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The Ukulele Orchestra of Great Britain strikes a chord with concept ensemble
The Ukulele Orchestra of Great Britain will celebrate 30 years of performing unique versions of pop, rock, classical and film theme songs with a concert at 7:30 p.m. Oct. 13 in Eisenhower Auditorium.
Image: photo provided by The Ukulele Orchestra of Great Britain
Heather Longley
Beginning in the late 1990s, celebrities such as Jake Shimabukuro, Zooey Deschanel and Eddie Vedder helped to reintroduce the ukulele to America. And then there's The Ukulele Orchestra of Great Britain. The band's been strumming away since 1985, when an idea to go lo-fi got out of hand.
"Now it's coming back into fashion," George Hinchliffe, orchestra co-founder and director, told The Telegraph back in 2009.
Seven years after Hinchliffe's comment and "30 plucking years" after the orchestra's inception, some might still find it hard to believe that an ensemble of ukulele players has achieved worldwide success. Orchestra members, in fact, jokingly accept part of the blame for the current "hot for uke" movement. It started with a small group of friends, including Hinchliffe and co-founder Kitty Lux, playing for fun.
From there, "the Ukes" performed in Germany and realized just how big they were in Japan, a country that embraces many aspects of Hawaiian culture.
"Then in about the year 2000, when broadband Internet arrived, people who did have left-field taste could stay much more in touch with each other," said orchestra musician Peter Brooke Turner. "A lot of ukulele players are quite techie people, and so there were, very soon, lots of people watching ukulele videos on YouTube. And since a lot of our stuff was put on YouTube, a lot of people watched it."
The whole process was do-it-yourself, a sentiment that echoed the instrument's history. The ukulele started out as the machete, a four-stringed chordophone taken to Hawaii by Portuguese laborers. Some enterprising immigrants with woodworking skills caught on to the instrument's growing novelty and opened their own ukulele shops, bringing the "bonsai guitar" to the masses by the mid 1880s.
An instrument for the people
From the beginning, even after Hawaiian royalty adopted it as their own, the ukulele was a people's instrument—an accessible, affordable and portable way to play music. A four-stringed instrument also might be easier for some people to learn and master than a guitar.
"We generally find that once you learn one or two chords, off you go, really, for playing a whole bunch of easy songs," Turner said.
In the mainland United States, Jazz Age soloists were the primary strummers of the ukulele. It endeared its way into jug, string and country music with its unmistakable, jangly pitch. The Ukulele Orchestra of Great Britain cites George Formby (the United Kingdom's highest-paid ukulele player in the 1930s and '40s) and Tiny Tim (famous for his performances on "Rowan & Martin's Laugh-In" in the 1960s) as influences even before they knew they wanted to create a band of ukes.
When a group of eight musicians pluck away at a tiny instrument, it's bound to make a big sound. Even with seven musicians on stage—Lux won't be joining the orchestra for the U.S. tour coming to the Center for the Performing Arts at Penn State Tuesday, Oct.13—the Ukes will sound fuller than one might expect. But the ensemble rarely performs with fewer than six members, lest it become an "unplugged" version of itself.
"Generally as long as we have enough of the original members, we can achieve some sort of a critical mass," Turner said.
Time for a uke-off?
The orchestra's success at playing everything from pop to classical seems to have helped create a cottage industry of ukulele ensembles. Anyone who sets trends knows that knockoffs are sure to follow. Decades after the orchestra's creation came ensembles with names like The Wellington International Ukulele Orchestra (New Zealand's own), The Ukulele Orchestra of the Western Hemisphere (from California) and The United Kingdom Ukulele Orchestra (based in Germany).
In a January interview with www.b-c-ing-u.com, Hinchliffe addressed how the concept of an orchestra composed of ukulele players, originally created for a laugh and a bit of irony, became a household term. He doesn't begrudge anyone for starting a ukulele ensemble but takes offense at copycats.
"We were the first ukulele orchestra and invented the term. The term 'ukulele orchestra' is now used by many groups who have been inspired by us to form their own ensembles. Some unscrupulous people have even passed themselves off as us and imitated our brand, our style, our lineup, our show," Hinchliffe said. "However, in general, we support all musicians, all ukulele players, indeed anyone who's been inspired by us to start their own ukulele orchestra, in whatever country. We don't like misrepresentation and duping of the public, however."
According to the BBC, a legal battle kicked off in September 2014. In July, a judge decided that the German orchestra "misrepresents to a substantial proportion of the public (in Great Britain) who recognize The Ukulele Orchestra of Great Britain as the trade name of a particular music group" and that the German counterparts' actions "caused damage to The Ukulele Orchestra of Great Britain's goodwill" over the original ensemble's reputation as performers.
"The judgment we got was quite satisfying," Turner said. "It was just very irritating, really, to be—I hesitate to use the word—copied."
"Copied" seems to be an understatement. The United Kingdom Ukulele Orchestra, the ensemble named in the 2014 lawsuit, went so far as to book a string of performance dates in the United States, an itinerary that seems to shadow the Ukes' dates and locations. In fact, a New Hampshire venue recently advertised The United Kingdom Ukulele Orchestra's concert using The Ukulele Orchestra of Great Britain's photo, promotional wording, press quotes and song lineup.
Perhaps the orchestras need to schedule a ukulele duel.
An appeal for enthusiasts
One aspect of the Ukes' appeal is its audience engagement.
"We always get a chance to go out and talk to people who are playing the ukulele," Turner said. "It's very important for us to keep in touch with the grassroots of the scene. We know that a lot of people started playing ukulele because they'd seen us at a concert."
For enthusiasts who need more ukulele, they can watch busking at Penn State by two local ukulele clubs prior to the Ukes' performance—PSUkulele at West Halls courtyard and the Allegheny Ukulele Kollective in the HUB-Robeson Center.
One might compare a ukulele orchestra of today with the drum circles of yesterday.
"It's not a pompous or pretentious instrument. It's very inclusive," Turner said. "It's very social, and the people who play it are always friendly."
For a musician who grew up in alternative, progressive and rock scenes in England, spending one's days playing around for fun with a group of like-minded friends, while selling out major concert halls, sounds like a dream come true.
"It's been a career I never envisioned happening," Turner said. "That's one of the wonders of this life, that I've ended up earning a living playing the ukulele."
Laura Sullivan
lls19@psu.edu
http://www.cpa.psu.edu
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Donald Trump Tax Plan: What's In It, And What Does It Mean For You?
Aaron Homer
Donald Trump's tax plan (or more likely, a list of guidelines and expectations that may or may not come to reality) was released this week, and here's a general analysis of what's in it and what it means for the average American.
As NPR reports, the Trump tax plan is not a piece of legislation to be voted on by Congress. Rather, it's more of a post-election campaign promise: that is, it's a list of "guidelines" that the president wants Congress to include in any tax legislation that he would be willing to sign.
20 Fascinating Facts You Never Knew About Donald Trump – https://t.co/AMzjugngAy pic.twitter.com/Icrd8Pgrnn
— Un-Silent Majority (@UnSilentNews) April 27, 2017
What's In It?
In a word (two of them, actually): tax cuts. Tax cuts across the board.
The cornerstone of the Trump tax plan is a huge reduction in the corporate income tax rate, cutting it down from 35 percent to 15 percent. Similarly, it also seeks to cut taxes on other businesses (that is, non-incorporated businesses who pay their taxes via income tax rather than corporate tax). Specifically, that spells pretty big tax cuts for small businesses as well as big corporations. And as for individuals, Americans at almost all income levels can expect a tax cut, including cuts described by NPR as "modest" for the middle class.
Businesses large and small can expect cuts via Donald Trump's tax plan. [Image by jacoblund/Thinkstock]
Another key point in Trump's tax plan is simplification. According to Trump's Treasury Secretary Steve Mnuchin, via Forbes, for most Americans, the arcane and interminable tax forms would be replaced by something much more simple.
"For most Americans, we think they should be able to do their taxes on a large postcard."
How Is It Going To Be Paid For?
In theory, slashing taxes means cutting government revenue, since there's less tax money coming in. The Trump administration hopes to offset that by explosive growth spurred on by the tax cuts.
Specifically, the Trump administration is hoping that its tax cuts would spur growth in the Gross Domestic Product (GDP) from its current levels of around 1.6 percent-1.8 percent per year to 3 percent, says Mnuchin.
"We fundamentally believe we can get to 3 percent sustained economic growth."
Also in the mix in discussion of the Trump tax plan is a border adjustment tax or BAT. In essence, it's an import tax on goods coming across the border. The downside to a BAT is that the tax would simply be passed on to consumers at the retail level, meaning that Americans would pay more at the check-out stand. That's why, for now, says NPR's Domenico Montanaro, it's off the table.
"That why it's being left out, because it's seen as too controversial right now. The last thing Trump wants is to have the narrative hijacked and turned into how he's proposing to increase the cost of everyone's avocados."
Who Would Benefit
Perhaps not surprisingly, one person who would benefit greatly from Donald Trump's tax plan is Donald Trump himself.
According to The Guardian, the Trump organization owns, manages, or is in one way or another connected to about 500 businesses. Cutting taxes on all of them would benefit Trump to the tune of tens of millions of dollars.
"Under current law, [Trump's business'] earnings can be taxed at up to the highest personal income rate of 39.6%; under Trump's plan, they would be taxed at 15%, equivalent to the personal tax rate of individuals earning under $37,950 a year. According to [a tax analyst], this alone could net Trump $65m a year."
As of this writing, Trump's tax plan is little more than a wish list and what form it takes when it is time to be signed by the chief executive remains to be seen.
Do you believe Donald Trump's tax plan is fair and will benefit the average American?
[Featured Image by Olivier Douliery-Pool/Getty Images]

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COLUMBIA – It wasn't a white Christmas, but many took advantage of Friday's warm weather.
Meg Gray walked her dogs around Stephens Lake Park late in the morning.
"We enjoy it more because it's really nice weather out and it's sunny," said Gray.
With a high of 51 degrees, the weather required lighter jackets than usual.
"It's really nice because I'm not bundled up to the forehead and everything," said Gray.
Others at the park walked, ran and played on the playground. Several families took the opportunity to take photos.
Gray said she is spending the rest of Christmas at a friend's house.
"We're going to a friend's and they have a bunch of dogs so the dogs will be playing in the backyard quite a bit probably," said Gray.


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The Shadowcasters Network is a group of passionate geeks who love making cool stuff about Shadowrun. Not only are we committed to creating the highest quality Shadowrun content for fans to enjoy, but we want to create new tools, amazing experiences, and bring Shadowrun to more people than ever before.
We're not just trying to accomplish this goal by pumping out more podcasts. There is already a satisfying menu of Shadowrun podcasts to choose from. We're focused on making our current products as good as they can possibly be. Quality over quantity. But more than podcasts, we want to extend the reach and influence of Shadowrun gaming. Imagine being able to play official Shadowrun: Missions games from the comfort of your home online? The Shadowcasters Network is creating an infrastructure to make that happen! What about tools and resources to enhance and enable your Shadowrun play experience? We're creating and bringing all these things under one curated umbrella so that Shadowrun fans can get it.


Side by side trial A1 and Kross. Which one is first, which second?
I can´t hear any major difference, even one is PCM and the other "Analog Modeling", that means, they should be able to sound same when using the traditional waveforms.
I did the same with Roland Gaia, DSI Mopho x4 and Minilogue.
Same result, no audible major difference except Minilogue, where the filter actually swallowed low end when adding resonance.
For me A1 beats many synths, also analog, because especially of the many additional waveforms, the mighty FX section, the four part multitimbrality and it´s 24 voices.
That´s why I prefer it over Prologue, DSI Prophets and even Moogs.
I still miss such a side by side comparison with a Moog, but I don´t expect big differences anymore. Moog´s analog ladder filter might be specific, but copying the simple circuit is no magic.
I´ve recently published some of my favorite Korg Kross patches and I can´t imagine, someone who could identify it as "non analog"
Last edited by Puukka on 03 Aug 2018, 08:05, edited 5 times in total.


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With an aim to promote adventure sports in Ladakh, World's highest rafting and kayaking competition was held on August 19 on Zanskar River. The event was organised by All Ladakh Rafting Operators (ALRO) in collaboration with All Ladakh Tour Operators Association (ALTOA). A total of 22 rafting teams and 62 kayakers paddled 14 km on Zanskar River to Sangam point, Nimmo. Nawang Rigzin Jora, MLA flagged off the team from Skorpochay, Chiling. Rafting team 'Mountain Fire' grabbed the winner title by covering the distance in 40 minutes 10 second. The runner-up team ITBP took 40 minutes 37 second. In kayaking, Rishi Rana and Shivam Gurung were the winners and runners-up respectively. JP Yadav, DIG, ITBP expressed his hope that the event will become an International level event just like Ladakh Marathon and Ice Hockey tournament. He said that there is a good scope for adventure sports in Ladakh where International players can also take part. Tsering Angchuk Nyaga, President, ALRO said that Zanskar River is one of the five best Rivers in India for white water rafting. He said that the competition which is organised for the first time in Ladakh will not only boost the tourism but also promote and encourage water sports activities in Ladakh. Talking about the growing River Rafting tourism in Ladakh, Tsetan Angchok, President, ALTOA said that Ladakh is a hub of adventure activities and tour operators in Ladakh are working hard to take it to a new height with limited infrastructure. He further appealed J&K Tourism department to organise certified training programme by inviting professional experts for adventurous water sports activities under Skill Development Programme and urged the local administration to add this event in the annual event of Ladakh calendar. He also informed about Rafting operators performing social responsibility by helping the administration during the rescue operation. Rinchen Nima Jora, Vice President, ALRO said that the idea of water sport was conceived to encourage local youths to explore the potential in water sports and participate in such competition. Goma Sonuwar from Nepal working with Aquafire Rafting Adventure was the only woman participant in the Kayaking competition.


Omarosa Manigault is getting hired.
A memorable contestant in the first season of "The Apprentice," Manigault is expected to join President-elect Donald Trump's White House staff, according to two people familiar with the decision. Her job is expected to focus on public engagement.
Manigault was one of Trump's most prominent African-American supporters during the campaign and has been working with his transition team. This will be her second tour of duty at the White House — she worked in the office of Vice President Al Gore during the Clinton Administration.
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Bands & Artists : KNIGHTMARE Releases: In Death's Shadow (Coming soon!)
MICKoWAR writes"
Following the release of the Unholy EP, Knightmare presents their first full length album, In Death's Shadow, prepared for the international release on December 1st. With humble roots as a power metal band, Knightmare's sound has evolved into an intricate and intense soundscape as they begin to take on the world from Australia.
Mixed by Ermin Hamidovic of Systematic Productions (Orpheus, Elm Street, Eye of the Enemy), and Mastered by Plec at Panic room studios in Sweden (Scar Symmetry, Watain), In Death's Shadow will certainly be a highly anticipated release.
The release will be supported by a music video being produced by the acclaimed team that have brought you (Manowar, Rhapsody, The Poodles, Gamma Ray, Doro and many more) to be Co-Directed by the internationally established Steve Ravic of Majestic Film and Hollywood filmmaker Neil Johnson of Morphius Film.
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7. Judgment (10:41)
Knightmare has already released a single track off the album, "Cazador De Hombres" as a taste of what's to come.
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Deputies are looking for whoever shot and damaged 75 insulators on PacifiCorp electricity transmission lines south of Kuna late last month.
The power lines are located in the desert about five miles south of Kuna/Mora Road, between Pleasant Valley and Swan Falls roads, in southern Ada County.
Officials with PacifiCorp first discovered a problem with the insulators on Nov. 21. The next day, employees sent to repair the power lines found about 75 insulators connected to the power lines were hit by shotgun rounds. Employees also found empty shotgun casings near the power lines.
PacifiCorp is offering a $5,000 reward for anyone who has information that leads to a conviction in the case. The amount and severity of damage to the power lines make this a felony malicious injury to property case.
Anyone with information should call Ada County dispatch at 377-6790.


Our prison justice system has been perverted (if not simply outlined) by the institutional racism that helps it and is supported by it — and it will take time to change that. Nevertheless it's not up for respectable debate that inequities produced both by the writing and enforcement of our legal laws are clearly established, or that those inequities have injured black people and black communities.
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Israeli forces have closed several entrances to East Jerusalem's flashpoint Al Aqsa Mosque complex amid ongoing clashes with Palestinian worshippers, a Palestinian official said on Tuesday.
"Dozens of Israeli soldiers stormed the al Aqsa compound and assaulted religious figures," Firas al Dibs, a spokesman for Jerusalem's Religious Endowments Authority, a Jordan-run agency tasked with overseeing the city's Muslim and Christian sites, said in a statement.
Israeli forces said they evacuated the compound after a Molotov cocktail damaged a security post.
Three people were arrested, Israelis said, while the Palestinian Red Crescent reported two people hurt.
Residents said police were also restricting Palestinian access to Jerusalem's Old City, where the site is located.
According to Dibs, Al Aqsa Mosque Director Omar Kiswani and Sheikh Wasef al Bakri, the acting supreme judge of Jerusalem's Islamic Courts, were among those assaulted by Israeli security forces.
Baton-wielding police, Dibs said, had attacked dozens of Muslim worshippers earlier Tuesday near the Al Aqsa compound's iconic Dome of the Rock Mosque.
"At least five Palestinians were arrested before being taken into custody for further investigation," he added.
In a subsequent statement, Dibs said that Israeli forces had expelled all Muslim worshippers from inside the mosque complex, during which at least 10 of the latter were assaulted.
He went on to assert that a fire that erupted inside the mosque complex, used to justify the expulsion of worshippers, had been started by police.
"Everything that happened was orchestrated by the police," he said.
The Palestinian Presidency, meanwhile, has condemned the reported escalations.
According to a statement published by Palestine's WAFA news agency, President Mahmoud Abbas is maintaining "intensive contacts" with all relevant parties in hopes of defusing the situation.
Tensions mounted in Jerusalem last month, when Israeli forces briefly sealed the Al Aqsa compound's Al Rahma Gate, located adjacent to the eastern wall of the Old City, sparking angry Palestinian demonstrations.
In the weeks since, the Israeli authorities have banned scores of Palestinians, including religious officials, from entering Al Aqsa, which for Muslims represents the world's third holiest site.
Israel occupied East Jerusalem, in which Al Aqsa is located, during the 1967 Arab-Israeli War. It annexed the entire city in 1980 in a move never recognised by the international community.


There are thousand of books on the subject of Celtic Christian Spirituality, and I have read only a minute smattering of them. The Following are some of the books I have read and some of the books I would like to read God willing, and if time will permit. However, it is mostly the books I have not read.


Afterward enjoy Cliff's apple cider donuts with hot cocoa, warm cider or coffee and warm up at our fire pit.
Direct from the North Pole, Santa and Mrs. Claus will be visiting our Christmas tree farm.
Santa and Wagon rides available from 11-2 on the Friday after Thanksgiving, Saturday and Sundays.
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of America (FFA)] will be there to assist you!
Of course bring your cameras as we have lots of great photo spots with majestic Sinnissippi Forest as your backdrop!


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Iron and steel sector capex costs - welded tube mill.
Capital investment cost figures compiled by Metals Consulting International, based on investments by IPSCO at the Regina Works in Canada; by Yalian Steel Pipe at Yalian in China; by the Essar Group at the Hazira Pipe Mill in India; by Samchai Steel at the Samchai Works in Thailand; by Welspun Gujarat at Little Rock in USA; and other recent modernisations. Figures shown above refer to a broad mix of pipe and tube plants including ordinary welded pipe for construction, scaffolding and furniture-building; linepipe, OCTG pipe, high pressure tube; and mechanical pipe.


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New Radiohead Song Featured in Paul Thomas Anderson's Inherent Vice
Paul Thomas Anderson's long-awaited new film inherent vice has finally been screened for the press (early reviews are raving), and with it came confirmation that the film features a never before released Radiohead song playing over the credits. Per the credits, the song is called "Spooks." Studious Radioheas fans will note that "Spooks" is a song the band's been playing live since 2006, perhaps a product of their In Rainbows sessions. It's been described by critics who saw the film as a surf-rock instrumental that nicely fits into the movie's hazy 1960s aesthetic.
We can't wait to hear it, but for now check out a video of Radiohead playing "Spooks" live:
http://youtu.be/xHdnLJ6fnE4
Joanna Newsom is in Paul Thomas Anderson's new film 'Inherent Vice'
Paul Thomas Anderson Has Directed a New Radiohead Music Video: Reports
Thom Yorke Performs New Radiohead Song "Give Up The Ghost"
New Radiohead Album Out This Year?
MP3s: Radiohead's 20 Best Cover Songs
Radiohead Extend North American Tour
New Album: Radiohead - TKOL RMX 1234567
New: Radiohead - "Identikit" & "Cut a Hole" (Live)
Radiohead share Paul Thomas Anderson-directed video for "The Numbers"
Radiohead will Guest on Special 1 Hour Colbert Report
Wolf Gang - "Pyramid Song" (Radiohead Cover)


Tower Hill Insurance Company - Royal Palm Insurance - Newman Insurance Agency, Inc.
Insurance scam rears its ugly head!
As if the economic and real estate situation isn't difficult enough! .
These homeowners insurance companies lure customers away from Citizens Insurance, evaluate the home and provide homeowners insurance.
Several weeks after they take the customer's money they notify the homeowner that they have cancelled the policy because the home is a risk.
Several weeks later they again offer insurance trough their subsidiary company but double or triple the premium.
Then during hurricane season they again notify the customer they have cancelled the policy because the home is at risk.
There may also be cases of discrimination involved.


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Chester Depot, VT
The 10 Best Thrift Shops in Vermont!
The Chester-Andover Family Center, Chester, VT
Just south of Chester lies a thrift shop, family center, and food shelf that was established in 1985. In 2012, the center expanded and partnered with Neighborhood Connections to provide a social…
Just south of Chester lies a thrift shop, family center, and food shelf that was established in…
The 7 Best Antique Stores in Vermont!
Stone House Antique Center, Chester, VT
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text:
\section{Introduction}\label{sec:intro}
The original idea of the Thomson problem\cite{thomson,Ashby1986} of 1904 was to find equilibrium positions of $N$ charges constrained to a spherical surface interacting with Coulomb's law.
More than 100 years later there is still interest in finding minimum energy configurations of such assemblies, see e.g. \cite{Whyte1952,Marx1970,Erber1991,Altschuler2005,Altschuler2006,Backofen2010,Lakhbab2012}.
There is also an interactive Java applet by Bowick et al.~\cite{thomsonApplet} from Syracuse university (NY) to find minimum configurations for the more general $r^{-n}$ potential using several different minimization algorithms.

In this paper, we generalize the Thomson problem to arbitrary curved non-self-penetrating parametrized two-dimensional surfaces that are embedded in three-dimensional Cartesian space. We present the mathematical details to study the motion and minimum energy configurations of an arbitrary number of charged particles constrained to these surfaces, and we briefly describe how to implement the resulting N-body simulation using the Compute Unified Device Architecture (CUDA) of todays graphics boards. The generalization to $r^{-n}$ potentials is left as exercise for motivated students having basic experiences with electrostatics and the Lagrangian formalism.

The structure of the paper is as follows. In Sec.~\ref{sec:trajectories} we briefly discuss the details for the parametrization of trajectories on curved surfaces. The Lagrangian for $N$ charged particles on this surface interacting via Coulomb is compiled in Sec.~\ref{sec:lagrangian}. From this Lagrangian, we derive in section \ref{sec:eom} the equations of motion for each particle. Additionally, we extend the equation of motion when there is also an external electric and magnetic field. In Sec.~\ref{sec:simvis} we discuss some implementation details for the N-body simulation and the subsequent visualization. Several examples and some feasible exercises are presented in sections~\ref{sec:examples} and ~\ref{sec:exercises}.

The source code to reproduce the examples in this paper is written in C/C++/CUDA and is freely available from (\url{http://www.vis.uni-stuttgart.de/chapacs}). It can be compiled on Linux and Windows systems.

\section{Particle constrained to a curved surface}\label{sec:trajectories}
The trajectory $\gamma$ of a particle that is constrained to a two-dimensional curved surface $S$ can be derived from the Lagrangian formalism using generalized coordinates that are adapted to the surface. These adapted coordinates are usually the parameters $(u^1,u^2)$ that are used as surface parametrization
\begin{equation}
  \mathbf{f}: \mathbb{U}\rightarrow\mathbb{E}^3,\quad (u^1,u^2)\mapsto\mathbf{f}\left(u^1,u^2\right)\in S\subset\mathbb{E}^3,
  \label{eq:surfFunction}
\end{equation}
where $\mathbb{U}$ is some open domain in $\mathbb{R}^2$ and $\mathbb{E}^3$ is the three-dimensional Euclidean space, see figure~\ref{fig:paramCurve}.
\begin{figure}[ht]

  \centering\includegraphics[scale=0.8]{figure1}
  \caption{Parametrized surface $S\subset\mathbb{E}^3$.}
  \label{fig:paramCurve}
\end{figure}

Instead of using the Cartesian coordinates $\mathbf{x}(t)$ for the particle's trajectory, we can now describe $\gamma$ also by means of a curve $\vec{u}(t)$ in $\mathbb{U}$,
\begin{equation}
   \gamma:\quad t\mapsto \mathbf{x}(t) = \mathbf{f}(\vec{u}(t))=\mathbf{f}\left(u^1(t),u^2(t)\right)
\end{equation}
where $t\in[t_i,t_f]$ denotes time. The velocity $\mathbf{v}(t)$ of the particle is then obtained by
\begin{equation}
  \mathbf{v}(t) = \mathbf{\dot{x}}(t) = \frac{d}{dt}\mathbf{x}(t) = \frac{d}{dt}\mathbf{f}\left(\vec{u}(t)\right) = \sum_{i=1}^2\pdiff{\mathbf{f}}{u^i}\frac{du^i}{dt} = \sum_{i=1}^2\pdiff{\mathbf{f}}{u^i}\dot{u}^i.
  \label{eq:timeDiffF}
\end{equation}

\section{Lagrangian for $N$ charged particles on a curved surface}\label{sec:lagrangian}
The Lagrangian for $N$ charged particles that are constrained to a two-dimensional curved surface can be obtained in the usual way,
\begin{equation}
  L = T - V,
\end{equation}
where $T$ is the total kinetic energy and $V$ is the total interaction energy. For $N$ particles the kinetic energy is
\begin{equation}
  T = \frac{1}{2}\sum_{A=1}^N M_A\left<\mathbf{\dot{x}}_A,\mathbf{\dot{x}}_A\right>,
  \label{eq:kinEnergy}
\end{equation}
where $M_A$ denotes the mass of particle $A$ and $\left<\cdot,\cdot\right>$ is the standard scalar product in $\mathbb{E}^3$. With \eref{eq:timeDiffF} we can write equation \eref{eq:kinEnergy} as
\begin{equation}
\fl T = \frac{1}{2}\sum_{A=1}^N M_A\sum_{i,k=1}^2\left<\pdiff{\mathbf{f}}{u^i}(\vec{u}_A),\pdiff{\mathbf{f}}{u^k}(\vec{u}_A)\right>\dot{u}_A^i\dot{u}_A^k = \frac{1}{2}\sum_{A=1}^N M_A\sum_{i,k=1}^2g_{ik}(\vec{u}_A)\dot{u}_A^i\dot{u}_A^k.
\end{equation}
Here, we have introduced the functions
\begin{equation}
   g_{ik}(\vec{u})=\left<\pdiff{\mathbf{f}}{u^i}(\vec{u}),\pdiff{\mathbf{f}}{u^k}(\vec{u})\right>
   \label{eq:metricCoeffs}
\end{equation}
on $\mathbb{U}$. They can be interpreted as the coefficients of the metric induced on $S$ expressed in the coordinate system $(u^1,u^2)$ (see e.g. \cite{doCarmo}).

The interaction energy is obtained as
\begin{equation}
  V = \sum_{\underset{A<B}{A,B=1}}^N V_{AB},
  \label{eq:fieldEnergy}
\end{equation}
where $V_{AB}$ is the interaction energy between particles $A$ and $B$. In the present case we are only concerned with the electrostatic (Coulomb) interaction,
\begin{equation}
  V_{AB} = \frac{1}{4\pi\epsilon_0}\frac{Q_AQ_B}{\|\mathbf{x}_A-\mathbf{x}_B\|} = \frac{1}{4\pi\epsilon_0}\frac{Q_AQ_B}{\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|}
\end{equation}
with $Q_A$ and $Q_B$ being the charges of particle $A$ or $B$, respectively.
Since $V_{AB}=V_{BA}$ and since we ignore the self-energy of a particle with itself, i.e. $V_{AA}=0$, we can write
\begin{equation}
  V = \frac{1}{2}\sum_{\underset{A\neq B}{A,B=1}}^N\frac{1}{4\pi\epsilon_0}\frac{Q_AQ_B}{\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|}.
\end{equation}
Thus, the total Lagrangian reads
\begin{equation}
L = \frac{1}{2}\sum_{A=1}^N M_A\sum_{i,k=1}^2g_{ik}(\vec{u}_A)\dot{u}_A^i\dot{u}_A^k - \frac{1}{2}\sum_{\underset{A\neq B}{A,B=1}}^N\frac{1}{4\pi\epsilon_0}\frac{Q_AQ_B}{\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|}.
\end{equation}

\section{Equations of motion}\label{sec:eom}
We obtain the equations of motion for particle $A$ by evaluating the Euler-Lagrange equations
\begin{equation}
 0 = \frac{d}{dt}\pdiff{L}{\dot{u}_A^i}-\pdiff{L}{u_A^i}\quad\mbox{for}\quad i=\{1,2\}.
 \label{eq:eulerLagrange}
\end{equation}
We compute the two terms separately. First, the partial derivative of $L$ with respect to $\dot{u}_A^i$ is
\begin{equation}
  \pdiff{L}{\dot{u}_A^i} = M_A\sum_{k=1}^2g_{ik}(\vec{u}_A)\dot{u}_A^k
\end{equation}
and its time derivative reads
\begin{equation}
  \frac{d}{dt}\pdiff{L}{\dot{u}_A^i}=M_A\left(\sum_{l,k=1}^2\pdiff{g_{ik}}{u^l}(\vec{u}_A)\dot{u}_A^l\dot{u}_A^k + \sum_{k=1}^2g_{ik}(\vec{u}_A)\ddot{u}_A^k\right).
  \label{eq:el1}
\end{equation}
Next, to compute the partial derivative of $L$ with respect to $u_A^i$ we need the partial derivative $\partial r^{-1}/\partial u_A^i$ with $r:=\|\mathbf{f}_A-\mathbf{f}_B\|=\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|=\sqrt{\left<\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B),\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\right>}$,
\begin{eqnarray}
   \pdiff{}{u_A^i}r^{-1} = -\frac{1}{2}r^{-3}\pdiff{}{u_A^i}\left<\mathbf{f}_A-\mathbf{f}_B,\mathbf{f}_A-\mathbf{f}_B\right> = -r^{-3}\left<\mathbf{f}_A-\mathbf{f}_B,\pdiff{\mathbf{f}_A}{u_A^i}\right>.
\end{eqnarray}
Hence,
\begin{eqnarray}
  \pdiff{L}{u_A^i} = \frac{1}{2}M_A\sum_{l,k=1}^2\pdiff{g_{lk}}{u^i}(\vec{u}_A)\dot{u}_A^l\dot{u}_A^k + \frac{Q_A}{4\pi\epsilon_0}\sum_{\underset{B\neq A}{B=1}}^N\frac{Q_B}{r^3}\left<\mathbf{f}_A-\mathbf{f}_B,\pdiff{\mathbf{f}_A}{u_A^i}\right>.
  \label{eq:el2}
\end{eqnarray}
Bringing equations \eref{eq:el1} and \eref{eq:el2} together and solving for $\ddot{u}_A^i$ we obtain
\begin{eqnarray}
 \fl \nonumber \ddot{u}_A^i &= -\frac{1}{2}\sum_{j,k,l=1}^2g^{ij}(\vec{u}_A)\left(2\pdiff{g_{jk}}{u^l}(\vec{u}_A)-\pdiff{g_{lk}}{u^j}(\vec{u}_A)\right)\dot{u}_A^l\dot{u}_A^k\\
 \fl  &\quad +\frac{Q_A}{4\pi\epsilon_0M_A}\sum_{\underset{B\neq A}{B=1}}^N\frac{Q_B}{\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|^3}\left<\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B),\sum_{j=1}^2g^{ij}\pdiff{\mathbf{f}}{u^j}(\vec{u}_A)\right>.
   \label{eq:eom}
\end{eqnarray}
Here, $g^{ij}$ are the matrix elements of the inverse of the matrix $g_{jk}$, so that we have the identity $\sum_{j=1}^2g^{ij}g_{jk}=\delta_k^i$ with $\delta_k^i$ being the Kronecker-$\delta$. The combination of derivatives of the metric at the beginning of \eref{eq:eom} is usually abbreviated by
\begin{equation}
  \frac{1}{2}\sum_{j,k,l=1}^2g^{ij}\left(2\pdiff{g_{jk}}{u^l}-\pdiff{g_{lk}}{u^j}\right)\dot{u}^l\dot{u}^k = \sum_{k,l=1}^2\Gamma_{lk}^i\dot{u}^l\dot{u}^k.
\end{equation}
The Christoffel symbols $\Gamma_{lk}^i$ have the geometric meaning of defining the notion of parallel displacement on the surface $S$. Note, that the functions $g_{ij}$, $g^{ij}$, and $\Gamma_{lk}^i$ can all be precomputed, once the parametrization $\mathbf{f}$ is given. They are intrinsic properties of the surface $S$. \ref{appsec:surfExp} lists the metric coefficients and Christoffel symbols for the sphere and the torus which we use in section \ref{sec:examples}.

If there is also an external electric or magnetic field, the interaction energy $V$ must be extended to
\begin{equation}
 V = \sum_{\underset{A<B}{A,B=1}}^N V_{AB} + \sum_{A=1}^N\left[Q_A\phi(\mathbf{f}(\vec{u}_A),t) - Q_A\left<\mathbf{\dot{f}}(\vec{u}_A),\mathbf{A}(\mathbf{f}(\vec{u}_A),t)\right>\right]
 \label{eq:lagrangian}
\end{equation}
with the electric potential $\phi$, and the magnetic vector potential $\mathbf{A}$. These potentials are related to their field values via $\mathbf{E}=-\mathbf{\nabla}\phi-\partial\mathbf{A}/\partial t$ and $\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}$.
The final equations of motion for particle $A$ then reads
\begin{eqnarray}
  \fl \nonumber \ddot{u}_A^i &= -\sum_{k,l=1}^2\Gamma_{lk}^i\dot{u}_A^l\dot{u}_A^k +\frac{Q_A}{4\pi\epsilon_0M_A}\sum_{\underset{B\neq A}{B=1}}^N\frac{Q_B}{\|\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B)\|^3}\left<\mathbf{f}(\vec{u}_A)-\mathbf{f}(\vec{u}_B),\sum_{j=1}^2g^{ij}\pdiff{\mathbf{f}}{u^j}(\vec{u}_A)\right>\\
   \fl &\quad + \frac{Q_A}{M_A}\left<\mathbf{E}+\mathbf{\dot{f}}(\vec{u}_A)\times\mathbf{B},\sum_{j=1}^2g^{ij}\pdiff{\mathbf{f}}{u^j}(\vec{u}_A)\right>.
   \label{eq:eomAll}
\end{eqnarray}

Here, we neglect that the charged particle motion itself yields a magnetic field that could influence the motion of the other particles. Furthermore, we do not take account of energy loss due to electromagnetic radiation caused by accelerated motion of the charged particles, but we add an artificial frictional term, see \ref{appsec:partFriction}.

\section{N-body simulation and visualization}\label{sec:simvis}
As long as the number of particles in an N-body simulation is in the order of a few thousands, we do not need any specific acceleration algorithm and/or approximation procedure, but we can calculate the N-body interaction by brute force: every particle interacts with every other particle. The computation time, however, increases quadratically with the number of particles. A first step to accelerate the computation is to handle each particle by a separate compute unit that has to integrate the equation of motion \eref{eq:eomAll} for this particle. The only prerequisite is that all compute units must have access to all particle positions and velocities which can be achieved using a shared memory system.
Today, virtually all standard home computers and even high-end smartphones have at least a dual core processor inside that have access to shared memory.
Parallelization of computation can then be realized, for example, by the programming interface \emph{OpenMP}\cite{openmp} that splits the computation into several threads.

Much higher parallelization can be achieved using the compute capability of modern graphics hardware. The \emph{Compute Unified Device Architecture} (CUDA) or the \emph{Open Compute Language} (OpenCL) offer a C-like programming interface in order to use the graphics processing units (GPUs) for general purpose computations.
Even without sophisticated algorithms for an efficient memory access, GPU computation leads to an enormous speed-up. Together with the Open Graphics Library (OpenGL) we can explore physical simulations at interactive frame rates.

The basic structure of our implementation is shown in figure~\ref{fig:progStructure}. The basic block is the GLUT~\cite{glut} main loop which acts on key strokes and mouse events, and which initiates rendering new frames.
\begin{figure}[ht]

  \centering\includegraphics[scale=0.7]{figure2}
  \caption{Basic structure of the program.}
  \label{fig:progStructure}
\end{figure}
The ``CUDA'' block includes all surface descriptions $(\mathbf{f},\partial\mathbf{f},g_{ij},\Gamma_{ij}^k)$ and calculates a single time step using either a standard Runge-Kutta second or fourth order method, see e.g. Press et al.~\cite{Press1992}. To calculate a time step, it uses the particle positions $u^k$ and velocities $\dot{u}^k$ stored within a \emph{Vertex Buffer Object (VBO)} which can be directly accessed by CUDA and OpenGL. The new particle positions can then be rendered directly or they can be mapped onto their corresponding surfaces. This mapping is realized by means of a \emph{Frame Buffer Object} (FBO). This FBO is an internal rectangular image (display) which, in our case, represents the domain $\mathbb{U}$. This image is then used to texturize the surface.

As an example, figure~\ref{fig:sphereTex} shows particles (yellow splats) projected onto a rectangular image (FBO) that is used as texture for the sphere. The inner gray rectangle covers the whole domain of the sphere $u^1=\varphi\in[0,2\pi)$, $u^2=\vartheta\in(0,\pi)$. The dark red border slightly expands the domain to prevent particle splats near the domain's boundary from being clipped, see figure~\ref{fig:splatArtefacts}.
The splats have to be distorted by means of the inverse metric of the surface to let them appear as circular splats when mapped onto the surface.
\begin{figure}[ht]

  \includegraphics[scale=0.3]{figure3.png}
  \caption{Particle rendering as Gaussian splats with radii $d\vartheta = ds/r$ and $d\varphi = ds/(r\sin\vartheta)$ for fixed value $ds$. This FBO image can be mapped onto a sphere, see figure~\ref{fig:splatArtefacts}. The inner gray rectangle covers the whole domain of the sphere $u^1=\varphi\in[0,2\pi)$, $u^2=\vartheta\in(0,\pi)$. The dark red border (color online) prevents particle splats near the domain's boundary from being clipped.}
  \label{fig:sphereTex}
\end{figure}

\begin{figure}[ht]

  \includegraphics[scale=0.2]{figure4a.png}
  \includegraphics[scale=0.2]{figure4b.png}
  \includegraphics[scale=0.2]{figure4c.png}
  \caption{Particle rendering as Gaussian splats without (left) and with (center) extended domain, compare figure~\ref{fig:sphereTex}. Without extended domain the splats are clipped. Right: wireframe view of the sphere.}
  \label{fig:splatArtefacts}
\end{figure}

The ``Surface'' block is responsible for drawing the surfaces themself. For that, we uniformly sample the domain using quads. These quads are then split into two triangles which are transformed by means of the surface function $\mathbf{f}$ of \eref{eq:surfFunction} within a so called vertex shader (see OpenGL Shading Language~\cite{opengl}). Figure~\ref{fig:splatArtefacts}~(right) shows the resulting wireframe of a sphere.

To keep the code simple, we do not have any sophisticated scene description language but implement each scene in a separate ``.inl''-file. A specific scene and its particular scene parameters have to be chosen at compile time. Each ``.inl''-file must have three functions: {\tt init\_Objects()} defines all surfaces and assigns IDs to them; {\tt set\_Particles()} registers the number of particles, their parameters (mass, charge, initial position, initial velocity), and the ID of the surface they belong to; {\tt set\_Supplement()} offers the possibility to change the camera parameters or the size of the window. The particle data could also be loaded from file when starting the program.

\section{Examples}\label{sec:examples}
In the following examples, we use the explicit second-order Runge-Kutta method without step size control and double precision floating-point numbers to integrate the equation of motion \eref{eq:eomAll}. The units in use are explained in \ref{appsec:units}. The divergence problem of close encounters due to the $1/\|\mathbf{f}_A-\mathbf{f}_B\|$ term is handled by modifying the Coulomb potential:
\begin{equation}
  \tilde{V}_{AB} = \frac{1}{4\pi\epsilon_0}\frac{Q_AQ_B}{\sqrt{\|\mathbf{f}_A-\mathbf{f}_B\|^2+\varepsilon^2}}\quad\mbox{with}\quad \varepsilon\ll 1.
\end{equation}
In principle, we should also take into account the magnetic field $\vec{B}$ generated by a moving charged particle which follows from the Biot-Savart law
\begin{equation}
  \vec{B} = \frac{\mu_0}{4\pi}\frac{Q_A\vec{v}_A\times\vec{r}}{r^3},
\end{equation}
where $\vec{r}$ is the relative distance of an other particle to particle $A$'s current position.
Then, from Lorentz force equation, the resulting acceleration of a particle $B$ with mass $M_B$ and current velocity $\vec{v}_B$ would be
\begin{equation}
  \vec{a} = \frac{Q_B}{M_B}\vec{v}_B\times\vec{B} = \frac{\mu_0e^2}{4\pi m_e}\frac{q_Aq_B}{m_B}\frac{\vec{v}_B\times(\vec{v}_A\times\vec{r})}{r^3},
  \label{eq:lorentz}
\end{equation}
where $\mu_0e^2/(4\pi m_e)\approx 2.81806\cdot 10^{-15}\unit{m}$. As we will see, this acceleration can be neglected.

\subsection{Particles on a sphere}
As a first example, we consider a Thomson problem situation where $N=128$ charged particles (electrons) are located on a sphere. For numerical reasons, we set the radius $r$ of the sphere such that the mutual acceleration $a$ between two particles due to Coulomb,
\begin{equation}
  a = \frac{e^2}{4\pi\epsilon_0m_e}\frac{1}{d^2}=\kappa\frac{1}{d^2},
\end{equation}
is roughly in the same order of magnitude as their distance $d\approx r$. For a sphere of radius $r=1\unit{m}$, this acceleration is $a\approx 253.27\unit{ms^{-2}}$. Furthermore, we cut out the poles of the sphere by limiting the colatitude angle $\vartheta$ to $\delta\leq\vartheta\leq\pi-\delta$ with $\delta\approx 0.01$.
To obtain a minimum energy configuration, we add an artificial linear friction term with frictional constant $\eta=50$, see \ref{appsec:partFriction}. As we will see, this is necessary because the energy loss due to electromagnetic radiation, which can be estimated by the Larmor formula
\begin{equation}
  P = \frac{e^2}{6\pi\epsilon_0c^3}a^2 \approx 5.708\cdot 10^{-54}\unit{\frac{Js}{m^3}}\cdot a^2,
  \label{eq:larmor}
\end{equation}
where we could use the centripetal acceleration $a=v^2/r$ for circular motion, is too small.

At the beginning of the simulation, the particles are randomly distributed and have zero initial velocity, see figure~\ref{fig:expSphere}. The field energy $W|_{t=0}/(\kappa m_e)\approx 8.634620\cdot 10^3~\unit{m^{-1}}$, see equation \eref{eq:fieldEnergy}, and the kinetic energy $T/m_e=0$.
When the simulation starts, the non-uniform distribution of the particles let them accelerate due to their mutual Coulomb interaction, and they reach a maximum kinetic energy of $T/m_e\approx 2.8467\cdot 10^5\unit{m^2s^{-2}}$ at $t\approx 0.0054\unit{s}$. Then, the mean velocity of a single particle is $v\approx 47.2\unit{ms^{-1}}$ and, thus, energy loss due to electromagnetic radiation and acceleration due to equation \eref{eq:lorentz} can be neglected.
At simulation time $t=2.5\unit{s}$, the field energy has dropped to $W/(\kappa m_e)\approx 7.393081\cdot 10^3\unit{m^{-1}}$ and the kinetic energy has reduced to $T/m_e\approx 2.433780\cdot 10^{-2}\unit{m^2s^{-2}}$, which is nearly impossible to observe visually, however.
After about $6.9$ seconds, the kinetic energy has dropped below $T/m_e=10^{-15}\unit{m^2s^{-2}}$.
\begin{figure}[ht]

  \includegraphics[scale=0.15]{figure5a.png}
  \includegraphics[scale=0.15]{figure5b.png}
  \includegraphics[scale=0.15]{figure5c.png}
  \includegraphics[scale=0.15]{figure5d.png}
  \caption{$N=128$ charged particles on a sphere at simulation times $t=\{0,0.0054,0.1,2.5\}\unit{s}$ with step size $\Delta t=10^{-4}\unit{s}$ and velocity-dependent friction $\eta=50$ (see \ref{appsec:partFriction}).}
  \label{fig:expSphere}
\end{figure}

In table~\ref{tab:E_sphere} we compare our minimum energy results for several number of particles $N$ with the literature values. Glasser and Every's~\cite{Glasser1992} estimation formula
\begin{equation}
  E(N) = \frac{N^2}{2}\left(1-aN^{-1/2}+bN^{-3/2}\right)
  \label{eq:eEstimate}
\end{equation}
with parameters $a=1.10461$ and $b=0.137$ from \cite{Morris1996} already gives a good approximation. The lowest energies for $110<N\leq 200$ from Morris et al.~\cite{Morris1996} are determined using a genetic algorithm.
\begin{table}[ht]
   \setlength{\tabcolsep}{0.35cm}
   \begin{tabular}{cccc}
     \hline\\[-1.1em]
     \hline\\[-1em]
     $N$ & $W/(\kappa m_e)$ & $W_N$ & $E(N)$\\ \hline\\[-0.8em]
     $112$ & $5.618044887\cdot 10^3$ & $5.618044882\cdot 10^3$ & $5.618079704\cdot 10^3$\\
     $128$ & $7.393007443\cdot 10^3$ & $7.393007443\cdot 10^3$ & $7.392951914\cdot 10^3$\\
     $161$ & $1.183308476\cdot 10^4$ & $1.183308474\cdot 10^4$ & $1.183308683\cdot 10^4$\\
     $200$ & $1.843885657\cdot 10^4$ & $1.843884272\cdot 10^4$ & $1.843881429\cdot 10^4$
   \end{tabular}
   \caption{Minimum field energies $W/(\kappa m_e)$ for $N$ particles with charge $q=e$, time step $\Delta t=10^{-4}\unit{s}$ and frictional constant $\eta=50$; compared with values $W_N$ taken from Morris et al.~\cite{Morris1996} and estimation given by \eref{eq:eEstimate}. The latter two values were calculated using $\kappa m_e=1$.}
   \label{tab:E_sphere}
\end{table}

Depending on the initial random configuration and on the integration time step, the simulation does not always reach the same minimum energy configuration. In that case, the particles have to be either randomly distributed again or they have to be given a small jerk by pressing a key. In the just discussed example, our worst minimum energy for $N=128$  was about $W/(\kappa m_e)\approx 7.393166 \cdot 10^3\unit{m^{-1}}$.

\subsection{Particles on a torus}
In the second example, we consider $N=1024$ particles $(q=e)$ on a torus with radii $R=2\unit{m}$ and $r=0.9\unit{m}$. As before, the particles are randomly distributed at the beginning of the simulation, see figure~\ref{fig:expTorus}. At $t\approx 30\unit{s}$, the kinetic energy has dropped below $T/m_e=2.134\cdot 10^{-4}\unit{m^2s^{-2}}$ and the field energy reads $W/(\kappa m_e)\approx 2.1760534\cdot 10^5\unit{m^{-1}}$.
\begin{figure}[ht]

  \includegraphics[scale=0.2]{figure6a.png}
  \includegraphics[scale=0.2]{figure6b.png}
  \caption{$N=1024$ charged particles on a torus at simulation times $t=\{0,5\}\unit{s}$ with step size $\Delta t=10^{-3}\unit{s}$ and velocity-dependent friction $\eta=200$.}
  \label{fig:expTorus}
\end{figure}
Expectedly, all particles move to the outer side of the torus. Again, to stress the validity of our code, we compare our minimum energy configurations with the literature, see table~\ref{tab:E_torus}. As can be seen, we are in good agreement with the literature values and sometimes we have found even lower energies.
\begin{table}[ht]
   \setlength{\tabcolsep}{0.35cm}
   \begin{tabular}{cccc}
     \hline\\[-1.1em]
     \hline\\[-1em]
     $N$ & $a=R/r$ & $W/(\kappa m_e)$ & $W_N$ \\ \hline\\[-0.8em]
      $20$ & $1.414$ & $1.029846718\cdot 10^2$ & $1.029846689\cdot 10^2$\\
      $20$ & $1.618$ & $1.098582569\cdot 10^2$ & $1.098582529\cdot 10^2$\\
     $100$ & $1.414$ & $3.082218463\cdot 10^3$ & $3.082217005\cdot 10^3$\\
     $100$ & $1.618$ & $3.295918820\cdot 10^3$ & $3.296043624\cdot 10^3$\\
     $415$ & $1.414$ & $5.660043959\cdot 10^4$ & $5.660070457\cdot 10^4$
   \end{tabular}
   \caption{Minimum field energies $W/(\kappa m_e)$ for $N$ particles with charge $q=e$, time step $\Delta t=0.0002\unit{s}$, and frictional constant $\eta=50$ on a torus with aspect $a=R/r$ and radius $R=1\unit{m}$; compared with values $W_N$ taken from \cite{thomsonApplet}.}
   \label{tab:E_torus}
\end{table}

The dynamic evolution of the $N=415$ example of table~\ref{tab:E_torus} can be read from figure~\ref{fig:torusEnergies}. At the beginning of the simulation, there is a very short peak of high kinetic energy due to the particles that are accelerated from rest. After less than $0.8$ seconds, the kinetic energy has dropped below $T/m_e=1\unit{m^2s^{-2}}$. Then, at least from the visual impression, the particles do not move any more. However, there is still kinetic energy in the system which does not dissipate uniformly. Only after about $10$ seconds, the particles slowly settle down and reach a minimum energy configuration (plateau in figure~\ref{fig:torusEnergies}, right).
\begin{figure}[ht]

  \includegraphics[scale=0.62]{figure7a}\quad
  \includegraphics[scale=0.62]{figure7b}
  \caption{Kinetic energy $T/m_e$ in $\unit{m^2s^{-2}}$ (left) and field energy $W/(\kappa m_e)$ in $\unit{m^{-1}}$ (right) depending on time in $\unit{s}$ for the torus simulation with $N=415$ particles, aspect $a=1.414$, step size $\Delta t=0.01\unit{s}$, and a frictional constant $\eta=50$. The loss of kinetic energy is also due to the dissipative RK2 method.}
  \label{fig:torusEnergies}
\end{figure}

\subsection{Two intertwined torii}
A more intricate example is shown in figure~\ref{fig:expTorii} where two torii are intertwined and the particles have either the same or opposite charges. The corresponding field lines are shown in figure~\ref{fig:expToriiFL}. They were started close to the charged particles in the direction of the outer surface normal and were integrated with a constant step size along the total electric field of all particles. Please note that the field lines end after $n=800$ steps by default. And, as the object surface is no equipotential surface, field lines could also penetrate the object surface non-orthogonally.
\begin{figure}[ht]

  \includegraphics[scale=0.237]{figure8a.png}
  \includegraphics[scale=0.237]{figure8b.png}
  \caption{$N=2\times 512$ particles $(q=\pm e)$ on two intertwined torii with radii $R=2\unit{m}$ and $r=0.9\unit{m}$ of either the same (left) or opposite (right) charge.}
  \label{fig:expTorii}
\end{figure}

\begin{figure}[ht]

  \includegraphics[scale=0.237]{figure9a.png}
  \includegraphics[scale=0.237]{figure9b.png}
  \caption{The same situation as in figure~\ref{fig:expTorii} but now with field lines (green) that were started close to the particle positions and perpendicular to the surface.}
  \label{fig:expToriiFL}
\end{figure}

\section{Exercises}\label{sec:exercises}
In the following we give a few suggestions for possible exercises that could be done using the ChaPaCS source code.

\begin{cpExercise}
  Consider the Thomson problem situation where $N=128$ charged particles $(q=e)$ are located on a sphere. How does the minimum field energy vary with the size of the radius?
\end{cpExercise}

\begin{cpResult}
  When the charged particles are located in a minimum energy configuration, their relative positions on the sphere will not change when the radius of the sphere is changed. As the distance $d$ between two points on a sphere of radius $r$ is $d=2r\sin(\alpha/2)$, where $\alpha$ is the angular separation between both points, the field energy scales inversely proportional with the radius of the sphere: $W_{r_1}/W_{r_2}=r_2/r_1$. To test this with ChaPaCS, start with a minimum field energy and store the positions to disk. Restart the program with an other radius and load the previously stored positions.
\end{cpResult}

\begin{cpExercise}
  Adapt the {\tt set\_Particles()} method of the single sphere example such that one half of the particles have $q=e$ and the other ones have $q=10e$. To circumvent numerical instabilities, you have to reduce the time step to $\Delta t=2\cdot 10^{-5}\unit{s}$ and you have to increase the frictional constant to $\eta=500$. What happens?
\end{cpExercise}

\begin{cpResult}
  Because of the partially higher charges, the field energy rises considerably from $W_1/(\kappa m_e)\approx 7.3930\cdot 10^3\unit{m^{-1}}$ to $W_2/(\kappa m_e)\approx 2.1709\cdot 10^5\unit{m^{-1}}$. Additionally, the symmetric arrangement of the particles breaks down.
\end{cpResult}

\begin{cpExercise}
   Determine the influence of the aspect $a=R/r$ in the torus example with $N=415$ particles on the minimum field energy.
\end{cpExercise}

\begin{cpResult}
   Unfortunately, the change of the minimum field energy cannot be calculated as easily as in the first exercise. When the aspect ratio is increased, the 'thickness' radius decreases and the particles move together which results in a higher field energy.\\[0.3em]
   \begin{tabular}{cccc}
     \hline\\[-1.1em]
     \hline\\[-1em]
      $a=R/r$ & $W/(\kappa m_e)$  \\ \hline\\[-0.8em]
      $1.45$ & $5.73353\cdot 10^4$\\
      $1.5$  & $5.83289\cdot 10^4$\\
      $1.7$  & $6.20165\cdot 10^4$\\
      $2.0$  & $6.68266\cdot 10^4$\\
      $2.5$  & $7.34052\cdot 10^4$\\
      $3.0$  & $7.87078\cdot 10^4$\\
      $5.0$  & $9.29907\cdot 10^4$
   \end{tabular}
\end{cpResult}

\begin{cpExercise}
   Determine a minimum energy configuration for $N=128$ charged particles $(q=e)$ on the sphere and save it to disk. Restart the program with this configuration. Set the frictional constant to zero and add an external electric field, $\tilde{\mathbf{E}}_y=100\unit{ms^{-2}}$. Slightly increase the electric field strength and discuss what happens.
\end{cpExercise}
\pagebreak

\begin{cpResult}
  At the beginning of the simulation, the particles are at rest. With the electric field turned on, the particles will be accelerated in the direction of the field. But because of the spherical constrained, the particle move together which increases the electric repulsion. Hence, the particles begin to oscillate.
  Due to numerical dissipation, the oscillation is damped and the particles come to rest after some time.
  If the field strength is increased, the oscillation amplitude becomes higher. Since the particles are constrained to the sphere and cannot linearly follow the field lines, the system becomes more and more chaotic. And since the step size is fixed, the numerical integration becomes unstable.
\end{cpResult}

\noindent {\bf Further exercises:}
\begin{itemize}
  \item Find the parametrization of an ellipsoid and determine the metric coefficients $g_{ij}$ and the Christoffel symbols $\Gamma_{ij}^k$. Expand the code to handle ellipsoids and find the minimum energy configurations.
  \item Construct a new scene with a small sphere hovering above a plane.
  \item Construct a new scene with four small spheres at the corners of a quad (quadrupole). Study the distribution of particles on the spheres and the overall field lines.
\end{itemize}

\section{Summary}
In this work we have developed the equations of motion of $N$ charged particles that are constrained to a curved two-dimensional surface but interact in three dimensions as usual. We have also given a short introduction how to implement the particle simulation using the compute capability of todays graphics boards. The source code (C/C++/CUDA) of the prototype implementation \emph{ChaPaCS} is freely available and can be easily extended by other curved surfaces.

\ack
This work was partially funded by Deutsche Forschungsgemeinschaft (DFG) as part of the Collaborative Research Centre SFB 716.

redpajama_set_name: RedPajamaArXiv


Canada's corporate image isn't looking so squeaky-clean in the World Bank's books — all thanks to SNC-Lavalin.
from bidding on its global projects under its fraud and corruption policy, 117 are from Canada — with SNC-Lavalin and its affiliates representing 115 of those entries, the World Bank said.
"As it stands today, the World Bank debarment list includes a high number of Canadian companies, the majority of which are affiliates to SNC Lavalin Inc.," said the bank's manager of investigations, James David Fielder.
As a result of the misconduct found during the probe, the Montreal-based engineering and construction firm, and its affiliates as per World Bank policy, were debarred in April 2013 for 10 years, as part of a settlement with SNC-Lavalin. And in one fell swoop, 115 Canadian firms were blacklisted by the World Bank, making Canada seemingly look like the worst offending country.
It's quite the jump from 2012, when no Canadian companies were barred.
Overall, Canada had 119 total companies on the list (including two permanently debarred since 1999). United States had the second largest number, with 44 companies on the debarment list (among which are several SNC-Lavalin companies), and Indonesia came in third with 43.
SNC-Lavalin did not respond to a request for comment.
The World Bank has been stepping up its fraud investigations in recent years, with four times more debarments than in 2012, and compared to the past seven years combined, said Mr. Coleman.
Canadian firms may be at higher risk of being sanctioned by the World Bank, he added, because Canadian authorities have not been as aggressively enforcing its anti-corruption laws as its U.S. counterparts.
"The result of that is that Canadian companies may have been lulled into a false sense of security, because their own national authorities were not closely scrutinizing their global operations, that nobody else would either," he said.
There were 89 SNC-Lavalin affiliates, however, that were instead given a conditional non-debarment by the World Bank for 10 years. This is essentially a probation of sorts that requires these companies to adhere to compliance standards, but are still allowed to bid on the World Bank's development projects around the world.
These sanctions were handed down instead of debarment, in part, because these companies had come forward to the bank had demonstrated concrete steps towards anti-corruption practices, said a World Bank spokesperson.
That is a sign that SNC-Lavalin has begun taking compliance more seriously. said Milos Barutciski, Bennett Jones senior partner and co-chair of the firm's international trade and investment practoce.


text:
Trump agrees to keep about 400 US troops in Syria, official says
President Donald Trump has agreed to keep about 400 US troops in Syria after the withdrawal of most US forces there this spring, a senior administration official said Friday.
Trump agreed on Thursday to keep about 200 US troops as part of planned multinational force that would maintain a safe zone in Northeastern Syria, the official said. This force would be in addition to the 200 troops the US is planning to maintain after the withdrawal at its base in at-Tanf, Syria, the official said.
The US and NATO allies are in the process of assembling what officials are calling a "monitoring and observer" force of roughly 800 to 1,500 troops to maintain a safe zone in Northeastern Syria. The goal of the force would be to maintain a buffer between Turkey and US-allied Syrian opposition forces to ensure that neither side carries out attacks on the other.
The US initially planned to only provide air support -- not ground troops -- to the observer force, but NATO allies objected and said they would not contribute troops toward such a mission without a US troop commitment, the official said.
It became clear to US officials on Monday that the observer force would never become a reality absent a US commitment of ground troops. After consulting with State and Defense Department officials, National Security Adviser John Bolton met with Trump on Thursday and convinced him to agree to commit "a couple hundred" troops, said a senior administration official, who briefed a small group of reporters on Friday on the plans on condition of anonymity.
The decision ticks up the number of troops that Trump is agreeing to maintain in Syria after he hastily announced his plans to withdraw US troops from Syria late last year, stunning US officials and allies and triggering the resignation of Secretary of Defense Jim Mattis. US officials and allies have sought to mitigate the risk to US-allied Syrian opposition forces and to ensure that the US can adequately respond to the potential resurgence of ISIS in the region.
Planning for the multinational observer force stems from an idea that Turkey proposed to Bolton and the chairman of the Joint Chiefs of Staff Gen. Joseph Dunford during a visit in January, amid concerns about the potential for conflict between Turkey and US-backed Kurdish forces. The Joint Chiefs has since been leading discussions with NATO allies to assemble the force.
Turkish officials have expressed concerns that US-backed Kurdish forces would carry out terrorist attacks in Turkey, and the US-backed forces have worried about the fret of a Turkish invasion.
The senior administration official made clear that the observer force would not be a United Nations-chartered peacekeeping force and would not face the same restrictive rules of engagement that those forces face, but declined to elaborate on what those rules of engagement would be. The forces would, however, be combat-ready, the official said.
The Joint Chiefs of Staff are continuing to hammer out discussions with NATO allies, but given that a commitment of US troops was the Europeans' main objection, US officials are optimistic that this force will be assembled.
The senior administration official briefed reporters on Friday after White House press secretary Sarah Sanders issued a brief one-line statement asserting that "a small peace keeping group of about 200 will remain in Syria for a period of time." That figure did not take into account the presence of another 200 troops expected to remain at al-Tanf.
US officials hope the multinational force in Northeastern Syria will minimize the risk of conflict between Turkey and US-backed forces in Syria.
US officials are aiming to withdraw most US troops from Syria by April, but a senior administration official said Friday that date could shift depending on how quickly the multinational observer force is assembled.


text:
A natural wonder of nature
Editorial 28. July 2021 880All-SeasonArticlesKirkjubæjarklausturNewsSouth IcelandSuðurland
East of Vestur-Skaftafellssýsla county, just west of Kirkjubæjarklaustur lies one of Iceland's natural wonders, Fjarðárgljúfur. It is believed that the glacial river Fjaðrá formed the gorge some 9000 years ago, but the river is believed to have been much bigger then. The canyon is 2 km / 1.2 mi long and at its highest about 100 m / 300 ft. A good hiking trail runs along the east side. Canadian pop star Justin Bieber made Fjaðrárgljúfur famous when he recorded a music video there in 2015. The gorge became so popular that it was closed to tourists for a while two years ago due to intrusion.
Skaftárhreppur 27/07/2021 20:31 135mm
The picture: Tourists take a look at the view from a viewing platform over Fjarðárglúfur in the last rays of the sun last night.
Photograph and text: Páll Stefánsson
Íslenska (Icelandic)


text: Q: Conditional Expectation constant If we have a probability space $(\Omega,\mathcal{A},P)$ and pairwise disjoint events, say $B_i$ for $1\leq i \leq n \leq \infty$ such that $\bigcup B_i = \Omega$ and if we further set $\mathcal{F}=\sigma(B_i,1\leq i\leq N)$ why is then the conditional expectation constant on $B_i \;\forall i$, $E[X\cdot 1_{B_i}|\mathcal{F}] = b_i$? Because it is $\mathcal{F}$-measurable?

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At least one recent Emerson graduate is happy that Boston has been chosen as a contender for the 2024 Summer Olympics.
Kelsey Doherty '14 and two other recent graduates, Donovan Birch '14, and Luke Fraser '14, did a class project in a sports communication class taught by Associate Professor Gregory Payne in the Spring 2014 semester that advocated bringing the Olympics to Boston.
2002 was the last time the Olympics were held in the United States, in Salt Lake City.
A mock Facebook page Emerson students established last spring as part of a sports communication class project proposing Boston host the 2024 Summer Olympics.
Doherty, Birch, and Fraser met with an executive at Suffolk Construction—a major supporter of the Boston Olympics bid—before formulating a research paper and multimedia presentation on why the Olympics should be held in the Hub.
Not everyone is happy with the idea of the Olympics coming to Boston—citing concerns about taxpayer costs, traffic, construction, and possible residential displacement.
Doherty, a Biddeford, Maine, native who majored in Marketing Communications and now works as a social media specialist for the Boston Red Sox, believes those concerns can be worked out.
"As a whole, it's a great opportunity for the city and the country," she said.
Payne's sports communication course focused on business and crisis management, conflict resolution, and how sports intertwines with politics. Students also received several guest lectures from sports journalist Tom E. Curran of Comcast SportsNet.
After attending six Olympic games, Payne said he's in favor of bringing them here.


Gunter Reski is easily overlooked – so far he has been rather sparsely taken up by important institutions and galleries – and yet he is one of the key figures within the German discourse on painting. This is mainly because of his writing: temperate, cryptic exhibition reviews in which he feels for a language to adequately describe painting. That the Bochum-born, fifty-something artist's own painting practice is far from insignificant, though, is once again highlighted by Organwanderung jetzt, his first exhibition with Nagel Draxler. In particular the exhibition highlights how Reski sees the separation between talking and writing about painting and painting itself as obsolete, manifested in the way the works are displayed as a gigantic, floor-to-ceiling pseudo-newspaper: every bit of wall is covered in text columns into which Reski inserts a mass of his own paintings and works on paper. The overloaded and erratic flood of images characteristic of a salon hang thereby gains an underlying composition, acquiring not only structure but also a certain logic.
Contentwise, the 'newspaper' is explained in the text as the creation of a single fictitious person who produces it, daily, against boredom: an eccentric monologue, which the unnamed author deems an absolute necessity. It is surely not too farfetched to read Reski's placement of his work within this format as an allegory of the contemporary painter. The wall newspaper serves not only as a simple display device but aims at the mutual annotation of text and image, and this naturally guides the gaze towards paintings closely linked to the narration: Staubfussel (Dustball, 2018), for example, an elegantly painted, highly magnified dustball with some entangled hairs, the writing proximately talking about a messy apartment. The painting is more than an illustration, though, linking directly to the painterly discourse of the Rhineland. The graceful, vague physicality of the dustball, with its mauve shadowing, is reminiscent of Gerhard Richter's cloud paintings; the hairs' wild lines recall certain works by Albert Oehlen. The thematically related painting Abflussfussel (Drain Lint, 2018) depicts a filthy tangle of hair, again blurred in a Richteresque manner, but more harshly and against the background of a specific blue that, to a trained eye, immediately connects to Markus Lüpertz.
Yet Reski doesn't do this primarily to plant references for an informed public. Rather he thematises style as such, by trolling and splashing together the stylistic signatures of other painters. He does position himself as a repertoire painter, though since he has been doing this for a long time, a number of styles are now noticeably Reskian. The grisaille Kühlschrank mit Heizung (Fridge with Heater, 2018), painted with heavy strokes, in which a heater jumps out of a fridge, is part of his sequence of broadly painted tautologies in kitchen spaces. Plus there's a multitude of typefaces, sometimes added to the painting as a quasi-architectural moment, as in Nudeln mit Teller (Pasta with Plate, 2013), on which a bunch of spaghetti hovers above a brown plate that belonged to the artist's dead mother, as the text framing it informs us. Sometimes the text is the only element, as in one piece of cardboard with three-dimensional letters mounted on it or the crumpled, unstretched canvases, whose sprayed folds of fabric reveal short texts such as 'I always hear you inhale' (Ich höre dich immer nur einatmen, 2016). Together with Reski's stylistic pluralism, this extension of writing into painting pleads for a style of painting that does not end in a single canvas but only reveals itself in an exchange. Since, for Reski, this exchange should be as linguistically precise as possible, he still clings to German as 'his' language. This, incidentally, may also be a reason why he is not so well known abroad.


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A surveillance balloon and Afghan soldiers playing football contributed to an RAF helicopter crash that killed five in 2015 in Kabul, a coroner in Oxfordshire has ruled.
Two Britons, Flt Lt Geraint Roberts and Flt Lt Alan Scott, were among the dead. Two others were American and the fifth, Gordon Emin, French. Five others were also injured.
The coroner's court while it circled the Nato landing site, which doubled as a football pitch, where soldiers were playing at the time. While waiting for the pitch to clear, the helicopter hit the cable and crashed. The coroner, Darren Salter, said the two Britons and the Frenchman had died from their injuries either on impact or soon afterwards.
He said it had been accidental, but the siting of the balloon close to the landing site contributed. There was also an unsatisfactory system of management in regard to communications and clearance of the site, he said, "resulting in the need to go around or overshoot, which increased the risk of a tether strike given its proximity".
"The helicopter was the second aircraft in a formation of two whose task it was to take personnel between helicopter landing sites in Kabul," Salter said. "The helicopter was due to land at the dual-use helicopter landing site at headquarters known as the soccer field.
RAF Sgt Simon Craig, in another helicopter, said people on the football pitch had not moved so his helicopter overshot. But the other helicopter flew over the pitch.
"Puma II began pitching and rolling and it was clear something had gone wrong. We heard the mayday call from Puma II and it was clear it had hit the ground," Craig said.
Flt Lt Scott's wife, Lisa, said: "My Al was kind, intelligent, determined and conscientious and we know he could have done great things had his life not been cut so tragically and unfairly short.
"The only solace we have from this is that Al died doing what he loved, which is a claim that very few people can make.
Emin's family is to pursue a compensation claim with the UK (MoD).
They said: "Obviously, it's been extremely painful for our family to hear the evidence about what happened to Gordon during this inquest.
"We always felt there were questions to be answered and this process has gone some way to doing that. Gordon was an exceptional, loving and caring man, and we will ensure his memory lives on through his two young children.
"It is our sincere wish that the recommendations highlighted this week are put into place to prevent other people having to cope with another tragic and avoidable loss of life.


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A spasm of the facial muscles following a tap on the facial nerve; seen in hypocalcemic tetany.
Chvostek sign is a sample topic from the Taber's Medical Dictionary.
Venes, Donald, editor. "Chvostek Sign." Taber's Medical Dictionary, 23rd ed., F.A. Davis Company, 2017. Nursing Central, nursing.unboundmedicine.com/nursingcentral/view/Tabers-Dictionary/732398/all/Chvostek_sign.
Chvostek sign. In: Venes D, ed. Taber's Medical Dictionary. 23rd ed. F.A. Davis Company; 2017. https://nursing.unboundmedicine.com/nursingcentral/view/Tabers-Dictionary/732398/all/Chvostek_sign. Accessed April 26, 2019.
Chvostek Sign [Internet]. In: Venes D, editors. Taber's Medical Dictionary. F.A. Davis Company; 2017. [cited 2019 April 26]. Available from: https://nursing.unboundmedicine.com/nursingcentral/view/Tabers-Dictionary/732398/all/Chvostek_sign.


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You are here: Home / Economy / Đà Nẵng attracts $542 million of foreign investment in H1
Đà Nẵng attracts $542 million of foreign investment in H1
July 10, 2019 vietnamnews Leave a Comment
A corner of Đà Nẵng City. The city attracted US$542 million in foreign capital in the first half of 2019. – Photo cafef.vn ĐÀ NẴNG – More than US$542 million of foreign capital was invested in the central city of Đà Nẵng in the first half of this year. Speaking at the 11th session of the 9th municipal People's Council in Đà Nẵng on Tuesday, Hồ Kỳ Minh, vice chairman of the city's People's Committee, attributed the results to investment attraction activities organised as part of the city's policy to focus on investment this year. He said Đà Nẵng recently held a conference with investors in which investment registration certificates were granted for eight projects worth over $492 million. Local authorities have also allowed research to proceed on 11 potential projects worth nearly $3.5 billion. "Despite such positive outcomes, the city has failed to meet the targets in some aspects," Minh said. For the remaining six months, the city plans to improve its business-investment promotions and management of investment in construction. It also will speed up the progress of major projects and enhance its urban and environmental management, among other tasks. Participants at the 11th session also reviewed socio-economic plans for the first half of the year and outlined measures to carry out tasks for the remainder of the year, resolving pressing issues such as waste, pollution, water shortages and land acquisition. – VNS
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