text:
\begin{document}
\author{Ronie Peterson Dario}
\title{Kaplansky Radical, Hilbert's Theorem 90 and Pro-2 Galois Groups}
\maketitle
\begin{abstract}
The Hilbert's Theorem 90 is proved for a field with non-trivial Kaplansky radical and the Galois group of its maximal $2$-extension as a finitely generated Elementary Type pro-2 group.
In the process, it is related the non-triviality of the Kaplansky radical with a particular decomposition of the Galois group as a free pro-$2$ product.
\end{abstract}
\tableofcontents
{\bf Keywords} Kaplansky radical, Hilbert's Theorem 90, Galois Theory, valuations

\section{Introduction}

Let $F$ be a field of characteristic not equal to $2$ and $\dtF$ its multiplicative group. The Hilbert's Theorem 90 assumes its simplest form when considered a quadratic extension $K=F(\sqrt{a})$. It states that the inverse image of the norm map $N: \dot{K} \to \dtF$ is $\dtF\dot{K}^2$, for $\dot{K}^2$ the subgroup of squares $\{\alpha^2, | \ \alpha\in\dot{K}\}$.

On the other hand, the Kaplansky radical of $F$ is denoted by $R(F)$ and it can be identified as the radical of the symmetric pairing
\begin{equation}\label{map:cup product} \cq \times \cq \to \twotorsion, \ \ ([a],[b])\mapsto [(F;a,b)] \end{equation}
where $(F;a,b)$ is the quaternion algebra generated by $i,j$ such that $i^2=a$, $j^2=b$, $ij=-ji$ and $\twotorsion$ is the set of all classes of finitely generated central simple algebras in the Brauer group of $F$ having order dividing $2$. The study of fields with a non-trivial radical, that is, $\Fs \subsetneqq R(F)$, is an important subject in the algebraic theory of quadratic forms over fields \cite[Chapter XII, \S6]{lam1}, \cite{kijima1}, \cite{becher-leep}, \cite{kula2}, \cite{cordes2}, \cite{dario3}. In this context, $R(F)$ is translated as the set of all $a\in \dtF$  such that the value group of the binary quadratic form $X^2 -aY^2$ is the entire multiplicative group $\dtF$.

It was Kijima and Nichi \cite{kijima1} who conjectured a version of the Hilbert's 90 Theorem for fields having a non-trivial Kaplansky, just replacing $\Fs$ by $R(F)$. Here we add the condition $(\dtF:\Fs)<\infty$. In this way,

\begin{claim}[H90-R]\label{claimh90} For $F$ a field of characteristic not equal to $2$, $(\dtF:\Fs)$ finite, let $K = F(\sqrt{a})$, $a\in \Fx\backslash \Fs$, be a quadratic extension of $F$ and $N:\dot{K} \to \dtF$ the norm map. Then $N^{-1}(R(F)) = \dtF R(K)$. \end{claim}

Cordes and Ramsey \cite[Theorems 3.10, 4.14]{cordes3} proved it under the hypothesis that $(\dtF: \Fs)$ is finite and $\twotorsion$ has exactly $2$ elements. Assuming an extra condition on the set of orders of $F$, Claim \ref{claimh90} was studied for quasi-Pythagorean fields \cite{kijima1}, \cite{kijima2}, \cite{iwakami}, that is, fields having the Kaplansky radical as the set of all non-zero sums of two squares. Without the assumption that $(\dtF:\Fs)$ is not finite, Claim \ref{claimh90} is not true in general. Indeed, using field valuations methods, Becher and Leep \cite[Theorem 4.8]{becher-leep} found a counter-example.

Denote by $F(2)$ the maximal $2$-extension of $F$, that is, the quadratic closure of $F$ inside a fixed separable closure of $F$. It is defined as the compositum of all finite Galois extensions of $F$ which degree is a power of $2$. The corresponding Galois group $\mathrm{Gal}(F(2);F)$ is denoted by $G_F(2)$. For $F$ a real closed field (e.g. $\R$) we have $G_F(2)\cong \Z/2\Z$. Otherwise, it is an infinite pro-$2$ group defined as the inverse limit of the Galois groups of all finite $2$-extensions of $F$.

In \cite{dario3} we study the interplay between field valuations, Kaplansky radical, and $G_F(2)$. Indeed, there we found a precise description for the Kaplansky radical of a field $F$ having a valuation compatible with $R(F)$ (see Example \ref{ex:pre-2-henselian-fields}). We call those fields {\it pre-$2$-henselian} and in \cite[Theorem 5.8]{dario2} (in Portuguese) it was shown that H90-R holds for such fields. By nature, this result depends heavily on valuations. The main idea in the proof is that $G_F(2)$ admits a particular decomposition, characterizing it as a very specific example of an ETG$_2$ group, as defined next.

The long-standing Elementary Type Conjecture for Galois Groups was inspired in the analogous for Witt rings, stated by Marshall \cite{marshall-book}. The version in Galois Theory states that if $G_F(2)$ is (topologically) finitely generated then it can be iteratively built from two operations in the category of pro-$2$ groups and basic Galois pro-$2$ groups. The operations are the free pro-$2$ product and the semi-direct product having the action described by the ciclytomic character \cite[Proposition 1.1]{engler3}. The basic groups are the Galois groups of finite, real closed or local fields. They are characterized as free pro-$2$ groups, $\Z/2\Z$, and (infinite) $2$-Demushkin groups, respectively. We call the resulting class of groups constructed by this way of the {\it class of Elementary Type pro-$2$ Galois Groups}, since all such groups are realizable as $G_F(2)$, for some field $F$. An element in this class is called an {\it ETG$_2$ group}. Jacob and Ware \cite{ware3} has described ETG$_2$ groups from Witt rings. A more general approach, also considering possibly non-realizable groups, can be found in \cite[Section 4]{minac18}.

In Theorem \ref{theorem90H} we prove H90-R (Claim \ref{claimh90}) for a field $F$ such that $G_F(2)$ is a ETG$_2$ group. From Galois Cohomology, the minimal number of (topological) generators of $G_F(2)$, denote by $d(G)$,  is equal to $\mathrm{dim}_{\F_2}\cq$. Therefore, we are assuming that $(\dtF:\Fs)$ is finite, that is, $G_F(2)$ is finitely generated.

Our result is a natural generalization of previously mentioned examples, as pre-$2$-henselian fields and quasi-Pythagorean fields, detailed in Example \ref{ex:pre-2-henselian-fields} and Example \ref{ex:quasi-pitag}, respectively. The same is true for examples constructed by Kula (Example \ref{ex:kula}) and Berman (Examples \ref{ex:berman1} and \ref{ex:berman2}). For the proof we make use of the Kurosh Subgroup Theorem for pro-$2$ groups (Theorem \ref{jkdfs7}) in order to find $G_2(F(\sqrt{a}))$ and $R(F(\sqrt{a}))$, $a\in \dtF\backslash\Fs$. It has two separated cases, according to $a\in \Fx\backslash R(F)$ or $a\in R(F)$, both detailed in Theorem \ref{djk7e}, cases $(1)$ and $(2)$.

In order to prove Theorem \ref{theorem90H} we also need to precisely decode non-triviality of Kaplansky radical from the Galois group $G_F(2)$. This subject has its own interest because important field arithmetic properties can be deduced from this group. For example, orders on $F$ come from the existence of torsion elements in $G_F(2)$. Recently, Koenigsmann and Strommen \cite{koe-strom-Q2} showed that the dyadic valuation on $F$ can be recovered if $G_F(2)\cong G_{\Q_2}(2)$.

Quasi-Pythagorean and pre-$2$-henselian fields have non-trivial Kaplansky radical and the Galois group $G_F(2)$ of those fields is described in Examples \ref{ex:quasi-pitag} and \ref{ex:pre-2-henselian-fields}, always assuming it finitely generated. Both are particular cases of the following.

\begin{claim}\label{claim-dec} Let $F$ be a field, $\mathrm{char} (F) \neq 2$, $\Fs \neq R(F) \neq \dtF$ and $(\dtF: \Fs)$  finite. There are two field extensions $L$ and $H$ of $F$ in $F(2)$ such that $G_2(F)$ admits a decomposition in a pro-$2$ free product $G_F(2)\cong G_L(2)\ast_2 G_H(2)$ satisfying:
\begin{itemize} \item[$(1)$] $G_L(2)$ is a free pro-$2$ group ($R(L) = \dot{L}$) of rank equal to $\mathrm{dim}_{\F_2}R(F)/\Fs$.
\item[$(2)$] The field $H$ has trivial Kaplansky radical. \end{itemize} \end{claim}

In Theorem \ref{decomposition} we prove it for a field $F$ such that $G_F(2)$ is a ETG$_2$ group. It includes the previous cited examples, as well Berman and Kula constructions (Examples \ref{ex:berman1}, \ref{ex:berman2} and \ref{ex:kula}).

If we assume the extra hypothesis $|\twotorsion|=2$ as the example of Cordes and Ramsey \cite{cordes3}, we have the so-called {\it one relator pro-$2$ Galois group}. Although Claim 1 is true for such field \cite[Theorems 3.10, 4.14]{cordes3}, in \cite{wulfel} it was raised question (which still not answered) if $G_F(2)$ has a decomposition like in Claim \ref{claim-dec}, with $H$ being a $2$-Demushkin field, as explained in Subsection \ref{sec:demushkin-fields}. If true, it would say that if $G_F(2)$ is an one-relator pro-$2$ group then it is ETG$_2$.

In the next section, we list other properties of $R(F)$, as well many examples. The relations between Kaplansky radical and basic groups, and with basic operations, are studied in Section \ref{sec:basic-groups-radical} and Section \ref{sec:basic-operations-and-radical}, respectively. In Section 5 we proved the decompositions results. Finally, H90-R for ETG$_2$ groups is proved in Section 6.

All fields will always be assumed to have characteristic not equal to $2$. Despite all the machinery in quadratic forms, another reason to exclude this case is that the Galois group $G_F(2)$ is a free pro-$2$ group.
Subgroups of profinite groups are assumed to be closed subgroups. homomorphisms are continuous.

\section*{Acknowledgments} I am deeply grateful to Antonio Jos\'e Engler, for great ideas for this paper, and to J\'an Min\'a\v{c}, for the endless mathematical discussions and all the inspiration to this work, partially written during my year at Western University. There I also had the pleasure of working with Federico Pasini and Marina Palaisti.

\section{The Kaplansky radical of a field}

Basic notations in this Section can be found in \cite[Chapter XII, \S6]{lam1}.

Let $F$ be a field of characteristic not equal to $2$ and $D_F\langle 1,-a\rangle$ be the value group of the quadratic form $X^2 - aY^2$ over $F$, or equivalently, the image of the norm map $N:F(\sqrt{a})^{\times}\to \dtF$.
We define the Kaplansky radical of $F$ as the set of all $a\in \dtF$ such that $N$ is onto, that is, $D_F\langle 1,-a \rangle= \dtF$. In the pairing (\ref{map:cup product}) remember that
\begin{equation}\label{zero}[(F;a,b)]=0 \in\twotorsion \Leftrightarrow a\in D_F\langle 1,-b\rangle \Leftrightarrow b \in D_F\langle1, -a\rangle.\end{equation}
Putting all together it follows that
\begin{equation}\label{radical-as-intersection}R(F) = \bigcap_{x\in \dtF}D_F\langle 1,x\rangle.\end{equation}
Denoting by $\sum \dtF^2$ all the non-zero finite square sums, we have a chain of subgroups
\begin{equation}\label{chain} \Fs \subseteq R(F) \subseteq D_F\lan 1,1\ran \subseteq \sum \dtF^2 \subseteq \dtF, \end{equation}
which collapses only for a quadratically closed field.

Most fields are indeed in the lower bound for the radical, that is, $R(F)=\Fs$. We will say that such a field has {\it trivial radical}.
It also means that the pairing (\ref{map:cup product}) is non-degenerated.

\begin{ex}\label{ex:indice-2} If $(\dtF:\Fs)=2$ we have $R(F)=\dtF$ or $R(F)=\Fs$. The first case occurs for finite fields and the formal Laurent series field $k((T))$, for $k$ a quadratically closed field. Remember that $F$ is a formally real field it has at least one total order, or equivalently,  $-1\notin\sum\Fs$. {\it Euclidean Fields} are formally real fields with $(\dtF:\Fs)=2$ and trivial radical.
\end{ex}
\begin{ex}\label{ex:fields-trivial-radical}[Fields having trivial radical]
\begin{itemize}
\item[$(a)$] {\it Formally real Pythagorean fields}. Remember that $F$ is Pythagorean if $\sum \dtF^2 = \Fs$. Note that a non-real Pythagorean field would be quadratically closed.
\item[$(b)$] {\it $2$-Demushkin fields}, e.g., the field $\Q_p$ of $p$-adic numbers, for $p$ a prime number. The radical is trivial by definition, see Section \ref{sec:demushkin-fields}.
\item[$(c)$] {\it Fields having rigid elements}: See Proposition \ref{prop:fields-with-rigid}. For instance, $2$-henselian fields which value group is not $2$-divisible (Example \ref{ex:2-henselian-fields}).
\item[$(d)$] {\it Real closed fields}: Euclidean fields (Example \ref{ex:indice-2}) such that all polynomials of odd degree have a root in the field, e.g., $\R$.
\end{itemize}
\end{ex}

The search for fields with non-trivial Kaplansky radical was motivated by the attempt of classify Witt rings of fields having eight squares classes. In order to finish the classification, Cordes \cite{cordes2} mentioned that it was necessary to build fields with non-trivial radical.

By using quadratic form schemes, Kula \cite{kula1} show that given positive integers $n$ and $m$, there is a field $F$ such that $(\dtF:R(F))=2^n$ and $(R(F):\Fs)=2^m$. In the same year, Berman \cite{berman} studied non-real extensions of Pythagorean fields to build more examples, also with the prescribed integers $n, m$. The answer to Cordes question appears in \cite{kula2}.

Kula and Berman examples are important particular cases of our results. For this reason we will see more details. There are two main results in \cite{berman}, both starting with formally real Pythagorean fields.

\begin{ex}\label{ex:berman1}\cite[Theorem 2.3]{berman}  Let $X(F)$ be the set of orders of the Pythagorean field $F$. Assuming that $X(F)$ has $2^n$ elements, $n\geq 2$, and $(\dtF:\Fs) = 2^{n+1}$, the field $K=F(\sqrt{-1})$ has $2^n$ square classes and $(\dot{K}:R(K))=4$.
\end{ex}

\begin{ex}\label{ex:berman2}\cite[Theorems 3.9 and 3.12]{berman} It starts with two particular Pytha\-gorean fields $F_1$ and $F_2$. The first one is called a {\it super Pythagorean field}: it has $2^n$ squares classes and $2^{n-1}$ orders, $n\geq 3$. The second one satisfies the so-called {\it Strong Approximation Property}: it has $2^m$ square classes and $m$ orders, $m\geq 2$. The later is called a {\it SAP Pythagorean field}. Then it was shown that there is a field $F$ such that $\dtF/\Fs \cong \dtF_1/(\dtF_1)^2\times\dtF_2/(\dtF_2)^2$ and for $K=F(\sqrt{-1})$ one has $\dot{K}/R(K)\cong \dtF_1/(\{\pm 1\}\dtF_1)^2$ and $R(K)/\dot{K}^2\cong\dtF_2/(\dtF_2)^2$.
 \end{ex}

The main example of Kula constructions has a different approach, as follows.

\begin{ex}\label{ex:kula}\cite{kula1} Let $L$ be a field with a finite number of square classes and $F$ a field such that $G_F(2)\cong G_{L}(2)\ast_2 G_{H}(2)$, where $R(L)=\dot{L}$ and $H$ is the field of the iterated Laurrent series $L((X_1))\ldots ((X_n))$. Then $(\dtF:R(F)) = 2^n(\dot{L}:\dot{L}^2)$. \end{ex}

Examples \ref{ex:berman1}, \ref{ex:berman2} and \ref{ex:kula} produced non formally real fields. On the other side, the main example of a real field having non-trivial radical is a {\it quasi-Pythagorean field}.
The definition of such field $F$ is the same as the condition $R(F)=D_F\langle 1,1\rangle$. It actually implies $D_F\langle 1,1\rangle =  \sum \Fs$ \cite[Corollary 6.5 (2), p. 452]{lam1}. Observe that if such field would be non-real, then $-1\in R(F)$ and by definition, $D_F\langle 1,1\rangle = \dtF$. Therefore, $R(F)=\dtF$. Excluding this case to be study in Section \ref{sec:free-2-galois}, we can  assume such fields as real fields.

\begin{ex}[Quasi-Pythagorean fields] \label{ex:quasi-pitag} Let $F$ be a (formally real) quasi-Pytha\-go\-rean field as above defined. For future reference, let us remember that Ware \cite[Corollary 1]{ware1} prove that $G_F(2)$ is the free pro-$2$ product of a free pro-$2$ group and a pro-$2$-group (topologically) generated by involutions, provided conditions that holds for $(\dtF:\Fs)$ finite.
\end{ex}

Another important class of fields with non-trivial Kaplansky radical is built from valuation theory. As we mention in the Introduction, pre-$2$-henselian fields were detailed in \cite{dario3}. Here, we postponed this example to Section 4.

\section{Basic groups and the Kaplansky radical} \label{sec:basic-groups-radical}

As mentioned in the Introduction, there are three kinds of basic pro-$2$ Galois groups. The only which is finite is $\Z/2\Z$. It corresponds to real-closed fields, or in general, Euclidean fields. They have trivial radical (Examples \ref{ex:indice-2} and \ref{ex:fields-trivial-radical}-d). It remains to study the other building blocks of the Galois group:

\begin{itemize}
	\item[$(a)$]  free pro-$2$ groups: described in the next subsection, corresponds to fields having the radical as the entire multiplicative group of the field.  This condition simplifies field arithmetic properties.
	\item[$(b)$] (infinite) $2$-Demushkin fields, in Subsection 3.2. The radical is trivial by definition and the field arithmetic properties are far more complicated.
\end{itemize}

It is convenient to mention that $\Z/2\Z$ also corresponds to a Demushkin field. Indeed, it is the only finite Demushkin group (See Definition \ref{def:2-demuskin}) \cite[Proposition 3.9.10]{neukirch}. For our work it is convenient to split in two cases.

\subsection{Free pro-$2$ Galois groups}\label{sec:free-2-galois}

Let $p$ be a prime number. The free pro-$p$ group $\mathcal{F}$ over the set $X$ is defined by a universal property, see \cite[Definition 3.5.14]{neukirch}, or directly built as a specific inverse limit using the ordinary free group over $X$ \cite[p. 41]{koch1}. The {\it rank} of $\mathcal{F}$ is the cardinality of $X$ and denoted by $rk(\mathcal{F})$.

\begin{ex}\label{ex:finite-fields} The free pro-$2$ group of rank $1$ is $\Z_2$, the additive group of  $2$-adic integers. If $F$ is a finite field, or $F=\C((T))$, then $G_F(2)\cong\Z_2$ \cite[p. 392]{ware3}.  In general, a free pro-$2$ group is realizable as a Galois pro-$2$ group $G_F(2)$ by taking free pro-$2$ products of $\Z_2$.\end{ex}

At the right end of the chain in (\ref{chain}) we have
\begin{equation}\label{free} R(F)=\dtF \Leftrightarrow  \twotorsion =\{0\} \Leftrightarrow  G_{F}(2) \text{  is a free pro-}2 \text{  group}.\end{equation}

The first equivalence follows from the (\ref{zero}) and (\ref{radical-as-intersection}). Indeed, one only has to observe that $R(F)=\dtF$ occurs iff the quaternion algebras $(F;a,b)$ splits, for all $a,b\in \dtF$, which is equivalent to $\twotorsion = \{0\}$, by Merkurjev's Theorem. The second equivalence follows from \cite[Corollary 3.8, p. 262 and Corollary 3.2, p. 255]{ribes1}.

In terms of quadratic forms (see \cite{lam1}), it means that every $2$-Pfister form is hyperbolic. If $IF$ is the fundamental ideal of even-dimensional quadratic forms classes in the Witt ring $W(F)$, it also means that $I^2F = \{0\}$. For the $u$-invariant of $F$ we have $u(F)\leq 2$.

\subsection{$2$-Demushkin Fields and pre-Hilbert fields}\label{sec:demushkin-fields}

Irving Kaplansky defined the radical studying $2$-Demushkin fields \cite{kaplansky1}. Back then, it was called Hilbert fields. See \cite[Chapter XII, \S6, p. 455-456]{lam1} for more details on Hilbert fields. They have trivial radical by definition, according to the item $(b)$ of the following.

\begin{defn}\label{def:2-demuskin} We say that $F$ is a $2${\it-Demushkin field} if the following three conditions are verified:
	\begin{itemize}
		\item[$(a)$] $\dtF/\Fs$ is a finite dimensional $\F_2$-vector space.
		\item[$(b)$] The pairing (\ref{map:cup product}) is non-degenerated ($F$ has trivial radical).
		\item[$(c)$] $\twotorsion\cong \Z/2\Z$, that is, only one quaternion algebra (up to isomorphism) is non-splited.
	\end{itemize}
Moreover, in this case, the Galois group $G_F(2)$ is called a $2${\it-Demushkin Galois group}.
\end{defn}

We have seen that $G_F(2)=\Z/2\Z$ is the only finite Demushkin group, realizable for a real-closed field, e.g. $\R$.

Demushkin groups are defined in general for a prime number $p$ and they are not necessarily realizable by an extension of fields \cite[Definition 3.9.9]{neukirch}. The structure of relations and generators of a Demushkin group was completely described by Labute and Serre and it is now very well understood \cite[Theorem 3.9.11 and 3.9.19]{neukirch}.

\begin{ex} Well known examples of $2$-Demushkin fields are finite extensions of the $p$-adic numbers, where $p$ is a prime number, as well the field of formal Laurent series $k((T))$, over a finite field $k$, $\mathrm{char}(k)\neq2$. \end{ex}

Condition $(a)$ of Definition \ref{def:2-demuskin} says that $G_F(2)$ is finitely (topologically) generated and by $(c)$, it has only one relation \cite[Corollary 3.9.5]{neukirch}.
However, condition $(b)$ does not have a clear interpretation in $G_F(2)$. In the absence of $(b)$ we say that $G_F(2)$ is an {\it one-relator} pro-$2$ group, so-called {\it pre-Hilbert field}. In \cite[p.356]{wulfel} it was raised the question if in this case $G_F(2)$ has a decomposition like Claim \ref{claim-dec}, with $H$ being a $2$-Demushkin field.

\section{Basic operations and the Kaplansky radical} \label{sec:basic-operations-and-radical}
In this section, we describe the behavior of the Kaplansky radical under the basic operations in the category of pro-$2$ groups. We start with free pro-$2$ products and then we do the same for semi-direct products.

\subsection{Free pro-$2$ products} \label{decomposition}
We need some results from Galois Cohomology, following notations from \cite{{neukirch}}.

Let $p$ be a prime number. The free pro-$p$ product of the pro-$p$ groups $G_1, \ldots, G_n$ is denoted by $G_1\ast_p \ldots \ast_p G_n$ and defined in \cite[Definition 4.1.1]{neukirch}.

Let $H^i(G, \F_p)$ be the $i$-th Galois cohomology group of the pro-$p$ group $G$ with coefficients in $\F_p$. By a well known result of Neukirch \cite[Satze 4.2, 4.3]{neukirch2}, $G$ is the free pro-$p$ product $G\cong G_1\ast_p \ldots \ast_p G_n$ iff the restriction map $\mathrm{res}_j: H^j(G, \F_p)\to  H^j(G_1, \F_p) \oplus \ldots \oplus H^j(G_n, \F_p)$ is an isomorphism for $j=1$ and a monomorphism for $j=2$.

In the next theorem it is translated for field theory and our case $p=2$ by choosing $G=G_F(2)$ and $G_i = G_{F_i}(2)$, for field extensions $F_i|F$ in $F(2)$, $i=1, \ldots, n$. We also need the isomorphisms
$$H^1(G;\F_2) \longrightarrow \dtF/\Ftwo, \ \  (a)\longmapsto a\Fs$$
$$H^2(G;\F_2) \longrightarrow \twotorsion, \ \  (a)\cup (b) \longmapsto [(a,b)]$$

\begin{thm}\label{teo:decomposition-neu-traduzido} Let $F_i$, $i=1, \ldots, n$, be field extensions of the field $F$ in $F(2)$. Then the Galois group $G_F(2)$ decomposes as the pro-$2$ free product $G_F(2) \cong G_{F_1}(2)\ast_2 \ldots \ast_2 G_{F_n}(2)$ if and only if the following two conditions are true
	\begin{itemize}
		\item[$(i)$] the inclusions $F\hookrightarrow F_i$, $i=1, \ldots, n$, induces an isomorphism $$\dtF/\Fs \to  \dtF_1/(\dtF_1)^2 \times \ldots \times \dtF_n/(\dtF_n)^2$$
		\item[$(ii)$] the scalar extensions $A\mapsto A\otimes_F F_i$, $i=1, \ldots, n$, induces a monomorphism $$\twotorsion \to  {}_2\mathrm{Br}(F_1) \times \ldots \times {}_2\mathrm{Br}(F_n).$$
	\end{itemize}
\end{thm}

For practice reasons, we will use the equivalences: $(i)\Leftrightarrow$ (I) and $(ii)\Leftrightarrow$  (II), as follows. The first one is immediate and for the other one it is enough to use the equivalences in (\ref{zero}).
\begin{itemize}
	\item[(I)] $\dtF\cap (\dtF_1)^2\cap\ldots\cap (\dtF_n)^2=\Fs$ and $\dot{F}_i = \dtF (\dot{F}_i)^2$, $i=1,\ldots, n.$
	\item[(II)] $\dtF \cap \displaystyle\bigcap_{i=1}^n D_{F_i}\langle 1, x\rangle = D_F\langle 1, x\rangle, \text{ for all }x\in \dtF.$
\end{itemize}

From (II) and the very definition (\ref{radical-as-intersection}) of $R(F)$ we have

\begin{thm}\label{radical-of-fre-product} If $F_1, \ldots, F_n \subset F(2)$ are $2$-extensions of the field $F$ such that $G_F(2) \cong G_{F_1}(2)\ast_2 \cdots \ast_2 G_{F_n}(2)$, then the Kaplansky radical of $F$ is \[R(F) = \dtF \cap \bigcap_{i=1}^n R(F_i).\]
	\end{thm}

As simple consequences of Theorem \ref{radical-of-fre-product} and (I), it follows:

\begin{cor}\label{cor:factors-free} In the same hypotheses of Theorem \ref{radical-of-fre-product}, if each $F_i$, $i=1, \ldots, n$, has trivial radical (respec. $R(F_i)=\dot{F}_i$, $i=1, \ldots, n$)  then $F$ has trivial radical (respec. $R(F)=\dtF$).\end{cor}

\begin{cor}\label{cor:free-factor-F_1} In Theorem \ref{radical-of-fre-product} suppose, in addition, that $G_{F_1}(2)$ is a free pro-$2$ group and $F_2, \ldots, F_n$ have trivial radical. Then
\begin{itemize}
\item[$(a)$] $R(F) = \dtF \cap (\dot{F}_2)^2\cap \ldots (\dot{F}_n)^2$.
\item[$(b)$] The inclusion $F\hookrightarrow F_1$ induces a monomorphism $R(F)/\Fs \longrightarrow \dot{F}_1/(\dot{F}_1)^2$.
\item[$(c)$] $(R(F):\Fs)=(\dot{F_1}:(\dot{F_1})^2)$. Therefore $rk(G_{F_1}(2)) = \mathrm{dim}_{\F_2} R(F)/\Fs$.
\end{itemize}
\end{cor}
\begin{proof} $(a)$ it follows from Theorem \ref{radical-of-fre-product} and $R(F_1)=\dot{F}_1$. \\
$(b)$ Note that $(\dot{F}_1)^2\cap R(F) = \Fs$, by condition $(I)$.\\
$(c)$ Let $H$ be the fixed field of $G_{F_2}(2)\ast_2\ldots\ast_2 G_{F_n}(2)$ in $F(2)$. Then $G_F(2)\cong G_{F_1}(2)\ast_2G_{H}(2)$. By Cororally \ref{cor:factors-free} and condition $(I)$, $H$ has trivial radical and $R(F)=\dtF\cap\dot{H}^2$. Now $\dot{H}/\dot{H}^2 = \dtF\dot{H}^2/\dot{H}^2\cong \dtF/(\dot{H}^2\cap\dtF)=\dtF/R(F)$. Thus $(\dot{H}:\dot{H}^2) = (\dtF:R(F))$. By Theorem \ref{teo:decomposition-neu-traduzido} $(i)$ we have $(\dot{F_1}:(\dot{F_1})^2)(\dot{H}:\dot{H}^2)=(\dtF:\Fs)$, which also is $(\dtF:R(F))(R(F):\Fs)$. Putting all together, $(R(F):\Fs)=(\dot{F_1}:(\dot{F_1})^2)$.
\end{proof}

\subsection{Semi-direct products and valuations} \label{sec:semi-direct}
Now let us turn to the second basic operation. While free pro-$2$ products can produce non-triviality for the radical, we will see that semi-direct products, in general, do the opposite, because it produces valuations rings and as consequence, the appearance of rigid elements.

Remember that we denote by $D_F\langle 1,a\rangle$ the group of values of the quadratic form $X^2+aY^2$ over the field $F$.

\begin{defn}\label{def:rigid-birigid-elements} A element $a\in \dtF\backslash\pm\Fs$ is called {\it rigid}
if $D_F\langle 1,a\rangle=\Fs\cup a\Fs$ and {\it birigid} if $a$ and $-a$ are rigids.
\end{defn}

Ware \cite{ware2} proved that the existence of enough rigid elements in $F$ allows us to build valuations with great properties, as we will see next. Along the same lines, more general versions were developed in \cite{arason1} and \cite{engler3}.

Now let us see the behavior of Kaplansky radical under the existence of at least one rigid element.

\begin{prop}\label{prop:fields-with-rigid} If the field $F$ has a rigid element $a\in \dtF$ then $(R(F):\Fs)\leq 2$ and it occurs one, and only one, of the following alternatives:
	\begin{itemize}
		\item[$(a)$] $\Fs = R(F)\neq \dtF$, that is, $F$ has trivial radical and $G_F(2)$ is not free.
		\item[$(b)$] $F$ is a formally real quasi-Pythagorean field and $R(F) = D_F\langle 1,1\rangle = \Fs \cup a\Fs$. Moreover, $G_F(2)\cong \Z_2\ast_2 \mathcal{H}$, where $\mathcal{H}$ is generated by involutions.
		\item[$(c)$] $R(F)=\dtF = \Fs\cup a\Fs$. In this case, $-1\in \Fs$ and $G_F(2)\cong \Z_2$.
	\end{itemize}
\end{prop}
\begin{proof} We have $R(F)\subseteq D_F\langle 1, a\rangle = \Fs\cup a\Fs$, since $a$ is rigid. Thus, $(R(F):\Fs)\leq 2$. If $a\notin R(F)$, then it occurs the first possibility. Let us assume that $a\in R(F)$. If $R(F)=\dtF$, then $-1\in \Fs$, because $a\notin\Fs$. Then we have option $(c)$. Now suppose $R(F)\neq\dtF$. By \cite[Proposition 6.3 (1), p.451]{lam1}, $a\in R(F)$ implies $D_F\langle 1, a\rangle = D_F\langle 1, 1\rangle$. Therefore $R(F)=D_F\langle 1,1\rangle=\Fs \cup a\Fs$. Finally, the description of $G_F(2)$ in $(b)$ follows from Example \ref{ex:quasi-pitag}.
\end{proof}

\begin{cor}\label{cor:birigid-implies-trivia-radical} If $(\dtF:\Fs)\geq 4$ and $F$ has a birigid element, then $F$ has trivial radical. \end{cor}
\begin{proof} It follows immediately from Definition \ref{def:rigid-birigid-elements} and the inclusion $R(F)\subseteq D_F\langle 1,a\rangle\cap D_F\langle 1,-a\rangle$. \end{proof}

In the following, all the basic definitions and results on field valuations can be found in \cite{engler2}.

By a {\it valued field} $(F,v)$ we mean a field $F$ having a valuation $v: F\to \Gamma_v\cup \{\infty\}$, where $\Gamma_v$ is the value group, the associated valuation ring $A_v = \{x\in F \ ; \ v(x)\geq 0\}$ has maximal ideal $m_v = \{x\in F\ ; \ v(x)>0\}$. The residue class field is $k_v=A_v/m_v$. Let us study briefly the compatibility conditions between the Kaplansky radical and valuations.

\begin{prop}\label{prop:radical-valued-field}\cite[Prop. 2.4]{dario3} Let $(F,v)$ be a valued field, $\mathrm{char}(k_v)\neq 2$ and
$R(F)\neq \dtF$.
	\begin{itemize}
		\item[$(a)$] If $R(F)\nsubseteq (1+m_v)\Fs$, then $(\Gamma_v:2\Gamma_v)\leq 2$. Moreover, $(\Gamma_v:2\Gamma_v)=2$ implies that $k_v$ is quadratically closed.
		\item[$(b)$] If $(1+m_v)\Fs\nsubseteq R(F)$, then $\Gamma_v=2\Gamma_v$.
\end{itemize} \end{prop}

\begin{ex}\label{ex:2-henselian-fields}($2$-henselian fields) Remember that the valued field $(F,v)$ is called {\it $2$-henselian} field if $v$ has a unique extension to $F(2)$, or equivalently, it holds the Hensel's Lemma for polynomials of degree $2$. If $\mathrm{char}(k_v)\neq 2$, it is also equivalent to the compatibility condition $1+m_v\subset \Fs$ \cite[Corollary 4.2.4]{engler2}. \end{ex}
\begin{prop}\label{prop:radical-of-2hens} Let $(F,v)$ be a $2$-henselian valued field, $\mathrm{char}(k_v)\neq 2$, $\Gamma_v\neq2\Gamma_v$, and $R(F)\neq \dtF$. Then $F$ has trivial radical.
\end{prop}
\begin{proof} Proposition \ref{prop:radical-valued-field} $(a)$ implies $R(F)=\Fs$, provided one of the following alternatives:
	\begin{itemize}
		\item[$(a)$]  $(\Gamma_v:2\Gamma_v)>2$
		\item[$(b)$] $(\Gamma_v:2\Gamma_v)=2$ and $k_v\neq k_v(2)$
	\end{itemize}
In our setup, the two conditions in $(b)$ do not occur together. Indeed, we would have $\dtF = A_v^{\times}\Fs\cup xA_v^{\times}\Fs$, for some $x\in \dtF$. If $\dot{k}_v=\dot{k}_v^2$, condition $1+m_v\subset \Fs$ implies $A_v^{\times}\subset \Fs$. Then $(\dtF:\Fs)=2$. In this case, $R(F)\neq\dtF$ implies $R(F)=\Fs$.
\end{proof}

Proposition \ref{prop:radical-of-2hens} also follows from the fact that $v(a)\in \Gamma_v\backslash 2\Gamma_v$ implies that $a$ is birigid and applying Corollary \ref{cor:free-factor-F_1}.

On the other hand, $2$-henselian valuations can be build from the existence of enough rigid elements \cite{ware2}, \cite[Theorem 2.2.7]{engler2}.
This approach was successfully applied in Galois Theory to identify $2$-henselian valuations in $G_F(2)$ \cite{engler4}, as well more generally \cite{engler3}. It will be useful in the next theorem.

Now let us remember semi-direct products in our context. More details can be found in \cite[Proposition 1.1]{engler3}.

Let $\mu_2$ the group of all roots of unity in $F(2)$ having order a power of $2$.
The action of $G_F(2)$ on $\mu_2$ induces the ciclytomic character $\chi: G_F(2)\to U_2$, where $U_2$ is the multiplicative group of $\Z_2$. Finally, for a integer $m\geq 1$, the character  $\chi$ induces the action of $G_F(2)$ onto $\Z_2^m$, defining the semi-direct product $\Z_2^m\rtimes G_F(2)$.

\begin{thm}\label{eng-nog-thm} Let $F$ be a field such that $G_F(2)=\Z_2^{m}\rtimes G_K(2)$, as defined above. Assume that $F$ is not a formally real Pythagorean field with $G_F(2)\cong \Z_2\rtimes \Z/2\Z$. Then $F$ has a $2$-henselian valuation $v$ such that $\Gamma_v\neq 2\Gamma_v$ and $\mathrm{char}(k_v)\neq 2$.
\end{thm}
\begin{proof} Since $\Z_2^{m}$ is a normal abelian subgroup of $G_F(2)$, it follows from \cite[Theorem 4.4]{engler4}. 	\end{proof}

\begin{cor}\label{cor:do-teorema-eng-nog} If $F$ is a field as in Theorem \ref{eng-nog-thm}, then $R(F)=\Fs$, that is, $F$ has trivial radical. \end{cor}
\begin{proof} It follows from Proposition \ref{prop:fields-with-rigid}. Observe that the field excluded in Theorem \ref{eng-nog-thm} has trivial radical, according to Example \ref{ex:fields-trivial-radical} (a). \end{proof}

To finish this section, let us see that from a $2$-henselian field we can build a field with non-trivial radical.

\begin{ex}\label{ex:pre-2-henselian-fields}[pre-2-henselian-fields] A valued field $(F,v)$ of $\mathrm{char}(k_v)\neq 2$ is called a {\it pre-$2$-henselian field} when it holds the compatibility condition $1+m_v\subset R(F)$. By Proposition \ref{prop:radical-valued-field} (b), if $R(F)\neq \dtF$ and $\Gamma_v\neq 2\Gamma_v$, then $R(F) = (1+m_v)\Fs$. Therefore, it has trivial radical iff it is $2$-henselian. By \cite[Theorem 4.2]{dario3} $G_F(2)\cong G_L(2)\ast_2(\Z_2^m\rtimes G_{k_v}(2))$, for $m=\mathrm{dim}_{\F_2}\Gamma_v/2\Gamma_v$ and $L$ a extension of $F$ in $F(2)$ such that $G_F(2)$ is a free pro-$2$ group and $\dot{L}/\dot{L}^2\cong R(F)/\Fs$. Conversely, a field admitting such decomposition for $G_F(2)$ is a pre-$2$-henselian field.\end{ex}

\section{The Kaplansky radical and ETG$_2$ groups}

Now we put all the previous information together to prove Theorem \ref{decomposition}, which corresponds to prove Claim \ref{claim-dec} for an ETG$_2$ group.

Remember that such a group is built from basic pro-$2$ groups: $\Z/2\Z$, (infinite) Demushkin groups and free pro-$2$ groups, and basic operations: free pro-$2$ products and semi-direct products, as the previous sections. The Elementary Type Conjecture states that every finitely generated pro-$2$ Galois group is ETG$_2$. See \cite[Section 4.3]{minac18} for details and a complete version.

\begin{thm}\label{thm:claim2-ETG2} Let $F$ be a field, $\mathrm{char}(F) \neq 2$, $\Fs \neq R(F) \neq \dtF$, and suppose that $G_F(2)$ is an ETG$_2$ group. There are two field extensions $L$ and $H$ of $F$ in $F(2)$ such that $G_2(F)$ admits a decomposition as the pro-$2$ free product $G_F(2)\cong G_L(2)\ast_2 G_H(2)$ satisfying:
	\begin{itemize} \item[$(1)$] $G_L(2)$ is a free pro-$2$ group ($R(L) = \dot{L}$) of rank equal to $\mathrm{dim}_{\F_2}R(F)/\Fs$.
		\item[$(2)$] The field $H$ has trivial Kaplansky radical. \end{itemize} \end{thm}
\begin{proof} The hypothesis  $\Fs \neq R(F)$ excludes the possibility of non-occurrence of the free pro-$2$ product operation in the elementary construction of $G_F(2)$. Note that in this case we would have $G_F(2) \cong \Z_2^m\rtimes G_1$, for some integer $m\geq 1$ and some ETG$_2$ pro-$2$ group $G_1$, which also does not decomposes as a free pro-$2$ product. Then the triviality of $R(F)$ comes from Theorem \ref{eng-nog-thm} and Corollary \ref{cor:do-teorema-eng-nog}. Now suppose that the free pro-$2$ product operation does occurs in the elementary construction of $G_F(2)$ and let us proceed by induction over $n = \mathrm{dim}_{\F_2} \dtF/\dtF^2$, the minimal number of (topological) generators of $G_F(p)$. The case $n=1$ is excluded by hypotheses on $R(F)$. There are six possible groups $G_F(2)$ for the case $n=2$, listed in \cite[Table 5.2, p. 393]{ware3}. Only the case $\Z_2\ast_2(\Z/2\Z)$ corresponds to a field $F$ such that $\Fs \neq R(F) \neq \dtF$. Now it is enough to take fixed fields and apply Corollary \ref{cor:factors-free}. For the general case remember that we have at least one occurrence of the pro-$2$ product operation. Thus, $G_F(2)\cong G_K(2)\ast_2 G_N(2)$, for $K,N$ field extensions in $F(2)$ having Galois groups as ETG$_2$ groups. Note that $K$ and $N$ cannot have trivial radical, because it would imply $F$ has trivial radical, according to Corollary \ref{cor:factors-free}. For the same reason, $K$ or $N$ must have the Galois group not free. Now, by Theorem \ref{teo:decomposition-neu-traduzido} $(1)$,  $d(G_F(2))=d(G_K(2))+d(G_N(2))$. Then we can apply the induction hypothesis over only two cases:
\begin{itemize}
	\item[$(a)$] $R(K)=\dot{K}$ and $R(N)=\dot{N}^2$ (the case $R(K)=\dot{K}^2$ and $R(N)=\dot{N}$ is analogous).
	\item[$(b)$] $\dot{K}^2\neq R(K)\neq\dot{K}$ and $\dot{N}^2\neq R(N)\neq\dot{N}$.
\end{itemize}
For the first case we just take $L=K$ and $H=N$. For case $(b)$ we have $G_K(2)\cong G_{L_1}(2)\ast_2 G_{H_1}(2)$ and $G_N(2)\cong G_{L_2}(2)\ast_2 G_{H_2}(2)$, with $H_1, H_2$ having trivial radical and $G_{L_i}(2)$ pro-$2$ free, $i=1,2$. Finally, choosing $L$ the fixed filed of $G_{L_1}(2)\ast_2 G_{L_2}(2)$, $H$ the fixed filed of $G_{H_1}(2)\ast_2 G_{H_2}(2)$, and applying Corollary \ref{cor:factors-free}, we conclude the prove. \end{proof}

We have listed several examples of fields having non-trivial Kaplansky radical. All of them are examples of Theorem \ref{thm:claim2-ETG2}.

\begin{thm}\label{theo:examples-are-elementary} If $F$ is a pre-$2$ henselian field, or a field described in Examples \ref{ex:berman1}, \ref{ex:berman2}, \ref{ex:kula} or \ref{ex:quasi-pitag}, then $G_F(2)$ admits the decomposition described in Theorem \ref{thm:claim2-ETG2}. \end{thm}
\begin{proof} \begin{itemize}
\item[$(a)$] (pre-$2$ henselian fields) As Example \ref{ex:pre-2-henselian-fields}, it is a valued field $(F,v)$ and $G_F(2)$ admits such decomposition choosing $H$ as a $2$-henselization of $(F,v)$ \cite[Theorem 4.2]{dario3}.
\item[$(b)$] (quasi-Pythagorean fields - Example \ref{ex:quasi-pitag}) Here we have $R(F) = D_F\langle 1,1\rangle$.  Ware \cite[Corollary 1]{ware1} proved $G_F(2)$ is the free pro-$2$ product of a free pro-$2$ group and a pro-$2$ group (topologically) generated by involutions, provided that $(\dtF:\Fs)$ is finite.
\item[$(c)$] Example \ref{ex:berman1} (Berman) It is a field $K=F(\sqrt{-1})$ with $(\dot{K}:R(K))=4$ and $F$ a Pythagorean field. For every $x\in\dot{K}\backslash R(K)$ we have $(D_K\langle 1,x\rangle:R(K))=2$. It means that such $x$ is $R(K)$-rigid \cite[Lemma 2.1]{dario3} and by \cite[Theorem 2.2 (1)]{dario3} $K$ is a pre-$2$ henselian field, since it is not formally real.
\item[$(d)$] Example \ref{ex:kula} (Kula) It is immediate from Example \ref{ex:2-henselian-fields}, since $L((X_1))\ldots((X_n))$ is a $2$-henselian field.
\item[$(e)$] Example \ref{ex:berman2} (Berman) Again we have $K=F(\sqrt{-1})$, but now $\dtF/\Fs \cong \dtF_1/(\dtF_1)^2\times\dtF_2/(\dtF_2)^2$, where $F_1$ is super Pythagorean and $F_2$ is SAP Pytha\-go\-rean. It was show in \cite[Theorem 4.8]{dario3} that $G_K(2)$ is isomorphic to $G_{F_1(\sqrt{-1})}(2)\ast_2G_{F_2(\sqrt{-1})}(2)\ast_2\Z_2$. Since $F_2$ is SAP, we have
that $G_{F_1(\sqrt{-1})}(2)$ is a free pro-$2$ group \cite[Remark 2]{ware1}. Thus, we choose the field $L$ in Theorem \ref{thm:claim2-ETG2} as the fixed field of $G_{F_2(\sqrt{-1})}(2)\ast_2\Z_2$. On the other side, by \cite{elmanlam2}, $H=F_1(\sqrt{-1})$ has trivial radical. 	\end{itemize} \end{proof}

It follows the translation of Theorem \ref{thm:claim2-ETG2} for the algebraic theory of quadratic forms. Let $W(F)$ denote de Witt ring of anisotropic quadratic forms over the field $F$
and $IF$ the fundamental ideal of even-dimensional quadratic forms classes in $W(F)$. The Witt ring has {\it elementary type} if it can be build from the Witt ring from basic Witt rings and basic operations in the category of Witt rings. The basic Witt rings correspond to those of finite, real closed, and local fields. The basic operations consists in direct product and group ring formation.

\begin{cor}\label{cor:witt-rings} Let $F$ be a field, $\mathrm{char}(F)\neq 2$,  such that $(\dtF:\Fs)$ is finite. If $W(F)$ has elementary type, then there are $L,H$ field extensions of $F$ in $F(2)$ such that $W(F)\cong W(L)\times W(H)$, with $I^2L=\{0\}$  and $H$ has trivial Kaplansky radical. \end{cor}

\section{The Hilbert's Theorem 90}

In order to prove Claim \ref{claimh90} (H90-R) for an ETG$_2$ group we need the Kurosh Subgroup Theorem for pro-$2$ groups (Theorem \ref{jkdfs7}).
A proof for a more general case can be found in \cite[Theorem 4.2.1, p.208]{neukirch} or \cite{bnw}.

Theorem \ref{jkdfs7} allows to find $G_2(F(\sqrt{a}))$ and $R(F(\sqrt{a}))$, $a\in \dtF\backslash\Fs$, described in Theorem \ref{djk7e}. It has two separated cases, according to $a\in \Fx\backslash R(F)$ or $a\in R(F)$.

\begin{thm}[Kurosh]\label{jkdfs7} Let $G$ be a pro-$2$ group and $H$ a open\footnote{ $H$ is open when it is closed and $G/H$ is finite.} subgroup of $G$. Suppose that $G\cong G_1\ast_2 G_2$, $G_1, G_2$  closed subgroups of $G$. Let $$G=\displaystyle\bigcup_{i=1}^n G_1a_iH \ \  \text{ and }\ \  G=\displaystyle\bigcup_{j=1}^m G_2b_jH, \ \ a_i, b_j\in G,$$ decomposition in double cosets of $G$ w.r.t.  $H, G_1$ and $H, G_2$, respectively. Then
$$H=\Big(\ast_{i=1}^n H\cap a_iG_1a_i^{-1}\Big)\ast \Big(\ast_{j=1}^m H\cap b_jG_2b_j^{-1}\Big) \ast \mathcal{F}$$ where $\mathcal{F}$  is a free pro-$2$ group of rank $1+(G: H) - (m+n)$.
\end{thm}

The existence of the fields $F_0$ and $F_1$ in the next theorem follows from Theorem \ref{thm:claim2-ETG2}.

\begin{thm}\label{djk7e} Let $F$ be a field of $\mathrm{char}(F)\neq 2$, $\Fs\neq R(F)\neq\dtF$ and $G_F(2)$ a ETG$_2$ group. Choose extensions $F_0, F_1\subseteq F(2)$ such that $G_F(2)\cong G_{F_0}(2)\ast_2 G_{F_1}(2)$, with $G_{F_0}(2)$ a free pro-$2$  group and $F_1$ having trivial radical. Let  $K=F(\sqrt{a})$, $a\in \dtF\backslash \Fs$. There is a extension $L_0\subseteq F(2)$ of $F$, with $G_{L_0}(2)$ a  free pro-$2$ group, such that
\begin{itemize}
\item[$(1)$] If $a\in \dtF\backslash R(F)$, then $G_K(2)\cong G_{L_0}(2)*_2G_{F_1(\sqrt{a})}(2),$ $R(K)= F_1(\sqrt{a})^2\cap \dtF$.
	\item[$(2)$] if $a\in R(F)$, then $G_K(2)\cong G_{L_0}(2) *_2 G_{F_1}(2)*_2  G_{F^{\sigma}_1}(2),$ and $R(K)= \dot{F}_1^2\cap (\dot{F}_1^{\sigma})^2 \cap \dtF$, where $G(K;F) = \{id, \sigma|_K \}$ and $F^{\sigma}_1 = \sigma(F_1)$.
\end{itemize}
\end{thm}
\begin{proof} It follows from Corollary \ref{cor:free-factor-F_1} (a) that $R(F) = \dot{F}_1^2\cap \dtF$.  \\
(1) Assume that $a\in \dtF \backslash R(F)$ and let us define $K_0=F_0(\sqrt{a})$ and $K_1= F_1(\sqrt{a})$.
$$\xymatrix{
                &   F(2)    &                \\
K_0 \ar@{-}[ur]     &           & K_1 \ar@{-}[ul]    \\
F_0 \ar@{-}[u]      & K \ar@{-}[uu] \ar@{-}[ul] \ar@{-}[ur]    & F_1 \ar@{-}[u]    \\
                & F \ar@{-}[u] \ar@{-}[ul] \ar@{-}[ur] & \\ }$$
\underline{First case}: $a\notin \dot{F}^2_0$. Then $K\not\subset F_0$ and $G_{F_0}(2) \not\subset   G_K(2)$.
As $a\notin R(F) = \dot{F}_1^2\cap \dtF$, it follows that $K_1 \neq F_1$ e $G_{K_1}(2)\subsetneqq G_{F_1}(2)$. Since $(G_F(2):G_K(2))= 2$, it follows that $G_F(2)=G_{F_0}(2)G_K(2) = 	G_{F_1}(2)G_K(2)$ are two trivial decompositions of $G$ in double cosets w.r.t. the pairs $G_K(2),G_{F_0}(2)$ and $G_K(2),G_{F_1}(2)$,  respectively. From Kurosh Subgroup Theorem (Theorem \ref{jkdfs7}) we have $$G_K(2) = \Big(G_{F_0}(2)\cap G_K(2)\Big)*_2 \Big( G_{F_1}(2)\cap G_K(2)\Big)*_2 \mathcal{F}\cong G_{K_0}(2)*_2G_{K_1}(2)*_2\Z_2,$$ with $\mathcal{F}$ a free pro-$2$ group of rank $1$, that is, $\mathcal{F}\cong \Z_2$. Now, it is enough to chose $L_0 \subseteq F(2)$ as the fixed field of the free pro-$2$ group $G_{K_0}(2)*_2\Z_2$.\\
\underline{Second case}: $a\in \dot{F}^2_0$. Then $G_{K_0}(2)= G_{F_0}(2)$, $G_F(2)=G_{F_1}(2)G_K(2)$ and we have the following decompositions in double cosets $$G_F(2)=G_K(2)\cup \tau G_K(2) = G_{F_0}(2)G_K(2)\cup G_{F_0}(2)\tau G_K(2),$$ where $\tau$ is an automorphism such that $\tau|_K$ is the only non-trivial $K$-automorphism fixing $F$. Again by Theorem \ref{jkdfs7},
$$G_K(2) = \Big(G_{F_0}(2)\cap G_K(2)\Big) *_2 \Big(\tau G_{F_0}(2) \tau^{-1}\cap G_K(2)\Big) *_2 \Big( G_{F_1}(2)\cap G_K(2)\Big).$$
The result now follows remembering that $G_{F_1}(2)\cap G_K(2) = G_{K_1}(2)$ and choosing $L_0$ as the fixed field of the free pro-$2$ group $(G_{F_0}(2)\cap G_K(2)) *_2 (\tau G_2(F_0) \tau^{-1}\cap G_2(K))$.
\\
\\
(2) Now let us assume $a\in R(F) = \dot{F}_1^2\cap \dtF$. The case $a\in \dot{F}^2_0$ does not occur anymore. Indeed, the decomposition $G_F(2)=G_{F_0}(2)*_2 G_{F_1}(2)$ implies the injectivity of $\mathrm{res}_1: \dtF /\Fs  \to \dot{F}_0 / \dot{F}_0^2 \times \dot{F}_1 / \dot{F}_1^2$, according to Theorem \ref{teo:decomposition-neu-traduzido}.  Then, $a\in \dot{F}^2_0\cap \dtF$ would imply $a\in \Fs$.   Therefore, $G_{F_0}(2)$ is not contained in $G_K(2)$ and we have $G_F(2)=G_{F_0}(2)G_K(2)$, since $(G_F(2):G_K(2))=2.$ By the other side, $G_{F_1}(2)\subset G_K(2)$, because $a\in \dot{F}_1^2$. Then $G_F(2)= G_K(2)\cup \sigma G_K(2) = G_{F_1}(2)G_K(2)\cup G_{F_1}(2)\sigma G_K(2)$.
$$\xymatrix{
	&   F(2)                    &                    \\
	F_0(\sqrt{a}) \ar@{-}[ur]   &   F_1^{\sigma} \ar@{-}[u] & F_1 \ar@{-}[ul]               \\
	F_0 \ar@{-}[u]              &   K \ar@{-}[u] \ar@{-}[ur] \ar@{-}[ul]                       &     \\
	F \ar@{-}[u] \ar@{-}[ur]                            &  &  }$$
By Kurosh Subgroup Theorem again,
$$G_K(2) =  \Big(G_{F_0}(2)\cap G_K(2)\Big)*_2  \Big(G_{F_1}(2)\cap G_K(2)\Big)*_2 \Big( \sigma G_{F_1}(2)\sigma^{-1} \cap G_K(2)\Big) =$$
$$ =  G_{F_0(\sqrt{a})}(2)*_2 G_{F_1}(2)*_2 G_{F_1^{\sigma}}(2).$$
In this case, $L_0 = F_0(\sqrt{a})$. \end{proof}

Finally, we are able to prove Claim \ref{claimh90} (H90-R) for an ETG$_2$ group. Once again, it applies for all examples in Theorem \ref{theo:examples-are-elementary}.

\begin{thm}[H90-R]\label{theorem90H} For $F$ a field of characteristic not equal to $2$ let $K = F(\sqrt{a})$, $a\in \Fx\backslash \Fs$, be a quadratic extension of $F$ and $N:\dot{K} \to \dtF$ the norm map.
If $G_F(2)$ is an ETG$_2$ group, then $N^{-1}(R(F)) = \dtF R(K)$.
\end{thm}
\noindent
{\bf Proof} The inclusion $\dtF R(K)\subseteq N^{-1}(R(F))$ is already automatic and does not depends on $G_F(2)$. Indeed, if  $r\in R(K)$, then $r\in D_K\lan 1, -x \ran$, for all $x\in K^{\times}$. By the Knebusch Norm Principle \cite[Theorem 5.1, p.206]{lam1},  $N(r)\in D_F\lan 1, -x\ran$, for all $x\in \Fx$. Therefore, $N(r)\in R(F)$. For the other inclusion let us choose, according to Theorem \ref{thm:claim2-ETG2}, extensions $F_0, F_1\subseteq F(2)$ such that $G_F(2)\cong G_{F_0}(2)\ast_2 G_{F_1}(2)$, with $G_{F_0}(2)$ a free pro-$2$  group and $F_1$ having trivial radical. It follows from Corollary \ref{cor:free-factor-F_1} (a) that $R(F) = \dot{F}_1^2\cap \dot{F}$.
\\
\underline{First case}: Assume $a\in \dtF\backslash R(F)$. Defining $K_1 = F_1(\sqrt{a})$, Theorem \ref{djk7e} (1) says $G_K(2)\cong G_{L_0}(2)*_2G_{K_1}(2)$ and $R(K)= \dot{K}_1^2\cap \dot{K}$. Take  $x\in \dot{K}$ such that $N(x) \in R(F) = \dot{F}_1^2\cap \dot{F}$ and denote $N_1: \dot{K}_1\to \dot{F}_1$ the correspondent norm application. By Hilbert'90 Theorem (announced in the first paragraph of the Introduction),  $x\in \dot{F}_1\dot{K}_1^2$. Remember that $\dot{F}_1=\dtF \dot{F}_1^2$, according to Theorem \ref{teo:decomposition-neu-traduzido}. Putting all together, $x\in \dtF \dot{K}_1^2 \cap \dot{K} = \dtF R(K)$.
\\
\underline{Second case}: Now assume $a\in R(F) = \dot{F}_1^2\cap \dot{F} = (\dot{F}_1^{\sigma})^2\cap \dot{F}$. Again we choose $x\in \dot{K}$ such that $N(x) \in R(F)$. Now we have $K\subset F_1$ and $\dtF_1 = \dtF\dtF_1^2$. Then $x=\alpha t^2$, for some $\alpha\in \dtF$ and $t\in \dtF_1$. It follows that $N(x) = \alpha N(t^2)=t^2\sigma(t)^2$. Then $t^2\in \dtF_1^{\sigma}$, because also are there $N(x)$ and $\sigma(t)^2$. Therefore $x=\alpha t^2 \in \dtF(\dtF_1^2\cap (\dtF_1^{\sigma})\cap \dot{K}) = \dtF R(K)$. \fds


TITLE: Differentiating milk from water
QUESTION [0 upvotes]: A milk seller has a milk of Rs.100/- per litre. In what ratio should water be mixed in that milk, so that after selling the mixture at Rs.80/- per litre, he may get a profit of 50%? 7:8/7:9/9:7/7:5

My attempt:
To get 50% profit, selling price should be Rs.150/- per litre (only milk). But we are selling at Rs.80/- per litre (milk+water). So, maybe water is 70% (150-80), and milk is 80%. Even answer is 7:8. But I am not feeling confident of this solution. It's appearing fuzzy. (and even non-sensical)

REPLY [1 votes]: Let's say he mixes milk to water in the ratio $x:(1-x)$. For $1L$ of mixture, his cost would be $Rs 100x$ and his selling price would be $Rs 80$. This would give a profit of $Rs (80 - 100x)$. His percentage profit would be $\frac{(80 - 100x)}{100x}$. Equating that to $50 \%$ (or half), you get:
$\frac{(80 - 100x)}{100x} = \frac 12$
$160 - 200x = 100x$
$300x = 160$
$x = \frac{160}{300} = \frac{8}{15}$
That's the proportion of the mixture that was milk, so the proportion that was water was $1 - \frac{8}{15} = \frac {7}{15}$ giving a ratio of water to milk of $7:8$


TITLE: In what sense does continuity hold for unbounded functional calculus?
QUESTION [0 upvotes]: Let $f : \mathbb{R} \to \mathbb{R}$ be a Schwartz function. Let $I$ be the identity operator on some separable Hilbert space $\mathcal{H}$ and $A$ be an unbounded self-adjoint operator on $\mathcal{H}$.
Then, $I+xA$ is self-adjoint for all $x \in \mathbb{R}$ and $f(I+xA)$ is a bounded operator on $\mathcal{H}$. However, is there any continuity or smoothness available for $f(I+xA)$ with respect to $x$?
For example, as $x \to 0$, does $f(I+xA)$ converge to $f(I)$ in any sense?

REPLY [2 votes]: Yes, $f(I+xA)$ converges to $f(I)$ in the strong operator topology, meaning that  $f(I+xA)\xi \to f(I)\xi $, for every vector $\xi $ in
$\mathcal H$.
The fact that $f$ is a Schwartz function is not necessary and here is a proof that works for every bounded continuous
function $f$.   Write
$$
  A=\int_{\mathbb R} \lambda  \,dP(\lambda )
  $$
by the spectral Theorem, so that
$$
  f(I+xA) = \int_{\mathbb R} f(1+x\lambda ) \,dP(\lambda ).
  $$
Next observe that $f(1+x\lambda ) \to  f(1)$, as $x\to 0$, pointwise on the variable $\lambda $.  Writing
$$
  g_x(\lambda ) = f(1+x\lambda ) - f(1)
  $$
we then have that   $g_x(\lambda )\to 0$, as $x\to 0$, pointwise on  $\lambda $.
Therefore, for every $\xi $ in $\mathcal
H$, one has that
$$
  \|g_x(A)\xi \|^2 =
  \langle g_x(A)\xi , g_x(A)\xi \rangle  =
  \langle g_x(A)^2\xi , \xi \rangle  =
  \int_{\mathbb R} g_x(\lambda )^2 \,d\langle P(\lambda )\xi , \xi \rangle ,
  $$
which converges to zero by the Lebesgue Dominated Convergence Theorem, and the fact that $g_x(\lambda )^2$ is
uniformly bounded.


text:
\begin{document}

\begin{center}
{\LARGE Putting the Prisoner's Dilemma in Context \vspace{0.7cm}}
\end{center}

{\large L. A. Khodarinova}$^{{\dag }}${\large \ and\ J. N. Webb
{\large \ \vspace{0.3cm}}

$^{{\dag }}$\textit{Magnetic Resonance Centre, School of Physics and
Astronomy, University of Nottingham, Nottingham, England NG7 2RD, e-mail:
LarisaKhodarinova@hotmail.com}
\vspace{1.0cm}

\textbf{Abstract.} The standard iterated prisoner's dilemma is an unrealistic model of social
behaviour because it forces individuals to participate in the interaction.
We analyse a model in which players have the option of ending their
association. If the payoff for living alone is neither too high nor too low
then the potential for cooperative behaviour is enhanced. For some parameter
values it is also possible for a polymorphic population of defectors and
conditional cooperators to be stable.

\section*{Introduction}

The iterated, or repeated, prisoner's dilemma is the most popular model of
social interactions \cite{AH}. Since its inception the basic model has been
modified in many ways (see Dugatkin \cite{D} for a review). However, in all
these versions it is assumed that the players must engage in the interaction
and have no opportunity to end it. This unrealistic feature is just one
facet of the more general assumption that one particular social interaction
may be considered in isolation from all others that an individual may face.

In this paper we use the framework of stochastic games \cite{FV} to consider
a version of the iterated prisoner's dilemma in which the players may choose
to discontinue their association. We assume that once the partnership has
been dissolved by one or more of the players, then each receives the same,
fixed, per-period payoff. This is probably the simplest way that an
interaction can be considered as being dependent on other situations in
which individuals find themselves during a complex and, at least partly,
social life. We will use the standard replicator dynamics \cite{TJ}\ to
investigate the effect that the existence of this outside option has on the
evolution of cooperative behaviour in a population of players.

\section*{The Model}

A general stochastic game has three major components: the set of states, the
games played in each of these states and the (possibly behaviour-dependent)
probabilities for transition between the states. In our model the states
represent the different contexts in which players may interact, so we will
refer to them as context games.

We consider an interaction described by the following multi-state,
stochastic game. There are three possible context-games (states) $G_{0},$ $
G_{1}$ and $G_{2}.$ The interaction starts with context-game $G_{0}.$ In
this game the players make the decision about whether or not they wish to
initiate or continue an association. The first player and the second player
choose between two possible actions: $A$=\textquotedblleft
associate\textquotedblright\ or $B$=\textquotedblleft break
up\textquotedblright . There are no payoffs directly associated with this
decision. Context-game $G_{1}$ represents some specific activity in which
the individuals can participate together. It is modelled by the prisoner's
dilemma and the players choose between the possible actions: $C$
=\textquotedblleft cooperation\textquotedblright\ or $D$=\textquotedblleft
defection\textquotedblright . Context-game $G_{2}$ can be considered as a
background state representing the situation when there is no interaction or
association between the players. There is only one possible action: $L$
=\textquotedblleft be alone\textquotedblright. \medskip

\begin{center}
\begin{tabular}{c}
$
\begin{array}{ll}
G_{0}\!: &
\begin{tabular}{|l|c|c|}
\hline
{\footnotesize P}$_{1}\setminus ${\footnotesize P}$_{2}$ & {\footnotesize A}
& {\footnotesize B} \\ \hline
\multicolumn{1}{|c|}{\footnotesize A} & \multicolumn{1}{|l|}{$\!\!\!
\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( 0,0\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
0,1,0\right] }\!\!\right. \!
\end{array}
\!\!\!$} & \multicolumn{1}{|l|}{$\!\!\!\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( 0,0\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
0,0,1\right] }\!\!\right. \!
\end{array}
\!\!\!$} \\ \hline
\multicolumn{1}{|c|}{\footnotesize B} & \multicolumn{1}{|l|}{$\!\!\!
\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( 0,0\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
0,0,1\right] }\!\!\right. \!
\end{array}
\!\!\!$} & \multicolumn{1}{|l|}{$\!\!\!\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( 0,0\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
0,0,1\right] }\!\!\right. \!
\end{array}
\!\!\!$} \\ \hline
\end{tabular}
\end{array}
\begin{array}{ll}
G_{1}\!: &
\begin{tabular}{|l|c|c|}
\hline
{\footnotesize P}$_{1}\setminus ${\footnotesize P}$_{2}$ & {\footnotesize C}
& {\footnotesize D} \\ \hline
\multicolumn{1}{|c|}{\footnotesize C} & \multicolumn{1}{|l|}{$\!\!\!
\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( r,r\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
1,0,0\right] }\!\!\right. \!
\end{array}
\!\!\!$} & \multicolumn{1}{|l|}{$\!\!\!\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( s,t\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
1,0,0\right] }\!\!\right. \!
\end{array}
\!\!\!$} \\ \hline
\multicolumn{1}{|c|}{\footnotesize D} & \multicolumn{1}{|l|}{$\!\!\!
\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( t,s\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
1,0,0\right] }\!\!\right. \!
\end{array}
\!\!\!$} & \multicolumn{1}{|l|}{$\!\!\!\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( p,p\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
1,0,0\right] }\!\!\right. \!
\end{array}
\!\!\!$} \\ \hline
\end{tabular}
\end{array}
$ \\
$
\begin{array}{ll}
G_{2}\!: & \rule[0.2in]{0in}{0.35in}
\begin{tabular}{|l|c|}
\hline
{\footnotesize P}$_{1}\setminus ${\footnotesize P}$_{2}$ & {\footnotesize L}
\\ \hline
\multicolumn{1}{|c|}{\footnotesize L} & \multicolumn{1}{|l|}{$\!\!\!
\rule[-0.14in]{0in}{0.35in}
\begin{array}{l}
\!\!^{\left( z,z\right) }\!\!\left/ \!_{\rule[-0.06in]{0in}{0.14in}\left[
0,0,1\right] }\!\!\right. \!
\end{array}
\!\!\!$} \\ \hline
\end{tabular}
\ \medskip
\end{array}
$ \\
$
\begin{array}{c}
\text{\rule{0in}{0.35in}Table1.The multi-state game.} \\
\text{In each state the per-period payoffs for the relevant action choices }
\\
\text{are given above and to the left of the diagonal line;} \\
\text{the behaviour-dependent probabilities of transition to the other
states } \\
\text{are given below the line.\rule[-0.3in]{0in}{0.18in}}
\end{array}
$
\end{tabular}
\end{center}

The actions chosen define both immediate payoffs to the individuals and\
future transition probabilities. The immediate payoffs collected by the
players are given in table~1. The first entry in each payoff pair contains
the payoff to the player $P_{1},$ who selects the row action, the second is
for the player $P_{2},$ who selects the column action. In this paper we are
considering an extension of the standard iterated prisoner's dilemma, for
which the following inequalities hold in $G_{1}$.
\begin{equation*}
t>r>p>s\geq 0.
\end{equation*}
Transition probabilities, which are determined by the choice of actions are
presented in table~1 as a set of three numbers. This set of numbers appears
in square brackets in each cell of the matrices. Here the first, second or
third number is, respectively, the probability that context-game $G_{0},$ $
G_{1}$ or $G_{2}$ is played at the next round. The probabilities are defined
by the following rules. If context-game $G_{0}$ is played and action $A=$
\textquotedblleft associate\textquotedblright\ is chosen by both players, at
the next round context-game $G_{1}$ is played; if action $B$
=\textquotedblleft break up\textquotedblright\ is chosen by at least one
player, context-game $G_{2}$ is played at the next round with probability~$
1. $ Whatever actions are chosen when context-game $G_{1}\ $is played,
context-game $G_{0}$ is played at the next round with probability~$1$. If
context-game $G_{2}$ is played, at the next round context-game $G_{2}$ is
played again with probability$~1.$

We assume that after playing context game $G_{0}$, players survive to play
game $G_{1}$or $G_{2}$ (as appropriate) with probability 1. After playing
context games $G_{1}$ or $G_{2}$ players survive to the next round with
probability $\beta $ $(0\leq \beta <1)$. In principle, these survival
probabilities could be different but, for simplicity, we will assume they
are equal.

As with the iterated prisoner's dilemma there is an infinite number of pure
strategies that could be considered. We will initially restrict our
attention to the following three strategies.

\begin{itemize}
\item Conditional cooperation (which we denote $\sigma _{C}).$ A player
following this strategy will initially ``Associate'' in $G_{0}$ then
``Cooperate'' in $G_{1}$; if this behaviour is reciprocated then the player
will continue to associate and cooperate; otherwise\ it will choose ``Break
up'' in $G_{0}$.

\item Defection (which we denote $\sigma _{D}$). A player following this
rather pathological strategy will ``Associate'' in $G_{0}$ and then
``Defect'' in $G_{1}$.

\item An unsociable strategy (which we denote $\sigma _{B}).$ A player
following this strategy will ``Break up'' in $G_{0}$. Strictly speaking this
is a set of strategies since any behaviour is allowed in $G_{1}$. However,
since we do not consider the possibility that players make errors, the
behaviour in $G_{1}$ does not affect payoffs. Consequently we ignore this
technicality.
\end{itemize}

\noindent The consequences of introducing a fourth strategy of unconditional
cooperation will be considered later.

\section*{Evolutionary Dynamics}

We set up the evolutionary dynamics by considering an infinitely large
population of individuals who adopt one of the three pure strategies. The
payoffs in the repeated game, $\pi (\sigma ,\sigma ^{\prime })$\ for
adopting strategy $\sigma $ against an opponent who adopts strategy $\sigma
^{\prime }$ are
\begin{equation}
A=\left[
\begin{array}{lll}
\pi (\sigma _{C},\sigma _{C}) & \pi (\sigma _{C},\sigma _{D}) & \pi (\sigma
_{C},\sigma _{B}) \\
\pi (\sigma _{D},\sigma _{C}) & \pi (\sigma _{D},\sigma _{D}) & \pi (\sigma
_{D},\sigma _{B}) \\
\pi (\sigma _{B},\sigma _{C}) & \pi (\sigma _{B},\sigma _{D}) & \pi (\sigma
_{B},\sigma _{B})
\end{array}
\right] =\frac{1}{1-\beta }\left[
\begin{array}{ccc}
r & s\left( 1-\beta \right) +\beta z & z \\
t\left( 1-\beta \right) +\beta z & p & z \\
z & z & z
\end{array}
\right] .  \label{Payoff_Matrix}
\end{equation}

Let $x_{1}$ and $x_{2}$ be the proportions of individuals who adopt $\sigma
_{C}$ and $\sigma _{D}$ respectively. The proportion of individuals using $
\sigma _{B}$ is then $1-x_{1}-x_{2}$. The standard replicator dynamics \cite
{TJ} is then two equations describing the evolution of a point $x=\left(
x_{1},x_{2}\right) $ in the domain
\begin{equation}
\Delta =\left\{ \left( x_{1},x_{2}\right) :(x_{1}\geq 0)\cap (x_{2}\geq
0)\cap (x_{1}+x_{2}\leq 1)\right\} .  \label{Simplex_2}
\end{equation}

Denote
\begin{eqnarray*}
a &=&\left( z-r\right) -\gamma \left( z-t\right) ;\quad b=\gamma \left(
z-s\right) -\left( z-p\right) ; \\
c &=&\left( z-r\right) ;\quad f=\gamma \left( z-s\right) \quad \text{where}
\quad \gamma =1-\beta .
\end{eqnarray*}
Then the Replicator Dynamics can be written as the following system of
equations.
\begin{equation*}
\begin{array}{l}
\dot{x}_{1}=x_{1}\left( cx_{1}^{2}+\left( f+c-a\right) x_{1}x_{2}+\left(
f-b\right) x_{2}^{2}-cx_{1}-fx_{2}\right)  \\
\dot{x}_{2}=x_{2}\left( cx_{1}^{2}+\left( f+c-a\right) x_{1}x_{2}+\left(
f-b\right) x_{2}^{2}+\left( a-c\right) x_{1}+\left( b-f\right) x_{2}\right)
\end{array}
\end{equation*}
Although this system is integrable for arbitrary choices of parameter values
\cite{B}, the general solution given in appendix A is not easy to work with.
We will now introduce a commonly used set of values for the prisoner's
dilemma context game $G_{1}$ to reduce the number of parameters, and we will
use the standard linearization approach to study how the solution depends on
the value of the outside option, $z$, and the survival probability, $\beta $
. Accordingly we set
\begin{equation*}
r=3,\qquad s=0,\qquad t=5,\qquad p=1.
\end{equation*}
The payoff matrix then becomes
\begin{equation*}
A=\frac{1}{1-\beta }\left[
\begin{array}{ccc}
3 & \beta z & z \\
5\left( 1-\beta \right) +\beta z & 1 & z \\
z & z & z
\end{array}
\right] .
\end{equation*}
and the Replicator Dynamics is as follows.
\begin{equation}
\begin{array}{l}
\dot{x}_{1}=\frac{x_{1}}{1-\beta }\left( \left( z-3\right) x_{1}^{2}+\left(
1-\beta \right) \left( 2z-5\right) x_{1}x_{2}+\left( z-1\right)
x_{2}^{2}+\left( 3-z\right) x_{1}+\left( \beta z-z\right) x_{2}\right)  \\
\dot{x}_{2}=\frac{x_{2}}{1-\beta }\left( \left( z-3\right) x_{1}^{2}+\left(
1-\beta \right) \left( 2z-5\right) x_{1}x_{2}+\left( z-1\right)
x_{2}^{2}+\left( 1-\beta \right) \left( 5-z\right) x_{1}+\left( 1-z\right)
x_{2}\right)
\end{array}
\label{RD2d}
\end{equation}

There are four fixed points for this Dynamics and a standard linearization
analysis produces the results shown in table 2. \medskip

\begin{center}
\begin{tabular}{c}
\begin{tabular}{|c|c|c|}
\hline
Point & eigenvectors & eigenvalues \\ \hline
$\left\{ 0,0\right\} $ & $
\begin{array}{c}
e_{1}=\left( 1,0\right)  \\
\medskip e_{2}=\left( 0,1\right)
\end{array}
$ & $
\begin{array}{c}
\lambda _{1}=0 \\
\medskip \lambda _{2}=0
\end{array}
$ \\ \hline
$\left\{ 1,0\right\} $ & $
\begin{array}{c}
e_{1}=\left( 1,0\right)  \\
\medskip e_{2}=\left( -1,1\right)
\end{array}
$ & $
\begin{array}{c}
\lambda _{1}=\frac{z-3}{1-\beta } \\
\medskip \lambda _{2}=\frac{2-5\beta +\beta z}{1-\beta }
\end{array}
$ \\ \hline
$\left\{ 0,1\right\} $ & $
\begin{array}{c}
e_{1}=\left( 1,-1\right)  \\
\medskip e_{2}=\left( 0,1\right)
\end{array}
$ & $
\begin{array}{c}
\lambda _{1}=\frac{\beta z-1}{1-\beta } \\
\medskip \lambda _{2}=\frac{z-1}{1-\beta }
\end{array}
$ \\ \hline
$\left\{ \frac{\beta z-1}{1+2\beta z-5\beta },\frac{\beta z+2-5\beta }{
1+2\beta z-5\beta }\right\} $ & $
\begin{array}{c}
e_{1}=\left( -1,1\right)  \\
\medskip e_{2}=\left( 1,\frac{\beta z-5\beta +2}{\beta z-1}\right)
\end{array}
$ & $
\begin{array}{c}
\lambda _{1}=\frac{\left( 1-\beta z\right) \left( \beta z-5\beta +2\right) }{
\left( 1-\beta \right) \left( 2\beta z-5\beta +1\right) } \\
\medskip \lambda _{2}=\frac{z\beta \left( 2-\beta \right) \left( z-5\right)
+3+z}{\left( 1-\beta \right) \left( 2\beta z-5\beta +1\right) }
\end{array}
$ \\ \hline
\end{tabular}
\\
$\text{\rule{0in}{0.35in}}$Table 2. Eigenvalues and eigenvectors for the
fixed points  \\
in the replicator dynamics system given by equations (\ref{RD2d}).$\text{
\rule[-0.3in]{0in}{0.18in}}$
\end{tabular}
\end{center}

Depending on the values of the parameters $z$ and $\beta $ we obtain
different solutions for the dynamics~(\ref{RD2d}). The $\beta -z$ parameter
space can be divided into 10 regions (see figure 1) which have qualitatively
different pictures of the dynamics (see figure~2). The main features of this
overall picture can be summarized as follows. If $z>3$ then the population
evolves towards a monomorphic state in which every player uses the
unsociable strategy $\sigma _{B}$. If $z<1$ then the picture resembles the
iterated prisoner's dilemma: if $\beta $ is small then defection is stable,
but if $\beta $ is large populations using either defection or conditional
cooperation are asymptotically stable and the population which arises
depends on the initial conditions. The most interesting dynamics occur when $
\beta $ is large and $1<z<3$ (labelled as regions VI to IX in figure 1). If $
\beta $ is large and $z<\frac{5\beta -2}{\beta }$ then conditional
cooperation is the only asymptotically stable behaviour, and in region VII
this is the endpoint of all trajectories which start in the interior of the
simplex. In region VI a polymorphic population is stable: a proportion of
players, $x$, use the conditional cooperative strategy and a proportion, $
1-x $, defect where $x=\frac{\beta z-1}{1+2\beta z-5\beta }$. In this
population the proportion of individuals that would be observed in
cooperative partnerships is $x^{2}$; the proportion of individuals involved
in partnerships for which mutual defection was the norm would be $(1-x)^{2};$
and a proportion $2x(1-x)$ of individuals would be living alone.

\begin{figure}[ht]
\centering
\begin{minipage}[b]{1.0\textwidth}
\centering
\unitlength=0.90pt
\begin{picture}(265.00,450.00)(0.00,0.00)
\linethickness{0.2mm}
\put(300.00,30.00){\makebox(0.00,0.00){$\beta$}}
\put(30.00,370.00){\makebox(0.00,0.00){$z$}}
\put(10.00,350.00){\makebox(0.00,0.00){$4.0$}}
\put(10.00,270.00){\makebox(0.00,0.00){$3.0$}}
\put(10.00,190.00){\makebox(0.00,0.00){$2.0$}}
\put(10.00,110.00){\makebox(0.00,0.00){$1.0$}}
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\put(220.00,190.00){\makebox(0.00,0.00){VII}}
\put(180.00,131.00){\makebox(0.00,0.00){VIII}}
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\end{picture}
\end{minipage}
\caption{Regions for the parameters $z$ and $\beta$ which lead to qualitatively
different pictures of the evolutionary dynamics.}
\end{figure}

\begin{figure}[ht]
\centering
\begin{minipage}[b]{0.3\textwidth}
\centering
\unitlength=0.37pt
\begin{picture}(260.00,260.00)(0.00,0.00)
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\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2i.eps}}
I
\end{minipage}
\hspace{0.03\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2ii.eps}}
II
\end{minipage}\\
\vspace{-0.5cm}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2iii.eps}}
III
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2iv.eps}}
IV
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2v.eps}}
V
\end{minipage}\\
\vspace{-0.5cm}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2vi.eps}}
VI
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2vii.eps}}
VII
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2viii.eps}}
VIII
\end{minipage}\\
\vspace{-0.5cm}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2ix.eps}}
IX
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.3\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.16\textheight, width=3.5cm]{fig2x.eps}}
X
\end{minipage}
\caption{Qualitative pictures of the dynamics in the different regions of
the $\protect\beta-z$ parameter space. The regions are labelled according to
figure 1. Point $a$ corresponds to a population which consists of 100\% of
players using strategy $\protect\sigma_{B}$. Point $b$ corresponds to a
population which consists of 100\% of players using strategy $\protect\sigma
_{C}$. Point $c$ corresponds to a population which consists of 100\% of
players using strategy $\protect\sigma_{D}$. Point $d$ corresponds to a
polymorphic population with players using either $\protect\sigma_{C}$ or $
\protect\sigma_{D}$. Asymptotically stable points are shown as solid
circles, the other fixed points are shown as open circles.}
\end{figure}

\section*{Introducing unconditional cooperators}

In region VII of the $\beta -z$ parameter space we have found that
conditional cooperation is asymptotically stable. It is pertinent to ask
whether this property would be destroyed if we allowed individuals to use
the ``sucker'' strategy of unconditional cooperation (which we denote $
\sigma _{S}$). Recall that in the iterated prisoner's dilemma, tit-for-tat
is not asymptotically stable due to the presence of unconditional
cooperators. Similarly, it is conceivable that the polymorphic population
which is stable in region VI could be destabilized by the introduction of a
strategy of unconditional cooperation.

We introduce a proportion $x_{3}$ of players who use the strategy of
unconditional cooperation, $\sigma _{S}$. These players associate in $G_{0}$
and cooperate in $G_{1}$ whatever their opponent does. (The proportion of
individuals using $\sigma _{B}$ is then $1-x_{1}-x_{2}-x_{3}$.) The new
payoff matrix is given by
\begin{equation}
A=\left[
\begin{array}{llll}
\pi (\sigma _{C},\sigma _{C}) & \pi (\sigma _{C},\sigma _{D}) & \pi (\sigma
_{C},\sigma _{S}) & \pi (\sigma _{C},\sigma _{B}) \\
\pi (\sigma _{D},\sigma _{C}) & \pi (\sigma _{D},\sigma _{D}) & \pi (\sigma
_{D},\sigma _{S}) & \pi (\sigma _{D},\sigma _{B}) \\
\pi (\sigma _{S},\sigma _{C}) & \pi (\sigma _{S},\sigma _{D}) & \pi (\sigma
_{S},\sigma _{S}) & \pi (\sigma _{S},\sigma _{B}) \\
\pi (\sigma _{B},\sigma _{C}) & \pi (\sigma _{B},\sigma _{D}) & \pi (\sigma
_{B},\sigma _{S}) & \pi (\sigma _{B},\sigma _{B})
\end{array}
\right] =\frac{1}{1-\beta }\left[
\begin{array}{cccc}
3 & \beta z & 3 & z \\
5\left( 1-\beta \right) +\beta z & 1 & 5 & z \\
3 & 0 & 3 & z \\
z & z & z & z
\end{array}
\right]  \label{RD3d}
\end{equation}

An analysis of the corresponding Replicator dynamics leads to the dynamics
shown in figures 3 and 4 for regions VI and VII respectively (see appendix B
for details). These figures show the dynamics for particular values of $z$
and $\beta $ but the pictures are qualitatively similar for any values of
these parameters in the appropriate range. From these figures we can see
that the polymorphic population remains asymptotically stable in region VI.
In region VII, the population of conditional cooperators is no longer
asymptotically stable. However, populations which consist of mixtures of
conditional and unconditional cooperators are the only end points of all
solution trajectories which start in the interior of the simplex.

\begin{figure}[ht]
\centering
\begin{minipage}[b]{0.45\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.3\textheight, width=7cm]{fig3.eps}}
Region VI: $z=\frac{25}{10}$ and $\protect\beta = \frac{75}{100}$.
\end{minipage}
\hspace{0.04\textwidth}
\begin{minipage}[b]{0.45\textwidth}
\centering
\subfigure{\includegraphics[totalheight=0.3\textheight, width=7cm]{fig4.eps}}
Region VII: $z=\frac{25}{10}$ and $\protect\beta =\frac{9}{10}$.
 \end{minipage}
\caption{Qualitative picture of the dynamics for the replicator system with
payoff matrix given by equation (\ref{RD3d}). Vertex $S$ corresponds to a
population which consists of 100\% of players using strategy $\protect\sigma
_{S}$. Vertex $C$ corresponds to a population which consists of 100\% of
players using strategy $\protect\sigma_{C}$. Vertex $D$ corresponds to a
population which consists of 100\% of players using strategy $\protect\sigma
_{D}$. The unlabelled vertex corresponds to a population in which all
players live alone.}
\end{figure}

\section*{Discussion}

A minimal version of the iterated prisoner's dilemma deals with a population
consisting of unconditional cooperators, unconditional defectors and
conditional cooperators (such as tit-for-tat). In that model there is a
threshold problem: cooperative behaviour only evolves if the initial
proportion of conditional cooperators exceeds some value \cite{VR, D}.
Although it is sometimes suggested that the always defect strategy is an ESS
or that the corresponding population is asymptotically stable, this is not
the case. If sufficiently many varied strategies are introduced then the
barrier can be removed \cite{NS}.

We have introduced an outside option into the iterated prisoner's dilemma,
which allows individuals to avoid being condemned to maintain an
unprofitable interaction of permanent mutual defection. This provides
another way of removing the barrier to the evolution of cooperative
behaviour. The requirement is that the payoff from the outside option should
be neither so poor that it is irrelevant nor so high that everyone opts for
a solitary existence. The existence of the outside option also admits a
range of parameter values for which a polymorphic population involving
defectors and conditional cooperators is asymptotically stable, even in the
presence of unconditional cooperators.

Some of the results we have obtained are similar to those obtained for
optional public good games, which are multi-player generalizations of the
prisoner's dilemma \cite{HMHS}. In these games, as in ours, making
participation voluntary enhances the possibilities for cooperation. One
difference between the two models is that in the optional public good game
rock-scissors-paper style cycles may occur. In our model, such cyclic
behaviour does not arise. However, in both models the fixed point
representing non-participatory behaviour may be non-hyperbolic. This leads
to periods of cooperative behaviour, but eventually the population returns
to a state in which everyone lives alone.

The iterated prisoner's dilemma is an unrealistic model of social
interactions because it treats one type of interaction between individuals
in isolation from all others. We have shown, by means of a relatively simple
example, that the methods of stochastic game theory can be employed to
overcome this restriction. The prisoners dilemma has also been criticized as
being an unrealistic model of social interactions on other grounds \cite{DMG}
. Our approach is not specific to the prisoner's dilemma. That context game
may be replaced by any other game or, indeed, a game which is randomly
selected with a known probability from a set of games \cite{K}. This allows
quite complex social behaviour to be analyzed.

\section*{Appendix A}

To integrate the Replicator Dynamics system (\ref{RD2d}) we make the
following coordinate substitutions.
\begin{equation*}
k=\frac{x_{2}}{x_{1}}\ \ \text{\ and\ }\ \ l=\frac{1-x_{1}-x_{2}}{x_{1}}
\end{equation*}
or
\begin{equation*}
x_{1}=\frac{1}{1+k+l}\text{ \ \ \ \ \ and \ \ \ \ \ }x_{2}=\frac{k}{1+k+l}.
\end{equation*}
Then we have
\begin{equation*}
\begin{array}{l}
\dot{k}=k\frac{a+bk}{1+l+k} \\
\dot{l}=l\frac{c+fk}{1+l+k}
\end{array}
.
\end{equation*}
Solution trajectories can be found by integrating
\begin{equation*}
\frac{dl}{dk}=\frac{l\left( c+fk\right) }{k\left( a+bk\right) }
\end{equation*}

\begin{equation*}
\frac{dl}{l}=\frac{c}{a}\frac{dk}{k}+\frac{af-bc}{a}\frac{dk}{\left(
a+bk\right) }
\end{equation*}

This can be done analytically to obtain
\begin{eqnarray*}
bc\ln \left| k\right| +\left( af-bc\right) \ln \left| a+bk\right| &=&ab\ln
\left| l\right| +C \\
\left| k\right| ^{bc}\left| a+bk\right| ^{\left( af-bc\right) } &=&C\left|
l\right| ^{ab}
\end{eqnarray*}
where $C$ is\ a constant that depends on the initial conditions. Finally,
substituting the expressions for $k$ and $l$ into the above formula, we find
that the solution trajectories are described by the expression.
\begin{equation*}
\left| \frac{x_{2}}{x_{1}}\right| ^{bc}\left| a+b\frac{x_{2}}{x_{1}}\right|
^{\left( af-bc\right) }=C\left| \frac{1-x_{1}-x_{2}}{x_{1}}\right| ^{ab}.
\end{equation*}

\section*{Appendix B}

The Replicator Dynamics with payoff matrix (\ref{RD3d}) is given by the
following system of equations.
\begin{equation*}
\begin{array}{l}
\dot{x}_{1}=\frac{x_{1}\left( \left( x_{1}+x_{3}\right) \left(
1-x_{1}-x_{3}\right) \left( 3-z\right) +\left( 2z-5\right) \left( \left(
1-\beta \right) x_{1}+x_{3}\right) x_{2}+\left( \left( z-1\right)
x_{2}-\left( 1-\beta \right) z\right) x_{2}\right) }{1-\beta } \\
\dot{x}_{2}=\frac{x_{2}\left( \left( x_{1}+x_{3}\right) ^{2}\left(
z-3\right) +\left( \left( 5-z\right) +\left( 2z-5\right) x_{2}\right) \left(
\left( 1-\beta \right) x_{1}+x_{3}\right) +\left( 1-z\right) \left(
1-x_{2}\right) x_{2}\right) }{1-\beta } \\
\dot{x}_{3}=\frac{x_{3}\left( \left( x_{1}+x_{3}\right) \left(
1-x_{1}-x_{3}\right) \left( 3-z\right) +\left( 2z-5\right) \left( \left(
1-\beta \right) x_{1}+x_{3}\right) x_{2}+\left( \left( z-1\right)
x_{2}-z\right) x_{2}\right) }{1-\beta }
\end{array}
\end{equation*}
The fixed points together with their associated eigenvectors and eigenvalues
are given in table 3. \medskip

\begin{center}
\begin{tabular}{c}
\begin{tabular}{|c|c|c|c|}
\hline
Population & Point & Eigenvectors & Eigenvalues \\ \hline
$100\%$ $\text{of}\ \sigma _{B}$ & $_{\left\{ 0,0,0\right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( 1,0,0\right) } \\
_{e_{2}=\left( 0,1,0\right) } \\
_{\medskip e_{2}=\left( 0,0,1\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=0} \\
_{\lambda _{2}=0} \\
\medskip _{\lambda _{3}=0}
\end{array}
$ \\ \hline
$100\%\text{ of}\ \sigma _{D}$ & $_{\left\{ 0,1,0\right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( -1,1,0\right) } \\
_{e_{2}=\left( 0,1,0\right) } \\
\medskip _{e_{2}=\left( 0,1,-1\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=\frac{\beta z-1}{1-\beta }} \\
_{\lambda _{2}=\frac{z-1}{1-\beta }} \\
\medskip _{\lambda _{3}=-\frac{1}{1-\beta }}
\end{array}
$ \\ \hline
$100\%\text{ of}\ \sigma _{C}$ & $_{\left\{ 1,0,0\right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( 1,0,-1\right) } \\
_{e_{2}=\left( -1,1,0\right) } \\
\medskip _{e_{3}=\left( 1,0,0\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=0} \\
_{\lambda _{2}=\frac{2-5\beta +\beta z}{1-\beta }} \\
\medskip _{\lambda _{3}=\frac{z-3}{1-\beta }}
\end{array}
$ \\ \hline
$100\%\text{ of}\ \sigma _{S}$ & $_{\left\{ 0,0,1\right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( 1,0,-1\right) } \\
_{e_{2}=\left( 0,1,-1\right) } \\
\medskip _{e_{3}=\left( 0,0,1\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=0} \\
_{\lambda _{2}=\frac{2}{1-\beta }} \\
\medskip _{\lambda _{3}=\frac{z-3}{1-\beta }}
\end{array}
$ \\ \hline
$
\begin{array}{c}
\alpha \%\text{ of }\sigma _{C} \\
\text{and} \\
\left( 1-\alpha \right) \%\text{ of}\ \sigma _{S}
\end{array}
$ & $_{\left\{ \alpha ,0,1-\alpha \right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( 1,0,-1\right) } \\
_{e_{2}=\left( \alpha \frac{\beta z-2\alpha \beta z+5\alpha \beta -2}{
2-5\alpha \beta +\alpha \beta z},1,\frac{\left( \alpha -1\right) \left(
2\alpha \beta z-5\alpha \beta +2\right) }{2-5\alpha \beta +\alpha \beta z}
\right) } \\
\medskip _{e_{3}=\left( 1,0,\frac{1-\alpha }{\alpha }\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=0} \\
_{\lambda _{2}=\frac{2-5\alpha \beta +\alpha \beta z}{1-\beta }} \\
\medskip _{\lambda _{3}=\frac{z-3}{1-\beta }}
\end{array}
$ \\ \hline
$
\begin{array}{c}
\frac{\beta z-1}{2\beta z-5\beta +1}\%\text{ of }\sigma _{C} \\
\text{and} \\
\frac{2-5\beta +\beta z}{2\beta z-5\beta +1}\%\text{ of}\ \sigma _{S}
\end{array}
$ & $_{\left\{ \frac{\beta z-1}{2\beta z-5\beta +1},\frac{2-5\beta +\beta z}{
2\beta z-5\beta +1},0\right\} }$ & $
\begin{array}{c}
_{e_{1}=\left( -1,1,0\right) } \\
_{e_{2}=\left( 1,\frac{2-5\beta +\beta z}{\beta z-1},0\right) } \\
\medskip _{e_{3}=\left( 1,\beta \frac{5-3z}{\beta z-1},\frac{2\beta z-5\beta
+1}{\beta z-1}\right) }
\end{array}
$ & $
\begin{array}{c}
_{\lambda _{1}=\frac{\left( 1-\beta z\right) \left( \beta z-5\beta +2\right)
}{\left( 1-\beta \right) \left( 2\beta z-5\beta +1\right) }} \\
_{\lambda _{2}=\frac{z\beta \left( 2-\beta \right) \left( z-5\right) +3+z}{
\left( 1-\beta \right) \left( 2\beta z-5\beta +1\right) }} \\
\medskip _{\lambda _{3}=\frac{-\beta z\left( 2-5\beta +\beta z\right) }{
\left( 1-\beta \right) \left( 2\beta z-5\beta +1\right) }}
\end{array}
$ \\ \hline
\end{tabular}
\\
$\text{\rule{0in}{0.35in}}$Table 3. Eigenvalues and eigenvectors for the
fixed points  \\
of the replicator system with payoff matrix given by equation (4).$\text{
\rule[-0.3in]{0in}{0.18in}}$
\end{tabular}
\end{center}

config: arxiv


TITLE: How to prove that a B-spline curve is a Bezier curve
QUESTION [2 upvotes]: Let P be a quadratic B-spline with three control points $ p_{0},p_{1},p_{2},  $ and knot vector $\tau = (0,0,0,1,1,1) $. How to prove explicitly that P is a Bezier curve?

REPLY [1 votes]: Yes, your approach is correct. Just use the usual (recursive) definition of b-spline basis functions. You will find that the b-spline basis functions of degree $1$ are $1-t$ and $t$. From these, we can compute the basis functions of degree 2. You will find that these are $(1-t)^2$, $2t(1-t)$, and $t^2$, which are the Bernstein polynomials of degree 2. Continue from there.
Another approach ...
You know that any b-spline curve is just a sequence of polynomial segments (i.e. Bezier curves), strung together end-to-end. There is one polynomial segment corresponding to each non-trivial knot span $[t_i, t_{i+1}]$, where $t_i < t_{i+1}$. In our case, there is only one non-trivial knot span, namely $[0,1]$, so the entire b-spline curve consists of just one polynomial segment. In other words, the b-spline curve is a Bezier curve.


text:
\section{Introduction}
Deep penetration of renewable energy generation is essential to ensure a sustainable future. Renewable energy resources, such as wind and solar, are intrinsically variable. Uncertainties associated with these intermittent and volatile resources pose a significant challenge to their integration into the existing grid infrastructure \cite{NRELWest2010}. More flexibility, especially in shifting energy supply and/or demand across time and network, is desired to cope with the increased uncertainties.

Energy storage provides the functionality of shifting energy across time. A vast array of technologies, such as  batteries, flywheels, pumped-hydro, and compressed air energy storages, are available for such a purpose \cite{Denholm2010, lindley2010naturenews}.

Furthermore, flexible or controllable demand provides another ubiquitous source of storage. Deferrable loads -- including many thermal loads, loads of internet data-centers and loads corresponding to charging electric vehicles (EVs) over certain time interval \cite{thermalStor1993} -- can be interpreted and controlled as \emph{storage of demand} \cite{ObRACC2013}.
Other controllable loads which can possibly be shifted to an earlier or later time, such as thermostatically controlled loads (TCLs), may be modeled and controlled as a storage with negative lower bound and positive upper bound on the storage level \cite{HaoSanandajiPoollaVincent2013}.

These forms of storage enable inter-temporal shifting of excess energy supply and/or demand, and significantly reduce the reserve requirement and thus system costs.

On the other hand, shifting energy across a network, \ie, moving excess energy supply to meet unfulfilled demand  among different geographical locations with transmission or distribution lines, can achieve similar effects in reducing the reserve requirement for the system.

Thus in practice, it is natural to consider these two effects together. Yet, it remains mathematically challenging to formulate a sound and tractable problem that accounts for these effects in electric grid operations. Specifically, due to the power flow and network constraints, control variables in connected buses are coupled. Due to the storage constraints, control variables in different time periods are coupled as well.
On top of that, uncertainties associated with stochastic generation and demand dramatically complicate the problem, because of the large number of recourse stages and the need to account for all probable realizations.

Two categories of approaches have been proposed in the literature. The first category is based on exploiting structures of specific problem instances, usually using dynamic programming. These structural results are valuable in providing insights about the system, and often lead to analytical solution of these problem instances.
However, such approaches rely heavily on specific assumptions of the type of storage, the form of the cost function, and the distribution of uncertain parameters. Generalizing results to other specifications and more complex settings is usually difficult, and consequently this approach is mostly used to analyze single storage systems.
For instance, analytical solutions to optimal storage arbitrage with stochastic price have been derived in  \cite{QRsimpleStorPes2012} without storage ramping constraints, and in \cite{MITrampStor} with ramping constraints. Problems of using energy storage to minimize energy imbalance are studied in various contexts; see \cite{SuEGTPS, RLDSACC} for reducing reserve energy requirements in power system dispatch, \cite{BitarRACC_colocated, Powell} for operating storage co-located with a wind farm, \cite{IBMload, DataCenter} for operating storage co-located with end-user demands, and \cite{StorDRLongbo} for storage with demand response.

The other category is to use heuristic algorithms, such as Model Predictive Control (MPC) \cite{XieEtAlWindStorMPC} and look-ahead policies \cite{NRELStorValue2013},  to identify sub-optimal storage control rules. Usually based on deterministic (convex) optimization, these approaches can be easily applied to general networks. The major drawback is that these approaches usually do not have any performance guarantee. Consequently,  it lacks theoretical justification for their implementation in real systems.
Examples of this category can be found in \cite{XieEtAlWindStorMPC} and references therein.

This work aims  at designing distributed online deterministic optimizations that solve the stochastic control problem with provable guarantees. It contributes to the existing literature in the following ways. First, we formulate the problem of  storage network operation as a stochastic control problem with general cost functions, which encapsulates a variety of problems with different types of storage as well as different uses of storage.
Second, we devise an online algorithm for the problem based on the theory of Lyapunov optimization,
and prove guarantees for its performance in terms of a bound of its sub-optimality. This converts the ``intractable'' stochastic control program to a sequence of tractable deterministic optimization programs.
The bound is useful not only in assessing the performance of our algorithm, but also in evaluating the performance of other sub-optimal algorithms when the optimal costs are hard to obtain. It can also be used to estimate the maximum cost reduction that can be achieved by {\it any} storage operation, thus provides understanding for the limit of a certain storage system.
To the best of our knowledge, this is the first algorithm with provable guarantees for the storage operation problem with general electric networks.
Finally, we derive task-based distributed implementation of the online algorithm using the alternating direction method of multipliers (ADMM).

The paper generalizes our prior work \cite{OMGarxiv} by modeling a networked storage system, and extending the online control algorithm to the network setting. Preliminary results related to the network setting have been presented in \cite{QCYR:acm}. The online optimization in this paper is different from that in \cite{QCYR:acm}, and the sub-optimality bound here is significant superior to the bound for the algorithm proposed in  \cite{QCYR:acm}. The aspect of distributed implementation is also new in this paper.

The rest of the paper is organized as follows. Section 2 formulates the problem of operating a storage network under uncertainty. Section 3 gives the online algorithm and states the performance guarantee. Section 4 discusses the distributed implementation of the online program. Numerical examples are then given in Section 5. Section 6 concludes the paper.


TITLE: Proof laplace trasform of $xf(x)$
QUESTION [0 upvotes]: I want to prove that $\mathcal L[xf(x)] = -\frac{d \tilde{f}(s)}{d t}$
my try
By definition $$\mathcal L[x(f(x)] = \int_0^{\infty} xf(x) e^{-sx}dx = \left[x\int f(x)e^{-sx}dx\right]_0^\infty-\int_0^\infty f(x)e^{-sx}dx$$
But now how to I proceed?

REPLY [1 votes]: $$
\frac{\mathrm{d}}{\mathrm{d}s}\mathcal{L}\left[f(x)\right]=\frac{\mathrm{d}}{\mathrm{d}s}\int_{0}^{+\infty}f(x)e^{-sx}\mathrm{d}x
$$
$$
\frac{\mathrm{d}}{\mathrm{d}s}\mathcal{L}\left[f(x)\right]=\int_{0}^{+\infty}f(x)\frac{\mathrm{d}}{\mathrm{d}s}(e^{-sx})\mathrm{d}x
$$
and
$$
\frac{\mathrm{d}}{\mathrm{d}s}(e^{-sx})=-xe^{-sx}
$$
$$
\frac{\mathrm{d}}{\mathrm{d}s}\mathcal{L}\left[f(x)\right]=-\int_{0}^{+\infty}f(x)xe^{-sx}\mathrm{d}x
$$
$$
\frac{\mathrm{d}}{\mathrm{d}s}\mathcal{L}\left[f(x)\right](s)=-\mathcal{L}\left[xf(x)\right](s)
$$