text:
Home/Local/Lisa Marie Presley is buried at Graceland, across from Elvis and next to her son
Lisa Marie Presley is buried at Graceland, across from Elvis and next to her son
Bradford Betz2 weeks ago
Her complicated relationship with sprawling Graceland was well known and now, sadly, the estate she owned is preparing to bury Lisa Marie Presley following her death last week at the age of 54.
The daughter of Elvis Presley will be buried alongside her son Benjamin Keough at 9am on Sunday at a public memorial open to the public – who have passed by the king's shrine by the millions for years.
The mother-of-four is being drawn into a conspiracy against her legendary father following her untimely death last week after suffering double cardiac arrest while attending the Golden Globes.
Indeed, during preparation for duty Tuesday, the LA County coroner announced that he would defer the results of her official cause of death pending the results of a toxicology test — which could be years.
Presley grew up fearing her own mortality after finding her father dead at age nine, which traumatized her. She often spoke of her own death, even referring to the Graceland estate's backyard as a "graveyard."
"How many people have a family plot in their backyard? How many people are reminded of their fate, of their mortality, every damn day? All the graves are lined up and there's a place waiting for me right next to my grandmother," she told Playboy in 2003.
Preparations are underway for Lisa Marie Presley's burial site, which will be positioned alongside her other relatives at Graceland along the backyard fountain
Despite her anxiety, the singer believed she had many years to live last Sunday, telling her boyfriend David Kessler, who works as a grief counselor, that she had "so much to do".
The couple were at the Graceland estate to celebrate Elvis' 88th birthday, and Presley, who rarely appears in public, told the admiring crowd, "You're the only ones who can get me out of the house."
'I'm not joking. I love you back and that's why I'm here," she said.
Preparations are underway for her burial site, which will be positioned alongside her other relatives at Graceland along the backyard fountain.
Several of the Presleys also died from heart disease, including their own father, who died in 1977 at the age of 42 from cardiac arrhythmias associated with ventricular fibrillation — a condition in which the heart beats abnormally and then stops.
Her grandmother, Gladys Love Presley, died of heart failure at the age of 46, and her grandfather, Vernon Elvis Presley, died of cardiac arrest at the age of 63.
Sources told TMZ that a stage and backdrop will be built on the front lawn of the mansion. It's unclear who will speak at the service, but it will be open to the public.
"In lieu of flowers, the family encourages anyone wishing to send to do so in the form of a donation to the Elvis Presley Charitable Foundation," Graceland's website reads.
Workers dig her grave, which is right next to her son and almost directly across from her father's (bottom left).
Presley, who died of cardiac arrest last week at the age of 54, will be buried this Sunday at 9 a.m. at the family home in Memphis
Additionally, 24-hour camera coverage at Graceland is now focused on the front building as preparations that began Tuesday continue.
Since her death, fans have come to lay flowers at the front gate of the property to honor the singer.
Presley died on January 12 after suffering a second cardiac arrest in the hospital. Her family signed a do not resuscitate order.
The 54-year-old only child of Elvis and Priscilla Presley was rushed to hospital Thursday morning after she was found collapsed at her home in Calabasas by her housekeeper.
Paramedics who got to her home managed to revive her and she got a pulse again, but she was brain dead on arrival at the hospital, TMZ reported Friday.
The family rushed to her bedside, where she was in an induced coma and on life support. When they learned that she was already brain dead, they signed a "do not resuscitate" order.
She suffered a second cardiac arrest and died.
Her mother Priscilla, 77, was at her bedside.
"It is with a heavy heart that I share the devastating news that our beautiful daughter Lisa Marie has left us," Priscilla Presley said in a statement Thursday night.
"She was the most passionate, strong and loving woman I have ever known."
Ex-husband Danny Keough – her first of four with whom she was living at the time of her death – had returned to the home shortly after her collapse and performed CPR until paramedics arrived.
Keough is the father of actress daughter Riley, 33, and son Benjamin, who took his own life in July 2020 at the age of 27.
Her fourth husband, Michael Lockwood, is the father of Presley's 14-year-old twin daughters, Harper Vivienne and Finlay, who have said they will not return to the home where their mother died.
Presley was at the Golden Globes two days before her death, celebrating Austin Butler's Best Actor Award for portraying her father, where she burst into tears during the actor's speech.
Since her death, fans have come to lay flowers at the property's gate to honor the singer
And on Sunday, the singer-songwriter told the doting crowd at her father's Memphis estate that they were the "only people who can get me out of the house" to celebrate his 88th birthday.
Fans said the King of Rock 'n' Roll's daughter, who hasn't posted to Instagram since August, looked "incredibly sad" and "really hurt" two years after the suicide of her son Benjamin, who is buried next to Elvis at Graceland.
Furthermore, Presley was so traumatized by her father's death – and the discovery of his body when she was nine – that she spent the next 35 years trying to be "cured" by Scientology, according to the claims of her former counselor Karen De La Carriere .
De La Carriere, known as the "Queen of Scientology," was a senior member of the church for 35 years.
Lisa Marie with her parents in 1970. Presley found her father dead when she was nine
She said that Priscilla Presley, who turned to Scientology to ease the grief of Elvis' sudden death in 1977, used the church to bring her daughter's behavior problems under control.
As a result, Lisa Marie, who was only 11 at the time, and her mother endured a "hostile relationship" and found themselves in "mortal throes" for the rest of their lives, De La Carriere claims.
The counselor became Presley's auditor – Scientology language for counselor – and said that Presley was placed in "obedience therapy" as a schoolgirl.
De La Carriere said Elvis would never have allowed that if he had lived.
https://www.dailymail.co.uk/news/article-11649681/Lisa-Marie-Presleys-grave-prepared-iconic-Memphis-home-dad-Elvis-plot.html?ns_mchannel=rss&ns_campaign=1490&ito=1490 Lisa Marie Presley is buried at Graceland, across from Elvis and next to her son
Bradford Betz
A fan runs onto the pitch to take a selfie with Manchester United star Casemiro
MTG Phyrexia: All Shall Be One: Full Map List


Aaron Daur, Jennifer Plourde, Samantha Strueby, Joshua Visser, and 544 other B.C. CPA students successfully passed the multi-day national Common Final Examination.
The exam, written in September 2016, was administered by the Chartered Professional Accountants Western School of Business.
"We welcome them to the ranks of B.C.'s chartered professional accountants," said Dave Hallinan, chair of Chartered Professional Accountants of B.C.
In order to become designated, students must complete rigorous course work, pass the exam, and fulfill relevant practical experience requirements.
The national exam ensures all Canadian chartered professional accountants meet the same high standards, which are recognized nationally and internationally. The school delivers chartered accountants professional education program to western Canada.


text:
est un astéroïde de la ceinture principale découvert par le programme LINEAR le  à Socorro, au Nouveau-Mexique.

Compléments

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Références

Objet céleste nommé par sa désignation provisoire
Astéroïde de la ceinture principale
Planète mineure découverte en 2009
Objet céleste découvert par LINEAR
Objet céleste découvert à l'observatoire Magdalena Ridge

redpajama_set_name: RedPajamaWikipedia


text:
Carr comes on to lead Panthers
ATLANTA -- The Carolina Panthers lost their starting quarterback and had trouble slowing Joey Harrington. Still, they managed to get out of Atlanta with a victory, largely because DeAngelo Hall lost his cool. The Falcons' Pro Bowl cornerback picked up three penalties for 67 yards on Carolina's tying drive, and David Carr came on in relief of Jake Delhomme to lead the Panthers to their final 10 points in a 27-20 victory.
Panthers story line
DeShaun Foster rushed for 122 yards and scored a couple of TDs, which helped the Panthers (2-1) overcome the loss of Delhomme late in the third. He came off after hurting his elbow on a previous possession but didn't think the injury was serious. Carr finished off the go-ahead touchdown drive, handing off to Foster for a 10-yard touchdown run, then guided the Panthers to a field goal that made it 27-17. Carolina plays the surprising 2-1 Bucs next week.
Falcons story line
Atlanta (0-3) is off to its worst start since 1999 when the Falcons lost their first four games coming off a trip to the Super Bowl. Indeed, Harrington made sure all those Atlanta fans wearing No. 7 jerseys had no reason to boo the quarterback who took over for Vick -- whose career is on the ropes -- by completing 31 of 44 for 361 yards and two touchdowns. The Falcons were held to under 100 yards rushing for the 3rd straight game.
*Falcons WR Roddy White caught seven passes for 127 yards and a touchdown.
*Falcons TE Alge Crumpler had six catches for 63 yards and a touchdown.
Injuries of note
*Panthers LB Dan Morgan sustained a hamstring injury.
*Falcons CB Lewis Sanders didn't return after taking a knock to the head.
Postgame chat
* "In hindsight, yeah, you have to try to control yourself." -- Falcons RB Warrick Dunn on Hall's penalties
*"It's our job as a team to pick our teammates up. We were not able to do that today." -- Harrington
* "I felt like I was on the bad end of the stick. I felt like a lot of calls could have gone either way. They all went against me." -- Hall
* "He took a cheap shot at me. That's the kind of player he is." -- Panthers WR Steve Smith, who was on the receiving end of Hall's penalties
Falcons fans can't blame Joey Harrington for this loss.

redpajama_set_name: RedPajamaCommonCrawl


text:
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  DEPENDS
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    vtkFiltersCore
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    vtkInteractionStyle
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    vtkRenderingFreeTypeOpenGL
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  )

redpajama_set_name: RedPajamaGithub


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Why are you here? What are your wishes and dreams? Where do you think you'll be in five years? Answer these and consistent resembling questions to create your personal development plan. It will help to revaluate and reconsiders your beliefs, principles and aspirations.
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Recollect your peak moments, how you've felt yourself on top of the world. Recall how you've got there and realize that albeit you can't go back in time, you can get a new achievement. Because you've already proven that you can overcome all obstacles if you're willing something passionately. Since when you have the power, your mind simply doesn't see them. Aren't you tired from depressions in which you sink every time when you bring back the memories of your failures? There's nothing you can do, just deal with it. But you still can escape future mistakes by learning your lessons from those experiences. Consider that you fail only once: when you don't get up after you fall. Other events are the part of your universe exploration as well as your triumphs.
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Delicious mini cupcakes full of bananas and rich chocolate to yield a bite of heaven.
Not sure there's anything more satisfying and delicious than a moist banana chocolate cake. But add some banana chocolate buttercream and top it with a pretty garnish, and this little sweetheart is a blue-ribbon winner. Good thing these pastries are mini sized!
Preheat oven to 350° F. Line mini muffin pans with liners and set aside.
In a large bowl, peel and mash bananas. Sprinkle with baking soda.
Add oil, eggs, orange juice and vanilla. Whisk until smooth.
Add flour and sugar and stir with gentle strokes until barely blended.
Use a tablespoon to fill muffin liners about 2/3 full. Bake for 7-9 minutes or until tops spring back when lightly pressed.
While cupcakes are cooling, beat butter, banana and sugar. Melt chocolate and add to mixture, beat well. Add powdered sugar, vanilla and enough milk to create a thick buttercream texture.
Pipe swirls onto each mini cupcake. Top with a slice of dried banana.


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text:
\section{Introduction}

Physics in the context of Lorentz-violating theories have been an important
branch for different theoretical and experimental research in recent years.
The standard model extension (SME) \cite{Colladay1,Colladay2,Coleman1} is
the general theoretical framework for studying Lorentz violation (LV)
effects, and it is build by adding Lorentz-violating terms in all sectors of
the minimal standard model. Particulary, the Abelian gauge sector of the SME
is the principal focus in searching for LV effects in physical systems \cit
{Jackiw}-\cite{Klink3}.

On the other hand, magnetic flux vortices have gained great interest since
Abrikosov's description for Type-II superconductors \cite{Abrikosov}, which
arise naturally from the non-relativistic limit of Ginzburg-Landau (GL)
theory \cite{GL}. In field theory, stable vortex configurations came up with
the seminal work by Nielsen and Olesen \cite{NO} whose study of the
Maxwell-Higgs (MH) model shows that electrically neutral vortex solutions
correspond to the ones obtained by Abrikosov. Lately, the existence of
electrically charged vortex solutions in Chern-Simons-Higgs (CSH) \cit
{CS,CSV} and Maxwell-Chern-Simons-Higgs (MCSH) \cite{MCS,Bolog} models was
verified.

The study of topological defects in the context of Lorentz-violating models
has been performed in some cases,for example, scalar systems \cite{Defects},
Abelian monopoles \cite{Monopole1}, tensor fields \cite{Seifert}, etc. The
research for explicit BPS vortex solutions in the context of SME was
recently performed in Refs. \cite{Carlison1,Carlison2,Lucas,cpt13,lucas2}.

The aim of this article is to investigate how LV introduced in Higgs sector
affects the structure of topological defects; particularly, we discuss
charged BPS vortices in Lorentz-violating MH models. It is easily observed
that Higgs's Lorentz-violating term introduces anisotropies in the
dispersion relations of the Higgs field. Consequently, these anisotropies
provide a Hamiltonian dynamics very similar to GL models describing
anisotropic and layered superconductors (e.g. high-$T_{c}$ superconductors)
\cite{ref_1,ref_2,ref_3,ref_4}. So, the main motivation for studying
topological defects in the context of Lorentz-violating theories comes from
the similitude with the mathematical structure describing the physics of new
high-$T_{c}$ superconductors. These new structures lead to a strong
anisotropy of magnetic properties and gives rise to new phenomena, such as
formation of two-dimensional vortices and existence of new type of vortex
lattices \cite{ref_2,ref_4}.

The manuscript begins by studying the MH model supplemented with a CPT-odd
or Carroll-Field-Jackiw (CFJ) term and then with a CPT-even term in gauge
sector. In both situations we have introduced a CPT-even and
Lorentz-violating term in the Higgs sector. As can be expected, for the
first model electrically charged configurations appear due to the CFJ term
in the same manner as in MCSH model. For the second model, electric and
magnetic sectors are coupled due to parity-odd coefficients presents in the
electromagnetic CPT-even and Lorentz-violating term. In the following
sections, we analyze in detail both LV scenarios for
Abrikosov-Nielsen-Olesen (ANO) vortices.

\section{TOPOLOGICAL CHARGED BPS VORTICES IN A CPT-ODD AND LORENTZ-VIOLATING
FRAMEWORK}

We consider the Abelian sector of the SME, which corresponds to the MH model
supplemented with Lorentz-violating terms \cite{Colladay2}. Our first
analysis is restricted to the photon sector, including only the CFJ term
\cite{Jackiw}, while the Higgs sector is modified by a CPT-even term,
\begin{eqnarray}
\mathcal{L} &=&-\frac{1}{4}F_{\alpha \beta }F^{\alpha \beta }-\frac{1}{4
\epsilon ^{\alpha \beta \rho \sigma }\left( k_{AF}\right) _{\alpha }A_{\beta
}F_{\rho \sigma }  \label{Lag_CFJ} \\
&&+\left\vert D_{\mu }\phi \right\vert ^{2}+\left( k_{\phi \phi }\right)
^{\mu \nu }\left( D_{\mu }\phi \right) ^{\ast }\left( D_{\nu }\phi \right)
-U\left( \left\vert \phi \right\vert \right) ,  \notag
\end{eqnarray
where $A_{\mu }$ is the Abelian gauge field, and $F_{\mu \nu }=\partial
_{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the electromagnetic strength
tensor. The four-vector $\left( k_{AF}\right) _{\alpha }$ is the CFJ
background with mass dimension +1. The coupling between the gauge field and
Higgs field $\phi $ is given by the minimal covariant derivative $D_{\mu
}\phi =\partial _{\mu }\phi -ieA_{\mu }\phi $, where $e$ is the electric
charge. The LV in the Higgs sector is ruled by the CPT-even real symmetric
and dimensionless tensor $\left( k_{\phi \phi }\right) ^{\mu \nu }$. The
stationary Gauss law i
\begin{equation}
\ \partial _{j}\partial _{j}A_{0}-\left( k_{AF}\right) _{i}B_{i}=-e\mathcal{
}_{0},  \label{gauss}
\end{equation
with $\mathcal{J}_{0}=\left[ 1+\left( k_{\phi \phi }\right) _{00}\right]
J_{0}-\left( k_{\phi \phi }\right) _{0i}J_{i}$, and $J_{\mu }=i\left[ \phi
\left( D_{\mu }\phi \right) ^{\ast }-\phi ^{\ast }D_{\mu }\phi \right] $ is
the conserved current density in absence of LV. Equation (\ref{gauss}) shows
clearly that the vectorial back-ground $\left( k_{AF}\right) _{i}$ is
responsible for the coupling between electric and magnetic sectors, allowing
the occurrence of electrically charged vortex solutions.

Similarly that happens in the MCSH model \cite{MCS,Bolog}, we must modify
the model in (\ref{Lag_CFJ}) by introducing a neutral scalar field $\Psi $
such that the new model supporting vortex solutions is
\begin{eqnarray}
\mathcal{L} &=&\mathcal{-}\frac{1}{4}F_{\alpha \beta }F^{\alpha \beta }
\frac{1}{4}\epsilon ^{\alpha \beta \rho \sigma }\left( k_{AF}\right)
_{\alpha }A_{\beta }F_{\rho \sigma }+\left\vert D_{\mu }\phi \right\vert ^{2}
\notag \\
&&+\left( k_{\phi \phi }\right) ^{\mu \nu }\left( D_{\mu }\phi \right)
^{\ast }\left( D_{\nu }\phi \right) +\frac{1}{2}\partial _{\mu }\Psi
\partial ^{\mu }\Psi  \\
&&-e^{2}\left[ 1+\left( k_{\phi \phi }\right) _{00}\right] \Psi
^{2}\left\vert \phi \right\vert ^{2}-U\left( \left\vert \phi \right\vert
,\Psi \right) ,  \notag
\end{eqnarray
and the potential $U\left( \left\vert \phi \right\vert ,\Psi \right) $
providing charged BPS vortices is
\begin{equation}
U=\frac{1}{2}\left[ ev^{2}-e\eta \left\vert \phi \right\vert ^{2}-\left(
k_{AF}\right) _{3}\Psi \right] ^{2},
\end{equation
where $\eta $ is defined in terms of the parity-even components of the
Lorentz-violating Higgs tensor
\begin{equation}
\eta =\sqrt{\left[ 1-\left( k_{\phi \phi }\right) _{\theta \theta }\right]
\left[ 1-\left( k_{\phi \phi }\right) _{rr}\right] }.
\end{equation
It can be observed that this potential provides two vacua states, one
symmetric $\left[ \phi =0,\,~\Psi =ev^{2}/\left( k_{AF}\right) _{3}\right] $
and another asymmetric, responsible for generating topological BPS vortices\
$\left[ \Psi =0\mathbf{,\ }\left\vert \phi \right\vert =v/\eta ^{1/2}\right]
$. The Higgs field vacuum expectation value gains contributions from
Lorentz-violating terms introduced in Higgs sector.

\subsection{BPS formalism}

We are interested to find self-dual equations or BPS \cite{BPS} which are
first-order differential equations whose solutions also solve the
second-order Euler-Lagrange equations guaranteeing minimum energy
configurations. To carry out the BPS procedure, we first calculate the stationary
energy density of the model,
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}B^{2}+\frac{1}{2}\left[ ev^{2}-e\eta \left\vert
\phi \right\vert ^{2}-\left( k_{AF}\right) _{3}\Psi \right] ^{2}  \notag \\%
[0.2cm]
&&+\left[ \delta _{jk}-\left( k_{\phi \phi }\right) _{jk}\right] \left(
D_{j}\phi \right) ^{\ast }\left( D_{k}\phi \right) +\frac{1}{2}\left(
\partial _{k}A_{0}\right) ^{2}  \label{ED1} \\[0.2cm]
&&+\frac{1}{2}\left( \partial _{k}\Psi \right) ^{2}+e^{2}\left[ 1+\left(
k_{\phi \phi }\right) _{00}\right] \left[ \left( A_{0}\right) ^{2}+\Psi ^{2
\right] \left\vert \phi \right\vert ^{2},  \notag
\end{eqnarray
which is positive definite as long as the matrix $\delta _{jk}-\left(
k_{\phi \phi }\right) _{jk}$ is positive definite and $\left( k_{\phi \phi
}\right) _{00}>-1$.

In a similar way as it is done in MCSH model \cite{CSV1,CSV2}, the next step
in the BPS procedure is to impose the condition ${\Psi =\mp {A}_{0},}$ in
the energy density (\ref{ED1}), which becomes
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}B^{2}+\frac{1}{2}\left[ ev^{2}-e\eta \left\vert
\phi \right\vert ^{2}\pm \left( k_{AF}\right) _{3}A_{0}\right] ^{2}  \notag
\\[0.2cm]
&&+\left[ \delta _{jk}-\left( k_{\phi \phi }\right) _{jk}\right] \left(
D_{j}\phi \right) ^{\ast }\left( D_{k}\phi \right)  \label{Hstat} \\[0.2cm]
&&+\left( \partial _{k}A_{0}\right) ^{2}+2e^{2}\left[ 1+\left( k_{\phi \phi
}\right) _{00}\right] \left( A_{0}\right) ^{2}\left\vert \phi \right\vert
^{2}.  \notag
\end{eqnarray}

We now introduce a modified vortex ansatz \cite{Carlison1} which considers
Lorentz-violating contributions
\begin{eqnarray}
\phi &=&\frac{v}{\sqrt{\eta }}g\exp \left( i\theta \frac{n}{\Lambda }\right)
,  \notag \\[-0.2cm]
&&  \label{newaz} \\
A_{\theta } &=&-\frac{1}{er}\left( a-\frac{n}{\Lambda }\right) \ \text{\ },\
\text{\ }A_{0}=\omega \left( r\right) ,  \notag
\end{eqnarray
where $n$ is called the winding number, which expresses the topological
character of fields. The parameter $\Lambda $ depends only on the
parity-even components of the matrix ruling LV in the Higgs sector
\begin{equation}
\Lambda =\sqrt{\frac{1-\left( k_{\phi \phi }\right) _{\theta \theta }}
1-\left( k_{\phi \phi }\right) _{rr}}}.
\end{equation
The scalar functions $g\left( r\right) $, $a\left( r\right) $ and $\omega
\left( r\right) $ are regular, satisfying the boundary conditions
\begin{eqnarray}
g\left( 0\right) &=&0\ \ ,\ \ a\left( 0\right) =\frac{n}{\Lambda }\ \ ,\ \
\omega ^{\prime }\left( 0\right) =0~,  \notag \\[-0.2cm]
&&  \label{BC} \\
g\left( \infty \right) &=&1\ \ ,\ \ a\left( \infty \right) =0\ \ ,\ \omega
\left( \infty \right) =0~,  \notag
\end{eqnarray
these will be explicitly established below in Sec. \ref{BC-cfj}.

Under (\ref{newaz}), the Gauss law is now written a
\begin{equation}
\omega ^{\prime \prime }+\frac{\omega ^{\prime }}{r}-\left( k_{AF}\right)
_{3}B=2e^{2}v^{2}\Lambda \Delta \omega g^{2},  \label{gauss2}
\end{equation
where the magnetic field is $B(r)=\displaystyle-\frac{a^{\prime }}
er}$, and the Lorentz-violating parameter $\Delta $ is
given b
\begin{equation}
\Delta =\frac{1+\left( k_{\phi \phi }\right) _{00}}{\Lambda \eta }>0\mathbf{
}
\end{equation}

By replacing the ansatz (\ref{newaz}) in (\ref{Hstat}) and by using Eq.
(\ref{gauss2}) the energy density becomes written as a sum of square terms
plus one term proportional to the magnetic field and a total divergence,
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}\left( \frac{{}}{{}}B\mp \left[ ev^{2}\left(
1-g^{2}\right) \pm \left( k_{AF}\right) _{3}\omega \right] \right) ^{2} \\%
[0.2cm]
&&\hspace{-0.5cm}+\frac{v^{2}}{\Lambda }\left( g^{\prime }\mp \Lambda \frac
ag}{r}\right) ^{2}\pm ev^{2}B\pm v^{2}\frac{\left( ag^{2}\right) ^{\prime }}
r}+\frac{\left( r\omega \omega ^{\prime }\right) ^{\prime }}{r}.  \notag
\end{eqnarray}

The energy will be minimized by equating quadratic terms to zero, obtaining
the self-dual equations
\begin{equation}
g^{\prime }=\pm \Lambda \frac{ag}{r},  \label{AD}
\end{equation
\vspace{-0.5cm}
\begin{equation}
B=-\frac{a^{\prime }}{er}=\pm ev^{2}\left( 1-g^{2}\right) +\left(
k_{AF}\right) _{3}\omega ,  \label{B}
\end{equation
both together Gauss's law in Eq. (\ref{gauss2}) describe electrically
charged configurations which will be referred as CPT-odd vortices. The first
of these equations, (\ref{AD}), has modifications due to LV present in the
Higgs sector.

Under BPS equations, the energy density reads
\begin{equation}
\mathcal{E}_{BPS}=\pm ev^{2}B\pm v^{2}\frac{\left( ag^{2}\right) ^{\prime }}
r}+\frac{\left( r\omega \omega ^{\prime }\right) ^{\prime }}{r}~,
\end{equation
whose integration under boundary conditions (\ref{BC}) provides the total
BPS energ
\begin{equation}
E_{BPS}=\pm ev^{2}\int d^{2}r~B=\pm ev^{2}\Phi =\pm 2\pi v^{2}\frac{n}
\Lambda }\ ,  \label{flux_cfj}
\end{equation
which remains proportional to the magnetic flux, which, besides being
proportional to the winding number, also depends explicitly on LV belonging
to the Higgs sector.

We use BPS equations and Gauss's law to express energy density as a sum of
positive terms
\begin{equation}
\mathcal{E}_{BPS}=B^{2}+2\Lambda v^{2}\left( \frac{ag}{r}\right)
^{2}+2e^{2}v^{2}\Lambda \Delta \left( \omega g\right) ^{2}+\left( \omega
^{\prime }\right) ^{2}~\text{,}
\end{equation
it is positive definite because $\Lambda ,\Delta >0.$

Also from Gauss's law in (\ref{gauss2}), the total charge of the self-dual
vortices i
\begin{equation}
Q=2\pi \int_{0}^{\infty }dr~r~q_{0}=\frac{\left( k_{AF}\right) _{3}}{e}\Phi
=2\pi \frac{\left( k_{AF}\right) _{3}}{e}\frac{n}{\Lambda },
\label{charge_cfj}
\end{equation
where $q_{0}=-2ev^{2}\Lambda \Delta \omega g^{2},$ is the electric charge
density. The result is that the total electric charge is proportional to
n/\Lambda $\ just as the magnetic flux is.

\subsection{Asymptotic behavior\label{BC-cfj}}

The asymptotic behavior is studied by solving BPS equations (\ref{AD}) and (\re
{B}) together with Gauss's law (\ref{gauss2}) at the limits $r\rightarrow0$
and $r\rightarrow\infty$.

At the origin, the field profiles has the following behavior:
\begin{eqnarray}
g\left( r\right) &=&Gr^{n}+~...~,  \label{g(0)} \\[0.2cm]
a\left( r\right) &=&\frac{n}{\Lambda }-\frac{1}{2}[ ev^{2} +(k_{AF})
_{3}\omega _{0}] er^{2}+~...~,  \label{a(0)} \\[0.2cm]
\omega \left( r\right) &=&\omega _{0}+\frac{1}{4}[ ev^{2}+( k_{AF})\omega
_{0}] ( k_{AF})_{3}r^{2}+...,  \label{A(0)}
\end{eqnarray
where $\omega _{0}=\omega \left( 0\right) $. Equation (\re
{a(0)}) describes the behavior of $a\left( r\right) $ close to origin, and it
justifies the new ansatz imposed in Eq. (\ref{newaz}). On the other hand,
Eq. (\ref{A(0)}) shows explicitly that the electric field at the origin must
be null, $\omega ^{\prime }\left( 0\right) =0$.

It can be shown this model supports ANO vortex solutions whose
characteristic behavior, at $r\rightarrow \infty $, for field profiles is
\begin{eqnarray}
1-g\left( r\right) &\sim &r^{-1/2}e^{-\beta r}\ \text{,}\   \notag \\[0.08in]
a\left( r\right) &\sim &r^{1/2}e^{-\beta r}\ \text{,}\   \label{anocc} \\%
[0.08in]
\omega \left( r\right) &\sim &r^{-1/2}e^{-\beta r}~\text{,}  \notag
\end{eqnarray
where $\beta $ is a real and positive parameter related to the spatial
extension of the vortex, given b
\begin{eqnarray}
\beta &=&\frac{1}{2}\sqrt{\left[ \left( k_{AF}\right) _{3}\right]
^{2}+2e^{2}v^{2}\Lambda \left( \sqrt{\Delta }+1\right) ^{2}}  \label{BETA} \\%
[0.2cm]
&&-\frac{1}{2}\sqrt{\left[ \left( k_{AF}\right) _{3}\right]
^{2}+2e^{2}v^{2}\Lambda \left( \sqrt{\Delta }-1\right) ^{2}}.  \notag
\end{eqnarray
This shows that for fixed $\left( k_{AF}\right) _{3}$, the asymptotic
behavior of the vortex profiles is ruled by the Lorentz-violating parameters belonging to
the Higgs sector. Under such circumstance we have two interesting
situations: The first one occurs when $\Delta =1$, $\Lambda =1$, and $\beta $
behaves as
\begin{equation}
\beta \rightarrow \frac{1}{2}\sqrt{\left[ \left( k_{AF}\right) _{3}\right]
^{2}+8e^{2}v^{2}}-\frac{1}{2}\left\vert \left( k_{AF}\right) _{3}\right\vert
.
\end{equation
Such a value is exactly the same ruling the asymptotic behavior of MCSH
vortices. The second interesting case occurs when $\Delta $ takes
sufficiently large values ($\Delta \gg \left( k_{AF}\right) _{3}$):
\begin{equation}
\beta \rightarrow ev\Lambda ^{1/2}\sqrt{2},
\end{equation
This value corresponds to the usual MH model with Lorentz violation only in
the Higgs sector, and it can be verified explicitly in Ref. \cite{Carlison1}.
On the other hand, by fixing the Lorentz-violating parameters of the Higgs sector and
sufficiently large values of $\left( k_{AF}\right) _{3}$, $\beta $ behaves a
\begin{equation}
\beta \rightarrow \frac{2e^{2}v^{2}}{\left( k_{AF}\right) _{3}}\Lambda \sqrt
\Delta }~,
\end{equation
which is the mass scale of a CSH model with Lorentz-violating terms only in the Higgs
sector.

\subsection{Numerical analysis}

In order to perform numerical analysis, we use dimensionless coordinates and
fields. We define the dimensionless coordinate $\rho =ev\Lambda ^{1/2}r$ and
perform the following field redefinitions
\begin{eqnarray}
& \displaystyle{g\left( r\right) \rightarrow g\left( \rho \right)
\;,\;\;a\left( r\right) \rightarrow \frac{a\left( \rho \right) }{\Lambda
\;,\;\;\omega \left( r\right) \rightarrow \frac{v}{\Lambda ^{1/2}}\omega
\left( \rho \right) ,} &  \notag \\[0.06in]
&\displaystyle{B\left( r\right) \rightarrow ev^{2}B\left( \rho \right)
\;,\;\;\mathcal{E}_{BPS}\left( r\right) \rightarrow \frac{v^{2}}{\Lambda
\mathcal{E}_{BPS}\left( \rho \right) ,}&  \label{redef} \\[0.08in]
&\displaystyle{\left( k_{AF}\right) _{3}\rightarrow ev\Lambda ^{1/2}\kappa .
&  \notag
\end{eqnarray}

With these new definitions, we rewrite the BPS equations and Gauss's law:
\begin{eqnarray}
&\displaystyle{g^{\prime }=\pm \frac{ag}{\rho }~,}&  \label{xxq1} \\[0.2cm]
&\displaystyle{-\frac{a^{\prime }}{\rho }=\pm \left( 1-g^{2}\right) +\kappa
\omega ~,}&  \label{xxq2} \\
&\displaystyle{\omega ^{\prime \prime }+\frac{\omega ^{\prime }}{\rho
+\kappa \frac{a^{\prime }}{\rho }=2\Delta \omega g^{2}~\text{.}}&
\label{xxq3}
\end{eqnarray
It can be observed that if $\Delta =1$, Eqs. (\ref{xxq1}-\ref{xxq3}) looks
exactly like the ones in MCSH model. The set of equations will be solved by
using the following boundary condition
\begin{eqnarray}
g\left( 0\right) &=&0\ \text{,}\ a\left( 0\right) =n\ \text{,}\ \omega
^{\prime }\left( 0\right) =0~\text{,} \\[0.2cm]
g\left( \infty \right) &=&1\ \text{,}\ a\left( \infty \right) =0\ \text{,}\
\omega \left( \infty \right) =0.  \label{bccfj}
\end{eqnarray}

Also the BPS energy density is expressed by
\begin{equation}
\mathcal{E}_{BPS}\left( \rho \right) =B^{2}+2\left( \frac{ag}{\rho }\right)
^{2}+2\Delta \left( \omega g\right) ^{2}+\left( \omega ^{\prime }\right)
^{2};  \label{defen}
\end{equation}
it will be positive definite whenever $\Delta>0$.

The dimensionless $\beta $ mass scale reads
\begin{eqnarray}
\beta &=&\frac{1}{2}\sqrt{\kappa ^{2}+2\left( \sqrt{\Delta }+1\right) ^{2}}
\label{BETA1} \\[0.15cm]
&&-\frac{1}{2}\sqrt{\kappa ^{2}+2\left( \sqrt{\Delta }-1\right) ^{2}}.
\notag
\end{eqnarray}

Below, we depict the profiles obtained from numerical solutions of Eqs. (\re
{xxq1}--\ref{xxq3}) under boundary conditions (\ref{bccfj}) for $\kappa =1$,
$n=1$ and some values for $\Delta $. Because the BPS energy density (\re
{defen}) is positive definite for $\Delta >0$, we consider two regions
0<\Delta <1$ (green lines) and $\Delta >1$ (red lines), in which the behavior
of solutions are different. There are two interesting values of $\Delta $
which allow to obtain two well-know models: The first one is $\Delta =1$
(solid black lines) whose profiles correspond to MCSH model, and the second
value is the limit $\Delta \rightarrow \infty ,$ when MH model (solid blue
lines) is attained.

Figures \ref{S_BPSx} and \ref{A_BPS} depict the profiles of the Higgs and vector
fields, respectively. It is observed that they reach the asymptotic values
rapidly for $\Delta >1$ and saturate more slowly for $\Delta <1,$ as is
expected from Eq. (\ref{BETA1}) for fixed $\kappa $. Also, it is verified
explicitly that the solutions for $1<\Delta <\infty $ (red lines) are
confined between the MCSH (solid black line) and MH (solid blue line)
models. Such that the profiles are narrower when $\Delta >1$, and the
maximum narrowing is attained when $\Delta \rightarrow \infty $ (solid blue
line) where the MH model is recovered. On the other hand, for $0<\Delta <1$, the
profiles get more spread out, occupying a larger area when $\Delta
\rightarrow 0$. In this limit, the vector potential has a very light decay,
by which we can infer that the magnetic field corresponding to this values
will have low intensity.

\begin{figure}[H]
\centering\includegraphics[width=8.25cm]{01_Higgs_mcfj.eps}
\caption{Scalar field $g(\protect\rho )$.}
\label{S_BPSx}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=8.25cm]{02_vetorial_mcfj.eps}
\caption{Vector potential ${a}(\protect\rho )$. }
\label{A_BPS}
\end{figure}

\begin{figure}[H]
\centering\includegraphics[width=8.25cm]{03_magnetico_mcfj.eps}
\caption{Magnetic field ${B}(\protect\rho )$.}
\label{B_BPS}
\end{figure}

Figure \ref{B_BPS} shows the profiles for the magnetic field which are lumps
centered at the origin. For increasing values of $\Delta $, the profiles are
narrower and have higher amplitude in such a way the maximum narrowing and
maximum intensity are reached when $\Delta \rightarrow \infty $ (solid blue
line). For $0<\Delta <1$, the profiles have less intensity and are more
spread when $\Delta \rightarrow 0$. Higher amplitudes are obtained for the
values $1<\Delta <\infty $; such profiles are located between the MCSH (
\Delta =1$, solid black line) and the MH profile ($\Delta \rightarrow \infty
$, solid blue line). At infinity, the magnetic field vanishes more rapidly as
$\Delta $ increases.

\begin{figure}[H]
\centering
\includegraphics[width=8.25cm]{04_escalar_mcfj.eps}
\caption{Scalar potential $\protect\omega (\protect\rho )$.}
\label{W_BPS}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=8.25cm]{05_eletrico_mcfj.eps}
\caption{Electric field $\protect\omega ^{\prime }(\protect\rho )$.}
\label{El_BPS}
\end{figure}

Figure \ref{W_BPS} depicts the profiles for the scalar potential. For $\Delta
>1$, the CFJ parameter is less relevant for increasing values of $\Delta $,
such that the profiles become smaller, and in the limit $\Delta \rightarrow
\infty $ (solid blue line), the profile overlaps the horizontal axis, implying
that vortices become electrically uncharged such as in the MH model. On the
other hand, in the interval $0<\Delta <1$, the scalar potential profiles are
more extended and with greater intensity at the origin. For this range of
\Delta $, the CFJ parameter becomes more relevant when $\Delta \rightarrow 0$.

\begin{figure}[H]
\centering
\includegraphics[width=8.25cm]{06_energia_mcfj.eps}
\caption{Energy density $\mathcal{E}_{BPS}(\protect\rho )$.}
\label{Energy_BPS}
\end{figure}

Figure \ref{El_BPS} describes the behavior for the electric field. For $n=1$,
the maximum electric field intensity is reached for $\Delta =1$ (solid black
line), corresponding to MCSH model. The electric field gets radially spread out
for decreasing values of $\Delta $. On the other hand, for $\Delta >1,$\ the
electric field is located closer to the origin and decays rapidly for
increasing values of $\Delta $. In the limit $\Delta \rightarrow \infty ,$
the electric field disappears, which result agrees with electrically
uncharged vortex models.

Figure \ref{Energy_BPS} describes BPS energy density profiles, which for $n=1$
are lumps centered at origin. For increasing values of $\Delta ,$\ we have
higher amplitudes at the origin which sharply decay, reaching zero rapidly, which
corresponds to well-located vortices. On the other hand, when $\Delta
\rightarrow 0$, amplitude decreases at the origin and reaches zero slowly,
implying a more extended, large energy density distribution but with
little intensity. The profiles of BPS energy density have the same behavior as the magnetic field profiles.

\section{TOPOLOGICAL CHARGED BPS VORTICES IN A CPT-EVEN AND LORENTZ-VIOLATING FRAMEWORK}

In this section, we investigate the formation of topological
electrically charged BPS vortices in a Lorentz-violating MH model, where
both the gauge and Higgs sector contain CPT-even and Lorentz-violating terms
\cite{Colladay2}. Specifically, we analyze the nonbirefringent Lorentz violating
coefficients of the electromagnetic sector of SME. This model is a natural
generalization of earlier studies performed in Refs.\cite{Carlison1,Carlison2}.
The Lagrangian density for this model describing charged vortices is
\begin{eqnarray}
\mathcal{L} &=&-\frac{1}{4}F_{\mu \nu }F^{\mu \nu }-\frac{1}{2}\kappa ^{\rho
\alpha }F_{\rho \sigma }F_{\alpha }^{\text{ \ }\sigma }+\left\vert D_{\mu
}\phi \right\vert ^{2}  \notag \\[0.2cm]
&&+\left( k_{\phi \phi }\right) ^{\mu \nu }\left( D_{\mu }\phi \right)
^{\ast }\left( D_{\nu }\phi \right) -U\left( \left\vert \phi \right\vert
\right) ~\text{.}  \label{Lag2}
\end{eqnarray
In stationary regime, Gauss's and Ampere's law are
\begin{eqnarray}
L_{ij}\partial _{i}\partial _{j}A_{0}+\epsilon _{ija}\kappa _{0i}\partial
_{j}B_{a} &=&-e\mathcal{J}_{0}~\text{,} \\[0.2cm]
{M}_{jbc}\partial _{b}B_{c}-\kappa _{0i}\partial _{j}\partial
_{j}A_{0}+\kappa _{0j}\partial _{j}\partial _{i}A_{0} &=&e\mathcal{J}_{i}
\text{,}
\end{eqnarray
respectively. The matrix $L_{ij}=\left( 1+\kappa _{00}\right)
\delta _{ij}-\kappa _{ij}$ is symmetric and positive definite
which carries parity-even and CPT-even Lorentz-violating coefficients. The density
\mathcal{J}_{0}$ is the same defined after Eq. (\ref{gauss}). In
Ampere's law, we have defined the tensor $M_{jbc}=\epsilon _{jbc}-\epsilon
_{jac\ }\kappa _{ab}-\kappa _{ja}\epsilon _{abc}$, and current
density $\mathcal{J}_{i}=J_{i}-\left( k_{\phi \phi }\right)
_{ij}J_{j}+\left( k_{\phi \phi }\right) _{0i}J_{0}$, with $J_{\mu }
 also defined after Eq. (\ref{gauss}). We observe that parity-odd
coefficients $\kappa _{0i}$ and $\left( k_{\phi \phi
}\right) _{0i}$ are responsible for electric and magnetic sector
coupling however some $\left( k_{\phi \phi }\right) _{0i}\ $ do not
contribute to the formation of vortex solutions.

As was shown in Ref. \cite{Carlison2}, the model (\ref{Lag2}) must be
modified to supports BPS solutions:
\begin{eqnarray}
\mathcal{L} &=&\mathcal{-}\frac{1}{4}F_{\mu \nu }F^{\mu \nu }-\frac{1}{2
\kappa ^{\rho \alpha }F_{\rho \sigma }F_{\alpha }{}^{\sigma }  \notag \\%
[0.2cm]
&&+\left\vert D_{\mu }\phi \right\vert ^{2}+\left( k_{\phi \phi }\right)
^{\mu \nu }\left( D_{\mu }\phi \right) ^{\ast }\left( D_{\nu }\phi \right)
\notag \\[-0.2cm]
&&  \label{Lag MHpara} \\
&&+\frac{1}{2}\left( 1+\kappa _{00}\right) \partial _{\mu }\Psi \partial
^{\mu }\Psi +\frac{1}{2}\kappa ^{\mu \nu }\partial _{\mu }\Psi \partial
_{\nu }\Psi  \notag \\[0.2cm]
&&-e^{2}\left[ 1+\left( k_{\phi \phi }\right) _{00}\right] \Psi
^{2}\left\vert \phi \right\vert ^{2}-U\left( \left\vert \phi \right\vert
,\Psi \right) ~\text{.}  \notag
\end{eqnarray}

The interaction term $U\left( \left\vert \phi \right\vert ,\Psi \right) $
given by
\begin{equation}
U\left( \left\vert \phi \right\vert ,\Psi \right) =\frac{\left( ev^{2}-e\eta
\left\vert \phi \right\vert ^{2}-\epsilon _{ij}\kappa _{0i}\partial _{j}\Psi
\right) ^{2}}{2\left( 1-s\right) }  \label{P}
\end{equation
contains a derivative coupling and a $\left\vert \phi \right\vert ^{4}-
potential, which provides BPS configurations. Above we have define the
parameter $s=\kappa _{11}+\kappa _{22}=\kappa _{rr}+\kappa _{\theta \theta }
.

\subsection{BPS formalism}

In order to implement the BPS formalism, we first write down the stationary
energy densit
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}L_{ij}\left( \partial _{i}A_{0}\right) \left(
\partial _{j}A_{0}\right) +\frac{1}{2}\left( 1-s\right) B^{2}  \notag \\%
[0.08in]
&&\hspace{-0.7cm}+\left[ \delta _{ij}-\left( k_{\phi \phi }\right) _{ij
\right] \left( D_{i}\phi \right) ^{\ast }\left( D_{j}\phi \right) +\frac{1}{
}L_{ij}\partial _{i}\Psi \partial _{j}\Psi  \label{E_Xa} \\[0.08in]
&&\hspace{-0.7cm}+e^{2}\left[ 1+\left( k_{\phi \phi }\right) _{00}\right]
\left[ \left( A_{0}\right) ^{2}+\Psi ^{2}\right] \left\vert \phi \right\vert
^{2}+U\left( \left\vert \phi \right\vert ,\Psi \right) .  \notag
\end{eqnarray
It is positive-definite whenever $s<1$,$\ \left( k_{\phi \phi }\right)
_{00}>-1$, and $\delta _{ij}-\left( k_{\phi \phi }\right) _{ij},$ a
positive-definite matrix.

By following a similar procedure to that used in the previous model, we
impose condition $\Psi =\mp A_{0},$ in the energy density (\re
{E_Xa}), which become
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}\left( 1-s\right) B^{2}+\frac{\left( ev^{2}-e\eta
\left\vert \phi \right\vert ^{2}\pm \epsilon _{ij}\kappa _{0i}\partial
_{j}A_{0}\right) ^{2}}{2\left( 1-s\right) }  \notag \\[0.08in]
&&\hspace{-0.7cm}+\left[ \delta _{ij}-\left( k_{\phi \phi }\right) _{ij
\right] \left( D_{i}\phi \right) ^{\ast }\left( D_{j}\phi \right)
\label{E_X} \\[0.08in]
&&\hspace{-0.7cm}+L_{ij}\left( \partial _{i}A_{0}\right) \left( \partial
_{j}A_{0}\right) +2e^{2}\left[ 1+\left( k_{\phi \phi }\right) _{00}\right]
\left( A_{0}\right) ^{2}\left\vert \phi \right\vert ^{2}.  \notag
\end{eqnarray}

Now we introduce the vortex ansatz (\ref{newaz}) to express Gauss's
law as follows
\begin{equation}
{\left( 1+\lambda _{r}\right) }\left( \omega ^{\prime \prime }+\frac{\omega
^{\prime }}{r}\right) -\kappa _{0\theta }\frac{\left( r{B}\right) ^{\prime
}{r}={2e}^{2}v^{2}\Lambda \Delta {\omega g^{2},}  \label{GD_3}
\end{equation
with $\lambda _{r}=\kappa _{00}-\kappa _{rr}$. Afterwards
rewriting the energy density (\ref{E_X}) by using the ansatz (\ref{newaz}), we
use Eq. (\ref{GD_3}) to express it in the following convenient form
\begin{eqnarray}
\mathcal{E} &=&\frac{1}{2}\left( 1-s\right) \left[ B\mp \frac{ev^{2}\left(
1-g^{2}\right) \mp \kappa _{0\theta }\omega ^{\prime }}{1-s}\right] ^{2}
\notag \\[0.2cm]
&&+\frac{v^{2}}{\Lambda }\left[ ~g^{\prime }\mp \Lambda \frac{ag}{r}\right]
^{2}\pm ev^{2}B\pm v^{2}\frac{\left( ag^{2}\right) ^{\prime }}{r}  \label{E2}
\\[0.2cm]
&&+(1+\lambda _{r})\frac{\left( r\omega \omega ^{\prime }\right) ^{\prime }}
r}-\kappa _{0\theta }\frac{\left( r\omega B\right) ^{\prime }}{r}.  \notag
\end{eqnarray
It is minimized by equating quadratic terms to zero, providing self-dual or
BPS equations
\begin{eqnarray}
&\displaystyle{g^{\prime }=\pm \Lambda \frac{ag}{r}~,}&  \label{GD_1} \\%
[0.2cm]
&\displaystyle{B=-\frac{a^{\prime }}{er}=\pm \frac{ev^{2}\left(
1-g^{2}\right) }{1-s}-\frac{\kappa _{0\theta }\omega ^{\prime }}{1-s}~}&,
\label{GD_2}
\end{eqnarray
both together with Gauss's law in Eq. (\ref{GD_3}) describe electrically neutral
configurations, which will be called as CPT-even vortices. The total
electric charge of the topological vortices is zero, as can explicitly
shown by integrating Gauss's law in Eq. (\ref{GD_3}) under boundary
conditions given in Eqs. (\ref{BCGD1}) and (\ref{BCGD2}):
\begin{equation}
Q={4\pi e}v^{2}\Lambda \Delta \int_{0}^{\infty }dr~r{\omega g^{2}=0.}
\end{equation}

Whenever the BPS equations are satisfy, the BPS energy density is
\begin{eqnarray}
\mathcal{E}_{BPS} &=&\pm ev^{2}B\pm v^{2}\frac{\left( ag^{2}\right) ^{\prime
}}{r}  \label{Eden} \\[0.2cm]
&&+(1+\lambda _{r})\frac{\left( r\omega \omega ^{\prime }\right) ^{\prime }}
r}-\kappa _{0\theta }\frac{\left( r\omega B\right) ^{\prime }}{r},  \notag
\end{eqnarray
whose integration under boundary conditions established in Eqs. (\ref{BCGD1
) and (\ref{BCGD2}) provides total BPS\ energy
\begin{equation}
E_{BPS}=\pm ev^{2}\int Bd^{2}x=\pm 2\pi v^{2}\frac{n}{\Lambda }~,
\label{flux_mhLV}
\end{equation
where the magnetic flux, such as in the previous model, besides being
proportional to the winding number, also depends explicitly on LV belonging
to the Higgs sector.

We use BPS equations and Gauss's law to express energy density as
\begin{eqnarray}
\mathcal{E}_{BPS} &=&\left( 1-s\right) B^{2}+2\Lambda v^{2}\left( \frac{ag}{
}\right) ^{2}  \notag \\[-0.2cm]
&& \\
&&+2e^{2}v^{2}\Lambda \Delta \left( g\omega \right) ^{2}+\left( 1+\lambda
_{r}\right) \left( \omega ^{\prime }\right) ^{2};  \notag
\end{eqnarray
it is positive-definite for $s<1$ and $\lambda _{r}>-1$.

\subsection{Boundary conditions}

By solving BPS\ equations\ (\ref{GD_1} and \ref{GD_2}) and Gauss's law (\re
{GD_3}) at $r=0$, we obtain
\begin{eqnarray}
g\left( r\right) &=&G_{n}r^{n}+\mathcal{\ldots },  \label{gg(r)} \\[0.2cm]
a\left( r\right) &=&{\frac{n}{\Lambda }}-\,{\frac{{e}^{2}{v}^{2}\left(
1+\lambda _{r}\right) ~}{2\Sigma }r}^{2}+\mathcal{\ldots },  \label{aa(r)} \\%
[0.2cm]
\omega \left( r\right) &=&\omega _{0}+{\frac{e{v}^{2}\kappa _{0\theta }}
\Sigma }}~r+\mathcal{\ldots },  \label{ww(r)}
\end{eqnarray
where $\omega _{0}=\omega \left( 0\right) $ and
\begin{equation}
\Sigma =\left( \kappa _{0\theta }\right) ^{2}+\left( 1-s\right) \left(
1+\lambda _{r}\right).  \label{sigma}
\end{equation
Therefore, the boundary conditions at the origin are
\begin{equation}
g\left( 0\right) =0,~a\left( 0\right) =\frac{n}{\Lambda },~\omega ^{\prime
}\left( 0\right) =\frac{e{v}^{2}\kappa _{0\theta }}{\Sigma }  \label{BCGD1}
\end{equation}

The model also supports ANO vortices, whose behavior when $r\rightarrow
\infty $ is given by Eq. (\ref{anocc}), so the boundary conditions for the
field profiles at infinity are
\begin{equation}
g\left( \infty \right) =1\ ,\ \ a\left( \infty \right) =0\ ,\ \ \omega
\left( \infty \right) =0,  \label{BCGD2}
\end{equation
but now the mass scale $\beta $ is
\begin{equation}
\beta =ev\Lambda ^{1/2}\sqrt{\frac{\beta _{1}\pm \beta _{2}}{\Sigma }},
\label{betax}
\end{equation
where
\begin{eqnarray}
\beta _{1} &=&\left( 1+\lambda _{r}\right) +\Delta \left( 1-s\right) , \\%
[0.3cm]
\beta _{2} &=&\sqrt{\left[ \left( 1+\lambda _{r}\right) -\Delta \left(
1-s\right) \right] ^{2}-4\Delta \left( \kappa _{0\theta }\right) ^{2}},
\end{eqnarray
and $\beta _{2},\;\beta _{2}$ are positive real numbers. For $\beta _{2}$, the
condition $4\Delta \left( \kappa _{0\theta }\right) ^{2}\leq \left[ \left(
1+\lambda _{r}\right) -\Delta \left( 1-s\right) \right] ^{2}$ must be
satisfied. The signals will be used as follows: $+\left( -\right) $ for
\left( 1+\lambda _{r}\right) -\Delta \left( 1-s\right) >0\left( <0\right) $.

When $\kappa _{0\theta }=0$, the Eq. (\ref{betax}) recovers the mass scale
of electrically uncharged BPS vortices,
\begin{equation}
\beta =ev\Lambda ^{1/2}\sqrt{\frac{2}{1-s}},
\end{equation
already obtained in Ref. \cite{Carlison1}.

On the other hand, when $\beta _{2}=0$, the parity-odd coefficient can be
expressed in terms of parity-even ones,
\begin{equation}
\kappa _{0\theta }=\pm\frac{\left\vert 1+\lambda _{r}-\Delta \left(
1-s\right) \right\vert }{2\sqrt{\Delta }},
\end{equation
and the mass scale become
\begin{equation}
\beta =\ ev\Lambda ^{1/2}\sqrt{\frac{4\Delta }{\left( 1+\lambda _{r}\right)
+\Delta \left( 1-s\right) }}.
\end{equation}

Another interesting possibility is to set $s=0$ and $\lambda _{r}=0$ in Eq.
\ref{betax}); i.e., we can consider null parity-even electromagnetic
coefficients, getting
\begin{equation}
\beta =ev\Lambda ^{1/2}\sqrt{\frac{1+\Delta \pm \sqrt{\left( 1-\Delta
\right) ^{2}-4\Delta \left( \kappa _{0\theta }\right) ^{2}}}{1+\left( \kappa
_{0\theta }\right) ^{2}}}\ ,  \label{betax2}
\end{equation
with signal $+\left( -\right) $ for $1-\Delta >0\left( <0\right) $,
respectively. This situation also provides ANO-like vortices whenever
\begin{equation}
\left( 1-\Delta \right) ^{2}\geq 4\Delta \left( \kappa _{0\theta }\right)
^{2}
\end{equation
is satisfied. We remark that in the absence of LV in Higgs sector ($\Delta
=1 $) (see Ref. \cite{Carlison2}), it is impossible to obtain ANO vortices
when $s$\ and $\lambda _{r}$ are nulls, because the mass scale (\ref{betax2})
becomes a complex number.

\subsection{Numerical analysis}

Under the same coordinate rescaling and field redefinitions as in (\re
{redef}) but with $\kappa _{0\theta }\rightarrow \kappa $, the dimensionless
BPS equations and Gauss's law are
\begin{eqnarray}
&\displaystyle{g^{\prime }=\pm \frac{ag}{\rho }~\ ,}&  \label{eqt2a} \\%
[0.2cm]
&\displaystyle{-\frac{a}{\rho }=\pm \frac{\left( 1-g^{2}\right) }{\left(
1-s\right) }-\frac{\kappa }{\left( 1-s\right) }\omega ^{\prime },}&
\label{eqt2b} \\[0.2cm]
&\displaystyle{\left( 1+\lambda _{r}\right) \left( \omega ^{\prime \prime }
\frac{\omega ^{\prime }}{\rho }\right) -{\kappa }\frac{\left( \rho {B
\right) ^{\prime }}{\rho }=2\Delta g^{2}\omega .}&  \label{eqt2c}
\end{eqnarray
We can see that for $\kappa \rightarrow -\kappa $, the solutions change as
g\rightarrow g$, $a\rightarrow a$, $\omega \rightarrow -\omega $.

On the other hand, the BPS energy density is given by
\begin{eqnarray}
\mathcal{E}_{_{BPS}}\left( \rho \right) &=&\left( 1-s\right) B^{2}+2\left(
\frac{ag}{\rho }\right) ^{2}  \notag \\[-0.3cm]
&&  \label{H_DEF_POSx} \\
&&+2\Delta \left( {g\omega }\right) ^{2}+\left( 1+\lambda _{r}\right) \left(
\omega ^{\prime }\right) ^{2},  \notag
\end{eqnarray
is defined positive providing that $s<1$, $\Delta >0$, $\lambda _{r}>-1$.

\subsubsection{A charged vortex configuration}

Now we consider the following configuration to solve numerically the
dimensionless Eqs. (\ref{eqt2a})-(\ref{eqt2c}):
\begin{equation}
\lambda _{r}=0~\ ,~\ s=0~\ ,~\ \kappa =\frac{\Delta -1}{2\sqrt{\Delta }}
~,~\ \Delta >0 .
\end{equation}
In this case, the boundary conditions are expressed as
\begin{eqnarray}
g\left( 0\right) &=&0,~a\left( 0\right) =n,~{\omega }^{\prime }\left(
0\right) ={\frac{2\left( \Delta -1\right) \sqrt{\Delta }}{\left( 1+\Delta
\right) ^{2}}},  \notag \\[-0.15cm]
&&  \label{bcsxx} \\[-0.15cm]
g\left( \infty \right) &=&1,~a\left( \infty \right) =0,~{\omega }\left(
\infty \right) =0.  \notag
\end{eqnarray
The mass scale $\beta $ is given b
\begin{equation}
\beta =2\sqrt{\frac{\Delta }{\left( 1+\Delta \right) }},  \label{betaMH}
\end{equation
and we note that for $\Delta \ll 1$, the defect reaches its asymptotic
values slowly. But when $\Delta \sim 1$, the behavior is close to the MH
profiles (see below, Figs. \ref{HIGGS}--\ref{ED}, solid black lines), because
$\beta \rightarrow \sqrt{2}$, MH mass scale. On the other hand, for $\Delta
\rightarrow \infty $, the mass scale reaches its maximum value $\beta
\rightarrow 2$ (see below, Figs. \ref{HIGGS}-\ref{MF},\ref{EF},\ref{ED},
solid blue lines).

Below, we depict the profiles obtained from numerical solutions of the Eqs.
\ref{eqt2a})-(\ref{eqt2c}) under boundary conditions (\ref{bcsxx}). Without
loss of generality, we set $n=1$. Because the BPS energy density is
positive definite for $\Delta >0$, we consider two regions $0<\Delta <1$
(green lines) and $\Delta >1$ (red lines) in which the behavior of solutions
are different. The solutions are always compared with profiles of MH model
(solid black lines).

\begin{figure}[H]
\centering\includegraphics[width=4.35cm]{01_Higgs_MHLV_a.eps}\!\!
\includegraphics[width=4.25cm]{01_Higgs_MHLV_b.eps}
\caption{ Higgs field ${g}(\protect\rho)$. Left-side represents the profiles
for $0\leq\Delta\leq 1$ and right-side for $\Delta\geq 1$.}
\label{HIGGS}
\end{figure}

Figure \ref{HIGGS} shows the profiles of the Higgs field. For $\Delta \ll 1,$ they
are very spread and reach their asymptotic value slowly. But when $\Delta
\rightarrow 1,$ they are narrower and attain the vacuum state more rapidly.
For $0\leq \Delta \leq 1,$ the profiles are limited by the usual MH solution
$\Delta =1$ (solid black line). For $\Delta >1$, the profiles are wider than
the MH one, but such increment is limited by the profile obtained when
\Delta \rightarrow \infty $ (solid blue line).

\begin{figure}[H]
\centering
\includegraphics[width=4.35cm]{02_vetorial_MHLV_a.eps}\!\
\includegraphics[width=4.25cm]{02_vetorial_MHLV_b.eps}
\caption{Vector potential ${a}(\protect\rho)$.}
\label{A}
\end{figure}

Figure \ref{A} depicts the profiles of the vector potential. For $0<\Delta <1,$
the profiles are wider when $\Delta \rightarrow 0,$ and narrower when
\Delta \rightarrow 1$. The maximum narrowing is attained in $\Delta =1,$
where MH model (solid black line) is recover. For $\Delta >1,$ the profiles
becomes wider when $\Delta $ increases, but the incremented width has a
limit that is attained when $\Delta \rightarrow \infty $ (solid blue line).

\begin{figure}[H]
\centering
\includegraphics[width=4.35cm]{03_magnetico_MHLV_a.eps}\!\
\includegraphics[width=4.25cm]{03_magnetico_MHLV_b.eps}
\caption{Magnetic field ${B}(\protect\rho)$.}
\label{MF}
\end{figure}

The magnetic field profiles are shown in Fig. \ref{MF}, whose amplitudes at
origin are given by $4\Delta /(\Delta +1)^{2}$. For the range $0<\Delta \leq
1$ (green lines), they are lumps centered at origin with increasing
amplitudes whenever $\Delta $ increases, whose maximum value is reached for
\Delta =1$ (MH model). For $\Delta >1,$ the amplitude decreases when $\Delta
$ increases continuously. An interesting fact is observed around $\Delta
\sim 6$: the lump format of the magnetic field begins to deform, transforming
in a ringlike profile. In the limit $\Delta \rightarrow \infty ,$ it has a
null value at the origin such that the ring profiles resemble those of CSH
and MCSH models.
\begin{figure}[H]
\centering
\includegraphics[width=6cm]{06_magnetico_MHLV_inv.eps}
\caption{Magnetic flux inversion.}
\label{MF INV}
\end{figure}
The model has localized magnetic flux inversion when $0\leq \Delta <1,$ such
as it is shown in Fig. \ref{MF INV}, where a zoom was performed for the
profiles with $0.01\leq \Delta \leq 0.5$. One can clearly observe the
localized magnetic flux inversion, which is more pronounced for values
\Delta <0.5$. For $\Delta >0.5,$ such inversion does not exist. This
peculiarity in this Lorentz-violating MH model was also observed in the absence
of LV in Higgs sector \cite{Carlison2}. Thus, we can say that such
localized magnetic flux inversion is a proper effect of CPT-even and
Lorentz-violating terms.

\begin{figure}[H]
\centering
\includegraphics[width=8.25cm]{04_eletrico_MHLV.eps}
\caption{Electric field ${\protect\omega}^{\prime}(\protect\rho)$.}
\label{EF}
\end{figure}

Figure \ref{EF} shows the electric field profiles, which are lumps centered at
origin, whose amplitudes are given by Eq. (\ref{bcsxx}). It is negative for
0<\Delta <1,$ taking its minimum value $-1/2,$ when $\Delta =3-2\sqrt{2}\sim
0.172,$ and it is positive for $\Delta >1,$ taking its maximum value $1/2,$
when $\Delta =3+2\sqrt{2}\sim 5.828$. The electric field is null when
\Delta \rightarrow 0,1,\infty $.

\begin{figure}[H]
\centering
\includegraphics[width=4.35cm]{05_energia_MHLV_a.eps}\!\
\includegraphics[width=4.25cm]{05_energia_MHLV_b.eps}
\caption{BPS energy density ${\mathcal{E}}(\protect\rho)$.}
\label{ED}
\end{figure}

Figure \ref{ED} presents the profiles for the BPS energy density, which are
similar to magnetic field ones due to the direct relation between energy and
magnetic flux. For $0<\Delta <1$, they are lumps centered at the origin whose,
amplitude increases when $\Delta $ increases, attaining its maximum value in
\Delta =1$, which reproduces the BPS energy density of the MH model. For
\Delta >1$, the lumps decrease in their amplitudes, but around $\Delta \sim 6,$
they transform in rings. Such ring-like behavior is maintained for all
values of $\Delta >6$, and their amplitudes at origin decrease whenever
\Delta $ increases continuously such that in the limit $\Delta \rightarrow
\infty ,$ the amplitude is $\sim 0.46$ (but for $n>1$, the amplitude at
origin is zero). The ringlike behavior resembling the CSH and MCSH
behaviors is an interesting effect of LV in the Higgs sector.

\section{REMARKS AND CONCLUSIONS}

We have performed a comprehensive analysis on the formation of charged BPS
vortices in the context of the SME. Two cases were studied separately. First,
we have discussed the Abelian MH model supplemented with the CFJ term in
electromagnetic sector and a CPT-even Lorentz-violating term in the Higgs sector. The second
case analyzed is also the Abelian MH model but this time provided with
CPT-even and Lorentz-violating terms in both the photon and Higgs sectors.
We have verified that for supporting charged BPS vortex solutions the
original models must be modified by introducing a neutral scalar field $\Psi
$, with appropriate dynamic. This procedure is similar to the one that
happens in MCSH model. Notwithstanding, there are some differences between
the CPT-odd and CPT-even charged vortices: In the CPT-odd case, both the
charge density and the total charge are non-null whereas, in the CPT-even
case, the charge density is nonzero, but the total electric charge is null.
Another important result involves the magnetic flux, which, besides being
proportional to the winding number, also depends explicitly in the
Lorentz-violating coefficients belonging to the Higgs sector, such as
presented in Eqs. (\ref{flux_cfj}) and (\ref{flux_mhLV}). This result was
firstly obtained in the context of uncharged BPS vortices in SME \cit
{Carlison1}, and it appears to be a general effect produced by the CPT-even
and Lorentz-violating terms of the Higgs sector.

One of the observations already made in Ref.\cite{Carlison1} is the change of
the vortex ansatz (\ref{newaz}) describing the profiles of the Higgs and
gauge fields. This is an important fact because the boundary condition
satisfied by gauge field at origin [see Eqs. (\ref{BCGD1}) and (\ref{BC})]
is modified by LV belonging to Higgs sector. Otherwise, the solutions of
self-dual or BPS equations will not be topological; neither will have finite
energy. Such change in gauge field boundary conditions are crucial for the
new characteristic presented by the quantized magnetic flux.

Our analysis shows that for fixed LV in photon sector, the LV in Higgs
sector increases the richness of BPS solutions in the context of SME. For
example, the presence of LV in Higgs sector allows to interpolate CPT-odd
vortices between very spread for $0<\Delta <1,$ to some more localized ones
when $\Delta \rightarrow 1$. For $1\leq \Delta <\infty ,$ profiles run from
the MCSH $\left( \Delta =1\right) $ to the MH ($\Delta \rightarrow \infty $)
solutions. For the CPT-even vortices, the profiles in the region $0<\Delta
\leq 1,~$are wider and more spread for decreasing $\Delta $. But now the
value $\Delta =1,$ represents the vortex solution of MH model. For $\Delta
>1$ and increasing $\Delta $, the vortex profiles are confined between the MH
model (solid black line) and the one obtained when ($\Delta \rightarrow
\infty $) (solid blue line). An interesting fact in CPT-even vortices is the
appearance of ringlike profiles (for magnetic field and BPS energy density)
for sufficiently large values of $\Delta $ parameter, as explicitly shown in
Figs. \ref{MF} and \ref{ED}, respectively. Another peculiarity is the
localized magnetic flux inversion, such as was also observed in the
absence of LV in Higgs sector \cite{Carlison2}. This property appears to
be a typical effect of CPT-even and Lorentz-violating terms in photon sector.

A remark on the subject of considering large values for
Lorentz-violation coefficients: In the context of standard model extension, the Lorentz-violating parameters
must be sufficiently small. However, as energy positivity is preserved
whenever $\Delta >0$, we can consider large values of $\Delta $ within an
effective model describing the electrodynamics of some type of material
media.

In this context we, can also comment about implication of
these Lorentz-violating models in superconductivity physics. The new
materials presenting high-temperature superconductivity are
anisotropic; most of them have a layered structure. The
simplest GL model is for an uniaxial superconductor whose free energy
density is written a
\begin{eqnarray}
\mathcal{F} &=&\frac{B^{2}}{8\pi }+\alpha \left\vert \phi \right\vert ^{2}
\frac{\beta }{2}\left\vert \phi \right\vert ^{2}  \label{ffee} \\
&&\hspace{-0.5cm}+\gamma _{jk}\left( i\hbar \partial _{j}\phi ^{\ast }-\frac
2e}{c}A_{j}\phi ^{\ast }\right) \left( -i\hbar \partial _{k}\phi -\frac{2e}{
}A_{k}\phi \right) ,  \notag
\end{eqnarray
where $B$ is the magnetic field produced by the gauge
field }$A_{k}$ and $\beta $ is a constant, but $\alpha
 depends in temperature changing its sign at $T=T_{c}$,
and $\gamma _{jk}$ is a constant tensor coupled to the crystal
axes.

It is possible to establish a connection between the
Lorentz-violating models whose energy densities are given by Eqs. (\re
{Hstat}), (\ref{E_X}) and the free energy (\ref{ffee}) describing a high-$
T_{c}$ superconductor. Such connection is provided by the term
\left[ \delta _{ij}-\left( k_{\phi \phi }\right) _{ij}\right] \left(
D_{i}\phi \right) ^{\ast }\left( D_{j}\phi \right) $, which allows
to relate the matrix  $\delta _{ij}-\left( k_{\phi \phi }\right) _{ij},
 with the tensor $\gamma _{jk}.$ It is clear that in both
LV models, the coefficients do not depend in temperature as
happens in phenomenological Ginzburg-Landau's models. The study of possible
contributions to superconductivity can be obtained by analyzing the
nonrelativistic limits of both LV models at finite temperature. Such
approach would allow to analyze Lorentz-violations effects or contributions
in London free energy, the Josephson effect, the interaction of vortices with
the crystalline structure (the vortex's intrinsic pinning), and resistivity, among
others. Advances in such interesting research would be developed in a future
manuscript.

Future developments can be oriented to analyze the influence of
Lorentz violation in the existence of topological defects in the non-Abelian
sector of the standard model-- i.e., magnetic monopoles and non-Abelian
vortices in the context of standard model extension. The results will be
reported soon.

\textbf{ACKNOWLEDGMENTS}

The authors thank CAPES, CNPq and FAPEMA (Brazilian agencies) for
financial support.

redpajama_set_name: RedPajamaArXiv


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A very intelligent friend of mine of markedly different political persuasions said the other day that he avoided "technical economic arguments" with me as I've just graduated with a degree in economics. I was rather sad to hear this.
The simple truth is that after doing a module called, say, 'Introduction to Economic Principles and Policy', you will not study very much more which will add greatly to your understanding of the subject. Beyond that, in an 'Intermediate Microeconomics' course for example, you are simply ladling mostly unnecessary algebra onto the subject.
This is actually some of the more accessible math involved in modern economics. Furthermore, the concepts it is dealing with, constant returns to scale and the per worker production function, are pretty straightforward. Yet many, including almost all university economics lecturers, will tell you that this is economics. It is, in fact, simply applied algebra – mathematics looking for a real world application. Economists eager to give their art the patina of science and mathematicians searching for real world relevance have combined to render economics impenetrable.
This trend also stems from the view of economics as the study of a mechanism. People may now laugh at the model of the economy A.W. Phillips built in the basement of the London School of Economics in 1949 with its gurgling pipes full of different coloured liquids representing money literally sloshing around an economy controlled by sluice gates. But it isn't conceptually different from the computerised models that are in use today guiding research and government policy.
These models often fail. If your model is based on erroneous assumptions, such as the creation of phantom capital called Quantitative Easing actually stimulating an economy, you will get erroneous outputs; Garbage In Garbage Out as they say. But there is a more fundamental problem. There is no exogenous 'thing' called 'The Economy' which can be quantified and controlled, there is only each of us doing what we do every day. That is why Ludwig von Mises called his great treatise on economics 'Human Action'. Or as Friedrich von Hayek rapped recently "The economy's not a car. There's no engine to stall. No experts can fix it. There's no "it" at all. The economy is us".
It is little wonder that even intelligent people feel themselves cowed and run in terror from the thicket of abstraction that shrouds modern economics. It needn't be like this. The great works of the discipline, those of Adam Smith or Carl Menger for example, managed to lay the foundations of the subject without it.
Do not leave economics to the abstract eggheads. Pick up Economics in One Lesson by Henry Hazlitt, Free to Choose: A Personal Statement by Milton Friedman or even Eat the Rich: A Treatise on Economics by P.J. O'Rourke: writers who, in these books and others, passed the economist Armen Alchian's test of whether they truly understood the subject – they could explain it to someone who doesn't know a darn thing about it.
Economics is about you. It is your subject. Reclaim and enjoy it.
Editor's note: the Cobden Centre bookstore is here.
I believe that it was von Mises who said that those who attempt to reduce economics to a mathematical equation are admitting that they are incapable of thinking as economists. I think he was right.
I have read a lot of books on economics and I was once a chemist and am well acquainted with mathematics. The example equation is just pure gibberish to me, and contributes nothing to the understanding of economics.
So, yes, I would love to see us return to the study of economics as the study of human nature and human action.
The fact that Austrian Economics does not focus on the math is part of what drew me to it in the first place. But more importantly it made sense. Economics is the study of man and how we interact with each other. It is not some convoluted formula like the one in this post. I have no idea on how to begin to tackle that problem. I will concede that the math does have its place and can help economists but that many economists put too much emphasis on it. Most of my economics background has come through reading. The Austrians present a very good case. I particularly like their focus on what we do not know. The "economy" is just too big to fit nicely into neat little formulas. It is too complex because it is made up on billions of people making decisions for themselves that ultimately affect others.


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text:
Q: Simple proof of Jordan normal form A lot of proofs in linear algebra use the fact that any square matrix can be written in Jordan normal form.

Unfortunately I can't see why this is the case, I didn't get what Wikipedia said and I just can't find a proof that I understand. Can you guys please tell me what is the simplest proof of it in your opinion ?

A: If $A$ is $n \times n$, let $f$ be the corresponding endomorphism of $V = \mathbf{C}^n$.
Then $V$ can be given a $\mathbf{C}[X]$-module structure by defining $P(X)\cdot v = P(f)(v)$.
A system of representatives for the irreducible elements of $\mathbf{C}[X]$ is given by $X - \lambda, \ \lambda \in \mathbf{C}$. By the structure theorem for fintely generated modules over a PID, $V$ is isomorphic to a finite direct sum of modules of the form $\mathbf{C}[X]/(X-\lambda)^k$. (A summand $\mathbf{C}[X]$ cannot occur because $V$ is finite-dimensional over $\mathbf{C}$.)
This expression of $V$ as a direct sum is also a direct sum of $\mathbf{C}$-vector subspaces, which will be the subspaces on which the Jordan blocks act. Moreover, since each summand is closed under multiplication by $X$, it is stable under $f$.
The only thing left to check is that $f$ acts as a Jordan block on each summand. Without loss of generality, we may assume that the summand is $\mathbf{C}[X]/(X-\lambda)^k$ and that $f$ is multiplication by $X$.
Then it is easy to check that the matrix of $f$ in the basis $((X-\lambda)^{k-1},(X - \lambda)^{k-2},\dots,X - \lambda, 1)$ is the $k \times k$ Jordan block with diagonal element $\lambda$.
Of course, all of this is close to trivial with the structure theorem, but every other proof I've seen of the existence of the Jordan decomposition amounts to rewriting a proof of the structure theorem in vector space language. So the most transparent thing is just to read a proof of the existence part of the structure theorem.

redpajama_set_name: RedPajamaStackExchange


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text:
Q: Missing rate per column per month in pandas Dataframe Let's take the following pd.DataFrame
>>> df = pd.DataFrame({
    'M'   : ['1', '1' , '3',  '6',  '6',  '6'],
    'col1': [None, 0.1, None, 0.2,  0.3,  0.4],
    'col2': [0.01, 0.1, 1.3,  None, None, 0.5]})

which creates
    M   col1  col2
0   1   NaN   0.01
1   1   0.1   0.10
2   3   NaN   1.30
3   6   0.2   NaN
4   6   0.3   NaN
5   6   0.4   0.50

I would now like to have the missing rate percentage per month per column. The resulting table should look like this
M   col1  col2
1   50.0   0.0
3  100.0   0.0
6    0.0  66.6

where the values in the cells in col1 and col2 state the missing rates per month for the column.
How can I do this?

A: You can use a groupby.mean on the boolean columns:
out = (df.drop(columns='M').isna()  # check if the value is missing
         .groupby(df['M'])          # for each M
         .mean().mul(100).round(2)  # get the proportion x 100
         .reset_index()             # index as column
      )

output:
   M   col1   col2
0  1   50.0   0.00
1  3  100.0   0.00
2  6    0.0  66.67

A: Code
df.set_index('M').isna().mean(level=0).mul(100).reset_index()

Result
   M   col1   col2
0  1   50.0   0.0
1  3  100.0   0.0
2  6    0.0   66.7


Did you know that over 50% of Charlotte-Mecklenburg School students grades K-8 are eligible for free or reduced lunch? This is a direct correlation to family income levels, which affect youths' involvement in extracurricular activities. Help us minimize this negative impact by becoming a part of Mecklenburg County's Dr. Charles L. Sifford Memorial Youth Sports Scholarship Fund Golf Tournament. This annual event, held in October, provides youth sports scholarship opportunities for deserving children in our communities. Get involved and be a part of something special.
Dr. Charles L. Sifford's legacy is treasured and valued in Mecklenburg County. He was an inspiration to many by becoming the first African-American to play on the PGA tour. He had to overcome many barriers to achieve this accomplishment and eventually win two PGA tour events. We hope to pass along his dedication and commitment to all county youth who will benefit from this scholarship fund by enabling them to participate in golf and other County recreational programs at a free or reduced cost.
Please join us for the Dr. Charles L. Sifford Memorial Youth Sports Scholarship Fund Golf Tournament. Our goal is to raise $10,000 for the Dr. Charles L. Sifford Memorial Youth Sports Scholarship Fund. Whether you play in the tournament, become a sponsoring partner, or simply make a donation, your involvement in this memorable event will have an immediate impact.
To register for the tournament please follow this link. Once you have completed the registration form, you will be contacted for payment.
For more information about the 2015 tournament, to learn more about becoming a sponsor or to make a donation to the Dr. Charles L. Sifford Memorial Youth Sports Scholarship Fund email Andreal.Shelton@MecklenburgCountyNC.gov or Preston.Buckman@MecklenburgCountyNC.gov.


text:
Running without fear: how to reduce the risk of heart attack and sudden death during aerobic exercise
By Kenneth H Cooper. 1985
Fitness, Sports and games
Presents the latest research concerning the relationship of exercise and coronary disease. Includes eleven "rules of risk" for heart disease as well as guidelines for reducing the risks. c1985.
Rock solid: my life in baseball's fast lane
By Alan Maimon, Tim Raines. 2017
Biography, Sports biography
For more than a decade, Tim Raines patrolled left field for the nascent Montreal Expos, alongside other superstars such as…
Andre "Hawk" Dawson, Gary Carter, and Steve Rogers. The seven-time all-star tore through the Expos' record books before moving on to the Chicago White Sox, New York Yankees, Oakland Athletics, Baltimore Orioles, and Florida Marlins, ultimately earning three World Series rings (one as a coach). But it wasn't always easy - Raines's performance dipped in 1982, and at the end of the season, he entered a substance abuse program for cocaine addiction. He would ultimately resume his career several times, culminating in the Expos retiring his number (#30), and in January 2017, being elected to the National Baseball Hall of Fame in Cooperstown, New York. 2017.
The final race: the incredible World War II story of the Olympian who inspired Chariots of fire
By Eric Eichinger. 2018
Eric Liddell was on top of the world after winning the gold in the 1924 Olympics. But in the years…
after, Eric faced unspeakable odds, and became an inspiring vision of what life means, even when the final hour comes. 2018.
The feeling of greatness: the Moe Norman story
By Jim O'Connor, Tim O'Connor. 1995
Biography, Sports biography, Canadian biography, Canadian non-fiction
This is the story of a relatively unknown sports personality who could not be part of the golf 'system'. While…
his life was unconventional, his golf is considered exceptional. An unpolished man, he is a legend in the amateur ranks in Canada, owning 33 course records from the 1950's. 1995.
The fighting fisherman: the life of Yvon Durelle
By Raymond Fraser. 1981
Biography, Sports biography, Canadian non-fiction, Sports and games
Biography of the Canadian boxer who won the Canadian and British titles, and went on to fight for the light-heavyweight championship of the world. 1981.
By Norman Mailer. 1975
Sports biography, Humour, Sports and games
Explosive, funny, perceptive account of the George Foreman-Muhammad Ali heavyweight boxing championship in Zaire, Africa. Some strong language and explicit descriptions of sex. 1975.
The essence of taiji qigong: the internal foundation of taijiquan (Martial Arts - Qigong Ser.)
By Jwing-Ming Yang. 1994
This text is for students who have learned a Taiji (Tai Chi) form and want to reach new levels of…
skill and ability. Regular Qigong practice accelerates the health benefits of Taiji; reducing stress, strengthening immune system and deepening awareness of breath and body coordination. 1994.
The hockey I love
By Vladislav Treti͡ìak, V Snegirev. 1977
Biography, Sports biography, Hockey
The Russian goalie relives various international hockey series and provides interesting thoughts and comments about team mates, coaches, opponents and his own future in Soviet sports. c1977.
The home team: fathers, sons & hockey
By Roy MacGregor. 1995
Sports biography, Sports and games, Hockey
MacGregor examines the relationship between father and son as it is lived through hockey. Using examples from famous hockey players,…
including stars like Gretzky, Messier, and Coffey, he describes the dynamic between generations. c1995.
The great Antonio
By Elise Gravel. 2016
Biography, Sports biography, Canadian biography, Canadian non-fiction, Canadian authors (Non-fiction)
He weighed as much as a horse; he once wrestled a bear; he could devour twenty-five chickens in one sitting.…
This whimsical book tells the story of Antonio Barichievich, the larger-than-life strongman who had muscles as big as his heart. Grades K-3. 2016.
The hall of the mountain king
By Howard H Snyder. 1973
Adventurers and explorers, Sports biography, Sports and games
A chilling record by a participant in the 1967 Wilcox Mount McKinley expedition, from which only five of the 12 climbers returned. 1973.
The greatest: Muhammad Ali
By Walter Dean Myers. 2001
Biography, Sports biography, Lifestyle, Sports and games
Account of Muhammad Ali's life and career, focusing on his professional accomplishments, principles, and fight against racial injustice. Describes his…
Olympic triumph, his transformation from the young Cassius Clay into Muhammad Ali, and his rise to fame as world heavyweight champion. Examines boxing's damaging effects. Grades 5-8 and older readers. Some descriptions of violence. 2001.
The goshawk (Wilder places)
By T. H White. 1951
Biography, Sports biography, Animals and wildlife
Chronicles the two-month battle of wills between the author, a novice falconer, and the hawk he is trying to tame…
and train for hunting. Provides daily accounts of progress, mistakes, and setbacks, as well as insights and self-discoveries. 1951.
The education of a coach
By David Halberstam. 2006
Bestsellers (Non-fiction), Biography, Sports biography
David Halberstam offers listeners this moving profile of NFL coaching legend Bill Belichick. Early in his life, Bill realized he…
had a special aptitude for football and teaching. During his 31 years in the league, he would parlay a lifetime of experience into one of the most successful careers in NFL history. Bestseller. 2006.
The complete book of massages
By Clare Maxwell-Hudson. 1988
Health and medicine, Hobbies and crafts
A comprehensive guide to massage techniques with clear step-by-step instructions, written and demonstrated by a leading international expert on massage.…
"Clare's massage course is a sensuous sun-filled treat. And the bonus is that afterwards you have a terrific, much-appreciated skill". 1988.
The complete yoga course: a personal yoga programme that will transform your daily life
By Howard Kent. 1997
Health and medicine, Fitness
A month-by-month course which builds confidence and enables even complete beginners to appreciate the benefits of yoga. Step-by-step instructions guide…
you through the course as you learn new postures and develop your own daily programme. The book also integrates the meditative side of yoga to build a calm, positive manner. 1997.
The boys in the boat: nine Americans and their epic quest for gold at the 1936 Olympics
By Daniel Brown. 2013
Movie and television tie-ins, Bestsellers (Non-fiction), Biography, Sports biography
Traces the story of an American rowing team from the University of Washington that defeated elite rivals at Hitler's 1936…
Berlin Olympics, sharing the experiences of their enigmatic coach, a visionary boat builder, and a homeless teen rower. Bestseller. 2013.
Non-fiction: Fitness, Hobbies and crafts, Sports biography
(-) Fitness (3837)
(-) Hobbies and crafts (3663)
(-) Sports biography (1891)
Gentle mysteries
Multi-cultural fiction
(-) Hobbies and crafts
(-) Sports biography


A Hearing is generally an exceptional event as opposed to a normal part of the Appeal procedures and a hearing date is only arranged if, following all submissions, the Commission determines that it requires further information which can only be gleaned from a Hearing. In such an event both parties will be contacted, provided with a copy of the final bundle of submissions, and a hearing date arranged at a mutually convenient time generally within 15 calendar days.
The Secretariat must be notified at least forty eight (48) hours prior to a scheduled Hearing of the names of any representative(s) appearing for either the Appellant or the respondent.
Hearings are conducted in the same manner as formal Court proceedings. Each of the parties will be sworn in and invited to make verbal representations on the final bundle of submissions. Parties will be expected to provide answers to any questions posed by the Commission which it considers that either the Appellant or respondent should be able to answer.
It should be noted that responses by either party seeking to claim lack of instruction will be treated by the Commission as failure to answer. Likewise, failure of either party to attend the Hearing, or provide their legal representative with the necessary information/instructions to answer the Commission's questions, will be treated as a failure to provide the Commission with the information necessary to determine the appeal.


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Vice President-Board
Secretry of Board
Treasurer of Board
Dr. Lenneal Henderson
Karen Collins,M.S, M.Ed
Dr. Brynne Schroeder
Anthony Hotakainen
I am so humbled and honored to work with such a dedicated, formidable, talented Board of Directors. I have had the distinct honor to have been supported by many exceptional young people too, working as unpaid interns, during production processes. They served in doing illustrations, marketing, and promotion of GLOBALLY MINDFUL, MOM'S CHOICE, MONTAIGNE Medal, AWARD WINNING Jelly Beans the Cheetah and Hope. They all made significant contributions and were examples of "springing into compassionate action!" They passionately supported the messages of Jelly Beans the Cheetah and Hope surrounding empathy, shared humanity-that we are one human FAMILY, and are connected to all, including the endangered cheetah! Any successes are as a result of this collaboration!
High school age student interns from Perpich and Brooklyn Center illustrated the first book! I am delighted that interns continue to support the Jelly Beans the Cheetah book series through marketing, graphic design, music arrangement, performance, production, recording, video production, editing, illustrations, and translations. MOM'S CHOICE AWARDED, FIVE STAR, EMPATHY AIRLINES: Jelly Beans the Cheetah and Hope, Book 2 was illustrated by college-aged Jacob Peterson and Sarah Williams as interns. Sarah earned six credits from Minneapolis College of Art and Design for the work she did, while Jacob (who worked on the first book) generously donated his time. Our team has included; college students and graduates from MCAD (Minneapolis College of Arts and Design) the University of Minnesota, St. Catherine's University (St. Paul) St. Cloud State University, and Teachers from St. Cloud and Minneapolis. Tina Inez (Graphic Design Intern) worked remotely and received credit from The Art Institute of Washington (Washington D.C./Baltimore areas).
CLICK on on album-image!
De'Arris Judkins (DRS) WEBSITE
Anthony Hotakainen MUSIC- YOUTUBE
Lisa Laubach, M.S.
Cornelius Rish, M.Ed
Mary Rathsabandith
De'Arris Wayne Judkins
Dr. Lugene Kennebrew
Dr. Gregory T. Williams
Interns:
Anjail (left) & Aaliyah (right)
Dr. Wendy Muhlhauser-Tingblad
Founder/Executive Director
Intern TEAM EMPATHY 2019 Book Launch!
Intern: Lauryn
De'Arris - Music
Assistant Executive Director &
Podcast and Tv Show Producer
Welcome Damian Kinsey & Deborah Ferrell to Board!!
Deborah Ferrell
Damian Kinsey
Purposeful education and outreach worldwide through technology(online) in person and through children's television and our other venues! The specific purpose of our 501 (c) (3) nonprofit is to provide education, training, and opportunity to learn about social and ecological justice. We model and teach equity and anti-racist, shared humanity and sustainability practices. These actions of love, EMPATHY, peace, service, and creative play are at the heart of building and repairing communities (including OUR worldwide community) affected and impacted by a lack of understanding of human diversity. Specifically, the nonprofit will provide education & training through Online workshops, events, written composition, creative/original works of art, theater, music, television, and ultimately, the creation of an institution of higher learning as avenues of spreading the mission & vision of SissyMarySue Education Fund!
You Can Be A Supporter Too!
We hope our formidable, talented Board Of Directors
along with Founder/ Executive Director
Dr. WendyMuhlhauser a.k.a SissyMarySue inspire you and your family, school, community or country to encourage empathy, equity and creative play with us!!
If you want to be added to the growing list of supporters please feel free to get ahold of us on CONTACT US or you can DONATE HERE!
Children's TV Show Sponsorship an interest ?
Thank you for all your generous donations!
Empowering Youth with Empathy
To Spring into Compassionate Action in their families, schools, communities & in service for our World!
Purposeful education and outreach worldwide through technology(online) in person and through children's television and our other venues!
The specific purpose of our 501 (c) (3) nonprofit is to provide education, training, and opportunity to learn about social and ecological justice. We model & teach equity and anti-racist, shared humanity and sustainability practices. These actions of love, EMPATHY, peace, service, and creative play are at the heart of building and repairing communities (including OUR worldwide community) affected and impacted by a lack of understanding of human diversity.
Specifically, the nonprofit will provide education & training through Online workshops, events, written composition, creative/original works of art, theater, music, television, and ultimately, the creation of an institution of higher learning as avenues of spreading the mission & vision of SissyMarySue Education Fund!
wmuhlhauser@email.fielding.edu
©2021 by SissyMarySue Education Fund


text:
Q: How To Expand Ubuntu 20.04 LTS Filesystem Volume on Hyper-V I created a Ubuntu 20.04 LTS virtual machine (VM) on Hyper-V and now I am running out of disk space and I need to increase the filesystem space.

A: This is an issue I've run into so many times, I've decided to leave a public trace on how to do it quickly.

*

*In Hyper-V, edit the virtual machine and go under "Hard Drive"

*Click on "Edit", select "Expand" and enter the desired size

*Reboot the virtual machine

*SSH into the virtual machine

*Run sudo lvdisplay to get the name of the logical volume

*Run sudo lvextend -l +100%FREE /dev/ubuntu-vg/ubuntu-lv to extend the volume to the maximum size available

*Run df -h to see the status of the filesystem free space

*Grab the name of your target filesystem (typically /dev/mapper/ubuntu--vg-ubuntu--lv)

*Run sudo resize2fs /dev/mapper/ubuntu--vg-ubuntu--lv to resize the filesystem

*Run again df -h and you should now see that your volume has been extended

That's it! Rebooting might not be a bad idea.

A: I tried existing solutions on 18.04 and found it doesn't work with a 'nothing to do' output.  Some further searching had similar-but-slightly-different steps and needing to merge a few different bits-and-pieces from other solutions on the web to make work.
Setup: Hyper-V; VHDX file as hard disk; Ubuntu-18.04
Steps:

*

*Expand the VDHX file via Hyper-V as mentioned in existing solutions and then inside the VM:

*fdisk -l
See which partition is the current Ubuntu setup - should be obvious based on size (in my case was sda3)

*growpart /dev/sda 3
Note the space as mentioned.

*pvresize /dev/sda3
This is the step which isn't mentioned in a lot of places; its the intemediate step that allows the logical volume extension step work.

*lvextend -l +100%FREE /dev/ubuntu-vg/ubuntu-lv
The /dev/ part can seen in the fdisk output from step 1.

*resize2fs  /dev/mapper/ubuntu--vg-ubuntu--lv
After the prep above, this step now works.  Takes a couple of moments and afterwards can verify with df -h that the partition is expanded.

A: For me, sudo lvdisplay does not work and returns nothing, for some reason. GParted also refused to start.
So I had to find other ways. Here is what brought luck:

*

*Start the VM again and install Guest Utils:

$ sudo apt install cloud-guest-utils

*

*If not using English, override locale settings to avoid issues with non-English locales:

$ LC_ALL=C

*

*Expand partition into the free space:

$ sudo growpart /dev/sda 5
# NB: space between `partition` and `id`!

*

*Resize:

$ sudo resize2fs /dev/sda5
# NB: *no space* between partition and id!

A: Ubuntu: Extend your default LVM space
Resize partition: sudo cfdisk
Extend PV physical volume: pvresize /dev/sda3
Extend logical volume: lvextend -l +100%FREE /dev/ubuntu-vg/ubuntu-lv
Resize: resize2fs /dev/mapper/ubuntu--vg-ubuntu--lv

A: Expand the Hyper-v hardrive, Under the Edit in Versual hard disk section.
Make sure the checkpoints are merged and deleted while the Virtual
Machine is OFF.
Boot Ubuntu system.
Command line:
sudo su -
fdisk -l
growpart /dev/sda 3
pvresize /dev/sda3
lvextend -l +100%FREE /dev/ubuntu-vg/ubuntu-lv
resize2fs  /dev/mapper/ubuntu--vg-ubuntu--lv
df -h


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