text:
\section{Introduction and Preliminaries}

A simple but widely-used transmission model in coding theory is the
Binary Symmetric Channel (BSC). The general assumptions for this
transmission channel (cf.\ \cite{CodingTheory}) are as follows:
\begin{itemize}
\item bitstrings (i.e.\ words consisting of the symbols $0$ and $1$) are
transmitted,
\item for each symbol the ``flip probability'' (i.e.\ the
probability that $0$ and $1$ is sent, but $1$ and $0$ is received,
respectively) is constant over all transmissions,
\item transmission errors occur independently.
\end{itemize}

Under these assumptions, the number of transmission
errors in an $n$-bit word can be modeled as a binomially distributed
random variable.

However, when investigating Physically Unclonable Functions (PUFs),
these assumptions do not apply. PUFs---and in particular
SRAM-PUFs---generally produce a noisy bitstring (response) for a given
input (challenge) where the flipping probability varies from bit to bit. Thus, a BSC
is no longer an exact model for the noise of SRAM-PUFs (cf.\
\cite{helperData}). Another approach to model PUF error behavior
(concentrating on considerations regarding entropy) is pursued in
\cite{entropyAnalysis}.

From a system security theoretic point of view, PUFs are very
interesting because they can be used to construct a challenge-response
mechanism without the need of having a ``master key''. Usually, the
master key is used to derive and verify responses to some given input
challenges. However, as PUFs basically are hardware challenge-response
mechanisms, a master key is not necessary. A general introduction to
the topic of PUFs can be found in \cite{PUFintroduction}.

In our setting, an SRAM-PUF consists of $n\in \mathbb{N}$ uninitialized
SRAM cells (cf.\ \cite{SRAMintroduction}). When
powering on the device, these cells either assume state $0$ or
$1$---and most of them do so in a very stable way. Based on a series
of measurements, we are able to identify a ``stable state'' for each
cell, which, in turn, is then used to compute the \emph{error
  probability} (or \emph{flipping probability}) for each cell. For
example, if a cell takes the state $1$ more often than the state $0$
in our measurements, we assess $1$ as the stable state of this
cell and $0$ is its error state. A query of all $n$ PUF bits is called
a \emph{PUF evaluation}. Otherwise, when investigating only a subset
of all PUF bits, then we speak of \emph{PUF responses}.

From a statistical point of view, the SRAM cells can be modeled by
means of independent Bernoulli distributed random variables $X_{j}
\sim \Ber(p_{j})$ for $j=1,2,\ldots, n$. That is, we have $\P(X_{j} =
1) = p_{j}$ and $\P(X_{j} = 0) = 1 - p_{j}$. The stability of an
SRAM-cell is then related to its error probability, where $p_{j} =
0.5$ means complete instability and $p_{j} = 0$ or $p_{j} = 1$ means
total stability. As we are only interested in the
stability of the cells, and not in the concrete values they assume,
our error probabilities can be bounded from above by $0.5$---this can
be enforced by choosing the stable state of the cell accordingly.

If these random variables are Bernoulli distributed with a
common error probability $p$, it is well known that the random number
of errors $E$ in a response of length $\ell$ follows a binomial
distribution with parameters $\ell$ and $p$, short $E\sim
\operatorname{Bi}(\ell, p)$. However, when investigating PUFs, the
situation is not quite as simple. Not all of the
SRAM cells are equally stable, which is indicated in
Figure~\ref{fig:errorAndWeight}, showing histograms for the error rate
and bit weight\footnote{The bit weight $w_{j}$
  is used to determine the stable states. It is computed
  separately for each bit as the proportion of ones among all
  measurements.} for a given set of SRAM-PUF measurements. One can see
that the majority of bits (about $80\%$) have an error rate of less
than $2\%$, while about $2.3\%$ of all SRAM cells show an error
probability greater than $40\%$, indicating strong instability of
these cells. Moreover, from the symmetry in the bit-weight histogram
one can see that the PUF actually is very balanced: there are
approximately the same number of bits assuming stable state $0$ as
there are assuming stable state $1$.

\begin{figure}[ht]
  \centering
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/PUFerrors.pdf}
  \end{minipage}\hfill
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/PUFweight.pdf}
  \end{minipage}
  \caption{SRAM cell stability: bit weights and error rates.}
  \label{fig:errorAndWeight}
\end{figure}

\begin{figure}[ht]
  \centering
  \includegraphics[width=0.9\linewidth]{figures/densityComparison.pdf}
  \caption{Goodness of fit: binomial vs.\ generalized binomial distribution.}
  \label{fig:densityComparison}
\end{figure}

In the case of varying error probabilities, the random number of
errors no longer follows a classical binomial distribution.
Instead, we apply a \emph{generalized binomial
  distribution}, which yields a much better fit than the binomial
distribution for the number of flipped cells per PUF evaluation. This
is illustrated in Figure~\ref{fig:densityComparison}, showing a fitted
binomial and generalized binomial distribution for the histogram of
the number of flipped cells in a SRAM-PUF response of size $n = 2^{16}$.

In this paper, we investigate various properties of the
generalized binomial distribution as the proper model for the noise of
SRAM-PUFs (cf.\ Section~\ref{sec:gbin}). Furthermore, we develop a suitable
statistical model for evaluating the overall noise behavior of
SRAM-PUFs on the basis of measurements from several SRAM-devices (cf.\
Section~\ref{sec:model}). An empirical Bayesian approach is employed
to assess the parameters of the underlying generalized binomial
distribution. Several estimation techniques for determining the
hyperparameters in the empirical Bayesian model are compared within a
simulation study in Section~\ref{sec:applications}.

Finally, we extend the proposed model to concrete measurements obtained within
the CODES\footnote{\url{https://www.technikon.com/projects/codes}}
research project and use these concrete measurements to discuss
stabilization methods for SRAM-PUF responses on the basis of error
correction schemes and order statistics.

\section{Some properties of the generalized binomial distribution}\label{sec:gbin}

In this section, we will discuss some properties of the generalized
binomial distribution which arises naturally when investigating the
number of flipped SRAM cells per SRAM-PUF response.

In general, the binomial distribution originates in the context of
Bernoulli trials. In \cite{fisz}, the author calls the trials
under which the ``generalized binomial distribution'' originates, \emph{Poisson
  trials}\footnote{Therefore, the generalized binomial
  distribution is also often called \emph{Poisson binomial
    distribution}.}. These are $n$ independent trials where the probability
that some event occurs in the $j$-th trial is $p_{j}$. Then, the
number of occurrences of the event within those $n$ trials follows a
generalized binomial distribution with parameter vector $\mathbf{p}
= (p_{1}, \ldots, p_{n})^{\top}$. Thus, we may define this distribution as
follows:

\begin{definition}[Generalized binomial distribution]\label{def:GBi}
  Let $p_{1}$, $p_{2}$, \ldots, $p_{n}\in [0,1]$. Assume that the
  random variables $X_{j}\sim \Ber(p_{j})$ are independent for
  $j\in \{1,2,\ldots,n\}$. Then, the random variable $X :=
  \sum_{j=1}^{n} X_{j}$ follows a \emph{generalized binomial
    distribution} with parameter vector $\mathbf{p} =
  (p_{1},p_{2},\ldots, p_{n})^{\top}$. For short, we write $X\sim\GBi(p_{1}, p_{2},
  \ldots, p_{n})$.
\end{definition}

There is a number of elementary properties for the generalized
binomial distribution which follow immediately from the definition.

\begin{proposition}[Elementary properties]\label{prop:elementaryProperties}
  Let $X\sim \GBi(p_{1}, p_{2}, \ldots, p_{n})$. Then the following
  statements hold:
  \begin{enumerate}
  \item[(a)] The support of the random variable $X$ is contained in
    $\{0,1,\ldots, n\}$.
  \item[(b)] The probability mass function of $X$ is given by
    \[ \P(X = k) = \sum_{\substack{S\subseteq \{1,2,\ldots, n\} \\ |S| = k}}
    \prod_{s\in S} p_{s}~~\cdot~~ \prod_{\mathclap{s\in \{1,2,\ldots, n\}\setminus S}} (1-p_{s}).  \]
  \item[(c)] Expectation and variance of $X$ are given by
    \[ \mathbb{E} X = \sum_{j=1}^{n} p_{j} \quad \text{ and }\quad \Var(X) =
    \sum_{j=1}^{n} p_{j}\cdot (1-p_{j}),  \]
    respectively.
  \item[(d)] The characteristic function $\varphi_{X}(t)$ of $X$ is
    given by
    \[ \varphi_{X}(t) = \prod_{j=1}^{n} \left(1-p_{j} + p_{j}\cdot
      e^{it}\right).  \]
  \item[(e)] The generalized binomial distribution is a generalization
    of the binomial distribution: for $p_{1} = p_{2} = \cdots = p_{n}
    =:p$, the random variable $X$ follows a binomial distribution with
    parameters $n$ and $p$.
  \end{enumerate}
\end{proposition}
\begin{proof}
  With the notation of Definition~\ref{def:GBi}, (a) follows
  directly as we know $\supp (X_{j}) \subseteq \{0,1\}$ for all $j\in
  \{1,2,\ldots, n\}$. Therefore, the support of the sum is contained
  in $\{0,1,\ldots, n\}$.\\
  Based on the independence of $X_{j}$, the
  described probability mass function as stated in (b) follows
  immediately from
  \begin{align*}
    \P(X = k) & = \sum_{\substack{S\subseteq \{1,2,\ldots, n\} \\ |S| = k}}
  \P\left(X_{s} = 1 \text{ for } s\in S \text{ and } X_{s} = 0 \text{
      for } s\in \{1,2,\ldots, n\}\setminus S\right) \\
  & = \sum_{\substack{S\subseteq \{1,2,\ldots, n\} \\ |S| = k}} \prod_{s\in S}
  p_{s} ~~\cdot~~ \prod_{\mathclap{s\in \{1,2,\ldots, n\}\setminus S}} (1-p_{s}).
  \end{align*}
  The remaining statements follow from the definition of $X$ as
  the sum of independently distributed random variables. We find
  \[ \mathbb{E} X = \mathbb{E} \left(\sum_{j=1}^{n} X_{j}\right) = \sum_{j=1}^{n} \mathbb{E}
  X_{j} = \sum_{j=1}^{n} p_{j},  \]
  by using the linearity of $\mathbb{E}$. We obtain $\Var(X_{j}) = p_{j}
  (1-p_{j})$ by using the linearity of the variance for independent
  random variables. Statement (d) is proved as
  the characteristic function of $X_{j}$ is $\varphi_{X_{j}}(t) =
  1-p_{j}+p_{j} e^{it}$ and because
  \[ \varphi_{X_{1}+X_{2}+\cdots +X_{n}}(t) = \prod_{j=1}^{n}
  \varphi_{X_{j}}(t)  \]
  holds for independent random variables $X_{1}$, \ldots,
  $X_{n}$. Finally, as the probabilities in (e) are all equal to $p$,
  $X$ is the sum of $n$ independent and identically distributed
  Bernoulli random variables---which is an alternative definition for
  the binomial distribution with parameters $n$ and $p$. This
  completes the proof.
\end{proof}

The \texttt{R} package ``\texttt{GenBinomApps}''
(cf.\ \cite{genBinomAppsPkg}) offers an
efficient implementation to compute the probability mass function
recursively. The theoretic background for this recursive computation
approach is covered in \cite{genBinomKurz}.

However, for very large dimensions of the probability vector, the
computation of the distribution function is quite expensive. In such
cases, approximation with a binomial distribution would be
desirable. In the following proposition we give some useful properties
of such an approximation.

\begin{proposition}[Binomial approximation]\label{prop:approx}
  Let $X\sim \GBi(p_{1},p_{2},\ldots, p_{n})$ and $Y\sim
  \operatorname{Bi}(n, p^{*})$. Then the following properties hold:
  \begin{enumerate}
  \item[(a)] The expectation of $X$ is equal to the expectation of $Y$
    if and only if $p^{*} = \overline{p} := \frac{1}{n} \sum_{j}
    p_{j}$.
  \item[(b)] If we have $p^{*} = \overline{p}$, then the inequality $\Var(X) \leq
    \Var(Y)$ holds for arbitrary parameters $p_{1}$, $p_{2}$, \ldots, $p_{n}
    \in [0,1]$ of $X$ and equality holds if and only if $p_{1} = p_{2}
    = \cdots = p_{n}$.
  \end{enumerate}
\end{proposition}
\begin{proof}
  The first statement follows immediately from
  Proposition~\ref{prop:elementaryProperties} and because of simple
  properties of the binomial distribution. We have $\mathbb{E} X =
  \sum_{j=1}^{n} p_{j}$ and $\mathbb{E} Y = n\cdot p^{*}$. Therefore, the
  relation
  \[ \mathbb{E} X = \mathbb{E} Y \quad \iff \quad p^{*} = \frac{1}{n} \sum_{j=1}^{n}
  p_{j}  \]
  follows immediately. The inequality from (b) can be proven by
  showing its equivalency to the Cauchy-Schwarz inequality. Let
  $\mathds{1}$ denote the $n$-dimensional vector of ones and
  $\mathbf{p} = (p_{1},p_{2},\ldots, p_{n})^{\top}$. Then we obtain
  \begin{align*}
    \Var(X) \leq \Var(Y) &\iff \sum_{j=1}^{n} p_{j}\cdot (1-p_{j})
    \leq n \cdot\overline{p}\cdot (1-\overline{p})\\ & \iff \sum_{j}
    p_{j}\cdot (1-p_{j}) \leq \bigg(\sum_{j} p_{j}\bigg)\cdot
    \bigg(1 - \frac{1}{n} \sum_{j} p_{j}\bigg)\\
    & \iff \sum_{j} p_{j} - \sum_{j} p_{j}^{2} \leq \sum_{j} p_{j} -
    \frac{1}{n} \bigg(\sum_{j} p_{j}\bigg)^{2} \\
    & \iff \bigg(\sum_{j} p_{j}\bigg)^{2} \leq n\cdot \sum_{j}
    p_{j}^{2}\\
    & \iff \left|\langle \mathds{1}, \mathbf{p}\rangle\right|^{2} \leq \langle
    \mathds{1},\mathds{1}\rangle \cdot \langle \mathbf{p}, \mathbf{p}\rangle,
  \end{align*}
  which is the Cauchy-Schwarz inequality for the
  Euclidean scalar product. Finally, equality holds in the
  Cauchy-Schwarz inequality if and only if the vectors $\mathds{1}$
  and $\mathbf{p}$ are linearly dependent, that is $\mathbf{p} =
  p\cdot \mathds{1}$. In this case, $p_{1} = p_{2} = \cdots = p_{n}  =
  p$ follows, which is equivalent to the fact that $X$ is binomially
  distributed with parameters $n$ and $p$.
\end{proof}

By property (b) of Proposition~\ref{prop:approx}, the variance of the
generalized binomial distribution equals the variance of the
approximating binomial distribution if and only if the distributions
coincide. In the following example, we investigate whether
the variance may be used to assess the goodness of fit for such an
approximation.

\begin{example}
  After performing some simulations (in which the input parameters of
  the generalized binomial distribution were generated from various beta
  distributions), we plotted the difference between the variances
  against the maximum error between the corresponding distribution
  functions. The result of one of these simulation batches ($N= 1000$
  generalized binomial distributions with $n=100$
  $\Be(1.5,1.8)$-distributed parameters each in the left plot, and
  following a $\Be(0.3,0.1)$-distribution in the right plot) is
  illustrated in Figure~\ref{fig:binomApprox}. The plots depict the
  relation between the difference of the variances and the maximum
  approximation error of the respective distribution functions
  mentioned above.

  It is interesting to see that there is a very strong correlation
  (with correlation coefficient greater than $0.95$) between
  the difference of the variances and the maximal approximation error
  in our simulations when the $p_{j}$ concentrate around a single
  value. In this case, the variance difference also is significantly
  lower (as can also be seen in Figure~\ref{fig:binomApprox}).
\end{example}

\begin{figure}[ht]
  \centering
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/approx-correl1.pdf}
  \end{minipage}
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/approx-correl2.pdf}
  \end{minipage}
  \caption{Binomial approximation of the generalized binomial
    distribution: variance vs.\ maximum approximation error.}
  \label{fig:binomApprox}
\end{figure}

The following considerations require the notion of \emph{compound
  distributions}.

\begin{definition}[Compound distribution]\label{def:compound}
  Let $\mathbf{X}$ be a random vector with probability density
  function $p_{\mathbf{X}}(\mathbf{x}\,|\,\boldsymbol{\theta})$ depending on
  some random vector $\boldsymbol{\theta}$. Furthermore, let
  $G_{\boldsymbol{\theta}}(\boldsymbol{\vartheta})$ be the
  distribution function of $\boldsymbol{\theta}$. Then the
  probability density of the compound distribution of $\mathbf{X}$ with
  respect to $G$ is given by
  \[ p_{\mathbf{X}}(\mathbf{x}) = \int_{\boldsymbol{\vartheta}}
  p_{\mathbf{X}}(\mathbf{x}\mid
  \boldsymbol{\vartheta})~dG_{\boldsymbol{\theta}}(\boldsymbol{\vartheta}).  \]
\end{definition}

Now, let us assume that the error probabilities $p_{j}$ are realizations of
some random variable $\pi$ with distribution function $F_{\pi}$ and $\supp(
\pi) \subseteq [0,1]$. A quite interesting question is, how the compound
distribution of the generalized binomial distribution with respect
to the parameters of the distribution of $\pi$ looks like. The following
theorem characterizes these compound distributions with respect to the
generalized binomial distribution.

\begin{theorem}[Compound distribution for the generalized binomial
  distribution]\label{thm:compound}
  Let $\pi_{1}$, $\pi_{2}$, \ldots, $\pi_{n}$ be independently and
  identically distributed random variables on $[0,1]$ with
  distribution function $F_{\pi|\boldsymbol{\vartheta}}$. Then the
  compound distribution $\GBi(\pi_{1},\pi_{2},\ldots, \pi_{n})$ under
  the parameter vector $\boldsymbol{\vartheta}$ is the binomial
  distribution $\Bi(n, \mathbb{E}(\pi\mid \boldsymbol{\vartheta}))$.
\end{theorem}
\begin{proof}
  It is easy to see that the expected value $\mathbb{E}(\pi\mid
  \boldsymbol{\vartheta})$ always exists and is contained in $[0,1]$,
  as the support of $\pi$ itself is contained in $[0,1]$. The expected
  value therefore is a valid second parameter for the binomial
  distribution.

  Now, let $X\mid \pi_{1},\ldots, \pi_{n}\sim \GBi(\pi_{1},\ldots,
  \pi_{n})$ and $\pi_{i}\stackrel{\text{iid}}{\sim} F_{\pi\mid
    \boldsymbol{\vartheta}}$. According to the definition of
  compound distributions, the probability mass
  function $p_{X}(k\mid \boldsymbol{\vartheta})$ is then determined as
  follows:
  \begin{align*}
    p_{X}(k\mid \boldsymbol{\vartheta}) & = \sum_{\substack{S\subseteq
        \{1,\ldots, n\}\\ |S|=k}} \prod_{s\in S}
    \left( \int_{p_{s}=0}^{1} p_{s}~dF_{\pi\mid \boldsymbol{\vartheta}}
    \right)\cdot \prod_{s\not\in S} \left( 1 - \int_{p_{s} = 0}^{1}
      p_{s}~dF_{\pi\mid \boldsymbol{\vartheta}} \right)\\
    & = \sum_{\substack{S\subseteq \{1,\ldots, n\}\\ |S|=k}} \prod_{s\in S}
    \mathbb{E}(\pi\mid \boldsymbol{\vartheta})\cdot \prod_{s\not\in S}
    (1-\mathbb{E}(\pi\mid \boldsymbol{\vartheta}))  = \binom{n}{k} \mathbb{E}(\pi\mid
    \boldsymbol{\vartheta})^{k}\cdot (1- \mathbb{E}(\pi\mid
    \boldsymbol{\vartheta}))^{n-k}.
  \end{align*}
  This is exactly the probability mass function of the $\Bi(n,
  \mathbb{E}(\pi\mid \boldsymbol{\vartheta}))$-distribution and therefore, proves
  the theorem.
\end{proof}

\begin{remark}
  \begin{enumerate}
  \item[(i)]  This result is in accordance with the result stated in property (b) of
    Proposition~\ref{prop:approx}: by forming the compound distribution,
    the variance also increases in general.
  \item[(ii)] In a Bayesian context, the integrand in
    Definition~\ref{def:compound} can be identified with the
    likelihood (parameterized by $\boldsymbol{\vartheta}$), the
    integrating function $G_{\boldsymbol{\theta}}$ with the prior
    distribution of $\boldsymbol{\theta}$ and the left-hand side
    $p_{\mathbf{X}}(\mathbf{x})$ portrays the marginal distribution of
    $\mathbf{X}$, also known as prior predictive distribution. The
    increase in the variance of the compound distribution then
    accounts for the parameter uncertainty in a natural way.
  \end{enumerate}
\end{remark}

\begin{corollary}[Properties of the compound distribution]
  Let $X\sim \GBi(\pi_{1},\pi_{2},\ldots, \pi_{n})$ with the
  identically and independently distributed random variables
  $\pi_{1}$, $\pi_{2}$, \ldots, $\pi_{n}$ on $[0,1]$ with distribution
  function $F_{\pi|\boldsymbol{\vartheta}}$. Then the expected value
  and variance of $X$ are given by
  \[ \mathbb{E} X = n\cdot \mathbb{E}(\pi\mid \boldsymbol{\vartheta})\quad \text{ and
  }\quad \Var(X) = n\cdot \mathbb{E}(\pi\mid \boldsymbol{\vartheta})\cdot
  (1-\mathbb{E}(\pi\mid \boldsymbol{\vartheta})).  \]
  Furthermore, the characteristic function of $X$ has the form
  \[ \varphi_{X}(t) = \left[1-\mathbb{E}(\pi\mid \boldsymbol{\vartheta}) +
    \mathbb{E}(\pi\mid \boldsymbol{\vartheta})\cdot e^{it}\right]^{n}.  \]
\end{corollary}
\begin{proof}
  These statements follow immediately from Theorem~\ref{thm:compound}
  and some elementary properties of the binomial distribution.
\end{proof}

We will use these results in Section~\ref{sec:applications} in order to
estimate the probability that the SRAM-PUF noise cannot be
``corrected'' properly in a response of given length.

\section{Statistical Model for overall SRAM-PUF noise behavior}\label{sec:model}

In this section, we discuss a suitable
statistical model for the noise behavior of a set of different
SRAM-PUF devices, where the model parameters can be estimated from a
series of simple PUF evaluations.  The model is based on an
exploratory statistical analysis carried out within the CODES research
project. Furthermore, we propose several methods for assessing the
model parameters.

\subsection{Model development}\label{sec:modelDevel}

We propose a Bayesian model for the noise behavior of SRAM-PUFs:
assume that we have $m_{\operatorname{dev}}$ SRAM-PUF devices, where
device $i$ has $c_{i}$ SRAM-cells for $i=1,\ldots,
m_{\operatorname{dev}}$. Then, we model the noise of the $i$-th device
as a vector of $c_{i}$ independently distributed random variables
$X_{ij}\sim \Ber(p_{ij})$ such that
\begin{align*}
  \P(\text{Cell } j \text{ in device } i \text{ flips}) & := \P(X_{ij}
= 1) = p_{ij}.
\end{align*}

Motivated by the results of an exploratory statistical analysis, we
further model these error probabilities to follow a scaled beta
distribution on the interval $[0,\sfrac{1}{2}]$.

\begin{definition}[Scaled beta distribution]
  Let $a$ and $b$ be real numbers and $a < b$. If the random variable
  $P$ follows a beta distribution with parameters $\alpha$ and
  $\beta$, $P\sim \Be(\alpha, \beta)$, the random variable $Q = a +
  (b-a)\cdot P$ follows a scaled beta distribution on the interval
  $[a,b]$ with parameters $\alpha$ and $\beta$. For short, we write $Q \sim
  \Be_{[a,b]}(\alpha, \beta)$.
\end{definition}

For the $i$-th device, we parametrize the beta distribution of the
parameters $(p_{ij})_{j=1}^{c_{i}}$ as
$\Be_{[0,\sfrac{1}{2}]}(2\delta_{i}\cdot K_{i}, (1-2\delta_{i})\cdot
K_{i})$, such that $\delta_{i}$ denotes the distribution's expected value and $K_{i}$ is a
shape parameter controlling the variance. In the next step, we assign
prior distributions to $\delta_{i}$ and $K_{i}$. More precisely, we
model $(\delta_{i})_{i=1}^{m_{\operatorname{dev}}}$ to
follow a scaled beta distribution (again scaled to the interval $[0,
\sfrac{1}{2}]$) with parameters $\alpha$ and $\beta$, and the shape
parameters $(K_{i})_{i=1}^{m_{\operatorname{dev}}}$ to follow a gamma
distribution with parameters $\kappa$ and $\lambda$. Altogether, we
have
\begin{align*}
  X_{ij}\,|\, p_{ij} & \sim \Ber(p_{ij}) &  \text{for }i = 1,2,\ldots,
  m_{\operatorname{dev}}, &\quad j= 1,2,\ldots, c_{i},\\
  (p_{ij}\,|\,\delta_{i}, K_{i})_{j=1}^{c_{i}} & \sim \Be_{[0,\sfrac{1}{2}]}(2\delta_{i}\cdot K_{i}, (1-2\delta_{i})\cdot
  K_{i}) & \text{for } i=1,2,\ldots, m_{\operatorname{dev}},\\
  (\delta_{i}\,|\,\alpha,\beta)_{i=1}^{m_{\operatorname{dev}}} & \sim
  \Be_{[0,\sfrac{1}{2}]}(\alpha, \beta),\\
  (K_{i}\,|\,\kappa,\lambda)_{i=1}^{m_{\operatorname{dev}}} &\sim \Ga(\kappa, \lambda).
\end{align*}

\begin{remark}
  We choose this model over a simplified model without assumed
  distributions for the parameters $\delta_{i}$ and $K_{i}$ primarily
  because of the control we have over the mean error rate, as well as
  to reflect that different devices may have varying mean error rates.
  Within this model, a variety of situations related to the SRAM-PUF
  production can be modeled and simulated.
\end{remark}

\subsection{Parameter estimation}\label{sec:paramEst}

Assuming we have $m_{\operatorname{dev}}$ SRAM-devices with $c_{i}$
SRAM-cells in the $i$-th device, the result of a series of $m_{i}$ measurements of
device $i$ is a vector $\mathbf{x}_{i}=(x_{i1},\ldots, x_{ic_{i}})^{\top}$,
where the component $x_{ij}$ denotes the number of measured error
states for the $j$-th cell of device $i$ and is $\Bi(m_{i},
p_{ij})$-distributed. Starting from these measurements, we wish to
estimate the parameters $\alpha$ and $\beta$ of the scaled beta
distribution $\Be_{[0,\sfrac{1}{2}]}(\alpha, \beta)$ modeling the mean
error rates $\delta_{i}$, and the parameters $\kappa$ and $\lambda$ of
the gamma distribution $\Ga(\kappa, \lambda)$ modeling the
distribution of the shape parameters $K_{i}$. Note that the component
$x_{ij}$ is a realization of a $\Bi(m_{i}, p_{ij})$ distribution.

\begin{remark}
  The joint posterior density function for our model is of the form
  \begin{multline*}
     f(\mathbf{p}, \boldsymbol{\delta}, \mathbf{K}, \alpha, \beta,
     \kappa, \lambda \mid \mathbf{x}) \propto \Bigg(\prod_{i=1}^{m_{\operatorname{dev}}}\Bigg(
  \prod_{j=1}^{c_{i}} f_{\Bi}(x_{ij}\mid p_{ij})\cdot
  f_{\Be}(p_{ij}\mid \delta_{i}, K_{i})\Bigg)\\
 \cdot
f_{\Be}(\delta_{i}\mid \alpha, \beta)\cdot f_{\Ga}(K_{i}\mid
\kappa,\lambda)\Bigg)\cdot f(\alpha,\beta,\kappa,\lambda),
  \end{multline*}
  where $f_{\Bi}(\,\cdot\,\mid p_{ij})$ denotes the density of the
  $\Bi(m_{i}, p_{ij})$ distribution, and
  $f_{\Be}(\,\cdot\,\mid \delta_{i},K_{i})$ and $f_{\Be}(\,\cdot\,\mid
  \alpha, \beta)$ denote the density of
  the $\Be_{[0,\sfrac{1}{2}]}(2\delta_{i}\cdot K_{i},
  (1-2\delta_{i})\cdot K_{i})$ and $\Be_{[0,\sfrac{1}{2}]}(\alpha,
  \beta)$ distribution, respectively. Moreover, $f_{\Ga}(\,\cdot\,|
  \kappa,\lambda)$ denotes the density of the
  $\Ga(\kappa,\lambda)$-distribution,
  $f(\alpha,\beta,\kappa,\lambda)$ denotes some joint prior of the
  parameters $\alpha$, $\beta$, $\kappa$ and $\lambda$, and
  $\mathbf{x}$ denotes the vector of all measurements. Due to the
  (practically) very large number of parameters the simulation based
  on this posterior is computationally intractable.

  To overcome this, we make use of an empirical Bayesian
  approach, meaning that we approximate the
  ``expensive'' posterior $f(\mathbf{p}, \boldsymbol{\delta},
  \mathbf{K}, \alpha, \beta, \kappa, \lambda \,|\, \mathbf{x})$ by the
  joint density $f(\mathbf{p}, \boldsymbol{\delta}, \mathbf{K}\mid
  \hat\alpha, \hat\beta, \hat\kappa, \hat\lambda, \mathbf{x})$ (which is also
  called ``pseudo posterior'') with empirically estimated parameters
  $\hat\alpha$, $\hat\beta$, $\hat\kappa$ and $\hat\lambda$.
\end{remark}

We estimate the parameters according to the model hierarchy:
\begin{itemize}
\item from the measurements $x_{ij}$, we estimate the flipping
  probabilities $p_{ij}$,
\item from the estimated flipping probabilities, we estimate the
  parameters $\delta_{i}$ and $K_{i}$ for $i = 1,2,\ldots,
  m_{\operatorname{dev}}$,
\item and from these estimated parameters, we estimate the
  hyperparameters $\alpha$, $\beta$, $\kappa$ and $\lambda$.
\end{itemize}
The estimates of the hyperparameters are then depending on the
estimation techniques used for the different parameter
layers. For example, possible approaches are the \emph{method of moments},
\emph{maximum likelihood estimation} (MLE), or the construction of a
Bayes estimator.

By MLE for the $p_{ij}$, we obtain $\hat p_{ij} =
\frac{x_{ij}}{m_{i}}$. In this case, this MLE-estimator coincides with the
estimator obtained by the method of moments. Another approach to estimate this parameter
(in the context of Bayesian statistics) is to choose the expected value of the
posterior obtained with respect to a scaled Jeffreys prior,
$p_{ij}\sim \Be_{[0,\sfrac{1}{2}]}(\sfrac{1}{2}, \sfrac{1}{2})$. The
posterior distribution is then given by
\[f(p_{ij}\,|\, x_{ij}) \propto \frac{p_{ij}^{x_{ij}}(1-p_{ij})^{m_{i}-x_{ij}}}{\sqrt{(2p_{ij})(1-2p_{ij})}}, \] forcing
us to compute the expectation numerically as
\[ \mathbb{E}(p_{ij}\mid x_{ij}) = \hat p_{ij} = \int_{0}^{\sfrac{1}{2}} p_{ij}\cdot
  f(p_{ij}\mid x_{ij})~dp_{ij},  \]
or to approximate it by Monte Carlo simulation. An advantage of this approach for
the estimation of the flipping probabilities is that it avoids an
underestimation of the flipping probabilities in the zero error
case. This is because the method of moments and MLE yield a flipping
probability of $0$ for cells without observed errors, which is not
realistic. Moreover, this approach allows us to compute sensible
credible intervals for  these probabilities, whereas the usual
confidence intervals based on the MLE would have zero lengths and thus
be meaningless.

Given the estimates of $p_{i1}$, $\ldots$, $p_{i c_{i}}$, we can
estimate the parameters $\delta_{i}$ and $K_{i}$, either again by MLE,
by the method of moments or by a Bayesian approach similar to the one above,
where we use the joint noninformative prior
\[p(\delta, K) \propto \frac{1}{K\cdot \sqrt{(2\delta)(1-2\delta)}}.\]

After estimating the $\delta_{i}$ and $K_{i}$, we may use these values
to obtain estimations for the hyperparameters $\alpha$, $\beta$,
$\kappa$ and $\lambda$. As we want to avoid proposing more priors for
these parameters, we will use either MLE or the method of moments.

\section{Results and Applications}\label{sec:applications}

On the basis of the statistical model proposed in the previous
section, we will determine a suitable method for parameter estimation
of this model in Section~\ref{sec:simStudy} by comparing possible
approaches in a simulation study. Afterwards, in
Section~\ref{sec:appData}, we will estimate the parameters of our
model (based on the superior estimation method from the simulation
study) for real measurements from the CODES project. Finally, in Section~\ref{sec:corrAndRed}, we will use
the posterior predictive distribution based on our real data to
evaluate approaches to correct and stabilize the SRAM-PUF responses.

\subsection{Simulation study}\label{sec:simStudy}

We are interested in comparing different parameter estimation methods
as discussed in Section~\ref{sec:paramEst} for the proposed
statistical model. In order to choose the ``best'' estimation method,
we will estimate these hyperparameters from simulated data with known
hyperparameters. The quality of these estimation methods will then be
compared by the value of a quadratic loss function for the parameter
vector: $L(\theta, \hat\theta) = \|\theta - \hat\theta\|^{2}$, where
$\theta\in \{(\alpha, \beta)^{\top}, (\kappa,\lambda)^{\top}\}$. Note
that we are especially interested in a good estimation of the
parameters $\alpha$ and $\beta$ of the beta distribution modeling the
mean error rates.

We will generate the simulation data ($m_{\operatorname{dev}}=20$
SRAM-devices with $c = 10000$ cells each and $m = 500$ simulations per
device) from the following parameters\footnote{These parameters are
  roughly based on parameters we used for testing in the CODES project.}:
\[ \alpha = 100,\quad \beta = 900,\quad \kappa = 800,\quad \lambda =
900.  \]

Concretely, there are $8$ methods of parameter estimation we will
compare. These methods originate from the different possibilities to
estimate the various parameter hierarchies. Let $x_{ij}$ denote the
number of assumed error states of the $j$-th cell in device $i$.
\begin{itemize}
\item The flipping probabilities $p_{ij}$ can be estimated by the
  method of moments (which, in this case, coincides with maximum
  likelihood estimation) by $\hat p_{ij} = \frac{x_{ij}}{m}$, or by
  computing the Bayes-estimator with respect to the Jeffreys
  $\Be_{[0, \sfrac{1}{2}]}(\sfrac{1}{2}, \sfrac{1}{2})$-prior.
\item The parameters $\delta_{i}$ and $K_{i}$ can be estimated either
  by the method of moments, yielding the estimators
  \[ \hat \delta_{i} =
  \frac{1}{c} \sum_{j=1}^{c} \hat p_{ij},\quad \hat K_{i} =
  \frac{2\hat\delta_{i} (1-2\hat\delta_{i})}{\frac{4}{c-1}\cdot
    \sum_{j=1}^{c} (\hat p_{ij} - \hat\delta_{i})^{2}} - 1,  \]
  by maximum likelihood estimation with the \texttt{R}-package
  \texttt{maxLik} (cf.\ \cite{maxLikPkg}), or by using a Bayesian
  approach and computing the mode of the joint (independence)
  posterior distribution subject to the Jeffreys $\Be_{[0, \sfrac{1}{2}]}(\sfrac{1}{2},
  \sfrac{1}{2})$-prior for $\delta_{i}$ and the non-informative
  $\frac{1}{\vartheta}$-prior for $K_{i}$.

\item Finally, the hyperparameters $\alpha$, $\beta$ from the proposed
  beta distribution of the $\delta_{i}$ and the parameters $\kappa$, $\lambda$ from
  the proposed gamma distribution of the $K_{i}$ can be estimated by
  the method of moments, which yields
  \[ \hat\alpha = 2\overline{\delta}\left(\frac{2\overline{\delta}
      (1-2\overline{\delta})}{4v_{\delta}} - 1\right),\quad
  \hat\beta = (1-2\overline{\delta})\left(\frac{2\overline{\delta}
      (1-2\overline{\delta})}{4v_{\delta}} - 1\right),\quad
  \hat\kappa = \frac{\overline{K}^{2}}{v_{K}},\quad \hat\lambda =
  \frac{\overline{K}}{v_{K}}, \]
  where $\overline{\delta}$, $v_{\delta}$, $\overline{K}$ and $v_{K}$
  denote the means and sample variances of the $\hat\delta_{i}$ and
  the $\hat K_{i}$, respectively---or by maximum likelihood
  estimation. As we want to refrain from proposing another set of
  priors for these parameters, we will not use Bayesian estimation for
  $\alpha$, $\beta$, $\kappa$ and $\lambda$.
\end{itemize}

In order to compare the various possible combinations of parameter
estimation methods, we used a quadratic loss function to measure the distance
from the original parameters. After performing $10000$ simulations,
and investigating the respective mean losses (which can be found in
Table~\ref{tab:simStudy}), we find that estimating the $p_{ij}$ and
the parameters $\delta_{i}$ and $K_{i}$ with Bayesian methods, as well
as the hyperparameters $\alpha$, $\beta$, $\kappa$ and $\lambda$ with
maximum likelihood estimation yields the lowest overall loss (where
the estimates of all four hyperparameters are taken into account) as
well as the lowest loss for just the parameters $\alpha$ and $\beta$
of the beta distribution. However, the lowest loss for the parameters
$\kappa$ and $\lambda$ of the gamma distribution originates from
estimating the $p_{ij}$ with Bayesian methods, but using the method of
moments to estimate everything else.

Note that even although the approach where we estimate $p_{ij}$ as
well as $\delta_{i}$ and $K_{i}$ by Bayesian means, and the remaining
parameters by MLE yields the lowest loss function with respect to the
parameters $\alpha$ and $\beta$, this estimator is rather conservative
with respect to the expected mean error rate $\mathbb{E}\delta =
\frac{1}{2}\cdot \frac{\alpha}{\alpha + \beta} = 0.05$: taking the
average value over the $\frac{1}{2} \cdot \frac{\hat\alpha}{\hat\alpha
+ \hat\beta}$ obtained in the simulation study (where $\hat\alpha$ and
$\hat\beta$ have been constructed as mentioned above) yields a value
of $0.0618$, which can be contributed to the high number of
observed very unstable bits. In order to
cover the occurrence of such bits, the expected average error rate is
increased in the MLE-estimation. Therefore, estimating
the parameters with this approach yields a model, which possesses
a certain ``robustness'' regarding a decline of the PUF's
stability. The most accurate approximation of the expected mean error
rate is obtained by estimating all parameters by the method of moments.

\begin{table}[ht]
  \centering
  \begin{tabular}[ht]{l|l|l|r|r}
    $\hat p_{ij}$ & $\hat\delta_{i}$, $\hat K_{i}$ & $\hat\alpha$,
    $\hat\beta$, $\hat\kappa$, $\hat\lambda$ & Mean loss ($\alpha$, $\beta$) & Mean
    loss ($\kappa$, $\lambda$) \\ \hline
    Moments/MLE & Moments & Moments & $10684999.5$ & $202115.0$ \\
    Moments/MLE & MLE & MLE & $6573128.9$ & $397729.0$ \\
    Bayes & Moments & Moments & $11077285.5$ & $\underline{200906.9}$ \\
    Bayes & Moments & MLE & $22138561.9$ & $236938.5$ \\
    Bayes & MLE & Moments & $15936758.9$ & $917774.9$ \\
    Bayes & MLE & MLE & $7831878.6$ & $750779.0$ \\
    Bayes & Bayes & Moments & $14402261.7$ & $968651.6$ \\
    \textbf{Bayes} & \textbf{Bayes} & \textbf{MLE} & $\underline{3042068.4}$ & $1895868.2$
  \end{tabular}
  \caption{Average loss function values from the simulation study
    ($10000$ simulations).}
  \label{tab:simStudy}
\end{table}

\subsection{Parameter estimation for real data}\label{sec:appData}

We are investigating measurements originating from
$m_{\operatorname{dev}} = 15$ different SRAM-PUF devices, each of them
with $c = 2^{16}$ SRAM cells. Note that our given
measurements were carried out on ASICs that have been manufactured in
TSMC $65 \si{nm}$ CMOS technology within a European multi-project
wafer run. The ASIC has been designed within the
UNIQUE\footnote{\url{http://www.unique-project.eu}} research project.

For each device, we have $340$
evaluations---however, the first $50$ measurements are discarded
because they were conducted during an aging process, which slightly
changed the behavior of the SRAM-PUFs. Afterwards, during the
remaining $m = 290$ measurements, the devices are stable again,
meaning that we will focus our analysis on these measurements.

For the parameter estimation, we will follow the results of the
simulation study, meaning that we will estimate the flipping
probabilities $p_{ij}$ and the parameters $\delta_{i}$ and $K_{i}$ by
the Bayesian approaches described above, and the four parameters
$\alpha$, $\beta$, $\kappa$ and $\lambda$ by maximum likelihood estimation.

This results in the following estimates:
\[ \hat\alpha = 9378.324\quad \hat\beta = 81409.79,\quad \hat\kappa = 7166.669,\quad
\hat\lambda = 3965.296. \]

In Figure~\ref{fig:paramDens}, we plotted histograms for the
(respectively) estimated $\delta_{i}$ and $K_{i}$, as well as the
densities of the proposed probability distributions.

\begin{figure}[ht]
  \centering
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/density-delta.pdf}
  \end{minipage}\hfill
  \begin{minipage}[ht]{0.49\linewidth}
    \includegraphics[width=1\linewidth]{figures/density-K.pdf}
  \end{minipage}
  \caption{Histograms and estimated densities for $\delta_i$ and $K_i$.}
  \label{fig:paramDens}
\end{figure}

Furthermore, we may compute the expected values for the parameters
$\delta$ and $K$, and construct credible intervals. From the estimated
parameters we obtain
\[ \mathbb{E} \delta = \frac{1}{2} \frac{\alpha}{\alpha + \beta}
\approx 0.05165,\quad \mathbb{E} K = \frac{\kappa}{\lambda} \approx 1.8073,  \]
and empirical $95\%$ credible intervals
\[ \delta\in [0.05066, 0.05264],\quad K\in [1.76574, 1.84943].  \]

\subsection{Error correction and reduction}\label{sec:corrAndRed}

In practice, we are interested in stabilizing the responses of a PUF
such that it can be used for system security related aspects like
constructing a challenge-response system without the need of storing a
master key. We want to present two general approaches (concentrating
on error correction and error reduction) for stabilizing these
responses and use our proposed model in order to evaluate their
effectiveness in specific examples.

Based on the parameters $\alpha$, $\beta$, $\kappa$ and $\lambda$
estimated in the previous section, we may investigate the
posterior-predictive distribution for the flipping probabilities. Its
density function is given by
\begin{equation*}
f(p\mid \hat\alpha,\hat\beta,\hat\kappa,\hat\lambda, \mathbf{x}) = \int_{\delta} \int_{K}
f_{\Be}(p\mid \delta, K)\cdot f_{\Be}(\delta\mid \hat\alpha,\hat\beta)\cdot f_{\Ga}(K\mid \hat\kappa,\hat\lambda)~dK~d\delta
\end{equation*}
for $p\in (0,\sfrac{1}{2})$ and $0$ otherwise. By simulation of a
sample of size $100000$, we obtain $\overline{p} = 0.05187$ as an
approximation for the expected value of the posterior-predictive
distribution under the empirically estimated parameters $\hat\alpha$,
$\hat\beta$, $\hat\kappa$ and $\hat\lambda$ from above.

We are interested in the number of errors in an $\ell$-bit SRAM-PUF
response, where the bit-wise error probabilities are distributed
according to the posterior-predictive distribution from above. For
fixed error probabilities, the quantity of errors follows a
generalized binomial distribution. Due to Theorem~\ref{thm:compound},
the resulting compound distribution is a $\Bi(\ell,
\overline{p})$-distribution. This distribution can now be used to
compute the probability that a given error correction mechanism fails.

\begin{example}
  Assume that an SRAM-PUF is embedded within a construction which
  allows the correction of up to $239$ bits in responses of length
  $\ell = 1953$. Following the model above, the expected number of errors is
  $101.3101$, and the probability that the PUF does not work properly
  (i.e.\ the probability that more than $239$ errors occur) is
  negligibly small (less than $10^{-20}$).
\end{example}

Instead of designing powerful mechanisms for error correction which are able to
compensate for the noise an SRAM-PUF produces, another approach is to
``ignore'' SRAM-cells for which a high flipping probability is
known or has been estimated. Assuming that we are investigating $\ell$
bit SRAM-PUF responses, it is an interesting question how the removal
of $r\ll \ell$ unstable bits influences the noise behavior.

In general, a good measure to judge the effect of ignoring the $r$
``worst'' bits is the average flipping probability of the
remaining cells, that is if $p_{1}$, \ldots, $p_{\ell}$ are the
respective flipping probabilities, and $p_{(1)}\leq \cdots \leq
p_{(\ell)}$ denote the related ordered probabilities, then we are
interested in $\frac{1}{\ell - r}\cdot \sum_{j=1}^{\ell - r}
p_{(j)}$. The assumption that these probabilities are realizations of
identical and independently distributed random variables leads us to
\emph{order statistics}.

\begin{definition}[Order statistics]
  Let $X_{1}$, \ldots, $X_{n}$ be identical and independently
  distributed random variables with respect to some distribution
  $X$. Then the \emph{ordered} random variables $X_{(k)}$ with
  $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ are called
  $k$-th smallest \emph{order statistic} of size $n$ with respect to $X$.
\end{definition}

The following theorem states a central result from the theory of order
statistics, a proof can be found in \cite{orderStatLit}.

\begin{theorem}[Order statistics and the beta distribution]
  The density function of the $k$-th smallest order statistic
  $U_{(k)}$ of size $n$ with respect to the uniform distribution on
  the interval $[0,1]$ is
  \[ f_{U_{(k)}}(u) = \begin{cases} \frac{n!}{(k-1)!\,
        (n-k)!}\cdot u^{k-1}\cdot (1-u)^{n-k} & \text{for } u\in
      (0,1),\\ 0 & \text{else,} \end{cases}  \]
  which is the density of a beta distribution with parameters $k$ and
  $n-k+1$. Therefore, we have $U_{(k)}\sim \Be(k, n-k+1)$.
\end{theorem}

By the technique of \emph{Probability Integral Transform}, this result
may be used to express the density function of an arbitrary
continuous random variable $X$ with distribution function $F_{X}$ and
density function $f_{X}$: in this case, we obtain
\[ f_{X_{(k)}}(x) = \frac{n!}{(k-1)!\, (n-k)!}\cdot
\left[F_{X}(x)\right]^{k-1}\cdot \left[1 - F_{X}(x)\right]^{n-k}\cdot
f_{X}(x)   \]
for the density function of the $k$-th smallest order statistic of
size $n$ with respect to the distribution of $X$.

\begin{remark}
  Note that as for $1\leq j < k \leq n$ the relation $X_{(j)}\leq
  X_{(k)}$ holds, the order statistics are \emph{not} independently
  distributed any more. This means that if we would like
  to compute some compound distribution of, for example, the
  generalized binomial distribution and these order statistics with
  respect to a (scaled) beta distribution (which could be used
  to predict the probability that an $\ell$-bit PUF response with the
  $r$ most unstable bits removed is still too noisy for correction),
  we would have to consider the respective joint densities (which can
  be found in \cite{orderStatLit}) when integrating over the respective
  parameters. As we focus on the scaled beta distribution, the arising
  integrals cannot be computed analytically (mainly because of the
  occurring products of incomplete beta functions)---however, by
  simulating the procedure, i.e.\ generating $N$ beta-distributed
  samples of size $\ell$ and removing the $r$ largest values, we may
  still give good approximations for some interesting parameters
  within specific examples.
\end{remark}

In the setting above, we are interested in the expected flipping
probability, averaged over all remaining SRAM-cells, that is
\[ \mathbb{E}\left[\frac{1}{\ell - r}\cdot \sum_{j=1}^{\ell - r}
  \pi_{(j)}\right] = \frac{1}{\ell -r}\cdot \sum_{j=1}^{\ell - r} \mathbb{E}
\pi_{(j)}.  \]
Thus, we primarily want to compute the expected value of the order
statistics. For the same reasons as mentioned in the remark above,
this is not possible analytically for arbitrary
distributions. However, for a special case of the scaled beta
distribution it is actually possible, and that is for $\Be(\alpha,
\beta)$ with $\alpha = 1$ or $\beta = 1$.

\begin{proposition}\label{prop:simplifiedOrderStat}
  Let $\pi_{1}$, \ldots, $\pi_{n}$ be independent and identically
  distributed random variables following a $\Be_{[a,b]}(\alpha,
  1)$-distribution. The expected value of the $k$-th smallest order
  statistic of size $n$ is then given by
  \[ \mathbb{E} \pi_{(k)} = a + (b-a)\cdot \frac{B(k + \sfrac{1}{\alpha}, n - k
    + 1)}{B(k, n-k+1)}.  \]
\end{proposition}
\begin{proof}
  As scaled beta distributed random variables are affine-linearly
  transformed beta distributed random variables (which also holds for
  the related order statistics), and as the expectation is a linear
  operator, we may concentrate on the case $\pi_{1}$, \ldots,
  $\pi_{n}\stackrel{\text{iid}}{\sim} \Be(\alpha, 1)$.

  The density function of these random variables is given by $f(x) =
  \mathds{1}_{(0,1)}(x)\cdot\alpha \cdot x^{\alpha-1}$, and thus the
  distribution function has the shape
  \[ F(x) = \begin{cases} 0 & \text{for } x\leq 0, \\
    x^{\alpha} & \text{for } 0 < x < 1, \\
    1 & \text{for } x\geq 1.
  \end{cases}  \]
  As mentioned above, by \emph{Probability Integral Transform}, the
  density of the $k$-th smallest order statistic of size $n$ with
  respect to the $\Be(\alpha, 1)$-distribution has the form
  \begin{align*}
    f_{\pi_{(k)}}(x) & = k \binom{n}{k}\cdot [F(x)]^{k-1}\cdot [1- F(x)]^{n-k}\cdot f(x)\\
    & = \mathds{1}_{(0,1)}(x)\cdot \alpha k \binom{n}{k} \cdot
    x^{\alpha k - 1}\cdot (1 - x^{\alpha})^{n-k}.
  \end{align*}
  The expected value thus reads
  \begin{equation*}
    \mathbb{E} \pi_{(k)}  = \alpha k \binom{n}{k}\cdot \int_{0}^{1} x^{\alpha
    k} (1-x^{\alpha})^{n-k}~dx,
  \end{equation*}
  which, after a change of variables $t = x^{\alpha}$, becomes
  \begin{equation*}
  \mathbb{E} \pi_{(k)}  = \frac{\int_{0}^{1} t^{k + \sfrac{1}{\alpha} - 1}
    (1-t)^{n-k}~dt}{B(k,n-k+1)}
   = \frac{B(k+\sfrac{1}{\alpha}, n-k+1)}{B(k,n-k+1)}.
  \end{equation*}
  Finally, by the transformation $\pi_{(k)}\mapsto a + (b-a)\cdot \pi_{(k)}$, the
  statement is proven.
\end{proof}
\begin{remark}
  An analogous statement holds for the $\Be_{[a, b]}(1,
  \beta)$-distribution. This follows directly from the fact that if
  $X$ follows a $\Be_{[a,b]}(\alpha, \beta)$-distribution, then the
  linearly transformed variable $Y = a+b - X$ follows a
  $\Be_{[a,b]}(\beta, \alpha)$-distribution.
\end{remark}
\begin{example}
  We want to investigate responses of length $n = 16$ of an
  SRAM-PUF embedded within an error correction scheme such that up to
  $3$ errors can be corrected. For the sake of simplicity, we will
  assume that the cell-wise error probabilities are distributed
  according to a $\Be_{[0,\sfrac{1}{2}]}(\sfrac{1}{9}, 1)$-distribution (such that the
  mean error rate is $\frac{1}{2}\cdot
  \frac{\sfrac{1}{9}}{\sfrac{1}{9} + 1 } = 0.05$). Note that by
  ``ignoring'' bits of the PUF responses, also the error correction
  scheme gets weakened: for every $2$ ignored bits, the correction
  capacity reduces by $1$. Table~\ref{tab:OS1} contains the expected
  values of the respective order statistics (computed along the lines of
  Proposition~\ref{prop:simplifiedOrderStat}).
  \begin{table}[ht]
    \centering
  \scalebox{0.89}{\begin{tabular}[ht]{c||cccccccc}
      $k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline
      $\mathbb{E} P_{(k)}$ & $2.45\cdot 10^{-7}$ & $2.45\cdot 10^{-6}$ &
      $1.35\cdot 10^{-5}$ & $5.38\cdot 10^{-5}$ & $0.00017$ & $0.00048$ &
      $0.00122$ & $0.00279$ \\ \hline \hline
      $k$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ & $15$ & $16$ \\
      \hline
      $\mathbb{E} P_{(k)}$ & $0.00594$ & $0.01189$ & $0.02260$ &
      $0.04110$ &  $0.07193$ & $0.12173$ & $0.2$ &
      $0.32$
    \end{tabular}}
    \caption{Order statistics -- expected values (simplified model).}
    \label{tab:OS1}
  \end{table}
  Furthermore, by simulation we are able to estimate the probability
  that a system failure (i.e.\ more errors than the correction scheme
  can handle) occurs when ignoring the $r$ most unstable cells. The
  results of this simulation (with $100000$ simulated PUFs) can be
  found in Table~\ref{tab:OS2}.
  \begin{table}[ht]
    \centering
    \scalebox{0.9}{\begin{tabular}[ht]{c||ccccccc}
      Ignored cells & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\ \hline
      Correction capacity & $3$ & $3$ & $2$ & $2$ & $1$ & $1$ & $0$ \\
      avg.\ sys.\ failure prob. & $0.00704$ & $0.00183$ &
      $0.00462$ & $0.00134$ & $0.00579$ & $0.00201$ & $0.02203$ \\
      max.\ sys.\ failure prob. & $0.49622$ & $0.35504$ &  $0.49410$ &
      $0.34375$ &  $0.52352$ &  $0.38755$ &  $0.67622$
    \end{tabular}}
    \caption{System failure probabilities (simplified model).}
    \label{tab:OS2}
  \end{table}
\end{example}

However, practically, the simplified model is not as precise as the
statistical model developed in the previous section. Therefore,
we investigate a similar example based on this more
sophisticated model from the previous section. In this case, all
parameters will have to be estimated by simulation.

\begin{example}
  We use the posterior-predictive distribution discussed at the
  beginning of this section to obtain flipping probabilities for the
  simulated SRAM-PUFs with $512$ cells each. We have plotted the
  results of this simulation (with $100000$ simulated SRAM devices) in
  Figure~\ref{fig:OS3}. Remarkably, the mean error rate can be
  reduced very quickly from slightly above $0.05$ to about $0.026$ by
  ignoring the $50$ most unstable bits per device (about $10\%$
  information loss). This demonstrates that the exclusion of unstable
  bits is a viable and practically relevant approach to increase the
  stability of an SRAM-PUF.

  \begin{figure}[ht]
    \centering
    \begin{minipage}[ht]{0.49\linewidth}
      \includegraphics[width=1\linewidth]{figures/OS-errorRates.pdf}
    \end{minipage}\hfill
    \begin{minipage}[ht]{0.49\linewidth}
      \includegraphics[width=1\linewidth]{figures/OS-expMax.pdf}
    \end{minipage}
    \caption{Order statistics and mean error rate.}
    \label{fig:OS3}
  \end{figure}
\end{example}

\section{Conclusion}

The design of error correcting mechanisms capable of correcting the
noise emitted by SRAM-PUFs requires a precise statistical analysis. In
this paper, we presented the framework for such an analysis by proposing
a statistical model which captures the noise behavior of a collection
of SRAM-PUFs (cf.\ Section~\ref{sec:model} and
Section~\ref{sec:appData}). In practice, such a model allows for a
certain flexibility when designing and conducting statistical tests in
the context of quality assurance---which is very important, as these
tests are very expensive in general.

The second tool developed in this paper in order to ascertain precise
predictions for the number of errors in SRAM-PUF responses is the
generalized binomial distribution (cf.\ Section~\ref{sec:gbin}). This
distribution, in combination with the posterior-predictive
distribution for the error probabilities obtained from our given
measurements, permits an evaluation and the design of sufficiently
strong error correction mechanisms (cf.\ Section~\ref{sec:corrAndRed}). Finally, we showed that by ignoring the
most unstable parts of SRAM-PUF responses, the mean error rate could
be reduced significantly. Thus, weaker and simpler error correction
mechanisms could be used.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


It is important to note that majority of homes have lawns. A number of people do not get time to maintain the lawns. It is good to consider that a number may have a busy schedule. This leaves a little time to check on the maintenance of the lawns. It is therefore important to consider hiring lawn care services. The lawn care services are available in a wide range. It is important to note that the services include weeding, feeding, aerating and mowing the lawns. Choosing the right lawn services may be difficult. It is important to note the reason that there are majority of companies offering lawn care services. It is important to note that the tips stated below highlights the tips to consider while hiring the best lawn care services. Find the best services for lawn care Toronto or garden design Toronto.
It is advisable to check on the rate of professionalism and experience. It is good to note the importance of hiring professional lawn attenders. The professionals have knowledge on the best mechanisms of eliminating weeds. It is important to note that they have knowledge of a number of chemicals used in order to control pests. They have knowledge about the right seasons for planting specific plants. It is important to note that they also have knowledge about the type of soils and their ability to maintain crops.
It is important to note on checking on the customer services before choosing. One should look for a company that is readily available. It is important to note that the lawn services should be provided from time to time. Select the services that does not require you to follow up inquiring on unattended lawn. It is important to note that some of the lawn companies are not organized. Go for a company that guarantees a timely service delivery. It is good to consider that customers should be provided with a date schedule on when the lawns are attended. It is important to note the essence of communication therefore one should choose a lawn company he can communicate to at any time.
It is important to check on the machinery of the company offering lawn care services. Hiring a lawn care company without proper machinery assures improper services. It is good to consider that one's lawn will be left unattended sometimes. It is important to note the reason that the company may lack machines that are good enough to perform different duties. A company in possession of all the required machines guarantees complete and fast services.
It is important for one to note on checking on the reputation of the company he is hiring. It is advisable to consider inquiring from the neighbors about the companies they have worked with. It is advisable to inquire about the service delivery of the company. Visit the websites to check on the customer reviews. When there are a lot of negative reviews from the customers about the company. Take caution because most of the reviews are genuine.


This archive was pointed out to the members of a discussion group. Basically, is is a bunch of junk mail. But WHAT junkmail... a Pantone catalog, a flight control module from Macdonnel Aircraft (I think), computer catalogs, an electron beam recorder, assorted scientific data recorders... my brother like the Heathkit catalog.


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Getting the gang back together proved so much fun last year, why not do it all again?
That's the spirit! Netflix announced today that Wet Hot American Summer: Ten Years Later will come to its streaming screens much sooner than later — with an eight-episode order set to premiere in 2017.
Netflix teased it perfectly, using a scene from the end of the original 2001 film in which Bradley Cooper's character suggests the 10-year reunion. Roll the clip!
Of course, the movie itself imagined what that reunion would look like in a bonus post-credits scene.
And the First Day at Camp eight-episode of prequel that debuted on Netflix in 2015 hilariously had fun with how much the actors and actresses had aged, even though they were now playing so much younger. Michael Showalter especially? Showalter and Michael Ian Black already turned today's announcement into a funny joke in itself.
David Wain and Showalter are executive producing Ten Years Later alongside Howard Bernstein, Jonathan Stern and Peter Principato.


1. 81% of people perform some type of online research before making a large purchase.
The importance of having positive writing on the web about your product or service can't be overstated. Creating content about your business will allow you to take control of your online space, show potential customers what you want them to see, overshadow negative reviews, and keep customers from bouncing. The more web content you have out there the more likely potential customers are to come across your writing. Consistently creating online content will build the availability of your writing on search engines. Content Cucumber is a content writing business that produces a blog or piece of web copy for you every business day -- hundreds of potential blogs every year!
2. 60% of marketers create at least one piece of content each day.
And there's a reason for this. Content marketing is the most effective way to get your business to show up organically on search engines. 71.33% of searches result in an organic click — It's no surprise that ads are less attractive to searchers. If you want your business to show up organically when relevant topics are searched, you need web content. It's important to note that advancements in technology have allowed search engines to understand what content is actually informative and well-written. You need both quantity and quality to show up organically, so invest in content writers that produce quality writing on a consistent basis.
3. Content marketing costs 62% less than traditional marketing and generates about 3 times as many leads.
This goes back to what was said about the majority of searches resulting in organic clicks. Search engine ads do drive conversions, but they just can't stack up to the attractiveness of an organic result. Web content is a long-term investment when compared to PPC ads, but it is critical to generating leads. The good thing is that the price per lead is astronomically lower with SEO marketing. Some of the highest-quality content writers will produce content at less than $25 per article.
4. 92% of B2B marketers leverage LinkedIN over all other social platforms.
Social media is undoubtedly important when it comes to marketing. If you are B2B marketing, LinkedIN is the platform for you. But no matter what the platform you are using, you won't get anywhere without making posts. Your posts also have to be well written and informative. Hiring a professional content writer to create posts is a great way to make sure that you present yourself and your business as innovative and professional. And because posting content of all kinds is very important, working with versatile, talented writers makes building your online presence efficient, both in terms of time and cost.
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Consider this for a moment: In all your schooling did anyone ever teach you how to study?
Today people are graduating school unable to read or write at a level adequate to holding a job or dealing with life. Illiteracy is a huge problem. It's not that subjects cannot be learned, it's that people often don't know how to learn them. Learning how to learn is the missing step in all education.
This technology provides an understanding of the basics of learning and supplies exact ways to overcome all the obstacles one can encounter during study.
Study technology is not speed reading or memory tricks. These have not been proven to increase literacy or improve one's ability to comprehend what was studied. Study technology shows how one should study in order to comprehend a subject so they can apply it.
Contained herein is only a small portion of the entire body of the study technology developed by Mr. Hubbard; however, this brief overview contains fundamentals which you can use to study more effectively. With this technology any subject can be learned by anyone.


Anxiety can be an extremely helpful thing. In fact, I would say for all of us to survive we need anxiety. There is something that prevents us from walking into the street when a car is coming. There is some reason why our ancestors were not eaten by saber tooth tigers. They were the ones who had enough anxiety that caused them to go and sleep in a safe place at night. Obviously, the people who didn´t are not around anymore. So anxiety can be a very helpful thing and it is important to remember that anxiety can be a helpful thing when it comes to our children because there is a sweet spot of anxiety. There is a certain amount of anxiety that will motivate them to study. But then if there is too much anxiety that studying becomes overwhelming, then they will stop. So there is a certain zone, and for every child this zone is different, where anxiety can be helpful. And we call that the sweet spot. Ideally, as a parent what you want to do is to try to keep your child in that sweet spot where they are motivated but not overwhelmed.


text:
MTV Networks, Film production companies of the United States, Companies established in 1995,
Nickelodeon films
Article Feedback 5
Movie Studios
Viacom subsidiaries
Nickelodeon Movies
Old logo
Nickelodeon Movies is the motion picture production arm of children's cable channel Nickelodeon, originally launched in 1995. Its first film was Harriet the Spy. It has produced family features and films based on Nickelodeon programs, as well as other adaptations and original projects. The films are released by Viacom division Paramount Pictures. It currently has 24 films.
On August 21, 2006, Nickelodeon Movies and MTV Films became full joint venture labels of the Paramount Motion Pictures Group.[1] This studio is noted to be one of the only few movie studios, based on TV cable channels.
With the exception of Angus, Thongs and Perfect Snogging, Fun Size, Teenage Mutant Ninja Turtles (2014), Teenage Mutant Ninja Turtles: Out of the Shadows, all of the films are family focused.
Its longtime partner is Klasky Csupo.
Wild Thornberrys Movie variant
Spongebob Squarepants Movie variant
logo with KC
Film list Edit
Harriet the Spy July 10, 1996 Paramount / Rastar $13 million $26,570,048
Good Burger July 25, 1997 Paramount / Tollin/Robbins Productions $9 million $23,712,993
The Rugrats Movie November 20, 1998 Paramount / Klasky Csupo $24 million $140,894,675
Snow Day February 11, 2000 Paramount / C.O.R.E. Digital Pictures $13 million $62,464,731
Rugrats in Paris: The Movie November 17, 2000 Paramount / Klasky Csupo $30 million $103,291,131
Jimmy Neutron: Boy Genius December 21, 2001 Paramount / O Entertainment $30 million $102,992,536
Clockstoppers March 29, 2002 Paramount / Valhalla Motion Pictures $26 million $38,793,283
Hey Arnold!: The Movie June 28, 2002 Paramount / Snee-Oosh $3–4 million $15,249,308
The Wild Thornberrys Movie December 20, 2002 Paramount / Klasky Csupo $35 million $60,694,737
Rugrats Go Wild June 13, 2003 $25 million $55,405,466
The SpongeBob SquarePants Movie November 19, 2004 Paramount / United Plankton Pictures $30 million $140,161,792
Lemony Snicket's A Series of Unfortunate Events December 17, 2004 Paramount / DreamWorks $142 million $209,073,645
Mad Hot Ballroom May 13, 2005 Paramount Classics / Just One Productions $500,000 $8,117,961
Yours, Mine and Ours November 23, 2005 Paramount / Metro-Goldwyn-Mayer / Columbia $45 million $72,028,752
Nacho Libre June 16, 2006 Paramount / Black & White Productions $35 million $99,255,460
Barnyard August 4, 2006 Paramount / O Entertainment $52 million $108,000,000
Charlotte's Web December 15, 2006 Paramount / Walden Media / The K Entertainment Company $85 million $144,877,632
The Spiderwick Chronicles February 14, 2008 Paramount / The Kennedy/Marshall Company $90 million $162,899,667
Hotel for Dogs January 16, 2009 DreamWorks (under Paramount Pictures) / Cold Spring Pictures / Donners' Company / Montecito Picture Company $75 million $117,000,198
Angus, Thongs and Perfect Snogging July 25, 2008 (UK)
March 12, 2009 (US) Paramount $1 million $13,835,569
Imagine That June 12, 2009 Paramount / di Bonaventura Pictures $55 million $22,985,194
The Last Airbender July 1, 2010 Paramount / Blinding Edge Pictures / The Kennedy/Marshall Company $150 million $319,713,881
Rango March 4, 2011 Paramount / GK Films / Blind Wink / Industrial Light and Magic $135 million $245,375,374
The Adventures of Tintin December 21, 2011 Paramount / Columbia / Amblin Entertainment / The Kennedy/Marshall Company / WingNut Films $135 million $372,623,016
February 6, 2015 Paramount / Paramount Animation / Nickelodeon Animation Studio / United Plankton Pictures $74 million $323,436,538
Teenage Mutant Ninja Turtles: Out of the Shadows June 3, 2016 Paramount / Platinum Dunes / Heavy Metal / Alibaba Pictures $135 million $245,623,848
Monster Trucks January 13, 2017 Paramount / Paramount Animation / Disruption Entertainment $125 million $63,721,721
Upcoming projects Edit
An original animated feature produced by Paramount Animation and Nickelodeon Movies in association with Ilion Animation Studios titled Wonder Park is scheduled to be released on March 15, 2019. The same year, a television series based on it will also debut on Nickelodeon.
On October 23, 2017, Paramount Pictures announced a live-action film adaptation of the Nick Jr. series Dora the Explorer. Produced by Paramount Players, it is expected to be released on August 2, 2019. This will mark the first film to be based on a Nick Jr. series.
On November 13, 2017, it was announced that a film adaptation and reboot of Are You Afraid of the Dark? was in the works at Paramount Players, with a release date set for October 11, 2019. It writer Gary Dauberman will be writing the screenplay, and Matt Kaplan will be producing.
A sequel to The SpongeBob Movie: Sponge Out of Water, titled The SpongeBob Movie: It's a Wonderful Sponge, will be released on July 17, 2020, with former writer Tim Hill co-writing and directing.
On July 16, 2018, Nickelodeon Movies announced a reboot of the Rugrats film series, with a live-action/CGI hybrid film currently in development. Produced by Paramount Players and written by David A. Goodman, it is scheduled to be released on November 13, 2020.
Harriet the Spy (1996)
Good Burger (1997)
The Rugrats Movie (1998)
Snow Day (2000)
Rugrats in Paris: The Movie (2000)
Jimmy Neutron: Boy Genius (2001)
Clockstoppers (2002)
Hey Arnold!: The Movie (2002)
The Wild Thornberrys Movie (2002)
Rugrats Go Wild (2003)
The SpongeBob SquarePants Movie (2004)
Lemony Snicket's A Series of Unfortunate Events (2004)
Mad Hot Ballroom (2005)
Yours, Mine & Ours (2005)
Nacho Libre (2006)
Barnyard (2006)
Charlotte's Web (2006)
The Spiderwick Chronicles (2008)
Angus, Thongs and Perfect Snogging (2008)
Hotel for Dogs (2009)
Imagine That (2009)
The Last Airbender (2010)
Rango (2011)
The Adventures of Tintin (2011)
Fun Size (2012)
Teenage Mutant Ninja Turtles (2014)
Teenage Mutant Ninja Turtles: Out of the Shadows (2016)
Monster Trucks (2016)
Rugrats Films The Rugrats Movie (1998), Rugrats in Paris: The Movie (2000), Rugrats Go Wild (2003)
SpongeBob Films The SpongeBob SquarePants Movie (2004), The SpongeBob Movie: Sponge Out of Water (2015), SpongeBob SquarePants 3 (2019)
Teenage Mutant Ninja Turtles Teenage Mutant Ninja Turtles (2014), Teenage Mutant Ninja Turtles: Out of the Shadows (2016)
Animated Films Jimmy Neutron: Boy Genius (2001), Hey Arnold!: The Movie (2002), The Wild Thornberrys Movie (2002), Barnyard (2006), Rango (2011), The Adventures of Tintin (2011)
Live Action Films Harriet the Spy (1996), Good Burger (1997), Snow Day (2000), Clockstoppers (2002), Lemony Snicket's A Series of Unfortunate Events (2004), Mad Hot Ballroom (2005), Your's Mine and Ours (2005), Nacho Libre (2006), Charlotte's Web (2006), The Spiderwick Chronicles (2008), Angus, Thongs and Perfect Snogging (2008), Hotel for Dogs (2009), Imagine That (2009), The Last Airbender (2010), Monster Trucks (2016)
↑ http://goliath.ecnext.com/coms2/gi_0199-5687434/MTV-and-Nickelodeon-to-Take.html
↑ /http://www.cinemablend.com/new/SpongeBob-SquarePants-Coming-Back-Theaters-2014-29689.html
↑ /http://www.hollywoodreporter.com/news/viacom-paramount-release-spongebob-movie-late-2014-philippe-dauman295695
↑ http://durancemagazine.org/2012/02/paramount-announces-that-a-second-spongebob-squarepants-movie-will-come-in-2014/
Nickelodeon Movies portal
Nickelodeon Movies website
Nickelodeon Movies at Nick.com
Nickelodeon Movies at the Internet Movie Database
Nacho Libre/Credits
Retrieved from "https://movies.fandom.com/wiki/Nickelodeon_Movies?oldid=161859"
MTV Networks
Film production companies of the United States

redpajama_set_name: RedPajamaCommonCrawl


When you have a mass tort lawyers – 3m earplugs issue, there are many benefits to choosing a local mass tort lawyers – 3m earplugs lawyer. In addition to supporting a member of your Illinois City community, you are also choosing a lawyer that is more convenient to travel to, being local to Illinois City, than if you were to choose a lawyer outside of the Illinois City area.
By hiring a reputable Illinois City mass tort lawyers – 3m earplugs lawyer, you can ensure that your case will be handled in the most professional manner possible. And finding a good, experienced local Illinois City mass tort lawyers – 3m earplugs lawyer will give you the comfort of knowing that if you should have another mass tort lawyers – 3m earplugs issue in the future, your lawyer is right around the corner.


I have come to the conclusion, not an original one, that this is THE biggest, most important tool you have, if you want to be a writer. It really almost comes down to the watershed between the serious writer (who wants to be published and keep selling, or who just wants to be serious) and the wannabe. If you write every day, you might still not be particularly serious, just habitual. However, if you don't write every day, don't ask yourself any other questions about why your writing career isn't happening.
It's probably good if you can write part of a story every day, but if you are writing anything daily… blogs, journaling, information, anything that involves coherent sentences being formed and word choice being made, then you are doing better than most people who say 'I want to be a writer.' Give yourself a pat on the back and keep at it, you have crossed the watershed!


Before to begin the paver project, clean the area and outline the area to be paved. You can lay out your paved area using a hose. Keep in mind to prevent pooling. The area needs to have a slope.
Make a plan and order your materials.
Learn how to plan and determine the area for a paver project.


At the heart of the Vexin Normand and the beautiful valley of the Epte, in a quiet hamlet, Claude, and Bernard welcome you in this cosy and charming studio. On their property, near the manor house, you will enjoy the beautiful landscaped garden. Entry and private space on the property. At ground level, the studio has a kitchen-dining area, a relaxation area and a very comfortable bedding (2 single beds can be installed in 1 or 2 beds).


Can you really receive healing energy from singing bowls? I'm sure you've seen them in your yoga class, or maybe you've even got your own singing bowl. They're becoming more and more mainstream in Western culture, but what's all the fuss? Can these pleasant sounds really help heal us from the inside out?
I just love it when a fashion trend comes along that I'm 100% on board with and love! Bright and beautiful patterns are popping up everywhere in fashion this year and I. AM. EXCITED! I'm going to share some of my favorite Etsy fashion artists with you and some of the new Zenned Out patterned jewelry pieces, as well as some tips for rocking all of these patterns.


text:
Власний розклад матриці (спектральний розклад) — у вигляді діагональної матриці з власними значеннями та матриці з власними векторами
 Сингулярний розклад — у вигляді діагональної матриці та двох унітарних матриць
 Полярний розклад — у вигляді добутку унітарної матриці та невід'ємної ермітової матриці
 QR-розклад — у вигляді добутку унітарної та трикутної матриці
 LU-розклад — у вигляді добутку нижньої трикутної матриці та верхньої трикутної матриці
 Розклад Холецького — представлення симетричної додатноозначеної матриці у вигляді добутку трикутної матриці з додатними елементами на діагоналі на себе транспоновану.
 Розклад Шура — у вигляді унітарної та трикутної матриці

Джерела

Теорія матриць

redpajama_set_name: RedPajamaWikipedia


Don't let careless mistakes leave you in dark. Call Your Chandler Electricians to get your Chandler Breaker Service and repair. We are so confident in our services that if any complication arrives from your breaker box till 3 years after we have put in our installation services, we will repair your system for free. Our trained and professional electricians are committed to provide 100% customer satisfaction.
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You no longer need to wait for weeks only for an electrician to show up at your house, or to fetch up additional electrical equipment's that add up to one giant services bill. Your Chandler Electricians arrive with their truck stocked with premium electrical breaker box parts to ensure efficiency of your Chandler breaker box installation service and repair. Moreover as an added bonus, if we arrive late for our scheduled appointment, we will gladly perform the after sale or repair service for free.
The Chandler breaker box service and repair is completed by our highly trained certified professional electricians because we understand the importance of professionalism in doing such a critical and important job that if performed improperly can be o dangerous. We perform exceptional services all thanks to the excellence of our technical skills. Our polite and friendly conduct, as well as professionalism ensures that we have a long term relationship with our client not just that of doing business.


Meray pass 160 GB ki HDD thi jo k awaz dena shoru ho gai he jiska mtlb he wo near to dead he. So, meray bhai ki aik 512GB wali hard disk pari thi wo me ne lga di lakin us me windows 7 installed he jiska mjhay koi experience nahi he. aur us me partitionning bhi ghlat ki hoi he. Koi partition chota he aur koi bra he. Aur aik aur bat me ne HDD check ki he us me windows 7 aur activator sb kuch mojod he. Meray kuch swalat hen koi bhai jawab de de.
Me windows 7 install krna chahta hon. Koi bhai mjhay sikha de ya iska koi tutorial pehle se ITD pr mojod he to bta de jis me activation ka triqa bhi mojod ho.
Koi bhai bta de k me HDD ko format kiay bghair kese partition ka size kam ya zidad kr skta hon ya kuch partitions ko merge kr skta hon?
Aur aik aur bat k is mege ya resize k doran kia mera data zaya ho jay ga??
There are 10 type of people in this world, Those who understand Binary and those who not.
Data Lost ho skta hai is se,..
app wo data kuch deir k liye windows wali drive me rkh lyn,,.
phr Partion magic se nai partitions banaye,..
pHr nai widnows 7 kr lyn,..
Bhut shukria. Bhai kia Partition magic portable windows 7 k liay mil skta he?
Tmam Members ka shukria. ITD pr win7 instllation ka koi tutorial nai he kia?
Thanks for help. Usman bhai ne bhut acha article likha he.
Kia me windows seven ka ghost bna skta hon kio k meray ilm k mutabiq ghost aik drive image hota he aur wo to kis bhi drive ka bnaya ja skta he. Kia ye ghost chale ga ya nahi?
Kia me XP aur win 7 ikathey aik hi pc pe chala skta hon?
All times are GMT +5. The time now is 01:04 PM.


This is the perfect time of year for retreating: after the stimulation of the holiday season, gathering to re-connect with our inner lives, and to reflect on the season.
This period of intensive practice is suitable for both beginning and experienced practitioners, and – as with all our retreats – will include systematic meditation instructions, guided Loving Kindness practices, Dharma Talks, individual practice discussions and a daily period of mindful yoga.
We will celebrate New Year's Eve with late evening sitting, together with a ceremony fitting the turning of the year, with the option of staying up till midnight.


Friday night I had the pleasure of sitting beside the Sainsbury support crew at the Asian Cup opener, they'd travelled all the way from WA. Trent Sainsbury's GrassRoots club is, Armadale FC from Perth's south eastern suburbs. It was Trent's 1st international in Australia and the crew couldn't have been prouder. we are proud to have Socceroo #574 as part of the GFP.


text:
#include <hxcpp.h>

#ifndef INCLUDED_openfl_display3D__Context3DCompareMode_Context3DCompareMode_Impl_
#include <openfl/display3D/_Context3DCompareMode/Context3DCompareMode_Impl_.h>
#endif
namespace openfl{
namespace display3D{
namespace _Context3DCompareMode{

Void Context3DCompareMode_Impl__obj::__construct()
{
	return null();
}

//Context3DCompareMode_Impl__obj::~Context3DCompareMode_Impl__obj() { }

Dynamic Context3DCompareMode_Impl__obj::__CreateEmpty() { return  new Context3DCompareMode_Impl__obj; }
hx::ObjectPtr< Context3DCompareMode_Impl__obj > Context3DCompareMode_Impl__obj::__new()
{  hx::ObjectPtr< Context3DCompareMode_Impl__obj > _result_ = new Context3DCompareMode_Impl__obj();
	_result_->__construct();
	return _result_;}

Dynamic Context3DCompareMode_Impl__obj::__Create(hx::DynamicArray inArgs)
{  hx::ObjectPtr< Context3DCompareMode_Impl__obj > _result_ = new Context3DCompareMode_Impl__obj();
	_result_->__construct();
	return _result_;}

int Context3DCompareMode_Impl__obj::ALWAYS;

int Context3DCompareMode_Impl__obj::EQUAL;

int Context3DCompareMode_Impl__obj::GREATER;

int Context3DCompareMode_Impl__obj::GREATER_EQUAL;

int Context3DCompareMode_Impl__obj::LESS;

int Context3DCompareMode_Impl__obj::LESS_EQUAL;

int Context3DCompareMode_Impl__obj::NEVER;

int Context3DCompareMode_Impl__obj::NOT_EQUAL;

int Context3DCompareMode_Impl__obj::_new( int a){
	HX_STACK_FRAME("openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_","_new",0xaeee8447,"openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_._new","openfl/display3D/Context3DCompareMode.hx",22,0x9103941b)
	HX_STACK_ARG(a,"a")
	HX_STACK_LINE(22)
	int tmp = a;		HX_STACK_VAR(tmp,"tmp");
	HX_STACK_LINE(22)
	return tmp;
}

STATIC_HX_DEFINE_DYNAMIC_FUNC1(Context3DCompareMode_Impl__obj,_new,return )

int Context3DCompareMode_Impl__obj::fromInt( int s){
	HX_STACK_FRAME("openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_","fromInt",0xc10b427f,"openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_.fromInt","openfl/display3D/Context3DCompareMode.hx",27,0x9103941b)
	HX_STACK_ARG(s,"s")
	HX_STACK_LINE(29)
	int tmp = s;		HX_STACK_VAR(tmp,"tmp");
	HX_STACK_LINE(29)
	return tmp;
}

STATIC_HX_DEFINE_DYNAMIC_FUNC1(Context3DCompareMode_Impl__obj,fromInt,return )

int Context3DCompareMode_Impl__obj::toInt( int this1){
	HX_STACK_FRAME("openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_","toInt",0x79c3588e,"openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_.toInt","openfl/display3D/Context3DCompareMode.hx",34,0x9103941b)
	HX_STACK_ARG(this1,"this1")
	HX_STACK_LINE(36)
	int tmp = this1;		HX_STACK_VAR(tmp,"tmp");
	HX_STACK_LINE(36)
	return tmp;
}

STATIC_HX_DEFINE_DYNAMIC_FUNC1(Context3DCompareMode_Impl__obj,toInt,return )

Context3DCompareMode_Impl__obj::Context3DCompareMode_Impl__obj()
{
}

bool Context3DCompareMode_Impl__obj::__GetStatic(const ::String &inName, Dynamic &outValue, hx::PropertyAccess inCallProp)
{
	switch(inName.length) {
	case 4:
		if (HX_FIELD_EQ(inName,"_new") ) { outValue = _new_dyn(); return true;  }
		break;
	case 5:
		if (HX_FIELD_EQ(inName,"toInt") ) { outValue = toInt_dyn(); return true;  }
		break;
	case 7:
		if (HX_FIELD_EQ(inName,"fromInt") ) { outValue = fromInt_dyn(); return true;  }
	}
	return false;
}

#if HXCPP_SCRIPTABLE
static hx::StorageInfo *sMemberStorageInfo = 0;
static hx::StaticInfo sStaticStorageInfo[] = {
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::ALWAYS,HX_HCSTRING("ALWAYS","\xcf","\xba","\x59","\xf9")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::EQUAL,HX_HCSTRING("EQUAL","\x94","\x07","\x71","\xf0")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::GREATER,HX_HCSTRING("GREATER","\xda","\x64","\x94","\x37")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::GREATER_EQUAL,HX_HCSTRING("GREATER_EQUAL","\x2f","\xd3","\x96","\xd6")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::LESS,HX_HCSTRING("LESS","\x59","\xdb","\x70","\x32")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::LESS_EQUAL,HX_HCSTRING("LESS_EQUAL","\xee","\xb2","\x55","\xe4")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::NEVER,HX_HCSTRING("NEVER","\x6c","\x76","\x1e","\x17")},
	{hx::fsInt,(void *) &Context3DCompareMode_Impl__obj::NOT_EQUAL,HX_HCSTRING("NOT_EQUAL","\x08","\x7c","\x82","\x7d")},
	{ hx::fsUnknown, 0, null()}
};
#endif

static void sMarkStatics(HX_MARK_PARAMS) {
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::__mClass,"__mClass");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::ALWAYS,"ALWAYS");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::EQUAL,"EQUAL");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::GREATER,"GREATER");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::GREATER_EQUAL,"GREATER_EQUAL");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::LESS,"LESS");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::LESS_EQUAL,"LESS_EQUAL");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::NEVER,"NEVER");
	HX_MARK_MEMBER_NAME(Context3DCompareMode_Impl__obj::NOT_EQUAL,"NOT_EQUAL");
};

#ifdef HXCPP_VISIT_ALLOCS
static void sVisitStatics(HX_VISIT_PARAMS) {
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::__mClass,"__mClass");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::ALWAYS,"ALWAYS");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::EQUAL,"EQUAL");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::GREATER,"GREATER");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::GREATER_EQUAL,"GREATER_EQUAL");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::LESS,"LESS");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::LESS_EQUAL,"LESS_EQUAL");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::NEVER,"NEVER");
	HX_VISIT_MEMBER_NAME(Context3DCompareMode_Impl__obj::NOT_EQUAL,"NOT_EQUAL");
};

#endif

hx::Class Context3DCompareMode_Impl__obj::__mClass;

static ::String sStaticFields[] = {
	HX_HCSTRING("ALWAYS","\xcf","\xba","\x59","\xf9"),
	HX_HCSTRING("EQUAL","\x94","\x07","\x71","\xf0"),
	HX_HCSTRING("GREATER","\xda","\x64","\x94","\x37"),
	HX_HCSTRING("GREATER_EQUAL","\x2f","\xd3","\x96","\xd6"),
	HX_HCSTRING("LESS","\x59","\xdb","\x70","\x32"),
	HX_HCSTRING("LESS_EQUAL","\xee","\xb2","\x55","\xe4"),
	HX_HCSTRING("NEVER","\x6c","\x76","\x1e","\x17"),
	HX_HCSTRING("NOT_EQUAL","\x08","\x7c","\x82","\x7d"),
	HX_HCSTRING("_new","\x61","\x15","\x1f","\x3f"),
	HX_HCSTRING("fromInt","\xa5","\xdd","\xfa","\x57"),
	HX_HCSTRING("toInt","\x34","\xbe","\x11","\x14"),
	::String(null()) };

void Context3DCompareMode_Impl__obj::__register()
{
	hx::Static(__mClass) = new hx::Class_obj();
	__mClass->mName = HX_HCSTRING("openfl.display3D._Context3DCompareMode.Context3DCompareMode_Impl_","\x48","\x5b","\xc7","\x5e");
	__mClass->mSuper = &super::__SGetClass();
	__mClass->mConstructEmpty = &__CreateEmpty;
	__mClass->mConstructArgs = &__Create;
	__mClass->mGetStaticField = &Context3DCompareMode_Impl__obj::__GetStatic;
	__mClass->mSetStaticField = &hx::Class_obj::SetNoStaticField;
	__mClass->mMarkFunc = sMarkStatics;
	__mClass->mStatics = hx::Class_obj::dupFunctions(sStaticFields);
	__mClass->mMembers = hx::Class_obj::dupFunctions(0 /* sMemberFields */);
	__mClass->mCanCast = hx::TCanCast< Context3DCompareMode_Impl__obj >;
#ifdef HXCPP_VISIT_ALLOCS
	__mClass->mVisitFunc = sVisitStatics;
#endif
#ifdef HXCPP_SCRIPTABLE
	__mClass->mMemberStorageInfo = sMemberStorageInfo;
#endif
#ifdef HXCPP_SCRIPTABLE
	__mClass->mStaticStorageInfo = sStaticStorageInfo;
#endif
	hx::RegisterClass(__mClass->mName, __mClass);
}

void Context3DCompareMode_Impl__obj::__boot()
{
	ALWAYS= (int)519;
	EQUAL= (int)514;
	GREATER= (int)516;
	GREATER_EQUAL= (int)518;
	LESS= (int)513;
	LESS_EQUAL= (int)515;
	NEVER= (int)512;
	NOT_EQUAL= (int)517;
}

} // end namespace openfl
} // end namespace display3D
} // end namespace _Context3DCompareMode

redpajama_set_name: RedPajamaGithub


The duo first worked together as producers on the Robin Williams-starrer Boulevard, the indie that marked the final dramatic onscreen appearance by the beloved actor. Also starring Kathy Baker, the film centered on a devoted husband in a marriage of convenience who is forced to confront his secret life. More recently, the pair penned The Adventurers Club for Fox, and up next, Belenzon is set direct their Black List script Clarity, which he co-wrote with Gelber.
They will continue to be repped by Hirsch Wallerstein.


I woke up to the gentle caressing of my head.
I noticed that I was on a bed when I woke up and was clenching onto the Hermit's Key for some reason.
Getting off the bed, I once again stretched myself and I checked the condition of my armor after recalling the battle I was in but it was not damaged at all.
Shisho and the others entered the room just as I was making my conjecture.
I recalled the hair-raising sensation once I was told that I fought a demon instead of something like a goblin king.
So I probably would have died if I slipped up? So they are that strong if I do not set up 「Sanctuary Circle」?
I could tell that Shisho was delighted from the bottom of his heart but recalling the scene of this person crying after I defeated the demon, perhaps he conducted the training under tremendous conflict within his heart.
Lionel showed a relieved expression seeing that I had recovered but he immediately said that after glancing over at Shisho and Shisho looked at me with a determined expression.
Then, Shisho abruptly sat down and prostrated to apologize to me.
My thought process grounded to a half for a split second.
I couldn't articulate well from the shock and somehow squeezed out a question.
Lionel replied to that instead of Shisho who was prostrating on the ground.
A prophecy from a Prophet of the guild … but I don't think he would normally believe it?
But since the mines between Meratoni and Grandol disappeared and the incidents with dragons, spirits, demons and reincarnated person happened one after another, I guess it's not unreasonable that he believed it.
So everyone did not tell me about it because he asked them to keep quiet about it?
I felt the story begin to add up.
It's true that it was a severe training but I couldn't wipe away the feeling that I ended up pressuring Shisho.
I spoke to everyone after helping Shisho up.
After becoming the guildmaster of the Adventurer's Guild branch, he's not been active as an adventurer for at least 7 years so it can't be helped that he had weakened from inactivity.
Since Shisho is the type to do what he's set his mind to thoroughly.
Kefin immediately followed up after Cathy's honest opinion but I could tell that he was trying to draw attention away from her.
Maybe they weren't enduring their laughter but actually killing their thoughts that he had gone crazy?
When I said that, Shisho gave out an awkward aura for some reason and absolutely did not make eye contact.
Lydia and Nadia broke the silence and there were no smiles on their faces as they were still worried even now.
My heart squeezed a little after seeing the expressions they made.
While noticing that Estia wasn't present, I asked about what happened after that.
I would have died if I made even a slight error. Feeling terrified at that thought, I changed the topic.
Shisho acted like he could not remember Estia.
Although Lionel and the others remembered her, their memories for these couple of days were hazy.
There's no doubt that the Darkness Spirit used quite some power.
Among them, only the Spirit-user Lydia properly grasped Estia presence.
The Darkness Spirit might have planted the suggestion in me to take my Hermit's Key from my item bag while I was asleep.
I vaguely sensed that Fornoir was the Light Spirit I saw but if the Darkness Spirit did not overdo it, there's a high chance I would still be unconscious.
Even though I didn't trust Estia, thinking of how Estia used her power for me to the point of collapse, I thought maybe I've been unconsciously looking at her with prejudice from the Empire slave and Darkness Spirit incidents.
Thanking Estia in my heart, I decided to also properly thank Fornoir and the Darkness Spirit.
I asked Shisho about the training since I had to end the training halfway.
Shisho stumbled for words for an instant but he happily said that with a smile.
Don't tell me he plans to cancel the training? Unfortunately, I don't have the intention to do that at all.
Then, if I can secretly drag him into the party as a helper, won't I be able to break through all the labyrinths that have dragons in them? I began envisioning that blueprint.
Even so, I totally don't feel like I can win in the mock battles that Shisho put together but thinking about the prophecy that Shisho would die protecting somebody, that somebody would probably be me.
If that's the case, once I become stronger, Shisho would probably be able to calm down.
I decided to show some filial piety to my mentor.
Shisho said that and took out a magic communication bead.
I've not heard of such logic? Shisho?
But I did not dispute it.
Since I was not able to hide my surprise that my perception abilities rose to Level III in just half a month.
As expected of a spartan training, it seems like it was quite a considerably useful special training.
After that, we talked for a bit before they invited me for dinner but I decided to refrain this time.
After checking that everyone had left the room, I opened the Hermit's stable and perhaps Fornoir was tired, I saw that she was sleeping.
I don't know if it has any effect but I activated 「Extra Heal」 and 「Purification」 before I closed the Hermit's stable.
After swearing to my heart to clarify my aims, I took out the Transformation Dresser from my item bag and took off my armor before continuing to eat the meals in my item bag until I had my fill.
After that, I took out Angel's Pillow, prepared to sleep early so that I could be prepared for tomorrow and proceeded to consolidate my mental image on Presence Perception and Magical Power Perception.


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