No two bikes are the same, so we tailor the servicing to fit your bike perfectly.
While all Ducati's are scheduled for servicing every 12 months, not everyone covers the same mileage every year, so we need to adjust the servicing accordingly.
If you have riden 3000 miles in 2 years you won't need as much done as a rider who has covered 12,000, equally 6000 miles round Brands Hatch is a long way so we are happy to adjust the service items to suit you and your Ducati.
All about us... We are, and always have been a workshop focused company. We are a small independent workshop that is fully authorised by Ducati to look after your bike.
All our workshop team are Ducati factory trained, very experienced and work only on Ducati motorcycles.
What does this mean to you? It means that your Ducati will be serviced to the highest possible standard by someone who not only knows how to carry out routine maintenance work, but knows exactly what to check on each specific model. We feel this is the only way to a well maintained Ducati.
Workshop collection and delivery service - FREE locally too.
Fully flexible servicing, we can advise and discuss just what YOUR Ducati needs, rather than just go by the recommended service list, this keeps your bill sensible too!
Split hourly rates: have two hourly rates, for simple jobs, tyre fitting, valeting, chain adjustments etc we charge HALF price, and our regular hourly rate is low too at just £65 per hour.
We are proud that our servicing prices are as sharp as when we were a small stand alone workshop, so you get more without paying more.
Modern bikes require an annual service and oil change.
Due every 12 months or when the 'Oil Service' light shows on your dash. The milage interval varies with each model, typically between 7,500 and 9,000 miles.
This service can be carried out within a working day, you are welcome to wait or use our courtesy bike.
Price varies with model but is typically £199.
Only due when your Ducati reaches the specified mileage.
This is the specialist service where in addition to the routine maintenance, we check and reset the Desmodronic valve system unique to Ducati motorcycles. We also repalce cambelts where required and carry out some of the more indepth and specilaist service items to the motor and chassis.
This service often takes more than a day to complete, so we offer a service courtesy bike or colletion and delivery.
Please call us on 01962 873984 and speak to our workshop who can advise accordingly.
This option excludes mileage only items, such as the front fork strip and refresh (only really required at 12000 mile intervals) so your bike gets just what it needs and you're not paying for service items not required.
It is rare that these motorcycles fit into the service schedule set out by Ducati, so we are completely flexible to our approach to what these bikes need. Call the workshop on 01962 873984 and we can advise what work is needed.
These are the some of the more specific service options available in the Moto Rapido workshop. Below are the most popular options for the dedicated Ducati owner, for the rider or racer who wants the very best out of their bike, developed from our experience with our own Ducati's on road and track, this is the kind work that our reputation is built on. Not available at your average Ducati dealer!
The desmodromic valve operating system used by Ducati is, as all race derived systems, particularly sensitive to the components being set perfectly, so worn rockers or camshafts, valves not seating properly or covered in exhaust carbon and unmatched shim tolerances all result in poor running, difficult starting, poor throttle response, less power and increased oil consumption.
This is often refered to as a 'top end blueprint' and is very popular with our track day customers and racers, setting the desmo valve system to exact tolerances, rather than factory built ones, simply makes your Ducati as it should be. We often do this service along with cam timing (see below) to get the best out of the desmo V twin engine.
We remove both cylinder heads, strip and professionally clean them completely, de-coke and pro clean the pistons, barrels and heads, check valve guide condition and replace if required (usually 24000 miles of road use) clean and lap the valves / valve seats, replace worn rockers, replace collets and stem seals. Then assemble each head and set the valve clearances exactly, clean barrels/cases and replace base gaskets and refit the heads.
This upgrade is a complete top end rebuild, and you will notice the difference, more power, smoother running and sharper throttle response are the ususal remarks from customers.
Testatretta Evo motors : These motors suffer far less than the previous models and its rare we see worn rockers and cams prior the service interval of 7500 (for road bikes).
As each motor requires different parts, these prices exclude the cost of gaskets, O rings, stem seals and valve collets etc that your bike would need. Other parts (valves, rockers, valve guides, cams, cam bearings etc) are in addition and we will advise accordingly. No additional labour is charged for fitting these as part of this service.
As we have the heads apart and on the bench for this service, upgrade Camshafts, race rocker springs, R/RS/Corse rockers, rocker covers etc are fitted free! and it is an ideal time for head porting / gasflowing !
This is a MAJOR Rebuild (sometimes and called 'Blueprinting') specifically designed by us to get your Ducati running and handling the very best it can.
Every motorcycle is built to a manufacturing tolerance rather than being hand build precisely. This service changes that, this is why we get such good results by carefully setting everything to the exact settings.
We are happy to advise and amend the list of service items on this and all our services to suit your bikes needs exactly - Just call the workshop to go through what your bike specifically needs. This is where our workshop really gets its reputation, there are many Ducati trained mechanics but few can strip and rebuild a motor to this specification. The same people who build our race motors, carry out this level of engine work. Precision and attention to the finest detail are everything.
Re assemble crank assembly and individually shim crank, gearbox and gearbox selection drum into cases - road or race settings.
A very worthwhile note… We can now remap the majority of Ducati ECU's and eprom chips in house. So if you have a special requirement for your 1098 / 848 / 999 etc we can now reamp these ECU's with upgraded fuel tables, ignition tables, alter RPM limits etc, just call (some more info CLICK HERE).
** Upgrade or performance parts marked are fitted free as part of this service, many other tuning parts (engine parts, exhuast systems etc) are also available with free fitting when we have your bike on the operating table for this service, speak to us about what you would like your Ducati to do and we can go through the options.
Again, a matter of getting your Ducati running just right. Perfect Cam timing is critical to getting the most out of the Ducati motor, often overlooked in favor of big exhaust kits or high compression pistons. this is an excellent way of getting more from your motor without spending fortunes.
We have many sets of cam timing, each designed for a particular task and model, more top end power, more mid range or more bottom end grunt.
The latest generation of motors (Testastretta superbike, monster and Multistrada) have adjustable cam pulleys fitted as standard so there is little expense, an ideal upgrade to scheduled services.
Previous models can be fitted with lightweight adjustable pulley sets too.
Prices for cam timing: Cam timing is charged at our regular service labour rate (£55.00 per hour) as some models take longer than others to set up. It takes approximately 4-5 hours, less if done during a service.


At DJB "Big enough to know. Small enough to care." are the words we live by. We value our clients and taking care of their unique needs is what drives our business. We have the expertise to provide financial consulting services related to any situation from complex mergers and acquisitions to assisting new entrepreneurs through the many facets of a small business. We do all of this while remaining personally involved with our clients and the communities where we work and live.


Donald Trump's recognition of Israeli sovereignty over the captured Golan Heights delighted Prime Minister Benjamin Netanyahu, but it may have put the kibosh on the U.S. president's own Middle East peace plan.
Trump is counting on Arab allies such as Saudi Arabia and Egypt to give his still-secret plan traction, but they'll be hard-pressed to support him after the Golan move. As Glen Carey reports, both countries are staunchly pro-Palestinian, and Trump's repeated actions in favor of Israel make what he calls the "deal of the century" an even tougher sell.
The decision on the Golan, which Israel captured from Syria in the 1967 Middle East war, is just the latest example of Trump's capricious foreign policy, which has repeatedly puzzled and angered U.S. allies.
In Washington this week, the North Atlantic Treaty Organization will celebrate its 70th anniversary, but the occasion will be a subdued affair as members question the president's commitment to the alliance, which he has suggested the U.S. abandon.
Coming on the heels of Trump's decision to relocate the U.S. Embassy to Jerusalem – which also sparked outrage in the Arab world – his Golan move adds to the potentially destabilizing impact of foreign policy pivots that have characterized the Trump administration and its "America First" stance.
Even staunch ally Saudi Arabia denounced Trump's Golan decision as a violation of international law that would have "negative effects" on peace efforts.
Angry electorates | Voters in Turkey, Ukraine and Slovakia served notice to the establishment this weekend. A recession fueled opposition gains in Turkey, where President Recep Tayyip Erdogan's party lost the capital, Ankara, and was pushed hard in Istanbul. A comedian beat out incumbent Petro Poroshenko in the first round of Ukraine's presidential election, while in Slovakia an anti-corruption campaigner took the presidency with a socially liberal campaign.
Brexit chaos | A long Brexit delay and a U.K. general election are looking more likely – especially if Prime Minister Theresa May and lawmakers can't agree on a way forward this week. May failed again to get her Brexit deal through Parliament on Friday, amid rowdy Westminster street protests by pro-divorce supporters. With just 11 days left until the European Union's deadline for the U.K. to come up with a plan or leave without a deal, members of Parliament today are themselves trying to design a Plan B.
Symbolic sale | Washington may finally sell Taiwan the warplanes it's long wanted as a fortification against China, one of a series of Trump's gestures of support for Taiwan in recent months – and one that would signal a new U.S. willingness to back the island. The possibility comes despite Trump nearing a trade deal with Chinese leader Xi Jinping, and as Xi's special envoy, Liu He, travels to Washington this week.
Power drain | Another day, another round of electricity outages and more protests. That's the bind Venezuelan President Nicolas Maduro is finding himself in as his administration seems unable to restore power. While he's blamed National Assembly leader Juan Guaido and the U.S., late yesterday he announced a 30-day rationing plan to deal with the crisis as the authorities try to fix a poorly maintained national grid in dire need of investment.
New Zealand Prime Minister Jacinda Ardern was welcomed by Chinese Premier Li Keqiang at the Great Hall of the People in Beijing today as the countries signed tax agreements aimed at shoring up relations following a series of incidents that cast doubt on the strength of their economic partnership.
And finally... The world's oldest continuous monarchy of Japan added the newest chapter to its tradition that dates back over 1,300 years of assigning a new name to an imperial era. Isabel Reynolds in Tokyo explains "Reiwa," which was chosen by a panel of experts and the government. The label was inspired by an eighth-century poem about plum blossoms and can mean an "auspicious calm." In newspapers and official documents, Reiwa will mark the year when the current emperor Akihito steps down and his son Naruhito ascends the Chrysanthemum Throne – an event planned for May 1.


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Richard Clayderman Returns To Malaysia for 40th Anniversary Concert
The Prince of Romance; Richard Clayderman returning to Malaysia once again and will be performing live for one-night only in Kuala Lumpur on 16th November (Friday) at the Plenary Hall, Kuala Lumpur Convention Centre. Performing since 1979, he has played over 2,500 concerts in 80 countries, playing in front of Malaysians 17 times.
Celebrating his return to Malaysia and in conjunction with his 40th Anniversary concert, Starline Productions together with Yamaha will be organizing a contest looking for a talented Malaysian musician. The winner will be the opening act for the concert performing in front of 2,000 people and enjoy an exclusive meet and greet session with the Prince of Romance himself.
The contest is open to everyone at the ages of 5 and above and must play a musical instrument. To take part, contestants must submit a public post that consist of a video of themselves performing through YouTube, Facebook or Instagram and include the hashtags #RCOpeningAct2018. Contestants can submit their entries today until 9th November 2018, with the winner announced on 12th November 2018.
Certified by the Guinness Book of Records as one of the most successful pianist in the world, Richard Clayderman is one of the best-selling pianists recording artist and concert performer in the world. He has to date sold over 150 million records and has gone on to achieve 267 Gold and 70 platinum status.
His first single Ballade our Adeline alone blew expectations away when it sold over 22 million copies in 38 countries capturing the attention of the world. Set to perform life for one night only and entertaining the Malaysia with his signature 'New Romantic' style, tickets are now available at www.MyTicket.asia with prices as follows; RM 538 (Platinum), RM 388 (Category 1), RM 288 (Category 2 & 3) and RM 208 (Category 4).
The full seating chart can be viewed in the website (www.myticket.asia). For more information please visit Starline Production on Facebook

redpajama_set_name: RedPajamaCommonCrawl


Insomniac Radio is your home for all genres of current dance music plus superstar DJ sets and festival music. Hear tracks by Jauz, Disclosure, Afrojack, and more!
Welcome to Factory 93, the home of house and techno on Insomniac Radio. Enter a virtual warehouse and listen to the stylings of artists like ANNA, BETON, Green Velvet, Solardo, Nicole Moudaber, Cirez D, Hot Since 82, and more!
Insomniac Radio is the channel that never sleeps! Up All Night is home to the music that gets you moving, regardless of the time of day. Hear artists like Bicep, Fakear, Peggy Gou, and more that are guaranteed to keep you up into the early hours of the morning!
Oakenfold is the DJ, remixer, and producer who broke house music in Britain in the late '80s after hosting a series of club nights. He pushed the new dance crossover onto the charts with hit productions by Happy Mondays and others, and formed the successful dance label Perfecto Records.
MAKJ brings you every week his favourite new tunes plus exclusive edits and previews of his upcoming releases.
Welcome to the Basscon Zone, Insomniac's one stop shop for all things Hard Dance! Hear tunes from artists like Gammer, Audiofreq, Kayzo, Coone, Darren Styles, and more!
It's Thursday, meaning we throw it back to the days of old on Insomniac Radio! Hear classic cuts and certified throwbacks from artists like Daft Punk, Porter Robinson, Zedd, Calyx & Teebee, and more, every Thursday on Insomniac Radio!
Welcome to the Dreamstate Zone, Insomniac Radio's ground zero for #TranceFam. Hear the biggest and best names in the Trance world, including Armin van Buuren, Estiva, ALPHA 9, maRLo, and more!


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NewYork-Presbyterian Ranks Among the Top 10 in the Nation According to U.S. News Media Group's 2010-11 Best Hospitals
Media Requests: [email protected]
#6 in the Nation, #1 Among Ranked Hospitals in New York Area
NewYork-Presbyterian Hospital, one of the country's largest and most comprehensive academic medical centers, is ranked #6 in the nation and #1 among ranked hospitals in the New York area, according to U.S. News Media Group's 2010-11 Best Hospitals. This marks the 10th consecutive year that NewYork-Presbyterian has been listed on U.S. News' prestigious "Honor Roll."
Best Hospitals 2010-11 ranks 152 medical centers nationwide in 16 specialties. The Honor Roll highlights 14 of these medical centers that are ranked at or near the top in at least six specialties based on reputation, death rates, procedure volume and the balance of nurses and patients.
NewYork-Presbyterian ranked in the following specialties: Cancer; Diabetes & Endocrinology; Ear, Nose & Throat; Gastroenterology; Geriatrics; Gynecology; Heart & Heart Surgery; Kidney Disorders; Neurology & Neurosurgery; Ophthalmology; Orthopedics; Pulmonology; Psychiatry; Rheumatology; and Urology.
Dr. Herbert Pardes, president and CEO of NewYork-Presbyterian Hospital, lists the top six reasons he believes that NewYork-Presbyterian is one of the nation's top six hospitals:
Top Six Reasons We're in the Top Six
We put patients first. From our world-class clinicians and cutting-edge treatment options to our culture of empathy and emphasis on quality and safety, NewYork-Presbyterian is committed to putting patients and their families first.
Our physicians are re-inventing medical practice. As the nation's only hospital affiliated with two leading Ivy League medical schools, NewYork-Presbyterian performed the world's first pediatric multivisceral surgery involving the removal and partial reimplantation of six abdominal organs. We also developed a breakthrough treatment that delivers chemotherapy directly into a malignant brain tumor.
We have some of the world's most advanced facilities and technologies. NewYork-Presbyterian opened the Ronald O. Perelman Heart Institute and the Vivian and Seymour Milstein Family Heart Center, with advanced operating rooms that allow clinicians to perform "hybrid" heart procedures combining surgery and interventional cardiology. We are the nation's first hospital to use the state-of-the-art Artis Zeego imaging system for advancing treatments for cardiovascular, neurologic and gastrointestinal conditions.
Our diverse team of specialists work together to provide world-class care. From clinicians, including nurse-Ph.D.s, Pharm.D.s and social workers, to translation and patient services professionals, NewYork-Presbyterian's unique team is committed to our patients and their families.
Patient safety and quality of care are foremost in our minds. The Hospital is recognized for its pioneering initiatives, including its award-winning Patient Safety Fridays program to ensure the highest quality and safety for the patient.
Financial stability provides a vision for the future. In the challenging and changing world of health care, NewYork-Presbyterian has a strong financial foundation that allows us to invest in new ways to serve patients and families.
"We are very pleased to again be counted among the outstanding institutions on the U.S. News Media Group's 2010-11 Best Hospitals 'Honor Roll.' With expertise across the full spectrum of medical specialties, our health care professionals are dedicated to providing high-quality, effective and empathetic care to patients and their families," says Dr. Herbert Pardes, president and CEO of NewYork-Presbyterian Hospital. "Through our affiliation with two of the nation's leading Ivy League medical schools — Weill Cornell Medical College and Columbia University College of Physicians and Surgeons — we are continually developing cutting-edge medical technologies and potentially life-saving therapies."
Dr. Steven J. Corwin, executive vice president and COO, adds, "This distinction is a tribute to the continued excellence in the care and service we deliver to our patients every single day. We applaud and thank our amazing staff for their commitment to teamwork, and to putting patients and families first."
U.S. News Media Group's 2010-11 Best Hospitals is available online at www.usnews.com/besthospitals and featured in the August print issue of U.S.News & World Report, available on newsstands July 27.
NewYork-Presbyterian Hospital
NewYork-Presbyterian Hospital, based in New York City, is the nation's largest not-for-profit, non-sectarian hospital, with 2,353 beds. The Hospital has nearly 2 million inpatient and outpatient visits in a year, including more than 220,000 visits to its emergency departments — more than any other area hospital. NewYork-Presbyterian provides state-of-the-art inpatient, ambulatory and preventive care in all areas of medicine at five major centers: NewYork-Presbyterian Hospital/Weill Cornell Medical Center, NewYork-Presbyterian Hospital/Columbia University Medical Center, NewYork-Presbyterian/Morgan Stanley Children's Hospital, NewYork-Presbyterian/The Allen Hospital and NewYork-Presbyterian Hospital/Westchester Division. One of the most comprehensive health care institutions in the world, the Hospital is committed to excellence in patient care, research, education and community service. NewYork-Presbyterian is the #1 hospital in the New York metropolitan area and is consistently ranked among the best academic medical institutions in the nation, according to U.S. News Media Group's Best Hospitals. The Hospital has academic affiliations with two of the nation's leading medical colleges: Weill Cornell Medical College and Columbia University College of Physicians and Surgeons. For more information, visit www.nyp.org.
Bryan Dotson 212-305-5587 [email protected]


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Andrew Zimmern: Filipino food is the 'next big thing'
Vegetarian mechado from Filipino restaurant Milkfish. Food personality Andrew Zimmern thinks Filipino fare will be trendy in the next two years.Today
June 12, 2012, 2:52 PM UTC
By Veronica Meewes
Few types of cuisine are hard to find anymore. Mexican and Tex-Mex are readily available, Indian buffets are standard fare, sushi just seems to keep growing in popularity, and Chinese, Thai, and Vietnamese are easily accessible. And food experts claim that Peruvian and Korean cuisine are the fare du jour.
So what's the next fad food? Andrew Zimmern, host of "Bizarre Foods" on the Travel Channel, has a theory: "I predict, two years from now, Filipino food will be what we will have been talking about for six months … I think that's going to be the next big thing," he told TODAY.com.
"I want to go on record — this is not something that's hot now somewhere and will get hot everywhere else," he said. "It's just starting. I think it's going to take another year and a half to get up to critical mass, but everybody loves Chinese food, Thai food, Japanese food, and it's all been exploited. The Filipinos combined the best of all of that with Spanish technique. The Spanish were a colonial power there for 500 years, and they left behind adobo and cooking in vinegar — techniques that, applied to those tropical Asian ingredients, are miraculous."
Andrew Zimmern, host of "Bizarre Foods."Today
Filipino cuisine has a variety of foreign influences. The impact of China is evidenced in their use of noodles (pancit), fried rice (sinangang) and spring rolls (lumpia), as well as the soy sauce and fish sauce found in many other dishes. Indonesian and Malaysian influence can be seen in the use of coconut milk and rice, particularly in desserts, as well as the use of chilis (though most Filipino food isn't very spice-heavy).
The Spanish were responsible for bringing bay leaves, tomatoes and garlic, as well as the technique of sautéing with olive oil. Longanisa is a sweet pork sausage (similar to the Spanish longaniza) which can be found in Filipino dishes. Other Spanish dishes often found on Filipino menus are flan, paella, and adobo, a method of braising meat in garlic, vinegar, peppercorns, and soy sauce.
Filipino food isn't on the radar of mainstream America, but Zimmern thinks that's going to change. "San Diego is now a big enough ethnic population of Filipinos that chefs are going there and seeing stuff. I think it'll creep up into Los Angeles and from there go around the rest of the country," he foresees.
Cristina Quackenbush is the head chef and proprietor of Milkfish, a popular Filipino pop-up restaurant found inside Marie's Bar, a Marigny neighborhood favorite in New Orleans. "I have grown up cooking Filipino food from my mother and learning homemade-from-scratch fare from my grandmother. She had 20 acres of land in which she had planted every fruit and vegetable you can think of!" she told TODAY.com.
Before relocating to New Orleans 12 years ago, Quackenbush lived briefly in San Diego, where she encountered most of the Filipino restaurants she's seen in the States. "I also found a little one in Tennessee once," she recalls. "I have not encountered any other than those! This is why I want to bring it to the forefront. It is such a wonderful cuisine that must be shared."
Quackenbush's menu at Milkfish is split into three categories: appetizers, small dishes and dinners. She also offers vegetarian twists on classic Filipino dishes, such as vegetarian mechado, which traditionally appears as a marinated beef dish. She describes her cuisine as the soul food of Southeast Asia. "I definitely think (Filipino food) is gaining popularity," she affirms. "I have never encountered anyone that I have fed that did not like it!"
Spam started being used in Filipino cooking during World War II and remains a popular ingredient. The following recipe is a typical breakfast dish, served with an over-easy egg. Try it and you might just be pleasantly surprised (I was!).
Spam fried rice from Milkfish in New Orleans.Veronica Meewes / Today
Spam fried rice
1 can of Spam, diced
4 cup of cooked, then cooled, rice
Diced green onions
¼ cup soy sauce (La Choy or Silver Swan is preferred)
Dice Spam into 1/4-inch cubes and sauté in canola oil until crisp on outside; add onions and garlic and cook until onions are translucent. Add cooled rice and 1/8 cup of oil. Mix rice thoroughly with Spam, onions and garlic. Add 1/8 cup of soy sauce and be sure sure to distribute evenly over mixture. Flatten out rice and let brown slightly. Stir again and flatten out and fry slightly. Make a hole in middle of rice and break egg into it. Let egg cook as you gradually incorporate it into the rice. Add rest of soy sauce and oil and stir fry until egg is cooked completely. Season with salt and pepper. Add diced tomatoes over top and fry an over-easy egg for the top to finish!
Veronica Meewes is an Austin-based freelance writer who will travel for food but always comes back for breakfast tacos. Follow her on Twitter @wellfedlife and visit her blog.
More from TODAY Food
Video: Andrew Zimmern's fave sandwiches from around the world
How to order from the 'secret' menu at Chinese restaurants
Ceviche is the new sushi: Peru's cuisine goes global


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Kevin Trent Boswell
The Music, Magick, Poetry and Madness of Kevin Trent Boswell
The Music Of Kevin Trent Boswell
Magus Music on Facebook
Magus & The Plastic Infinity
Conjure Work
Conjure Work on Facebook
Conjure Sound
antiverse – more poetry of Kevin Trent Boswell
Uncategorized Posts
Tag: thrash metal
Punk Rock, Climate Extinction
Some Punk Rock for the Existential Climate Crisis
This song is written in the vein of old school punk rock and thrash metal, bands like the Sex Pistols, Misfits, Motorhead, etc. It's deliberately noisy and brash. I didn't try to use proper singing technique on the track; it's not supposed to be pretty.
"I Wasn't Using It" is brand new; it's not on the new album that is becoming available on June 8th. However, it's available for download to patrons on the Patreon page.
It helps a lot when you like 👍 and subscribe ✅ on YouTube. Ring that stupid little bell 🔔 and click "all," so you get notifications. Share the video if you like it. If you don't, it's okay; nobody will be around to hear it, soon enough 🤷♂
Don't feel bad the stores are empty
Or that food won't grow anymore
It's not your fault; you had no other planet
On which you could wage your war
Don't feel bad that you broke it;
No, really, it's OK
I really wasn't using it
It was an old world anyway
I don't need no oceans
Arctic ice or stupid bees
It was really nice of you
To cut down all those ugly trees
I don't mind the heat
Going up past 105°
Anything to help with industry
I don't really need to stay alive
I really wasn't using
The world is on fire
Disaster at the door
But I'm running out of plastic
Can you please make me some more?
Seriously, man, it's OK
It's not like I was using the thing
©2022 Kevin Trent Boswell
New Album on June 8th
Available June 8th on Amazon, iTunes, Spotify, and many other major streaming services
Get exclusive bonus content at the Patreon page.
Magus72 on Patreon – the music, poetry, and madness of Kevin Trent Boswell
Author Kevin Trent "Magus" BoswellPosted on June 1, 2022 Categories MusicTags angry music, climate change, climate crisis, climate emergency, dark, death, existential, extinction, global warming, guitar, independent musicians, music video, new music, original music, punk rock, thrash metal, underground artists, underground bandsLeave a comment on Punk Rock, Climate Extinction
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You can hear more of the music of Magus (Kevin Trent Boswell) at: CDBaby at The Plastic Infinity, Conjure Sound. For the esoterically minded, see Conjure Work.
The Flagship album is available for streaming on most major music services, such as iTunes, Apple Music, Spotify, Amazon, SoundCloud, iHeart Radio, YouTube Music, Deezer,
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Kevin Trent Boswell Create a free website or blog at WordPress.com.


The sports industry has an expected value of over $73 billion by 2019, and a master's in sports management online will help you succed in this soaring field. With positions in advertising, talent representation, and front offices, opportunities in sports are nearly as vast as the industry itself. As the industry grows, so too does the need for qualified, innovative people. With a degree in sports management, sports and business intersect to turn one's passion for the game into a career. According to the North American Society for Sport Management, 243 postsecondary institutions offer undergraduate or graduate programs in the field. Additionally, the U.S. Bureau of Labor Statistics (BLS) reports that spectator sports is the third biggest employer of talent agents in the nation, with individuals earning an average of $104,530 annually. What's more, the BLS projects 10% growth over the next decade (2016-2026) for marketing managers, public relations specialists, and athletic directors. Now is a perfect time to consider joining the lucrative and exciting field of sports management.
University of Florida Distance Learning ranks #1 on The 25 Best Online Master's in Sports Management Degree Programs!
Located in Gainesville, the University of Florida welcomes students of all backgrounds with over 170 distance education degree programs, including an online master of science in sports management. This comprehensive graduate track combines experiential learning based on research and practicum opportunities with the flexibility of remote, asynchronous coursework delivered through the Canvas learning platform. Students develop the administrative, marketing, and physiological skills needed to help keep athletes healthy and contribute to an organization's growth.
The curriculum requires students to complete 36 credits; most distance learners graduate in two years. Required coursework includes sports marketing, sociology, and leadership, as well as a research core in human performance methods and evaluation. After completing 27 credits, degree candidates must also complete a capstone experience that consists of writing self-reflective essays. This degree track stands out from many other online master's programs in sports management in that it offers several specialization options, including athlete development, sport law, or high performance training. These concentrations provide students with more control over their degree plan, preparing them for specific careers. UF does not require students to complete a practicum, but the school does provide placement assistance for students who want to pursue opportunities with any of the school's institutional partners or organizations in their communities. The program admits students four times each year.
The University of Texas at Austin ranks #2 on The 25 Best Online Master's in Sports Management Degree Programs!
To earn this master's in sports management, online students complete 30 credits of coursework. Required topics include organizational behavior, legal issues in sports, and facility and event management. Faculty deliver online courses in accelerated, six-week sessions, which students can take year-round. There is no thesis requirement. However, as one of the top public research schools in the country, UT Austin provides ample assistance to students who elect to pursue self-directed academic and professional projects. This assistance comes in the form of faculty support, funding resources, and access to state-of-the-art software and technologies.
Georgetown University ranks #3 on The 25 Best Online Master's in Sports Management Degree Programs!
Georgetown University is located in Washington D.C. and regularly earns recognition for its academic rigor and the accessibility of its distance education programs, which include an online master's in sports industry management. Through a curriculum that emphasizes global perspectives in the sports field, students develop the theoretical knowledge and practical skills needed to pursue careers in collegiate athletics, facility operations, professional franchises, and even in the nonprofit sector. Georgetown does not require any on-campus residencies, letting students earn their degree entirely online in fewer than two years.
This online master's in sports management program totals 33 credits. Students take fundamental courses in business and marketing, while also pursuing advanced topics like applied ethics, social responsibility, and globalization in sports. In addition to individualized attention in the classroom, students enjoy professional guidance, mentoring, and internship support. The curriculum culminates with a capstone project, which students complete with the help of a faculty advisor and a professional sponsor. Degree candidates present this work to a panel of industry experts. All online classes consist of 7.5-week sessions, except for the 15-week capstone experience.
Northeastern University ranks #4 on The 25 Best Online Master's in Sports Management Degree Programs!
Founded in 1898, Northeastern University is a research and academic institution that offers distance education programs through its Global Network, including an online master's in sports leadership. Grounded in experiential learning and ethical leadership, this program prepares students for careers in sports-related marketing, communication, and management. Prospective students should note that this is a hybrid track. While most courses can be accessed online asynchronously through Blackboard, some required classes involve in-person attendance at the Boston campus over summer sessions. Part-time students generally earn their master's in two years, while those enrolled full-time can finish in just one.
The degree plan consists of 45 quarter credits in classes such as sports law, media relations, and organizational management. Students take additional coursework to pursue either a professional sports or collegiate athletics administration concentration. Learners also complete six elective credits. The Northeastern program culminates in either an internship or capstone experience. Both options provide students with industry experience through hands-on skill development and networking opportunities, which they subsequently document and analyze in a web portfolio. The capstone experience requires intensive research and a final paper; this choice may be more suitable for students already working in the sports industry.
Washington State University ranks #5 on The 25 Best Online Master's in Sports Management Degree Programs!
Washington State University was established in 1890 as the state's original land grant college. Today, WSU offers over 79 graduate degree programs, including distance education options like its online master of arts in education - sport management. Through a curriculum that blends theory, research, and practical experience, students develop the business and instructional skills needed to join the ever-growing sports industry. Graduates go on to work in positions as community program directors, franchise marketing specialists, and corporate managers.
Degree candidates choose from either a thesis or non-thesis curriculum, both of which feature 34 credits. Coursework covers sport media and communication, administrative concepts, values and ethics for educational leaders, and at least nine elective credits. All students also complete core research requirements that teach them the fundamentals of conducting qualitative inquiry and analyzing statistics. Non-thesis students finish their studies with a comprehensive examination, while their counterparts carry out original research that tackles a challenge in the field. Thesis students also defend their ideas in a culminating event open to the public. WSU delivers asynchronous online courses through Blackboard.
Northwestern University ranks #6 on The 25 Best Online Master's in Sports Management Degree Programs!
Based in Evanston, Illinois, Northwestern University offers eight graduate degree programs to distance learners. Among these options is an online master of arts in sports administration, which strongly emphasizes the business aspects of the industry, from creating strategic and financial plans to assessing performance using statistical information. Students may earn their degree entirely online or they can opt for a hybrid experience that integrates flexible on-campus classes. The university provides internship opportunities across the U.S. as well as resources through its career advancement office. Northwestern graduates go on to occupy positions that include corporate consultants, sports agents, and team executives.
This online master's in sports management program consists of 11 courses, covering topics like legal and ethical issues, labor relations, and intercollegiate sports management. The degree plan also contains a perfunctory leadership class that teaches students communicative strategies and their applications in a professional setting. Distance learners enjoy exclusive access to a sports analytics specialization that covers the programming and mathematical modeling skills needed to evaluate individual players, entire teams, and related administrative initiatives. Students round out their education by completing either a thesis project or a capstone class. Faculty deliver online, asynchronous courses through the Canvas learning management system.
Texas A&M University ranks #7 on The 25 Best Online Master's in Sports Management Degree Programs!
Established in 1876 as the state's first public institution of higher learning, Texas A&M University enrolls approximately 62,500 students annually. The school offers over 170 graduate degree programs, including distance education pathways like its online master of science in sport management. With three entry dates in April, October, and February, as well as no thesis requirement, this program provides students with a flexible, comprehensive education that prepares them to work as administrators and leaders.
To earn their master's in sports management, online students complete 36 credits, including a research core and advanced classes like personnel management, revenue generation, and diversity in sports organizations. The curriculum also allows students to take electives to explore additional topics in the field or complete courses in related subjects such as marketing, international business, or education. Degree candidates finish their studies with an internship where they develop practical skills and reflect on their professional experience by writing journal entries and a final paper. Distance learners can complete this internship with one of the university's business partners or with an organization in their area. Those already working in the sports industry may opt to fulfill this requirement through their employer, although not in the same capacity as their current job.
Texas A&M University - College Station is regionally accredited by the Southern Association of Colleges and Schools Commision on Colleges (SACSCOC).
University of Massachusetts–Amherst ranks #8 on The 25 Best Online Master's in Sports Management Degree Programs!
The University of Massachusetts is located in Amherst and enrolls nearly 7,000 graduate students in 77 degree programs. These include distance education opportunities like the school's online master of business administration with a focus in sport management. With a curriculum based in the field's best practices and current trends, this program prepares learners for careers in all aspects of the sports industry, from facilities operations and program management to consumer sales and market data analytics. Outside the classroom, students can access career advice and professional development opportunities through the university's large network of alumni and corporate partnerships.
The University of Massachusetts - Amherst is regionally accredited by the New England Association of Schools and Colleges (NEASC). The Isenberg School of Management receives specialized accreditation from the Association to Advance Collegiate Schools of Business (AACSB).
Virginia Commonwealth University ranks #9 on The 25 Best Online Master's in Sports Management Degree Programs!
Liberty University Online ranks #10 on The 25 Best Online Master's in Sports Management Degree Programs!
Based in Lynchburg, Virginia, Liberty University boasts a robust distance education platform with over 289 programs, including an online master of science in sport management. With fully remote coursework and nine specialization choices (including thesis and non-thesis options), students can work and pursue their degree simultaneously. The curriculum combines industry best practices with business and marketing skills, enabling graduates to pursue work as event planners, athletic marketers, facility managers, and sports broadcasters.
The degree plan totals 36 credits, which most students complete in fewer than two years. Required classes cover topics in social issues, marketing and public relations, and financial management for sport industry professionals. Students take advanced courses in their specialization, choosing from tourism, sports administration, outdoor adventure sport, coaching and athletic administration, or a generalist track. Learners do not need to complete a practicum, but they can use departmental resources to access internship opportunities. The program culminates with a comprehensive exam. Faculty deliver asynchronous, online classes through Blackboard. Students can also enroll in accelerated, eight-week courses to hasten graduation.
Troy University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Founded in 1887 as a normal school, Troy University is now a comprehensive institution that offers graduate students 40 distance education programs, including an online master of science in sport and fitness management with a sport management concentration. The learning structure of this program combines seminars, lectures, and practical experiences to propel students into leadership roles. Graduates can go on to work for recreation departments, Division I athletic programs, and professional sport organizations.
The degree plan totals 36 credits and students may choose either a thesis or non-thesis track. In addition to required courses in statistical analysis and research methods, degree candidates pursue advanced topics like consumer behavior, revenue generation, and sociological aspects in sports management. With the aid of a faculty advisor, thesis students conduct research and propose solutions to challenges in the industry. Alternatively, non-thesis students take more elective classes. Distance learners access asynchronous, 16-week courses through the Canvas platform and gain access to resources through a digital student union where they can connect with other learners from across the globe.
Drexel University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Located in Philadelphia, Pennsylvania, Drexel University stands as one of the largest private institutions of higher education in the U.S., hosting over 200 academic programs. These offerings include distance education opportunities, such as the school's online master of science in sport management. Designed for industry professionals and newcomers alike, this program trains students to plan, implement, and assess sports programs and carry out related administrative tasks. Offered as a part-time study plan, most degree candidates finish in fewer than two years and move on to careers in areas like corporate sponsorship, player personnel and draft analysis, and visual sports journalism.
The 45-quarter-credit curriculum includes required courses related to negotiations for leaders, sports economics, and managing the total enterprise. The degree plan also consists of 12 elective credits, which students utilize to craft a focus area related to their career interests. Additionally, to earn their sports management master's degree, online learners complete an experiential component, choosing from a senior seminar, internship, independent study, or research project/thesis. Although all aspects of the program can be completed remotely, students have the option of attending a four-day summer campus residency where they engage with guest speakers and network with peers and industry professionals.
American University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
American University delivers distance education coursework through Blackboard, offering six graduate certificates and degrees as well as several individual and non-degree classes. Included among these offerings is an online master of science in sports analytics and management, which focuses on the data-driven nature of organizational planning and leadership. With a curriculum grounded in emerging technologies and intercultural communications, students develop the skills needed to pursue the next generation of sport industry careers, from performance analysis and interactive spectatorship to e-sports gaming and virtual reality.
Totalling 30 credits, AU's online master's in sports management program contains required courses in data-driven decision making, project cost-benefit analysis, and client communications and professional consulting. Students round out their education with an elective class and a capstone experience. The latter tests students' strategic thinking and understanding of applied research and assessment, culminating in a written business plan and presentation. To facilitate professional development, AU also offers distance learners career-relevant webinars and practicum opportunities. Learners may also seek out guidance and resources through the school's adult career coaching center, which provides job search, resume review, and networking assistance.
Temple University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Temple University stands as a leader in the sports administration field, offering related degree programs since 1908. Today, distance learners can access graduate-level academics through the school's online executive master of science in sport business program. Tailored for professionals with experience in the industry, this degree track teaches students the advanced strategic management and marketing skills needed to become leaders in their organizations. Part-time students can earn their master's in two years, while full-time participants can graduate in as few as 13 months. Temple graduates go on to work at sporting good companies, sport governance organizations, and athletic representation agencies.
The 30-credit degree plan covers financial management, global business strategies, and understanding sport consumers. The program does not contain any internship requirements or campus residencies, enabling students to complete all work remotely. Faculty deliver online courses through the WebEx learning platform. While most classes feature asynchronous instruction, distance learners also participate in live weekly sessions that allow them to discuss course materials and build relationships with their teachers and peers. During these live sessions students engage in small group projects where they solve practical challenges faced by sport managers.
Western Carolina University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Based in Cullowhee, North Carolina, Western Carolina University enrolls over 11,000 students annually, many of whom earn their degrees remotely through one of the school's 16 distance education programs. These online offerings include a comprehensive online master of science in sport management that trains students to lead teams, administer programs, and execute and assess strategic projects. This program emphasizes flexibility with no prerequisite requirements and fully asynchronous online coursework. Full-time students can earn their degrees in one year. Part-time learners typically finish in two years, but there is no required minimum course load as long as students complete the program within six years.
University of Southern Mississippi ranks among The 25 Best Online Master's in Sports Management Degree Programs!
With students from all 50 states and 72 countries around the world, the University of Southern Mississippi's academic programming has a global reach. Distance learners can select from 30 graduate programs, including an online master of science in sport management. This program develops critical thinking and problem solving skills, exposing students to research and best practices in the sport industry. Graduates can work as talent scouts, operations managers, and franchise marketing specialists. Distance learners may earn their master's entirely online through asynchronous courses delivered by Canvas or through a hybrid experience that mixes in traditional classes on the Hattiesburg campus. Most students finish the program in fewer than two years.
Georgia Southern University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Old Dominion University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Saint Leo University Online ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Adelphi University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
The 36-credit degree plan consists of foundation, core, and specialization classes. Covered topics include research methods, emergent technologies, and sport management personnel and program evaluation. Students pick from either a collegiate administration or sport marketing specialization. The former focuses on facilities management and media relations specific to athletic departments, while the latter emphasizes product branding, corporate sponsorship, and promotional campaigns. In lieu of a residency, the program culminates with a capstone course centered on in-depth research of contemporary challenges in the sport industry. The university provides online, personalized academic advising and online tutoring services. Program participants also gain access to an alumni network that helps them find internships and pursue industry certifications.
Gonzaga University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Gonzaga's online master's in sports management program contains no residency requirements. Students take all of their eight-week courses remotely in an asynchronous format facilitated by the Canvas platform. Individuals typically earn their degree in two years. Totalling 31 credits, the curriculum includes sport and athletic promotion, ethical and legal aspects, and facility management. Distance learners also fulfil an internship experience where they engage in at least 175 hours of hands-on work experience and complete accompanying writing assignments. Internships can be carried out at an approved organization in their community or with one of Gonzaga's partners, like the Seattle Mariners. Degree candidates round out their education with a capstone project where they create a final portfolio of work, which is presented in front of peers, faculty, and industry professionals.
DePaul University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
DePaul University offers four graduate-level distance education programs, including an online master of science in sport, fitness, and recreation leadership. Designed for sport industry professionals seeking to advance in their careers, this program emphasizes data-driven decision making, ethical management techniques, and the holistic care of individuals in recreation and fitness settings. From a main campus based in Chicago, Illinois, DePaul provides distance learners with flexible, asynchronous coursework through the Desire2Learn platform. Students also undergo hands-on skill development by completing two residencies. Graduates stand prepared for leadership positions in corporate fitness centers, university athletic programs, and professional sport franchises.
Mississippi College ranks among The 25 Best Online Master's in Sports Management Degree Programs!
Located in the city of Clinton, Mississippi College enrolls over 5,000 students every year, making it the state's largest private institution of higher education. MC offers 19 graduate programs for distance learners, including an online master of science in athletic administration. Through a core curriculum and specialization options that focus on strategic and financial management, students develop the skills needed to lead successful sport programs at secondary schools and college departments. Faculty deliver online courses asynchronously through the Moodle platform. Most students earn their degrees in fewer than two years.
University of Northern Colorado ranks among The 25 Best Online Master's in Sports Management Degree Programs!
West Virginia University ranks among The 25 Best Online Master's in Sports Management Degree Programs!
An online master's in sports management combines theoretical instruction with practical experience, preparing graduates for management and administrative roles at all levels. With a master's in sports management online degree, students learn effective management techniques such as understanding team dynamics, motivating people, and making sound decisions. Core courses of a master's in sports management cover the same ground as an advanced management degree, but from the perspective of the sports industry. A typical curriculum includes classes in marketing, economics, accounting, and communication, with additional courses in psychology, social issues, and sports ethics.
Students in an online master's in sports management program tailor their curriculum by focusing on areas such as leadership, global business, collegiate sports, or analytics. The latter has become increasingly important as professionals use data to inform on- and off-field moves. A leadership focus, meanwhile, prepares graduates to lead from the front office as a general manager, athletic director, or marketing director. An internship is also an important component of an online master's in sports management program, as it gives students the chance to gain important industry experience and explore a potential career track.


text:
\section{Introduction}
\label{section:1}
The three-player-game (3PG) is an interesting mathematical quiz. A 3PG involves three players and consists of a series of rounds, each zero-win round involves two out of the three players, at the end of the round the loser quits and the third player enters the ring and another round starts. The game terminates if all six win-lose relationships appear. During each round, two players win with equal probability.

For example, let Alice, Bob and Carole be three players. Alice and Bob play the first round, then Alice loses. Therefore the second round involves Bob and Carole. If Carole loses then Alice has to return to the ring. The game proceeds until every player has beaten the other two players for at least once.

One should take the win-lose situation of each round as the underlying infinite probability space and the number of rounds as a random variable. The target is to compute the expectation of this random variable.

We solve this question from scratch and give some additional analysis on generalization.

\section{Solving 3PG by Reduction}
\label{section:2}
\subsection{Formulation and Reduction Rules}
It is straightforward to observe that the sufficient statistics of any round in the game is the occurance of all possible win-lose relationships sofar and the current players in the ring. That is to say, at each stage of the game, the current situation is comprehensively described by six bits (indicate whether or not the six win-lose relationship appear) and one ternary bit (indicate the current players in the ring). We denote all possible states in the following form:
$$\mathcal{S}=\left\{\left(b_{1}b_{2}b_{3}b_{4}b_{5}b_{6}|t_{1}t_{2}\right)\right\},$$
where each $b_{i}\in\left\{0,1\right\}$ denotes whether a win-lose relationship appears or not. Whereas $t_{1}$ and $t_{2}$ in $\left\{1,2,3 \right\}$ are the indices for the players in battle. From $i=1,\cdots,6$, $b_{i}$ denotes the win-lose relationship of:
$$\text{Player 1 beats Player 2},$$
$$\text{Player 2 beats Player 1},$$
$$\text{Player 1 beats Player 3},$$
$$\text{Player 3 beats Player 1},$$
$$\text{Player 2 beats Player 3},$$
$$\text{Player 3 beats Player 2}.$$
For each $S\in\mathcal{S}$, let $f(S)$ be the expectation of the number of rounds of a modified 3PG begins with $S$. Now we are asked to compute, w.l.o.g.,:
$$f(000000|\left\{1,2\right\}).$$

It is natural to use conditional expectation to introduce the reduction rule, we begin with the definition:
$$\mathbb{E}[X]=\mathcal{P}(A)\mathbb{E}[X|A]+\mathcal{P}(\overline{A})\mathbb{E}[X|\overline{A}],$$
where $X$ is any r.v. and $A$ is an event. Let $X$ be \emph{the number of rounds of a 3PG begins with $S$} and $A$ be the event that $t_{1}$ wins over $t_{2}$ w.r.t. $S$, we have:
$$\mathbb{E}[X]=f(S),$$
$$\mathcal{P}(A)=\mathcal{P}(\overline{A})=\frac{1}{2},$$
$$\mathbb{E}[X|A]=1+f(S_{1}),$$
$$\mathbb{E}[X|\overline{A}]=1+f(S_{2}),$$
where $S$ reduces to state $S_{1}(S_{2})$ if $A$ does (not) happen. This expansion yields:
\begin{equation}
\label{equation:1}
f(S)=1+\frac{1}{2}\cdot f(S_{1})+\frac{1}{2}\cdot f(S_{2}),
\end{equation}
where $S_{1}$ and $S_{2}$ are two results from $S$ given $t_{1}$ or $t_{2}$ wins. \eqref{equation:1} would become the basis of reduction which finally solves the 3PG. We are going to see a dozen of examples in the coming section.

The reduction terminates at the basic states
$$f(111111|\left\{t_{1},t_{2}\right\})=0,$$
where $t_{1},t_{2}$ can be any players.

We define the \emph{order} of a state by the number of 0s in its binary parts. For example, $(000000|\left\{1,2\right\})$ is a state of order six.

The number of states $|S|$ is 192, however, some states is not going to appear with the root state $(000000|\left\{1,2\right\})$. Moreover, one is encouraged to evoke symmetry to further reduce computation. We define two states $S_{A}$ and $S_{B}$ to be symmetric if there exist a permutation $\pi$ on $\left\{1,2,3\right\}$ such that
$$\pi(S_{A})=S_{B},$$
where applying $\pi$ on a state $S\in\mathcal{S}$ changes both the order of binary indicators and the name of players in the ring. For $b$ indicates the relationship Player 1 beats Player 2 in $S_{A}$, its values also indicates whether the relationship Player $\pi (1)$ beats Player $\pi (2)$ in $S_{B}$. That is to say, $\pi$ on $\left\{1,2,3\right\}$ introduces a permutation on $\left\{1,2,3,4,5,6 \right\}$, for example, $\pi =(1,2)$ introduces (using group algebra notation):
$$(1,2)(3,5)(4,6)$$
While $\pi=(2,3)(1,2)$ introduces:
$$(1,4,5)(2,3,6)$$
on the first six binary bits on $\mathcal{S}$. Finally,
$$t_{B,i}=\pi (t_{A,i}),i=1,2.$$

For example, let
$$S_{A}=(101000|\left\{1,3 \right\})$$
$$S_{B}=(101000|\left\{1,2 \right\})$$
then $S_{A}$ and $S_{B}$ are symmetric by adopting
$$\pi=(2,3).$$
Naturally, symmetry is an equivalent relationship, and for symmetric states $S_{A}$ and $S_{B}$ we have:
$$f(S_{A})=f(S_{B}),$$
since their difference is only a matter of naming. This observation helps to reduce the number of states significantly. However, there is hardly any method to examine whether two states are symmetric other than checking all possible permutations.

\subsection{Preparations and Preprocessing}
Having obtained \eqref{equation:1}, one might eagerly argue that a simple recursion program would trivially solve the task:

Define:
$$\text{Compute}(S)=1+\frac{\text{Compute}(S_{1})}{2}+\frac{\text{Compute}(S_{1})}{2},$$
$$\text{Compute}(111111|\left\{t_{1},t_{2} \right\})=0.$$
Return $\text{Compute}(000000|\left\{1,2\right\})$.

With $\text{Compute}(\cdot)$ as an algorithmatic realization of $f(\cdot)$. Computing $\text{Compute}(000000|\left\{1,2\right\})$ is then done automatically by spanning a recursion tree (let $S_{1},S_{2}$ be two children of the node represents $S$), during which dynamic programming might help to reduce computation time \cite{cormen2009introduction}. However, this method is not determined to success since \eqref{equation:1} does not ensured that a state $S$ itself does not appear in the computing tree spanned by with $S$ as the root, which fact deadlocks this paradigm. At certain states, it is necessary to use the linear relationship between their expectations to solve $f(\cdot)$ and a naive recursion is far from enough.

This fact, together with the last observation from the previous section, indicates that instead of \emph{passively} spanning a computing tree, we should better \emph{aggressively} compute the leave states (those states with few 0s in their binary part) at first.

Before actually conducting reduction from $(000000|\left\{1,2\right\})$, we conduct preprocessing by computing $f(S)$ for some elementary states (states with small orders) beforehand. These computations are collected into a series of gradual propositions.

\textbf{Proposition A:}
$$f(111110|\left\{2,3 \right\})=4,$$
$$f(111110|\left\{1,2 \right\})=6,$$
$$f(111110|\left\{1,3 \right\})=6,$$

\textbf{Proof:} Let $x,y,z$ denote these three values respectively, we have according to \eqref{equation:1}:
$$x=1+\frac{y}{2},$$
$$y=1+\frac{z}{2}+\frac{x}{2},$$
$$z=1+\frac{y}{2}+\frac{x}{2}.$$
That is to say
$$
\begin{pmatrix}
x \\
y\\
z
\end{pmatrix}=
\begin{pmatrix}
0 & \frac{1}{2} & 0\\
\frac{1}{2} & 0 & \frac{1}{2}\\
\frac{1}{2} & \frac{1}{2} & 0\\
\end{pmatrix}
\begin{pmatrix}
x \\
y\\
z
\end{pmatrix}+
\begin{pmatrix}
1 \\
1\\
1
\end{pmatrix}
$$
This gives $x=4,y=6,z=6$ as the only solution. \qed

Proposition A finishes the computation of all states $S$ with five 1s in their binary part, i.e., all states of order one. Technically, let the only component as 0 be $t_{A}$ beats $t_{B}$ in a state $S=(\cdots|\left\{t_{1},t_{2}\right\})$, then if $\left\{t_{1},t_{2} \right\}=\left\{t_{A},t_{B} \right\}$ then the expectation of the corresponding state $f(S)$ is 4, otherwise it is 6. We now use this as the block of building estimation for states with order two.

Let us begin with states with two vacant relationships $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{A}$.

\textbf{Proposition B:}
$$f(111100|\left\{2,3 \right\})=7.$$

\textbf{Proof:} Applying \eqref{equation:1} to this state:
$$
\begin{aligned}
f(111100|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(111110|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111101|\left\{1,3\right\}).
\end{aligned}
$$
Applying Proposition A finishes the proof. \qed

\textbf{Proposition C:}
$$f(111100|\left\{1,3\right\})=9.$$

\textbf{Proof:}
Applying \eqref{equation:1} to this state:
$$
\begin{aligned}
f(111100|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(111100|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111100|\left\{2,3\right\}).
\end{aligned}
$$
Now considering $\pi =(2,3)$ then we have:
$$f(111100|\left\{1,3\right\})=f(111100|\left\{1,2\right\}),$$
combining this with Proposition B finishes the proof.\qed

Now if the only two vacant relationships are $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{A}$, we are ready to read the expectation of the state $S(\cdots|\left\{t_{1},t_{2}\right\})$. If $\left\{t_{1},t_{2}\right\}=\left\{t_{A},t_{B}\right\}$ then $f(S)$ is 7, else it is 9.

We then proceed to states with vacant relationships
\begin{itemize}
\item $t_{A}$ beats $t_{B}$ and $t_{A}$ beats $t_{C}$,
\item $t_{A}$ beats $t_{B}$ and $t_{C}$ beats $t_{B}$,
\item $t_{A}$ beats $t_{B}$ and $t_{B}$ beats $t_{C}$.
\end{itemize}

\textbf{Proposition D:}
$$f(010111|\left\{2,3 \right\})=8,$$
$$f(010111|\left\{1,2 \right\})=7,$$

\textbf{Proof:} Let $x,y$ denote $f(010111|\left\{2,3 \right\})$ and $f(010111|\left\{1,2 \right\})$, according to \eqref{equation:1} (one easily notes that $(010111|\left\{1,2 \right\})$ is symmetrical to $(010111|\left\{1,3 \right\})$):
$$x=1+\frac{y}{2}+\frac{y}{2},$$
$$y=1+\frac{f(110111|\left\{1,3 \right\})}{2}+\frac{x}{2}.$$ Applying Proposition A finishes the proof. \qed

\textbf{Proposition E:}
$$f(101011|\left\{2,3 \right\})=9,$$
$$f(101011|\left\{1,2 \right\})=8,$$

\textbf{Proof:} Let $x,y$ denote $f(101011|\left\{2,3 \right\})$ and $f(101011|\left\{1,2 \right\})$, using symmetry and Proposition A as in the proof of Proposition D:
$$x=1+y,$$
$$y=1+\frac{y}{2}+\frac{6}{2}.$$
This finishes the proof.\qed

\textbf{Proposition F:}
$$f(011101|\left\{1,2\right\})=\frac{38}{5},$$
$$f(011101|\left\{2,3\right\})=\frac{36}{5},$$
$$f(011101|\left\{1,3\right\})=\frac{42}{5}.$$
\textbf{Proof:} Let $x,y,z$ denote these three values, using \eqref{equation:1} and Proposition A:
$$x=1+\frac{6}{2}+\frac{y}{2},$$
$$y=1+\frac{4}{2}+\frac{z}{2},$$
$$z=1+\frac{x}{2}+\frac{y}{2}.$$
Solving this system yields
$$x=\frac{38}{5},y=\frac{36}{5},z=\frac{42}{5}.$$
\qed

Proposition B-F finish computing states of order two.

The number of states with three 1s/of order three is larger. There are at least $\frac{3*\binom{6}{3}}{3\!}=10$ unsymmetric states. Although unnecessary for the following sections, one is encouraged to compute all 13 independent states with three appeard win-loss relationships.

We are now ready to begin from $f(000000|\left\{1,2\right\})$ and hope that compution meets with the Propositions A-F before at an early stage of computation.

\subsection{The Main Reduction}
Attempting to solve this problem by reduction, applying \eqref{equation:1} onto $f(000000|\left\{1,2 \right\}).$:
$$
\begin{aligned}
f(000000|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(100000|\left\{1,3\right\})\\
&+\frac{1}{2}\cdot f(010000|\left\{2,3 \right\}).
\end{aligned}
$$

From now on, let $S_{1}$ be the state where the player with the smaller index winning the current round. Adopting
$$\pi =(1,2),$$
then $(100000|\left\{1,3\right\})$ and $(010000|\left\{2,3 \right\})$ are symmetric, so:
\begin{equation}
\label{equation:2}
f(000000|\left\{1,2\right\})=1+f(100000|\left\{1,3\right\}).
\end{equation}
Therefore we are left with the problem of computing $f(100000|\left\{1,3\right\})$. Now
\begin{equation}
\label{equation:3}
\begin{aligned}
f(100000|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(101000|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(100100|\left\{2,3 \right\}).
\end{aligned}
\end{equation}
We first address $f(101000|\left\{1,2\right\})$ and then return to {\color{red}$f(100100|\left\{2,3\right\})$}. Since
$$
\begin{aligned}
f(101000|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(101000|\left\{1,3\right\})\\
&+\frac{1}{2}\cdot f(111000|\left\{2,3 \right\}).
\end{aligned}
$$
However, let
$$\pi =(2,3),$$
we conclude that
$$f(101000|\left\{1,2 \right\})=f(101000|\left\{1,3\right\}),$$
thus
\begin{equation}
\label{equation:4}
f(101000|\left\{1,2 \right\})=2+f(111000|\left\{2,3\right\}).
\end{equation}

Keep reducing:
$$
\begin{aligned}
f(111000|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(111010|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(111001|\left\{1,3 \right\}).
\end{aligned}
$$
We are now meeting two states with four 1s and two 0s, $f(111010|\left\{1,2\right\})$ is symmetric to $f(010111|\left\{2,3\right\})$, hence its value is 8 according to Proposition D, while $f(111001|\left\{1,3\right\})$ addresses a state symmetric to $f(011101|\left\{2,3 \right\})$, whose value is $\frac{36}{5}$ according to Proposition F. Pluggin them back into \eqref{equation:4} gives:
$$f(101000|\left\{1,2\right\})=\frac{53}{5}.$$

To return to \eqref{equation:3}, we still need to compute $f(100100|\left\{2,3\right\})$.
\begin{equation}
\label{equation:5}
\begin{aligned}
f(100100|\left\{2,3\right\})=1&+\frac{1}{2}\cdot f(100110|\left\{1,2 \right\})\\
&+\frac{1}{2}\cdot f(100101|\left\{ 1,3\right\}).
\end{aligned}
\end{equation}
For
$$
\begin{aligned}
f(100110|\left\{1,2\right\})=1&+\frac{1}{2}\cdot f(100110|\left\{1,3 \right\})\\
&+\frac{1}{2}\cdot f(110110|\left\{2,3 \right\}).
\end{aligned}
$$

Where as one can easily observe the symmetry between $(100110|\left\{1,2\right\})$ and $(100110|\left\{1,3\right\})$, we have:
$$f(100110|\left\{1,2\right\})=2+f(110110|\left\{2,3\right\})=\frac{46}{5}.$$
The last term remained is $f(100101|\left\{1,3\right\})$. We begin with
$$
\begin{aligned}
f(100101|\left\{1,3\right\})=1&+\frac{1}{2}\cdot f(101101|\left\{1,2\right\})\\
&+\frac{1}{2}\cdot f(100101|\left\{2,3 \right\})\\
=\frac{9}{2}&+\frac{1}{2}\cdot f(100101|\left\{2,3\right\}).
\end{aligned}
$$
Finally, we have:
$$f(100101|\left\{2,3\right\})=\frac{24}{5}+\frac{1}{2}\cdot f(100101|\left\{1,3\right\}).$$
This gives:
$$f(100101|\left\{1,3\right\})=\frac{46}{5}.$$
Pluggin them into \eqref{equation:5} yields:
$$f(100100|\left\{2,3\right\})=\frac{51}{5}.$$
Now \eqref{equation:3} yields:
$$f(100000|\left\{1,3\right\})=11.4.$$
At length, pluggin this into \eqref{equation:2} yields:
$$f(000000|\left\{1,2\right\})=12.4.$$

Hitherto we have finished all the reduction.

\subsection{Simulation Results}
As for an empirical verification of the result, a straightforward Monte Carlo simulation was conducted (with 1,000 samples) and the result is shown as Figure. \ref{figure:1}, the mean of the number of rounds is 12.4287.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth] {./1.jpeg}
\caption{Monte Carlo simulation for 3PG with 1,000 samples.}
\label{figure:1}
\end{figure}

\section{The Second Order Analysis}
Given the expectation of the number of rounds in any states:
$$\left\{f(S):S\in\mathcal{S} \right\},$$
it is straightforward to compute the variance of the number of rounds by reduction. The bridge is:
$$\text{var}[X]=\text{var}[\mathbb{E}[X|Y]]+\mathbb{E}[\text{var}[X|Y]].$$
Now let $X$ be the random variable that denotes the number of rounds in the current state and $Y$ be the indicator of the current competition. Let $g(S)$ be the variance of the number of rounds of a 3PG begins from the state $S$, we have:
$$g(S)=\frac{(f(S_{1})-f(S_{2}))^{2}}{2}+\frac{g(S_{1})+g(S_{2})}{2}.$$
Thus given $\left\{f(S):S\in\mathcal{S} \right\}$ it is straightforward to compute $\left\{g(S):S\in\mathcal{S} \right\}$ (repeat what has been done in the sections before, \emph{reversely computing along the martingle!}) and deduce the variance of the number of rounds in a 3PG.

For example, consider the variances of the number of rounds for states $S_{1}=(111110|\left\{2,3 \right\})$, $S_{2}=(111110|\left\{1,2 \right\})$, $S_{3}=(111110|\left\{1,3 \right\})$. Using Proposition A, we have:
$$\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}=
\begin{pmatrix}
0 & \frac{1}{2} & 0 \\
\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & 0
\end{pmatrix}
\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}
+
\begin{pmatrix}
18 \\
2 \\
2
\end{pmatrix}.
$$
Which yields:
$$\begin{pmatrix}
g(S_{1}) \\
g(S_{2}) \\
g(S_{3})
\end{pmatrix}=
\begin{pmatrix}
40 \\
44 \\
44
\end{pmatrix}.
$$

\section{The Probabilistic Framework}
The analysis so far is hardly relied on the probability space. The reason behind is that it is hard to establish the equivalence between an element in the probability space and the value of the random variable \cite{alon2004probabilistic}. Considering:
$$\Omega=\left\{+,- \right\}^{\infty}.$$
Where $+/-$ denotes the player with larger/smaller index winning the current round. To compute $\text{Pr}(X=n)$, where $X$ is the random variable that counts the number of rounds until termination. One has to find the number of $\left\{+,-\right\}^{n}$ sequences where all win-lose relationship appears until the final round. Although it is efficient to transcript a $\left\{+,-\right\}^{n}$ sequence into win-lose relationship sequences, it is hard to write down (be it exists) a tractable necessary and sufficient condition for $X=n$.

However, we could use the solution of 3PG to answer questions yielded from a more probabilistic perspective. For example:

\emph{Building up a string s with three characters \{`''a'',''b'',''c''\}, s begins with ''a'' and each character is followed by one different character with equal probability, s terminates until all six pairs appears in the string. What is the expected length of s?} This question is isomorphic to 3PG.

\section{Generalization}
Having finished the analysis of 3PG, we now proceed to a genelization study. The problem is, is it possible to find the asymptotical behavior of the solution to $n$-PG? The generalization of 3PG to $n$-PG is not unique, e.g., each round can still involve two players, and one random player enters the next round instead of the loser, or one can adopt ternary logic to mark the result of battles.

We study the general $n$-PG with two players participating each round, and a player is randomly (uniformly and independently) chosen to replace the loser of the current round in the ring.

First we try to address the states of order one, w.l.o.g., let the vacant relationship be Player 1 beats Player 2. There are four independent (unsymmetric) states with $\left\{1,2\right\}$, $\left\{1,3\right\}$, $\left\{2,3\right\}$ and $\left\{3,4 \right\}$ as the current pair of players in the ring (assuming $n \geq 4$). Let $x,y,z,w$ denote the corresponding expectations, then we have:
$$
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}=
A_{1}
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}+
\begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix},
$$
with:
$$A_{1}=
\begin{pmatrix}
0 & 0 & \frac{1}{2} & 0\\
\epsilon & \frac{1}{2} & \epsilon & \frac{1}{2}\\
\epsilon & \epsilon & \frac{1}{2} & \frac{1}{2}\\
0 & \epsilon & \epsilon & 1-2\epsilon
\end{pmatrix},$$
where
$$\epsilon=\frac{1}{2(n-2)}.$$
Since we have:
$$
\begin{pmatrix}
x\\
y\\
z\\
w
\end{pmatrix}=(I-A_{1})^{-1}\cdot
\begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix},
$$
as:
$$(I-A_{1})^{-1}=\sum_{i=0}^{\infty}A_{1}^{i},$$
the only task remained is to track the spectral radius of $A_{1}$ \cite{meyer2000matrix}, the trick here is to apply the Gerschgorin theorem to the last row of $A_{1}$, with yields that the largest eigenvalue of $A_{1}$ (assumed to be real) is no less than:
$$\lambda=1-4\epsilon.$$
Therefore the spectral radius of $(I-A)^{-1}$ is no less than:
$$\frac{1}{1-\lambda}\sim O(n).$$
So is the order of $x,y,z,w$.

Moving to states of order two is a similar case, let the independent states be $x',y',\cdots,w'$, we have:
$$
\begin{pmatrix}
x'\\
y'\\
\cdots\\
w'
\end{pmatrix}=A_{2}
\begin{pmatrix}
x'\\
y'\\
\cdots\\
w'
\end{pmatrix}+
\begin{pmatrix}
c_{x}\\
c_{y}\\
\cdots\\
c_{w}
\end{pmatrix},
$$
Where elements in $(c_{x},c_{y},\cdots,c_{w})^{\text{T}}$ are constants with value 1 or a multiple of $x,y,z,w$ that has been evaluated before. Hence the order of elements in $(c_{x},c_{y},\cdots,c_{w})^{\text{T}}$ is at most $O(n)$.

To measure the spectral radius of $A_{2}$, we resort to a similar line of reasoning: let $w$ be the expectation of the state where the current players on the ring is different from those players involved in the vacant win-lose relationships. Then the final row of $A_{2}$ has $1-\epsilon_{2}$ as the last component, where
$$\epsilon_{2}=\frac{4}{n-2}$$
in the most probable case. This yields the fact that the order of the spetral radius of $(I-A_{2})^{-1}$ be $O(n)$, hence the order of the expectations of states of order two turns out to be $O(n^{2})$.

In general, for states of order $\phi$, let $w_{\phi}$ be the expectation of the state where the current players on the ring are free from those $\phi$ vacant pairs, let $A_{\phi}$ be the transition matrix at that order. There are at most $2\phi$ players involve with the vacant pairs, hence the entry on the right-bottom most side of $A_{\phi}$ is at most:
$$1-\frac{2\phi}{n-2}.$$
That is to say, the spectral radius of $(I-A_{\phi})^{-1}$ is of order:
$$O\left(\frac{n}{\phi}\right).$$
Finally, counting all states of order $\phi=0,1,\cdots,n$, we have the order of the solution of an $n$-PG be:
$$O\left(\frac{n^{n}}{n!}\right)=O\left(\frac{\text{e}^{n}}{\sqrt{n}}\right).$$

Analogously, the variance for general $n$-PG can be approximated using the same framework. The $f(\cdot)$ for states of order $\phi$ is of order:
$$\frac{1}{\phi}\left(\frac{n\text{e}}{\phi} \right)^{\phi}.$$
We have that $g(\cdot)$ for states of order $\phi$ (denoted by $g_{\phi}$)is determined by the larger term in $f^{2}_{\phi}$ and $g_{\phi-1}$, so at least:
$$g_{\phi}\geq O\left(\frac{n}{\phi}\cdot f^{2}_{\phi}\right),$$
which is far less than the order of $f^{2}_{\phi}$, therefore we conjecture that the order of the variance in $n$-PG is:
$$O\left(\frac{\text{e}^{2n}}{n} \right).$$

In fact, we observe that the estimation on the expectation is possibly a rather slack one, this is due to the following facts:
\begin{itemize}
\item The estimation based on the Gerschgorin theorem on $A$ might significantly increase the spectral radius of $(I-A)^{-1}$.
\item The spectral radius might be involved with negligible terms so the speed of growth declines.
\end{itemize}
The simulation results of general $n$-PGs are illustrated as in Figure. \ref{figure:2}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth] {./2.jpeg}
\caption{Monte Carlo simulation for $n$-PG with 100 samples for each $n$.}
\label{figure:2}
\end{figure}

From which we might optimistically conjecture that the growth of the expectation of the number of rounds is only of order $n^{2}$, but there seems to be a vacancy in establishing this result.

\section{Conclusion}
This paper address the 3PG question. We attack this question with dynamic programming, highlight the necessary tricks that signicantly reduce redundant computation and analyze the ideas behind. The general case is also proposed and a rough bound is derived.

\section{Acknowledgement}
Haoran Ye for provided the 3PG question, Runbo Ni and his colleagues provided an early version of solution.

\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv


text:
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