The first category is fear of the wolf's outlaw and predatory qualities. In many cases, the wolf is associated with predatory danger, something wild that could destroy vulnerable farm animals and home life; such as exploitative predatory energy, or sexual energy gone awry. Remember the days when a sexually dangerous man was referred to as a "wolf"? Think of the wolf in Little Red Riding Hood, who disguised himself in order to pursue the innocent. If you enter this experience of the wolf, it is indeed an unfairly powerful creature that preys on the weak, the innocent, and those without adequate defenses.
There is also the suggestion that somehow the domestication and containment of animals makes them easy prey for the wolf. (Thus, what we do to be civilized or "contained" in life, may make us easier prey for those who walk on the wild side, or to be savaged by our own inner wolves.) This association to the wolf is appears more common to men who feel the anguish of wanting to protect their families against the myriad hazards of life; which, ultimately, cannot be adequately anticipated. In this sense many men try to keep vigil against the "wolf" energy in the world; anything that might threaten those they would love and protect. The wolf in these dreams may also refer to the wolf or wild energy within the dreamer, which may feel as if it could pose a threat to the security of domestic life, unless he "guards against it."
In some instances women dream of fearful wolves that seem intent on destroying them. In most cases these monstrous wolves have been associated with fears of falling prey to "wild" energy in some way. These dreams can be related to traumatic events in the past; or they may arise when a disturbing element is sensed in a co-worker or acquaintance. Because our survival instincts are easily aroused and often weave strong themes into our dreams, even a minor quality of unwholesomeness in waking life can be illustrated by the pursuing wolf in our dreams.
The second category of wolf dream is quite different. In these dreams the wolf is the symbol of noble wildness, freedom from societal restraints, and elemental reliance on instincts and nature. This is the women-who-run-with-the-wolves image, and it appears in the dreams of anyone who is beginning to acknowledge the powerful and natural side of themselves. This wolf often recalls a sense of power, wisdom and life-in-the-moment. There is an almost elemental freedom that temporarily is lost to us during the socialization process; but our vivid aliveness is never truly gone. Typically when an individual begins to feel stirrings of their "natural" self, or a power based on aliveness rather than external achievement they may dream of a wolf that is both awesomely powerful yet strangely welcoming to them.


text:
Canucks' Linden wants to work with struggling Edler rather than trade him
By Cam TuckerJun 4, 2014, 1:25 AM EDT
Alex Edler struggled mightily last season. That's stating the obvious. He never seemed to find his way under head coach John Tortorella, who was fired after one season in Vancouver.
A suspension early in the season and an injury contributed to the difficulties of the 28-year-old Swedish defenseman in 2013-14. He finished the season with a league-worst minus-39 rating, although the validity of that statistic has come under fire in recent years with the rise on analytics. His contributions offensively were also well below what he's done in the past, with 22 points in 63 games. That was his same point total from the 45 games he played during the 2013 lockout-shortened season.
His name has also come up in trade speculation (that includes last summer and at this year's March deadline). He's about to enter the second year of a six-year, $30 million contract that includes a no-trade clause, as per Capgeek.com.
But it seems that with the newly forming Trevor Linden era, the plan is keep Edler and continue to try and develop his game back rather than deal him.
"He'd be the first to admit he didn't have a good year, and part of our job is to recapture the Alex Edler that we all know he can be. He has the skill-set that every team wants and we just have to get him back where he should be," said Linden, as per The Province newspaper in Vancouver.
"There isn't a team in the league that wouldn't have its hand up for the opportunity to have an Alex Edler. It's not in our consideration. We want to work with Alex; we have to get his game back, and I know he's committed to doing that."
Follow @CamTucker_Metro

redpajama_set_name: RedPajamaCommonCrawl


text:
Any Price (3)	 Under $100 (1)	 $100-499 (2)
ACI 318-11 Building Code Requirments for Structural Concrete & Commentary, 2011.
ACI, 01/01/2011. The code portion of this document covers the proper design and construction of buildings of structural concrete. The code has been written in such form that it may be adopted by reference in a general building code and earlier editions have been widely used in this manner. Among... More about ACI 318-11 Building Code Requirments for Structural Concrete & Commentary, 2011
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ACI, 01/01/2014. Referenced by the 2016 California Building Code: The code portion of this document covers the proper design and construction of buildings of structural concrete. The code has been written in such form that it may be adopted by reference in a general building code and earlier editions have... More about ACI 318-14 Building Code Requirments for Structural Concrete & Commentary, 2014
Working with Concrete.
Rick Arnold
Taunton Press, 04/01/2003. From the Publisher: Concrete is magical mud. This mixture of sand, stone, Portland cement and water is an essential ingredient in just about any building project. Whether you're pouring a concrete walkway or staking out the excavation for a basement foundation, doing the job right demands a.... More about Working with Concrete
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text:
Duke (90)
Hierarchical Path: Seven Years War (Main Page) >> Navies >> British Navy >> Duke (90)
The original ship was built at Portsmouth Dockyard and launched in 1678 under the name of "Vanguard". In November 1703, she sank in the Medway, she was then raised in 1704, rebuilt at Chatham Dockyard and relaunched in 1710.
The ship was rebuilt once more at Chatham Dockyard and launched in April 28 1739 under the name of "Duke".
from September 1755: captain George Murray
from July 1756: captain Samuel Graves
from 1757: captain Thomas Hanway
from 1758: captain James Norman
from 1759 to 1762: captain Thomas Graves
from November 1762 to 1763: captain Thomas Warwick
The ship was stricken off of the fleet and broken up at Plymouth (completed on August 13 1769).
In 1757, the ship carried the flag of rear-admiral Henry Harrison.
In 1758, the ship carried the flag of commodore Piercy Brett and served in Anson's fleet.
In 1759, the ship served in Hawke's fleet. On November 20, she part in the decisive battle of Quiberon. This British victory eliminated any serious threat from the French navy for the rest of the war.
In 1761 and 1762, the ship was assigned to harbour service at Portsmouth. In 1762, she carried the flag of vice-admiral Philip Durell.
Lower gundeck 26 x 32-pdrs
Middle gundeck 26 x 18-pdrs
Upper gundeck 26 x 9-pdrs
Quarterdeck 10 x 6-pdrs
Crew 750 men
Length at gundeck 174 ft (53.0 m)
Width 50 ft (15.2 m)
Depth 20 ft 6 in (6.20 m)
Displacement 1625 tons (1651 tonnes)
Blasco, Manuel, 3 Decks Wiki – British 2nd Rates
HMS Vanguard (1678)
Retrieved from "http://kronoskaf.com/syw/index.php?title=Duke_(90)&oldid=5620"


text:
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text:
\section{Introduction}

In the past fifteen years, the remarkable original proposal
of Refs.~\cite{R1}, relaying a potential physical relevance to Hamiltonians
respecting parity ($\p$) and time-reversal ($\tp$) symmetries,
has received considerable attention~\cite{R2}.
This proposal has highlighted the interest in considering (as operators potentially
bearing real spectra) Hamiltonians that are invariant
under these fundamental symmetries as an extension of the more
standardly considered self-adjoint Hamiltonian operators of quantum mechanics.
While for a decade since their inception, these notions were studied intensely at the linear
level, especially in the mathematical community (see e.g.
the review of~\cite{R2}), more recently it was realized
that linear optics~\cite{Muga} could present an ideal
playground for the realization of such non-Hermitian settings
(i.e., in ``open'' systems bearing gain and loss but in a $\PT$-symmetric form).
In particular, discrete systems~\cite{Kulishov,dimer_OL} with balanced gain and loss have been suggested as reduced models of the non-Hermitian optics,  obeying remarkable properties of waveguiding  and giving origin to the
blossoming field of discrete $\PT$-symmetric optics. $\PT$ symmetry was
thus first studied experimentally in the optical experiments of~\cite{salamo,dncnat,scherer,pertsch}. As a natural extension of optical applications,  it was suggested in Ref.~\cite{ziad} to consider nonlinear optical systems whose linear limit is $\PT$ symmetric, and in particular it  was shown that such systems with a periodic potential support stable solitons.

The simplest basic element of the discrete $\PT$-symmetric optics is a dimer with one site subjected to dissipation and another site subjected to gain~\cite{Kulishov,dimer_OL}. As a natural application of this system, one can consider a coupler with one active and one lossy waveguide~\cite{dncnat}. When the nonlinear effects are included, one deals with a $\PT$-symmetric nonlinear coupler, i.e. mathematically with a $\PT$-symmetric nonlinear dimer. Such a dimer with
a Kerr-type nonlinearity was intensively studied showing remarkable properties. In particular, it was shown in ~\cite{Ramezani} that such a nonlinear coupler is an integrable system allowing for a solution in the form of
quadratures.
The effect of nonlinear suppression of the periodic time reversals   and the
beam switching to the waveguide with gain was reported in~\cite{sukh1}
(a similar switching effect can also be implemented with
$\PT$-symmetric impurities inserted in an otherwise conservative
coupler~\cite{R30add3}).

Further studies of the nonlinear discrete optical systems were performed in a
number of directions. We mention a few of these in what follows.
First, including one more coordinate (in addition to the evolution one; in optical applications such systems could be seen as coupled {planar} waveguides with balanced gain and loss) made it relevant to consider the dynamics of bright~\cite{bright} and dark~\cite{dark} solitons, breathers~\cite{breathers}, as well as instabilities and rogue waves~\cite{rogue}. Another extension of nonlinear dimer activity is related to the inclusion of $\PT$-symmetric defects in discrete nonlinear systems. In the latter context, problems such as nonlinear wave
scattering~\cite{dmitriev1} (see also~\cite{jennie})
and the lifting of the degeneracy of
discrete vortices~\cite{leykam} were considered. Finally, a nonlinear dimer or more generally the so-called nonlinear ``oligomers'' (i.e., few site configurations) introduced in~\cite{pgk} (see also~\cite{uwe}), were shown to
allow for the existence of continuous families of nonlinear modes~\cite{konorecent3}. Among these, a $\PT$-symmetric quadrimer model naturally appears in the description of light propagation in a birefringent coupler~\cite{LZKK}.  Discrete solitons in different types of infinite $\PT$-symmetric waveguide arrays
were studied numerically~\cite{dmitriev2} and analytical proofs for their
existence have been proposed using the anticontnuum limit~\cite{KPZ} and
via analysis of the modes bifurcating from the linear limit~\cite{tyugin}. Solitons in a necklace of coupled   dispersive  waveguides were reported in \cite{necklace}.

Most of the above investigations have taken place at the level of
the well-known Kerr-type nonlinearity.  Existence of nonlinear modes and integrals of motion in $\PT$-symmetric systems with  more
general cubic nonlinearities  was investigated in~\cite{ZK_nonlin}.
Nevertheless, another type
of nonlinearity of particular interest to optics is the quadratic
one~\cite{buryak}. In the latter context,
switching in two parallel waveguides was studied in
\cite{Schiek94}. The  intensity-dependent switching in lithium niobate directional couplers was subsequently first observed in \cite{Schiek96}.
 The particular case of a dimer with one nonlinear and one linear waveguide
was considered in \cite{Assanto}.
Furthermore, nonlinear localized  modes in  arrays with a quadratic
nonlinearity are considered in numerous
works~\cite{Mak97,Peschel98,Miller,Darmanyan}.
Experimentally~\cite{Pertsch}, the fundamental modes of a second harmonic are
strongly  confined, so the coupling constant between second harmonic modes in
different waveguides
is very small and sometimes  can even
be neglected. Discrete solitary waves in this configuration
were systematically probed in~\cite{Iwanow}. The plane waves and
localized modes
in this case were considered in~\cite{Kevrekidis}. Extensions in the
case of two-dimensional states including discrete vortices were
proposed in~\cite{pgk2}, while the mobility of the solitary waves
in both one and two dimensions was explored in~\cite{pgk3}.

Recently the studies of solitons in quadratically nonlinear
media were extended to $\PT$-symmetric systems.
More specifically, the existence and stability of solitons for
localized potentials in quadratic media
was explored in~\cite{Mor1}. In the work of~ \cite{Mor2},
the effect of periodic  $\PT$-symmetric potentials on $\chi^{(2)}$ solitons was described.

It is on that direction of exploring the interplay of
quadratic nonlinearity and $\PT$-symmetric potentials that
the present work is focused. It is appreciated that even
 {in the   case of two waveguides,}
this combination offers
a significant level of complexity, as well as a number of
features that are absent in the cubic Kerr nonlinearity case.
In particular,  due to quadratic nonlinearity and the  particular structure of  eigenvectors of the  underlying linear $\PT$-symmetric operator, continuation of the linear eigenvectors into the nonlinear domain is performed in two different ways, depending on whether the first or the second harmonic is vanishing in the linear eigenvector. We develop perturbative formal expansions that enable us
to capture  analytically these  two different types of bifurcations  of the nonlinear modes from the linear solutions. Next, we employ numerical computations and  observe the symmetry
breaking, as well as saddle-node bifurcations and
identify the stability characteristics
of the
solutions. In the case of instability,
the dynamical evolution of the configurations is explored.

Our presentation is structured as follows. In section~\ref{linear}, upon
introducing the general model, we focus on
its linear analysis. In section~\ref{sec:bif}, we introduce
nonlinearity and analytically
explore how it affects the linear  modes
(i.e., consider the bifurcations of nonlinear modes from linear ones).
In section~\ref{sec:nonlin}, we corroborate our analytical considerations with
detailed numerical results identifying    the nonlinear modes and
their stability for different values of
the propagation constant
and the gain/loss strength parameters. For the unstable modes,
we touch upon their dynamical evolution in section~\ref{sec:dyn}. Finally,
in section~\ref{sec:concl}, we summarize our findings and present some
interesting directions for future studies.

\section{The Model and its linear Analysis}
\label{linear

The prototypical setup of equations describing the $\PT$ symmetric coupler with  quadratic nonlinearity
reads as follows:
\begin{subequations}
\label{basic_equations}
\begin{eqnarray}
i\dot{u}_{1} &=& k_1 u_2-2u_1^* v_1 + i\gamma_1 u_1,\\
i\dot{v}_{1} &=& k_2 v_2-u_1^2-q v_1 + i\gamma_2 v_1,\\
i\dot{u}_{2} &=& k_1 u_1-2u_2^* v_2 - i\gamma_1 u_2,\\
i\dot{v}_{2} &=& k_2 v_1-u_2^2-q v_2 - i\gamma_2 v_2.
\end{eqnarray}
\end{subequations}
Each waveguide contains two harmonics: the fundamental field (first harmonic)  $u_j$ and the second harmonic $v_j$, $j=1,2$, which are nonlinearly coupled. The linear coupling
between the first harmonics is characterized by the parameter $k_1$, while that of the second harmonics by
$k_2$. Both $k_1$ and $k_2$ will be considered positive. The gain (loss) strength
in the two arms of the dimer is given by the parameters $\gamma_{j}>0$ ($\gamma_{j}<0$), for the first ($j=1$) and second ($j=2$) harmonics, respectively.
In what follows, we will explore different parameter values of
$(\gamma_1,\gamma_2)$ to get a systematic sense of the model phenomenology.
The overdot in (\ref{basic_equations}) denotes the derivative with respect to the evolution
variable which, here, we will denote as $t$ (although in the optical
realm it represents the propagation distance $z$).

Being interested in the stationary modes, we make use of the ansatz
\begin{eqnarray}
\label{lin_ansatz}
\left(
\begin{array}{c}
u_1 \\ v_1 \\u_2 \\ v_2
\end{array}
\right)=e^{-i\Lambda Et} \bw, \quad\mbox{where}\quad \Lambda=\left(
\begin{array}{cccc}
1&0&0&0
\\
0&2&0&0\\
0&0&1&0
\\ 0&0&0&2
\end{array}
\right),
\end{eqnarray}
 $E$ is the propagation constant, and $\bw$ is $4\times 1$ constant column vector. This reduces (\ref{basic_equations}) to the eigenvalue problem
\begin{equation}
\label{eq:stat}
E\Lambda\bw = H\bw + F(\bw) \bw,
\end{equation}
with the  respective linear
operator given by
\begin{equation}
\label{eq:H} {H}=\left(\!
         \begin{array}{cccc}
           i\gamma_1 & 0 & k_1 & 0\\
           0 & i\gamma_2 - q & 0 & k_2\\
           k_1 & 0 & -i\gamma_1 & 0 \\
           0 & k2 & 0 & -i\gamma_2 -q
         \end{array}
       \!\right)
\end{equation}
and the nonlinear part described by the matrix-function
\begin{equation}
\label{F}
F(\bw)=-\left(\!\!
         \begin{array}{cccc}
           0 & 2(w^{(1)})^* & 0 & 0\\
           w^{(1)} & 0 & 0 & 0\\
           0 & 0 & 0 & 2(w^{(3)})^* \\
           0 & 0 & w^{(3)} & 0
         \end{array}
       \!\right),
\end{equation}
where $w^{(j)}$ are used for the entries of vector $\bw$.

It is easy to check that $H$ is $\PT$-symmetric with respect to the action of the
parity operator
\begin{eqnarray}
\label{eq:P} {\p}=\left(\!\!
         \begin{array}{cccc}
           0 & 0 & 1 & 0\\
           0 & 0 & 0 & 1\\
           1 & 0 & 0 & 0 \\
           0 & 1 & 0 & 0
         \end{array}
       \!\right),
\end{eqnarray}
and time reversal operator ${\cal T}$ performing the complex
conjugation (along with $t \rightarrow -t$): $H\PT = \PT H$.

In the linear case [when one neglects  all the
nonlinear terms, i.e $F(\bw)\bw$] the problem decouples into two $\PT$-symmetric
dimers: the first one is composed of fields $u_1$ and $u_2$ and
another one of the fields $v_1$ and $v_2$. Now the eigenvalue problem (\ref{eq:stat}) is reduced to
$H\tbw=\Lambda\tE\tbw$, where we use tilde in order to refer to eigenvectors and eigenvalues of the linear problem. Solutions of the latter equation can be found from the spectrum of the operator $\Lambda^{-1}H$.  The computation yields the following linear eigenvalues
\begin{eqnarray}
\label{E} \tE_{1,2}=\pm\sqrt{k_1^2-\gamma_1^2}, \quad \tE_{3, 4}
=\frac12\left(-q\pm\sqrt{k_2^2-\gamma_2^2}\right).
\end{eqnarray}
The eigenvectors associated with the eigenvalues $\tE_{1,2}$  can be
written down as follows:
\begin{eqnarray}
\label{eq:w12} \tbw_1= \left(\!\!
         \begin{array}{c}
           e^{i\theta_1/2}\\
           0\\
           e^{-i\theta_1/2}\\
           0
         \end{array}
       \!\right), \quad \tbw_2= \left(\!\!
         \begin{array}{c}
           ie^{-i\theta_1/2}\\
           0\\
           -ie^{i\theta_1/2}\\
           0
         \end{array}
       \!\right),
\end{eqnarray}
while for $\tE_{3,4}$ one has
\begin{eqnarray}
\label{eq:w34} \tbw_3= \left(\!\!
         \begin{array}{c}
          0\\
            e^{i\theta_2/2}\\
          0\\
            e^{-i\theta_2/2}
         \end{array}
       \!\right), \quad \tbw_4= \left(\!\!
         \begin{array}{c}
           0\\
           ie^{-i\theta_2/2}\\
           0\\
           -ie^{i\theta_2/2}
         \end{array}
       \!\right),
\end{eqnarray}
where $\theta_{1,2} = \arctan\left(
\frac{\gamma_{1,2}}{\sqrt{k_{1,2}^2 - \gamma_{1,2}^2}}\right)$. Introducing the inner product as $\langle \bw_1, \bw_2 \rangle = \bw_1^\dag \bw_2$ (hereafter $\bw^\dag = (\bw^T)^*$ is the Hermitian conjugation), we observe that the  linear eigenvectors
obey the following relation:
\begin{equation}
\langle \tbw_j^*,\Lambda\tbw_p\rangle=\tbw_j^T\Lambda\tbw_p=0, \quad j\neq p.
\label{ortt}
\end{equation}
Notice however that  $\langle \tbw_j^*,\Lambda\tbw_j\rangle \ne 0$.

Generally speaking, the existence of two different gain/loss coefficients in $\PT$-symmetric lattices  results in the existence of different ``phases'' \cite{konorecent3,ZK_nonlin} featuring different linear properties of the model. In the case at hand the linear part of the system can belong to one of the
{  four} phases: (i) unbroken (or \textit{exact}) $\PT$
symmetry when all the four eigenvalues  (\ref{E}) are real; this phase
corresponds to the rectangle given by the inequalities   $|\gamma_{1}| < k_{1}$ and $|\gamma_{2}| < k_{2}$; (ii)
 $\tE_{1,2}$ are complex conjugates while   $\tE_{3,4}$ remain   real
which corresponds to $|\gamma_1|>k_1$ and  $|\gamma_2|<k_2$;
 (iii) \textit{vice versa},
$\tE_{1,2}$ are real while $\tE_{3,4}$ are complex conjugates; this
corresponds to   $|\gamma_1|<k_1$  and $|\gamma_2|>k_2$;
 (iv)
all four  eigenvalues have nonzero imaginary part, i.e.
$|\gamma_{1}|>k_{1}$ and $|\gamma_{2}|>k_{2}$. Phases (ii)-(iv) correspond to the
broken $\PT$ symmetry. On the plane ($\gamma_1,\gamma_2)$ there exist four \textit{quadruple points}
corresponding to the corners of the above mentioned rectangle: $(\gamma_1,\gamma_2)=(\pm k_1, \pm k_2)$. These are the exceptional points  where all
four phases touch.

If the $\PT$ symmetry is unbroken, i.e.  $|\gamma_{1}| < k_{1}$ and $|\gamma_{2}| < k_{2}$, then the choice of
the eigenvectors in Eqs.~(\ref{eq:w12})--(\ref{eq:w34})  makes
them    $\PT$ {\em invariant}, i.e. $\PT \tbw = \tbw$. One can see that in the absence of the degeneracy, i.e. at all $\tE_j$ different, in
the stationary linear regime the total energy is concentrated in only one
harmonic of each waveguide. Namely, the field is guided only in the first (second) harmonic, i.e.  $v_{1,2}=0$ ($u_{1,2}=0$), for the
eigenvectors corresponding to the eigenvalues $\tE_{1,2}$ ($\tE_{3,4}$).
This will prove rather critical in some of the considerations
that follow
 (especially as regards the continuation of nonlinear
modes from the linear limit).
Notice that the linear modes $\tbw_3$ and $\tbw_4$  at the same time solve the full (i.e. the nonlinear) system (\ref{basic_equations}) because in the absence of the energy guided in the fundamental mode, the system is effectively linear: $F(\tbw_{3,4})=0$.

\section{Bifurcations of nonlinear modes from the linear eigenstates}
\label{sec:bif}
In the previous section we computed solutions of the linear problem which can be formally obtained from the full nonlinear problem (\ref{eq:stat})  by neglecting the nonlinear term $F(\bw)\bw$. In this section we look for solutions of the full  nonlinear problem bifurcating from the linear solutions. To this end  we construct formal small parameter expansions around the linear  eigenvectors (\ref{eq:w12}) and (\ref{eq:w34}).  However, properties of the eigenvectors $\tbw_{1,2}$ and $\tbw_{3,4}$ are essentially different:  the latter couple of linear eigenvectors simultaneously solve the full nonlinear problem  because the nonlinear operator vanishes at them, i.e., $F(\bw_{3,4})=0$. However for the eigenvectors $\tbw_{1,2}$ one has  $F(\bw_{1,2})\ne 0$. This suggests that the formal expansions for the
nonlinear modes bifurcating from   $\tbw_{1,2}$ and $\tbw_{3,4}$ should be constructed in different ways.

\subsection{Nonlinear modes bifurcating from $\tbw_{1,2}$}

Since  $F(\bw_{1,2})\ne 0$, one can expect that the linear solutions  $\tbw_{1,2}$ can approximate nonlinear modes only in a situation when the nonlinear term $F(\bw)\bw$  is negligible in Eq.~(\ref{eq:stat}).  This  situation takes place if one considers  nonlinear modes $\bw$ of small amplitude, i.e. at $\|\bw\| \to 0$ which is usually referred to as the {\em linear limit}  (here $\|\tbw\|$ stands for a norm of the vector $\tbw$, which could be, say, the Euclidean one). Therefore, let us search  for small-amplitude nonlinear modes bifurcating from the linear solutions $\tbw_{1,2}$. We assume that $\PT$ symmetry is unbroken and all the eigenvalues (\ref{E}) are distinct from each other  and  introduce  the following formal expansions for the nonlinear modes $\bw_j$ bifurcating from the $j$th linear eigenstate ($j=1,2$):
\begin{eqnarray}
\label{expan_w}
\bw_j  = \vep\tbw_j + \vep^2 \bW_j^{(1)} + \vep^3\bW_j^{(2)}+\cdots,\\
\label{expan_E}
E_j  =   \tE_j + \vep e_j^{(1)} +  \vep^2 e_j^{(2)}+\cdots.
\end{eqnarray}
Here $\vep$ is a small positive  formal parameter, and    $\bW_j^{(1,2,\ldots)}$ and $e_j^{(1,2,\ldots)}$ are the vectors and the coefficients to be determined.
The expansion (\ref{expan_w}) and the definition (\ref{F}) imply that $F(\bw_j) = \vep F(\tbw_j) + \vep^2 F(\bW_j^{(1)}) + \cdots$.

 Since the linear eigenvectors $\tbw_p$ ($p=1,2,3,4$) constitute a complete basis, we can search  for the correction  $\bW_j^{(1)}$ in the form
\begin{eqnarray}
\label{exp_W}
\bW_j^{(1)}=\sum_{p=1, \, p\neq j}^4 c_p \tbw_p
\end{eqnarray}
(notice that in the latter equation we set $c_j=0$ which can always be achieved by means of renormalization of the small parameter $\vep$).
Substituting the introduced expansions into the nonlinear problem (\ref{eq:stat}), in the $\vep^2$-order we have
\begin{equation}
\label{ord2}
\tE_j\Lambda\bW_j^{(1)}+e_j^{(1)}\Lambda\tbw_j = H\bW_j^{(1)} + F(\tbw_j)\tbw_j.
\end{equation}

An unusual property of the case at hand is that the nonlinearity is orthogonal to the states $\tbw_j^*$ and $\tbw_{3-j}^*$:
\begin{eqnarray}
\langle\tbw_j^*,F(\tbw_j)\tbw_j\rangle=\langle\tbw_{3-j}^*,F(\tbw_j)\tbw_j\rangle=0.
\end{eqnarray}
Thus applying $\tbw_j^\dag$ and $\tbw_{3-j}^\dag$ to the both sides of (\ref{ord2}), using that $H^\dag=H^*$ and $\Lambda^\dag=\Lambda$ and accounting for  (\ref{ortt}), one readily finds that $e_j^{(1)}=0$ and $c_{3-j}=0$. Finally, applying $\tbw_3^\dag$ and $\tbw_4^\dag$, we compute the coefficients $c_{3,4}$ explicitly which yields the following expression:
\begin{equation}
\label{W1}
\bW_j^{(1)}=\frac{\langle\tbw_3^*,F(\tbw_j)\tbw_j\rangle\tbw_3}{(\tE_j-\tE_3)\langle\tbw_3^*,\Lambda\tbw_3\rangle} +\frac{\langle\tbw_4^*,F(\tbw_j)\tbw_j\rangle\tbw_4}{(\tE_j-\tE_4)\langle\tbw_4^*,\Lambda\tbw_4\rangle}.
\end{equation}

Proceeding to the next order of the expansions, at $\vep^3$ we obtain
\begin{eqnarray}
\label{W2}
\tE_j\Lambda\bW_j^{(2)} + e_j^{(2)}\Lambda\tbw_j &=& H\bW_j^{(2)}+F(\tbw_j)\bW_j^{(1)},
\end{eqnarray}
where we have used (\ref{W1}) and the property $F(\tbw_{3})=F(\tbw_{4})=0$ yielding
$F(\bW_j^{(1)})=0$.
Thus applying $\tbw_j^\dag$ to (\ref{W2}), we compute (recall $j=1,2$)
\begin{equation}
\label{W3}
e_j^{(2)}=\frac{1}{\langle\tbw_j^*,\Lambda\tbw_j\rangle}\sum_{p=3,4}\frac{\langle\tbw_p^*,F(\tbw_j)\tbw_j\rangle \langle\tbw_j^*,F(\tbw_j)\tbw_p\rangle}{(\tE_j-\tE_p)\langle\tbw_p^*,\Lambda\tbw_p\rangle}.
\end{equation}
Formulas (\ref{W1}) and (\ref{W3}) determine  the required terms of the expansions.

\subsection{Nonlinear modes bifurcating from $\tbw_{3,4}$}
\label{sec:bif34}
As it was mentioned above in our specific case $F(\tbw_{3,4})=0$ and thus the linear eigenvectors $\tbw_{3,4}$ at the same time solve the original nonlinear model (\ref{eq:stat}).
This means that the nonlinear modes (if any) bifurcating from $\tbw_{3,4}$  should in general possess finite nonzero   amplitude at the point of the bifurcation. In other words,   the bifurcations of nonlinear modes $\bw$   occur not from the linear limit (which is understood as $\|\bw\|\to0$), but rather from a finite amplitude solution (see also the discussion in~\cite{Mor1,Mor2}).
This readily
suggests that the small parameter expansions for nonlinear modes bifurcating from $\tbw_{3,4}$ should be looked for as follows ($j=3,4$):
\begin{eqnarray}
\label{expan_w34}
\bw_j  = \alpha_j\tbw_j + \vep  \bW_j^{(1)} + \vep^2\bW_j^{(2)}+\cdots,\\
\label{expan_E34}
E_j  =   \tE_j + \vep e_j^{(1)} +  \vep^2 e_j^{(2)}+\cdots,
\end{eqnarray}
where we have introduced the proportionality coefficients $\alpha_{j}$, which must be defined from   requirement of consistency of the asymptotic expansion. Notice that if $\alpha_j \ne 0$, then at the point of bifurcation (i.e. at $\vep=0$),  the power of the first   harmonic goes to zero, i.e. $|w_j^{(1,3)}| = 0$, while  $|w_j^{(2,4)}|$ stay finite.

Now one   has $F(\bw_j) = \vep F(\bW_j^{(1)}) + \vep^2 F(\bW_j^{(2)}) + \cdots$.
Substituting the introduced expansions into the nonlinear problem (\ref{eq:stat}), we observe that it is automatically satisfied in the leading order $\vep^0$.  In the $\vep^1$-order for
$j=3,4$ we obtain
\begin{equation}
\label{eq:vep1w34}
\alpha_j e_j^{(1)}\Lambda\tbw_j + \tE_j\Lambda\bW_j^{(1)} = H\bW_j^{(1)} + \alpha_j F(\bW_j^{(1)})\tbw_j.
\end{equation}
Employing again representation (\ref{exp_W}) [recall that $F(\bw_{3,4})=0$] we find $F(\bW_j^{(1)})=c_1 F(\tbw_1) + c_2 F(\tbw_2)$. Applying $\tbw_{7-j}^\dag$ to   both sides of (\ref{eq:vep1w34}) and using properties (\ref{ortt}) and
\begin{eqnarray}
\langle\tbw_{j}^*,F(\tbw_{1,2})\tbw_j\rangle=\langle\tbw_{7-j}^*,F(\tbw_{1,2})\tbw_j\rangle=0,
\end{eqnarray}
  we find that $c_{7-j}=0$. Next, we apply $\tbw_j^\dag$ and using the same arguments find that $e_j^{(1)}=0$. Finally, we apply $\tbw_{1}^\dag$  and $\tbw_{2}^\dag$ which yields the following system:
\begin{eqnarray}
c_{1} \langle\tbw_{1}^*,\Lambda \tbw_{1} \rangle (\tE_j -  \tE_{1})
 -\alpha_j c_1  \langle\tbw_{1}^*, F(\tbw_{1}) \tbw_j\rangle
 \nonumber \\%
 - \alpha c_2  \langle\tbw_{1}^*, F(\tbw_{2})  \tbw_j \rangle = 0,
 \\[2mm]
 c_{2} \langle\tbw_{2}^*,\Lambda \tbw_{2} \rangle (\tE_j -  \tE_{2})
  -\alpha_j c_1  \langle\tbw_{2}^*, F(\tbw_{1}) \tbw_j\rangle
  \nonumber \\%
  - \alpha_j c_2  \langle\tbw_{2}^*, F(\tbw_{2})  \tbw_j \rangle = 0.
\end{eqnarray}
For a given $j=3,4$ the latter equation form  a homogeneous linear system with respect to the coefficients $c_{1}$ and $c_{2}$. The compatibility condition of this system results in a \textit{quadratic} equation with respect to $\alpha_j$ whose roots give admissible values of $\alpha_j$ in  (\ref{expan_w34}). If $\alpha_j$ is chosen to satisfy the compatibility condition,  then  coefficients of $c_1$ and $c_2$ are given up to a multiplier, which however can be scaled out by means of renormalization of the small parameter $\vep$.

In order to determine $e_j^{(2)}$ one can proceed to the  $\vep^2$-order, which yields
\begin{eqnarray}
\alpha_j e_j^{(2)}\Lambda\tbw_j + \tE_j\Lambda \bW_j^{(2)} =
\hspace{3cm}
\nonumber \\%
 H\bW_j^{(2)}  + \alpha_j F(\bW_j^{(2)} ) \tbw_j + \alpha_j F(\bW_j^{(1)} )\bW_j^{(1)}.
\end{eqnarray}
After applying $\tbw_j^\dag$ one finds
\begin{equation}
 e_j^{(2)} = \frac{1}{\alpha_j \langle\tbw_{j}^*,\Lambda \tbw_{j} \rangle}\sum_{p=1}^2\sum_{q=1}^2 c_pc_q  \langle\tbw_{j}^*, F(\tbw_p)\tbw_{q} \rangle,
\end{equation}
which gives   the leading order   correction to the propagation constant.

\subsection{Discussion}

The results of two previous subsections show that   due to the interplay between the quadratic nonlinearity and the  particular structure of   eigenvectors of the  underlying linear problem, continuation of the linear eigenvectors into the nonlinear domain occurs  in two different ways, depending on whether the second or the first harmonic is vanishing in the linear eigenvector.  In the first case, when the total energy of the linear eigenvectors $\tbw_{1,2}$ is fully concentrated in the first harmonic, the bifurcations of nonlinear solutions from the linear ones resemble the standard Kerr nonlinearity case: each linear eigenstate gives birth to one family of nonlinear modes, and at the point of
the bifurcation ($\vep=0$) the amplitude of the nonlinear modes is zero, gradually increasing when one passes from $\vep=0$ to small nonzero $\vep$. Therefore, the bifurcations occur from the linear limit.

However, in the second situation, when the linear eigenvectors $\tbw_{3,4}$ have a
vanishing first harmonic contribution and a nonvanishing second harmonic
one, the bifurcations of nonlinear modes  occur in another way.  Now, at the point of bifurcation ($\vep=0$), the nonlinear modes generically bear a finite nonzero amplitude due to the presence of the additional term with the coefficient $\alpha_j$. Moreover, since the possible values of $\alpha_j$ are given by the quadratic equation, each linear eigenstate   in general gives birth  to at least two physically distinct families of nonlinear modes. Finally we notice that we have computed values of   $\alpha_j$ for several  choices of model parameters, obtaining in this way  analytical prediction for the   amplitude of   nonlinear modes at the point of   bifurcation ($\vep=0$).
For all cases that we checked  we   observed that  the analytically  predicted  amplitude of the bifurcating nonlinear modes agrees with that obtained from direct numerical results that follow in Sec.~\ref{sec:nonlin}.

\section{Fully Nonlinear Modes}
\label{sec:nonlin}
\subsection{Theoretical Setup}

Going beyond the consideration of the nonlinear modes  described by the
expansions (\ref{expan_w})--(\ref{expan_E}) and (\ref{expan_w34})--(\ref{expan_E34}), let us now turn to the set of all stationary  nonlinear modes  obeying the system (\ref{eq:stat}).
We observe that the latter   system has a considerable wealth of solutions of which we provide a representative set in what follows. In particular, we focus on nonlinear solutions
preserving the symmetry pertinent to the linear part, i.e. to the $\PT$-invariant modes obeying $\PT \bw = \bw$. Using for such modes  the amplitude-phase decomposition we rewrite the stationary solution $\bw$ introduced by (\ref{lin_ansatz}) in the form
\begin{eqnarray}
\bw=\left(
\begin{array}{c}
Ae^{i\phi_1}
\\
Be^{i\phi_2}
\\
Ae^{-i\phi_1}
\\
Be^{-i\phi_2}
\end{array}
\right),
\end{eqnarray}
where $A$ and $B$ are real stationary amplitudes and $\phi_{1,2}$
are stationary phases.
This ansatz reduces (\ref{eq:stat}) to the system of stationary equations as follows
\begin{subequations}
\begin{eqnarray}
       E& = &k_1e^{-2i\phi_1} - {2}Be^{i(\phi_2 -2\phi_1)} + i\gamma_1,\\
    {2}E& = &k_2e^{2i\phi_2} - (A^2/B) e^{i(\phi_2 -2\phi_1)} - q  - i\gamma_2
\end{eqnarray}
\end{subequations}
(where it is assumed that $B\neq 0$). Further splitting to real and imaginary parts yields four equations:
\begin{subequations}
\label{eq:sys4}
\begin{eqnarray}\label{nmsys}
E &=& k_1\cos(2\phi_1) -  {2}B\cos(\phi_2 - 2\phi_1),\\
0 &=& -k_1\sin(2\phi_1) -  {2}B\sin(\phi_2 - 2\phi_1) + \gamma_1,\\
 {2}E &=& k_2\cos(2\phi_2) - (A^2/B)\cos(\phi_2 - 2\phi_1)-q,\\
0 &=& k_2\sin(2\phi_2) - (A^2/B)\sin(\phi_2 - 2\phi_1) - \gamma_2.
\end{eqnarray}
\end{subequations}

If we take $A$, $B$, $\phi_{1,2}$ as four unknowns in the system
(\ref{eq:sys4}), then  one can expect that there exists one or
several solutions for any given $E$. Therefore, we can speak about
{\em continuous families} of nonlinear modes.
In order to visualize these families, one can
introduce the quantity
\begin{eqnarray}
\label{energy}
U=\langle \bw,\Lambda\bw \rangle=\bw^\dag\Lambda\bw
\end{eqnarray}
which corresponds to the Manley-Rowe invariant (of the conservative system $\gamma_{1,2}=0$ where it is a conserved quantity). In the case at hand we have
$U= 2(A^2+ 2 B^2)$.
Then the families of the nonlinear modes can be displayed as
dependencies on the $(E,U)$ plane  as the functions $U$ \textit{vs} $E$.

\subsection{Numerical results}
Several examples  are presented in Fig.~\ref{fams} where  we first
address the situation $\gamma_{1,2}=0$ corresponding to the
conservative limit of the problem and then consider the effect of
nonzero $\PT$-symmetric components $\gamma_{1,2}$ for some
representative value pairs.
\begin{figure
{\includegraphics[width=1.0\columnwidth]{fig01.eps}
\caption{(Color online)  Families of
nonlinear modes on the plane $U$   \textit{vs} $E$.
Values of parameters are given as
  $k_1=1$, $k_2 =2$, $q =
0.5$, and  $\gamma_{1,2}=0$ [panel (a)];  $\gamma_1=0.1$ and $\gamma_2
= 0.5$  [panel (b)];  $\gamma_1=0.1$ and $\gamma_2=0.9$  [panel (c)]. Stable and  modes correspond to the blue (solid)  and  red (dashed) fragments of the curves. Insets (b1) and (c1) provide better resolution for some bifurcational features   not visible well in the main plots.}
\label{fams}
\end{figure}

The nonlinear system (\ref{eq:sys4}) features several interesting properties. First, as it was predicted above
there exist families bifurcating from the linear eigenstates $\tbw_{1,2}$  [see
Eqs.~(\ref{E}) and (\ref{eq:w12})], one family bifurcating from each eigenstate.  In accordance with the expansions (\ref{expan_w})--(\ref{expan_E}), at the points of the bifurcations ($\vep=0$) one has $E=\tE_{1,2}$ and $U=0$. The modes obeying expansions (\ref{expan_w34})--(\ref{expan_E34}) have also been found in our numerics. For such modes at $\vep=0$ one has  $E=\tE_{3,4}$ and $U=4\alpha_{3,4}^2$   with $\alpha_{3,4}$ being solutions of the quadratic equation introduced in Sec.~\ref{sec:bif34} [in order to obtain the latter equality we used Eqs.~(\ref{eq:w34}) for the explicit form of  eigenvectors $\tbw_{3,4}$]. In all the considered cases  we have found two distinct families bifurcating either from $\tE_3$ or from $\tE_4$.
{Notice however, that the two families bifurcating from $\tE_4$ for $\gamma_{1,2}=0$ are not distinguishable in Fig.~1(a). This is because for each given $E$ the modes belonging to  those families are mutually complex conjugate. Therefore, these solutions have the same $U$ characteristic. They also have the identical stability properties (see below for the discussion on stability) which allows us to consider only one family of those  two in what follows.}

It is also important to indicate that in all cases addressed in Fig.~\ref{fams}, the values of
the parameter $\alpha_{3,4}$ allowing for the bifurcations from $\tE_{3,4}$ are distinct from zero.   Therefore,  the  value  $U=4\alpha_{3,4}^2$ corresponding to the point of bifurcation   is also distinct from zero.  The latter comment is relevant because in all   three panels of Fig.~\ref{fams} one can observe that one of the  emerging at $E=\tE_3$ families approaches closely the horizontal axis $U = 0$. However, as the above analysis clearly indicates, the relevant bifurcation point is still distinct from $U = 0$.

In terms of gross
features of the bifurcation diagrams, we observe that the families
are extended to the domains of either
positive or negative $E$, and some families feature a
parabolic-shaped pattern. We also notice
that an apparent
pitchfork bifurcation existing in   panel (a) with
$\gamma_{1,2} =0$ breaks into a pair of ``fold
points'', as we deviate from the Hamiltonian limit. Additional
such fold points can be observed e.g. in   panel (c) of
Fig.~\ref{fams} for $E\approx 0.85$, or in the   panel (b)
 for $E\approx -1.63$ [see inset (b1)]. However, it is evident that most families in the figure either come from $-\infty$ or asymptote towards $+ \infty$,
for large values of $U$.

In Fig.~\ref{fams} we also address stability of the obtained modes
by means of identifying of the spectrum of the
linearization of the original equations around each family of
stationary modes. One can observe that the stability situation may be fairly
complicated with the same family  having alternating domains of
stability and instability (these will be analyzed in detail below
for the parameters used).
The linearization spectrum contains a
double zero eigenvalue (due to the global phase invariance of the full model).
Instability can be caused either by a pair
of purely real eigenvalues in the spectrum (one of them is
responsible for instability) or by a quartet of complex
eigenvalues (two of which correspond to unstable modes).

Quite remarkably, nonlinear modes (including stable ones) can be
also found in the regime where
the $\PT$ symmetry of the underlying linear problem
is broken. This is a feature that nonlinearity has been
shown to sustain even in the cubic case, in particular for a quadrimer setting (see e.g. the relevant discussion of~\cite{pgk,konorecent3}).
Two examples are shown in Fig.~\ref{fams-broken}. The
respective parameters correspond to the underlying linear problem
belonging to two different ``phases'' of the broken $\PT$ symmetry
(see Sec.~\ref{linear}). Namely, the system belongs to the phase
(ii) for  panel (a) while   panel (b) corresponds to the
phase (iii).  Respectively,  in the situation of panel (a) the eigenvalues $\tE_{1,2}$  are complex (with nonzero imaginary parts) and do not allow for bifurcations of nonlinear modes. However, the eigenvalues $\tE_{3,4}$ are still real and give birth to the families of solutions (two families emerge at  $\tE_3$ and
at $\tE_4$). \textit{Vice versa},  in panel (b) the eigenvalues $\tE_{3,4}$ are complex but one can observe families bifurcating from the  real eigenvalues $\tE_{1,2}$ (one family from each eigenvalue).
Solutions arising from $\tE_{1,2}$ are unstable in the vicinity of the bifurcations, but  sufficiently strong
nonlinearity in this case is critical for enabling dynamical stability.
Practically, it also appears that the modes where the Manley-Rowe
invariant has a positive slope have a wider stability interval, although
a more quantitative observation along these lines is, presently, absent.

\begin{figure
 {\includegraphics[width=1.\columnwidth]{fig02.eps}
 \caption{(Color online)  Families of
nonlinear modes on the planes $U$   \textit{vs} $E$   when the $\PT$ symmetry of the underlying linear
problem is broken.  Panel (a): regime (ii) with
$\gamma_1 = 1.1$ and $\gamma_2 =
0.5$;  panel (b): regime (iii) with
$\gamma_1 = 0.1$, $\gamma_2 = 2.1$ [see Sec.~\ref{linear} for definition of regimes (ii) and (iii)]. All other
parameters are the same as in Fig.~\ref{fams}. Notice that eigenvalues $\tE_{1,2}$ are complex in panel (a), while $\tE_{3,4}$ are complex in panel (b).}
\label{fams-broken}
\end{figure}

We now turn to a more systematic analysis of the existence and stability
properties of the different  families of solutions identified previously, for
reasons of completeness.
Fig.~\ref{chisq_k12q5g15} illustrates the situation where $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.5$.
There are eight families of solution in this case, denoted by different symbols.
Their eigenvalues for the respective parameters of
existence are shown in the case of three different choices
of $E$ in Fig.~\ref{chisq_k12q5g15_stab}.

\begin{figure
\scalebox{0.28}{\includegraphics{chisq_k12q5g15.eps}}
\caption{(Color online) Existence and stability properties of nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.5$.
The four panels denote the solution amplitude
(top left), phase differences between adjacent nodes
(top right),  real and imaginary parts (second row) of eigenvalues.
For a detailed explanation of the different families, see the text.}
\label{chisq_k12q5g15}
\end{figure}

\begin{figure}[htp]
\scalebox{0.27}{\includegraphics{chisq_k12q5g15_e15.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g15_e5.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g15_e18.eps}}
\caption{(Color online) Eigenvalues of the linearization problem of nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.5$. The same notation
has been used as in Fig.~\ref{chisq_k12q5g15}.}
\label{chisq_k12q5g15_stab}
\end{figure}

\begin{itemize}
\item The family denoted by blue circles arises from $E=\tE_3 \approx 0.72$ and continues monotonically  increasing
its Manley-Rowe diagnostic upon increase of $E$ to infinity.
It always has two pairs of purely imaginary and
 one pair of real eigenvalues, which give rise to its instability.

\item The brown pentagram family exists for all the considered
values of $E$. The amplitudes of both harmonics reach their minimum
(within the parabolic shape of the family reported previously) in
the interval $E \in [0.2,0.3]$, but at different
 points. This family has three pairs of purely imaginary eigenvalues, two of which collide at $E\approx -2.13$ and turn into a complex quartet.
 At $E \approx -0.65$, the complex quartet collides on the imaginary axis and
splits anew into two pairs of imaginary eigenvalues, restabilizing the
waveform. The larger  of the two imaginary
 pairs subsequently meets the largest imaginary eigenvalues
and the collision yields a complex quartet within the short
parametric interval of $E \in [-0.20,-0.19]$ (hereafter boundaries of the intervals are given approximately). The remaining (lowest frequency)
pair  collides with the spectral plane origin and turns into a real pair
at $E\approx -0.2$. This pair of eigenvalues becomes imaginary again shortly at
$E\approx 0.2$ and collides
  with its former partner at $E\approx 0.41$ to form a complex quartet. This complex quartet once again splits into two purely imaginary pairs
  at $E\approx 1$. As a result, the brown pentagrams family is stable for all $E$ except on $[-2.13,-0.65], [-0.2,0.2], [0.41,1]$.
From the above, the substantial complexity of the family stability properties
should be rather evident.

\item The green pluses and the red crosses arise together from a saddle-node bifurcation at $E\approx 0.3$. The green pluses family is essentially
 stable except when $E$ is within a small interval of $[0.47,0.48]$,
where two out of three pairs of purely imaginary eigenvalues collide
yielding a Hamiltonian-Hopf bifurcation and a
 complex quartet and the reverse path renders the eigenvalues
purely imaginary again. This family terminates at $E=\tE_3 \approx 0.72$  with  the first harmonic amplitude vanishing.

\item The red crosses family bifurcates from the same point as the green pluses,
however it does not terminate. It is   unstable only on an interval
of $E \in [0.39,0.89]$ due to a complex quartet.

\item The magenta stars family arises from the linear limit at $E = \tE_1 \approx 0.99$ and
exists always thereafter. It has three pairs of purely imaginary eigenvalues, too. Two of them turn into a complex quartet within the
small interval $[1.38,1.43]$ and make the family unstable in this interval.

\item The cyan squares family comes from $-\infty$ having a real pair and two purely imaginary pairs of eigenvalues. This branch is stable only after $E\approx -1.3$ where the real pair turns purely imaginary; subsequently the branch terminates at $E=\tE_4 \approx -1.22$
(with  the first harmonic amplitude vanishing).

\item The orange diamonds and the black hexagrams   emerge from a saddle-node bifurcation at $E\approx -1.63$. The orange diamonds constitute
 the only family that is always stable, having three pairs of purely
imaginary eigenvalues. This family terminates at the linear limit of
$E=\tE_2 \approx -0.99$.

\item The black hexagrams start at the same point as the orange diamonds but terminates at $E=\tE_4 \approx -1.22$. It always has two pairs of purely imaginary and one pair of real eigenvalues. Hence it is generically unstable.
\end{itemize}
As general comments we can infer that, arguably, the most robust families
and ones that will generically exist are the ones emerging from the
eigenvalues  $\tE_{1,2}$ of the linear limit. The other families may have intervals of
stability but also often suffer oscillatory or real instabilities and
are subject to saddle-center bifurcations (although e.g., the family
starting from $\tE_1 \approx 0.99$ also has a small interval of instability, and  the generically stable family
starting from $\tE_2 \approx -0.99$ is subject to a saddle-node bifurcation).

\begin{figure
\scalebox{0.28}{\includegraphics{chisq_k12q5g19.eps}}
\caption{(Color online)  Existence and stability properties of nonlinear modes with similar settings as in Fig.~\ref{chisq_k12q5g15} but for $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.9$.}
\label{chisq_k12q5g19}
\end{figure}

\begin{figure
\scalebox{0.27}{\includegraphics{chisq_k12q5g19_e15.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g19_e5.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g19_e14.eps}}
\caption{(Color online)  Eigenvalues of the linearization problem of nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.9$.}
\label{chisq_k12q5g19_stab}
\end{figure}

Fig.~\ref{chisq_k12q5g19} and Fig.~\ref{chisq_k12q5g19_stab} show us the solution profiles and their eigenvalues under the parameter $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.9$. In this case there are seven families. The black hexagrams family of Fig.~\ref{chisq_k12q5g15} does not exist any more. We briefly summarize the difference in each family in the following compared with the previous ones.
\begin{itemize}
\item The red crosses family now arises from a saddle-node bifurcation with the green pluses at $E\approx 0.36$ and terminates into another saddle-node bifurcation with the blue circles families at $E \approx 0.85$. It now has a pair of purely imaginary and a complex quartet eigenvalues. The latter one reshapes into two pairs of purely imaginary eigenvalues at $E\approx 0.74$, and one of
them becomes real at
$E\approx 0.84$. Hence, it is unstable except on the interval $[0.74,0.84]$.

\item The blue circle branch is still unstable but now exists from $E=\tE_3 \approx 0.64$ to $E\approx 0.85$.

\item The green pluses branch
now exists from $E\approx 0.36$ to $\tE_3 \approx 0.64$. It is essentially stable except when $E$ is between $[0.4,0.46]$.

\item The brown pentagrams still exist for all $E$ and bear similar eigenvalues as in Fig.~\ref{chisq_k12q5g15}. In this case, the
branch is stable except on $[-2.06,-0.61],[-0.22,-0.17],[-0.1,0.07],[0.37,0.97]$.

\item The magenta stars family is similar as in Fig.~\ref{chisq_k12q5g15},
again bifurcating from the linear limit and
now being stable in the exception of the interval $E \in [1.27,1.45]$.

\item The unstable cyan squares family still comes from $-\infty$, but now it is always unstable and terminates at $\tE_4=-1.14$.

\item The orange diamonds family exists from $\tE_4\approx -1.14$ to $\tE_2 \approx -0.99$. It is unstable until $E\approx -1.1$ and becomes stable thereafter.
\end{itemize}

\begin{figure
\scalebox{0.28}{\includegraphics{chisq_k12q5g00.eps}}
\caption{(Color online)  Existence and stability properties
of nonlinear modes with similar settings as Fig.~\ref{chisq_k12q5g15} but for $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0,\ \gamma_2 = 0$.}
\label{chisq_k12q5g00}
\end{figure}

\begin{figure
\scalebox{0.27}{\includegraphics{chisq_k12q5g00_e15.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g00_e5.eps}}
\scalebox{0.27}{\includegraphics{chisq_k12q5g00_e18.eps}}
\caption{(Color online) Eigenvalues of the linearization problem of nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0,\ \gamma_2 = 0$.}
\label{chisq_k12q5g00_stab}
\end{figure}

For comparison purposes, we also consider the
Hamiltonian case  $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0,\ \gamma_2 = 0$.
Fig.~\ref{chisq_k12q5g00} and Fig.~\ref{chisq_k12q5g00_stab} show the six families of nonlinear modes in this case.
\begin{itemize}
\item The blue circle family is similar to the one in Fig.~\ref{chisq_k12q5g15}, i.e.   arises from $E=\tE_3 = 0.75$
and is always unstable. It possesses a pair of real and two pairs of purely imaginary eigenvalues for all $E$ where it exists.

\item The brown pentagrams now exist only up to $\tE_3=0.75$.
This branch is stable except on $[-2.17,-0.66], [-0.17,0.24]$, where it has a
complex quartet of eigenvalues.

\item The red crosses now bifurcate from the brown pentagrams at $E\approx 0.25$ and persist beyond the point.
It is this bifurcation that apparently splits into two fold points in
the two cases considered previously.
The red crosses are unstable only on the interval $[0.41,1]$,
where a complex quartet of eigenvalues comes
from two pairs of purely imaginary ones
colliding at $E\approx 0.41$ and returning to the imaginary axis at $E\approx 1$.

\item The magenta stars family still arises from the linear limit at $\tE_1=1$. However, it now always has three pairs of purely imaginary eigenvalues and hence is
stable wherever it exists.

\item The cyan squares family is similar to the one in Fig.~\ref{chisq_k12q5g15}, too. It is unstable, comes from $-\infty$, and terminates at $\tE_4=-1.25$.

\item The orange diamonds branch now also exists from $-\infty$ and terminates at $\tE_2=-1$. It is always stable in this case, too, again verifying the robustness
of the families that emerge from the linear limit.
\end{itemize}

\section{Dynamics of the system}
\label{sec:dyn}
Finally, from the point of view of numerical results,
we have also performed   direct numerical simulations of the propagation
dynamics of the quadratically nonlinear $\PT$-symmetric dimer.
These simulations allow us to obtain a feeling about the
dynamical implications of the instabilities presented above.

\begin{figure
\subfigure[\ blue circles]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u1e15.eps}}
\subfigure[\ brown pentagrams]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u2e5.eps}}}\\
\subfigure[\ green pluses]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u3e47.eps}}
\subfigure[\ red crosses]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u4e5.eps}}}\\
\subfigure[\ magenta stars]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u5e14.eps}}
\subfigure[\ cyan squares]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u6e15.eps}}}\\
\subfigure[\ orange diamonds]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u7e15.eps}}
\subfigure[\ black hexagrams]{\scalebox{0.28}{\includegraphics{chisq_k12q5g15_u8e15.eps}}}\\
\caption{(Color online) Dynamical plots in a semilogarithmic scale
for the $y$-variable (denoting the amplitudes of the fundamental and the
second harmonic) for different  nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.5$. The family considered and the value
of the propagation constant are depicted explicitly in each panel.}
\label{chisq_k12q5g15_dyn}
\end{figure}

In Fig.~\ref{chisq_k12q5g15_dyn}, we show the dynamics of the nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.5$, which corresponds to Fig.~\ref{chisq_k12q5g15}. We choose different values of $E$ for the different families, usually in order to simulate their
typical unstable behavior under a small perturbation by numerical errors
up to $10^{-7}$ [however, as an exception for the orange diamonds e.g. of
panel (g), we only confirm their generic stability].
In panel (a),(b) and (d), we pick $E=1.5$ for the
blue circles family, $E=0.5$ for the brown pentagrams family and the
red crosses
family, where all of them are unstable. In all three cases here,
the amplitudes of the first waveguide (which features gain)
grow exponentially fast after some
oscillation. The amplitudes of the second waveguide (which sustains loss)
 keep oscillating but also appear to increase in comparison to their
initial values. In panel (c), all the amplitudes of the green pluses family
are relatively constant for a long evolution interval and oscillating
around their initial values, due to its short-living complex quartet of eigenvalues at $E=0.47$. Panel (e) shows the amplitudes of the two waveguides of the
magenta star family which are oscillating quasi-periodically in a similar way at $E=1.4$. Panel (f) and (h) illustrate the instability of cyan squares and black hexagrams where the amplitudes of both harmonics of the first waveguide grow exponentially at about $t=50$ while the amplitudes of the second waveguide do not appear to grow indefinitely
(but contrary to the cubic case, they are also not observed to
systematically decay~\cite{tyugin}). The stable dynamics of the orange
diamonds family at $E=-1.5$ is also plotted in panel (g). Generally, for the
unstable families, we infer either a growth in the first waveguide coupled
with a bounded oscillation in the second waveguide, or a bounded
evolution in both waveguides.

\begin{figure
\subfigure[\ blue circles]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u1e8.eps}}
\subfigure[\ brown pentagrams]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u2e5.eps}}}\\
\subfigure[\ green pluses]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u3e45.eps}}
\subfigure[\ red crosses]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u4e5.eps}}}\\
\subfigure[\ magenta stars]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u5e14.eps}}
\subfigure[\ cyan squares]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u6e2.eps}}}\\
\subfigure[\ orange diamonds]{\scalebox{0.28}{\includegraphics{chisq_k12q5g19_u7e114.eps}}
\caption{(Color online)  Dynamical plots in a semi-logarithmic scale
for the $y$-variable (denoting the amplitudes of the fundamental and the
second harmonic) for different  nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0.1,\ \gamma_2 = 0.9$.}
\label{chisq_k12q5g19_dyn}
\end{figure}

Fig.~\ref{chisq_k12q5g19_dyn} shows similar dynamic plots corresponding to
the families plotted in Fig.~\ref{chisq_k12q5g19}.
Here all of the blue circles, brown pentagrams, green pluses, and red crosses in panel (a)--(d) are unstable and present similar features as before, with
unbounded growth in the one waveguide (but no decay of amplitude on the
second).
The amplitudes of the magenta stars still oscillate quasi-periodically
around their initial values. In panels (f) and (g), both amplitudes of the
first waveguide grow exponentially. The amplitudes of the second waveguide in cyan squares decay a little and then feature a weak oscillation around their
initial values, whereas for the
orange diamonds family they grow a little and then
feature a similar weak oscillation.

\begin{figure}[tph]
\subfigure[\ blue circles]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u1e15.eps}}
\subfigure[\ brown pentagrams]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u2e15.eps}}}\\
\subfigure[\ red crosses]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u3e5.eps}}
\subfigure[\ magenta stars]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u4e15.eps}}}\\
\subfigure[\ cyan squares]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u5e15.eps}}
\subfigure[\ orange diamonds]{\scalebox{0.28}{\includegraphics{chisq_k12q5g00_u6e15.eps}}}
\caption{(Color online)  Dynamical semi-logarithmic plots of nonlinear modes with $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0,\ \gamma_2 = 0$.}
\label{chisq_k12q5g00_dyn}
\end{figure}

Fig.~\ref{chisq_k12q5g00_dyn} shows the dynamics of the Hamiltonian
case under the parameters $k_1=1$, $k_2 =2$, $q = 0.5$, $\gamma_1=0,\ \gamma_2 = 0$ that corresponds to Fig.~\ref{chisq_k12q5g00}. Since $\gamma_1=0,\ \gamma_2 = 0$, neither of the two waveguides has a gain or loss profile. Shown in
panels (a)--(c), are all the amplitudes of the harmonics of the the blue
circle, brown pentagram, and red cross families which are oscillating around their initial values, with no trend of indefinite growth or decay, just as expected by the absence
of $\PT$-symmetric terms. In panel (e), the cyan squares family amplitudes now
weakly oscillate
periodically
(but in a way preserving as
they should the Manley-Rowe invariant). Panels (d) and (f) show the stable
dynamics of the magenta stars and the orange diamonds families in this case,
i.e., confirming the dynamical robustness of the families stemming from the
linear limit.

\section{Conclusions and future work}
\label{sec:concl}
In the present work, we considered systematically the features of stationary states of a prototypical $\PT$-symmetric quadratically nonlinear dimer (or coupler).  We  explored different parametric regimes in the two-dimensional plane of gain and loss (for the first and the second harmonic) and in each considered case we identified families of nonlinear modes and addressed the
stability and dynamics of the solutions. We found
 numerous unexpected features that distinguish this system
e.g. from its more well studied sibling, namely the cubic $\PT$-symmetric
nonlinear dimer.

We have started our analysis by considering  the spectrum of the underlying linear problem and found that its eigenvalues always have total energy fully  concentrated either in the first or in the second harmonic of the waveguide.  Turning to the full nonlinear problem, we have established that the found linear solutions give birth to nonlinear modes which (in the vicinity of the bifurcation from linear eigenstate) can be described by means of the small-parameter formal expansions. We have further  revealed  two types of the bifurcations of nonlinear modes from the linear solutions. Namely, the nonlinear modes continued from the linear eigenvectors with total energy concentrated in the first harmonic, have zero  value of the  Manley-Rowe  characteristic at the point of bifurcation. On the other hand, the nonlinear modes arising from the linear eigenvectors with total energy concentrated in the second harmonic bifurcate with finite nonzero  Manley-Rowe  characteristic. Moreover, in the latter case there can exist   two physically distinct  families bifurcating from the same linear state.  These findings were at first quantified by the perturbative formal expansions  which are shown to acquire different forms for the two above-mentioned situations. Then we  confirmed the analytical predictions via  numerical computations of  the
full nonlinear system determining its nonlinear modes.  We have addressed several representative sets of the system parameters and numerically computed continuous families of nonlinear modes as functions of the propagation constant.

Further, we have numerically examined the stability of the identified
families.
Generally, the stability
was found to have rather complex properties but some gross
features could still be discerned such as the systematic robustness
of the modes that emerged from the linear solutions. The dynamics
also features differences from the cubic case, such as the fact
that the lossy waveguide does not typically appear to have a vanishing
amplitude (when growth occurs on the gain side).

Finally, there are numerous directions that one can consider for
future study. On the one hand, one can address simple extensions
of the present dimer, such as the case with  competition
between the signs of $\gamma_1$ and $\gamma_2$ with gain in the
first harmonic but loss in the second (or vice versa). In the way
of extensions to models with more degrees of freedom, one can envision
chains of such dimers at the lattice level (whereby discrete solitary
waves and their properties can be considered)
or even continuum extensions  of the dimer in transverse
continuous directions. In that case, the dimer considered herein
would constitute the limit of homogeneous solutions along such
transverse directions. Some of these possibilities are currently
under investigation and will be reported  in future publications.

\acknowledgments

The work of DAZ and VVK was supported by FCT (Portugal)
through the grants PTDC/FIS-OPT/1918/2012 and
PEst-OE/FIS/UI0618/2011. PGK gratefully acknowledges
support from the US NSF under grant CMMI-1000337,
from the US AFOSR under grant FA9550-12-1-332, and
from the Binational Science Foundation under grant
2010239.

redpajama_set_name: RedPajamaArXiv


text:
Seven Sisters 7: The Missing Sister by Lucinda Riley
Publication Date: May 2021
They'll search the world to find her.
From the Sunday Times number one bestselling author Lucinda Riley,The Missing Sister is the seventh instalment in the multimillion-copy epic series The Seven Sisters.
The six D'Apliese sisters have each been on their own incredible journey to discover their heritage, but they still have one question left unanswered: who and where is the seventh sister?
They only have one clue - an image of a star-shaped emerald ring. The search to find the missing sister will take them across the globe - from New Zealand to Canada, England, France and Ireland - uniting them all in their mission to complete their family at last.
In doing so, they will slowly unearth a story of love, strength and sacrifice that began almost one hundred years ago, as other brave young women risk everything to change the world around them.
PRAISE FOR THE SEVEN SISTERS SERIES
'The Seven Sisters series is heart-wrenching, uplifting and utterly enthralling'
Lucy Foley
'Well researched and compelling ... on an epic scale' Sunday Express
'There's something magical about these stories' Prima
'Addictive storytelling' Woman & Home
'A masterclass in beautiful writing' The Sun


text:
By Rachael in Caring For The Elderly
Also Known As: Waltraud Wagner, Maria Gruber, Irena Leidolf and Stephanija Meyer
Born: 1960, 1964, 1962 and 1940 respectively
Died: –
Active: 1983 to 1989
Method of Disposal: Morphine, Insulin or Rohypnol Overdose; Drowning
Consequence: Imprisonment
Unfortunates:
At least 49 patients at Lainz General Hospital (aka the Death Pavilion or Murder Station of Lainz)
Waltraud Wagner became a nurse because she wanted to help people, to make them comfortable and ease their pain and suffering. She chose to work at Lainz General Hospital's Pavilion 5, where many of the hospital's elderly and infirm resided. A lot of the patients in this particular Pavilion were suffering from terminal illnesses. Consequently, it was a relatively high, but not entirely unexpected, death rate.
It was perhaps inevitable that, during her working life in Pavilion 5, Waltraud would encounter residents who felt they would rather leave this life than face another day of pain. She would ordinarily help the patient through their fear and denial, by talking to them or supplying them with pain relief.
But one day she was confronted with an appeal that was so impassioned that she was moved to give a second thought to the idea of assisting another person to die. After all, the point of her job was to provide palliative care for the residents, usually until death finally came to call. Surely it was better if the patient chose when they expired, rather than wait. Really, if death was inevitable, what did it matter when said death occurred?
Having decided that five minutes of internal dialogue was enough time to completely evaluate all the pros and cons of what she was about to do, Waltraud obtained a lethal dose of morphine from the medicine store. As the morphine worked its magic, the look of serenity on her charge's face was all Waltraud needed to see to know she had done the right thing. She finished her shift feeling more energised than ever.
Over the next few days, Waltraud relived in her mind what she had done. She realised that it was a very powerful gift she had been given, to be able to take away someone's pain like that. She also discovered, after performing a few more acts of kindness on other terminally ill patients, that the calling was bigger than one person could handle.
She needed to impart her wisdom on others.
Over the course of the next four years Waltraud was joined by three 'recruits': Maria Gruber, Irene Leidolf and Stephanija Meyer. With Waltraud as their mentor, each woman was taught the proper techniques that go into a successful mercy killing. Her students were understandably slow to begin with, but by the end of 1987 they had all perfected the art and were 'assisting' at a steady pace. Along with the morphine, the group eventually added Insulin and Rohypnol to their list of tools, just to mix things up every now and then.
It may have been that the medical supplies had rather dwindled after a while; for Waltraud decided that a new technique would be included in their repertoire. This procedure was called the "Water Cure". It was a simple process really, requiring only a large jug of water to be poured into the mouth of the patient by one nurse while another held the patient's nose and chin. It was heralded as a 'natural' alternative to their existing practice. This was due to the fact that elderly patients often 'naturally' had water on the lungs when they died. It also removed the need to complete pesky administration forms explaining the rapid depletion of medical stock.
This new method had come at a crucial time, as Waltraud had widened the definition of 'patient in need'. It now included any patient who wet the bed, refused to take their medicine, or pressed the nurse's button just when they'd popped the kettle on. Waltraud classed these as 'annoyances', and the Water Cure was employed as a rule whenever an 'annoyance' arose. To ensure that the procedure was followed correctly Waltraud always carried out the Water Cure herself, with the help of one of the other 'students'.
As is the usual practice, rumours began to circulate about the regularity with which the undertaker was attending the hospital's Pavilion 5. If the four women noticed that the abnormally high death rate on their ward had attracted the attention of other staff, they didn't let on. Waltraud and her followers would often meet at a local pub, toasting their success. During these meetings they would reminisce about their past accomplishments, recalling in minute detail how they had helped their most recent patient towards the light.
As it happened, a doctor was sitting nearby. He was familiar with the jargon they were using and, upon realising what he was hearing, promptly sprinted to the nearest police station. The police quickly began an investigation. Surely the doctor must be mistaken, they reasoned. This is a state-run hospital; there was no chance anything like this could have gone on for so long.
Apparently, there was. After six weeks, all four women were arrested and immediately began pointing fingers. Mostly at Waltraud.
Waltraud admitted to all the killings that she could remember (she got as far as 39 before her memory escaped her). Stephanija helpfully added several more to that list. Not to be outdone, Irene and Maria insisted that there were hundreds more. Over the course of the next 18 months, all four suspects threw up more and more imaginative patient numbers (including a few figures that clearly exceeded the total patient population of the hospital at the time). Finally, in late 1990, Waltraud revised her patient list to a mere ten.
To which the Judge and jury replied, "We can make up numbers too." Waltraud and Irene were sentenced to life imprisonment, while Stephanija and Maria each received 15 years.
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text:
Lheure en France s'étend sur . La France est ainsi le pays qui couvre le plus de fuseaux horaires au monde.

Situation actuelle

Le territoire français, outre-mer compris, couvre  en hiver, ou 12 si on exclut la Terre-Adélie. En été, il couvre dans tous les cas un fuseau horaire de plus, donc respectivement 14 ou 13. De ce fait, la France est le pays qui couvre le plus de fuseaux horaires au monde.

Heure en métropole
L'heure légale en France métropolitaine est, l'hiver, l'heure normale d'Europe centrale (HNEC) c'est-à-dire celle du fuseau horaire UTC+01:00 ; l'été c'est l'heure avancée d'Europe centrale (HAEC) c'est-à-dire celle du fuseau horaire UTC+02:00. La France métropolitaine vit à cette heure avancée du dernier dimanche de mars (à 2 heures, HNEC) au dernier dimanche d'octobre (à 3 heures, HAEC).

Heure outre-mer
La France d'outre-mer vit dans les fuseaux horaires suivants:

Histoire

Origines

Dans l'Antiquité, la Gaule comprenant la France actuelle est une province romaine qui vit sous l'influence de Rome. Les heures, quand elles sont employées, sont des heures temporaires.

Dans la période médiévale, l'Église met en place des heures temporaires simplifiées et grossières, les heures canoniales. Avec l'apparition de l'horloge mécanique, elles sont remplacées progressivement par les heures équinoxiales ou égales, données par le cadran solaire classique qui sert de référence pour l'horloge. Le temps ainsi compté, implicitement, devient le temps civil.

Au début du , le temps du cadran solaire et celui de l'horloge vont se différencier : le cadran classique donnant le temps vrai et l'horloge le temps moyen. Ce dernier peut être lu sur des méridiennes de temps moyen qui deviennent référence pour montres et horloges. On trouve également des outils de mesure personnels portatifs comme l'Anneau de paysan qui donne l'heure solaire en temps moyen avec une précision d'environ un quart d'heure.

En 1826, plus d'un siècle plus tard, Paris adopte pour ses horloges le temps moyen et l'heure moyenne qui en découle.

1891 à l'heure de Paris
Au , le besoin d'unifier l'heure sur tout le territoire se fait sentir, notamment avec l'essor du chemin de fer et l'impossibilité de régler les horaires des trains sur des heures locales variant d'un point géographique à un autre. Pour la régularité du service des trains, les chemins de fer prennent l'heure d'origine des réseaux français, c'est-à-dire celle de Paris, mais, pour favoriser les voyageurs non ponctuels, on décide que l'heure du chemin de fer serait en retard de cinq minutes sur l'heure de Paris. Progressivement, au cours de la seconde moitié du , les villes françaises adoptent le double affichage horaire ou l'heure de Paris. À Rennes, par exemple, desservie par le train depuis 1857, l'horloge de l'hôtel de ville indique l'heure locale et celle de la capitale à l'aide de deux aiguilles des minutes : l'une dorée (heure locale, c'est-à-dire l'heure solaire vraie), l'autre noire. En 1880, s'alignant sur une pratique qui se généralise dans les grandes villes (Bordeaux, Nantes), la capitale bretonne adopte l'heure parisienne pour tous les cadrans publics de son territoire.

L'uniformisation de l'heure sur l'ensemble du territoire national est imposée par la loi du  qui fixe l'heure légale en France métropolitaine et en Algérie à l'heure du temps moyen de Paris. Cette harmonisation est l'un des marqueurs de l'adaptation de la France à la révolution industrielle et économique en cours à cette époque.

1911 à l'heure de Greenwich — presque

Par la loi du , la France adopte l'heure du méridien de Greenwich, abrégé GMT en anglais, ainsi que le découpage de la journée en  au lieu de deux fois . Ces usages avaient été recommandés en  à la Conférence internationale de Washington.

Notons que le texte de loi ne mentionne pas explicitement la ville anglaise de Greenwich. En effet, l'heure de référence reste l'« heure de Paris » à laquelle on soustrait  et , pour correspondre à l'heure GMT.

Le méridien de Paris étant à 2°20'13,82" à l'est de celui du Greenwich, le décalage horaire entre le méridien de Paris et celui de Greenwich est en réalité de 9 minutes et  (arrondies à 21 secondes), à raison d'une variation de l'angle horaire du Soleil moyen de . En toute rigueur, l'heure légale française telle qu'elle est définie retarde donc de près d'un dixième de seconde sur le temps moyen de Greenwich.

Ce changement d'heure a nécessité la mise en œuvre de procédures, en particulier pour les compagnies de chemin de fer.

1916 à l'heure de la guerre

Pendant la Première Guerre mondiale, les territoires français sont à l'heure allemande ou française selon qu'ils sont ou non occupés par les troupes allemandes.

L'heure dite « d'été » est légalisée par la « loi ayant pour objet d'avancer l'heure légale » du . Celle-ci fixe dans son article unique que . Le lendemain, , son décret d'application dispose que .

Puis, le , André Honnorat fait passer à l'Assemblée nationale, par  , une nouvelle loi prolongeant les modalités de la précédente afin de diminuer la consommation nationale d'énergie.

1923 aux heures d'hiver et d'été

La loi du  aménage le changement d'heure saisonnier. Désormais la France métropolitaine vit à l'heure moyenne de Greenwich avancée d'une heure l'été, en coordination avec les nations alliées voisines. Ainsi jusqu'à la Seconde Guerre mondiale, l'heure passe à UTC±00:00 le premier samedi d'octobre, à minuit, et UTC+01:00 un samedi compris entre le dernier de mars et le troisième d'avril, à 23 heures.

1940 à l'heure allemande
À partir du printemps 1940, l'occupant allemand fait passer la zone occupée de la France à l'heure allemande pour des raisons pratiques. Au fur et à mesure que l'armée d'occupation avance, les horloges, elles, sont avancées d'une heure. La zone occupée passe donc à l'heure d'Europe centrale, pendant que la zone libre reste à l'heure de Greenwich.

En , aucune des deux zones ne repasse à l'heure d'hiver. En territoire occupé, on reste donc à UTC+2, et pour le reste du pays à UTC+1.

La France métropolitaine est ainsi divisée entre deux fuseaux horaires. Cela engendre d'importantes difficultés pour les transports ferroviaires, civils et militaires. Pour les régler, le gouvernement à Vichy décide de synchroniser à partir du  l'heure de la zone libre sur celle des autorités d'occupation. Après l'entrée des troupes allemandes et italiennes en zone libre le , les changements d'heure saisonniers reprennent de façon régulière suivant l'heure du Reich.

1944 à l'heure de l'Europe continentale
Quand les troupes alliées occidentales débarquent en France en  , leurs montres sont réglées sur la double heure d'été britannique, donc sur GMT+2 tout comme les forces d'occupation qui sont à l'heure d'été d'Europe centrale.

Le  la zone occupée passe à l'heure d'hiver allemande (GMT+1), suivie le  par la partie du territoire libérée que dirige le Gouvernement provisoire de la République française. Le , l'ensemble des alliés occidentaux repassent à l'heure d'été allemande (GMT+2).

Après la Libération, le Gouvernement provisoire de Charles de Gaulle décrète le  que le retour à l'heure légale (GMT) sera effectué en 2 temps, le  et le , et l'abandon du changement d'heure saisonnier pour 1946. Le , l'heure d'été française (GMT+1) est rétablie. Il prévoit un rétablissement de l'heure d'hiver française (GMT) pour le  mais, le , l'article 2 du décret est annulé. La France est restée en heure d'été française (GMT+1) jusqu'en 1976.

La plupart des pays d'Europe continentale partagent ainsi l'heure d'Europe centrale, . C'est aussi le cas de l'Espagne, qui est la plus décalée par rapport à l'heure solaire.

Après la création d'un bureau national de métrologie et à la suite de l'adoption du temps universel coordonné, le gouvernement de la Cinquième République crée en 1973 une « Commission nationale de l'heure », chargée d'étudier « ses propositions tendant à modifier les dispositions législatives ou réglementaires en la matière ».

1976 à l'heure du choc pétrolier

Le décret du  rétablit le changement saisonnier de l'heure. À la suite du premier choc pétrolier, le gouvernement compte par cette mesure réduire la facture énergétique. Son application est fixée pour la période du  au . La mesure est alors présentée comme provisoire. Mais elle est toujours en vigueur au siècle suivant.

La loi de 1911 est abrogée par le décret du  qui fait du temps universel coordonné (UTC mis en place en 1972) la base du temps légal en France, abandonnant définitivement la référence à l'heure de Paris : les horloges publiques et TDF diffusent ainsi UTC + 1 heure (heure d'hiver) ou UTC + 2 heures (heure d'été).

À titre de point de repère, la France et une bonne partie de l'Europe vivent donc en hiver à l'heure de Prague, et en été à l'heure d'Odessa. Comme l'heure moyenne de Greenwich (maintenant l'UTC) est le fuseau horaire « naturel » de la France métropolitaine, l'heure de l'Europe centrale (UTC+1) utilisée par la métropole peut être considérée comme une forme d'heure avancée en hiver, tandis que l'heure d'été de l'Europe centrale (UTC+2) est en quelque sorte une heure doublement avancée.

De 1981 à 1995, le passage à l'heure d'été a lieu le dernier dimanche de mars, et le passage à l'heure d'hiver le dernier dimanche de septembre. À partir de 1996, le passage à l'heure d'hiver a lieu le dernier dimanche d'octobre.

1998 harmonisation européenne
En 1998, la période de l'heure d'été est harmonisée dans toute l'Union européenne par le directive 97/44/CE, qui fixe le passage à l'heure d'été le dernier dimanche de mars et le passage à l'heure d'hiver, le dernier dimanche d'octobre ; l'harmonisation est pérennisée par la directive 2000/84/CE.

En aout 2018, la Commission européenne lance une consultation publique pour recueillir l'avis des citoyens de l'Union européenne sur le maintien de l'alternance entre l'heure d'été et l'heure d'hiver (dite « normale »).

Sur les 4,6 millions d'internautes participants, 80 % se prononcent pour la fin du changement d'horaire en gardant plutôt l'horaire d'été sur l'année. Cette consultation mobilise essentiellement des internautes allemands (2/3 des votants), pays où le sujet est sensible, c'est-à-dire où la population se sent concernée.

Le , la Commission européenne propose une directive aux termes de laquelle . Pour entrer en vigueur, cette proposition doit être approuvée par le Parlement européen et le Conseil au plus tard en .

Réunis en conseil des ministres, les États membres ont reporté la suppression du changement d'heure à 2021, craignant qu'une mise en œuvre trop rapide ne transforme l'espace européen en patchwork et ne déstabilise le transport aérien et le marché unique. Mi-2020, aucune décision n'avait encore été prise, les pays européens ayant d'autres priorités.

En France, du  au , les Français peuvent voter sur le site de l'Assemblée nationale en faveur du maintien ou de l'abrogation de l'heure d'été, et dans le second cas exprimer leur préférence pour l'heure méridienne (UTC±00:00), l'heure d'hiver actuelle (UTC+01:00) ou l'heure d'été actuelle (UTC+02:00).
Le mercredi  une des questions de la consultation est modifiée :

La seconde question reste :
« Selon vous, en cas de suppression du changement d'heure, quel fuseau horaire la France métropolitaine doit-elle choisir ?
UTC+0 ou actuellement heure d'hiver de l'Europe occidentale (ex : Portugal et Royaume-Uni en hiver)
UTC+1 ou actuellement heure d'hiver de l'Europe centrale (ex : France métropolitaine, Allemagne, Italie et Espagne en hiver)
UTC+2 ou actuellement heure d'été de l'Europe centrale (ex : France métropolitaine, Allemagne, Italie et Espagne en été) ».

Résultats de la consultation ( exprimées) :
 61,16 % répondent avoir une expérience négative ou très négative du changement d'heure ;
 83,71 % répondent souhaiter mettre fin au changement d'heure 2 fois par an ;
 59,17 % choisissent de toujours rester à l'heure d'été ;
 51,29 % choisissent UTC+2 (actuelle heure d'été en France), 36,71 % choisissent UTC+1 (heure d'hiver en France), et 12 % préfèrent UTC±0 (heure « méridienne »).

Notes et références

Notes

Références

Voir aussi

Textes officiels
  ayant pour objet l'adoption de l'heure temps moyen de Paris comme heure légale en France et en Algérie, dans .
  portant modification de l'heure légale française, pour la mettre en concordance avec le système universel des fuseaux horaires, dans .
  fixant l'heure légale française, dans .

Bibliographie
 .
 .
 .

Articles connexes
 Fuseau horaire
 Heure allemande
 Heure d'été
 Liste des fuseaux horaires

France
Société française

redpajama_set_name: RedPajamaWikipedia


text:
Gillingham Football Club badge - Link to home
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NELSON: "WE'RE IN IT TOGETHER"
Gills keeper insists its a team effort...
Gills goalkeeper Stuart Nelson is adamant the squad will not let their effort levels drop as they look to maintain their five point gap at the top of npower League Two.
Nelson has only conceded 10 goals so far in the league this season but insists it's a team effort and every player knows their job.
He said: "It's a good squad and probably the best squad I have been involved in. It doesn't matter who plays, people come in and they won't let anyone down.
"If you have got boys who are willing to learn and listen, work hard on the training ground and do what they have been told to do you will get the rewards and so far this season that is what's happened.
"I've had good starts before but have fallen away so I'm quite aware what could happen but this squad is good enough not to let that happen. As long as we keep doing what we have practised and work hard then we will keep getting results.
"It doesn't matter who is in front of me, they have the same shirt on and they put in the same amount of effort.
"If there was a problem when certain people come in and didn't put 100% in then I probably would have a problem but there isn't one because everyone does the job whether its Callum, Flanners [Tom Flanagan], Adam Barrett, Connor or Andy Frampton – it doesn't make a difference.
"To me they are all wearing Gillingham shirts and we are all in this together."
Stuart was asked whether it would be easy to take Plymouth lightly considering their position in the league, but Nelson dismissed that thought and said each game must be taken on its individual merit.
He said: "We wouldn't let ourselves, the manager has his say but part of our success so far this season is that we take each game as it comes. It's an old cliché but that's what we have done and Plymouth is so different.
"We will go away from home, away from our supporters who have been so good here at Priestfield. It's a long trip and we have to take that as it comes – we have 90 minutes of football to play down there and we'll strive to get the result that we need."
Nelson says he isn't concerned where the team are in the league at this moment in time and is focussing on getting as many points as possible.
He added: "If we do what we are capable of doing this season then we won't be far off and at the moment we are doing that. We have only played 16 games so it's early and it counts for nothing.
"The points we are leading by can go so quick if we take our foot off the pedal, start cutting corners and only give 99% and the players who come in haven't listened.
"Maybe in the last five games of the season I will look at where we are but at the moment it counts for nothing."
Finally Stuart talked about the strong unity that the goalkeepers have at the club and how it is the entire club that keeps clean sheets, not just himself between the sticks.
He said: "We all work hard and the gaffer picks the keeper he wants to play and that's how it is here. We are a tight unit of keepers; we all train together, work hard and encourage each other and those boys support me and I support them.
"Hopefully they accept my advice as much as I'll accept theirs. If they see something that can help me on a Saturday they tell me and I'll take it on board.
"Likewise if I see something that can help them then I will take them. You can't go through your career without taking any advice that's given to you and it's up to you whether you want to take it on board.
"I know Danny Kedwell and Deon Burton have kept nine clean sheets, I don't need to go on but every fan in the Rainham End has done it too.
"I have my own personal targets as a keeper. They are private but I'll tell you at the end of the season."


text:
Aquarium Product Reviews
Tank Set Up & Maintenance
Aquarium Shrimp
Aquarium Snails
Dwarf Lionfish Care and Breeding Guide
By Adam Edmond
Dendrochirus Brachypterus commonly known in the aquarium trade simply as Dwarf Lionfish is a marine water aquarium fish that is a native of Indo-Pacific waters. The fish has in the recent years gained much media prominence for due to its invasion of the tropical waters of the Atlantic. Today you will find many more of the fish in the within the Caribbean reef than they are even in their native waters of the Indo-Pacific. The fish goes by many other names such as the shortfin turkeyfish and shortfin turkeyfish. The fish remains popular with many aquarists due to its hardy nature and of course it's beautiful stripped body.
It is a slow swimmer but it is known to be a perfect hunter that will munch small animals that come across its mouth. It is a solitary fish that can comfortably be kept alone in a tank and still manage to survive. If you need to keep it with other tank mates, it is best to keep it with other fish that are aggressive because any docile fish will find it rough with the Dwarf fish.
Appearance of Dwarf Lionfish
Tank requirements
Setting up Tank
The fish exhibits a beautifully stripped body with amazing fins and frills making it a favorite of many aquarists. The scale structure that the fish has usually makes it to appear somewhat indistinct to other larger fish or animals that prey on it. It has got a small size making it easy to be kept even in moderate aquariums. In the wild it usually grows to around 15cm in length. As for coloration, the fish comes in a variety of colors based on the geographic location of the fish. Among some of the colors you can find the fish exhibiting includes; deep red color morph, rare silver color, with striking yellow dorsal and pectoral fin. You however should not expect striking colors from fish you will get from the pet shops.
The Dwarf Lionfish is native to the Indo-Pacific region where it inhabits saltwater although it has spread to areas that were once not it native areas. It is now found in the Caribbean waters where it is considered as an invasive species. Naturally, the fish inhabits coastal areas that are shallow featuring reed-encrusted rocky outcrops where it stays in hiding in order to waylay unsuspecting crustaceans.
These fish have got an attractive fuzzy appearance with a one-of-a-kind scale morphology together with striking patterns that are used as a means of camouflage when hunting.
This fish does not need to be housed in a very large tank because it is not that big and also it isn't an active swimmer. Please note that in the wild the fish prefers to lay in wait for its small food rather having to chase it around.
If you need to keep this fish in your home aquarium, it will be best if you replicate for it the rocky habitat that it is used to in the wild. If you can manage to provide for it cave-like structures in the tank with overhangs, then the fish can really feel more at home! Please note that the fish will prefer to spend most of its time lurking in between the rocky caves. You shouldn't expect much movement from the fish!
To have a detailed look into the basic tank requirement for the Dwarf Lionfish please keep reading this article below to find out more:
As for aquarium lighting, just make sure that you don't make it too much but rather subdued lighting is desirable. The fish have been found to suffer from stress under very strong lighting!
The aquarium that the fish can be housed in need not to be very large because it is not an active swimmer and therefore will not need lots of room to swim around. They are also not that large and therefore just a small aquarium can do.
The minimum tank capacity that is suitable for this fish should be at least 150 liters for a single adult fish.
Ensure good water quality for this fish otherwise it simply jus stop feeding and then eventually die.
The pH range needs to be kept between 8.1 and 8.4.
As for the tank temperature it is best to try and mimic that which is as close to that of its habitat as possible. So in this case setting the temperature to a range of 74 degrees and 82 degrees Fahrenheit.
It is best to make the tank feel safe and secure for your fish as much as possible because it will refuse to feed if it feels threatened.
Feeding the Dwarf Lionfish should not be much of a hustle for you because it known to be a predatory fish that feeds on small sea crustaceans. In captivity you can offer the fish live brine shrimp as well as small crabs. While the fish is not a very active swimmer, it will readily eat small invertebrates that it will be offered.
Because the fish is rather too choosy with foods that it is offered it, it is best to have live shrimp ready with you that you will feed during its first days in the aquarium.
It takes long for the fish to finally accept frozen meaty foods and therefore as the new owner you will need to be extra patient with it.
The most difficult part of rearing the Dwarf Lionfish in captivity is getting it to breed and many aquarists have had difficulty achieving this. This is not to say that it will totally fail to spawn in captivity. The male and the female of the fish can still be placed in the same tank by an experienced aquarist and spawning with egg fertilization can still take place. However, it must be noted that this is something that novices cannot achieve and therefore if you are still new in the hobby perhaps working with an expert is desired. You will however enjoy the beauty and elegance of the fish when you introduce it in your home aquarium.
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Stockholm, 27 June 2016 – GP Bullhound, the technology investment bank, today announced that Oskar Herdland has joined its Investment Banking team as Director, Equity Capital Markets.
Based in Stockholm, Oskar will lead GP Bullhound's equity capital markets activities. Previously, Oskar was CEO and Head of Equities at SEB Enskilda in New York.
Prior to his role of CEO and Head of Equities at SEB Enskilda in New York, Oskar worked with institutional equity sales on Wall Street. Before this, he covered the tech sector in Silicon Valley after starting his career as an equity analyst for Enskilda Securities based in Stockholm in the 90s. He graduated from Uppsala University and UCF.
Passionate about technology, innovation and entrepreneurs, Oskar has spent over 15 years in the United States, first in Silicon Valley and San Francisco, then in New York for more than a decade. He has been involved in numerous equity transactions including IPOs, private placements, new issues, and secondaries, in the United States as well as Scandinavia.


Hardcover, 8.25 x 8 in. / 298 pgs / illustrated throughout.
This first monograph on one of Czechoslovakia's most important designers, Libuse Niklová (1934–1981), explores the rich and delightful world of her plastic toys, featuring nearly 500 color photographs and archival reproductions. Her whistling accordion-pleated Tomcat, and inflatable Buffalo and Elephant have been treasured possessions for several generations of children in Czechoslovakia, occupying a unique position in twentieth-century product design, and many of them are firmly established icons in the Czech design canon. This fun, accessible book gathers her oeuvre into three chapters: "Plastic Playthings 1954–1964," "Accordion and Vernian Toys 1963–1967" and "Inflatable Toys 1963–1981." Full-color reproductions of the toys are punctuated with quotes from Niklová and her colleagues, and with sketches and archival photographs. Also included are a biography and a booklet, inserted toward the rear, listing all of her works.
Featured image, of some of Libuse Niklová's plastic playthings (1954-1964), is reproduced from Libuse Niklová.
FORMAT: Hbk, 8.25 x 8 in. / 298 pgs / illustrated throughout.


text:
air scare
Passengers and crew subdued a man who tried to break into the cockpit on a flight to Hawaii
byBonnie Kristian
Donna Basden/Associated Press
The passengers and crew of a flight from Los Angeles to Hawaii on Friday subdued a Turkish national who attempted to break into the cockpit. Two U.S. fighter jets then arrived to escort the plane to its destination, where the man was arrested by federal agents.
Identified as Anil Uskanil, 25, the unruly passenger attempted to force his way past a drink cart and through the first class cabin. "It took seconds" to subdue him, said Lee Lorenzen, a passenger who helped flight attendants restrain Uskanil. "He was pushing against the cart and a bunch of guys grabbed him. They found some duct tape. There were pillows and blankets. And they taped him to his chair."
Uskanil had a run-in with police at the Los Angeles airport before the flight, too. He was arrested on suspicion of misdemeanor trespassing after attempting to go through a door leading to the tarmac, then released with a court date. No explosives were found on the plane after it landed, and Uskanil's visa to the United States has been revoked. He is expected to face charges that would land him in prison for at least a decade.