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text:
\section{Introduction\label{sec:intro}}

In the last decade, the possible appearance of the inhomogeneous chiral phase in the QCD phase diagram has been studied
\cite{Nakano:2004cd,Tatsumi:2014wka,Schon:2000qy,Thies:2003kk,Nickel:2009wj,Nickel:2009ke,Basar:2008ki,Basar:2009fg,Kojo:2009ha,Kojo:2011cn,Muller:2013tya,Buballa:2014tba,Carignano:2010ac,Carignano:2014jla,Carignano:2012sx,Karasawa:2013zsa,Maedan:2009yi,Abuki:2013vwa,Abuki:2013pla,Abuki:2011pf,Frolov:2010wn,Gubina:2012wp,Ebert:2014woa},
where the  quark condensate is spatially modulated.
For the analysis of the  inhomogeneous chiral phase, the chiral order parameter,
$M({\bf x})=-2G\left[ \left< \overline \psi \psi\right> +i\left< \overline \psi i\gamma^5\tau^3\psi\right>\right]$,
has been used.
Using the effective models of QCD including the Nambu-Jona-Lasinio  model or the Schwinger-Dyson approach,
there appears the inhomogeneous chiral phase in the vicinity of the chiral transition and its critical point is changed to be the Lifshitz point \cite{Nickel:2009ke}.
The dual chiral density wave (DCDW) or the real kink crystal (RKC) has been  often used as a typical condensate with one dimensional spatial modulation.
DCDW is a plane wave configuration, $ M({\bf x}) =me^{iqz}$,
while RKC is a multi-soliton configuration,  $ M({\bf x}) =\frac{2m\nu}{1+\sqrt{\nu}}{\rm sn}\left( \frac{2mz}{1+\sqrt{\nu}}, \nu\right) $,
without the phase degree of freedom.
In Ref. \cite{Nickel:2009wj}, it has been shown that one or two dimensional modulation can be embedded in 1+3 dimensions  by using the Lorentz boost.
The general solutions have been obtained by using the NJL$_2$ model in 1+1 dimensions \cite{Basar:2008ki}, which is called complex kink crystal.
Higher dimensional modulations has been also considered in some studies \cite{Basar:2008ki,Basar:2009fg,Carignano:2012sx,Abuki:2011pf,Kojo:2011cn}.

In Refs. \cite{Nakano:2004cd,Nickel:2009wj,Nickel:2009ke,Muller:2013tya},
it has been shown that the inhomogeneous chiral phase can appear at low temperature and moderate density region as an intermediate phase prior to the chiral transition.
Such inhomogeneous phase has been also studied in condensed matter physics, e.g. spin or charge density wave and the FFLO state of the superconductivity \cite{Overhauser:1962zz,Pierls:1955,Fulde:1964zz,larkin:1964zz}.
The appearance of the inhomogeneous chiral phase in the QCD diagram has been extensively studied by using the various approaches,
but there are few works about the external field, isospin asymmetry, and current quark mass.
These effects should be very important in realistic situations, especially for compact stars.
The effect of the current quark mass has been studied in some papers \cite{Nickel:2009wj,Maedan:2009yi,Karasawa:2013zsa} and change of the phase diagram has been figured out.

In this paper we consider the inhomogeneous chiral phase in the presence of the external magnetic field to figure out some magnetic properties.
The effect of the magnetic field is theoretically and phenomenologically interesting and important,
since quark matter is put into the strong magnetic field in compact stars or in heavy-ion collision process \cite{Skokov:2009qp}.
QCD in the external magnetic field has recently attracted great attention,
and it has been shown that the magnetic field gives rise to various phenomena such as chiral magnetic effect \cite{Fukushima:2008xe},
magnetic catalysis \cite{Gusynin:1995nb,Suganuma:1990nn,Klevansky:1989vi}, magnetic inhibition \cite{Bali:2011qj,Fukushima:2012kc,Braun:2014fua,Mueller:2015fka}.
Lattice QCD simulations have been also performed to study the properties of the QCD vacuum in response to the magnetic field \cite{Bali:2011qj};
the effect of the magnetic field on the chiral transition or deconfinement has been studied at chemical potential $\mu=0$.

The property of the inhomogeneous chiral condensate in the magnetic field has been first studied by Frolov et al. \cite{Frolov:2010wn}.
They have found that the DCDW phase develops in a wide density region at $T=0$ under the magnetic field,
and that some peculiar behaviours of the amplitude and of the wavevector can be seen due to the de Haas-van Alphen effect \cite{deHaas1936,Ebert:1999ht}.
However, they did not take into account the possibility of the RKC suggested to be favored in the absence of the magnetic field \cite{Nickel:2009wj}.

In this paper, we study the QCD phase diagram in the Nambu-Jona-Lasinio (NJL) model \cite{Nambu:1961tp,Hatsuda:1994pi,Klevansky:1992qe} in the magnetic field, taking into account both of the condensates.
We introduce the a new type of the condensate called hybrid chiral condensate (HCC) ,
$
 M({\bf x}) = \frac{2m\nu}{1+\sqrt{\nu}}{\rm sn}\left( \frac{2mz}{1+\sqrt{\nu}},\nu \right)e^{iqz}
$, which smoothly connects both DCDW and RKC by changing the modulus $\nu$ or the wavevector $q$, and demonstrate that the magnetic field favors the phase modulation:
it is found through the analysis of the thermodynamic potential that the wavevector $q$ takes a nonzero value in the presence of the magnetic field,
and thus DCDW and RKC coexist as HCC in the weak magnetic field at moderate densities.
We shall see that the phase degree of freedom in HCC plays an important role in the presence of the magnetic field.
The energy spectrum of the quark field becomes asymmetric in the presence of the magnetic field, which gives rise to anomalous quark number \cite{Tatsumi:2014wka}.
Such spectral asymmetry is closely connected with chiral anomaly and moves the Lifshitz point to zero chemical potential $\mu=0$ .

We consider only the case of isospin symmetric matter ($\mu_u=\mu_d$) in the chiral limit($m_c=0$) for simplicity.
In  Sec. II we briefly summarize the general framework to deal with the inhomogeneous chiral phases in the presence of the magnetic field.
We introduce HCC in Sec.III.
Spectral asymmetry in the HCC phase and some topological features are also discussed there.
The phase diagram is presented in Sec.IV in the $B-\mu$ plane. Sec.V is devoted to concluding remarks.
The proper-time regularization method is given Appendix A.
Some details about spectral asymmetry are presented in Appendix B and C.
An expansion of the thermodynamic potential with respect to $B$ is given in Appendix D .

\section{Model and Energy spectrum\label{sec:model}}
Here we briefly summarize the general framework to get the quark energy spectrum in the inhomogeneous chiral phase in the presence of the magnetic field.
First, we consider the case of $N_f=N_c=1$ for simplicity.
The case of three colors and two flavors is also calculated in the same way.
Taking the magnetic field $\bf B$ along , e.g., $z$ axis, the Lagrangian reads
\begin{eqnarray}
 L &=& \overline \psi\left[ -i\gamma^\mu D_\mu  +\frac{1+\gamma^5}{2}M +\frac{1-\gamma^5}{2}M^*\right] \psi -\frac{\left| M\right|^2}{4G},
\end{eqnarray}
within the mean-field approximation,  where $\psi$ is 4-dimensional spinor,
 $M$ is the order parameter of chiral transition, $M({\bf x})=-2G\left[ \left< \overline \psi \psi\right> +i\left< \overline \psi i\gamma^5\tau^3\psi\right>\right]$,
 ,$D_\mu$ is covariant derivative $D_\mu = \partial_\mu -ieA_\mu$.
We consider one dimensional modulation along the $z$ axis as well, $M({\bf x})=M(z)$.
It is assumed that magnetic field is uniform and parallel to modulation of the order parameter.
We shall see later that this orientation should be most favorable due to the topological aspects \cite{Tatsumi:2014wka}.
We combined Nickel's method \cite{Nickel:2009wj} and Frolov's method \cite{Frolov:2010wn} for obtaining the energy spectrum.

We choose the Landau gauge, $A^\mu=(0,{\bf A}),{\bf A}=(0,xB,0)$, and assume $eB>0$.
The Hamiltonian then renders
\begin{eqnarray}
H_D &=& {\bf \alpha\cdot\Pi}+\gamma^{0}M\frac{1+\gamma^{5}}{2}+\gamma^{0}M^{*}\frac{1-\gamma^{5}}{2},
\end{eqnarray}
where ${\bf \Pi}$ is kinetic momentum, $\Pi_i = -i\partial_i + eA_i$.
The Hamiltonian satisfies the commutation relation, $[H_D,-i\partial_{y}]=[H_D,(\alpha_{\perp}\cdot\Pi_\perp)^{2}]=0$,
where ${\bf \alpha}_\perp=(\alpha_x,\alpha_y,0)$ and ${\bf \Pi}_\perp=(\Pi_x,\Pi_y,0)$ .
The eigenspinor of $-i\partial_{y}$ and $(\alpha_{\perp}\cdot\Pi_\perp)^{2}$ can be written as
 \begin{eqnarray}
\psi_{n,k}=\frac{1}{\sqrt{2\pi}}\left(eB \right)^{1/4}e^{iky}\left(\begin{array}{ccc}    c_{1}(z)u_{n-1}(\eta)\\
 ic_{2}(z)u_{n}(\eta) \\
   c_{3}(z)u_{n-1}(\eta)\\
 ic_{4}(z)u_{n}(\eta)\\
\end{array}
\right),
\end{eqnarray}
where $\eta=x\sqrt{eB}+k/\sqrt{eB}$,  $u_{n}(\eta)$ is the Hermite function \cite{Sokolov1986} which satisfies
$
 \left(\frac{\partial}{\partial\eta}+i\eta \right)u_n(\eta)=\sqrt{2n}u_{n-1}(\eta)
 $
 and
 $
 \left(\frac{\partial}{\partial\eta}-i\eta \right)u_{n-1}(\eta)=-\sqrt{2n}u_{n}(\eta),
$
and $n=0,1,2,....$ denotes the discrete Landau levels.
Using this eigenspinor, the Hartree-Fock equation $H_D\psi = {\cal E} \psi $ is reduced to
 \begin{eqnarray}
\left(\begin{array}{cccc}   -i\partial_z & 0 & M(z) & \sqrt{2eBn} \\
  0&i\partial_z &\sqrt{2eBn} & M^{*}(z)\\
  M^{*}(z)&\sqrt{2eBn} &i\partial_z  & 0 \\
\sqrt{2eBn}  & M(z)& 0 & -i\partial_z \\
\end{array}
\right)
\left(\begin{array}{c}    c_{1}(z)\\
 c_{2}(z) \\
   c_{3}(z)\\
 c_{4}(z)\\
\end{array}
\right)
=
{\cal E}
\left(\begin{array}{c}     c_{1}(z)\\
 c_{2}(z) \\
   c_{3}(z)\\
 c_{4}(z)\\
\end{array}
\right).
\label{eq:3dh}
\end{eqnarray}
for $n=1,2,...$, and
 \begin{eqnarray}
\left(\begin{array}{cc}
  i\partial_z & M^{*}(z)\\
 M(z) & -i\partial_z \\
\end{array}
\right)
\left(\begin{array}{c}
 c_{2}(z) \\
 c_{4}(z)\\
\end{array}
\right)
=
{\cal E}
\left(\begin{array}{c}
 c_{2}(z) \\
 c_{4}(z)\\
\end{array}
\right).
\label{eq:1dh}
\end{eqnarray}
for $n=0$.
The latter equation (\ref{eq:1dh}) resembles the Bogoliubov-de Gennes (BdG) equation in 1+1 dimensions,
while the former equation (\ref{eq:3dh}) is the same form as the one  without magnetic field.
Thus the energy spectrum for these equations can be obtained once the corresponding one is given in the absence of the magnetic field.
The energy spectrum in the case of $B=0$ is simply written as ${\cal E}=\lambda_{\pm} \sqrt{1+p_{\perp}^2/{\lambda_\pm^2}}$ with
the perpendicular component of the momentum, $p_{\perp}$ \cite{Nickel:2009wj} ,
where $\lambda_+$ is the eigenenergy  of the 1+1 dimensional Hartree-Fock equation \cite{Basar:2008ki},
  \begin{eqnarray}
\left(\begin{array}{cc}
i\partial_z & M(z)\\
M(z)^* & -i\partial_z
\end{array}
\right)\psi &=& \lambda_+\psi
,
 \end{eqnarray}
and $\lambda_-$ is defined as the eigenenergy for the complex conjugate transformation : $M({\bf x})\to M({\bf x})^*$.
Since Eq.~ (\ref{eq:3dh}) has a similar form to the usual Dirac equation with momentum $p_\perp$, the eigenvalue can be simply obtained by replacing $p_\perp$ by $\sqrt{2eBn}$.
 Thus we obtain the energy spectrum for Eqs.(\ref{eq:3dh}) and (\ref{eq:1dh});
\begin{eqnarray}
{\cal E}_{n,\zeta} &=&
\left\{
\begin{array}{ll}
\lambda_{\zeta}\sqrt{1+\frac{2eBn}{\lambda_{\zeta}^{2}}} & n=1,2....\\
\lambda_{\zeta=+} & n=0,
\end{array}
\right.
\label{eq:energy}
\end{eqnarray}
where $\lambda_\zeta$ is asymmetric respect to $\lambda_\zeta=0$ if complex conjugate symmetry is broken, $M({\bf x})\not=M({\bf x})^*$\cite{Tatsumi:2014wka}.

These results can be easily generalized to the case of $N_f=2$.
Assuming that the ground state is the charge eigenstate,
\begin{eqnarray}
M = -2G[\langle\overline{\psi}\psi\rangle+i\langle\overline{\psi}i\gamma^{5}\tau^{3}\psi\rangle]\\
\langle\overline{\psi}i\gamma^{5}\tau^{1}\psi\rangle= \langle\overline{\psi}i\gamma^{5}\tau^{2}\psi\rangle=0,
\end{eqnarray}
the NJL Lagrangian with three colors and two flavors is  written as
\begin{eqnarray}
\nonumber L &=& \overline{\psi}(i\gamma^{\mu}D_{\mu}-M\frac{1+\tau^3\gamma^{5}}{2}-M^{*}\frac{1-\tau^3\gamma^{5}}{2})\psi- \frac{\mid M\mid^{2}}{4G},
\end{eqnarray}
which is flavor diagonal, so that we can calculate the energy spectrum for each flavor.

Thermodynamic potential is now written as
\begin{eqnarray}
\Omega[\mu,T,B;\Delta(z)]
&=& \frac{\left<\left| \Delta (z)\right|^2 \right>}{4G}
-TN_c\sum_f\frac{|e_fB|}{2\pi}\sum_{n,\zeta}\sum_{\lambda_\zeta} {\rm ln}\left[ {\rm 2cosh}(\frac{\lambda_{\zeta}\sqrt{1+\frac{2 |e_fB|n}{\lambda_{\zeta}^2}}-\mu}{2T})   \right] \\
\nonumber &=& \frac{\left<\left| \Delta (z)\right|^2 \right>}{4G}
-TN_c\sum_f\frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \left[\sum_{\lambda_\zeta} \delta(\lambda - \lambda_{\zeta}) \right]{\rm ln}\left[ {\rm 2cosh}(\frac{\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}-\mu}{2T})   \right]\\
 &=& \frac{\left<\left| \Delta (z)\right|^2 \right>}{4G}
-TN_c\sum_f\frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \rho_\zeta(\lambda){\rm ln}\left[ {\rm 2cosh}(\frac{\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}-\mu}{2T})   \right],
\label{eq:tp1}
\end{eqnarray}
where $\rho_\zeta(\lambda)$ is the density of states, $\rho_\zeta(\lambda)= \sum_{\lambda_\zeta}\delta(\lambda - \lambda_{\zeta})$.

\section{Hybrid Chiral Condensate in the External Magnetic Field \label{sec:hybrid}}

We introduce the hybrid chiral condensate (HCC) which has the properties of both of DCDW and RKC,
\begin{eqnarray}
M(z)&=& \frac{2m\nu}{1+\sqrt{\nu}}{\rm sn}\left(\frac{2mz}{1+\sqrt{\nu}};\nu \right)e^{iqz},
\end{eqnarray}
and is characterized by three parameters; $m,q,\nu$.
It is reduced to the pure DCDW in one limit, $\nu\rightarrow 1$, while to the pure RKC in the other limit, $q\rightarrow 0$.
Thus HCC is the minimum configuration which includes both of DCDW and RKC.
Note that HCC is simply given by the product of the two types of the condensate,
but it satisfies the BdG equation within the NJL$_2$ model.
In Refs. \cite{Basar:2008ki,Basar:2009fg}, Basar et al. have found the general form of the condensate in $1+1$ dimensions,
\begin{equation}
M(z)=-m\, e^{i q z}\, A\,\frac{\sigma(m A\, z+i{\bf K}^\prime -i\theta/2)}
{\sigma(m A\, z+i{\bf K}^\prime)\sigma(i\theta/2)}
\,\exp\left[im A\, z \left(-i\,\zeta(i\theta/2)+i\,{\rm ns}(i\theta/2)\right)+i\,\theta \eta_3/2\right]
 \quad
 \end{equation}
characterized by four parameters; $m,q,\nu,\theta$,
where $A=A(\theta,\nu)=-2i{\rm sc}(i\theta/4;\nu){\rm nd}(i\theta/4;\nu)$, and $\sigma$ and $\zeta$ are Weierstrass sigma and zeta functions, and $\eta_3= \zeta(i{\rm K}')$.
When $\theta=2{\bf K}(\nu)$, this condensate becomes HCC.
It can be easily seen that the energy spectrum in the HCC phase $\lambda_\zeta$ is uniformly shifted by $\zeta q/2$ from the RKC one, $\lambda_\zeta\rightarrow \lambda_\zeta-\zeta q/2$.
 Accordingly, the density of states is given as
\begin{eqnarray}
\rho_{\zeta}(\lambda) &=&\rho_{\rm RKC}({\lambda-\zeta q/2})\nonumber\\
&=&\frac{1}{\pi}\frac{|(\lambda-\zeta q/2)^2 + m^2c|}{\sqrt{((\lambda-\zeta q/2)^2-m^2)((\lambda-\zeta q/2)^2-m^2{\nu'})}} ,
\label{eq:density}
\end{eqnarray}
 by using the density of states for RKC, $\rho_{\rm RKC}(\lambda)$ given in \cite{Basar:2008ki}, where
 $c=(1-\nu-2{\bf E(\nu)/K(\nu)})/(1+\sqrt{\nu})^2$, $\nu'=(1-\sqrt{\nu})^2/(1+\sqrt{\nu})^2$ and ${\bf E(\nu),K(\nu)}$ are the complete elliptic integrals.

\begin{figure}[htb]
\includegraphics[width=10cm]{HCC.eps}
\caption{ Profile of HCC on the base manifold, which is the direct product of  the horizontal $z$ axis and the vertical chiral circle given by the scalar and pseudo-scalar condensates.}
\label{fig:HCC}
\end{figure}

Putting the density of states (\ref{eq:density}) in Eq.(\ref{eq:tp1}), we have the thermodynamic potential, which is decomposed into the vacuum contribution, the medium contribution and the thermal contribution:
\begin{eqnarray}
\label{eq:potential}
{\Omega}[\mu,T,B;m,\nu,q] &=& \frac{m^2}{4G}\left( 1-\frac{{\bf E}(\nu)}{{\bf K}(\nu)}\right) +\Omega _{\rm vac} + \Omega _{\mu} +\Omega _{T} \\
\Omega _{\rm vac} &=&-\frac{1}{2}N_c \sum_{f} \frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \rho_\zeta(\lambda)|\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}| \\
\Omega _{\mu}&=&-\frac{1}{2}N_c \sum_{f} \frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \rho_\zeta(\lambda)
 \left[ |\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}-\mu|-|\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}| \right]\\
 \Omega _{T} &=&-TN_c \sum_{f} \frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \rho_\zeta(\lambda){\rm ln}\left[ 1+ {\rm exp}(-\frac{|\lambda\sqrt{1+\frac{2|e_fB|n}{\lambda^2}}-\mu|}{T})   \right]
\end{eqnarray}
Here the vacuum contribution $ \Omega _{\rm vac}$ is divergent, so that we use the proper time regularization.
\begin{eqnarray}
\nonumber \Omega _{\rm vac} &=& N_c \sum_{f,\zeta} \frac{|e_fB|}{16\pi^{3/2}}\int_{1}^{\infty}\frac{d\tau}{\tau^{3/2}}{\rm coth}(\tau|e_fB|)
\int d\lambda\rho_{\zeta}(\lambda){\rm exp}\left(-\tau \lambda^2\right)
\end{eqnarray}

\section{Spectral asymmetry and anomalous quark number density\label{sec:sa}}
The fermion number is given by
\begin{eqnarray}
\label{eq:number}
N_B &=&-\frac{1}{2}\eta_H + VN_c \sum_{f} \frac{|e_fB|}{2\pi}\sum_{n,\zeta}\int d\lambda \rho_\zeta(\lambda)\left[\frac{\theta({\cal E})}{1+e^{({\cal E}-\mu)/T}}+\frac{\theta({\cal -E})}{1+e^{-({\cal E}-\mu)/T}}\right],
\end{eqnarray}
where the first term is the fermion number from {\it spectral asymmetry} \cite{Niemi:1984vz} characterized by the Atiyah-Patodi-Singer $\eta$-invariant \cite{Niemi:1984vz,Atiyah1973} which is written as
\begin{eqnarray}
\nonumber \eta_H &=& VN_c \sum_{f}\frac{|e_fB|}{2\pi}\left[ \sum_{\lambda>0}1-\sum_{\lambda<0}1\right]\\
&=& VN_c \sum_{f}\frac{|e_fB|}{2\pi}\lim_{s \to +0} \int_{-\infty}^{\infty}d\lambda \rho_+(\lambda){\rm sign}(\lambda)\left|\lambda\right|^{-s},
\end{eqnarray}
in our case.
Here, we have used the fact that spectral asymmetry appears only in the spectrum of the lowest Landau level ($n=0$),
and the higher Landau levels ($n \not = 0$) have no contribution to spectral asymmetry.
Note that the spectrum becomes symmetric without magnetic field.
The second term in Eq.(\ref{eq:number})  counts the number of states for the given $\mu$ with the Fermi-Dirac distribution functions and  is usual number density:
all  the Landau levels contribute to this term.
Considering quark number density is related to the thermodynamic potential through the thermodynamic relation (\ref{eq:potential}): ${N_B}/{V} = -{\partial\Omega}/{\partial\mu}$,
 we can explicitly verify that this derivative of thermodynamic potential and Eq.(\ref{eq:number}) are equivalent  by using Eq.(\ref{eq:potential}).

\begin{figure}[htb]
\begin{center}
\subfigure[DCDW ($m=0.5,\nu=1,q=0.4$)]{\includegraphics[width=5cm,angle=0]{spectrum1.eps}}
\hspace{5mm}
\subfigure[RKC ($m=0.5,\nu=0.3,q=0$)]{\includegraphics[width=5cm,angle=0]{spectrum2.eps}}
\hspace{5mm}
\subfigure[HCC ($m=0.5,\nu=0.3,q=0.4$)]{\includegraphics[width=5cm,angle=0]{spectrum3.eps}}
   \end{center}
\caption{The behaviour of $\rho_+(\lambda)$.
$E_1=-m+q/2$,~$E_2=-m\nu'+q/2$,~$E_3=m\nu'+q/2$,~$E_4=m+q/2$.
DCDW has the gap between $\lambda=-m+q/2$ and $\lambda=m+q/2$, and has no mid-gap states.
The spectrum of DCDW is asymmetric with respect to 0.
 The spectrum of RKC is symmetric and has the mid-gap states.
 In the case of HCC, spectrum  is asymmetric and has  the mid-gap states.
}
\label{fig:spectrum}
\end{figure}

The density of states of the lowest Landau level (LLL) is schematically shown in Fig. \ref{fig:spectrum}.
When the Hamiltonian is symmetric for complex conjugation operation $M({\bf x})\to M({\bf x})^*$, the spectrum is symmetric.
So in the case of RKC or homogeneous condensate, number density becomes usual number density.
In the case of HCC or DCDW, the contribution of  $\eta$-invariant is nonzero in the presence of the magnetic field.
The  $\eta$-invariant has been evaluated to give
\begin{equation}
\eta_H=-VN_c \sum_{f}\frac{|e_fB|}{2\pi}\frac{q}{\pi}
\label{eta-DCDW}
\end{equation}
for DCDW in the case of $-m+q/2<0$ \cite{Tatsumi:2014wka}.
It has a topological origin, and is equal to the expression given by chiral anomaly \cite{Son:2007ny} (see Appendix \ref{sec:spectral2}).
It is straightforward to evaluate the $\eta$-invariant for the case of HCC (Appendix \ref{etainv}),
\begin{equation}
\eta_H=-VN_c \sum_{f}\frac{|e_fB|}{2\pi}\left(\frac{q}{\pi}-N_{\rm midgap}\right),
\end{equation}
for the case of $-m+q/2<0$, where the second term is the contribution from the mid-gap states.
In particular, for $m > q/2 > m\nu'$, it equals to the number of nodes of HCC, $N_{\rm nodes}=m/[(1+\sqrt{\nu}){\bf K}(\nu)]$, independent of $q$.
Note that this is the same form as in the DCDW phase, except the number of nodes of the condensates.
 For the general case, the $\eta$-invariant can be written as
$
\eta_H=VN_c \sum_{f}\frac{|e_fB|}{2\pi}\left[ -\frac{q}{\pi} +\frac{2m}{\pi}{\rm Re}\left({\bf F}(q/2;\nu^\prime) +c{\bf F}(q/2;\nu^\prime)-{\bf E}(q/2;\nu^\prime) \right)\right],
$
where ${\bf F}(x;\nu^\prime)$ and ${\bf E}(x;\nu^\prime)$ are the incomplete elliptic integrals of first and second kind. Note that $\eta_H$ is reduced to the DCDW one (\ref{eta-DCDW})
as $q\rightarrow 0$, where the energy spectrum is reduced to the one of RKC and symmetric about zero.

For low chemical potential, $m+q/2>\mu$, usual number density is zero, and the inhomogeneous phase is forbidden by the Lorentz symmetry of the vacuum.
On the other hand, spectral asymmetry gives nonzero number density, and the appearance of the inhomogeneous phase is allowed in the presence of the magnetic field.
The contribution of spectral asymmetry is taken in the thermodynamic potential as the term, $\mu \eta_H /2$.
Since this term includes the linear term in $q$ and the order parameters are determined by the stationary conditions for the thermodynamical potential,
 $q=0$ is never the optimal point. In other words the RKC phase itself does not appear in the QCD phase diagram in the magnetic field.

In \cite{Tatsumi:2014wka}, spectral asymmetry has been also calculated using the derivative expansion and $\eta_H=N_c\sum_f |e_f|{\bf B\cdot q}/2$ has been obtained for DCDW.
Consequently the ${\bf q}\cdot{\bf B}$ term should appear in the thermodynamic potential by way of the thermodynamic relation, and $\bf q$ is favored to be parallel to $\bf B$.
For HCC, the derivative expansion can't be directly applied because the condensate has nodes and the premise that the amplitude is much larger than the wavevector breaks down.
However, we can manage to evaluate the $\eta$-invariant by separating the small nodal region to find $\eta_H=N_c\sum_f |e_f|{\bf B\cdot q}/2+N_{\rm nodes}$ for $m\nu'<q/2$,  which suggests
$\bf q$ is most favored to be parallel to $\bf B$ as well in the HCC phase.

\section{Phase diagram\label{sec:phase}}
For obtaining the phase diagram, we numerically search the minima of the thermodynamic potential with respect to the order parameters; $m$, $q$ and $\nu$  for given values of the magnetic field $B$ and chemical potential $\mu$.
In this paper, we show the phase diagram at zero temperature.
We use $G\Lambda^2 = 6.35$ and $\Lambda = 660$MeV which reproduce $f_\pi = 93$MeV and the constituent quark mass $\simeq 330$MeV in the vacuum.

 \subsection{RKC phase in the magnetic field}
First, we consider the effect of the magnetic field on the RKC phase.
Without the magnetic field, the RKC phase is energetically more favorable than the DCDW phase in the framework of the NJL model \cite{Nickel:2009wj}.
For RKC, the order parameter $M(z)$ is real and there appears no spectral asymmetry.

The phase diagram of the RKC phase is essentially unchanged, while ,as we shall see later, that of the DCDW phase is significantly changed in the presence of the magnetic field.
Since the chiral condensate is neutral, one may expect that there is little effect of the magnetic field on the inhomogeneous chiral phase.
Although it holds for the RKC phase, some anomalous effect coming from spectral asymmetry plays an important role in the DCDW phase.
Figure \ref{fig:rkcphase} shows the phase diagram of the RKC phase in the presence of the magnetic field.
We can see some oscillation of the  phase boundary with respect to the magnetic field, which comes from the Landau quantization and related to the de Haas-van Alphen effect \cite{deHaas1936,Ebert:1999ht}.
This oscillation is also observed in the case of the homogeneous chiral condensate within the NJL model \cite{Inagaki:2003yi}.

Figure \ref{fig:rkcorder} shows the order parameters as functions of chemical potential in different magnetic fields.
At  $\sqrt{eB} = 60 {\rm MeV}$, the order parameters behave like continuous functions of the chemical potential, and are very similar to those in the absence of the magnetic field.
At  $\sqrt{eB} = 120{\rm MeV}$, the order parameters exhibit some discontinuous jumps, since the thermodynamic potential  has some local minima as a function of $m$ and $\nu$ in this region.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=9cm,angle=0]{rkc_phase.eps}
   \end{center}
\caption{Phase diagram for the RKC phase.
}
\label{fig:rkcphase}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[$\sqrt{eB}=60 {\rm MeV}$]{\includegraphics[width=7cm,angle=0]{rkc_60.eps}}
\hspace{9mm}
\subfigure[$\sqrt{eB}= 120{\rm MeV}$]{\includegraphics[width=7cm,angle=0]{rkc_120.eps}}
\vspace{12mm}
   \end{center}
\caption{Order parameters as functions of chemical potential in different magnetic field,
where $k$ is the wavevector of RKC : $k=2m/[(1+\sqrt{\nu}){\bf K}(\nu)]$
}
\label{fig:rkcorder}
\end{figure}

\subsection{HCC phase in the magnetic field}
Here, we consider the phase diagram by introducing the HCC, which includes both  features of DCDW and RKC,
and use the following approximation instead of fully evaluating the thermodynamic potential.
When the magnetic field is much weak, the approximation
\begin{eqnarray}\label{eq:approx}
\Omega(B) &\simeq& \Omega(B=0)+eB\Omega^{(1)}
\end{eqnarray}
is valid, where the first order correction is written in
\begin{eqnarray}
eB\Omega^{(1)} &=&1/2\left( \Omega_{LLL,q}-\Omega_{LLL,-q} \right)
\end{eqnarray}
(See Appendix:\ref{sec:expansion}).
This term is an odd function of $q$, and vanishes at $q=0$,
so that this term does not appear in the thermodynamic potential for the RKC phase.
Only LLL contributes to $\Omega^{(1)}$, while the higher Landau levels ($n\neq 0$) and LLL contribute to the second and higher order terms.
We checked the validity of this expansion for some $eB$ by comparing the numerical results with Eq.~(\ref{eq:approx}) and the full expression Eq.~(\ref{eq:potential}).
Consequently we have found that the phase structure is almost unchanged for $\sqrt{eB} < 0.2\Lambda \simeq 120{\rm MeV}$.

The magnetic properties of the DCDW phase in the external magnetic field has been studied by Frolov et al. \cite{Frolov:2010wn}.
They have shown that the DCDW phase is always favorable than that of the homogeneous chiral condensate in the presence of $B$.
As is already stated in the previous section, it is shown that the mechanism of superiority of the DCDW phase is related to spectral asymmetry of LLL states.

The phase diagram is shown in Fig.\ref{fig:phase}.
In this figure, A denotes the weak DCDW phase, B the HCC phase ,C the strong DCDW phase, and D the chiral-restored phase.
The triple points appear at $(\mu,\sqrt{eB}) \sim (320{\rm MeV},110{\rm MeV})$ and $(\mu,\sqrt{eB}) \sim (350{\rm MeV},{\rm 30MeV})$.
For the limit $eB \to 0$ , the weak DCDW phase is reduced to the homogeneously chiral-broken phase and the HCC phase to the RKC phase.
In the $eB \not = 0$ region, the order parameter is always finite, and
there is no homogeneously chiral-broken phase nor the RKC phase if the magnetic field has nonzero strength.
The phase boundary between the chiral-broken and restored phases moves to higher $\mu$ as the magnetic field becomes stronger.
Thus the magnetic field expands the chiral-broken phase mainly due to the phase degree of freedom.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm,angle=0]{phase.eps}\\
\vspace{12mm}
   \end{center}
\caption{Phase diagram at T=0.
A: Weak DCDW phase
B: HCC phase
C: Strong DCDW phase
D: Chiral-restored phase.
When $eB \to 0$ , the weak DCDW phase becomes the homogeneously chiral-broken phase smoothly, and the HCC phase becomes the RKC phase.
The filled circles represent triple points.}
\label{fig:phase}
\end{figure}

Figure \ref{fig:surface} shows the energy surface of the thermodynamic potential in the $\xi-q$ plane,
where $m$  is set to be the optimal value for given $\xi,q$ with $\xi = 16^{(1-1/\nu)}$.
Without the magnetic field, there are two local minima corresponding to DCDW and RKC in the energy surface of the thermodynamic potential;
RKC is energetically more favored than DCDW.
The minimum with the homogeneous chiral condensate is smoothly changed to that of RKC, so this phase transition is  of the second order.
There is a competition between RKC and DCDW, and there appears no phase in which both of phase and amplitude modulations are large.
Once turning on  the magnetic field,  both minima of RKC and DCDW move to the larger $q$ direction, which is caused by spectral asymmetry.

\begin{figure}[htb]
\begin{center}
\subfigure[$\sqrt{eB}=0,\mu=0.48\Lambda$]{\includegraphics[width=5cm,angle=0]{1_0_480.eps}}
\subfigure[$\sqrt{eB}=0,\mu=0.485\Lambda$]{\includegraphics[width=5cm,angle=0]{1_0_485.eps}}
\subfigure[$\sqrt{eB}=0,\mu=0.49\Lambda$]{\includegraphics[width=5cm,angle=0]{1_0_490.eps}}
\subfigure[$\sqrt{eB}=0,\mu=0.5\Lambda$]{\includegraphics[width=5cm,angle=0]{1_0_500.eps}}
\subfigure[$\sqrt{eB}=0.18\Lambda,\mu=0.48\Lambda$]{\includegraphics[width=5cm,angle=0]{1_018_480.eps}}
\subfigure[$\sqrt{eB}=0.18\Lambda,\mu=0.485\Lambda$]{\includegraphics[width=5cm,angle=0]{1_018_485.eps}}
\subfigure[$\sqrt{eB}=0.18\Lambda,\mu=0.49\Lambda$]{\includegraphics[width=5cm,angle=0]{1_018_490.eps}}
\subfigure[$\sqrt{eB}=0.18\Lambda,\mu=0.5\Lambda$]{\includegraphics[width=5cm,angle=0]{1_018_500.eps}}
   \end{center}
\caption{Energy surface of thermodynamic potential at different $eB$ and different $\mu$}
\label{fig:surface}
\end{figure}

It has been discussed that the mechanism of emergence of  the inhomogeneous chiral phase is the Fermi surface nesting \cite{Nakano:2004cd}.
Complete nesting is realized in 1+1 dimensions, when the wavenumber of condensate $k_c$ and the Fermi wavenumber $k_F$ have  the relation $2k_F=k_c$.
In 1+3 dimensions, the nesting effect is incomplete, but its reminiscence is left in the DCDW phase \cite{Nakano:2004cd}: $k_c$ is large, $k_c\sim O(2k_F)$.
The situation is a little changed in the HCC phase.
Using  an approximation $M(z) \simeq m{\rm cos}(kz)e^{iqz}$, we can decompose the order parameter into two different component:
$M(z) = \frac m2(e^{i(q+k)z}+e^{i(q-k)z}) = \frac m2(e^{iq_+z}+e^{iq_-z})$.
The $q_+$ and $q_-$ can not satisfy the nesting relation simultaneously if $k \not = 0$ and $q\not =0$.
Thus HCC can not satisfy the nesting relation, and there is no HCC phase without  the magnetic field.
In the magnetic field, spectral asymmetry contributes to the emergence of  the HCC phase.
Note that non-zero value of $q$ is favored by some topological effects in this case, different from the nesting effect.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=7cm,angle=0]{h=0.eps}
\hspace{9mm}
\includegraphics[width=7cm,angle=0]{h=010.eps}
\hspace{9mm}
\includegraphics[width=7cm,angle=0]{h=018.eps}
\vspace{12mm}
   \end{center}
\caption{Order parameters as functions of chemical potential in different magnetic field.
From left to right:
(a) $eB=0$
(b) $\sqrt{eB}=060{\rm MeV}$
(c) $\sqrt{eB}=120{\rm MeV}$
}
\label{fig:order}
\end{figure}

The order parameters are shown as functions of chemical potential in Fig.\ref{fig:order}.
Under no magnetic fields $q$ is zero everywhere, so that there are the homogeneously chiral-broken phase  ($\xi=0$),
the RKC phase ($\xi\neq 0$), and the chiral-restored phase ($\xi=1$) whose result is consistent with Nickel's result.

With  increasing the magnetic field the DCDW phase  ($\xi=0,q\neq 0$), the HCC phase ($\xi\neq 0,q\neq 0$)
and the chiral-restored phase  ($\xi=1$) appear.
In the low $\mu$ region, $q$ is small compared to other order parameters, so we call the DCDW phase in the low $\mu$ region "weak DCDW phase."
We call the DCDW phase in the high $\mu$ region "strong DCDW phase," which is similar to the usual DCDW phase with the wavevector $q$ being sufficiently large.
In other words we may say that the weak DCDW phase is driven by the topological effect due to spectral asymmetry, while the strong DCDW phase by the nesting effect.
We can see the second order phase transition between the weak DCDW phase to the HCC phase, where the order parameters are  continuously changed.
At the phase transition between  the HCC phase to  the strong DCDW phase, the order parameters  change discontinuously.
Thus this phase transition is of the first order.

As we have already seen in the RKC phase, the order parameters should exhibit the de Haas-van Alphen effect \cite{deHaas1936,Ebert:1999ht} as a function of  the magnetic field.
In our results, the corresponding effect cannot be seen, since we have discarded the contribution of the higher Landau levels in our approximation.
If  full order contributions of the magnetic field is taken into account, the de Haas-van Alphen effect can appear.
Anyway, the oscillation of the order parameter should be very small at the weak $B$ region , where one may expect the HCC phase.
On the other hand, only the DCDW phase appears in the high $B$ region, and the phase diagram becomes the same with Frolov's results \cite{Frolov:2010wn}.

\section{ Summary and Concluding remarks\label{sec:concl}}
In this paper, we have studied chiral phase transition in the external magnetic field $\bf B$,
taking account of a new type of the inhomogeneous condensate called hybrid chiral condensate (HCC).
HCC is then a self-consistent solution within the NJL model under the mean-field approximation, and exhibits both features of DCDW and RKC.
We have seen that the quark energy spectrum becomes asymmetric about zero due to the phase degree of freedom of DCDW, and there appear the mid-gap states
due to the solitonic  property of RKC.
Generally speaking, spectral asymmetry  plays very important roles for appearance of inhomogeneous phase in the magnetic field \cite{Tatsumi:2014wka}.
In  some case,  the contribution of spectral asymmetry is equivalent to  manifestation of chiral anomaly.
We have explicitly evaluated the Atiyah-Patodi-Singer $\eta$-invariant for the case of HCC.
The energy spectrum has a gap and $\eta_H$ is given by the sum of the one given by the states above and below the gap and the one given by the mid-gap states;
the former does not depend on the modulus parameter in HCC to give the same form as in the DCDW case,
and the latter is related to the number of nodes of HCC.

 We have studied the phase diagram of the inhomogeneous chiral phase in the $\mu-B$ plane at $T=0$ for two cases.
First we have explored the pure RKC phase and found that the phase diagram is little affected by the magnetic field,
while some oscillation due to the de Haas-van Alphen effect can be slightly observed.
Next, we have discussed the full phase diagram, taking into account HCC.
Since spectral asymmetry implies that the phase modulation is always favored in the magnetic field,
independent of $\mu$,
the phase diagram consists of three regions besides the chiral-restored phase: the strong DCDW phase, the weak DCDW phase, and the HCC phase.
The strong DCDW phase resembles the pure DCDW phase, but the appearance of the weak DCDW phase is attributed to the anomalous quark number caused by spectral asymmetry;
actually they disappear as the magnetic field is turned off.
Note that pure RKC phase never appears once the magnetic field is turned on, and is replaced by the HCC phase.

In this paper, we have considered only the flavor symmetric case, $\mu_u=\mu_d$ for simplicity,
while $u$ and $d$ quarks  should have different number in a realistic situation due to different electric charge.
 Actually cold catalyzed matter develops inside neutron stars, where charge neutrality and chemical equilibrium should be established.
Thus nonzero isospin chemical potential ($\mu_{\rm I}\equiv\mu_u-\mu_d\neq 0$) is very important in the magnetic field.
In Refs. \cite{Ebert:2014woa,Abuki:2013vwa,Abuki:2013pla},
they have been studied the phase diagram taking into account isospin chemical potential in the absence of the magnetic field.

We  have also considered  cold quark matter ($T=0$), while it may be interesting to study how thermal effect modifies our findings  .
Actually it has been recently discussed that the external magnetic field suppresses chiral condensates at finite temperature
 \cite{Bali:2011qj,Fukushima:2012kc,Mueller:2015fka}.
This subject is to be discussed elsewhere.

We have discussed the phase diagram of the inhomogeneous chiral phase in the chiral limit $m_c=0$,
while it is known that the current quark mass $m_c$ suppresses the inhomogeneous phase \cite{Nickel:2009wj,Karasawa:2013zsa,Maedan:2009yi,Abuki:2011pf}.
For the case with the magnetic field it has been supposed that the effect of current quark mass defeat the effect of spectral asymmetry and that chiral condensate is homogeneous at low $\mu$ and low $B$ region \cite{Tatsumi:2014wka}.

Finally it should be worth mentioning that HCC may have some implications in the context of the FFLO state of superconductivity \cite{Fulde:1964zz,larkin:1964zz}.
Very recently an evidence of the LO state has been reported, and the Andreev bound states are emphasized as a hallmark \cite{Mayaffre:2014}.
Usually this subject has been separately discussed for the FF state or the LO state.
Since it has been shown that there is a duality relation between superconductivity and spontaneous breaking of chiral symmetry in 1+1 dimensions \cite{Thies:2003zr},
the FFLO state with one dimensional modulation may be similarly treated to our subject;
the Andreev bound states then correspond to the mid-gap states.
Note that HCC satisfies the BdG equation and  can give the pairing function connecting the LO and FF states smoothly.
The coexistence of the FF and LO states in the quasi-one dimensional system has been  discussed as an appearance of time crystal phase in which time translation symmetry is spontaneously broken \cite{Yoshii:2014fwa}.

\section*{Acknowledgement}
We thank N. Yamanaka and R. Yoshiike and T.-G. Lee for useful discussions. This work is partially supported by Grants-in-Aid for Scientific Research on Innovative Areas
through No. 24105008 provided by MEXT.

redpajama_set_name: RedPajamaArXiv


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Hiking is a popular activity in West Virginia, which is also known as the "Mountain State". Many trails exist which are easily hiked by a beginner or seasoned pro. One such trail is the Bluestone Turnpike Trail in the southern part of West Virginia. It winds along the banks of the Bluestone River for 8 miles and connects Pipestem State Park to Bluestone State Park. This is a well-shaded and easy trail for the entire family to experience. Another favorite hiking destination is the popular Spruce Knob Summit. With over 60 trails, this area is an outdoor enthusiast's delight. Spruce Knob, with an elevation of 4,863 feet, is the highest point in West Virginia. This trail is better suited to the experienced hiker.
West Virginia is home to some of the finest whitewater rafting in the world. Every year, people travel from all over the country to navigate the world-class rapids on the Gauley River. On the Gauley, people can experience anywhere from class III to class V rapids. A 28-mile section, with a drop of more than 668 feet, provides a quality rafting experience. Another popular river for whitewater rafting is the New River. The New River is home to a 14-mile section that has an approximate drop of 240 feet. This river is excellent for both the beginner and the pro.
Pipestem Resort State Park is one of the most picturesque parks in West Virginia. The park consists of two lodges, 26 fully equipped cottages, camping sites, two restaurants, a golf course, riding stables, and both indoor and outdoor swimming pools. The main lodge is McKeever Lodge. It consists of 113 rooms, a conference center, a restaurant, an indoor heated pool, a sauna, a game room, and an exercise room. Another lodge at Pipestem State Park is Mountain Creek Lodge. It is only accessible by an aerial tramway ride which goes 3600 feet down into the Bluestone Gorge. This lodge has 30 rooms and is only open from the middle of May until the end of October. Other activities to enjoy are fishing, horseback riding, hiking, golfing, and swimming.
Canaan Valley Resort is another favorite West Virginia State Park. In addition to 23 cabins and 34 campsites, Canaan Valley Resort features a 160-room lodge. There is also a fitness center, indoor and outdoor pools, a golf course, a conference center, a restaurant, and a ski area. Many cross-country ski trails and hiking trails are available. There are 43 slopes and trails for skiers and snowboarders to enjoy.
Whether it's hiking, whitewater rafting, skiing, or camping, West Virginia is an outdoor enthusiast's dream. With so many outdoor activities to choose from, a West Virginia vacation will be a fun-filled adventure for any family.


LEGENDARY England goalkeeper Gordon Banks has passed away from cancer at the age of 81, his former club Stoke have confirmed.
Banks won the Three Lions' only ever World Cup in 1966 on home soil and famously pulled off a physics-defying save from a header by Brazil legend Pelé four years later in Mexico.
The Sheffield native had been undergoing spells of chemotherapy for several months and was visited by a number of former teammates in recent weeks as his health took a turn for the worst.
He passed away peacefully overnight with his loved-ones at his bedside, his family said.
"It is with great sadness that we announce that Gordon passed away peacefully overnight," they said in a statement released via Stoke City.
Banks is survived by his wife Ursula and their three children, Julia, Robert and Wendy.
Widely regarded as one of the greatest footballers in his position of all time, Banks was named FIFA goalkeeper of the year six times and earned 73 caps for England.
He made 628 club appearances during a 15-year career in the Football League, playing for Chesterfield, Leicester City and Stoke.
A League Cup winner with both the Foxes (1964) and the Potters (1972), Banks was won FWA Footballer of the Year in 1972, while also being named in FIFA's World Cup All-Star Team in 1966 after playing a key role in England's iconic 4-2 triumph at Wembley in the final against West Germany.


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text:
Red Bull Bragantino, commonly known as Bragantino, is a Brazilian football club based in Bragança Paulista, São Paulo. It competes in the Série A, the top tier of the Brazilian football league system, as well as in the Campeonato Paulista Série A1, the highest level of the São Paulo state football league.

The club was known as Clube Atlético Bragantino, before club administration was taken over by Red Bull GmbH in 2020 who renamed the club and changed its colours from its traditional black and white to red and white.

Although the partnership began in April 2019, during the 2019 Campeonato Brasileiro Série B (second division of Brazilian football), the team was called 'Bragantino' and Red Bull appeared only as a sponsor. In the 2020 season the name changed to 'Red Bull Bragantino'.

In their first season, they were champions of the 2019 Série B being promoted to the Série A and qualifying for the 2020 Copa do Brasil round of 16.

History

On 8 January 1928 former Bragança Futebol Clube members founded the Clube Atlético Bragantino.

In 1949 the club played in the Campeonato Paulista Second Division for the first time. In 1965 Bragantino was promoted to the Campeonato Paulista First Division for the first time. In 1966, however, the club was relegated to the Campeonato Paulista Second Division.

In 1988 Bragantino was the Campeonato Paulista Second Division champion. In 1989 the club was promoted to the Campeonato Brasileiro Série A for the first time after winning the Campeonato Brasileiro Série B. In 1990 Bragantino defeated the Novorizontino to win the Campeonato Paulista First Division. The final was nicknamed the caipira final (final caipira, in Portuguese language).

In 1991 the club was the Campeonato Brasileiro Série A runner-up. In the final, Bragantino was defeated by São Paulo. In 1992 the club competed in the Copa CONMEBOL, debuting in international competitions. In 1993 Bragantino competed in the Copa CONMEBOL for the second time. In 1995 the club was relegated to the Campeonato Paulista Second Division. In 1996 Bragantino competed in the Copa CONMEBOL for the third time.

In 1998 the club was relegated to the Campeonato Brasileiro Série B. In 2002, after a poor campaign, Bragantino were relegated to the Campeonato Brasileiro Série C. In 2005 the club was promoted to the Campeonato Paulista First Division. In 2007 Bragantino won the Campeonato Brasileiro Série C thus being promoted to the following year's Campeonato Brasileiro Série B.

In April 2019 Bragantino signed a deal with Red Bull GmbH handling over management of all their football-related assets. From 2020 a new logo and the name of Red Bull Bragantino was introduced.

Stadium

Red Bull Bragantino's stadium is the Estádio Nabi Abi Chedid, built in 1949 with a maximum capacity of 21,210 people. The stadium, owned by the Clube Atlético Bragantino, honors Nabi Abi Chedid, a former president of the club and father of present president Marco Antônio Abi Chedid. It had previously been named the Estádio Marcelo Stéfani, in honor of Marcelo Stéfani, a player and former president of the club. As Estádio Marcelo Stéfani, the stadium was also known by the nickname Marcelão. The name change was effected on 6 January 2009 amid criticism from the Bragança Paulista population.

Season records

{|class="wikitable"
|-bgcolor="#efefef"
! Season
! Div.
! Pos.
! Pl.
! W
! D
! L
! GS
! GA
! Pts.
! style="width:15%;"|Copa do Brasil
!colspan=2|CONMEBOL
|-
!align=center colspan="13"|As Clube Atlético Bragantino
|-
|align=center|2002
|align=center rowspan="1"|Série B
|align=center|26
|align=center|25
|align=center|4
|align=center|5
|align=center|16
|align=center|23
|align=center|54
|align=center|17
|align=center|First round
|align=center colspan="2"|DNP
|-
|align=center|2003
|align=center rowspan="1"|Série C
|align=center|6
|align=center|12
|align=center|7
|align=center|2
|align=center|3
|align=center|20
|align=center|15
|align=center|23
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2004
|align=center colspan="9"|DNP
|align=center colspan="1"|DNP
|align=center colspan="2"|DNP
|-
|align=center|2005
|align=center colspan="9"|DNP
|align=center colspan="1"|DNP
|align=center colspan="2"|DNP
|-
|align=center|2006
|align=center colspan="9"|DNP
|align=center colspan="1"|DNP
|align=center colspan="2"|DNP
|-
|align=center|2007
|align=center rowspan="1"|Série C
|align=center bgcolor="gold"|1
|align=center|32
|align=center|16
|align=center|7
|align=center|9
|align=center|46
|align=center|32
|align=center|55
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2008
|align=center rowspan="1"|Série B
|align=center|7
|align=center|38
|align=center|16
|align=center|9
|align=center|13
|align=center|47
|align=center|41
|align=center|57
|align=center|Second round
|align=center colspan="2"|DNP
|-
|align=center|2009
|align=center rowspan="1"|Série B
|align=center|9
|align=center|38
|align=center|15
|align=center|8
|align=center|15
|align=center|52
|align=center|51
|align=center|53
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2010
|align=center rowspan="1"|Série B
|align=center|8
|align=center|38
|align=center|13
|align=center|14
|align=center|11
|align=center|52
|align=center|37
|align=center|53
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2011
|align=center rowspan="1"|Série B
|align=center|6
|align=center|38
|align=center|16
|align=center|10
|align=center|12
|align=center|65
|align=center|53
|align=center|58
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2012
|align=center rowspan="1"|Série B
|align=center|14
|align=center|38
|align=center|12
|align=center|8
|align=center|18
|align=center|45
|align=center|53
|align=center|44
|align=center|DNP
|align=center colspan="2"|DNP
|-
|align=center|2013
|align=center rowspan="1"|Série B
|align=center|16
|align=center|38
|align=center|13
|align=center|8
|align=center|17
|align=center|37
|align=center|43
|align=center|47
|align=center|Second round
|align=center colspan="2"|DNP
|-
|align=center|2014
|align=center rowspan="1"|Série B
|align=center|16
|align=center|38
|align=center|13
|align=center|7
|align=center|18
|align=center|45
|align=center|55
|align=center|46
|align=center|Round of 16
|align=center colspan="2"|DNP
|-
|align=center|2015
|align=center rowspan="1"|Série B
|align=center|6
|align=center|38
|align=center|19
|align=center|3
|align=center|16
|align=center|56
|align=center|66
|align=center|60
|align=center|Second round
|align=center colspan="2"|DNP
|-
|align=center|2016
|align=center rowspan="1"|Série B
|align=center|19
|align=center|38
|align=center|8
|align=center|8
|align=center|22
|align=center|30
|align=center|54
|align=center|32
|align=center|Third round
|align=center colspan="2"|DNP
|-
|align=center|2017
|align=center|Série C
|align=center|7
|align=center|18
|align=center|4
|align=center|9
|align=center|5
|align=center|16
|align=center|19
|align=center|21
|align=center|Second round
|align=center colspan="2"|DNP
|-
|align=center|2018
|align=center|Série C
|align=center|4
|align=center|22
|align=center|9
|align=center|8
|align=center|5
|align=center|25
|align=center|18
|align=center|35
|align=center|Third round
|align=center colspan="2"|DNP
|-
|align=center|2019
|align=center|Série B
|align=center bgcolor="gold"|1
|align=center|38
|align=center|22
|align=center|9
|align=center|7
|align=center|64
|align=center|27
|align=center|78
|align=center|DNP
|align=center colspan="2"|DNP
|-
!align=center colspan="13"|As Red Bull Bragantino
|-
|align=center|2020
|align=center|Série A
|align=center|10
|align=center|38
|align=center|13
|align=center|14
|align=center|11
|align=center|50
|align=center|40
|align=center|53
|align=center|First round
|align=center colspan="2"|DNP
|-
|align=center|2021
|align=center|Série A
|align=center|6
|align=center|38
|align=center|14
|align=center|14
|align=center|10
|align=center|55
|align=center|46
|align=center|56
|align=center|Third round
|align=center|CS
|align=center bgcolor="silver"|Runners-up
|-
|align=center|2022
|align=center|Série A
|align=center|14
|align=center|38
|align=center|11
|align=center|11
|align=center|16
|align=center|49
|align=center|59
|align=center|44
|align=center|Third round
|align=center|CL
|align=center|Group stage
|-
|}

Current squad

Reserve team

Out on loan

Personnel

Current staff

Honours

International

 Copa Sudamericana
 Runners-up: 2021

Domestic

Campeonato Brasileiro Série A:
 Runners-up: 1991*

Campeonato Brasileiro Série B:
 Winners (2): 1989*, 2019*

Campeonato Brasileiro Série C:
 Winners: 2007*

State

Campeonato Paulista:
 Winners: 1990*

Campeonato Paulista Série A2: 2
 Winners (2): 1965*, 1988*

Campeonato Paulista Série B2: 1
 Winners: 1979*

 Campeonato Paulista do Interior: 1
 Winners: 2020

*Titles before 2020, as Clube Atlético Bragantino

See also
 RB Leipzig
 New York Red Bulls
 FC Red Bull Salzburg
 FC Liefering
 Red Bull Ghana
 Red Bull Brasil

References

External links
 Official Website

Football clubs in São Paulo (state)
1928 establishments in Brazil
Association football clubs established in 1928
Red Bull sports teams
Clube Atlético Bragantino

redpajama_set_name: RedPajamaWikipedia


German language beginners may easily get besonders, besonderes, and Besonderes confused, since they sound nearly the same and have only minor differences in spelling. Once you learn the grammar behind the different spellings, however, you should be able to easily distinguish the reasons for the spelling differences.
So was zum Beispiel läuft besonders gut.
Something like this, for instance, is going particularly well.
The adjective besondere (only written with "-s" in its nominative neuter form) is usually translated as "special" or "particular." It is always written in lower case except when it is the first word in a sentence. It will, like all German adjectives, have different endings depending on whether it's in the nominative, dative, accusative, or genitive case, and depending on whether the noun it modifies is feminine, masculine or neuter, and singular or plural.
Einen Wettlauf der besonderen Art liefern sich der britische Milliardär Richard Branson und Hollywood-Regisseur James Cameron.
A race of a special kind is being carried out by the British billionaire Richard Branson and Hollywood director James Cameron.
The word Besonderes, always written capitalized and with the extra "e" before the last "s," is a nominalized adjective, which is an adjective that has been turned into a noun. Note that nominalized adjectives in German are not necessarily so in English! In German, it is usually preceded by either etwas or was (meaning "something" in this context) or the word nichts (meaning "nothing").
In vielen Familien ist es an Weihnachten Tradition, dass es etwas Besonderes zum Essen gibt.
In many families, it is a tradition at Christmas that there is something special to eat.
To review: besonders is an adverb, besondere (with possible declensions ending with -r, -s, -n, -m) is an adjective, and Besonderes is a nominalized adjective. For more in-depth information on how German turns non-nouns into nouns, read this paper. Then visit Yabla German and search for examples of the above words as spoken in a real world context.


PMT Northampton is much more than just an average music shop; Professional Music Technology is East Midland's largest music store showcasing the very latest musical instruments and recording equipment.
We have gone to great lengths to ensure that every PMT music store is a musical instrument shopping experience all of its own and our Northampton store is no exception. Shop for guitars, drums, pianos or synths and if you need help, our friendly PMT Experts are there to give you the very best advice possible.
As well selling guitars and instruments, we also host to a number of guitar workshops, drum clinics & musical events through the year. Here, budding musicians can pick up tips from some the biggest names in the industry or be treated to sneak previews of some the newest music equipment on the market.
The downstairs showroom displays around over 500 electric and acoustic guitars and basses from the world's top brands, including Gibson, Fender, PRS, Ibanez, Rickenbacker, Taylor, Epiphone, Yamaha, Martin, Schecter, Tanglewood, Faith, Takamine and Gretsch, which range in prices suited from beginners, all the way to professional guitar players.
PMT Northampton is also an authorised Epiphone dealer, stocking a wide range of electric and acoustic Epiphone guitars for you to try out. So, if you're looking for a great selection of Epiphone guitars in Northampton, call in today to try some out for yourself!
There are also hundreds of amplifiers and guitar pedals to choose from too, from TC Electronic, BOSS, Wampler, Electro-Harmonix, Jim Dunlop and Strymon – all of which can be tried out in the music shop's private demo booth.
So you like to tinkle the ivories, eh? That's okay, as PMT Northampton is also a very fine piano shop. In fact, it's a keyboardists wonderland filled with the very best arranger keyboards, digital pianos and synthesizers from Roland, Yamaha, Korg, Nord, Novation, Moog and Arturia … and if that doesn't satisfy the Jon Lord in you, we've got a keytar or two in-store too.
Live sound and recording enthusiasts with love our recording and PA department, our range is vast and we carry stock of the best hi-tech brands of today, including Bose, RCF, Native Instruments, Ableton, Shure, Sennheiser, QSC, KRK, Pioneer, Focusrite, M-Audio, Akai, Rode and more.
And if you're new to computer-based recording products, bring in your laptop and the PMT experts will install your new software or audio interface for you and make sure that you leave the shop with a ready-to-go system.
Shoppers looking for a drum shop in Northampton will definitely not be disappointed when visiting PMT Northampton, as this shop has a whole floor that is entirely dedicated to drum kits and percussion instruments!
Visit the house of drums for collector's snares; acoustic drum kits from Natal, Drum Workshop Inc., Gretsch, Yamaha and Mapex; the latest electronic drum kits by Roland, Yamaha and Alesis; the Zildjian and Sabian Walls of cymbals and thousands of drum accessories.
Not only do we stock a wide range of professional level drum kits here in PMT Northampton, but we can also help you find the perfect kids drum kit for your budding rock stars. We have a selection of junior drum kits and can help create the perfect acoustic or electric drum kit for your children. Whether your child is in elementary school or secondary school, we'll find the right kit to suit all musical genres and ages. Speak to our in-house drum experts today about creating the perfect kids drum set for your child.
Here at Professional Music Technology Northampton, we specialise in complete Church sound system installation packages as well as installed live sound solutions for halls, Churches and venues of all sizes.
Our team of PA experts can advise you on anything to do with audio systems for Church set ups. Whether you need advice on what the best PA system for Church sound systems may be, which wireless microphones would suit your PA or need a complete Church sound system installation from the ground up, we can help you every step of the way.
Our team have many years of experience working within the local community and supplying bespoke Church audio system packages for Churches of all sizes. We offer complete installation packages and aftercare ensuring to install each aspect of your PA system in such a way so to maintain a visually clean area and take great care to ensure your congregation enjoys crystal clear sound at all times.
Contact us at Professional Music Technology Northampton on via phone for more information. We can create a bespoke Church Audio system to suit the needs of your congregation, whether you need the best wireless microphone system, an audio mixer or a complete Church PA system.
In the city: PMT Northampton located in the heart of Northampton town on Bridge Street.
By bus: The nearest bus stop is on Bridge Street, opposite the Plough Hotel, which is a 1-minute walk away from the shop. The buses that stop there are Moulton Park 7, Blackthorn 8, Parklands 10, King's Heath 12, Horton 14, Weston Favell Centre 19, Sunnyside 19a, Northampton 22, 33A, 57, 86, 89, & X89.
By train: It is a 10-minute walk from Northampton train station to PMT.
By car: If you're coming to the shop by car, the shop is just over 3 miles from Junction 15 of the M1, with a free car park at the back of the shop.
Fantastic shop with awesome customer service! Oli, Lee, Jason and Tommy - great lads, helped a lot.
Super helpful! Even when a dumb drummer like me walks into the guitar section to buy some birthday stuff for a guitarist friend. Thanks for the help!
PMT Northampton is my local store and I have always had a very good experience there. Thanks to all staff i.e. Jay, Oli & Lee. Special thanks to Jay, who sorted me out when purchasing my new Gibson Les Paul when not feeling that great. Overall, I am treated very well and we are all are on first name terms, which is great! Thanks again to all staff - Brian.
I spoke to Ben recently and he sure is a swell guy. He really knows his stuff and helped me out massively planning my dream drum kit. It's nice and tidy and it was a really refreshing atmosphere upstairs and I walked away with a few bits I really needed to tide me over until I can afford the real deal. Thanks, PMT Northampton.
Went in today with my 14 year old son who wants to learn the guitar. They gave him a free lesson of about 30-45 minutes and were so helpful and friendly. They advised us on the best start up guitar. Excellent service = one very happy son. Definitely recommend this shop.
I must comment on how helpful and friendly Luke, Heath and Lee were when I visited to buy another amp. Nothing was too much trouble and I didn't feel under any pressure to buy. I had all the time I needed to try out various amps in the showroom, which I have to say, is really well laid out with no end of stuff.
PMT Northampton is a friendly, helpful and knowledgeable store. Ben and Joe in the percussion department have been brilliant advising me on all things drum and cymbal related. Great guys too. Highly recommended company to deal with.
Called into Northampton store Saturday AM for my son to try some Epiphone Guitars, good range available to test. Really relaxed and helpful service. No pressure and good advice given. We ordered the guitar from your online service as decided on a different colour from those in stock. Online service also good.
Called in to see if they any Bass Guitar machine heads in stock as one of mine on my Peavey Millenium had snapped the night before. Although they had none in stock they had replaced some on another instrument recently and still had the old ones. They kindly gave me one of these free of charge! I now won't miss my practice session. Fantastic bunch of people and have earned my (paying) business in the future. Thank you!
Good, friendly atmosphere in the shop at the bottom of Bridge Street, Northampton. I would recommend anyone to visit and peruse. 11 out of 10!
Bought a Korg minilogue and received fantastic service. Price matched and total patience as I tried to decide what to buy. Advice was spot on for my needs, so much so, I went back a few weeks later and bought another keyboard.
Absolutely fantastic service... I would 100% recommend a visit. Decided to move away from HiFi and its snake oil. Thought I would look into Pro-Audio as a replacement. From start to finish, I received excellent advice, as well as being able to try out quite a bit of kit until I found a solution which worked for me. Again, the staff made this experience super easy, and nothing was a problem. Ended up purchasing Adam Audio A77X's, an Adam Sub15, plus a pair of PMC Result6 to add extra depth. This has resulted in a solution which is quite frankly breathtaking. So far I have not heard a hifi under £20k which even comes close! Nor did the above cost anywhere as near as much. Overall a fantastic experience from start to finish!


On behalf of Darryl A. Goldberg posted in Federal Crimes on Friday, February 2, 2018.
In 2015, U.S. Senator from New Jersey Robert Menendez was charged with 18 counts of corruption, which included allegations of fraud, conspiracy and bribery. Seven of these counts--all bribery offenses--were connected to Mendenez's relationship with his long-time friend, Dr. Salomon Melgen. Melgen purportedly bought Menendez a lavish trip to Paris and donated to his political campaign in exchange for Mendendez's assistance with a medical dispute, a port security contract and securing U.S. visas for Melgen's girlfriends.
When the case went to trial last year, the jury was unable to reach a unanimous verdict, resulting in a mistrial. Menendez's lawyers then pushed for acquittal--which the court granted last week for the seven counts of bribery. It ruled that the prosecution produced insufficient evidence of the gifts and donations having a direct impact on the senator's political actions.
· Many counts of the original charge have already been dropped.
· 10 out of 12 jury members voted in favor of acquittal in the first trial.
· The judge in the original case has already invalidated a portion of the prosecution's case.
Senator Menendez is the first sitting U.S. Senator to face allegations of federal bribery in over 30 years. His seat is up for reelection in November, and he is currently expected to win.
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After working as an assistant director with Vidhu Vinod Chopra for a few years, Sanjay Leela Bhansali made his debut directorial Khamoshi: The Musical, which was a box-office dud but got a lot of appreciation from critics. He shot to fame with Hum Dil De Chuke Sanam, which got several awards. His other successful work includes Devdas and Black.


The New Humanitarian | What's happening at Kerem Shalom?
What's happening at Kerem Shalom?
Kerem Shalom is currently the only entry point for commercial and humanitarian goods from Israel into Gaza, and as such has become a life-line, but how does it work?
A small crossing without the facilities to allow large numbers of trucks to enter, Kerem Shalom lies inside Israel, 3km from the Gazan town of Rafah on the Egyptian border.
The only people authorised by Israel to physically access Kerem Shalom are employees of Shaiber Company, a private Gaza company with permission from Israel to access Kerem Shalom to retrieve imports. The company declined to comment.
The Gaza authorities cannot access Kerem Shalom. Israeli truck drivers dump pallets of goods in an open area and withdraw, and then Palestinian truck drivers from Gaza enter and retrieve the goods, according to senior Hamas official Ghazi Hamad and Gaza traders.
Israeli authorities say they are using Kerem Shalom after closing the much larger and better equipped Karni crossing because of Palestinian attacks on Karni - which left several Israeli soldiers dead - several years ago.
Kerem Shalom is completely controlled by Israel. Fifteen trucks containing carnations are the total exports from Gaza this year.
There is also no contact between the Gaza finance ministry and relevant ministries under the PA in Ramallah, said Shaer.
There is no lighting and containers are prohibited on the Palestinian side of Kerem Shalom by Israel, according to the logistics cluster led by the World Food Programme.
"We cannot operate Kerem Shalom at night. If we put lights there, snipers can use this to fire against the crossing," Israeli Defence Ministry spokesman Shlomo Dror told IRIN.
Empty Palestinian trucks are lined up from Rafah to Kerem Shalom, waiting to pick up any goods the Israeli authorities may allow through.
Once the Gazan trucks are loaded they trundle off to Rafah where Gaza government officials begin their intake process. There is no scanning equipment; officials check the documentation for each truck and its contents.
Palestinian traders pay taxes to Israel, and the Gaza government collects 14.5 percent value added tax (VAT) on the goods, Shaer said.
Gaza and the West Bank are treated as part of the same customs envelope by Israel, which collects the customs taxes and is supposed to remit them monthly to the PA in Ramallah, according to the Paris Protocol signed in conjunction with the Oslo Accords in 1994.
A major trader in Gaza City who preferred anonymity said he was importing mainly electronics via Kerem Shalom "since shoes and clothing are not allowed to enter".
"The government in Ramallah coordinates with the Israeli authorities so I can get pharmaceuticals into Gaza," said Nader Shurrab, owner of Gaza Central Company, the largest pharmaceutical importer in Gaza.
There are five crossings into Gaza - four from Israel and one from Egypt. Rafah and Erez are passenger crossings; Nahal Oz is only for fuel; the Karni conveyor belt is only for wheat and animal feed.
As of mid-April the Israeli authorities have been allowing 32 types of products to enter Gaza, said the Palestine Trade Centre (PalTrade) in Gaza.
According to the Coordinator of Government Activities in the Territories (COGAT - under the Israeli Defence Ministry), in April, 92 percent of goods allowed to enter Gaza via Kerem Shalom were food and hygiene products.
Construction material, spare parts, agricultural inputs including livestock, and IT equipment are on the banned list, according to OCHA.
The 22 March Israeli cabinet decision to lift restrictions on food items allowed to enter Gaza has yet to be implemented, reports the logistics cluster.
"The lack of imports has halted over 90 percent of Gaza industries and created massive shortages of food and basic supplies," said Hamas's Hamad , who failed to mention the dozens of illegal tunnels between Egypt and Gaza, which however, provide only limited supplies due in part to a clampdown by the Egyptian authorities and damage by Israeli security forces .
"The Israeli authorities allow 80-110 trucks per day to enter Gaza via Kerem Shalom," according to Hamad, though prior to June 2007, when Hamas won elections in Gaza, about 475 trucks entered Gaza from Israel daily, according to OCHA.
The Israeli standpoint is somewhat different: "The amount of trucks containing humanitarian aid has been consistent in March and April. There are no shortages of humanitarian aid or other supplies in Gaza… The situation can change, but it depends on Hamas, if they recognise that caring for civilians is their responsibility," Dror told IRIN.


text:
HOMEPAGE MONEY
The disadvantages of the hydraulic metal press
By: Maria Kielmas
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Kim Steele/Photodisc/Getty Images
Metal presses are machine tools that shape pieces of metal by applying pressure. They work the metal through a force that acts over a distance -- or stroke -- and forges the metal into shape. Presses can be powered by hydraulics, electricity, compressed air (pneumatics) or steam. Hydraulic presses are built in many different sizes and are used in all types of manufacturing industry, ranging from automobile plants to jewellery making.
Hydraulic circuit
A hydraulic press delivers force to shape metal through fluid pressure. The fluid – usually oil or an oil and water emulsion -- is kept in a chamber and its pressure can be increased or decreased by using pumps. A typical hydraulic circuit consists of a motor, a cylinder, piston, pumps, pipes and valves. The fluid is poured into the cylinder below the piston. The fluid's pressure increase makes the piston rise. The fluid is pumped out from below the piston decreasing the pressure in that space. Simultaneously, fluid is pumped in above the piston, increasing the pressure there and forcing the piston down. The downward force is used to work the metal.
Leaks of fluid are the most common problem associated with hydraulic presses. Even a small leak can cause a large mess. Leaks can show up in the hydraulic fluid lines and fittings that are not properly tightened. They can be caused by operator error and poor maintenance. Older presses are more prone to leaks as their components become worn. Water-based hydraulic fluids can cause erosion of metal components and affect the system's seals.
Heat develops in a hydraulic system as the fluid is forced through pumps, tubing and valves. The system also generates heat when the fluid moves from a high pressure to a low pressure area without performing any work, such as the pressure drop. A temperature build up above 82C (180F) can degrade and oxidise the hydraulic fluid, make it more viscous and deposit sludge at the bottom of the cylinder. This lessens the system's lubrication, blocks relief valves and can seize up operations. High temperatures will also damage seals and increase the frequency of leaks.
Operational speed
Hydraulic presses are not high-speed machines. The cylinder action and fluid pressure build up is a slow process – it takes about a second -- compared with mechanical or pneumatic presses whose cycles take fractions of a second. Hydraulic metal presses cannot be used for rapid production line processing of small metal components that require fast strokes.
The pumps used in hydraulic metal presses are bulky and have high energy consumption. These have to provide the high pressure for fluids in the cylinder. Insufficient pressure can cause irregularities in the metal processing. If a batch of work has to be repeated, even more energy is consumed. When combined with deterioration in machinery due to temperature and pressure fluctuations and leaks in hydraulic pipes, the energy efficiency of this system falls even further.
The Library of Manufacturing: Hydraulic and mechanical presses
Press Master: Common hydraulic press problems and how to avoid them
Machinery Lubrication: Symptoms of common hydraulic problems and their root causes
Chesterton: Hydraulic presses need effective sealing
Sieb & Meyer: It doesn't always have to be hydraulics
Gates: Introduction to hydraulics
Ganoskin: Practical, inexpensive hydraulic pressing and die forming for the artist metalsmith
Mobil: Hydraulic system care and maintenance
Based in London, Maria Kielmas worked in earthquake engineering and international petroleum exploration before entering journalism in 1986. She has written for the "Financial Times," "Barron's," "Christian Science Monitor," and "Rheinischer Merkur" as well as specialist publications on the energy and financial industries and the European, Middle Eastern, African, Asian and Latin American regions. She has a Bachelor of Science in physics and geology from Manchester University and a Master of Science in marine geotechnics from the University of Wales School of Ocean Sciences.

redpajama_set_name: RedPajamaCommonCrawl