text:
\section{Introduction}\label{intro}
The genetic code arose through evolution.
We think of it as being universal, optimal, and highly redundant~\cite{Sella2006}.
The mechanics of the evolution of life on Earth means that all organisms share the same genetic code in which each codon produces the same amino acid.
We call this the \textbf{standard genetic code}.
(While there are deviations from the standard genetic code~\cite{exception}, these involve only a handful of codons being coded differently and will not be the subject of our investigations here.)
Because of selection pressures, the standard genetic code has self optimized to minimize errors in translation and  transcription~\cite{1inamillion,Freeland2003,Haig1991}.
From theoretical considerations, \citet{woese} showed that the standard genetic code is related to a property called the \textit{polar requirement}, which has subsequently been corroborated by experiment~\cite{Mathew2008} and shown to be highly optimal when considering one type of error: point mutations~\cite{butler}.
It can be considered as part of an abstract chemical property of the genetic code~\cite{sella}.
Following~\citet{nigel} and~\citet{vestigian}, in this paper, we consider the code's optimality in terms of these features.

\citet{sella} attempted to model the evolution of the genetic code.
This was done through considering the coevolution between the genetic code and the encoding of a protein within a closed model system.
This allows for complex dynamics between mutations of messages
and selection on proteins in order to minimize the lethal effects of these mutations.
This minimizes the errors through mutations and allows protein networks to develop to promote a higher likelihood of survival.
We seek to understand whether the algorithm based on this coevolutionary model can be phrased as a purely physical problem of dynamical evolution.
To address this we must first discuss the algorithm.

The algorithm described by~\citet{nigel} models the evolution of the genetic code through \textbf{horizontal gene transfer (HGT).}
This allows organisms to exchange genetic information via DNA through transferring the segments of a genome to each other within the same generation through various mechanisms.
This is used in concert with the Code Message Coevolution Model dynamics described by~\citet{sella} to obtain an iterative discrete time algorithm.
Implementation of the algorithm demonstrates convergence of the genetic code to a highly optimised and near universal solution.
This solution is an attractor.
Horizontal gene transfer is crucial to achieving this solution.
This result provides a testable model for understanding the standard genetic code.

The dynamics from the coevolution model along with the additional communication incorporated by the iterative discrete time algorithm governs the evolution of genetic code states (genetic code configurations) of which there is a finite number.
This allows us to treat the algorithm as a discrete dynamical system, as it is the time dependent dynamics of coevolving matrices which represent the genetic code state.
We will consider this algorithm as if it were a system out of equilibrium for which there is the emergence of an attractor solution in the space of genetic code mappings~\cite{strogatz2000nonlinear}.

The behavior we establish exemplifies the notion of \textit{universality}.
In statistical physics, systems with a large number of degrees of freedom exhibit universality in a scaling limit.
Historically, this idea originated in the theory of phase transitions and was made mathematical precise through the renormalization group.
(See, for example,~\cite{Wilson:1974mb,Wilson:1993dy,Goldenfeld:1992qy}.)
Starting from specific initial conditions, by integrating out degrees of freedom (\textit{e.g.}, through coarse-graining), we flow to a fixed point.
At the fixed point, the theory is scale invariant.\footnote{
This is a simplification.
At a fixed point of the renormalization group, the theory enjoys a larger symmetry, \textit{conformal invariance}, which includes scale or dilatation invariance.}
Very different physical systems can flow to the same fixed point in that correlation functions of local operators behave in an identical manner.
Such theories are said to belong to the same universality class.
Input parameters dictate how the system converges to universality.
It is therefore natural to examine how variations on these parameters influence the universality of the solution.
We will use approximate scale invariance as a tool to assess how close we are to universality and to diagnose features of the universal solution.

In this article, we investigate a mechanism for the origin of the genetic code that leads to universal behavior at late times.
We initially describe the model with considerations of universality as in \citet{nigel} by defining it in such a way that all entities within the algorithm converge to a single solution.
We then consider universality in a more formal manner through statistical mechanics.

Let us begin by stating the main results.
The first requirement for the convergence of the genetic code is a trivial observation that we make rigorous:
there must be more codons than amino acids.
The second requirement is that we must demand horizontal gene transfer to optimize the setup.
This corroborates the claims of~\citet{nigel}.
We also discover that universality in terms of scaling in the solution depends on the rate of mutations.
 It is largely independent of mistranslations of the genetic code.

The organization of this paper is as follows.
In Section~\ref{sec:two}, we recast the biological algorithm into a computational algorithm to which we can apply the principles of dynamical systems.
We reproduce the results from~\citet{nigel} and show that the initial conditions do not influence the algorithm's ability to flow to a near universal solution.
Our new results are in Section~\ref{sec:three}.
In particular, in Section~\ref{subsec:31}, we correct minor errors in the literature.
In Sections~\ref{subsec:32}--\ref{subsec:34}, we vary all the parameters in the model in order to examine their influence on the attractor mechanism.
In Section~\ref{subsec:36}, we discuss automorphisms and scaling in the genetic code.
We then illustrate the mechanism for universality in terms of scaling with an example.
In Section~\ref{subsec:38}, we re-examine universality and express this behavior in terms of the homogeneity of the fitness function.
We find that the universal solution is characterized by the rate of mutations and is largely independent of the mistranslation rate.
In Section~\ref{sec:four}, we conclude with a summary and directions for future work.
Finally, the Appendices collect the results of various experiments less central to the argument than those discussed in the main text.

\section{Modeling framework}\label{sec:two}
\numberwithin{equation}{section}

We begin with a precise rephrasing of the problem addressed by~\citet{nigel} into one of computation.
Emphasis will be on representing the aspects of the mathematical problem while minimizing the amount of biology introduced.

\subsection{Basic definitions}\label{subsec:21}

We model the set of codons making up DNA geometrically using a Hamming metric~\cite{baylis2017error}.

\begin{definition}\label{codons}
Let $i_b \in i $ be a set of elements (\textbf{bases}) forming an alphabet of length $|i|$.
We define a \textbf{codon} as a sequence of $n$ bases such that $c \in \mathcal{C}:=\{i_{b,1},\dots , i_{b',n} \}$.
The number of possible codons is $|\mathcal{C}| = |i|^n$.
\end{definition}
For codons in the standard genetic code, we have $|i|=4$ (A,C,G,T) and $n=3$ meaning that $|\mathcal{C}|=64$.
We define the set of codons, $\mathcal{C}$, lexicographically.
Note that there is an associated symmetry with a Hamming metric~\cite{automorph}.

We can next define the structure of the genetic code:
\begin{definition}\label{code}
Denoting the set of animo acids $a \in \mathcal{A}$ such that we have $|\mathcal{A}|$ amino acids, the mapping from codon space to amino acid space, $G : \mathcal{C} \rightarrow  \mathcal{A}$ is the \textbf{genetic code}.
We represent the map $G$ as matrix $\Delta_{c,a}$ with dimensions $ |\mathcal{C}| \times |\mathcal{A}|$ such that:
\begin{equation}
\Delta_{c,a}= \left\{
\begin{array}{lcl}
1 && \text{if}\ G(c) = a ~,\\
0 && \text{otherwise} ~.
\end{array} \right.
\label{eq:code}
\end{equation}
\end{definition}

We refer to $\Delta_{c,a}$ as the \textbf{delta matrix}.
This matrix defined in \eqref{eq:code} has one entry per row (as each codon can only map to a single amino acid) and no empty columns (we assume that every amino acid has been mapped to).
The map $G$ is surjective but not injective.
In Nature, we encounter $20$ amino acids in the genetic code.\footnote{
For the purposes of calculation, we treat the stop codons as mapping to a dummy amino acid, so in our language $|\mathcal{A}|=21$ in the standard genetic code.}
The codon TTT maps to the amino acid phenylalanine, for example.
This matrix can therefore be considered as a non-square, row-stochastic, binary matrix with no empty columns.
Notice that when summing over the columns of a row-stochastic matrix, we get $1$.
These properties place constraints on the information flow for optimization~\cite{bialek2012biophysics}.
Note that due to these constraints, we also must have $ |\mathcal{C}|\geq |\mathcal{A}|$.
Using the inclusion/exclusion principle~\cite{Allenby:2010:CIC:1942905}, the number of possible configurations of the delta matrix $\Delta_{c,a}$ is
\begin{equation}
\#_{\text{config}}= \sum_{j=0}^{|\mathcal{A}|}  (-1)^j  \binom{|\mathcal{A}|}{j}(|\mathcal{A}|-j)^{|\mathcal{C}|} \ .
\label{eq:config}
\end{equation}

During the map $G$, errors occur with a given probability:
\begin{equation}
\text{prob}(c \rightarrow a )= \sum_{c'} L_{c,c'}\Delta_{c',a}\ , \qquad
L_{c,c'}(\ell)=\left\{
\begin{array}{lcl}
\frac{\ell}{(n(|i|-1))} && \text{dist}(c,c')=1 ~, \\
1-\ell && \text{dist}(c,c')=0 ~, \\
0 && \text{otherwise} ~. \
\end{array}\right.
\label{eq:2.3}
\end{equation}
The parameter $\ell\in[0,1] \subset \mathbb{R}$.
The distance is defined using the Hamming metric such that:
\begin{equation}
\text{dist}(c,c') := \#(c_j\neq c'_j)_{j=1,\dots, n} \ .
\label{eq:2.4}
\end{equation}
The distance is the number of bases $i_b$ that differ between two codons $c$ and $c'$.
Here, $\ell$ represents a parameter for the probability of error, and $L_{c,c'}(\ell)$ is a bistochastic matrix, \textit{viz.}, a symmetric, non-negative matrix whose rows and columns sum to $1$.
The matrix $L_{c,c'}(\ell)$ is used in order to only consider nearest neighbors in codon space.
The number of codons with $\text{dist}(c,c')=1$ is given by  $n(|i|-1)$ as there are $n$ positions which can have $|i|-1$ different values.
We can encode this information in a \textit{Hamming graph} in which the $\mathcal{C} = |i|^n$ possible codons correspond to vertices and an edge joins vertices whose corresponding codons that differ by a single letter --- \textit{i.e.}, those codons at Hamming distance $1$.

In this algorithm the genetic code $G$ rearranges itself in order to minimize the likelihood that probabilistic nature of the map causes a differing amino acid $a$ to appear when mapping from codon space~\cite{Freeland2003,Haig1991}.
In this model we consider two forms of errors, both of which only occur on nearest neighbors ($\text{dist}(c,c')=1$).
They are the following:
\begin{enumerate}
\item \textbf{Mistranslation}: When a single base $i_b$ is read incorrectly.
We will denote this $T_{c,c'}$ and take $\ell \rightarrow \nu$, where $\nu$ is the rate of mistranslation.
\item \textbf{Point mutations}: A single base $i_b$ changes before being read.
We will denote this $M_{c,c'}$ and $\ell \rightarrow \mu$ where $\mu$ is the rate of mutations.
There are various kinds of point mutations.
\end{enumerate}
For simplicity, we will neglect excisions or insertions of bases.

\subsection{Fitness}\label{subsec:22}
Information is translated from genome to proteome.
For our purposes, these are sequences of codons and amino acids, respectively.
In particular,
\begin{definition}\label{genome}
A sequence $S_G = \{c_1, \ldots, c_M\}$ of length $M$ is called a \textbf{genome}, where each codon $c_x \in \mathcal{C}$ has a position $x$ in the sequence $\{1,\dots, M\}$.
A target amino acid $s(x)$ is the mapping under the genetic code $G$ of the codon at position $x$ to the amino acid $s(x) \in \mathcal{A}$.
The image under the genetic code map $G$ of the genome sequence $S_G$, gives a sequence $S_P =\{s_1, \ldots, s_M\}$ of target amino acids called the \textbf{proteome}, which is a subsequence of a protein.
\end{definition}
We denote the target amino acid $s(x)$ as $s$ in order to abbreviate notation.
The definition we quote above is a slight simplification of~\cite{nigel,vestigian,sella} as in this algorithm, we assume $s \equiv a \in \mathcal{A}$.
As the amino acids at each position in the sequence are indistinguishable \cite{nigel}, we can store details of the proteome and genome within the following objects:
\begin{definition}\label{codeusage}
A vector $L_s$ specifies the frequency of the target amino acid $s$ in a proteome sequence of length $M$.
The codon usage matrix $U_{c,s}$ specifies the frequency of a codon $c$ within a target amino acid $s$.
\end{definition}
Crucially, we encode all necessary information about the genome and proteome within these two objects without having to go through the respective sequences analytically.
The two matrices are both column stochastic, \textit{i.e.},
\begin{equation}
\sum_{s}^{|\mathcal{A}|} L_{s}= 1 ~, \qquad \sum_{c}^{|\mathcal{C}|} U_{c,s} = 1_s ~.
\label{eq2.5}
\end{equation}
\newpage
The notion of distance in amino acid information space is structurally ambiguous (not well defined like a Hamming metric).
Due to this we can define the topological distance between amino acids by the following $a_d \in [0,1]$, which can be randomly generated.
The notion of distance is normalized.
Using this we can define a \textbf{fitness matrix} as:
\begin{equation}
W_{a,s}= \Phi^{|a_d - s_d|} ~.
\label{eq2.6}
\end{equation}
As in~\citet{sella}, $\Phi$ is a parameter used to consider how an abstract physicochemical distance between amino acids scales into the fitness.
This makes the fitness matrix \eqref{eq2.6} some measure of how \enquote{useful} each arbitrary amino acid $a$ is instead of the target amino acid $s$.
Since $0 < \Phi \leq 1$, this is a positive symmetric matrix.
By considering the probability of mistranslations and the entire genome we can describe an overall fitness score~\cite{nigel}:
\begin {equation}
f=\prod_c \prod_s \{ \sum_{c'} \sum_{a}T_{c,c'} \Delta_{c',a}W_{a , s} \}^{ L_{s} U_{c,s}} \ .
\label{eq2.7}
\end{equation}
This product is taken component wise.

To measure how well a delta matrix performs, we define the \textbf{optimality score} $O$ as:
\begin{equation}
O= \sum_c \sum_{c'} (N_{c, c'}  \{ \sum_a \sum_b \Delta_{c,a}S_{a,b}\Delta_{c',b}^T\}) ~,
\label{eq2.8}
\end{equation}
which measures the average amino acid similarity between neighboring codons.
We define amino acid similarity as $S_{a,b}=\sum_s |W_{a , s}-W_{s,b}|$.
In~\eqref{eq2.8}, $N_{c, c'}$ is $1$ if two codons are nearest neighbors ($\text{dist}(c,c')=1$) and zero otherwise~\cite{nigel,vestigian}.
Note that it is a tautology that two isomorphic genetic codes give the same optimality score.
We return to this point in Section~\ref{subsec:36} below.

\subsection{The algorithm}\label{subsec:23}
Based on these mathematical preliminaries, we consider the following algorithm~\cite{nigel}.
\begin{enumerate}
\item \textbf{Construction:}
We can construct a set of $N$ objects each with their own genetic code $G$ and therefore delta matrix $\Delta_{c,a}$ and their own codon usage matrix $U_{c,s}$.

\item \textbf{Mixing:} We randomly select one object as the acceptor $A$ and a random subset $K$ of $N$ as the donors ($k \in K\subset N$) and run them through the iteration:
\begin{equation}\
( 1 - H)  U_{c,s}^A +  \frac{H}{K}\sum_{k \in K} U_{c,s}^{(k)} \to   U_{c,s}^{A} ~.
\label{eq2.7}
\end{equation}
$H$ represents the fraction of the genetic code due to horizontal gene transfer ($H\in[0,1]$).

\item \textbf{Fitness maximization:} We attempt an elementary code change to the delta matrix $\Delta_{c,a}$.
We do this by assigning one codon to a new amino acid.
This is done by reallocating a unit entry in $\Delta_{c,a}$ to a different position within that row of the matrix.
We accept the new code if and only if it preserves or increases the fitness score $f$, which has been calculated using the new $U_{c,s}^{A}$.
Otherwise, we keep the original delta matrix $\Delta_{c,a}$, if there are no new possibilities.

\item \textbf{Mutational equilibrium:} We can derive a new codon usage matrix $U_{c,s}^{A}$ from the new delta matrix $\Delta_{c,a}$ uniquely at mutational selection equilibrium.
We first derive a fitness matrix with respect to codons $F_{c,s}=\sum_{a}\Delta_{c,a}W_{a,s}$.
Using the Perron--Frobenius theorem, we calculate the column stochastic eigenvector corresponding to the largest eigenvalues, for the following matrix ($Q^s$):
\begin{equation}
Q_{c,c'}^s =\sum_{c''}M_{c,c''}\delta_{c'',c'}F_{c'',s} ~,
\label{eq:2.8}
\end{equation}
where $\delta_{c'',c'}$ is a Kronecker delta so that we consider the $s^{th}$ column of the matrix $F_{c'',s}$ as a diagonal matrix.
The index $s$ here is fixed and not a free index.
Each column stochastic eigenvector of $Q_{c,c'}^s$ corresponds to the $s^{th}$ column of $U_{c,s}^{A}$.
We normalize the eigenvector so that it is column stochastic) by setting the sum of elements to unity.

\item \textbf{Repetition:} We repeat steps $2$ through $4$ for $t$ time steps.
\end{enumerate}

\paragraph{Experimental setup:}
In this model there are $12$ parameters to generate and define.
These are tabulated as follows.
\begin{itemize}
\item \textbf{Space structure:}
$|i|$ and $n$ for the codon space,
$|\mathcal{A}|$ and $a_d$ for amino acid space,
and $L_s$ target amino acid frequency;
\item \textbf{Innovation pool structure:}
$N$ number of objects,
$K$ number of donors per iteration, and
$H$ fraction of genome that is similar due to horizontal gene transfer;
 \item \textbf{Noise and fitness parameters:}
 $\nu$, $\mu$, and $\Phi$;
 \item \textbf{Number of time steps:} $t$.
 \end{itemize}

\begin{figure}[H]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.5\linewidth]{H0first.png}
\includegraphics[width=0.5\linewidth]{H04first.png}
\end{tabular}
\caption{Evolution of optimality score for (a) $H=0$ (\textcolor{red}{red}) and (b) $H=0.4$ (\textcolor{blue}{blue}).
Both graphs were produced using the following parameters: $|i|=4$, $n=3$ giving $|\mathcal{C}|=64$, $|\mathcal{A}|=20$, $N=80$, $K=1$, $\nu=0.01$, $\mu=10^{-4}$, and $\Phi=0.99$.
$L_s$ and $a_d$ are the same for both graphs.}
\label{fig:one}
\end{figure}

We generate  $L_s$ and $a_d$ randomly.
The parameters $|i|$, $n$, $|\mathcal{A}|$, $N$, and $K$ are positive integers while $H$, $\nu$, $\mu$, and $\Phi$ take any value in the interval $[0,1]$.
We reproduce the results in~\citet{nigel} using their parameters as quoted in Figure~\ref{fig:one}.
The initial delta matrices $\Delta_{c,a}$ are identical at the start, as done in \citet{nigel}
The results in Figure~\ref{fig:one} show that when modeling without horizontal gene transfer (Figure \ref{fig:one} (a), red), the delta matrices $\Delta_{c,a}$ optimize themselves, but do not converge to a universal solution.
This is shown by the optimality scores, $O$, ranging from $0.7$ to $1.25$ and not changing after $1500$ time steps.
When including horizontal gene transfer (Figure \ref{fig:one} (b), blue) we get a set of optimality scores, $O$, that optimize on average more than without horizontal gene transfer (red, $H=0$).
The results display the attractor mechanism.
This is because the optimality score falls in a smaller range (between $0.75$ and $1$) and fluctuations continuing at the the $t=5000$ time step.

The time taken to produce these results was very large as we use $|i|=4$, $n=3$, and $|\mathcal{A}|=20$ giving us $64\times20$ matrices.
To perform a more careful analysis, we consider a toy model by reducing the matrix dimensions to $27\times9$.
This corresponds to the parameters $|i|=3$, $n=3$ (so $|\mathcal{C}|=27$) and $|\mathcal{A}|$=9.
We also set up the algorithm so that each entity has its own unique delta matrix $\Delta_{c,a}$ such that they all start with different initial optimality scores in order to see if the scores will still converge.
These results are in Appendix~\ref{app:a}.
They show some convergence after $5000$ time steps.
This suggests that the set of $\Delta_{c,a}$ can be arbitrary in order for optimised attractor mechanism to emerge.
This also points to the existence of an attractor mechanism at work for $H\ne 0$.

When performing the analysis all initial delta matrices $\Delta_{c,a}$ will be identical.
We will vary a single parameter from the set $\{|i|, n, |\mathcal{A}|, N, K, H, \nu, \mu, \Phi \}$ while keeping the others  fixed.
We generate $a_d$ and $L_s$ randomly for all runs.
We will take ten runs for each parameter and take the average.
The standard deviation will be used to analyze the spread (to measure the rate of convergence).
We use the standard deviation as it allows us to measure the spread of optimality scores in an intuitive manner, similar to the mean code distance used in \citet{nigel} as we highlight further in Section~\ref{subsec:36}.
We will take the average of the standard deviation over the ten runs.
These results are discussed in Sections~\ref{subsec:32}--\ref{subsec:34} below, where we consider universality to be represented by the ability for all entities within the algorithm to converge to a single solution.
The remainder of Section~\ref{sec:three} is devoted to re-defining and re-examining universality using statistical mechanics and the renormalisation group.

\section{Results and analysis}\label{sec:three}

\subsection{The model of Sella and Ardell}\label{subsec:31}

In an insightful paper, \citet{sella} develop a Code Message Coevolution Model that describes the impact of message mutation on the fitness of the genetic code.
The authors observe that at mutational equilibrium, there is a balance between mutations in messages and selection on proteins.
This model has been summarized in~\citet{ardell}.
The process involves calculating the column stochastic eigenvector corresponding to the largest eigenvalues as described in step~4 (mutational equilibrium) of the algorithm described in Section~\ref{subsec:23}.
We note for completeness and reproducibility two minor errata.
Example~A from~\citet{sella} consists of a model with the following setup.
There is a ring of five codons mapping to a ring of five amino acids with $\Delta_{c,a}$ being the identity matrix.
Note that we will denote the eigenvector as $U_{c,s}$, however, $s$ is fixed and is not a free index (as in our step~4).
To reproduce the results, we take $\Phi = 0.8^5$ and $\mu=0.01$.\footnote{
We are grateful to D.~Ardell for communications on this point.}
The resulting eigenvalues and eigenvectors are given in Table~\ref{tab:one}.
Note that at machine precision the eigenvectors sum to one ($\sum_c U_{c,s}=1$) as required.

\begin{table}[H]
\begin{center}
\par\medskip
\textbf{Eigenvalues and column stochastic eigenvectors for a set of $\Phi$ and $\mu$ }
\begin {tabular}{c c c c}
&
Scaling for an abstract
&
Rate of mutations $\mu$
&
Largest eigenvalue $\lambda^s$
\\
&
physicochemical between amino acids $\Phi$
\\ \hline
\textbf{1}
&
0.8
&\
0.1
&
0.9549
\\
\textbf{2}
&
$0.8^5$
&
0.01
&
0.9808
\end{tabular}
\begin{tabular}{c c}
&
Corresponding eigenvector ${U_{c,s}}^T$
\\[1ex] \hline
\textbf{1}
&
$\begin{bmatrix}
0.2635 & 0.2134 &  0.1549 &  0.1549 &  0.2134
\end{bmatrix}$
\\[1ex]
\textbf{2}
&
$\begin{bmatrix}
0.9058 & 0.0461 & 0.0011 & 0.0011 &0.0461\\
\end{bmatrix}$
\end{tabular}
\caption{Row \textbf{1} gives the resulting eigenvalues and eigenvectors for the parameters stated in the paper.
Row \textbf{2} provides the eigenvalues and eigenvectors from the paper using the corrected parameters.
Note these eigenvectors are for a given value of $s$. }
\label{tab:one}
\end{center}

\end{table}

\subsection{Varying parameters for space structure}\label{subsec:32}

Recall that $|i|$ counts the number of nucleotides.
These are the letters that comprise a DNA sequence.
Taking a codon to consist of an $n$ nucleotide sequence, the number of possible codons is then $|\mathcal{C}| = |i|^n$.
These codons describe $|\mathcal{A}|$ amino acids.
With four bases as in the standard genetic code, there are $64$ three base sequences corresponding to possible codons.
These codons correspond to $20$ amino acids, so we have a many to one map.
In this subsection, we report on experiments involving varying parameters corresponding to the spatial structure of the map.

\newcounter{blah}
\setcounter{blah}{1}

\paragraph{Experiment~\theblah: Varying the number of nucleotides}${}$\\
We test the fitness optimization for the cases $|i|= 3 ,4 ,5$.
We do not take $|i|=2$ as this gives $|\mathcal{C}|=8<|\mathcal{A}|=9$, breaking one of the constraints on the delta matrix.
We do not take $|i| \geq 6$ either as this produces a matrix that is at minimum $216\times 9$, which takes significant processing time to iterate.
The results are given in Figure~\ref{fig:base}.
For all values of $|i|$, Figure~\ref{fig:base} (a) displays an optimised solution with an attractor mechanism converging to a near universal solution
When varying $|i|$ we find the value of the optimality score, $O$, increases proportionally as shown in Figure~\ref{fig:base} (b).
This makes sense as increasing $|i|$ increases $|\mathcal{C}|$ meaning we sum over more elements to get the optimality score $O$.
The rate of convergence decreases as indicated by the error bars increasing proportionally with $|i|$ in Figure~\ref{fig:base} (b).
This is as expected as larger matrices should take longer to find the universal solution.

\begin{figure}[h!!!]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.47\linewidth]{stdevoi.png}
&
\includegraphics[width=0.5\linewidth]{aviover10.png}
\\
&
\end{tabular}
\caption{Figure \ref{fig:base} (a) shows the average time evolution of the standard deviation of the optimality score for a given $|i|$ over ten runs.
Figure \ref{fig:base} (b) shows the average final optimality score for a given $|i|$ over ten runs.
The initial parameters are the same for all runs: $n=3$, $|\mathcal{A}|=9$, $N=80$, $K=1$, $H=0.4$, $\nu=0.01$, $\mu=10^{-4}$, and $ \Phi=0.99$.
The error bars show the average one standard deviation spread of final optimality scores over the ten runs (to measure the rate of convergence).}
\label{fig:base}
\end{figure}

\stepcounter{blah}

\paragraph{Experiment~\theblah: Varying the length of a codon}${}$\\
For the length of a codon $n$, we take $n=2, 3, 4$.
We do not take $n=1$ or $n\geq5$ for the same reasons as when varying $|i|$.
The results are given in Figure~\ref{fig:sequence}.
They display the same pattern as when varying $|i|$, because we are increasing $|\mathcal{C}|$ again.
When $n=2$, we get $|\mathcal{C}|=|\mathcal{A}|=9$ which means $\Delta_{c,a}$ forms a permutation matrix.
This matrix cannot be changed in step~3 of the algorithm discussed in Section~\ref{subsec:23} (fitness maximization) as we cannot reassign a single codon $c$ to a new amino acid $a$ without being left with an empty column.
The delta matrix, $\Delta_{c,a}$, cannot therefore evolve, giving a single flat line for $n=2$ as seen in Figure~\ref{fig:sequence} (a).
This implies that we require $|\mathcal{C}| > |\mathcal{A}|$ for the algorithm to work.
We take note of the smallness of the standard deviations in Figure~\ref{fig:sequence} (b).

\begin{figure}[H]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.47\linewidth]{stdevon.png}
&
\includegraphics[width=0.5\linewidth]{avnover10.png}
\end{tabular}
\caption{Figure \ref{fig:sequence} (a) shows the average time evolution of the standard deviation of the optimality score for a given $n$ over ten runs.
Figure \ref{fig:sequence} (b) shows the average final optimality score for a given $n$ over ten runs.
The initial parameters are the same for all runs: $|i|=3$, $|\mathcal{A}|=9$, $N=80$, $K=1$, $H=0.4$, $\nu=0.01$, $\mu=10^{-4}$, and $ \Phi=0.99$.
The error bars show the average one standard deviation spread of final optimality scores over the ten runs (to measure the rate of convergence).}
\label{fig:sequence}
\end{figure}

\stepcounter{blah}

\paragraph{Experiment~\theblah: Varying the number of amino acids}${}$\\
The result of this experiment is that variations on $|\mathcal{A}|$ display convergence.
It is important to consider that the randomly generated values for topological amino acid distance $a_d$ and site frequency $L_s$ will also vary, as they are generated with consideration on $|\mathcal{A}|$.
The results for this are given in Figure~\ref{fig:acid}.
There appears to be an upwards trend in Figure \ref{fig:acid} (b).
This result is intuitively expected as increasing the number of amino acids increases the number of terms we sum over; however, further investigation is needed to confirm this.
This should be done keeping randomly generated variables fixed where possible.

\begin{figure}[h!!!]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.5\linewidth]{stdevoa.png}
&
\includegraphics[width=0.5\linewidth]{avaover10.png}
\end{tabular}
\\
\caption{Figure~\ref{fig:acid} (a) shows the average time evolution of the standard deviation of the optimality score for a given $|\mathcal{A}|$ over ten runs.
Figure~\ref{fig:acid} (b) shows $|\mathcal{A}|$ (number of amino acids) against the average final optimality score, averaged over ten runs.
The initial parameters are the same for all runs: $|i|=3$, $n=3$, $N=80$, $K=1$, $H=0.4$, $\nu=0.01$, $\mu=10^{-4}$, and $ \Phi=0.99$.
The error bars in show the average one standard deviation spread of final optimality scores over the ten runs (to measure the rate of convergence).}
\label{fig:acid}
\end{figure}

\stepcounter{blah}

\subsection{Varying parameters for the innovation pool structure}\label{subsec:33}
Recall that the algorithm from Section~\ref{subsec:23} begins by constructing $N$ objects each with a genetic code $G$.
At each time step, one of these objects receives a fragment of genome from $K$ donors selected from the set of objects.
The parameter $H$ computes the fraction of the recipient genome due to horizontal gene transfer.

\pagebreak
\paragraph{Experiment~\theblah: Varying $N$}${}$\\
The results for varying $N$ is shown in Figure~\ref{fig:entity} in Appendix~\ref{app:b}.
The number of entities $N$ is taken from $10$ to $100$ in steps of $10$.
The results when varying the number of entities show a linear relationship between number of entities $N$ and final average final optimality score as seen in Figure~\ref{fig:entity} (b).

\stepcounter{blah}

\paragraph{Experiment~\theblah: Varying $K$}${}$\\
When varying the number of donors $K$ from $1$ to $5$, we find that it does not affect the algorithm's dynamics as seen in Figure~\ref{fig:donor} in Appendix~\ref{app:b}.
This makes sense as we are always adding the same fraction $H$ to the acceptor codon usage $U_{c,s}^A$.
We should note there does appear to be a slight upwards trend in average final optimality score as shown in Figure~\ref{fig:donor} (b).
Further investigation should be undertaken with larger value of $K$ such that $K\approx N$.

\stepcounter{blah}

\paragraph{Experiment~\theblah: Varying $H$}${}$\\
We take $H$, the fraction of the genome similar due to horizontal gene transfer from $0$ to $1$ in increments of $0.1$.
The results are shown in Figure~\ref{fig:hgt} below.
Figure~\ref{fig:hgt} (a) that for $H=0$, there is no convergence as expected, while for $H=0.2$ the results begin to converge but at a very slow rate.
Looking at the Figure~\ref{fig:hgt} (b), it is clear that $0.4  \leq H \leq 0.7 $ gives the minimal optimality score and the smallest error bars.
This implies the final results have been substantially optimised and have converged to a greater extent via the attractor mechanism to a near universal solution.
When $H\geq 0.8$ , the final scores, $O$, are less optimal and have converged less
This is probably  due to a change from \enquote{mixing} to \enquote{swapping} of codon usage matrices $U_{c,s}$, preventing optimal communication.
Results from Figure~\ref{fig:hgt} suggest that maximum mixing occurs around $H=0.6$.

\begin{figure}[h!!!]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.5\linewidth]{stdevoH.png}
&
\includegraphics[width=0.5\linewidth]{avHover10.png}
\end{tabular}
\caption{The average final optimality score for a given $H$.
Figure~\ref{fig:hgt} (a) shows the average time evolution of the standard deviation of the optimality score for a given $H$ over ten runs.
Figure~\ref{fig:hgt} (b) shows the average final optimality score for a given $H$ over ten runs.
The initial parameters are the same for all runs: $|i|=3$, $n=3$, $|\mathcal{A}|=9$, $N=80$, $K=1$, $\nu=0.01$, $\mu=10^{-4}$, and $ \Phi=0.99$.
The error bars show the average one standard deviation spread of final optimality scores over ten runs (to measure the rate of convergence).}
\label{fig:hgt}

\end{figure}

\stepcounter{blah}

\paragraph{Experiment~\theblah: Time evolution of $H$}${}$\\
The parameter $H$ is best considered a variable that decreases with time \cite{Woese8742, nigel}.
This is due to better translation of the model allowing evolution of a protein network with more specific interactions to occur \cite{nigel}.
To model this, we define $H$ in the following manner:
\begin{equation}
H(t)=H_0e^{-kt} ~.
\label{eq:timeh}
\end{equation}

In this equation, $H_0$ is the initial fraction of horizontal gene transfer similar ($0\leq H_0\leq1$), and $k$ is a constant.
Initially, $H_0$ is set to $1$.
Setting $k=10^{-3}$ gives a number that is approximately zero at, say, $t=5000$.
The results in Section~\ref{sec:three} indicate that we expect convergence to occur after $5000$ iterations.
For this reason, we set $k=10^{-4}$ so that $H(t=5000)$ is relatively far from zero.
The resulting dynamics is given in the Figure~\ref{fig:timeH} and Appendix \ref{app:evo}, where the degree of convergence is significantly improved in several runs, to all prior results.
The majority of the trials have fully converged and can be considered universal.
Note that the solutions converge to different values, due to $a_d$ being randomly generated.
We can see that only Figure \ref{fig:timeH} (d), does not completely converge.
The rate of convergence seem to to be fairly similar to the runs when using large constant $H$.
This makes sense as this is a stochastic process meaning the rate of convergence should vary between runs.
However, the average rate of convergence significantly improves in the cases where universality manifests within this time frame.
Note that $H(t)$ sits in the optimal range suggested in Section~\ref{subsec:33} for approximately the last $1500$ time steps.
\vspace{0.2cm}

\begin{figure}[h!!!]
\centering
\begin{tabular}{c c}
\includegraphics[width=0.5\linewidth]{timevarH5.png}
&
\includegraphics[width=0.5\linewidth]{timevarinH1.png}
\\
\includegraphics[width=0.5\linewidth]{timevarinH4.png}
&
 \includegraphics[width=0.5\linewidth]{timevarH6.png}
\end{tabular}

\caption{Evolution of optimality score with a time evolving parameter $H$.
Four runs showing the evolution of optimality score for a time evolving $H$ (according to \eqref{eq:timeh}).
We use: $|i|=3$, $n=3$, $|A|=9$, $N=80$, $K=1$, $\nu=0.01$, $\mu=10^{-4}$, $ \Phi=0.99$, $k=10^{-4}$, and $H_0=1$.}
\label{fig:timeH}
\end{figure}

\stepcounter{blah}

\pagebreak
\subsection{Varying error and fitness parameters}\label{subsec:34}
Recall that the parameters $\nu$, $\mu$, and $\Phi$ take values in the interval $[0,1]$.
The parameter $\nu$ measures the rate of mistranslation, when a single base is misread.
The parameter $\mu$ measures the rate of mutation, when a single base is changed.
The parameter $\Phi$ characterizes  how an abstract physicochemical distance between amino acids scales into the fitness.

\paragraph{Experiment~\theblah: Varying $\nu$ and $\mu$}${}$\\
The plots for these variations are in Appendix~\ref{app:c}.
It can be seen that variations on $\nu$ have no effect on a given optimality score.
As we are trying to minimize the effects of errors from $\nu$ and $\mu$, there is less requirement to optimize as they decrease.
This can be seen in Figure~\ref{fig:error} (b) in Appendix~\ref{app:c}.
As these parameters decrease the optimality score increases.
Note we take $\nu$ and $\mu$ from $1$ to $10^{-4}$ on a log scale.
We do not try $\nu,\mu=0$ as this implies there is no need to optimize the code as no errors can occur.

\stepcounter{blah}

\paragraph{Experiment~\theblah: Varying $\Phi$}${}$\\
As described by \citet{vestigian}, $\Phi$ is a scale for the fitness for one amino acid substitution.
This implies that it should not affect the rate of convergence directly.
However, it will affect the score converged to.
To examine this we reduce the fitness score $f$ to a function of $\Phi $ and $\nu$ in order to consider their role in the algorithm given by Figure~\ref{fig:phi} (a) in Appendix~\ref{app:c}.
The rest of the values are randomly generated.
The variations of $\Phi$ are proportional to $f$ as expected.

\subsection{Re-defining universality within a genetic code model}\label{subsec:36}
The framework we have described so far shows that there is some degree of convergence via an attractor mechanism
With horizontal gene transfer turned on ($H\ne 0$), we have an attractor.
While the details of the solution depend in part on the initial conditions assigned to parameters in the model, the model exhibits near universality at late times.
This is demonstrated by the converging behaviour of the optimality scores $O$ of all entities
We aim to refine the concept of universality.
To do this we must first understand genetic code configurations and the possible symmetries associated with them.
We will then analyze the fitness landscape of all genetic code configurations in order to to see if this function can be scaled homogeneously.
To re-examine universality further, we  mainly consider the model provided by \citet{sella} while also incorporating the fitness function provided by \citet{nigel}.

Note that for this section we will also work with the fitness in the following form:
\begin {equation} \label{eq3.2}
\log f=\sum_c \sum_s L_{s} U_{c,s} \log ( \sum_{c'} \sum_{a}T_{c,c'} \Delta_{c',a}W_{a , s} ) ~.
\end{equation}
The logarithm simplifies algebraic manipulations.

Using Definition~\ref{code}, we represent each genetic code configuration using a delta matrix $\Delta_{c,a}$.
We can also calculate total number of configurations using~\eqref{eq:config}.
This framework allows us to consider the genetic code mapping as a surjective mapping from a Hamming graph (of codons) to a random graph (of distance between amino acids in an abstract topological information space).
These graphs have automorphisms due to labeling which we will highlight clearly in an upcoming example.
The automorphisms imply that certain genetic code configurations (and therefore delta matrices $\Delta_{c,a}$) are isomorphic to each other, meaning that they represent the same genetic code map $G$ even though they have different delta matrices $\Delta_{c,a}$.
Considering the random graph is randomly generated, we \textit{a priori} assume that no automorphisms exist within the amino acid graph.
Note this is only true for $|\mathcal{A}| > 2$, as $|\mathcal{A}|=1$ is trivial and $|\mathcal{A}|=2$ has an inherent symmetry in swapping the labels.
Now the codons graphs as setup as a Hamming graph.
Hamming graphs are known for having automorphisms~\cite{automorph}.
Due to there being a certain number of automorphisms for the Hamming graph for a given $|i|$ and $n$, we quotient~\eqref{eq:config} by the number of symmetries to get the number of unique codes.
As previously noted in Section \ref{subsec:22}, these automorphisms imply that two isomorphic genetic codes should yield the same optimality score.

\subsubsection{Example}\label{subsec:37}
In order to understand the isomorphisms, we will consider an example.
Put $|i|=2$, $n=2$, and $|\mathcal{A}|=3$.
This means $|\mathcal{C}|=4$ giving delta matrices with dimensions $4\times3$.
Using~\eqref{eq:config} we get $\#_{\text{config}}(|\mathcal{C}|=4,|\mathcal{A}|=3)=36$.
We represent this map in the following format:
\begin{equation}
\label{eq:hamming}
\begin{tikzpicture}
\filldraw[black] (0,3) circle (2pt) node[anchor=east]  {01};
\filldraw[black] (3,0)circle (2pt)  node[anchor=west]  {10};
\filldraw[black] (0,0)circle (2pt)  node[anchor=east]  {00};
\filldraw[black] (3,3)circle (2pt)  node[anchor=west]  {11};

\draw[black,thick] (0,3) -- (0,0);
\draw[black,thick] (0,3) -- (3,3);
\draw[black,thick] (3,0) -- (0,0);
\draw[black,thick] (3,0) -- (3,3);

\draw[->,thick] (4,1) -- (8,1);
\filldraw[black] (6,1)node[anchor=south]  {G};

\filldraw[black] (9,2) circle (2pt) node[anchor=east]  {a};
\filldraw[black] (13,2)circle (2pt)  node[anchor=west]  {b};
\filldraw[black] (11,0)circle (2pt)  node[anchor=north]  {c};
\draw[blue,thick] (9,2) -- (13,2);
\draw[blue,thick] (11,0) -- (13,2);
\draw[blue,thick] (9,2) -- (11,0);
\end{tikzpicture}
\end{equation}
In~\eref{eq:hamming}, $G: \{00, 10, 11, 01\}  \mapsto \{c, c, b, a\}$.
We see that the Hamming graph on the left hand side is isomorphic under relabeling~\cite{automorph}.
In particular, if we relabel $0\longleftrightarrow 1$, the genetic code map would not change.
This configuration has a symmetry factor $18$.
Taking the quotient of the number of configurations with the symmetry factor suggests that there are only two unique configurations of $\Delta_{c,a}$ for $|\mathcal{C}|=4$ and $|\mathcal{A}|=3$.
Said another way, in this example, there are $4\choose 2$ ways of selecting a pair of codons that are mapped by $G$ to the same amino acid.
Taking into account the repetition, there are $3!$ (the order of $S_3$) ways of mapping the codons to the amino acids.
The product of these terms gives the $36$ configurations.
Taking into account the isomorphisms, we pick out the odd and even elements of the permutation group $S_3$ as our distinguished configurations.

We will now calculate~\eqref{eq3.2} for all configurations of $\Delta_{c,a}$.
We do this for a given value of $\Phi$ and generate plots in the $\mu$--$\nu$ phase space plane (error space).
In Figure~\ref{fig:splotsone}, we show results for $\Phi= 0.99, 0.5, 0.1, 0.01$.
We expect each unique genetic code configuration to correspond to a unique surface in error space.
\begin{figure}
\centering
\begin{tabular}{c c}
\includegraphics[width=0.5\linewidth]{223logfphi099.png}
&
\includegraphics[width=0.5\linewidth]{223logfphi05.png}
\\
\includegraphics[width=0.5\linewidth]{223logfphi01.png}
&
\includegraphics[width=0.5\linewidth]{223logfphi001.png}
\end{tabular}

\caption{Four surface plots of $\log f$ in $\mu$--$\nu$ phase space for $\Phi= 0.99$ (a), $0.5$ (b), $0.1$ (c), $0.01$ (d).
We have set $|\mathcal{C}|=4$ and $|\mathcal{A}|=3$. Each colored surface corresponds to a unique genetic code configuration.}
\label{fig:splotsone}
\end{figure}

\pagebreak
From Figure~\ref{fig:splotsone}, we see that there are two unique surfaces for a given value of $\Phi$.
This is due to there being two unique configurations of $\Delta_{c,a}$.
Within this phase space, there is a curve on which the surfaces interact.
These curves are a critical locus for which the ability of a code to produce a maximum $\log f$ changes.
As $\Phi$ varies, the shape of the surfaces change and critical locus changes.
In the case for $\Phi=0.99$ (Figure~\ref{fig:splotsone} (a)), the critical locus is essentially independent of $\nu$.
The dependence on $\nu$ for the surfaces and the critical locus grow as $\Phi$ decreases.
In order to verify this, we take a polynomial fit to the critical locus for this model.

Taking the ansatz,
\be
\log(f)_\text{fit} = a + b_1\nu + b_2\mu + c_1\nu^2 + c_2\nu\mu + c_3\mu^2 ~,
\ee
for $\Phi=0.99$ the surfaces have polynomials of the form:
\bea
\log(f)_\text{fit} &=& -84.8470+0.0801 \nu-0.0618 \nu^2+22.0655 \mu-15.2344 \mu^2 ~, \label{surf1} \\
\log(f)_\text{fit} &=& -84.0745+0.0522 \nu-0.0404 \nu^2+26.1504 \mu-23.5190 \mu^2 ~, \label{surf2}
\eea
with $R^2 > 0.99995$.
Taking the difference between~\eref{surf1} and~\eref{surf2}, we get the critical locus
\begin{equation}\label{critline}
-0.7725 + 0.0278\nu - 0.0214\nu^2 + 4.0849\mu  +8.2846\mu^2=0 ~.
\end{equation}
As inferred from Figure~\ref{fig:splotsone}, the dependence on $\nu$ is negligible as the coefficients are two or three orders of magnitude smaller than the coefficients for terms involving $\mu$.
For any value of $\Phi$, the coefficient of the $\mu\nu$ cross term ${\cal O}(10^{-10})$.
Thus, at $\Phi\approx 1$,
\begin{equation}\label{eq:assume}
\frac{\partial \log f}{\partial \nu}=0 ~.
\end{equation}
This relation does not necessarily hold for smaller values of $\Phi$ for which we report results in Appendix~\ref{app:e}.
Here, the coefficients of $\nu$ are on a similar magnitude to those for $\mu$.
This implies that $\Phi$ influences the effects of mistranslations $\nu$ in an inversely proportional manner.
For the results in the prior sections we use $\Phi=0.99$ for all runs as in~\cite{nigel}.
This is due to the fact that the effects of mistranslations are more likely to be non-lethal.
Note that as $\Phi$ and $\mu$ are related through eigenvectors and therefore not linearly related.

Note we also have results for $|i|=4$, $n=1$ and$|\mathcal{A}|=3$ such that we have another case with $|\mathcal{C}|=4$ and $|\mathcal{A}|=3$ but with different automorphisms in Appendix~\ref{app:e}.
For this we find a unique configuration of genetic codes and therefore no critical locus.
We also find that the dependence on $\nu$ increases and $\Phi$ decreases as before.

\subsection{Re-examining universality}\label{subsec:38}
In order to understand the universality of the model in a more formal manner, we examine it in terms of Widom scaling~\cite{widom}.
In particular, we look for homogeneous behavior as a signal of scale invariance on a critical locus.
Consider the logarithm of the fitness function, $\log f$.
As above, $f(\nu,\mu)$ is a function of the rate of mistranslations and the rate of mutations.
Now, homogeneity of $\log f$ demands that
\begin{equation}\label{eq:condition}
\log f(\kappa\nu,\kappa\mu)=\kappa^{\beta}\log f(\nu,\mu) ~,
\end{equation}
where $\kappa\in \mathbb{R}$ is a scale and $\beta$ is the degree of homogeneity.
As $\nu,\mu \in[0,1]$, we require that $\kappa\nu, \kappa\mu \in[0,1]$.
In particular, when $\kappa=1$,
\begin{equation}\label{eq:homog}
\nu \frac{\partial \log f}{\partial \nu}+ \mu\frac{\partial \log f}{\partial \mu}=\beta \log f(\nu,\mu) ~.
\end{equation}
This is the content of Euler's homogeneous function theorem.

However, if we consider the case of $\Phi \approx 1-\epsilon$, where $\epsilon \ll 1$, then contributions from $\nu$ become negligible such that we can apply~\eref{eq:assume}, leaving us to calculate $\frac{\partial \log f}{\partial \mu}$.
From~\ \eref{eq3.2}, the only part of the $\log f$ that depends on $\mu$ is $U_{c,s}$.
Therefore, we must calculate $\frac{\partial U_{c,s}}{\partial \mu}$.

Suppose $A$ is a real symmetric matrix with eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$ such that $\mathbf{v}_i^T \mathbf{v}_i = 1$.
The Perron--Frobenius theorem ensures that the matrix $A$ has a unique real eigenvalue with a magnitude larger than that of any other eigenvalue and a corresponding eigenvector with positive components.
Then
\be
\partial\mathbf{v}_i = (\lambda_i \mathbf{1} - A)^+ (\partial A) \mathbf{v}_i ~,
\ee
where $X^+$ denotes the Moore--Penrose inverse of $X$~\cite{Petersen2008}.
In defining $U_{c,s}$, we have normalized so that $ \sum_{c} U_{c,s} = \mathbf{1}_s$.
 As $Q_{c,c'}^s$ is a symmetric and real matrix, we therefore only need to rescale $U_{c,s}\rightarrow U_{c,s}'$ such that $\sum_c U_{c,s}' \cdot U_{c,s}'=1$ for any given $s$.
By doing this we can differentiate $U_{c,s}'$:

\begin{equation}\label{eq:order}
\hspace{-0.25cm}
\beta= \frac{\sum_c \sum_s \mu( (\lambda_s^\text{max} \mathbf{1}-Q_{c,c'}^s)^+(\sum_{c''}\frac{\partial M_{c,c''}}{\partial \mu}\delta_{c'',c'}F_{c'',s}) U_{c,s}')'L_{s}}{\sum_c \sum_s U_{c,s}L_{s}} ~,
\end{equation}
where
\begin{equation}\label{eq:mdiff}
\frac{\partial M_{c,c''}}{\partial \mu}=\left\{
\begin{array}{lcl}
1/(n(|i|-1)) && \text{if  dist}(c,c')=1 ~, \\
-1 && \text{if dist}(c,c')=0 ~, \\
0 && \text{otherwise} ~. \
\end{array}\right.
\end{equation}

We have derived the degree of scaling in the limit $\Phi\to 1$.
This implies that the model is approximately homogeneous in the regime that we have worked in, with degree specified by~\eref{eq:order}.
We see that the order is dependent on the mutations $\mu$, implying that the universality of the code arise due to species having similar mutational errors.
We regard the scaling behavior as a phenomenological observation about the solution near an approximate fixed point of the renormalization group.

We emphasize that when investigating what happens as we tweak the parameters of the model, we calculate the standard deviation to establish which results are significant when we measure the optimality score.
In assessing Widom scaling, we look at the degree of homogeneity, not the degree of optimality; we do this using the fitness score, which is the function actually being maximized in this algorithm, not the optimality score.
The $L_{s}$ terms act as a weighting for each amino acid $s$.
We have used a scaling argument in order to justify universality of the algorithm.
The fact that the degree of homogeneity is independent of $\nu$, the rate of mistranslations, in the $\Phi\to 1$ limit, does not imply that the final optimality score is independent of mistranslations.
As seen in Experiment~8 in Section~\ref{subsec:34}, it is not: decreasing $\nu$ increases the optimality score.
We should emphasise that for smaller values of $\Phi$, \eref{eq:order} will not hold. This is because \eref{eq:assume} begins to break down for values of $\Phi$ outside the condition $\Phi \approx 1$.
We can amend \eref{eq:order} to consider results around $\Phi=0.9$ by incorporating the addition on an error term through considerations of the first term of \eref{eq:homog}. This approximation does not apply for smaller values of $\Phi$, however, since, as suggested by \citet{sella}, we expect $\Phi$ to be relatively large.

\section{Conclusion and prospects}\label{sec:four}
In this paper, we have argued that with generic initial conditions, there is a late time near universality resulting from the flow of the theory to an attractive solution, \textit{viz.}, the standard genetic code.
The convergence via the attractor mechanism to a near universal solution relies on the mechanism of horizontal gene transfer~\cite{nigel}, which corresponds to setting a parameter $H$ to a non-zero value.
We varied the parameters of the model and found that all variations still display this convergence, except for $H=0$.
This demonstrates the robustness of the model.
We also found that for $0.3< H < 0.7$ we obtain near universal solutions with the greatest degree of optimisation, with $H>0.7$ not being as effective due to some transition from \enquote{mixing} to \enquote{swapping}.
Taking $H$ as a decreasing time-dependent function vastly improves the convergence in comparison to constant values of $H$.
We found that increasing the number of codons, $|\mathcal{C}|$, increases the optimality score $O$.
By limiting the fitness function to a regime that we work in ($\Phi=0.99$), we are able to make approximations that lead to homogeneity in $\log f$, where $f(\nu,\mu)$ is a fitness function depending on the rate of mistranslation and the rate of point mutation.
We derive an expression for the degree of homogeneity, $\beta$.
In the limit $\Phi\to 1$, $\beta$ depends strongly on the mutation rate and negligibly on the mistranslation rate.
We conclude that the rate of point mutations is the crucial factor in driving arbitrary initial conditions to the attractor solution that optimizes fitness of the genetic code.
The point mutation rate is determined by the eigenvalue of the linearized renormalization group transformation around the fixed point for the dynamics.

Improvements to make the algorithm more accurate for biology would involve exploring how to incorporate stop codons which do not code for amino acids into the model as something more than a dummy amino acid.
We should also consider that mutations and horizontal gene transfer do not occur at the same rate as suggested by  both occurring at each iteration.
Some work to estimate a timescale for this model possibly by considering rate of error as the sum of all errors ($\nu+\mu$) and relating this to  the measured rate of error in, for example, a kinetic proofreading model~\cite{alon2019introduction}.
We also suggest that additional factors and steps should be incorporated in order to guarantee a universal solution is converged to every run.

Variations on the dimensions, $|\mathcal{C}|$ and $|\mathcal{A}|$, which count the number of codons and the number of amino acids, respectively, display convergence to a universal result.
This could have applications to synthetic biology  where codes with up to $8$ bases have been created~\cite{Hoshika884}.
These codes should also converge to a universal genetic code given enough time.
An open question is to determine what sets the initial conditions.
Why did life on Earth evolve to make use of four base pairs in DNA, three base pairs per codon, and $20$ amino acids?

We have focused on a single basin of attraction, whereas there could be others.
The basin of attraction may be determined by biochemistry inputs.
We can imagine, for example, a different basin of attraction in which the solvent is ammonia, methane, or hydrogen fluoride instead of water.
We can also imagine biochemistry organized around silicon instead of carbon.
The molecular realization of the genetic code would be different based on these other inputs, but we expect that the same principles apply, and these other hypothetical genetic codes would also evolve to a universal solution based on the principle of horizontal gene transfer.

Broadly speaking, we have argued that the concept of universality from statistical physics applies to biological systems like the genetic code.
The thermodynamic limit arises from a large $N$ number of degrees of freedom in the entities studied.
The dynamical system is driven to an attractor solution as a result of interactions, in this case horizontal gene transfer.
We have considered a mechanism for horizontal gene transfer and by tweaking its parameters identified which ones are the most important.
The existence of approximate homogeneity offers evidence for universality.
We would like to interrogate how general this setup is and whether it is useful for studying other complex systems.

Indeed, like thermodynamics and evolution itself, we wish to consider horizontal gene transfer as an organizing principle in Nature.
Different solutions to a theory or different possible initial conditions can exchange information with each other through complex processes.
Dynamics can then flow the system to a late time attractor solution that is independent of specific parameters of the model.
As a proving ground for this hypothesis, we can consider the vacuum selection problem in quantum gravity.
String theory, a promising candidate framework for marrying gravitation with quantum theory, generically predicts a landscape of vacua, one of which is our Universe with the Standard Models of particle physics and cosmology as phenomenological features that explain dynamics at small and large scales.
(See, for example,~\cite{Denef:2006ad, jkm, Yang} for related reviews.)
These vacua inevitably arise as a consequence of a simple observation --- we live in four spacetime dimensions whereas the consistency of the theory demands ten, and there is no unique way to reduce the number of dimensions.
Because they are unobserved, the extra dimensions predicted by string theory comprise a compact geometry with special properties.
The moduli space of string compactifications is believed to be connected.
We can calculate the degree of fine tuning necessary to support certain cosmological structures and the astrophysical and chemical preconditions necessary for life~\cite{Adams:2019kby}.
Rather than making an explicitly anthropic argument~\cite{Weinberg:1988cp, Polchinski:2006gy}, we can model a dynamics for vacuum selection which incorporates a mechanism analogous to horizontal gene transfer to lead to universal and optimal structures as an attractive fixed point.
Thus, instead of arguing that low energy observables such as the cosmological constant are distributed randomly across the landscape, horizontal gene transfer, by driving the system to the attractor value, may obviate aspects of the measure problem.
Developing and testing this hypothesis within the string theory framework is work in progress.

\section*{Acknowledgements}
We are grateful to David Ardell, Nigel Goldenfeld, Sujay Nair, and Kalin Vetsigian for feedback and insightful discussions.
YHH is indebted to the Science and Technology Facilities Council, UK, for grant ST/J00037X/1.
VJ thanks the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation for support.
DM thanks the Julian Schwinger Foundation and the US Department of Energy (DE-SC0020262) for support.

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HomeNASCAR
NASCARXfinity Series
Austin Cindric Holds Off Ty Gibbs at Pocono For Fourth Xfinity Win of 2021
By Toby Christie
Screengrab from NBCSN broadcast
Austin Cindric led a race-high 26 laps and he bested Ty Gibbs by just over 3-tenths of a second to score his fourth win of the 2021 NASCAR Xfinity Series season in Sunday's Pocono Green 225 at Pocono Raceway.
As it looked like Cindric had things all sewed up, Gibbs began to close in the closing laps. Cindric chalks that up to running in the Xfinity Series being a challenge.
"I didn't know what happened to the 66 there. I went from a three second lead to a half a second lead," Cindric said after climbing from his No. 22 Team Penske Ford Mustang. "This thing is never easy. I had to avoid two wrecks today."
Cindric has now amassed 27 Playoff points through 16 races, and looks well on his way to potentially defending his 2020 NASCAR Xfinity Series championship.
For Gibbs, 18, this is his fourth top-two finish in just eight-career NASCAR Xfinity Series starts. After another good run, he's still learning and is very thankful for his opportunity to be running in the Xfinity Series.
"I'm very thankful to be here," Gibbs said. "Congrats to Austin Cindric and the whole Penske Group over there, but I can't thank everyone enough. This is a blast and a dream come true running the Xfinity Series."
Justin Allgaier finished in third behind Cindric and Gibbs. Noah Gragson came home fourth and AJ Allmendinger was fifth.
Daniel Hemric, who was leading with less than 20 laps to go, had a speeding penalty on his final pit stop. Hemric would rebound to finish sixth, but it was another potential win that slipped through his fingers.
Brandon Jones, Jeb Burton, Josh Berry and Myatt Snider rounded out the top-10 finishers.
On lap 23, a hard crash broke out when Justin Haley was tagged by Sam Mayer, who was making his NXS debut. The contact sent Haley careening into the inside wall and back across the track. Ryan Vargas had nowhere to go and clocked Haley's car.
Video: Justin Haley and Ryan Vargas Crash Hard at Pocono After Contact From Mayer Triggers Crash
Luckily, Haley and Vargas both walked away without injury.
While Haley was okay after the violent crash, Justin Allgaier will replace Haley in the Spire Motorsports No. 77 car in Sunday's NASCAR Cup Series race.
Justin Allgaier spoke to Spire executive after Xfinity race. He will drive the No. 77 Cup car today. Out of abundance of caution, Haley will not race after wreck today. pic.twitter.com/wb4Fa08GSp
— Bob Pockrass (@bobpockrass) June 27, 2021
Video: Riley Herbst's Day Ends After Crashing In Tunnel Turn
Results: 2021 NASCAR Xfinity Series Pocono Green 225 at Pocono
Toby Christiehttp://tobychristie.com
Toby is the Founder, and Editor-in-Chief of TobyChristie.com. Toby is also the co-host of The Final Lap Weekly Podcast. Additionally, he is a NMPA (National Motorsports Press Association) award-winning writer, and has followed the sport as a fan since 1993.
Inspection Complete: Austin Cindric Officially Wins NASCAR Xfinity Series Race at Pocono - TobyChristie.com June 27, 2021 At 4:27 pm
[…] Austin Cindric Holds Off Ty Gibbs at Pocono For Fourth Xfinity Win of 2021 […]

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International Relations and Security Studies (2022)
Like most states, New Zealand depends for its security and prosperity on regional and international stability. Its political, economic and societal well-being can be affected by, for example, major power relations; global financial crises; and the rise of violent extremist non-state actors and the responses to them. As a good international citizen, New Zealand seeks to play a role in the international institutions that try to uphold international law and order and to contribute to international stabilisation operations in various countries. Given the significance of such issues, International Relations and Security Studies provides students with the opportunity to pursue further study in various aspects of international relations and security, including aspects of international law and trade, and to focus on particular regions of concern (notably, South-East Asia). A wide range of different length dissertations and theses can be undertaken, allowing students to focus on areas of special interest to them.
International Relations and Security Studies is available as a minor.
To complete a minor in International Relations and Security Studies, students must complete at least 60 points in papers listed for the minor, including at least 30 above 100 level. Students must complete both IRSST103 and IRSST206.
Prescriptions for the PGCert(IRSSt), PGDip(IRSSt) and MA
To complete the Postgraduate Certificate in International Relations and Security Studies, students must complete at least 60 points, including one of IRSST502, IRSST506 or POLSC537.
To complete the Postgraduate Diploma in International Relations and Security Studies, students must complete at least 120 points from the following papers, including at least two of IRSST502, IRSST506 or POLSC537.
To be eligible to be considered for enrolment in the MA in International Relations and Security Studies, a student must have completed either:
a) a BA or BSocSc with a major in Political Science (or equivalent) and have gained at least a B average in the 300 level papers, or for a qualification considered by the Academic Board to be equivalent, or
b) a BA(Hons) or BSocSc(Hons) in Political Science (or equivalent) with at least second class honours
(second division), or a Postgraduate Diploma in International Relations and Security Studies (or equivalent) and have gained at least a B average, and
c) have satisfied the prerequisites for graduate study in the subject(s) being presented for the Degree, at levels considered appropriate by the Academic Board.
Completion requirements for the MA in International Relations and Security Studies vary according to admission criteria:
All students must take at least two of IRSST502, IRSST506 or POLSC537 (unless these have already been taken as part of an earlier qualification) and complete either a dissertation or a thesis.
Students admitted under a) above must complete 180 points from approved 500 level papers, including either a 120 point thesis, or a 90 point thesis or a 60 point dissertation.
Students admitted under b) above must complete a 120 point thesis, a 90 point thesis and a further 30 points from approved 500 level papers, or a 60 point dissertation and a further 60 points from approved 500 level papers.
Occurrence / Location
IRSST103 Introduction to International Relations 15.0 22A (Hamilton) & 22A (Online)
This paper provides an introduction to the study of international relations in an era of globalisation. It covers the historical background, key concepts and theories, case studies, and contemporary developments in the study of world politics.
POLSC103 Let's Get Political 15.0 22A (Online) & 22A (Tauranga)
Politics affects every aspect of our lives. This paper is an introduction to understanding political issues, helping you make your voice heard in debates about our future as a society.
IRSST206 International Relations: The Security Agenda 15.0 22A (Hamilton)
This paper introduces students to International Relations' dynamic and evolving security agenda and covers some of the most urgent contemporary security concerns.
POLSC203 The Rhetoric of Collective Memory 15.0 22A (Online) & 22A (Tauranga)
Why do social groups decide to keep certain memories of the historical past alive while repressing others? Students tackle this question by analysing museums, monuments and popular culture.
POLSC224 Terrorism, Violence and the State 15.0 22B (Hamilton)
Terrorism is an increasingly prominent global phenomenon. We will clarify it conceptually and situate it historically with theory and case studies in situating it in comparative political and International Relations analysis.
POLSC229 US Politics 15.0 22B (Online) & 22B (Tauranga)
Students in this course explore the foundational elements of the United States, its governing institutions, the extent of public influence within the system, and major contemporary policy issues that the country faces.
IRSST301 International Relations: The Context, Theory and Practice of New Zealand Foreign Policy 15.0 22A (Hamilton)
This course delivers a comprehensive historical overview of New Zealand foreign policy from the beginning of the 20th century, and considers the role government departments play in the creation and implementation of policy.
IRSST315 International Politics of Asia 15.0 22B (Online)
It provides an introduction to, and analysis of, critical issues in Asia. Issues may include some, or all, of the Kashmir dispute; the situation on the Korean Peninsula; the South China Sea disputes; and tensions across the Taiwan Strait.
IRSST331 Cyber Security and the Global Politics of the Internet 15.0 22B (Online)
This course explores the global politics of cyber security and the internet. It covers the major political/strategic/theoretical debates around enhancing the security of computer systems and how states and non-state actors are responding.
POLCY318 Global Environmental Politics and Policy 15.0 22A (Hamilton)
This paper explores contemporary debates on environmental politics, sustainability, justice, and environmental governance. It examines the political processes underpinning the search for cooperative solutions to environmental dilemmas, including climate change.
POLSC301 Visual Communication: Politics, Propaganda, Protest 15.0 22A (Online) & 22A (Tauranga)
This paper introduces students to the power of visual communication. The theoretical knowledge gained is applicable to politics and beyond.
IRSST502 Security Issues in South East Asia 30.0 22B (Hamilton)
After a brief conceptual and historical introduction to security in the region, the paper goes on to consider a range of contemporary security issues. These include ethnic conflict, militant Islam, maritime security, and relations with the major extra-regional powers.
IRSST506 International Relations and Global Security 30.0 22A (Hamilton)
This paper teaches and utilises major international relations theories to consider pressing global issues. This includes the causes of war and peace, globalisation, the Cold War, the widening security agenda, terrorism, the environment, inequality, new technologies, and international institutions.
IRSST507 Cyber Security and Cyber Warfare 15.0 22B (Online)
This course examines the political, strategic, legal and economic debates around cyber security and cyber warfare. Particular emphasis is placed on how militaries, national security agencies, and law enforcement are adapting to online threats and incorporating cyber in their strategic planning.
LEGAL576 The Laws of Armed Conflict & International Humanitarian Law 30.0 22B (Hamilton) & 22B (Tauranga)
This paper is designed to provide the postgraduate student with an advanced understanding of the basic principles of the Laws of Armed Conflict and International Humanitarian Law through an examination of the principles of international law that are to be applied before, during, and after armed conflict.
POLSC537 Environmental Politics and Public Policy 30.0 22A (Hamilton)
This paper introduces students to the complexities and challenges of environmental policy processes, and develops skills in scholarly research on environmental policy making. A significant focus of the paper is on analysing and understanding the complex intersections of institutions, culture, science and technology, and systems of ...
POLSC590 Directed Study 30.0 22A (Hamilton), 22B (Hamilton), 22D (Hamilton) & 22X (Hamilton)
Students have the opportunity to pursue a topic of their own interest under the guidance of academic staff.
POLSC591 Dissertation 30.0 22A (Hamilton), 22B (Hamilton), 22D (Hamilton) & 22X (Hamilton)
A report on the findings of a theoretical or empirical investigation.
POLSC592 Dissertation 60.0 22X (Hamilton)
POLSC593 Political Science Thesis 90.0 22X (Hamilton)
An externally examined piece of written work that reports on the findings of supervised research.
POLSC594 Political Science Thesis 120.0 22X (Hamilton)
2022 Catalogue of Papers information current as of : 10 January 2022 12:10pm


A great Video about Coffee and a lot of Passion, Thanks Andy.
This entry was posted in Food and tagged andy lösch, coffee, coins, four barrel, jeremy, tellason, the mill, video, vimeo on 4. September 2013 by Alex.


We've been fans of the Weekend Box Club for a very long time. There is nothing more exciting that a box of activities popping through your letterbox just in time for the weekend.
The Little Man was thrilled to try out the Winter Weekend Box.
As usual the letterbox sized parcel contains the familiar little packages in their themed colours. It also contained a booklet, a sticker height chart, some stickers and the extra bits and pieces needed for the activities.
The colourful booklet contained some fun facts, jokes, a word search, instructions for all the activities, an animal quiz, two colouring pages and a little comic.
The Blue bag from Sammy the Squirrel contained a Salted Snowflake Activity.
The Little Man enjoyed this activity of making 3D snowflakes using glitter glue, salt and paint. If I'm honest, I liked the white salty snowflakes best before we put on the paint. But the Little Man enjoyed watching the paint run along the salt lines.
In the Red bag we had a recipe for Rocky Road Snow-bites.
These were really fun and simple to make and the Little Man did it all by himself. The end result was four extremely tiny chocolates with biscuits and marshmallows. It would have been nice to have them a little bigger. I wanted to make two bigger ones, but the Little Man was adamant that we stuck to making four.
In the green bag was an activity to make an Icy Tealight Holder.
We messed this up because we forgot to go back for the balloon in the freezer. By the time we did it was way too frozen. I have seen some lovely photographs from successes with this activity though, we will definitely get another balloon and try again.
The activity required filling a balloon with water and popping it in the freezer for 4 hours. It needs to be frozen on the outside but still wet on the inside. At a week point you make a hole big enough for a tealight candle. Then watch it burn and melt outside in the garden.
We had lots of fun making these pop up penguins. Everything was supplied apart from the scissors. We had to cut out penguin shapes and wings from black card. Then flippers and beaks from the orange card. We stuck these together and then made the paper spring by folding strips of paper.
The penguins where decorated the penguins with the beaks and beady eyes and the end result was really cute. We were even provided with a shiny pebble because male penguins put a pebble in a female's nest as a proposal.
The Winter Time Weekend Box was very entertaining and contained plenty to keep the Little Man entertained for quite a while.
If you would like to spend more time doing activities with your kids, and less time resourcing the activities and getting the supplies, then The Weekend Box is ideal. Kid's love the excitement of receiving a parcel too.
What often falls in winter but never gets hurt?
Disclosure: The Weekend Winter Box was sent to us for free so we could review it. All words, pictures, thoughts and opinions are my own.


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Home»Business & Policy»Trade
Canadian Meat Council wants TPP agreement wrapped up
Group calls for earliest possible completion, ratification and implementation of trade agreement.
By	Canadian Meat Council
November 1, 2017, Ottawa, ON – The Canadian Meat Council (CMC) released its submission to the Government of Canada pertaining to the "Consultations on Canada's discussions with the remaining members of what was previously the Trans-Pacific Partnership".
The Council's submission called for the "earliest possible completion, ratification and implementation" of the TPP. Failure to do so, according to the Council, would result in a devastating blow to the Canadian Meat Industry, which represents a significant segment of Canada's economy and jobs.
The submission highlights the important part in which international trade represents for the Canadian livestock and meat sector. In 2016, $6.2 billion of meat products was exported to more than 100 countries around the globe. Twenty-eight per cent or $1.7 billion was exported to TPP-11 members, particularly Japan.
According to the CMC, "under TPP-11, Canadian meat exports to Japan are projected to increase by $500 million." Without the speedy implementation of the TPP, Canada risks losing a critical competitive advantage to other large exporters such as the European Union.
The Canadian meat industry amid these current negotiations, is reminded of the costly decline it experienced as a result of stalled Canadian negotiations with South Korea.
The CMC notes that "not only did Canadian meat exports collapse by 56 per cent after competitors gained preferential access to the Korean market, the Canadian market access disadvantage will endure during the remaining years of the 15-year implementation period of the Canada-Korea Free Trade Agreement.
The cost of Canada's non-participation in the TPP could spell disaster for the meat industry, particularly, in the meat packing and processing sector which is typically a major or primary employer in towns and cities across Canada.
The CMC affirms that abstention or indecision by Canada should not be accepted as a viable option.
The Council also notes that should the ratification or implementation of the TPP by other countries be delayed, Canada should immediately re-invigorate negotiations of the Canada-Japan Economic Partnership Agreement to counter the increasing loss of Canadian competitiveness to other countries.
Canada could ratify new NAFTA even if U.S. tariffs stay put: Trudeau
USMCA deal harmful for Canadian turkey farmers
USMCA secures greater U.S. access to Canada chicken sector
More in this category: « Canada seeking TPP changes on supply management	| Federal ministers reconnect with pro NAFTA allies in U.S. as new stage nears »


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'Killer Elite' Trailer: Statham Like a Hurricane
Posted on Thursday, June 23rd, 2011 by Russ Fischer
Here's the trailer for Killer Elite, which is the Jason Statham-iest Jason Statham movie yet. The muscular star plays a killer (elite) whose mentor (Robert De Niro) is kidnapped by another killer (Clive Owen). Cue a rescue attempt and a great many instances of people talking about just what a badass Jason Statham is. Gary McKendry directs this one, which should be an instant ticket purchase for anyone who loved The Mechanic. Check the trailer below.
This is not a remake of the late Sam Peckinpah film The Killer Elite (notice this title has no definite article) though this looks like it could be that film's high-energy cousin. Guess that's where the symphony-enhanced version of 'Rock You Like a Hurricane' by the Scorpions comes in. I do like seeing Robert De Niro in something like old Robert De Niro mode, though.
Killer Elite opens on September 23, 2011.
An ex-special ops agent is lured out of retirement to rescue his mentor. To make the rescue, he must complete a near-impossible mission of killing three tough-as-nails assassins with a cunning leader.
Yahoo has the HD feed.
'Angel of Mine' Trailer: Noomi Rapace is a Mourning Mother on the Edge
'The Informer' Trailer: Joel Kinnaman Becomes an FBI Snitch for Rosamund Pike
Robert De Niro Becomes the Bagel Boss in Weird UK Bagel Commercial
'Ophelia' Trailer: Daisy Ridley Shows a New Side to Shakespeare's 'Hamlet'
/Tweeted, Action/Adventure, Movie Trailers, Clive-Owen, Dominic Purcell, Gary McKendry, Jason-Statham, Killer Elite, Robert-De-Niro, Yvonne Strahovski


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Q: Different approaches to load the data from Hadoop(on-premise) to Azure Data Lake I have a requirement to load the data from Hadoop (which is on-premise Hortonworks cluster) to Azure Data Lake every 8 hours. Can you let me know- what the different approaches that can be tried here in implementing the solution ?

*

*ADF [ We can create a pipeline activity and schedule them ]
Anything else ?

Thanks,
Aravind

A: We have provided high-level options at this link. Search for: "Data stored in on-premise or IaaS Hadoop clusters".
However, there will be fine-tuning needed based on:
1. How much data you want to move, # of files, size of files etc.
    Smaller sized files will take longer to transfer.
2. Network connectivity from onprem to Azure
    If you have ExpressRoute available, you will have better experience. If you have only public internet connectivity, you will be bottlenecked by network and not the tool you use.
Thanks,
Sachin Sheth
Program Manager, Azure Data Lake.

redpajama_set_name: RedPajamaStackExchange


At Hilton Coylumbridge you will find the grandeur of the Scottish Highlands right on your doorstep and leisure facilities to suit the whole family, available throughout the year. Within the 65-acre grounds there are two indoor heated pools, tennis courts, children's adventure play- ground, archery facilities and a choice of restaurants and bars. Skiing, sailing lessons and clay pigeon shooting can all be arranged. The lodges are of an unusual design and furnished to a high standard. This Gold Crown Resort embodies the romance and rugged spirit of the Scottish highlands.


Rebecca came over with Spencer and once both the babes went for naps, the rest of us went outside to exercise.
Setting up seven different moves to rotate in doing. Not enough were in the shade but we managed to survive and then it was lunch time on the deck since Alaina wanted a 'picnic'.
Mom made her wonderous sandwiches and we ate them along with chips, iced tea and then cookies.
It was back and forth between the pool and playing badminton for most of the afternoon.
Pro was making pasta and shrimp for supper and I wanted the spicy shrimp that Becca had made before, so I got the recipe from her and worked on preparing that.
The usual after supper routine is clean up the kitchen, get the coffee brewing, sit around the table visiting and then slowly either kid duty calls a few away or the heat calls us to sit outside.
All in all it's usually a relaxed evening spent together.
Pro and I had to go into town to check if Prince Albert happened to have a place that would renew our passports.
Back at the house groceries were unpacked, chats chatted and such activities as life usually requires done.
Eventually we headed to the outdoors as the weather had been staying very hot.
There were attempts of playing badminton between child relocating and or snacking, then a couple of moments sitting 'poolside'.
It was so hot inside that we set up the small oven out on the deck to bake the eggplant in and farmers sausage was barbecued for our supper.
Eeny, Meeny, Miney, Moe, If I Touch The Cables, Will I Glow?


On 01/12/2018, Alaska Wildlife Troopers in Soldotna, issued a citation to Jedidiah Johnston, 38 of Soldotna, for taking an over limit of deer in GMU 8. Johnston turned himself into AWT after not wanting to falsify harvest reports. Johnston was given a mandatory court appearance in the District Court at Kodiak on February 22nd, 2019 at 0830 hours.


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Hello, I would like more information about the 2015 Ram 1500, stock# 9020.
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Unveiling the OnCloud 50 Companies to Watch
AlwaysOn Network, 2/12/15
The private companies on the sixth annual AlwaysOn OnCloud 50 Companies to Watch (formerly OnDemand 50 Companies to Watch) are taking the initiative to bring the world's businesses and enterprises into the cloud.
AlwaysOn's 50 Companies to Watch list of companies are different then the more established OnCloud Top 100 companies in that they are generally earlier-stage, still emerging as top contenders, and just beginning to be reckoned within the cloud market. With creative ways in approaching the highly competitive sector, the OnCloud 50 Companies to Watch leverage their technologies and apply fresh, strategic thinking in building robust and sustainable businesses.
The AlwaysOn editorial team, along with partners at GSV Capital, KPMG, Highland Capital, Kleiner Perkins Caufield & Byers, Bridge Bank, BMC Software, DFJ, Morgan Stanley, Salesforce.com, JMP Securities, InterWest Partners, HP, Bullpen Capital, IVP, Bessemer Venture Partners, Crosslink Capital, Accel Partners, Sapphire Ventures (SAP), USVP, IBM Ventures, Focus Ventures, Microsoft, Jafco Ventures, NEA, RWI Ventures, Jefferies, Alsop Louie Partners, Golden Seeds, Samsung Ventures, AT&T, Mohr Davidow Ventures, Emergence Capital and industry experts across the globe, went out into the entrepreneurial ecosystem to find the 50 private companies that are bringing the cloud to the enterprise and the enterprise to the future.
Vital role of the cloud sector
The year's OnCloud 50 Companies to Watch list represents a sophisticated ecosystem that is finding ways to deal with market segmentation, big data warehousing, virtualized networks, cyber-security, and myriad infrastructure-related issues. At the same time, data continues to find itself front and center, as software becomes ever more sophisticated and nimble, dealing with everything from specialized industries to global spanning businesses.
Ellen Rubin selects Silver Winner for CEO of the Year
See the Award
The 25 Most Influential Women in Cloud in 2015
Who Rocked the 2014 NEVYs – This Year's Winners


What do we mean when we say there is no salvation outside the Catholic Church?
One common mistake is the thought that I can come and go from the Catholic Church kind of like Musical Chairs. Which ever church I end up in will be the instrument of my salvation. If you have true knowledge of The Faith, and you choose to leave The Church for, shall we say "man-made reasons", you could be in a lot of hot water.
The particular situation I want to address is leaving for the wrong reasons. Lets say I know my faith very well. I know in my heart of hearts that it is The True Faith. That is, the one set up by Jesus Christ Himself as the ordinary way to salvation, then I am sinning by cutting myself off from the graces found in The True Faith.
Here is the most common example I have seen.
Bob is a good Catholic. He makes frequent use of the sacraments and is part of a prayer group and bible study group too. Bob knows his faith and is very happy in it. Bob believes in his "heart of hearts" that the Catholic Faith is the way to salvation.
Bob meets Tyra. She is a (insert your favorite non Catholic faith in here). Tyra is beautiful ( or as my teens say…She is a Hotty (SP?) Hottie (SP?). Tyra seems to be physically and emotionally what Bob has been searching for all his life. Tyra invites Bob to go to her church with her.
Bob is strong though and asks her to come to his church and attend Mass. Bob also is careful to instruct Tyra not to receive Holy Communion (since she is not currently in union with the Church) Tyra is reluctant to even go to Mass but promises to go if Bob will reciprocate and go with her to her church. Bob sees no harm in this because he is a strong faithful Roman Catholic. He is sure he can handle going to her church as a part of getting to know her better.
OK, so far no problem. Many Catholic people do this very thing. It is not a sin to go to the other church (if it is a Christian Denomination) as long as you fulfill your Sunday mass obligation as well. That means you go to mass on the same Sunday you attend the other church. Also, you must not go to Communion in this non Catholic Church because what you would be doing would say that the non Catholic church also has the True Presence of Jesus Christ – Body, Blood, Soul, and Divinity – which is a claim even the non Catholic Church would not be making. Basically you are free to go to the other church as an observer.
You should not assume that you will hear Catholic theology preached at this type of service. You should not base any of your faith on anything you hear preached there unless you are certain it is theologically sound. That means run it by a priest or a good Catholic theologian before you assume it is part of your Roman Catholic Faith.
Now back to Bob and his Hotty. Over a period of weeks Bob starts getting tired of going to both a service and a Mass every weekend. He and Tyra have become very close they are even talking about marriage. Bob starts thinking that it is a little silly to go to both churches and besides he has heard some great new stuff at Tyra's church. He starts thinking how the Holy Spirit is in all Christians ( which we would agree with) so why make a distinction? Her (Insert one of the following: Pastor, Minister, Deacon, Priest, Priestess, Master, Dali(SP?), Mulah (Sp?), Rabbi, Leader, etc. ) is just as spirit filled as Fr. Ed, his Roman Catholic parish priest is. Maybe even more so. This (leader of her church) can really preach up a storm. Besides Fr. Ed has pretty boring homilies and he is old too. Tyra's (church leader) seems to have a great relationship with God (or the Supreme Being whatever that is…) and it is easier to go to her church anyway. It is closer and the people are friendlier too.
On top of all this Tyra is starting to get tired of Bob's need to go to both. She has even stopped attending Mass with him and waits at his apartment watching TV while he is gone. That way she can keep him up on the scores of any football he might miss. Bob really enjoys Tyra's company and does not like to be away from her for too long. Bob slowly rationalizes that he could just as easily fulfill his Sunday obligation by going to Tyra's church. After all God is omnipresent (he is everywhere) right???
Bob chooses Tyra and her church over his because it is just plain easier.
Fr. Ed notices that Bob has stopped coming to mass. He sees Bob at the store buying groceries and asks if everything is OK.
Bob feels uncomfortable and thinks it is a personal matter now. He does not want to offend Fr. Ed by telling him the reason he has not been to mass.
Bob decides to leave The Faith rather than being a hypocrite!
Bob is now in the area that the Catechism addresses when it says that "there is no salvation outside the Catholic Church".
A few months later Bob wonders why he was ever Catholic anyway. It's probably just because his parents were Catholic, or because he was baptized as an infant and not given the fundamental option to choose Jesus as His personal Savior.
A few years later Bob is a (Deacon, Master, Dali, Rabbi, etc.) in Tyra's church, now his church too, and he tells his congregation how he used to be Catholic and how wrong they are. He tells his church that they don't have to go to a priest to be forgiven, just go directly to God. He wonders why for all those years he worshiped statues (which we don't do by the way) by lighting candles before them, or prayed a Rosary to Mary!
Bob actually feels free now and wants to help other members of his family, who are still (in his mind), "blinded by their faith in the Pope", to read the bible, to think for themselves, and get the answers they need.
OK, so this was just a story. I tell you this "story" because it almost happened to me.
Satan will not come at you with a red tail and a pitchfork. He is far too crafty for that. He will come slowly and sweetly making the path to heaven seem like a lot of bother. While his way is easier and more fun!
Mortuum Mundo! Vivum In Christo!


text:
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Darts Jigsaw Puzzle Game
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Darts Puzzle Details:
About: Darts is a traditional pub game, commonly played in the United Kingdom, across the Commonwealth, the Netherlands, Germany, Belgium, France, Republic of Ireland, the Scandinavian countries, the United States, and elsewhere. The dartboard may have its origins in the cross-section of a tree. An old name for a dartboard is "butt"; the word comes from the French word but, meaning "target". Initially the missiles were simply cut down arrows or crossbow bolts.The first purpose made darts were manufactured in one piece from wood; wrapped with a strip of lead for weight and fitted with flights made from split turkey feathers. The standard dartboard is divided into 20 numbered sections, scoring from 1 to 20 points, by wires running from the small central circle to the outer circular wire. Circular wires within the outer wire subdivide each section into single, double and triple areas. There are many games that can be played on a dartboard, but darts generally refers to a game whereby the player throws three darts per visit to the board with the goal of reducing a fixed score, commonly 501 or 301, to zero, with the final dart landing in either the bullseye or a double segment.
Puzzle Of The Day On: 05/Aug/2016
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