text:
from . import utils

import functools

def watch_login(status_code=302, msg='',
                get_username=utils.get_username_from_request):
    """
    Used to decorate the django.contrib.admin.site.login method or
    any other function you want to protect by brute forcing.
    To make it work on normal functions just pass the status code that should
    indicate a failure and/or a string that will be checked within the
    response body.
    """
    def decorated_login(func):
        @functools.wraps(func)
        def wrapper(request, *args, **kwargs):
            # if the request is currently under lockout, do not proceed to the
            # login function, go directly to lockout url, do not pass go,
            # do not collect messages about this login attempt
            if utils.is_already_locked(request):
                return utils.lockout_response(request)

            # call the login function
            response = func(request, *args, **kwargs)

            if request.method == 'POST':
                # see if the login was successful
                if status_code == 302:  # standard Django login view
                    login_unsuccessful = (
                        response and
                        not response.has_header('location') and
                        response.status_code != status_code
                    )
                else:
                    # If msg is not passed the last condition will be evaluated
                    # always to True so the first 2 will decide the result.
                    login_unsuccessful = (
                        response and response.status_code == status_code
                        and msg in response.content.decode('utf-8')
                    )

                # ideally make this background task, but to keep simple,
                # keeping it inline for now.
                utils.add_login_attempt_to_db(request, not login_unsuccessful,
                                              get_username)

                if utils.check_request(request, login_unsuccessful,
                                       get_username):
                    return response

                return utils.lockout_response(request)

            return response

        return wrapper
    return decorated_login

redpajama_set_name: RedPajamaGithub


Beyond Golf opened its doors in 2009 in effort to bridge the long seasonal gap to Midwest golfers. Throughout the last decade Beyond Golf has evolved into much more through providing a fun atmosphere all year round. In 2012, the first major addition was our kitchen and we haven't looked back since. We've turned a corner of the bar into a fully functioning kitchen whipping up fresh meals to hungry guests seven days a week. From house smoked meats to freshly sliced cheeses and vegetables we have designed our menu to be a unique spin on your not so typical bar food. Our dedication to fresh, quality ingredients led us to adding over a dozen different liquor infusions to our bar menu. Along with all your standard drink choices, our mixologists will build you a unique and flavorful cocktail incorporating these house infusions. Also, with a strong emphasis on local and regional craft beers Beyond Golf added a 24 tap beer system back in 2017. Our goal is to provide our guests with a dozen of our favorite staple beers along with another dozen rotating lines that are carefully selected to keep your life interesting. At Beyond Golf our goal is to constantly improve every aspect of the business to bring you the best place to enjoy a beverage, a meal, and/or a round of indoor golf.
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Lewis Carroll sent a great message in a passage he wrote: "If you don't know where you are going, any road will take you there" and it couldn't be a more accurate description for those who are keeping their career on auto-pilot. Overseeing your career is managing it, choosing the direction you want to go rather than coasting through your work life without much time spent directing your path.
People who are fresh in the job market, often send a mixed signal when talking about their career focus. Some will describe their next steps as simply looking for a new job because any job will do, while others are looking for a job within the same industry and the same title.
When taking a closer look, both groups for the first time might be faced with having to manage their career, something they never spent that much time thinking about until a job search forced them too.
An auto-pilot career can feel comfortable and less stressful at the time, compared to those who have planned career goals to achieve. I mean think about this, someone in your family's history probably had a long-term career where they went to work every day, did a great job and in turn their employer laid out their career map for them. A comfy employer relationship that over time planned out their career without much direction on their part.
Today's marketplace is built on change and adjustments, companies must adapt or risk losing business. Your career is the same way, it's built on change with you keeping a watchful eye and adapting to new opportunities.
Placing your career in the hands of someone else is a red danger flag blowing in the wind.
The good news, you can take your career off auto-pilot anytime by recognizing a need for change.
There are three prominent signs that your career path is in danger of drifting with the first sign resembling a dread. It often starts with a gut-feeling of dread that happens on Sunday night. The night before you start the workweek, where you dread going to work, it's often referred to as the Monday' morning blues.
The second sign happens when people ask you about your career interests and your mind goes blank. If you can't manage to talk about what you like and want to continue to do in the future – it's a sign you have not given your career much thought.
The third sign occurs during talks with your manager or performance reviews where you are asked about your career goals and all you can come up with is doing the same thing but adding some new words to make it sound different.
There are more than three signs that your career could be on falling into an auto-pilot state of mind but the ones mentioned desire your attention.
It's never too late to start taking back your career, the most important step is identifying your goals and staying focused. Start describing your ideal job, what would it look like? Your answer is a good clue, if it's a fuzzy reply without much detail then a career map would benefit you.
Career mapping gives you a snapshot of where you have been and where you want to go in the future. Begin by writing down where you are right now and where you want your career to go. Identify your various job roles, the skills you have enjoyed the most and what you want to do more of in the future.
Look at your career in terms of both vertical and horizontal experiences, an internal promotion within your employer from manager to director could be a vertical move. A horizontal move could be the same position with a different employer. Both are great options to consider when setting the direction of your career.
Taking your career off auto-pilot could indicate you need to learn new skills, obtained specific certifications or ask to take on different projects. Any action toward a career goal based on your plan is a great step in the right direction.
What are some of the things you did to take charge of your career?


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text:
Simone and Her Books
Books, Reading, and everything in between
Book bloggers and friends
The House of Broken Angels by Luis Alberto Urrea
May 9, 2018 May 29, 2018 by Simone
The American Dream. Many people talk about it, but no one really knows what it means. Some people work their entire lives to try and achieve it and some people succeed. Sadly, sometimes it takes death to figure out that not only you achieved the American Dream, but excelled beyond it.
Here's more about the book
In his final days, beloved and ailing patriarch Miguel Angel De La Cruz, known affectionately as Big Angel, has summoned his entire clan for one last legendary birthday party. But as the party approaches, his mother, nearly one hundred, dies herself, leading to a farewell doubleheader.
Across one bittersweet weekend in their San Diego neighborhood, the revelers mingle among the palm trees and cacti, celebrating the lives of Big Angel and his mother, and recounting the many tales that have passed into family lore, the acts both ordinary and heroic that brought them to a fraught and sublime country and allowed them to flourish in the land they have come to call home. The story of the De La Cruzes is the American story. This indelible portrait of a complex family reminds us of what it means to be the first generation and to live two lives across one border. Teeming with brilliance and humor, authentic at every turn, The House of Broken Angels is Luis Alberto Urrea at his best, and it cements his reputation as a storyteller of the first rank.
The novel was bookended with a death and another death. The first death was Mama America, Big Angel's mother where the family gathers in San Diego to mourn. The second death is Big Angel's own death. However, his death isn't a significant part of this book. The most significant part of this book is Big Angel's birthday party, which happened the day after Mama America's funeral.
I really liked this novel. I didn't think I would at first because novels about the grip of life and what it all means tend to force anxiety on me. But even in the short conversations about death and the meaning of life, there was a sense that this was not about it.
I think it has to do with the fact that it was centered around a huge family gathering with Big Angel thinking about his life and what he's accomplished. The story followed all of Big Angel's family members in three generations. You read the bustling setting of a family gathering while each character has their own mini-story contributed to the dynamic of the family. If you've ever read This is Where I Leave You by Jonathan Tropper, then you'll definitely get those vibes from this book.
Each family member had a different trait and like many families, they represent one or another diverse issue. There were the women that did everything for their husbands and made sure to take care of the entire family. There were the male characters who dabbled in drugs and gang violence. There was even a character who was only half-Mexican and you get to hear the kind of discrimination he comes across not only from his peers but from his family. There's a little something for everyone here.
I don't know, I feel like I saw a lot of my grandfather in Big Angel. My grandfather was definitely my family's patriarch. When my mom and her sisters would argue, he'd be the one mediating the arguments and keeping everyone calm. He was the silent force behind my family and ever since he passed, things have been so different.
Reading about Big Angel's final birthday was like remember my grandfather at one of our last big family gatherings. For some reason, my family is so interconnected. My best friends are my sister and my cousins. My family meet on random Sundays after church just to get lunch and be with each other. We are each other's rocks and we're always chatting with each other. Only a few people dared to even leave the Tri-State area and their loss has been so hard on my grandmother.
Similarly to Big Angel, my grandfather ran a coffee shop, a deli, a donut shop, and so many other small businesses to help his family stay afloat. It was exactly how Big Angel started out while he was learning computer science. He had jobs here and there, but this wasn't the end all. He wanted more and he craved more and just like my grandfather, he got more. When my grandparents got married, they didn't even exchange rings and when my grandfather could finally afford it, he bought my grandmother the most extravagant gifts. For example, my grandmother threw away a vintage Chanel quilted purse because it was too old. We still chastise her for throwing away something most women covet.
Also, I love Big Angel because of his push to be more than just an illegal immigrant from Mexico. I always tell myself that someone will hire me for a job because of my skillset and not because I match some quota. Big Angel did exactly that not only defying the stereotypes behind Mexican workers, but also assuring me and many other first and second generation Americans that with hard work you can still achieve your dreams.
He was such a proud character and you can tell that by the way he prides in his house. The descriptions weren't the best and the house wasn't in the nicest neighborhood, but Big Angel was proud to say he owned a house on American soil. I mean, that has got to be the American Dream right there.
Sadly the really confusing part was all the folks and even sometimes the writing style. Overall, I loved the way this book was written. It felt like Urrea inhabited each character as he described the scenes of Big Angel's birthday party, but it wasn't as streamlined as I would have preferred. One minute you're seeing things through the eyes of Little Angel and then the next you're reading from Paz's point of view, and then Lalo, and then Minnie. It was kind of confusing, but once you got a swing of things you can see how the multiple narratives moved with the same stride as the massive birthday party being held.
Little, Brown (March 6, 2018)
Buy The House of Broken Angels on Amazon
I received a copy of this book from Little, Brown for free in exchange for an honest review. My opinions have not been influenced by the publisher or the author.
Simone and Her Books is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com. This in no way affects my opinion of the above book.
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redpajama_set_name: RedPajamaCommonCrawl


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13:35 – 14:15 Personal Data Protection vs Blockchain Regulation: Can they co-exist or are they mutually exclusive?
14:45 – 15:25 How Traditional Venture Capital is being Disrupted by Blockchain?
16:25 - 17:05 From a Global Perspective: Where is the future of blockchain and why?
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text:
\section{Introduction}
\label{sec:level1}
One of the most exciting subject in theoretical nuclear physics is the
double beta decay, especially due to the neutrino-less ($0\nu\beta\beta$)
process
\cite{Hax,Fas1,Fas2,Suh}. Indeed, its discovery would answer a
fundamental question whether
neutrino is a Majorana or a Dirac particle. The theories devoted to the
description of this process suffer of the lack of reliable tests for the
nuclear matrix elements. O possibility to overcome such difficulties would
be to use the matrix elements which describe realistically the rate of
$2\nu\beta\beta$ decay. In this context many theoretical work have been
focussed on $2\nu\beta\beta$ process. Most formalisms are based on the
proton-neutron quasiparticle random phase approximation (pnQRPA)
which includes the particle-particle ($pp$) channel in the two body interaction.
Since such an interaction is not considered in the mean field equations the
approach fails at a critical value of the interaction strength, $g_{pp}$.
Before this value is reached, the Gamow-Teller transition amplitude
($M_{GT}$) is decreasing rapidly and after a short interval is
becoming equal to
zero. The experimental data for this amplitude is reached for a value of
$g_{pp}$ close to that one which vanishes $M_{GT}$ and also close to
the critical value.
Along the time, the instability of the pnQRPA ground state was considered
in different approaches. The first formalism devoted to this feature
includes anharmonicities through the boson expansion technique
\cite{Rad1,Rad2,Suh1,Grif}.
Another method is the renormalized pnQRPA procedure (pnQRRPA)
\cite{Toi} which keeps
the harmonic picture but the actual boson is renormalized by effects coming
from the terms of the commutators algebra, which are not taken into
account in the standard pnQRPA approach.

In a previous paper\cite{Rad3} we have proved that the pnQRRPA procedure does not
include the additional effects in an consistent way. Indeed, if the
commutators of two quasiparticle operators involves the average of
monopole terms then these terms should be considered also in the
commutators of the scattering terms. If one does so, new degrees of freedom
are switched on and a new pnQRPA boson can be defined. This contains,
besides the standard two quasiparticle operators, the proton-neutron
quasiparticles scattering terms. If the amplitude of the scattering term
is dominant comparing it to the  other amplitudes, the
pnQRRPA phonon describes a new nuclear state.

The aim of this paper is to show that such a mode appears in a natural way
within a time dependent treatment. The present approach points out new
properties of the new proton-neutron collective mode. We use a schematic
many-body Hamiltonian which for a single j-shell is exactly solvable.
In this way the approximations might be judged by comparing the predictions of the actual model with the
corresponding exact results. Since the semi-classical treatment is the proper way
to determine the mean field, one expects that the present approach
is suitable to account for ground state correlations in a consistent way
 and therefore some of the
drawbacks mentioned in a previous publication \cite{Rad3}, like the breaking down
of the fully renormalized RPA before the standard RPA breaks down, are removed.
To understand better the virtues of the present model we compare
its predictions with the results obtained in a renormalized RPA approach and
a boson expansion formalism. Since the semi-classical methods have, sometimes,
intuitive grounds we aim at obtaining a clear interpretation for the new
proton-neutron mode.  It is well known that the breaking down of the RPA
approach is associated to a phase transition. In this respect the semi-classical
formalism is a suitable framework to define the nuclear phases which are bridged
by the Goldstone mode. Above arguments justify our option for a semi-classical treatment
and also sketch a set of expectations.

This project is achieved according to the following plan:
In Section II, we describe the model Hamiltonian. The main features of the
fully renormalized RPA approach, presented in a previous paper, are
briefly reviewed. A time dependent variational principle is formulated
in connection with a truncated quasiparticle Hamiltonian, in Section III. This Hamiltonian
is the term of the model Hamiltonian which determines the equations of
motion for the quasiparticle proton-neutron scattering terms, in the
de-coupling regime. The classical equations of motion
and their solutions are presented in Section IV.
The new $pn$ collective mode is
alternatively described through the renormalized RPA approach and boson expansion formalism
in Section V.
Numerical results are
analyzed in Section VI while the final conclusions are given in Section VII.
\section{The model Hamiltonian. Brief review of frn-RPA }
\label{sec:level2}
Since we are not going to describe realistically some experimental data
but to stress on some specific features of a heterogeneous many
nucleon system with proton-neutron interaction, we consider a schematic
Hamiltonian which is very often \cite{Sam,Rad4} used to study the
single and double beta Fermi transitions:

\begin{eqnarray}
H & = &
\sum_{jm}{(\varepsilon_{pj}-\lambda_{p})c^{\dag}_{pjm}c_{pjm}}
+\sum_{jm}{(\varepsilon_{nj}-\lambda_{n})c^{\dag}_{njm}c_{njm}}\nonumber\\
&&-\frac{G_{p}}{4}\sum_{jm,j'm'}{c^{\dag}_{pjm}
c^{\dag}_{\widetilde{pjm}}c_{\widetilde{pj'm'}}c_{pj'm'}}
-\frac{G_{n}}{4}\sum_{jm,j'm'}{c^{\dag}_{njm}c^{\dag}_{\widetilde{njm}}
c_{\widetilde{nj'm'}}c_{nj'm'}}\nonumber\\
&&+\chi\sum_{jm,j'm'}{c^{\dag}_{pjm}c_{njm}c^{\dag}_{nj'm'}c_{pj'm'}}
-\chi_{1}\sum_{jm,j'm'}{c^{\dag}_{pjm}c^{\dag}_{\widetilde{njm}}
c_{\widetilde{nj'm'}}c_{pj'm'}}.
\end{eqnarray}
$c^{\dag}_{\tau jm}(c_{\tau jm})$ denotes the creation (annihilation) of a
$\tau(=p,n)$ nucleon in a spherical shell model state $|\tau;nljm\rangle
=|\tau jm\rangle$ with $\tau$ taking the values $p$ for protons and $n$
for neutrons, respectively. The time reversed state corresponding to
$|\tau jm\rangle$ is $|\tau{\widetilde{jm}}\rangle=(-)^{j-m}|\tau j-m\rangle$

For what follows it is useful to introduce the quasiparticle ($qp$)
representation, defined by the Bogoliubov-Valatin (BV) transformation:
\begin{eqnarray}
a^{\dag}_{pjm}& = & U_{pj}c^{\dag}_{pjm}-V_{pj}c_{\widetilde{pjm}},\;\;
a_{pjm} =  U_{pj}c_{pjm}-V^{*}_{pj}c^{\dag}_{\widetilde{pjm}}, \nonumber\\
a^{\dag}_{njm}& = & U_{nj}c^{\dag}_{njm}-V_{nj}c_{\widetilde{njm}},\;\;
a_{njm} = U_{nj}c_{njm}-V^{*}_{nj}c^{\dag}_{\widetilde{njm}} .
\end{eqnarray}
which quasi-diagonalizes the first four terms, i.e in the new
representation they are replaced by a set of independent quasiparticles of
energies:
\begin{equation}
E_{\tau}=\sqrt{(\epsilon_{\tau}-\lambda_{\tau})^{2}+\Delta_{\tau}^{2}}.
\end{equation}
In the new $qp$ representation, the model Hamiltonian, denoted by $H_q$,
describes a set of
independent quasiparticles, interacting among themselves through a two body
interaction determined by the images of the $\chi$ and $\chi_1$ terms
through the BV transformation.

Various many-body approaches have been tested by using not the $qp$ image
of $H$ but another Hamiltonian derived from $H$ by ignoring the
scattering $qp$ terms:
\begin{eqnarray}
B^{\dag}(jpn)&=&\sum_{m}a^{\dag}_{pjm}a_{njm}, \nonumber\\
B(jpn)& = &\sum_{m}a^{\dag}_{njm}a_{pjm}.
\end{eqnarray}
and restricting the space of single particle states to a single $j$-state.
Thus, the model Hamiltonian contains, besides the terms for the $qp$
independent motion, a two body term which is quadratic in the two
quasiparticle operators $A^{\dag}, A$:
\begin{equation}
A^{\dag}(jpn)=\sum_{m}a^{\dag}_{pjm}a^{\dag}_{\widetilde{njm}},
\;\;A(jpn)=(A^{\dag}(jpn))^{\dag}.
\end{equation}

In a previous publication\cite{Rad3}, we showed that going beyond the quasiparticle
random phase
approximation (pnQRPA) through a renormalization procedure, a new degree of
freedom is switched on, which results in having a renormalized pnQRPA boson
operator as a superposition of the operators $A^{\dag}(jpn), A(jpn)$ and
scattering terms $B^{\dag}(jpn),B(jpn)$. This picture differs from the
standard $pnQRRPA$ approach, where the boson operators involve only the
 operators $A^{\dag}$ and $A$, and is conventionally called as fully
renormalized RPA ($frn-RPA$). Obviously, when the amplitudes of
scattering terms are dominant, one deals with a new kind of collective $pn$
excitation.

In order to define clearly the distinct features of the new proton-neutron
($pn$) mode revealed in the present paper a brief description of the
results obtained in a previous publication \cite{Rad3} is necessary.
The equations of motion associated to the many-body Hamiltonian, written in
terms of quasiparticle operators, are determined by the commutators algebra
of the two quasiparticle ($A^{\dag}, A$) and scattering $(B^{\dag},B)$
operators defined by eqs. (2.5) and (2.4) respectively.
Within the $frn-RPA$, the exact commutators are approximated as follows:
\begin{eqnarray}
\left[A(jpn),A^{\dag}(jpn)\right ] &=& C^{(1)}_{jpn},
\nonumber \\
\left[B(jpn),B^{\dag}(jpn)\right ] &=& C^{(2)}_{jpn},
\nonumber\\
\left[A(jpn),B^{\dag}(jpn)\right ] &=&\left[A(jpn),B(jpn)\right ] = 0.
\end{eqnarray}

The terms $C^{(1)}_{jpn}, C^{(2)}_{jpn}$ appearing in the r.h.s. of the
above equations are the averages of the corresponding exact commutators, on
the correlated ground state $|0>$:
\begin{equation}
C^{(1)}_{jpn}=\langle0|1-\hat{N}_{jn}-\hat{N}_{jp}|0\rangle,\: \:
C^{(2)}_{jpn}=\langle0|\hat{N}_{jn}-\hat{N}_{jp}|0\rangle.
\end{equation}
with $\hat{N}_{j\tau}$ standing for the $\tau$ (=p,n) quasiparticle number
operator in the shell j.
The normalized operators
\begin{eqnarray}
\bar{A}^{\dag}(jpn)=\frac{1}{\sqrt{C^{(1)}_{jpn}}}A^{\dag}(jpn),\:
\bar{A}(jpn)=\left(\bar{A}^{\dag}(jpn)\right)^{\dag},
\nonumber\\
\bar{B}^{\dag}(jpn)=\frac{1}{\sqrt{|C^{(2)}_{jpn}|}}B^{\dag}(jpn),\:
\bar{B}(jpn)=\left(\bar{B}^{\dag}(jpn)\right)^{\dag},
\end{eqnarray}
satisfy bosonic commutation relations and thereby their equations of
motion are linear:
\begin{equation}
\left[H_q,\left(\matrix{\bar{A}^{\dag}(jpn) \cr
                        \bar{A}(jpn)        \cr
                         \bar{B}^{\dag}(jpn) \cr
                        \bar{A}(jpn)}\right)\right]
=\sum_{j,j^{\prime}}T^{j,j^{\prime}}\left(\matrix{\bar{A}^{\dag}(jpn) \cr
                        \bar{A}(jpn)        \cr
                         \bar{B}^{\dag}(jpn) \cr
                        \bar{B}(jpn)}\right).
\end{equation}
The matrix $T^{j,j^{\prime}}$ depends on the U and V coefficients as well as
on the strengths $\chi, \chi_1$ of the two body interactions.
The $frn-RPA$ approach defines  a linear combination of the basic operators
$\bar{A}^{\dag}(jpn), \bar{A}(jpn), \bar{B}^{\dag}(jpn), \bar{A}(jpn)$,
\begin{equation}
\Gamma^{\dag}=\sum_{j}\left[X(j)\bar{A}^{\dag}(jpn)+Z(j)D^{\dag}(jpn)
-Y(j)\bar{A}(jpn)-W(j)D(jpn)\right],
\end{equation}
so that the following commutation relations with its
hermitian conjugate operator and the model Hamiltonian hold:
\begin{eqnarray}
\left[\Gamma,\Gamma^{\dag}\right] &=& 1,
\\
 \left[H_q,\Gamma^{\dag}\right] &=& \omega \Gamma^{\dag}.
\end{eqnarray}
The operators $D^{\dag}(jpn)$ are identical with ${\bar{B}}^{\dag}(jpn)$
or $\bar{B}(jpn)$ depending on whether the sign of $C^{(2)}_{jpn}$ is plus or
minus. The equation (2.12) provides a set of homogeneous equations- called
the $frn-RPA$ equations- for the amplitudes $X,Y,Z,W$:
\begin{eqnarray}
\left(\matrix{{\cal A} & {\cal B}\cr
              -{\cal B}  &-{\cal A}}\right)\left(\matrix{X\cr Z\cr Y\cr
W}\right) =\omega \left(\matrix{X\cr Z\cr Y\cr W}\right),
\end{eqnarray}
while the equation (2.11) yields the normalization equation
\begin{equation}
\sum_{j}(X^2(j)+Z^2(j)-Y^2(j)-W^2(j))=1.
\end{equation}
The $frn-RPA$ matrices depend on the renormalization constants $C^{(1)},
C^{(2)}$ which, at their turn, depend on the phonon amplitudes. Therefore,
the equations (2.12) and (2.7) should be self-consistently solved.

In ref. \cite{Rad3} the $frn-RPA$ equations have been solved both for a proton-neutron
dipole-dipole interaction, needed for the description of the
double beta Gamow-Teller decay and for a proton-neutron monopole-monopole interaction
used in the calculation of the rates of the double beta Fermi decay.
Equations obtained in the two cases have some common features which,
for what follows, are worth being enumerated.

\noindent
1) The dimension of the $frn-RPA$ matrix is twice as large as that of the
standard RPA and consequently new solutions show up.

\noindent
2) The solutions characterized by that the largest phonon amplitude is of
type Z define a new class of proton-neutron excitations.

\noindent
3) Due to the attractive character of the two body interaction in the
particle-particle ($pp$) channel, the lowest new state has an energy which
is smaller than the minimal absolute value of the relative energy of the
proton and neutron quasiparticle partner states, related by the operators
$B^{\dag}(jpn), B(jpn)$.

\noindent
4) For the N=Z nuclei, this minimal value is vanishing and therefore the
lowest mode becomes spurious or in other words saying a new symmetry
is open. The new symmetry corresponds to the restriction
$C^{(2)}_{jpn}=0$, i. e. the average of the third component of the
isospin operator is vanishing.
This means that the system is invariant to rotations around any axes in
the ($X,Y$) plane of the isospin space associated to the (jpn) orbits.

\noindent
5) Important quantitative effects are expected for heavy nuclei having the
proton and neutron Fermi energies lying far apart from each other.

\noindent
6) The presence of the additional states influences also the structure of
the states lying close to those predicted by the standard RPA. Indeed, the
actual normalization condition for the phonon amplitudes implies new
values for the X and Y weights. Consequently, the strengths for $\beta^-$
and $\beta^+$ transitions are shared by the "old"-lying close to the
standard RPA states- and the ``new'' states- for which the amplitudes Z
are dominant.

\noindent
7) The standard RPA approach is based on the quasi-boson approximation and
therefore it ignores some important dynamic effects (only the terms
$A^{\dag}A^{\dag}, A^{\dag}A, AA$ are considered in an approximative
manner) and moreover
the Pauli principle is violated.
By contrast, within the $frn-RPA$ all the terms of the model Hamiltonian are
taken into account. Also the Pauli principle is, to a certain extent,
restored. Due to this feature, large corrections to the double beta
transition amplitude as well as to the Ikeda sum rule are expected by
changing the RPA to the frn-RPA.

\noindent
8) The equations of motion for the $A^{\dag },A$ and $B^{\dag}, B$
operators are coupled by the terms $A^{\dag}B^{\dag}, A^{\dag}B,
AB^{\dag}, AB$ involved in the quasiparticle Hamiltonian. These terms are
multiplied by the factors $ U_pV_nU_{p^{\prime}}U_{n^{\prime}},
 U_pV_nV_{p^{\prime}}V_{n^{\prime}},  V_pU_nU_{p^{\prime}}U_{n^{\prime}},
 V_pU_nV_{p^{\prime}}V_{n^{\prime}}$ in the $ph-ph$ interaction (the
$\chi$ term)
and by  $U_pU_nU_{p^{\prime}}V_{n^{\prime}},
U_pU_nV_{p^{\prime}}U_{n^{\prime}},  V_pV_nU_{p^{\prime}}V_{n^{\prime}},
V_pV_nV_{p^{\prime}}U_{n^{\prime}}$ in the $pp-hh$ interaction (the $\chi_1$
term), respectively. Note that the coupling terms change the number of
either proton or neutron quasiparticles by two units. The terms
bringing the main contribution to the equations of motion for the
operators $A^{\dag}, A$ commute with $ \hat{N}_{jp}-\hat{N}_{jn}$
but not with $ \hat{N}_{jp}+\hat{N}_{jn}$. By contrary the terms
having the dominant contribution to the equations of motion for the
operators $B^{\dag},B$ commute with $ \hat{N}_{jp}+\hat{N}_{jn}$
and not with $ \hat{N}_{jp}-\hat{N}_{jn}$. None of the two operators,
$ \hat{N}_{jp}-\hat{N}_{jn}$, $ \hat{N}_{jp}+\hat{N}_{jn}$, commutes with
the coupling terms.
Retaining from the $\chi$-interaction the $pp-hh$ terms (those multiplied
by $U_pU_nV_{p^{\prime}}V_{n^{\prime}}$) and from the $\chi_1$ interaction
only the $ph-ph$ terms (those proportional to
$U_pV_nU_{p^{\prime}}V_{n^\prime} $ ) the equations of motion  for the
operators $B^{\dag},B$ are decoupled from those for $A^{\dag}$ and $A$.
One may conclude that the new mode is determined by a combined effect
coming from the $pp-hh$ and $ph-ph$ terms belonging to the $\chi$ and
$\chi_1$ interactions, respectively.

\noindent
9) In the particle representation the $frn-RPA$ phonon operator is a linear
superposition of $ph, hp, pp$ and $hh$ operators.

\noindent
10) In the limit of large $pp$ and negligible $ph$ interactions, the
amplitudes $Z$ can be analytically calculated. The result is that $Z$ is
proportional to either $U_pV_n$ or $V_pU_n$, depending on whether the sign
of $E_p-E_n$ is plus or minus, respectively. When this amplitude prevails
over the other ones, the corresponding mode describes a neutron-hole
proton-particle (or a proton-hole neutron-particle) excitation of the
mother nucleus $(N,Z)$. Therefore the state is associated to the $(N-1,Z+1)$
(or to the $(N+1,Z-1)$) nucleus. In this case the state might be reached by exciting the ground state
through the transition operator $c^{\dag}_pc_n$ (or $c^{\dag}_nc_p$), which is typical for the
$\beta^-$ ( or $\beta^+$) decay. Since the double beta decay is conceived  as taking place through two
successive $\beta^-$ transitions, one expects that this process is also influenced by considering
this new state as an intermediate state characterizing the odd-odd neighboring nucleus.

\noindent
11) When the $pp$ interaction is small the amplitude Z is proportional to
$U_pU_n$ if $E_p>E_n$ or to $V_pV_n$ in the case $E_p<E_n$.
The new mode characterizes the nucleus $(N+1,Z+1)$ in the first case and
the nucleus ($N-1,Z-1$) in the second situation. The transition operators
 which could excite
these states are obviously of the types $c^{\dag}_pc^{\dag}_n$ and $c_nc_p$, respectively.

Note that the restriction of the phonon operator to the scattering terms
resembles the standard RPA boson operator written in the particle
representation. This comparison has, however, only a formal value since in
the quasiparticle representation there is no Fermi energy and therefore
one cannot speak about quasiparticle-quasihole excitations. Similar
features are met in solid state physics for the description of electron
excitations in narrow energy bands, spin waves and
plasma oscillations \cite{Hub}. In
nuclear physics, the scattering terms have been also considered but not
for proton-neutron excitations. Indeed, using the thermal response theory,
Tanabe \cite{Tan} studied the charge conserving phonons in nuclear systems
at a finite temperature. It seems that the contribution of the scattering
terms to the charge conserving bosons, does not survive at vanishing
temperature \cite{Hat}. Moreover, the
dispersion relation for the mode energy cannot be obtained from a
linearized set of equations as it is required by the spirit of the RPA
approach.

\section{Semi-classical treatment}
\label{sec:level3}

As we already mentioned, the scope of the present paper is to study the
$pn$ mode caused by the quasiparticle scattering terms within a
semi-classical approach. In this formalism the renormalization condition
(2.6) is missing and therefore the harmonic motion of the new degrees of
freedom hinges on a more physical ground.
Moreover, we address the question whether this mode survives when the
non-scattering terms are switched off. Thus, it is worth to know if such a
mode appears only when the scattering terms accompany the two
quasipatricle operators or it might be determined by the scattering terms
alone.

From the brief presentation of the previous Section it is clear that the
mode does not appear within the RPA approach. Indeed, it occurred within
the $frn-RPA$ after a consistent renormalization was performed (i. e. not
only the operators $A^{\dag}, A$ where renormalized but also $B^{\dag}$
and $B$). If that mode is a signature of the higher RPA formalisms, then it
should also appear within the semi-classical formalism as well as in the
boson expansion framework. As we shall see the semi-classical approach is
able to predict the mode even in the harmonic approximation, the mode
being associated with the small oscillations of the system around a static
correlated ground state. Moreover, the semi-classical frame is expected to
allow us an intuitive interpretation of this new type of excitation.

We recall that the higher order corrections to the standard RPA approach are
frequently studied, with different purposes, using a single j case and
ignoring the scattering terms. The procedure has the advantage that the
resulting Hamiltonian is exactly solvable. Therefore the quality of
the adopted approximations may be tested by comparing the predictions
with the corresponding exact results.

 To touch the goal of the present paper
we adopt a similar point of view.
Indeed, if the coupling terms (mentioned at the point 8 of the previous
section) are ignored, the equations of motion for the scattering operators are
decoupled. Moreover the motion of these operators is determined also by an
exactly solvable Hamiltonian, which reads:

\begin{equation}
H^{(q)}_{pn}=E^{'}_{p}{\hat N}_{p}+E^{'}_{n}{\hat N}_{n}+\lambda_{1}
B^{\dag}(pn)B(pn)+\lambda_{2}\big(B^{\dag2}(pn)+B^{2}(pn)\big),
\end{equation}
where the following notations have been used:
\begin{eqnarray}
E_p^{\prime}&=&E_p+(\chi_1-\chi)V_p^2,\nonumber\\
E_n^{\prime}&=&E_n+(\chi+\chi_1)V^2_p(V_n^2-U_n^2),\nonumber\\
\lambda_1&=&\chi(U_p^2U_n^2+V_p^2V_n^2)-\chi_1(U_p^2V_n^2+V_p^2U_n^2),
\nonumber\\
\lambda_2&=&-(\chi+\chi_1)U_pU_nV_pV_n,\\
{\hat N}_{\tau}&=&\sum_{m}a^{\dag}_{\tau jm}a_{\tau jm}.
\end{eqnarray}
Also, to simplify the notation we omitted the quantum number $j$ for
the operators $B^{\dag}(jpn), B(jpn)$ as well as for the $U, V$
coefficients and quasiparticle energies.
This model Hamiltonian will be studied within a time dependent variational
formalism. Therefore, some static and dynamic properties will be described
by solving the equations provided by the time dependent variational
principle (TDVP)\footnote{Throughout this paper the units of $\hbar=1$ are used}:

\begin{equation}
\delta \int_0^t {\langle\Psi|H^{(q)}_{pn}-i\frac{\partial}{\partial t'}|\Psi\rangle}
\,dt' =0.
\end{equation}
If the variational state $|\Psi\rangle$ spans the whole Hilbert space
describing the many-body system, solving the equation (3.4) is equivalent
to solving the time dependent Schroedinger equation, which would be a very
difficult task. In the present paper,
the trial function is taken as:
\begin{equation}
|\Psi\rangle = exp[zB^{\dag}(pn)-z^{*}B(pn)]|NT\; -T\rangle,
\end{equation}
where $|NTT_{3}\rangle$ denotes the common eigenfunction of the quasiparticle
total number (${\hat N}$), the quasiparticle isospin squared
(${\hat T}^{2})$, and its
z-axis projection ($T_{z}$) operators, respectively.
$z$ is a complex function of time and $z^*$, the corresponding complex conjugate function.
We justify this choice by the symmetry properties of the
model Hamiltonian. Indeed, let us note first that $H^{(q)}_{pn}$ commutes
with the
quasiparticle total number operator. Moreover, it can be written in terms
of the quasiparticle total number operator and generators of the SU(2) isospin algebra

\begin{eqnarray}
\tau_{+1} & = & -\frac{1}{\sqrt 2}B^{\dag}(pn),\nonumber\\
\tau_{-1} & = & \frac{1}{\sqrt 2}B(pn),\nonumber\\
\tau_{0} & = & \frac{1}{2}(\hat N_{p}-\hat N_{n}).
\end{eqnarray}
Due to this property of $H^{(q)}_{pn}$, the function
$|\Psi\rangle$, which is a coherent state for the SU(2) group, is the
most suitable for a semi-classical treatment.

Before closing this section we would like to write the trial function
in a form which suits better the further purposes. Using the Cambel Hausdorff
factorization\cite{Kir} for the exponential function, as explained in Appendix A,
one obtains:
\begin{eqnarray}
|\Psi\rangle & = & {\cal N} e^{\alpha B^{\dag}(pn)}|NT\;-T\rangle,
\nonumber\\
{\cal N}& = &(1+\alpha^{*}\alpha)^{-T}.
\end{eqnarray}
where $\alpha$ depends on the polar coordinates, $z=\rho e^{i\varphi}$:
\begin{equation}
\alpha=\tan(\rho)e^{i\varphi}.
\end{equation}
\section{Equations of motion}
\label{sec:level4}
In order to write the equations of motion provided by the TDVP (3.4),
we need the matrix element of $H^{(q)}_{pn}$ as well as of the time derivative operator,
$\frac{\partial}{\partial t}$. These can be evaluated by direct calculation, using the
expressions (3.5) when the average of $H^{(q)}_{pn}$ is considered and (3.7) for the
classical action. The result is:
\begin{eqnarray}
\langle\Psi|H^{(q)}_{pn}|\Psi\rangle& =& -T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})
+\frac{N}{2}(E_{p}^{\prime}+E_{n}^{\prime})+2T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})
\frac{\alpha^{*}\alpha}{1+\alpha^{*}\alpha}\nonumber\\
&&+2T(2T-1)\left[\lambda_{1}\frac{\alpha^{*}\alpha}
{(1+\alpha^{*}\alpha)^{2}}+\lambda_{2}\frac{\alpha^{*2}+
\alpha^{2}}{(1+\alpha^{*}\alpha)^{2}}\right],
\nonumber\\
\langle\Psi|\frac{\partial}{\partial t}|\Psi \rangle &=&
T\frac{\alpha^{*}\stackrel{\bullet}{\alpha}-\stackrel{\bullet}{\alpha}^{*}
\alpha}{1+\alpha^{*}\alpha}.
\end{eqnarray}
Considering $\alpha,\alpha^*$ as classical phase space coordinates, the
TDVP equation (3.4) yields the following classical equations of motion,
describing the nuclear system:

\begin{eqnarray}
\frac{\partial{\cal H}}{\partial\alpha}&=&-2i
\frac{T\stackrel{\bullet}{\alpha}^{*}}{(1+\alpha^{*}\alpha)^{2}}, \nonumber\\
\frac{\partial{\cal H}}{\partial \alpha^{*}}&=&2i
\frac{T\stackrel{\bullet}{\alpha}}{(1+\alpha^{*}\alpha)^{2}}.
\end{eqnarray}
Here ${\cal H}$ denotes the classical energy function:
\begin{equation}
{\cal H}=\langle\Psi|H^{(q)}_{pn}|\Psi\rangle.
\end{equation}
In order to quantize the classical trajectories satisfying the equations
(4.2) as well as to have an one to one correspondence between the classical
and quantal behaviors of the nucleon system, it is convenient to chose
those conjugate variables which bring the equations of motion in a
canonical Hamilton form. A possible choice of the coordinates with the
above mentioned property is
\begin{eqnarray}
r&=&\frac{2T}{1+\alpha^{*}\alpha},\\
\psi&=&-\frac{1}{2i}(\ln \alpha-\ln \alpha^{*})=-\varphi.
\end{eqnarray}
Indeed, in the new variables the classical equations read:
\begin{eqnarray}
\frac{\partial{\cal H}}{\partial r}&=&-\stackrel{\bullet}{\psi},\nonumber\\
\frac{\partial{\cal H}}{\partial \psi}&=&\stackrel{\bullet}{r}.
\end{eqnarray}
with the classical energy:
\begin{eqnarray}
{\cal H}&=&T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})+\frac{N}{2}
(E_{p}^{\prime}+E_{n}^{\prime})
\nonumber\\
&&-(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})r+\frac{2T-1}{2T}r(2T-r)
(\lambda_{1}+2\lambda_{2}\cos2\psi).
\end{eqnarray}
Note that $r$ has the significance of a generalized coordinate while $\psi$
that of generalized linear momentum.
Due to the generalized momentum $\psi $, the equations motion are not
linear and therefore analytical solutions are not obtainable.
The equations can however be approximatively solved if they are linearized around the
minimum point of the energy function:
\begin{equation}
\stackrel{\circ}{r}=T\left[1-\frac{E_{p}^{\prime}-E_{n}^{\prime}+
2\lambda_{1}}{(2T-1)(\lambda_{1}-2\lambda_{2})}\right],
\;\;
\stackrel{\circ}{\psi}=\frac{\pi}{2}.
\end{equation}
In order that the minimum exits, it is necessary that the generalized
coordinates satisfy a
consistency condition, required by the definition range of $r$:
\begin{equation}
0\leq\stackrel{\circ}{r}\leq2T.
\end{equation}
By means of (4.8), this provides a constraint for the strengths of the two
body interactions.
The linearized equations, written in terms of the deviations
\begin{equation}
q=r-\stackrel{\circ}{r},\;  p=\psi-\stackrel{\circ}{\psi},
\end{equation}
are of harmonic type:

\begin{eqnarray}
-\stackrel{\bullet}{p}=-2\frac{2T-1}{2T}(\lambda_{1}-2\lambda_{2})q,
\nonumber\\
\stackrel{\bullet}{q}=4\frac{2T-1}{T}\stackrel{\circ}{r}
(2T-\stackrel{\circ}{r})\lambda_{2}p.
\end{eqnarray}

These describe a harmonic motion for the conjugate coordinates, with the
angular frequency:

\begin{equation}
\omega=2\frac{2T-1}{T}[-\lambda_{2}(\lambda_{1}-2\lambda_{2})\stackrel{\circ}
{r}(2T-\stackrel{\circ}{r})]^{\frac{1}{2}}.
\end{equation}
The condition that $\omega$ is a real quantity brings an additional constraint for
the strength parameters $\chi,\chi_1$:

\begin{equation}
\chi\geq\chi_1\left(\frac{U_pV_n-V_pU_n}{U_pU_n+V_pV_n}\right)^2.
\end{equation}
As we said already before, the schematic model has the advantage, over the
realistic formalisms, that allows us to compare the approximative solutions
with the exact one. For the particular Hamiltonian used in the present paper,
the exact eigenvalues can be obtained by diagonalization in the basis
{$|NTM\rangle$}. Indeed, in this basis the model Hamiltonian has the
following non-vanishing matrix elements:
\begin{eqnarray}
\langle NTM|H^{(q)}_{pn}|NTM\rangle &=&\frac{1}{2}(E_p^{\prime}+E_n^{\prime})N
+(E_p^{\prime}-E_n^{\prime})M+
\lambda_1(T+M)(T-M+1),\nonumber\\
\langle NTM+2|H^{(q)}_{pn}|NTM\rangle &=&\lambda_2\left[(T-M-1)(T-M)(T+M+1)(T+M+2)
\right]^{\frac{1}{2}},
\nonumber\\
\langle NTM|H^{(q)}_{pn}|NTM+2\rangle &=&\langle NTM+2|H^{(q)}_{pn}|NTM\rangle.
\end{eqnarray}

\section{The renormalized RPA and boson expansion}
\label{sec:level5}

Within the RPA approach, the renormalization of the quasiparticle mean
field due to the two quasiparticle interactions is usually ignored.
Therefore the Hamiltonian considered is:

\begin{equation}
H_{qp}=E_{p}{\hat N}_{p}+E_{n}{\hat N}_{n}+\lambda_{1}
B^{\dag}(pn)B(pn)+\lambda_{2}\big(B^{\dag2}(pn)+B^{2}(pn)\big),
\end{equation}
The operators $B^{\dag},B$ satisfy the commutation relation:
\begin{equation}
[B(pn),B^{\dag}(pn)]={\hat N}_n-{\hat N}_p.
\end{equation}
If the r.h. side of the above equation is replaced by its
average on the ground state,
\begin{equation}
C=\langle  0|{\hat N}_n-{\hat N}_p|0\rangle
\end{equation}
which is to be determined, then the operators
$B,B^{\dag}$ become bosons, after the following renormalization
\begin{equation}
{\widetilde {B}}^{\dag}(pn)=\frac{1}{\sqrt{C}}B^{\dag}(pn),\;
{\widetilde {B}}(pn)=\frac{1}{\sqrt{C}}B(pn),
\end{equation}
if $C$ is positive, while for negative $C$ the renormalized operators are:
\begin{equation}
{\widetilde {B}}^{\dag}(pn)=\frac{1}{\sqrt{|C|}}B(pn),\;
{\widetilde {B}}(pn)=\frac{1}{\sqrt{|C|}}B^{\dag}(pn).
\end{equation}
Suppose, for the time being, that $C>0$. If that is not the case
the corresponding calculations can be worked out in a similar way.
The equations of motion for the renormalized operators are:
\begin{eqnarray}
\left[H_{qp},{\widetilde {B}}^{\dag}(pn)\right]& = & (E_p-E_n+\lambda_1C)
{\widetilde{B}}^{\dag}(pn) +2\lambda_2C{\widetilde{B}}(pn),
\nonumber \\
\left[ H_{qp},{\widetilde{B}}(pn)\right]& =& -2\lambda_2C{\widetilde{B}}^{\dag}(pn)
-(E_p-E_n+\lambda_1C){\widetilde{B}}(pn) .
\end{eqnarray}
Since the equations are linear in ${\widetilde {B}}^{\dag}(pn)$ and
${\widetilde{B}}(pn)$, one can define the phonon operator
\begin{equation}
\Gamma^{\dag}=X\widetilde {B}^{\dag}(pn)-Y\widetilde {B}(pn),
\end{equation}
with the amplitudes determined such that the following equations are
fulfilled:
\begin{eqnarray}
\left[H_{qp},\Gamma^{\dag}\right]&=&\omega\Gamma^{\dag},\nonumber\\
\left[\Gamma,\Gamma^{\dag}\right]&=&1.
\end{eqnarray}
The first equation provides the dispersion equation for the mode energy
\begin{equation}
\omega=\left[(E_p-E_n+\lambda_1C)^2-4\lambda_2^2C^2\right]^{\frac{1}{2}},
\end{equation}
while the second one the normalization relation for phonon amplitudes:
\begin{equation}
X^2-Y^2=1.
\end{equation}
The renormalized RPA vacuum is defined by
\begin{equation}
\Gamma|0\rangle=0.
\end{equation}
The solution of the above equation is:
\begin{equation}
|0\rangle=e^{-\frac{1}{8}(\frac{Y}{X})^2}e^{\frac{Y}{2X}{\widetilde
{B}}^2} |NT,-T\rangle.
\end{equation}
Then the renormalization constant $C$ can be exactly evaluated:
\begin{equation}
C=2T-2+\frac{2}{X^2}.
\end{equation}
Since $T\ge1$, the constant C is always positive.
The equations of motion allow us to express the amplitude Y in terms of X:
\begin{equation}
Y=\frac{1}{2\lambda C}[\omega-(E_p-E_n+\lambda_1C)]X,
\end{equation}
which together with the normalization condition (5.10) determines fully the
amplitudes X and Y in terms of C and $\omega$. Inserting the result for $X$
into the equation (5.13),
one obtains an equation for C as a function of $\omega$.
This and eq.(5.9) form a set of two nonlinear equations for the unknowns
$\omega$ and $C$.

As we mentioned before, another way to improve the RPA treatment is to use
the boson expansion concept. Through this procedure, the $SU(2)$
algebra, with the fermionic generators $\tau_{\pm 1}, \tau_0$ defined by
eq.(3.6), is mapped to
a boson $SU(2)$ algebra, generated by $\hat {T}_{\pm 1}, \hat {T_0}$.
Denoting by $b^+,b$ a pair of boson operators, the
$SU(2)$ algebra generators $\hat {T}_{\pm 1},\hat {T}_0$ can be constructed
as function of $b^+$ and b. The resulting expressions are conventionally
called as the boson expansion of the fermionic generators, respectively.
There are three distinct boson mappings for the fermionic SU(2) algebra found
by Holstein-Primakoff \cite{Hol}, Dyson \cite{Dy} and one of the present
authors (A. A. R.)\cite{Rad5}, respectively.
For the present purpose here we use the Holstein-Primakoff (HP) expansion:
\begin{eqnarray}
\hat{T}_{+1}&=&-\sqrt{T}b^+\left(1-\frac{b^+b}{2T}\right)^{\frac{1}{2}},
\nonumber\\
\hat{T}_{-1}&=&\sqrt{T}\left(1-\frac{b^+b}{2T}\right)^{\frac{1}{2}}b,
\nonumber\\
\hat{T}_0&=&b^+b-T.
\end{eqnarray}
By a direct calculation it can be checked that, by this mapping, to the
operator $\tau ^2$ it corresponds a C-number:
\begin{equation}
\hat{T}^2=T(T+1).
\end{equation}
The fermion Hamiltonian $H_{qp}$ commutes with the quasiparticle
total number and the same is true for the generators $\tau_{\pm 1},\tau_0$.
Therefore the image of the quasiparticle total number operator through
the  HP mapping is invariant against any rotation in the isospin space and
consequently, according to the above equation, is a C-number.
Apart from an additive constant, the image of $H_{qp}$ through the HP
boson expansion is:
\begin{equation}
H^{(b)}_{qp}=(E_p-E_n)\hat{T}_0-2\lambda_1\hat{T}_{+1}\hat {T}_{-1}
+2\lambda_2(\hat {T}^2_{+1}+\hat {T}^2_{-1}).
\end{equation}
Making use of eqs. (5.15), the boson mapping of $H_{qp}$ is a infinite
series in the bosons $b^+, b$, due to the square root operators.
Expanding the square root operators and truncating the result at the
second order in bosons, the boson Hamiltonian becomes:
\begin{equation}
H^{(b)}_{qp;2}=(E_p-E_n+2\lambda_1T)b^+b+2\lambda_2T({b^+}^2+b^2).
\end{equation}
For a limited range of the interaction strength, this Hamiltonian can be
diagonalized through a canonical transformation:
\begin{eqnarray}
b^+&=&UB^++VB,
\nonumber\\
b&=&UB+VB^+,
\nonumber\\
1&=&U^2-V^2.
\end{eqnarray}
The restriction that the "dangerous" terms have a vanishing strength
yields the expression for the transformation coefficients and the
coefficient, $\omega_1$, of the diagonal term $B^+B$:
\begin{eqnarray}
\left(\matrix{U\cr V}\right)&=&\frac{1}{\sqrt{2}}\left[\mp
1+\frac{|E_p-E_n|}{\sqrt{(E_p-E_n+2\lambda_1T)^2 -16\lambda_2^2T^2}}
\right]^{\frac{1}{2}},
\nonumber\\
\omega_1&=&\left[(E_p-E_n+2\lambda_1T)^2
-16\lambda_2^2T^2\right]^{\frac{1}{2}}.
\end{eqnarray}
Comparing the expressions of $\omega_1$ (5.20) and $\omega$ (5.9),
one sees that the two energies are identical for the limiting case of
$X=1$, which is met when $\lambda_2=0$ (see eqs. (5.9) and (5.14)).

At this stage it is worthwhile to make the following remarks:
a) When the HP boson expansion of the model Hamiltonian is truncated at
the second
order terms in bosons, the quasiparticle total number operator is no
longer a C number. Therefore the contribution of this term should have
been considered in a consistent manner.
Moreover the truncation is justified only for large values of the total
isospin T.
 b) The same inconsistency appears
in the calculation of the renormalization constant C. Indeed the
expression (5.14) is exact and therefore includes all contributions coming
from the infinite boson series of the correlated ground state, given by
(5.14).
c) Since the boson mapping (5.15) is an unitary transformation, the
exact eigenvalues of $H_{qp}$ are reproduced by diagonalizing the boson
expanded Hamiltonian $H^{(b)}_{qp}$.
For the second order truncated Hamiltonian, the canonical transformation
breaks down at a critical value of the attractive interaction strength.
However the diagonalization procedure is able to find the eigenvalues for
any strength of the attractive interaction. The resulting energies exhibits a phase transition
(the first derivative has a jump) at the critical value of the strength.
If the second branch of the energy curve could also be approximated by an
harmonic mode, describing small oscillations of the classical system around
a stationary state, this is still an open question \cite{Rad4}.
d) The HP boson representation provides  for the harmonic
mode the interpretation of an wobbling motion of the system around the
total isospin. e) The HP boson expansion is justified (in the sense that
some eigenvalues of the truncated Hamiltonian are close to the
corresponding exact ones) when the rotation axis in the isospin space is
close to the quantization axis (z axis), which is usually taken as the axis
to which the maximum ``moment of inertia'' corresponds. If the angle between the
rotation axis and z-axis is large the harmonic energy may collapse.
In this case the quantization axis should be chosen as one of the X and Y axes
depending on the magnitude of the strength of the $\tau_x^2$ and
$\tau_y^2$ terms from the quasiparticle Hamiltonian $H^{(q)}_{pn}$.
In this case the boson representation suitable for the low order
description should be of Dyson type \cite{Rad6}. The harmonic approximation
for the new representation describes also a wobbling motion
of a frequency equal to the square root of the product of the inverse of the non-maximal
moments of inertia normalized to the inverse of the maximal moment of inertia.

\section{Numerical results}
\label{sec:level6}
The formalism described in the previous sections, has been applied to the
case $j=\frac{19}{2}$. On the proton level, 6 protons are distributed
while in the neutron level, 14 neutrons. Alike nucleons interact with each
other through pairing forces whose strength are $G_p=0.2 {\rm MeV}$ and $G_n=0.4$
MeV. From the pairing equations it results the following expression for
the quasiparticle energy:
\begin{equation}
E_{\tau}=\frac{1}{2}G_{\tau}\Omega,\; \Omega=\frac{2j+1}{2}.
\end{equation}
With the data specified above the result for the quasiparticle energy is:
\begin{equation}
E_p=2\; {\rm MeV},\;\;E_n=1\;\;{\rm MeV}.
\end{equation}
According to our previous study, the renormalized RPA ground state
involves a small number of quasiparticles. For example, for a small
strength of the particle-particle interaction, the quasiparticle
total number is about 2 while for large values of the above mentioned
strength the number may reach the value 4. Due to this behavior of the
correlated ground state we considered for the isospin carried by the
quasiparticles in the ground state, alternatively the values 1 and 2.
Although these values vary with increasing the particle particle strength
we kept them constant.

The numerical analysis refers to the dependence of the energy $\omega$
of the new nuclear mode, on the strengths of the $ph$ and $pp$ monopole
interactions, $\chi, \chi_1$. Aiming at showing how good is the
semi-classical approach for this new type of pn excitation, we calculated
also the
exact eigenvalues of the model Hamiltonian, by diagonalizing the associated
matrix (4.14) within the basis $|NTM\rangle$.
The results are shown in Fig. 1 and Fig. 2.
From Fig. 1, one notices that the harmonic mode collapses for a critical
value of the attractive interaction $\chi_1$.
This critical value is certainly depending on the repulsive interaction
strength. The larger is that strength the larger the critical value.
In Fig. 1, we have also plotted the normalized energy for the first excited
state. There are intervals for $\chi_1$ where the energy of the harmonic
mode approximates reasonably well the exact excitation energy. Moreover, for
two values of the strength parameter, the exact solutions are precisely
reproduced. For the case T=2, the two energies, exact and $\omega$, are
the same for $\chi_1=0$ and $\chi_1=0.3$ for $\chi=1$  and $\chi=0.5$
respectively, but the curves are going apart for the first part of interval
and then converge to an intersection point close to the critical value.

The peculiar feature of $\omega$ as a function of
$\chi_1$, which distinguishes it from the standard RPA modes,
consists of its non-monotonic behavior with respect to the increase of
the strength of the $pp$ interaction. The reason is that in the common
cases the mean field is constant when the two body interaction is varied,
while here by changing $\chi_1$ we change also the minimum point for energy
and therefore another mean field is obtained. It is interesting to notice
that although the ph interaction,  the $\chi$ term, is kept constant,
the change of the mean field is equivalent to an increase of the effective
$ph$ interaction until $\omega$ reaches the maximum value from where the
attractive component of the two body interaction prevails.

In Fig. 2, the energies $\omega$ and the normalized
energy of the first excited state are shown as function of $\chi$, the
strength parameter of the $ph$ interaction. Both energies are
monotonically increasing with the increase of the interaction strength.
In contrast to what happens in the case of $\chi_1$ dependence, here
the change of the mean field by changing the energy minimum does not change
the repulsive character of the $\chi$ interaction. The agreement between
$\omega$ and the exact energy of the first excited state is reasonable good.

In Fig. 3. the energies characterizing the harmonic mode predicted by the
renormalized RPA and semi-classical method are plotted as function of
$\chi_1$. Also, the exact energy of the first excited state is presented.
Although they have different trends, the semi-classical and renormalized
RPA energies are not far from each other for $\chi<0.45$. At the
critical value $\chi$=0.57 the energy yielded by the
renormalized RPA is going very fast to zero. This behavior is specific
to the present model where only the scattering terms are considered.
Indeed, if the phonon operator includes both the two quasiparticle and
scattering terms, the corresponding mode collapses for larger $\chi_1$.
In the semi-classical treatment this happens only for very large $\chi_1$
since the static ground state is changed by increasing $\chi_1$.
The result obtained with the truncated HP boson expanded Hamiltonian
(see eq. (5.20)) is very close to the result shown in Fig. 3 for the
renormalized RPA procedure.

Comparing the results from Fig. 1a and Fig. 3, we remark on
the following features.
While the renormalized RPA energy collapses at a relatively small value of
$\chi_1$, the mode energy predicted by the semi-classical formalism
vanishes for a very large $\chi_1$, far beyond the realistic value, which is
$\chi_1=\chi$. This feature is a consequence of
changing the static ground state with $\chi_1$.
The energy behavior provided by the semi-classical method is also
different from that predicted by the standard renormalized pnQRPA
(see for example ref. 9) where the mode energy is a monotonic function of
$\chi_1$ and goes asymptotically to zero.

In this context we recall that the $frn-RPA$ breaks down \cite{Rad3}
before the standard RPA does, and that happened due to the fact that the
lowest $frn-RPA$ energy is that associated with the new collective mode.
From the present calculations one sees that {\it this is not true within the
semi-classical approach and therefore including the scattering terms in
the expression of the phonon operator does not prevent the treatment of
the many-body system
for a realistic value of the $pp$-interaction strength.}

The vanishing energies for the new mode, shown in Figs. 1a and 3 suggest
that a phase transition occurs according to the corresponding formalisms.
As we already mentioned this is clearly revealed if one diagonalizes the
Hamiltonian given be eq. (5.18) \cite{Sam,Rad4}. In the renormalized RPA
procedure the new phase is determined by a new minimum of the classical
energy associated to $H^{(b)}_{pn;2}$, reflecting the fact that the
$\lambda_2$ term is the dominant one for these values of $\chi_1$.
In the full-line and dotted-line curves of Fig. 1a, the corresponding energies
also vanish at certain critical values which result in having again a phase transition.
This is reflected in the
curve obtained by exact calculations, by the fact that the energy is minimum for the critical strength.
The increasing branch shown by the exact calculations (corresponding to
the second nuclear phase) might be semi-classically described by changing
the trial function, involved in the time dependent variational equations,
by rotating it (in the isospin space) with an angle which corresponds to the
orientation of the axis of maximum ``moment of inertia''.

It is remarkable that the far intersection points of the curves obtained
by semi-classical and exact calculations respectively, are lying close to
the critical values of the semi-classical description. Also, the first
intersection point is not far from the critical value of the
renormalized RPA treatment. In the classical treatment this feature is
well known \cite{Ghe}. Indeed, in the above quoted reference it is shown,
for a triaxial rotor cranked on an arbitrarily oriented axis, that for certain
critical values of the strength parameters, the period of the harmonic orbits
is equal to the period characterizing the motion on the closed exact orbit.

\section{Conclusions}
\label{sec:level7}
The main result of this paper refers to the existence of an harmonic mode
determined by the scattering quasiparticle terms, which are usually
neglected in the standard RPA approach.

The new mode is described within a time dependent variational formalism with an exactly solvable
many-body Hamiltonian. The variational state is a coherent state for the underlying symmetry
group, which is the SU(2) group.
A pair of classical canonical conjugate coordinates, which bring the equations of
motion to the Hamilton form, is found. The classical energy has an interesting structure.
It is quadratic in coordinate but highly non-linear in the conjugate momentum.
Therefore one finds first the stationary point which minimizes the energy, and then linearizes the
equations of motion around the minimum point in the classical phase space. The solution for the linearized equations is
harmonic and its time period determines the energy of the new mode. Despite the fact the classical system has an harmonic
motion, the mode does not exist in the standard RPA approach. In this sense one may say that the present description corresponds to a
''renormalized RPA''. However as we have seen, by comparing the corresponding predictions, the renormalization involved in
the semi-classical description is completely different from the renormalization described in Section V as well
as from the boson expansion method.

It is known the fact that the topological structure of the energy surface depends on the strength parameters involved in the model Hamiltonian.
Thus, in the parameters space one can define several regions, each of them corresponding to a distinct nuclear phase.
Having this in mind, we studied  the behavior of the new mode energy when the strength parameter for the $pp$
interaction ($\chi_1$) is varied. A particular feature for the semi-classical description is that the energy is not monotonic
decreasing function of $\chi_1$, but it increases in the first part of the interval, reflecting that here the $ph$ two body interaction prevails,
reaches a maximum value, then decreases and finally vanishes. This property is caused by that for each $\chi_1$
a new ground state is determined. This aspect is missing in both the renormalized RPA and boson expansion procedures.

Since the model Hamiltonian resembles the triaxial rotor which was semi-classically studied by one of the present authors (A. A. R)
in refs. \cite{Rad6,Ghe}, the interpretation of the new mode is imported from there. Thus, the new mode describes a wobbling motion around
a given total isospin.

The vanishing energy is a sign for a phase transition. In the first phase the rotation axis, in the isospin space
lies close to the z-axis, which has the maximum moment of inertia in the region of small $\chi_1$, while for $\chi_1$
larger than the critical value (where the energy vanishes), the rotation axis lies closer to the (X,Y) plane in the isospin space.
While the first phase may be described by a HP boson expansion formalism, for the second phase the Dyson boson representation is the proper one [20].
In the semi-classical approach, the new phase might be described by changing the trial function associated to the first phase,
through a rotation which brings the z-axis to the actual axis of maximal ``moment of inertia''.

The occurrence of the phase transition can be noticed also in the curve showing the exact first excitation energy as function of $\chi_1$.
Indeed at the critical value of $\chi_1$, this curve exhibits a minimum.

Another critical values of $\chi_1$ are those where the mode energy is equal to the exact excitation energy produced by the diagonalization procedure.
For these values the linearization does not affect at all the period of the exact closed classical orbit. As a matter of fact, for $\chi_1$ lying close to
these points the linearization are best justified. It is interesting to notice that these values of $\chi_1$
 lie however close to the values where the phase transitions in the
semi-classical treatment (the far intersection point) and the renormalized RPA approach (the near intersection point) take place.
This observation allows us to conclude that
the semi-classical approach works very well for the values of $\chi_1$ where the renormalized RPA breaks down and that the interval where
the linearization procedure does not work, ending with the critical value where the semi-classical energy vanishes, is very narrow.

The energy of the new mode vanishes for a value of $\chi_1$ which is far beyond the physical value ($\chi_1=\chi$).
In this way the drawback of the $frn-RPA$, of breaking down earlier than the standard RPA does, is removed.

How could  the new state be populated? We identified the transition operators which could excite the new state from the ground state.
The conclusion is that these state can be seen either in a $\beta^-$ (or $\beta^+$) decay or in a deuteron transfer reaction experiment.

The coupling of this mode to other collective states will be studied in
a subsequent paper using a realistic interaction and a large model space
for the single particle motion.

\vskip0.5cm
{\bf Acknowledgement}
\vskip0.5cm
\noindent
One of us (B. C.) wants to thank Prof. Amand Faessler for hospitality
in Institute of Theoretical Physics of Tuebingen University where
a part of this work was performed. A. A. R. thanks Prof. Faessler for
reading the manuscript and valuable remarks concerning the structure of
the variational function.

\section{Appendix A}
\label{sec:level8}
Here to derive the factorization of the trial function
$|\Psi\rangle$. To this purpose we address the following more general
question. Which are the t-functions $A(t), B(t), C(t)$ satisfying the
equation
\begin{equation}
e^{t[(z\sum_{m}a_{pm}^{\dag}a_{nm}-z^{*}\sum_{m}a_{nm}^{\dag}a_{pm})]}
=e^{A(t)a_{p}^{\dag}a_{n}}e^{C(t)(\hat N_{p}
-\hat N_{n})}e^{B(t)a_{n}^{\dag}a_{p}},
\end{equation}
with the initial conditions
\begin{equation}
A(0)=B(0)=C(0)=0
\end{equation}
and t a real parameter. Once we solve this problem
the needed factorization is obtain from (4.1) for $t=1$.
Taking the first derivative of the eq.(6.1), with respect to t,  and
identifying the coefficients of the similar operators one obtains the
following system of differential equations for the three unknown functions,
$A(t), B(t), C(t)$ :

\begin{eqnarray}
\stackrel{\bullet}{z} &=& \stackrel{\bullet}{A}-2A
\stackrel{\bullet}{C}-\stackrel{\bullet}{B}A^{2}e^{-2C(t)},\nonumber\\
0&=& \stackrel{\bullet}{C}+\stackrel{\bullet}{B}Ae^{-2C(t)},\nonumber\\
-\stackrel{\bullet}{z}^* &= &\stackrel{\bullet}{B}e^{-2C(t)}.
\end{eqnarray}
Eliminating the functions $B, C$ from these equations, one obtains the following equation for
$A(t)$.
\begin{equation}
\stackrel{\bullet}{z}=\stackrel{\bullet}{A}-A^{2}\stackrel{\bullet}{z}^*
\end{equation}
which admits the solution:
\begin{equation}
A(t)=\tan(\rho t)e^{i\varphi}.
\end{equation}
Here the polar coordinates $(\rho,\varphi)$($z=\rho e^{i\varphi}$) have been used.
Inserting the result for $A(t)$ in the eq. (6.3), the equations for the remaining
functions can be easily integrated. The result is:
\begin{eqnarray}
C(t)&=&-\ln \big( \cos(\rho t) \big)\nonumber\\
 B(t)&=&\tan(\rho t) e^{-i\varphi}.
\end{eqnarray}

For the sake of simplifying the writing, hereafter the following
notation will be used:
\begin{equation}
\alpha=A(1)
\end{equation}
Using these results the trial function can be written as:

\begin{eqnarray}
|\Psi\rangle & = & e^{-2C(1)T}e^{A(1)B^{\dag}(pn)}|NT\;-T
\rangle\equiv {\cal N} e^{A(1)B^{\dag}(pn)}|NT\;-T\rangle
\end{eqnarray}
where $\cal N$ denotes the normalization factor:
\begin{equation}
{\cal N} = e^{-2C(1)T}=e^{2 \ln(\cos \rho)T}={(1+|\alpha|^2)}^{-T}.
\end{equation}
\nopagebreak

redpajama_set_name: RedPajamaArXiv


A compare and contrast essay, also known as a comparison essay, talks about how two ideas or objects differ and how they are similar. Some essays may only talk about similarities, while others may only talk about differences.
How can the answer be improved? Using comparisoncontrast for all kinds of writing projects. Sometimes you may want to use comparisoncontrast techniques in your own prewriting work to get ideas that you can later use for an argument, even if comparisoncontrast isnt an official requirement for the paper youre writing.
How to Write Compare and Contrast Essay: Defining This Type of Paper This type of assignment is an academic paper, which depicts 2 or more similar yet different things by focusing on what they have in common and what makes them different. The compare and contrast essay, also called the comparison and contrast essay, requires the writer to compare the differences and similarities between two or more items. The context will vary depending on the nature of the essay. To write a comparecontrast essay, youll need to make NEW connections andor express NEW differences between two things.
The key word hereis NEW! Choose 2 things that could go in the same category, but are also quite different. Sep 03, 2018 Compare and contrast essays are often assigned to students because they promote critical thinking, analytical reasoning and organized writing.
A compare and contrast A compare and contrast essay outline example such as comparing and contrasting two dogs, Molly and Morgan may sound something similar to this: Do opposite poles attract? Well, the principle of attraction states that they do.
TIP Sheet WRITING A COMPARECONTRAST PAPER. A compare and contrast essay examines two or more topics (objects, people, or ideas, for example), comparing their similarities and contrasting their differences.


text:
Home News Restrictions on indoor and outdoor dining and bars in Maryland by county...
Restrictions on indoor and outdoor dining and bars in Maryland by county due to COVID-19 – Updated December 30th
GET THE LATEST UPDATES ON THIS TOPIC HERE:
Restrictions on indoor and outdoor dining and bars in Maryland by county due to COVID-19 – Updated January 13th
Here is a listing (that I updated on December 30th at 11:15am) of restrictions by county in Maryland on indoor and outdoor dining and bars due to the COVID-19 pandemic:
Statewide:
November 10th – INDOOR DINING CAPACITY REDUCED TO 50%. The governor issued an emergency order reducing indoor operations for bars and restaurants from 75% to 50%. Bars and restaurants in the State of Maryland are permitted to be open for seated and distanced service only, with strict capacity restrictions (Read More)
November 17th – Governor Hogan Institutes 10 PM Closure for Bars and Restaurants (Read More)
Baltimore City:
December 9th – All food service establishments will be closed to indoor and outdoor dining starting on Friday (December 11th) at 5 p.m (Read More)
Anne Arundel County:
Update December 30th – Anne Arundel County Executive Issues Executive Order to Maintain Indoor Dining at 25% Capacity. Today's order allows indoor dining at 25 percent capacity for establishments that collect information for contact tracing purposes and follow all other safety protocols. Outdoor dining is permitted, provided that tents have at least half of the number of sides up at all times. (Read More)
UPDATE December 16th – An Anne Arundel County judge agreed Wednesday to issue a temporary restraining order to block the county executive's order that banned indoor dining. As a result of the judge's order and until a decision is made in the Dec. 28 hearing, the county's restaurant restrictions revert to those found in Executive Order No. 38, which limits indoor dining capacity to 25% of maximum capacity. (Read More)
December 10th – Effective December 16th Foodservice Establishments will be restricted to carryout, delivery or curbside service only. No indoor or outdoor seating is permitted. (Read More)
Prince George's County:
December 10th – Starting at 5 p.m. on Wednesday, December 16, indoor dining at restaurants must close. Indoor dining is also prohibited at banquets, receptions and meeting rooms in hotels, conference centers and similar establishments. Outdoor dining will be allowed at 50% capacity, and restaurants can still offer curbside and takeout services. (Read More)
Montgomery County:
December 9th – The new Executive Order 139-20 will restrict restaurant service to outdoor dining only (Outdoor dining will be prohibited between 10 p.m. and 6 a.m., per an existing Executive Order from Maryland Governor Larry Hogan.). Carry-out, delivery and drive-through service will be allowed to continue. The Executive Order has been sent to the County Council for its review. The Council can approve or disapprove the order but cannot change it. (Read More)
December 10th – The Montgomery County Council will introduce, hold a public hearing, and vote on a resolution and Board of Health Regulation that would approve Executive Order 139-20 at 1 p.m. on Tuesday, December 15, 2020. If approved by the Council, Executive Order 139-20 would be effective December 15, 2020 at 5 p.m. (Read More)
December 15th – The Montgomery County Council approved County Executive Marc Elrich's Executive Order 139-20AM2 today that suspends indoor dining, reduces indoor sports gatherings to no more than 10 people and reduces the maximum capacity of retail establishments. The new restrictions go into effect today at 5 p.m. (Read More)
Randi Bass (Reporter & Weekend Morning Anchor for @WDVMTV) reports the following:
The updates above are the latest I have seen for bars and restaurants in Maryland. Here are some statements from organizations on this topic:
NFIB Maryland (a member-driven organization advocating for small and independent businesses in Washington, D.C. & all 50 states) issued the following press release on this topic around 4:30pm on December 10th:
Local officials, including the county executives of Anne Arundel, Montgomery, and Prince George's counties, recently announced additional restrictions on retail, personal services, recreational businesses, and bars and restaurants saying they are necessary to stem the flow of the COVID-19 virus. These restrictions include the elimination of indoor dining, further restrictions in capacity at indoor establishments, and the full closure of various facilities. The National Federation of Independent Business (NFIB) in Maryland, which represents thousands of small businesses in the state, reacted to the news. The following comments are from NFIB's Maryland State Director, Mike O'Halloran:
"This is extremely difficult news for hundreds of small businesses in these jurisdictions. Owners are doing everything they can to protect their employees, customers, and clients. But every time businesses are forced to close, however temporary officials think it might be, or face restrictions which make it impossible to operate, those closures and restrictions mean more businesses will shut down for good, more jobs will be lost forever, and the unemployment numbers will grow.
We are at a point of no return for so many small businesses. They have gone to great lengths and great expense to follow every health requirement when it comes to PPE, physical installations, social distancing, and restrictions on customers and operations. Business owners did that to stay open and preserve jobs. But these latest restrictions will likely mean the end for many small businesses especially with financial assistance already exhausted."
On December 9th the Restaurant Association of Maryland released a statement on the announcement from Baltimore City shutting down indoor and outdoor dining:
"Restaurants are frustrated and angry that their livelihoods, and the livelihoods of their employees, have been taken away from them. Restaurants need significantly more financial relief to ensure that their employees can make ends meet and that the restaurant can continue to pay rent and be able to reopen fully when restrictions are lifted. Mayor Scott has taken significant and drastic actions in the interest of public health, and he now needs to take the same significant and drastic actions to ensure that 1,400 City restaurants also survive this pandemic."
I will continue to try and track restrictions from other counties if new restrictions are put in place…as always I will watch closely for announcements here in Howard County.
Indoor dinning
Previous articleHandgun used in home invasion / robbery in Columbia on December 9th
Next articleHoward County General Hospital receives state accreditation to train community health workers
The Board of Education of Anne Arundel County voted to implement a hybrid learning program no later than March 1, 2021; health and safety...
Governor Hogan Announces Expansion of COVID-19 Vaccine Eligibility for Marylanders 65 and Over


The most summerish accessory is here and of course its personalized just for you!
The Charm Anklet with Pineapple is our newest trend, goes perfect with your legs and with some coconut water by the beach!
The perfect item to take on you next holiday, all eyes will be on you!
The pineapple shape has some cut out lines and your initial.
Great gift to have with your bffs!
24mm x 10mm / 0.9" x 0.4"


Our Set of the Month for April is Godfrey Phillips M36 Characters Come to Life issued in 1938. The set presents the stars in roles for which they were famous such as William Powell as Ziegfeld (of Follies fame), Flora Robson as Queen Elizabeth in "Fire Over England", Errol Flynn as Major Vickers in "The Charge of the Light Brigade" and Charles Laughton in "The Private Life of Henry VIII". The pictures leap out at you with superb clarity and there is an informative text on the back. Illustrated here are Joan Crawford as Peggy O'Neal in "The Gorgeous Hussey", Clark Gable as Parnell in "Parnell", a gilded Claudette Colbert as "Cleopatra" in Cleopatra and Raymond Massey as Philip of Spain in "Fire Over England" .These are an important addition to any cinema collection.
This set of Godfrey Phillips M36 Characters Come To Life 1938 in Finest Collectable Condition is catalogued at £36.00.


text:
Contact Us|中文
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Home > Foreign Reports
China looks for closer DPRK ties with a new Tumen River bridge
( KBS )
There is a new bridge in the northeastern Chinese city of Hunchun, Jilin province taking shape across the Tumen River, which separates China from the DPRK, set to open May next year, and all bets are on it to increase trade between the two countries a great deal, according to the ROK's Korean Broadcasting System, on Nov 8.
The 800-meters bridge will connect Hunchun with Rajin and is said to be costing 1.37 billion yuan. It is expected to speed up the transportation of goods through an area that now handles a large amount of trade between the two countries.
The KBS went on to speculate on China's reasons for paying for the new bridge, by saying, "China sees a competitive relationship with Russia in the Rajin-Songbong region where it is also doing some investment."
Hunchun is in an envious geographical location thanks to its clear view of a railway bridge connecting Russia and the DPRK across the Tumen River, and its own close access. Another ROK news outlet, in a recent report on Hamgyong province, DPRK, to the north of Tumen River, hinted that Russian petrol is taking a larger market share in the area than China's so DPRK is beginning to look to Russia for its oil products.
The Russian Ministry for the Development of the Far East, which coordinates development in the region, has said that they plan to put $25 billion into DPRK's infrastructure, including railways, tunnels and bridges, over a 20-year period and to take out coal and nonferrous metals as reimbursement.
The new Chinese bridge will replace a much older one, built back in 1938, which is a tourist attraction, Hunchun locals said.
Written by Wang Zhen and edited by Roger Bradshaw
Jilin social media contest
Jilin province and Two Sessions
Scenic high-speed rail route
A trip along Jilin-Tumen-Hunchun high-speed railway line
Fall in love with Jilin
A wonderful encounter between Jilin and me
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text:
I voted for Theresa May's Brexit deal but now she must compromise
Nicky MorganConservative MP for Loughborough and former Secretary of State for EducationWednesday 16 Jan 2019 9:09 am
We are prepared to change the law to ensure that a 'no deal' outcome cannot unfold (Photo: PA/ Getty)
After last night's historic defeat, for the prime minister's draft Withdrawal Agreement and political declaration on a future framework for a relationship with the EU, the question on everyone's lips is, 'what next?'.
The prime Minister gave an initial answer immediately after the vote saying there would be more involvement by senior MPs to try to identify what is needed to achieve a deal which can be supported by the majority of MPs.
This assumes that the government wins the 'no confidence' motion tabled by the Labour Party last night. With the support of the DUP, which they have pledged, and all Conservative MPs then we can be certain that the 'no confidence' motion will not succeed.
Part of answering 'what next?' is to note that the House of Commons made clear last week that it would not support a 'no deal' outcome to Brexit. And we are prepared to change the law to ensure that a 'no-deal' outcome cannot unfold.
But the best way to prevent a 'no-deal', and to end the uncertainty for businesses and individuals, particularly EU citizens living in the UK, is to put a deal in place, which is why I supported the draft Withdrawal Agreement. Given that the agreement has now been voted down it must be the case that all deal options are now back on the table.
To find a way through the prime minister and the government must now show maximum flexibility and signal clearly that they are prepared to move away from the hard red lines which have led us to this impasse.
Many of us have been saying for months that a consensus can be found in Parliament around some form of access to the single market and a customs arrangement which would help to address the complicated issue of the Irish border.
I am a supporter of Common Market 2.0. The prime minister was right to prioritise frictionless trade, protecting the 'just in time' supply chains of our manufacturers and introducing reforms to our immigration system to give ministers more control. And she is also right to prioritise delivering the referendum result – which was for the UK to leave the EU, but it did not stipulate how the UK was to leave.
I hope that the prime minister and the cabinet will be able to demonstrate, very swiftly, the leadership necessary to show a way forward which gets to a deal which MPs can support and approve.
Time is short. The 29th March 2019 is not far away.
But, if that cannot happen then the EU Withdrawal (No. 2) Bill (or, rather more snappily, the 'No to no-deal' Bill) which I have been working on with fellow MPs Sir Oliver Letwin and Nick Boles, with cross-party support from Yvette Cooper, Hilary Benn and Norman Lamb, provides an alternative route.
This gives the government some space to come up with a plan but also tasks a committee of senior MPs to do the same too. If MPs approve that plan then it must be adopted by the Brexit Secretary and pursued with the EU. If that doesn't happen, or the plan stipulates that an extension to the Article 50 period must be requested, then that is what the Government must do.
A deal with the EU is needed to avoid the disruption which will be caused by the UK leaving the EU without a transition period.
Leadership requires decisions to be made but it also requires genuine listening, the ability to compromise and flexibility.
If all those elements are put in place by the prime minister then together the government and parliament can find a way through the Brexit chaos.
MORE: How newspapers reacted to Theresa May's humiliating Brexit defeat
MORE: Can the UK cancel Brexit and stay in the EU?
MORE: Brexit 'must be delayed as matter of urgency to protect jobs'
BrexitEU referendumEuropean UnionPoliticsTheresa May
House prices stall as Brexit hits buyers' confidence over summer
Boris Johnson's no deal Brexit 'will be taken to court'
Boris Johnson admits he doesn't know his own Brexit plan


text:
Home News Scotland remains silent during Remembrance Sunday
Scotland remains silent during Remembrance Sunday
Caption for the image
Two minutes of silence was observed in Edinburgh by veterans
Scotland's people have stopped to think about those who gave their lives for our country.
At 11:00, Remembrance Day saw thousands of people fall silent to reflect for two minutes.
Near the Royal Mile, there were hundreds who gathered in Edinburgh for the service to remember.
In honor of those who have fought in the past conflict, wreaths were laid on the Stone of Remembrance.
Among the dignitaries who participated in proceedings were Nicola Sturgeon, First Minister, and Alister Jack (the Scottish Secretary).
What's Remembrance Sunday?
Why is the Scottish poppy so special?
Similar events have been held all over the country since they were cancelled by the coronavirus pandemic.
This is on the 100th anniversary celebration of the poppy's national symbol of remembrance.
PoppyScotland which manages the Scottish Poppy Appeal said that there was a 28% decrease in donations because they couldn't do face-to–face collection.
Gordon Michie, the head of fundraising said that there has been a tremendous response to this year's centenary appeal.
He said, "It was especially satisfying to see such determination by our volunteers to return to face-to–face collections again after so many were forced home last year due to the pandemic."
Jack Ransom from Largs, Ayrshire was one of those who placed a wreath in the capital. He served in World War Two.
After being captured by Singapore authorities in 1942, he was made a prisoner of war and was sent to Burma Railway.
A special service was held in Musselburgh's Northesk Parish Church to remember Helen Burnett Wood who died in 1917.
Helen was born in 1888. She immigrated to Illinois at the age of 21 and began her career as a telephone operator.
In May 1917, she served with the US Base Hospital Number 12 as a volunteer and was then deployed to France.
She was on board the SS Mongolia with American medical personnel, when the accident occurred the next day.
A ship's gun exploded while conducting a practice drill. A hail of shell fragments fell across the deck and killed Edith Ayres, a 29-year old Scot, as well as Edith Ayres, an army nurse.
The Scotsman reported the news of her death. Annie, her sister, also emigrated from Britain to the US later said: "I didn't want Helen go. But she stated that if her brothers were willing to risk their lives in Britain for her, then she would be open to risking hers for America."
At the request of the Evanston History Centre, the American flag to Helen Wood was raised above Washington DC's Capitol Building this fall.
Jenny Thompson is the director of education. She teamed up to Simon Fairnie (a local historian) to send it to Musselburgh and he returned a Saltire flag.
Ms Thompson stated: Helen Wood's tale resonates with more people each year. Helen Wood was one of the first American casualties in the First World War. She was a female soldier who had volunteered for service and was an immigrant from the US.
"We've been fascinated with her story for many decades and we wanted more to link our two communities and to discover the common history."
Remembrance poppy
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Radio Pendimi is an Islamic radio station in albanian language, which began its work in October 2006. Radio Pendimi aims the spread of religional knowledge and methodology to all muslims of the region and those who reserach after the truth. Radio Pendimi does this by its educative materials and analysation of global occurrences by a deep religional view.
Through all this it depends on help of Allah subhanehu ue teala, then on Muhamed sal-lallahu alejhi ue selem guidance and in tradiotion of the three very first generation of muslims. The accomplishment of which is achieved by its professional staff which is engaged in this activity. You can find Radio Pendimi's materials in our web page now, after a short period of time.
Radio Pendimi tries to eliminate the partisanship confrontation and personal divergencies. Through all this, Radio Pendimi tries to clear the religional basis and truths as they are. Radio Pendimi is an Islamic radio station, where the truth is cleared and the consciousness is revived. Maybe we won't be able to tell the truth all the time but with the permission of Allah we will never tell the useless.


Some 45 Member Associations from across Asia learned their opponents for the upcoming AFC U-16 Championship 2018 qualifiers on Friday following the official draw at AFC House.
Defending champions Iraq, have been grouped with India, Palestine and Nepal who will host the qualifiers in Group D, while the Islamic Republic of Iran - last year's runners-up - are scheduled to meet Kyrgyz Republic, Afghanistan, Lebanon and Bhutan in Group C.
Elsewhere, two-time champions and 2016 semi-finalists DPR Korea, will face Hong Kong, Chinese Taipei, Macau and Brunei Darussalam in Group F, with Japan, who also reached last year's semi-finals drawn in Group J, along with Singapore, Malaysia and Guam.
The ten (10) group winners and five (5) best second-placed teams will advance to the final stage of the tournament, along with the host Member Association. The AFC U-16 Championship 2018 qualifiers are set to take place from September 16 to 24, 2017.
The Indian U-16 National Team will play two matches as part of the India-Egypt Friendship Cup 2017 in Cairo, Egypt to prepare for the qualifiers . The first of which kicks-off on April 22, 2017 against Egypt's Al-Ahly Club in Nasr City, Cairo, before the Colts lock horns against Smouha Club on April 23, 2017 in Alexandria.


Thanks to Emma Burford for this inspirational guest post and for making me feel hungry!
2 years ago when on holiday reading a magazine I came across an advert for My Little Wrapper promoting their mini business opportunity. I just love chocolate and design and so the thought of creating special personalised chocolate wrappers for all sorts of occasions and being able to make a little money in the process seemed really appealing.
On returning from holiday I looked into it and discussed it with my friend Lily. We both have 3 children and our youngest were just about to start school. So we decided to jointly start up our own personalised chocolate wrapping service covering our immediate London and South East area.
It has been hard work there is no denying, but we are great perfectionists and love what we do…We both attend wedding fairs, networking events, and other local fairs promoting our businessand have diversified along the way by supplying Hen Night Sashes and Personalised Ribbons,Balloon Creations with a twist from Julie at Balloon Baboon and have also added Organic Fair-trade Chocolate to our portfolio.
We have learnt a lot and now know what works and what doesn't and we've had a lot of fun and met many wonderful people who have helped us along our chocolate journey. In the age of the internet having a web presence is a must! We designed and built our own website Say it with Chocolate and added a Paypal shop which has gone from strength to strength.
Recently we sourced a lovely Chocolatier as we were constantly being asked if we could supplyorganic fair-trade chocolate. We have now launched our Organic Fair-trade Range on our website andit is proving to be really popular for our more discerning customers.
ChocoSpell consists of 33 x 10 gram Organic Fair-trade Chocolate Bars hand-made andhand-wrapped with your own personalised message and symbols added to the wrappers andcomes complete with gift box and gift tag which can be posted within the UK all for £24.95.
If you are interested in becoming a ChocoSpell supplier and running your own small Organic Fair-trade business or even adding it to your current business range do get in touch!
We are looking to franchise this product UK wide at a very reasonable start up cost. You don't even have to have a printer we will supply the printed wrappers and chocolate kit ready for you to wrap and sell.
Unlike the cost we incurred ourselves some 2 years ago which did take the best part of a a year and lots of hard work to recoup we feel that ChocoSpell would give Mums who are just starting out a much more affordable product to market!
I will be attending the Business Mums Conference in Brighton on the 5th October and will take a few samples along so do come and say hello if you are going, hope to see you there.