text:
\section{Introduction}

The electroweak sector of the standard model
harbours unstable non-perturbative classical solutions: sphalerons
\cite{Manton:1983nd,Klinkhamer:1984di,Kleihaus:1991ks,Kunz:1992uh}.
The static sphaleron solution of Weinberg-Salam theory represents
the top of the energy barrier between topologically inequivalent vacua.
Since the standard model does not absolutely conserve
baryon number \cite{'tHooft:1976up,Ringwald:1989ee}, at finite temperature
baryon number violating processes can arise because of
thermal fluctuations of the fields
large enough to overcome the energy barrier between
distinct vacua.
The rate for baryon number violating processes
is then largely determined by a Boltzmann factor,
containing the height of the barrier at a given temperature
\cite{McLerran:1993rv,Rubakov:1996vz,Klinkhamer:2003hz,Dine:2003ax}.
The sphaleron itself carries baryon number $Q_{\rm B}=1/2$
\cite{Klinkhamer:1984di}.

At finite weak mixing angle the static electroweak sphaleron
possesses a large magnetic moment but does not carry electric charge.
As argued before \cite{Saffin:1997ae}
and demonstrated recently in nonperturbative studies
\cite{Radu:2008ta,Kleihaus:2008cv},
the addition of electric charge
leads
to a non-vanishing Poynting vector and thus a finite angular momentum
density of the system.
Consequently
a branch of spinning electrically charged sphalerons arises.
Since these carry non-vanishing baryon number as well,
they also entail baryon number violating processes.

Beside the sphaleron,
the non-trivial topology of the configuration space of Weinberg-Salam theory
gives rise to further unstable classical solutions.
A superposition of $n$ sphalerons, for instance,
can lead to static axially symmetric solutions, multisphalerons,
which carry baryon number $Q_{\rm B}=n/2$ and
whose energy density is torus-like
\cite{Brihaye:1994ib,Kleihaus:1994yj,Kleihaus:1994tr}.
A superposition of a sphaleron and an antisphaleron,
on the other hand, can give rise to a bound
sphaleron-antisphaleron system, in which a sphaleron and an
antisphaleron are located at an equilibrium distance
on the symmetry axis
\cite{Klinkhamer:1985ki,Klinkhamer:1990ik,Klinkhamer:1993hb}.
Such a sphaleron-antisphaleron pair has vanishing baryon number,
$Q_{\rm B}=0$, since the antisphaleron carries $Q_{\rm B}=-1/2$.
The sphaleron-antisphaleron pair therefore does not mediate
baryon number violating processes.

Recently,
the sphaleron-antisphaleron pair solutions have been generalized,
to form sphaleron-antisphaleron chains,
where $m$ sphalerons and antisphalerons are located
on the symmetry axis, in static equilibrium
\cite{Kleihaus:2008gn},
in close analogy to the monopole-antimonopole chains
encountered in the Georgi-Glashow model
\cite{Kleihaus:2003nj}.
When multisphalerons and -antisphalerons
are considered instead, vortex ring solutions arise,
when $n \ge 3$ \cite{Kleihaus:2008gn},
where the Higgs field vanishes not (only) on isolated points
on the symmetry axis but (also) on one or more rings,
centered around the symmetry axis
\cite{Kleihaus:2003xz,Kleihaus:2004is}.

Here we consider
the addition of electric charge
to such multisphaleron and sphaleron-antisphaleron systems.
As for the simple sphaleron,
the non-vanishing Poynting vector leads to a finite angular momentum
density for these configurations.
Thus branches of rotating electrically charged sphalerons
emerge from the respective static electrically neutral
configurations.
We construct these solutions explicitly and discuss their properties.
The angular momentum and the electric charge of the solutions
are proportional \cite{VanderBij:2001nm,Radu:2008ta,Kleihaus:2008gn}.

In section 2 we present the action, the Ansatz and the
boundary conditions.
In section 3 we consider the
relevant physical properties and, in particular,
derive the linear relation
between angular momentum and electric charge.
We present and discuss the numerical results in section 4.
These include global properties of the solutions,
such as their energy, their angular momentum, their charge
and their magnetic moments,
but also local properties, such as their
energy density, their angular momentum density
and the modulus of their Higgs field.
Moreover, we discuss the equilibrium conditions
for the solutions.
We give our conclusions in section 5.

\section{Action and Ansatz}

We consider the bosonic sector of Weinberg-Salam theory
\begin{equation}
{\cal L} = -\frac{1}{2} {\rm Tr} (F_{\mu\nu} F^{\mu\nu})
-  \frac{1}{4}f_{\mu \nu} f^{\mu \nu}
- (D_\mu \Phi)^{\dagger} (D^\mu \Phi)
- \lambda (\Phi^{\dagger} \Phi - \frac{v^2}{2} )^2
\
\label{lag1}
\end{equation}
with su(2) field strength tensor
\begin{equation}
F_{\mu\nu}=\partial_\mu V_\nu-\partial_\nu V_\mu
            + i g [V_\mu , V_\nu ]
 , \end{equation}
su(2) gauge potential $V_\mu = V_\mu^a \tau_a/2$,
u(1) field strength tensor
\begin{equation}
f_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu
 , \end{equation}
and covariant derivative of the Higgs field
\begin{equation}
D_{\mu} \Phi = \Bigl(\partial_{\mu}
             +i g  V_{\mu}
             +i \frac{g'}{2} A_{\mu} \Bigr)\Phi
 , \end{equation}
where $g$ and $g'$ denote the $SU(2)$ and $U(1)$ gauge coupling constants,
respectively,
$\lambda$ denotes the strength of the Higgs self-interaction and
$v$ the vacuum expectation value of the Higgs field.

The gauge symmetry is spontaneously broken
due to the non-vanishing vacuum expectation
value of the Higgs field
\begin{equation}
    \langle \Phi \rangle = \frac{v}{\sqrt2}
    \left( \begin{array}{c} 0\\1  \end{array} \right)
 , \label{Higgs} \end{equation}
leading to the boson masses
\begin{equation}
    M_W = \frac{1}{2} g v   , \ \ \ \
    M_Z = \frac{1}{2} \sqrt{(g^2+g'^2)} v  \, , \ \ \ \
    M_H = v \sqrt{2 \lambda} \,  .
\end{equation}
$ \tan \theta_{\rm w} = g'/g $ determines
the weak mixing angle $\theta_{\rm w}$,
defining the electric charge $e = g \sin \theta_{\rm w}$.
We also denote the weak fine structure constant $\alpha_{\rm W}=g^2/4\pi$.

To obtain stationary rotating solutions of the bosonic sector
of Weinberg-Salam theory,
we employ the time-independent axially symmetric Ansatz
\begin{equation}
V_\mu\, dx^\mu
  = \left( B_1\, \frac{\tau^{(n,m)}_r}{2g}
         + B_2\, \frac{\tau^{(n,m)}_\theta}{2g} \right) \, dt
            -n\sin\theta\left(H_3 \frac{\tau^{(n,m)}_r}{2g}
            + H_4 \frac{\tau^{(n,m)}_\theta}{2g}\right)\, d\varphi
+\left(\frac{H_1}{r}\, dr +(1-H_2)\, d\theta \right)
  \frac{\tau^{(n)}_\varphi}{2g}
  , \label{a1} \end{equation}

\begin{equation}
A_\mu\, dx^\mu = \left( a_1\, dt + a_2\, \sin^2 \theta \, d\varphi \right)/g'
  , \end{equation}
and
\begin{equation}
\Phi = i\,
      \left( \phi_1 \, \tau^{(n,m)}_r
           + \phi_2  \tau^{(n,m)}_\theta \right)
    \frac{v}{\sqrt2} \left( \begin{array}{c} 0\\1  \end{array} \right)
  . \end{equation}
where
\begin{eqnarray}
\tau^{(n,m)}_r & = & \sin m\theta (\cos n\varphi \tau_x + \sin n\varphi \tau_y)
           + \cos m\theta \tau_z \ , \ \
\nonumber \\
\tau^{(n,m)}_\theta & = & \cos m\theta (\cos n\varphi \tau_x + \sin n\varphi \tau_y)
           - \sin m\theta \tau_z \ , \ \
\nonumber \\
\tau^{(n)}_\varphi & = & (-\sin n\varphi \tau_x + \cos n\varphi \tau_y)
\ , \ \ \nonumber
\end{eqnarray}
$n$ and $m$ are integers,
and $\tau_x$, $\tau_y$ and $\tau_z$ denote the Pauli matrices.

The two integers $n$ and $m$ determine the type of configuration,
that is put into rotation.
For $n=m=1$ the solutions correspond to rotating sphalerons.
Rotating multisphaleron configurations arise for $n>1$ and $m=1$.
For $n=1$ and $m>1$ rotating sphaleron-antisphaleron
pairs ($m=2$) or sphaleron-antisphaleron chains arise,
and for $n \ge 3$ rotating vortex ring solutions.

The ten functions $B_1$, $B_2$, $H_1,\dots,H_4$, $a_1$, $a_2$,
$\phi_1$, and $\phi_2$ depend on $r$ and $\theta$, only.
With this Ansatz the full set of field equations reduces to a system
of ten coupled partial differential equations in the independent variables
$r$ and $\theta$. A residual $U(1)$ gauge degree of freedom is
fixed by the condition $r\partial_r H_1 - \partial_\theta H_2=0$
\cite{Kleihaus:1991ks}.

Requiring regularity and finite energy, we impose
for odd $m$ configurations the boundary conditions
\begin{eqnarray}
r=0: &  &
B_1 \sin m \theta + B_2 \cos m \theta =0  , \
\partial_r \left( B_1 \cos m \theta - B_2 \sin m \theta\right) =0  , \
H_1=H_3=H_4=0 , \ H_2=1 , \
\nonumber \\
&  &
\partial_r a_1 =0  , \ a_2=0  , \
\phi_1=0 , \ \phi_2 = 0
\nonumber \\
r\rightarrow \infty: &  &
B_1 = \gamma \cos m \theta  , \ B_2 = \gamma \sin m \theta  , \
H_1=H_3=0 , \ H_2=1-2m  ,  \ H_4=\frac{2\sin m\theta}{\sin\theta} , \
\nonumber \\[-0.7ex]
& &
a_1=\gamma  , \ a_2 = 0  , \
\phi_1=1 , \ \phi_2 = 0 , \
 {\rm where } \ \gamma = const.
\nonumber \\[0.5ex]
\theta = 0: &  &
\partial_\theta B_1 =0  , \ B_2 =0  , \
H_1=H_3=0 , \ \partial_\theta H_2=\partial_\theta H_4=0 , \
\partial_\theta a_1 = \partial_\theta a_2 = 0  , \
\partial_\theta \phi_1 = 0  , \ \phi_2=0  ,
\end{eqnarray}
where the latter hold also at $\theta=\pi/2$,
except for $B_1=0$ and $\partial_\theta B_2 =0$.
For even $m$ configurations the same set of boundary conditions
holds except for
\begin{eqnarray}
r=0: &  &
\phi_1 \sin m \theta + \phi_2 \cos m \theta =0  , \
\partial_r \left( \phi_1 \cos m \theta - \phi_2 \sin m \theta\right) =0
\nonumber \\
\theta=\pi/2: & &
\partial_\theta B_1=0 , \ B_2 = 0 \ , \partial_\theta H_3 = 0 , \ H_4=0 .
\end{eqnarray}

\section{Global charges of sphaleron-antisphaleron systems}

We now address the global charges of the sphaleron-antisphaleron systems,
their mass, their angular momentum, their electric charge,
and their baryon number.
The mass $M$ and angular momentum $J$ are defined in terms of
volume integrals of the respective components
of the energy-momentum tensor. The mass is obtained from
\begin{equation}
M =- \int T_t^t d^3 r ,
\label{Mint}
\end{equation}
while the angular momentum
\begin{equation}
J = \int T_\varphi^t d^3 r
= \int \left[ 2 {\rm Tr} \left( F^{t\mu} F_{\varphi\mu} \right)
 + f^{t\mu} f_{\varphi\mu} + 2 \left( D^t \Phi \right)^\dagger
 \left( D_\varphi \Phi \right)   \right] d^3 r
\label{Jint}
\end{equation}
can be reexpressed with help of the
equations of motion and the symmetry properties of the Ansatz
\cite{VanderBij:2001nm,Kleihaus:2002tc,Volkov:2003ew,Radu:2008pp}
as a surface integral at spatial infinity
\begin{equation}
J =  \int_{S_2} \left\{
 2 {\rm Tr} \left( \left(V_\varphi - \frac{n\tau_z}{2g}\right) F^{r t} \right)
 +
 \left(A_\varphi - \frac{n}{g'}\right) f^{r t}
 \right\} r^2 \sin \theta d \theta d \varphi .
\label{Joint}
\end{equation}
The power law fall-off of the $U(1)$ field
of a charged solution allows for a finite flux
integral at infinity and thus a finite angular momentum.
Insertion of the asymptotic expansion for the $U(1)$ field
\begin{eqnarray}
a_1= \gamma - \frac{\chi}{r} + O \left(\frac{1}{r^2}\right) , \nonumber\\
a_2= \frac{\zeta}{r} + O \left(\frac{1}{r^2}\right) ,
\label{asymp}
\end{eqnarray}
and of the analogous expansion for the $SU(2)$ fields
then yields for the angular momentum
\begin{equation}
\frac{J}{4 \pi} =

\frac{n\chi}{g^2} + \frac{n\chi}{{g'}^2}
= \frac{n\chi}{g^2 \sin^2 \theta_{\rm w}} = \frac{n\chi}{e^2} .
\label{Joint2}
\end{equation}

The field strength tensor ${\cal F}_{\mu\nu}$ of the
electromagnetic field ${\cal A}_{\mu}$,
\begin{equation}
{\cal A}_{\mu}=\sin \theta_{\rm w} V^3_\mu + \cos \theta_{\rm w} A_\mu ,
\label{emfield}
\end{equation}
as given in a gauge where the Higgs field asymptotically tends to
Eq.~(\ref{Higgs}),
then defines the electric charge $\cal Q$
\begin{equation}
{\cal Q} =  \int_{S_2}
  {^*}{\cal F}_{\theta\varphi} d\theta d\varphi
 =4\pi\left\{
 \frac{\sin \theta_{\rm w} \chi}{g} + \frac{\cos \theta_{\rm w} \chi}{g'}
 \right\}
 = 4\pi\frac{\chi}{e}
  , \label{Qel} \end{equation}
where the integral is evaluated at spatial infinity.
Comparison of Eqs.~(\ref{Joint2}) and (\ref{Qel}) then
yields a linear relation between the angular momentum $J$
and the electric charge $\cal Q$ \cite{Radu:2008ta,Kleihaus:2008cv}
\begin{equation}
 J = \frac{n \cal Q}{e}
\label{JQrel}
 . \end{equation}
This relation
corresponds to the relation for
monopole-antimonopole systems without magnetic charge
\cite{Kleihaus:2007vf}.
The magnetic moment $\mu$ is obtained from
the asymptotic expansion Eq.~(\ref{asymp}), analogously to the
electric charge,
\begin{equation}
\mu= \frac{4\pi\zeta}{e} \ .
\end{equation}

Addressing finally the baryon number $Q_{\rm B}$,
its rate of change is given by
\begin{equation}
 \frac{d Q_{\rm B}}{dt} = \int d^3 r \partial_t j^0_{\rm B}
= \int d^3 r \left[ \vec \nabla \cdot \vec j_{\rm B}
 + \frac{1}{32 \pi^2} \epsilon^{\mu\nu\rho\sigma} \, \left\{
 g^2 {\rm Tr} \left(F_{\mu\nu} F_{\rho\sigma} \right)
 + \frac{1}{2} {g'}^2 f_{\mu\nu} f_{\rho\sigma} \right\} \right]  .
\end{equation}
Starting at time $t=-\infty$ at the vacuum with $Q_{\rm B}=0$,
one obtains the baryon number of a sphaleron solution at
time $t=t_0$ \cite{Klinkhamer:1984di},
\begin{equation}
 Q_{\rm B} =
\int_{-\infty}^{t_0} dt \int_S \vec K \cdot d \vec S
+  \int_{t=t_0} d^3r K^0
  , \end{equation}
where the $\vec \nabla \cdot \vec j_{\rm B}$ term is neglected,
and the anomaly term is reexpressed in terms of the
Chern-Simons current
\begin{equation}
 K^\mu=\frac{1}{16\pi^2}\varepsilon^{\mu\nu\rho\sigma}
 \left\{ g^2 {\rm Tr}\left( F_{\nu\rho}V_\sigma
 - \frac{2}{3} i g V_\nu V_\rho V_\sigma \right)
 + \frac{1}{2} {g'}^2 f_{\nu\rho}A_\sigma \right\}
  . \end{equation}
In a gauge, where
\begin{equation}
V_\mu \to \frac{i}{g} \partial_\mu \hat{U} \hat{U}^\dagger   , \ \ \
\hat{U}(\infty) = 1   ,
\end{equation}
$\vec K$ vanishes at infinity.
Subject to the above ansatz and boundary conditions
the baryon charge
of the sphaleron solution \cite{Kleihaus:1994tr,Kleihaus:2003tn} is then
\begin{equation}
Q_{\rm B}= \int_{t=t_0} d^3r K^0  = \frac{n \ (1-(-1)^m)}{4} \, .
\label{Q_B}
\end{equation}

\section{Results and discussion}

We have solved the set of ten coupled non-linear
elliptic partial differential equations numerically \cite{FIDISOL},
subject to the above boundary conditions
in compactified dimensionless coordinates,
$x = \tilde r/(1+ \tilde r)$, with $\tilde r = gvr$.
Restricting to $M_H=M_W$,
and employing the physical value for the mixing angle $\theta_{\rm w}$,
we have performed a systematic study
of the rotating sphaleron-antisphaleron systems
with $1 \le m \le 6$ and $1 \le n \le 6$.

Starting from a given static neutral solution
for a sphaleron-antisphaleron system characterized by
the integers $n$ and $m$,
we have constructed the corresponding branch of
rotating solutions, by slowly increasing
the asymptotic value of the parameter
$\tilde \gamma = \gamma/gv$,
which specifies the boundary conditions for the
time components of the gauge fields.
The rotating branch ends
when the limiting value $\tilde \gamma_{\rm max}=1/2$
is reached \footnote{The
exponent arises as a combination from the mass term,
obtained from the Higgs vaccuum expectation value,
and the non-Abelian gauge field interaction term,
and yields asymptotically an exponential decay with
decay constant proportional to $\sqrt{1-4\tilde \gamma^2}$.
}.
Here some of the gauge field functions
no longer decay exponentially.
This therefore precludes localized solutions
for larger values of $\tilde \gamma$.

\subsection{Multisphalerons}

We first present our results
for the
rotating multisphaleron solutions,
i.e., the branches of rotating solutions with $m=1$ and $n>1$.
We exhibit in Fig.~\ref{f-1}a
the asymptotic gauge field parameter
$\tilde \gamma = \gamma/gv$ versus the
scaled angular momentum of multisphaleron solutions
consisting of $n$ superposed sphalerons with $1\le n \le 6$.
Since the angular momentum $J$ increases with $n$,
we exhibit the angular momentum per sphaleron
$J/n$, choosing units of $J_0=4 \pi/g^2$.
As $\tilde \gamma$ increases from zero to
its maximal value $\tilde \gamma_{\rm max}=1/2$,
the angular momentum increases monotonically.
Consequently the solutions have maximal spin
at $\tilde \gamma_{\rm max}$.

\begin{figure}[h!]
\lbfig{f-1}
\begin{center}
\hspace{0.0cm} (a)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1_gamma0_over_j_by_n.eps}
\hspace{0.5cm} (b)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1_chi_over_j_by_n.eps}
\\
\hspace{0.0cm} (c)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1_e_by_n_over_j_by_n.eps}
\hspace{0.5cm} (d)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1_mubymu0_by_n_over_j_by_n.eps}
\end{center}
\caption
Properties of multisphaleron solutions ($m=1$, $n=1,\dots,6$):
(a) the asymptotic value of the $U(1)$ field $\tilde \gamma = \gamma/gv$,
(b) the $U(1)$ charge parameter $\chi$,
(c) the mass per sphaleron $E/n$ (in units of $E_0=4\pi v/g$),
(d) the magnetic moment per sphaleron $\mu/n$
(in units of $\mu_0=e/\alpha_{\rm W} M_{\rm W}$)
versus the angular momentum per sphaleron $J/n$
(in units of $J_0=4 \pi/g^2$)
}
\end{figure}

Considering the increase of the angular momentum with
the number of sphalerons $n$,
we note that this increase is faster than linear.
Thus the value of the maximal angular momentum
per sphaleron, $J_{\rm max}/n$ increases with $n$.
The more sphalerons a configuration
consists of, the more angular momentum the constituents
can contribute.

Let us next consider the linear relation
(\ref{JQrel}) between the charge $\cal Q$ and
the angular momentum $J$.
According to this relation,
we should obtain a single straight line,
when exhibiting the charge
versus the scaled angular momentum per sphaleron, $J/n$.
We demonstrate this in Fig.~\ref{f-1}b,
where we exhibit the charge parameter $\chi$
(which is proportional to the charge $\cal Q$)
versus the scaled angular momentum per sphaleron, $J/n$.
We indeed observe a single straight line,
which extends the further the greater $n$.
Since the charge parameter has been extracted from the
asymptotic fall-off of the $U(1)$ function $a_1$,
whereas the angular momentum has been obtained from
the volume integral of the angular momentum density $T^t_\varphi$,
this agreement reflects the good numerical quality of the
solutions.

In Fig.~\ref{f-1}c
we present the energy of these solutions,
which has been obtained from
the volume integral of the energy density $-T^t_t$.
For multisphalerons consisting of $n$ sphalerons one expects
that their energy should be of the order of $n$ times
the energy of a single sphaleron,
thus $E/n$ should be roughly constant.
The deviations of the energy per sphaleron $E/n$
from the energy of the single sphaleron
can then be attributed to the interaction of the $n$ sphalerons
and be interpreted in terms of the binding energy
of these multisphaleron configurations.
For the employed value of the Higgs mass, we note that
the static solutions with $n=2-4$ represent bound states,
since $E/n$ is smaller than the energy of a single sphaleron,
whereas the static solutions with $n>4$ are slightly unbound
\cite{Kleihaus:1994tr}.

When charge is added to these configurations and the
solutions begin to spin, their energy increases monotonically
with their angular momentum.
The increase of the energy per sphaleron $E/n$
with the angular momentum per sphaleron $J/n$ is strongest
for the branch of single sphaleron solutions.
The more sphalerons a configuration consists of,
the weaker is the increase of its energy per sphaleron $E/n$
with increasing angular momentum per sphaleron $J/n$.
Thus charge and rotation contribute relatively less
to the total energy for these ``many sphaleron'' configurations
(e.g.~only 8\% for $n=6$ as compared to 30\% for $n=1$).
Consequently, the rotating multisphaleron configurations
turn into bound states beyond some critical value
of the angular momentum.

Sphalerons possess a large magnetic moment $\mu$.
For multisphalerons consisting of $n$ sphalerons one
then expects
that their magnetic moment should be roughly $n$ times
the magnetic moment $\mu$ of a single sphaleron.
As seen in Fig.~\ref{f-1}d, where we exhibit
the magnetic moment per sphaleron $\mu/n$ versus
the angular momentum per sphaleron $J/n$,
this first guess is somewhat crude.
For static configurations,
the interaction between the sphalerons
gives rise to a systematic (almost linear) increase of
the magnetic moment per sphaleron $\mu/n$ with
the number of sphalerons.

When charge and thus angular momentum
is added to these configurations,
their magnetic moment increases monotonically
with increasing angular momentum.
Again, this increase is strongest
for the branch of single sphaleron solutions,
and the increase is the weaker
the more sphalerons a configuration consists of.

Having discussed the global properties of these solutions,
we now turn to their local properties.
In particular, we address the effect of
the presence of charge and rotation
on the energy density $-T^t_t$,
and on the modulus of the Higgs field $|\Phi|$.
As an example, we illustrate the energy density $-T^t_t$,
the magnitude of the Higgs field $|\Phi|$,
and the angular momentum density $T^t_\varphi$
for an almost maximally rotating
multisphaleron solution ($m=1$, $n=3$, $\tilde \gamma =0.499$)
in Fig.~\ref{f-2}.

\begin{figure}[h!]
\lbfig{f-2}
\begin{center}
\hspace{0.0cm} (a)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1n3_0.4990_ed.eps}
\hspace{0.5cm} (b)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m2n3_0.4990_ed.eps}
\\
\hspace{0.0cm} (c)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1n3_0.4990_higgs.eps}
\hspace{0.5cm} (d)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m2n3_0.4990_higgs.eps}
\\
\hspace{0.0cm} (e)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m1n3_0.4990_jd.eps}
\hspace{0.5cm} (f)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m2n3_0.4990_jd.eps}
\end{center}
\vspace{-0.5cm}
\caption{\small
The energy density $-T^t_t$ (top),
the modulus of the Higgs field $|\Phi|$ (middle),
and the angular momentum density $T^t_\varphi$ (bottom)
are exhibited for an $m=1$, $n=3$ multisphaleron solution (left)
and an $m=2$, $n=3$ sphaleron-antisphaleron system (right)
close to maximal rotation.
}
\end{figure}

In multisphalerons, the region with large energy density
is torus-like, where its maximum is forming a ring in
the equatorial plane.
When we add electric charge and angular momentum
to the static configuration,
we observe that
the energy density is spreading further out,
while at the same time its overall
magnitude is reduced.
Such a spreading of the energy density with increasing charge is
also seen in dyons, for instance,
and we attribute this effect to the presence of charge
and the associated repulsion.
Indeed, this spreading
becomes quite pronounced for large values of the charge.
The expected effect of the presence of angular momentum,
on the other hand,
is a centrifugal shift of the energy density.
Indeed, we observe, that with increasing angular momentum
the torus-like region of large energy density
moves further outward to larger values of $\rho$.

The modulus of the Higgs field of the multisphaleron
solutions has a single node at the origin,
from where it starts to increase linearly
in the direction of the symmetry axis,
to reach its vacuum expectation value at infinity.
In the equatorial plane, in constrast,
the modulus of the Higgs field starts to
inrease from the origin much slower (i.e., only with power $\rho^n$).
As the configurations are endowed with
charge and rotation, the Higgs field hardly changes.
Indeed, the effect of charge and rotation on
the modulus of the Higgs field is barely noticable,
even at the maximal strength.

The angular momentum density for the multisphaleron solutions
is torus-like and centered in
the equatorial plane analogous to the energy density.
However, the region of large angular momentum density
is located further outwards at larger values of $\rho$,
while it vanishes on the symmetry axis.

\subsection{Sphaleron-antisphaleron systems}

Let us now turn to sphaleron-antisphaleron systems.
For $n=1$ they represent sphaleron-antisphaleron chains,
where $m$ sphalerons and antisphalerons are located
on the symmetry axis, in static equilibrium.
For $n=2$ the chain is formed
from  $m$ multisphalerons and -antisphalerons,
thus the modulus of the Higgs field
still possesses only isolated nodes on the symmetry axis.
However, as $n$ increases further, the character of the solutions
changes, and new types of configurations appear,
where the modulus of the Higgs field vanishes on rings
centered around the symmetry axis.
Therefore we refer to these solutions as vortex ring solutions.
We note, that the precise evolution of the isolated nodes
on the symmetry axis and the vortex rings in the bulk
with increasing $n$ is sensitive
to the value of the Higgs mass \cite{Kleihaus:2008gn}.

In the following we demonstrate the effect of charge
and rotation on these sphaleron-antisphaleron systems
by focussing on the configurations with $n=3$ and
$m=1,...,6$.
The static solutions have been constructed before \cite{Kleihaus:2008gn}.
For the chosen parameters, the modulus of the Higgs field
of the static $m=2$ configuration
vanishes only on a ring in the equatorial plane.
The static $m=3$ configuration
has a node at the origin and in addition two rings,
located symmetrically above and below the $xy$-plane.
The static $m=4$ configuration has only two symmetrical vortex rings.
For $m=5$, the two symmetrical vortex rings
are supplemented by a node at the origin,
and for $m=6$ two symmetrical vortex rings and two inner nodes
on the symmetry axis are present.

Again we first discuss the global properties of these solutions,
following the above scheme.
Since only the integer $n$ (but not $m$) enters the
relation between the charge and the angular momentum,
we continue to show all quantities versus
the scaled relative angular momentum $J/n$.
We exhibit in Fig.~\ref{f-3}a
the dependence of the asymptotic gauge field parameter
$\tilde \gamma = \gamma/gv$ on the
scaled relative angular momentum $J/n$
for the configurations with $n=3$ and $m=1,...,6$.
As before, at the maximal value $\tilde \gamma_{\rm max}=1/2$
the solutions have maximal angular momentum.
Moreover, we observe that
the maximal value of the relative angular momentum
$J_{\rm max}/n$ increases with $m$.
Thus the higher the number of constituents a configuration
(i.e., the product $mn$),
the more angular momentum each of the constituents
can contribute.

\begin{figure}[h!]
\lbfig{f-3}
\begin{center}
\hspace{0.0cm} (a)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{n3_gamma0_over_j_by_n.eps}
\hspace{0.5cm} (b)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{n3_chi_over_j_by_n.eps}
\\
\hspace{0.0cm} (c)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{n3_e_by_mn_over_j_by_n.eps}
\hspace{0.5cm} (d)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{n3_mubymu0_by_mn_over_j_by_n.eps}
\end{center}
\caption
Same as Fig.\ref{f-1}
for sphaleron-antisphaleron systems ($m=1,\dots,6$, $n=3$).
}
\end{figure}

In Fig.~\ref{f-3}b we demonstrate
that the linear relation (\ref{JQrel})
between the charge $\cal Q$ and the angular momentum $J$
is well satisfied also by the numerically constructed
sphaleron-antisphaleron systems.
We again obtain a single straight line,
when exhibiting the charge
versus the relative angular momentum $J/n$,
which extends the further the greater $n$.

Addressing the energy of these solutions,
we expect that it should be roughly proportional
to the number $mn$ of constituents of the configurations.
We therefore consider the relative energy $E/mn$,
i.e., the energy per constituent.
The deviations of $E/mn$
from the energy of a single sphaleron
can then be attributed to the interaction of the sphalerons
and antisphalerons in the system
and be interpreted in terms of their binding energy.
We present the relative energy $E/mn$ of these solutions
in Fig.~\ref{f-3}c.
We note that the binding energy increases with an
increasing number of constituents.
Also, the increase of the energy per constituent $E/mn$
with $J/n$ is the stronger the smaller the number of constituents.
Charge and rotation contribute therefore relatively less
to the total energy for the ``many constituents'' configurations.

Addressing finally the magnetic moment $\mu$ of these
solutions, we also consider
the magnetic moment per constituent $\mu/mn$.
As seen in Fig.~\ref{f-3}d,
the interaction between the constituents
gives rise to a decrease of
the magnetic moment per constituent $\mu/mn$ with
increasing $m$.

Now we turn to the local properties of these
sphaleron-antisphaleron systems.
As examples, we exhibit the $m=2$, $n=3$
system in Fig.~\ref{f-2} (right column)
and the $m=3$, $n=3$ and $m=4$, $n=3$ systems in Fig.~\ref{f-4},
where we again display the energy density,
the modulus of the Higgs field and
the angular momentum density
for solutions close to maximal rotation.

\begin{figure}[h!]
\lbfig{f-4}
\begin{center}
\hspace{0.0cm} (a)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m3n3_0.4990_ed.eps}
\hspace{0.5cm} (b)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m4n3_0.4990_ed.eps}
\\
\hspace{0.0cm} (c)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m3n3_0.4990_higgs.eps}
\hspace{0.5cm} (d)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m4n3_0.4990_higgs.eps}
\\
\hspace{0.0cm} (e)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m3n3_0.4990_jd.eps}
\hspace{0.5cm} (f)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m4n3_0.4990_jd.eps}
\end{center}
\vspace{-0.5cm}
\caption{\small
Same as Fig.\ref{f-2}
for the sphaleron-antisphaleron systems $m=3$, $n=3$ (left)
and $m=4$, $n=3$ (right).
}
\end{figure}

Only in sphaleron-antisphaleron pairs and chains with $n=1$
the regions with large energy density
are localized around the Higgs field nodes
on the symmetry axis.
In all other sphaleron-antisphaleron systems,
the regions with large energy density are torus-like,
where a configuration may possess several such tori.
The location and number of these tori
depends on the numbers $m$ and $n$ and on the parameters,
just as the nodes of the Higgs field are
determined by these quantities.

As for the multisphalerons,
the effect of the presence of electric charge
is that the energy density spreads further out,
while at the same time its overall
magnitude reduces.
Likewise, the effect of the presence of angular momentum
is a centrifugal shift of the energy density.
With increasing angular momentum
the torus-like regions of large energy density
move further outward to larger values of $\rho$.

The modulus of the Higgs field of the
sphaleron-antisphaleron systems changes very little
and is barely noticable,
even when the systems carry maximal charge
and angular momentum.

The angular momentum density of the sphaleron-antisphaleron systems
is also characterized by the presence of tori.
But it contains tori of large positive angular momentum density
as well as negative angular momentum density.
The tori of the angular momentum density are spatially related to the tori
of the energy density.
In particular, the location of the positive tori is associated with
the location of the tori of the energy density,
with the negative tori inbetween.

\subsection{Equilibrium condition}

Finally, we would like to address the question of the
equilibrium of such composite configurations
as sphaleron-antisphaleron pairs and more general
sphaleron-antisphaleron systems.
As discussed previously \cite{Aharonov:1992wf,Beig:2008qi,Beig:2009jd},
a necessary condition for the equilibrium
of such axially symmetric configurations is
\begin{equation}
\int_S T_{zz} dS =0
\label{Tzz}
\end{equation}
where $T_{zz}$ is the respective component
of the stress energy tensor and $S$ is the equatorial plane.
When this condition is satisfied, the net force between
the constituents in the upper and in the lower
hemisphere vanishes, thus yielding equilibrium.
If $T_{zz}$ vanishes everywhere in the equatorial plane,
this condition is met trivially,
if on the other hand $T_{zz}$ does not identically vanish,
the various contributions to the surface integral
(\ref{Tzz}) must precisely cancel each other.

To understand how the equilibrium condition is satisfied
in these sphaleron-antisphaleron systems,
we have extracted the $T_{zz}$ component of the
stress energy tensor.
We illustrate
$T_{zz}$ for two rather different configurations in Fig.~\ref{f-5}.
In Fig.~\ref{f-5}a
we display $T_{zz}$
for the static sphaleron-antisphaleron pair
in the upper hemisphere.
In the equatorial plane $T_{zz}$ appears to almost vanish.
We therefore focus on the equatorial plane in Fig.~\ref{f-5}c.
Here $T_{zz}$ is small, but finite (except when it changes sign).
To gain further insight into how the equilibrium
results from the various forces present in the system,
we consider the contributions from the respective parts of the Lagrangian
separately.
We exhibit these also in Fig.~\ref{f-5}c.
We note, that
the positive contribution from the $SU(2)$ gauge field part
almost cancels the negative contributions from the $U(1)$ and Higgs
parts, yielding in total a $T_{zz}$ which is almost but not quite vanishing
in the equatorial plane.
In the inner region the total $T_{zz}$ is slightly positive, while in the
outer region it is slightly negative, yielding
together a vanishing surface integral (\ref{Tzz}),
within the numerical accuracy.

The situation is similar for the sphaleron-antisphaleron
chain, consisting of four constituents ($m=4$, $n=1$).
Also, the inclusion of rotation does not change this
overall behaviour of these types of solutions.
For most other systems, however, $T_{zz}$ does not nearly vanish
in the equatorial plane.
This is exhibited examplarily in Fig.~\ref{f-5}b
for the fast rotating sphaleron-antisphaleron system with
$m=4$, $n=3$. The features of $T_{zz}$
seen here, are very typical, and hardly change with rotation,
since the effect of rotation is basically a slight shift in magnitude.
However, while $T_{zz}$ is rather large in the equatorial plane
for these configurations,
its positive and negative contributions to the surface integral
do cancel as required for equilibrium.

\begin{figure}[h!]
\lbfig{f-5}
\begin{center}
\hspace{0.0cm} (a)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m2n1_0.0000_tzz.eps}
\hspace{0.5cm} (b)\hspace{-0.6cm}
\includegraphics[height=.25\textheight, angle =0]{m4n3_0.4990_tzz.eps}
\\
\hspace{0.0cm} (c)\hspace{-0.6cm}
\includegraphics[height=.23\textheight, angle =0]{m2n1_0.0000_tzzparts_xy.eps}
\hspace{1.0cm} (d)\hspace{-0.6cm}
\includegraphics[height=.23\textheight, angle =0]{m4n3_0.4990_tzzparts_xy.eps}
\end{center}
\vspace{-0.5cm}
\caption{\small
The full stress-energy component $T_{zz}$
(upper)
and its $SU(2)$, $U(1)$, Higgs covariant derivative
and Higgs potential parts in
the equatorial plane (lower)
for the sphaleron-antisphaleron systems
$m=2$, $n=1$ $J=0$ (left)
and $m=4$, $n=3$, $J\approx J_{\rm max}$ (right).
}
\end{figure}

\section{Conclusions}

We have considered
sphaleron-antisphaleron pairs, chains and vortex ring solutions
in Weinberg-Salam theory,
which are characterized by two integers $n$ and $m$.
Starting from the respective neutral electroweak configurations,
we have obtained the corresponding
branches of rotating electrically charged solutions.
These branches exist up to maximal values of the charge
and angular momentum,
beyond which localized solutions are no longer possible.

The angular momentum $J$ and the charge $\cal Q$ of these
sphaleron-antisphaleron systems are proportional
$$J = n {\cal Q} /e.$$
Their energy and binding energy increase with increasing rotation,
and so does their magnetic moment.
With increasing charge
the energy density of the configurations spreads further out,
while its overall magnitude reduces.
At the same time the effect of the rotation
is a centrifugal shift of the energy density tori
to larger radii.

We have also addressed the equilibrium condition (\ref{Tzz})
for these sphaleron-antisphaleron systems.
In all systems, it is the surface integral that vanishes
to give equilibrium, and not the
stress-energy tensor component $T_{zz}$ by itself.
However, for the sphaleron-antisphaleron pair
(and other even $m$ chains)
the stress-energy tensor component $T_{zz}$
almost vanishes in the equatorial plane.
In these configurations the positive contribution from the
$SU(2)$ part almost cancels the negative
contributions from the $U(1)$ and Higgs parts,
thus yielding an almost vanishing total $T_{zz}$
in the equatorial plane.

The next step will be to study the fermion modes
in the background of such rotating electroweak configurations
to understand their relevance for baryon number violating
processes \cite{Kunz:1993ir}.
Moreover it will be interesting to include the effect of gravitation
\cite{Ibadov:2008hj} to obtain rotating gravitating regular configurations
as well as black hole solutions.

{\bf Acknowledgement}:
We gratefully acknowledge discussions with G.~Gibbons,
E.~Radu and M.~S.~Volkov.
R.I.~acknowledges support by the Volkswagen Foundation,
and B.K.~support by the DFG.

redpajama_set_name: RedPajamaArXiv


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text:
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Her podcast is called "Little Mind Chats" her goal was for Gen Alpha and Gen Z to know what's happening now that they could do more of the good and less of the bad
#145 The Magnificent 7 with Dr. Bob Rakowski
It is the 365th episode of Mi365 Podcast. The first podcast I had was also with the same person we have today. Get to know more about Dr. Bob Rakowski and The Magnificent 7. There are no shortcuts to health as discussed by Dr. Rakowski and know the effects of the food we eat. One interesting concept that he discussed is the powerful nature's pharmacy and superfoods.


Sending money online, whether you're splitting the check at lunch, paying your share of the bills, or buying something via Craigslist, is easier now than it's ever been. There are tons of services that make it easy to transfer money with a phone number or email address. This week we're looking at five of the best, based on your nominations.
What's The Best Way to Pay Other People Online?
Square Cash is fast, flexible, available for iOS and Android, and lets you send money to other individuals in minutes. All you need for an account is a smartphone and an email address, and of course, the debit card you'd like to connect your Square Cash account to, and that's about it. From there you can send and deposit money easily. You can even create what the service calls "$cashtags," or custom names that identify you and make it easy for other people to use Square Cash to send you money—you can add it to an email signature or put it on a business card or whatever. Best of all, money transfers with Square Cash are free, meaning they don't take a cut of your transaction, and they don't charge you fees for sending or receiving money. If you're a business user, they take a 1.5% cut of your transactions, but individual users don't pay a dime.
Those of you who supported Square Cash's nomination glowed about it—some of you praised the service for their speedy, easy money transfers, and many of you highlighted the fact that the money is often in your bank account within minutes, instead of waiting days to clear like other services. Some of you said that the fact that they let you do all of this for free is akin to magic, but you weren't complaining. A few of you rightfully pointed out that Square Cash is US only, which makes it off limits to a lot of people, but for those people who can use it, it's a windfall. One of you even mentioned you use it to send money even though you and your spouse have accounts at the same bank! Read more in its nomination thread here.
If you're looking for the company that really started the whole "send payment from your phone easily" craze, it's probably Venmo. The app (available for iOS and Android) is extremely popular, and makes it easy to send payments to anyone with a phone number or email address, completely free. Sending money from your Venmo wallet or a connected debit card or bank account is free, and receiving money is free—you only pay (a 3% fee) if you want to send money from a credit card or non-major debit card (you can see what's "major" and what isn't on their pricing page.) Once you have an account, sending money is as easy as sending cash you already have in your Venmo account (you can use it as a wallet if you choose) or through a linked bank account or debit card. Those transactions can be public or private (and the public ones can be hilarious), and you can either cash out the money you have in Venmo to your bank account, or leave it there for future money transfers. Money you cash out is usually in your bank account within a business day.
Many of you praised Venmo for having the largest userbase of many of these services (even though it's technically US only), and for being big enough that you know you can offer to send someone money via Venmo and they either have an account or have heard of it. Venmo has had its security problems, as we noted in this post on money transfer security, which more than a few of you called out as reasons to think twice about using the service, but others said that it was secure enough for you, and the anti-fraud protections your bank offers made you feel at ease. Others of you praised it for being fast and easy and for letting you keep a wallet of funds, so not every transaction has to go back through your bank, and others mentioned that it's got that "all the kids use it" air about it that means it's easy to find people—especially younger people—who use it. it's also important to mention that Venmo is owned by PayPal, another contender in this roundup. One of you said Venmo is "like their fun teen brand," and that sounds about right. Read more in its nomination thread here.
Google Wallet definitely lets you send money to people quickly and easily, that's true, but the service is much much more than that. All your recipient needs is a Gmail address (and to live in the US, although the service is experimenting with international users), and you can send them cash from your Google Wallet account, or from a linked debit card, bank account, or credit card. Google Wallet is available for Android and iOS, and in addition to being an easy way to send or receive money, if you have a phone with NFC, you can use it for contactless payments in brick-and-mortar stores, to check out when shopping online, and even use the Google Wallet card as a physical card (backed by Mastercard) that's tied to your Google Wallet account. Using Google Wallet is fast and easy, and money cashed out from your account to a bank account or debit card takes about a day or so to land, if that much. Plus, it's free to send money from your wallet, debit card, or bank account (it only costs money if you use a credit card), and it's free to receive money as well. Best of all, since all a person needs is a Gmail account to use Google Wallet, odds are they already have it as a payment option, even if they don't actively use it.
Those of you who nominated Google Wallet pointed out that those instant cash transfers are great, and the ability to easily send money to people combined with the fact that you can use Google Wallet to pay for things at retailers or online makes it a logical do-everything option if you want one app that can handle just about everything. Plus, like we said—all the other person needs is a Gmail account and they have Google Wallet. You don't have to get them to sign up for anything, just ask for their email address. Combined with the fact that Google just announced Android Pay, which may either be a replacement for or improvement on Google Wallet, the option is about to get even more useful. You can read more in its nomination thread here.
PayPal has a long and interesting history. From being one of our worst companies for customer service and being bad enough to inspire a whole article discussing PayPal alternatives, the service seems to have at least woken up to its bad rap in recent years, even if it's still not completely out of the headlines for holding people's money and freezing accounts when they're most needed. PayPal is ubiquitous, and while its subsidiary, Venmo, has a lot of mindshare, almost everyone has a PayPal account. The service lets you send money in a couple of clicks via the web or its mobile apps (available for Android and iOS), keep your money in your PayPal account like a wallet, or even use PayPal to check out at tons of online retailers, or just about anyone else with an online shop that doesn't want to handle payment processing themselves. Sending money with PayPal is easy and free for personal transactions as long as you use your PayPal account, or an authorized debit card or bank account (fees come into play when you use credit cards), and as long as you're sending money within the United States (PayPal does support international money transfers, there are just fees involved, along with the currency conversion.) When you pay for goods or services using your PayPal account, the person providing the service gets hit with a 2.9% + $.30 fee for your transaction (which they may or may not pass back along to you.) PayPal's fees can be confusing, but you can read a breakdown of them here. In short, just make sure your transactions are personal, and not labeled as for goods or services. Once you have the money in your PayPal account, cashing out to a bank account usually takes several days (sometimes up to five business days or more), so be ready for that.
Those of you who nominated PayPal did so because of two reasons mostly: Either a: It's everywhere and everyone you know has an account, or b: It's the only one of these kinds of services you've heard of. Few of you said you actually liked using PayPal, but you did like the fact that it's easy enough to use, available everywhere, supports international money transfers (unlike most of the other contenders here), and people who don't know anything about banking or sending money online recognize it as a long-standing and well-known name. You can read more of the whole back and forth in its nomination thread here.
Surprisingly, you don't have to have an account at Chase Bank to use Chase Quick Pay, but if you do, it really helps. If someone sends you money through Chase Quick Pay, you can sign up for an account and use it from then on, but if you want to use it yourself and you're not a Chase customer, you can actually sign up and link another bank account or debit card to the service (however, one person in the transaction does have to have a Chase checking account to get started—essentially, you can use it to receive money if you're not a Chase customer, but you can't use it to send.) Sending and receiving money is free, and sending just requires a phone number or an email address. Getting your money is fast, and usually takes just a few minutes, and if you're a Chase customer, your money is available instantly. If you're not a Chase customer, it usually takes a little longer, but not much. The service is US only, and there are no fees to send or receive money.
Those of you who nominated Chase Quick Pay praised it for being fast and easy, and for even making it somewhat easy to send money to non Chase customers, but equally many of you said that the process is difficult for those non-Chase customer to accept money, and that the Chase mobile banking app is only useful for Chase customers, so those people without Chase checking accounts but with Chase Quick Pay accounts are still kind of out in the cold. More than a few of you said that if Chase made it easier for others to use the service, and to access their money. You can read more in its nomination thread here.
No honorable mentions this week, as the nominations dropped off pretty quickly after these five. We should, however, give a nod to Bitcoin though, since a few people did nominate it as an anonymous and private way to send and manage funds—although how easy and effortless it is depends heavily on the wallet application you're using for the purpose. We don't have the space here to really get into the ins and outs of Bitcoin, but if you'd like to learn more, our primer to Bitcoin and what you can do with it is a great place to start. If ain't nobody got time for that, this two-minute video should do the trick.
Title photo made using Olga Lebedeva (Shutterstock).


Bronchitis is an inflammation of the primary airway (bronchi) to your lungs. It triggers a cough, shortness of breath and chest tightness. Coughing often brings up yellow or greenish mucous. There are 2 primary types of bronchitis: intense and persistent.
Severe bronchitis is usually caused by viruses, and it may occur together with or following a cold or other respiratory infection.
Bacteria such as viruses can be spread out from individual to person by coughing.
They can likewise be spread out if you touch your mouth, nose, or eyes after entering into contact with breathing fluids from an infected person.
People are inclined to believe that some matter found here that is relating to Bronchitis Severe Bronchitis is false.
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Treatment for intense bronchitis: Your kid's age, general health, and medical history Extent of the condition Your child's tolerance for specific medications, treatments, or treatments As we got to composing on Bronchitis Symptoms, we discovered that the time we were offered to compose was insufficient to compose all that there is to write about Bronchitis Manifestations! So large are its resources.
Expectations for the course of the condition Water Consume a minimum of 8 glasses of water or other fluids every day. This will assist to keep mucous thin, making it simpler to cough up. Prevent sugar and dairy products due to the fact that they have the tendency to deteriorate the immune system and trigger excessive mucous production. With individuals wishing to learn more about Bronchitis, it has supplied the required incentive for us to write this fascinating article on Bronchitis!
The cough might be dry or might produce phlegm. Substantial phlegm production suggests that the lower respiratory tract and the lung itself might be infected, and you may have pneumonia. We have not in fact turned to roundabout means of getting our message on Bronchitis Acute Bronchitis through to you. All the matter here is authentic and to the point.
Getting enough rest so your body has actually the energy needed to fight the infection. Usually, you feel much better earlier if you rest breathe easy salt pipe salt inhaler you have intense bronchitis. It is the normal design of writers to include additional info with the objective of extending the length of an article. However, we have actually offered a short and succinct article with just required info on Chronic Bronchitis Symptoms Cough.
When the main air passageways in your lungs (bronchial tubes) are swollen, they frequently produce large quantities of tarnished mucus that turns up when you cough. If this continues for more than 3 months, it is referred to as chronic bronchitis. Mucus that isn't really white or clear usually implies there's a secondary infection. The more fascinating a post, the more takers there are for the post. So we have actually made it a point to make this post on Bronchitis Cough as interesting as possible!
It is more common throughout the denver seminary often establishes after an upper respiratory disease such as influenza (influenza) or a cold triggered by an infection such as coronavirus, adenovirus, or a rhinovirus. Respiratory syncytial infection (RSV) might be a cause, especially in grownups older than About 10% of the time, severe bronchitis is triggered by germs. We were rather indecisive on where to stop in our writings of Bronchitis Cough. We simply went on composing and writing to offer a long post.
Persistent bronchitis is discovered in higher-than-normal rates amongst coal miners, grain handlers, metal molders, and other individuals who are continuously exposed to dust. But the chief cause is heavy, long-term cigarette smoking, which aggravates the bronchial tubes and causes them to produce excess mucus.
Some cases, the doctor will advise using a medicine called a bronchodilator that makes breathing easier.
People with persistent bronchitis ought to stop cigarette smoking and prevent second-hand smoke.
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Bronchitis is a very common breathing illness and it can take place in anybody, despite age. However, bronchitis has a higher incidence in smokers, individuals with breathing insufficiency and people with weak body immune system. The issue with bronchitis is that it generates symptoms that have an unspecific character. The signs of bronchitis are likewise characteristic to other respiratory health problems and in most cases they can be misleading in developing the appropriate diagnose.
Bronchitis treatments for chronic types of the illness often include bronchodilators such as albuterol and ipratropium.
These are breathed in medications that eliminate the excess mucous accountable for blockage of the respiratory system and shortness of breathing.
Bronchitis treatments can likewise include steroids for reinforcing the body's defenses against germs and infections.
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The signs of bronchitis continue for more than 2 days, it is highly advised to see a medical professional.
He will carry out a health examination or laboratory tests in order to expose the nature of the disease.
The physician will only prescribe a specific bronchitis treatment if the health problem is severe and caused by infection with bacteria.
Patients that are following bronchitis treatments should avoid exposure antioch university santa barbara toxins (smoke, dust, chemicals, astringent substances, vapors of alcohol or vapors of gas). Likewise, self-medication must be prevented, as unsuitable medications can trigger a stress of the disease. Prevent taking prescription antibiotics without the doctor's consent and keep in mind not to offer aspirin to little kids. For grownups, aspirin or acetaminophen can momentarily alleviate fever and other signs, however it is extremely important to see a physician if the symptoms intensify. The first impression is the best impression. We have actually written this short article on Bronchitis in such a way that the impression you get will definitely make you want to read more about it!
Although they aren't very safe, long-lasting bronchitis treatments with antibiotics are recommended for getting rid of persistent kinds of the health problem. The issue with bronchitis treatments that involve using prescription antibiotics is that they can compromise the organism in time, making it more susceptible to other infections. In spite of their performance in fighting malign germs, bronchitis treatments with antibiotics likewise destroy internal benign bacteria that belong to the immune system. We have consisted of the history of Bronchitis Treatment here so that you will learn more about its history. It is just through it's history can you find out more about Bronchitis Treatment.
Even if the disease is correctly diagnosed, recommending the very best bronchitis treatment is a tough task for doctors. Antibiotics are typically part of bronchitis treatments. Nevertheless, if bronchitis is brought on by infections, it is thought about types bronchitis treatments with prescription antibiotics are ineffective in conquering the health problem. In some viral forms of severe bronchitis, medical treatment isn't really needed at all, as the disease clears on itself in a couple of weeks. Even if the disease is caused by germs, bronchitis treatments with antibiotics can cause a lot of damage to the organism if they are administered constantly. Don't be amazed if you find anything unusual here about Bronchitis Medical. There has been some fascinating and uncommon feats here worth reading.
It is triggered by an extreme swelling or the blocking of the air tubes in the lungs. Bronchitis generally takes place in 2 kinds: intense or chronic. Intense bronchitis is the lesser condition of the 2 but it can still last for a number of weeks and might lead to pneumonia if it is not looked after right now. Persistent bronchitis, on the other hand, can ultimately result in more major heart issues.
The most essential thing to do is to stop smoking and prevent secondhand smoke. Avoid contamination by remaining inside your home during smog notifies and avoid using sprays for cleansing, insect repelling, or deodorizing. These consist of a large number of potentially damaging ingredients that can damage your lung tissues if utilized frequently. If you are delicate to dairy products, prevent them because they can increase mucus development. When you notice the start of a cough, get lots of rest and use moderate heat on your chest and back before going to sleep. Consume at least 8 to 10 glasses of clear water every day and eat foods that are high in vitamin A, C, E and zinc. Foods that are high in these nutrients consisting of eggs, chicken, pecans, citrus fruits, sunflower seeds, broccoli, sardines and avocados.
Bronchitis can be recognized by the following symptoms: frequent and uncomfortable cough with phlegm, high fever, sore throat, chest discomfort, extreme sounds when breathing and chills. Intense bronchitis is the common result of an infection like the influenza or a without treatment cold. Chronic bronchitis is generally triggered by a lung irritant like cigarette smoke, allergic reactions, dust and all other kinds of air contamination. It is only because that we are rather proficient on the subject of Bronchitis Common that we have actually ventured on composing something so prominent on Bronchitis Common like this!
You can take a number of vitamins to promote healing of bronchitis as well as preventing it. Vitamin A in big dosages taken daily for approximately Thirty Days can promote much faster healing of irritated mucous membranes and strengthens the entire immune system. Vitamin C brought with it can boost your resistance also and assists build up better collagen and mucus linings in the lungs. Vitamin E is a powerful antioxidant that safeguards your lungs from air contamination. Zinc works with vitamin A to heal lung membranes and other tissues. An herb called astragalus taken daily in small dosages can likewise help in reducing symptoms associated with chronic bronchitis. Although there was a lot of change in the composing designs of we independent authors, we have developed a final product on Bronchitis Patients worth reading!
Is a common error to take antihistamines, prescription antibiotics, or decongestants for bronchitis. These drugs do not eliminate swollen windpipes and can even dry the mucus linings of the lungs. They can likewise make the mucous so thick that coughing can be extremely hard and agonizing. Antibiotics don't generally help because the majority of bronchitis is not triggered by bacterial infections. Antibiotics hurt your possibilities of getting better by exterminating friendly micro-organisms and ensuring germs strains more antibiotic-resistant. A typical natural antidote for bronchitis is to attempt drinking half a cup of horehound tea 3 times a day. This can assist to thin and release mucous that accumulates in the lungs of bronchitis patients. When a child reveals a flicker of comprehending when talking about Acute Bronchitis, we feel that the goal of the meaning of Acute Bronchitis wheezing spread out, being accomplished.


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Guidance on changes to banking reporting requirements
The PRA has made a number of changes to banking regulatory reporting requirements, and this statement explains how these changes are being implemented in the CRR reporting modules.
Statement on supervisory benchmarking exercise
Regulatory reporting banking sector
PRA Capital Requirements Regulations (CRR)
Following PS22/21 'Implementation of Basel Standards', the PRA has incorporated the entire body of the UK version of COREP and FINREP requirements, which is aligned to the European Banking Authority (EBA) Taxonomy 3.0, into PRA rules to create a single source for reporting requirements for firms.
New data item: COR016 Global Systemically Important Institutions (G-SII)
The PRA aligned with the reporting changes contained in the European Banking Authority (EBA) Taxonomy 3.0 for global systemically important institutions (G-SII). The PRA has designated this module as COR016 G-SII, which will be submitted via RegData in line with other COREP and FINREP data.
Change of data item code: COR017 Net Stable Funding Ratio (NSFR)
The PRA also aligned with the reporting changes contained in the EBA's Taxonomy 3.0 for the net stable funding ratio (NSFR) to ensure that firms report to the PRA under a consistent taxonomy, and so that implementation efforts are proportionate to the PRA's new data needs. The PRA has modified the data item code from COR003 NSFR to COR017 NSFR. Firms will see this change reflected in their RegData submission schedule.
Removal of FSA045 from schedules
The PRA has amended PRA reporting to remove the existing duplicative reporting requirements identified. The COREP internal ratings-based (IRB) credit risk template C08.03 on the breakdown by probability of default (PD) ranges has the potential to be duplicative, as it contains information very similar in nature to the PRA's FSA045 template on IRB portfolio risk. Accordingly, the PRA will retire FSA045 from Saturday 1 January 2022.
New data item: LVR001 Leverage Ratio
In PS21/21 'The UK leverage ratio framework', the PRA made changes to the Leverage Ratio reporting requirements which will take effect on Saturday 1 January 2022. This has been introduced in version 3.5.0 of the Bank of England Banking XBRL taxonomy as a new module, LVR001 Leverage Ratio. Firms should submit these data via the BEEDS portal. Firms that need support on submitting via BEEDS should contact BEEDSQueries@bankofengland.co.uk.
As noted in PS21/21, firms will no longer be expected to submit FSA083 Leverage returns (via TDCM). Furthermore, the Financial Services Act 2021 (Prudential Regulation of Credit Institutions and Investment Firms) (Consequential Amendments and Miscellaneous Provisions) Regulations 2021 will revoke, among other things, COREP leverage reporting requirements, and therefore firms will also no longer be expected to submit COR001b leverage returns (via RegData).
COR014 and COR015 Remuneration
Firms are required to submit Remuneration Benchmarking and High Earners Reports to the PRA, as set out in Chapters 17 and 18 of the Remuneration Part of the PRA Rulebook, respectively. As part of the EBA Taxonomy 2.10, the Remuneration module became reportable for the first time in XBRL format effective from Thursday 31 December 2020. However, the PRA became aware of issues with the EBA XBRL Remuneration reporting templates, for which the EBA released a patch. At the time, the PRA decided not to implement the patch in order to minimise the burden placed on firms and instead reverted back to the XML-based REP004 and REP005 reporting templates for submission of 2020 data. Since then, the necessary changes have been implemented in the EBA Taxonomy 3.0 and firms are expected to submit in the required XBRL format depending on their year-end date.
Firms with a year-end date up to Friday 31 December 2021 will submit REP004 and REP005 XML returns.
Firms with a year-end date on/after Saturday 1 January 2022 are expected to submit XBRL reportable templates Remuneration Benchmarking and HighEarners, which are designated as COR014 and COR015 respectively.
Pause of Supervisory Benchmarking Portfolio
As set out in a PRA statement in May 2021 on the Supervisory Benchmarking exercise relating to capital internal models, the technical standards to report this information are outdated, and in relation to market risk are no longer applicable under UK law. As a result, firms will not be required or expected to submit any data for the 2022 and 2023 benchmarking exercise. This includes Credit Risk, IFRS 9 and Market Risk data (also known as COR009a, COR009b and COR010).
If firms have any questions regarding this statement, they should contact: PRA.Firmenquiries@bankofengland.co.uk.


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Mark James Bentley (born 7 January 1978) is an English former professional footballer who is now a first-team coach at Southend United.

Playing career
Born in Hertford, Bentley started his playing career with Aveley before moving on to Enfield in August 1998. He later moved to Aldershot Town, Gravesend & Northfleet (where he won Player of the Year in 2002–03) and Dagenham & Redbridge. On 15 January 2004, he stepped up into The Football League when he signed for Southend United, where he made 93 league appearances including 23 as a sub. He also scored 12 league goals including five in his last season at Roots Hall and was involved in two consecutive promotions. He was offered a new one-year contract to stay at Southend, but opted to go for the security of a two-year deal at Gillingham, signing for them on 22 May 2006. He made his debut in the 2–1 home win over Huddersfield Town on 5 August and scored his first goal away to Bradford City in a 4–2 defeat on 12 August.

Bentley was runner up in the club's 2006–07 player of the year award behind Andrew Crofts and signed a two-year contract extension in September 2007. After coming second in the Player of the Year awards for the 2009–10 season, he signed a new one-year contract for new manager Andy Hessenthaler with a clause stating that if he played a certain number of games it would be extended for another season, taking him to the end of the 2011–12 season. In March 2011, Bentley signed for Conference National club Cambridge United on a one-month loan from Gillingham. On 11 May 2011, Bentley was released by Gillingham after serving five seasons with the club, after the club failed to gain promotion from League Two. He subsequently joined Hayes & Yeading United. Following Jamie Hand's departure to Luton Town in early September, Bentley was appointed club captain.

In June 2013, Bentley signed for Conference South club Boreham Wood. Bentley signed for Wealdstone in October 2013, to help with the club's push for promotion to the Conference South. Bentley left the club by mutual consent, having made ten league appearances, in the summer of 2014.

Coaching career
Bentley was appointed as player and assistant-manager alongside his role as a coach at Grays Athletic following the departure of manager Jody Brown and a number of coaches. He was then promoted to player/caretaker-manager in January 2015, before being given the job until the end of the 2014–15 season. Bentley received widespread attention in March 2015 when he put himself on as a substitute in a game: he both scored and saved a penalty kick, helping turn a 2–0 deficit into a 3–2 win.

On 1 October 2018, Bentley was appointed as the new head coach of Canvey Island.

On 20 October 2021, Bentley was appointed as a first-team coach at former club Southend United with his former teammate Kevin Maher being appointed as head coach. At the time of leaving Canvey Island, his side were top of the Isthmian Premier Division North.

References

External links
Mark Bentley player profile at gillinghamfootballclub.com

1978 births
Living people
People from Hertford
English footballers
English football managers
Association football midfielders
Association football player-managers
Aveley F.C. players
Enfield F.C. players
Aldershot Town F.C. players
Ebbsfleet United F.C. players
Dagenham & Redbridge F.C. players
Southend United F.C. players
Gillingham F.C. players
Cambridge United F.C. players
Hayes & Yeading United F.C. players
Boreham Wood F.C. players
Wealdstone F.C. players
English Football League players
National League (English football) players
Isthmian League players
Grays Athletic F.C. players
Grays Athletic F.C. managers
Canvey Island F.C. managers
Southend United F.C. non-playing staff
Isthmian League managers
Association football coaches

redpajama_set_name: RedPajamaWikipedia


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Pret A Manger announces ambition to double size of business within five years
Death Wish Coffee Co. launches Coffee Notes with Band Halestorm
Qargo Coffee is ready to empower the creators in Great Lakes State
Panera Bread ignites 2022 Menu with new Thai Chicken Soup
Sca announces its qualified sponsors for the 2022-2025 World Barista Championship
First ever World of Coffee event in the Middle East opened yesterday in Dubai
LONDON, UK – Pret A Manger has today announced plans to double the size of its business within the next five years, backed by a new £100 million net "war chest" from JAB and Pret Founder Sinclair Beecham. The plans, which include expanding into new international markets, growing its shop estate and investing in new digital capabilities, are the next phase of Pret's transformation plan – a plan which shifts the business from "following the skyscraper", to bringing Pret to more people across the UK and beyond.
Pret first embarked on its transformation programme during the summer of 2020, as sales fell significantly due to the impact of the COVID-19 pandemic. In accounts filed this week at Companies House, revenue for 2020 was £299m, down compared to 2019 (£708m). Pret's operating loss for the year after tax and total comprehensive income was £256.5m.
Over the last month however, Pret's recovery has gathered pace, with trade now approaching pre-pandemic levels, once again making profit and a major recruitment drive underway. The number of employees now working across the business has grown 28% since the start of 2020, with over 6,000 employees in the UK alone.
At the heart of Pret's transformation plan has been a significant shift in the way the business serves customers, serving delicious food and coffee through new digital and retail channels. A year ago this month, Pret launched the UK's first Coffee Subscription service, with customers able to enjoy up to five hot drinks every day for £20 a month. The move came alongside new retail partnerships with Tesco, Sainsburys, Amazon and other leading retailers to sell bake-at-home frozen, Granolas, coffee and barista coffee beans for customers to enjoy at home.
As Pret A Manger continues its transformation, the business will continue to invest in its customer loyalty programme, while also putting a renewed focus on menu development and product innovation. The launch of Pret's recent autumn menu with the Chicken & Chilli Burrito and Naked Chicken Katsu Baguette marked the return of Pret's famous record of new recipe launches and product innovation for customers, following a period of a slimmed down menus through the height of the pandemic.
As well as continuing to develop its menu and transform Pret's digital footprint, the business will expand its shop estate in the UK, with more than 100 shops due to be opened in the next two years, largely located in regional and suburban areas.
The first major franchise partnership was signed over the summer, with further agreements expected later this year. Many of the new shops will be located in transport hubs and motorway service stations, further building on Pret's existing partnership with independent forecourt operator, MFG, and motorway services operator, Moto.
As the company expands, Pret will roll out a recruitment programme across markets, aiming to hire at least [3,000] Team Members and Baristas by the end of 2023. This follows news last week that Pret was increasing pay by at least 5% for its UK shop Team Members, in addition to the £1 Mystery Shopper bonus and free food on shift.
Commenting on the expansion plans, Pret A Manger 's CEO, Pano Christou, said:
"Last year we were in the eye of the storm during the height of the pandemic. Now we have the chance to build a bright new future for Pret.
"What the pandemic has shown us is that even at the darkest moments, more people want to experience Pret – whether that's customers outside of London and other big cities, new franchise partners who want to work with us here and overseas, investors in our business, or people who want to grow their careers here and be part of what we're trying to build.
"It's been an incredibly tough two years, but we have an incredibly opportunity ahead. Last year, we delivered more change than in thirty years of Pret history. As we move into the next phase of our transformation, we want to keep the same pace of innovation, but use it drive new growth.
I would like to thank all the Pret Team as we would not be in this position without all of their efforts through Covid. I also want to thank JAB and Sinclair Beecham our founder for supporting the Pret team and I know everyone involved with Pret is excited for what the future holds."
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Buildings under construction, bridges under construction, roads under construction; it seems impossible not to notice them. Nevertheless, for security reasons, it is recommended to warn passers-by. The purpose of 'Under Construction' signs is not only to warn them of the dangers inherent in the work being done, it is also a question of simply warning them of the inconveniences that they will have to endure until the end of the works: noise, dust and others.
If parents live near the construction site, 'Under Construction' signs will warn them of the need to keep their children out of the building site. People with allergies, for example or insomnia can take the necessary measures to avoid unpleasant fallout on their health.
Car drivers and truck drivers will take these warnings into consideration and adopt a reasonable speed when passing the site under construction. 'Under Construction' signs play a preventive role, while improving safety measures both inside and outside the building site. Warn all the people around of dangers and at the same time, protect your employees as well as the site on which construction or reconstruction work is taking place. Choose one of the classic signs or customize your own sign.
Examples of Under construction signs - Choose one of our example signs and adapt it to your requirements.


text:
namespace Anycmd.Edi.Service.Tests
{
    using Microsoft.VisualStudio.TestTools.UnitTesting;
    using ServiceStack;
    using System;

    [TestClass]
    public class TempTest
    {
        [TestMethod]
        public void StructTest()
        {
            Guid id = Guid.NewGuid();
            StructValue s = StructValue.Create(id, "Name1");
            StructValue s1 = new StructValue().PopulateWith(s);
            Assert.AreEqual(Guid.Empty, s1.Id);
            Assert.AreEqual(null, s1.Name);
            var nameProperty = s1.GetType().GetProperty("Name");
            nameProperty.SetValue(s1, "Name1");
            Assert.AreEqual(null, s1.Name);
            s1.Name = "Name1";
            Assert.AreEqual("Name1", s1.Name);
        }

        public struct StructValue
        {
            public static StructValue Create(Guid id, string name)
            {
                return new StructValue
                {
                    Id = id,
                    Name = name
                };
            }

            public Guid Id { get; private set; }
            public string Name { get; set; }
        }
    }
}

redpajama_set_name: RedPajamaGithub


Big Blue Filter Housings - Pure Aqua, Inc.
Pure Aqua's 10" and 20" Big Blue filter housings are constructed of a durable reinforced polypropylene or clear Styrene-Acrylonitrile (SAN). Pure Aqua's line of Big Blue filter housings are available in many styles and connections sizes. Pure Aqua's housings will accommodate a wide range of 4.5" diameter cartridges. The reinforced polypropylene cap offers an optional pressure-relief button on the inlet side to relieve pressure inside the housing when changing filter cartridges.
NOTE: The housings are available with and without the pressure relief, please add PR to the end of a part number for pressure relief option.


The thought of training salespeople can send shivers through the strongest of warriors. Why does sales training have such a bad rep? Contrary to popular opinion, it's not the salesperson's fault—it's ours. As trainers, we need to understand our audience and enable accordingly. And salespeople are generally relationship people and storytellers.
When we bust out a slide deck to engage an audience of gregarious salespeople, it's a recipe for a lullaby. Instead, we need to make certain those extroverted brains absorb vital information. When I think to some of the best learning experiences or presentations I've seen, there is one thing in common: the presenter told an amazing story.
Here is how to construct your training in the same way.
Every great story has a hero who we want to see live "happily ever after." If you are a sales person, your hero is your buyer. If you are training salespeople, your hero is sales. When you keep the story centered on the hero, everything falls into place—from the language you use, to the main objective of the story. It is about THEM, not about what we want them to accomplish in the training.
The hero has to achieve something by the end of the learning experience in order to live "happily ever after." This is your end goal (your objective). When I develop a sales training package, I start with the end goal in mind. Then, I work backward from the goal to flesh out the beginning and middle of the story.
Think about any awesome movie you've watched, like one of my personal favorites, Star Wars (Episode 4). There is a clear Beginning (you meet Luke and learn about the Force), a Middle (he goes on a journey to becoming a Jedi and a bunch of crazy, dangerous stuff ensues), and an End (he becomes a powerful Jedi and saves the Universe from the evil Empire). There is no reason why your training can't follow the same flow for your sales heroes!
Any training involves an element of change. Indeed, you want the people in the room to undergo a change in order to be better than they are now. But change is hard! Instead of shying away from that fact, embrace it in your training session and develop materials that will allow each participant to actively undergo that transformation through practice, role playing, and activities.
The best thing about adopting a storytelling approach to sales training is how it makes the sales reps feel. Hopefully, by the end of the learning experience, they have a sense of empowerment—like they overcame great obstacles to accomplish an awesome goal. I encourage anyone in a sales enablement role to give it a try…just don't let those salespeople know your storytelling secret!
Melissa Madian is vice president of sales enablement for Vision Critical. Her expertise is in applying the principles of modern marketing to the development of the skills, knowledge, behaviors, and processes of sales teams so they excel while growing personally and professionally. In September of 2013, Melissa was featured as the ATD Sales Enabler of the Month.


text:
The Copiale cipher is an encrypted manuscript consisting of 75,000 handwritten characters filling 105 pages in a bound volume. Undeciphered for more than 260 years, the document was cracked in 2011 with the help of modern computer techniques.  An international team consisting of Kevin Knight of the University of Southern California Information Sciences Institute and Viterbi School of Engineering, along with Beáta Megyesi and Christiane Schaefer of Uppsala University in Sweden, found the cipher to be an encrypted German text.  The manuscript is a homophonic cipher that uses a complex substitution code, including symbols and letters, for its text and spaces.

Previously examined by scientists at the German Academy of Sciences at Berlin in the 1970s, the cipher was thought to date from between 1760 and 1780.  Decipherment revealed that the document had been created in the 1730s by a secret society called the "high enlightened (Hocherleuchtete) oculist order" of Wolfenbüttel, or Oculists. The Oculists used sight as a metaphor for knowledge.

The manuscript is in private collection. A parallel manuscript is kept at the Staatsarchiv Wolfenbüttel.

The Copiale cipher includes abstract symbols, as well as letters from Greek and most of the Roman alphabet. The only plain text in the book is "Copiales 3" at the end and "Philipp 1866" on the flyleaf. Philipp is thought to have been an owner of the manuscript. The plain-text letters of the message were found to be encoded by accented Roman letters, Greek letters and symbols, with unaccented Roman letters serving only to represent spaces.

The researchers found that the initial 16 pages describe an Oculist initiation ceremony.  The manuscript portrays, among other things, an initiation ritual in which the candidate is asked to read a blank piece of paper and, on confessing inability to do so, is given eyeglasses and asked to try again, and then again after washing the eyes with a cloth, followed by an "operation" in which a single eyebrow hair is plucked.

Substitution cipher
The Copiale cipher is a substitution cipher. It is not a 1-for-1 substitution but rather a homophonic cipher: each ciphertext character stands for a particular plaintext character, but several ciphertext characters may encode the same plaintext character. For example, all the unaccented Roman characters encode a space. Seven ciphertext characters encode the single letter "e". In addition, some ciphertext characters stand for several characters or even a word. One ciphertext character ("†") encodes "sch", and another encodes the secret society's name.

Decryption method

A machine translation expert, Knight approached language translation as if all languages were ciphers, effectively treating foreign words as symbols for English words.  His approach, which tasked an expectation-maximization algorithm with generating every possible match of foreign and English words, enabled the algorithm to figure out a few words with each pass.  A comparison with 80 languages confirmed that the original language was likely German, which the researchers had guessed based on the word "Philipp," a German spelling.  Knight then used a combination of intuition and computing techniques to decipher most of the code in a few weeks.  Megyesi later realized that a particular symbol meant "eye", and Schaefer connected that discovery to the Oculists.

The Oculists
The Oculists were a group of ophthalmologists led by Count Friedrich August von Veltheim, who died in April 1775.  The Philipp 1866 Copiales 3 document, however, appears to suggest that the Oculists, or at least Count Veltheim, were a group of Freemasons who created the Oculist society in order to pass along the Masonic rites which had recently been banned by Pope Clement XII.

See also
List of ciphertexts
 Voynich manuscript

References

External links

History of cryptography
Classical ciphers