text:
\section{Introduction}
\IEEEPARstart{O}{ne} of computation models for a Boolean function $\mathrm{f}:\{-1,1\}^n\rightarrow\{-1,1\}$  is a Boolean  circuit or formula ~\cite{BooleanBook}. A  \emph{circuit}  is a directed acyclic graph in which nodes of in-degree zero  are either the Boolean constants or variables, nodes of in-degree $k\geq1$ are logical gates, computing Boolean functions  of $k$ arguments, and nodes of out-degree zero  correspond to the circuit outputs. If   a circuit has only a single output and  the output of each gate is used as an input to at most one gate then this circuit  is called a  \emph{formula}. In circuits,  as in any other model of computation,  the computational complexity and effects of noise are  important questions~\cite{Hajnal}.

The circuit complexity of a Boolean function is the minimum number of gates  (circuit \emph{size})  or the minimum  \emph{depth}\footnote{The depth of a circuit  is  the number of gates on the longest path from an input node to the output node } of a circuit, constructed from a particular set of gates, which computes this function.  However,  to find a circuit representation of a Boolean function  with a bounded  size or depth   is  a difficult problem~\cite{BooleanBook}. One approach to this problem  is to study complexity of \emph{typical} Boolean functions computed by random formulas~\cite{Brodsky}.

The two most studied methods of generating random formulas use random tree generation and a growth process as their core procedures. In the first method, a rooted $k$-ary tree is sampled from the uniform distribution of all rooted $k$-ary trees; the leaves of this random tree are then labelled by reference to the Boolean variables and internal nodes are labelled by the Boolean gates. This method was used to investigate the complexity of typical functions computed by  random AND/OR formulas~\cite{Lefmann,Chauvin,Gardy} and allowed to obtain a close relation between  the probability $\mathrm{P}[\mathrm{f}]$ of a random formula to compute a Boolean function $f$ and its size (complexity). However, it seems that this probability distribution is biased towards very low complexity functions~\cite{Chauvin}.

The second method uses the following growth process: Firstly, one defines the initial probability distribution $\mathrm{P}_{0}[\mathrm{f}]$ over the set $\mathrm{F}^0$  of simple Boolean functions of $N$ variables. Secondly, and in further steps, the functions chosen from the distributions $\mathrm{P}_{t}[\mathrm{f}]$  defined in previous steps are combined by Boolean gates: $\mathrm{F}^{t+1}=\{\alpha(\mathrm{f}_1,\ldots,\mathrm{f}_k); \mathrm{f}_j\in \mathrm{F}^t \textrm{ for } j=1,2,\ldots,k\}$. This process can be seen as a growth of $k$-ary balanced trees and was first used by Valiant to obtain an upper bound on the size of monotone  formulas computing the majority function~\cite{Valiant}.   Savick\'{y} recently showed for one of these processes, for $\mathrm{P}_{0}[\mathrm{f}]$ that is uniform on  some set of Boolean functions $\mathrm{F}^0$ and under very broad conditions on $\alpha$\textcolor{blue}{,} the probability  $\mathrm{P}_{t}[\mathrm{f}]$    tends to the uniform distribution over all Boolean functions of $N$ variables when $t\rightarrow\infty$~\cite{Savicky}.  The convergence rates of the Savick\'{y}'s process and its variants with different gates and initial conditions were studied in~\cite{Brodsky}.

Another important question in the circuit theory  is a reliable computations of Boolean functions in the presence of noise. One of the first to study the effect of noise in computing systems was von Neumann who attempted to explain the robustness of biologically-inspired computing circuits~\cite{VonNeumann}.  His model represented neural activities by a circuit (or formula) composed of $\epsilon$-noisy Boolean gates. The  $\epsilon$-noisy gate is designed to compute a Boolean function $\alpha:\{-1,1\}^k\rightarrow\{-1,1\}$,  but for each input $\sigma\in\{-1,1\}^k$  there is an error probability $\epsilon$ such that $\alpha(\sigma)\rightarrow-\alpha(\sigma)$. To simplify the analysis, error-probability is taken to be independent for each gate in the circuit. Clearly, a noisy circuit ($\epsilon>0$) cannot perform any given computation in a deterministic manner: for any circuit-input there is a non-vanishing probability that the circuit will produces the wrong output.  The maximum of this error probability $\delta$ over all circuit-inputs determines the \emph{reliability} of the circuit.  In his paper, von Neumann showed that reliable computation ($\delta<1/2$) is possible for a sufficiently small $\epsilon$~\cite{VonNeumann} and demonstrated how reliability of a Boolean noisy circuit can be improved by using constructions based only on $\epsilon$-noisy gates.

There had been little development in the analysis of noisy computing systems until the seminal work of Pippenger~\cite{Pippenger:RC} who addressed the problem from an information theory point of view. He showed that if a noisy $k$-ary formula is used to compute a Boolean function $f$ with the error probability $\delta<1/2$, then (i) there is an upper bound for the gate-error $\epsilon(k)$ which is strictly less than $1/2$ and (ii)
there is a lower bound for the formula-depth $\hat{d}(k,\epsilon,\delta)\ge d$, where $d$ is the depth of a noiseless formula computing $f$. In comparison to its noiseless counterpart, a noisy formula that computes reliably has greater depth due to the presence of restitution-gates, implying longer computation times~\cite{Pippenger:RC}.

A number of papers have followed and extended Pippenger's results. For instance, similar results were derived for circuits by Feder~\cite{Feder:RC}, who also improved the bounds obtained by Pippenger for formulas. The exact noise thresholds for $k$-ary Boolean formulae were later determined for odd $k$~\cite{Hajek:MTN,Evans:MTNK} and  for  formulas constructed from $2$-input NAND gates~\cite{Evans:MTN}; the latter was recently suggested as the exact noise threshold for general $2$-input gate formulas~\cite{Unger}.

Results derived for noisy Boolean formulas in~\cite{Hajek:MTN,Evans:MTNK} rely on a specific construction which uses $\epsilon$-noisy  \emph{majority} gates. The noiseless variant of this gate performs the majority-vote function\footnote{We use the definition $\sgn[x]\!=\!1$ for $x>0$, $\sgn[x]\!=\!-1$ for $x<0$ and  $\sgn[0]\!=\!0$ throughout this paper.} $\sgn[\sum_{i=1}^k S_j]$ on the binary inputs $S_j\in\{-1,1\}$ and naturally the number of these inputs $k$ is \emph{odd}. In contrast to previous work, in this paper we concentrate on the possibility of reliable computation in \emph{randomly generated} Boolean formulas. As a first step towards this goal, we study the effects of $\epsilon$-noise on the formulas generated in the Savick\'{y}'s growth process.

\section{Noisy growth processes and main results}
\begin{figure}[!t]
\setlength{\unitlength}{1mm}

\begin{picture}(50,30)

\put(-2,12){\begin{tabular}{ l c c c r }
   $\xi\!=\!$& \textcolor{red}{-1}& 1 & \ldots &\textcolor{red}{-1}\\
  $\mathrm{f}_1\!=$& 1& 1 & \ldots &-1\\
    $\mathrm{f}_2\!=$ & 1& -1 & \ldots &1 \\
  \vdots & \vdots & \vdots & \ldots &\vdots \\
    $\mathrm{f}_k\!=$ & 1 &-1 & \ldots &-1 \\
\end{tabular}}

\put(41,9){$\alpha$}

\put(49,9){\begin{tabular}{ l c c r }
   \textcolor{red}{-1}& 1 & \ldots &$\textcolor{red}{1}=\!\mathrm{f}$
  \end{tabular}}
\thicklines
\put(37, 19.5){\vector(1, -2){4}}
\put(37, 13.5){\vector(1, -1){4}}
\put(37, 0){\vector(1, 2){4}}

\put(43, 9.5){\vector(1, 0){7}}

\put(37, 26){\line(1, 0){8}}
\put(45, 26){\line(0, -1){16.5}}

\put(43.6, 8.5){$\times$}
\end{picture}

\caption{Noisy growth process. i) Boolean functions $\mathrm{f}_1,\ldots,\mathrm{f}_k$ (represented by binary strings of length $2^N$) are sampled randomly and independently from the distribution $\mathrm{P}_t[\mathrm{f}]$. ii) These functions are then used to compute a new Boolean function $\mathrm{f}$ via the gate $\alpha$. At each step of this computation noise (represented by the binary string $\xi$) inverts the output of $\alpha$ (this operation is represented by the $\times$ symbol) with probability $\epsilon$. In this figure the first and the last bits of the function $\mathrm{f}$ (in red) are inverted by noise. Repeating operations i) and ii) many times gives rise to an ensemble described by the  distribution $\mathrm{P}_{t+1}[\mathrm{f}]$.    \label{fig:1}
 }
\end{figure}
Let us introduce noise into the formulas generated by  Savick\'{y}'s growth process. In order to do this we note that the noiseless case, as described in the Introduction, can be  also seen as a computation, performed by gate $\alpha$, of a new Boolean function $\mathrm{f}$ from $k$ Boolean functions $\mathrm{f}_1,\ldots,\mathrm{f}_k$. These functions, represented by binary $\pm1$ strings (or vectors) of length $2^N$, are drawn randomly and independently from the same distribution~\cite{Savicky}. However,  each computation at the gate $\alpha$  may be corrupted by noise that inverts the result of this computation with  probability $\epsilon$ (see Figure \ref{fig:1}). Averaging the process over many realizations of noise  leads us to the equation
\begin{eqnarray}
\mathrm{P}_{t+1}[\!\;\mathrm{f}\!\;]&=&\sum_{\{\mathrm{f}_j\}}\prod_{j=1}^k \big\{\mathrm{P}_t[\;\mathrm{f}_j\;]\big\}\label{eq:ProbOfF}\\
&&\times\prod_{i=1}^{2^N}\frac{\mathrm{e}^{\beta \mathrm{f}^i\alpha(\mathrm{f}_1^i,\ldots,\mathrm{f}_k^i)}}{2\cosh\beta  \alpha(\mathrm{f}_1^i,\ldots,\mathrm{f}_k^i)}\nonumbe
\end{eqnarray}
which gives us the probability of a Boolean function $\mathrm{f}$ being computed by the noisy formulas of depth $t+1$. Here for convenience we have introduced the inverse ``temperature'' parameter $\beta=1/T$ which is related  to the noise parameter $\epsilon$ via the equality $\tanh\beta=1-2\epsilon$.  The limits $\beta\rightarrow0/\infty$ correspond to completely random/deterministic cases.

Without noise ($\beta\rightarrow\infty$) the equation  (\ref{eq:ProbOfF}) reduces to
\begin{eqnarray}
\mathrm{P}_{t+1}[\!\;\mathrm{f}\!\;]&=&\sum_{\{\mathrm{f}_j\}}\prod_{j=1}^k \big\{\mathrm{P}_t[\;\mathrm{f}_j\;]\big\}\label{eq:ProbOfFnoNoise}\\
&&\times\prod_{i=1}^{2^N} \delta\left[\mathrm{f}^i;\alpha(\mathrm{f}_1^i,\ldots,\mathrm{f}_k^i)\right],\nonumber
\end{eqnarray}
where we use $\delta[x;y]$ to denote Kronecker delta. Equation~(\ref{eq:ProbOfFnoNoise}) was studied in the original Savick\'{y}'s work~\cite{Savicky} and subsequent studies~\cite{Brodsky} where the stationary distribution  $\mathrm{P}_\infty[\!\;\mathrm{f}\!\;]=\lim_{t\rightarrow\infty}\mathrm{P}_t[\!\;\mathrm{f}\!\;]$ of the noiseless process (\ref{eq:ProbOfFnoNoise}) was studied with the initial conditions $\mathrm{P}_{0}[\!\;\mathrm{f}\!\;]\;\;\;=\frac{1}{\vert\mathrm{F}^0\vert}\sum_{\mathrm{g}\in\mathrm{F}^0}\prod_{i=1}^{2^N}\delta[\mathrm{f}^i;\mathrm{g}^i] $ for  different initial sets $\mathrm{F}^0$  of simple Boolean functions (constants, identities, etc.) and different gates $\alpha$. Depending on these parameters the stationary distribution is  either concentrated on a single function, i.e. $\mathrm{P}_\infty[\!\;\mathrm{f}\!\;]=\prod_{i=1}^{2^N}\delta\left[\;\mathrm{f}^I;\mathrm{g}^i\right]$ or on some set of functions $\mathrm{F}$, i.e. $\mathrm{P}_\infty[\!\;\mathrm{f}\!\;]=  \frac{1}{\vert\mathrm{F}\vert}\sum_{\mathrm{g}\in\mathrm{F}}\prod_{i=1}^{2^N}\delta[\mathrm{f}^i;\mathrm{g}^i]$. There are also cases when for $t\rightarrow\infty$ the distributions $\mathrm{P}_t[\!\;\mathrm{f}\!\;]$ and $\mathrm{P}_{t+1}[\!\;\mathrm{f}\!\;]$ are distinct.

Our main contribution to these studies is the following result for the noisy process (\ref{eq:ProbOfF}).
\begin{theorem}\label{theorem:1}
For a \emph{balanced} gate\footnote{The gate is balanced when it has an equal number of $+1$'s and $-1$'s in its output.} $\alpha$  the stationary distribution  $\mathrm{P}_{\infty}[\mathrm{f}]=\frac{1}{2^{2^N}}$ is a stable and unique solution of the process (\ref{eq:ProbOfF}) when $\epsilon>\epsilon(k)=\frac{1-b(k)}{2}$, where $b(k)\!\equiv \!\left\{2^{k\!-\!1}/k\binom{k\!-\!1}{(k\!-\!1)/2};\; 2^{k\!-\!2}/(k\!-\!1)\binom{k\!-\!2}{(k\!-\!2)/2}\right\}$, with $k\geq3$, for $k$ odd and even respectively
\end{theorem}
\begin{IEEEproof}
In order to show this, we first use the result,  derived in  Appendix \ref{section:m}, that the distribution $\mathrm{P}_{t+1}[\mathrm{f}]$ can be represented via its moments $\mathrm{m}^S(t+1)=\sum_{\hat\mathrm{f}}\mathrm{P}_{t+1}[\hat\mathrm{f}]\prod_{i\in S}\hat\mathrm{f}^i$, where $S$ is a subset of the set $[2^N]=\{1,\ldots,2^N\}$,   and $\mathrm{P}_{t+1}[\mathrm{f}]$ is given by
	%
\begin{eqnarray}
\mathrm{P}_{t+1}[\mathrm{f}]=\frac{1}{2^{2^N}}\left(1+\sum_{S\subseteq[2^N]}\mathrm{m}^S(t+1)\prod_{i\in S}\mathrm{f}^i\right),
\end{eqnarray}
where the $n$-th moment is  governed by the equation
\begin{eqnarray}
&&\mathrm{m}^{i_1,\ldots,i_n}(t+1)\label{eq:p-m}\\
&&=\tanh^n(\beta)\sum_{\{\mathrm{f}_j^i\}}\prod_{j=1}^k\Bigg\{\nonumber\\
&&\times \frac{1}{2^{n}}\left[1+\sum_{S\subseteq\textbf{I}}\mathrm{m}^S(t)\prod_{i\in S}\mathrm{f}^i_j\right]\Bigg\}\nonumber\\
&&\times\prod_{i\in \textbf{I}}\alpha(\mathrm{f}_1^{i},\ldots,\mathrm{f}_k^{i})\nonumber
\end{eqnarray}
with $\textbf{I}=\{i_1,\ldots,i_n\}$. Thus the $n$-th moment at $t+1$ is a function of only the $n$-th and lower order moments at $t$.

Let us assume now that $\tanh(\beta)<b(k)$ then by the Lemma \ref{lemma:2}, with $n=1$, the first moments of the distribution $\mathrm{P}_{t+1}[\mathrm{f}]$ are vanishing as $t\rightarrow\infty$. But then, by applying the same lemma to the order $n\geq2$ moments, we conclude that all moments are vanishing  as $t\rightarrow\infty$.
\end{IEEEproof}

\begin{figure}
\setlength{\unitlength}{1.0mm}
\begin{picture}(50,50)
\put(0,0){\epsfysize=50\unitlength\epsfbox{epsilon_k.eps}}
\put(1,35){$\epsilon(k)$
\put(39,-3){$k$}
\end{picture}
\caption{Upper bound for reliable computation by noisy $k$-ary random formulas. \label{fig:2}}
\end{figure}

In addition to its direct interpretation that above $\epsilon(k)$ (see Figure  \ref{fig:2}) the noisy process (\ref{eq:ProbOfF}) is \emph{ergodic} and has only one stationary solution, the result of Theorem \ref{theorem:1}  also has consequences for computation  in noisy random formulas.  A feature of noisy formulas\textcolor{blue}{,} which is  essential for reliable computation\textcolor{blue}{,} is their greater depth due to the presence of correcting $\epsilon$-noisy gates~\cite{Pippenger:RC}. This correction operation can be seen as a procedure which reduces the entropy, but in our case of very deep ($t\rightarrow\infty$) random formulas the entropy is at its maximum when $\epsilon>\epsilon(k)$. Thus any computation, even as simple as computing identity function, can not be performed reliably in this regime.

For odd $k$ our result for the bound $\epsilon(k)$ is exactly equal to the  \emph{exact} threshold for reliable computation by general $k$-ary formulas~\cite{Hajek:MTN,Evans:MTNK}. It is not clear however if this threshold is also exact, i.e. any Boolean function can be computed for $\epsilon\in(0,\epsilon(k))$ with the error $\delta<1/2$, for randomly generated formulas. For even $k>2$ this threshold is not known, but our result suggests that for balanced gates $\alpha$ it can not exceed the bound $\epsilon(k)$ of Theorem \ref{theorem:1}. Furthermore as  $k\rightarrow\infty$ the $\epsilon(k)$ approaches $1/2$ as $1/2-\epsilon(k)=O(1/\sqrt{k})$, this is follows from the Stirling's approximation of $b(k)$, which is in agreement with the bound computed in~\cite{Evans:SPNC} for general formulas.

The results of Lemma \ref{lemma:1}, which are used in the proof of Theorem \ref{theorem:1}, can be also exploited to show that reliable computation in randomly generated formulas is possible. This can be shown as follows. Suppose that  gate $\alpha$ in the process (\ref{eq:ProbOfF}) is the same as the one studied in Lemma \ref{lemma:1}. Assume that the initial distribution $\mathrm{P}_{0}[\mathrm{f}]$ is such that the stationary distribution $\mathrm{P}_{\infty}[\mathrm{f}]$ of the noiseless  process (\ref{eq:ProbOfFnoNoise}) is concentrated on only one Boolean function, i.e. all formulas compute the same function.  This implies that for any input $I\in\{-1,1\}^N$ all formulas simultaneously provide an output of $+1$ or $-1$. Then in the presence of noise the average formula errors in its output with the probability $(1-\mathrm{m}_I(\infty))/2$, where $\mathrm{m}_I(\infty)$ is the stationary solution of equation (\ref{eq:m}) corresponding to this input. From the analysis in Lemma \ref{lemma:1} follows  that the maximum of this error over all inputs is $\delta=(1-\mathrm{m}(\infty))/2$ and is bounded away from $1/2$ when  $\epsilon<\epsilon(k)$. Furthermore, the output error can be reduced by decreasing  $\epsilon$ or by increasing $k$. Thus in this regime any Boolean function can be computed with any desired accuracy.

\section{Computation of the lower bound values\label{section:ub}}
In this section we compute the values of lower bounds appearing in  Theorem \ref{theorem:1}. In order to do this we  choose a balanced gate $\chi(\sigma)$ from the set of gates $\sgn\left[\sum_{j=1}^k \sigma_j\right]+\mathbf{1}\left[\sum_{j=1}^k \sigma_j=0\right]\gamma(\sigma)$, where $\gamma(\sigma)\in\{\!-\!1,1\}$ is such that $\sum_\sigma\mathbf{1}\left[\sum_{j=1}^k \sigma_j=0\right]\gamma(\sigma)\!=\!0$, and consider the first moments $\mathrm{m}_i(t)=\sum_{\mathrm{f}}\mathrm{P}_{t}[\;\mathrm{f}\;]\;\mathrm{f}^i$. These are governed by the equations
\begin{eqnarray}
\mathrm{m}(t+1)&=&F_{\chi}(\mathrm{m}(t))\label{eq:m}\\
&=&\tanh(\beta)\sum_{\ell=0}^k \binom{k}{\ell}\sgn[2\ell-k]\nonumber \\
&&\times\left[\frac{1+\mathrm{m}(t)}{2}\right]^{\ell}\left[\frac{1-\mathrm{m}(t)}{2}\right]^{k-\ell}\nonumber
\end{eqnarray}
\begin{lemma}\label{lemma:1}
For $k\geq3$ the function $F_\chi(\mathrm{m})$, defined in the equation (\ref{eq:m}), has the following properties: i) if $\tanh\beta\leq b(k)$ then $\mathrm{m}>F_\chi(\mathrm{m})$ for $\mathrm{m}\in(0,1]$ and $F_\chi(\mathrm{m})>\mathrm{m}$ for $\mathrm{m}\in[-1,0)$; ii) if $\tanh\beta> b(k)$ then $ \exists\; \mathrm{m}^*\neq 0$ such that $\mathrm{m}^*=F_\chi(\mathrm{m}^*)$,
where $b(k)$ is defined in the Theorem \ref{theorem:1}.
\end{lemma}
\begin{IEEEproof}
This lemma follows from  the equalities $F_\chi(\pm1)=\pm\tanh\beta, F_\chi(0)=0$ (this can be shown by direct substitution) and the fact that $F_\chi(\mathrm{m})$ is a strictly increasing function, which is also convex and concave on the intervals $(-1,0)$ and $(0,1)$, respectively (to show this we study properties of $F_\chi(\mathrm{m})$ in the Appendix~\ref{section:F}). Then i) is true because $\frac{\mathrm{d} F_\chi}{\mathrm{d}\mathrm{m}}\vert_{\mathrm{m}=0}<1$ when $\tanh\beta< b(k)$ and  ii) is true because  of $\frac{\mathrm{d} F_\chi}{\mathrm{d}\mathrm{m}}\vert_{\mathrm{m}=0}\geq1$ when $\tanh\beta\geq b(k)$.
\end{IEEEproof}

\section{Moments of $\mathrm{P}_{t}[\;\mathrm{f}\;]$\label{section:moments}}
Let us consider  equation (\ref{eq:p-m}) for an $n$-th moment $\mathrm{m}$. Assuming that all lower order moments  vanish allows us to write this equation in a very simple form
\begin{eqnarray}
\mathrm{m}(t+1)&=& F_\alpha(\mathrm{m}(t))\label{eq:p-m-reduced}\\
&=&\tanh^n(\beta)\sum_{\{\sigma_j^i\}}\prod_{j=1}^k\Bigg\{\nonumber\\
&&\times \frac{1}{2^{n}}\left[1+\mathrm{m}(t)\prod_{i=1}^n\sigma^i_j\right]\Bigg\}\nonumber\\
&&\times\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i}).\nonumber
\end{eqnarray}
For a balanced gate $\alpha$  the point $\mathrm{m}=0$ is a stationary solution  of the above equation and  has the following property.

\begin{lemma}\label{lemma:2}
The point $\mathrm{m}=0$ is a stable and unique solution of (\ref{eq:p-m-reduced}) when $\tanh^n(\beta)<b(k)$.
\end{lemma}
\begin{IEEEproof}
In order to prove this we first show that
\begin{eqnarray}
&&\tanh(\beta)^{n-1}\mathrm{F}_\chi(m)\\
&&=\tanh^n(\beta)\sum_{\{\sigma_j^i\}}\prod_{j=1}^k \frac{1}{2^{n}}\left[1+\mathrm{m}\prod_{i=1}^n\sigma^i_j\right]\nonumber\\
&&\times\sgn\left[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\right],\nonumber
\end{eqnarray}
where $F_\chi(m)$ is defined in (\ref{eq:m}). This can be shown by a direct calculation as follows
\begin{eqnarray}
&&\tanh^n(\beta)\sum_{\{\sigma_j^i\}}\prod_{j=1}^k \frac{1}{2^{n}}\left[1+\mathrm{m}\prod_{i=1}^n\sigma^i_j\right]\\
&&\times\sgn\left[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\right]\nonumber\\
&&=\tanh^n(\beta)\frac{1}{2^{n-1}}\sum_{\{\sigma_j^i\}}\prod_{j=1}^k \left[\frac{1+\mathrm{m}\sigma^1_j}{2}\right]\nonumber\\
&&\times\sgn\left[\sum_{j=1}^k\sigma^1_j\right]=\tanh(\beta)^{n-1}\mathrm{F}_\chi(m).\nonumber
\end{eqnarray}
In the above the first equality was obtained by applying transformation $\sigma_j^1\rightarrow\prod_{i=1}^n\sigma^i_j$ and the last equality followed from comparing this result with the right hand side of equation (\ref{eq:m}).

Next, for a balanced gate $\alpha$  we compute the difference $\Delta(\mathrm{m})=\tanh^{n-1}(\beta)\mathrm{F}_\chi(\mathrm{m})-\mathrm{F}_\alpha(\mathrm{m})$ in  Appendix~\ref{section:deltam}. The result of this computation is that $\Delta(\mathrm{m})\geq0$ and $\Delta(\mathrm{m})\leq0$ on the intervals $\mathrm{m}\!\in\![0,1)$ and $\mathrm{m}\!\in\!(-1,0]$, respectively, from which the bounds $\tanh^{n-1}(\beta)\mathrm{F}_\chi(\mathrm{m})\geq \mathrm{F}_\alpha(\mathrm{m})$ and $\tanh^{n-1}(\beta)\mathrm{F}_\chi(\mathrm{m})\leq \mathrm{F}_\alpha(\mathrm{m})$ on the same intervals follow. The behaviour of $\tanh^{n-1}(\beta)\mathrm{F}_\chi(\mathrm{m})$ with respect to the inverse temperature $\beta$ is the same as of $\mathrm{F}_\chi(\mathrm{m})$, which we described in Lemma \ref{lemma:1}, but with the $\tanh(\beta)$ being replaced by the $\tanh^n(\beta)$.

\end{IEEEproof}

\section{Conclusion}
The paper extends previous work~\cite{Hajek:MTN,Evans:MTNK} on the reliability of computation in Boolean formulas and generation of random Boolean functions~\cite{Savicky,Brodsky}, by investigating the properties of formulas constructed by a random growth process whereby computing elements, primarily $k$-ary balanced gates, are subject to $\epsilon$-noise.

We show that the noisy growth process is ergodic above the noise bound $\epsilon(k)$ and hence the formulas generated by it are unreliable. We also show that  formulas constructed from majority-like gates, which saturate this bound, can be used for computing any Boolean function when $\epsilon<\epsilon(k)$. Our earlier work, which uses methods of non-equilibrium statistical physics, suggests that the same noise bound  also applies to the noisy feed-forward~\cite{noisyPRE} and recurrent Boolean networks~\cite{PhilMag}.

The current analysis  is restricted to reliable computation in a growth process that uses only \emph{balanced} gates\footnote{The results of this paper can be easily extended to the \emph{distributions} over  balanced gates~\cite{PhilMag}.} and produces (without noise) only \emph{one} Boolean function;  but we envisage that it can be extended to study more general scenarios of non-balanced gates and a  richer distributions of Boolean functions~\cite{Brodsky}.

\appendices

\section{Moment representation of $\mathrm{P}_{t}[\;\mathrm{f}\;]$\label{section:m}}
The probability distribution $\mathrm{P}_{t}[\;\mathrm{f}\;]$ can be represented via its moments.  In order to find this representation we can use the identity $\sum_{\hat\mathrm{f}}\delta[\hat\mathrm{f};\mathrm{f}]=1$ to write $\mathrm{P}_t[\mathrm{f}]=\sum_{\hat\mathrm{f}}\delta[\hat\mathrm{f};\mathrm{f}]\mathrm{P}_t[\hat\mathrm{f}]$. Then, because $\delta[x;y]=\frac{1}{2}(1+xy)$ for $x,y\in\{-1,1\}$, we obtain
\begin{eqnarray}
\mathrm{P}_t[\mathrm{f}]&=&\sum_{\hat\mathrm{f}}\mathrm{P}_t[\hat\mathrm{f}]\prod_{i=1}^{2^N}\frac{1}{2}\left(1+\hat\mathrm{f}^i\mathrm{f}^i\right)\label{def:ProbOfFviAm}\\
&=&\frac{1}{2^{2^N}}\left(1+\sum_{\hat\mathrm{f}}\mathrm{P}_t[\hat\mathrm{f}]\sum_{S\subseteq[2^N]}\prod_{i\in S}\hat\mathrm{f}^i\mathrm{f}^i\right)\nonumber\\
&=&\frac{1}{2^{2^N}}\left(1+\sum_{S\subseteq[2^N]}\mathrm{m}^S(t)\prod_{i\in S}\mathrm{f}^i\right),\nonumber
\end{eqnarray}
where $\mathrm{m}^S(t)=\sum_{\hat\mathrm{f}}\mathrm{P}_t[\hat\mathrm{f}]\prod_{i\in S}\hat\mathrm{f}^i$ are the moments of $\mathrm{P}_t[\mathrm{f}]$.

Let us now derive equation governing evolution of  the $n$-th moment $\mathrm{m}^{i_1,i_2,\ldots,i_n}(t)$. This  can be obtained by multiplying both sides of  equation~(\ref{eq:ProbOfF}) by the monomial $\prod_{i\in \textbf{I}}\mathrm{f}^i$, where $\textbf{I}=\{i_1,\ldots,i_n\}$, and taking the sums over $\mathrm{f}$ as follows
\begin{eqnarray}
&&\sum_\mathrm{f}\mathrm{P}_{t+1}[\!\;\mathrm{f}\!\;]\prod_{i\in \textbf{I}}\mathrm{f}^i\label{def:m}\\
&&=\sum_\mathrm{f}\sum_{\{\mathrm{f}_j\}}\prod_{j=1}^k \big\{\mathrm{P}_t[\;\mathrm{f}_j\;]\big\}\nonumber\\
&&\times\prod_{i=1}^{2^N}\frac{\mathrm{e}^{\beta \mathrm{f}^i\;\!\alpha(\mathrm{f}_1^i,\ldots,\mathrm{f}_k^i)}}{2\cosh\beta  \alpha(\mathrm{f}_1^i,\ldots,\mathrm{f}_k^i)}\prod_{\ell\in \textbf{I}}\mathrm{f}^\ell\nonumber\\
&&=\tanh^n(\beta)\sum_{\{\mathrm{f}_j^i\}}\prod_{j=1}^k \big\{\mathrm{P}_t[\;\mathrm{f}_j^{i_1},\ldots,\mathrm{f}_j^{i_n}\;]\big\}\nonumber\\
&&\times\prod_{i\in \textbf{I}}\alpha(\mathrm{f}_1^{i},\ldots,\mathrm{f}_k^{i})\nonumber\\
&&=\mathrm{m}^{i_1,\ldots,i_n}(t+1),\nonumber
\end{eqnarray}
where in the above we have used the property $\tanh(-x)=-\tanh(x)$. Finally, using the moment representation (\ref{def:ProbOfFviAm}) in the above, we obtain
\begin{eqnarray}
&&\mathrm{m}^{i_1,\ldots,i_n}(t+1)\\%\label{eq:p-m}
&&=\tanh^n(\beta)\sum_{\{\mathrm{f}_j^i\}}\prod_{j=1}^k\Bigg\{\nonumber\\
&&\times \frac{1}{2^{n}}\left[1+\sum_{S\subseteq\textbf{I}}\mathrm{m}^S(t)\prod_{i\in S}\mathrm{f}^i_j\right]\Bigg\}\nonumber\\
&&\times\prod_{i\in \textbf{I}}\alpha(\mathrm{f}_1^{i},\ldots,\mathrm{f}_k^{i}).\nonumber
\end{eqnarray}

\section{Derivation of $F_{\chi}$ and analysis of its properties\label{section:F}}
Here we first derive the function $F_\chi(\mathrm{m})$ then we study its properties. Let us first compute the sum
\begin{eqnarray}
&&\sum_{\{\mathrm{f}_j\}}\prod_{j=1}^k \left[\frac{1+\mathrm{f}_j\;\mathrm{m}}{2}\right]\Bigg\{\sgn\left[\sum_{j=1}^k \mathrm{f}_j\right]\\
&&+\mathbf{1}\left[\sum_{j=1}^k \mathrm{f}_j=0\right]\gamma(\mathrm{f}_1,\ldots,\mathrm{f}_k)\Bigg\}\nonumber\\
&=&\sum_{\{\mathrm{f}_j\}} \left[\frac{1+\mathrm{m}}{2}\right]^{(\sum_{j=1}^k\mathrm{f}_j+k)/2} \left[\frac{1-\mathrm{m}}{2}\right]^{(k-\sum_{j=1}^k\mathrm{f}_j)/2}\nonumber\\
&&\times\Bigg\{\sgn\left[\sum_{j=1}^k \mathrm{f}_j\right]+\mathbf{1}\left[\sum_{j=1}^k \mathrm{f}_j=0\right]\gamma(\mathrm{f}_1,\ldots,\mathrm{f}_k)\Bigg\}\nonumber\\
&=&\sum_{\ell=0}^k \binom{k}{\ell}\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\sgn[2\ell-k]~,\nonumber
\end{eqnarray}
in the definition (\ref{def:m}) for the specific choice of $\alpha\equiv\chi$. This result leads  to the  function $F_\chi(\mathrm{m})$ used in  equation (\ref{eq:m}).

We are interested in how the function  $F_\chi(\mathrm{m})$ behaves on the interval  $\mathrm{m}\in[-1,1]$ and how this behaviour is affected by the parameter $\tanh\beta$. In order to find this out  we first rewrite $F_\chi(\mathrm{m})$ as follows
\begin{eqnarray}
&&F_{\chi}(\mathrm{m})\\
&&=\tanh(\beta)\nonumber\\
&&\times\sum_{\ell=0}^k \binom{k}{\ell}\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\nonumber\\
&&\times\left\{\mathbf{1}[2\ell-k>0]-\mathbf{1}[2\ell-k<0]\right\},\nonumber
\end{eqnarray}
but
\begin{eqnarray}
&&\sum_{\ell=0}^k \binom{k}{\ell}\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\\
&&=\sum_{\ell=0}^k \binom{k}{\ell}\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\nonumber\\
&&\times\left\{\mathbf{1}[2\ell-k\geq0]+\mathbf{1}[2\ell-k<0]\right\}\nonumber\\
&&=1\nonumber
\end{eqnarray}
so
\begin{eqnarray}
&&F_{\chi}(\mathrm{m})\\
&&=\tanh(\beta)\Bigg(1-2\sum_{\ell=0}^{\tilde{k}} \binom{k}{\ell}\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~\times\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\nonumber\\
&&- \mathbf{1}[k\equiv0\!\!\!\!\pmod{2}]\nonumber\\
&&~~~~\times\binom{k}{k/2}\left(\left[\frac{1+\mathrm{m}}{2}\right]\left[\frac{1-\mathrm{m}}{2}\right]\right)^{k/2}\Bigg),\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\tilde{k}&=&\mathbf{1}[k\equiv0\pmod{2}](k/2-1)\nonumber\\
&+&\mathbf{1}[k\equiv0\pmod{1}](k-1)/2.\nonumber
\end{eqnarray}

Now we use the above representation of $F_{\chi}(\mathrm{m})$ to compute
\begin{eqnarray}
&&\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})\\
&&=\tanh(\beta)\sum_{\ell=0}^{\tilde{k}} \binom{k}{\ell}\nonumber\\
&&\times\Bigg((k-\ell)\left[\frac{1+\mathrm{m}}{2}\right]^{\ell} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell-1}\nonumber\\
&&-\ell\left[\frac{1+\mathrm{m}}{2}\right]^{\ell-1} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\ell}\Bigg)\nonumber\\
&&+ \tanh(\beta)\frac{k}{4}\mathbf{1}[k\equiv0\!\!\!\!\pmod{2}]\binom{k}{k/2}\nonumber\\
&&\times\mathrm{m}\left(\left[\frac{1+\mathrm{m}}{2}\right]\left[\frac{1-\mathrm{m}}{2}\right]\right)^{k/2-1}\nonumber\\
&&=\tanh(\beta) \binom{k}{\tilde{k}+1}(\tilde{k}+1)\nonumber\\
&&\times\left[\frac{1+\mathrm{m}}{2}\right]^{\tilde{k}} \left[\frac{1-\mathrm{m}}{2}\right]^{k-\tilde{k}-1}\nonumber\\
&&+ \tanh(\beta)\frac{k}{4}\mathbf{1}[k\equiv0\!\!\!\!\pmod{2}]\binom{k}{k/2}\nonumber\\
&&\times\mathrm{m}\left(\left[\frac{1+\mathrm{m}}{2}\right]\left[\frac{1-\mathrm{m}}{2}\right]\right)^{k/2-1}\nonumber
\end{eqnarray}
So, using definition of $\tilde k$,   we obtain
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})&=&\tanh(\beta) \binom{k}{(k+1)/2}\!\!\left(\frac{k+1}{2}\right)\label{eq:dF1}\\
&&\times\left(\left[\frac{1+\mathrm{m}}{2}\right] \left[\frac{1-\mathrm{m}}{2}\right]\right)^{(k-1)/2}\nonumber
\end{eqnarray}
for $k$ odd and
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})&=&\tanh(\beta) \binom{k}{k/2}\left(\frac{k}{4}\right)\label{eq:dF2}\\
&&\times\left(\left[\frac{1+\mathrm{m}}{2}\right] \left[\frac{1-\mathrm{m}}{2}\right]\right)^{k/2-1}\nonumber
\end{eqnarray}
for $k$ even.

Thus $\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})>0$ on for all $\mathrm{m}\in(-1,1)$ and hence $F_\chi(\mathrm{m})$ is a strictly increasing function. Furthermore, the function $F_\chi(\mathrm{m})$  at the point $\mathrm{m}=0$ changes its slope from $\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})\vert_{\mathrm{m}=0}<1$ to $\frac{\mathrm{d}}{\mathrm{d}\mathrm{m}}F_\chi(\mathrm{m})\vert_{\mathrm{m}=0}\geq1$ at  $$\tanh(\beta)=2^{k-1}/k\binom{k-1}{(k-1)/2}$$ for $k$ odd and $$\tanh(\beta) =2^{k-2}/\binom{k-2}{(k-2)/2}(k-1)$$ for $k$ even.

Let us now compute the second derivative of $F_\chi(\mathrm{m})$. Differentiating  equations~(\ref{eq:dF1}) and (\ref{eq:dF2})  with respect to $\mathrm{m}$ gives us
\begin{eqnarray}
\frac{\mathrm{d}^2}{\mathrm{d}\mathrm{m}^2}F_\chi(\mathrm{m})&=&-\mathrm{m}\tanh(\beta)\label{eq:ddF1}\\
&&\times \binom{k}{(k+1)/2}\left(\frac{k+1}{2}\right)\frac{(k-1)}{4}\nonumber\\
&&\times\left(\left[\frac{1+\mathrm{m}}{2}\right] \left[\frac{1-\mathrm{m}}{2}\right]\right)^{(k-1)/2-1}\nonumber
\end{eqnarray}
for $k$ odd and
\begin{eqnarray}
\frac{\mathrm{d}^2}{\mathrm{d}\mathrm{m}^2}F_\chi(\mathrm{m})&=&-\mathrm{m}\tanh(\beta)\label{eq:ddF2}\\
&&\times \binom{k}{k/2}\left(\frac{k}{4}\right)\frac{(k-2)}{4}\nonumber\\
&&\times\left(\left[\frac{1+\mathrm{m}}{2}\right] \left[\frac{1-\mathrm{m}}{2}\right]\right)^{k/2-2}\nonumber
\end{eqnarray}
for $k$ even. We note that both are of the form $\frac{\mathrm{d}^2}{\mathrm{d}\mathrm{m}^2}F_\chi(\mathrm{m})=-\mathrm{m} G(\mathrm{m})$, where $G(\mathrm{m})>0$ for all $\mathrm{m}\in(-1,1)$. Thus the function  $F_\chi(\mathrm{m})$ is strictly convex and concave on the intervals  $(-1,0)$ and $(0,1)$ respectively.

\section{Computation of $\Delta(\mathrm{m})$ and analysis of its properties\label{section:deltam}}
Let us first define the average $\left\langle \cdots\right\rangle_{\sigma\vert \mathrm{m}}=\sum_{\{\sigma_j^i\}}\prod_{j=1}^k \frac{1}{2^n}\left[1+\mathrm{m}\prod_{i=1}^n\sigma^i_j\right](\cdots)$ and shorthand notations $\left\{\mathbf{1}_+[\sigma],\mathbf{1}_-[\sigma],\mathbf{1}_0[\sigma]\right\}$ for the indicator functions $\{\mathbf{1}[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\!>\!0],\mathbf{1}[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\!<\!0],\mathbf{1}[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\!=\!0]\}$. Then the rescaled difference $\Delta(\mathrm{m})/4\tanh^n(\beta)$ can be computed as follows
\begin{eqnarray}
&&\frac{\Delta(\mathrm{m})}{4\tanh^n(\beta)}\label{eq:deltaM1}\\
&&=\frac{1}{4}\left\langle\sgn\left[\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\right] -\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\right\rangle_{\sigma\vert \mathrm{m}}\nonumber\\
&&=\frac{1}{4}\Bigg\langle \mathbf{1}_+\left[\sigma\right]-\mathbf{1}_-\left[\sigma\right]\nonumber\\
&&-\left(\mathbf{1}_+\left[\sigma\right]+\mathbf{1}_-\left[\sigma\right]+\mathbf{1}_0\left[\sigma\right]\right)\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\Bigg\rangle_{\sigma\vert \mathrm{m}}\nonumber\\
&&=\frac{1}{2}\Bigg\langle
\mathbf{1}_+\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!-\!1\right]\nonumber\\
&&-\mathbf{1}_-\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!+\!1\right]\nonumber\\
&&-\frac{1}{2}\mathbf{1}_0\left[S\right]\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\Bigg\rangle_{\sigma\vert \mathrm{m}}.\nonumber
\end{eqnarray}

In the above we can use the identity
\begin{eqnarray}
&&\prod_{j=1}^k \frac{1}{2^n}\left[1+\mathrm{m}\prod_{i=1}^n\sigma^i_j\right]=\\
&&\left[\frac{1\!+\!\mathrm{m}}{2^n}\right]^{\frac{k+\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j}{2}}\left[\frac{1\!-\!\mathrm{m}}{2^n}\right]^{\frac{k-\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j}{2}}\nonumber
\end{eqnarray}
to obtain
\begin{eqnarray}
&&\frac{\Delta(\mathrm{m})}{4\tanh^n(\beta)}\label{eq:deltaM2}\\
&&=\frac{1}{2}\left(\left[\frac{1\!+\!\mathrm{m}}{2^n}\right]\left[\frac{1\!-\!\mathrm{m}}{2^n}\right]\right)^{\frac{k}{2}}\nonumber\\
&&\times\Bigg\{\sum_{\{\sigma_j^i\}}\left[\frac{1\!+\!\mathrm{m}}{1\!-\!\mathrm{m}}\right]^{\frac{\vert\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\vert}{2}}\nonumber\\
&&\times\mathbf{1}_+\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!-\!1\right]\nonumber\\
&&-\sum_{\{\sigma_j^i\}}\left[\frac{1\!-\!\mathrm{m}}{1\!+\!\mathrm{m}}\right]^{\frac{\vert\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\vert}{2}}\nonumber\\
&&\times\mathbf{1}_-\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!+\!1\right]\nonumber\\
&&-\frac{1}{2}\sum_{\{\sigma_j^i\}}\mathbf{1}_0\left[\sigma\right]  \prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\Bigg\}.\nonumber
\end{eqnarray}
Now because $\alpha$ is a balanced gate we have the following identity
\begin{eqnarray}
&&\sum_{\{\sigma_j^i\}} \prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\label{eq:zero}\\
&&=\sum_{\{\sigma_j^i\}}\Bigg(\mathbf{1}_+\left[\sigma\right]+\mathbf{1}_-\left[\sigma\right]+\mathbf{1}_0\left[\sigma\right]\Bigg)\nonumber\\
&&~~~~\times\Bigg(\mathbf{1}\left[ \prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!+\!1\right] \nonumber\\
&&~~~~-\mathbf{1}\left[ \prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!-\!1\right]\Bigg)\nonumber\\
&&=\sum_{\{\sigma_j^i\}}\Bigg(\frac{1}{2}\mathbf{1}_0\left[\sigma\right]\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\nonumber\\
&&~~~~+\mathbf{1}_-\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!+\!1\right]\nonumber\\
&&~~~~-\mathbf{1}_+\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!-\!1\right]\Bigg)\nonumber\\
&&=0.\nonumber
\end{eqnarray}

Adding the above representation of zero to the terms inside the curly brackets in equation (\ref{eq:deltaM2}) leads  to the final result
\begin{eqnarray}
&&\Delta(\mathrm{m})\\
&&=2\tanh^n(\beta)\left(\left[\frac{1\!+\!\mathrm{m}}{2^n}\right]\left[\frac{1\!-\!\mathrm{m}}{2^n}\right]\right)^{\frac{k}{2}}\nonumber\\
&&\times\Bigg\{\sum_{\{\sigma_j^i\}}\left(\left[\frac{1\!+\!\mathrm{m}}{1\!-\!\mathrm{m}}\right]^{\frac{\vert\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\vert}{2}}-1\right)\nonumber\\
&&~~~~\times\mathbf{1}_+\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!-\!1\right]\nonumber\\
&&+\sum_{\{\sigma_j^i\}}\left(1-\left[\frac{1\!-\!\mathrm{m}}{1\!+\!\mathrm{m}}\right]^{\frac{\vert\sum_{j=1}^k\prod_{i=1}^n\sigma^i_j\vert}{2}}\right)\nonumber\\
&&~~~~\times\mathbf{1}_-\left[\sigma\right]\mathbf{1}\left[\prod_{i=1}^n\alpha(\sigma_1^{i},\ldots,\sigma_k^{i})\!=\!+\!1\right]\Bigg\}.\nonumber
\end{eqnarray}
From the above it is clear that $\Delta(\mathrm{m})\geq0$ for $\mathrm{m}\!\in\![0,1)$ and $\Delta(\mathrm{m})\leq0$ for $\mathrm{m}\!\in\!(-1,0]$.

\section*{Acknowledgment}

This work is supported by the EU FET project STAMINA (FP7-265496)  and the Leverhulme trust grant F/00 250/H.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

redpajama_set_name: RedPajamaArXiv


The Kirk-Habicht Company has been in business since 1907. The company was incorporated by Charles E. Kirk, and William C. Habicht. In the early years, the company produced brass parts - ship's telegraphs, speaking tubes, rails, etc. - and other hardware for ships docked in the Baltimore harbor. In 1947, George and Jennie Habicht bought the remaining stock from Charles Kirk.
In 1957, Dave Habicht joined his parents full time. At that time, The Company Produced 50% brass hardware and 50% springs. With the increasing mechanization of the industrial operations, springs became more important and a greater percent of their product line. The Kirk-Habicht Company was the only spring company in Baltimore and served the needs of all the local industries.
Dave took over the management of the company in 1967 at the time of his father, George's death. In 1969, the company acquired their first automatic spring coiler. In 1970, the City of Baltimore relocated the company to make room for the Inner Harbor revitalization.
In 1982 Mark Habicht began working full time at the company. He was the fourth generation of Habichts to work with the company. Very rarely does a company remain family owned and managed for four generations. In 1999, Mark became President and supervises all the day to day operations. The company moved to its present location in 1996.
The Kirk-Habicht Company continually strives to produce the finest quality springs at the most competitive prices. Our first priority is to provide our customers with the best service and the best product that is available. We look forward to another generation of continued growth.
To provide to our customers a quality product through fair, honest, and ethical business practices.
To insure customer satisfaction through personal communications and rapid response to resolve our customers questions, problems and product requirements.
To be responsive to customer needs especially during emergencies.
To continually train ourselves and our employees to produce quality and competitively priced products and to make our employees more flexible and efficient.


text:
You are here: Home / Articles / Comparison between curved-front fitting and straight-front fitting / Figural art that supports the look of straight front seam tailoring
Figural art that supports the look of straight front seam tailoring
Boucher, Francois. 20,000 Years of Fashion: The History of Costume and Personal Adornment. New York: Harry N. Abrams
p. 203: Boccaccio of the Duc de Berry, c. 1410, Paris, ms. fr. 598 fol. 49v; woman wearing fancy over-gown with pendant flaps lined in fur.
p.204: Italian breviary, c. 1380, ms. lat. 577, fol. 380; multiple women
p. 205: "Virgin and Child", aka "Agnes Sorel" by Jean Fouquet c. 1460 (or 1480); famous 'baseball boobs' portrait; though the picture clearly shows a straight front seam closure, it also shows a six-part construction, which provided additional tailoring opportunities.
p. 206: Queen Jeanne, panel from Narbonne, c. 1373–1378; Queen Jeanne wears a sideless surcotte over her gown, but the height of her bosom could indicate a straight-front-seam construction method.
Camille, Michael. The Medieval Art of Love. London: Harry N. Abrams, 1998.
p. 76 Lovers go shopping from Le Chevalier Errant, Paris, c. 1410; woman in blue gown on the right; side-view.
Pognan, Edmond, ed. Boccaccio's Decameron: 15th Century Manuscript. Fribourg–Geneva: Productions Liber S.A., 1978.
Throughout this text there are multiple instances of bust shaping that could be achieved with the straight-front-seam method. Note: Although the editor dates this manuscript to 1330-1340, I have a hard time believing that it was done much past 1410, as the styles appear dead-on for that earlier time period.
The Belles Heures of Jean, Duke of Berry. New York: Metropolitan Museum of Art, 1958.
book-plate 30: A woman praying in fancy overgown — though painted by the Limbourg brothers who also painted April, May, and June of the Tres Riches Heures (cited as examples of silhouettes achievable with the curved-front-seam method), the bust shaping in this one picture strongly resembles the push-up-and-shelf effect of the straight-front-seam method.
Spencer, Judith, trans. The Four Seasons of the House of Cerruti. New York: Facts on File Publications, 1984.
Notable for its many large-busted female figures, this manuscript is peppered with many fine examples of gowns that can be recreated using the curved-front seam method. To be fair, it also portrays gown styles that more closely resemble the look of the straight-front-seam method.
Sronkova, Olga. Gothic Fashions in Women's Dress. Prague: Artia, 1954.
p. 167: Bible of King Wenceslas IV, fol. 69, c. end of the 14th century; Woman on the left with 'frounced' headdress wears a particularly good example of a gown that achieves the high lift and shelf-like cleavage of the straight-front-seam method.
Thomas, Marcel. The Golden Age: Manuscript Painting at the Time of Jean, Duke of Berry. New York: George Braziller, 1979
p. 42: The Tacuinum Sanitatis fol. 25 "Orgeat (Aqua Ordei)"; a lady in close-fitting beige gown (side-laced) with dagged 'angel' sleeves in the Italian style of the very late 14th century and turn of the 15th century.

redpajama_set_name: RedPajamaCommonCrawl


text:
Archive for the 'Supreme Court' Category
Devious Dickie Blumenthal At It Again!
As if it were not enough prevarication already with his lies about Vietnam service, newly elected Connecticut junior senator Richard Blumenthal is now embellishing his contribution to Roe v Wade while clerking for Justice Blackmun. According to Juff Dunetz at Yid With Lid:
"…Senator Richard Blumenthal who, during his campaign was exposed for lying about serving in Vietnam, got caught in a brand new lie. This time he mislead people about his tenure as clerk for Supreme Court Justice Harry Blackmun. During a press conference about Abortion rights held by Senate Democrats, Blumenthal claimed he was right in the middle of the Roe vs Wade decision which was written by Blackmun in 1973.
I'm new to the Senate but I'm not new to this battle. Since the days of Roe v. Wade, when I clerked for Justice Blackmun, as a state legislator, as attorney general, I have fought this battle.
The problem is, Blumenthal clerked for Supreme Court Justice Harry Blackmun in 1974, the year after Blackmun wrote Roe v. Wade decision, in 1973 Blumenthal was busy not being in Vietnam." (emphasis added) Read more at Yid With Lid.
Tags:blumenthal lies, ct senator blumenthal, fabulist blumenthal, justice blackmun, roe v wade
Posted in Campaign '10, Connecticut News, Left Lunacy, National Politics, Richard Blumenthal, Supreme Court | Leave a Comment »
Obama Wants Sotomayor to Provide Some Justice
She's Brilliant, She's Qualified, I Want to See Her Confirmed, I Want Her Walking Up Those Marble Steps and Starting to Provide Some Justice!
Thus spake Barack Obama, about Sonia Sotomayor, to the Los Angeles elite where he also touted his accomplishments so far. (here's the video)
…and the justice to be provided by Ms. Sotomayor is Obama's justice. As Patrick J. Buchanan points out:
When you think about it, Sonia Sotomayor is the perfect pick for the Supreme Court — in Barack Obama's America.
Like Obama, himself a beneficiary of affirmative action, she thinks "Latina women," because of their life experience, make better judicial decisions than white men, that discrimination against white men to advance people of color is what America is all about, that appellate courts are "where policy is made" in the United States.
To those who believe the depiction of our first Hispanic justice as an anti-white liberal judicial activist, hearken to her own words.
Speaking at Berkeley in 2001, Sonia told her audience, "I would hope that a wise Latina woman with the richness of her experience would more often than not reach a better conclusion (as a judge) than a white male who hasn't lived that life."…. Speaking at Duke in 2005, Sotomayor declared: "(The) court of appeals is where policy is made. I know this is on tape, and I should never say that because we don't make law I know." She and the audience joined in the laughter.
Who were they laughing at? Americans who still believe the role of judges is to apply the Constitution as the Framers intended and to interpret the law as written by our elected legislators.
In Barack Obama's America, that is so yesterday.
Sadly this is no laughing matter. Barack Obama believes the U.S. Constitution is an impediment, an obstacle, to the attainment of his socialist agenda. So bye bye U.S. Constitution and hello another activist judge in our highest court. Sotomayor is a racist political hack destined for a lifetime sinecure, one of nine justices who are sworn to uphold the U.S. Constitution.
Day by Day Cartoon: Chris Muir
h/t Day by Day Cartoon
Also see Michelle Malkin
Tags:barack obama, constitutional crisis, obama socialism, obama socialist, Socialism, sotomayor.pat buchanan, Supreme Court
Posted in barack obama, Conservatism, Left Lunacy, National Politics, Oaths of Office, Socialism, Supreme Court, The Constitution, The Presidency | 4 Comments »
State Sovreignty: Sam. Adams on Despotism
I hope Congress, before they adjourn, will take into very serious Consideration the necessary Amendments of the Constitution. Those whom I call the best—the most judicious & disinterested Fœderalists, who wish for the perpetual Union, Liberty & Happiness of the States and their respective Citizens many of them, if not all are anxiously expecting them—They wish to see a Line drawn as clearly as may be, between the federal Powers vested in Congress and the distinct Sovereignty of the several States upon which the private and personal Rights of the Citizens depend. Without such Distinction there will be Danger of the Constitution issuing imperceptibly, and gradually into a Consolidated Government over all the States, which, altho it may be wished for by some, was reprobated in the Idea by the highest Advocates for the Constitution as it stood without amendmts. I am fully persuaded that the People of the United States being in different Climates—of different Education and Manners, and possest of different Habits & Feelings under one consolidated Governmt. can not long remain free, or indeed under any Kind of Governmt. but Despotism.
Samuel Adams in a letter to Elbridge Gerry 1789
Found at ConSource
Patriot Samuel Adams
Tags:central government, despotism, Socialism, State Sovreignty, Supreme Court, u s constitution
Posted in Conservatism, National Politics, State Sovreignty, Supreme Court, The Constitution, U.S. History | Leave a Comment »
Leo Donofrio's Biggest Gamble: The Lawyer as Artist….
….the Defense Attorney who is defending human beings against the assaults of Satan*
I say there's nothing wrong with tireless self promotion as long as there is some ingenuity and talent to back it up. Such is the case with Leo C. Donofrio, most recently in the blogger news (not much in the main stream news anyway) with his lawsuits before the SCOTUS regarding the eligibility of Barack H. Obama. I had heard of his skill at poker. He took top prizes at two "hold 'em" tournaments earlier this year.
Donofrio's biggest gamble however is his attempt to remove ineligible candidates (Obama, McCain, Calero) from the presidential ballot for failing to meet the "natural born" test. His first suit "Donofrio v. Wells" was conferenced by the Justices on Dec 5 but they declined to order oral arguments. The next test comes tomorrow (Dec 15) in the Wrotnoski v. Bysiewicz when the Court will announce if it will hear that case. Today, on his blog, Natural Born Citizen, we learn that Donofrio is also an artist ,"damn it to hell"!
From Olberman at MSNBC and right on down the line with very few exceptions, I've been labeled a tin foil hat sporting freak show loony toon… and for what? I questioned whether a man – who admits his birth status was governed by the laws of Great Britain – is a "natural born Citizen" under our Constitution.
Well, I don't really give a rats booty product what the media says. It's all lies and spin anyway. But what does bother me is that they've insulted my artistic abilities. I am an artist, damn it to hell!
He goes on to spin a conspiracy yarn he calls "Obama Has a Twin". He has also written a song with that title which he performs on his MySpace page where he is called The Paraclete. One reference to Paraclete on Wikipedia defines the term as "as the Defense Attorney who is defending human beings against the assaults of Satan (the Prosecuting Attorney, the Accuser, the fomenter of violence)."* He also plays in a band called Schizo Fun Addict.
Donofrio is an excellent writer and researcher and somewhat of a sleuth. His article titled The Mystery of Southern Pines Golf Club is an interesting read, even if you don't care about golf.
I think I will come down on the side of those, and there are many, who praise his legal talent and his gutsiness in taking on a potentially huge constitutional issue. One can only wish him well in that regard, self promoter or not.
Tags:barack obama, big gambles, constitution, leo donofrio, obam eligible, obama citizen, poker, poker champ, scotus, Supreme Court
Posted in barack obama, Campaign '08, CT Election Law, john mccain, National Politics, Supreme Court, The Constitution, The Presidency | 13 Comments »
Oh No! Glass Ceiling Repaired: Hillary Takes Pay Cut
We reported earlier (12/04/08) that Hillary Clinton's nomination as Secretary of State was in jeopardy for constitutional reasons:
There is a long history of controversy and litigation regarding the provision in the U.S. Constitution prohibiting members of Congress from subsequently attaining to federal offices offering increased monetary compensation due to the largess of the Congress in which they sat. The most recent example is that of Hillary Clinton who has been nominated to the office of Secretary of State. In Clinton's case the salary for the position was increased, not by Congress, but via an executive order by President Bush as a COL increase.
In any event the Constitutional provision comes in to play: (CNS reports)…the Senate's senior member and staunchest constitutional advocate on the Democratic side of the aisle, Robert Byrd of West Virginia, is exploring whether Sen. Hillary Clinton (D-N.Y.) is eligible to become secretary of state in the Obama administration. (CNS reporting again)
Since the bill passed both houses of Congress unanimously, we are guessing that Senator Byrd chose to approve this end run around the constitution.Here is James Taranto at WSJ online writing today:
As if she hasn't already suffered enough indignities at the hands of the unfairer sex, Hillary Clinton will make "about $4,700 less as secretary of state than her predecessor, Condoleezza Rice," the Associated Press reports:
Congress late Wednesday lowered the salary for the nation's top diplomat to keep Mrs. Clinton's nomination from running afoul of the Constitution.An obscure section on compensation for public officials, the Emoluments Clause, says that no member of Congress can be appointed to a government post if that job's pay was increased during the lawmaker's current term.In other words, Clinton, D-N.Y., might have been ineligible to serve in the post because she was serving in Congress when Rice's salary was raised to its current level of $191,300. So late Wednesday, the House and Senate quietly rolled the secretary of state's salary back to $186,600, its level in January 2007 when Clinton began her second Senate term.So as a result of the emoluments clause–written by dead white males–Mrs. Clinton will be earning less than a man would for the same job. She'll even be earning less than another woman now makes! (emphasis added)
Tags:Add new tag, glass ceiling, hillary clinton eligibility, senator robt byrd, u s constitution
Posted in Campaign '08, hillary clinton, National Politics, Supreme Court, The Constitution, Uncategorized | Leave a Comment »
Leo Donofrio: "The Propaganda Ponies of the Fourth Estate"
The Connecticut version of the Obama "natural born citizen" cases is before the Supreme Court for conferencing today. The title is, Wrotnowski v. Bysiewicz, docketed here. Cort Wrotnowski is a Greenwich CT attorney store owner (see comment below) and Susan Bysiewicz is Connecticut's Secretary of the State. Leo Donofrio is the New Jersey attorney whose own case Donofrio v. Wells was declined a hearing before SCOTUS last week. Donofrio has assisted Wrotnowski in strengthening this case.
In his blog, Natural Born Citizen, Donofrio takes the press to task for their failure to recognize what this suit is about: namely, the removal of the ineligible candidates (Obama, McCain, Calero) from the ballot because they fail the "natural born citizen" test and not because of their various places of birth. Donofrio fires off this salvo:
This week has been quite enlightening as to the blatantly obvious fact that our "Fourth Estate" press corps have been transmogrified into propaganda ponies polly wanna crackering whatever may be handed down to them from "The One Corporation – your source for everything…" (cue eery theme tune). They don't report the news anymore. No. Now they tell you what they want the news to be. There's a huge difference.
For the record, my law suit was brought to remove three candidates from the ballots – three candidates who have big Constitutional issues as to their eligibility.
At the time of his birth, Obama was a British/Kenyan citizen by descent of his father. Because I pointed out pesky international laws which governed his citizenship due to the fact that a father has every legal right in the world to have the laws of his nation apply to his son, I have been labeled a conspiracy freakoid of nature.
Never mind that I included demands for Panama John McCain and the Nicaraguan born Roger Calero to also be removed from our ballots. No, they don't want to talk about that do they – because it would blow the "he's just another Obama hater" mantra clear out of play.
A citizen (me) raised the Constitutional issue of first impression as to the meaning of "natural born Citizen" in Article 2, Section 1, of the United States Constitution – that ultimate pesky legal document for those who would rather "be" the law instead of following it.
What are the Fourth Estate propagandists worried about? Thou doth protest too much. Me thinks so. Why? Because the law is against their man – it indicates Barack Obama is not a natural born Citizen of the United States. And most of the media pundits have basically agreed by default. I say this because when yelling and mocking the issue, their main argument is not that the law is on their side (they know it isn't), but rather that the law shouldn't be discussed at all.
The gist of Donofrio's argument regarding Barack Obama:
Like it or not, rich or poor, great or strong, Democrat or Republican, Obama was born under the jurisdiction of Great Britain via Kenya. There is nothing conspiratorial about saying that. Obama has it posted on his own web site. It's this very definition which I included in Cort's Wrotnowski's brief. Here's what it says at Obama's web portal, Fight The Smears:
When Barack Obama Jr. was born on Aug. 4,1961, in Honolulu, Kenya was a British colony, still part of the United Kingdom's dwindling empire. As a Kenyan native, Barack Obama Sr. was a British subject whose citizenship status was governed by The British Nationality Act of 1948. That same act governed the status of Obama Sr.'s children. (Emphasis added.)
There it is. Obama is telling you his status was "governed" by a foreign jurisdiction. This is no theory. This is a fact.
Tags:connecticut law, cort wrotnowski, leo donofrio, obama citizenship, obama natural born, scotus
Posted in barack obama, Campaign '08, CT Election Law, john mccain, Main Stream Media, National Politics, Supreme Court, The Constitution, The Presidency, Uncategorized | 16 Comments »
WROTNOWSKI v. BYSIEWICZ Moves Forward, Supplemental Brief Tomorrow
Attorney Leo C. Donofrio reports in an update this evening that the SCOTUS docket item WROTNOWSKI v. BYSIEWICZ is scheduled for conferencing among the justices on Friday Dec 12. This was submitted by Justice Scalia. Recent discoveries about President Chester A. Arthur and one of his Supreme Court appointees may have a bearing on the present case according to Donofrio. Therefore:
Tomorrow, Dec. 9 – Cort Wrotnowski will submit a supplemental brief concerning the newly discovered ineligibility of twenty-first President Chester Arthur due to his having been born as a British subject. This is relevant to the case at hand in that Justice Gray – who wrote the seminal opinion in United States v. Wong Kim Arc – was appointed by Chester Arthur.
We reported on the Arthur business yesterday (The Liar Who Became President). Donofrio's complete update is here. He concludes his report:
Leo Donofrio will accompany Cort Wrotnowski to Washington D.C. tomorrow and both will be available for comment at 11:00 AM on the steps of the Supreme Court. This is not a rally, protest or vigil. If the media would like to discuss this historical brief and the issues discussed above, Donofrio and Wrotnowski will be available to answer any questions thereto.
Tags:constitution, cort wrotnowski, natural born citizen, obama eligibility, obama fraud, Supreme Court, susan bysiewicz
Posted in barack obama, Campaign '08, National Politics, Supreme Court, The Constitution, The Presidency | 2 Comments »
Donofrio v. Wells Not Heard, Wrotnowski v. Bysiewicz Still Docketed
The Barack Obama citizenship case emanating from courts here in Connecticut is still docketed at SCOTUS. Cort Wrotnowswki v Susan Bysiewicz, CT Secretary of State, raised much the same issues as his own case (Donofrio v. Wells) reports Leo C. Donofrio this morning.The U.S. Supreme Court declined to accept the Donofrio suit for oral arguments. Regarding the Wrotnowski case Donofrio states:
I don't know if it's significant that Cort's case was not denied at the same time as mine. His case argues the same exact theory – that Obama is not a natural born citizen because he was a British citizen at birth…..
If Cort's application is also denied then the fat lady can sing. Until then, the same exact issue is before SCOTUS as was in my case. Cort's application before SCOTUS incorporates all of the arguments and law in mine, but we improved on the arguments in Cort's quite a bit as we had more time to prepare it.
I was in a rush to get mine to SCOTUS before election day, which I did do on Nov. 3.
Cort's case has a much cleaner lower court procedural history….
Regarding his own case Donofrio says:
The main stream media should stop saying SCOTUS refused to hear the case. It was distributed for conference on Nov. 19. They had the issue before them for for sixteen days. Yes, they didn't take it to the next level of full briefs and oral argument. But they certainly heard the case and read the issues. The media is failing to acknowledge that. The case and issues were considered. Getting the case to the full Court for such consideration was my goal. I trust the Supreme Court had good reason to deny the application. Despite many attempts to stop their full review, my case was placed on their desks and into their minds. Please remember that. It's important for history to record that.
Tags:cort wrotnowski, ct secretary of state, leo donofrio, obama citizenship, obama eligibility, susan bysiewicz
Court Won't Hear Donofrio v Wells….
This just in from Yahoo News via Drudge…
WASHINGTON – The Supreme Court has turned down an emergency appeal from a New Jersey man who says President-elect Barack Obama is ineligible to be president because he was a British subject at birth.
The court did not comment on its order Monday rejecting the call by Leo Donofrio of East Brunswick, N.J., to intervene in the presidential election. Donofrio says that since Obama had dual nationality at birth — his mother was American and his Kenyan father at the time was a British subject — he cannot possibly be a "natural born citizen," one of the requirements the Constitution lists for eligibility to be president.
Donofrio also contends that two other candidates, Republican John McCain and Socialist Workers candidate Roger Calero, also are not natural-born citizens and thus ineligible to be president.
At least one other appeal over Obama's citizenship remains at the court. Philip J. Berg of Lafayette Hill, Pa., argues that Obama was born in Kenya, not Hawaii as Obama says and the Hawaii secretary of state has confirmed. Berg says Obama also may be a citizen of Indonesia, where he lived as a boy. Federal courts in Pennsylvania have dismissed Berg's lawsuit.
Tags:donofrio v wells, obama case, obama citizenship, scotus, Supreme Court
Posted in Campaign '08, National Politics, Race Relations, Supreme Court, The Constitution, The Presidency, Uncategorized | 4 Comments »
The Liar Who Became President ….
Once upon a time there was a politically ambitious man who told many lies about himself and his family. We shall call him Mr. X. During one of his early political campaigns a political enemy circulated rumors that Mr. X was born in a foreign country and therefore ineligible for high elective office. Mr. X denied the charge saying his father was a British subject but that he (X) was born here.
Mr. X claimed that his mother had always lived in the U.S. which was untrue as she had lived in another country with her husband and had (a child) there. Mr. X realized early on that he was not a "natural born citizen" because he knew that he was born before his father was naturalized. X told lie upon lie about his father's age, the year the father came to America, even his own age; and he destroyed most of his papers all in the effort to conceal the fact that he was ineligible for high office. Who is Mr. X?
As strange as this may sound, we are not discussing Barack Obama here. Mr. X is Chester A. Arthur, the 21st President of the United States who succeeded to the Presidency upon the assassination of President James Garfield. Due to the diligent research of Leo C. Donofrio with the assistance of Arthur biographer Greg Dehler we learn that Chester A. Arthur was a usurper, never eligible for POTUS. Talk about an amazing confluence of events: just as we are ready to install the ineligible Barack Obama as POTUS we learn that this has happened before. As Donofrio says in his report on the matter; "… it's no precedent to follow."
The political enemy mentioned in the lead paragraph was, in reality, one Arthur P. Hinman (see footnote below)*, who may have been hired by the Democrats to smear Arthur. Interestingly, the charge of ineligibilty for POTUS brought by Hinman against Arthur was true, but for the wrong reason: Chester A. Arthur was a British subject at birth (just as Obama was) but by virtue of his birth prior to his father's naturalization rather than birth on foreign soil. As in Obama's case, who acknowledges on his website that he had dual citizenship at birth, we find the evidence in plain sight but too late discern its meaning.
President Chester A. Arthur
*More interesting but entirely without foundation was the Hinman myth circulated in 1880 and 1881. This story asserted that Elder Arthur had three sons: William Chester Alan Arthur, born at the home of his mother's parents in Dunham, Province of Quebec; Chester Abell Arthur, born at Fairfield; and William Arthur, Jr., born at Hinesburgh, Vermont. When William Arthur, Jr., was born, the oldest son dropped the William and retained the names Chester Alan, as he could do because of the death in infancy of his brother, Chester Abell. He later, according to the Hinman story, appropriated the birth record of the second son in order to sustain his American citizenship. No death record existed to prove this substitution because the father had sold the infant's body to a medical school! On the basis of these allegations, the American public were assured that Arthur was a British subject and in consequence disqualified for the Vice Presidency or Presidency. It was a political maneuver, and, as such, ineffective. (from "Chester A. Arthur-A Quarter-Century of Machine Politics" by George F. Howe)
Tags:chester arthur, constitutional crisis, greg dehler, leo donofrio, obama fraud, obama ineligible, obama usurper
Posted in barack obama, Campaign '08, National Politics, Supreme Court, The Constitution | 26 Comments »


How Does The Facebook Newsfeed Really Work?
Facebook Pages today are faced with a much tougher challenge of keeping their boat afloat owing to the evolution of the Facebook newsfeed algorithm. Facebook weighs in a lot of quality factors, which collectively decide how many people can a post reach. This, coupled with the ever-intensifying competition for feed space means that Pages have to fight for every inch, like literally! Following up on our recent post about Negative Feedback and its impact on Reach, we'll discuss the whole newsfeed ranking process in a bit more detail.
Not reaching most of its fans is any page's biggest concern. Facebook wants brands to pay for organic reach. While some pages who can afford to, take the bait, others like Eat24 aren't as compromising, and end up ditching their entire Facebook campaign. One big reason as to why this is happening is, Facebook is trying to filter out the 'bad guys' since there's surplus content but a lack of space.
The number of pages liked by people has increased, yet the newsfeed stays the same. This means more competition. According to Facebook, an average user might have 1500 posts eligible to appear in their feed each day. As people continue to like more pages, organic reach for each of these pages decreases. This is confirmed by the fact that even though there was a 50% increase in the amount of pages liked by people in the last year, there's also a roughly 50% decrease in the organic reach for these pages.
All Pages and people are not treated equally because Facebook's goal is to show people the most engaging posts out of all the ones they could see each day. Facebook's ability to earn money showing ads and pursue its mission to connect the world hinges on people coming back because they see interesting content there and don't get bored. This puts Facebook in the very tough position of choosing what content gets shown and what doesn't. Of course, Facebook cannot show all the posts in a reverse-chronological manner due to the sheer spam and noise that could drown out important messages.
This means the more successful a post is, and the more popular its creator is amongst everyone and the potential viewer, the more likely that viewer is to see the post. The fact that someone Liked a Page or added a friend at some point over the years doesn't matter nearly as much. It's whether the Page continues to be interesting to everyone and to any specific potential viewer of their posts.
Basically, Facebook wants you to spend money on it - as much money as you can. Ads are the only way to beat the system. Facebook used used to tell companies to buy Likes as a long-term investment. But now, that investment has gone for most companies. For them to regain that chunk of money, or at leat a part of it, companies now have to spend even more money. Luring them into a trap - sound business strategy for Facebook, not so much for the companies out there.
What do you think about Facebook? Should one continue spending time and money on Facebook? Is it worth the effort, or will it be worth the effort a few years from now, given what Facebook has done with companies? Let's hear what you think!


text:
MENU[et_social_follow icon_style="slide" icon_shape="rounded" icons_location="left" col_number="auto" outer_color="dark"]
The Observatory
About the Observatory
Legal Responsibility & Privacy Policy
EJO Network
AJO Arabic
AJO French
Media Economics
Newsroom Management
Media & Politics
Specialist Journalism
The New Barons. Power, Politics, and Media
August 13, 2013 • Media and Politics, Specialist Journalism • by Michał Kuś, Stephan Russ-Mohl, Adam Szynol
Media barons – media owners who use their newsrooms to exercise political pressure and power – have become more influential in many Central and Eastern European countries. The most recent mega deals in the U.S. and in Germany indicate that the barons are returning to the West too.
Current media markets are turbulent. Journalism is expensive and it is becoming harder to turn a profit. Successful business models are yet to be built. If media companies no longer generate profits, anyone who owns one must have a different motivation for their investment. In such instable circumstances, little wonder that media barons are reappearing.
What is a media baron? Canadian media researcher David Taras has defined him as a "powerful media owner who sees himself or herself as […] part business person, part journalist, and part politician. The influence of media barons comes not from their imposing commercial pressure on news reporting but from the extent to which they enter […] the fray directly as journalistic impresarios and political partisans. They have bought newspapers and own TV stations precisely because they are ardent political combatants."
The phenomenon of media barons has to be looked at alongside the issue of media ownership concentration. When profits at companies controlled by families or individual owners begin to fall, owners may well be tempted to sell. More powerful or wealthier owners of rival media groups can turn the situation to their advantage, both politically and economically.
The reappearance of media barons has also been observed recently in the United States. Long before Jeff Bezos bought the Washington Post and the billionaire John Henry the Boston Globe, the New York Times' media expert David Carr cited several examples of media barons. He argues that the phenomenon is closely related to the crisis in the U.S. newspaper industry, which has halved in size in the last seven years.
"If most newspapers are an uneconomical proposition incapable of sustaining profits,…who is left to own them?" he asks. "Rich guys," is his answer, and he adds more precisely, "Not merely the well off, but the kind of men who long ago separated themselves from humdrum economic realities of life. Sure, there are other expensive hobbies, but how many antique cars or 19th century landscapes can you own?" Carr foresees that some of these new owners will potentially use their "collectibles" for political goals or purposes serving their other business interests.
The situation in Europe is much more complicated, because media barons' origins and appearances are more heterogeneous than in the U.S. In some countries like Italy, France, or the U.K., "old" media barons have been part of the media landscape for decades. Tycoons like Rupert Murdoch have used their media outlets to back one political party or another, others like Silvio Berlusconi have abused their media to install themselves in powerful political positions. Before they died, Jesús de Polanco in Spain, Hans Dichand in Austria and decades earlier Axel Springer and Rudolf Augstein in Germany or Alfred Harmsworth (Lord Northcliffe), and Robert Maxwell in the U.K. were archetypal media barons – finding ways to exert power discreetly, and occasionally flexing their muscles openly.
In comparison, the media barons in Eastern Europe are newcomers. But they have become so numerous and powerful there that the phenomenon is now associated with the post-communist countries. Media barons became influential political actors in the early 1990s, some of them immediately after the decay of the Soviet Empire.
Post-Communist Russia birthed several media barons, many of whom later lost their power because they clashed with Vladimir Putin's regime. Media barons have also become powerful in Balkan countries like Serbia and Albania, or in Romania and Bulgaria where they could establish themselves in media markets at the beginning of the countries' democratic transformation. Other local tycoons entered the scene after Western investors like Mecom in Poland, the WAZ Group in Serbia, moved out.
In Europe, media barons can be clustered in three groups: "hidden", "high profile," and "potential" media barons. The major examples of "hidden media barons" come from the Western Balkans. In many cases the real owners of media outlets cannot be identified as they are hiding themselves behind companies founded outside the country. A recent report by the Serbian Anti-Corruption Council points out that out of 30 of the country's most significant media outlets there were 18 outlets whose actual owners were unknown. To list one example: Zeljko Mitrovic owns TV Pink and a 4.95% share of TV Avala. "Additionally," explains Serbian media researcher Veselin Kljajic (University of Belgrade), "the Austrian company Greenberg Invest is perceived to be closely related to Mitrovic, with its ownership of 48 % of TV Avala. Shares. This fuels public speculation that the station is actually in majority ownership of Mitrovic."
In Albania, media ownership is slightly more transparent, but there is still a public belief that "legal owners are not the real owners", says Artan Fuga, professor of communications at the University of Tirana. "The overall number of newspapers and television channels is very large in proportion to the population, and the advertising market is still rather small. This makes it hard to run a viable media business." Owners use the media and its power mainly "to exert pressure and to gain favors," says Remzi Lani, director of the Albanian Media Institute.
Compared to the Western Balkans, countries like Romania – as well as the Czech Republic and Ukraine – have "high profile media barons" who are powerful and visible. Media ownership is highly concentrated, but transparent.
Romania might be called the "Italy of the East". A significant number of persons and families listed among the richest Romanians own important media outlets, even if the media business is not the core of their business activities. Among its media barons Dinu Patriciu and Dan Voiculescu need to be mentioned. Patriciu is the only Romanian billionaire to appear on the Forbes rich list, a former member of parliament for the Liberal Democratic Party, and owns Adevarul-Holding, a conglomerate which owns print media as well as Adevarul TV.
Voiculescu, ranked second in a list of the most influential media people in Romania and a founding member of the Conservative Party of Romania. Though his daughters Corina and Camelia now control the Intact Media Group and its five TV channels, four newspapers, as well as magazines and other media assets, "its newsrooms remain closely connected to Voiculescu's political ambitions", states Alina Vasiliu, professor of journalism at Andrei Saguna University of Constanta.
In the Czech Republic, Jaromir Soukup, owner of Empresa Media and the media agency Médea, refers to himself as a Czech "Berlusconi." He supported the Green Party, the Social Democrats, and most recently the former Prime Minister Jan Fischer for his presidential run in 2013. Another businessman who mixes politics, business, and media is 58-year-old Andrej Babiš, who earned millions through his agrochemical company Agrofert. His current expansion into the media business involves the 2012 launch of a wide regional net of free weekly newspapers called 5+2.
In Ukraine the process of economic liberalization went hand in hand with growing corruption and the emergence of financial-industrial groups called "clans" closely connected with the political establishment. Media outlets became attractive assets for these groups. In effect, the richest Ukrainian oligarchs, like Rinat Akhmetov and Victor Pinchuk, are also the biggest media owners.
Despite popular belief, not all media barons are Eastern oligarchs: several "high profile" media barons have always been operating successfully in Western Europe. In Italy Silvio Berlusconi has for decades been the most powerful of all of them. However, his main competitor Carlo De Benedetti and several other Italian family-owned business empires have always held newspapers – not for the profit's sake, but for their returns in political influence. In Italy this phenomenon is called the "editoria impura" (the impure publishing business).
In France, Martin Bouygues, owner of the fifth biggest construction company in Europe, holds a 44 percent share in TF1, the biggest French TV station. Bouygues was the best man of former French president Nicolas Sarkozy and his first wife Cécilia, as well as the godfather of their son Louis. Serge Dassault, supplier of fighter jets for the French army and a senator for the right wing party UMP (Union pour un mouvement populaire), is considered to be the new tycoon dominating the French media market. Dassault owns the Figaro Group which publishes the biggest national daily Le Figaro (300,000 copies circ.) and the leading French television weekly TV Magazine (6 million copies circ.)
In the U.K. the most visible mogul is Rupert Murdoch, owning The Times (daily circulation 400,000) and The Sun (2.6 million copies daily), and a 40 percent share of BskyB, owner of Sky News, through News UK (formerly News International). His use of the media to influence politics and political figures has been asserted – but has been difficult in the past to prove. Recently the Leveson Inquiry, investigating the phone hacking scandal of the News of the World, provided more evidence of his links to the government.
In Western Europe and in Poland there are many "potential" media barons: rich media owners who certainly might become media barons, but so far they have not exhibited such political ambitions. For example, Germany and Switzerland are countries with few active media barons. However, media ownership is becoming more concentrated and the mainstream media is less profitable than it was: changing conditions that create a breeding grounds for media barons and a potential threat to democracy.
The two largest publishing companies in Germany, Bertelsmann and Springer – with total revenues in 2012 reaching €16.6 billion (Bertelsmann AG) and €3.3 billion (Springer AG) – are now both controlled by heiresses: Liz Mohn and Friede Springer are two of the world's most influential female media entrepreneurs: Both women are difficult to imagine as "media baronesses," they are private people who do not seek out the limelight.
Nevertheless, both media companies are important participants in the political power play. In the case of Bertelsmann, the foundation which is closely intertwined with the company is not only a think tank, but has become a political powerhouse of its own. The most powerful political figure within the Springer Group are possibly not the owners or the CEO, but the editor of Bild-Zeitung, Kai Diekmann; a man who can change the fortunes of a politicians' career in Germany.
In Switzerland, populist politician Christoph Blocher has recently invested in the Basler Zeitung, turning him into a media baron in a move that has been greeted with some consternation. However, the power balance in the country would only be seriously affected if one of the family members of the two largest private media companies, Ringier and Tamedia, ever showed signs of having political ambitions.
In Poland we are dealing as well with "potential" media barons after the Western investors Orkla Media and Mecom withdrew from the market. The new regional media owners are currently wary of being too politically involved, and tend to balance out their support among different political groups. The bigger media owners Zygmunt Solorz-Żak and the Walter and Wejchert families, who own two private nationwide TV stations, Polsat and TVN, seek political balance and the creation of a good environment for their business, rather than visibility and influence and the political process.
While Western European investors have been moving out of Eastern Europe, at least one Russian oligarch "invaded" successfully the West: Alexander Lebedev became a significant player in the U.K. newspaper market and another potential media baron after purchasing the London Evening Standard (an evening paper that once had a cover price but is now a free sheet with more than 0.5 million circulation) and The Independent (circ. about 70,000 daily) and Sunday Independent (circ. about 100,000). He and his son Evgeny Lebedevs have since launched a new newspaper "i" aimed at the middle market (circ. about 300,000).
Media barons come in Europe in many shapes and sizes. The way they operate, the objectives they hope to achieve, vary from country to country. Political culture, market size, wealth, all play a part in the way they operate, as does the flow of foreign capital, particularly in eastern European countries.
It is not easy to identify clear patterns in their behavior. But it is fair to conclude that media barons have gained power in a significant number of European countries. An increasing concentration of media ownership and decreasing profitability in the traditional media have all played a part in the emergence of the new wave of media investors with political interests.
It cannot be assessed accurately yet whether this applies also to Jeff Bezos and Warren Buffett, who has recently integrated several newspapers in the U.S. into his empire.
The Swiss and the German examples and the American developments provide a scenario of what might happen next: If the profitability of newspapers continues to decrease as rapidly as in the U.S., more of them will become insolvent like the Frankfurter Rundschau or be closed down like the Financial Times Deutschland, and other once prestigious "old" outlets like the Basler Zeitung may easily become prey to "new" populist barons like Blocher. These new tycoons may not exercise uncontrollable political influence like Silvio Berlusconi in Italy, but in a highly concentrated media market a dangerous liaison of populist media owners and populist politicians can endanger the checks and balances of democracy. We have seen this tasking place in Eastern Europe. It could happen in Western Europe as well.
This article was created with contributions from Tina Bettels, Hana Biriczova, Philip Di Salvo, Natascha Fioretti, Cristina Gelan, Jonila Godole, Milica Jevtić, Olena Kutovenko, Marko Nedeljkovic, Dariya Orlova, Meera Selva and Rrapo Zguri.
A shorter version of this article was published in the Neue Zürcher Zeitung.
Photo credit: Chris Devers / Flickr CC
Tags: Albanian media, Czech Media, Czech Republic, David Taras, European Media, French Media, German Media, Jeff Bezos, Journalism, Media barons, media markets, media owners, Media Ownership, media ownership concentration, Newspapers, Polish Media, Political influence, post-communist Eastern Europe, post-communist media, Post-Communist Transition, Romanian Media, Rupert Murdoch, Swiss Media, U.K. Media, U.S. Media, Ukrainian Media, Washington Post
Michał Kuś, Stephan Russ-Mohl, Adam Szynol
Michal Kuś is assistant profesor in the the Department of Communication and Journalism (Institute of Political Science, University of Wroclaw). His research focuses on media systems, with special attention to media control, ownership and deregulation, relations between media and politics and political communication.Stephan Russ-Mohl is the Director of the European Journalism Observarory since October of 2003; Professor of Journalism and Media Management at the Università della Svizzera italiana, Lugano since 2002. Frequent contributor to Neue Zürcher Zeitung (Zurich), columnist for Die Furche (Vienna), Schweizer Journalist (Zurich) and Werbewoche (Zurich).Adam Szynol is assistant profesor in the the Institute of Journalism and Social Communication (University of Wroclaw). He is also an editor and adviser of the Polish EJO website. His research interest focuses on Polish press market, with an emphasis on ownership and market concentration.
» Media and Politics, Specialist Journalism » The New Barons. Power, Politics, and...
Crowdfunding campaigns give boost to independent Czech media
NewsGuard's "news trust" ratings rolled out to Europe
Power and pin money: Time for a new...
Can Innovative Funding Models Help East European Media...
How do the European media cover migration?
The filter bubbles that divide the Hungarian media
Mobile journalism: a catalyst for digital transition
Bulgarian public service media under threat
Engaged journalism takes off in Romania
How Data Journalism Is Taught In Europe
13 Things Newspapers Can Learn From Buzzfeed
Why Journalism Needs Scientists (Now)
Digitalisation: Changing The Relationship Between Public Relations And Journalism
Can Robots Do Public Interest Journalism?
The Rise of the Drone-Journalists
Political Discourse
in the Albanian Media
Can Germany's New Law End 'Hate Speech' on Social Networks?
The Color of Terrorism
Facebook Launches Campaign To Block Hate Speech On Social Media
Archives Select Month January 2020 (4) December 2019 (5) November 2019 (2) October 2019 (2) September 2019 (6) August 2019 (4) July 2019 (4) June 2019 (7) May 2019 (3) April 2019 (4) March 2019 (4) February 2019 (3) January 2019 (6) December 2018 (6) November 2018 (7) October 2018 (11) September 2018 (5) August 2018 (6) July 2018 (7) June 2018 (8) May 2018 (7) April 2018 (8) March 2018 (12) February 2018 (8) January 2018 (8) December 2017 (10) November 2017 (6) October 2017 (8) September 2017 (6) August 2017 (4) July 2017 (4) June 2017 (9) May 2017 (7) April 2017 (4) March 2017 (7) February 2017 (8) January 2017 (7) December 2016 (6) November 2016 (6) October 2016 (9) September 2016 (7) August 2016 (6) July 2016 (4) June 2016 (11) May 2016 (9) April 2016 (10) March 2016 (9) February 2016 (10) January 2016 (7) December 2015 (6) November 2015 (7) October 2015 (10) September 2015 (9) August 2015 (7) July 2015 (9) June 2015 (13) May 2015 (12) April 2015 (12) March 2015 (9) February 2015 (9) January 2015 (10) December 2014 (7) November 2014 (10) October 2014 (10) September 2014 (9) August 2014 (4) July 2014 (6) June 2014 (8) May 2014 (8) April 2014 (6) March 2014 (10) February 2014 (9) January 2014 (8) December 2013 (11) November 2013 (10) October 2013 (7) September 2013 (10) August 2013 (6) July 2013 (10) June 2013 (7) May 2013 (7) April 2013 (7) March 2013 (13) February 2013 (12) January 2013 (10) December 2012 (8) November 2012 (9) October 2012 (5) September 2012 (5) August 2012 (6) July 2012 (8) June 2012 (7) May 2012 (8) April 2012 (3) March 2012 (8) February 2012 (9) January 2012 (7) December 2011 (4) November 2011 (10) October 2011 (6) September 2011 (3) August 2011 (5) July 2011 (3) June 2011 (4) May 2011 (4) April 2011 (3) March 2011 (6) February 2011 (6) January 2011 (6) December 2010 (6) November 2010 (8) October 2010 (11) September 2010 (6) August 2010 (4) July 2010 (5) June 2010 (11) May 2010 (8) April 2010 (13) March 2010 (20) February 2010 (8) January 2010 (6) December 2009 (4) November 2009 (8) October 2009 (1) September 2009 (4) August 2009 (1) July 2009 (1) June 2009 (4) May 2009 (1) April 2009 (2) March 2009 (2) February 2009 (6) December 2008 (8) November 2008 (1) October 2008 (3) September 2008 (1) August 2008 (5) July 2008 (4) June 2008 (8) May 2008 (4) April 2008 (2) March 2008 (1) February 2008 (2) December 2007 (1) November 2007 (2) October 2007 (2) September 2007 (1) July 2007 (2) June 2007 (1) May 2007 (1) April 2007 (1) March 2007 (9) November 2006 (1) October 2006 (1) August 2006 (2) June 2006 (1) May 2006 (2) April 2006 (2) March 2006 (1) February 2006 (2) December 2005 (2) November 2005 (1) October 2005 (1) September 2005 (1) August 2005 (1) July 2005 (1) May 2005 (1) April 2005 (1) March 2005 (2) February 2005 (1) January 2005 (1) December 2004 (2) October 2004 (2) September 2004 (2) August 2004 (2) July 2004 (4) June 2004 (2) May 2004 (5) April 2004 (3) March 2004 (4) February 2004 (3) December 2003 (2) November 2003 (1) October 2003 (3) July 2003 (2) April 2003 (1) March 2003 (1) February 2003 (3) January 2003 (1) August 2002 (1) February 2002 (1)
Columbia Journalism
European Journalism Centre
Financial Times media pages
Guardian Media
ICFJ
Journalism Research News
Journalism.co.uk
Politico Europe – "On Media" blog
© 2020 European Journalism Observatory – EJO |
Powered by Wordpress. Designed by Themnific™. Adapted and customized by EJO and USI Web Services
We use cookies only to help us understand how the site is used. We do not collect personal data, nor do we sell data to third parties, or use cookies for marketing purposes.
If you continue to use this site we assume that you are happy with this. To learn more, please look at our Privacy Policy.OkPrivacy policy


text:
Vezin-le-Coquet (bret. Gwezin) – miejscowość i gmina we Francji, w regionie Bretania, w departamencie Ille-et-Vilaine.

Według danych na rok 1990 gminę zamieszkiwało 3268 osób, a gęstość zaludnienia wynosiła 416 osób/km² (wśród 1269 gmin Bretanii Vezin-le-Coquet plasuje się na 158. miejscu pod względem liczby ludności, natomiast pod względem powierzchni na miejscu 915.).

Bibliografia

Miejscowości w departamencie Ille-et-Vilaine

redpajama_set_name: RedPajamaWikipedia


Extraversion has been shown to be important to the perception of leadership. That is true. And most of us believe that extraverts are better leaders. Yet a recent study by Grant, Gino and Hofmann suggests that there are times when extraversion in a leader might decrease group performance.
So what is extraversion? It is the tendency to seek out status, to be dominant, outgoing and talkative. Typically we think of extroverts when we think of leaders.
The research team wondered if follower proactivity/passiveness would impact the effectiveness of a leader's extroverted style.
What is proactivity? Proactivity is when employees take advance actions in order to implement organizational change. Actions such as speaking up or giving voice, taking charge or influencing those above them.
In their first study they measured the results from a nationally franchised pizza delivery service, controlling for location. They considered whether the local manager was extraverted or introverted, and whether the employees as a group were proactive or passive. They found that "extraverted leadership predicted higher store performance (profits) when employees were passive, but lower store performance when employees were proactive." (p. 536).
A second experimental study followed, with university students placed into groups, with an assigned leader and followers. Two of the followers were "confederates" of the researchers. The students' task was to fold as many t-shirts as possible in 10 minutes. Leaders of the group read a passage about either extroverted or introverted leaders prior to starting the task.
The confederates (additional followers) were randomly assigned to a passive behaviour, doing nothing or to an active behaviour, suggested a better method of folding.
The results were interesting. When the confederates were passive, the extraverted leaders groups performed better than when the confederates were proactive.
However, the results reversed themselves for the introverted leaders, that is the groups with introverted leaders performed better when the confederates were proactive, less well when the confederates were passive.
The authors think that the explanation for this behaviour might lie in the theory of complimentary behaviours. For example, extroverted leaders, who seek the centre of attention and status, might see proactive behaviours as threatening. Introverted leaders might be better listeners and better engaged with group processes. As a result, employees might believe that the leader is receptive or not receptive to proactive behaviour, changing their performance as a result.
Their research suggests that the perception of leaders receptivity to input might impact motivation and performance. That is, that leaders who are believed to be high in receptivity tend to have better group performance than leaders who are low in receptivity. Extraverted leaders were believed to be lower in receptivity to input than introverted leaders.
Why is this important? Current popular leadership discussions are encouraging both extraverted leadership styles (including theories of charismatic leadership and transformational leadership) and, at the same time, encouraging proactive follower behaviours. This research suggests that doing both of these things at the same time might be incompatible with better performance.
Less extraverted leaders can actively facilitate more proactive behaviours among employees, by being more receptive to employees speaking up, by encouraging them to take charge, to develop new work methods, and to exercise upward influence.
Highly extraverted leaders can improve group performance by determining which situations calls for proactive behaviour, and practice a less dominating style, becoming more reserved in those situations.
Training future extroverted leaders to identity, encourage and reward proactive behaviours.
As a recovering extravert, I know I'm trying to be more reserved and receptive to others ideas. How about you?
Is there a correlation between a person's extroversion and his ability to be receptive?
In the definitions of extroversion that I quickly looked up I did not see "tendency to seek out status, to be dominant." Rather I saw assertive and enthusiastic … obtaining gratification from what is outside the self.
So a 'healthy' extrovert would by definition be receptive.
If a person was self absorbed and only concerned about themselves, then of course they are not receptive and probably not a good leader. Whether they were introverted or extroverted seems like a separate (and non connected) issue.
I would argue the world needs strong leaders who are interested in the opinions of others.
And in the post-industrial age, self leadership and the ability to work within teams will be most valued.
This is a fascinating exercise in thinking but I believe the original flawed definition of extroversion sets up a premise where two independent concepts are incorrectly correlated.
You missed an important distinction in your summary of the "Healthy Extrovert". Yes, they are indeed receptive. What's missing? It is what they are receptive to. Extrovert leaders are not, usually, looking for a better way of getting something done or creating a better mouse trap so they are usually in conflict with those who take that approach – even if undeclared. What extroverts are looking for is confirmation from the outside that they are doing it right and this is true in leadership, singing and public speaking as well as most other parts of living.
The introvert is a different animal altogether and that doesn't necessarily mean better. However, the introvert is more of a receptive animal because their confirmation of doing it right or don't know if it's right is internal.
The two styles are like Yin & Yang. They work best together balancing each other out. The study would seem to confirm that.
Since everyone has both of these traits, strong leadership can only come from a "Healthy introvert/extrovert" who can and will listen to the opinions of others while correcting or re-aligning their views where necessary.
Thanks, Skotia, for your thoughtful reply. "Healthy" and "open" and "balance" are all things we agree. And understanding the differing styles, the Yin & Yang, are what makes leaders great. You point is well taken.


The Dwyer MM10 digital multimeter is a compact, low cost instrument ideal for general electrical testing and troubleshooting. This full function multimeter measures DC and AC voltage to 600V, DC current to 10A and resistance to 20 MΩ. In addition to volts, amps and ohms, this meter includes audible continuity, diode, LED and transistor hFE tests. Rotary switch allows you to select from any of the twenty ranges quickly and easily. The 3.5 digit LCD features automatic polarity correction, low battery and overload indication. Rugged, high impact case comes equipped with built-in tilt-stand for versatile, hands-free operation. Overload protected to 600 VAC/600 VDC for voltage ranges and to 500 VAC/500 VDC for resistance, diode and continuity test ranges. Current ranges are protected by 0.5A/250V, replaceable fuse.


The word Solitaire is used to describe a single stone diamond or gemstone set into an Engagement ring.
4 Claw Classic Solitaire Diamond Ring: The White Gold Solitaire features a luminous round cut bri..
Classic 4 Claw Solitaire Diamond Ring: This Classic Solitaire is a remarkable piece that combines..
Classic Claw Set Diamond Solitaire Ring: Delicately hewn from white gold and set with a round bri..
The Cushioned Diamond Solitaire Ring: The Cushioned Diamond Solitaire features a dazzling brillia..
Diamond Solitaire Ring: The White Gold Solitaire features a luminous round cut brilliant diamond,..
Lightning Diamond Solitaire Ring Set in an elegant curve of white gold, the round brilliant cut d..
Shoulder Split Band Solitaire Ring: Elegantly crafted from white gold, the Shoulder Split Band So..
Solitaire Diamond Classic Ring: The White Gold Solitaire features a luminous round cut brilliant ..
Split Band Solitaire Diamond Ring The Split Band Solitaire Diamond ring features a remarkable spl..
SQUARE 0.30CT CLASSIC SOLITAIRE: Featuring an exquisite square princess cut diamond in a delicate..
SQUARE 0.56CT CLASSIC SOLITAIRE: This Square Classic Solitaire features a breath-taking square pr..
SQUARE DIAMOND CLASSIC SOLITAIRE: Beautifully crafted from timeless white gold and set with a squ..
SQUARED SOLITAIRE DIAMOND RING: The Squared Solitaire Diamond Ring features an arresting design i..
TAPERED SOLITAIRE DIAMOND RING: Featuring a stunning brilliant cut diamond in an elegantly tapere..
TENSION SET 0.62CT SPLIT RING: The Tension Set Split Ring has been crafted from exquisite white g..