text:
Tod & Tara Leiweke
Tod and Tara Leiweke
Seattle Sounders FC announced the addition of Tod and Tara Leiweke to the club's Ownership Group on October 5, 2020. The Leiwekes join the Rave Green's Seattle-focused family of team owners, bringing their rich knowledge and experience from the world of professional sports, as well a special passion for the greater Seattle community and Puget Sound region.
"It is a proud day for Sounders FC, as we officially welcome the Leiweke family as partners in our club's future," said Sounders FC Majority Owner Adrian Hanauer. "When Tod and Tara returned to Seattle two years ago to lead the Kraken, I was thrilled on a professional level to have Tod back in the Seattle sports scene, but also, more personally, to welcome back some of the finest, most caring and genuine people I've ever had the pleasure to know. Given Tod's history with the club, I couldn't have been more enthusiastic when the Leiwekes expressed interest in reinvolving themselves with Sounders FC on a deeper level. Their ownership stake was finalized earlier this year, though we chose at the time to delay the official announcement due to the gravity of the issues being faced throughout the world. While these issues certainly remain, as a club we felt it was time to share the news."
No stranger to the Emerald City, Leiweke's roots in Seattle sports run deep. The St. Louis native led the NFL's Seattle Seahawks from 2002-2010 as Chief Executive Officer, during which time he spearheaded the re-invention of the "12s" concept, still celebrated and synonymous with Seahawks support across the globe. Leiweke also oversaw Sounders FC's expansion into MLS in 2009, helping the club make its wildly successful jump into the top tier of American soccer.
Following his first stint in Seattle, Leiweke served as CEO of the NHL's Tampa Bay Lightning from 2010-2015 before joining the National Football League as Chief Operating Officer, the righthand position to Commissioner Roger Goodell. Leiweke returned to the Pacific Northwest in 2018, hired to serve as CEO of the then-unnamed Seattle NHL expansion franchise.
Now dubbed the Seattle Kraken, Leiweke is set to lead Seattle's newest professional sports team as it hits the ice in 2021. Alongside wife Tara, the seasoned sports executive further deepened his Seattle sports roots earlier this year when deciding to rejoin Sounders FC, this time in an ownership capacity.
"As a family, we have been privileged to enjoy an incredible journey through the professional sports world, but anyone that knows us understands that Seattle is home, and our fondness for Sounders FC runs deep," said Tod and Tara Leiweke. "Our friendship with Adrian dates back to a time when the idea of launching Sounders FC in Major League Soccer was just beginning. The launch of the club was a truly special moment in our lives, and seeing the organization thrive from the very beginning is something we take a lot of pride in. In our time away from Seattle, we always followed this great club, and to be able to return to it now is significant for us."
With their previous experience assisting Sounders FC in its initial launch, the Leiwekes have a strong foundational knowledge of the club's close ties to the community and its fans. They join the rest of the Rave Green's ownership contingent with a passion for continuing to build a successful franchise that represents the city and wider region.
Last summer, Sounders FC announced the addition of 11 families to the team's Ownership Group. The new investors were led by former Microsoft executive Terry Myerson and his wife Katie, and included Seattle icons such as Ciara and husband Russell Wilson, Macklemore and wife Tricia Davis, and many more, creating a diverse roster of partners with direct ties to Seattle and a wealth of experience in sports, entertainment, technology, broadcast, finance and beyond. Following that initial news, the club then announced the addition of a 12th family to the Ownership Group, as current President of Business Operations Peter Tomozawa and wife Donna LeDuc officially joined the organization.
The Leiwekes join this group of diverse partners, adding to Sounders FC's longstanding ownership contingent of Adrian Hanauer, Lenore Hanauer, Drew Carey, Jody Allen and Paul Barry. More information on the club's full Ownership Group can be found via SoundersFC.com.
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text:
HOMEPAGE HOBBIES
Disadvantages to a parallel circuit
By: April Kohl
Updated on: May 13, 2017
How to Interpret Circuit Diagrams
Factors Affecting Electrical Conductivity
How to calculate output voltage
How to Calculate a Current Through a Resistor
What is a 3-Phase Power Supply?
Maria Toutoudaki/Photodisc/Getty Images
Parallel circuits are circuits in which there are multiple paths running alongside one-another. In many cases this will be due to either multiple power sources flowing to a single output, or one power source running to multiple outputs. By splitting the circuit along several lines in this way, parallel circuits are able to achieve things series circuits cannot, but they also come with several disadvantages.
Limited voltage
No matter whether you have several power sources or just one, the voltage across a parallel circuit will remain the same. This is because the potential difference, which is the source of voltage, is split across the circuit as a whole; so increasing the power will not alter the voltage.
Variable current
In a parallel circuit, the energy from the power source is split across the circuit. This results in different levels of current flowing, depending on where in the circuit you place an ammeter to check it. This prevents parallel circuits from being used anywhere a constant current is required throughout.
Complex resistance
When branches are added to a parallel circuit, the voltage is equalised across the whole circuit, meaning the current flow must change to compensate. This has a knock-on effect on the resistance in the circuit as a whole and results in a lower resistance in the circuit when more resistors are added in new branches. The only way to increase resistance is to add resistors in series with one another and on existing branches.
GCSE Physics: Current in Parallel Circuits
Based in the United Kingdom, April Kohl has been writing since 1992, specializing in science and legal topics. Her work has appeared on the Second Life News Network website and in British Mensa's "LSQ" magazine. Kohl holds a Bachelor of Science in physics from Durham University and a diploma in English law from the Open University.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Issue with QGIS modeler raster calculator I'm trying to apply a formula to scale an NDVI map from a WMS with QGIS modeler, but it keeps throwing me an error. I tried to do this without the modeler and it works.
Here are the steps I want to accomplish in the QGIS modeler:

*

*Apply the algorithm "map to raster" to my NDVI map (previously added to my project)  with a predefined resolution (since I'm working with small areas, I set it to 1 unit map per pixel)

*With the algorithm "raster calculator", apply the recommended formula from the website I downloaded the NDVI, to the NDVI generated in the previous step. The formula is: (1 * BandaTriada - 100) / 100, where BandaTriada means my "OUTPUT@1"

At this point I already have the NDVI as I want it to be. All that's left is adding my NDVI style to the raster layer generated, with the  set style to raster layer algorithm.
The script gets stuck while performing the raster calculator part - the error I'm getting is:
Versión de QGIS: 3.10.4-A Coruña
Revisión del código de QGIS: 10bc9f2ff2
Versión de Qt: 5.11.2
Versión de GDAL: 3.0.4
Versión de GEOS: 3.8.1-CAPI-1.13.3
Versión de PROJ: Rel. 6.3.1, February 10th, 2020
Procesando algoritmo...
Algoritmo 'modelo' comenzando…
Parámetros de entrada:
{ 'Extension' : '312455.56408728764,313833.46447932476,4606153.619686341,4607207.818906171 [EPSG:25831]', 'NDVImapWMS' : 'crs=EPSG:25831&dpiMode=7&format=image/png&layers=ox3dndvi2019&styles&url=http://geoserveis.icgc.cat/icc_ortoxpres/wms/service?', 'qgis:rastercalculator_1:NDVI scaled' : 'TEMPORARY_OUTPUT' }

Preparar algoritmo: qgis:rasterize_1
Ejecutando Map to raster [1/3]
Parámetros de entrada:
{ EXTENT: '312455.56408728764,313833.46447932476,4606153.619686341,4607207.818906171 [EPSG:25831]', LAYER: 'crs=EPSG:25831&dpiMode=7&format=image/png&layers=ox3dndvi2019&styles&url=http://geoserveis.icgc.cat/icc_ortoxpres/wms/service?', MAKE_BACKGROUND_TRANSPARENT: False, MAP_THEME: '', MAP_UNITS_PER_PIXEL: 1, OUTPUT: 'C:\\Users\\...\\AppData\\Local\\Temp/processing_ca25d8d8ef7c438194bb38a19eb91be7/24fdd4b2efac401d814bf96c14b56787/OUTPUT.tif', TILE_SIZE: 1024 }
OK. La ejecución tardó 20.074 s (1 salidas).
Preparar algoritmo: qgis:rastercalculator_1
Ejecutando Raster calculator [2/3]
Parámetros de entrada:
{ CELLSIZE: 1, CRS: 'ProjectCrs', EXPRESSION: '(1 * \"\'Capa de salida\' from algorithm \'Map to raster\'@1\" - 100) / 100', EXTENT: 'C:/Users/.../AppData/Local/Temp/processing_ca25d8d8ef7c438194bb38a19eb91be7/24fdd4b2efac401d814bf96c14b56787/OUTPUT.tif', LAYERS: '', OUTPUT: 'TEMPORARY_OUTPUT' }
An error occurred while performing the calculation
Se encontró un error al ejecutar Raster calculator
Se encontró un error al ejecutar Raster calculator
La ejecución falló después de 20.24 segundos

Cargando las capas resultantes
Las siguientes capas no se generaron correctamente.

C:\Users\...\AppData\Local\Temp/processing_ca25d8d8ef7c438194bb38a19eb91be7/79264905edc3416891cf30ddd1d56e06/OUTPUT.tif

Puede comprobar el «Panel de mensajes del registro» en la ventana principal de QGIS para encontrar más información sobre la ejecución del algoritmo.

I have to say that I also had to tweak the "cellsize" and "CRS" options from the raster calculator, since I was getting an error aswell if I lefted it without any value or CRS. Not sure if it helps:
No reference layer selected nor CRS provided
No reference layer selected nor cellsize value provided

redpajama_set_name: RedPajamaStackExchange


text:
#SOPHFIFEST
Where Soph meets Sci-Fi
CONTACT SOPH
Day: 10 December 2020
Cosmic Candy [2020, SciFi London 2020]
by B.B. Chee December 10, 2020 December 10, 2020 SciFi London 2020 / sophfifest2020Leave a Comment on Cosmic Candy [2020, SciFi London 2020]
"You must not do the things I do…"
Cosmic Candy almost didn't happen due to budget constraints but I'm so glad it did. This film – a first time feature film for director Rinio Dragasaki – follows protagonist Anna. She works in a supermarket which stocks a product called 'Cosmic Candy' which is "a potent version of Space Dust/Pop Rocks that sends its user into a dreamlike, hallucinogenic state". In real life, Anna's world is drab and seems to be orderly which is illustrated beautifully through the almost militarily organised shelves at the shop and the shelves in her home, but behind the curtain Anna is spiralling. She cannot function in real life without some sort of drug.
One day her whole world is turned upside down by a young girl called Persa who has been abandoned by her father. He has supposedly told her he will be back for her. Anna is faced with caring for this child who pulls her more and more to reality and brings to the light her idiosyncrasies.
I really liked Anna's character. She was a pretty complicated, dysfunctional person who just wasn't capable of dealing with normality and had this childlike personality. You could imagine that she was a teenager and not a 30-something year old but she is also capable of change, and that's what is so refreshing about her character. I noticed that as the film progressed you see that the shelves and the tidiness around Anna gets more chaotic as she unravels, and I really liked that device.
The visuals for this film are really really good. It's trippy and vibrant and shows you the inner workings of Anna's mind while she is on Cosmic Candy. However, it's not all unicorns and rainbows. There's this repeated paranoia element within the trippyness, which if you have taken drugs and had a bad trip you will probably recognise. There are multiple moments in the film where Anna is at the centre of a larger audience's ridicule, i.e. pointing, laughing and so on, but it seems like it's probably all in her mind. It's definitely the stuff that nightmares are made of, and seems to stem from this thing of wanting to be accepted when in reality she is kind of a loner (though some of that may be of her own choosing).
I liked the inclusion of (I think???) Debussy's music through these weird celestial keyboard sounds which perfectly matches the trippy vibe of the movie. I also enjoyed the mix of ethereal, theramin sounds, and then also the very quaint, fun pop and electro songs, like the one about french toast, or computer song with its lyrics "I program my feelings away". Computer humour snarf! 🙂 I see what you did there!!!
I noticed what might be an Easter Egg but I wasn't able to ask the Director this. There is one scene where Anna is at her home. This is after she starts to spiral. I noticed a spinning tractricoid toy, like the ones in the film Inception. In Inception, the device is used so that protagonists can determine whether they are in reality or if they are in a dream (within a dream within a dream). If the toy doesn't fall like gravity dictates, they know they are in a dream state and indeed in Cosmic Candy, you don't see it fall, so I felt like that was a device to illustrate the dreamlike state that Anna was in. Its inclusion whether intentional or not made me really second guess if any of it was real as the story progressed. Maybe Anna was in a dream the whole time? Maybe the girl wasn't real? Maybe the girl… dun dun daaaa WAS HERSELF?!
The film cascades with the ending sequence where Persa is due to act in a school play. The play is about Manto Mavrogenous, who was a revolutionary Greek commander and a woman (yeehar!!) so there are definite Feminist messages in the film. For those who don't know, Manto was part of a secret society called Filiki Eteria which was a movement attempting to overrun the Ottoman empire which massacred the Greeks. I was really psyched to see this featured in the movie for my own personal reasons. My dad is Black (half Trinidadian/half Chinese) but my mum is actually Greek. Her heritage is of the Pontic Greeks who emigrated from the mainland to a mountainous region to the North of what is now Turkey called Pontus. Over many centuries of the Ottoman rule, Greeks were forced to leave Pontus to neighbouring countries for safety and things came to a head in the first world war where Pontic Greeks were taken from Pontus and ritually massacred or forced to take part in something called a 'Death March', which is where you are made to walk to concentration camps and those who have died along the way are left. My own great grandmother and grandmother (and her siblings) were actually caught in the middle of this but managed to escape from what would have been certain death, apparently into a forest and from there into mainland Greece (and similarly my great grandfather and grandfather escaped eastward to Georgia). So the strength that they sing of about the fierceness of Greek women in this story is something that I feel very strongly in my own family history, and Anna definitely channels that strength for herself. Brava! Anyways, I digress.
So all in all, I really enjoyed this film. I wasn't sure at first what made it sci fi but then any world which would legalise hallucinogenics must be an alternate reality (lol). It's such a compelling story, with this underlying unease like something is very wrong and the visuals blew my MIND. Check it out!
You can watch Cosmic Candy here, but you will need to buy a film or festival pass.
Tagged : cosmic candy / dream / dreams / Greece / Greek / Greek film / hallucinogenic / Manto Mavrogenous / Rinio Dragasaki / scifi london / SciFi London 2020 / sophfifest / sophfifest2020 / vibrant
Mirror Human [2020, SciFi London 2020]
by B.B. Chee December 10, 2020 December 10, 2020 SciFi London 2020 / sophfifest2020Leave a Comment on Mirror Human [2020, SciFi London 2020]
There is a LOT going on in this film and I feel like I didn't do it justice watching it (my concentration wasn't 100% in it), so I think I need to re-watch this.
This film follows three different interlocking stories around genetically modified replicas/androids called Transcendants. Their lives are tragically short and it explores their complicated stories. From the offset we find out that the world is quite barren and that these androids have been created with excellent genes for a variety of reasons, but at some point they become unacceptable to human society and the Mirror project (which is what they are called) is suspended.
There's this evident sense of otherness for the Transcendants compared to humans. At one point someone says that they are "the epitome of the human condition", that they are "not like us" and they are not capable of loving in the way that humans do. However, we see very quickly that they definitely are, for instance through Xiao-Sun (the Boxer) and Tien's relationship which is so endearing. The interesting thing about Xiao-Sun as well is that as the android, he is totally being used by the system/the man as a form of entertainment for the masses. You see him being physically beaten for others' financial gain and it's obvious from this that Transcendants are considered the lowest of the low. Like they're disposable.
The film explores what it is to be human. There's one part of the film where one of the androids says "once you have a name, you can call yourself human" (the androids are assigned numbers) and that sentence really made me think is that the only thing that sets us apart? Perhaps it's the self awareness of our identities? I was reading recently about why we can't remember the first years of our lives (don't ask me why, I just was!) and that our earliest memories don't start to form until approximately three onwards. I believe it's the New Zealand Maori who have the earliest record of childhood memories which is something like from 2.5 years old onwards, and that's simply because their culture places importance on indigenous and family history. There's this theory in all of this that our memories only form when we have the language to do so, that memories are connected to language and not feelings/emotions and so when I heard them say the sentence about humanity and your name, it reminded me of that theory. Because having a name is more than about identity, it's about connection to memories and history and kin and collectivity. This is the rabbit hole I went down having watched this film…
Liang (who calls himself the delivery driver) and his best friend Jay who is an android have a great dynamic which I enjoyed. Jay says that he has perfect genes yet he cannot beat Liang in a race, and Liang says that fear of death is what pushes him to drive faster. I feel like this is something that you see in sports people a lot. In order to compete at the highest level, to run your fastest, swim your farthest and so on, you have to feel like you are putting everything of yourself into it. Like you are pushing your body to the absolute limit so that idea definitely resonated. Perhaps androids aren't capable of doing that, or perhaps it's that the Transcendants sole desire is to live so why would they push that far (to the point of imminent death)? Also, such a cute little Star Wars reference (BB8) through the little robot friend of Liang's.
I enjoyed the soundtrack which was lead by very elegant piano bits and it swung between very whimsical and then super determined sounding.
There was one bit of the film which I didn't quite understand, which was that at parts people were wearing breathing masks. When the Boxer died, they all turned to face the camera with these masks on and it was acknowledged that he was no longer alive. I might have missed that but I wasn't sure what the ecological reason was behind that. Who were the people with the masks? Was the earth unlivable, or was it a specific area? And there was an unspeaking, smiling girl with pigtails handing out masks. Was she a Transcendant? What I got from that was that she was, and that her purpose was to sit in an area that would be harmful to breathe if you were human, but because she's like this disposable android her job is to do just that and hand out masks. I might totally have my wires crossed about that though… I really wish that there was a Q&A for this film because I would have liked to question this.
So all in all, this film definitely gave me some things to think about but I'm not sure how I feel about the film. Nice soundtrack. I liked the concept. I found it a little too complicated for a single watching, so I'll definitely need to watch it again. I liked the dynamics between the characters and I know I will be thinking about this film for a long while. Really polished film, seemed like it had high production values and I really felt the tragic nature of the android's lives. If you have any thoughts, would love to hear from you readers!
Check out Mirror Human here. To watch it, you'll need to buy a film or festival pass.
Tagged : androids / clones / genetic modification / Mirror Human / scifi london / SciFi London 2020 / sophfifest / sophfifest2020 / Zhang Linzi
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text:
\section{Introduction}

Experiments on a variety of condensed matter systems such as under-doped high-temperature superconductors, strongly correlated quasi-two dimensional organic compounds like $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$\cite{Kanoda} and the triangular lattice antiferromagnet CsCuCl$_4$ \cite{Coldea} all point to the possibility of realizing genuinely new phases of matter outside of standard paradigms such as fermi liquid theory of normal metals, spin-wave theory of magnetically ordered states, and the BCS theory of superconductivity. There has been a great deal of activity in the recent past aimed at developing theoretically consistent descriptions of such genuinely new phases of matter---examples include various proposals for topologically ordered liquid states of strongly-correlated systems, and their gauge-theoretic description \cite{Read_Sachdev,Wen,Senthil0,Motrunich_Senthil, Balents_Fisher_Girvin,Motrunich0, Hermele_Balents_Fisher}.

A generic feature of these `exotic' phases is that their low lying excitations are described in terms of quasiparticles that carry unusual quantum numbers and are best thought of as fractions of the usual electron or magnon excitations. These fractionalized quasiparticles interact with each other by emergent gauge forces, and have an independent existence only in such exotic phases where the gauge forces scale to zero at large distances effectively `deconfining' the fractionalized quasiparticles.

In closely related work, Senthil~{\it et. al.}~\cite{Senthil_etal} have also developed a theory of a class of second order quantum phase transitions that fall outside of the rubric of Landau's theory of phase transitions. In the usual Landau theory, the transition is described by analyzing fluctuations of the order parameter field that encodes the sharp distinction between the two phases. In certain situations where the phases on either side of the transition break different symmetries,  Senthil {\it et. al.} demonstrated that this usual description fails, in that it would erroneously rule out the possibility of a direct second order transition in the generic case.

Such a direct transition is actually possible, and its properties are best understood in terms of new fractionalized degrees of freedom (that can be thought of as fractions of the quasiparticle excitations of  the adjoining phase) that are liberated at precisely the critical point~\cite{Levin_Senthil}.
This has sparked interest~\cite{Lee, Melko0,Melko1,Melko2,Sandviklatest} in theoretical studies of relatively simple model systems where one can check for the presence of such unusual phases of matter and  non-landau `deconfined' quantum phase transitions with numerical studies.

In recent work~\cite{Damle_Senthil}, two of the present authors argued that a $S=1$ antiferromagnet with isotropic exchange $J$ and strong uniaxial single-ion anisotropy $D$ on the kagome lattice would support an unusual spin-nematic state in zero field, and a spin-density wave magnetization plateau state for a range of non-zero magnetic fields along the easy-axis. Both phases and the transition between them can be modeled by a Hamiltonian for hard-core bosons with unfrustrated hopping $t$ and frustrated nearest-neighbour repulsive interactions $U$\cite{Damle_Senthil}; the
nematic state corresponds to a ordinary superfluid state of bosons at half-filling, while the magnetization plateau state corresponds to a $1/3$ filled charge-density wave insulator  that occupies a lobe in the $\mu$-$U/t$ phase diagram (where $\mu$, the chemical potential for bosons, is linearly related to the applied magnetic field, and $U/t \sim D^2/J^2$).

As the two phases break different symmetries, the transition between them has the possibility of being an unusual non-Landau transition~\cite{Isakov_Kim0}. Numerical studies of the transition close to the tip of the lobe~\cite{Damle_Senthil,Isakov_Kim1} reached the tentative conclusion
that the transition between the two phases may indeed be a direct second-order transition with unusual exponents, although more recent work  on larger sizes seems to see  faint signatures of a very weak first-order transition close to the tip~\cite{Isakov_Kim1}.

\section{The Model and schematic phase diagram}

Motivated in part by this intriguing transition seen on the Kagome lattice, and its possible relevance to the physics of anisotropic antiferromagnets,  we consider here a closely analogous model of hard-core bosons on the checkerboard (planar pyrochlore) lattice, with the aim of shedding further light on the physics of such systems (As our work was nearing completion, other work on similar {\it fermionic} models~\cite{Fulde1,Fulde2,Shtengel} and somewhat different SU(2) symmetric
spin models~\cite{Nussinov1,Nussinov2} has also appeared in the literature).

The checkerboard or planar pyrochlore lattice is made up of four site plaquettes which can be thought of as projections of a tetrahedron onto a plane, and are centered at the sites of an underlying square net (Fig~\ref{lattice}). The bosonic particles live on the vertices of this `tetrahedron' (which lie on the centers of the links of the square net). The Hamiltonian for our boson hubbard model reads

\begin{eqnarray}
\label{microham}
H = -t\sum_ {\langle ij\rangle}(b^{\dagger}_{i}b_{j} + h.c.)~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \\
\mbox{~~~~~~~~~~~~~}+ U\sum_{p}(n_{1p}+n_{2p}+n_{3p}+n_{4p}-f)^2
\end{eqnarray}

where $\langle ij\rangle$ refers to the nearest neighbour links of the checkerboard lattice, $b^{\dagger}_i$ ($b_{i}$) creates (destroys) a particle on site $i$, $t > 0$ denotes the nearest neighbour hopping, and $U > 0$ denotes the repulsion energy between bosons on the same `tetrahedral' plaquette $p$.
Note that $n_{1p}, n_{2p}, n_{3p}$ and $n_{4p}$ are the boson number operators of the sites of the $p^{th}$ such tetrahedral plaquette (which can equally well be indexed by the site of the underlying square lattice around which it is centered), and the
parameter $f$ serves as a reduced chemical potential that tunes the particle density in the ground state.

Before we begin a detailed discussion of our numerical results, it is useful to delineate the gross features of the phase diagram using fairly general and reliable arguments. To this end, we begin in the limit of vanishingly small repulsive interaction $U \ll t$.
As $U/t\rightarrow 0$, the hopping term in the Hamiltonian dominates and its unfrustrated nature immediately leads us to the conclusion that the ground state is a featureless superfluid. The opposite limit when $U/t\rightarrow\infty $ and the potential energy term dominates is best understood by first going over into the `atomic' limit of isolated tetrahedra ($t=0$) and then analyzing the properties of the perturbation expansion in $t/U$. In the atomic limit, the density on each tetrahedron is controlled by the reduced chemical potential $f$:

For $0 < f < 1/2$, the ground state has no particles, while for $1/2 < f < 3/2$, the preferred density is one particle per tetrahedron
(which translates to a density of $\rho=1/4$), and for $3/2 < f < 5/2$, each tetrahedral plaquette is half-filled. Further, the physics for $f >2 $ can be mapped exactly to the physics at $4-f$ by a canonical transformation (specific to {\em hard-core} bosons) which is best written in
terms of the equivalent pseudospin-1/2 operators $S^x-iS^y \equiv b$, $S^z \equiv n-1/2$: $S^y \rightarrow -S^y$, $S^z \rightarrow -S^z$, $S^x \rightarrow S^x$. Thus the density as a function of $f$ consists of a series of plateaus with
a preferred number of particles per tetrahedral plaquette, and in view of the above, it suffices to consider the physics of the $\rho=1/4$
and $\rho=1/2$ plateaux to obtain a complete account of the physics.
\begin{figure}
\includegraphics[width=\hsize]{lattice1.eps}
      \caption{(Color online). a) Periodic Checkerboard lattice and the square lattice whose bonds pass through the checkerboard sites. The basic corner-sharing units of this lattice are `tetrahedral' plaquettes centered at the sites of the square net (shown in b)). The presence of a boson on a site of the checkerboard lattice can be represented as a `dimer'
 covering the corresponding link of the underlying square lattice. In the plaquette ordered state, red squares resonate via the ring-exchange process (shown in c)) while the other links are empty ($n_{i}$ = 0). The dotted bonds indicate higher kinetic energy in the plaquette state. In the alternate columnar state, dimers cover all green edges but not red ones.}
      \label{lattice}
  \end{figure}

A crucial feature of these plateaus (except the trivial cases with no particles and a fully filled lattice) is that they represent properties of a manifold of degenerate ground states with macroscopically large degeneracy. For concreteness, we focus here on the plateau with one particle per tetrahedron (our results for the $\rho=1/2$ plateau will be discussed separately).  A convenient representation of the ground state manifold is to associate a dimer on a link of the underlying square lattice  with the presence of a particle on the corresponding site on the checkerboard lattice. Each state in the ground state manifold then corresponds to a perfect dimer cover of the underlying square lattice.

The $t$ term introduces quantum tunneling between these dimer states, which lifts the classical degeneracy. In the dimer language, the leading order effective Hamiltonian in the $U/t\rightarrow\infty $ limit is a quantum dimer model with a ring-exchange term (Fig~\ref{lattice}) of $O(t^2/U)$ that flips two parallel horizontal (vertical) dimers on any {\it flippable} plaquette to vertical (horizontal) dimers, and acts entirely within the classical ground state manifold.
\begin{figure}
\includegraphics[width=0.9\hsize]{phasediagram.eps}
      \caption{Schematic phase diagram for the transition from superfluid state to Mott insulator state at $\rho$ = 1/4. Scans I and II are discussed in text.}
      \label{phasediagram}
  \end{figure}

Quantum dimer models with ring-exchange on individual plaquettes of two dimensional bipartite lattices (here a square lattice) quite generally have crystalline ground states in which the spatial arrangement of dimers break lattice symmetry~\cite{Sachdev_Vojta}. In the square lattice case, one expects either a plaquette or columnar type Valence Bond Solid (VBS), in which there is crystalline order at $\vec{Q}_1 = (\pi,0)$ and $\vec{Q}_2 = (0,\pi)$ ~\cite{Syljuasen} (lattice directions defined in Fig~\ref{lattice}). Thus, at large but finite $U/t$, one has a Mott insulating lobe in the $f-U/t$ phase diagram with density pinned to $\rho$ = 1/4 for a finite range of values around $f \sim 1$.

As has been discussed earlier by Fisher~{\it et. al.} in their seminal analysis~\cite{Fisher_Fisher} of superfluid insulator transitions in bosonic lattice models, the contour of fixed density $\rho = 1/4$ is expected to terminate on the tip of the lobe, while all contours with density different from this value skirt around the Mott insulating lobe. Consequently,  the tip of the lobe of the $\rho$ = 1/4 Mott insulator represents a special point along the locus of superfluid - Mott insulator transitions---indeed, the low-energy theory at the tip is expected to have an additional particle-hole symmetry arising from identical excitation energies for quasi-particle and quasi-hole excitations.

This sort of reasoning leads to the schematic phase diagram shown in Fig~\ref{phasediagram}, but is silent on the nature of the quantum phase transition lines between the Mott insulating lobes and the superfluid state.  Since one of the phases is expected to break spatial lattice translation symmetry, and the other superfluid state breaks the internal $U(1)$ symmetry associated with particle number conservation, a Landau-type description of the competition between these two very different order parameters would generically predict (along the $\rho=1/4$ contour) either a first order phase transition, or an intermediate regime with both phases or an intermediate regime with neither order.

An intermediate phase with neither order would correspond to a liquid phase of bosons in two dimensions at zero temperature. As no consistent scenario for such a phase is known in two dimensions
in the present context, we reach the tentative conclusion that such a phase is unlikely. An intermediate phase with both orders would correspond to a lattice supersolid, and several examples of this are known in the literature~\cite{Supersolid1,Supersolid2,Supersolid3,Supersolid4,Supersolid5,Supersolid6}.  A direct second order phase transition between the two broken symmetry phases would require ``fine-tuning'' to a multicritical point in a Landau type description. However, as has been shown recently~\cite{Senthil_etal} by Senthil {\it et al.}, conventional Landau theory can fail in certain closely related bosonic models in which quantum mechanical Berry phase effects produce a direct second order phase transition.
\begin{figure}
\includegraphics[width=\hsize]{plateau_new.eps}
      \caption{(Color online). Vertical scan (II) showing mott insulator state at $U/t = 3.50$ and $T = t/10$ for size $L =24$. Here $m^2$ = $|\psi_{c}|^2$, with $\psi_c$ defined as in Eqn~({\protect{\ref{def_of_orderparameters}}}). Note that the magnitude of the
      bragg peak in the kinetic energy correlator is also non-zero in the plateau state; this is exemplified by the
      data shown here (${\mathrm{KE}}_{00}$) on the bragg peak in the correlator of kinetic energy along  bond-type labeled $0$
      in Fig~{\protect{\ref{lattice}}}. Error bars are smaller than the symbol sizes in all the cases.}
      \label{plateau}
      \end{figure}

 \section{Numerical study}
  Our goal here is to study this model more closely using the well-documented stochastic series expansion (SSE) Quantum Monte Carlo (QMC) method~\cite{SSEprb,SSEpre,SSEmath} to access the phase diagram (Fig~\ref{phasediagram}), and then compare our numerical results with the expectations built on the previous work of Senthil {\it et. al.}. For this purpose, we have performed a detailed study along two scans in the $U/t$-$f$ phase diagram: Scan I varies $U/t$ at a fixed value of $f$---this value has been chosen after several preliminary exploratory scans to ensure that  it intersects the locus of phase transitions {\it{close}} to the tip of the lobe. The second scan (II) was performed at fixed $U/t$ such that one encounters an insulating phase for an appreciable range of $f$, and can probe the nature of the ordered state in somewhat more detail at a point relatively deep in the insulator. In addition, to probe
 nature of the particle-hole symmetric transition in more detail, we have
  also performed a detailed study of (crystalline) order-parameter statistics for a sequence of critical points approaching
  the tip of the lobe.
\begin{figure}
\includegraphics[width=0.8\hsize]{extrapolations.eps}
\caption{(Color online). Finite size extrapolations establishing the presence of superfluid and charge density wave ordered phases. Here $f = 0.875$ and $T = t/15$. Error bars are smaller than the symbol sizes.}
\label{extrapolations}
\end{figure}

Most of our data is on $L \times L$ samples (where $L \times L$ is the number of unit cells) where $L$ ranges from 20 to 48. We use standard SSE estimators to calculate the superfluid density $\rho_{s}$, the equal time correlator of the density $S_{\alpha \beta}(\bf{q})$ = $\langle n_{\alpha\tau}({\bf{q}}) n_{\beta\tau}(-{\bf{q}}) \rangle$ and the static structure factor of the density at zero frequency $\chi_{\alpha\beta}(\bf{q})$ =  $\langle \int{d\tau n_{\alpha\tau}({\bf{q}}) n_{\beta0}}({-\bf{q}}) \rangle$ where $n_{\alpha\tau}({\bf{q}})$ = $(1/L^2) \sum_{i} n_{\alpha \tau}({\bf{r}}_i)exp(i{\bf{q}}{\bf{r}}_i)$, $n_{\alpha \tau}({\bf{r}}_i)$ being the boson number operator at imaginary time $\tau$, and all site types ($\alpha$ = 0,1)  are assigned the coordinates of the  corresponding site (i) of the square net (Fig~\ref{lattice}). We also calculate the static correlators for the `kinetic energy' $K_{l}$ = $(b^{\dagger}_i b_j + h.c.)_l$ on the link $l$, $S^{\alpha \beta}_K ({\bf{q}})$ = $\langle \int d\tau K_{\alpha \tau}({\bf{q}}) K^{\dagger}_{\beta 0}({\bf{q}}) \rangle$, where $K_{\alpha \tau}({\bf{q}})$ = $(1/L^2) \sum_{i} K_{\alpha \tau}({\bf{r}}_i) exp(i {\bf{q}} {\bf{r}}_i)$, $K_{\alpha \tau}({\bf{r}})$ being the kinetic energy operator at imaginary time $\tau$ (all bond types ($\alpha$ = 0,1,2,3) are assigned the coordinates of the  corresponding site $(i)$ of the square net as in Fig~\ref{lattice}).

To verify the presence of the Mott insulator lobe in the phase diagram, we performed vertical scans at constant $U/t$ for different values of $f$. One particular scan (Scan II) at $U/t$ = 3.5 is shown here (Fig~\ref{plateau}). The scan clearly shows the existence of the mott insulator over an appreciable region in the phase diagram.
As is clear from Fig~\ref{extrapolations}, the insulating phase has Bragg peaks in the structure factor $S_{00}({\bf{q}})$  [$S_{11}({\bf{q}})$] and the static susceptibily $\chi_{00}({\bf{q}})$ [$\chi_{11}({\bf{q}})$] at wavevectors $(\pi,0)$ [$(0,\pi)$] which tend to a non-zero value in the the $L \rightarrow \infty$ limit.  In contrast, there are no appreciable Bragg peaks in the cross-correlator or off-diagonal static susceptibility $S_{01}({\bf{q}})$  [$\chi_{01}({\bf{q}})$].  The wavevectors of these peaks point to the presence of either plaquette and columnar order in the insulator.

To obtain a better characterization of this ordering, we measure two complex order parameters sensitive to the spatial symmetry breaking
\begin{eqnarray}
\psi_c &=& n_{0 \tau}(\vec{Q}_1) + i n_{1 \tau}(\vec{Q}_2) \nonumber \\
\psi_k &=& (K_{2\tau}(\vec{Q}_1) + K_{3 \tau}(\vec{Q}_1) - K_{0
 \tau}(\vec{Q}_1) - K_{1 \tau}(\vec{Q}_1)) \nonumber \\ &+&  i(K_{1 \tau}(\vec{Q}_2) + K_{2 \tau}(\vec{Q}_2) - K_{0 \tau}(\vec{Q}_2) - K_{3 \tau}(\vec{Q}_2)) \nonumber \\
 \label{def_of_orderparameters}
\end{eqnarray}
where $\vec{Q}_1 = (\pi, 0)$ and $\vec{Q}_2 = (0 , \pi)$.

This spatial symmetry breaking can now be analyzed using a Landau theory written in terms of the order parameters $\psi_c$ and $\psi_k$.  For our purposes here, it suffices to focus on just the ``potential energy'' part of the landau theory, and leave out all terms involving spatial
gradients. Terms in the landau theory are constrained by the requirement of invariance under the symmetries of the system, and the relevant symmetry operations in our case are the discrete operations of the space group of the square lattice, {\it i.e.} $\pi/2$ rotations, translations, reflections and inversion. The action of these on the order parameters is given in Table~\ref{tab:sym}.
\begin{table}[htbp]
  \centering
\vspace{0.5cm}
  \begin{ruledtabular}
  \begin{tabular}{l|c|c|c}
    Operation & Coordinates & $\psi_c$ transformation & $\psi_k$ transformation \\
 \hline
$R_{\pi/2}$ & $x_i \rightarrow \epsilon_{ij} x_j$ & $\psi_c \rightarrow i \psi_c$ & $\psi_k \rightarrow i \psi_k$ \\
$T_y$ & $y \rightarrow y+1$ & $\psi_c \rightarrow \psi_c^{*}$ & $\psi_k \rightarrow \psi_k^{*}$ \\
$T_x$ & $x \rightarrow x+1$ & $\psi_c \rightarrow -\psi_c^{*}$ & $\psi_k \rightarrow -\psi_k^{*}$\\
${\cal R}_x$ & $x \rightarrow -x$ & $\psi_c \rightarrow -\psi_c^{*}$ & $\psi_k \rightarrow -\psi_k^{*}$\\
${\cal R}_y$ & $y \rightarrow -y$ & $\psi_c \rightarrow \psi_c^{*}$ & $\psi_k \rightarrow \psi_k^{*}$\\
${\cal I}$ & $r \rightarrow -r$ & $\psi_c \rightarrow -\psi_c$ & $\psi_k \rightarrow -\psi_k$
\end{tabular}
\end{ruledtabular}
\caption{Transformations of the order parameters $\psi_c$ and $\psi_k$ on the square lattice under the discrete symmetry generators of the lattice. Here $i = 1,2=x,y$ is a spatial index.}
\label{tab:sym}
\end{table}

Writing all terms consistent with the symmetries of the model (up to fourth order), we obtain the Landau energy functional
\begin{eqnarray}
{\it{L}}_{pot} & =& {\it{F}}(|\psi_c|^2 , |\psi_k|^2) + C_{\theta_c}(\psi_c^4 + \psi_c^{*4}) + C_{\theta_k}(\psi_k^4 + \psi_k^{*4})\nonumber \\
&+& C_{\theta_{ck}}(\psi_c \psi_k^* + \psi_c^* \psi_k) \; , \nonumber \\
\end{eqnarray}
where $F(x,y)$ is an analytic function of its arguments.
The nature of ordering (plaquette versus columnar) is determined by the signs of the coefficients $C$ of terms in the landau free energy which fix the relative as well as absolute phases of the two order parameters. Our results on the statistics of the phases of the various order parameters (shown in Fig~\ref{landau} for a point deep in the insulating phase) can be modelled by taking $C_{\theta_{ck}}$ to be negative. However, the signs of $C_{\theta_c}$ and $C_{\theta_k}$ cannot be inferred from these histograms as no statistically significant bias is apparent in the value of the absolute phase.
\begin{figure}
\includegraphics[width=\hsize]{landau.eps}
      \caption{(Color online). Histograms of the relative phase of the two order parameters and the absolute phase of the kinetic energy order parameter characterizing the insulator state at $U/t$ = 3.50, $f$ = 0.875 and $T$ = $t$/15 for size $L$ = 28.}
      \label{landau}
  \end{figure}

As the absolute phase is too weakly pinned to usefully discriminate between the plaquette and columnar ordering possibilities, we need to rely on indirect evidence to obtain the nature of the ordering pattern: In this regard, the presence of clearly discernible Bragg peaks in the kinetic energy correlators that survive in the thermodynamic limit is suggestive of plaquette type order (corresponding to a positive sign for the coefficients $C_{\theta_c}$ and $C_{\theta_k}$) since the simplest caricatures of columnar
ordering have essentially no kinetic energy ordering associated with the spatial symmetry breaking.
Indeed, the nature of Bragg peaks seen in the real part of the static correlators of kinetic energy, $S_{K}^{\alpha \beta}(\vec{q})$, is completely consistent with expectations based on the simplest variational wavefunction for
the plaquette insulator. More specifically, the plaquette ordered variational wavefunction denoted pictorially in
Fig~\ref{lattice} leads directly to the following behaviour: $Re(S_{K}^{00}(\vec{q})), Re(S_{K}^{11}(\vec{q})), Re(S_{K}^{22}(\vec{q})), Re(S_{K}^{33}(\vec{q}))$ have maxima at $(\pi , 0)$ and $(0, \pi)$, $Re(S_{K}^{01}(\vec q)), Re(S_{K}^{23}(\vec q))$ have maximum at $(\pi,0)$ and minimum at $(0,\pi)$, $Re(S_{K}^{03}(\vec q)), Re(S_{K}^{12}(\vec q))$ have minimum at $(\pi,0)$ and maximum at $(0,\pi)$ and $Re(S_{K}^{02}(\vec q)), Re(S_{K}^{13}(\vec q))$ have minima at $(\pi,0)$ and $(0,\pi)$ (the other correlators can be obtained from the above ones trivially). Precisely
this behaviour is seen in the QMC data shown in Fig~\ref{kebragg}.
\begin{figure}
\includegraphics[width=\hsize]{KEpeaks.eps}
\caption{(Color online). Plots of $Re(S_{K}^{00}(\vec{q}))$,$Re(S_{K}^{02}(\vec{q}))$, $Re(S_{K}^{01}(\vec{q}))$ and $Re(S_{K}^{12}(\vec{q}))$ for $L = 28$, $U/t = 3.190$, $f=0.875$ and $T = t/15$. The presence of the extrema at wavevectors $(\pi,0)$ and $(0,\pi)$ with the correct signs is consistent with the distribution of K.E. shown in Fig~\ref{lattice}. Error bars are smaller than the symbol sizes.}
\label{kebragg}
 \end{figure}

  \begin{figure}
\includegraphics[width=\hsize]{dipolar.eps}
\caption{(Color online). Static susceptibilities $\chi_{00}({\bf{q}})$ and $\chi_{11}({\bf{q}})$ for $L = 28$, $U/t = 3.180$, $f = 0.875$ and $T = t/15$. In both panels $q_x$ and $q_y$ vary from $0$ to $2\pi$. The white regions are because values of $\chi_{00}$ and $\chi_{11}$ $>$ 0.001 are set to zero.}
\label{dipolar1}
\end{figure}

 In the superfluid phase, there are no Bragg peaks that survive the thermodynamic limit, even in the immediate vicinity of the transition to the insulator. However, equal time correlator  and static structure factor both show an interesting `dipolar' structure in the superfluid phase near the transition (Fig~\ref{dipolar1}). These dipolar correlations do not seem to be associated with the quantum phase transition itself, and persist into the insulating state as well.
 It is therefore instructive to look more closely at the static susceptibilities $\chi_{00}({\bf{q}})$ and $\chi_{11}({\bf{q}})$ and ask whether the dipolar nature is just due the strong local constraint introduced by a large $U$ term.

 More specifically, if we are at $f \sim 1$ and temperatures high enough to wash out the coherent quantum dynamics associated with the kinetic energy term, one would expect static density correlators to match (up to prefactors) the predictions for equal time quantities of a classical dimer model on the square lattice (viewing each particle as a dimer on the corresponding link of the underlying square lattice). The dipolar structure seen at lower temperatures can then be viewed as remnants of this essentially classical physics as the system crosses over into a coherent quantum regime.

To check this scenario, we fit the static structure factor to a functional form closely related to the classical result expected for the equal time correlator of classical dimers on the square lattice:
\begin{eqnarray}
\chi_{00}({\bf{q}}) &=& \frac{a(\cos^2(q_y/2)+\Delta^2)}{\cos^2(q_x/2) + \cos^2(q_y/2)+\Delta^2} \nonumber \\
\chi_{11}({\bf{q}}) &=& \frac{a(\cos^2(q_x/2)+\Delta^2)}{\cos^2(q_x/2) + \cos^2(q_y/2)+\Delta^2}
\label{fiteqn}
\end{eqnarray}
For the critical state of fully-packed dimers, one expects $\Delta^2=0$, and we allow for a non-zero $\Delta^2$ to account for slight density deviations from quarter filling.

As is clear from Fig~\ref{fitdipolar}, the dipolar structure in the higher temperature data fits remarkably well to our functional form, with a reasonably small value of $\Delta^2 \approx 0.0388$ reflecting a slight deviation of the density from quarter filling. Furthermore, consistent with our expectation above, the low temperature ($\beta t = 15$) data of Fig~\ref{fitdipolar} also displays qualitatively similar features, although the classical form (Eqn~\ref{fiteqn}) does not fit the data as well.
\begin{figure}
\includegraphics[width=\hsize]{susfits.eps}
      \caption{(Color online). Top panel: The static susceptibility $\chi_{00}({\bf{q}})$ plotted along three cuts in momentum space for $L = 28$, $U/t = 3.180$, $f = 0.875$ and $T = t/15$. Bottom panel: The same plotted at $U/t=4$, $f=0.875$ and higher temperature $T = t$, showing an excellent fit to the dipolar form Eqn~({\protect{\ref{fiteqn}}}) with parameters $a= 0.00033$ and $\Delta^2 = 0.0388$. Error bars are smaller than the symbol sizes.}
   \label{fitdipolar}
  \end{figure}

We  now turn our attention to the nature of the quantum phase transition at the tip of the $\rho = 1/4$ lobe.
 We perform Scan I (Fig~\ref{phasediagram}) at a constant aspect ratio of $\beta/L$ = $15/28$. We also compare data at different sizes at a range of inverse temperatures $\beta$ ranging from 5 to 15 to ascertain whether the $T \rightarrow 0$ limit is reached for a given size. The value of $f=0.875$ for this scan was chosen after some trial and error to satisfy two criteria: Firstly, this appears to be close to  the value at which the insulator has maximum extent. Secondly, the system is close to being particle-hole symmetric in the transition region for this value of $f$ (see Fig~\ref{phsymmetry}).
\begin{figure}
\includegraphics[width=\hsize]{quarterfill.eps}
      \caption{(Color online). Histograms of the density deviation from quarter-filling in the transition region of scan I for size $L=28$. Deviations
      are quantified in terms of the total number of particles by which the system deviates from quarter-filling, and the different
      curves are labeled by the corresponding values of $U/t$ at constant $f=0.875$ and $T=t/15$.}
      \label{phsymmetry}
  \end{figure}

To determine the location of the critical point quantitatively (under the assumption of a scale invariant second-order critical point), we calculate the Binder cumulant $g_{m}$ and $\rho_s L$ for different system sizes at a constant aspect ratio of $\beta/L$ = 15/28. For a continuous phase transition, scaling predicts that these two quantities may be written as $g_m$ = $F_{g_m}(L^{1/ \nu} ((t/U)_c - (t/U)), \beta/L^z)$ and $\rho_s L^z$ = $F_{\rho_s}(L^{1/ \nu} ((t/U)_c - (t/U)), \beta/L^z)$ near the critical point at sufficiently low temperature, where $F_{g_m}$ and $F_{\rho_s}$ are the appropriate scaling functions. Hence if the transition is continuous with $z$ = 1, both the quantities should exhibit the same crossing point in data taken for various $L$ at a fixed aspect ratio $\beta /L$ when plotted versus $U/t$. The reasonably sharp crossings and their locations indicate that
there may be a continuous transition at $U/t \approx 3.178 \pm 0.002$ (to be precise, we obtain a transition at $U/t \approx 3.180$ based on the crossing of $\rho_sL$, while the binder cumulant of the order parameter $m^2$ yields a crossing point at $U/t \approx 3.176$ (see Fig~\ref{scanscaling1})). The crossings shift very little when we consider the constant temperature runs at $T$ = $t$/15 (suggesting that we have reached the asymptotic low temperature regime at these sizes). However, it appears finite-size effects may be more significant, since we do notice a slight tendency of the superfluid and density wave transitions to move closer to each other if we look at the crossings of data at successively larger sizes. Our conclusion based on available data at these
smaller sizes is thus in favour of a single transition near the tip of the lobe, with no appreciable region of phase coexistence.
\begin{figure}
\includegraphics[width=0.9\hsize]{binders.eps}
      \caption{(Color online). Crossing of the Binder cumulant $g_m$ = $1 - \langle m^4 \rangle/(3\langle m^2 \rangle^2)$ where $m^2$ = $|\psi_c|^2$, and $\rho_s$L for different system sizes at $f$ = 0.875 and $\beta/L$ = 15/28. Error bars are smaller than the symbol sizes.}

      \label{scanscaling1}
  \end{figure}

To further explore the possibility that the transition is actually continuous and not first order, we do a scaling collapse of $\rho_s L$, $\chi(Q)$ and $S(Q)$ (see Fig~\ref{collapse2}), where $\chi(Q)$ = ($\chi_{00}(\vec{Q}_1)$ + $\chi_{11}(\vec{Q}_2)$)/2 and $S(Q)$ = ($S_{00}(\vec{Q}_1)$ + $S_{11}(\vec{Q}_2)$)/2 (the averaging is done to improve statistics). For a direct continuous transition, $\chi(Q)$ = $L^{-\eta} F_{\chi}(L^{1/ \nu}((t/U)_c - (t/U)), \beta/L^z)$ and $S(Q)$ = $L^{-\eta - z} F_{S}(L^{1/ \nu}((t/U)_c - (t/U)), \beta/L^z)$ where the critical exponents are the same as in the $\rho_s L^z$ scaling relation. These scaling relations can also be used to check whether $z$ = 1 by extracting $\eta$ from the scaling collapse of $\chi (Q)$ and then using it for collapsing the $S(Q)$ data. From the scaling collapse of the data, we estimate that $\nu \approx 0.46 \pm 0.05$ and $\eta \approx -0.25 \pm 0.12$ where the error bars are obtained by attempting scaling collapse of the data with different values of the exponents. The scaling collapses are certainly consistent with a direct transition with $z$ = 1, especially since the value of $\nu$ obtained from the superfluid and bragg peak data very nearly coincide within their error bars.
\begin{figure}
\includegraphics[width=0.9\hsize]{scaling_collapse.eps}
      \caption{(Color online). Scaling collapse of the static structure factor, equal time correlator and superfluid density at $f = 0.875$ and $\beta/L$ = 15/28. Error bars are smaller than the symbol sizes.}
      \label{collapse2}
  \end{figure}

While our scaling collapse does seem reasonable, the negative value of $\eta$ obtained from these fits leads  us to speculate that the transition for this particular value of $f=0.875$ may actually be very weakly first order---this is because a negative value of $\eta$ is ruled out on quite general grounds at any conformally  invariant critical point. To confirm this suspicion, we look at the histograms of the charge density wave order parameter $m^2$. They show a characteristic double-peaked structure when one goes to large system sizes indicating that the transition is weakly first order (see Fig~\ref{firstorderproof}).
\begin{figure}
\includegraphics[width=\hsize]{firstordernature.eps}
      \caption{(Color online).  Distribution of the order parameter $m^2$ for different system sizes at $U/t=3.1775$, $f=0.875$ and $\beta/L=15/28$. The double-peaked structure becomes more prominent at larger system sizes, as does the deviation from particle-hole
      symmetry.}
      \label{firstorderproof}
  \end{figure}
Thus, based on available evidence, we conclude that the transition close to the tip of the lobe is very weakly first order. The first order nature becomes more apparent at larger sizes $L \ge 48$. It should be noted  that it is precisely at these large sizes that the deviation from particle-hole symmetry at criticality also becomes more evident.
  This correlation in the scaling with size of deviation from particle-hole symmetry and double-peaked nature
  of the order parameter histogram might suggest that the weak first order jump is tied intimately to the slight deviation from particle-hole symmetry.

  In order to test this hypothesis, we have located a series of critical points along the phase boundary in the vicinity
  of the tip, and obtained detailed statistics of the charge-density wave order parameter for this sequence of critical
  points. If this hypothesis were valid, one would expect that the double-peak in the order parameter histogram
  at the largest size would become successively less pronounced and extrapolate to zero upon approaching
  the tip of the lobe. Simultaneously, the deviation from quarter filling would also extrapolate to zero.
\begin{figure}
 \includegraphics[width=\hsize]{latest2.eps}
     \caption{(Color online). Distribution of the order parameter $m^2$ and histogram of deviation of the density
      from quarter filling for $L=48$, $\beta/L=15/28$. The legend indicates the value of $f$ for each of
     these critical points.}
      \label{firstorderattip}
  \end{figure}

  However, our numerical results {\em do not} support this hypothesis. This is clear from Fig~\ref{firstorderattip}, where
  we show data at three critical points for our largest system size $L=48$: Of these, one is hole-rich, the other particle rich, and the third is
  perfectly particle-hole symmetric to within our error bars. As is clear from the accompanying histogram
  of the charge-density wave order parameter, there is no significant smearing of the double-peak
  structure upon approaching the particle-hole symmetric transition.

 Thus, our numerical results lead us to conclude that although the transition at the tip
 appears to be `close' to a landau forbidden second order transition, it is actually a very weakly first
 order transition as far as we can tell with the largest sizes and lowest temperatures available to us.
 As we now demonstrate below, an analytical study of the problem leads us to
 the conclusion that the effective low energy theory for the particle-hole symmetric transition
 is precisely the easy-plane NC{\it{CP}}$^1$ model of Ref~\cite{Senthil_etal}. Our conclusion of a weakly
 first-order transition then suggests that more work is needed to fully understand the physics
 of this universality class.

\section{Mapping to a gauge theory}
\begin{figure}
\includegraphics[width=0.9\hsize]{theory.eps}
      \caption{(Color online). a)The four degenerate ground states associated with the plaquette VBS state, which may be associated with four different orientations of a $Z_4$ order parameter $\psi_c$.Also shown in b) and c) are two types of $Z_4$ vortices in the plaquette state. The blue dotted lines represent the four domain walls. The encircled site is the core of the vortex.}
      \label{defect}
  \end{figure}
We now develop a theory for the phase transition of the model in terms of the defects of the Mott
insulator (MI) state at quarter filling, being careful to distinguish between transitions with
full low energy particle-hole symmetry, and those without. The insulator phase is characterized by a $Z_4$ clock order parameter $\psi_c$. In Fig~\ref{defect} (a), we show the four plaquette ground states which may be associated with the four orientations of the $Z_4$ order parameter (similarly, the four equivalent columnar states correspond to the four `columnar' choices of the phase of $\psi_c$).

Because the order parameter is discrete, the natural topological defects are domain walls
between different orientations of the phase of $\psi_c$,  and four of them can terminate at a point to create a point-like $Z_4$ vortex excitation. There are two types of vortices present, depending on whether the core of the vortex is hole-like with a negative charge deviation at that
site (Fig~\ref{defect} (b)) or particle-like with a positive charge deviation (Fig~\ref{defect} (c)).
Translating the entire valence bond pattern by one lattice spacing reverses the direction of winding of the phase of the VBS order parameter. Thus, particle (hole)-like vortices have opposite windings on the two sublattices. Moreover, a particle-rich vortex has a phase winding which is
opposite to a hole-rich vortex located on the same sublattice.

Keeping these two points in mind, we define the charge of a vortex with its core
at site $r_i$ of the underlying square
lattice as $n_{r_i} \equiv (n_{1i}+n_{2i}+n_{3i}+n_{4i} - 1)$ for $r_i \in A$ sublattice,
and $n_{r_i} \equiv -(n_{1i}+n_{2i}+n_{3i}+n_{4i} - 1)$ for $r_i \in B$ sublattice. Note
that $n_{r_i}$ is non-zero only at the core of the vortices in any vortex configuration. Further,
since the total boson charge may be expressed as
\begin{equation}
Q= \frac{1}{2} \sum_i \left(n_{1i}+n_{2i}+n_{3i}+n_{4i} \right) \; ,
\end{equation}
we note that the vortices represent {\it fractionally charged excitations} carrying
quanta of charge equal
to one half the elementary boson charge. The mechanism for this is essentially
the same as in many
other examples discussed earlier in the literature ({\it e.g.} the Motrunich-Senthil
model realization of a $Z_2$ deconfined phase~\cite{Motrunich_Senthil}). However, in
contrast to earlier work on $Z_2$ deconfined phase, these fractionally charged vortex excitations are expected to be strongly interacting in our model (see below),  and we do {\it not} expect them to have an independent existence in the insulating phase. Nevertheless, as we will see below, they are convenient degrees of freedom in terms of which one may hope to describe the phase transition to a superfluid state, following the approach of Ref. \onlinecite{Levin_Senthil}.

In addition, we define $\tilde{n}_{r_i r_j}$ as the charge on the links of the square lattice where $r_i,r_j$ are nearest neighbours on the underlying square lattice, and identify it with the boson number at the center of the link. We also find it convenient to soften the hard-core constraint on the boson number, and instead capture the same physics through a strong on-site Hubbard potential. Finally, it is also useful to work with a rotor description of the charge degrees of freedom (measured from a background value corresponding to the quarter filled insulating state).
We can then describe the microscopic Hamiltonian in terms of rotor `angular momenta'  [$n_{r_A},n_{r_B},\tilde{n}_{r_A r_B}$] , and their conjugate angle variables [$\varphi_{r_A},\varphi_{r_B},\tilde{\varphi}_{r_A r_B}$].
Writing the effective Hamiltonian in terms of these variables, we obtain
\begin{eqnarray}
H &=& U\sum_{A}(n_{r_A}+1-f)^2 + U\sum_{B}(-n_{r_B}+1-f)^2 \nonumber \\
 &-&2t\sum_{r_A r_{A'}}\cos(\varphi_{r_A} - \varphi_{r_{A'}} + \tilde{\varphi}_{r_B r_A} - \tilde{\varphi}_{r_B r_{A'}})\nonumber \\
&-&2t\sum_{r_B r_{B'}}\cos(\varphi_{r_B} - \varphi_{r_{B'}} + \tilde{\varphi}_{r_A r_{B'}} - \tilde{\varphi}_{r_A r_B}) \nonumber \\
&+&V\sum_{r r'}\tilde{n}_{rr'}^2
\end{eqnarray}
along with the local constraints (due to the overcomplete set of variables used)
\begin{eqnarray}
\sum_{r_{B'}}\tilde{n}_{r_{A}r_{B'}} - 1 &=& n_{r_A} \nonumber \\
\sum_{r_{A'}}\tilde{n}_{r_{B}r_{A'}} - 1 &=& -n_{r_B}
\end{eqnarray}

As mentioned earlier, a large positive $V$ is introduced to make
the physics qualitatively similar to the hard-core boson case.
This Hamiltonian and the local constraints define a compact $U(1)$
gauge theory coupled to matter. To make this mapping explicit, we
define a compact vector potential $\vec{a}$ on the links of the
square lattice such that $\vec{a}_{r_A r_B} = \tilde{\varphi}_{r_A
r_B}$ and $\vec{a}_{r_A r_B} = -\vec{a}_{r_B r_A}$. The conjugate
electric field vector is then defined as $\vec{e}_{r_A r_B} =
\tilde{n}_{r_A r_B}$,  and $\vec{e}_{r_A r_B} = -\vec{e}_{r_B
r_A}$. The Hamiltonian in these new variables reads \begin{eqnarray}
\label{gaugerep}
H &=& U\sum_{A}(n_{r_A}+1-f)^2 + U\sum_{B}(-n_{r_B}+1-f)^2 \nonumber \\
&-&2t\sum_{r_A r_{A'}}\cos(\varphi_{r_{A'}} - \varphi_{r_A} - \int_{A}^{A'}\vec{a}\cdot d\vec{l})\nonumber \\
&-&2t\sum_{r_B r_{B'}}\cos(\varphi_{r_{B'}} - \varphi_{r_B} - \int_{B}^{B'}\vec{a}\cdot d\vec{l})\nonumber \\
&+&V\sum_{rr'}e_{rr'}^2 \; , \end{eqnarray} while the constraint can now be
written as $(\triangledown \cdot \vec{e})_r = n_r + \epsilon_r$,
where $\epsilon_A = 1$ and $\epsilon_B = -1$. Thus, our effective
model can be reformulated as a compact ($\vec{a}$ is an angle)
$U(1)$ lattice gauge theory coupled to two types of matter fields
$(n_{r_A},n_{r_B})$ in the presence of a background (static)
staggered charge pattern.

The transformation properties of the
fields $\phi,n, a_{\alpha}, e_{\alpha}$ under discrete lattice
symmetries are readily obtained (see Table \ref{tab:psggt1}) and
play an important role in the discussion below. The global $U(1)$
symmetry associated with boson number conservation is realized by
the phase rotation
\begin{equation}
e^{i\phi} \rightarrow e^{i\alpha \epsilon_r} e^{i\phi}
\end{equation}
with $\alpha$ a constant. Correspondingly the total conserved
boson number is given by
\begin{equation}
Q = \frac{n_{r_A} - n_{r_B}}{2}
\end{equation}
Thus $e^{i\varphi_{r_A}}$ creates a state with total boson number
$1/2$ while $e^{i\varphi_{r_B}}$ creates a state with total boson
number $-1/2$.

\begin{table}[htbp]
  \centering
\vspace{0.5cm}
  \begin{ruledtabular}
  \begin{tabular}{l|c|c|c|c}
    Operation &  $\phi$  & $n$ & $a_{\alpha}$ & $e_{\alpha}$ \\
 \hline
 $T_x$ & $-\phi$ & $-n$ & $-a_{\alpha}$ & $-e_{\alpha}$ \\
 $R_{\pi/2}$ & $-\phi$ & $-n$ & $-\epsilon_{\alpha\beta} a_{\beta}$ & $-\epsilon_{\alpha\beta} e_{\beta}$ \\
$R_y$ & $ -\phi $ & $-n $ & $-a_{\alpha}$ & $-e_{\alpha}$\\
${\cal T}$ & $-\phi$ & $n$ & $-a_{\alpha}$ & $e_{\alpha}$
\end{tabular}
\end{ruledtabular}
\caption{Transformations of the fields on the square lattice under
the various independent discrete symmetry operations. $T_x$ is
translation by a unit cell in the $x$-direction. $R_{\pi/2}$ is
$\pi/2$ rotation about the center of a plaquette. $R_y$ is
reflection about the y-axis through midpoint of a horizontal bond.
${\cal T}$ is time reversal and is antiunitary. All other symmetry
operations may be built up from these. } \label{tab:psggt1}
\end{table}

This compact gauge theory is somewhat non-standard because the $A$
type particles can never convert to the $B$ type particles and
vice-versa, which makes it a bit inconvenient to analyse. To
overcome this technical difficulty, we define a theory with two kinds of matter
fields $(n_1,n_2)$ (and their conjugate variables
$(\varphi_1,\varphi_2)$), both of which can live on the $A$ and
$B$ sublattices, and hop between the $A$ and $B$ sublattice. The
original restriction that particles of one type only live on the corresponding
sublattice can be captured by a term of the kind
$(V\sum_{r_B}n_1^2 + V\sum_{r_A}n_2^2)$ with $V$ large and
positive as before. We expect that these changes preserve the universal
properties of our original theory.

Now, the charge at each site is $(n_{1r} - n_{2r})$, and the
constraint becomes $(\triangledown \cdot \vec{e})_r = n_{1r} +
n_{2r}+ \epsilon_r$. The electric field is however still related
to the microscopic boson density $n_{rr'}$ as $n_{rr'} =
\epsilon_r \vec{e}_{rr'}$ so that the electric field correlators
give direct information about the density correlations in our
microscopic model. Further simplification arises when we approach the
transition at the tip of the MI lobe. Because of particle-hole
symmetry at the tip, linear terms in $n_1,n_2$ vanish. As a result
it is enough for our purposes to consider the following simplified
theory: \begin{eqnarray}
H &=& U\sum_r (n_1^2 + n_2^2) - t'\sum_{r,\mu}\cos(\triangle_{\mu}\varphi_1 - a_{\mu}) \nonumber \\
&-&t'\sum_{r,\mu}\cos(\triangle_{\mu}\varphi_2 - a_{\mu}) + V\sum_{r,\mu}e_{\mu}^2 \nonumber \\
&-&K\sum_{\Box}\cos(\triangle \times \vec{a}) + V\sum_{r_B}n_1^2 + V\sum_{r_A}n_2^2 \; ; \nonumber \\
&& (\triangledown \cdot \vec{e})_r = n_{1r} + n_{2r}+ \epsilon_r
\end{eqnarray} where $\mu$ stands for the two lattice directions $\hat{x}$
and $\hat{y}$. The $K$ term has been added to make the theory look
like a more standard gauge theory and represents a ring-exchange
term in the dimer language. Such a term of this form would not
change any of the universal properties of the theory, and is expected to be dynamically generated
at low energies. The
transformation properties of these new fields under lattice
symmetries are also readily obtained and we list them below in
Table \ref{tab:psggt2}. As before global $U(1)$ symmetry is
realized by the phase rotation
\begin{eqnarray}
e^{i\phi_1} & \rightarrow & e^{i\alpha}e^{i\phi_1} \nonumber \\
e^{i\phi_2} & \rightarrow & e^{-i\alpha}e^{i\phi_2}
\end{eqnarray}

\begin{table}[htbp]
  \centering
\vspace{0.5cm}
  \begin{ruledtabular}
  \begin{tabular}{l|c|c|c|c}
    Operation &  $\phi_s$  & $n_s$ & $a_{\alpha}$ & $e_{\alpha}$ \\
 \hline
 $T_x$ & $-\phi_{\bar{s}}$ & $-n_{\bar{s}}$ & $-a_{\alpha}$ & $-e_{\alpha}$ \\
 $R_{\pi/2}$ & $-\phi_{\bar{s}}$ & $-n_{\bar{s}}$ & $-\epsilon_{\alpha\beta} a_{\beta}$ & $-\epsilon_{\alpha\beta} e_{\beta}$ \\
$R_y$ & $ -\phi_{\bar{s}} $ & $-n_{\bar{s}} $ & $-a_{\alpha}$ & $-e_{\alpha}$\\
${\cal T}$ & $-\phi_s$ & $n_s$ & $-a_{\alpha}$ & $e_{\alpha}$
\end{tabular}
\end{ruledtabular}
\caption{Transformations of the fields $(\phi_s,n_s, a_{\alpha},
e_{\alpha})$ on the square lattice under the various independent
discrete symmetry operations. Here $s = 1,2$ and $\bar{s} = 2$ if
$s = 1$ and vice-versa.} \label{tab:psggt2}
\end{table}

This model may now be analysed by a duality transformation. The results are rather similar to dual versions of boson models at
half-filling on a square lattice. In particular there are two vortex species $\psi_1$ and $\psi_2$ (corresponding to vortices in $\phi_1$ and $\phi_2$ respectively).
These two vortices move independently on the dual square lattice. The universal critical properties of the transition may then be described in a continuum
description of this dual theory.
The oscillating background gauge charge (due to the $\epsilon_r$ in the Gauss law constraint)
ensures that the leading term that allows for mixing of $\psi_1$ and $\psi_2$ vortices in the continuum dual action is
\begin{equation}
S_{mix} = -\int d^2x d\tau \lambda \left[\left (\psi_1^* \psi_2 \right)^4 + c.c \right]
\end{equation}
The full action then reads
\begin{eqnarray}
S & = & S_v + S_{mix} \nonumber \\
S_v & = & \int d^2x d\tau \sum_s |\left(\partial_{\mu} -i A_{\mu}\right)\psi_s|^2 + r|\psi_s|^2 \nonumber \\
& & +u_{d}|\psi_1|^2|\psi_2|^2 + u\left(|\psi_1|^2 + |\psi_2|^2 \right)^2 \nonumber \\
& & + \kappa \left(\epsilon_{\alpha\beta\gamma} \partial_\beta A_{\gamma} \right)^2
\end{eqnarray}
Here $A_{\mu}$ is the dual gauge field whose magnetic field strength is the original boson density. This is exactly the same critical theory as that derived in
Ref. \onlinecite{courtney} to describe the superfluid-VBS transition of bosons at half-filling on a square lattice.

We thus expect that the present model lies in the same universality class as that transition. As argued in Ref. \onlinecite{Senthil_etal}, this is the easy plane NC{\it{CP}}$^1$ universality class and is described by the action $S_v$. The term $S_{mix}$ has been argued to be irrelevant at the corresponding fixed point. This low energy theory $S_v$  was
studied in Ref~\onlinecite{mv} by Motrunich and Vishwanath. Their analysis concluded that the theory was
self-dual and has a single continuous phase transition with unusual exponents that were
estimated in their work based on a detailed numerical study of a particular lattice regularization
of the theory. However, in more recent work, Kuklov {\em et. al.} have studied~\cite{Kuklov_etal} a different lattice regularization and
argued in favour of a weakly first order transition. Since the present work also finds a similar very weakly
first order transition, more work appears to be needed to understand the physics of this universality class in greater
detail.

Finally, moving away from the tip of the Mott lobe corresponds to turning on a non-zero chemical potential in the original boson model. The transition is then expected to be either first order  or to proceed through two independent transitions with an intermediate supersolid phase. Although the present analysis cannot decide between these two possibilities without additional
input specific to the microscopic model being considered, our numerics provides evidence in favour of a first order
transition.

\section{Conclusions}
We have thus obtained convincing numerical evidence for a four-fold lattice symmetry breaking Mott insulator
at large boson repulsion in a simple model of hard-core bosons on the planar pyrochlore (checkerboard) lattice at quarter filling. We have also
studied the nature of the quantum phase transition from the superfluid state of the weakly interacting
system to the lattice symmetry broken Mott insulator. Our analytical results above suggest  a direct
transition at the tip of the $\rho = 1/4$ insulating lobe---more precisely, this transition would be
expected to be a continuous transition in the universality class of the easy-plane NC{\it{CP}}$^1$ model considered by Motrunich and Vishwanath in Ref~\onlinecite{mv}

In our numerical work, we have studied the corresponding transition close to the tip of the lobe.
Our numerical evidence points to  a direct very weakly first order transition close to the tip of the lobe.
A detailed extrapolation of the strength of the first-order jump suggests that the transition remains
very weakly first order even at the particle-hole symmetric tip.
Thus, the present study underlines the need for further work to fully understand the physics
of the easy-plane NC{\it{CP}}$^1$ universality class, particularly with a view towards understanding
the very weakly first order nature of the transition seen in our work and in previous studies~\cite{Kuklov_etal}
of models that belong to this universality class.

\section{Acknowlegment}
We are grateful for access to computational resources at the Tata Institute of Fundamental Research and acknowledge support from the DAE Outstanding Investigator Program (TS) and DST SR/S2/RJN-25/2006 (KD).
.

redpajama_set_name: RedPajamaArXiv


We're regularly asked about what #cycling clothing and equipment to buy. But we also love a bargain. In this short video, Katie Butler (https://www.instagram.com/katiekookaburra1) compares two very similar base layers (with two very different price tags!).
What do you think? Is the price tag REALLY worth it?


text:
Fortnite creator Epic Games has been fined $775 million for enticing children to make online purchases
Epic Games, makers of the blockbuster video game Fortnite, on Monday agreed to pay $775 million ($520 million) to the U.S. Federal Trade Commission.
The company admitted to violating children's privacy laws and tricking minors and adults into making unintentional online purchases.
The FTC said the comparisons from one of the biggest names in video games set records, blaming Epic for knowingly targeting Fortnite players under the age of 13 and exposing children to harassment and trauma in chat rooms.
READ MORE GAMING NEWS HERE
In its complaint, the FTC said Epic was aware that many children were playing Fortnite and collected their information without parental consent, as required by US law.
"Even when Epic became aware that certain Fortnite players were under the age of 13, Epic took no steps to comply with (US law)," the FTC complaint reads.
The FTC also accused Epic of setting online chat defaults that allowed children and teens to play with strangers and exposed them to potential harassment.
"Children and youth have been subjected to bullying, threats, harassment and dangerous and psychologically traumatizing issues such as suicide during Fortnite," the FTC said.
For these violations, Epic agreed to pay $275 million and is now prohibited from enabling chats unless parents of users under the age of 13 or teenage users give their direct consent.
In a separate complaint, Epic was accused of engaging in so-called dark patterns, the practice of tricking users into making unwanted purchases or opting into certain settings without their knowledge.
Among other things, the FTC said that Epic "allows children to file tortious charges without parental involvement."
The commission also claimed that Epic intentionally made refund or cancellation requests difficult to complete and penalized users who questioned payments.
The commission said the company agreed to pay $245 million in consumer refunds to resolve the complaint.
In a statement, Epic said that "no developer makes a game with the intention of ending up here." "We accepted this agreement because we want Epic to be at the forefront of consumer protection and provide the best experience for our players," it added.
Epic said it has made changes since the breaches to "meet the expectations of our players and regulators" and that the practices referenced in the FTC's complaints are not consistent with how Fortnite is currently operating
Originally released as "Fortnite," creator Epic Games has been fined $775 million for tricking kids


In the midsummer of a year, July offers infinite passion with hot temperature. In such fervent atmosphere, Grand Hotels Media has been more enthusiastic towards the readers, aiming to bring the most colorful and exciting hotel news with zest.
In the cover story of this issue, we had an exclusive interview with Mr. Anthony Meng, Cluster General Manager of Hilton Garden Inn Lijiang & Hilton Garden Inn Shangri-La. He brought us into the wonderland of beauty and showcased the unique culture and feature of the two hotels.
We also paid a visit to Hilton Shanghai Hongqiao and interviewed Mr. Joseph Zitnik, General Manager of the hotel, bringing the latest news about the hotel as the host venue for the F&B Master event. As a leading MICE hotel in the market, the 1700 sqm pillar-less grand ballroom as well as experienced team members give Hilton Corporate great confidence that the event could be an extraordinary and successful one. Besides, we also had a dialogue with Mr. Wesley Wang, General Manager of Himalayas Qingdao Hotel, presenting the most authentic hotel interiors from design concept, recreation facility, culture and spirit.
In this issue, Grand Hotels Media also had the great honor to invite Ms. Lu Chun Feng, Dim Sum chef of InterContinental Shanghai Ruijin as the special guest of Classical Private Cuisine, a program access to the latest five-star hotel cuisine and culture which is jointly launched by Grand Hotels Media and FM94.7.
At the end of the year, we will embrace the 11th Grand Hotels Media Award Ceremony. Since early July, we have started to track the event from the Best Hotel with a number of releasing news about awards such as TOP Stylish Hotels Award, Best Cover Of The Year, Chef Awards and Outstanding Leadership Award. Who will be the final winner? Who will be the most shining one? Please look forward to our follow-up stories.
Last but not at least, the charity auction event of Grand Hotels Media is also in full swing. More detailed information can be found in inside page. Readers can also get known of the special cuisine, wonderful event, holiday resort of different hotels in this season in our magazine, mastering the first-hand hotel news while remaining within doors.


The Rocky Mountains. Now this is Serenity.
Posted in Travel & Exploration, Uncategorized and tagged as calm, peace, Photography, Rocky Mountain National Park, serenity, solitude, Weekly Photo Challenge.
Thank you. Hard to take a bad photo surrounded by such magnificence.
They are beautiful but so, I think, are almost all mountains.
Thank you. I did so love it there!
Tks Julie. A wonderful place. We're going back again this summer to climb a BIG mountain.
It certainly is – but I'm guessing it's pretty snow-covered at the moment!
I'd say you're probably right!


The following list is a summary of the many services offered.
The services offered can stop at design or can progress through to site works.
It is entirely up to you, so the appointment can be tailored to your requirements.


Amrita M Replied on March 1, I have a DL version, not a W and as per Wids comment, he said he was able to get the firmware installed; how? The site given below provides the information that the drivers for the printer is being developed, but should have come out by July , you may visit the link given below and check if that is of help. Although there "supposedly" are no drivers for that printer for Win7, what I've done in the past is install the Vista drivers. I downloaded the Latest driver from Konica Website and everything is working well.
Kpnica must log in or register to reply here. Best thing about this new service is that you are never placed on hold and get to talk to real repairmen in the US. I just got my magicolor from a state sell today. Thanks very much, if you can help me with this.
I bet the interface may be semi-generic as Linux can print to it. How is the Gold Competency Level Attained? If you chose to 'open' rather than save, koniva this step since the zip file will already be open.
I have a DL version, not a W and as per Wids comment, he said he was able to get the firmware installed; how? As a matter of fact, I'm going to get my old DL this afternoon returned to me, I let my daughter use it for awhile and am going to try the same thing. In reply to Amrita M's post on March 1, I have done the update on my DL and have the correct firmware, according to the print-out I get. Showcase New media New comments Search media. Another batch file is 'TCPupdate.
Jace Romine New Member Jun 1, What's new New posts New media New media comments Latest activity.
As a Gold Certified Independent Software Vendor ISVSolvusoft is able to provide the highest level of customer satisfaction through delivering top-level software 22300dl service solutions, which have been subject to a rigourous and continually-audited approval process by Microsoft. Connect the Magicolor Desklaser to the PC using a parallel cable. The site given below provides the information that the drivers for the printer is being developed, but should have come out by Julyyou may visit the link given below kknica check if that is of help.
Posted by takok14 on Oct 14, Click "Add" to insert your video.
Not finding what you are looking for? How satisfied are you with this reply? Have you tried the drivers? The printer shows no signs of activity and it's been about ten minutes. Cannot get Vista business to print to magicolor dl i can connect visa the printer with ie8 tried many different drivers and driver install methods.
Tell us some more! It works fine from my wife's XP computer, but I had to switch to a Windows 7 operating system. The "w" in the model number makes it completely different than the DL kinica.
TCPUpdate ip-to-printer I did this and it worked great. Konica Minolta's site indeed has Vista drivers for it.
There are multiple files in that zip and one of them is a batch file for the parallel update. The firmware installation continues The PC screen will display the following message: Solvusoft is recognized by Microsoft as a leading Independent Software Vendor, achieving the highest 2300sl of completence and excellence in software development.
Lately during the filling cycle water hammer is occurring. Turn on the printer.


text:
Financial Times: Bulgaria needs stability before uncertainty
March 25, 2013 by Cup&Cross
Filed under Featured, News
A caretaker government in Sofia will do its utmost to steady the tiller before May's snap elections, following several weeks of street protests that toppled the previous administration and plunged Bulgaria into political uncertainty. But what happens after the poll is anybody's guess. Many in Sofia's political elite seem reluctant to grasp the poisoned chalice of leadership and their capacity to satisfy the demands of a restive and inchoate popular movement is limited.
On Wednesday, President Rosen Plevneliev ended weeks of speculation by naming Marin Raykov, Bulgaria's ambassador to France, as caretaker prime minister until the May 12 elections. Raykov is a career diplomat and was deputy foreign minister under a right-of-centre government between 1998 and 2001 and again in 2009-2010 under Boyko Borisov, whose resignation as prime minister last month led to the political void which Raykov must temporarily fill. Raykov also served as an ambassador and a foreign policy chief under the Bulgarian Socialist Party (BSP), the main opposition force. The new premier will be backed by a cabinet including three deputy prime ministers – all women – Bulgaria's first female interior minister and a selection of other "experts" drawn from various parts of national life, particularly academia. There are also some promotions from within Borisov's GERB party.
Bulgarian journalists have been quick to pick up on Raykov's family background. His father, Rayko Nikolov, was a senior ambassador in the Communist era, including to the UN, Yugoslavia and France. Nikolov's reported close links to the intelligence services come as a surprise to no-one. But of more interest is the allegation, made by the French press some years ago, that he was responsible for recruiting French politicians to work for the Soviet Union, including Charles Hernu, who went on to be defence minister in the 1980s.
Raykov's political career on the "reformist" right but with roots among the Communist nomenklatura have led to suspicions that he is what Bulgarians gnomically call a "hidden lemon" – someone who is not what they appear to be. But Raykov's diplomatic record suggests that he is robustly on the pro-European, rather than pro-Kremlin, side of Bulgarian politics, according Ivo Indzhev, a well-regarded Bulgarian blogger. And after all, one can't choose one's parents.
Dimitar Bechev, head of the Sofia office of the European Council on Foreign Relations, a think-tank, says Raykov is a sound choice. "This government won't do much beyond organise the elections," he says. "Raykov is solid – linked to the 1990s right, but with enough credibility with all parties. He brings predictability and reassurance, as well as credibility for outsiders. He is aware of European policy issues and shows that Bulgaria will play by the rules."
With the temporary administration in place, attention shifts to the election and what is likely to be a confused aftermath. Polls suggest GERB and the BSP are level pegging, with the largely Muslim-backed DPS and Bulgaria for the Citizens, led by former European Commissioner Meglena Kuneva, also likely to make it to the next parliament. Confusingly, BSP leader and ex-prime minister Sergey Stanishev has said he will not be premier again, while Kuneva has indicated that her party will not join a coalition.
Anthony Georgieff, a Bulgarian journalist and a vocal critic of Borisov, says the most likely scenarios as the mainstream opposition backs away from government are a discredited GERB being returned to power, considerably weakened, or the strengthening of hardline left- and right-wing forces unpalatable to Bulgaria's EU partners. The far-right party Ataka, until recently seen as on the wane, seems to be recovering, and ultranationalists have been a significant presence at many demonstrations, along with well as anti-capitalist malcontents.
"Bulgaria's moral crisis is reflected in the mainstream politicians refusing to take part in any coalition government of the future, and in the general public voting with their feet," he says. Bechev is more upbeat, saying that Stanishev and Kuneva's statements should be "taken with a pinch of salt". He outlines a number of more moderate outcomes, including a grand coalition with GERB but without Borisov, and a BSP-DPS partnership, possibly bringing in Bulgaria for the Citizens, not dissimilar to the government from 2005 to 2009.
Whether any of these permutations will have the political will, economic resources and administrative capacity to address the demands of the street protesters – and the many Bulgarians at home who have lost faith in the political elite's ability to deliver real change – is questionable. Demonstrations that started over power prices now encompass grievances such as corruption, authoritarianism, government links to organised crime, and privatisation. Most fundamentally, they may be about low incomes and lack of prospects in the EU's poorest country – something that Raykov has said that he will look to address, within his capabilities. But with the eurozone, Bulgaria's main trading and investment partner, still in crisis, and the legacy of years of maladministration at home, immediate change for the better seems unlikely.
"Unlike other crises Bulgaria has had since it shook off Communism in 1989, the current one has no easy answers because there is little if anything to look forward to, Bulgaria now being a full member of the EU and NATO," says Georgieff. The death by self-immolation of three demonstrators since protests began certainly indicates the desperation that many Bulgarians feel. But some are hopeful that the street movement can put pressure on the next government to deliver the transparency and accountability that has been lacking from its predecessors.
Tags: bulgarias, caretaker, descends, promises, spell, stability, uncertainty
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* The 1/64 version is not a raced version and does not have the black tape around the hood, as pictured.
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Q: Origin of the convolution theorem I am a chemist, with some interest in signal processing. Sometimes, we use the deconvolution process to remove the instruments response from the desired signals. I am looking for the earliest reference which proposed the convolution theorem which is often utilized in signal processing (i.e., convolution becomes a multiplication in the Fourier domain).
The Earliest Known Uses of the Word of Mathematics websites gives lot of details on the word convolution, but who was the first person to specifically show the above mentioned property- the connection of Fourier transforms with convolution?
Here is the history of Convolution Operation: https://pulse.embs.org/january-2015/history-convolution-operation/
However, a mathematician privately disagreed with that historical account given in this article.
Thanks.

A: The general theorem that Fourier's $\mathscr F:L^2(G)\to L^2(\hat G)$ maps convolution to product and vice versa is in Weil (1940, p. 113), for any locally compact abelian $G$ with dual $\smash{\hat G}$. But the special cases of the real line, the circle group (with $(f,g,h)(t)=$ $\sum (a_n,b_n,c_n)e^{int}$), and the integers:
$$
\begin{align}
&\mathbf{(R)}\qquad h(t)=\textstyle\int_{-\infty}^\infty f(s)g(t-s)\,ds &&\Rightarrow &&\textstyle\mathscr F(h)=\mathscr F(f)\mathscr F(g)\\
&\mathbf{(T)}\qquad h(t)=\smash[t]{\textstyle\int_0^{2\pi} f(s)g(t-s)\,ds} &&\Rightarrow &&c_n=a_nb_n\\
&\mathbf{(Z)}\qquad h(t)=f(t)g(t) &&\Rightarrow &&\textstyle c_n=\sum_k a_kb_{n-k}\\
\end{align}
$$
were known earlier:

*

*Dieudonné (1981, p. 195) and Mackey (1980, p. 628) attribute $\mathbf{(R)}$ to Lyapunov (1900, 1901?); it's also in Hausdorff (1901, p. 169), Poincaré (1912, p. 207), Lévy (1925, p. 184; 1928, p. 79), etc. (They state results in terms of added independent random variables, not repeating the step that $h(t)dt$ is the image of the product measure $f(s)ds\times g(t)dt$ under addition $\mathbf{R\times R\to R}$.)

*Kahane and Lemarié-Rieusset (1995, p. 19) attribute $\mathbf{(T)}$ to Fourier (1822, p. 259) where it is rather implicit; it's explicit in Young (1912, p. 32).

*Burkhardt (1901, p. 84; 1914, p. 947) finds $\mathbf{(Z)}$ in Cauchy (1844, p. 1125) and earlier in Euler (1760), for both trigonometric polynomials ("product-to-sum", pp. 176-186) and some series (p. 200); it's also in Pringsheim (1886, p. 158), Hurwitz (1902, p. 369), Lebesgue (1906, §52), etc.
(And indeed, the history in Domínguez (2015, p. 46) cheerfully ignores all of the above.)

Added: Adams (2009) contains translations of Lyapunov's (1900, 1901). This should be consulted, as e.g. Fischer (2011, p. 201) contradicts Dieudonné and Mackey by saying: "Lyapunov never used general concepts such as inversion formula or correspondence between convolution of distributions and products of characteristic functions" (and I am indeed not really seeing $\mathbf{(R)}$ in (1900)).


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It was supposed to be a boring trip to a boring village to exorcise a boring demon. The bandits on the road had other ideas. Caliban and Slate, trying to build something like a normal life in the aftermath of the war.


At a half pass eleven, we went to our first destination.. cosmetics shop. I need to buy facial wash and nite cream, hehee. After that, we went to Pacet but through other way, not from my house. We're gonna took a picture and ate "Mie Ayam Timbul". I like that chicken noodle, bcoz there are some meatballs inside.. Hmm Yummmy. And it has beautiful scenery absolutely. Accross the place, we can see rice field and Welirang mountain in Pacet.
We continued our trip after the rain stopped. We planned to go to Kawasan Wisata Pacet while we ate. There are Welirang Mountain, hot tubs, and waterfall. Unfortunately again.. Rain starts to fall and we couldn't continue our trip, huhuuu..


465 people have done this repair successfully ! Why not you?
This guide is for replacing the back camera on the iPhone 6.
Before disassembling the back camera of your iPhone 6, you have to securely turn it off.
With a pair of tweezers, gently remove the black tape covering the back camera as shown on the pictures.
The sticker is located on the left side of the back camera.
Then use a nylon spudger to disconnect the back camera as shown on pictures 3 and 4.
Use the cruciform screwdriver again and remove the two screws circled in red.
Those screws maintain the back camera to the read panel of the iPhone.
Use tweezers again and remove the back camera bracket.
Use the nylon spudger, gently pry and unclip the back camera.
Remove the back camera with tweezers.
The back camera of your iPhone 6 is now disassembled.


On February 7th, we drove north to Whitney just outside Algonquin Park, and stayed at a resort over night. We returned home the next day. Both coming and going we stopped off in Algonquin Park.
Of course we started from home (#1) and drove up through Algonquin Park (2) to Whitney (3) where the resort was located.