text:
\begin{document}

\title{A semi-good frame with amalgamation and tameness in $\lambda^+$}

\author{Adi Jarden}
\email[Adi Jarden]{jardenadi@gmail.com}
\address{Department of Computer Science and Mathematics \\ Ariel University Center of Samaria \\ Ariel, Israel}
\address{Department of Mathematics \\ Bar-Ilan University \\ Ramat-Gan, Israel}

\begin{abstract}
We introduce a connection between tameness and non-forking frames. We assume the existence of a semi-good non-forking $\lambda$-frame, $(\lambda,\lambda^+)$-tameness and amalgamation in $\lambda^+$ and present sufficient additional conditions for the existence of a good non-forking $\lambda^+$-frame.

Moreover, we improve results of \cite{jrsi3} about independence and dimension.
\end{abstract}

\maketitle

\section{Introduction}
In \cite{shh}.III Shelah presented an axiomization of a non-forking relation in the context of AECs. But this non-forking relation relates to models of a specific cardinality, $\lambda$, only. Shelah presented a way to extend a non-forking relation to models of cardinality $>\lambda$ and proved that several axioms preserved.

The extension, uniqueness and symmetry are the problematic axioms. But uniqueness for models of cardinality $>\lambda$ is actually tameness. It is known that we can derive extension from uniqueness in the smaller cardinalities. So our main challenge is to get symmetry.  Some years ago, we conjectured that the symmetry holds too, but it is still an open question.

In \cite{bo complete...}, Boney presented a variant of tameness that is a sufficient condition for symmetry. Later, Sebastien  proved a non-structure theorem assuming the failure of symmetry in a similar context.  It should be checked, if we can apply the non-structure theorem of Sebastien.

Here, we present a new sufficient condition for symmetry: we show that a variant of the continuity property for independence (studied in \cite{jrsi3}) is a sufficient condition and prove it under reasonable hypothesis.

In \cite{shh}.III and \cite{jrsh875} we derive good $\lambda^+$-frames too. Let us compare the main results of those papers with the main result in the current paper. The advantages of the current paper are:
\begin{enumerate}
\item We do not restrict our selves to the saturated models, \item we do not restrict the relation $\preceq$ to $\preceq^{NF}_{\lambda^+}$ and \item we do not assume that $I(\lambda^{++},K)<2^{\lambda^{++}}$.
\end{enumerate}

The disadvantages of the current paper are the following hypotheses:
\begin{enumerate}
\item the amalgamation property in $\lambda^+$ and
\item tameness.
\end{enumerate}

\section{Non-forking frames}

In \cite{shh}.III, Shelah introduced the notion of a good (non-forking) $\lambda$-frame. It is an axiomatization of non-forking. In Definition \cite[2.1.3]{jrsh875}, good frames generalized to semi-good frames: the stability hypothesis is weakened. From now we assume:
\begin{hypothesis}\label{hypothesis 1}
$\frak{s}=(K,\preceq,\dnf,S^{bs})$ is a semi-good non-forking frame.
\end{hypothesis}

\begin{comment}

 \begin{definition} \label{2.1a} \label{definition good frame}
\label{definition of a good frame}
$\frak{s}=(K,\preceq,S^{bs},\dnf)$ is a \emph{good $\lambda$-frame} if:\\
(0)
\begin{enumerate}[(a)]
\item $(K,\preceq)$ is an AEC. \item $LST(K,\preceq) \leq
\lambda$.
\end{enumerate}

(1)
\begin{enumerate}[(a)]
\item $(K_\lambda, \preceq \restriction K_\lambda)$
satisfies the joint
embedding property. \item $(K_\lambda, \preceq \restriction K_\lambda)$ satisfies the amalgamation property. \item There is no $\preceq$-maximal model in $K_\lambda$.\\
\end{enumerate}

(2) $S^{bs}$ is a function with domain $K_\lambda$, which
satisfies the following axioms:
\begin{enumerate}[(a)]
\item $S^{bs}(M) \subseteq S^{na}(M)=\{tp(a,M,N):M\prec N \in
K_\lambda,\ a\in N-M\}$. \item It respects isomorphisms: If
$tp(a,M,N) \in S^{bs}(M)$ and $f:N \to N'$ is an isomorphism, then
$tp(f(a),f[M],N') \in S^{bs}(f[M])$. \item Density of the basic
types: If $M,N \in K_\lambda$ and $M \prec N$, then there is $a
\in N-M$ such that $tp(a,M,N) \in S^{bs}(M)$. \item Basic
stability: For every $M \in K_\lambda$, the cardinality of
$S^{bs}(M)$ is $\leq \lambda$.
\end{enumerate}

(3) the relation $\dnf$ satisfies the following axioms:
\begin{enumerate}[(a)]
\item $\dnf$ is a set of quadruples $(M_0,M_1,a,M_3)$ where
$M_0,M_1,M_3 \in K_\lambda$, $a \in M_3-M_1$ and for $n=0,1$
$tp(a,M_n,M_3) \in S^{bs}(M_n)$ and it respects isomorphisms: If
$\dnf(M_0,M_1,a,M_3)$ and $f:M_3 \to M_3'$ is an isomorphism, then
$\dnf(f[M_0],f[M_1],f(a),M_3')$. \item Monotonicity: If $M_0
\preceq M^*_0 \preceq M^*_1 \preceq M_1 \preceq M_3 \preceq
M_3^*,\ M^*_1\bigcup \{a\} \subseteq M^{**}_3 \preceq M^*_3$, then
$\dnf(M_0,M_1,a,M_3) \Rightarrow \dnf(M^*_0,M^*_1,a,M^{**}_3)$.
From now on, `$p \in S^{bs}(N)$ does not fork over $M$' will be
interpreted as `for some $a,N^+$ we have $p=tp(a,N,N^+)$ and
$\dnf(M,N,a,N^+)$'. See Proposition ~\ref{dnf not as a sign}.
\item Local character: For every limit ordinal $\delta<\lambda^+$
if $\langle M_\alpha:\alpha \leq \delta+1 \rangle$ is an increasing
continuous sequence of models in $K_\lambda$, and
$tp(a,M_\delta,M_{\delta+1}) \in S^{bs}(M_\delta)$, then there is
$\alpha<\delta$ such that $tp(a,M_\delta,\allowbreak
M_{\delta+1})$ does not fork over $M_\alpha$. \item Uniqueness of
the non-forking extension: If $M,N \in K_\lambda$, $M \preceq N$,
$p,q \in S^{bs}(N)$ do not fork over $M$, and $p\restriction
M=q\restriction M$, then $p=q$. \item Symmetry: If $M_0,M_1,M_3
\in K_\lambda$, $M_0 \preceq M_1 \preceq M_3,\ a_1 \in M_1,\
tp(a_1,M_0,\allowbreak M_3) \in S^{bs}(M_0)$, and
$tp(a_2,M_1,M_3)$ does not fork over $M_0$, \emph{then} there are
$M_2,M^*_3 \in K_\lambda$ such that $a_2 \in M_2,\ M_0 \preceq M_2
\preceq M^*_3,\ M_3 \preceq M^*_3$, and $tp(a_1,M_2,M^*_3)$ does
not fork over $M_0$. \item Existence of non-forking extension: If
$M,N \in K_\lambda$, $p \in S^{bs}(M)$ and $M\prec N$, then there
is a type $q \in S^{bs}(N)$ such that $q$ does not fork over $M$
and $q\restriction M=p$. \item Continuity: Let $\delta<\lambda^+$
and $\langle M_\alpha:\alpha \leq \delta \rangle$ an increasing
continuous sequence of models in $K_\lambda$ and let $p \in
S(M_\delta)$. If for every $\alpha \in \delta,\ p\restriction
M_\alpha$ does not fork over $M_0$, then $p \in S^{bs}(M_\delta)$
and does not fork over $M_0$.
\end{enumerate}
\end{definition}

As in some previous papers (complete...), we do not assume stability but only basic \emph{almost} stability:
\begin{definition} \label{2.1b}
$\frak{s}=(K^\frak{s},\preceq^\frak{s},S^{bs,\frak{s}},\dnf^\frak{s})=(K,\preceq,S^{bs},\dnf)$
is a \emph{semi}-good $\lambda$-frame, if $\frak{s}$ satisfies the
axioms of a good $\lambda$-frame except that instead of assuming
basic stability, we assume that $\frak{s}$ satisfies \emph{basic almost
stability}, namely, for every $M \in K_\lambda$ $S^{bs}(M)$
is of cardinality at most $\lambda^+$.

$\frak{s}$ is said to be a \emph{semi-good frame} if it is a
semi-good $\lambda$-frame for some $\lambda$.
\end{definition}

\end{comment}

\begin{remark}\label{local character alomost implies continuity}
By \cite{jrsh940}, without loss of generality, for each $M \in K_\lambda$ $S^{bs}(M)=S^{na}(M)$, namely, the basic types are the non-algebraic types.

\end{remark}

\begin{comment}
Recall (from \cite{she46}):
\begin{definition}\label{2.2}
Let $p_0 \in S(M_0), p_1 \in S(M_1)$. We say that $p_0,p_1$ are
\emph{conjugate} if for some $a_0,M_0^+,a_1,M_1^+,f$, the
following hold:
\begin{enumerate}
\item For $n=0,1$, $tp(a_n,M_n,M_n^+)=p_n$. \item $f:M_0^+ \to
M_1^+$ is an isomorphism. \item $f \restriction M_0:M_0 \to M_1$
is an isomorphism. \item $f(a_0)=a_1$.
\end{enumerate}
\end{definition}

\begin{proposition}\label{2.3}
Assume that $p_0,p_1$ are conjugate and the types $p_1,p_2$ are
conjugate. Then the types $p_0,p_2$ are conjugate.
\end{proposition}

\begin{proof}
Compose the isomorphisms.
\end{proof}

\begin{definition}\label{f(p)}
Let $p=tp(a,M,N)$. Let $f$ be a bijection with domain $M$. Define
$\emph{f(p)}:=tp(f(a),f[M],f^+[N])$, where $f^+$ is an extension
of $f$ (and the relations and functions on $f^+[N]$ are defined
such that $f^+:N \to f^+[N]$ is an isomorphism).
\end{definition}

\begin{remark}
The definition of $f(p)$ in Definition \ref{f(p)} does not depend
on the representative $(M,N,a) \in p$.
\end{remark}

\begin{definition} \label{2.4}
Let $\frak{s}$ be a semi-good $\lambda$-frame. We say that
$\frak{s}$ \emph{satisfies the conjugation property} when:
$K_\lambda$ is categorical and if $M_1,M_2 \in K_\lambda$, $M_1
\preceq M_2$ and $p_2 \in S^{bs}(M_2)$ is the non-forking
extension of $p_1 \in S^{bs}(M_1)$, then the types $p_1,p_2$ are
conjugate.
\end{definition}

By Claim 2.18 in \cite{shh}.II:
\begin{fact}[The transitivity proposition] \label{2.7} \label{s is
transitive}\label{transitivity}

Suppose $\frak{s}$ is a semi-good $\lambda$-frame. Then: If $M_0
\preceq M_1 \preceq M_2,\ p \in S^{bs}(M_2)$ does not fork over
$M_1$ and $p\restriction M_1$ does not fork over $M_0$, then $p$
does not fork over $M_0$.
\end{fact}

By Claim 2.16 in \cite{shh}.II:
\begin{fact} \label{axiom i is redundant}
Suppose
\begin{enumerate}
\item $\frak{s}$ satisfies the axioms of a semi-good
$\lambda$-frame. \item $n<3 \Rightarrow M_0 \preceq M_n$. \item
For $n=1,2$, $a_n \in M_n-M_0\ and \ tp(a_n,M_0,M_n) \allowbreak
\in S^{bs}(M_0)$.
\end{enumerate}
Then there is an amalgamation $(f_1,f_2,M_3)$ of $M_1,M_2$ over
$M_0$ such that for $n=1,2$ $tp(f_n(a_n),f_{3-n}[M_{3-n}],M_3)$
does not fork over $M_0$.
 \end{fact}

\end{comment}

\begin{comment}
\begin{proof} Suppose for $n=1,2$ $M_0\prec M_n \wedge tp(a_n,M_0,M_n)
\in S^{bs}(M_0)$. Take $N_1,f_1$ such that:
\begin{enumerate}
\item $M_1 \preceq N_1$. \item $f_1:M_2 \to N_1$ is an embedding
over $M_0$. \item $tp(f_1(a_2),M_1,N_1)$ does not fork over $M_0$.
\end{enumerate}

\begin{displaymath}
\xymatrix{M_2 \ar[r]^{f_1} & N_1 \ar[rr]^{id} && N_2 \ar[r]^{id}
\ar[d]^{f_2} & N_3\\
&& N_2^* \ar[r]^{id} \ar[ru]^{id} & f_2[N_2] \ar[ru]^{id}\\
M_0 \ar[r]^{id} \ar[uu]^{id} \ar[rru]^{id} & M_1 \ar[uu]^{id} }
\end{displaymath}

By Axiom e (the symmetry axiom), there are models $N_2,N_2^* \in
K_\lambda$ such that:
\begin{enumerate}
\item $N_1 \preceq N_2 \in K_\lambda$. \item $M_0 \bigcup
\{f_1(a_2)\} \subseteq N_2^* \preceq N_2$. \item
$tp(a_1,N_2^*,N_2)$ does not fork over $M_0$.
\end{enumerate}

By existence of non-forking extension we can find $N_3,f_2$ such
that:
\begin{enumerate}
\item $N_2 \preceq N_3$. \item $f_2:N_2 \to N_3$ is an embedding
over $N_2^*$. \item $tp(a_1,f_2[N_2],N_3)$ does not fork over
$N_2^*$.
\end{enumerate}
So by Proposition \ref{s is transitive}, $tp(a_1,f_2[N_2],N_3)$ does not fork over $M_0$. So
since $M_0 \preceq f_2 \circ f_1[M_2] \preceq f_2[N_2]$ by Axiom b
(monotonicity) $tp(a_1,f_2 \circ f_1[M_2],N_3)$ does not fork over
$M_0$. As $f_2 \restriction N_2^*=id_{N_2^*}$,
$f_2(f_1(a_1))=f_1(a_1)$.
\end{proof}
\end{comment}

\begin{comment}
By the following fact (from \cite{jrsh875}), $K$ is stable in $\lambda$.
\begin{fact} \label{4.0}\label{stability}
Suppose $\frak{s}$ satisfies conditions 1 and 2 of a semi-good
$\lambda$-frame (so actually the relation $\dnf$ is irrelevant).
\begin{enumerate}
 \item Suppose:
\begin{enumerate}
\item $\langle  M_\alpha:\alpha \leq \lambda^+ \rangle$ is an
increasing continuous sequence of models in $K_\lambda$. \item
There is a stationary set $S \subseteq \lambda^+$ such that for
every $\alpha \in S$ and every model $N$, with $M_\alpha \prec N$,
there is a type $p \in S^{bs}(M_\alpha)$ which is realized in
$M_{\lambda^+}$ and in $N$.
\end{enumerate}
Then $M_{\lambda^+}$ is full over $M_0$ and is saturated in
$\lambda^+$ over $\lambda$. \item Suppose:
\begin{enumerate}
   \item $\langle  M_\alpha:\alpha \leq \lambda^+ \rangle$ is an increasing continuous
sequence of models in $K_\lambda$.
   \item For every $\alpha \in \lambda^+$ and every $p \in
S^{bs}(M_\alpha)$, there is $\beta \in (\alpha,\lambda^+)$ such
that $p$ is realized in $M_\beta$.
\end{enumerate}
Then $M_{\lambda^+}$ is full over $M_0$ and $M_{\lambda^+}$ is
saturated in $\lambda^+$ over $\lambda$. \item There is a model in
$K_{\lambda^+}$ which is saturated in $\lambda^+$ over $\lambda$.
\item $M \in K_\lambda \Rightarrow |S(M)| \leq \lambda^+$ (we know
that $|S^{bs}(M)| \leq \lambda^+$, but the point is that $|S(M)| \leq
\lambda^+$).
\end{enumerate}
\end{fact}
\end{comment}

\begin{comment}

In our work we study 1-types only. But what is
the relation between almost-stability in the sense of 1-types and
almost stability in the sense of n-types? For example, we do not
answer the following question:
\begin{question}
Is there an AEC $(K,\preceq)$ such that:
\begin{enumerate}
\item $LST(K,\preceq) \leq \lambda$. \item $(K_\lambda,\preceq
\reduction K_\lambda)$ satisfies the amalgamation property. \item
$(K,\preceq)$ is almost stable  in lambda. \item is not
n-type-stable for some finite n?
\end{enumerate}
\end{question}
Of course in such an AEC there is no saturated model.

\end{comment}

\subsection{Non-forking with larger models}
We recall \cite[Definition 2.6.1]{jrsh875}, where we extend the non-forking relation to include models of
cardinality greater than $\lambda$.

\begin{definition}\label{preparation for forking for big models}
\label{2.9} $\dnf^{\geq \lambda}$ is the class of quadruples
$(M_0,a,M_1,M_2)$ such that:
\begin{enumerate}
\item $\lambda \leq ||M_i||$ for each $i<3$.\item $M_0 \preceq M_1
\preceq M_2$ and $a \in M_2-M_1$. \item For some model $N_0 \in
K_\lambda$ with $N_0 \preceq M_0$ for each model $N \in
K_\lambda$, $N_0 \bigcup \{a\} \subseteq N \preceq M_1 \Rightarrow
\dnf(N_0,a,N,M_2)$.
\end{enumerate}
\end{definition}

\begin{definition}\label{forking for big models}
Let $M_0,M_1$ be models in $K_{\geq \lambda}$ with $M_0 \preceq
M_1$ and $p \in S(M_1)$. We say that $p$ does not fork over $M_0$,
when for some triple $(M_1,M_2,a) \in p$ we have $\dnf^{\leq
\lambda}(M_0,a,M_1,M_2)$.
\end{definition}

\begin{remark}
We can replace the quantification `for some' ($M_1,M_2,a$) in
Definition \ref{forking for big models} by `for each'.
\end{remark}

\begin{definition}\label{basic for big models}
Let $M \in K_{>\lambda},\ p \in S(M)$. $p$ is said to be
\emph{basic} when there is $N \in K_\lambda$ such that $N \preceq
M$ and $p$ does not fork over $N$. For every $M \in K_{>\lambda},\
S^{bs}_{> \lambda}(M)$ is the set of basic types over $M$.
Sometimes we write $S^{bs}_{\geq \lambda}(M)$, meaning $S^{bs}(M)$
or $S^{bs}_{> \lambda}(M)$ (the unique difference is the
cardinality of $M$).
\end{definition}

The following fact  is an immediate consequence of \cite[Theorem 2.6.8]{jrsh875}.

\begin{fact}
If $\frak{s}^+$ satisfies basic stability, uniqueness, extension and symmetry then it is a good non-forking frame.
\end{fact}

\begin{comment}
By the following fact (an immediate consequence of \cite[Theorem 2.6.8]{jrsh875}), every non-forking relation in
$(K_\lambda, \preceq \restriction K_\lambda)$ can be lifted to
$K_{\geq \lambda}$ with many properties preserved. We prove that density, monotonicity, transitivity, local character and continuity are preserved.
\begin{fact} \label{2.10}\label{lifting}
Let $\frak{s}$ be a semi-good $\lambda$-frame.
\begin{enumerate}
\item Density: If
$M\prec N,\ M \in K_{\geq \lambda}$,
 then there is $a
\in N-M$ such that $tp(a,M,N) \in S^{bs}_{\geq \lambda}(M)$. \item
Monotonicity: Suppose $M_0 \preceq M_1 \preceq M_2,\ n<3
\Rightarrow M_n \in K_{\geq \lambda},\
||M_2||\allowbreak>\lambda$. If $p \in S^{bs}_{\geq \lambda}(M_2)$
does not fork over $M_0$, then
\begin{enumerate}
\item $p$ does not fork over $M_1$. \item $p\restriction M_1$ does
not fork over $M_0$.
\end{enumerate}
\item Transitivity: Suppose $M_0,M_1,M_2 \in K_{\geq \lambda}$ and
$M_0 \preceq M_1 \preceq M_2$. If $p \in S^{bs}_{\geq
\lambda}(M_2)$ does not fork over $M_1$, and $p\restriction M_1$
does not fork over $M_0$, then $p$ does not fork over $M_0$. \item
Local character: let $\delta$ be a limit ordinal. If $\langle M_\alpha:\alpha \leq \delta \rangle$ is
an increasing continuous sequence of models in $K_{>\lambda}$, and
$p \in S^{bs}_{>\lambda}(M_\delta)$ then there is $\alpha<\delta$
such that $p$ does not fork over $M_\alpha$. \item Continuity:
Suppose $\langle M_\alpha:\alpha \leq \delta+1 \rangle$ is an
increasing continuous sequence of models in $K_{\geq\lambda}$. Let
$c \in M_{\delta+1}-M_\delta$. Denote
$p_\alpha=tp(c,M_\alpha,M_{\delta+1})$. If for every
$\alpha<\delta,\ p_\alpha$ does not fork over $M_0$, then
$p_\delta$ does not fork over $M_0$.

\end{enumerate}
\end{fact}
\end{comment}

From now on we add the following hypothesis:
\begin{hypothesis}\label{hypothesis 2}
\begin{enumerate}
\item The amalgamation property in $\lambda^+$ holds and
\item $(K,\preceq)$ satisfies $(\lambda,\lambda^+)$-tameness.
\end{enumerate}
\end{hypothesis}

\begin{theorem}\label{the main theorem}\label{symmetry implies good-frame}
Suppose Hypotheses \ref{hypothesis 1} and \ref{hypothesis 2}. If $\frak{s^+}$ satisfies symmetry then it is a good non-forking $\lambda^+$-frame.
\end{theorem}

\begin{proof}
By Propositions \ref{uniqueness in s^+},  \ref{extension in s^+} and \ref{stability in s^+}.

\begin{proposition}\label{uniqueness in s^+}
$\frak{s}^+$ satisfies uniqueness.
\end{proposition}

\begin{proof}
By uniqueness for $\frak{s}$ and tameness.
\end{proof}

\begin{proposition}\label{extension in s^+}
$\frak{s}^+$ satisfies extension.
\end{proposition}

\begin{proof}

It is sufficient to prove that if $M$ is a model of cardinality $\lambda$, $N$ is a model of cardinality $\lambda^+$, $M \preceq N$ and $p \in S^{bs}(M)$ then there is a non-forking extension of $p$ to a type over $N$. Take a filtration $\langle M_\alpha:\alpha<\lambda^+ \rangle$ of $N$ with $M_0=M$. Let $N_0$ be a model of cardinality $\lambda$ such that $M_0 \preceq N_0$ and for some $a \in N_0-M_0$ $tp(a,M_0,N_0)=p$. We choose by induction on $\alpha<\lambda^+$ a model $N_\alpha$ and an embedding $f_\alpha:M_\alpha \to N_\alpha$ such that:
\begin{enumerate}
\item $f_0$ is the identity from $M$ to $N_0$,
\item if $\alpha=\beta+1$ then $f_\beta \subseteq f_\alpha$ and $tp(a,f_\alpha[M_\alpha],N_\alpha)$ does not fork over $f_\beta[M_\beta]$ (it is possible by the extension property in $\frak{s}$) and \item if $\alpha$ is a limit ordinal then $N_\alpha=\bigcup_{\beta<\alpha}N_\alpha$ and $f_\alpha=\bigcup_{\beta<\alpha}f_\beta$.
\end{enumerate}
Note that if $\alpha$ is limit then by continuity in $\frak{s}$, $tp(a,f_\alpha[M_\alpha],N_\alpha)$ does not fork over $f_\beta[M_\beta]$ for each $\beta<\alpha$.

Define $N_{\lambda^+}=:\bigcup_{\alpha<\lambda^+}N_\alpha$ and $f=:\bigcup_{\alpha<\lambda^+}f_\alpha$.
Since $\dnf$ is closed under isomorphisms, it is sufficient to prove that $tp(a,f[N],N_{\lambda^+})$ does not fork over $M$. Let $M'$ be a model of cardinality $\lambda$ with $M \preceq M' \preceq f[N]$. For some $\alpha<\lambda^+$, we have $M' \subseteq f[M_\alpha]$. But $tp(a,f[M_\alpha],f[N])$ does not fork over $M$. Now use monotonicity of non-forking.
\end{proof}

\begin{proposition}\label{stability in s^+}
$\frak{s}^+$ satisfies basic stability.
\end{proposition}

\begin{proof}
By basic stability in $\frak{s}$ and tameness.

\end{proof}

This completes the proof of Theorem \ref{the main theorem}
\end{proof}

\section{Symmetry}

\begin{comment}

By the following proposition, the symmetry goes down.
\begin{proposition}
If $\lambda<\mu<\kappa$ and $\frak{s}^\kappa$ satisfies symmetry then $\frak{s}^\mu$ satisfies symmetry.
\end{proposition}

\begin{proof}
\begin{displaymath}
\xymatrix{& N_1 \ar[r]^{id} & N_2 \ar[rr]^{id} && N_4 \\
N_0 \ar[ru]^{id} \ar[rrr]^{id} &&& N_3 \ar[ru]^{id} \\ \\
& M_1 \ar[r]^{id} \ar[uuu]^{id} & M_2 \ar[rr]^{id} \ar[uuu]^{id} && M_4 \ar[uuu]^{id} \\
M_0 \ar[ru]^{id} \ar[rrr]^{id} \ar[uuu]^{id} &&& M_3 \ar[ru]^{id} \ar[uuu]^{id} }
\end{displaymath}

Let $M_0,M_1,M_2$ be models of cardinality $\mu$ with $M_0 \preceq M_1 \preceq M_2$. Let $a_1$ be an element in $M_1$ such that $tp(a_1,M_0,M_1)$ is basic and let $a_2$  be an element in $M_2$ such that $tp(a_2,M_1,M_2)$ is basic and does not fork over $M_0$. We should find models $M_3,M_4$ of cardinality $\mu$ such that the following hold:
 \begin{enumerate}
 \item $M_0 \preceq M_3 \preceq M_4$ \item $M_2 \preceq M_4$ \item $a_2 \in M_3$ \item $tp(a_1,M_3,M_4)$ does not fork over $M_0$.
 \end{enumerate}

We have to increase the models, such that we will be able to use symmetry in $\kappa$. Let $N_0$ be a model of cardinality $\kappa$ with $M_0 \preceq N_0$. Let $N_1$ be a model of cardinality $\kappa$ which is an amalgam of $N_0,M_1$ over $M_0$ such that the type of $a_1$ over the image of $N_0$ does not fork over $M_0$. Without loss of generality, $N_0 \preceq N_1$ and $M_1 \preceq N_1$, because we can replace $N_0$ by its image under an embedding to $N_1$ (we use invariance of $\dnf$). Let $(f,id_{M_2},N_2)$ be an amalgamation of $N_1,M_2$ over $M_1$ such that $||N_2||=\kappa$ and $tp(a_2,N_1,N_2)$ does not fork over $M_1$. Without loss of generality, $f$ is inclusion, because we can replace the models $N_0,N_1$ by their images, $f[N_0],f[N_1]$ (we use again invariance of $\dnf$).

Now $tp(a_2,N_1,N_2)$ does not fork over $N_0$. So by symmetry in $\frak{s}^\kappa$, we can find models $N_3,N_4$ such that the following hold:
 \begin{enumerate}
 \item $N_0 \preceq N_3 \preceq N_4$ \item $N_2 \preceq N_4$ \item $a_2 \in N_3$ \item $tp(a_1,N_3,N_4)$ does not fork over $N_0$.
 \end{enumerate}

Since $LST(K,\preceq) \leq \lambda \leq \mu$, we can find a model $M_3$ of cardinality $\mu$ such that $M_3 \preceq N_3$ and $M_0 \cup \{a_2\} \subseteq M_3$. Similarly, we can find a model $M_4$ of cardinality $\mu$ such that $M_4 \preceq N_4$ and $M_3 \cup M_2 \cup \{a_2\} \subseteq M_3$.

By transitivity, $tp(a_1,N_3,N_4)$ does not fork over $M_0$. Therefore by monotonicity, $tp(a_1,M_3,M_4)$ does not fork over $M_0$. Obviously, $M_0 \preceq M_3$.
\end{proof}

We want to present an assumption under which we can prove that symmetry goes up.
\end{comment}

Recall (from \cite{jrsi3}):
\begin{definition}\label{independence between elements}
The sequence $\langle a,b \rangle$ is independent in $(M,N)$ means that $\{a,b\} \subseteq N-M$ and for some $M_1,M_2$, we have $M \preceq M_1 \preceq M_2$, $N \preceq M_2$, $a \in M_1$, $tp(a,M,M_1)$ is basic, and the type $tp(b,M_1,M_2)$ does not fork over $M$.
\end{definition}

Using the independence teminology, we can reformulate symmetry as follows: for every $M,N,a,b$ the sequence $\langle a,b \rangle$ is independent in $(M,N)$ if and only if the sequence $\langle b,a \rangle$ is independent in $(M,N)$.

In \cite{jrsi3}, independence is defined for sequences of infinite length too, but since it is not used in the current paper, the reader may ignore the following definition and replace $\beta^*$ by $2$ in Definition \ref{definition of the continuity of serial independence}. Anyway, for future applications, we study the more general case.
\begin{definition} \label{1.6} \label{definition of independence}
\mbox{}

(a) $\langle M_\alpha,a_\alpha:\alpha<\alpha^* \rangle ^\frown
\langle M_{\alpha^*} \rangle$ is said to be \emph{independent}
over $M$ when:
\begin{enumerate}
\item $\langle M_\alpha:\alpha \leq \alpha^* \rangle$ is an
increasing continuous sequence of models in $K_\lambda$. \item $M
\preceq M_0$. \item For every $\alpha<\alpha^*$, $a_\alpha \in
M_{\alpha+1}-M_\alpha$ and the type
$tp(a_\alpha,M_\alpha,M_{\alpha+1})$ does not fork over $M$.
\end{enumerate}

(b) $\langle a_\alpha:\alpha<\alpha^* \rangle$ is said to be
\emph{independent} in $(M,M_0,N)$ when $M \preceq M_0 \preceq N$,
$\{a_\alpha:\alpha<\alpha^*\} \subseteq N-M$ and for some
increasing continuous sequence $\langle M_\alpha:0<\alpha \leq
\alpha^* \rangle$ and a model $N^+$ the sequence $\langle
M_\alpha,a_\alpha:\alpha<\alpha^* \rangle ^\frown \langle
M_{\alpha^*} \rangle$ is independent over $M$, $N \preceq N^+$ and
$M_{\alpha^*} \preceq N^+$.
\end{definition}

\begin{definition}\label{definition of the continuity of serial independence}
\emph{The $\lambda^+$-continuity of serial independence property} is the following property:
Let $\beta^*<\lambda^+$, $M \in K_{\lambda^+}$, $M \preceq N \in K_{\lambda^+}$ and let $\langle M_\alpha:\alpha<\lambda^+ \rangle$ be a filtration of $M$. If $\langle a_\beta:\beta<\beta^* \rangle$ is independent in $(M_\alpha,N)$ for each $\alpha<\lambda^+$ then it is independent in $(M_{\lambda^+},N)$.
\end{definition}

In the following proposition we can weaken the assumption, so that it will refer to sequences of length $\beta^*=2$ only.
\begin{proposition}\label{continuity implies symmetry}
If the $\lambda^+$-continuity of serial independence property holds then symmetry holds. So $\frak{s}^+$  is a good $\lambda^+$-frame.
\end{proposition}

From now on, our goal is to prove the $\lambda^+$-continuity of serial independence property under sufficient conditions.

\section{A Non-Forking Relation on Models}
By Theorem \ref{continuity of serial independence}, the $\lambda^+$-continuity of serial independence property holds if there is a `non-forking' relation $NF$ on models (a relation $NF$ satisfying $\bigotimes_{NF}$, see Definition \ref{definition of non-forking relation on models}).

In \cite{shh}.III, Shelah defined a non-forking relation $NF$ on models, which is based on the non-forking relation $\dnf$. In \cite{jrsh875}, we presented an equivalent definition of $NF$, such that limit models are not mentioned. The new definition is easier to work with and can be applied even when stability in $\lambda$ does not hold.

In Definition \ref{NF}, we list axioms for a relation $NF$ and denote `the relation $NF$ satisfies the list of the axioms' by $\bigotimes_{NF}$. Fact \ref{if the uniqueness triples satisfy the existence property then NF}, we present sufficient conditions for the existence of a relation $NF$ satisfying $\bigotimes_{NF}$ and respecting $\frak{s}$.

In Definitions \ref{definition of widehat{NF}},\ref{definition of preceq^{NF}} we present two relations, that are based on $NF$.

Lemma \ref{we can use NF} and Proposition \ref{widehat{NF} respects independenc} are the key points to prove the $\lambda^+$-continuity of serial independence property.

\begin{definition}\label{definition of non-forking relation on models}
Let $NF \subseteq \ ^4K_\lambda$. \emph{$\bigotimes_{NF}$} means that the following hold:
\begin{enumerate}[(a)]
\item If $NF(M_0,M_1,M_2,M_3)$ then for each $n\in \{1,2\}$
$M_0\leq M_n\leq M_3$ and $M_1 \cap M_2=M_0$. \item Monotonicity:
if $NF(M_0,M_1,M_2,M_3)$, $N_0=M_0$ and for each $n<3$ $N_n\leq
M_n\wedge N_0\leq N_n\leq N_3, (\exists N^{*})[M_3\leq N^{*}\wedge
N_3\leq N^{*}]$ then $NF(N_0 \allowbreak ,N_1,N_2,N_3)$. \item
Extension: For every $N_0,N_1,N_2 \in K_\lambda$, if for each
$l\in \{1,2\}$ $N_0 \leq N_l$ and $N_1\bigcap \allowbreak
N_2=N_0$, then for some $N_3 \in K_\lambda$,
$NF(N_0,N_1,N_2,N_3)$. \item Weak Uniqueness: Suppose for $x=a,b$,
$NF(N_0,N_1,N_2,N^{x}_3)$. Then there is a joint embedding of
$N^a,N^b \ over \ N_1 \bigcup N_2$. \item  Symmetry: For every
$N_0,N_1,N_2,N_3 \in K_\lambda$, $NF(N_0,N_1,N_2,N_3)
\Leftrightarrow NF(N_0,N_2,N_1,N_3)$. \item Long transitivity: For
$x=a,b$, let $\langle  M_{x,i}:i\leq \alpha^* \rangle$ an
increasing continuous sequence of models in $K_\lambda$. Suppose
that for each $i<\alpha^*$, $NF(M_{a,i},\allowbreak
M_{a,i+1},M_{b,i},\allowbreak M_{b,i+1})$. Then
$NF(M_{a,0},M_{a,\alpha^{*}},M_{b,0},M_{b,\alpha^{*}})$. \item $NF$ is closed under isomorphisms: if $NF(M_0,M_1,M_2,M_3)$ and
$f:M_3 \to N_3$ is an isomorphism then
$NF(f[M_0],f[M_1],f[M_2],f[M_3])$.
\end{enumerate}
\end{definition}

\begin{definition}\label{NF respects the frame}
Let $NF$ be a relation such that $\bigotimes_{NF}$ holds. The relation $NF$ \emph{respects the frame} $\frak{s}$ means that if $NF(M_0,M_1,M_2,M_3)$, $a \in M_1-M_0$ and $tp(a,M_0,M_1)$ is basic then $tp(a,M_2,M_3)$ does not fork over $M_0$.
\end{definition}

By Theorem \cite[5.5.4]{jrsh875}
(and Definitions \cite[5.2.1,5.2.6]{jrsh875}):
\begin{fact} \label{jrsh875.5.15}
If the class of uniqueness triples satisfies the existence property
then there is a (unique) relation $NF \subseteq \ ^4K_\lambda$ satisfying $\bigotimes_{NF}$ and respecting the frame $\frak{s}$.
\end{fact}

From now on we assume:
\begin{hypothesis}\label{hypothesis for bar{NF}}
\begin{enumerate}
\item $\frak{s}$ is a semi-good non-forking frame, \item there is a non-forking relation $NF$ repecting $\frak{s}$.
\end{enumerate}
\end{hypothesis}

We define a notion for: a model of size $\lambda$ is independent from a model of size $\lambda^+$ over a model of size $\lambda$ in a model of size $\lambda^+$.
\begin{definition}\label{definition of widehat{NF}}\label{5.14} Define a 4-ary relation $\widehat{NF}$ on $K$ by $$\widehat{NF}(N_0,N_1,M_0,\allowbreak M_1)$$ when the following hold:
\begin{enumerate}
\item $N_0,N_1$ are of cardinality $\lambda$, \item $M_0,M_1$ are of cardinality $\lambda^+$,
\item There are filtrations $\langle
N_{0,\alpha}:\alpha<\lambda^+ \rangle,\ \langle
N_{1,\alpha}:\alpha<\lambda^+ \rangle$ of $M_0,M_1$ respectively, such that
$NF(N_{0,\alpha},N_{1,\alpha},N_{0,\alpha+1},N_{1,\alpha+1})$ holds for every $\alpha<\lambda^+$.
\end{enumerate}
\end{definition}

\begin{fact} [basic properties of $\widehat{NF}$]
\label{5.15}\label{the widehat{NF}-properties} \mbox{}
\begin{enumerate}[(a)] \item Disjointness: If
$\widehat{NF}(N_0,N_1,M_0,M_1)$ then $N_1 \bigcap M_0=N_0$. \item
Monotonicity: Suppose $\widehat{NF}(N_0,N_1,M_0,M_1),\ N_0 \preceq
N^{*}_1 \preceq N_1,\ N_1^* \bigcup M_0 \allowbreak \subseteq
M^*_1 \preceq M_1$ and $M_1^* \in K_{\lambda^+}$. Then
$\widehat{NF}(N_0,N^{*}_1,M_0,M^*_1)$. \item Extension: Suppose
$n<2 \Rightarrow N_n \in K_\lambda,\ M_0 \in K_{\lambda^+},\ N_0
\preceq N_1,\ N_0 \preceq M_0,\ N_1 \bigcap M_0=N_0$. Then there
is a model $M_1$ such that $\widehat{NF}(N_0,N_1,\allowbreak
M_0,M_1)$. \item Weak Uniqueness: If $n<2 \Rightarrow
\widehat{NF}(N_0,N_1,M_0,M_{1,n})$, then there are $M,f_0,f_1$
such that $f_n$ is an embedding of $M_{1,n}$ into $M$ over $N_1
\bigcup M_0$. \item Respecting the frame: Suppose

$\widehat{NF}(N_0,N_1,M_0,M_1)$ and $tp(a,N_0,M_0) \in \allowbreak S^
{bs} \allowbreak (N_0)$. Then $tp(a,N_1,M_1)$ does not fork over
$N_0$.
\end{enumerate}
\end{fact}

Now we define a relation $\preceq^{NF}_{\lambda^+}$ on
$K_{\lambda^+}$, that is based on the relation $\widehat{NF}$:
\begin{definition} \label{6.1}\label{6.4 in april}
Suppose $M_0,M_1 \in K_{\lambda^+}$, $M_0 \preceq M_1$. Then $M_0
\preceq^{NF}_{\lambda^+} M_1$ means that $\widehat{NF}(N_0,N_1,M_0,M_1)$ for some $N_0,N_1 \in
K_\lambda$.
\end{definition}

\begin{fact} \label{6.2}\label{6.5}\label{preceq^{NF}-properties}
$(K_{\lambda^+},\preceq^{NF}_{\lambda^+})$ satisfies the following
properties:
\begin{enumerate}[(a)]
\item Suppose $M_0 \preceq M_1,\ n<2 \Rightarrow M_n \in
K_{\lambda^+}$. For $n<2$, let $\langle
N_{n,\epsilon}:\epsilon<\lambda^+ \rangle$ be a representation of
$M_n$. Then $M_0 \preceq^{NF}_{\lambda^+}M_1$ iff there is a club
$E \subseteq \lambda^+$ such that $(\epsilon<\zeta \wedge
\{\epsilon,\zeta\} \subseteq E) \Rightarrow
NF(N_{0,\epsilon},N_{0,\zeta},N_{1,\epsilon},N_{1,\zeta})$. \item
$\preceq^{NF}_{\lambda^+}$ is a partial order. \item If $M_0
\preceq M_1 \preceq M_2$ and $M_0 \preceq^{NF}_{\lambda^+} M_2$
then $M_0 \preceq^{NF}_{\lambda^+} M_1$. \item
$(K_{\lambda^+},\preceq^{NF}_{\lambda^+})$ satisfies Axiom c of
AEC in $\lambda^+$, i.e.: If $\delta \in \lambda^{+2}$ is a limit
ordinal and $\langle M_\alpha:\alpha<\delta \rangle$ is a
$\preceq^{NF}_{\lambda^+}$-increasing continuous sequence, then
$M_0 \preceq^{NF}_{\lambda^+} \bigcup \{M_\alpha:\alpha<\delta \}$
and obviously it is $\in K_{\lambda^+}$. \item $K_{\lambda^+}$ has
no $\preceq^{NF}_{\lambda^+}$-maximal model.

\end{enumerate}
\end{fact}

\begin{remark}\label{remark preceq^{NF}}
Let $M_1,M_2$ be models of cardinality $\lambda^+$ with $M_1 \preceq M_2$. Then $M_1 \preceq^{NF}_{\lambda^+}M_2$ if and only if for every two filtrations $\langle M_{1,\alpha}:\alpha<\lambda^+ \rangle$ and $\langle M_{2,\alpha}:\alpha<\lambda^+ \rangle$ of $M_1$ and $M_2$ respectively, for some club $E$ of $\lambda^+$ for every $\alpha \in E$ we have $\widehat{NF}(M_{1,\alpha},M_{2,\alpha},M_1,M_2)$.
\end{remark}

\begin{proof}
One direction holds by definition. We prove the hard direction. Suppose $M_1 \preceq^{NF}_{\lambda^+}M_2$. Let $\langle M_{1,\alpha}:\alpha<\lambda^+ \rangle$ and $\langle M_{2,\alpha}:\alpha<\lambda^+ \rangle$ be two filtrations of $M_1$ and $M_2$ respectively. By Fact \ref{preceq^{NF}-properties}(a), for some club $E$ of $\lambda^+$, for every $\epsilon,\zeta \in E$ if $\epsilon<\zeta$ then $NF(M_{1,\epsilon},M_{1,\zeta},M_{2,\epsilon},M_{2,\zeta})$. Let $\alpha \in E$. Then the filtrations $\langle M_{1,\epsilon}:\epsilon \in E-\alpha \rangle$ and $\langle M_{2,\epsilon}:\epsilon \in E-\alpha \rangle$ wittness that $\widehat{NF}(M_{1,\alpha},M_{2,\alpha},M_1,M_2)$.
\end{proof}

\begin{lemma}\label{we can use NF}

For every two models $M,M^+$ of cardinality $\lambda^+$: $$M \preceq M^+ \Leftrightarrow M \preceq^{NF}_{\lambda^+}M^+.$$

\end{lemma}

\begin{proof}
By Fact \ref{preceq^{NF}-properties}(c), it is sufficient to find a model $N \in K_{\lambda^+}$ such that $
M \preceq ^{NF}_{\lambda^+}N$ and $M^+ \preceq N$.

Without loss of generality, $M \neq M^+$ (otherwise, $N=:M$ does).
Let $A$ be the class of pairs, $(M_1,M^+_1)$ of models of cardinality $\lambda^+$ with $M_1 \preceq M^+_1$.
Define a strict partial order, $<_A$
on $A$, by: $(M_1,M^+_1) <_A (M_2,M^+_2)$ when the following hold:
\begin{enumerate}
\item $M_1 \preceq^{NF}_{\lambda^+} M_2$,
\item $M^+_1 \preceq M^+_2$,
\item $M_2 \cap M^+_1 \neq M_1$.
\end{enumerate}

\begin{displaymath}
\xymatrix{M_2 \ar[r]^{id} & M^+_2 \\
M_1 \ar[r]^{id} \ar[u]^{\preceq^{NF}_{\lambda^+}} & M^+_1 \ar[u]^{id}
}
\end{displaymath}

It is sufficient to find a pair $(N,N^+) \in A$ such that $(M,M^+)<_A(N,N^+)$ and $N=N^+$, becuase it yields $M^+ \preceq N$. So by the following claim, it is sufficient to find a pair $(N,N^+) \in A$ such that $(M,M^+)<_A(N,N^+)$ and $(N,N^+)$ is a  $<_A$-maximal pair in $A$.
\begin{claim}\label{importance of maximal}
Let $(N,N^+) \in A$. If $(N,N^+)$ is $<_A$-maximal then $N=N^+$.
\end{claim}

\begin{proof}
Let $(N,N^+)$ be a pair in $A$ with $N \neq N^+$. We should prove that $(N,N^+)$ is not $<_A$-maximal. By density of basic types (in $\lambda^+$), for some $a \in N^+-N$ the type $tp(a,N,N^+)$ is basic. So there is $N^- \in K_\lambda$ such that $N^- \preceq N$ and $tp(a,N,N^+)$ does not fork over $N^-$. For some $N^-_1 \in K_\lambda$ and some $b \in N^-_1$ we have $tp(b,N^-,N^-_1)=tp(a,N^-,N^+)$.
\begin{displaymath}
\xymatrix{b \in N^-_1 \ar[rr]^{f} && N_1 \ar[r]^{g} & N_1^+ \\
N^- \ar[rr]^{id}  \ar[u]^{id} && N \ar[r]^{id} \ar[u]^{\preceq^{NF}_{\lambda^+}}   & N^+ \ni a \ar[u]^{id}
}
\end{displaymath}
By Fact \ref{the widehat{NF}-properties}(c), for some amalgamation $(id \restriction N,f,N_1)$ of $N^-_1$ and $N$ over $N^-$ we have $\widehat{NF}(N^-,N,f[N^-_1],N_1)$. Since the relation $\widehat{NF}$ respects $\frak{s}$, $tp(f(b),N,N_1)$ does not fork over $N^-$. By uniqueness of non-forking (in $\lambda^+$), $tp(a,N,N^+)=tp(f(b),N,N_1)$. Therefore there is an amalgamation $(id \restriction N^+,g,N_1^+)$ of $N^+$ and $N_1$ over $N$ with $g(f(b))=a$. Now we have $$(N,N^+)<_A(g[N_1],N_1^+)$$ (because $N^+ \preceq N_1^+$, $N \preceq^{NF}_{\lambda^+} g[N_1]$ and $a \in g[N_1] \cap N^+-N$).
\end{proof}

\begin{claim}\label{<_A satisfies axiom c}
If $\langle (M_\alpha,M^+_\alpha):\alpha<\delta \rangle$ is a $<_A$-increasing continuous sequence of pairs in $A$ then $(M_\alpha,M^+_\alpha)<_A(\bigcup_{\alpha<\delta}M_\alpha,\bigcup_{\alpha<\delta}M^+_\alpha)$ for each $\alpha<\delta$.
\end{claim}

\begin{proof}
By smoothness, $\bigcup_{\alpha<\delta}M_\alpha \preceq \bigcup_{\alpha<\delta}N_\alpha$. By \cite[Theorem complete...]{jrsh875}, $M_{\lambda^+} \preceq^{NF}_{\lambda^+} \bigcup_{\alpha<\delta}M^\alpha$. By the definition of AEC, $M_{\lambda^{+}+1} \preceq \bigcup_{\alpha<\delta}N_\alpha$.
\end{proof}

Now we can complete the proof of the lemma, using Claim \ref{importance of maximal}. For the sake of a contradiction, assume that there is no $<_A$-maximal pair. We choose by induction on $\alpha<\lambda^{++}$ a pair $(M_\alpha,M^+_\alpha) \in A$ such that for every $\alpha<\lambda^{++}$, $(M_\alpha,M^+_\alpha) <_A (M_{\alpha+1},M^+_{\alpha+1})$ and for every limit $\alpha<\lambda^{++}$, $M_\alpha=\bigcup_{\beta<\alpha}M_\beta$ and $M^+_\alpha=\bigcup_{\beta<\alpha}M^+_\beta$ (so by Claim \ref{<_A satisfies axiom c}, $(M_\beta,M^+_\beta)<_A(M_\alpha,M^+_\alpha)$ for each $\beta<\alpha$). Define $M_{\lambda^{++}}=:\bigcup_{\alpha<\lambda^{++}}M_\alpha$. The sequences $\langle M_\alpha:\alpha<\lambda^{++} \rangle$ and $\langle M^+_\alpha \cap M_{\lambda^{++}}:\alpha<\lambda^{++} \rangle$ are filtrations of $M_{\lambda^{++}}$. So for some $\alpha<\lambda^{++}$ (actually, for a club of $\alpha$'s) we have $M_\alpha=M^+_\alpha \cap M_{\lambda^{++}}$. So $$ M_\alpha \subseteq M^+_\alpha \cap M_{\alpha+1} \subseteq M^+_\alpha \cap M_{\lambda^{++}}=M_\alpha.$$
Therefore $M^+_\alpha \cap M_{\alpha+1}=M_\alpha$, which is impossible, because $(M_\alpha,M^+_\alpha) <_A (M_{\alpha+1},M^+_{\alpha+1})$. This contradiction shows that there is a $<_A$-extension $(N,N^+)$ of $(M,M^+)$, such that $(N,N^+)$ is $<_A$-maximal. So by Claim \ref{importance of maximal}, $N=N^+$. So $M \preceq^{NF}_{\lambda^+}$ and $M^+ \preceq N^+=N$. Lemma \ref{we can use NF} is proved.
\end{proof}

\begin{proposition} \label{a rectangle of models with NF}
Let $\alpha^*,\beta^*$ be two ordinals $\leq \lambda^+$ and let $\langle M_{a,\alpha}:\alpha<\alpha^* \rangle$ and $\langle M_{b,\alpha}:\alpha<\beta^* \rangle$ be two increasing continuous sequence of models of cardinality $\lambda$ such that $M_{a,0}=M_{b,0}$. Then there is a `rectangle of models' $\{M_{\alpha,\beta}:\alpha<\alpha^*,\beta<\beta^* \}$ and a sequence $\{f_\beta:\beta<\beta^* \}$ such that for every $\alpha<\alpha^*$ and $\beta<\beta^*$ the following hold:
\begin{enumerate}
\item $M_{\alpha,\beta} \in K_\lambda$, \item if $\alpha+1<\alpha^*$ then $M_{\alpha,\beta} \preceq M_{\alpha+1,\beta}$, \item if $\beta+1<\beta^*$ then $M_{\alpha,\beta} \preceq M_{\alpha,\beta+1}$, \item if $\alpha$ is a limit ordinal then $M_{\alpha,\beta}=\bigcup_{\alpha'<\alpha}M_{\alpha',\beta}$, \item if $\beta$ is a limit ordinal then $M_{\alpha,\beta}=\bigcup_{\beta'<\beta}M_{\alpha,\beta'}$,
\item $f_\beta$ is an isomorphism of $M_{b,\beta}$ onto $M_{0,\beta}$, \item if $\beta=0$ then $M_{\alpha,\beta}=M_{a,\alpha}$ and $f_\beta$ is the identity on $M_{a,0}=M_{b,0}$, \item $f_{\beta'} \subseteq f_\beta$ for every $\beta'<\beta$, \item if $\beta$ is a limit ordinal then $f_\beta=\bigcup_{\beta'<\beta}f_{\beta'}$, \item $NF(M_{\alpha,\beta},M_{\alpha,\beta+1},M_{\alpha+1,\beta},M_{\alpha+1,\beta+1})$, unless $\alpha+1=\alpha^*$ or $\beta+1=\beta^*$.
\end{enumerate}
\end{proposition}

\begin{displaymath}
\xymatrix{M^a_{\alpha+1} \ar[r]^{=} & M_{\alpha+1,0}
\ar[rrr]^{id} &&& M_{\alpha+1,\beta} \ar[r]^{id} & M_{\alpha+1,\beta+1} \\
M^a_{\alpha} \ar[r]^{=} & M_{\alpha,0} \ar[u]^{id}
\ar[rrr]^{id} &&& M_{\alpha,\beta} \ar[r]^{id} \ar[u]^{id} & M_{\alpha,\beta+1} \ar[u]^{id} \\ \\
M^a_{0} \ar[r]^{=} &  M_{0,0} \ar[uu]^{id}
\ar[rrr]^{id} &&& M_{0,\beta} \ar[r]^{id} \ar[uu]^{id} & M_{0,\beta+1} \ar[uu]^{id} \\
& M^b_0 \ar[u]^{=} &&& M^b_\beta \ar[u]^{f_\beta} & M^b_{\beta+1} \ar[u]^{f_{\beta+1}}
}
\end{displaymath}

\begin{proof}
We $\{M_{\alpha,\beta}:\alpha<\alpha^*\}$ and $f_\beta$ by induction on $\beta$.

\emph{Case a:} $\beta=0$. In this case, see Clause (7).

\emph{Case b:} $\beta$ is a limit ordinal. In this case, by Clause (5), we must choose $M_{\alpha,\beta}=\bigcup_{\beta'<\beta}M_{\alpha,\beta'}$ and by Clause (9), we must choose $f_\beta=\bigcup_{\beta'<\beta}f_{\beta'}$. Fix $\alpha<\alpha^*$. By Clauses (3) and (5) of the induction hypothesis, the sequences $\langle M_{\alpha,\beta'}:\beta'<\beta \rangle$ and $\langle M_{\alpha+1,\beta'}:\beta'<\beta \rangle$ are increasing and continuous. So by smoothness $M_{\alpha,\beta} \preceq M_{\alpha+1,\beta}$. Similarly, $f_\beta[M^b_\beta] \preceq M_{0,\beta}$. It remains to show that if $\alpha$ is limit then $M_{\alpha,\beta}=\bigcup_{\alpha'<\alpha}M_{\alpha',\beta}$. On one hand, if $x \in M_{\alpha',\beta}$ for some $\alpha'<\alpha$ then by the induction hypothesis, $x \in M_{\alpha',\beta'}$ for some $\beta'<\beta$. But $M_{\alpha',\beta'} \subseteq M_{\alpha,\beta'} \subseteq M_{\alpha,\beta}$. On the other hand, if $x \in M_{\alpha,\beta}$ then $x \in M_{\alpha,\beta'}$ for some $\beta'<\beta$. Therefore $x \in M_{\alpha',\beta'}$ for some $\alpha'<\alpha$. But $M_{\alpha',\beta'} \subseteq M_{\alpha',\beta}$ compelte...

\emph{Case c:} $\beta=\gamma+1$. complete...
\end{proof}

\begin{proposition} \label{widehat{NF} with independence}
If the sequence $\langle N_\alpha,a_\alpha:\alpha<\alpha^* \rangle ^\frown \langle N_{\alpha^*} \rangle$ is independent over $N_0$ and $N_0 \preceq N_0^* \in K_{\lambda^+}$ then for some $N_1^* \in K_{\lambda^+}$ and some embedding $f:N_0^* \to N_1^*$, the sequence $\langle a_\alpha:\alpha<\alpha^* \rangle$ is independent in $(f[N_0^*],N_1^*)$ and  $\widehat{NF}(N_0,N_{\alpha^*},f[N_0^*],N_1^*)$.
\end{proposition}

\begin{proof}
For every $\beta<\lambda^+$ and every $\alpha \leq \alpha^*$, we choose $N_{\alpha,\beta} \in K_\lambda$ and an embedding $f_{\alpha,\beta}:N_{\alpha,\beta} \to N_{\alpha,\beta+1}$ such that the following hold:
\begin{enumerate}
\item if $\alpha_1<\alpha_2 \leq \alpha^*$ and $\beta_1 \leq \beta_2$ then complete...
\end{enumerate}

\end{proof}

\begin{proposition}\label{widehat{NF} implies independence}\label{widehat{NF} respects independence}
Let $\alpha^*<\lambda^+$. If $\widehat{NF}(M_0^-,M_1^-,M_0,M_1)$ and $\langle a_\alpha:\alpha<\alpha^* \rangle$ is independent in $(M_0^-,M_1^-)$ then it is independent in $(M_0,M_1)$.
\end{proposition}

\begin{proof}
The idea is to find an amalgam $M_2$ of $M_1^-$ and $M_0$ over $M^-_0$ such that $\widehat{NF}$ holds and the sequence is independent in $(M_0,M_2)$. It is sufficient since the relation $\widehat{NF}$ satisfies weak uniqueness (by complete...).

We elavorate.
By the definition of independence, there is an increasing continuous sequence $\langle N_\alpha:\alpha \leq \alpha^* \rangle$ of models of cardinality $\lambda$ such that $N_0=M_0^-$, $M_1^- \preceq N_{\alpha^*}$ and the sequence $\langle M_\alpha,a_\alpha:\alpha<\alpha^* \rangle ^\frown \langle M_{\alpha^*} \rangle$ is independent over $M_0^-$. By Proposition \ref{widehat{NF} with independence}, for some $N_1^* \in K_{\lambda^+}$ the sequence $\langle a_\alpha:\alpha<\alpha^* \rangle$ is independent in $(M_0,N_1^*)$ and  $\widehat{NF}(N_0,N_{\alpha^*},M_0,N_1^*)$.

But $\widehat{NF}(M_0^-,M_1^-,M_0,M_1)$. Therefore by the weak uniqueness of $\widehat{NF}$ (Fact \ref{complete...}) complete...

complete...
\end{proof}

\begin{theorem}\label{continuity of serial independence}\label{if the class of uniqueness triples satisfies the existence property then continuity}
The $\lambda^+$-continuity of serial independence property holds.
\end{theorem}

\begin{proof}
Let $\beta^*<\lambda^+$, $M_1 \in K_{\lambda^+}$, $M_1 \preceq M_2 \in K_{\lambda^+}$ and let $\langle M_{1,\alpha}:\alpha<\lambda^+ \rangle$ be a filtration of $M_1$. Suppose $\langle a_\beta:\beta<\beta^* \rangle$ is independent in $(M_{1,\alpha},M_2)$ for each $\alpha<\lambda^+$. We have to prove that $\langle a_\beta:\beta<\beta^* \rangle$ is independent in $(M_1,M_2)$.

By Lemma \ref{we can use NF}, $M_1 \preceq^{NF}_{\lambda^+}M_2$. Let $\langle M_{2,\alpha}:\alpha<\lambda^+ \rangle$ be a filtration of $M_2$. By Remark \ref{remark preceq^{NF}}, there is a club $E$ of $\lambda^+$ such that for every $\alpha \in E$, $\widehat{NF}(M_{1,\alpha},M_{2,\alpha},M_1,M_2)$.   Define $J=:\{a_\beta:\beta<\beta^*\}$. $J \subseteq M_2$. Since $|J|<\lambda^+$, for some $\alpha \in E$ we have $J \subseteq M_{2,\alpha}$. But

$\widehat{NF}(M_{1,\alpha},N_{2,\alpha},M_1,M_2)$. So by Proposition \ref{widehat{NF} implies independence}, $\langle a_\beta:\beta<\beta^* \rangle$ is independent in $(M_1,M_2)$.
\end{proof}

\begin{proposition}\label{continuity implies symmetry}
Suppose:
\begin{enumerate}
\item $\frak{s}=(K,\preceq,S^{bs},\dnf)$ is a semi-good $\lambda$-frame \item $(K,\preceq)$ satisfies the amalgamation property in $[\lambda,\mu]$ \item $(K,\preceq)$ satisfies $(\lambda,\mu)$-tameness \item $\frak{s}$ satisfies the continuity of serial independence property.
\end{enumerate}
Then $\frak{s}^{\mu}$ satisfies symmetry.
\end{proposition}

\begin{proof}

Suppose $N_0,N_1,N_2$ are models of cardinality $\mu$, $N_0 \preceq N_1 \preceq N_2$, the type $tp(a,N_0,N_1)$ is basic and $tp(b,N_1,N_2)$ does not fork over $N_0$. Let $\langle M_{0,\alpha}:\alpha<\lambda^+ \rangle$, $\langle M_{1,\alpha}:\alpha<\lambda^+ \rangle$, $\langle M_{2,\alpha}:\alpha<\lambda^+ \rangle$ be filtrations of $N_0,N_1,N_2$ respectively such that $M_{0,\alpha} \preceq M_{1,\alpha} \preceq M_{2,\alpha}$ for each $\alpha<\lambda^+$. Without loss of generality, the types $tp(a,N_0,N_1)$, $tp(b,N_0,N_2)$ do not fork over $M_{0,0}$, $a \in M_{1,0}$ and $b \in M_{2,0}$.

The sequence $\langle a,b \rangle$ is independent in $(M_{0,0},M_{0,\alpha},M_{2,\alpha})$ for each $\alpha<\lambda^+$.
So by symmetry in $\frak{s}$, the sequence $\langle b,a \rangle$ is independent in $(M_{0,0},M_{0,\alpha},M_{2,\alpha})$ for each $\alpha<\lambda^+$. By Theorem \ref{continuity of serial independence} (the $\lambda^+$-continuity of serial independence property), $\langle b,a \rangle$ is independent in $(N_0,N_2)$.
\end{proof}

\begin{corollary}\label{corollary continuity implies good-frame}
Suppose:
\begin{enumerate}
\item $\frak{s}=(K,\preceq,S^{bs},\dnf)$ is a semi-good $\lambda$-frame \item $(K,\preceq)$ satisfies the amalgamation property in $[\lambda,\mu]$
\item $(K,\preceq)$ satisfies $(\lambda,\lambda^+)$-tameness
\item $\frak{s}$ satisfies the continuity of serial independence property.
\end{enumerate}
Then $\frak{s}^{+}$ is a good $\lambda^+$-frame.
\end{corollary}

\begin{proof}
By Proposition \ref{continuity implies symmetry} and Theorem \ref{symmetry implies good-frame}.
\end{proof}

\begin{proof}
By \ref{continuity implies symmetry}, symmetry holds. complete...
\end{proof}

In complete... we generalize the following corollary, eliminating the tameness assumption:
\begin{corollary}
Suppose:
\begin{enumerate}
\item $\frak{s}=(K,\preceq,S^{bs},\dnf)$ is a semi-good $\lambda$-frame \item $(K,\preceq)$ satisfies the amalgamation property in $[\lambda,\mu]$
\item $(K,\preceq)$ satisfies $(\lambda,\mu)$-tameness
\item the class of uniqueness triples satisfies the existence property.
\end{enumerate}
Then $\frak{s}^{\mu}$ is a good $\lambda$-frame.
\end{corollary}

\begin{proof}
By Theorem \ref{if the class of uniqueness triples satisfies the existence property then continuity} and  Corollary \ref{corollary continuity implies good-frame}
\end{proof}

\section{Combining Uniqueness Triples with the Amalgamation Property in $\lambda^+$}

\begin{theorem}
If
\begin{enumerate}
\item the class of uniqueness triples satisfies the existence property,
\item the amalgamation property in $\lambda^+$ holds.
\end{enumerate}
The $\frak{s}^+$ is a good non-forking $\lambda^+$-frame.
\end{theorem}

We give two proofs. The first one is similar to the proof  Corollary \ref{complete...}, using a replacement of $(\lambda,\lambda^+)$-tameness.
\begin{proof}
By complete... it is sufficient to prove uniqueness.
\end{proof}

Now we give a second proof, using results from \cite{jrsh875}:
\begin{proof}
By \cite[complete...]{jrsh875}, it is sufficient to prove that the relation $\preceq^{NF}_{\lambda^+}$ satisfies smoothness. But by Lemma \ref{we can use NF}, the relation $\preceq^{NF}_{\lambda^+}$ is equivalent to $\preceq \restriction K_{\lambda^+}$, so by the definition of AEC, it satisfies smoothness.
\end{proof}

\section{Solving the Goodness$^+$ Question}
Some years ago Shelah asked whether for every successful good $\lambda$-frame, $$M_1 \preceq M_2 \Leftrightarrow M_1 \preceq^{NF}_{\lambda^+} M_2$$
holds for every two models $M_1,M_2$ which are saturated in $\lambda^+$ over $\lambda$.

In \cite{shh}.last,complete... he proved it under the assumption that there are no many models of cardinality complete..
Here we solve this open question:
\begin{corollary}
complete...
\end{corollary}

\begin{proof}
Since we restrict ourselves to the saturated models in $\lambda^+$ over $\lambda$, we have categoricity in $\lambda^+$. But we assume $I(\lambda^{++},K)<2^{\lambda^{++}}$. Hence, by Fact \ref{complete...}, we have amalgamation in $\lambda^+$. So by Lemma \ref{we can use NF}, the relations $\preceq \restriction K_{\lambda^+}$ and $\preceq^{NF}_{\lambda^+}$ are equivalent.
\end{proof}

\section{Dimenstion}
In \cite{jrsi3} we proved that if the continuity property holds then the dimension is well-behaved. First note that the continuity property in \cite{jrsi3} relates to independence of sets, while the continuity property here relates to serial independence. In this sense, we study here a stronger continuity property.

While in \cite{jrsi3}, we study $\frak{s}$ (relating to models of cardinality $\lambda$), here we study $\frak{s}^+$. While in \cite{jrsi3}, we assume that the class of uniqueness triples (in $\lambda$) satisfies the existence property, we do not assume the same for $\lambda^+$ (but assume the same assumption). This cause a difficulty.

But by the $\lambda^+$-continuity of serial independence property, we can prove that the dimension is semi-well-behaved in the following sense:
\begin{theorem}
Suppose: complete..., and the class of uniqueness triples in $\lambda$ satisfies the existence property. Let $M,N$ be models of cardinality $\lambda^+$ with $M \preceq N$ and let $J_1,J_2$ be two maximal independent sets in $(M,N)$. Then one of the following hold:
\begin{enumerate}
\item $J_1,J_2$ are finite sets, \item $|J_1|+|J_2|=\lambda^+$ or \item $|J_1|=|J_2|$.
\end{enumerate}
\end{theorem}

\begin{proof}
complete...
\end{proof}


TITLE: Proof check for condition on being measurable
QUESTION [2 upvotes]: I was wondering if the following proof works to prove the following statement:
Prove the following are equivalent for $E \subseteq \mathbb{R}$, where $m$ denotes the Lebesgue outer measure.
(i) $E$ is measurable (definition taken to be satisfying the Caratheodory criterion).
(ii) $m(B) = m(B \cap E) + m(B \cap E^c)$ for all Borel sets $B$.
(iii) $m(U) = m(U \cap E) + m(U \cap E^c)$ for all open sets $U$.
Proof:
(i) $\Rightarrow$ (ii), (i) $\Rightarrow$ (iii), (ii) $\Rightarrow$ (iii) are all trivial. (iii) $\Rightarrow$ (ii) follows from the fact that the $\sigma$-algebra generated by the open sets is equal to the Borel algebra. To see (iii) $\Rightarrow$ (i), cover $A$ by open intervals $I_k$ such that $m(A) + \epsilon \geq \sum_k I_k $, then we have:
$$
m(A) + \epsilon \geq m(\cup_k I_k) = m(\cup_k I_k \cap E) + m(\cup_k I_k \cap E^c)
$$
by our hypothesis, but then since $A \subseteq \cup_k I_k$ we have that $m(\cup_k I_k \cap E) \geq m(A \cap E)$ and likewise for $E^c$ by monotonicity, thus giving:
$$
m(A) + \epsilon \geq m(A \cap E) + m(A \cap E^c)
$$
Letting $\epsilon \rightarrow 0$ thus gives the inequality and hence the result (reverse inequality trivially follows from countable sub-additivity). This completes the proof.
I'm fairly new to measure theory so any feedback, corrections or help would be greatly appreciated!

REPLY [1 votes]: You have to delete the sentence 'iii) implies ii) follows from the fact that the sigma algebra generated by the open sets is equal to the Borel sigma algebra'. This is not clear and it is not required. [For this to work you have to prove that the class of all sets for which the equation in ii) holds is a sigma algebra].  Once you prove that iii) implies i) you also have  iii) implies ii) because i) implies ii). Otherwise, your proof is fine.


text:
section \<open>Partial Function Package Setup\<close>
theory Refine_Pfun
imports Refine_Basic Refine_Det
begin

text \<open>
  In this theory, we set up the partial function package to be used
  with our refinement framework.
\<close>

subsection \<open>Nondeterministic Result Monad\<close>

interpretation nrec:
  partial_function_definitions "(\<le>)" "Sup::'a nres set \<Rightarrow> 'a nres"
  by unfold_locales (auto simp add: Sup_upper Sup_least)

lemma nrec_admissible: "nrec.admissible (\<lambda>(f::'a \<Rightarrow> 'b nres).
  (\<forall>x0. f x0 \<le> SPEC (P x0)))"
  apply (rule ccpo.admissibleI)
  apply (unfold fun_lub_def)
  apply (intro allI impI)
  apply (rule Sup_least)
  apply auto
  done

(*thm fixp_induct_option

lemma fixp_induct_nrec:
  fixes F :: "'c \<Rightarrow> 'c" and
    U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a nres" and
    C :: "('b \<Rightarrow> 'a nres) \<Rightarrow> 'c" and
    P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"
  assumes mono: "\<And>x. nrec.mono_body (\<lambda>f. U (F (C f)) x)"
  assumes eq: "f \<equiv> C (nrec.fixp_fun (\<lambda>f. U (F (C f))))"
  assumes inverse2: "\<And>f. U (C f) = f"
  assumes step: "\<And>f x z. (\<And>x. U f x = U f x; Q x z) \<Longrightarrow> Q (U (F (C f))) z"
 (* assumes defined: "RETURN y \<le> U (C f) x"*)
  assumes Q: "\<And>x z. Q x z \<longleftrightarrow> z = U f x \<and> U f x \<le> SPEC (P x)"
  shows "Q x z" oops
  (*using step defined
    nrec.fixp_induct_uc[of U F C, OF mono eq inverse2 nrec_admissible]
  unfolding Q oops (*
  by blast*)*)

lemma fixp_induct_nrec':
  fixes F :: "'c \<Rightarrow> 'c" and
    U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a nres" and
    C :: "('b \<Rightarrow> 'a nres) \<Rightarrow> 'c" and
    P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"
  assumes mono: "\<And>x. nrec_mono (\<lambda>f. U (F (C f)) x)"
  assumes eq: "f \<equiv> C (nrec_ffix (\<lambda>f. U (F (C f))))"
  assumes inverse2: "\<And>f. U (C f) = f"
  assumes step: "\<And>f x0. (\<And>x0. U f x0 \<le> SPEC (P x0))
    \<Longrightarrow> U (F f) x0 \<le> SPEC (P x0)"
  assumes defined: "RETURN y \<le> U f x"
  shows "P x y"
proof -
  note defined
  also have "\<forall>x0. U f x0 \<le> SPEC (P x0)"
    apply (rule nrec.fixp_induct_uc[of U F C, OF mono eq inverse2
      nrec_admissible])
    using step by blast
  hence "U f x \<le> SPEC (P x)" by simp
  finally show "P x y" by auto
qed
*)
(* TODO/FIXME: Induction rule seems not to work here !
    Function package expects induction rule where conclusion is a binary
    predicate as free variable.
*)

declaration \<open>Partial_Function.init "nrec" @{term nrec.fixp_fun}
  @{term nrec.mono_body} @{thm nrec.fixp_rule_uc} @{thm nrec.fixp_induct_uc}
  (*SOME @{thm fixp_induct_nrec}*) (NONE)\<close>

lemma bind_mono_pfun[partial_function_mono]:
  fixes C :: "'a \<Rightarrow> ('b \<Rightarrow> 'c nres) \<Rightarrow> ('d nres)"
  shows
  "\<lbrakk> monotone (fun_ord (\<le>)) (\<le>) B;
    \<And>y. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. C y f) \<rbrakk> \<Longrightarrow>
     monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. bind (B f) (\<lambda>y. C y f))"
  apply rule
  apply (rule Refine_Basic.bind_mono)
  apply (blast dest: monotoneD)+
  done

subsection \<open>Deterministic Result Monad\<close>

interpretation drec:
  partial_function_definitions "(\<le>)" "Sup::'a dres set \<Rightarrow> 'a dres"
  by unfold_locales (auto simp add: Sup_upper Sup_least)

lemma drec_admissible: "drec.admissible (\<lambda>(f::'a \<Rightarrow> 'b dres).
  (\<forall>x. P x \<longrightarrow>
    (f x \<noteq> dFAIL \<and>
    (\<forall>r. f x = dRETURN r \<longrightarrow> Q x r))))"
proof -
  have [simp]: "fun_ord ((\<le>) ::'b dres \<Rightarrow> _ \<Rightarrow> _) = (\<le>)"
    apply (intro ext)
    unfolding fun_ord_def le_fun_def
    by (rule refl)

  have [simp]: "\<And>A x. {y. \<exists>f\<in>A. y = f x} = (\<lambda>f. f x)`A" by auto

  show ?thesis
    apply (rule ccpo.admissibleI)
    apply (unfold fun_lub_def)
    apply clarsimp
    apply (drule_tac x=x in point_chainI)
    apply (erule dres_Sup_chain_cases)
    apply auto
    apply (metis (poly_guards_query) SUP_bot_conv(1))
    apply (metis (poly_guards_query) SUP_bot_conv(1))
    apply metis
    done
qed

declaration \<open>Partial_Function.init "drec" @{term drec.fixp_fun}
  @{term drec.mono_body} @{thm drec.fixp_rule_uc} @{thm drec.fixp_induct_uc}
  NONE\<close>

lemma drec_bind_mono_pfun[partial_function_mono]:
  fixes C :: "'a \<Rightarrow> ('b \<Rightarrow> 'c dres) \<Rightarrow> ('d dres)"
  shows
  "\<lbrakk> monotone (fun_ord (\<le>)) (\<le>) B;
    \<And>y. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. C y f) \<rbrakk> \<Longrightarrow>
     monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. dbind (B f) (\<lambda>y. C y f))"
  apply rule
  apply (rule dbind_mono)
  apply (blast dest: monotoneD)+
  done

end

subset_name: curated


TITLE: Finding n from a Poisson Approximation
QUESTION [0 upvotes]: Q. In a certain population, 1 per cent of sheep are brown. How large must a
      random sample be to ensure that the probability of including at least brown sheep
      is at least 0.95? (Assume that sampling is with replacement, and use a
      Poisson approximation.)

I am familiar with using the Poisson distribution to obtain a probability, but this seems to be asking me to do the reverse and I can't wrap my head around it. My thought process was:
success = brown sheep
Pr(>=1 success out of n trials) >= 0.95
x >= 1
lambda = np = 0.01n
Solve for n using the Poisson equation. But this is algebraically tricky (at my level anyway) and this question was worth very few marks. What am I missing?
Thanks so much.

REPLY [0 votes]: Suppose you are drawing $n$ sheeps randomly, with replacement. If $X$ be the no. of brown sheeps in your random sample, then $X\sim \text{Bin}(n, 0.01).$ And we wish to find (minimum) $n$ such that $\Pr(X\ge 1) \ge 0.95,$ i.e. $\Pr(X=0)\le 0.05.$ It seems that $n$ should be large enough, so we may approximate the distribution of $X$ by a Poisson distribution with mean $\lambda = 0.01n.$ Now, $\Pr(X=0)=e^{-\lambda},$ so we require $e^{-0.01n}\le 0.05$ which gives $n\ge -\frac{\ln(0.05)}{0.01}\approx 299.57,$ i.e. $n\ge 300.$


TITLE: On a substitution to solve a high order differential equation (exercise).
QUESTION [0 upvotes]: I have solved Linear first order differential equations, I know how to separate variables, for higher orders I know the the substitutions that apply to the bernoulli equation and homogeneous equations.
The exercise asks to solve:
$$y^{(4)}+ 4y^{(2)}+4y = cos2t - sin2t$$
I am given a hint that : $$r^4+4r^2 + 4r^0 = 0 \implies (r^2+2)^2 = 0 \implies r = \pm i \sqrt{2}$$
Could I have a reference to how to solve these kind of differential equations? Or am I just missing a trick that makes this a first order?
I tried thinking about substituting Eulers formula in the left hand side but I am missing an $i$.

REPLY [1 votes]: hint: use the fact that $e^{i\sqrt{2}t}=\cos(\sqrt{2}t)+i\sin(\sqrt{2}t)$


text:
\section{\sc\bf Preview of the basic notions}

  \subsection{Admissible and Carter diagrams, solid and dotted edges}

  \subsubsection{The primary root system}
    \label{sec_fulcrum}
    \index{primary root system}
    \index{root system ! - primary}
   We consider a root system $\varPhi$, the finite Weyl group $W$ acting on $\varPhi$,
   the diagram Dynkin $\Gamma_{D}$ associated with $\varPhi$ and the corresponding Cartan matrix {\bf B}.
   The quadratic form $\mathscr{B}$ is associated with the Cartan matrix {\bf B} and $\mathscr{B}$
   determines the inner product $(\cdot,\cdot )$  on the linear space $V$ spanned
   by simple roots of $\varPhi$. Inside the root system $\varPhi$ we will consider different
   root subsets and root subsystems, so we call $\varPhi$ the {\it primary root system}.
   There exist root subsets which can be embedded into different primary root systems. For example,
   $\varPhi(A_7) \subset \varPhi(A_8), \varPhi(D_8), \varPhi(E_8)$.
   In this paper, besides the quadruple of objects $\{\varPhi, W, \Gamma_{D}, {\bf B}\}$ we consider also
   a certain its generalization, the  quadruple $\{\mathcal{P}, W^{\vee}, \Gamma, B_{\Gamma}\}$,
   where $\mathcal{P}$ (resp. $W^{\vee}$, resp. $B_{\Gamma}$) is the {\it partial root system},
   (resp. partial Weyl group, partial Cartan matrix) and $\Gamma$ is the Carter diagram.
   These notions will be explained shortly. The generalized quadruple is our object of interest.
   And all this is studied in the area which is the primary root system $\varPhi$.

   \subsubsection{The admissible diagram}
  \label{sec_adm_diagr}
  \index{root system}

 In $1972$, R.~Carter introduced {\it admissible diagrams} to classify conjugacy classes in a finite Weyl group $W$.
 These diagrams are also used to characterize elements of the Weyl group, see definition in \S\ref{sec_Carter}.
 Each element $w \in W$ can be expressed in the form
 \begin{equation}
   \label{any_roots_0}
    w  = s_{\tau_1} s_{\tau_2} \dots s_{\tau_k}, \text{ where } \tau_i \in \varPhi,
 \end{equation}
 and $s_{\tau_i} \in W$ are reflections
 corresponding to not necessarily simple roots $\tau_i \in \varPhi$.

 Carter proved that $k$ in the decomposition \eqref{any_roots_0} is
 the smallest if and only if the subset of roots $\{\tau_1, \tau_2, \dots, \tau_k\}$ is linearly independent; such a
 decomposition is said to be {\it reduced}.
  We denote by $l_C(w)$ the smallest value $l$ in any expression like
 \eqref{any_roots_0}, we have $l_C(s_{\tau_1} s_{\tau_2} \dots s_{\tau_l}) = l$.
 The set $S = \{\tau_1,\dots,\tau_l\}$ consists of linearly
 independent and \underline{not necessarily simple} roots, see \cite[Lemma $3$]{Ca72}.
 We always have $l_C(w) \leq l(w)$, where $l(w)$ is the smallest value
 $k$ in any expression like \eqref{any_roots_0} such that all roots $\alpha_i$ are \underline{simple}.

 \begin{lemma}{\cite[Lemma 3]{Ca72}}
  \label{lem_lin_indep}
   Let $\alpha_1, \alpha_2, \dots, \alpha_k \in \varPhi$.
   Then $s_{\alpha_1} s_{\alpha_2} \dots s_{\alpha_k}$ is reduced
   if and only if $\alpha_1, \alpha_2, \dots, \alpha_k$ are linearly
   independent. \qed
 \end{lemma}

 We associate with the element $w \in W$ (and with conjugacy class $C(w)$ containing $w$) the diagram
 $\Gamma$ with nodes corresponding to roots in \eqref{any_roots_0}, nodes $\alpha$ and $\beta$  are joined
 by  $n_{{\alpha}{\beta}}\cdot n_{{\beta}{\alpha}}$ bonds, where
\begin{equation}
   \label{n_sr}
       n_{{\alpha}{\beta}} = 2\frac{(\alpha,\beta)}{(\alpha,\alpha)}, \quad
       n_{{\beta}{\alpha}} = 2\frac{(\beta,\alpha)}{(\beta,\beta)}.
 \end{equation}
Number $n_{{\alpha}{\beta}}$ and $n_{{\beta}{\alpha}}$  are integer, and $n_{{\alpha}{\beta}} \cdot n_{{\beta}{\alpha}} \in \{0, 1, 2, 3\}$,
see \cite[Ch.6, \S1, ${n^o}$3]{Bo02}.

\index{admissible diagram}
\index{diagram! - admissible diagram}
 A diagram $\Gamma$ is said to be {\it admissible}, see \cite[p. 7]{Ca72}, if
\begin{equation}
 \label{eq_def_adm}
 \begin{split}
 & (a)  \text{ The nodes of } \Gamma \text{ correspond to a set of linearly independent roots in } \varPhi. \\
 & (b) \text{
  If a subdiagram of } \Gamma \text{ is a cycle, then it contains an even number of nodes. }
 \end{split}
\end{equation}

 Surprisingly,
 the admissible diagrams can contain cycles though the extended Dynkin
 diagram $\widetilde{A}_l$ cannot be a part of any admissible diagram, see Lemma \ref{lem_must_dotted}.
 It turned out that the cycles in admissible diagrams essentially differ from the cycle $\widetilde{A}_l$.
 Namely, in such a cycle, there exist necessarily two pairs of roots: A pair with a
 positive inner product and a pair with a negative inner product. This does not happen for $\widetilde {A}_l$.

 \subsubsection{Solid and dotted edges}
  \label{sec_solid_dotted}
 \index{dotted ! - edge}
 \index{solid ! - edge}
 This observation motivated us to distinguish such pairs of roots:
 Let us draw the {\it dotted} (resp. {\it solid}) edge $\{\alpha, \beta\}$ if
 $(\alpha, \beta) > 0$ (resp. $(\alpha, \beta) < 0$), see Fig. \ref{E6a1_exam_linkages}.
 The diagrams with properties of admissible diagrams and containing
 dotted edges are said to be {\it Carter diagrams}. Up to dotted edges,
 the classification of Carter diagrams coincides with the
 classification of admissible diagrams.

  Recall that $(\alpha, \beta) > 0$
 (resp. $(\alpha, \beta) < 0$) means that the angle between
 roots $\alpha$ and $\beta$ is acute (resp. obtuse).
  Recall that, for the Dynkin diagrams, all angles between simple roots
  are obtuse, all edges are solid, and no special designation is necessary.
  A {\it solid edge} indicates an obtuse angle between roots exactly as for simple roots
  in the case of Dynkin diagrams. A {\it dotted edge} indicates an acute angle between the
  roots considered.

 \subsubsection{The $\Gamma$-associated root subset}
   \label{sec_associated}
   \index{$S$, $\Gamma$-associated root subset}
   \index{$\Gamma$-associated root subset}
   \index{primary root system}
   \index{root system ! - primary}

  For any Carter diagram $\Gamma$ and a primary root system $\varPhi$,
  let $\Gamma'$ be a Dynkin diagram such that $\rank(\Gamma') = \rank(\Gamma) + 1$,
  $\varPhi(\Gamma') \subseteq \varPhi$, where $\varPhi(\Gamma')$ is the root system associated with $\Gamma'$.
  Consider a subset $S \subset \varPhi(\Gamma')$ of linearly independent roots with following properties:
  Vertices of $\Gamma$ are in one-to-one correspondence with roots of $S$;
  solid (resp. dotted) edges of $\Gamma$ are in one-to-one correspondence with pairs
  $\{\alpha, \beta\}$, where $\alpha, \beta \in S$, such that $(\alpha, \beta) < 0$
  (resp. $(\alpha, \beta) > 0$. The root subset $S$ is said to be {\it $\Gamma$-associated}.

  \begin{remark}
  {\rm (i) Emphasize that whenever we say that $S$ is the $\Gamma$-associated root subset,
 we mean that elements of $S$ are taken from $\varPhi(\Gamma') \subseteq \varPhi$ , i.e.,
 they are taken from the primary root system $\varPhi$, see \S\ref{sec_fulcrum}.

 (ii) Not necessarily for every Carter diagram $\Gamma$ there exists a set $S$
 with a given properties. For example, there is no $E_6$-associated root subset in the primary root subset $\varPhi = \varPhi(D_n)$
 for any integer $n$, see Lemma \ref{lem_E6_not_in_Dn}.

 }
  \end{remark}

 \subsubsection{The Carter diagrams}
  \label{sec_Carter}

 \index{Carter diagram}
 \index{diagram! - Carter diagram}
 \index{$\Gamma$-associated set of roots}
 \index{$\Gamma$-associated element}
 \index{$S$-associated element}
 \index{$S$ (set of linearly independent roots)}
 \index{$S_{\alpha}$ (subset $\{\alpha_i \mid i = 1,\dots,k\}$)}
 \index{$S_{\beta}$ (subset $\{\beta_j \mid j = 1,\dots,h\}$)}
 \index{Weyl group}
  Any admissible diagram $\Gamma$ is said to be a {\it Carter diagram}
  if any edge connecting a pair of roots $\{\alpha, \beta\}$ with
  inner product $(\alpha, \beta) > 0$ (resp. $(\alpha, \beta) < 0$)
  is drawn as dotted (resp. solid) edge. There is no edge for the inner product $(\alpha, \beta) = 0$. Let
\begin{equation}
   \label{eq_alpha_bet}
     S = \{\alpha_1, \alpha_2, \dots, \alpha_k, \beta_1, \beta_2, \dots, \beta_h \}
 \end{equation}
 be any $\Gamma$-associated set (of not necessarily simple roots),
 where roots of the set $S_{\alpha} := \{\alpha_i \mid i = 1,\dots,k\}$ (resp. $S_{\beta} := \{\beta_j \mid j = 1,\dots,h\}$) are
 mutually orthogonal.
 According to (\ref{eq_def_adm}$(a)$), there exists the set \eqref{eq_alpha_bet} of linearly independent roots.
 Thanks to (\ref{eq_def_adm}$(b)$), such a partition into the sum of two mutually orthogonal sets $S_{\alpha}$
 and $S_{\beta}$ is possible. The partition \eqref{eq_alpha_bet} is said to be {\it bicolored partition}.
 Let $w = w_1 w_2$ be the decomposition of $w$ into the product of two involutions.
 By \cite[Lemma 5]{Ca72} each of $w_1$ and $w_2$ can
 be expressed as a product of reflections corresponding to mutually
 orthogonal roots as follows:
 \begin{equation}
   \label{two_invol}
      w = w_1{w}_2, \quad \text{ where } \quad
      w_1 = s_{\alpha_1} s_{\alpha_2} \dots s_{\alpha_k}, \quad
      w_2 = s_{\beta_1} s_{\beta_2} \dots s_{\beta_h}.
 \end{equation}
 Since $S$ is linearly independent, the decomposition
 \eqref{two_invol} is reduced, see Lemma \ref{lem_lin_indep}, and $k + h = l_C(w)$.
 The decomposition \eqref{two_invol}
 is said to be a {\it bicolored decomposition}.
 The subset of roots corresponding to $w_1$ (resp. $w_2$) is said to be {\it $\alpha$-set} (resp. {\it $\beta$-set}):
 \index{bicolored decomposition}
\begin{equation}
   \label{two_sets}
       \alpha\text{-set} = \{ \alpha_1, \alpha_2, \dots, \alpha_k \}, \quad
       \beta\text{-set} =  \{ \beta_1, \beta_2, \dots, \beta_h \}.
 \end{equation}

 \index{$w = w_1{w}_2$, bicolored decomposition of $w$; $w_1$, $w_2$ - involutions}
 \index{$\alpha$-set, subset of roots corresponding to $w_1$}
 \index{$\beta$-set, subset of roots corresponding to $w_2$}
 \index{$\alpha$-label, coordinate out of $\alpha$-set}
 \index{$\beta$-label, coordinate out of $\beta$-set}
 \index{label ! - $\alpha$-label}
 \index{label ! - $\beta$-label}
 \index{$k$, number of coordinates in $\alpha$-set (= number of $\alpha$\text{-labels})}
 \index{$h$, number of coordinates in $\beta$-set (= number of $\beta$\text{-labels})}
 \index{$l$, number of vertices in the Carter diagram $\Gamma$, $l = k + h$}

 Up to the difference between solid and dotted edges,
 the Carter diagram $\Gamma$ is the admissible diagram.
 Any coordinate of a given vector of linkage labels from an $\alpha$-set (resp. a $\beta$-set)
 we call an {\it $\alpha$-label} (resp. a {\it $\beta$-label}).
 Let $L \subset V$ be the linear subspace spanned by root subsets \eqref{eq_alpha_bet}.
 We denote by $\Pi_w$ the corresponding root basis, whose elements
 are not necessarily simple roots:
 \index{$\Pi_w$, root subset associated with the bicolored decomposition of $w$}
 \begin{equation}
   \label{root_subset_L}
       \Pi_w = \{ \alpha_1, \alpha_2, \dots, \alpha_k, \beta_1, \beta_2, \dots, \beta_h \}.
 \end{equation}

\subsubsection {Three classes of Carter diagrams}
   \label{sec_3_class}
 \index{$W$, Weyl group}
 \index{Weyl group ! - $W$}
 \index{class of diagrams ! - $\mathsf{C4}$}
 \index{class of diagrams ! - $\mathsf{DE4}$}
 \index{${\bf B}$, Cartan matrix associated with a Dynkin diagram}
 \index{Cartan matrix ! - ${\bf B}$}
 \index{label ! - linkage label vector}
 \index{class of diagrams ! - simply-laced connected Carter diagrams}
 \index{$\Gamma$, Carter diagram}
 \index{uniqueness of the associated conjugacy class}
 \index{uniqueness theorem}

  We consider three classes of simply-laced connected Carter diagrams:
  Denote the class of diagrams containing $4$-cycle $D_4(a_1)$ by $\mathsf{C4}$;
  the class of diagrams without cycles and containing $D_4$ (as a subdiagram) by $\mathsf{DE4}$,
  i.e., $\mathsf{DE4}$ is the class consisting of Dynkin diagrams  $E_6$, $E_7$, $E_8$ and $D_l$ for $l \geq 4$;
  the class of Dynkin diagrams $A_l$ by $\mathsf{A}$.
  Conjugate elements in the Weyl group $W$ are associated with the same Carter diagram $\Gamma$.
  For any $\Gamma \in \mathsf{C4} \coprod \mathsf{DE4}$, the {\it uniqueness of the associated conjugacy class} takes place,
  see \cite[Theorem 6.5]{St10}.

\begin{remark}
  \label{two_class_Al}
{\rm
    Note that the uniqueness theorem does not hold for $A_l$.
    There are two conjugacy classes of type $A_5$ in $W(E_7)$ and in $W(D_6)$,
    two conjugacy classes of type $A_7$ in $W(E_8)$, see \cite[p. 31]{Ca72} and \cite[Lemma 27]{Ca72}.
    Moreover, for the Carter diagram $A_3$, there exist three conjugacy classes in $W(D_4)$.
   } \qed
\end{remark}

\subsection{Linkage diagrams and linkage systems}

 \subsubsection{The partial Cartan matrix $B_{\Gamma}$}
    \label{sec_partial_B}
    \index{$B_{\Gamma}$, partial Cartan matrix}
    \index{Cartan matrix ! - partial Cartan matrix $B_{\Gamma}$}

  Similarly to the Cartan matrix associated with Dynkin diagrams, we determine the Cartan matrix
  for each Carter diagram $\Gamma$ as follows

 \begin{equation}
   \label{canon_dec_2}
   B_{\Gamma} :=
      \left (
        \begin{array}{cccccc}
         (\alpha_1, \alpha_1) & \dots & (\alpha_i, \alpha_k) &
         (\alpha_1, \beta_1)  & \dots &  (\alpha_1, \beta_h) \\
         \dots                & \dots & \dots &
         \dots                & \dots & \dots \\
         (\alpha_k, \alpha_1) & \dots & (\alpha_k, \alpha_k) &
         (\alpha_k, \beta_1)  & \dots &  (\alpha_k, \beta_h) \\
         (\beta_1, \alpha_1)  & \dots & (\beta_1, \alpha_k) &
         (\beta_1, \beta_1)   & \dots & (\beta_1, \beta_h) \\
         \dots                & \dots & \dots &
         \dots                & \dots & \dots \\
         (\beta_h, \alpha_1)  & \dots & (\beta_h, \alpha_k) &
         (\beta_h, \beta_1)   & \dots & (\beta_h, \beta_h) \\
        \end{array}
      \right ),
 \end{equation}
 where $S = \{\alpha_1,\dots,\alpha_k, \beta_1, \dots, \beta_h\}$. Here, subsets of roots
 $\alpha$-set and $\beta$-set, see \eqref{two_sets}, match
 the bicolored partition \eqref{eq_alpha_bet} and bicolored decomposition of a certain element $w \in W$
 corresponding to $\Gamma$. We also write $S = \{ \tau_1,\dots,\tau_{k+h} \}$ if the
 bicolored partition like \eqref{eq_alpha_bet} does not matter.

 \index{quadratic form ! - $\mathscr{B}_{\Gamma}$, associated with the partial Cartan matrix $B_{\Gamma}$}
 \index{$\mathscr{B}_{\Gamma}$, quadratic form associated with the partial Cartan matrix $B_{\Gamma}$}
 \index{$S = \{\tau_1,\dots,\tau_l\}$, $\Gamma$-associated root subset}
 \index{Cartan matrix ! - inverse $B^{-1}_{\Gamma}$ of the partial Cartan matrix}
 \index{$(\cdot, \cdot)$, symmetric bilinear form associated with ${\bf B}$}
 \index{symmetric bilinear form $(\cdot, \cdot)$}
 \index{$(\cdot, \cdot)_{\Gamma}$, symmetric bilinear form associated with $B_{\Gamma}$}
 \index{symmetric bilinear form $(\cdot, \cdot)_{\Gamma}$}
 \index{$S$-associated subspace}
 We call the matrix $B_{\Gamma}$ a {\it partial Cartan matrix} corresponding to
 the Carter diagram $\Gamma$.
 The partial Cartan matrix $B_{\Gamma}$ is well-defined since $(\tau_i, \tau_j)$ in \eqref{canon_dec_2}
 do not depend on the choice of $\Gamma$-associated root subset.
 If $\Gamma$ is a Dynkin diagram,
 the partial Cartan matrix $B_{\Gamma}$ is the Cartan matrix
 associated with $\Gamma$.
 The matrix $B_{\Gamma}$ is positive definite (Proposition \ref{restr_forms_coincide}).
 The symmetric bilinear form associated with
 the partial Cartan matrix $B_{\Gamma}$ is denoted by $(\cdot, \cdot)_{\Gamma}$ and the
 corresponding quadratic form is denoted by $\mathscr{B}_{\Gamma}$. Let $L \subset V$ be the
 subspace spanned by the root subset $S$, the subspace $L$ is said to be {\it S-associated subspace}.
 For $S = \{\tau_1,\dots,\tau_l\}$, we write $L =[\tau_1,\dots,\tau_l]$.

\begin{remark}{\rm
 \label{rem_values}
 In all cases where we consider the partial Cartan matrix $B_{\Gamma}$ for simply-laced Carter diagrams,
 we assume that $(\tau_i, \tau_j)$ takes values in $\{-1, 0,  1 \}$, not in $\{-\frac{1}{2}, 0, \frac{1}{2}\}$.
 In other words, we consider the bilinear form $(\cdot, \cdot)$ obtained by doubling the usual bilinear form
 related to the Cartan matrix. Then linkage labels introduced in \S\ref{sec_linkage_diagr} are integer.
 }
\end{remark}

\subsubsection{The linkage diagrams}
  \label{sec_linkage_diagr_0}
  \index{Dynkin labels}
  \index{label ! - Dynkin labels}
  \index{label ! - linkage label vector}
  \index{$\gamma$, linkage root}
  \index{linkage root}
  \index{class of diagrams ! - $\Gamma$, Carter diagrams}
  \index{$\Gamma$, Carter diagram}
  \index{connection diagram}

  We consider a class of diagrams, called {\it linkage diagrams}, which
  constitute a subclass of the class of connection diagrams, see \S\ref{sec_connection},
  and generalize the class of Carter diagrams, see \S\ref{sec_Carter}.
  Any linkage diagram is obtained from a Carter diagram $\Gamma$ by adding one extra root $\gamma$,
  with its bonds, so that the roots corresponding to vertices of $\Gamma$ together with $\gamma$ form some
  linearly independent root subset. The extra root $\gamma$ added to the Carter diagram $\Gamma$ is said to be
  a {\it linkage root}, see \S\ref{sec_linkage}.  Any linkage diagram constructed in this way may
  also be a Carter diagram but this is not necessarily so.  The following inclusions hold:

 \begin{equation*}
   \fbox{$\begin{array}{c}
      \text{ Dynkin } \\
      \text{ diagrams of CCl\footnotemark[1] }
   \end{array}$} \quad\subset\quad
   \fbox{$\begin{array}{c}
      \text{ Carter } \\
      \text{ diagrams }
   \end{array}$} \quad\subset\quad
   \fbox{$\begin{array}{c}
      \text{ Linkage } \\
      \text{ diagrams }
   \end{array}$} \quad\subset\quad
   \fbox{$\begin{array}{c}
      \text{ Connection } \\
      \text{ diagrams }
   \end{array}$}
 \end{equation*}

 \footnotetext[1]{The Dynkin diagrams in this article appear in two ways:
 (1) associated with some irreducible root system (customary use);
 (2) representing some conjugacy class (CCl), i.e, the Carter diagram
 which looked like a Dynkin diagram.
 In a few cases Dynkin diagrams represent two (and even three!) conjugacy classes, see Remark \ref{two_class_Al}.}

  With every linkage diagram we associate the vector of {\it linkage labels}.
  The linkage labels are similar to the Dynkin labels,
  see \cite{Sl81}, which are the {\lq\lq}numerical labels{\rq\rq} introduced by Dynkin in \cite{Dy50}
  for the study of irreducible linear representations of the semisimple Lie algebras, \cite{GOV90}, \cite{KOV95}, \cite{Ch84}.

  For any simply-laced Carter diagram $\Gamma$, a linkage label
  takes one of three values $\{-1, 0, 1 \}$. There is one-to-one correspondence between
  linkage diagrams obtained from the given simply-laced Carter diagram $\Gamma$ and vectors of linkage labels
  taking coordinates from the set $\{-1, 0, 1 \}$.
  For this reason, we use terms \underline{linkage labels} and  \underline{linkage diagrams} as
  synonyms. Some linkage diagrams and their linkage labels for the Carter diagram $E_6(a_1)$
  are depicted in Fig. \ref{E6a1_exam_linkages}.
 \index{linkage diagrams ! - examples for $E_6(a_1)$}
 \begin{figure}[h]
\centering
\includegraphics[scale=0.7]{E6a1_exam_linkages_newd.eps}
 \caption{\hspace{3mm}Examples of linkage diagrams and vectors of linkage labels for $E_6(a_1)$.}

\label{E6a1_exam_linkages}
\end{figure}
  The linkage diagrams are the main characters of this paper. {\bf We give a complete description of linkage diagrams constructed
  for every simply-laced Carter diagram.}
  By abuse of notation, this description is essentially based on the answer to the following question:
 ~\\

  {\it What linkage roots can be added to a given irreducible linearly independent root subset,
  so that the resulting set would also be an irreducible linearly independent root subset?}
 ~\\
 ~\\
 It turns out that the answer to this question is very simple within the framework of the
 quadratic form associated with the partial Cartan matrix $B_{\Gamma}$, see \S\ref{sec_criterion}.

\subsubsection{The linkage systems}
 The group $W^{\vee}_S$, named the {\it dual partial Weyl group}, acts
 on linkage diagrams, see \S\ref{sec_dual_partial}.
 Under the action of $W^{\vee}_S$ the set of
 linkage diagrams (= vectors of linkage labels) constitute the
 diagram called the {\it linkage system} similarly to the weight system
 in the representation theory of semisimple Lie algebras, \cite[p. 30]{Sl81}.
 We denote the linkage system associated with the Carter diagram $\Gamma$ by $\mathscr{L}(\Gamma)$.

\subsection{Dynkin extensions, partial root systems and linkage system components}
   \label{sec_D_ext_1}

\subsubsection{$A$-, $D$-, $E$-types of Dynkin and Carter diagram}
  The Dynkin diagram $A_l$, where $l \ge 1$ (resp. $D_l$, where $l \ge 4$; resp. $E_l$, where $l = 6,7,8$) is said to be the
  {\it Dynkin diagram of $A$-type}
  (resp. {\it $D$-type}, resp. {\it $E$-type}). The Carter diagram $A_l$, where $l \ge 1$ (resp. $D_l$, $D_l(a_k)$,
  where $l \ge 4$, $1 \leq k \leq \big [ \tfrac{l-2}{2} \big ]$; resp. $E_l$, $E_l(a_k)$, where $l = 6,7,8$, $k = 1,2,3,4$)
  is said to be the {\it Carter diagram of $A$-type} (resp. {\it $D$-type}, resp. {\it $E$-type}).

\subsubsection{Partial root system}
   For a $\Gamma$-associated root subset $S$, we denote by $\varPhi(S)$ the subset of roots of $\varPhi$
   \underline{linearly dependent} on roots of $S$. The subset of roots $\varPhi(S)$ is said to be
   a {\it partial root system}.
   There are examples of two non-conjugate $\Gamma$-associated root subsets
   $S_1$ and $S_2$, see \S\ref{sec_example_4cycles}.
   This is a reason why I prefer denote the partial root system by $\varPhi(S)$, not by $\varPhi(\Gamma)$.

\subsubsection{Covalent Carter diagrams}
  \label{sec_coval_list}
 Carter diagrams of the same type and the same index are said to be {\it covalent} Carter diagrams.
 The corresponding root systems (usual or partial) are said to be {\it covalent} root systems.
 For example, the $E$-type covalent diagrams of index $6$ are $\{E_6, E_6(a_1), E_6(a_2)\}$ and
 the $E$-type covalent diagrams of index $7$ are $\{E_7, E_7(a_1), E_7(a_2), E_7(a_3), E_7(a_4))\}$,
 the $D$-type covalent diagrams of index $l$ are
          $\{D_l, D_l(a_1), D_l(a_2), \dots, D_l(a_{\big [ \tfrac{l-2}{2} \big ]}\}$.
~\\

 We are interested in the following covalent pairs of diagrams $\{\widetilde\Gamma, \Gamma\}$, where $\Gamma$ is a Dynkin diagram
 and $\widetilde\Gamma$ is a Carter diagram covalent to $\Gamma$:
   \begin{equation}
     \label{eq_pairs_root_syst_0}

     \begin{split}
      & \{D_l(a_k), D_l\} \text{ for } l \geq 4 \text{ and } 1 \leq k \leq \big [\tfrac{l-2}{2} \big ], \\
      & \{E_6(a_k), E_6\} \text{ for }  k = 1,2, \\
      & \{E_7(a_k), E_7\} \text{ for }  k = 1,2,3,4.
     \end{split}
   \end{equation}

   Let $\{\widetilde\Gamma, \Gamma\}$ be a covalent pair of Carter diagrams out of list \eqref{eq_pairs_root_syst_0}
   and $\widetilde{S}$ (resp. $S$) be $\widetilde\Gamma$-associated (resp. $\Gamma$-associated) subset,
   $\mathcal{P}$ (resp. $\mathcal{R}$) be the partial root system (resp. root system) spanned by $\widetilde{S}$ (resp. $S$).

   In \S\ref{sec_transition}, we show that there exists the invertible linear map $M : \mathcal{P} \longmapsto \mathcal{R}$,
   Moreover, $\widetilde\tau$ is a root of $\mathcal{P}$ if and only if $\widetilde\tau$ is a root of $\mathcal{R}$
   (Theorem \ref{prop_bijection_root_syst}). For any covalent pair \eqref{eq_pairs_root_syst_0}, we have
   $\mid \mathcal{R} \mid = \mid \mathcal{P} \mid.$ (Corollary \ref{corol_biject}).

  \begin{table}[H]

  \Small
  \centering
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular} {|c|c|c|c|}
  \hline
      $\Gamma$ &  Number of  &
      \multicolumn{2}{c|}{$E$-components}   \cr
      \cline{3-4}
        &  components   &
      $\mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla}$) &  Number of \cr
        &   &  &  linkage diagrams \\
    \hline
       $E_6$, $E_6(a_1)$, $E_6(a_2)$ &   $2$  &  $\frac{4}{3}$  & $2\times27 = 54$  \\
    \hline
       $E_7$, $E_7(a_1)$, $E_7(a_2)$, $E_7(a_3)$, $E_7(a_4)$ &   $1$  &   $\frac{3}{2}$  & $56$ \cr
   \hline
  \end{tabular}
  ~\\ \vspace{2mm}
  \centering
  \begin{tabular} {|c|c|c|c|c|}
  \hline
     $\Gamma$ &  Number of  &
      \multicolumn{3}{c|}{Number of linkage diagrams,}  \cr
       &  components   & \multicolumn{3}{c|}{and $p = \mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla})$}    \cr
      \cline{3-5}
         &    & $D$-components   & $E$-components &
          \hspace{1.8mm} In all \hspace{1.8mm} \cr
      \cline{3-4}
          &     & $p = 1$ & $p = \frac{l}{4}$ & \\
    \hline

       $D_4$, ~$D_4(a_1)$ &   $3$   & $3 \times 8 = 24$ & - &  $24$ \\
    \hline
       $D_5$, ~$D_5(a_1)$ &   $3$   &  $10$ & $2\times16 = 32$   & $42$ \\
    \hline
       $D_6$, $D_6(a_1)$, $D_6(a_2)$  &   $3$ & $12$  & $2\times32 = 64$  &    $76$ \\
    \hline
       $D_7$, $D_7(a_1)$, $D_7(a_2)$ &   $3$   & $14$ & $2\times64 = 128$  &  $142$ \\
    \hline
       $D_l$, ~$D_l(a_k)$, $l > 7$ &   $1$  & $2l$ & -  &  $2l$ \\
   \hline
\end{tabular}
  ~\\ \vspace{2mm}
  \centering
\begin{tabular} {|c|c|c|c|c|}
  \hline
    $\Gamma$
     & \multicolumn{4}{c|}{Number of linkage diagrams, and the number $p = \mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla})$}  \cr
        \cline{2-5}
    &  $A$-component        & $D$-component         &  $E$-component         &
     \hspace{6.0mm}In all \hspace{6.0mm}  \cr

      \cline{2-4}
          &    $p = \frac{l}{l+1}$ & $p = 2\frac{l-1}{l+1}$    &  $p = 3\frac{l-2}{l+1}$   &      \\
  \hline
        $A_3$     &  $2 \times 4 = 8$ & $6$  &  -     & $14$ \\
  \hline
        $A_4$     &  $2 \times 5 = 10$ & $2 \times 10 = 20$  &  -     & $30$ \\
  \hline
        $A_5$     &  $2 \times 6 = 12$ & $2 \times 15 = 30$  & $20$  & $62$ \\
  \hline
        $A_6$     &  $2 \times 7 = 14$ & $2 \times 21 = 42$  & $2 \times 35 = 70$  & $126$ \\
  \hline
        $A_7$     &  $2 \times 8 = 16$ & $2 \times 28 = 56$  & $2 \times 56 = 112$
                        & $184$ \\
  \hline
        $A_8$     &  $2 \times 9 = 18$ & $2 \times 36 = 72$  & -  & $90$ \\
  \hline
        $A_l$, $l > 8$  &  $2\times(l+1)$  & $2 \times \frac{l(l+1)}{2} = l(l+1)$   &  - &
         $(l+1)(l+2)$   \\
   \hline
  \end{tabular}
  \vspace{2mm}
  \caption{\hspace{1mm}The $ADE$ components of the linkage system $\mathscr{L}(\Gamma)$,
  the number of linkage diagrams in every component and values $\mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla})$.}
  \label{tab_val_Bmin1}
  \end{table}

\subsubsection{Dynkin extensions}
  \label{sec_Dynkin_ext0}

   We show in Corollary \ref{cor_exist_Dynkin_ext} that for a Carter diagram  $\Gamma$ (rank($\Gamma$) $< 8$) and any
   $\Gamma$-associated root subset $S$ \underline{there exists the Dynkin diagram $\Gamma'$} such that $\rank(\Gamma') = \rank(\Gamma) + 1$
   and $\varPhi(S) \subset \varPhi(\Gamma')$, where $\varPhi(S)$ is the partial root system
   and $\varPhi(\Gamma')$ is the root system associated with $\Gamma'$.
   This pair $\{\Gamma, \Gamma'\}$ is said to be the {\it Dynkin extension  of the Carter diagram $\Gamma$} and
   is denoted by $\Gamma <_D \Gamma'$.

  For any pair $\{\widetilde\Gamma, \Gamma\}$ out of list \eqref{eq_pairs_root_syst_0},
  we have $\mid \mathscr{L}(\widetilde\Gamma) \mid ~=~ \mid \mathscr{L}(\Gamma) \mid$ (Corollary \ref{corol_Estim_2}).

\subsubsection{Root stratum}
  \label{sec_Dynkin_ext}
 \index{regular extension $\Gamma < \widetilde{\Gamma}$}
 \index{$\Gamma < \widetilde{\Gamma}$, regular extension}
 \index{$\Gamma \stackrel{\alpha}{<} \widetilde{\Gamma}$, regular extension}

  Consider the connection diagram $\widetilde{\Gamma}$, see \S\ref{sec_connection},
  obtained from a certain Carter diagram $\Gamma$ by adding only one vertex $\alpha$,
  where $\alpha$ is connected to $\Gamma$ at $v$ points, where $v=1,2$ or $3$. If $\widetilde{\Gamma}$ is also a Carter diagram, this
  extension $\Gamma$ to $\widetilde{\Gamma}$ is said to be a {\it regular extension} and is denoted by
 \begin{equation*}
  \Gamma < \widetilde{\Gamma} \quad \text{ or } \quad  \Gamma \stackrel{\alpha}{<} \widetilde{\Gamma}.
 \end{equation*}
   The Dynkin extension is not necessary regular.  For example, $D_n(a_k) <_D D_{n+1}$
   is a Dynkin extension but $D_n(a_k)$ is not a subdiagram of $D_{n+1}$.
 ~\\

   The subset of roots $\varPhi(\Gamma')\backslash\varPhi(S)$ is said to be the {\it root stratum};
   it consists of roots of $\varPhi(\Gamma')$ \underline{linearly independent} of roots of $S$.

\subsubsection{Linkage system components}
  \label{sec_link_syst_comp}
 \index{ $\varPhi(S)$, partial root system}
 \index{root system ! - partial}
 \index{partial root system}
 \index{conjugation ! - of bases of the root system}
 \index{conjugation ! - non-conjugate $\Gamma$-associated root subsets $S_1$ and $S_2$}
 \index{root stratum}
 \index{root system}
  \index{linkage system component}
  \index{stratum size}
  Let $\mathscr{L}_{\Gamma'}(\Gamma) := \{\gamma^{\nabla} \mid \gamma \in \varPhi(\Gamma')\backslash\varPhi(S)\}$.
  Here, $\gamma^{\nabla}$ is the vector of linkage labels associated with $\gamma$.
  The set $\mathscr{L}_{\Gamma'}(\Gamma)$ is said to be the {\it linkage system component}
 (or, {\it $\Gamma'$-component} of $\mathscr{L}(\Gamma)$).
  The linkage system $\mathscr{L}(\Gamma)$ is the union of the sets $\mathscr{L}_{\Gamma_i}(\Gamma)$
  taken for all Dynkin extensions of $\Gamma$:
  \index{linkage diagrams ! - in $\mathscr{L}_{\Gamma'}(\Gamma)$}
 \begin{equation*}
     \mathscr{L}(\Gamma) \quad = \quad  \bigcup_{\Gamma <_D \Gamma_i} \mathscr{L}_{\Gamma_i}(\Gamma).
 \end{equation*}

 There are several Dynkin extensions and, accordingly,
 several linkage system components for the given Carter diagram $\Gamma$.
 The number of roots in the root stratum $\varPhi(\Gamma')\backslash\varPhi(S)$
 is said to be the {\it stratum size}. We have
 \begin{equation*}
   \mid \varPhi(\Gamma')\backslash\varPhi(S) \mid  ~=~  \mid\varPhi(\Gamma')\mid - \mid\varPhi(S)\mid
     \text{ and }
   \mid \mathscr{L}_{\Gamma'}(\Gamma) \mid  \quad \leq \quad  \mid\varPhi(\Gamma')\mid - \mid\varPhi(S)\mid,
 \end{equation*}
 see \S\ref{sec_part_root_and_comp}. From the latter inequality we get the following proposition:
 ~\\
 ~\\
  {\bf Proposition} (Proposition \ref{prop_Estim})
     {\it For any Carter diagram $\Gamma$, we have $\mid \mathscr{L}(\Gamma) \mid \leq \mathscr{E}$, where
     $\mathscr{E}$ is given by Table \ref{tab_val_Estim}.}
 ~\\

  It is checked in \S\ref{sec_stratif_D}, \S\ref{sec_stratif_E} and \S\ref{sec_stratif_A} that
  there exists at least $\mathscr{E}$ linkage diagrams in the linkage system $\mathscr{L}(\Gamma)$.
  Together with Proposition \ref{prop_Estim}, we get the following
  ~\\
  ~\\
 {\bf Corollary} (Corollary \ref{corol_Estim})
     {\it For any Carter diagram $\Gamma$, we have $\mid \mathscr{L}(\Gamma) \mid = \mathscr{E}$,
     see Table \ref{tab_val_Estim}.}
~\\

   For each Carter diagrams $\Gamma \in  \mathsf{C4} \coprod \mathsf{DE4} \coprod \mathsf{A}$,
   the values $\mathscr{B}^{\vee}_{\Gamma}(\gamma^{\nabla})$, the number of components, and number of
   linkage diagrams are presented in Table \ref{tab_val_Bmin1}, see Theorem \ref{th_full_descr}.
 ~\\

 Every component of $\mathscr{L}(\Gamma)$
 is determined by the type of Dynkin extension $\Gamma <_D \Gamma'$. There are $A$-, $D$-, $E$-types of
 Dynkin extensions. The corresponding components of the linkage systems are called $A$-, $D$-, $E$-components.
 For description of components of the linkage system for each Carter diagram $\Gamma$, see
 Table \ref{tab_val_Bmin1} and Theorem \ref{th_full_descr}.
 ~\\

config: arxiv


TITLE: Math Problem Help || Trig (Updated picture)
QUESTION [1 upvotes]: So I got the following math problem I solved but I'm not confident with my answer. My answer was 32.
The question is to find the area of the following:
http://i.gyazo.com/c00c5afc60c33ebc934333e2b0d755a6.png
UPDATED PICTURE BELOW (Because the old one was hurting people's eyes)
http://i.gyazo.com/a046efd5e588148ac5146329a5be438e.png

REPLY [2 votes]: First simplify the square roots: $\sqrt{96}=4\sqrt 6$, and $\sqrt{24}=2\sqrt 6$.
Second, recognize that the top and bottom sides of the quadrilateral are parallel, so this is a trapezoid. You have the bottom base, so all you need for the area are the top base and the height.
Third, you see a $30°$-$60°$-$90°$ triangle in the lower right hand corner (if you add an auxiliary line from the top right corner of the trapezoid). The height of the trapezoid is the leg opposite the $60°$ angle, and the top base is the bottom base minus the leg opposite the $30°$ angle. You know the hypotenuse, $2\sqrt 6$. You can then find the legs of that triangle. (I get the bottom leg as $\sqrt 6$ and the height as $3\sqrt 2$. That makes the upper base $3\sqrt 6$.)
Can you finish up now? I get the area as $21\sqrt 3$.
$$A=\frac{b_1+b_2}2h$$
$$=\frac{3\sqrt 6+4\sqrt 6}2\cdot 3\sqrt 2$$
$$=21\sqrt 3$$


TITLE: How you'd show that $f$ is not continuous?
QUESTION [1 upvotes]: How do you show that
$$f(x)=\begin{cases}2,&x < c\\1,&x\geq c\end{cases}$$ is not continuous at $c$ by using $\epsilon$-$\delta$ - formalism like here?

REPLY [3 votes]: Suppose $f$ were continuous.  Let $\epsilon=0.1$, and $\delta$ be chosen so that $|f(x)-f(c)|<\epsilon$ for all $|x-c|<\delta$. But $|(c-\delta/2)-c|<\delta$ and $|f(c-\delta/2)-f(c)|=|2-1|=1>\epsilon$.  Contradiction.

REPLY [2 votes]: Hint: You must show that there exists an $\epsilon>0$ so that no matter what $\delta>0$ you choose, $0<\lvert x-c\rvert<\delta$ is not enough to say that $\lvert f(x)-f(c)\rvert<\epsilon$.
That is, there exists an $\epsilon>0$ so that for any $\delta>0$, you can find a point $x$ which is within $\delta$ of $c$ but satisfies $\lvert f(x)-f(c)\rvert\geq\epsilon$.
You can find points as close to $c$ as you want which have $f(x)=2$, so that $\lvert f(x)-f(c)\rvert=1$.  How should you choose $\epsilon$?

REPLY [2 votes]: You need to show that there exists some $\epsilon>0$ such that for any $\delta>0$, you can find some $x$ with $|x-c|< \delta$ so that $|f(x)-f(c)| \ge \epsilon$.
In this particular case, this means finding some $\epsilon>0$ such that for any $\delta>0$, you can find some $x$ with $|x-c|< \delta$ so that $|f(x)-1| \ge \epsilon$.
If you draw a picture it may help find appropriate $\epsilon$ and $x$.

REPLY [2 votes]: Reword the definition:
$f$ is not continuous at $x_0$ if there exists an $\epsilon > 0$ such that for every $\delta > 0$, there exists an $x$ such that $|x-x_0| < \delta$ but $|f(x) - f(x_0)| > \epsilon$.
Now, let $x_0 = c$, take $\epsilon = 1/2$ (other choices will work), and then show that for every $\delta > 0$, there exists an $x$...