 2y & 0
\\
0 & 0 & 1
\end{array}
\right) $, being not $\mathcal{O}_{S}$-isomorphic to $V$, as
$\langle x,y \rangle$ is not principal.
\end{example}

\subsection{$|S|>1$}

If $|S|>1$ then an $\mathcal{O}_{S}$-regular quadratic space $V$ will possess multiple ($2^{|S|-1}$) genera,
and if $\text{rank}(V) = 3,4$ some of them may be anisotropic, i.e., contain anisotropic representatives only.

\begin{example}
\label{|S|=2}
Let $C$ be the projective line defined over $\mathbb{F}_{3}$, so
$K=\mathbb{F}_{3}(t)$, and let $S=\{t,t^{-1}\}$, so $\mathcal{O}_{S}=
\mathbb{F}_{3}[t,t^{-1}]$, being the ring of regular functions on the
multiplicative group $\underline{\mathbb{G}}_{m}$, thus it is a PID. The
ternary $\mathcal{O}_{S}$-space $V = \langle1,-1,-t \rangle$ with
$q(x,y,z) = x^{2}-y^{2}-tz^{2}$ is isotropic, e.g., $q(1,1,0)=0$. It is
properly isomorphic over $\mathcal{O}_{S}(i)$ (being a scalar extension
of $\mathcal{O}_{S}$ thus an \'{e}tale one), by $\text{diag}(1,i,-i)$
to the anisotropic form $V'=\langle1,1,t \rangle$. Both are
$\mathcal{O}_{S}$-unimodular as $\det(q)=t \in\mathcal{O}_{S}^{\times}$,
but belong to two distinct genera (there are exactly
$2^{|S|-1}=2$ such, cf. Proposition~\ref{Wittexactness}).
As $\mathcal{O}_{S}$ is a PID,
$\text{Pic}~(\mathcal{O}_{S})=1$, and so
according to Theorem~\ref{isotropicclassification} there is only one class in
$\text{Cl}_{S}(\underline{\text{\textbf{O}}}_{V})$.
The hyperbolic plane $\langle 1,-1 \rangle$ has trivial Witt invariant,
and it is orthogonal to $\langle -t \rangle$ in $V$, thus $w(q)=0$ (cf. \cite[IV, Prop.~8.1.1, 1), 3)]{Knu}), so $[q]$ corresponds to the
trivial element in $_{2}{\mathrm{Br}}(\mathcal{O}_{S})$. To compute the
Clifford algebra of $(V',q')$, we choose the natural basis
\begin{equation*}
\{e_{1}=(1,0,0), \ e_{2}=(0,1,0), \ e_{3}=(0,0,1)\}.
\end{equation*}
The embedding $i:V' \hookrightarrow\text{\textbf{C}}(q')$ satisfies the
relations $i(v)^{2}=q'(v) \cdot 1 \ \forall v \in V'$ which imply:
\begin{equation*}
i(v) i(u) + i(u) i(v) = B_{q'}(u,v) \ \ \forall u,v \in V'.
\end{equation*}
Since $\{e_{i}\}_{i=1}^{3}$ are orthogonal, this means that:
\begin{equation*}
i(e_{i})i(e_{j}) = - i(e_{j})i(e_{i})
\ \
\forall1 \leq i \neq j \leq3,
\end{equation*}
so we may choose (as $q'$ is anisotropic the obtained quaternion algebra
is not split):
\begin{equation*}
i(e_{1}) = \left(
\begin{array}{c@{\quad}c}
1 & 0
\\
0 & -1
\end{array}
\right) , \ i(e_{2}) = \left(
\begin{array}{c@{\quad}c}
0 & 1
\\
1 & 0
\end{array}
\right) , \ i(e_{3}) = \left(
\begin{array}{c@{\quad}c}
0 & \sqrt{-t}
\\
-\sqrt{-t} & 0
\end{array}
\right) .
\end{equation*}
Then $\{1,i(e_{1})i(e_{2}),i(e_{2})i(e_{3}),i(e_{1})i(e_{3})\}$ is a
basis of $\text{\textbf{C}}_{0}(q')$ (cf. \cite[V.~\S3]{Knu}):
\begin{equation*}
\text{\textbf{C}}_{0}(q') = \left\langle 1, \ \left(
\begin{array}{c@{\quad}c}
0 & 1
\\
-1 & 0
\end{array}
\right) , \ \left(
\begin{array}{c@{\quad}c}
-\sqrt{-t} & 0
\\
0 & \sqrt{-t}
\end{array}
\right) , \ \left(
\begin{array}{c@{\quad}c}
0 & \sqrt{-t}
\\
 \sqrt{-t} & 0
\end{array}
\right) \right\rangle
\end{equation*}
and $w(q') = [\text{\textbf{C}}_{0}(q')]$ is the non-trivial element in
$_{2}{\mathrm{Br}}(\mathcal{O}_{S})$.
\end{example}

{\bf Acknowledgements:}
I thank L.~Gerstein, B.~Kahn,
B.~Kunyavski\u{\i} and R.~Parimala for valuable discussions concerning
the topics of the present article. I would like to give special thanks
to P. Gille, my postdoc advisor in Camille Jordan Institute in the
University Lyon 1 for his fruitful advice and support.

redpajama_set_name: RedPajamaArXiv


text:
Electric longboard folds down, fits in knapsack
By Ben Coxworth
Electric longboard folds down,...
Linky folded down and ready to bag
Linky Innovation
Linky's wireless remote control unit
The accompanying bag is being made in three versions – two knapsacks and one travel bag – that accommodate varying amounts of extra "stuff" besides the board itself
Linky's brushless permanent magnet motor is located in the front (driving the front wheels), and is powered by a 100-Wh lithium-ion battery pack
The 4.9-kg (10.8-lb) Linky features a main structure made from a high-strength polymer, combined with a deck consisting of carbon fiber on the bottom and bamboo on top
Linky is designed to be easy to tote when not in use
Longboards are a popular type of skateboard for commuting, as their extra length allows them to track well in a straight line. Therefore, it comes as no surprise that we're seeing an increasing number of electric longboards hitting the market. One problem with them, however, is that their long form factor makes them awkward to carry around when not in use. That's why Linky was created. It's a folding electric longboard, that fits inside its own bag.
Developed by a group of Italian entrepreneurs, the 4.9-kg (10.8-lb) Linky features a main structure made from a high-strength polymer, combined with a deck consisting of carbon fiber on the bottom and bamboo on top.
Its brushless permanent magnet motor is located in the front (driving the front wheels), and is powered by a 100-Wh lithium-ion battery pack. One 60-minute charge should be good for a range of 15 km (9 miles) at a top speed of 30 km/h (18 mph). Additionally, because the battery is calibrated to meet IATA regulations, Linky can reportedly be taken aboard airliners as carry-on luggage.
Speed and braking are controlled using a wireless handheld remote, while riding modes (Eco, Cruise and Sport) are selected between using an iOS/Android app. An LED headlight can also be activated, when riding on dark streets.
The accompanying bag is being made in three versions – two knapsacks and one travel bag – that accommodate varying amounts of extra "stuff" besides the board itself.
If you're interested, Linky is currently the subject of a Kickstarter campaign. A pledge of €899 (about US$1,000) will presently get you a board with the basic bag, assuming all goes according to plans. Hopefully it will have better luck than the Caseboard, a folding electric skateboard that incorporated its own carrying case, and that failed to meet its funding goal.
Linky can be seen in action, in the following video.
Source: Linky
UPDATE (Oct. 21/16): Linky has now moved its crowdfunding campaign to Indiegogo.
LINKY The Foldable Electric Longboard | Full product presentation video
Urban TransportFoldingKickstarterElectric Skateboards
Ben Coxworth
Based out of Edmonton, Canada, Ben Coxworth has been writing for New Atlas since 2009 and is presently Managing Editor for North America. An experienced freelance writer, he previously obtained an English BA from the University of Saskatchewan, then spent over 20 years working in various markets as a television reporter, producer and news videographer. Ben is particularly interested in scientific innovation, human-powered transportation, and the marine environment.
AlexandraKorey May 26, 2016 05:50 AM
Thanks for sharing the news about Linky! Great article. We're counting on doing a lot better than Caseboard, because Linky does fold, but it has the sporty feel of a regular longboard. So you're not encumbered by the case or bag while you're riding, and you can go FAST! Thanks again Alexandra - Linky campaign manager
Dutch EV is more than an ebike, less than a car
Snoek may just be the fastest velomobile you can buy
CityQ is like a four-wheeled electronic-drive ebike
Cake teams up with Dometic for electric food delivery box


text:
Shifti:History
From Shifti
Revision as of 17:39, 4 November 2014 by ShadowWolf (Talk | contribs) (Reverted edits by RandomDSdevel (talk) to last revision by JonBuck)
Shifti (the Wiki for Transformation Stories) was born in July, 2007. This is known by quite a few people. What most people do not understand is the reason that it was born. It can be summed up simply - one man got a bit angry. But why is that summation correct? And who was this "angry man"?
1 Pre-History
3 Current
Shifti is a distant relative of the Transformation Story Archive, which, sadly, has closed it's doors to new admissions. That site is where most of the administrators of Shifti first found quality Transformation Fiction and it's sister-project, the TSA-Talk Mailing List, is how most of them met. TSA-Talk played a huge role in the birth of Shifti.
Pre-History
Around 1997 ShadowWolf first found the TSA-Talk mailing list and the Transformation Story Archive. While he never remained subscribed for long (his current subscription - dating back to 2004 - is the longest he's been continually subscribed), it proved to be a source of motivation for his own writing. But it would be several years before he actually shared anything he wrote, preferring, instead, to read stories posted to the list and finding that he had a very long road ahead of him.
Then, in 2003, the Transformation Story Archive stopped updating. Hoping that the archive was just seeing a lull in activity, ShadowWolf noted the cessation in updates and moved on with his life. He eventually managed to get his writing abilities polished with the help of good editors - like Cubist and Michael Bard - but was still left feeling like the community had lost something when the Transformation Story Archive stopped updating.
In 2005 Bryan started a Wiki on the old menagerie.tf server. This Wiki was named "Shifti" and, though it was based on the MoinMoin engine, is the direct ancestor of the modern Shifti. It never saw a lot of author traffic - whether due to lack of exposure or because the MoinMoin engine was hard for people to get used to - and only lasted a few months (the server was moved across the country to become Lapinia and the database never recovered).
In late 2006 a very vocal conversation started on TSA-Talk about how the Transformation Story Archive hadn't updated in three years. After a while the thread died down when several people (ShadowWolf included) reported that Thomas Hassan and others were working on a massive overhaul of the site.
This calm was short-lived - by May of 2007 it had resurfaced. At this point ShadowWolf had been working on an entirely custom code-base for a story archive for nearly five months. His progress was slow - the requirements he'd come up with for the site were very complex. When the new thread started Bryan said that he'd been thinking of restarting his Wiki project ("Shifti") but he would base it on a code-base he was familiar with from his work editing Wikipedia.
Angered by his own stupidity at wanting to write the code-base completely from scratch as much as he was by a perceived lack of any action on the parts of the people actually complaining, ShadowWolf downloaded a copy of the MediaWiki engine. After making sure his server had all the tools needed to support the Wiki engine, he created a database for it, created a special user and installed the software. It was then that Bryan was contacted and privately "handed the keys". A few weeks later Shifti went public. Within a week Viqsi had registered the domain-name shifti.org and donated it to the project.
In early 2008 the router providing the gateway between Shifti and the Internet began to experience problems. This was corrected with the purchase of a new router and Shifti went back to its steady growth. Within three months of that Shifti's server began experiencing problems because of the amount of traffic. Jon Sleeper, a long time "list uncle" (administrator) of TSA-Talk and founding administrator of Shifti donated a computer to the site. When the donated computer was brought online the problems with the site timing out soon disappeared, though it took some work to get rid of the bugs that had crept into the small network that the machines are on.
Starting in Late January 2009 a very vocal thread started on TSA-Talk about the merits of Shifti. That thread started to spiral out of control, but by the end it was clear that there were some people that were not happy with an "archive" that was an open Wiki. It also brought to light that there were people that believed Shifti was setup as direct competition for the Transformation Story Archive.
Shifti was created with a different goal in mind. It was created to solve a problem - with the Transformation Story Archive being essentially defunct, there was no longer a single site dedicated to TF fiction. There were numerous sites that specialized in one form of Transformation - but the TSA had long been the only site that covered all the numerous branches. With it no longer updating, there was no place to look for new stories - this was the reason Bryan started the original Shifti. And while it was simply anger at people complaining instead of doing that motivated ShadowWolf to setup the current version of Shifti, it cannot be doubted that somewhere underneath the anger he also wanted to correct that lack.
Retrieved from "https://shifti.org/index.php?title=Shifti:History&oldid=18064"
Stories By Author
Stories By Setting
Stories By TF
Stories By Genre
TSA-Talk
Shifti Writer's School
Stories that need Editing
Stories that need a Critique
About Shifti


According to an annual official report on departures and overstays released today, a total of 1.4 million Indians travelled to the US on various visas last year, out of which over 30,000 stayed in the country despite their visas running out.
The Department of Homeland Security (DHS) submitted the report to the Congress on foreign visitors who entered the US on various visas, including business, tourists, students and visitors exchange, as non-immigrant visitors. The report covers about 96 per cent of the total non-immigrant visitors.
More than 50 million non-immigrants were expected to depart from the US in 2016, said the report. Of these, 739,478 overstayed their admission, resulting in a total overstay rate of 1.47 per cent. Of the more than 739,000 overstays, DHS said 628,799 were suspected "in-country" overstays, which means there is no record of these foreign nationals leaving the US. The rest left the country after the expiry of their lawful admission period, which is called out-of-country overstay.
According to the figures, more than 1.4 million Indians who entered the US were expected to leave the country in 2016, but more than 30,000 overstayed. Of these 30,000, a little over 6,000 Indian nationals left the US after the expiry of their visas, the report said. In 2016, more than one million Indians who came to the US for business, tourism or pleasure were expected to leave the country. Of these, 17,763 have overstayed in the country, it said.
Among the overstayed are 2,040 Indians who departed the US only after the expiry of their visas. Similarly, in 2016, as many as 9,897 Indian students or exchange scholars were expected to depart by the end of the year of whom 4,575 overstayed their legal period.
As per available records, 1,561 Indian students and exchange visitors left the country after their visas expired, while 3,014 of them have overstayed in the country, the report said.
There were also 339,076 Indians in various other categories who were expected to leave the country in 2016. Of these, 8,061 overstayed, including 2,402 Indians who left the US after their visas expired.
According to the report, more than 881,000 Indians on business or pleasure were expected to leave the country in 2015. Of these 14,348 Indians overstayed, including 1,463 who left the US after the expiry of their visas.


Trade in your favorite tote for a too-big-to-fit-through-the-door bow backpack.
It's currently Couture fashion week in Paris, AKA the most wonderful time of the year, where the fashion is fash-un and street style takes maximalism to new levels. Normcore has never met Couture week, and if you enjoy even a hint of fashion freakery, this is your moment.
Each season we see things that blow our minds (remember that insane butterfly headpiece Frances McDormand wore to the Met Ball? That was Valentino Couture) and this season, we were dumbstruck by Givenchy's backpack/bow hybrid.
The odds are good that we'll see someone actually wear one to fashion week, and we're 100 percent ready for it.
The styles were so large and wide, they look like they'd very barely fit through a doorway — if at all. Nevermind on a packed subway car — this is the sort of impractical, delightful, fantastical fashion that we absolutely live for.


text:
Q: Combining numeric columns with dataframe I have int type columns with dataframe.
I want to make a new column that will hold text representation of the row.
df['text'] = df[['project', 'from', 'to', 'score']].apply(lambda x: '_'.join(x), axis=1)

I am getting following error TypeError: ('sequence item 1: expected string or Unicode, int found', u'occurred at index 0')
How I can add casting to string with my text generation function ?

A: Try cast to str by astype:
df['text'] = df[['project', 'from', 'to', 'score']]
                .apply(lambda x: '_'.join(x.astype(str)), axis=1)

Or:
df['text'] = df[['project', 'from', 'to', 'score']]
                 .astype(str)
                 .apply(lambda x: '_'.join(x), axis=1)

redpajama_set_name: RedPajamaStackExchange


text:
India vs Pakistan T20 World Cup: How Virat Kohli got three byes after being bowled on free hit - dead ball controversy, rules explained
India vs Pakistan, T20 World Cup: Despite fans and former cricketers citing a dead ball, Virat Kohli was awarded three byes after he was bowled off a freehit delivery during India's chase on Sunday.
23 October, 2022 18:54 IST
India's Virat Kohli is bowled no a free-hit ball during from Pakistan spinner Mohammad Nawaz in the final over of the chase. | Photo Credit: AFP
India's thrilling four-wicket win over Pakistan in the 2022 T20 World Cup Super 12 at the Melbourne Cricket Ground witnessed moments of controversy before Virat Kohli's unbeaten 82 helped his team seal the chase on Sunday.
With 13 required off three balls, Kohli struck a six over deep mid-wicket off a waist-high full toss which was deemed to be a contentious no-ball call by the umpires, leaving Pakistan skipper Babar Azam and Co. confused. It was followed by a wide which extended the free hit ball.
Matters flared up soon when Kohli was bowled off the free hit delivery, and he along with Dinesh Karthik, reeled in three byes as the ball trickled down towards third-man. The Pakistan team was then involved in a brief discussion with the umpires who eventually adjudged the runs in India's favour.
Why was no ball not reviewed, then how can it not be a dead ball when Kohli was bowled on a free hit. #INDvPAK#T20worldcup22pic.twitter.com/ZCti75oEbd
— Brad Hogg (@Brad_Hogg) October 23, 2022
Former cricketers including Australia's Brad Hogg and several Pakistan fans objected that a batter bowled off a free hit would result in a dead ball that prevents further scoring from the same delivery.
T20 World Cup: India beats Pakistan to break Australia's record for most wins across formats in a calendar year
Can a batter run off a freehit despite being bowled?
While allegations of a dead ball hit India's sensational chase, the MCC's Laws of Cricket states that: "20.1.1 The ball becomes dead when,
20.1.1.1 it is finally settled in the hands of the wicket-keeper or of the bowler.
20.1.1.2 a boundary is scored. See clause 19.7 (Runs scored from boundaries).
20.1.1.3 a batter is dismissed. The ball will be deemed to be dead from the instant of the incident causing the dismissal."
Since a free hit only allows four modes of dismissals - handled the ball, hit the ball twice, obstructing the field and run out - and bowled not being one of them, India was awarded the bye as the ball was still in play. It left India requiring two runs off two deliveries in the chase.
While Karthik was stumped off the penultimate delivery, Ravichandran Ashwin sealed the win with a lofted shot over mid-off off the last ball from left-arm spinner Mohammad Nawaz.
WATCH: Vinayak Mane decodes Suryakumar Yadav's recipe for success
IND vs ZIM Preview: Zimbabwe stands between India and semis
John Buchanan weighs in on Rohit, Warner's slump in form; predictions for T20 World Cup final
Rodrygo


text:
Q: Is countable infinity part of the natural numbers? A hopefully relatively simple question: is countable infinity an element of the set of natural numbers? A related question here has been answered with the statement that (classic) infinity is not part of the natural numbers, but what about countable infinity?

Context for this question: A definition of a Reproducing Kernel Hilbert Space (RKHS) I have come across is as follows:
The reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ is the closure of the linear span $\{f:f(x)=\sum_{i=1}^{m} a_ik(x,x_i), \space a_i \in \mathbb{R}, \space m \in \mathbb{N}, x_i \in \mathcal{X}\}$ equipped with inner products $\left< f,g \right>_{\mathcal{H}}=\sum_{ij}a_ib_jk(x_i,x_j)$ for $g(x)=\sum_{i=1}^{m} b_ik(x,x_i)$.
Many examples or RKHSs have infinitely many dimensions, so we would require $m=\infty \in \mathbb{N}$. Feel free to ignore this context if it is not required for your answer.

A: No, you do not need $m = \infty$. Note that you are taking the closure of the linear span, which takes care of the elements requiring an infinite sum for their representation.


text:
#ifndef QWINTHUMBNAILTOOLBAR_H
#define QWINTHUMBNAILTOOLBAR_H

#include <QtCore/qobject.h>
#include <QtCore/qscopedpointer.h>
#include <QtWinExtras/qwinextrasglobal.h>

QT_BEGIN_NAMESPACE

class QWindow;
class QWinThumbnailToolButton;
class QWinThumbnailToolBarPrivate;

class Q_WINEXTRAS_EXPORT QWinThumbnailToolBar : public QObject
{
    Q_OBJECT
    Q_PROPERTY(int count READ count STORED false)
    Q_PROPERTY(QWindow *window READ window WRITE setWindow)

public:
    explicit QWinThumbnailToolBar(QObject *parent = 0);
    ~QWinThumbnailToolBar();

    void setWindow(QWindow *window);
    QWindow *window() const;

    void addButton(QWinThumbnailToolButton *button);
    void removeButton(QWinThumbnailToolButton *button);
    void setButtons(const QList<QWinThumbnailToolButton *> &buttons);
    QList<QWinThumbnailToolButton *> buttons() const;
    int count() const;

public Q_SLOTS:
    void clear();

private:
    Q_DISABLE_COPY(QWinThumbnailToolBar)
    Q_DECLARE_PRIVATE(QWinThumbnailToolBar)
    QScopedPointer<QWinThumbnailToolBarPrivate> d_ptr;
    friend class QWinThumbnailToolButton;

    Q_PRIVATE_SLOT(d_func(), void _q_updateToolbar())
    Q_PRIVATE_SLOT(d_func(), void _q_scheduleUpdate())
};

QT_END_NAMESPACE

#endif // QWINTHUMBNAILTOOLBAR_H


Those who have services with Heart Internet won't have failed to notice the furore recently with an outage which lasted days (well 22 hours actually), caused considerable downtime for users with websites going dark & VPS's being unresponsive, & of course a subsequent Twitter mob baying for Heart's blood. So, what happens when your ISP goes dark?
← How to protect yourself online.