text:
\section{Introduction}

Colloidal particles have various industrial applications.  Their
tendency to flocculate sometimes causes problem in
applications.  Charging colloidal particles is an effective way to
prevent the flocculation by electrostatic repulsions.  However,
such charge effects are easily screened by salt and polar solvents in
the surrounding environment.  Grafting polymers onto the colloid
surface offers another way to prevent the flocculation.  Steric
repulsion induced by the grafted polymers makes colloidal particles repel
each other.

Grafting polyelectrolytes onto the colloid surface offers an even more
efficient way to prevent the flocculation of the colloids in polar
solutions \cite{napper83}.  The stabilization effect arises from both
steric and electrostatic repulsion.  Since the polyelectrolytes can
trap their own counterions in their vicinity, the grafting of
polyelectrolytes generates a layer of locally enhanced counterion
concentration.  The stabilization effect obtained by grafting
polyelectrolytes is thus less sensitive to the salinity of the
surrounding environment than the one obtained by solely charging the
colloids.

Grafted neutral polymers have been studied by Alexander\cite{alexander77}
and de Gennes\cite{degennes80}.  When polymer chains densely grafted onto
an impenetrable planar surface are immersed in a good solvent, the
excluded volume effects strongly stretch the chains and they
form a blob structure.  This particular distribution of
densely grafted polymers is called a ``brush''.  In this neutral
brush regime, they found that the density profile
of the brush is essentially flat and they predicted a linear dependence of
the brush height on the polymerization index and some other scaling
for the grafting density.

Later, self-consistent field theories were developed for neutral
polymer brushes\cite{edwards65}-\cite{netz03}.
In these papers, it was shown that polymer
chains are not uniformly stretched, and
the density profile of the brush is not flat but parabolic.

Polyelectrolyte brushes which consist of charged polymer chains
densely grafted onto a surface have been extensively studied by theories
\cite{miklavic88}-\cite{witte06},
simulations\cite{granfeldt90}-\cite{kumar05} and
experiments\cite{watanabe92}-\cite{ahrens04}.
The free energy of the polyelectrolyte brush consists of three terms in
general: The osmotic pressure of counterions, the Coulomb interactions between
the charges, and the excluded volume interactions of monomers.

Pincus \cite{pincus91} has proposed a scaling theory for the
polyelectrolyte brush assuming that its equilibrium height is
determined by balancing the osmotic pressure of the counterions
with the Gaussian elasticity of the chain.  Scaling laws were presented
for two regimes: (1) the osmotic regime: When the polyelectrolyte brushes
are densely grafted and have a large value of their charge fraction
(strongly charged case), most of the counterions are captured
inside and in the vicinity of the brush due to the strong Coulomb
attraction.  The brush height is roughly equal to the thickness of
the counterion layer and it is independent of the grafting density.
(2) Pincus regime: When the chains are sparsely grafted and have small
values of their charge fraction (weakly charged case), the Coulomb
attraction is relatively weak and the counterions are distributed
far beyond the brush region.  The counterion swelling pressure is reduced
by the fraction captured inside the brush.

The polyelectrolyte brush has been further investigated with the box model
where a flat density distribution is assumed, and a
collapsed regime was introduced \cite{csajka01,csajka01b}.  In this
regime, attractive forces between grafted polyelectrolytes occur
due to the correlations of the monomers and counterions.  The
brush height is determined by balancing the above attractive
forces and the repulsive forces due to the excluded volume effects.

Zhulina \textit{et al.}\cite{zhulina92} developed a self-consistent
analytical model and considered the effect of the excluded volume
interactions in the osmotic regime.  In the osmotic regime, for good
solvents, they obtained a scaling relation which does not agree with that of the
box model.
This work was generalized to poor solvents \cite{ross92,zhulina92}.
Borisov \textit{et al.}  \cite{borisov94} introduced a quasi-neutral
or Alexander regime where the excluded volume effects dominate over
electrostatic effects. The quasi-neutral regime has
the same scaling relation as the neutral brush except for the
dependence on the charge fraction of the chain.  The brush height is
determined by a balance between the excluded volume effects and the
chain elasticity.

Self-consistent field theories have been developed for
polyelectrolytes by several authors.  Borukhov \textit{et al.}
applied a self-consistent field theory to semidilute solutions of
polyelectrolytes and polyampholytes \cite{borukhov98}.  Wang
\textit{et al.} applied it to the interface of phase-separated
polyelectrolyte solutions\cite{wang04,wang05}.  Witte \textit{et al.}
applied it to the mixed polyelectrolyte brushes\cite{witte06}.

The phase diagram for various values of the charge fractions and grafting
densities has been obtained from scaling
theories\cite{borisov94,csajka01} or
simulations\cite{seidel03}.
The Bjerrum length $l_{\mathrm{B}}$ is a characteristic length scale
at which the electrostatic interactions between monovalent ions and
the thermal energy become equal.  When $l_{\mathrm{B}}^3 > v$, where $v$ is the
excluded volume parameter, the electrostatic interactions are relatively
strong.  This situation is called the 'strong coupling case'.
When $l_{\mathrm{B}}^3 < v$, the electrostatic interaction is relatively
weak.  This situation is called the 'weak coupling case'.  Scaling theories
\cite{csajka01} and molecular dynamics simulations\cite{csajka01b} both predict
that a collapsed regime appears in the strong coupling case.

In the present work we formulate the self-consistent field theory for
a salt-free polyelectrolyte brush in the presence of counterions
based on a theory developed  by Orland and
Schick\cite{orland96} for the neutral polymer case combined
with a theory developed by Netz and Orland\cite{netz00} for charged
electrolytes.  We apply it to
polyelectrolyte brushes grafted onto an impenetrable planar surface.
The chain
configurations are obtained by solving some modified Edwards equations,
including electrostatic interactions.  The electrostatic
potential is obtained by solving the nonlinear Poisson-Boltzmann
equation.

We solve these self-consistent field equations numerically in the weak
coupling case and obtain the monomer density profile, counterion distribution,
and total charge distribution.
These results are obtained for various charge fractions and we study the
scaling relations resulting from our model.

\section{The Partition Function for Polyelectrolyte Brushes}

We consider $M$ charged polymers of polymerization index $N$ grafted to a planar
surface of area $A$. We assume that the
grafting surface is impenetrable so that all polymers are
confined to a half-space.  The monomer
positions are denoted by $\{\mathbf{R}_k (s)\}$,
where $s \in [0,N]$ is the curvilinear abscissa
along a chain of $N$ monomers and $k=1,\cdots,M$ is the label of the $M$
chains. In addition, we assume that the $s=0$ extremities of the chains are grafted to the surface,
at positions $\mathbf{R}_k (0)=( \mathbf{u}_k,z_k(0)=0)$,
where $z$ denotes the axis perpendicular
to the grafting plane.
We consider $N_c$ counterions and denote their positions $\mathbf{r}_i $, where
$i=1,\cdots,N_c$. The canonical partition function $Z_{N_c}$
for this system can be written as
\begin{eqnarray}
Z_{N_c}&=&\frac{1}{N_c!}\left[\prod_{i=1}^{N_c}\int d\mathbf{r}_\mathrm{i} \right]
\left[\prod_{k=1}^{M}\int_{\mathbf{R}_k (0)=(\mathbf{u}_k,z_k(0)=0)}
\mathcal{D} \mathbf{R}_k (s) \right]
\exp \left(-\frac{3}{2a^2}\sum_{k=1}^{M}\int_0^N ds \dot{\mathbf{R}}_k^2(s)
\right) \nonumber \\
& &
\times \exp \left(
-\frac{1}{2 k_B T} \int d\mathbf{r}d\mathbf{r}^\prime
\hat{\rho}_{c}(\mathbf{r})
v_c(\mathbf{r},\mathbf{r}^\prime)
\hat{\rho}_{c}(\mathbf{r}^\prime)
\right) \nonumber \\
& &
\times \exp \left(
-\frac{v}{2}\int d\mathbf{r} \hat{\rho}_\mathrm{m} ^2(\mathbf{r})
\right),
\label{eq:1}
\end{eqnarray}
where $\int \mathcal{D}g$ is the functional integral over the function
$g(\mathbf{r})$. The first term in the exponent represents the elasticity
of the Gaussian chain, where
$\dot{\mathbf{R}}_k (s)$
is the derivative of $\mathbf{R}_k (s)$ with respect to the curvilinear abscissa $s$
and $a$ is the Kuhn length. The grafting of the chains is imposed by the boundary conditions.

The second term represents the electrostatic interaction where
$k_B T$ is the thermal energy,
\begin{equation}
v_c(\mathbf{r},\mathbf{r}^\prime)=
\frac{1}{4\pi \epsilon}\frac{1}{|\mathbf{r}-\mathbf{r}^\prime|}
\end{equation}
is the Coulomb interaction, $\epsilon$ is the dielectric constant of the
solvent and $\hat{\rho}_c(\mathbf{r})$ is the local charge density including
all charges in this system (charged monomers and counterions):
\begin{eqnarray}
\hat{\rho}_c(\mathbf{r})&=&
\sum_{k=1}^{M}\int_0^N ds \ q_k (s) \delta(\mathbf{r}-\mathbf{R}_k(s))
+\sum_{i=1}^{N_c}qe \delta (\mathbf{r}-\mathbf{r}_i),
\end{eqnarray}
where $q_k (s)$ is the variable denoting the charge carried by
the monomer $s$ along the chain $k$. The charge of the counterions is denoted by  $qe$.
Without loss of generality we assume that all polymers are negatively charged.
We use the $smeared$ model for the charge of the chains.
For a polyelectrolyte with a fraction $f$ of its monomers negatively charged,
this model assumes that each monomer carries a uniform fractional charge
$-fe$, where $-e$ is the electron charge. Namely, $q_k(s)=-fe$ for
any monomer $s$ on any chain $k$.
The condition of electroneutrality is then expressed as
\begin{equation}
\int d\mathbf{r} \hat{\rho}_c(\mathbf{r})=0,
\end{equation}
which yields $N_c=fMN$.

The third term in equation (\ref{eq:1}) represents
 the excluded volume interactions between monomers, where $v$
is the excluded volume parameter and
\begin{equation}
\hat{\rho}_\mathrm{m} (\mathbf{r})=
\sum_{k=1}^{M}\int_0^N ds \delta(\mathbf{r}-\mathbf{R}_k(s))
\label{eq:3}
\end{equation}
is the local monomer concentration.

The operator inverse of the Coulomb interaction for a system
with homogeneous dielectric constant $\epsilon$ is given by
\begin{equation}
v_c^{-1}(\mathbf{r},\mathbf{r}^\prime)=-\epsilon \nabla^2
\delta(\mathbf{r}-\mathbf{r}^\prime).
\end{equation}
We introduce two fields $\phi_1,\phi_2$ to perform the Gaussian transformations
\begin{eqnarray}
& &
\exp \left(
-\frac{1}{2k_B T} \int d\mathbf{r}d\mathbf{r}^\prime
\hat{\rho}_{c}(\mathbf{r})
v_c(\mathbf{r},\mathbf{r}^\prime)
\hat{\rho}_{c}(\mathbf{r}^\prime)
\right) \nonumber \\
& &=\int \mathcal{D}\phi_1 \exp\left\{
-\frac{k_B T}{2}\int d\mathbf{r}d\mathbf{r}^\prime \phi_1(\mathbf{r})
v_c^{-1}(\mathbf{r},\mathbf{r}^\prime)\phi_1({\mathbf{r}^\prime})
+i\int d\mathbf{r} \hat{\rho}_c({\mathbf{r}}) \phi_1({\mathbf{r}})\right\},
\end{eqnarray}
\begin{equation}
\exp\left[-\frac{v}{2}\int d\mathbf{r}\hat{\rho}_\mathrm{m} (\mathbf{r})^2 \right]
=\int \mathcal{D}\phi_2 \exp \left[-\frac{1}{2v}
\int d\mathbf{r}\phi_2^2 (\mathbf{r})-i\int d \mathbf{r}\hat{\rho}_\mathrm{m}(\mathbf{r})
\phi_2 (\mathbf{r})
\right],
\end{equation}
and the partition function reads
\begin{eqnarray}
Z_{N_c}&=&\frac{1}{N_c!}\int \mathcal{D}\phi_1 \mathcal{D}\phi_2
\exp
\left[
-\int d\mathbf{r} \frac{\left[\nabla \phi_1 (\mathbf{r}) \right]^2}{8\pi l_B q^2}
-\frac{1}{2v}\int d\mathbf{r}\phi_2^2 (\mathbf{r})
\right]
\left[\int d\mathbf{r}e^{-i\phi_1 (\mathbf{r})} \right]^{N_c} \nonumber \\
& &\prod_{k=1}^M
\left[
\int _{ \mathbf{R} (0)=( \mathbf{u}_k,0) }\mathcal{D}\mathbf{R}(s)
\exp\left[
-\int_0^N ds \left\{
\frac{3}{2a^2}\dot{\mathbf{R}}^2(s)-i\frac{f}{q}\phi_1 (\mathbf{R}(s))+
i\phi_2 (\mathbf{R}(s))
\right\}
\right]
\right],
\end{eqnarray}
where $l_B=e^2/4\pi \epsilon k_B T$ is
the Bjerrum length.

The partition function is brought into a more tractable form by going to the
grand canonical ensemble
\begin{equation}
Z_\lambda=\sum_{N_c=0}^\infty \lambda^{N_c} Z_{N_c},
\label{eq:4}
\end{equation}
where $\lambda$ is the fugacity which is related to the particle chemical
potential $\mu$ by $\lambda=e^{\mu/k_B T}$.

The grand canonical partition function can be written as
\begin{equation}
Z_\lambda=\int \mathcal{D}\phi_1 \mathcal{D}\phi_2
\exp \left[-H\left[\phi_1,\phi_2 \right] \right],
\end{equation}
with the Hamiltonian
\begin{equation}
\label{ham}
H\left[\phi_1,\phi_2 \right]=\int d\mathbf{r}
\left(
\frac{[\nabla \phi_1 (\mathbf{r})]^2}{8\pi l_B q^2}
+\frac{1}{2v}\phi_2^2 (\mathbf{r}) -\lambda e^{-i\phi_1(\mathbf{r})}
\right)
-\sum_{k=1}^M \ln Q_k \left[\phi_1,\phi_2 \right],
\end{equation}
where
\begin{equation}
Q_k \left[\phi_1,\phi_2 \right]=
\int  _{ \mathbf{R} (0)=( \mathbf{u}_k,0) }\mathcal{D}\mathbf{R}(s) \exp \left[
-\int_0^N ds \left\{
\frac{3}{2a^2}\dot{\mathbf{R}}^2(s)-i \frac{f}{q}\phi_1(\mathbf{R}(s))
+i\phi_2(\mathbf{R}(s))
\right\}
\right].
\end{equation}
As follows from the definition of the grand canonical partition function, equation (\ref{eq:4}),
the expectation value of the number of counterions is
given by
\begin{equation}
\langle N_c \rangle=
\lambda \frac{\partial \ln Z_\lambda}{\partial \lambda}
=\lambda \left\langle
\int d\mathbf{r}e^{-i\phi_1(\mathbf{r})} \right\rangle.
\label{eq:7}
\end{equation}

In the limit $M \to \infty$, if the grafting points on the surface are random independent variables, we can further simplify eq.(\ref{ham}). Indeed, according to the central limit theorem, we have
\begin{equation}
\sum_{k=1}^M \ln Q_k \left[\phi_1,\phi_2 \right] \approx_{M \to \infty} M \int d \mathbf{u} P (\mathbf{u})  \ln Q \left[\phi_1,\phi_2 \right] + O(\sqrt{M}),
\end{equation}
where $P (\mathbf{u})$ is the probability distribution of the grafting point $\mathbf{u}$ and
\begin{equation}
Q \left[\phi_1,\phi_2 \right]=
\int  _{ \mathbf{R} (0)=( \mathbf{u},0) }\mathcal{D}\mathbf{R}(s) \exp \left[
-\int_0^N ds \left\{
\frac{3}{2a^2}\dot{\mathbf{R}}^2(s)-i \frac{f}{q}\phi_1(\mathbf{R}(s))
+i\phi_2(\mathbf{R}(s)).
\right\}
\right].
\end{equation}

In the following, we will assume a uniform grafting density, that is
\begin{equation}
P (\mathbf{u})=\frac{1}{A},
\end{equation}
where $A$ is the total grafting area. If we define $\sigma=A/M$ as the average grafting area per chain, we have
\begin{equation}
\label{part}
Z_\lambda=\int \mathcal{D}\phi_1 \mathcal{D}\phi_2
\exp \left[-
\int d\mathbf{r}
\left(
\frac{[\nabla \phi_1 (\mathbf{r})]^2}{8\pi l_B q^2}
+\frac{1}{2v}\phi_2^2 (\mathbf{r}) -\lambda e^{-i\phi_1(\mathbf{r})}
\right)
+\frac{1}{\sigma}\int_A d \mathbf{u}  \ln Q \left[\phi_1,\phi_2 \right]
\right].
\end{equation}
\section{Mean-field Approximation}
In order to evaluate the partition function (\ref{part}), we use the saddle-point method to determine the fields $\phi_1$ and $\phi_2$ which most contribute to the functional integral.
We assume that these two fields possess translational
symmetry in the $(x,y)$ directions, parallel to the grafted plane (remember that $z$ is the direction normal to the plane).
The Hamiltonian reads
\begin{equation}
H\left[\phi_1,\phi_2 \right]=A \int dz
\left(
\frac{1}{8\pi l_B q^2}\left(\frac{d\phi_1(z)}{dz} \right)^2
+\frac{1}{2v}\phi_2^2 (z) -\lambda e^{-i\phi_1(z)}
\right)
-M \ln Q \left[\phi_1,\phi_2 \right],
\end{equation}
where
\begin{equation}
Q \left[\phi_1,\phi_2 \right]=
\int_{z(0)=0} \mathcal{D}z(s) \exp \left[
-\int_0^N ds \left\{
\frac{3}{2a^2}\dot{z}^2(s)-i \frac{f}{q}\phi_1(z(s))
+i\phi_2(z(s))
\right\}
\right].
\end{equation}

The saddle-point method provides the mean-field approximation to the system.
The mean-field $\Phi_1(z),\Phi_2(z)$ are given by the stationarity conditions
\begin{equation}
\left. \frac{\delta H[\phi_1,\phi_2]}{\delta{\phi_1(z)}} \right|_{\phi_1=-i\Phi_1}=0,
\end{equation}
\begin{equation}
\left. \frac{\delta H[\phi_1,\phi_2]}{\delta{\phi_2(z)}} \right|_{\phi_2=-i\Phi_2}=0.
\end{equation}
The mean-field approximation to the Hamiltonian $H_{\mathrm{mf}}$ is simply
\begin{equation}
H_{\mathrm{mf}}\equiv H[i\Phi_1,i\Phi_2].
\end{equation}
Note that $\Phi_1(z)$ is identified as the electrostatic potential.

The mean-field equations are thus
\begin{equation}
\frac{1}{4\pi l_B q^2}\frac{d^2 \Phi_1(z)}{dz^2}
+\lambda e^{-\Phi_1(z)}-\frac{1}{\sigma}
\frac{\delta \ln Q}{\delta \Phi_1(z)}=0,
\label{eq:mf1}
\end{equation}
\begin{equation}
\Phi_2(z)=-\frac{v}{\sigma}\frac{\delta \ln Q}{\delta \Phi_2(z)}
\equiv \frac{v}{\sigma}\rho_\mathrm{m}(z),
\label{eq:6}
\end{equation}
where $\sigma=A/M$ is the average area per chain and $\rho_\mathrm{m}(z)$ is
the local density of monomers.

The density of monomers can be written as
\begin{equation}
\rho_\mathrm{m} (z)=\frac{1}{Q}\int \mathcal{D}z(s)
\int_0^N ds \delta(z-z(s))
\exp \left[
-\int_0^N ds \left\{
\frac{3}{2a^2}\dot{z}^2(s)-\frac{f}{q}\Phi_1(z(s))
+\frac{v}{\sigma}\rho_\mathrm{m}(z(s))
\right\}
\right].
\end{equation}
The density is normalized so that
\begin{equation}
\int_0^\infty \rho_\mathrm{m}(z)dz=N.
\end{equation}

We take the grafting plane to be at $z=0$ and
introduce the monomer probability amplitudes
\begin{equation}
\psi(z,s)=\langle z| e^{-sH}|0 \rangle ,
\end{equation}
\begin{equation}
\psi^\dagger(z,s)=\int_0^\infty
dz^\prime \langle z^\prime | e^{-sH}| z \rangle
\end{equation}
where we use the standard quantum-mechanical notations
\begin{equation}
\langle z^\prime | e^{-sH}| z \rangle ,
=
\int_{z(0)=z}^{z(s)=z^\prime}
\mathcal{D}z(s) \exp \left[
-\int_0^s ds^\prime \left\{
\frac{3}{2a^2}\dot{z}^2(s^\prime)-\frac{f}{q}\Phi_1(z(s^\prime))
+\frac{v}{\sigma}\rho_{\mathrm{m}}(z(s^\prime))
\right\}
\right].
\end{equation}
Note that as before in the paper, the curvilinear abscissa  $s$ is defined so that its values at the
grafted and free ends are $0$ and $N$ respectively.

The probability amplitudes $\psi(z,s),\psi^\dagger (z,s)$ satisfy the
modified diffusion equations
\begin{equation}
\left[
\frac{\partial}{\partial s}-\frac{a^2}{6}\frac{\partial^2}{\partial z^2}
-\frac{f}{q}\Phi_1(z)+\frac{v}{\sigma}\rho_\mathrm{m}(z)
\right] \psi(z,s)=0 ,
\label{eq:8}
\end{equation}
\begin{equation}
\left[
\frac{\partial}{\partial s}-\frac{a^2}{6}\frac{\partial^2}{\partial z^2}
-\frac{f}{q}\Phi_1(z)+\frac{v}{\sigma}\rho_\mathrm{m}(z)
\right] \psi^\dagger(z,s)=0.
\label{eq:9}
\end{equation}
with the initial conditions $\psi(z,0)=\delta(s)$ and $\psi^{\dagger} (z,0)=1.$
Because the grafting plane is impenetrable,$\psi(z,s),\psi^\dagger (z,s)$ must satisfy
the boundary conditions $\psi(0,s)=0,\psi^\dagger(0,s)=0$.

The density of monomers
is given by
\begin{equation}
\rho_\mathrm{m}(z)=
\frac{\int_0^N ds \psi(z,s)\psi^\dagger (z,N-s)}
{\int_0^\infty dz^\prime \psi(z^\prime,N)},
\label{eq:11}
\end{equation}
and equation (\ref{eq:mf1}) can be written as
\begin{equation}
\frac{1}{4\pi l_B q^2}\frac{d^2 \Phi_1(z)}{dz^2}
+\lambda e^{-\Phi_1(z)}-\frac{f}{q}\frac{1}{\sigma}\rho_\mathrm{m}(z)=0.
\label{eq:mf11}
\end{equation}
This is just the Poisson-Boltzmann equation for all the charges in the system
 (polymers and counterions). In the following,  we will consider only monovalent counterions, that is $q=1$.

The counterion density per unit area of the grafting plane is given by
\begin{equation}
\rho_{\mathrm{ci}}(z)=\lambda e^{-\Phi_1(z)}.
\end{equation}
From equation (\ref{eq:mf11}) one obtains
$
\int_0^\infty \rho_{\mathrm{ci}}(z)dz=\langle N \rangle/A.
$
The condition of electroneutrality is given by
\begin{equation}
\int_0^\infty \rho_{\mathrm{ci}}(z)dz =\frac{fN}{\sigma}.
\label{eq:12}
\end{equation}

Equations (\ref{eq:8}) to (\ref{eq:12}) are the self-consistent field equations describing
the polyelectrolyte brush in the presence of counterions.
We assume that the grafting plane is electrically neutral. The electroneutrality of the
system is enforced by setting the field strength to zero at the boundaries
\begin{equation}
\left. \frac{d \Phi_1 (z)}{dz} \right|_{z=0}
=\left. \frac{d \Phi_1(z)}{dz}\right|_{z=\infty}=0.
\label{eq:13}
\end{equation}
The total charge density $\rho_{\mathrm{c}}(z)$ per unit area of the grafting plane
is given by
\begin{equation}
\rho_{\mathrm{c}}(z)=-\frac{f}{\sigma}\rho_{\mathrm{m}}(z)+\rho_{\mathrm{ci}}(z).
\label{eq:15}
\end{equation}
The total charge distribution $\rho_{\mathrm{c}}(z)$ satisfies the condition of
the electroneutrality
\begin{equation}
\int_0^\infty \rho_{\mathrm{c}}(z) dz=0.
\label{eq:14}
\end{equation}

\section{Numerical Solutions of the Self-Consistent Field Equations and Discussions}

In this section we present numerical solutions of the self-consistent field
equations as a function of the values of the polymerization index $N$,
grafting area per chain $\sigma$, excluded volume parameter $v$,
charge fraction $f$, and Bjerrum length $l_{\mathrm{B}}$.
In this paper, we vary $f$ from $0$ to $1.0$,
while we fix the values of parameters $N=30, v=1.0
a^3, l_{\mathrm{B}}=0.1a, \sigma=10a^2$, otherwise stated.
This corresponds to the weak coupling case $ l_{\mathrm{B}}^3 < v$
where the Coulomb interaction is relatively weak.
To investigate the grafting density dependence, we vary
$\sigma$ from $5 a^2$ to $20 a^2$.  To investigate the polymerization index
dependence, we vary $N$ from $30$ to $70$.

\subsection{Structure of the Brush}

In Figure \ref{figure1} we show the monomer density profiles
$\rho_{\mathrm{m}}(z)$ for the above fixed values and various values of $f$
including the monomer density profile of the neutral brush ($f=0$).
The brush height increases with increasing $f$.

The impenetrability of the grafting plane induces a depletion area near
the grafting surface in the monomer density profiles and pushes the maximum
of the density away from the grafting surface.
The depletion area is also observed for the neutral brush\cite{scheutjens79}.
The density maximum gets closer to the grafting plane with increasing $f$.
This is due to the strong Coulomb interaction between the monomers and the
counterions for large $f$.  The monomer density decreases
exponentially when the distance from the grafting plane $z$ becomes
large.

Taniguchi \textit{et al.}\cite{taniguchi03} obtained similar results
but the behavior of the monomer density near the grafting plane is
somewhat different from our results.

We measure the brush height $h_{\mathrm{m}}$ and the thickness of the counterion layers
$ l_{\mathrm{ci}} $ as the first moments of each density profiles:
\begin{equation}
h_{\mathrm{m}}=\frac{\int_0^\infty z \rho_{\mathrm{m}}(z)dz}
{\int_0^\infty  \rho_{\mathrm{m}}(z)dz},
\end{equation}
\begin{equation}
l_{\mathrm{ci}}=\frac{\int_0^\infty z \rho_{\mathrm{ci}}(z)dz}
{\int_0^\infty \rho_{\mathrm{ci}}(z)dz}.
\end{equation}
We show the values of $2h_{\mathrm{m}}$ and $2l_{\mathrm{ci}}$ for each values of $f$
in Table \ref{table1}.  We multiply by a factor $2$ because the height
for the uniform density distribution corresponds to twice of the first moment.

In Figure \ref{figure2}a we plot $h_{\mathrm{m}}$ and $f$ on a log-log scale.
For large values of $f \ge 0.5$, the brush height behaves as
$h_{\mathrm{m}} \sim f^\alpha$, with exponent $\alpha = 0.41$.
In Figure \ref{figure2}b we plot $h_{\mathrm{m}}$ and $1+f$ on a log-log scale.
Figure \ref{figure2}b shows the scaling relation
$ h_{\mathrm{m}} \sim (1+f)^\beta $ with exponent $\beta = 1.17$ for small values
of $f \le 0.5$.

We investigate the dependence of the brush height $h_{\mathrm{m}}$ on the other parameters
$N$ and $\sigma$.
Figure \ref{figure3} shows the scaling relation $h_{\mathrm{m}} \sim N^\gamma$
with exponent $\gamma = 1.0$ for $f=0.3,0.5,1.0$ respectively.
Figure \ref{figure4} shows the scaling relation $ h_{\mathrm{m}} \sim (1/\sigma)^\delta $ with exponent $\delta = 0.06, 0.13, 0.18, 0.23$ for $f=1.0, 0.5, 0.3, 0.0$ respectively.

We briefly review the scaling relation predicted by the scaling theory and other SCF theory.
For the strongly charged case, Pincus\cite{pincus91} predicted that the brush height $h$
behaves as
\begin{eqnarray}
&& h \sim N a f^{1/2} , \label{eq:r1} \\
&& \alpha=1/2, \gamma=1, \delta=0 , \nonumber
\end{eqnarray}
in the osmotic regime.

Zhulina \textit{et al.}  \cite{zhulina92} found that the scaling relation in
the osmotic regime changes due to excluded volume effects to
\begin{eqnarray}
&& h \sim N a f^{2/5} , \label{eq:r2} \\
&& \alpha=2/5, \gamma=1, \delta=0 . \nonumber
\end{eqnarray}
The scaling relations (\ref{eq:r1}), (\ref{eq:r2}) indicate
that the brush height $h$ is independent of the grafting density $1/\sigma$.

Recent study for the osmotic regime shows that the brush height $h$ slightly
increases with increasing $1/\sigma$\cite{naji03,seidel03,romet04,ahrens04}.
The term `nonlinear osmotic regime' is used for this weak dependence on $1/\sigma$. ($\delta=1/5$ for weak coupling and $f=1$)

Pincus\cite{pincus91} also obtained the brush height
in the Pincus regime
where counterions are distributed beyond the brush
behaves as
\begin{eqnarray}
&& h \sim N^3 a^2 f^2 /\sigma ,   \\
&& \alpha=2, \gamma=3, \delta=1 .  \nonumber
\label{eq:r3}
\end{eqnarray}

The quasi-neutral regime\cite{alexander77,degennes80,borisov94,csajka01}
appears
both for the strongly and weakly charged case.  The
brush height is determined by balancing the repulsive force due to the
excluded volume effect and the attractive force due to the chain
elasticity.  The scaling relation is identical to the neutral brush
case except for the dependence on $f$ for the strongly charged case.
The scaling relation for the strongly charged case is
\begin{eqnarray}
&& h \sim (1+f)^{2/3} N (a^2 v/\sigma)^{1/3} , \\
&& \beta=2/3, \gamma=1, \delta=1/3 .  \nonumber
\label{eq:r4}
\end{eqnarray}
For the weakly charged case, the scaling relation is exactly the same
as that for the neutral brush:
\begin{eqnarray}
&& h \sim N (a^2 v/\sigma)^{1/3} ,  \\
&& \alpha=0, \gamma=1, \delta=1/3 .  \nonumber
\label{eq:r5}
\end{eqnarray}

We compare our numerical results with the other theoretical
predictions.  First we investigate the results for $f \ge 0.5$.  Our
numerical results $\alpha = 0.41$ is in good agreement with
$\alpha=2/5$ of (\ref{eq:r2}) for the osmotic regime in the good
solvent.  Our numerical results $\gamma=1.0$ for $f=0.5,1.0$ is
consistent with (\ref{eq:r2}).  The brush height scales linearly with
$N$.  Our results show that the brush height $h_{\mathrm{m}}$ depends
weakly on the grafting density $1/\sigma$ with an exponent
$\delta=0.06,0.13$ for $f=1.0,0.5$ respectively.  The scaling relation
(\ref{eq:r2}) predicts $\delta=0$.  These comparisons show that our
numerical results for $f \ge 0.5$ corresponds to the osmotic regime.
The slight increase of the brush height for increasing the grafting
density is consistent with a recent study of the nonlinear osmotic
regime.\cite{naji03,seidel03,romet04,ahrens04}.

Next we investigate our results for $f \le 0.5$.
Our results of $\beta=1.17$
does not agree with the value $\beta=2/3$ of the scaling relation (\ref{eq:r4}) for
the quasi-neutral regime.  Our results of $\gamma=1.0$ for $f=0.3$
is consistent with (\ref{eq:r4}).  Our results of $\delta=0.18,0.23$
for $f=0.3,0.0$ differ from the exponent $\delta=1/3$ of (\ref{eq:r4}).

The neutral brush is characterized by a single stretching parameter\cite{netz98,netz03}
\begin{equation}
b=N \left(\frac{3 v^2}{2 a^2 \sigma^2}\right)^{1/3}.
\end{equation}
For large value of  $b \ge 10$, the monomer density profile
is well approximated by the semi classical parabola density profile\cite{milner88}
and the brush height satisfies the scaling relation (\ref{eq:r5}).
However the deviations from the parabolic profile become important when the values of
$N,1/\sigma,v$ decrease and the stretching parameter becomes $b \le 10$.
Our values of the parameters
$ N=30,v=1.0 a^3,5 a^2 \le \sigma \le 20 a^2 $ correspond to
$ 11.7 \ge b \ge 4.7 $.
Therefore, the disagreement between the calculated value of the exponent $\delta$ (0.06, 0.13, 0.18, 0.23)
and the value of the scaling prediction (\ref{eq:r5}) ($\delta=1/3$) is not surprising.

When $f$ decreases, the value of the exponent $\delta$ becomes close
to the value for the neutral case.  Roughly speaking
our results for small value of $f \le 0.3$ correspond to the quasi-neutral regime of the
phase diagram\cite{borisov94, csajka01, seidel03}.

Our value for the exponent $ \beta=1.17 $ does not agree with the value $\beta=2/3$
of the scaling relation (\ref{eq:r4}).
When (\ref{eq:r4}) was derived, only excluded volume interactions were considered and
the osmotic pressure of counterions was neglected.
A more realistic value of the exponent $ \beta $ should be derived when the
excluded volume interaction and the osmotic pressure of counterions are
equally treated.

Our numerical results show a
crossover phenomena
between the osmotic regime and the quasi-neutral regime.
Scaling theories assumes that one of the two repulsive effects is dominant and balance it to the elasticity of the Gaussian chain.
But in many regimes the two effects are comparable; The behavior of
the brush does not change abruptly and a smooth transition between the two regime is expected.
Our numerical results for $0.4 \le f \le 0.5$ correspond to this crossover region.

\subsection{Distribution of Counterions and Charges}

Next we consider the distribution of counterions and charges.
We first plot the distribution of counterions $\rho_{\mathrm{ci}}(z)$
for various values of $f$ in Figure \ref{figure5}.
We observe a depletion area of $\rho_{\mathrm{ci}}(z)$ near the grafting surface.
This depletion effect is clear for $f \ge 0.3$.

At large distance from the wall, the monomer density decreases faster than the counterion density. Therefore there are no more monomers in the solution, and the counterions see a virtual charged wall with an effective charge density $\sigma_{\rm eff}$ at position $r_{\rm eff}$.

We study the thickness of the counterion layer.
In Table \ref{table1}, we show the values of $2 l_{\mathrm{ci}}$ for each values
of $f$.

We fit the counterion distribution at large distance
by the GC profile
\begin{equation}
f(z)=\frac{1}{2 \pi l_{\mathrm B} (z-L_{\mathrm{f}})^2},
\label{eq:r6}
\end{equation}
where $L_{\rm f}$ is a fit parameter.

For $z \gg L_{\mathrm{f}}$, eq.(\ref{eq:r6}) is equivalent to the
distribution of counterions in a system which consists of a virtual charged plane located at $r_{\rm eff}$ with charge density $-\sigma_{\mathrm{eff}}e$ in the presence of
monovalent counterions.
The parameter $r_{\rm eff}$ is set to be the minimum value which satisfies the condition
\begin{equation}
\int_0^{r_{\rm eff}} \rho_c(z) dz = - \int_{r_{\rm eff}}^{\infty} f(z)dz,
\end{equation}
which expresses the fact that the total charge to the left of $r_{\rm eff}$ is equal to that of the counterions on the right of $r_{\rm eff}$.
$\sigma_{\rm eff}$ is determined by
\begin{equation}
\sigma_{\rm eff}= \frac{1}{2 \pi l_{\rm B} (r_{\rm eff} - H_{\rm m})}.
\end{equation}

The values of $r_{\rm eff}, \sigma_{\mathrm{eff}}$ are presented in
Table \ref{table1}.  The effective charge density $\sigma_{\mathrm{eff}}e $ is almost
equal to the integrated amount of charges of counterions for
$z \ge r_{\rm eff}$.  The total amount of charges of counterions
is $\sigma_{\mathrm{s}}\equiv f N/\sigma$.
From $\sigma_{\mathrm{eff}}$ we define the effective Gouy-Chapman (GC) length as
$l_{\mathrm{eff}}=1/2\pi l_{\mathrm{B}}\sigma_{\mathrm{eff}}$.
This effective GC length  $l_{\mathrm{eff}}$ provides a definition of the
thickness of the counterion layer outside the virtual wall.
We may thus take $ L_{\mathrm{ci}} \equiv r_{\rm eff}+l_{\mathrm{eff}} $ as an alternative definition of the
thickness of the counterion layer.

In Figure \ref{figure7} and Figure \ref{figure8},
we plot the absolute value of the negative charge distribution
$ f N/\sigma \rho_{\mathrm{m}}(z) $, the positive charge distribution $\rho_{\mathrm{ci}}(z)$,
the total charge distribution $\rho_{\mathrm{c}}(z)$, and the fitting GC solution (\ref{eq:r6}).
In Figure \ref{figure9} and \ref{figure10}, we plot $ \sigma_{\mathrm{eff}}/\sigma_{\mathrm{s}}, L_{\mathrm{ci}}/r_{\mathrm{eff}}$.

For large values of $f \ge 0.5$, the ratio $\sigma_{\mathrm{eff}}/\sigma_{\mathrm{s}}$ is smaller
than $ 0.10 $ and a large part of the negative charge of the brush is cancelled
by the positive charge of counterions. This strong screening of the negative
charges of the brushes can be verified by comparing Figure \ref{figure6} and \ref{figure7}.
For decreasing $f$, the screening of the brush charge
becomes weaker and the counterions gradually escape from the brush.
This behavior of counterions is verified in Figure \ref{figure8}.

From Table \ref{table1}, we see that for $f\ge0.2$, the value of $\sigma_{\mathrm{eff}} \approx 0.12$
is almost constant.
As was already mentioned above, far away from the wall, the polyelectrolyte brush can be viewed as
a charged plane located at $r_{\rm eff}$ with charge
density $-\sigma_{\mathrm{eff}} e$.
The charge density $-\sigma_{\mathrm{eff}} e$ is the `effective' charge of the brush which
is not cancelled by the counterions.
Our numerical results show that there is an upper limit for this effective charge.

Next, in Figure \ref{figure6}, we plot the total charge distributions $\rho_{\mathrm{c}}(z)$
for various values of $f$.
Figure {\ref{figure6}} shows that the values of $\rho_{\mathrm{c}}(z)$
near the grafting plane are always positive for any $f$.
This is due to the depletion of the monomer density near the grafting plane.
The amount of positive charges of counterions is larger than that
of the negative charges of the brushes.
The minimum of $\rho_{\mathrm{c}}(z)$, which is negative, is located near the grafting
plane and an electric dipole layer is generated.

The depth of the negative minimum of $\rho_{\mathrm{c}}(z)$ increases with $f$ and a
small peak in the absolute value of the positive charge $(fN/\sigma)
\rho_{\mathrm{m}}(z)$ appears for $f=1.0$  (see Figure {\ref{figure7}}).
As can be seen in  Figure {\ref{figure7}}, when $f$ increases,  in order to enforce locally charge neutrality, the monomer and counterion densities become closer. However, the depletion layer of the polymer near the wall is still present.
This region is filled by the counterions, and this accumulation of positive charges is compensated by a peak in the polymer density.
Taniguchi \textit{et al.} \cite{taniguchi03}
obtained similar results. However the behavior of the
depth of their negative minimum is somewhat different from our results.

\section{Conclusion}

We formulated the self-consistent field theory for a planar polyelectrolyte brush
in a salt free solution in the presence of counterions.
We numerically solved the self-consistent field equations for the weak coupling case
and obtained the monomer density profile, the distribution of counterions,
and the total charge distribution.
We also studied the scaling relation for the brush height.
For large values of the charge
fraction, our results correspond to those of the osmotic regime.
We observe a weak dependence of the brush height on the grafting density.
This is consistent with a recent study of the nonlinear osmotic regime.
For small values of the charge
fraction, our results correspond to those of the quasi-neutral regime.
We find a crossover phase between the osmotic regime and
the quasi-neutral regime.

We studied the distribution of the total charge, and found that an
electric dipole layer appears near the grafting area.  We
studied the distribution of counterions, and fit it
by the Gouy-Chapman theory for a virtual charged wall at the location
$r_{\rm eff}$ with an effective charge density $\sigma_{\mathrm{eff}}$ .
We calculated the integrated amount of the charge of counterions
outside the brush and the thickness of the counterions outside the
brush by introducing an effective GC length ($l_{\mathrm{eff}}$).  For
large values of the charge fraction, the charges of the monomers are
strongly screened by the charge of the counterions, and the
counterions are trapped inside the brush.  For smaller values of the
charge fraction, the screening effects are weak and the counterions
extend far beyond the brush.  Far away from the grafted surface, the
polyelectrolyte brush can be viewed as a virtual charged plane with an
effective charge density $\sigma_{\mathrm{eff}}$.  We found that this
effective charge density has an upper limit and the rest of the
polymer charges are cancelled by the charges of the counterions.

\newpage


text:
Czternasty Doktor (ang. Fourteenth Doctor) – postać fikcyjna związana z brytyjskim serialem science fiction pt. Doktor Who, w którą wciela się brytyjski aktor David Tennant.

Występy telewizyjne
Debiut Czternastego Doktora miał miejsce 23 października 2022 r. w Potędze Doktor – trzecim z serii odcinków specjalnych serialu, emitowanych w 2022 r. i połączonych postacią Trzynastej Doktor (granej przez Jodie Whittaker). W finale Potęgi Doktor Trzynasta regeneruje się w postać przypominającą Dziesiątego Doktora (inkarnację, którą w latach 2005–2010 grał David Tennant).

Czternasty Doktor będzie występował w trzech odcinkach specjalnych, emitowanych w listopadzie 2023 r. w związku z 60. rocznicą premiery serialu. Na ekranie towarzyszyć mu będą postacie kojarzone z czasami Dziesiątego Doktora – Donna Noble i Wilfred Mott.

Casting
W maju 2022 r. ogłoszono, że kolejnym Doktorem zostanie szkocki aktor pochodzenia rwandyjskiego, Ncuti Gatwa. Używano wówczas wobec niego określenia Czternasty Doktor, co sugerowało, że będzie bezpośrednim następcą Jodie Whittaker. Jego debiut miał mieć miejsce w listopadzie 2023 r. David Tennant miał wówczas towarzyszyć Gatwie jako Dziesiąty Doktor.

Dopiero po premierze Potęgi Doktor obecny showrunner serialu, Russell T Davies, potwierdził, że Ncuti Gatwa wcieli się dopiero w rolę Piętnastego Doktora.

Przypisy

Doktor (Doktor Who)

redpajama_set_name: RedPajamaWikipedia


Concorde is solidly bred to jump on both sides. Aside from his sire, Voltaire, his dam, Flyer, is by the legendary Marco Polo.
Nordrhein-Westfalen at Aachen and they were Champion of the Netherlands in 1994, and placed 2nd in the Grand Prix of Amsterdam, Darmstadt, Lanaken, Helsinki and Gothenburg.
For three years in a row, Concorde was the sire with the most offspring at the World Breeding Championships for Young Jumping Horses. In the year 2001 alone, Concorde produced 17 jumpers who entered Grand Prix ranks, including Audi's Jikke, Audi's Kathja, Conquest II, Larino, Viktor, Jacquet, Humphrey and Isabella. In 2003, three of his progeny qualified to go to Las Vegas for the World Cup final – they were Marius Claudius, Conquest and Viktor.
Concorde was selected as KWPN Horse of the Year in 2004 and was the youngest stallion ever to be awarded 'preferent' due to his exceptional progeny. He already has over thirty licensed sons in the European studbooks, and was one of the first KWPN stallions approved for Holstein. He is also proving a very popular sire in France.
In Bernard le Courtois' ranking of the top 75 jumping stallions of 2006, Concorde finishes in 8th place win 23 international winners. On the WBFSH standings for 2009, Concorde is 7th.


text:
Back to Help Center
What Are Anti-dumping and Countervailing Duties?
Customs and Compliance
Anti-dumping and countervailing duties are established to protect the U.S. manufacturing industry from foreign manufacturers.
Anti-dumping (AD) and countervailing duties (CVD) are intended to protect the U.S. manufacturing industry from foreign manufacturers.
Dumping occurs when foreign manufacturers sell goods in the U.S. at less than fair value. Countervailing cases are when a foreign government provides enough subsidies and tax benefits for their manufacturers to sell their goods more cheaply than U.S. manufacturers.
Both of these situations, dumping and countervailing, lead to foreign undercutting of U.S. manufacturers' prices.
How are these duties determined?
Anti-dumping duties are calculated at a company-specific level, where the actual duty amount makes up for the gap between foreign manufacturer pricing and fair market value.
Countervailing duties are determined on a country-specific level, and the duty rates counteract the subsidy or foreign government assistance's value to exactly level the playing field.
Petitioning
The U.S. International Trade Commission and the U.S. Department of Commerce oversee anti-dumping and countervailing duties.
Manufacturers or companies can file petitions to these two bodies if they find a foreign manufacturer selling under fair market value or a foreign government subsidizing manufacturers. These petitions will need to include:
Description of imported goods, the exporters, and the importers
Subsidy and price information
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More details on these duties and petitioning can be found at the U.S. ITC website:
http://www.usitc.gov/trade_remedy/documents/handbook.pdf
The Commission and Department will review the petition and conduct investigations to determine its validity. They will then instruct U.S. Customs to assess duties accordingly.
Are your products impacted by these duties?
AD/CVD duties are dependent on the country of origin and type of product. For a comprehensive and official list of goods affected, visit the International Trade Administration's website here.
Importing From Canada Into the US
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Importing Coffee into the United States

redpajama_set_name: RedPajamaCommonCrawl


text:
Return to Bios
Jarvis V. Hollingsworth
General Counsel, Secretary
Jarvis V. Hollingsworth serves as General Counsel and on the Executive Committee of Kayne Anderson Capital Advisors, L.P.
Prior to joining Kayne in 2019, Mr. Hollingsworth was in the private practice of law, most recently at Bracewell LLP in Houston, Texas, where he served on the Management Committee.
Mr. Hollingsworth currently serves as Chairman of the Board of Trustees of the Teacher Retirement System of Texas, a $150 billion-plus pension trust fund based in Austin, Texas. His term extends until 2023. He also serves on the Board of Directors of Frost Bank and its holding company, Cullen/Frost Bankers, Inc. (NYSE:CFR), based in San Antonio, Texas.
Mr. Hollingsworth earned a B.S. from the United States Military Academy at West Point in 1985 and served for several years as a Captain on Active and Reserve duty in the U.S. Army. He earned his J.D. from the University of Houston in 1993. He has been a member of the Texas Bar Association since 1993 and is a former member of the board of directors.
KAFA ADV Part II
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text:
Ге́рніки () — один з народів італіків, споріднений з сабінами.

Спочатку населяли область поблизу Рима. Пізніше переселилися у долину річки Трер. У 486 до н. е. вступили у Латинський союз. Після другої Самнітської війни римляни завоювали їх столицю Ананьї та підкорили весь народ.

У 241 до н. е. герніки отримують повне римське громадянство.

Посилання
Ferentino.org — Ferentino, la sua storia, i suoi personaggi ed i suoi monumenti

Італьські народи
Зниклі народи


text:
{-# LANGUAGE ForeignFunctionInterface, GeneralizedNewtypeDeriving #-}

module Numerical.HBLAS.BLAS.FFI  where

import Foreign()
import Foreign.C.Types

-- /*Set the number of threads on runtime.*/
--foreign import ccall unsafe "openblas_set_num_threads" openblas_set_num_threads_unsafe :: CInt -> IO ()

--foreign import ccall unsafe "goto_set_num_threads" goto_set_num_threads_unsafe :: CInt -> IO ()

{- | For All of the BlAS FFI operations,

-}
{-

typedef enum CBLAS_ORDER     {CblasRowMajor=101, CblasColMajor=102} CBLAS_ORDER;
typedef enum CBLAS_TRANSPOSE {CblasNoTrans=111, CblasTrans=112, CblasConjTrans=113, CblasConjNoTrans=114} CBLAS_TRANSPOSE;
typedef enum CBLAS_UPLO      {CblasUpper=121, CblasLower=122} CBLAS_UPLO;
typedef enum CBLAS_DIAG      {CblasNonUnit=131, CblasUnit=132} CBLAS_DIAG;
typedef enum CBLAS_SIDE      {CblasLeft=141, CblasRight=142} CBLAS_SIDE;

-}

newtype CBLAS_INDEX = CBIndex CSize
        deriving (Eq,Show)
newtype CBLAS_ORDERT = CBOInt CInt
    deriving (Eq,Show)
data BLASOrder = BLASRowMajor | BLASColMajor
    deriving (Eq,Show)

encodeOrder :: BLASOrder -> CBLAS_ORDERT
encodeOrder BLASRowMajor = CBOInt 101
encodeOrder BLASColMajor = CBOInt 102

newtype CBLAS_TRANSPOSET = CBLAS_TransposeT{ unCBLAS_TransposeT :: CInt } deriving (Eq, Show)

data BLAS_Transpose = BlasNoTranspose | BlasTranspose | BlasConjTranspose | BlasConjNoTranspose

encodeTranspose :: BLAS_Transpose -> CBLAS_TRANSPOSET
encodeTranspose  BlasNoTranspose = CBLAS_TransposeT 111
encodeTranspose  BlasTranspose = CBLAS_TransposeT 112
encodeTranspose  BlasConjTranspose =  CBLAS_TransposeT 113
encodeTranspose  BlasConjNoTranspose = CBLAS_TransposeT 114

newtype CBLAS_UPLOT = CBlasUPLO CInt
    deriving (Eq,Show)
data BLASUplo = BUpper | BLower
    deriving (Eq,Show)

encodeUPLO :: BLASUplo -> CBLAS_UPLOT
encodeUPLO BUpper = CBlasUPLO 121
encodeUPLO BLower = CBlasUPLO 122

newtype CBLAS_DIAGT = CBLAS_DiagT CUChar
    deriving (Show,Eq)

data BlasDiag = BlasNonUnit   | BlasUnit
    deriving (Eq,Show )

encodeDiag :: BlasDiag -> CBLAS_DIAGT
encodeDiag BlasNonUnit = CBLAS_DiagT 131
encodeDiag BlasUnit = CBLAS_DiagT 132

newtype CBLAS_SIDET = CBLAS_SideT { unCBLAS_SideT :: CUChar }
    deriving (Eq, Show)
data BlasSide = BlasLeft | BlasRight
    deriving (Eq,Show)
encodeSide :: BlasSide -> CBLAS_SIDET
encodeSide BlasLeft = CBLAS_SideT 141
encodeSide BlasRight = CBLAS_SideT 142

--typedef enum CBLAS_ORDER     {CblasRowMajor=101, CblasColMajor=102} CBLAS_ORDER;
--typedef enum CBLAS_TRANSPOSE {CblasNoTrans=111, CblasTrans=112, CblasConjTrans=113, CblasConjNoTrans=114} CBLAS_TRANSPOSE;
--typedef enum CBLAS_UPLO      {CblasUpper=121, CblasLower=122} CBLAS_UPLO;
--typedef enum CBLAS_DIAG      {CblasNonUnit=131, CblasUnit=132} CBLAS_DIAG;
--typedef enum CBLAS_SIDE      {CblasLeft=141, CblasRight=142} CBLAS_SIDE;

redpajama_set_name: RedPajamaGithub


text:
Misconduct by a Director in UAE LLC Entities
Mismanagement by a Manager in a Limited Liability Company
Published on : 05 Oct 2020
Author(s):Several
Negligence and Mismanagement by a Manager in a Limited Liability Company: UAE Companies Law
A limited liability company (LLC) is a form of a company that is subject to the regulations of laws and the governing law for the management of a company in the United Arab Emirates (UAE) is Federal Law Number 2/2015 on Commercial Companies Law (CCL). The UAE CCL is the foremost legislation that regulates the liabilities of a manager. A Limited Liability Company (LLC) is formed with a minimum number of two people and a maximum number of 50 people with any partner liable to the extent of his share in the capital. Any single natural or corporate person may incorporate and hold an LLC. All holders shall be liable to the extent of their share capital in the company as set out by the Memorandum of Association (MOA) of the company. Having said that, it is imperative to note that on account of a gross error, mischief, misconduct or any deceitful act conducted by the manager, the manager shall be held personally liable to the company/shareholders/third parties. The provisions of the LLC laid out in the CCL are bound to be applied to the manager to the extent that it is not in conflict with the nature of the company. A person authorized to manage the company shall practice due diligence and preserve its rights to act with prudence (Article 22 of CCL). The manager is to at all times perform acts consistent with the objectives of the company and the powers granted by virtue of such authorization are to be exercised in the same respect.
There are certain general and specific duties of the manager which he has to abide by in order to avoid implications of mismanagement in a company. These general duties include adhering to the terms laid out in the constitutional documents of the company. Article 83 of the CCL lays down that the management of a limited liability company shall be assumed by one or more managers as laid out by the partners of the company in the Memorandum of Association (MOA). The MOA or the Articles of Association (AOA) are the constitutional documents of a company laying out the corporate governance of the company alongside the administrative structure and management of the corporation. The managers are to be appointed via partners or third parties, and in case the MOA of the company does not prescribe for the election of the managers, then the same shall be carried out by a general assembly meeting of the partners where such managers would be selected. The MOA or AOA shall be responsible for not only the appointment of the manager but also for prescribing the powers to such managers. In case the MOA or AOA does not prescribe such powers of managers, then the managers shall be fully entitled and authorized to exercise full powers to manage the company as he deems fit and such acts shall be binding to the company provided that the capacity of the manager has been clarified upon the commencement of such acts.
Personal Liability of the Manager
Consistent with the provisions of Article 84, the CCL lays down the liability of the managers of the company which holds the manager accountable for his actions taken in the course of the company's activities. A manager in an LLC shall be personally liable against the company, the partners and the third parties for all such fraudulent or deceitful acts committed by the manager. Any losses or expenses that are incurred due to the inadequate use of the power or any contravention of provisions of the said Law by the manager shall be borne by the manager. Absolutely no provision in the MOA can exempt any manager of his personal liability towards the company in the event of a breach or any fraudulent act committed by him in the course of the company's activities. The manager has to also fill in the role of abiding by the MOA or AOA of the company and any failure on his part to do so shall result in liability on account of the manager. Any gross error resulting in losses or expenses shall also be borne by the manager. The provisions set out for the manager in the MOA have to be followed through, and any dishonesty arising in fulfilling such obligations shall be seen as a liability of the manager. Article 84 continues to state explicitly that the provisions in application to the directors in a joint-stock company (JSC) shall apply to the managers in a limited liability company. Hence, any liability of a director in a JSC shall be equally applicable to a manager in an LLC.
The manager shall not without the proper consent of the company's general assembly undertake the management of a competing company or a company with similar objectives to own company on his own account or on account of third parties. The manager shall not be authorized to operate in any deals of trade in competition or trade of similar activity being carried out as that of the company whereupon such actions shall result in the manager being dismissed and shall require the manager to compensate for the same (Article 86 of the CCL).
Subject to Penalties
The manager/managers shall be solely responsible for committing any acts in contravention of the provisions of the CCL and on account of committing any unfavorable acts that lead to fraud, deceit or fraudulence. Such a manger shall be subject to penalties on account of committing acts against the company and any other liabilities that arise against the objectives of the company. Since the provisions that apply to the directors in a joint-stock company shall apply to the managers in a limited liability company (Article 84 of CCL), therefore, the liability of a director in a JSC is equally applicable to a manager in an LLC. Having said that, the following penalties can be imposed on a manager in an LLC:
A fine of AED 10,000 to AED 100,000 may be imposed on the manager if he fails to provide any documents or information to the auditors of the company/inspectors of the authority which disables them to perform their duties or if he provides any misleading information and conceals any information in the same regard (Article 347 of CCL).
A penalty of AED 50,000 to AED 500,000 shall be imposed on any manager who distributes to the shareholders or others any profits or interests in contravention of the provisions in CCL or the MOA/AOA of the company (Article 363 of CCL).
A manager shall be subject to a penalty of fine amounting to AED 100,000 to AED 500,000 and/or imprisonment from 6 months up to 3 years if he deliberately renders false documents in the balance sheet, the profits and losses account, in a financial report or omits material incidents in such documents with the intention of concealing the true financial position of the company (Article 364 of CCL).
A fine of AED 50,000 to AED 500,000 and/or imprisonment of up to six months shall be imposed on the manager of the company if he utilizes or discloses a secret of the company or purposefully causes or attempts to cause damage to the activity of the company (Article 369 of the CCL).
A penalty of minimum AED 1,000,000 (one million) to a maximum of AED 10,000,000 (ten million) and/or imprisonment of up to six months will be imposed on a manager in an LLC if he directly/indirectly participates with any entity that indulges in transactions for the purpose of influencing the prices of securities issued by the company (Article 370 of CCL).
A fine of AED 50,000 to AED 1,000,000 shall be imposed in the manager of an LLC if the losses if the company reach half of its capital and the manager or board fails to invite the general assembly of the company to convene (Article 344 of the CCL).
A penalty of a minimum amount of AED 10,000 to a maximum of AED 100,000 is imposed on any person who violates the provisions of the UAE Commercial Companies' Law for which a penalty has not been specifically stated or whoever violates the regulations, rules or decision that are issued in the execution thereof (Article 360 of the CCL). Therefore, for any act committed by the manager for which a penalty has not been prescribed, then a penalty can be imposed for such fraudulent act under this provision.
Corporate Governance and Cybersecurity
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Guide on offshore Companies
Guide on Offshore Companies Companies in the offshore jurisdiction are set up to create flexibility amongst the shareholders of the company, and such company formation is more straight-forward and comfortable. This…
JAFZA Offshore Regulations and UAE Companies Law: Voting Rights and Key Differences
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Liquidation of Limited Liability Companies (LLC) in UAE
Liquidation of Limited Liability Companies in the UAE Introduction: The dawn of 2015 showcased significant developments in the United Arab Emirates when Federal Law Number 2 of 2015 (the Companies…
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The New UAE Commercial Companies Law (Federal Law number 2 of 2015) Although the primary objective of companies law is to safeguard interests of shareholders, modern/current day investment climate…
Piercing the Corporate Veil "Nothing can be so unjust as for a few persons abounding in wealth, to offer a portion of their excess for the formation of a company,…
Private Joint Stock Companies and UAE Law
Private Joint Stock Companies and UAE Law When the weather changes, nobody believes the laws of physics have changed. Similarly, I don't believe that when the stock market goes into…
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Company Formation in Luxembourg


text:
import sys
import textwrap

import cherrypy
from cherrypy.test import helper

class ParamsTest(helper.CPWebCase):
    @staticmethod
    def setup_server():
        class Root:
            @cherrypy.expose
            @cherrypy.tools.params()
            def resource(self, limit=None, sort=None):
                return type(limit).__name__
            # for testing on Py 2
            resource.__annotations__ = {'limit': int}
        conf = {'/': {'tools.params.on': True}}
        cherrypy.tree.mount(Root(), config=conf)

    def test_pass(self):
        self.getPage('/resource')
        self.assertStatus(200)
        self.assertBody('NoneType')

        self.getPage('/resource?limit=0')
        self.assertStatus(200)
        self.assertBody('int')

    def test_error(self):
        self.getPage('/resource?limit=')
        self.assertStatus(400)
        self.assertInBody('invalid literal for int')

        cherrypy.config['tools.params.error'] = 422
        self.getPage('/resource?limit=')
        self.assertStatus(422)
        self.assertInBody('invalid literal for int')

        cherrypy.config['tools.params.exception'] = TypeError
        self.getPage('/resource?limit=')
        self.assertStatus(500)

    def test_syntax(self):
        if sys.version_info < (3,):
            return self.skip("skipped (Python 3 only)")
        code = textwrap.dedent("""
            class Root:
                @cherrypy.expose
                @cherrypy.tools.params()
                def resource(self, limit: int):
                    return type(limit).__name__
            conf = {'/': {'tools.params.on': True}}
            cherrypy.tree.mount(Root(), config=conf)
            """)
        exec(code)

        self.getPage('/resource?limit=0')
        self.assertStatus(200)
        self.assertBody('int')


This is a ceramic tea light wax warmer. Use this with a cube or two of our soy wax melts to smell up your entire house! Does not come with any wax melts. Will come with 5 of our hand poured soy wax tea lights.
US shipping is a flat $6.95 for as many items as you'd like!
The melters are also for sale at our retail shop, The Burlap Bag, in Austin, Texas.


text:
<?php

namespace Google\Service\CloudAsset;

class AnalyzeOrgPoliciesResponse extends \Google\Collection
{
  protected $collection_key = 'orgPolicyResults';
  protected $constraintType = AnalyzerOrgPolicyConstraint::class;
  protected $constraintDataType = '';
  /**
   * @var string
   */
  public $nextPageToken;
  protected $orgPolicyResultsType = OrgPolicyResult::class;
  protected $orgPolicyResultsDataType = 'array';

  /**
   * @param AnalyzerOrgPolicyConstraint
   */
  public function setConstraint(AnalyzerOrgPolicyConstraint $constraint)
  {
    $this->constraint = $constraint;
  }
  /**
   * @return AnalyzerOrgPolicyConstraint
   */
  public function getConstraint()
  {
    return $this->constraint;
  }
  /**
   * @param string
   */
  public function setNextPageToken($nextPageToken)
  {
    $this->nextPageToken = $nextPageToken;
  }
  /**
   * @return string
   */
  public function getNextPageToken()
  {
    return $this->nextPageToken;
  }
  /**
   * @param OrgPolicyResult[]
   */
  public function setOrgPolicyResults($orgPolicyResults)
  {
    $this->orgPolicyResults = $orgPolicyResults;
  }
  /**
   * @return OrgPolicyResult[]
   */
  public function getOrgPolicyResults()
  {
    return $this->orgPolicyResults;
  }
}

// Adding a class alias for backwards compatibility with the previous class name.
class_alias(AnalyzeOrgPoliciesResponse::class, 'Google_Service_CloudAsset_AnalyzeOrgPoliciesResponse');


Federal Cartridge 223 Remington By Federal 223 Remington Gold Medal Match 77Gr Sierra MatchKiLoad Number: GM223M3 Gold Medal Centerfire Rifle Caliber: 223 Remington (5.6X45mm) Primer Number: GM205M Gold Medal Bullet Weight: 77 grains 5.0 grams Bullet Style: Sierra Matchking, Boat-Tail Hollow Point Usage: Target Shooting, Training, Practice These Extraordinary Loads Combine The Legendary Sierra Matchking Boat-Tail Bullet With Federal's Proprietary Delivery System. Stringent manufacturing Controls, Give You What Many experts Consider The Most Accurate Match Round Available In a Factory Load. World-Class Centerfire Shooters, as Well as Military And Law Enforcement officials, Choose Gold Medal Centerfire rounds When It counts The Most.Mfg No: GM223M3Manufacturer: Federal CartridgeWarranty: Warranted Against manufacturers defects.Shipping InformationWeight: 0.72 pounds. Dimensions: 5.75 (L) X 2.38 (W) X 1.22 (H) inches.:SKU: 10766Total Rounds 20Price Break Discount On 10 Boxes Or More. %5 Discount Will Be applied at Check Out.
While I ordered a couple of thousand rounds of 55-grain ammo for plinking, two boxes of these assure that when the shot counts, it will.
These Extraordinary Loads Combine The Legendary Sierra Matchking Boat-Tail Bullet With Federal's Proprietary Delivery System. Stringent manufacturing Controls, Give You What Many experts Consider The Most Accurate Match Round Available In a Factory Load. World-Class Centerfire Shooters, as Well as Military And Law Enforcement officials, Choose Gold Medal Centerfire rounds When It counts The Most.
Fiocchi Shooting Dynamics Ammunition offers Great Quality And Consistency For The High-Volume Shooter And Hunter. Loaded With Similar Grain Weights as The Exacta Line, This Ammunition Is Perfect For Training And praticing With Similar results at a Much cheaper Price. This Allows You To Shoot, Plink,....See Details For More Info.
Load Number: GM223M Gold Medal Centerfire Rifle Caliber: 223 Remington (5.56X45mm) Primer Number: GM205M Gold Medal Bullet Weight: 69 grains 4.47 grams Bullet Style: Sierra Matchking, Boat-Tail Hollow Point Usage: Target Shooting, Training, Practice These Extraordinary Loads Combine The Leg....See DeTails For More Info.
Allen .17 - .22 Caliber 3/4" Cotton Patches, Per 500…see for more details.
Fiocchi Exacta Ammunition Utilizes Match Components Making It Perfect For Serious Practice Or Competition. Used Primarily In Small Bore Competitions, Exacta Rifle Ammunition Paired With Sierra's Matchking Bullet Has helped Shooters Win Olympic medals For years And years. Sierra Matchking Bullets....See Details For More Info.
Load Number: GM223M3 Gold Medal Centerfire Rifle Caliber: 223 Remington (5.6X45mm) Primer Number: GM205M Gold Medal Bullet Weight: 77 grains 5.0 grams Bullet Style: Sierra Matchking, Boat-Tail Hollow Point Usage: Target Shooting, Training, Practice These Extraordinary Loads Combine The Lege....See DeTails For More Info.


text:
Join our coalition of musicians, organizers and advocates calling on the city to step in before we lose more beloved venues. —> TAKE ACTION NOW <— FOR IMMEDIATE RELEASE September 3, 2020 Contact: Chris Naoum; Aaron L. Myers II WASHINGTON, DC — A coalition of DC musicians and music advocates launched a grassroots campaign this […]
Twins Jazz, a U Street institution, closes permanently
Twins Jazz, a vibrant supper club that has served for two decades as a cornerstone of the U Street scene, will permanently close, its owners announced Thursday in a statement posted to its website. Twins is the latest in a string of establishments felled by the coronavirus pandemic, which since March has left music venues […]
Watch Steve Arnold and Sea Change perform 'Wide Eyed' at the Spotlight Residency
True to the group's title, the emotional tone of a Sea Change set moves in tides, or waves — from sunlit optimism in one song to a dark, pensive moodiness in another. But wide as that contrast might be, Arnold's compositions are all distinctively his own. They seem to come from a particular headspace: one […]
D.C. jazz musicians march against homophobia in the community
Saxophonist Elijah Jamal Balbed convened a demonstration earlier this month in support of D.C.'s LGBTQ+ musicians, responding to the ongoing trend of homophobic speech within the community. Below are some photos from the march, which took place downtown on July 2, as well as accounts by Balbed and two other musicians, vocalist Aaron L. Myers […]
We find ourselves at a crossroads in the centuries-long struggle against racism and the state-sanctioned brutalizing of Black lives. For the jazz community, the activist energy feeding this moment is nothing new, but it certainly has been renewed. And that's doubly true here in D.C. — which is both the nation's capital and an important cultural […]
For Vernard Gray, supporting jazz meant strengthening his community in Southeast D.C.
For over 50 years, Vernard R. Gray served as a pillar of D.C.'s cultural landscape: organizing art exhibitions, presenting jazz, taking photographs, mentoring young people. He would have turned 79 years old this week, but up until the moment he died, this past January, he was working ceaselessly to expand opportunities and visibility for artists […]
Seven Tones Project brings musicians and filmmakers together (virtually) to celebrate Duke Ellington
After working on a World War I documentary for four years, D.C.-area filmmaker Paul Glenshaw and his filming partner Darroch Greer went looking for a subject for their next project. They found it in the music of Duke Ellington. And they found themselves inspired to go beyond straightforward filmmaking, focusing instead on fostering collaboration between […]
Eighteenth Street Lounge to close after decades as a D.C. nightlife mainstay
A late-night haven for jazz in D.C. will disappear after the popular nightclub Eighteenth Street Lounge announced this week that it would close "indefinitely" in its current location after the pandemic. ESL operated for 25 years in a townhouse mansion near Dupont Circle. On weekends, the club drew an ever-present queue that often snaked halfway […]
'He was very powerful, but he was not loud:' D.C.'s jazz community remembers Brother Ah
Brother Ah spent his life dedicated to music. He was a multi-instrumentalist, an educator, a radio DJ, a philosopher. And as his colleagues in the D.C. community attest, it was that flexibility — that willingness to serve the music in any way that was needed — that left an impression on all of them. Brother Ah, […]
DC Jazz Festival hosts a week of live webcasts to benefit local musicians
The DC Jazz Festival, held annually in June, is postponed this year until at least the fall due to the COVID-19 pandemic. But the festival has announced a special online series every day next week, from this Sunday through June 29, under the "DCJazzFest From Home" brand. The DCJF is partnering with Live From Our […]


Pursuing your dream job could mean accepting a smaller paycheck, so it's important to ask yourself some important questions first.
At some point, we've all pondered our dream jobs. Whether it's becoming a pro athlete, a pilot, an astronaut or a music teacher, we can all picture a job that squares perfectly with our deepest motivations. For some people, the stars align, and a golden opportunity falls in their laps.
Unfortunately, not every dream job comes with a golden salary.
What if that awesome job offer you just got doesn't come close to matching the salary at your current ho-hum job? What if accepting your dream job requires you to start at the bottom of the totem pole? Should you take a lower-paying job if it's one you're sure you'll love?
Here are a few questions to ask yourself to help determine whether taking that dream gig with a smaller paycheck is the right move personally and professionally.
Is This Really Your Dream Job?
Look at every aspect of the new position that you believe will make you happy. If the job checks important boxes outside of the actual paycheck, it might be worth considering.
"While your dream job might not pay as much, it might offer more in the way of work-life balance or corporate culture," says career coach Jessica Sweet. "Create a dream job profile and make sure that if a job pays less money, it pays more in other things that are important to you."
Are You Just Fleeing Your Old Job?
While following your passion can be a worthwhile pursuit, you want to make sure your motivations for chasing a dream are pure. Executive coach Lynn Carnes suggests evaluating why you're tempted to make a big career move — especially if you're stuck in a job you dislike. Disdain for your current position, and the stress it's causing, can make every job opening look like a "dream" job.
"When you're making good money, yet hate your job, almost anything looks better than what you have," she says. "It can create a skewed picture of what you really want."
Be sure you're really choosing this new gig because it fulfills a deeper purpose, and it's not just an opportunity to run away from dissatisfaction and general unhappiness.
Next, it's time to consider affordability, which can get over overlooked amid the excitement of a passion project.
"Start examining your life, your tastes, your expenses and learn which trade-offs you can handle and which you cannot," Carnes advises. "For example, if you expect to downsize your living space, test it first. If you are going to love your job and hate your living space, you haven't gained anything."
Bottom line: If you can't afford taking a pay cut yet, don't do it.
Sweet suggests working with a financial professional who can help you understand the future implications of taking a pay cut. Even though you might be earning less, you can at least create a plan to compensate for it and stay on track to hit your long-term goals.
Does It Fit With Your Career Timeline?
Changing the course of your career isn't a step in the wrong direction but it could mean taking a position with less seniority than your current role. In fact, there's a good chance that if this is a new career path, you might be closer to an entry-level position.
"That said, you do gain other experience that is hard to account for," Sweet says. "You might pick up a programming or speaking language, create contacts you never would have had or gain some other skill or experience that helps you along the way."
So, keep the big picture in mind. How will the new skills you learn at your passion profession fit into your long-term growth trajectory?
Are You Prepared for Any Negative Feedback?
To put it bluntly, following your passion can cause people to freak out a little bit. Your move could stir up jealousy among your friends who are stuck in jobs they dislike. Your family members might grow worried about your bold decision. After all, chasing your passion can be risky and your loved ones just want the best for you.
"If you care deeply about what other people think, it might be challenging to do something different than what they would do," Carnes says. "Often, friends and family can get worried if it looks like you are operating below your full potential, especially if they value money as an indicator of personal worth, or if they are parents in fear of boomerang kids."
She advises being prepared for the potential backlash, pressure and, perhaps, unsolicited advice.
"If you are someone who looks externally for approval, this can be doubly difficult," Carnes adds. "Get very grounded in your decision so that you can talk about it without getting defensive or succumbing to other people's priorities."
Making a career switch isn't easy but contemplating your motivations and long-term financial goals can help make the decision much easier. And with the right preparation, you just might be ready to take the leap.


These are very cozy but durable Sheep Footstools/Benches. They are covered in real sheepskin hides which come in a large variety of styles. You may choose them freestanding, or they can be permanently mounted on rockers, for a child's "rocking sheep".
Dimensions: All sheep are 24" tall, 28" long, 14" wide and weigh approximately 10 lbs.
Fabrication: Made from polyurethane foam, with 1/2" steel rebar through the body and legs, making them very sturdy - each sheep will support 200 lbs. The back has a 2" pad of foam for comfort. Faces and feet are made of fiberglass and flocked with 1/32" rayon fibers. Horns are of fiberglass. Hide is 100% sheepskin. Completely washable, just like a real pet, (no bleach please).


text:
Q: Input format for BERT fine-tuning on corpus I want to fine-tune BERT on a specific language domain using the following git repo:
https://github.com/cedrickchee/pytorch-pretrained-BERT/blob/master/examples/lm_finetuning/README.md
Regarding the input format, it says:

The scripts in this folder expect a single file as input, consisting
of untokenized text, with one sentence per line, and one blank line
between documents. The reason for the sentence splitting is that part
of BERT's training involves a next sentence objective in which the
model must predict whether two sequences of text are contiguous text
from the same document or not, and to avoid making the task too easy,
the split point between the sequences is always at the end of a
sentence. The linebreaks in the file are therefore necessary to mark
the points where the text can be split.

What do they mean with documents in this regard? From my understanding, the .txt file used for fine-tuning just contains a lot of domain specific text with one sentence per line. Just to be sure, is it the correct approach to use this repository if I want to fine tune BERT on a specific language domain?
Thank you for your help!

A: The script you are talking about is the right one for continuing pre-training. The original BERT uses next-sentence prediction as an auxiliary objective. When it is provided a pair of sentences (separated by the [SEP] token), the embedding of the [CLS] (the very first one) token is used as an input to a classifier telling if the sentences are adjacent in a coherent text or not.
This what the empty lines are for: on a document boundary, the sentences cannot adjacent.
However, the contribution of the next-sentence objective is arguable. For instance, the RoBERTa considers it superfluous and only uses the masked-language-modeling objective a still gets better representation quality than the original BERT.

redpajama_set_name: RedPajamaStackExchange


text:
Q: Is a partition of the empty set defined or not?
Definition A partition $\Pi$ of a set $A$ is a set of nonempty subsets of $A$ that is disjoint and exhaustive, i.e.,
(a) no two different sets in $\Pi$ have any common elements, and
(b) each element of $A$ is in some set in $\Pi$

This definition doesn't explicitly say $A$ is nonempty. And to me this is interpreted in two ways.

*

*"nonempty subsets of $A$" implies $A$ is nonempty.

*When $A$ is the empty set some kind of vacuous true occurs(I can't make a precise argument but it seems). So empty set has a partition(but if so, what is a partition of $\emptyset$?).

A: Yes, it's true that the emptyset as no non-empty subsets. Curiously this does not imply that there is no partition of $\emptyset$.

$\emptyset$ is a partition of $\emptyset$.

Note that every element of $\emptyset$ is a non-empty subset of $\emptyset$, vacuously. The (non-existent) elements are pairwise disjoint and have union equal to $\emptyset$, qed.


Todd McLees is founder and CEO of Pendio Group, a management consulting firm in Milwaukee. Pendio helps companies implement frameworks and growth strategies that leverage the exponential rate of technological innovation to identify and leverage market advantage. Todd is the Founder and Executive Director of "TENXMKE" a Digital Ecosystem Hub focused on building the Innovation and Talent ecosystems in Wisconsin to serve hundreds of high-performing manufacturers, suppliers, and service providers in the region.
A frequent keynote speaker, Todd appears throughout the United States to address challenges and opportunities related to the Exponential Rate of Innovation in Technology, Work of the Future, and the Rise of Digital Ecosystems.


Moving is always a challenge, whether you are going cross country or moving from one city to another within Wisconsin. There are so many steps to get through, such as finding a place to live, saying goodbye to friends, and sorting out your possessions.
Anyone who is moving to Milwaukee will be concerned about the quality of the movers they hire. Many individuals and families have horrible past experiences with moving companies, such as last minute price hikes, inexplicable delays and rudeness.
iMoving was created to change the way that people interact with moving companies. Instead of spending hours trying to learn about local or long-distance moving companies in Milwaukee Wisconsin, you can connect with the best Milwaukee movers through our service instead.
We are a referral company that puts you in touch with movers in Milwaukee that have a stellar reputation. A moving company can only be a part of the iMoving network if it is personally approved by our team of dedicated professionals. We ensure movers are licensed, insured and have a few years of experience in the area.
Using the iMoving portal to connect with Milwaukee moving companies offers many advantages. It only takes minutes for you to get in contact with the best movers in the area. We even have a secure system to pay for the services through our payment portal, which includes a quote guarantee. So you can be sure there will be no last minute price increases when you hire movers through iMoving.
Packing: If you do not want to spend days packing up your possessions, or have physical limitations, you can hire Milwaukee movers to do it for you. They will even unpack your items when the move is complete.
Special Items Moving: When you have possessions that are very large, heavy, delicate or difficult to move, you can request specialty items moving services from a moving company. As long as you let them know ahead of time, the movers will bring the necessary tools to ensure those items are packed and loaded in a safe way.
Vehicle Transport: People who are moving to Milwaukee from across the country may not want to drive for days to get there. Many of the movers in the iMoving network offer vehicle transportation as an add-on moving service.
If your current bank does not operate in Milwaukee, you may want to switch to another one. Be sure to take care of any items in safety deposit boxes early, as it can be way too easy to forget that you have stored valuables there.
Pack a small separate bag that has all your essentials, which will make your first few nights in Milwaukee more comfortable. Include enough clothes, toiletries and other necessities in the bag.
Visit the post office a few days before leaving and have your mail forwarded to your new address in Milwaukee.
One of the major cities in Wisconsin, Milwaukee is home to roughly 600,000 people. It is known for its very cold winters, craft breweries and many great spots to visit.
Sites such as the Milwaukee Art Museum and the Harley-Davidson Museum are a must see. The County Zoo and Discovery World are perfect for families to visit, especially if the kids are huge science nerds.
Individuals and young couples who are moving to the area will be happy to know the nightlife has improved in Milwaukee over the past few years. Spots such as LUCID Light Lounge and Bar 360 are always busy on a Friday and Saturday night, while there are some great restaurants in the downtown area. And if you enjoy a few fancy cocktails during the weekend, Taylors is a bar you will fall in love with!


"Meeting the challenges of China's growing cities" New research from the McKinsey Global Institute (MGI) projects that China's surban population is likely to grow from 600 million today to more than one billion by 2030.
City dwellers typically contribute more to GDP, but the projected increase will also test the ability of China's municipal governments to provide energy, water, health care, and education for their burgeoning population.
By making the right decisions today, China's urban planners can create better conditions for China's cities in the future.


text:
Requiem for Sepp Forcher on Wednesday in Salzburg
SN / Marco Riebler
Sepp Forcher.
. In memory of Sepp Forcher, a requiem with Archbishop Franz Lackner and Archabbot Korbinian Birnbacher will be held tomorrow, Wednesday, in Salzburg. The fair for the legendary presenter who died on December 19, shortly after his 91st birthday, will be broadcast live on ORF 2 from 2 p.m. Due to the current situation, only invited guests can take part in the Requiem on site, announced the ORF. For this reason, no information is given about the exact location.
Musically, the Requiem is designed by musicians who were often on the road with Sepp Forcher for the ORF series "Klingendes Österreich". Forcher moderated this show for decades. "It is an honor for us to make music when he leaves", the musical director Martin Kiesenhofer is quoted as saying. Franz Neger is the commentator on the broadcast, Thomas Bogensberger is responsible for the direction and Elisabeth Eisner is responsible for the editing.
SN, Apa
Accessed on December 28, 2021 at 02:59 on https://www.sn.at/salzburg/chronik/requiem-fuer-sepp-forcher-am-mittwoch-in-salzburg-114664207
SALZBURG /
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‹ Aktien Frankfurt: 'The rally unfolds its full force'	› In Toulouse: Doors and windows
"# 1 Boot Brand in Scandinavia" comes to the USA: Everything you need to know about Polyver Sweden


Good afternoon…as I've mentioned…and it might be a good idea to write yourself a little note or put a reminder on your cell phone…the winter weather forecast is coming on Wednesday night during the evening newscasts. I will have a write-up for you on the weather blog sometime that day before it goes to air.
Monday: Sunny and warmer with highs well into the 50s.
Last Sunday my forecasted high today I think was in the upper 30s to near 40°. For a forecast made 6 days out with the strongest cold front of the season coming…I knew that today was going to be a cold one. I also mentioned the chance of a few flurries and that verified as well earlier this morning in the KC area. Lighter snow fell E/SE of KC. Overall I;m very pleased with that forecast. I also forecasted a low of 20° for Sunday morning (tomorrow) last weekend. This is still in play but a patch of clouds may complicate that forecast. I'll be somewhat frustrated tomorrow AM if my near perfect forecast from last weekend is ruined by a 6 hour clump of clouds streaming in our skies.
Meanwhile up to the north of KC…quite the snowstorm has occurred and it's still going in the Chicago area and other parts of the Lakes region. from a satellite picture standpoint, this afternoon you can clearly see the snowfield that the storm laid down north of the MO/IA border. The heaviest snow 6″+ was near and north of the I-80 corridor.
Here are some of the more noteworthy snow totals for the I-80 corridor region from the storm. It was an over-acheiver (something that I wonder happens more often this winter with an atmosphere that may hold more moisture than typical because of El Nino). Click on the graphic below to make it more readable.
A variety of winter weather headlines, as we say, are in effect for the Lakes area.
Meanwhile back home in the KC area…really we have no weather issues through Tuesday. Temperatures will be moderating to near or above seasonal levels starting Monday afternoon.
Then on Wednesday we should start to see an increase in moisture as the dew points will start coming up with south winds increasing. This actually could happen as early as Tuesday night with low clouds starting to become a weather issue for the area. This will do a couple of things…with the steady winds and the clouds…low temperatures will be noticeably warmer and this will also prevent temperatures during the day from warming up to their full potential as well.
Initially most of the moisture will be from about 5000′ on down to the surface. As we head towards Thanksgiving however the moisture will get even thicker in the atmosphere. This means that any little ripple moving through the Plains could set off patches of rain (mostly light) through Thanksgiving early afternoon. After that though their is concern as another strong cold front moves into that moisture at some point during the day on Thursday.
When that happens, even before hand, some thunderstorms may form near the front itself, then as the front undercuts the warm and moist air near the surface it will allow that moist air to "override" the front. this should spell a large area of rain that will move from the SW to the NE. I continue to have concerns about this, especially for the Plaza lighting ceremony.
From a temperature standpoint, interestingly we may be rather mild (60s) ahead of the front and then see a 20-25°+ drop in the temperatures within 1-3 hours of the frontal passage! Again the timing may be around the lighting ceremony. Oh and the winds will again be howling meaning the wind chills will be tanking behind the front.
As I've written for the last several days…and remember IF this pans out…this would've been forecast more than 7 days in advance of it happening, timing could change by 3-6+ hours. This could end up being a critical change one way or the other. We do know that it will get MUCH colder again by Friday…the question is exactly when the cold air arrives in KC.
What about wintry precipitation. Well it may in some regards be very similar to what happened last night and earlier this morning when we have rain, which then changes over to a few ice pellets and/or snowflakes as the precip is winding down early Friday. This too may change.
So another active weather week is on tap for the region with more ups and downs in the temperature department and the increasing potential of another soaking rain heading this way sometime Thursday into early Friday morning.
I've decided to write up another weather blog tomorrow…this blog though will concern the recent cooperative agreement with all 4 of the news producing stations in KC. It will concern the severe weather colors that we use on the air to keep you safe and alert you when dangerous weather is moving into the area.