text:
\section{Abstract}

We will show that almost all nonassociative relation algebras are symmetric and integral (in the sense that the fraction of both labelled and unlabelled structures that are symmetric and integral tends to \(1\)), and using a Fra{\"i}ss{\'e} limit, we will establish that the classes of all atom structures of nonassociative relation algebras and relation algebras both have \(0\)--\(1\) laws. As a consequence, we obtain improved asymptotic formulas for the numbers of these structures and broaden some known probabilistic results on relation algebras.

\section{Introduction}

The calculus of relations is a branch of logic that was developed in the nineteenth century, largely due to the work of Augustus De Morgan, Charles Peirce, and Ernst Schr{\"o}der; see \cite {morg}, \cite{peirce}, and \cite{schroder}, for example. By the beginning of twentieth century, the calculus of relations was considered to be an important branch of logic. Indeed, in \cite{russell}, Bertrand Russell stated that ``the subject of symbolic logic is formed by three parts: the calculus of
propositions, the calculus of classes, and the calculus of relations.'' The calculus of relations even played a role in the birth of model theory. Indeed, Leopold L{\"o}wenheim stated and proved the earliest known version of the L{\"o}wenheim-Skolem Theorem as a result on the calculus of relations; for more details, see \cite{bad}. Interest in the field mostly faded until the publication of \cite{og}, where Alfred Tarski defined an abstract algebraic counterpart to the calculus of relations, namely relation algebras. Tarski and many of his students took an interest in these algebras, which~lead to relation algebras becoming a fairly popular area of research that is still active.

The idea of defining the probability of a property holding in a class of finite structures as a limit is due to Rudolf Carnap (see \cite{carnap}), but similar ideas appeared earlier. In~\cite{mtprob}~and~\cite{mtnumb}, Ronald Fagin looked at the probability of certain sentences holding in the classes of all relational structures of a given type as well as asymptotic formulas for the numbers of these structures. The study of \(0\)--\(1\) laws was initiated by  Glebski\u{\i}, Kogan, Liogon'ki\u{\i}, and Talanov in the nineteen seventies; see \cite{og1} and \cite{og2}. The study of conditions that guarantee the existence of \(0\)--\(1\) laws for first-order and monadic second-order properties were studied extensively by Kevin Compton in \cite{comp1} and \cite{comp2}. In the context of relation algebras, there has been surprisingly little research published on probabilistic results of this nature.  In~\cite{fira},  Roger Maddux finds an asymptotic formula for the number of nonassociative relation algebras in which the identity is an atom and the number of integral relation algebras, and shows that almost all of these algebras are rigid and satisfy any finite set of equations that hold in all representable relation algebras. In the present article, we show that almost all finite nonassociative relation algebras are symmetric and have \(e\) as an atom, and establish a \(0\)--\(1\) law for the class of all atom structures of finite  nonassociative relation algebras. Combining these results with the results of Maddux, we obtain a simple asymptotic formula for both the number of nonassociative relation algebras and the number of relation algebras, and show that almost all finite nonassociative relation algebras are integral relation algebras.

\section{Preliminaries}

We begin by giving a formal definition of labelled and unlabelled probabilities of properties. We will mostly follow the approach taken in \cite{2kinds}. Note that by \(\bb{N}\) we mean \(\{1,2,\dots\}\).

\begin{definition}[Probabilities]
Let \(\c{K}\) be a class of finite structures of a finite signature \(F\) that is closed under isomorphism and has no upper bound on the size of its members.  For all \(n \in \bb{N}\), let \(\c{U}_n\) be a set with precisely one representative from each isomorphism class of \(n\)-element structures from \(\c{K}\) and let \(\c{L}_n\) be the set of all structures in \(\c{K}\) with universe~\(\{1,\dots,n\}\). Let \(P\) be some property of \(F\)-structures that is invariant under isomorphisms (for example, a~first-order property). Let \(s \colon \bb{N} \to \bb{N}\) be the increasing sequence of values of \(n\) with \(\c{U}_n \neq \varnothing\). If the limit
\[
\lim_{n \to \infty} \frac{|\{ \bf{A} \in \c{U}_{s(n)} \mid \bf{A} \models P\}|}{|\c{U}_{s(n)}|}
\]
exists, we call it the \emph{unlabelled probability of \(P\)} and denote it by \(\pru(P,\c{K})\). If the limit
\[
\lim_{n \to \infty} \frac{|\{ \bf{A} \in \c{L}_{s(n)} \mid \bf{A} \models P\}|}{|\c{L}_{s(n)}|}
\]
exists, we call it the \emph{labelled probability of \(P\)} and denote it by \(\prl(P,\c{K})\). If \(\pru(P,\c{K}) = 1\), we say that \emph{almost all structures in \(\c{K}\) satisfy \(P\)}.
\end{definition}

We will mostly work with classes where there are elements of every possible cardinality, so the sequence \(s\) will be the identity sequence.

The result we will need from Freese \cite{2kinds} is stated below; this result is stated for algebras, but the proof also works for relational structures.  Similar results appear in earlier articles, such as Fagin \cite{mtnumb}.

\begin{proposition}\label{rigid}
Let \(\c{K}\) be a class of similar finite structures of a finite signature \(F\) that is closed under isomorphism and has no upper bound on the size of its members, let \(R\) be the property of being rigid (i.e., having a trivial automorphism group), let \(P\) be a property \(F\)-structures that is invariant under isomorphism, and assume that we have \(\prl(R,\c{K}) = 1\). If one of \(\pru(P,\c{K})\) and \(\prl(P,\c{K})\) exists, then both quantities exist and are equal.
\end{proposition}

Now we are in the position to recall the definiton of an almost sure theory.

\begin{definition}[Almost sure theory]
Let \(\c{K}\) be a class of finite structures of a finite signature \(F\) that is closed under isomorphism and has no upper bound on the size of its members. We call the set of all first-order sentences \(\sigma\) with \(\prl(\sigma,\c{K}) = 1\) the \emph{almost sure theory of \(\c{K}\)}.
\end{definition}

Next, we will give a brief introduction to relation algebras. For a more extensive introduction, we refer the reader to Hirsch and Hodkinson~\cite{rabg}, Maddux~\cite{madduxRA}, and Givant~\cite{intRA}. We begin by defining relation algebras. To match the notation used in lattice theory, we will use \(\j\) and \(\wedge\) rather than \(+\) and \(\cdot\). This allows us to use a more group theoretic notation by using \(\cdot\) and \(e\)  rather than \(;\) and \(1'\). We will assume that unary operations are applied first and use multiplicative notation~for~\(\cdot\). Therefore \(x\b z \wedge y'\) means \(((x \b) \cdot z) \wedge (y ')\), for example. We call \(d \mathrel{\mathop:}= e'\) the \emph{diversity element}. We use \(\approx\) for logical equality, as in universal algebra. To avoid reusing symbols, we wil use \(\curlyvee\) for logical disjunction, \(\curlywedge\) for logical conjunction, and \(\neg\) for logical negation. Where it is possible, we will usually drop superscripts on operations.

\begin{definition}[Relation algebras]
An algebra \(\bf{A} = \inn{A;\j,\wedge,\cdot,{'},{\b}, 0,1,e}\) is called a \emph{nonassociative relation algebra} iff \(\inn{A;\j,\wedge,{'},0,1}\) is a Boolean algebra, \(e\) is an identity element~for~\(\cdot\), and the \emph{triangle laws} hold, i.e., we have
\[
xy \wedge z = 0 \iff x\b z \wedge y = 0 \iff zy\b \wedge x = 0,
\]
for all \(x,y,z \in A\). The class of all nonassociative relation algebras will be denoted by \(\sf{NA}\). An algebra \(\bf{A} \in \sf{NA}\) is called a \emph{relation algebra} iff \(\cdot\) is associative.  The class of all relation algebras will be denoted by \(\sf{RA}\). An algebra \(\bf{A} \in \sf{NA}\) is said to be \emph{symmetric} iff \(\bf{A} \models x\b \approx x\).
\end{definition}

We extend ideas from Boolean algebra to these algebras in the obvious way. For example, an atom of a nonassociative relation algebra is an atom of its Boolean algebra reduct.

Some basic properties of these algebras are summarised in the following result.

\begin{proposition}\label{basicRA}
Let \(\bf{A} \in \sf{NA}\).
\begin{enumerate}
\item
\(\bf{A} \models x(y \j z) \approx xy \j xz\) and \(\bf{A} \models (x \j y)z \approx xz \j yz\).
\item
\(\bf{A} \models (x \j y)\b \approx x\b \j y\b\).
\item
\(\bf{A} \models 0\b = 0\), \(\bf{A} \models 1\b = 1\), \(\bf{A} \models e\b = e\), and \(\bf{A}\models d\b = d\).
\item
\(\bf{A} \models x\b\b \approx x\).
\item
If \(a\) is an atom, then \(a\b\) is an atom.
\item
If \(\bf{A} \models e \approx 0\), then \(\bf{A}\) is trivial.
\end{enumerate}
\end{proposition}

Based on Proposition \ref{basicRA}, the operations of a complete atomic (and, in particular, a finite) nonassociative relation algebra are completely determined by their values on its atoms. Since a finite \(\bf{A}\in \sf{NA}\) has \(\log_2(|A|)\) atoms, this means these algebras are determined by a small subset of its elements. This motivates the following definitions. When \(e\) is an atom, it is sometimes convenient to include it in the signature rather than a unary relation.
\begin{definition}[Atom structure]
Let \(\bf{A}\) be a complete atomic nonassociative relation algebra. We call \(\At(\bf{A}) \mathrel{\mathop:}= \inn{\at(\bf{A}); f_\bf{A}, I_{\bf{A}}, T_{\bf{A}}}\) the \emph{atom structure} of \(\bf{A}\), where \(\at(\bf{A})\) denotes the set of all atoms of \(\bf{A}\), \(f_\bf{A}\) is defined by \(x \mapsto x\b\), \(I_{\bf{A}} \mathrel{\mathop:}= \{ a \in \at(\bf{A}) \mid a \le e\}\), and \(T_\bf{A} \mathrel{\mathop:}= \{ (a,b,c) \in \at(\bf{A})^3 \mid ab \ge c\}\). If \(e\) is an atom, we put \(\At_e(\bf{A}) \mathrel{\mathop:}= \inn{\at(\bf{A}); f_\bf{A}, e,T_\bf{A}}\).
\end{definition}

It turns out that these structures can be axiomatised.

\begin{definition}[\(\sf{FAS}\), \(\sf{FSIAS}\), and \(\sf{FSIAS}_e\)]
Let \(\sf{FAS}\) denote the class of all finite structures of the signature \(\{f,T,I\}\) (where \(f\) is a unary operation symbol, \(I\) is a unary relation symbol, and \(T\) is a ternary relation symbol) that satisfy
\begin{enumerate}
\item[\textup{(P)}]
for all \(a,b,c \in U\), we have \((f(a),c,b) , (c,f(b),a) \in T\) whenever \((a,b,c) \in T\),
\item[\textup{(I)}]
for all  \(a,b \in U\), we have \(a = b\) if and only if \(i \in I\) with \((a,i,b) \in T\).
\end{enumerate}
We call a \(\{f,T,I\}\)-structure \(\bf{U}\) \emph{integral} iff we have \(|I| = 1\), and \emph{symmetric} iff \(\bf{U} \models f(x) \approx x\). The class of all symmetric and integral members of \(\sf{FAS}\) will be denoted by \(\sf{FSIAS}\). Now, let \(\sf{FSIAS}_e\) be the class of all finite structures of the signature \(\{f,e,T\}\)  (where \(f\) is a unary operation symbol, \(e\) is a nullary operation symbol, i.e., a constant, and \(T\) is a ternary relation symbol) that satisfy \(f(x) \approx x\) and
\begin{enumerate}
\item[\textup{(IP)}]
for all \(a,b,c \in U\), we have \((f(a),c,b) , (c,f(b),a) \in T\) whenever \((a,b,c) \in T\),
\item[\textup{(II)}]
for all  \(a,b \in U\), we have \(a = b\) if and only if there is some \((a,e,b) \in T\).
\end{enumerate}
\end{definition}

The above definition abuses language slightly; a non-trivial \(\bf{A} \in \sf{NA}\) is called \emph{integral} iff  \(xy = 0\) implies that \(x=0\) or \(y = 0\), which is equivalent to \(e\) being an atom when \(\bf{A} \in \sf{RA}\), but not in general. We refer to Maddux \cite{madduxRA} and Maddux \cite{varcontRA} for further details.

\begin{proposition}\label{atcm}
\(\sf{FAS}\) is precisely the class of all atom structures of finite members~of~\(\sf{NA}\), \(\sf{FSIAS}\) is precisely the class of all atom structures of finite, integral, and symmetric members of \(\sf{NA}\). There are bijective correspondences between the sets of isomorphism classes from:
\begin{enumerate}
\item
the class of finite members of \(\sf{NA}\) and \(\sf{FAS}\);
\item
the class of finite, integral, and symmetric members of \(\sf{NA}\) and \(\sf{FISAS}\);
\item
\(\sf{FSIAS}\) and \(\sf{FSIAS}_e\).
\end{enumerate}
\end{proposition}

Next, we introduce the notion of a cycle (from Maddux \cite{fira}) which can be used to define and describe these structures.

\begin{definition}[Cycles]
Let \(\bf{U}\) be a \(\{f,T,I\}\)-structure and let \(a,b,c \in U\). We call \((a,b,c)\), \((f(a),c,b)\), \((b,f(c),f(a))\), \((f(b),f(a),f(c))\), \((f(c),a,f(b))\), and \((c,f(b),a)\) the \emph{Peircean transforms} of \((a,b,c)\). The set of all of these triples is called a \emph{cycle}, and is denoted by  \([a,b,c]\).  We call \((a,b,c)\) an \emph{identity triple} iff  \(I \cap \{a,b,c\} \neq \varnothing\), and a \emph{diversity triple} otherwise. We call \([a,b,c]\) an \emph{identity cycle} iff it contains an identity triple, and a \emph{diversity cycle} otherwise. We call \((a,b,c)\) \emph{consistent} iff \((a,b,c) \in T\), and \emph{forbidden} otherwise. We call \([a,b,c]\) \emph{consistent} iff  \([a,b,c] \subseteq T\), and \emph{forbidden} iff \([a,b,c] \cap T = \varnothing\). We call \(a\) an \emph{identity atom} iff \(a \in I\) and a \emph{diversity atom} otherwise. We extend these ideas to \(\{f,e,T\}\) in the obvious way.
\end{definition}

The following result from \cite{fira} illustrates the connection between cycles and the axioms for \(\sf{FAS}\) and \(\sf{FSIAS}_e\).

\begin{proposition}\label{asprop}
\begin{enumerate}
\item
Let \(\bf{U}\) be an \(\{f,T,I\}\)-structure.
\begin{enumerate}
\item
The following are equivalent:
\begin{enumerate}
\item
\(\bf{U}\) satisfies \textup{(P)};
\item
for all \(a,b,c \in U\), the cycle \([a,b,c]\) is either consistent or forbidden.
\end{enumerate}
\item
The following are equivalent:
\begin{enumerate}
\item
\(\bf{U}\) satisfies \textup{(I)};
\item
for all \(a, b \in U\), we have \(a = b\) if and only if \([a,i,b]\) is consistent, for some \(i \in I\).
\end{enumerate}
\item
If \(\bf{U}\) is integral, then the following are equivalent:
\begin{enumerate}
\item
\(\bf{U}\) satisfies \textup{(I)};
\item
\(f(e) = e\) and \(\{[a,e,a] \mid a \in U\}\) is the set of consistent identity cycles, where \(e\) is the unique element of \(I\).
\end{enumerate}
\end{enumerate}
\item
Let \(\bf{U}\) be a \(\{f,e,T\}\)-structure.
\begin{enumerate}
\item
The following are equivalent:
\begin{enumerate}
\item
\(\bf{U}\) satisfies \textup{(IP)};
\item
for all \(a,b,c \in U\), the cycle \([a,b,c]\) is either consistent or forbidden.
\end{enumerate}
\item
The following are equivalent:
\begin{enumerate}
\item
\(\bf{U}\) satisfies \textup{(II)};
\item
\(f(e) = e\) and \(\{[a,e,a] \mid a \in U\}\) is the set of consistent identity cycles.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{proposition}

Based on Proposition~\ref{atcm} and Proposition~\ref{asprop}, once given a finite set \(U\), some \(e \in U\), and an involution \(f \colon U \to U\) with \(f(e) = e\), each \(\bf{U} \in \sf{FAS}\) that is an expansion of \(\inn{U;f,\{e\}}\) is completely determined by which cycles are consistent or forbidden. Using this observation, it is possible count the number of atom-structures of a given (finite) size. Indeed, in \cite{fira}, Maddux obtains asymptotic formulas using this method. The results we will need from \cite{fira} are summarised in the following results.

\begin{proposition}\label{cyclecount}
Let \(U\) be an \(n\)-element set, for some \(n \in \mathbb{N}\), let \(e \in U\), let \(f\) be an involution of \(U\) with \(f(e) = e\), and let \(s \mathrel{\mathop:}= |\{a \in U \mid f(a) = a\}|\).
\begin{enumerate}
\item
There are \(s-1\) diversity cycles with \(1\) triple; ones of the form \([a,a,a]\).
\item
There are \((n-s)/2\) diversity cycles with \(2\) triples; ones of the form \([a,a,f(a)]\), where \(f(a) \neq a\).
\item
There are \((s-1)(n-2)\) diversity cycles with \(3\) triples; ones of the form \([a,b,b]\), where \(f(a)=a\) and \(a \neq b\).
\item
There are \((n-1)((n-1)^2-3s+2)/6 + (s-1)/2\) diversity cycles with \(6\) triples.
\item
There are \(Q(n,s) \mathrel{\mathop:}= (n-1)((n-1)^2+3s-1)/6\) diversity cycles in total.
\item
There are \(P(n,s) \mathrel{\mathop:}= (s-1)! ((n-s)/2)! 2^{(n-s)/2}\) automorphisms of \(\inn{U;f,\{e\}}\).
\end{enumerate}
\end{proposition}

Before we state the next result, we will pause to recall the definition of representability.

\begin{definition}[\sf{RRA}]
Let \(E\) be an equivalence relation over a set \(D\). We call the structure \(\inn{\raise0.545ex\hbox{{\scalebox{1.0925}[1.0925]{\(\wp\)}}}(E); \cup, \cap, {\mid}, {^c}, {^{-1}}, \varnothing, E, \mathrm{id}_D}\) the \emph{proper relation algebra} on \(E\), where  \(\raise0.545ex\hbox{{\scalebox{1.0925}[1.0925]{\(\wp\)}}}(E)\) be the powerset (i.e., set of all subsets) of \(E\), \(\mid\) is relational composition, \({^{c}}\) is set complement relative to \(E\), \({^{-1}}\) is relational inverse, and \(\mathrm{id}_D\) is the identity relation on \(D\). Thus, for each \(R,S \subseteq E\),
\begin{align*}
R \mid S &= \{ (x,z) \in D^2 \mid (x,y) \in R, (y,z) \in S, \rm{ for some } y \in D\}, \\
R^{-1} &= \{ (y,x) \in D^2 \mid (x,y) \in R\}, \\
\mathrm{id}_D & = \{ (x,y) \in D^2 \mid x = y \}.
\end{align*}
A relation algebra is said to be \emph{representable} iff it embeds into a proper relation algebra. The class of representable relation algebras will be denoted by \(\sf{RRA}\).
\end{definition}

The problem of determining whether every relation algebra is representable was the main focus of research into relation-type algebras for many years, until it was solved in the negative by Lyndon in \cite{lyndon}. Therefore one might be interested in the asymptotics of the fraction of representable relation algebras. Maddux took the first steps in this direction in \cite{fira}.

\begin{proposition}\label{firasumm}
\begin{enumerate}
\item
Almost all integral labelled structures in \(\sf{FAS}\) are rigid.
\item
If \(E\) is the conjunction of a finite set of equations that hold in all members of \(\sf{RRA}\), then \(E\) holds in almost all finite elements of \(\sf{NA}\) in which \(e\) is an atom. In particular, almost all nonassociative relation algebras in which \(e\) is an atom are relation algebras.
\end{enumerate}
\end{proposition}

\begin{proposition}\label{asymp}
For all \(n,s \in \bb{N}\) with \(s \le n\) and \(n-s\) even, let \(F(n,s)\) be the number of isomorphism classes of \(n\)-atom integral relation algebras with \(s\) atoms satisfying \(x \b = x\). Then \(2^{Q(n,s)}/P(n,s)\) is an asymptotic formula for \(F(n,s)\), in the sense that, for all \(\varepsilon > 0\), there exists \(N\in \bb{N}\) such that if \(n,s \in \bb{N}\) with \(n > N\), \(s \le n\), and \(n-s\) even, then
\[
\Bigg| 1 - \frac{F(n,s)P(n,s)}{2^{Q(n,s)}} \Bigg| < \varepsilon.
\]
Further, the same statement holds for nonassociative relation algebras in which \(e\) is an atom.
\end{proposition}

We conclude with a reminder of  Fra{\"i}ss{\'e} limits, which were defined by Roland Fra{\"i}ss{\'e} in \cite{fraisse}. We will mostly follow Hodges \cite{lsmt}. Firstly, we recall some definitions.

\begin{definition}[Age]
Let \(\bf{A}\) be a structure. The \emph{age of \(\bf{A}\)} is the class of all finitely generated structures that embed into \(\bf{A}\).
\end{definition}

\begin{definition}[HP, JEP, and AP]
Let \(\c{K}\) be a class of similar structures. We say that \(\c{K}\) has the \emph{hereditary property} (HP) iff \(\c{K}\) is closed under forming finitely generated structures. We say that \(\c{K}\) has the \emph{joint embedding property} (JEP)  iff, for all \(\bf{A}, \bf{B} \in \c{K}\), there is some \(\bf{C} \in \sc{K}\) that both \(\bf{A}\) and \(\bf{B}\) embed into. We say that \(\c{K}\) has the \emph{amalgamation property} (AP)  iff, for all \(\bf{A},\bf{B},\bf{C} \in \c{K}\) and embeddings \(\mu \colon \bf{A} \to \bf{B}\) and \(\nu \colon \bf{A} \to \bf{C}\), there is some \(\bf{D} \in \c{K}\) and embeddings \(\mu' \colon \bf{B} \to \bf{D}\) and \(\nu' \colon \bf{C} \to \bf{D}\) such that \(\mu' \circ \mu = \nu' \circ \nu\).
\end{definition}

\begin{definition}[Homogeneity]
Let \(\bf{A}\) be a structure. We call \(\bf{A}\)  \emph{ultrahomogenous} iff every isomorphism between finitely generated substructures of \(\bf{A}\) extends to an automorphism of \(\bf{A}\). We call \(\bf{A}\) \emph{weakly homogeneous} iff, for all finitely generated structures \(\bf{B}\) and \(\bf{C}\) of \(\bf{A}\) with \(\bf{B} \le \bf{C}\) and all embeddings \(\mu \colon \bf{B} \to \bf{A}\), there is an embedding \(\nu \colon \bf{C} \to \bf{A}\) extending \(\mu\).
\end{definition}

The following result shows that these definitions coincide

\begin{proposition}\label{homogequiv}
A finite or countable structure is ultrahomogeneous if and only if it is weakly homogeneous.
\end{proposition}

The main result on these structures is the following existence and uniqueness result, known as Fra{\"i}ss{\'e}'s Theorem.

\begin{proposition}[Fra{\"i}ss{\'e}'s Theorem]\label{fraisse}
Let \(S\) be a countable signature and let \(\c{K}\) be a class of at most countable \(S\)-structures, that has the \textup{HP}, \textup{JEP}, and \textup{AP}. Then there is an \(S\)-structure \(\bf{F}\) (called a \emph{Fra{\"i}ss{\'e} limit of \(\c{K}\)}), unique up to isomorphism, such that \(\bf{F}\) is at most countable, \(\c{K}\) is the age of \(\bf{F}\), and \(\bf{F}\) is ultrahomogeneous.
\end{proposition}

To state the last result of this section, we need to recall some definitions.

\begin{definition}[Uniform local finiteness]
Let \(\c{K}\) be a class of similar structures. We say that \(\c{K}\) is \emph{uniformly locally finite} iff there is a function \(f \colon \bb{N}\to \bb{N}\), such that, for all \(\bf{A} \in \c{K}\), each \(n \in \bb{N}\), and every subset \(S\) of \(A\) with \(|S| \le n\), the substructure of \(\bf{A}\) generated by \(S\) has cardinality at most \(f(n)\).
\end{definition}

\begin{proposition}\label{qeomega}
Let \(S\) be a finite signature, let \(\c{K}\) be a uniformly locally finite class of \(S\)-structures with the HP, JEP, and AP, and at most countably many isomorphism types of finitely generated \(S\)-structures, and let \(\bf{F}\) be a Fra{\"i}ss{\'e} limit of \(\c{K}\). Then the first-order theory of \(\bf{F}\) is \(\aleph_0\)-categorical and has quantifier elimination.
\end{proposition}

\section{Main results}\label{mains}

First, we show that almost all finite nonassociative relation algebras are symmetric and have \(e\) as an atom. The proof and the observation that almost all nonassociative relation algebras in which \(e\) as an atom are symmetric were discovered independently by the author, but this result was also conjectured by Roger Maddux in a private communication.

\begin{theorem}\label{aasi}
Almost all members of \(\sf{FAS}\) are in \(\sf{FSIAS}\).
\end{theorem}

\begin{proof}
Let \(n \ge 5\), let \(U\) be an \(n\)-element set, and let \(1 \le i < n\). Clearly, there are \(\binom{n}{i}\) ways~to~select \(i\) identity atoms. Now, let \(0 \le p \le \lfloor (n-i)/2 \rfloor\). There are at most  \(\binom{n-i}{2}^p\) involutions of \(U\) with \(p\) non-fixed pairs, i.e., with sets of the form \(\{u, f(u)\}\) with \(u \neq f(u)\), since \({\binom{n-i}{2}}^p\) is the number of \(p\) \emph{independent} selections of \(2\)-element sets of diversity atoms. Based on Proposition \ref{asprop}. 1. (b), there are \((2^i - 1)^n\) possible ways of selecting identity cycles, since each element of \(U\) must appear in at least one of the \(i\) cycles of the given form. Lastly, based on Proposition \ref{cyclecount}. 5., there are \(2^{Q(n-i+1,n-i+1-2p)}\) ways to select diversity cycles; the number of choices of diversity cycles in a \((n-i+1)\)-element structure satisfying \(|I|=1\) and a \(n\)-element structures such that \(|I| = i\) and \(|U \setminus I| = n-i\)  is clearly the same. Hence, by Proposition~\ref{basicRA}(5), the fraction of members of \(\sf{FAS}\) with universe \(\{1,\dots,n\}\) that belong to \(\sf{FSIAS}\) is bounded below by
\begin{align*}
&\frac{n2^{Q(n,n)}}{\sum_{i=1}^{n} \sum_{p = 0}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)}} \\
= & \frac{1}{(\sum_{i=1}^{n} \sum_{p = 0}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)})/n}
\end{align*}

We have
\begin{align*}
& \quad\,\,  \frac{1}{n}\sum_{i=1}^{n} \sum_{p = 0}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)} \\
& = \frac{1}{n} \sum_{p=0}^{\lfloor (n-1)/2 \rfloor} n \binom{n-1}{2}^p 2^{Q(n,n-2p)-Q(n,n)} \\
& \quad\,\, +  \frac{1}{n}\sum_{i=2}^{n-1} \sum_{p = 0}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)}.
\end{align*}
Clearly,
\[
\binom{n-1}{2}^p \le \bigg(\frac{n^2}{2}\bigg)^p.
\]
Now,
\begin{align*}
Q(n,n-2p) - Q(n,n) &= \frac{1}{6}(n-1)((n-1)^2 + 3(n-2p) - 1) - \frac{1}{6}(n-1)((n-1)^2 + 3n -1) \\
&= \frac{1}{6}(n-1)((n-1)^2 +3n-6p-1 -((n-1)^2 +3n-1)) \\
&= \frac{1}{6}(n-1)(3n -6p - 3n) \\
&= (1-n)p,
\end{align*}
hence
\begin{align*}
\bigg( \frac{n^2}{2} \bigg)^p 2^{Q(n,n-2p)- Q(n,n)} &= \bigg( \frac{n^2}{2} \bigg)^p 2^{(1-n)p} \\
&= \bigg(\frac{n^2}{2^n}\bigg)^p.
\end{align*}
Since \(n \ge 5\), we have \(0 < n^2/2^n < 1\), so the formula for a geometric sum gives
\begin{align*}
\frac{1}{n} \sum_{p=0}^{\lfloor (n-1)/2 \rfloor} n \binom{n-1}{2}^p 2^{Q(n,n-2p)-Q(n,n)}&\le \sum_{p=0}^{\lfloor (n-1)/2 \rfloor} \bigg(\frac{n^2}{2^n}\bigg)^p \\
&= \frac{1 - (n^2/2^n)^{\lfloor (n-1)/2 \rfloor +1}}{1 - n^2/2^n}.
\end{align*}
Using basic limits, \(n^2/2^n\) and \((n^2/2^n)^{\lfloor (n-1)/2 \rfloor +1}\) tend to \(0\), and so
\[
 \lim_{n \to \infty} \bigg(\frac{1 - (n^2/2^n)^{\lfloor (n-1)/2 \rfloor +1}}{1 - n^2/2^n}\bigg) = 1.
\]

Define \(S(m) \mathrel{\mathop:}= Q(m,m)\), for each \(m \in \bb{N}\). We have
\begin{align*}
S(m)  &= \frac{1}{6}(m-1)((m-1)^2+3m-1) \\
&= \frac{1}{6}(m-1)(m^2-2m+1+3m-1)\\
&= \frac{1}{6}(m-1)(m^2+m) \\
&= \frac{1}{6}(m^3-m),
\end{align*}
for each \(m \in \bb{N}\). Using the formula for a difference of cubes, we get
\begin{align*}
S(n-i+1) - S(n) & = \frac{1}{6}((n-i+1)^3 - (n-i+1) - n^3+n)\\
& = \frac{1}{6}((n-i+1-n)((n-i+1)^2 + n(n-i+1) + n^2) + i - 1) \\
& = \frac{1}{6}(-(i-1)(n^2 -2(i-1)n + (i-1)^2 + n^2 - (i-1)n + n^2) + i-1)\\
& = \frac{1}{6}(-3(i-1)n^2 + 3(i-1)^2n - (i-1)^3 + i-1).
\end{align*}
In particular,
\begin{align*}
i = 2 & \implies S(n-i+1) - S(n) = - \frac{1}{2}n^2+\frac{1}{2}n, \\
i = 3 & \implies S(n-i+1) - S(n) = -n^2+2n-1, \\
i = 4 & \implies S(n-i+1) - S(n) = - \frac{3}{2}n^2+ \frac{9}{2}n -4.
\end{align*}
If \(1 \le i < n\) and \(1 \le p \le \lfloor (n-i)/2 \rfloor\), then \(\lfloor (n-i)/2 \rfloor \le n\),  \(\binom{n}{i} \le n^n\), \(\binom{n-i}{2}^p \le (n^2)^n = n^{2n}\), \((2^i-1)^n \le 2^{in}\), and \(Q(n-i+1,n-i+1-2p) \le S(n-i+1)\). Over the interval \([1,\infty)\), \(x \mapsto (x^3-x)/6\) is increasing, hence \(S(n-i+1)-S(n)\) is maximised when \(i\) is minimised. Since \(n \ge 5\), we have \(n-1 \ge 4\) and \(2^n > 1\). Hence, using the formula for a geometric sum, we get
\begin{align*}
& \quad \,\, \frac{1}{n}\sum_{i=2}^{n} \sum_{p = 0}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)} \\
& \le \frac{1}{n} \sum_{i=2}^{n} \sum_{p=0}^{\lfloor (n-i)/2 \rfloor} n^n  n^{2n}  (2^i-1)^n 2^{S(n-i+1)-S(n)} \\
& \le \frac{1}{n} \sum_{i=2}^{n} n  n^{3n}  (2^i-1)^n 2^{S(n-i+1)-S(n)} \\
& = \sum_{i=2}^{n} n^{3n} (2^i-1)^n 2^{S(n-i+1)-S(n)} \\
& = n^{3n} 3^{n} 2^{-n^2/2+n/2} + n^{3n}7^{n} 2^{-n^2 + 2n -1} + \sum_{i=4}^{n} n^{3n} (2^i-1)^n 2^{S(n-i+1)-S(n)} \\
& \le n^{3n} 3^{n} 2^{-n^2/2+n/2} + n^{3n}7^{n} 2^{-n^2 + 2n -1} + \sum_{i=4}^{n} n^{3n} 2^{in} 2^{-3n^2/2+9n/2-4} \\
& \le n^{3n} 3^{n} 2^{-n^2/2+n/2} + n^{3n}7^{n} 2^{-n^2 + 2n -1} + n^{3n}2^{-3n^2/2+9n/2-4} \sum_{i=0}^{n} (2^n)^i \\
& = n^{3n} 3^{n} 2^{-n^2/2+n/2} + n^{3n}7^{n} 2^{-n^2 + 2n -1} + n^{3n}2^{-3n^2/2+9n/2-4} \frac{2^{n(n+1)}-1}{2^n-1}.
\end{align*}
We have
\[
 n^{3n} 3^{n} 2^{-n^2/2+n/2} = 2^{3n\log_2(n) + \log_2(3)n-n^2/2+n/2},
\]
which clearly tends to \(0\). Similarly,
\[
n^{3n} 7^n  2^{-n^2+2n+1} = 2^{3n\log_2(n)+\log_2(7)n -n^2+2n-1},
\]
which tends to \(0\). Lastly,
\begin{align*}
n^{3n}2^{-3n^2/2+9n/2-4} \frac{2^{n(n+1)}-1}{2^n-1} &= 2^{3n\log_2(n) - n^2/2+11n/2-4} \frac{2^{-n^2-n}(2^{n^2+n}-1)}{2^n-1} \\
& = 2^{3n\log_2(n)-n^2/2 +11n/2 -4} \frac{1-2^{-n^2-n}}{2^{n}-1}.
\end{align*}
Now, it is clear that \(2^{3n\log_2(n)-n^2/2+9n/2-4}\) tends to \(0\) and \((1-2^{-n^2-n})/(2^n-1)\) tends to \(0\), hence the term above has limit \(0\). Combining these results with basic limits, we get
\[
\lim_{n\to \infty} \Bigg( \frac{1}{n}\sum_{i=1}^{n} \sum_{p = 1}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)}\Bigg) \le 1,
\]
and so
\[
\lim_{n \to \infty} \Bigg(\frac{1}{(\sum_{i=1}^{n} \sum_{p = 1}^{\lfloor (n-i)/2 \rfloor} \binom{n}{i} \binom{n-i}{2}^p (2^i - 1)^n 2^{Q(n-i+1,n-i+1-2p)-Q(n,n)})/n}\Bigg) \ge 1.
\]
By the Squeeze Principle, the fraction of members of \(\sf{FAS}\) with universe \(\{1,\dots,n\}\) in \(\sf{FSIAS}\) tends to \(1\). Combining these results, Proposition~\ref{rigid}, Proposition~\ref{atcm}, and Proposition~\ref{firasumm}(1), we find that almost all members of \(\sf{FAS}\) belong to \(\sf{FSIAS}\), which is what we wanted.
\end{proof}

Combining this with Proposition \ref{atcm} and Proposition \ref{firasumm}. 2., we obtain the following.

\begin{corollary}\label{symint}
Almost all finite nonassociative relation algebras are symmetric integral relation algebras.
\end{corollary}

Using Proposition \ref{asprop} and Proposition \ref{asymp}, we obtain the following.

\begin{corollary}
\(2^{Q(n,n)}/(n-1)!\) is an asymptotic formula for the number of \(n\)-atom nonassociative relation algebras. The same formula holds if we add the assumption of associativity, symmetry, or \(e\) being an atom.
\end{corollary}

Next, we aim to establish a \(0\)--\(1\) law for \(\sf{FAS}\). Based on Proposition~\ref{aasi}, it will be enough to establish a \(0\)--\(1\) law for \(\sf{FSIAS}\). We essentially follow the method outlined in the introduction of Bell and Burris in \cite{bellburr}. For the completeness proof required for this method, we will make use of a Fra{\"i}ss{\'e} limit. It will be convenient to work with the class \(\sf{FSIAS}_e\) rather than \(\sf{FSIAS}\), then translate the result, as \(\sf{FSIAS}\) does not have the HP. First, we show that a limit exists.

\begin{lemma}
\(\sf{FSIAS}_e\) has a Fra{\"i}ss{\'e} limit.
\end{lemma}

\begin{proof}
By Theorem \ref{fraisse}, it is enough to show that \(\sf{FSIAS}_e\) has the HP, JEP, and AP.

By definition, \(\sf{FSIAS}_e\) is the class of finite members of a universal class. Based on this,  \(\sf{FSIAS}_e\) is closed under forming substructures, so \(\sf{FSIAS}_e\) clearly has the HP.

For the AP, let \(\bf{S},\bf{V},\bf{W} \in \sf{FSIAS}_e\) and let \(\mu \colon \bf{S} \to \bf{V}\) and \(\nu \colon \bf{S} \to \bf{W}\) be embeddings. Without loss of generality, we can assume that \(V \cap W = S\) and \(\mu\) and \(\nu\) are inclusion maps. Thus, we can define \(\bf{U} \mathrel{\mathop:}= \inn{U; f^\bf{U},e, T^\bf{U}}\), where  \(U \mathrel{\mathop:}= V \cup W\),  \(f^\bf{U}\) is given by
\[
f^\bf{U}(x) = \begin{cases}  f^\bf{V}(x) & \rm{if } x \in V \\ f^\bf{W}(x) & \rm{if } x \in W, \end{cases}
\]
and \(T^\bf{U} \mathrel{\mathop:}= T^\bf{V} \cup T^\bf{W}\).  Let \(a,b,c \in U\) and assume that \((a,b,c) \in T^\bf{U}\). By construction, we~have (\(a,b,c \in V\) and \((a,b,c) \in T^\bf{V}\)) or (\(a,b,c \in W\) and \((a,b,c) \in T^\bf{W}\)). In the first case,

\noindent \((f^\bf{U}(a),c,b), (c,f^\bf{U}(b),a) \in T^\bf{U}\), since \(T^\bf{V} \subseteq T^\bf{U}\), \(f^\bf{U}{\upharpoonright}_V = f^\bf{V}\), and \(\bf{V}\) satisfies (IP). Similarly, we have \((f^\bf{U}(a),c,b),(c,f^\bf{U}(b),a) \in T^\bf{U}\) in the second case, so \(\bf{U}\) satisfies (IP). Let \(a \in U\). If \(a \in V\), then we have \((a,e,a) \in T^\bf{U}\), since \(T^\bf{V} \subseteq T^\bf{U}\) and \(\bf{V}\) satisfies (II). Similarly, \((a,e,a) \in T^\bf{U}\) when \(a \in W\). Since \(U = V \cup W\), it follows that \((a,e,a) \in T^\bf{U}\) in every case. Lastly, let \(a,b \in U\) such that \((a,e,b) \in T^\bf{U}\). By construction, (\(a,b \in V\) and \((a,e,b) \in T^\bf{V}\)) or (\(a,b \in W\) and \((a,e,b) \in T^\bf{W}\)). Since \(\bf{V}\) and \(\bf{W}\) satisfy (II), we have \(a = b\), so (II) holds. Since \(\bf{V}\) and \(\bf{W}\) both satisfy \(f(x) \approx x\), it follows that \(f^\bf{V}\) and \(f^\bf{W}\) are both identity maps. By construction, \(f^\bf{U}\) is an identity map, so \(\bf{U} \models f(x) \approx x\). By definition, \(U = V  \cup W\), hence \(|U| \le |V|+|W|\). Thus, \(U\) is finite. Based the above results, we have \(\bf{U} \in \sf{FSIAS}_e\). Clearly, the inclusion maps \(\imath_V \colon V \to U\) and \(\imath_W \colon W \to U\) are embeddings and \(\imath_V \circ \mu =  \imath_W \circ \nu\). Combining these results, we find that \(\sf{FSIAS}_e\) has the AP, which is what we wanted to show.

Clearly, \(\sf{FSIAS}_e\) contains a trivial structure. This structure embeds into all \(\bf{A} \in \sf{FSIAS}_e\), so the JEP follows from the AP. Thus, \(\sf{FSIAS}_e\) has the HP, JEP and AP, as required. \end{proof}

\begin{definition}[\(\bf{L}_\sf{SI}\), \(T_\sf{SI}\), and \(S_\sf{SI}\)]
Let \(\bf{L}_\sf{SI}\) be a Fra{\"i}ss{\'e} limit of \(\sf{FSIAS}_e\),  let \(T_\sf{SI}\) be the first-order theory of \(\bf{L}_\sf{SI}\), and let \(S_\sf{SI}\) be the almost sure theory of \(\sf{FSIAS}_e\).
\end{definition}

The elements of \(\sf{FSIAS}_e\) are symmetric, so generated substructures contain at most one extra element, namely \(e\). So, by Proposition \ref{qeomega}, we have the following.

\begin{corollary}
\(T_\sf{SI}\) is \(\aleph_0\)-categorical and has quantifier elimination.
\end{corollary}

Next we introduce what Bell and Burris call extention axioms in \cite{bellburr}. The sentences essentially assert that a substructure can be extended by a single point in all possible ways. Here \(\neg^0\) and \(\neg^1\) mean no symbol and \(\neg\), respectively.

\begin{definition}
Let \(A_\sf{SI}\) be the set of first-order sentences of the form
\[
\forall x_1,\dots, x_m\colon \bigcurlywedge_{i=1}^m x_i \not\approx e  \to \exists y \colon y \not\approx e \curlywedge \Bigg( \bigcurlywedge_{i=1}^m y \not\approx x_i \Bigg)\curlywedge \neg^c T(y,y,y) \curlywedge{}
\]
\[ \Bigg( \bigcurlywedge_{i = 1}^{m} \neg^{c_i} T(x_i,y,y) \Bigg) \curlywedge \Bigg( \bigcurlywedge_{1 \le i \le j \le m} \neg^{c_{ij}} T(x_i, x_j, y) \Bigg),
\]
where \(m \in \omega\) and \(c, c_i, c_{ij} \in \{0,1\}\), for all \(1 \le i \le j \le m\).
\end{definition}

\begin{lemma}\label{axfraisse}
Let \(\bf{L}\) be countable model of \textup{(II)}, \textup{(IP)}, and \(f(x) \approx x\). Then \(\bf{L} \cong \bf{L}_\sf{SI}\) if and only if \(\bf{L} \models A_\sf{SI}\).
\end{lemma}

\begin{proof}
For the forward direction, assume that \(\bf{L} \cong \bf{L}_\sf{SI}\). Then \(\bf{L}\) is a Fra{\"i}ss{\'e} limit of \(\sf{FSIAS}_e\), hence the age of \(\bf{L}\) is \(\sf{FSIAS}_e\) and \(\bf{L}\) is ultrahomogeneous.  Let \(n \in \omega\), let \(c,c_i, c_{ij} \in \{0,1\}\), for all \(1 \le i \le j \le n\), and let \(u_1,\dots, u_n \in L\setminus \{e^\bf{L}\}\). Let \(\bf{U}\) denote the substructure of \(\bf{L}\) generated by \(U \mathrel{\mathop:}= \{u_1,\dots,u_n\}\). Fix some element \(v \notin U\) and define \(\bf{V} \mathrel{\mathop:}= \inn{V; f^\bf{V}, e^\bf{V}, T^\bf{V}}\), where \(V \mathrel{\mathop:}= U \cup \{e^\bf{F},v\}\), \(f^\bf{V} = \mathrm{id}_V\), \(e^\bf{V} = e^\bf{L}\), and \(T^\bf{V}\) is given by
\[
 \begin{cases} T^\bf{U} \cup [v,e^\bf{L},v]\cup [v,v,v]\cup \Big(\bigcup \{ [u_i,v,v] \mid c_i = 0\} \Big)\cup \Big(\bigcup\{ [u_i, u_j, v] \mid c_{ij} = 0\}\Big) & \rm{if } c = 0 \\
 T^\bf{U} \cup [v,e^\bf{L},v] \cup \Big(\bigcup \{ [u_i,v,v] \mid c_i = 0\} \Big)\cup \Big(\bigcup\{ [u_i, u_j, v] \mid c_{ij} = 0\}\Big) & \rm{if } c = 1.
\end{cases}
\]
Since the age of \(\bf{L}\) is \(\sf{FSIAS}_e\), we must have \(\bf{U} \in \sf{FSIAS}_e\), so it is easy to see that \(\bf{V} \in \sf{FSIAS}_e\). Again, since the age \(\bf{L}\) is \(\sf{FSIAS}_e\), there is a substructure \(\bf{W}\) of \(\bf{L}\) that is isomorphic~to~\(\bf{V}\). Let~\(\mu \colon \bf{V} \to \bf{W}\) be an isomorphism. Then \(\mu \circ\imath_U \) is an isomorphism from \(\bf{U}\) to the substructure of \(\bf{L}\) generated by \(\mu[U]\). As \(\bf{L}\) is ultrahomogeneous, \(\mu \circ\imath_U \) extends to an automorphism, say \(\nu\). Then, by construction, \(\nu^{-1}(\mu(v))\) is the witness to the sentence from \(A_\sf{SI}\) given by \(n\), \(c\), and each \(c_i\), and \(c_{ij}\), when choosing  \(x_i = u_i\), for each \(1 \le i \le n\). Thus, \(\bf{L} \models A_\sf{SI}\).

Conversely, assume that \(\bf{L} \models A_\sf{SI}\). To show that \(\bf{L} \cong \bf{L}_\sf{SI}\), we need to show that \(\sf{FSIAS}_e\) is the age of \(\bf{L}\) and that \(\bf{L}\) is ultrahomogeneous. As \(\bf{L}\) is a symmetric model of (IP) and (II), the age of \(\bf{L}\) is a subset of \(\sf{FSIAS}_e\). Assume, for a contradiction, that this inclusion is proper. Let \(\bf{U}\) be an element of minimal size in \(\sf{FSIAS}\) that is not in the age of \(\bf{L}\). Clearly, \(|U| > 1\). Now, let \(u \in U \setminus \{e^\bf{U}\}\) and let \(\bf{V}\) denote the substructure of \(\bf{U}\) generated by \( V \mathrel{\mathop:}= U \setminus \{u\}\). By our minimality assumption, \(\bf{V}\) embeds into \(\bf{L}\). Let \(\mu \colon \bf{V} \to \bf{L}\) be such an embedding. Let \(m \mathrel{\mathop:}= |V|-1\), let \(\{v_1,\dots,v_m\}\) be an enumeration of \(V\setminus \{e^\bf{V}\}\), let
\[
c \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [u,u,u] \subseteq T^\bf{U}\\ 1 & \rm{if } [u,u,u] \nsubseteq T^\bf{U}, \end{cases}
\]
let
\[
c_i \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [v_i,u,u] \subseteq T^\bf{U} \\ 1 & \rm{if } [v_i,u,u] \nsubseteq T^\bf{U}, \end{cases}
\]
for all \(1 \le i \le m\), and let
\[
c_{ij} \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [v_i,v_j,u] \subseteq T^\bf{U} \\ 1 & \rm{if } [v_i,v_j,u] \nsubseteq T^\bf{U}, \end{cases}
\]
for all \(1 \le i \le j \le m\). As \(\bf{L} \models A_\sf{SI}\),  \(\bf{L}\) satisfies the sentence defined by \(c\) and each \(c_i\) and \(c_{ij}\), so there is a witness, say \(y\), for the choice of \(x_i \mathrel{\mathop:}= \mu(v_i)\), for all \(1 \le i \le m\). By construction, the substructure of \(\bf{L}\) generated by \(\mu[V] \cup \{y\}\) is isomorphic to \(\bf{U}\). Thus, \(\bf{U}\) embeds into \(\bf{L}\), so \(\bf{U}\) is in the age of \(\bf{L}\), contradicting our assumption.

Lastly, based on Lemma \ref{homogequiv}, it will be enough to establish that \(\bf{L}\) is weakly homogeneous.  Let~\(\bf{U} \le \bf{V}\) be finitely generated substructures of \(\bf{L}\) and let \(\mu \colon \bf{U} \to \bf{L}\) be an embedding. Note that since \(\bf{L}\) is symmetric, \(e\) is the only new element that can be generated by a subset.  If \(\bf{U} = \bf{V}\), then we are done. Assume that \(\bf{U} \neq \bf{V}\) and fix some \(v \in V \setminus U\). Let \(m \mathrel{\mathop:}= |U|-1\), let \(\{u_1,\dots,u_m\}\) be an enumeration of \(U\setminus \{e^\bf{U}\}\), let

\[
c \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [v,v,v] \subseteq T^\bf{U}\\ 1 & \rm{if } [v,v,v] \nsubseteq T^\bf{U}, \end{cases}
\]
let
\[
c_i \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [u_i,v,v] \subseteq T^\bf{U} \\ 1 & \rm{if } [u_i,v,v] \nsubseteq T^\bf{U}, \end{cases}
\]
for all \(1 \le i \le m\), and let
\[
c_{ij} \mathrel{\mathop:}= \begin{cases} 0 & \rm{if } [u_i,u_j,v] \subseteq T^\bf{U} \\ 1 & \rm{if } [u_i,u_j,v] \nsubseteq T^\bf{U}, \end{cases}
\]
for all \(1 \le i \le j \le m\).  As \(\bf{L} \models A_\sf{SI}\),  \(\bf{L}\) satisfies the sentence defined by \(c\) and each \(c_i\) and \(c_{ij}\), so there is a witness, say \(y\), for the choice of \(x_i = \mu(u_i)\), for all \(1 \le i \le m\). By construction, the map \(\nu \colon U \cup \{v\} \to L\) given by
\[
\nu(x) = \begin{cases} \mu(x) & \rm{if } x \in U \\ y & \rm{if } x = v \end{cases}
\]
embeds the substructure of \(\bf{V}\) generated by \(U \cup \{v\}\) into \(\bf{L}\). By assumption, \(\bf{V}\) is finite, hence \(\mu\) can be extended to an embedding \(\nu \colon \bf{V} \to \bf{L}\) by repeating this construction. Thus, \(\bf{L}\) is weakly homogeneous, which is what we wanted to show.
\end{proof}

Based on Theorem \ref{fraisse}, we have the following.

\begin{corollary}
Together, \textup{(IP)}, \textup{(II)}, \(f(x) \approx x\), and \(A_\sf{SI}\) form a \(\aleph_0\)-categorical and therefore complete theory.
\end{corollary}

\begin{lemma}
\(A_\sf{SI} \subseteq S_\sf{SI}\).
\end{lemma}

\begin{proof}
Let \(n \in \bb{N}\),  let \(m \in \omega\), let \(c, c_i, c_{ij} \in \{0,1\}\), for all \(1 \le i \le j \le m\), and let \(\sigma\) be the sentence from \(A_\sf{SI}\) defined by these parameters. Clearly, given non-identity elements \(x_1,\dots,x_m, y \in \{1,\dots,n\}\) such that \(y \neq x_i\), for all \(1 \le i \le n\), at most
\[
1+m+\frac{m^2+m}{2} = \frac{m^2+3m+2}{2}
\]
cycles are forced to be consistent for \(\sigma\) to be satisfied for these given choices. Thus, the fraction of structures failing the sentence with these choices is below \(1-2^{-(m^2+3m+2)/2}\). There are \((n-1)^m\) ways to select \(x_1,\dots,x_m\), \(n-m-1\) ways to select \(y\) given \(x_1,\dots,x_m\), and the choices of \(y\) are independent once \(x_1,\dots,x_n\) are selected. Based on these results, the fraction of structures not modelling \(\sigma\) is bounded above by
\[
(n-1)^m (1-2^{-(m^2+3m+2)/2})^{n-m-1}.
\]
Clearly, \((n-1)^m \le n^{m} = 2^{m\log_2(n)}\) and \(n-m -1 < n\), so this quantity is below
\[
2^{m \log_2(n) + n\log_2(1-2^{-(m^2+3m+2)/2}) }.
\]
Since \(1-2^{-(m^2+3m+2)/1} < 1\), we have \(\log_2(1 -2^{-(m^2+3m+2)/2}) < 0\), hence
\[
\lim_{n \to \infty} 2^{m \log_2(n) + n\log_2(1-2^{-(m^2+3m+2)/2}) } = 0.
\]
By the Squeeze Principle, the fraction of structures not modelling \(\sigma\) tends to 0. Thus, \(A_\sf{SI} \subseteq S_\sf{SI}\), as claimed.
\end{proof}

So, based on Proposition \ref{fraisse} and Lemma \ref{axfraisse}, we have the following.

\begin{corollary}\label{01e}
\(S_\sf{SI}\) is \(\aleph_0\)-categorical, and therefore complete. Thus, \(\sf{FSIAS}_e\) has a \(0\)--\(1\) law.
\end{corollary}

This result can be translated to a result for \(\sf{FSIAS}\).

\begin{corollary}
\(\sf{FSIAS}\) has a \(0\)--\(1\) law.
\end{corollary}

\begin{proof}
Let \(\bf{U} \in \sf{FSIAS}\),  let \(e^\bf{U}\) be the unique element of \(I\), let \(\bf{U}_e \mathrel{\mathop:}= \inn{U; f^\bf{U}, e^\bf{U}, T^\bf{U}}\), let \(\sigma\) be a \(\{f,T,I\}\)-sentence, and let \(\sigma_e\) be the \(\{f,e,T\}\)-sentence obtained from \(\varphi\) by replacing all occurences of \(I(x)\), for some  \(x\), with \(x \approx e\). By construction,  \(\bf{U} \models \varphi\) if and only if \(\bf{U}_e \models \varphi_e\). As there is a one-to-one correspondence between isomorphism classes in \(\sf{FSIAS}\) and \(\sf{FSIAS}_e\), this observation and Corollary~\ref{01e} tell us that \(\sf{FSIAS}\) has a \(0\)--\(1\) law, as required.
\end{proof}

Hence, by Theorem \ref{aasi}, we have the following.

\begin{corollary}\label{01}
\(\sf{FAS}\) has a \(0\)--\(1\) law.
\end{corollary}

\section{Further work}

Perhaps the most obvious open problem in this area is problem of determining whether almost all nonassociative relation algebras are representable; this problem is mentioned by Maddux in \cite{perspecRA} and by Hirsch and Hodkinson in \cite{rabg}. A possible first step to solving this problem could be solving the corresponding problems for the classes of feebly and qualitatively representable algebras introduced by Hirsch, Jackson, and Kowalski in \cite{quali}.

\begin{problem}
Determine whether almost all nonassociative relation algebras are feebly, qualitatively, or (strongly) representable.
\end{problem}

Determining whether or not Corollary~\ref{01} extends to the classes of nonassociative relation algebras and relation algebras would also be an interesting problem.

\begin{problem}
Determine whether \(\sf{NA}\) and \(\sf{RA}\) have \(0\)--\(1\) laws.
\end{problem}

\section{Acknowledgements}

This is a pre-print of a paper contributed to the Journal of Symbolic Logic. The final authenticated version is available at: \url{https://doi.org/10.1017/jsl.2021.81}. The author would like to thank Roger Maddux for suggesting the result from Corollary~\ref{symint}.

redpajama_set_name: RedPajamaArXiv


text:
// Copyright (c) Microsoft. All rights reserved.
// Licensed under the MIT license. See LICENSE file in the project root for full license information.

/*
 * This header implements bond::detail::any class which provides similar functionality to
 * the boost::any which (1) uses user-defined CTTI instead of a built-in RTTI and (2) uses
 * user-defined storage size for small object optimization.
 */

#pragma once

#include <bond/core/config.h>

#include "mpl.h"

#include <bond/core/config.h>

#include <boost/assert.hpp>
#include <boost/static_assert.hpp>
#include <boost/utility/enable_if.hpp>

#include <cstdint>
#include <type_traits>

namespace bond
{
namespace detail
{

template <template <typename> class TypeId, std::size_t Size>
class any
{
    BOOST_STATIC_ASSERT(Size >= sizeof(void*));

public:
    any() BOND_NOEXCEPT
        : _id{}
    {}

    any(const any& other)
    {
        emplace_uninitialized(other);
    }

    template <typename T>
    any(const T& value)
    {
        emplace_uninitialized(value);
    }

    ~any()
    {
        destroy();
    }

    any& operator=(const any& other)
    {
        return assign(other);
    }

    template <typename T>
    any& operator=(const T& value)
    {
        return assign(value);
    }

    bool empty() const BOND_NOEXCEPT
    {
        return _id == 0;
    }

    explicit operator bool() const BOND_NOEXCEPT
    {
        return !empty();
    }

    template <typename T>
    T* cast() BOND_NOEXCEPT
    {
        return TypeId<T>::value == _id ? functions::template table<T>::unsafe_cast(*this) : nullptr;
    }

    template <typename T>
    const T* cast() const BOND_NOEXCEPT
    {
        return TypeId<T>::value == _id ? functions::template table<T>::unsafe_cast(*this) : nullptr;
    }

    bool operator==(const any& other) const
    {
        return _id == other._id && (empty() || _functions.compare(*this, other));
    }

    bool operator!=(const any& other) const
    {
        return !(*this == other);
    }

private:
    using storage = typename std::aligned_storage<Size>::type;

    struct functions
    {
        void (*destroy)(any& x);

        void (*assign)(any& x, const any& other);

        void (*emplace_uninitialized)(any& x, const any& other);

        bool (*compare)(const any& x, const any& y);

        template <typename T, bool IsSmall>
        struct impl;

        template <typename T>
        struct impl<T, true>
        {
            static T* unsafe_cast(any& x) BOND_NOEXCEPT
            {
                BOOST_ASSERT(TypeId<T>::value == x._id);
                return static_cast<T*>(x.data());
            }

            static const T* unsafe_cast(const any& x) BOND_NOEXCEPT
            {
                BOOST_ASSERT(TypeId<T>::value == x._id);
                return static_cast<const T*>(x.data());
            }

            static void emplace_uninitialized(any& x, const T& value)
            {
                new (x.data()) T{ value };
            }

            static void destroy(any& x)
            {
                unsafe_cast(x)->~T();
            }
        };

        template <typename T>
        struct impl<T, false>
        {
            static T*& unsafe_cast(any& x) BOND_NOEXCEPT
            {
                BOOST_ASSERT(TypeId<T>::value == x._id);
                return *static_cast<T**>(x.data());
            }

            static const T* unsafe_cast(const any& x) BOND_NOEXCEPT
            {
                BOOST_ASSERT(TypeId<T>::value == x._id);
                return *static_cast<const T* const*>(x.data());
            }

            static void emplace_uninitialized(any& x, const T& value)
            {
                *static_cast<T**>(x.data()) = new T{ value };
            }

            static void destroy(any& x)
            {
                T*& ptr = unsafe_cast(x);
                delete ptr;
                ptr = nullptr;
            }
        };

        template <typename T>
        struct table : impl<T, (sizeof(T) <= sizeof(storage))>
        {
            BOOST_STATIC_ASSERT(TypeId<T>::value != 0);

            using base = impl<T, (sizeof(T) <= sizeof(storage))>;

            template <typename U = T, typename boost::enable_if<std::is_copy_assignable<U> >::type* = nullptr>
            static void assign(any& x, const T& value)
            {
                *base::unsafe_cast(x) = value;
            }

            template <typename U = T, typename boost::disable_if<std::is_copy_assignable<U> >::type* = nullptr>
            static void assign(any& x, const T& value)
            {
                // TODO: Cache allocated buffer and reuse.
                base::destroy(x);
                emplace_uninitialized(x, value);
            }

            static void emplace_uninitialized(any& x, const T& value)
            {
                base::emplace_uninitialized(x, value);
                x._id = TypeId<T>::value;  // Update the id if emplace_uninitialized did not throw.
            }

            static functions make() BOND_NOEXCEPT
            {
                return
                {
                    base::destroy,
                    [](any& x, const any& other) { assign(x, *base::unsafe_cast(other)); },
                    [](any& x, const any& other) { emplace_uninitialized(x, *base::unsafe_cast(other)); },
                    [](const any& x, const any& y) { return *base::unsafe_cast(x) == *base::unsafe_cast(y); }
                };
            }
        };
    };

    const void* data() const BOND_NOEXCEPT
    {
        return &_storage;
    }

    void* data() BOND_NOEXCEPT
    {
        return &_storage;
    }

    /// The precondition is that *this is uninitialized (or destroyed).
    template <typename T>
    void emplace_uninitialized(const T& value)
    {
        functions::template table<T>::emplace_uninitialized(*this, value);
        _functions = functions::template table<T>::make();
    }

    /// The precondition is that *this is uninitialized (or destroyed).
    void emplace_uninitialized(const any& other)
    {
        if (!other.empty())
        {
            other._functions.emplace_uninitialized(*this, other);
            _functions = other._functions;
        }
        else
        {
            _id = {};
        }
    }

    template <typename T>
    any& assign(const T& value)
    {
        if (!try_assign_same(value))
        {
            // TODO: Cache allocated buffer and reuse.
            destroy();
            emplace_uninitialized(value);
        }

        return *this;
    }

    template <typename T>
    bool try_assign_same(const T& value)
    {
        if (_id == TypeId<T>::value)
        {
            functions::template table<T>::assign(*this, value);
            return true;
        }

        return false;
    }

    bool try_assign_same(const any& other)
    {
        if (_id == other._id)
        {
            if (!empty())
            {
                _functions.assign(*this, other);
            }

            return true;
        }

        return false;
    }

    void destroy()
    {
        if (!empty())
        {
            _functions.destroy(*this);
            _id = {};
        }
    }

    storage _storage;
    functions _functions;
    std::uint32_t _id;
};

template <typename T, template <typename> class TypeId, std::size_t Size>
inline T* any_cast(any<TypeId, Size>* x) BOND_NOEXCEPT
{
    BOOST_ASSERT(x);
    return x->template cast<T>();
}

template <typename T, template <typename> class TypeId, std::size_t Size>
inline const T* any_cast(const any<TypeId, Size>* x) BOND_NOEXCEPT
{
    BOOST_ASSERT(x);
    return x->template cast<T>();
}

} // namespace detail

} // namespace bond

redpajama_set_name: RedPajamaGithub


text:
Home Reports British MP accuses Saudi Arabia of violating human rights
British MP accuses Saudi Arabia of violating human rights
Al- Thawra Net
Helena Kennedy, a member of the House of Lords of the United Kingdom, has on Monday discussed a report by human rights groups referring to serious human rights violations in Saudi Arabia.
Kennedy said that the murder of Saudi journalist Jamal Khashoggi, inside his country's consulate in Istanbul, was orchestrated by senior officials, including Saudi Crown Prince Mohammed bin Salman, as confirmed by a report of the High Commissioner for Human Rights, Agnes Callamard.
Saudi Arabia has carried out 134 executions this year, 37 of which were against political opponents, who often were collectively executed. Aside from this, Saudi Arabia practices illegal detentions, enforced disappearances and imprisonments, Kennedy emphasized.
In early October, Khashoggi was killed inside the Saudi consulate in Istanbul, in a case that has shaken international public opinion and sparked widespread condemnation.
A few weeks ago, the UN High Commissioner for Human Rights published a 101-page report prepared by the UN Special Rapporteur on extrajudicial executions.
In it, the Commissioner blamed Saudi Arabia over the deliberate killing of Khashoggi, asserting that there was credible evidence that senior Saudi officials, including Crown Prince Mohammed bin Salman, were involved.
Previous articleQasef-K2 drones carry out retaliatory attacks on King Khalid Airbase in Khamis Mushayt
Next articleArmy and Popular Committees Captures 40 Saudi Mercenaries in Assir
Yemeni Armed Forces to announce details of military strike against United Arab Emirates
Over 25,000 Yemenis so far fell victim to Saudi-led aggression's cluster bombs
Security Council's statement on UAE military ship biased in favor of influential countries: FM

redpajama_set_name: RedPajamaCommonCrawl


Maxime Tétard, Romain Rachlin, Cyril Taïeb and François Dubois form the French graphic design studio Les Graphiquants.
Creative directors Maxime and Romain opened the studio after graduating from l'École nationale supérieure des Arts Décoratifs in 2008 to preserve and offer the most faithful vision of their creative intentions.
Recipients of several awards, they have built a solid reputation and following in the cultural and artistic domains, notably through a long-established collaboration with the Pinault Collection.
Les Graphiquants' pronounced originality and strong, yet meticulously calculated esthetic proposition, leaps out of traditional graphic design expectations and culture, offering a large display of commercial possibilities. Photography has become an increasing feature of their expanding portfolio and serves as yet another testament to their diverse talents.
Les Graphiquants dress projects and brands with distinctiveness and vision, gifting them with a curated and deliberate communication force.


But no political leaders did.
Prince Harry and Barack Obama go way back—but despite their hanging out together at Kensington Palace and interviewing each other on a BBC radio program, the former U.S. president and first lady did not receive an invitation to the royal wedding. But don't worry, no other political leaders did either, which means British Prime Minister Theresa May, President Trump, and Melania Trump will not be in attendance.
Palmer added that no one would be invited based on a position they held, and that Prince Harry and Meghan Markle will personally know everyone who attends their wedding.
Daily Mail royal reporter Rebecca English confirmed that the Obamas did not receive an invitation. This morning she tweeted: "President Obama & his wife, Michelle, will NOT be guests at Prince Harry & Meghan Markle's wedding next month—and nor will Donald Trump. The couple have decided not to invite any political leaders both from the UK & abroad, which means that even Br PM Theresa May is uninvited."
President Obama & his wife, Michelle, will NOT be guests at Prince Harry & Meghan Markle's wedding next month - and nor will Donald Trump.
The couple have decided not to invite any political leaders both from the UK & abroad, which means that even Br PM Theresa May is uninvited.
A source told English that it was not necessary for Prince Harry to have such a formal guest list because he is not a direct heir to the throne. English added that the Obamas "are very much looking forward to seeing them soon."
Hopefully we'll see another reunion—maybe a double date?—again soon after the honeymoon.


You are currently browsing the tag archive for the 'Dayasiri Jayasekara' tag.
I was privileged, a couple of weeks back, to attend the release of the Northern Education Sector Review Report at a ceremony held at Vembadi Girls School. I had last been at Vembadi in 2008, when the then Commander of the Special Forces in Jaffna, General Chandrasiri, arranged what was termed a Future Minds Exhibition. It was at the height of the war, but the General had already begun to plan for the future, and sensibly so for he stressed the need for the development of human resources.
I was struck by the irony now, with the controversy over his continuation as Governor. I will look at that issue elsewhere, but here I will dwell on the fact that the Provincial administration had invited him as Chief Guest, to be given the first copy of the report, and all the speeches made were in a spirit of cooperation. In particular the chair of the committee that had prepared the report, the distinguished athlete Nagalingam Ethirveerasingham, still described as the Olympian, emphasized that the recommendations of the Review were all within the framework of National Policy.
That having been said, the Review is masterly, in clearly identifying many of the problems we face, and suggesting simple remedies. But obvious though many of the pronouncements are, I fear that such an essentially sensible work could not have been produced in any other Province.
There are many reasons for this. I do not think there is any essential intellectual difference between those in the North and others in the country, but I do believe that the urgency of the problem with regard to education is better understood in the North. After all it was simplistic tampering with the education system that first roused deep resentments in the younger generation in the North (Prabhakaran's batch were the first victims of standardization), and the incapacity or unwillingness of successive governments since then to provide remedies has entrenched bitterness. And whereas Chandrasiri way back in 2008 understood the importance of action in this field, and entitled his Exhibition accordingly, he has since had to serve a political dispensation that cares nothing for the mind.


I've seen all of you excelling mothers out there with your homemade snow cream, handmade crafts and smocked little painters at easels happily painting the cold day away. Well done. Be sure to tell your troop leader you earned your Pinterest badge.
Don't mind me. I bitterly say that out of jealousy and shame.
An entire bag of blocks gets dumped in the middle of the kitchen floor.
Yes, that happened. I'm letting it happen.
I watch as he feeds the dog cheese from his highchair.
Yep. I just let it happen.
I didn't make it enough of a "Yes Day. " I didn't pick my battles. I let it get to me. Of course Yes Days can't happen every day. I would raise spoiled little assholes that would still be demanding I open applesauce pouches for them after college. But, sometimes? On some snow days? For my sanity? Yes. It's okay to have a Yes Day. It's even okay to toast to Yes Days with a drink. Cheers to spring being a few weeks away.
I'm all about the yeses this week. And last week. And whenever they're needed.
And this is why in Michigan, from about December-March my child watches far too much television and has far too many snacks. F U COLD AND SNOW!!!!


text:
Q: Shield UI Grid Hidden Columns ruins Cell alignments I have a shield-ui grid with a hidden column. Once I modify the data source ie perform a filter or do a sort my column sizes get messed up.
My problem can be seen on the demo website as well
Just sort the grid no problem, then hide a column and sort again and you will see the issue
https://demos.shieldui.com/aspnet/grid-columns/columns-show-hide
jQuery(function($) {

    $("#grid").shieldGrid({
        dataSource: {
            data: gridJson
        },
        sorting: {
            multiple: true
        },
        paging: {
            pageSize: 5,
            pageLinksCount: 4
        },
        selection: {
            type: "row",
            multiple: false,
            toggle: true
        },
        columns: [
            { field: "ServiceMarketingId", title: "ServiceMarketingId", width:"20%" },
            { field: "Code", title: "Code" , width:"20%"},
            { field: "Name", title: "Name", width:"20%" },
            { field: "MarketingName", title: "MarketingName", width:"20%" },
            { field: "Description", title: "Description" , width:"20%"}
        ],
        resizing: true

    });

    var dataSource = $("#grid").swidget().dataSource,
        timeout;

    $("#searchInput").on("keyup", function() {

        var val = $(this).val();
        console.log(val);
        clearTimeout(timeout);
        timeout = setTimeout(function() {
            dataSource.filter =  dataSource.filter = {
                or: [
                     { path: "Code", filter: "contains", value: val  },
                     { path: "Name", filter: "contains", value: val }
                ]
            }
            dataSource.read();
        }, 300);
    });

    $("#grid").swidget().hideColumn("ServiceMarketingId");

});

A: The Shield UI developers are working on fixing this bug and a fix should be available shortly.

redpajama_set_name: RedPajamaStackExchange


text:
Economist, author, journalist, documentary-maker and broadcaster.
David's capacity to empower everyone from business leaders to the ordinary 'Joe Soap' with greater knowledge and confidence is key to his success not just as a journalist, author, lecturer, broadcaster and documentary maker, but also as an international conference and keynote speaker of considerable repute.
Whether writing or speaking, David's style is informal and confident, designed to get you thinking – usually a bit differently – about how the world economy functions as an ecosystem. His particular skill lies in telling individual business sectors and industrial groups how best to deal with trends in growth, currencies and financial markets.
Born and based in Dublin, David studied economics at Trinity College Dublin and the College of Europe in Bruges, Belgium, before working as an economist first in the Irish Central Bank and later as Chief European Economist at UBS and Head of Emerging Markets Research at Banque Nationale de Paris.
He is well known for his prescience on economic matters at home and abroad. He was the first economist to foresee the 2008 crash of the Irish economy and its disastrous fallout, and also the first to predict a strong pro-Brexit vote in the U.K.
David's talent and ability are widely recognised. He was appointed a Young Global Leader by the WEF at Davos, serves as adviser to several major global investment funds and, currently, is Adjunct Professor at Trinity College Dublin in Global Economics.
He also written four bestsellers, presented award-winning documentaries and even brought economics to the stage (at Dublin's Abbey Theatre). His talents extend to festivals too, with the now-famous Kilkenomics, an eclectic combination of economics and stand-up comedy that counts as the world's only economics festival; and the Dalkey Book Festival.
As part of his mission to make economics accessible, David is extremely active on social media, with almost a quarter of a million followers on Twitter alone. He writes two economics columns a week, is a regular contributor to the Financial Times, and publishes his own daily global financial markets newsletter, GlobalMacro360.com.
Partnering with Udemy, David has an new online economics course – "Economics without Boundaries". His series of animated economics videos – Punk Economics – has also, to date, been viewed by 700,000 people, while his weekly column in the Irish Times is one of the most widely read in Irish media.
Renaissance Nation: Why Ireland has been the most Successful Economy in the Western World in the past 30 years
Brexit: The Implications
Globalisation: Trade and the Future of the World Economy
What Happens when Global Interest Rates Rise: Financial Markets, Financial Fragility and the Next Great Crash
Embracing Disruption: Finding Opportunities in Crises
What people say about David McWilliams
"On behalf of the National Recruitment Federation and the World Employment Conference we would like to thank you very much for taking part in the World Employment Conference 2018: Managing the Extended Workforce . We enjoyed your presentation which was thought provoking and highly relevant to current discussions in the field. The initial feedback from delegates has been very positive and I am sure your contribution will have stimulated ongoing reflection and discussion among delegates."
World Employment Conference
"I can honestly say that the feedback was extremely positive and if you saw our photos of the attendees and their reaction to David's speech you would see how much they enjoyed it. Certainly in my discussions with our guests the concept identified by David of the "Good Room" really resonated with them. It is a real skill/art form to speak without any prompts for over an hour in such an engaging way so please send David on our thanks for his contribution."
W&R Barnett Ltd
"We were most fortunate in having you as our guest speaker on the occasion of this celebration. It was a great success and we were thrilled with the feedback from our guests. All in attendance found you most entertaining and some people were talking about you for days afterwards…again David, thank you very much and continued success for the future."
McCarthy and David Kiely, Directors Jennings O'Donovan & Partners Consulting Engineers
Would you like to share this info about David McWilliams?


text:
Wild men with tatts arise to star in new feature film
HANDY TALENT: A still from the film Rise that was shot in Ipswich and at the Borallon prison using local extras. Contributed
by jgould
Joel Gould Journalist Joel is a journalist with 20 years of experience and since February, 2011 he has been the late reporter at The Queensland Times. Joel specialises in longer features and as the late reporter he chases all the breaking stories that unfold in the evenings. A die hard rugby league fan, Joel has been pushing hard for the Western Corridor bid to be admitted into the NRL.
2nd Sep 2014 5:00 AM
THE wild men of Ipswich with hearts of gold get their chance to shine in the largely Ipswich-shot film Rise.
Having a plethora of tattoos can be a drawback to getting some employment gigs, but not in the case of the Mack Lindon-directed movie, which was shot on location at Borallon Correctional Centre.
The film's co-producer, Maya Weidner, said there was plenty of quality material to work with when auditions were held in the Ipswich mall for prisoners for the jail shots.
"We needed the look. We wanted tattoos, and the more natural tattoos people had the less make-up we had to do on them," she said.
"At these auditions we had bikies come in and guys that had their sleeves, necks and faces tattooed. A lot of people have preconceived ideas, but the beautiful thing was that the biggest bikies and the scariest looking guys were the sweetest ones to work with on set.
"And that is what our whole film is about… changing the perceptions of people."
The city's reputation as the Hollywood of south-east Queensland is growing by the day and the film is another in the long line of movies to be filmed in the district.
The film was shot in the park at South St, in Limestone St and at North Ipswich Primary School.
Weidner said the support of the Ipswich City Council was a massive boost.
"We got in touch with Stephanie Hoffmann at the council, who is the film liaison officer," she said.
"She was beautiful. She had us come in for a meeting and would go out of her way to organise things for us.
"We felt like we were a priority, and it is lovely to feel that way.
"We had days where there were 150 extras from Ipswich that came out to the set.
"That local support was just beautiful, and we knew we would get that in Ipswich as opposed to anywhere else."
The film, based on the shocking real life experience of the director who was falsely accused of rape and had his conviction overturned, is still one of hope.
"I am a young female who chose to work on this film… and even though the themes are dark, it is about hope and overcoming," Weidner said
"It is a story about somebody who is falsely imprisoned, and what they do to get out of that and the relationships they build."
Lindon's part is played by Nathan Wilson, a QUT fine arts acting graduate.
Martin Sacks - well known for his acting in Blue Heelers, Wentworth and a host of other productions - plays the father figure role of an old criminal in the prison. Marty Rhone plays an old 1970s pop star confined to a wheelchair with a brain injury.
Lindon said Ipswich provided the ideal atmosphere for a director to work.
"Mayor Paul Pisasale's attitude and the way he welcomes the industry with open arms is massive," he said.
"In other areas there is a lot of red tape you have to go through, whereas he is welcoming to the creative arts. That trickles down to all levels of council.
"We used the old magistrates' court... and Borallon was an amazing facility that we were able to access for a three-week period."
The Brisbane premiere of the film is on November 4 at Palace Cinemas.
*See risemovie.com.au
borallon correctional centre
premium_icon Council slammed after workers fined 'half a week's wage'
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This adorable plush toy purifies corrupted souls. It will truly become a spiritual friend if always kept close.
Phantom Howl Massive Dark damage to a foe.
Slight chance to inflict Lethal HitInstant death for the victim.
Great Mystic mainhand, useful for farming in Angel Halo.
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A 30 round magazine, left, and a 10 round magazine, right, rest below an AR-15 rifle.
A federal judge in San Diego struck down California's ban on high-capacity gun magazines on Friday.
The ruling dealt a sizable blow to a law passed by the state legislature in 2016, which banned possession of gun magazines containing more than 10 bullets.
On Friday, U.S. District Judge Roger T. Benitez ruled that the law violated the 2nd Amendment rights of gun owners and issued an injunction barring the state from enforcing it.
To discuss what comes next is San Diego Union Tribune reporter Greg Moran.


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Posted on March 16, 2010 at 8:22 pm by Shanna Bright / Business, Humor, Lifestyle / 0
I received another e-mail today from one of the owners addressed to "Shana". I have politely and formally asked three times now to spell my name correctly and frankly, it's getting on my nerves. I know my name is unique, but that extra N really isn't so difficult. It's not like my name is Shanikwatara, but spelled with a Q but no U only on the day of the full or new moon. I think addressing one of your top-level executives incorrectly is rude and disrespectful. I've been told it's not that big of a deal and it's not the end of the world. Don't worry about it?
I was set to write a different blog entry on this topic, one which focused at the heart of the issue – why my name is not being spelled correctly – but I landed on Google and did a search for the name Shanna and Shana. Wow! Both have some cool results! So forget the fact that my name continues to be misspelled at work – look at what I found!
It turns our that the name Shanna was famous in Shanna the She-Devil of Marvel Comics. I think that's hilarious!!! Look at how awesome she is – Dang! Wish I was that voluptuous! According to Wikipedia:
Shanna the She-Devil is a fictional jungle adventuress in the Marvel Comics universe. (Thank goodness she's fictional!)
Shanna O'Hara Plunder is the daughter of a diamond miner named Gerald O'Hara. Born in Africa, she spent the majority of her childhood growing up in the jungles of Zaire. At the age of six, her father went to kill a rogue leopard that belonged to her mother, Patricia O'Hara. While hunting for the leopard, Shanna's father accidentally killed her mother. This traumatic incident led to Shanna's lifelong crusade against the use of firearms. After the incident, Shanna moved back to the United States to live with relatives. Shanna grows up to become an accomplished Olympic athlete, specializing in competitive swimming and track and field. She then became a licensed veterinarian.
My goodness! I don't think I can compare to that! What a woman!
I also remember a really trashy romance novel I had for years on my bookshelf with the title Shanna. Here's the description from Amazon:
Behind the foreboding walls of Newgate Prison, a pact is sealed in secret — as a dashing and doomed criminal consents to wed a beautiful heiress . . . in return for one night of unparalleled pleasure.
In the fading echoes of hollow wedding vows, a promise is broken — as a sensuous free-spirit flees to a lush Caribbean paradise, abandoning the handsome stranger she married to the gallows. But Ruark Beauchamp's destiny is now eternally intertwined with his exquisite, tempestuous Shanna's. And no iron ever forged can imprison his magnificent passion . . . and no hangman's noose will deny him the ecstasy that is rightfully his.
Very sexy stuff. Written by trashy romance novelist Kathleen E. Woodiwiss in 1977, whose other titles include best sellers Ashes in the Wind, Forever in Your Embrace and Moonlight Enchantress. (photo below) Goodness gracious!
Also for your reference, Wikipedia lists:
Shanna is an American feminine given name, and may refer to:
Shanna Collins (21st century), American actress
Shanna Compton (21st century), American poet
Shanna Crossley (born 1983), American professional basketball player
Shanna McCullough (born 1960), American pornographic actress
Shanna Moakler (born 1975), American model, actress and reality television star
Shanna Reed (born 1956), American dancer and actress
Shanna Slone (21st century), American singer
Shanna Swendson (21st century), American author
Shanna the She-Devil, a fictional jungle adventuress in the Marvel Comics universe
Shanna Wylie (How can you call her baby) Waiting to exhale cd
That's quite a list! But with S.H.A.N.N.A. posing such a challenge for some, I thought I would look up S.H.A.N.A to see if this version of my name had any fame attached to it. And what do you know – Shana is a really awesome Japanese animae character! Look at how cute she is (main photo above)! Shakugan no Shana means Burning-Eyed Shana. Her existence is a bit more complicated than the She-Devil:
The female lead of the show and a Flame Haze. Initially she is concentrated solely on her duty as a Flame Haze, going as far as to not bearing even a name. She sees Yuji only as a Torch, an object, marginally interesting because of his status as a Mystes. However, she slowly opens up to Yuji, coming to think of him as a person, despite being a Torch, and eventually comes to love him. Until she met Yuji, she was identified by only her Flame Haze title "Flame-Haired Blazing-Eyed Hunter," or her nodachi, Nietono no Shana (贄殿遮那, "Vairocana of the Offering Room"). Yuji, in his need to humanize her, chose to name her "Shana" after her sword.
Shana has a habit of saying "Urusai! Urusai! Urusai!" which means "Shut up! Shut up! Shut up!" whenever Yuji asks her questions or makes remarks that disturb her original point of view. But in reality they really care for each other. She has a great fondness for melon bread. Shana later comes to the startling conclusion that she loves Yuji, and even confesses it to him in the last episode of the first anime series when she thought she was going to die, but he does not hear what she said. Shana is continually torn between her mission to protect the balance, and her love for Yuji. Due to a decision she made near the end of the first season of the anime version, the closeness they once had begun to erode as Yuji kept a distance from her because of it. Yuji adopts a viewpoint similar to Shana's in the beginning of the series. However, Yuji comes to realize that he was wrong, and their relationship seems to be restored.
I love that she shouts "urusai" all the time. Poor little Shana seems to have issues with men. I totally identify with her! ;p
While this complicated little j-heroine seems to be the only noteworthy Shana on the internet (save the town Shana in Tehran, Shana Banana who makes banana bread and Shana Morrison who is apparently a "famous" singer), I rather like her image and her feisty little character. Considering my splendid five years in Japan and my connection with special eyes, I think I might just "go with the flow" and start using Shana with one N. Of course the minute I do this, the owner will probably purposely start spelling it with two, just for kicks.
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Uhhh yeah. I've done it again. Last time.
Woke up this morning with a funky feeling in my tummy. As I walked through the house to let the dogs out all I could think of were the various random chores I really wanted to get done. And how much I didn't want to go to work. A phone call was made, and here I am. Comprising a list.
That's quite the list. Not all has to be completed today but I'd like to get a jump of most of that. Oh yeah…setup the new blog. I just arrived here at wordpress.com and I love it. So much so, in fact, that I'm graduating to my own domain with it. A friend has offered me a deal I can't refuse with his hosting company and so I'm migrating. It won't happen for a day or two due to your standard DNS issues…but it's in the works.
In other news, my ankle is terribly swollen. I suspect that I was bit by a spider the other day when I was out hiking; in fact, I know I was bit by something. The reaction has been pretty severe, though. My ankle hurts and is a solid 50% larger than normal. This is complete with a loss in range of motion. Bummer, because tonight is a Splash-N-Dash at Tempe Town Lake.
Alright. Time for some coffee to help kickstart this "free day." Heidi's back tomorrow and I gotta get this place looking decent.


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While we were up in Minnesota, my beloved picked up a copy of David Carr's The Night of the Gun. I finished it last night. Carr, the media columnist for The New York Times, has written an amazing, harrowing story of how he survived drug addiction. Clearly written in response to the James Frey debacle, Carr's memoir directly challenges the notion that anyone's memories, much less those of former addicts, are reliable. Rather than simply write what he recalled, Carr decided to "report it out" - to go back and interview people from his past, dig up old arrest reports, rehab records, anything that might tell him what actually happened. Much of what he discovers surprises him; for example, he discovers a fifth stint in rehab when he'd always told himself it was only four.
All of this is interesting enough, but one of the most striking things for me is the book's setting: Twin Cities journalism in the 1980s and 1990s. Carr wrote for, and eventually edited, The Twin Cities Reader. In reading his description of The Reader's drab, suburban highway office park digs, I realized I had been there on a story the summer I interned for a local TV station. I recall thinking that the place looked more like a day care center than an office; toys everywhere, nobody really appearing to do any work. Carr's description of life at The Reader does little to dissuade that initial impression.
Not only did I recognize locations and names of local Twin Cities journalists, musicians and newsmakers, I found reference to two of my former college teachers (one of whom is now incarcerated). And, in classic St. Small fashion, I realized about two-thirds of the way through the book that Carr is a cousin of one of my best friends from high school.
This memoir about tricks of memory called up a lot of them for me. I'll end with this eerie past meets present moment. Carr's book mentioned Deborah Howell, who during much of the 80s was an editor at the St. Paul Pioneer Press; I knew Deborah because she worked for my dad. Just as I was reading the section of the book that mentions Howell, she was killed in a car accident while on vacation with her husband in New Zealand. Carr's touching reminiscence on his blog points out that the Pioneer Press won a Pulitzer in 1988 for a pioneering feature that Howell edited called "AIDS in the Heartland."
All this reading about Twin Cities journalism, plus Howell's death, caused me to remember something I hadn't thought about in years: I was in the newsroom when that Pulitzer was officially announced. I had gone to my dad's office to have our picture taken; I had won a small college scholarship from Knight-Ridder, the parent corporation, and they wanted a picture of us for the house organ. I don't know if it was happenstance or if Dad rigged it this way, but when we were done he said, "Hang around and come to the newsroom with me. Something exciting is going to happen." And so the news came across the wires (were they actually still called wires then? I don't know). People cheered. Champagne flowed. Speeches were made. And Jacqui Banaszynski and Jean Pieri got their due.
A random memory, an event I hadn't thought about in years. Somewhere my parents have a copy of that picture; I'm standing next to my dad, smiling, wearing a white and blue striped sweater.


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Alighting on serious risks
Wayne Beglan and Sam Fowles examine the Planning Court's recent judgment in an important judicial review concerning the correct approach of planning decision-makers in the context of sites presenting serious health and safety risks.
In (1) Valero Logistic UK Ltd (2) Greenergy Terminals Ltd v Plymouth City Council [2021] EWHC 1792 (Admin) Mrs Justice Thornton held that the Council had taken sufficient account of the risks posed and accordingly, the Court was not entitled to revisit the planning merits of the Council's decision. This case is a further demonstration of the need to avoid an overly prescriptive, legalistic, and hypercritical approach to risk assessment in Planning Committee discussions and Planning Officer reports.
The Defendant local planning authority, Plymouth City Council, granted planning permission for a development comprising a change of use from a private helipad to commercial heliport at a site in Plymouth. The Claimants operated distilled fuel storage depots some 125 metres away from the helipad. They sought to challenge the grant of planning permission in light of the potentially catastrophic consequences of a helicopter crashing onto highly flammable fuel. The fuel depots were "establishments" regulated under the Control of Major Accident Hazard Regulations 2015 ("COMAH sites") due to the serious risks to human health and the environment presented by the products handled and stored there.
The Interested Party, Halo Aviation Ltd, applied for planning permission on 20 May 2020. The application envisaged two or three commercial helicopter passenger landings per week, subject to evening and weekend limitations. Part of the proposed flight path involved helicopters flying over the bow of vessels discharging fuel. The Interested Party therefore submitted an agreement between it and the statutory harbour authority, as well as a specimen permission from the Civil Aviation Authority permitting landing and taking off for helicopters in a congested area.
Article continues below...
The Planning Officer sent the Interested Party's risk assessment and the proposed flight paths to the Civil Aviation Authority and sought its views. The Planning Officer, in recommending the grant of planning permission subject to planning conditions, observed that the proposal provided "tangible betterment, ensuring flights are carried out in a structured and considerate way by an operator who is fully accountable to the Civil Aviation Authority". Following a meeting of the Planning Committee, planning permission was granted subject to 23 conditions, including the establishment of a direct line of communication between the heliport and the neighbouring COMAH sites to increase awareness of aircraft activity. Limitation on the number of flight paths and landings per year were also imposed.
The Claimants argued that:
The Council failed to consider the risks posed by the proximity of the development to COMAH sites because the officer and planning committee relied on the aviation regulation scheme (and Civil Aviation Authority as the regulatory body) to ensure that the proposed heliport was operated safely.
The Council misconstrued the "fallback position" as "unrestricted and unregulated" helicopter flights from the site (under private ancillary use and/or permitted development) because the existing helipad fell within the curtilage of the existing building thereby taking it outside the General Permitted Development Order 2015 (Sch. 2, Pt. 4, Para. b).
It was irrational for the Council to rely on other regulatory regimes when granting permission. In particular, given the proposed aerodrome would be "unlicensed", the CAA would have insufficient power to regulate the risks.
The proposed development was not in accordance with the development plan and no other material consideration justified granting permission.
The decision was irrational.
Mrs Justice Thornton found:
On Ground 1: It was clear that the Planning Officer and Committee Members recognised that the risks to the COMAH sites from a helicopter crash were a principal issue in their consideration of the planning application. Both the officer's report and committee meeting shows consideration of the regulatory scheme, the CAA's role, the flight paths, and types and class of helicopter. It is well-established that the reports of Planning Officers must not be subject to "hypercritical analysis" and it was noted that the Planning Committee "are not specialist risk assessors".
On Ground 2: Thornton J declined to allow this argument on the basis that the point had been raised at a comparatively late stage of the judicial proceedings, and that it was at any rate a speculative submission: the most which could be said was that the helipad "seems" to be within the curtilage of a building.
On Ground 3: The Claimant was correct to say that the CAA could not regulate the proposed development as a licensed aerodrome under the Air Navigation Order 2016. The proposed development was, however, in a "congested area". This brought it within the CAA's regulatory ambit under Regulation 5 of the Rules of the Air Regulations 2015. The Council was, therefore, entitled to rely on the CAA and the relevant regulatory scheme to ensure that the proposed aerodrome remained safe in relation to the COMAH sites.
Grounds 4 and 5 were essentially parasitic on the first three grounds.
Mrs Justice Thornton accordingly dismissed the claim.
This case provides important guidance for both lawyers and local authorities. It identifies the standard to which local planning authorities must conform when making decisions about development that is covered by other regulatory schemes and the approach that should be taken to development in the vicinity of COMAH sites.
Wayne Beglan and Sam Fowles acted on behalf of the successful local planning authority. Philip Coppel QC and Olivia Davies acted on behalf of the Claimants. All are barristers at Cornerstone Barristers.
This article represents the perspective of the Defendants. The claimants reserve their right to appeal.
(c) HB Editorial Services Ltd 2009-2020


Join us for a live music performance with Larry and Elaine Conger, mountain dulcimer artists. This is a FREE (w/ admission) event. Learn more about Larry and Elaine here.
Live music events are weather dependent.


TIPTON — Encompass Credit Union can now serve anyone who lives, works, worships or attends school in Tipton, Clinton or Howard counties.
Daylight saving time will begin Sunday, March 11 at 2 a.m. Indiana residents are encouraged to include changing their smoke alarm batteries when changing their clocks forward an hour.
Hoosier homeowners, renters, businesses and private non-profit organizations that sustained uninsured damage caused by severe storms and flooding starting on or after February 15 can report damage online at: http://bit.ly/2018rainfall.
The Howard County Sheriff Department and Kokomo Police Departments are joining law-enforcement agencies across Indiana this March to increase dangerous and impaired driving patrols for the NCAA Tournament and St. Patrick's Day.


During the 19th and 20th centuries, water filters for domestic water production were generally divided into slow sand filters and rapid sand filters (also called mechanical filters and American filters). While there were many small-scale water filtration systems prior to 1800, Paisley, Scotland is generally acknowledged as the first city to receive filtered water for an entire town. The Paisley filter began operation in 1804 and was an early type of slow sand filter. Throughout the 1800s, hundreds of slow sand filters were constructed in the UK and on the European continent. An intermittent slow sand filter was constructed and operated at Lawrence, Massachusetts in 1893 due to continuing typhoid fever epidemics caused by sewage contamination of the water supply. The first continuously operating slow sand filter was designed by Allen Hazen for the city of Albany, New York in 1897. The most comprehensive history of water filtration was published by Moses N. Baker in 1948 and reprinted in 1981.
In the 1800s, mechanical filtration was an industrial process that depended on the addition of aluminum sulfate prior to the filtration process. The filtration rate for mechanical filtration was typically more than 60 times faster than slow sand filters, thus requiring significantly less land area. The first modern mechanical filtration plant in the U.S. was built at Little Falls, New Jersey for the East Jersey Water Company. George W. Fuller designed and supervised the construction of the plant which went into operation in 1902. In 1924, John R. Baylis developed a fixed grid backwash assist system which consisted of pipes with nozzles that injected jets of water into the filter material during expansion.
Filters use sieving, adsorption, ion exchanges and other processes. Unlike a sieve or screen, a filter can remove particles much smaller than the holes through which the water passes.
Types of water filters media filters, screen filters, disk filters, slow sand filter beds, rapid sand filters and cloth filters.
Point-of-use filters for home use include granular-activated carbon filters (GAC) used for carbon filtering, metallic alloy filters, microporous ceramic filters, carbon block resin (CBR), microfiltration and ultrafiltration membranes. Some filters use more than one filtration method. An example of this is a multi-barrier system. Jug filters can be used for small quantities of drinking water. Some kettles have built-in filters, primarily to reduce limescale buildup.
Three organizations are accredited by the American National Standards Institute, and each one of them certify products using ANSI/NSF standards. Each ANSI/NSF standard requires verification of contaminant reduction performance claims, an evaluation of the unit, including its materials and structural integrity, and a review of the product labels and sales literature. Each certifies that home water treatment units meet or exceed ANSI/NSF and Environmental Protection Agency drinking water standards. ANSI/NSF standards are issued in two different sets, one for health concerns (such as removal of specific contaminants (Standard 53, Health Effects) and one for aesthetic concerns (Aesthetic Effects, such as improving taste or appearance of water). Certification from these organizations will specify one or both of these specific standards.
NSF International: The NSF Water treatment Device Certification Program requires extensive product testing and unannounced audits of production facilities. The goal of this program is to provide assurance to consumers that the water treatment devices they are purchasing meet the design, material,and performance requirements of national standards.
Underwriters Laboratories: Underwriters Laboratories, Inc., is an independent, accredited testing and certification organization that certifies home water treatment units which meet or exceed EPA and ANSI/NSF drinking water standards of contaminant reduction, aesthetic concerns, structural integrity, and materials safety.
Water Quality Association:The Water Quality Association is a trade organization that tests water treatment equipment, and awards its Gold Seal to systems that meet or exceed ANSI/NSF standards for contaminant reduction performance, structural integrity, and materials safety.
Filters that use reverse osmosis, those labeled as "absolute one micron filters," or those labeled as certified by an American National Standards Institute (ANSI)- accredited organization to ANSI/NSF Standard 53 for "Cyst Removal" provide the greatest assurance of removing Cryptosporidium. As with all filters, follow the manufacturer's instructions for filter use and replacement.
Water filters are used by hikers, by aid organizations during humanitarian emergencies, and by the military. These filters are usually small, portable and light (1-2 pounds/0.5-1.0 kg or less), and usually filter water by working a mechanical hand pump, although some use a siphon drip system to force water through while others are built into water bottles. Dirty water is pumped via a screen-filtered flexible silicon tube through a specialized filter, ending up in a container. These filters work to remove bacteria, protozoa and microbial cysts that can cause disease. Filters may have fine meshes that must be replaced or cleaned, and ceramic water filters must have their outside abraded when they have become clogged with impurities.
These water filters should not be confused with devices or tablets that are water purifiers, some of which remove or kill viruses such as hepatitis A and rotavirus.
The term water polishing can refer to any process that removes small (usually microscopic) particulate material, or removes very low concentrations of dissolved material from water. The process and its meaning vary from setting to setting: a manufacturer of aquarium filters may claim that its filters perform water polishing by capturing "micro particles" within nylon or polyester pads just as a chemical engineer can use the term to refer to the removal of magnetic resins from a solution by passing the solution over a bed of magnetic particulate. In this sense, water polishing is simply another term for whole house water filtration systems.


In this tour, you will visit two major sightseeing spot in Nagasaki. Glover Garden is one of the most famous sightseeing spot in Nagasaki and it became as World Heritage in 2015. It is perched on Minamiyamate which is located at hillside overlooking the harbor in Glover Garden. Glover House, built in 1863, is the oldest remaining colonial style building in Japan and has been designated as an Important Cultural Asset.
After visiting Glover Garden, you will Inasayama Observatory. Mt.Inasa is a 333 meter high mountain that offers great views of the Nagasaki`s city center and one of the new three major night views in Japan. The round mountaiontop observatory offers an amazing view that is virtually a 360 degree aerial take on the city and its beautiful surrounding that changes from to season. It takes about five minutes to reach to the top of the mountain through the ropeway.