What's the Difference between this course, and the New Freelance Writer's Starter Pack: SEO Copywriting & the Guide to Killing It on Upwork?
1. First: The obvious difference is that course is offered IN CONJUNCTION WITH another course (the one on how to get jobs on Upwork, taught by Laura Pennington).
We joined forces to offer these together because they were a natural fit. As I termed it, "You Learn (SEO), then Earn (via Upwork)." You can learn more about how that all came together in the courses reviews offered on InkwellEditorial.com here, and on Laura's site, SixFigureWritingSecrets.com, here.
2. Second: This course is a stripped down version of the full course. What do I mean? Specifically, no reviews of materials are offered like in the other version. Also, there's no Facebook support group.
3. Third: While you do get two free ebooks offered with this course, the "How to Create an Online Course" class is not offered, as well as none of the freebies offered in Laura's course, of course.
All of the in-depth training is the same, just without a lot of the freebies and back-end support. Reviewing materials is time-consuming and as I do it all myself (no one on my staff does), I wanted to find a way to offer the training to those who wanted it, but who felt like they could take it and run with it without a lot of hand-holding -- as many have done.
In April (2017), I migrated all of Inkwell Editorial's e-courses to the Techable platform. It's easier to update the courses with fresh and/or bonus material, and give students immediate access to it. Because of all of this, enrollment has picked up.
In order to give the best student experience possible and keep the quality high (this is always paramount), the price of all of the SEO writing courses will be going up at some point.
No matter when you enroll or which course you enroll in, you will always have access to that particular course. If it's updated with bonus material, you get it. If it's updated because of some new SEO changes that come down the pike, you get it.
You pay one price, one time and the course is yours forever, to access whenever and from wherever you want. That's the way it should be, and that's the thing I love most about offering Inkwell Editorial's courses on this platform.
P.S.: Get 50% Off the "How to Create an Online Course" Class. See the next section to grab your coupon code. A discount on the Upwork course is offered too if you sign up for this class and decide to take that one at a later date. And, thank you for being a student in Inkwell Editorial's school.


that day I had a lunch with a friend and on the table i had yellow napkins. at that point i had an idea to make head accessory out of paper tissues.
to make a bow tie use the same method like before, just instead of headband use a rope. for adults u can also use thumbtack.


Claus Spreckels (1828–1908) was perhaps the most successful German-American immigrant entrepreneur of the late-nineteenth century; he was one of the ten richest Americans of his time.
The career of the "sugar king" of California, Hawaiʻi and the American West consisted of building and breaking monopolies in sugar, transport, gas, electricity, real estate, newspapers, banks and breweries.
The first industry in which Spreckels succeeded was quite typical for German immigrants: beer brewing. In the spring of 1857, together with his brother Peter Spreckels and Claus Mangels, among others, he founded the Albany Brewery, the first large-scale producer of beer in San Francisco.
Though profitable, he sold his beer operation in 1863 and switched to a new field that would make him rich: sugar. That year, he started the Bay Sugar Refining Company, but sold it three years later.
In 1878, through his friendship with King Kalākaua, Claus Spreckels secured a lease of 40,000-acres of land on Maui and by 1882 he acquired the fee simple title to the Wailuku ahupuaʻa.
That same year, Spreckels founded the Hawaiian Commercial Company, which quickly became the largest and best-equipped sugar plantation in the islands.
The Spreckelsville Mill was actually four mills in one complex (it was located just to the northeast of the present Kahului Airport, near the intersection of Old Stable Road and Hana Highway.) The town of Spreckelsville built up around it.
Part of the production innovation was the use of electric lights; the first recorded onshore use of electric lighting in Hawaiʻi was at Mill Number One of the Spreckelsville Plantation on Maui on Aug. 21, 1881.
To satisfy the curiosity of people anxious to see the "concentrated daylight," Capt. Coit Hobron ran a special train from Kahului, and King Kalākaua, Widow Queen Emma and Princess Ruth were among those who came to view the lights.
Sugar is a thirsty crop and Spreckels built the Haiku Ditch that spanned thirty miles and delivered fifty million gallons of water daily, irrigating twenty times as much land as had previously been irrigated.
Looking to upgrade from the mule and oxen means of moving sugar to the mill (as well as reduce costs,) Spreckels built a narrow-gauge railroad to haul the sugar from the plantation to the mill.
By 1881, twenty miles of iron track were completed. The rail line also transported the processed sugar to Maui's major port, Kahului. By 1885, Spreckelsville had forty-three miles of railroad, four engines and 498 cars for hauling cane.
Needing transportation to move his Hawaiʻi sugar for refining on the continent, he formed JD Spreckels & Bros. shipping line in 1879, which was incorporated as the Oceanic Steamship Company in 1881.
It was the first line to offer regular service between Honolulu and San Francisco, and his sons managed to reduce travel time immensely. While the sailing ship Claus Spreckels made a record run of less than ten days in 1879, by 1883 the new steam vessel Mariposa needed less than six days.
Spreckelsville was the largest sugar estate in the world by 1892.
The late-1890s saw internal family conflicts. Spreckels lost control of HC&S and in 1898; it became a part of Alexander & Baldwin Co. Following the 1948 merger of HC&S and Maui Agriculture Co., HC&S became a division of Alexander & Baldwin.
Hawaiʻi served as only one of the venues for the Spreckels holdings. During the 1880s and early 1890s, he bought and built up several blocks of office buildings in San Francisco.
I have added other images related to Claus Spreckels in a folder of like name in the Photos section on my Facebook and Google+ pages.


text:
Twisted Light Holds the key to the Future of Wireless Data Transmission
Rechelle Ann Fuertes Oct 30, 2017 at 12:19 pm GMT
Rechelle Ann Fuertes
Oct 30, 2017 at 12:19 pm GMT
Panuwatccn / Shutterstock.com
Scientists just found a way to transmit data way faster by using 'twisted' light.
Fiber optic cabling is considered as the fastest form of data transmission available today. However, the high price and complex deployment that comes with using fiber optics keep this innovation from reaching more people. In fact, even tech giants like Google have difficult times building their consumer base because of these hindrances.
But now, the future of data transmission might change for the best more quickly, thanks to a team of researchers from United Kingdom, Germany, New Zealand, and Canada. According to reports, scientists found a way to transmit data without the use of cables. Apparently, the team made use of 'twisted light' to efficiently transmit wireless data.
So, what is twisted light and how could it change the future of data transmission?
Researchers discovered that twisted light could be used to transmit data!Click To Tweet
Twisted Light: How Scientists Twisted Photons
New research into 'optical angular momentum (OAM) suggests that existing difficulties with using twisted light in open spaces could be fixed.
Optical angular momentum or orbital angular momentum is the component of angular momentum of a light beam that is dependent on the field spatial distribution and not polarization. By letting photons (individual particles of light) pass through a special type of hologram, something similar to that on a credit card, the photons acquire OAM and become twisted.
Now, the numerous intertwined twists in the twisted photons allow them to carry additional data. This is a far cry from the photons found in current digital communications which use only ones and zeroes to transmit information.
The capability of twisted light to carry extra information could only mean one thing: optical angular momentum could potentially be used in creating higher-bandwidth communications technology.
But, while OAM has already been used to carry data through cables, transmitting twisted light across open spaces is no easy feat.
Scientists must take into consideration several factors such as the changes in atmospheric pressure of open spaces that could affect and scatter the light beams, potentially causing the loss of spin information.
To guarantee the efficiency of the technology, the researchers tested it over a 1.6-km long free-space link in Germany, ensuring that it would pass through fields and streets near skyscrapers to simulate an urban setting and the atmospheric turbulence that affects spatial information transfer.
(A) Our 1.6-km free-space link is over the city of Erlangen in Germany. The optical beam is transmitted over both buildings, roads, and wooded areas as pictured. Image is courtesy of Google Map data 2016. (B) A fiber-coupled optical linearly polarized source with a wavelength of 809 nm is collimated and illuminates a spatial light modulator (SLM) | Advances Science | http://advances.sciencemag.org
The field test yielded new challenges that could help scientists improve the technology before making it commercially available. In a paper published by the researchers in the journal Science Advances, Dr. Martin Lavery, head of the Structured Photonics Research Group at University of Glasgow and lead author of the study, was quoted as saying:
"In an age where our global data consumption is growing at an exponential rate, there is mounting pressure to discover new methods of information carrying that can keep up with the huge uptake in data across the world.
A complete, working optical angular momentum communications system capable of transmitting data wirelessly across free space has the potential to transform online access for developing countries, defense systems, and cities around the world."
Dr. Lavery went on to say that free space optics is a potential solution that could give us the bandwidth of fiber without the need for physical cables.
"This study takes vital steps forward in the journey towards high dimensional free space optics that can be a cheaper, more accessible alternative to buried fibre optics connections."
The researchers also added that with this new development in OAM, scientists could now re-think their approaches to channel modeling and adaptive optics systems.
The study conducted by Dr. Lavery and his colleagues from the Max Planck Institute for the Science of Light and Institute of Optics, and the University of Otago, the University of Ottawa, and the University of Rochester addresses the challenges associated with developing adaptive optics for quantum information transfer that were not observed before.
Experts deem this latest discovery as a giant step towards a new age of free space optics that could someday replace fiber optic cables as the most functional mode of data transmission in urban environments and remote sensing systems.
Will twisted light be able to replace fiber optics anytime soon? What can you say about this latest development in spatial communication? Share your thoughts with us in the comment section below!
Let Rechelle Ann Fuertes know how much you appreciate this article by clicking the heart icon and by sharing this article on social media.
Rechelle is the Managing Editor of the EDGY blog. She's an experienced SEO content writer, researcher, social media manager, and visual artist. She enjoys traveling and spending time anywhere near the sea with her family and friends.
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redpajama_set_name: RedPajamaCommonCrawl


text:
Lionel Messi: I can convince him to stay, says Joan Laporta
By Nicholas Hughes
Updated 9:51 AM - January 11, 2021
Lionel Messi has left his future at Barcelona wide open, and presidential candidate Joan Laporta believes he is the man to get the Argentine star to stay at the club.
Laporta, who was President when Messi made his debut in 2004, believes that if he is re-elected then he has the tools to urge the six-time Ballon d'Or winner to remain at the Camp Nou.
Speaking to Goal, Laporta owed this confidence to his experience and prior relationship with Messi, that he says the No.10 recognises.
Jan 11, 2010: The Best
Jan 11, 2021: Still The Best pic.twitter.com/S9aGTSnpie
— FC Barcelona (@FCBarcelona) January 11, 2021
"I tell the truth, I am the one with the most experience and determination, as well as the most credibility in his eyes," he said.
"It fills me with pride that he tells me that everything I told him was fulfilled."
Laporta said this would put him in a position above all presidential candidates to converse with Messi and come to an agreement over both financial and sporting aspects of the Argentine's contract.
"If I am president of Barcelona…I will see what I can offer him within the possibilities of the club in terms of finances and in what interests Leo from a sporting perspective," he said.
"He wants to win the Champions League again with Barca and we have to make him see that this will be possible."
Lionel Messi's Barcelona stats in 2020/21
Statistic Messi
Goals 11
Dribbles P90 4.2
Big chances created 12
Via FootballCritic
The 33-year-old is now permitted to negotiate terms with other clubs for a free transfer in the summer with his contract at the Nou Camp running out in less than six months.
Messi has recently hinted that he may end up staying at Barcelona, warding off interest from the likes of PSG, Manchester City and clubs in the MLS.
And as well as his own lure, Laporta is hopeful the opening of the brand new Camp Nou, set to coincide with the club's 125th anniversary in 2024, will persuade Messi to stay.
"I wish we could celebrate the 125th anniversary at the new Camp Nou and with Leo with us," he said.
"It would be very nice to get to this anniversary with Messi, it would be good final fireworks."
Aston Villa transfer news: Sanson could leave for England, Marseille admit
'This is the Moise Kean of Juventus' – Everton ace back to himself at PSG
Timo Werner: Chelsea's answer to Arsenal flop Nicolas Pepe?
O'Shea: Ferdinand a better signing than Ronaldo
Laporta also insisted it is "very important" among his priorities to hold onto Messi if he were to be re-elected, admitting he is the "best in the world and surely in history".
But the presidential candidate did admit that the club will have to move on swiftly if it doesn't go to plan.
"If it doesn't happen, we will have to move on and sooner or later we will have to accept it," he said.
Messi has registered six goals and two assists in Barca's last five La Liga games as Ronald Koeman's team ride a run of eight games unbeaten at present, having them back up into third on the table.
← Camavinga to discuss 'pros and cons' of leaving Rennes this summer Ramos leaving would be a mistake for player and club, says former Real Madrid director →
Chelsea striker Timo Werner is in danger of being labelled one of the flops of the summer after a terrible run of form at Stamford Bridge.
Aston Villa have been heavily linked with a move for Marseille's Morgan Sanson for several months – and it could be coming to fruition.
Moise Kean's disappointing spell at Everton has been explained by a former Juventus colleague, who says he is back to his best at PSG.
Former Manchester United defender John O 'Shea believes that Rio Ferdinand remains the best piece of business that Sir Alex Ferguson did throughout his time at the club.
Arsenal Transfer News: Koeman denies Barcelona star's transfer request 1.1k views
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O'Shea: Ferdinand a better signing than Ronaldo 1k views


One of the biggest issues I have had the past year is controlling my sugar intake. I used to never crave sugar, ever, but the past year working in the office has been detrimental to my health. I have spiraled into this dark binging behavior of indulging in a sweet treat(s) almost everyday and I'm now seeing the repercussions.
The employees here are absolutely huge on sweets. Their primary duty in the office is to make sure our supply never gets low. Everyday I will find a box of Russian chocolates, Lindt truffles, Medovik honey cake, cupcakes, muffins, brownies, Gharadelli Squares.
My bad habit started the minute I came in on December 8, 2014. I was in really good shape and I had this great amount of energy. I rarely ate refined sugar, so the minute I was greeted with an unlimited supply of cupcakes and chocolate truffles, I looked at it as a way to spoil myself. But soon enough I became condition to treating myself with Lindt Truffles whenever I was bored or craved it , I would take one, which turned into 2 or 3 or 5 a day.. I was eventually eating an average of 4-5 a day, and I began to eat them because it was habit and when the candy got switched. to Ptichye Moloko, a stable Russian chocolate, I got even more immersed in my binge eating.
These were the new addiction for me. They are small individual chocolate-covered candy filled with soft meringue that come in different flavors, like lemon, marshmallow, raspberry, vanilla, etc. The problem with this is that they were so small and light I could easily eat 10 or more in one sitting without realizing, and 10 of those adds up to at least 60 grams of sugar. Plus sitting for about 7- 8 hours a day, I was asking for trouble.
Let me remind you i was averaging I'm sure an easily 70 grams a day thats about 5 Krispy Kreme donuts., something my body was not used to prior to my job, and my days are spent about 7 hours sitting plus 2 hours commuting ( 1 hour each way from and to work.) The repercussions have been hitting me pretty hard. My body wasn't taking this in lightly. I was tired, sluggish, and breaking out.
These treats can literally be found almost everyday if not certainly every week. Who can say no to mini Misha Cupcakes =( . I averaged three in one sitting, every time they were available to me.
Beside the physical changes that creeped up one me, I felt alot worst on the inside. I was so sluggish, so tired, I always wanted to crash towards the end of the day, ALWAYS THIRSTY and the most alarming part of all this my periods were becoming so unbearable. I couldn't go to work when the time came. This was not like me. I was never on the pill, but I noticed that when it came for the time of the month I would break out in sweat, get extremely light headed and get loss of vision hours before my menstruation. On top of that i had unbearable breast pain. In September 2015 I decided I couldn't take this any much longer , so I went to my gynecologist .
He requested I get my thyroid and vitamin b levels checked and sugar. I tested negative for thyroid, but I became positive with diabetes 2 at my age It was the last thing I expected because I considered myself primarily pretty healthy. It was just a wake up call to stop being so negligent to myself. I am not in any way paranoid over this diagnosis I am a bit shocked, it came so to me so quickly, but it does run very strong on both sides of my families. My grandmother , father, and three of my aunts have it.
So after the diagnosis it all made sense, the painful periods, the permanent acne, the lethargy, swelling in my mid section and who knows maybe my drastic hair loss ( that will be in another blog post) were all related to my elavated sugar levels.
Increased my daily sugar intake by about 200% daily value.
Sat 8 hours a day for 5 days. Getting up maybe every 1-2 hours to get water.
Reduced my water intake. Drank an average of 3-4 cups daily. My water to sugar ratio was extremely low.
I barely excreted sweat, I was literally walking maybe 10 minutes day. To my car, to my office, back to my car. Once I did get home all I wanted to do was sit because of how tired I felt.
I hate the office life.
It will be one of the toughest battles I will have because sugar is literally like a drug to me. If I get left alone just want to eat all the sweets I see and then I feel so guilty for doing it and obsesses over it.
I know this is going to be a lifestyle change not a temporary one. I will update you on my status in a future blog post.


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text:
After delay, FIFA to intensify stadium monitoring
FIFA will intensify its monitoring of the stadiums being prepared for the upcoming Confederations Cup after local organizers delayed the finish date of the venue hosting the opening match in Brasilia.
Football's governing body also reiterated Tuesday that such delays ''will not be tolerated'' ahead of the 2014 World Cup.
A day after the delay announced in Brasilia and less than two months before the Confederations Cup, FIFA met with local officials and the local 2014 World Cup organizing committee in the nation's capital and announced it will make a new inspection visit at the Estadio Nacional in two weeks.
The stadium was expected to be opened on Sunday, but problems with the pitch forced organizers to change the date to May 18, just three days before FIFA takes over all six Confederations Cup venues.
''FIFA and the LOC demonstrated concern with the tight deadlines and the short period available for the test events that guarantee the operational success of events with the size of the FIFA Confederations Cup,'' FIFA and the local organizing committee said in a joint statement.
''The daily and on-site monitoring of the stadium construction works and the hiring and assembling of the temporary structures by the LOC operational teams already working in the six FCC host cities will be also intensified,'' they said.
Officials in Brasilia blamed excessive rain in the region for the disruption of the installation schedule of the pitch. FIFA made it clear that it won't accept these types of excuses ahead of the World Cup.
''FIFA and the LOC highlight one more time that it is exactly because of natural challenges faced in big construction works - like the heavy rains observed in Brasilia - that the stadiums must be ready enough time before and stress again that delays like the ones observed will not be tolerated for the (World Cup) stadiums,'' the statement said.
FIFA and the local organizing committee noted that the World Cup involves a ''significantly higher number of host cities, teams, fans, matches, journalists and TV audience.''
''This means that the flexibility observed in the deadlines for the FIFA Confederations Cup will not be the same for the FIFA World Cup, when no exception will be made and all FIFA World Cup stadiums must be delivered by end of December 2013 with no compromises to be made by FIFA,'' the statement said.
FIFA originally wanted all Confederations Cup stadiums ready by last December but made an exception in Brazil because of delays with several venues, including at the Maracana Stadium, which will host the tournament's final in Rio de Janeiro on June 30. Local organizers then set April 15 as the deadline for the stadiums, but only three made it.
FIFA said it would be ideal to have three test events in each of the six venues, but Secretary General Jerome Valcke recently admitted that at this stage two would be fine.
Brasilia Gov. Agnelo Queiroz, who participated in Tuesday's meeting, guaranteed that the Estadio Nacional will be ready for the test events scheduled for May 18 and May 26.
''We are not going to risk the stadium delivery because of the problem with the pitch,'' he said.
The stadium is set to open the Confederations Cup with a match between Brazil and Japan.
Follow Tales Azzoni at http://twitter.com/tazzoni


text:
\section{Introduction}
\label{sec1}

The recent detection of gravitational waves (GWs)
from merging black holes (BHs) and neutron stars (NSs)
by the LIGO and Virgo collaborations
\cite{Abbott:2016blz,Abbott:2016nmj,TheLIGOScientific:2017qsa}
has opened new opportunities
to test gravitational and fundamental physics
in the extremely high density and/or high curvature regions.
The near-future detection of GWs will be able to test modifications of general relativity (GR)
in strong gravity regimes
in terms of the existence of the hairy BHs \cite{Berti:2015itd,Herdeiro:2015waa}
and the universal relations for NSs \cite{Doneva:2017jop}.

Although the data of LIGO and Virgo
are highly consistent
with the theoretical GW waveforms predicted from coalescing BHs and NSs in GR
so far,
they have not excluded the possibility of modified gravity theories
and/or the existence of
other more exotic compact objects yet,
and the future GW measurements would be able to test them
more precisely \cite{Berti:2018cxi,Berti:2018vdi,Cardoso:2017cqb}.
One of the candidates of more exotic compact objects is a boson star (BS)
which is a gravitationally bound nontopological solitonic object
in a bosonic field theory.
If the existence of the BSs could be verified through the near-future GW observations,
it may also give us a direct evidence of extra degrees of freedom
in modified gravity theories.
The BSs are characterized by
the two conserved quantities,
namely,
the Arnowitt-Deser-Misner (ADM) mass $M$
and
the Noether charge $Q$ associated with the global $U(1)$ symmetry of the field space.
$M$ and $Q$
correspond to the gravitational mass and the number of particles
inside a BS system, respectively,
and a BS is gravitationally bound
if it possesses the positive binding energy, $\mu_\phi Q-M>0$,
where $\mu_\phi$ is the mass of the scalar field.
The BS solutions have been first constructed
in the Einstein-scalar theory
with the mass term $\mu_\phi^2 |\phi|^2/2$ \cite{Kaup:1968zz,Ruffini:1969qy,Friedberg:1986tp,Jetzer:1991jr}.
The radial perturbation analysis
about the BS solutions \cite{Gleiser:1988rq,Jetzer:1991jr,Lee:1988av,Gleiser:1988ih}
has revealed that
the critical solutions dividing the stable BSs and the unstable ones
correspond to
those with the maximal ADM mass and Noether charge \cite{Gleiser:1988ih,Hawley:2000dt}.
GW signatures of the binary BSs
would be distinguishable from those of the binary BHs and NSs
as the consequence of
the different tidal deformabilities \cite{Sennett:2017etc}.

The maximal ADM mass depends on the potential of the scalar field \cite{Schunck:2003kk}.
In the Einstein-scalar theory
only with the mass term $\mu_\phi^2 |\phi|^2/2$,
it is of ${\cal O} (M_{pl}^2/\mu_\phi)$,
where $M_{pl}=\sqrt{\hbar c/G}= 1.221\times 10^{19} {\rm GeV}/c^2$ is the Planck mass
(in the rest we work in the units of $c=\hbar=1$),
which is much smaller than the Chandrasekhar mass
for fermions with the mass $\mu_\phi$,
of ${\cal O} (M_{pl}^3/\mu_\phi^2)$,
by assuming that $\mu_\phi\ll M_{pl}$.
In the theory with
both the quartic order self-interaction $\lambda_\phi |\phi|^4/4$
as well as
the mass term $\lambda_\phi |\phi|^2/2$,
it becomes of the same order as the Chandrasekhar mass
for the ferminons with the mass $\mu_\phi$,
of ${\cal O} (\sqrt{\lambda_\phi}M_{pl}^3/\mu_\phi^2)$ \cite{Colpi:1986ye}.

The BS solutions exist not only in the Einstein-scalar theory
but also in the Einstein-Proca theory \cite{Brito:2015pxa,Garcia:2016ldc,Brihaye:2017inn}.
In the Einstein-Proca theory with the mass term $\mu^2 \bar{A}^\mu A_\mu/2$,
where $A_\mu$ is the vector field and
$\bar{A}_\mu$ is the complex conjugate of $A_\mu$,
the properties of the vector BSs are quite similar
to those of the BSs in the Einstein-scalar theory with the mass term \cite{Brito:2015pxa}.
The critical solution dividing the stable and unstable vector BSs
corresponds to the solution with the maximal ADM mass and Noether charge,
and the maximal ADM mass of a vector BS is of ${\cal O}(M_{pl}^2/\mu)$.

The question which we addreess in this paper
is
how the maximal ADM mass and Noether charge of the BS solutions
in the Einstein-Proca theory
with the quartic order self-interaction of the vector field
$\lambda (\bar{A}^\mu A_{\mu})^2/4$
as well as the mass term $\mu^2 \bar{A}^\mu A_{\mu}/2$
are related to the mass and coupling constant of the vector field, $\mu$ and $\lambda$,
and
whether they are similar to those
in the Einstein-scalar theory
with the quartic order self-interaction.
The solutions of nontopological solitions
in the complex vector field theories with the nonlinear self-interaction potential
in the flat and curved spacetimes
have been presented in Refs.  \cite{Loginov:2015rya,Brihaye:2016pld,Brihaye:2017inn}.
While the results presented in this paper
have some overlap with those presented in Ref. \cite{Brihaye:2017inn},
we focus more on the role of the quartic order self-interaction
in the self-gravitating vector BS backgrounds,
and derive the quantitative dependence
of the physical properties of the BSs on $\lambda$.

The paper is constructed as follows:
In Sec. \ref{sec2},
we introduce the Einstein-Proca theory with the mass and
the quartic order self-interaction,
and derive a set of equations to find the structure of the vector BSs.
In Sec. \ref{sec3}, we numerically construct the vector BS solutions
and make arguments about their properties.
The last Sec. \ref{sec4} is devoted to giving a brief summary and conclusion.

\section{Vector boson star solutions}
\label{sec2}

\subsection{Theory}
\label{sec2a}

We consider the Einstein-Proca theory with the quartic order self-interaction
as well as the mass term;
\begin{align}
\label{action}
S&=\int d^4x \sqrt{-g}
\left[
 \frac{1}{2\kappa^2}R
-\frac{1}{4} F^{\mu\nu}{\bar F}_{\mu\nu}
\right.
\nonumber \\
&
\left.
-\frac{1}{2} \mu^2 A^\mu {\bar A}_\mu
-\frac{1}{4}\lambda \left(A^\mu {\bar A}_\mu\right)^2
\right],
\end{align}
where
the greek indices $(\mu,\nu,...)$ run the four-dimensional spacetime,
$g_{\mu\nu}$ is the metric tensor,
$g^{\mu\nu}:=\left(g_{\mu\nu}\right)^{-1}$,
$g={\rm det} (g_{\mu\nu})$ is the determinant of $g_{\mu\nu}$,
$R$ is the scalar curvature associated with $g_{\mu\nu}$,
$A_\mu$ is the complex vector field
and $\bar{A}_\mu$ is the complex conjugate of $A_\mu$,
$F_{\mu\nu}:= \partial_\mu A_\nu-\partial_\nu A_\mu$
is the field strength
and
$\bar{F}_{\mu\nu}:= \partial_\mu \bar{A}_\nu-\partial_\nu \bar{A}_\mu$
is its complex conjugate,
$\kappa^2:= 8\pi G$ with $G$ being Newton's constant,
$\mu$ is the mass of the complex vector field,
and
the dimensionless coupling constant
$\lambda$ measures the strength of the quartic order self-interaction of the vector field.
Throughout the paper we set $c=\hbar=1$, and
in these units
the Planck mass $M_{pl}$ is given by $M_{pl}=1/\sqrt{G}$.
Because of the different sign convention,
$\lambda$ in this paper corresponds
to $(-\lambda)$ in Refs. \cite{Brihaye:2016pld,Brihaye:2017inn}.

Varying the action \eqref{action} with respect to $g_{\mu\nu}$,
we obtain the gravitational field equations of motion (EOM)
\begin{align}
\label{grav_eq}
0={\cal E}_{\mu\nu}
&:= T^{(A)}_{\mu\nu}-\frac{1}{\kappa^2}G_{\mu\nu},
\end{align}
where
\begin{align}
\label{tf}
T^{(A)}_{\mu\nu}
&:=
 \frac{1}{2}
\left( F_{\mu\rho} {\bar F}_\nu{}^\rho
+ {\bar F}_{\mu\rho} F_\nu{}^\rho
\right)
-\frac{1}{4} g_{\mu\nu}F^{\rho\sigma} {\bar F}_{\rho\sigma}
\nonumber\\
&
+
\frac{\mu^2}{2}
\left[
   A_\mu {\bar A}_\nu
+ A_\nu {\bar A}_\mu
-
g_{\mu\nu} \bar{A}^\rho A_\rho
\right]
\nonumber\\
&+
\frac{\lambda}{2}
\left[
\left(\bar{A}^\rho  A_\rho\right)
 \left(
A_\mu {\bar A}_\nu
+ A_\nu {\bar A}_\mu
\right)
-\frac{1}{2}
g_{\mu\nu}
\left(\bar{A}^\rho A_\rho\right)^2
\right],
\end{align}
represents the energy-momentum tensor of the vector field.
Similarly,
varying the action \eqref{action} with respect to
$A_\mu$ and $\bar{A}_\mu$,
we obtain the EOM for the vector field
\begin{align}
\label{vector_eq}
0= {\cal F}_\nu
&:=
 \nabla^\mu F_{\mu\nu}
-\left[
 \mu^2
+\lambda \left(\bar{A}^\rho  A_\rho\right)
\right]
A_\nu,
\end{align}
and its complex conjugate $\overline{\cal F}_\nu=0$,
respectively.
Acting the derivative $\nabla^\nu$ on Eq. \eqref{vector_eq},
we obtain the constraint relation
\begin{align}
\label{gauge}
0={\cal G}&:=
\nabla^\nu
\left\{
\left[
 \mu^2
+\lambda \left(\bar{A}^\rho  A_\rho\right)
\right]
A_\nu
\right\}.
\end{align}
Similarly, $\nabla^\nu\overline{\cal F}_\nu=:\bar{\cal G}=0$.
Note that
in the theory \eqref{action}
there is the global $U(1)$ symmetry,
namely the symmetry
under the transformation $A_\mu\to e^{i \alpha} A_\mu$ where $\alpha$ is a constant,
and the associated Noether current
is given by
\begin{align}
\label{noether}
j^\mu=\frac{i}{2}
\left(
 \bF^{\mu\nu} A_\nu
-F^{\mu\nu} \bA_\nu
\right),
\end{align}
which satisfies the conservation law, $\nabla_\mu  j^\mu=0$.

\subsection{Static and spherically symmetric spacetime}
\label{sec2b}

We consider a static and spherically symmetric spacetime
\begin{align}
\label{metric_ansatz}
g_{\mu\nu}dx^\mu dx^\nu
&=-\sigma(r)^2
\left(1-\frac{2m(r)}{r}\right) d{\hat t}^2
\nonumber\\
&+
   \left(1-\frac{2m(r)}{r}\right)^{-1}
dr^2
 + r^2 d\Omega_2^2,
\end{align}
where $\hat t$ and $r$ are the time and radial coordinates,
$d\Omega_2^2$ is the metric of the unit two sphere,
and
$m(r)$ and $\sigma(r)$
depend only on the radial coordinate $r$.
Correspondingly,
we consider the ansatz for the vector field \cite{Brito:2015pxa}
\begin{align}
\label{vector_ansatz}
A_\mu dx^\mu
&=e^{-i{\hat\omega} {\hat t}}
 \left(
     a_0(r)d{\hat t}
   +i a_1 (r)dr
 \right),
\end{align}
where $a_0(r)$ and $a_1(r)$ depend only on $r$.
For the vector BS solutions,
the frequency ${\hat\omega}$ is assumed to be real and positive,
such that the vector field neither grows nor decays in time.
The ansatz \eqref{vector_ansatz}
is compatible with the static and spherically symmetric spacetime \eqref{metric_ansatz},
as it does not give rise to the explicit time dependence of the energy-momentum tensor.
In order to find $m$, $\sigma$, $a_0$, and $a_1$ numerically,
we rewrite EOMs \eqref{grav_eq}, \eqref{vector_eq} and \eqref{gauge}
into a set of the evolution equations
with respect to $r$.

\subsection{Evolution equations}
\label{sec2c}

From our ansatz \eqref{metric_ansatz} and \eqref{vector_ansatz},
the nontrivial components of the gravitational EOMs \eqref{grav_eq}
are given by
\begin{align}
\label{grav_eq0}
&
{\cal E}^{\hat t}{}_{\hat t}=0,
\quad
{\cal E}^r{}_r=0,
\quad
{\cal E}^i{}_i=0,
\end{align}
where the indices $(i,j,\cdots)$ run the directions of the two sphere.
Similarly,
the nontrivial components of the vector field EOM \eqref{vector_eq}
are given by
\begin{align}
\label{vector_eq0}
&{\cal F}_{\hat t}=0,
\quad
{\cal F}_r=0.
\end{align}
The complex conjugates of Eq. \eqref{vector_eq0}
give the same equations as Eq. \eqref{vector_eq0}
and need not be considered separately.
All these equations are related by
\begin{align}
\label{bianchi}
\nabla_\mu
{\cal E}^\mu{}_r
&=
-\frac{1}{2}
  g^{{\hat t}{\hat t}}
 \left(
{\cal F}_{\hat t}
\times
\bar{F}_{{\hat t}r}
 +
\bar{\cal F}_{\hat t}
\times
F_{{\hat t}r}\right)
\nonumber\\
&+\frac{1}{2}
\left(
{\cal G} \bA_r
+
 {\bar{\cal  G}} A_r
\right).
\end{align}

First, ${\cal E}^{\hat t}{}_{\hat t}=0$ and ${\cal E}^r{}_r=0$
can be arranged to give the evolution equations for $m$ and $\sigma$
in the $r$ direction  as
\begin{subequations}
\label{ms_evolv}
\begin{align}
m'&= F_m\left[a_0,a_0',a_1,m,\sigma \right],
\\
\sigma'&= F_\sigma\left[a_0,a_1,m,\sigma\right],
\end{align}
\end{subequations}
where a prime denotes the derivative with respect to $r$,
and $F_m$ and $F_\sigma$
are the nonlinear combinations of the given variables as
\begin{widetext}
\begin{subequations}
\label{fmfsigma}
\begin{align}
F_m
&:=\frac{\kappa^2}{8(r-2m)^2\sigma^4}
\left[
-3r^4\lambda a_0^4
+2r^2 a_0^2 \left(r\mu^2 +\lambda a_1^2 (r-2m)\right)
 (r-2m) \sigma^2
+\lambda a_1^4 (r-2m)^4\sigma^4
\right.
\nonumber\\
&
\left.
+2r a_1^2 (r-2m)^2 \sigma^2
\left(r{\hat \omega}^2+\mu^2 (r-2m)\sigma^2\right)
-4r^2 {\hat\omega} a_1 (r-2m)^2\sigma^2a_0'
+2r^2 (r-2m)^2\sigma^2a_0'^2
\right],
\\
F_\sigma
&:=
\frac{\kappa^2}{2 (r-2m)^3\sigma^3}
\left[
-r^2\lambda a_0^2
+
\left(
r\mu^2 +\lambda a_1^2 (r-2m)
\right)
(r-2m)
\sigma^2
\right)
\left(
r^2a_0^2 +a_1^2 (r-2m)^2\sigma^2
\right].
\end{align}
\end{subequations}
\end{widetext}
Similarly,
${\cal F}_{\hat t}=0$ and  ${\cal G}=0$
can be arranged to be
\begin{subequations}
\label{a0a1_evolv}
\begin{align}
a_0''&= {\cal H}^{-1}F_0\left[a_0,a_0',a_1,m,m',\sigma,\sigma'\right],
\\
a_1'&= {\cal H}^{-1} F_1\left[a_0,a_0',a_1,m,m',\sigma,\sigma' \right],
\end{align}
\end{subequations}
where $F_0$ and $F_1$ are regular combinations of the given variables
which are too involved to be shown explicitly,
and
\begin{align}
\label{h}
{\cal H}
:=
-r^2 \lambda a_0^2
+
\left(
r\mu^2 +3\lambda a_1^2 \left(r-2m\right)
\right)
 \left(r-2m\right)
\sigma^2.
\end{align}
Substituting Eqs. \eqref{ms_evolv} into the left-hand side of Eqs. \eqref{a0a1_evolv}
and eliminating $m'$ and $\sigma'$,
we obtain
\begin{subequations}
\label{a0a1_evolv2}
\begin{align}
a_0''&={\cal H}^{-1}
 \tilde{F}_0\left[a_0,a_0',a_1,m,\sigma\right],
\\
a_1'&={\cal H}^{-1}
 \tilde{F}_1 \left[a_0,a_0',a_1,m,\sigma \right],
\end{align}
\end{subequations}
where ${\tilde F}_0$ and $\tilde F_1$ are regular combinations of the given variables
which are also too involved to be shown explicitly.

Equations \eqref{ms_evolv} and \eqref{a0a1_evolv2}
are integrated from a point sufficiently close to $r=0$
with the boundary conditions given in Sec. \ref{sec2d}.
For a given set of parameters,
if ${\cal H}$ vanishes at some finite radius,
Eqs. \eqref{ms_evolv} and \eqref{a0a1_evolv2}
cannot be integrated further beyond this point,
and then no vector BS solution exists.

\subsection{Boundary conditions}
\label{sec2d}

Solving Eqs. \eqref{ms_evolv} and \eqref{a0a1_evolv2}
in the vicinity of $r=0$,
we find
\begin{subequations}
\label{bc}
\begin{align}
a_0
&= f_0
- \frac{f_0}{6\sigma_0^2}
\left(
 f_0^2\lambda
-\mu^2\sigma_0^2
+{\hat \omega}^2
\right)
  r^2
+{\cal O} \left(r^4\right),
\\
a_1
&=-
\frac{f_0 {\hat\omega}}{3\sigma_0^2}
r
+{\cal O} \left(r^3\right),
\\
m
&=
\frac{\kappa^2f_0^2}{24\sigma_0^4}
\left(
-3f_0^2\lambda
+2 \mu^2 \sigma_0^2
\right)
r^3
+{\cal O} \left(r^5\right),
\\
\sigma
&=\sigma_0
+
\frac{\kappa^2 f_0^2}{4\sigma_0^3}
\left(
-f_0^2 \lambda +\mu^2 \sigma_0^2
\right)
r^2
+{\cal O} \left(r^4\right).
\end{align}
\end{subequations}
Equation \eqref{bc} evaluated at $r=r_1$ which is sufficiently close to $r=0$
gives the boundary conditions
to integrate Eqs. \eqref{ms_evolv} and \eqref{a0a1_evolv2}
numerically
from $r=r_1$
to a sufficiently large value of $r$.

For $\lambda=0$,
${\cal H}$ defined in Eq. \eqref{h}
never crosses $0$
and the vector BS solutions exist
for an arbitrary value of
the central amplitude of the temporal component of the vector field, $f_0>0$.
For $\lambda\neq 0$,
expanding ${\cal H}$ in the vicinity of $r=0$
with Eq. \eqref{bc},
we obtain
\begin{align}
\label{calh}
{\cal H}
=
\left(
-f_0^2\lambda +\mu^2 a_0^2
\right)
r^2
+
{\cal O} (r^4).
\end{align}
For $\lambda>0$,
even if ${\cal H}<0$ in the vicinity of $r=0$,
as the ${\cal O}(r^4)$ terms in Eq. \eqref{calh} become important,
${\cal H}$ starts to increase and cross $0$.
Thus, a regular BS solution can be obtained only for
\begin{align}
\label{bound_amp}
0<f_0\leq f_{0,{\rm crit}}:= \frac{\mu\sigma_0}{\sqrt{\lambda}},
\end{align}
which was numerically confirmed,
where $f_{0,{\rm crit}}$ corresponds to the critical central amplitude
of the temporal component of the vector field.
On the other hand,
for $\lambda<0$,
even if ${\cal H}>0$ in the vicinity of $r=0$,
the higher order corrections to Eq. \eqref{calh}
make ${\cal H}$ decrease and cross $0$.
Thus, also for $\lambda<0$,
we numerically confirmed the existence of the critical central amplitude
of the temporal component of the vector field $f_{0,{\rm crit}}$,
although it cannot be expressed analytically.

For ${\hat\omega}$ chosen to be the correct lowest eigenvalue
of the vector BS
for a given set of parameters,
$m$ and $\sigma$ exponentially approach constant values,
$m_\infty>0$ and $\sigma_\infty>0$,
respectively,
while
$a_0$ and $a_1$ exponentially approach $0$ as $e^{-\sqrt{\mu^2-{\hat\omega}^2/\sigma_\infty^2}r}$.
Thus,
the metric exponentially approaches the Schwarzschild form
\begin{align}
\label{metric}
ds^2
&\to
-\sigma_\infty^2
  \left(
   1-\frac{2m_\infty}{r}
  \right)d{\hat t}^2
\nonumber\\
&+
\left(1-\frac{2m_\infty}{r}\right)^{-1}
dr^2
+r^2d\Omega_2^2,
\end{align}
where the proper time measured
by the observer at $r=\infty$ is given by
$t=\sigma_\infty {\hat t} $,
and
correspondingly
the proper frequency $\omega$ is given by
\begin{align}
\label{hatom}
{\omega}:=\frac{{\hat\omega}}{\sigma_\infty}.
\end{align}
The condition for the exponential fall-off
properties $e^{-\sqrt{\mu^2-\omega^2}r}$
requires $\omega <\mu$.
In the limit $f_0\to 0$,
$\omega\to \mu$,
and the Minkowski solution is obtained.

By the rescalings of
\begin{align}
\label{dimless}
&
{\hat\omega}\to \frac{{\hat\omega}}{\mu},
\quad
r\to r \mu,
\quad
m \to \mu m,
\quad
\sigma \to \sigma,
\nonumber\\
&
a_0 \to \kappa a_0,
\quad
a_1\to \kappa a_1,
\quad
\lambda
\to
\frac{\lambda}
{\mu^2\kappa^2},
\end{align}
the evolution equations \eqref{ms_evolv} and \eqref{a0a1_evolv2}
can be rewritten into the form without $\mu$ and $\kappa$,
and in the rescaled equations the strength of the self-interaction is measured by
the dimensionless coupling constant
\begin{align}
\label{Lambda}
\Lambda :=
\frac{\lambda}
{\mu^2\kappa^2}
=\frac{M_{pl}^2}{(8\pi)\mu^2}  \lambda.
\end{align}
For the numerical analysis,
we may set $\mu=\kappa=1$ and
as the result $\lambda=\Lambda$,
as it is  straightforward
to give back the dependence
of the physical quantities on $\mu$ and $\kappa$,
once the vector BS solutions are numerically obtained for $\mu=\kappa=1$.
In addition,
as $\sigma_0$ corresponds to the degree of freedom of the time rescaling
without loss of generality,
we may also set $\sigma_0=1$.
Thus,
only the remaining physical parameters are $f_0$ and $\Lambda$.

In Figs. \ref{figms} and \ref{figa0a1},
$(m,\sigma)$ and $(a_0, a_1)$
are shown as the functions of $r$
for $\Lambda=10$ and $f_0=1/(10\kappa)$, respectively.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{ms.eps}
\caption{
$\mu m$ and $\sigma$ are shown as the functions of
$\mu r$ for $\Lambda=10$ and $f_0=1/(10\kappa)$.
The black and green curves correspond to
$\mu m$ and $\sigma$, respectively.
}
  \label{figms}
\end{center}
\end{figure}
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{a0a1.eps}
\caption{
$\kappa a_0$ and $\kappa a_1$ are shown as the functions of
$\mu r$
for $\Lambda=10$ and $f_0=1/(10\kappa)$.
The red and blue curves correspond to
$\kappa a_0$ and $\kappa a_1$, respectively.
}
  \label{figa0a1}
\end{center}
\end{figure}
Note that
the dimensionless quantities introduced in \eqref{dimless}
are different from those introduced in Ref. \cite{Brihaye:2017inn}
where in the dimensionless equations of motion
$a_0$ and $a_1$ are rescaled
to the dimensionless quantities,
$(\sqrt{|\lambda|}/\mu) a_0$ and $(\sqrt{|\lambda|} /\mu) a_1$,
respectively.
While Ref. \cite{Brihaye:2017inn} focused on
the importance of the coupling to gravity
measured by the dimensionless parameter
$\alpha:=(\kappa^2\mu^2)/\lambda[=1/\Lambda]$
on the nontopological soliton backgrounds in the flat spacetime,
we focus more on the role of the quartic order self-interaction of the
vector field
on the self-gravitating vector BS backgrounds,
and derive the quantitative dependence
of the physical properties of BSs on the coupling constant $\lambda$.
Thus,
although there are similarities of our results to those in Ref. \cite{Brihaye:2017inn},
we make arguments about the properties of the vector BSs from the different perspectives.
In contrast to the case of the scalar BSs,
in the case of the vector BSs
$a_0$ has a single node
before approaching $0$.

\subsection{ADM mass, Noether charge, and binding energy}
\label{sec2e}

We then evaluate the conserved quantities characterizing the vector BSs.
The first is the ADM mass
\begin{align}
\label{adm}
M:=\frac{m_\infty}{G}=M_{pl}^2 m_\infty,
\end{align}
which is associated with the time translational symmetry.
The second is
the Noether charge associated with the global $U(1)$ symmetry,
which is given by
integrating $j^{\hat t}$ in Eq. \eqref{noether}
over a constant-${\hat t}$ hypersurface
\begin{align}
\label{charge}
Q
=\int_\Sigma d^3x \sqrt{-g} j^{\hat t}
= 4\pi \int_0^\infty
dr
\frac{
r^2a_1
\left(
 {\hat\omega} a_1
-a_0'
\right)}
{\sigma}.
\end{align}
A BS is gravitationally bound when $\mu Q-M>0$,
and we then define the relative binding energy
\begin{align}
\label{binding}
b:=\frac{B}{M}=\frac{\mu Q}{M}-1.
\end{align}

In order to discriminate various compact objects
and classify their physical properties,
it is important to define the effective compactness of the vector BS
\begin{align}
\label{compactness}
{\cal C}:=\frac{GM}{{\cal R}}
    =\frac{m_\infty}{{\cal R}},
\end{align}
where the effective radius
of it is given by \cite{Schunck:2003kk}
\begin{align}
\label{radius}
{\cal R}
&
:=
\frac{1}{Q}
\int_\Sigma d^3x
\sqrt{-g}
 \left(r j^{\hat t}\right)
=
\frac{4\pi}{Q}
 \int_0^\infty
dr
\frac{
r^3a_1
\left(
 {\hat\omega} a_1
-a_0'
\right)}
{\sigma}.
\end{align}

\subsection{Comparison with the Einstein-scalar theory}
\label{sec2f}

Before proceeding to the physical properties of the vector BSs,
we briefly review the case of
the Einstein-scalar theory with the quartic order self-interaction
\begin{align}
\label{action2}
S&=\int d^4x \sqrt{-g}
\left[
 \frac{1}{2\kappa^2}R
-\frac{1}{2} g^{\mu\nu}\partial_\mu \phi \partial_\nu \bar{\phi}
\right.
\nonumber\\
&\left.
-\frac{1}{2} \mu_\phi^2 |\phi|^2
-\frac{1}{4}\lambda_\phi |\phi|^4
\right],
\end{align}
where
$\mu_\phi$ and $\lambda_\phi$ are
the mass and coupling constant of the complex scalar field, respectively.
In the case without the self-interaction, $\lambda_\phi=0$,
the maximal ADM mass is given
as \cite{Jetzer:1991jr,Schunck:2003kk},
\begin{align}
M_{\rm max}
\simeq 0.633 \frac{M_{pl}^2}{\mu_\phi},
\end{align}
which is much smaller than
the Chandrasekhar mass for fermions with the same mass.
On the other hand,
in the case of $\lambda_\phi \left(M_{pl}/\mu_\phi\right)^2 \gg 1$,
no upper bound on the central amplitude of the scalar field  exists,
and the maximal ADM mass is given as
\cite{Colpi:1986ye},
\begin{align}
\label{mmax}
M_{\rm max}
\simeq 0.062\sqrt{\lambda_\phi}\frac{M_{pl}^3}{\mu_\phi^2}.
\end{align}
Moreover,
as shown in Fig. \ref{figa0a1}
the vector BS solution
has a single node in $a_0$,
while the scalar BS solution has no node in $|\phi|$.
In Sec. \ref{sec3}
we obtain
the corresponding relation to Eq. \eqref{mmax}
for the vector BSs
in the presence of the self-interaction $\lambda (\bar{A}^\mu A_\mu)^2/4$.

\section{Properties of Vector Boson Stars}
\label{sec3}

In this section, we make arguments about the properties
of the vector BS solutions in the theory \eqref{action}
obtained numerically.

\subsection{ADM mass and Noether charge}
\label{sec3a}

In Fig. \ref{figmqf},
the ADM mass $M$ and
the Noether charge multiplied by $\mu$, $\mu Q$,
are shown as the functions of $f_0$
for several values of $\Lambda\geq 0$ defined in Eq. \eqref{Lambda}.
We numerically confirmed
that there exists the critical central amplitude of
the temporal component of the vector field,
$f_{0,{\rm crit}}$,
and
that it agrees with Eq.  \eqref{bound_amp}.
The behavior for $0<\Lambda\lesssim 1$
is qualitatively similar to that for $\Lambda=0$
except for the existence of $f_{0,{\rm crit}}$;
namely
$M$ and $Q$
take the local maximal values at some intermediate value of $f_0$.
For $\Lambda>1$,
$M$ and $Q$
monotonically increase for increasing values of $f_0$,
and their maximal values
are obtained from the limit of $f_0\to f_{0, \rm crit}$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mqf.eps}
\caption{
$M$ and $\mu Q$
are shown as the functions of $f_0$ for $\Lambda>0$.
The red and blue curves
correspond to $M$ and $\mu Q$, respectively.
The solid, dashed, dot-dashed, and dotted curves
correspond to
$\Lambda=20, 10, 1, 0.1$, respectively.
$(M, \mu Q$) and $f_0$ are shown in
$M_{pl}^2/\mu$ and $M_{pl}/\sqrt{8\pi}[=1/\kappa]$,
respectively.
}
  \label{figmqf}
\end{center}
\end{figure}
On the other hand,
although we do not show the plots of $M$ and $\mu Q$
as the functions of $f_0$
for $\Lambda<0$,
the values of $M$ and $Q$
become smaller
than those for $\Lambda=0$
for the same values of $f_0$.
For any value of $\Lambda<0$,
we numerically confirmed that
there is also the critical amplitude of the temporal component of the vector field $f_{0,\rm crit}$,
which cannot be expressed analytically
and becomes smaller for larger $|\Lambda|$.
The maximal values of $M$ and $Q$
correspond to their local maximal values
obtained at the intermediate value of $f_0<f_{0,{\rm crit}}$.
Note that
for all the values of $\Lambda$,
$M=Q=0$ and
the Minkowski solution with
the vanishing vector field $A_\mu=0$
is obtained for $f_0=0$.

In Fig. \ref{figmo0},
$M$ is shown as the function of $\omega$ defined in Eq. \eqref{hatom}
for $\Lambda>0$.
While for $\Lambda=0$
the well-known spiraling behavior is observed
as shown in Ref. \cite{Brito:2015pxa}
and hence $M$ is the multivalued function of $\omega$,
$M$ eventually increases and becomes the single-valued function of $\omega$
for $0.1 \lesssim \Lambda\lesssim 1$.
Furthermore,
for $\Lambda\gtrsim 1$,
$M$ becomes the monotonically decreasing function of $\omega$.
Note that a quantitatively very similar behavior is obtained for $\mu Q$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mo0.eps}
\caption{
$M$ is shown as the function of $\omega/\mu$
for $\Lambda\geq0$.
The red, blue, magenta, green, and black curves
correspond to the cases of
$\Lambda=20, 10, 1, 0.1, 0$, respectively.
$M$ is shown in $M_{pl}^2/\mu$.
}
  \label{figmo0}
\end{center}
\end{figure}

In Fig. \ref{figmqo},
$M$ and $\mu Q$
are compared for $\Lambda=10,1,0.1$, respectively.
For all the cases,
the solutions with the maximal values of $M$ and $Q$
satisfy $M<\mu Q$, namely, $b>0$ in Eq. \eqref{binding},
and these vector BSs are gravitationally bound.
For $\Lambda \lesssim 0.1$,
$b<0$ for the smaller values of $\omega$
which are obtained from
the values of $f_0$ very close to
the critical value $f_{0,\rm crit}$, $f_0\lesssim f_{0,{\rm crit}}$.
For $\Lambda\approx 1$,
the values of $M$ and $\mu Q$ obtained from the limit of $f_0\to f_{0,{\rm crit}}$
become comparable to the local maximal values of $M$ and $\mu Q$
obtained at the intermediate value of $f_0<f_{0,{\rm crit}}$,
respectively.
For $\Lambda\gtrsim 1$,
we obtain
$b>0$ for all values of $\omega$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mqo.eps}
\caption{
$M$ and $\mu Q$
are compared as the functions of $\omega/\mu$
for $\Lambda>0$.
The red and blue curves
correspond to $M$ and $\mu Q$, respectively.
The solid, dashed, and dotted curves
correspond to
$\Lambda=10, 1, 0.1$, respectively.
$M$ and $\mu Q$ are shown in
$M_{pl}^2/\mu$.
}
  \label{figmqo}
\end{center}
\end{figure}

In Fig. \ref{figmom},
$M$ is shown as the function of $\omega$
for $\Lambda\leq 0$.
As $|\Lambda|$ increases,
$M$ decreases for a fixed value of $f_0$.
Because of the existence of the critical central amplitude
of the temporal component of the vector field $f_{0,\rm crit}$,
for larger values of $|\Lambda|$
the spiraling behavior is eventually resolved.
The maximal value of $M$ is obtained for $f_0<f_{0,\rm crit}$.
The quantitatively similar behavior is obtained for $\mu Q$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mom.eps}
\caption{
$M$ is shown as the function of $\omega/\mu$
for $\Lambda\leq 0$.
The red, blue, green, and black curves
correspond to the cases of
$\Lambda=-10, -1, -0.1, 0$, respectively.
$M$ is shown in $M_{pl}^2/\mu$.
}
  \label{figmom}
\end{center}
\end{figure}
In Fig. \ref{figmqom},
$M$ and $\mu Q$ are compared for $\Lambda<0$.
The maximal values of $M$ and $\mu Q$
always satisfy $b>0$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mqom.eps}
\caption{
$M$ and $\mu Q$
are compared as the functions of $\omega/\mu$
for $\Lambda<0$.
The red and blue curves
correspond to $M$ and $\mu Q$, respectively.
The solid, dashed, and dotted curves
correspond to
$\Lambda=-10, -1,-0.1$, respectively.
$M$ and $\mu Q$ are shown in
$M_{pl}^2/\mu$.
}
  \label{figmqom}
\end{center}
\end{figure}

We speculate that
the disappearance of the spiraling behavior
is due to the combination of the two effects
which were argued so far.
The first is the existence of the critical central amplitude of
the temporal component of the vector field $f_{0,{\rm crit}}$,
as discussed in Sec. \ref{sec2d}.
As $|\Lambda|$ increases,
$f_{0,{\rm crit}}$ decreases,
and hence the allowed region of the central amplitude,
$0< f_0\leq  f_{0,{\rm crit}}$, shrinks.
The second is the overall enhancement of the ADM mass and the Noether charge,
due to the stronger self-interaction for $\Lambda>0$,
as seen in Figs. \ref{figmo0} and \ref{figmqo}.

\subsection{Maximal ADM mass and Noether charge}
\label{sec3b}

In Fig. \ref{figmqmax},
the maximal values of $M$ and $\mu Q$,
which
from now on are denoted by $M_{\rm max}$ and $\mu Q_{\rm max}$,
respectively,
are shown as the functions of $\Lambda$.
They are monotonically increasing
for increasing $\Lambda$.
As already seen in Sec. \ref{sec3a},
$M_{\rm max}$ and $\mu Q_{\rm max}$
obtained for $\Lambda\lesssim 1$
have a different physical origin
from those obtained for $\Lambda\gtrsim1$,
which
can be observed as the break around $\Lambda\simeq 1$.
From $\Lambda \gg 1$,
$M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to the values of $M$ and $Q$
from the limit of $f_0\to f_{0,\rm crit}$,
respectively.
From the data of $M_{\rm max}$ and $\mu Q_{\rm max}$
for $20\leq\Lambda\leq 50$,
the fitting formulas
\begin{subequations}
\label{fitting}
\begin{align}
M_{\rm max}
&\approx
\frac{M_{pl}^2}{\mu}
\left[
1.383
-0.005099 \frac{\sqrt{\lambda}M_{pl}}{\mu}
\right.
\nonumber\\
&
\left.
+
\left(
-0.03967
+0.005675\frac{\sqrt{\lambda}M_{pl}}{\mu}
\right)
\ln
\left(\frac{\lambda M_{pl}^2}{\mu^2}\right)
\right],
\\
\mu Q_{\rm max}
&\approx
\frac{M_{pl}^2}{\mu}
\left[
1.500
-0.001989 \frac{\sqrt{\lambda} M_{pl}}{\mu}
\right.
\nonumber\\
&
\left.
+
\left(
-0.05386
+0.005694  \frac{\sqrt{\lambda}M_{pl}}{\mu}
\right)
\ln
\left(\frac{\lambda M_{pl}^2}{\mu^2}\right)
\right],
\end{align}
\end{subequations}
are obtained,
respectively,
which can also
fit the data of $M_{\rm max}$ and $\mu Q_{\rm max}$ for $50<\Lambda\leq 250$
very well.
Thus,
for a sufficiently large value of $\Lambda\gg 1$,
$M_{\rm max}$ and $\mu Q_{\rm max}$ become of
${\cal O}[\sqrt{\lambda}M_{pl}^3/\mu^2 \ln (\lambda M_{pl}^2/\mu^2)]$,
which is different from the case of the BS solutions
in the Einstein-scalar theory \eqref{mmax}.
Note that
for $\Lambda=0$
we recover
\begin{align}
M_{\rm max}\approx 1.058M_{pl}^2/\mu ,
\quad
\mu Q_{\rm max}\approx 1.088M_{pl}^2/\mu,
\end{align}
obtained in Ref. \cite{Brito:2015pxa}.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{mqmax.eps}
\vspace{0.3cm}
  \includegraphics[height=5.0cm,angle=0]{mqmaxp.eps}
\caption{
$M_{\rm max}$ and $\mu Q_{\rm max}$
are shown as the functions of $\Lambda$.
The red and blue points correspond
to $M_{\rm max}$ and $\mu Q_{\rm max}$, respectively.
They are shown in $M_{pl}^2/\mu$.
The lower panel is the enlarged display of the upper one
around $\Lambda=1$.
}
  \label{figmqmax}
\end{center}
\end{figure}

\subsection{Binding energy, compactness, and stability}
\label{sec3c}

In Fig. \ref{figbmax},
the relative binding energy defined in Eq. \eqref{binding}
for the vector BS solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$,
$b_{\rm max}:=\mu Q_{\rm max}/M_{\rm max}-1$,
is shown as the function of $\Lambda$.
The black curve $(\Lambda<1)$ corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$,
while
the red curve $(\Lambda>1)$
corresponds to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$.
Note that for all the values of $\Lambda$,
$b_{\rm max}>0$,
and hence the vector BS solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
are always gravitationally bound.
For $\Lambda\lesssim 1$,
$b_{\rm max}$ is an increasing function of $\Lambda$,
while for $\Lambda\gtrsim 1$
it rapidly increases but eventually
approaches the constant value $\simeq  0.056$.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{bmax.eps}
\caption{
$b_{\rm max}:=\mu Q_{\rm max}/M_{\rm max}-1$
is shown as the function of $\Lambda$.
The black curve corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$,
while
the red curve
corresponds to the case
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
are obtained from the limit of $f_0\to f_{0,{\rm crit}}$.
}
  \label{figbmax}
\end{center}
\end{figure}

In Fig. \ref{figcmax},
the compactness ${\cal C}$ defined in Eq. \eqref{compactness}
for the vector BS solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
is shown as the function of $\Lambda$.
The black curve corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0<f_{0,{\rm crit}}$,
while
the red curve
corresponds to the case
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
are obtained from the limit of $f_0\to f_{0,{\rm crit}}$.
The clear discontinuity on the value of ${\cal C}$
 exists around $\Lambda=1$.
For $\Lambda\gtrsim 1$,
${\cal C}$ is always larger than $0.2$,
but cannot exceed $0.32$.
Since photon spheres
could be formed for a spherically symmetric compact object
whose compactness is greater than $1/3=0.333\cdots$,
no photon spheres would be formed
around the vector BSs.
For $\Lambda<1$ including negative values,
${\cal C}$ is less than $0.2$
and gradually decreases
as $|\Lambda|$ increases.
Note that the compactness \eqref{compactness}
was defined with the effective radius
${\cal R}$ defined in \eqref{radius}
outside which the vector field $A_\mu$ does not completely vanish
and the spacetime geometry is not precisely given by the
vacuum Schwarzschild solution.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{cmax.eps}
\caption{
${\cal C}$ for the solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
is shown as the function of $\Lambda$.
The black curve corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$,
while
the red curve
corresponds to the case
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
are obtained from the limit of $f_0\to f_{0,{\rm crit}}$.
}
  \label{figcmax}
\end{center}
\end{figure}

In order to see
how the effective compactness depends
on the definition of the effective radius,
${\cal C}$ defined in Eq. \eqref{compactness}
is compared
with another definition of the effective compactness,
for example,
\begin{align}
\label{compactness2}
\tilde{\cal C}:=\frac{GM}{\tilde {\cal R}}
    =\frac{m_\infty}{\tilde {\cal R}},
\end{align}
for another definition of the effective radius
\begin{align}
\tilde{\cal R}
:=
\left(
\int_0^\infty dr\, r^3 \rho^{(A)}
\right)
\Big{/}
\left(
\int_0^\infty dr\, r^2 \rho^{(A)}
\right),
\end{align}
where
$\rho^{(A)}:=-T^{(A){\hat t}}{}_{\hat t}$ [see Eq. \eqref{tf}]
is the energy density of the vector field \cite{Schunck:2003kk}.
In Fig. \ref{figcrmax},
the ratio $\tilde{\cal C}/{\cal C}[={\cal R}/\tilde{\cal R}]$
for the solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
is shown
as the function of $\Lambda$.
The black curve $(\Lambda<1)$ corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$,
while
the red curve $(\Lambda>1)$
corresponds to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$.
We find
that for all values of $\Lambda$,
$\tilde{\cal C}/{\cal C}>1$ and hence $\tilde{\cal R}/{\cal R}<1$,
but the deviation from unity is at most 9\%, namely,
$(\tilde{\cal C}/{\cal C})_{\rm max}-1<0.09$.
The maximal deviation from unity
arises for $\Lambda\simeq 1$,
where $\tilde{\cal C}/{\cal C}\simeq1.087$.
As $|\Lambda|$ increases,
$\tilde{\cal C}/{\cal C}$ decreases toward unity.
Thus,
in most cases
the difference between ${\cal C}$ and $\tilde{\cal C}$
is not so quantitatively significant.
But for the solutions with $\Lambda\simeq 1$,
$\tilde{\cal C}$ takes the maximal value
$\tilde{\cal C}> 0.34$ which exceeds $1/3$,
while ${\cal C}<0.32$.
Therefore,
the ambiguity in the definition of
the effective compactness
around $\Lambda\simeq 1$
makes it unclear
whether
photon spheres
can be formed
around the most compact vector BSs
in the presence of the quartic order self-interaction,
or not.
For the more precise comparison with the other compact objects
such as the BHs or NSs
more careful analyses are requested,
and the possible formation of the photon spheres
should be judged by explicitly analyzing null geodesics
around the vector BSs,
which will be left for a future study.
\begin{figure}[h]
\unitlength=1.1mm
\begin{center}
  \includegraphics[height=5.0cm,angle=0]{crmax.eps}
\caption{
$\tilde{\cal C}/{\cal C}[={\cal R}/\tilde{\cal R}]$
for the solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
is shown as the function of $\Lambda$.
The black curve corresponds
to the cases
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
correspond to their local maximal values
obtained at the intermediate value of $f_0< f_{0,{\rm crit}}$,
while
the red curve
corresponds to the case
in which $M_{\rm max}$ and $\mu Q_{\rm max}$
are obtained from the limit of $f_0\to f_{0,{\rm crit}}$.
}
  \label{figcrmax}
\end{center}
\end{figure}

As we mentioned in Sec. \ref{sec1},
for $\Lambda=0$
the vector BS solution with $M_{\rm max}$ and $\mu Q_{\rm max}$
corresponds to the critical solution
which divides the stable and unstable vector BS solutions \cite{Brito:2015pxa},
as in the case of the scalar BS solutions \cite{Gleiser:1988ih,Hawley:2000dt}.
As mentioned previously,
for all the values of $\Lambda$
the vector BS solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
satisfy $M_{\rm max}<\mu Q_{\rm max}$,
and they are always gravitationally bound.
For $\Lambda\lesssim 1$
the solutions with $M_{\rm max}$ and $\mu Q_{\rm max}$
obtained from their local maximal values
are also expected to be dynamically stable,
and for $\Lambda\gtrsim 1$
those with $M_{\rm max}$ and $\mu Q_{\rm max}$
obtained from the limit of $f_0\to f_{0,{\rm crit}}$
would also be dynamically stable.
Thus,
for all values of $\Lambda$
the vector BS solution with $M_{\rm max}$ and $\mu Q_{\rm max}$
is expected to be stable.

\section{Conclusion}
\label{sec4}

In this paper,
we have investigated the BS solutions
in the Einstein-Proca theory with the quartic order self-interaction
as well as the mass \eqref{action}.
While the properties of the BS solutions
in the Einstein-Proca theory with the mass $\mu^2 \bar{A}^\mu A_\mu/2$
have a lot of similarities
with those of the BS solutions
in the Einstein-scalar theory with the mass $\mu_\phi^2|\phi|^2/2$,
we have found that
once the quartic order self-interaction $\lambda (\bar{A}^\mu A_\mu)^2/4$
is included into the action,
the properties of the vector BS solutions
become very distinct
from those of the scalar BS solutions
with the quartic order self-interaction $\lambda_\phi |\phi|^4/4$.

First,
we have formulated
the basic equations
to find the BS solutions in the Einstein-Proca theory.
Assuming the static and spherically symmetric metric ansatz \eqref{metric_ansatz}
and the monochromatic oscillation of the vector field in time \eqref{vector_ansatz},
the EOM could be rewritten
into a set of the evolution equations in the radial direction
\eqref{ms_evolv} and \eqref{a0a1_evolv2}.
Then, the boundary conditions for the metric and vector field variables
were derived by solving the evolution equations
in the vicinity of the center.
For the frequencies chosen to be the eigenvalues of the BS solutions,
Eqs. \eqref{ms_evolv} and \eqref{a0a1_evolv2}
were able to be numerically integrated,
and
the metric exponentially approaches the Schwarzschild form \eqref{metric},
while the two components of the vector field exponentially approach $0$.

The clear difference between the cases of the scalar and vector fields
appearing
in the presence of the quartic order self-interaction
was
that in the case of the Einstein-Proca theory
there is the critical amplitude of the temporal component
of the vector field at the center,
above which no vector BS solution could be obtained.
Moreover,
it was found that the qualitative behavior of the vector BSs was different across $\Lambda\simeq 1$,
where $\Lambda$ is the dimensionless coupling constant
defined in Eq. \eqref{Lambda}.
For $\Lambda\lesssim 1$,
including the negative values of $\Lambda$,
the behavior of the BS solutions was very similar to the case of $\Lambda=0$.
In this case,
the maximal values of the ADM mass and Noether charge
correspond to their local maximal values
obtained at the intermediate value of $f_0<f_{0,{\rm crit}}$,
and the compactness defined in Eq. \eqref{compactness} could not exceed $0.20$.
On the other hand,
for $\Lambda> 1$,
they
could be obtained from the critical central amplitude
of the temporal component of the vector field,
and
the compactness was always greater than $0.20$
but could not exceed $1/3=0.333\cdots$,
below which photon spheres would be absent.
However,
for the most compact vector BS solutions obtained for $\Lambda\simeq 1$
the ratio of the two different definitions of the effective compactness
\eqref{compactness} and \eqref{compactness2}
was close to $1.09$,
and it is still unclear
whether the photon spheres could be formed around them or not,
which will require further studies.
For $\Lambda\gg 1$
the maximal values of the ADM mass and Noether charge
could be fitted by the formulas \eqref{fitting},
which were of ${\cal O}[\sqrt{\lambda}M_{pl}^3/\mu^2 \ln (\lambda M_{pl}^2/\mu^2)]$,
and slightly larger than the Chandrasekhar mass for the fermions with the same mass $\mu$.

There are a lot of remaining issues,
e.g.,
the stability analysis against the radial and nonradial perturbations,
the implications for the future gravitational wave observations,
and
the BS solutions
in more general class of the complex vector-tensor theories.
They will be left for future work.

\section*{Acknowledgements}
This work was supported by FCT-Portugal through Grant No. SFRH/BPD/88299/2012.
We thank
the Yukawa Institute for Theoretical Physics, Kyoto University
for the hospitality
during the workshop ``Gravity and Cosmology 2018 '' (YITP-T-17-02)
and the  symposium ``General Relativity $-$ The Next Generation --'' (YKIS2018a).

redpajama_set_name: RedPajamaArXiv


text:
<?php

namespace Db\DbBundle\Entity;

use Doctrine\ORM\Mapping as ORM;
use Doctrine\Common\Collections\ArrayCollection;
use Symfony\Component\Validator\Constraints as Assert;
use Symfony\Bridge\Doctrine\Validator\Constraints\UniqueEntity;

/**
 * @ORM\Entity
 * @ORM\Table("Player")
 * @UniqueEntity("name")
 */
class Player extends Entity
{
	/**
	 * @var int
	 *
	 * @ORM\Id
	 * @ORM\Column(type="integer")
	 * @ORM\GeneratedValue(strategy="AUTO")
	 */
	protected $id;

	/**
	 * @var string
	 *
	 * @ORM\Column(type="string", length=100, unique=true)
	 *
	 * @Assert\NotBlank()
	 * @Assert\MinLength(3)
	 * @Assert\MaxLength(100)
	 * @Assert\Regex("/^[\w]*$/")
	 */
	protected $name;

	/**
	 * @var int
	 *
	 * @ORM\Column(type="integer", nullable=false)
	 *
	 * @Assert\NotBlank()
	 */
	protected $mana = 0;

	/**
	 * @var int
	 *
	 * @ORM\Column(type="integer", nullable=false)
	 *
	 * @Assert\NotBlank()
	 */
	protected $intelligence = 0;

	/**
	 * @var int
	 *
	 * @ORM\Column(type="integer")
	 *
	 * @Assert\NotBlank()
	 */
	protected $experience_points = 0;

	/**
	 * @var int
	 *
	 * @ORM\Column(type="integer", nullable=false)
	 *
	 * @Assert\NotBlank()
	 */
	protected $dexterity = 0;

	/**
	 * @var int
	 *
	 * @ORM\Column(type="integer", nullable=false)
	 *
	 * @Assert\NotBlank()
	 */
	protected $strength = 0;

	/**
	 * @var User
	 *
	 * @ORM\ManyToOne(targetEntity="User", inversedBy="players", cascade={"persist"})
	 */
	protected $user;

	/**
	 * @var ArrayCollection
	 *
	 * @ORM\OneToMany(targetEntity="Item", mappedBy="player")
	 */
	protected $item;

	/**
	 * @var UserClass
	 *
	 * @ORM\ManyToOne(targetEntity="UserClass")
	 */
	protected $class;

	/**
	 * @var Race
	 *
	 * @ORM\ManyToOne(targetEntity="Race", inversedBy="Player")
	 */
	protected $race;

	/**
	 * @var ArrayCollection
	 *
	 * @ORM\ManyToMany(targetEntity="Quest", mappedBy="Player")
	 */
	protected $quest;

	/**
	 * @ORM\OneToOne(targetEntity="Place")
	 * @ORM\JoinColumn(name="user_id", referencedColumnName="id")
	 */
	protected $place;

	/**
	 * Constructor.
	 */
	public function __construct()
	{
		$this->item = new ArrayCollection();
		$this->quest = new ArrayCollection();
	}

	public function __toString()
	{
		return $this->getName();
	}

	public function getSummaryIntelligence()
	{
		$sum = 0;
		foreach ($this->getItem() as $item) {
			if ($item->getWeared()) {
				$sum += $item->getIntelligence();
			}
		}
		$sum += $this->getIntelligence() + $this->getClass()->getIntelligence() + $this->getRace()->getIntelligence();
		return $sum;
	}

	public function getSummaryMana()
	{
		$sum = 0;
		foreach ($this->getItem() as $item) {
			if ($item->getWeared()) {
				$sum += $item->getMana();
			}
		}
		$sum += $this->getMana() + $this->getClass()->getMana() + $this->getRace()->getMana();
		return $sum;
	}

	public function getSummaryDexterity()
	{
		$sum = 0;
		foreach ($this->getItem() as $item) {
			if ($item->getWeared()) {
				$sum += $item->getDexterity();
			}
		}
		$sum += $this->getDexterity() + $this->getClass()->getDexterity() + $this->getRace()->getDexterity();
		return $sum;
	}

	public function getSummaryStrength()
	{
		$sum = 0;
		foreach ($this->getItem() as $item) {
			if ($item->getWeared()) {
				$sum += $item->getStrength();
			}
		}
		$sum += $this->getStrength() + $this->getClass()->getStrength() + $this->getRace()->getStrength();
		return $sum;
	}
}

redpajama_set_name: RedPajamaGithub


How to Find the Most Suitable Basketball Jerseys for Your Basketball Team Without question, the fans can easily recognize their favorite basketball teams because of the jerseys they wear during their games. Keep in mind that the jersey play an important role on your basketball team. Not only that the jerseys serve as the most usual source of identification for the team, on the other hand, it also makes sure that the players are completely comfortable when they play so that they can give their best during the game. For instance, if your basketball will be playing under tropical conditions, then it is definite that you would want a jersey cloth made from a refined quality in order to take in all the perspiration. Always keep in mind that is is very vital to zero in on the most suitable quality of jersey composing all the accessories since it will enable the performance of the players in a great way. And if the players are not given with the basketball jersey that fit them so well, then it is pretty obvious that the discomfort will surely affect their overall performance. The basketball jersey is usually composed of the player's number and name as well as the logo for them team and sports.
What are the things to consider in choosing basketball jerseys?
And due to the nature of the game, the sleeveless jerseys are usually in demanding in the basketball players. It is important for the players not to end up having a half sleeve or a full sleeve jersey because of the very reason that they are required to jump in order to shoot the ball with their own hands. How should you choose the color of the jersey? It is definite that the color for the jersey can be chosen after having a thorough discussing with the team members. But then again, keep in mind that each and every person may have their own opinions. Different colors have different importance and different players would certainly want their jersey to reflect on their various virtues. And because of this, it is highly advisable to pick a color that is the answer for the consensus of most team members. other things you should know about jersey. There are a great deal of sportswear companies that are offering personalized jersey at very competitive prices. Make certain that you discover these companies. It is vital that the clothing they offer are not just comfortable and appealing but are also customized as per your design and color tastes. As a result, be sure to choose the right company.


text:
Q: Is it possible to use apple's HLS on tvOS I want to add live channel streaming to my tvOS app. the plan was to use apple's HLS protocol.
I tried apple's HLS code example from:
https://developer.apple.com/videos/play/wwdc2016/504/
I have added import AVFoundation then added this code.
func setupAssetDownload() {
   let hlsAsset = AVURLAsset(url: assetURL)
   let backgroundConfiguration = URLSessionConfiguration.background(
      withIdentifier: "assetDownloadConfigurationIdentifier")
   let assetURLSession = AVAssetDownloadURLSession(configuration: backgroundConfiguration,
      assetDownloadDelegate: self, delegateQueue: OperationQueue.main())
   // Download a Movie at 2 mbps
   let assetDownloadTask = assetURLSession.makeAssetDownloadTask(asset: hlsAsset, assetTitle: "My Movie",
      assetArtworkData: nil, options: [AVAssetDownloadTaskMinimumRequiredMediaBitrateKey: 2000000])!
   assetDownloadTask.resume()
}

xcode gave me 'AVAssetDownloadURLSession' is unavailable
when i try the same code on iOS project it recognizes the AVAssetDownloadURLSession object.
according to apple documentation AVFoundeation supports HLS on tvOS.
what might be the issue ?

A: Currently in tvOS 10, HLS playback is indeed supported, for example when loading a stream like AVURLPlayer(url: "https://example.com/123.m3u8").
Unfortunately, AVAssetDownloadURLSession is not (yet) available right now. The Apple documentation mentioning HLS support is referring to the above.

redpajama_set_name: RedPajamaStackExchange


Let us help you to deliver your inquiry about the topic of Yogurt maker to the supplier(s) quickly.
Pictures from Internet About "Yogurt maker"
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Unravel the mystery as you build, buy and flip houses for big profits!
Elizabeth once again needs help dealing with her eccentric family and you're just the person for the job. Discover each of the hidden clues to unlock the mysterious secrets of the Graves family as you build, buy and flip houses in each of their eerie neighborhoods. With three campaigns to play, it won't take you very long to discover that some family secrets simply can't stay buried forever.


text:
E21 - Consumption; Saving; Wealth
Credit Conditions and Consumption, House Prices and Debt: What Makes Canada Different?
Staff Working Paper 2015-40 John Muellbauer, Pierre St-Amant, David Williams
There is widespread agreement that, in the United States, higher house prices raise consumption via collateral or possibly wealth effects. The presence of similar channels in Canada would have important implications for monetary policy transmission.
Content Type(s): Staff Research, Staff Working Papers Topic(s): Credit and credit aggregates, Domestic demand and components, Economic models, Financial Institutions, Financial stability, Financial system regulation and policies, Housing, Transmission of monetary policy JEL Code(s): E, E0, E02, E2, E21, E4, E44, G, G2, G21, R, R2, R21, R3, R31
On the Welfare Cost of Rare Housing Disasters
Staff Working Paper 2015-26 Shaofeng Xu
This paper examines the welfare cost of rare housing disasters characterized by large drops in house prices. I construct an overlapping generations general equilibrium model with recursive preferences and housing disaster shocks.
Content Type(s): Staff Research, Staff Working Papers Topic(s): Asset Pricing, Economic models, Housing JEL Code(s): E, E2, E21, E4, E44, G, G1, G11, R, R2, R21
Exploring Differences in Household Debt Across Euro Area Countries and the United States
Staff Working Paper 2015-16 Dimitris Christelis, Michael Ehrmann, Dimitris Georgarakos
We use internationally comparable household-level data for ten euro area economies and the United States to investigate cross-country differences in debt holdings and the potential of debt overhang.
Content Type(s): Staff Research, Staff Working Papers Topic(s): Credit and credit aggregates, Econometric and statistical methods, International topics JEL Code(s): D, D1, D12, E, E2, E21, G, G1, G11
Does Financial Integration Increase Welfare? Evidence from International Household-Level Data
Staff Working Paper 2015-4 Christian Friedrich
Despite a vast empirical literature that assesses the impact of financial integration on the economy, evidence of substantial welfare gains from consumption risk sharing remains elusive. While maintaining the usual cross-country perspective of the literature, this paper explicitly accounts for household heterogeneity and thus relaxes three restrictive assumptions that have featured prominently in the past.
Content Type(s): Staff Research, Staff Working Papers Topic(s): International financial markets, International topics, Labour markets, Recent economic and financial developments JEL Code(s): E, E2, E21, F, F3, I, I3, I31


text:
Q: Using Query to get the dates from a cell Google sheets Context
Following on from this question, I am trying to make this more dynamic. I want to input data in a cell 'D1' to then make Query bring back the data from another spreadsheet. Below is a depiction of what this looks like.

This is the formula I tried:
=QUERY(Query(ImportRange("ID", "AvevaGroupPrice"),"offset 1", 0), "where Col1 >= Date " &D1)
I used this format because eventually I want to create a dynamic date range where I type the respective dates I need in the first two 'D' rows.
Problem
How can I make this work using data in cell D1?

A: Based on the Query Language Documentation, when you include date keyword you need to use a date with yyyy-mm-dd format
In your current function, the date was converted to a number when appended to the string.
Sample: ="date "&D1
Result: date 43830
You need to use TEXT() to convert the date to yyyy-mm-dd format.
Sample: ="where Col1 >= Date '"&TEXT(D1,"yyyy-mm-dd")&"'"
Result: where Col1 >= Date '2019-12-31'

(UPDATE)
You can also use TO_TEXT() to convert the Cell Value to a text value.
Sample: ="where Col1 >= Date '"&To_text(D1)&"'"
Result: where Col1 >= Date '2019-12-31'

Your Formula:
=QUERY(Query(ImportRange("ID", "AvevaGroupPrice"),"offset 1", 0), "where Col1 >= Date '"&TEXT(D1,"yyyy-mm-dd")&"'")
or
=QUERY(Query(ImportRange("ID", "AvevaGroupPrice"),"offset 1", 0), "where Col1 >= Date '"&To_text(D1)&"'")


Earlier this month, a new cryptocurrency was created, as Bitcoin's blockchain was forked to create Bitcoin Cash (BCC) – a fork of Bitcoin that shares all historical Bitcoin transactions, but all future dealings will remain independent from one another.
You'd be forgiven for wondering why exactly BCC is getting such attention given that thousands of cryptocurrencies are now available. However, given that the currency was created by forking Bitcoin core, it now means that Bitcoin investors now have potential access to an equal amount of BCC in the coming months – akin to free money providing that BCC retains any kind of value.
BCC was forked out of the Bitcoin core following months of debate within the community of Bitcoin investors who have talked at length about how to scale this type of cryptocurrency to a wider user base.
The answer was to expand the maximum Blocksize limit parameter of the Bitcoin codebase, increasing it from 1MB – enough for around a quarter of a million daily transactions – to 8MB – catering for almost ten times as many processed transactions per day; with more transactions translating into quicker payments and reduced fees.
How much is Bitcoin Cash worth today?
At the time of writing, BCC is actually worth a decent amount of money on paper. Some exchanges appear to be trading BCC at $400 per coin, which actually ranks it among the top-five cryptocurrencies by market cap in August 2017.
The problem at the moment is that it's not that simple to sell BCC on the exchanges. Although some of the trading exchanges have already included BCC to trade, liquidity is extremely poor, which may be skewing the true value of BCC per coin at present.
The lack of liquidity is largely due to the fact that trading exchanges aren't accepting BCC deposits at present, leaving many investors with BCC that they can't utilise or sell until it can be transferred. On techcrunch.com, many cryptocurrency experts are predicting is that the price of BCC will tank significantly once liquidity increases as investors seek to make cash in on their BCC coins as soon as they can.
Meanwhile, other financial bloggers such as Wolf Richter, of wolfstreet.com, believe that Bitcoin itself and other cryptocurrencies could fall in value, with significant volatility across all exchanges at present. Richter claims its volatility does not make Bitcoin a "usable currency" and believes a crash is quite possible "someday".
A crash would by no means by a disaster for investors, who may profit long-term from being able to buy at rock-bottom rates at the exchanges and sell much further down the line. Sites such as ForexBonus.com.au are worth keeping an eye on, particularly for financial traders based in Oceania, to receive rebates and bonuses trading Bitcoin and Bitcoin cash for the very first time with well-regulated platforms.
With lots of daily volatility across all major cryptocurrencies at present, it's a great time to be a Bitcoin trader. Providing you understand technical analysis of the markets, you don't even need to understand the techno babble!


text:
Bengöl kan syfta på ett antal insjöar i Sverige:

Bengölen, Blekinge, sjö i Karlskrona kommun,
Bengölen, Östergötland, sjö i Linköpings kommun,
Lilla Bengölen, sjö i Norrköpings kommun,
Stora Bengölen, sjö i Norrköpings kommun,

Listor över Sveriges insjöar baserat på namn

redpajama_set_name: RedPajamaWikipedia


An in-depth and intimate executive leadership program designed for emerging female leaders just like you.
Where Mentors and Mentees share and collaborate.
The Outcome: Create a network of real relationships.
Participate in Leadership Programs, Think Tanks, and Mentor Roundtables.
The Result: Authentic dialogue with a community of peers.
Be matched with a Coach to hone your leadership skills and market your talent.
Your Reward: Excel in a job you love.


text:
Solar & Wind Articles, Stories & News
Capital Dynamics acquires 300 MW Texas wind farm
Once commissioned, Green Pastures will generate enough electricity to power 110,000 homes.
2nd Annual Energy Storage North America Conference and Expo Doubles 2013 Attendance
Sets New Standard for Industry Participation and Deal-making
Wind power needs subsidy to compete in US, says Siemens
The US wind power industry still needs subsidy to compete with fossil fuels, according to Germany's Siemens, one of the world's largest wind turbine manufacturers. Lisa Davis, who took over in August as the head of Siemens' global energy business, told the Financial Times that although wind power in the US was close to "grid parity" – the level at which it becomes competitive with other sources of electricity – "we're not there yet". Her comments come as the US industry is urging Congress to reinstate the tax credit for wind generation, which expired at the end of last year. The American Wind Energy Association has warned that the industry could face falling investment and employment if the credit is not restored quickly. Lazard, the investment bank, calculated recently that in parts of the US with strong wind, it could be a cheaper source of power than gas-fired generation, after steep declines in the cost of turbines. However, Ms Davis argued that costs still needed to be cut further before wind could compete on equal terms with gas. "We've not yet got to the point where it's truly self-sustaining," she said. "We've got to focus on cost competitiveness."
A solar cell that stores its own power: World's first 'solar battery' runs on light and air
Is it a solar cell? Or a rechargeable battery? Actually, the patent-pending device invented at The Ohio State University is both: the world's first solar battery. In the October 3, 2014 issue of the journal Nature Communications, the researchers report that they've succeeded in combining a battery and a solar cell into one hybrid device. Key to the innovation is a mesh solar panel, which allows air to enter the battery, and a special process for transferring electrons between the solar panel and the battery electrode. Inside the device, light and oxygen enable different parts of the chemical reactions that charge the battery. The university will license the solar battery to industry, where Yiying Wu, professor of chemistry and biochemistry at Ohio State, says it will help tame the costs of renewable energy. "The state of the art is to use a solar panel to capture the light, and then use a cheap battery to store the energy," Wu said. "We've integrated both functions into one device. Any time you can do that, you reduce cost." He and his students believe that their device brings down costs by 25 percent.
SPI 2014 - APS Now Shipping New YC1000 Microinverter, a Groundbreaking 277/480, 3-phase Unit for Large-scale Commercial Applications
APS, a global leader in solar microinverter technology, will showcase its new YC1000 microinverter for the domestic market at Solar Power International, Oct. 20-23 in Las Vegas.
Southern California Startup Is Changing The Way We Purchase Solar
Pick My Solar provides bids, evaluation and advocacy to make shopping solar easy.
Energy Storage North America Opens With Record Participation
Honors Top Storage Projects and Industry Champions With ESNA Innovation Awards
Canadian Solar Completes the Sale of Three Solar Power Plants to TransCanada
10/02/14, 12:31 PM | Solar & Wind | Canadian Solar
The William Rutley project closed on September 29, 2014 followed by Liskeard 3 & 4 on September 30, 2014
EDF Renewable Energy and Microsoft Corporation Break Ground on the Pilot Hill Wind Project
More than 100 people attended the ceremony including U.S. Rep. Robin Kelly, D-Ill., who called it a historic day for the state of Illinois and the nation.
NRG Energy Acquires Pure Energies Group, Adding Stronger Online Platform for Customer Access to NRG's Rapidly Expanding Residential Solar Business
Pure Energies builds on recent acquisitions to empower customers to go solar in an easy and streamlined way
Abengoa Yield's Solana Solar Power Plant in Arizona Wins an Energy Storage North America Innovation Award
Solana's molten-salt thermal energy storage system is able to generate power at maximum capacity for six hours without solar radiation. ---- Solana is the first commercial plant in the United States to use this new thermal energy storage technology.
SunShare and Hyland Hills Water World Sign one of the Nation's Largest Community Solar Agreements
Denver area's water park signs up to be powered by SunShare's Solar Gardens
Renewables sector welcomes Scottish contribution to marine energy research
10/02/14, 12:17 PM | Solar & Wind | Scottish Renewables Forum
Scottish Enterprise to contribute £450,000 to European research programme
GLOBAL CLEAN ENERGY INVESTMENT SUSTAINS ITS RECOVERY
Figures for the third quarter show another year-on-year rise, with a solar boom in China the biggest single contributor
Green Energy Ohio tour features GEM Energy's Anthony Wayne Solar Array
The 2.1-MW Anthony Wayne Solar Array, developed by GEM Energy, is featured on the Green Energy Ohio tour of clean energy and green design projects. GEM Energy will host an open house at the solar array from 11 a.m. to 1 p.m. Friday, October 3, at 671 Spencer St, Toledo, Ohio.
Records 8131 to 8145 of 25658
OMG Roofing Products - PowerGrip Universal
Eliminate Solar Rack Ballast and Reduce Rooftop Material Handling. PowerGrip™ Universal (PGU) is a commercial roof mount system designed to secure solar racks and other equipment to any type of commercial roofing system. PGU provides a secure connection directly to the roof deck or structural members and is designed to reduce or eliminate ballast in solar racking systems, so there's less weight on the roof and less material handling. Made of heavy-duty cast aluminum, PGU units include a base ring which is secured through the roofing cover and into the structural roof deck, and a waterproof top plate.


From I5 S take exit 234B to merge onto OR-99E/Pacific Blvd SE. Continue on Pacific Blvd for 2.2 miles then take the US-20 ramp to Albany City Center. Turn right onto Lyon Street then left onto W First Ave, we are in the third block down on the right at 317 W First Ave, Suite 101.
From I5 N take exit 233 and merge onto US-20 W Continue on US-20 W for 2 miles, then take the US-20 Ramp to Albany City Center. Turn right onto Lyon St then left onto W First Ave, we are in the third block down on the right at 317 W First Ave Suite 101.
Take Hwy US-20 E/Albany-Corvallis Hwy into Albany. Turn right onto W First Ave we are in the second block down on the right at 317 W First Ave, Suite 101.
We strive to give our customers as much attention and comfort as we can, as well as provide them with the best service possible.
If you've had an amazing experience, we would love to hear abut it or feel free to share your questions or suggestions on how we can make your experience at Level 3 better in any way.