text:
Q: Does there exist a closed form for the sum $\sum_{n=1}^{\infty} \frac{e^{tn}}{n^2}$ where $t<0$.? Does there exist a closed form for the sum $\sum_{n=1}^{\infty} \frac{e^{tn}}{n^2}$ where $t<0$.?
I was trying to form an arithmetico-geometric progression but ended futile.

A: As mentioned in the comments, this is known as the dilogarithm, also known as Spence's function. There are many different representations given on the polylogarithm, and various identities are shown on the same page.
$$\operatorname{Li}_2(e^t)=\sum_{n=1}^\infty\frac{e^{nt}}{n^2}$$


i. The Indian Parliament passed the Right of Children to Free and Compulsory Education (Amendment) Bill, 2018 that seeks to do away with the no-detention policy in schools. The legislation is significant as it brings accountability in the elementary education system.
i. The Indian Parliament passed the National Council for Teacher Education (Amendment) Bill, 2018, following the approval of the Rajya Sabha. The Lok Sabha had already passed the Bill. The Bill seeks to amend the National Council for Teacher Education (NCTE) Act, 1993.
i. U.S. President Donald J. Trump signed into law the Asia Reassurance Initiative Act (ARIA), which passed the U.S. Senate earlier in December. According to the White House, the act establishes a multifaceted U.S. strategy to increase U.S. security, economic interests, and values in the Indo-Pacific region.
i. Arunima Sinha, the first female amputee to have conquered Mt Everest, created yet another record as she became world's first woman amputee to climb Mount Vinson, the highest peak in Antarctica.
ii. Apart from Mount Everest, the 30-year-old has climbed the highest peaks of five continents and plans to do so in all seven.
i. HDFC Mutual Fund has pipped ICICI Prudential MF to become the country's largest asset management company after a gap of over two years.
i. Syndicate Bank and SBI Life Insurance signed a bancassurance pact aiming to offer a comprehensive financial planning solution to its customers. The pact was signed between Syndicate Bank MD & CEO, Mrutyunjay Mahapatra and MD and CEO of SBI Life, Sanjeev Nautiyal.
ii. Through the pact, Syndicate Bank will provide penetration in the market with its 3,000branches and SBI Life's diverse range of protection, wealth creation and savings insurance products will be brought to the table to the customers.
Syndicate Bank Headquarters: Manipal, Karnataka, Tagline: Faithful and Friendly.
i. Markets regulator Securities and Exchange Board of India (SEBI) has allowed custodial services in the commodity derivatives market. The move is aimed at enabling the participation of institutional investors such as mutual funds and portfolio managers in the commodity derivatives market.
ii. Under the new framework, existing custodians will be permitted to add commodities as an asset class and provide physical delivery of both the securities and commodities.
i. Renowned Bengali writer and Sahitya Akademi awardee Dibyendu Palit passes away due to age-related ailments. He was 79-year-old.
ii. Palit was conferred the Sahitya Akademi for his novel 'Anubhab' in 1998. He was awarded the 'Bankim Purashkar' for 'Dheu' in 1990 and 'Ananda Purashkar' in 1984 for 'Sahojoddha'.


text:
http://tvtropes.org/pmwiki/pmwiki.php/Creator/ZachAguilar
Creator / Zach Aguilar
Zach Aguilar (born 1998) is one of the newer voice actors in the block best known for his role as Slaine Troyard from Aldnoah.Zero.
He first started his career as an on-camera actor, but realized that he had much more to offer and could play many more roles without being limited by his appearance. Zach has had extensive training from some of the most recognized voice over artists in the industry.
In 2014, Zach won the Magi 2 voice acting auditions at Anime Expo hosted by Bang Zoom! Entertainment and Aniplex USA. At 16 years old, he was one of the youngest people to ever win and was cast in Magi: The Kingdom of Magic.
Notable roles by Zach Aguilar
Additional voices from BloodLad, Gargantia on the Verdurous Planet and Kill la Kill
Arthur Pendragon from The Seven Deadly Sins
Choppo from Magi: The Kingdom of Magic (first anime role)
Colt from Hunter × Hunter
Genos from One-Punch Man
Gyro from One Piece
Harunobu Nikaidou from 3-gatsu no Lion
Horikawa Kunihiro from Katsugeki/Touken Ranbu
Koichi Hirose in JoJo's Bizarre Adventure: Diamond is Unbreakable
Marcus Lithos in Sword Gai
Sieg in Fate/Apocrypha
Slaine Troyard from Aldnoah.Zero
Talken from Sword Art Online II
Takaki Uno from Mobile Suit Gundam: Iron-Blooded Orphans
Takato from Charlotte
Tropes relating to Zach Aguilar
Star-Making Role: Slaine Troyard from Aldnoah.Zero and Genos from One-Punch Man.
Yuko Mizutani
Names to Know in Anime
Laura Bailey

redpajama_set_name: RedPajamaCommonCrawl


Sutherland Hussey Architects in Edinburgh has completed a feasibility study for this new harbour at Portree on the Isle of Skye.
The project, christened "Lump and Line", will include lifeboat launch facilities, a visitor centre, a "mega-hide" for the RSPB and a home for the An Tuireann Arts Centre. The rock-like formation, which will also house a museum, has been designed to reflect the surrounding mountains. It is also hoped that deep-water moorings alongside the new harbour wall will attract international cruiseships and leisure boats.


text:
Mr Barfoot – owner
"I love Sweetings, which I bought from the wife of my very best friend, Graham Needham, who, unfortunately, after a very successful spell running it from 1979, died in 1993. His wife took over ownership and she sold it to me in 2000. I am the fifth owner in its history and we are an institution in the City of London. We've gone through two World Wars and several financial upsets that life has thrown at us, but because of our reputation customers have stayed loyal to Sweetings."
Sweetings is probably the oldest fish and oyster restaurant in London.
The original Sweetings opened in 1830 as John S. Sweetings, Fish and Oyster Merchant, in Lad Lane, Islington.
Additional premises at 159 Cheapside and 17 Milk Street were opened several years later.
Theses "Very Superior Oyster Rooms" sold mainly lobsters, salmon, Dover soles and oysters.
At that time Sweetings also supplied top quality fish, game and venison to hotels, restaurants and country residences from their shop in Lad Lane.
The present premises at 39 Queen Victoria Street have now been occupied for more than 100 years.
Designed by Suely Castro


ARTIFACT: This is a 1982 pair of cased silver life saving medals from Newburyport, MA. These were awarded to Theodore Nutting for saving four men on a capsized boat at Newburyport Bar (Plum Island) on either June 26th or 28th, 1892.
CONDITION: 7- (Very Fine-): The cases are dirtied and worn, but the coins are in excellent condition.


NEW YORK, NY (March 27, 2017)—Girl Scouts of the USA (GSUSA) spotlights its legacy of girls' civic engagement by showcasing its Citizen badges at the 43rd annual Macy's Flower Show. Revealed yesterday, the interactive garden display at Macy's Herald Square illustrates how communities can flourish when people lend a hand, a nod to Girl Scouts' longstanding commitment of encouraging girls to take the lead in their communities, the government, public policy, and the democratic process. The floral display symbolizes how communities bloom when everyone rolls up their sleeves and gets involved—like a Girl Scout.
The exhibit, on view through April 9 at Macy's Herald Square's recently renovated "One Below" floor, features a working spigot that visitors can pump to water a giant flower, allowing it to blossom, as well as levers that move to reveal the Citizen badges a Girl Scout earns when she engages in civic action. The badges (including Public Policy, Behind the Ballot, and Inside Government) foster vital skills for girls ages 7–18, such as how to take part in a civil debate, make group decisions, and get involved in government. Through unique Girl Scout programming, girls learn to stand up for what they believe in, identify issues they care about, and take the lead like a G.I.R.L. (Go-getter, Innovator, Risk-taker, Leader)™ to make the world a better place.
Girl Scouts continue to use these crucial skills in their adult lives: Girl Scout alumnae make up the majority of women in leadership positons in the current U.S. government, including 76 percent of women senators, 54 percent of women representatives, and four of the five women governors. According to the Girl Scout Research Institute report Girl Scouting Works: The Alumnae Impact Study (PDF), alumnae are also more civically engaged and likely to vote than their non–Girl Scout peers. And the longer women were Girl Scouts, the higher their level of civic participation.
A nonpartisan organization, Girl Scouts is committed to ensuring that girls have a voice and are empowered to use it to be actively involved in their communities. The organization was founded in 1912, when women in the United States did not even have the right to vote. When the U.S. Congress passed the 19th Amendment, granting women this right, Girl Scouts gathered outside polling places to support women as they cast their first ballots. Empowering girls to step up and forge their own life paths is at the center of everything Girl Scouts does.
Today, girls continue to advocate for equality, those less fortunate, and much more because of Girl Scouts. And countless civically engaged women who found their voice in Girl Scouts persistently take action in support of issues that matter to them—for instance, expanding access to high-quality education, championing efforts to protect the environment, and mobilizing grassroots efforts to improve communities.
Local councils Girls Scouts of Northern New Jersey, Girl Scouts of Central and Southern New Jersey, and Girl Scouts of the Jersey Shore also showed their support for the Macy's Flower Show by participating in Family Fun Day on March 26. These Girl Scouts volunteered to read stories to children and led various hands-on activities throughout the day. To learn more about how Girl Scouts gives girls opportunities to create positive change in their communities, or to join, volunteer, or donate, visit www.girlscouts.org.
We're 2.7 million strong—1.9 million girls and 800,000 adults who believe girls can change the world. It began over 100 years ago with one woman, Girl Scouts' founder Juliette Gordon "Daisy" Low, who believed in the power of every girl. She organized the first Girl Scout troop on March 12, 1912, in Savannah, Georgia, and every year since, we've made her vision a reality, helping girls discover their strengths, passions, and talents. Today we continue the Girl Scout mission of building girls of courage, confidence, and character, who make the world a better place. We're the preeminent leadership development organization for girls. And with programs for girls from coast to coast and across the globe, Girl Scouts offers every girl a chance to do something amazing. To volunteer, reconnect, donate, or join, visit www.girlscouts.org.
Girl Scouts of the USA spotlights its legacy of girls' civic engagement by showcasing its Citizen badges at the 43rd annual Macy's Flower Show. To learn more about how Girl Scouts gives girls opportunities to create positive change in their communities, or to join, volunteer, or donate, visit www.girlscouts.org.


text:
Flinton – wieś w Anglii, w hrabstwie East Riding of Yorkshire. Leży 15 km na północny wschód od miasta Hull i 256 km na północ od Londynu.

Przypisy

Wsie w hrabstwie East Riding of Yorkshire

redpajama_set_name: RedPajamaWikipedia


In this post you can see the results to polished a granite floor in the lift. You can see the difference.
New work… And new floor. This is the result of crystallized and non slip treatment. And remember never use wax to clean the marble floors.
Another satisfied customer… You can see the results Remember you, free quote.


Invented in the first decades of the 19th century, photography (by way of the camera) seemed able to capture more detail and information than traditional media, such as painting and sculpting. Photography as a usable process goes back to the 1820s with the development of chemical photography. The first permanent photoetching was an image produced in 1822 by the French inventor Nicéphore Niépce, but it was destroyed by a later attempt to duplicate it. Niépce was successful again in 1825. He made the first permanent photograph from nature (his View from the Window at Le Gras) with a camera obscura in 1826.
Because his photographs took so long to expose (eight hours), he sought to find a new process. Working in conjunction with Louis Daguerre, they experimented with silver compounds based on a Johann Heinrich Schultz discovery in 1816 that a silver and chalk mixture darkens when exposed to light. Niépce died in 1833, but Daguerre continued the work, eventually culminating with the development of the daguerreotype in 1837. Daguerre took the first ever photo of a person in 1838 when, while taking a daguerreotype of a Paris street, a pedestrian stopped for a shoe shine, long enough to be captured by the long exposure (several minutes). Eventually, France agreed to pay Daguerre a pension for his formula, in exchange for his promise to announce his discovery to the world as the gift of France, which he did in 1839.
A latticed window in Lacock Abbey, England, photographed byWilliam Fox Talbot in 1835. Shown here in positive form, this is the oldest known extant photographic negative made in a camera.
Meanwhile, Hercules Florence had already created a very similar process in 1832 in Brazil, naming it Photographie, and English inventor William Fox Talbot had earlier discovered another means to fix a silver process image but had kept it secret. After reading about Daguerre's invention, Talbot refined his process so that portraits were made readily available to the masses. By 1840, Talbot had invented the calotype process, which creates negative images. Talbot's famous 1835 print of the Oriel window in Lacock Abbey is the oldest known negative in existence.
John Herschel made many contributions to the new methods. He invented the cyanotype process, now familiar as the "blueprint". He was the first to use the terms "photography", "negative" and "positive". He discovered sodium thiosulphate solution to be a solvent of silver halides in 1819, and informed Talbot and Daguerre of his discovery in 1839 that it could be used to "fix" pictures and make them permanent. He made the first glass negative in late 1839.
In March 1851, Frederick Scott Archer published his findings in "The Chemist" on the wet plate collodion process. This became the most widely used process between 1852 and the late 1860s when the dry plate was introduced. There are three subsets to the collodion process; the Ambrotype (positive image on glass), the Ferrotype or Tintype (positive image on metal) and the negative which was printed on albumen or salt paper.
Many advances in photographic glass plates and printing were made in through the 19th century. In 1884, George Eastman developed the technology of film to replace photographic plates, leading to the technology used by film cameras today.


text:
Никола́й Па́влович Кривома́зов (20 сентября 1947, станица Мальчевская, Ростовская область — 15 марта 2012, Москва) — российский журналист, главный редактор журнала «Русская водка», ответственный секретарь газеты «Правда», писатель и сценарист. Регулярно и с симпатией писал об алкоголе как напитке и социальном явлении, проблематике алкогольного рынка, выпускал периодическое издание, посвящённое горячительной теме. Издавал также журнал «Гражданин».

Биография
После окончания Каменского музыкально-педагогического училища был направлен по распределению в Якутию. Два года работал учителем. Закончил Якутский университет и сценарное отделение ВГИКа. С 1968 года занимался профессиональной журналистикой. Работал в якутской районной газете «Ленские маяки», иркутской областной газете «Советская молодёжь». С 1977 года — собственный корреспондент по Красноярскому краю, Уралу, Иркутску, ряда центральных газет — «Комсомольская правда» (именно в этой газете была опубликована вызвавшая общественный резонанс его статья «Рагу из синей птицы» с резкой критикой группы «Машина времени»), затем «Социалистическая индустрия» и «Правда».

В 1989 году переведён на работу в центральный аппарат «Правды», где с 1992 по 1994 год работал ответственным секретарём, куда его выдвинул главный редактор Геннадий Селезнёв. В 1995—1998 годах — обозреватель «Российской газеты». Писал репортажи из воюющей Сербии. В 1998 году создал Издательский дом «Гражданин». Автор ряда книг и сценариев к фильмам, среди которых — боевик «Взбесившийся автобус», единственный советско-израильский художественный фильм.

Издавал несколько журналов, среди которых особый успех имел журнал «Русская водка», в котором алкогольная тема была раскрыта в разных точек зрения. В одном из номеров в качестве эпиграфа Кривомазов употребил фразу Горького: «Пьяных не люблю. Пьющих понимаю. Непьющих опасаюсь».

Автор повести «Пропало лето» о старшеклассниках, которые пытаются решить сложные проблемы экологии.

В первых числах февраля 2012 года Кривомазов внезапно почувствовал острое недомогание и был госпитализирован. Врачи поставили диагноз — саркома лимфатических узлов. 15 марта 2012 года он скончался в московской больнице. Отпевание прошло в храме Троицы в Останкине. Похоронен на  (участок № 69).

Николай был дважды женат, вдову зовут Людмила Анатольевна Кривомазова. В браке родились два сына: Алексей, живёт в Иркутске; Павел (умер в 1990-х годах). В семье Кривомазова выросли также двое приёмных сыновей от первого брака Людмилы — Михаил и Евгений.

Примечания

Журналисты газеты «Правда»
Главные редакторы России
Выпускники Якутского государственного университета
Выпускники сценарного факультета ВГИКа


How do I tag references?
In addition to creating lists and nested sublists to organize your library, you can now add one or more searchable Keyword tags to any reference.
To create a Keyword, just click the + in the Keywords field in the right panel, under the abstract.
Any keywords you create will be listed in the left menu panel, under the Keywords section. All tags become links - clicking on a tag will take you to the list with all references containing the same tag.
It's also quick and easy to search for any tag using the general Search bar at the top.
To remove a keyword from a reference, just delete the hashtag from the Keywords field. A tag that no longer has associated references will be automatically removed from the left menu.
How do I organize my articles in ReadCube Papers?
What are Lists & how do I make them in ReadCube Papers?
Can I mark an article as a favorite in ReadCube Papers?


Appears on all the coffee shop lists of all my Berlin coffee-loving friends. It's a little out of the way (but only a little) in a leafy neighbourhood. The coffee is very good, and the space is pleasant. There's a hearty collection of coffee and design magazines on the front windowsill.


In this class, my students made powerful, fully-functional bows and arrows from scratch, using traditional woodworking methods. We used red oak and many hardwoods, going beyond the cheap softwoods (like pine) used for most student projects. I then instructed the students in archery using the bows they created and we formed a DIY archery team.


Who is Skytells Framework For?
Fast, Powerful and Easy to use!
Built-in Ready HTML5 ( 404, 505, 403 ) Pages.
Built-In Embed System for Social Networks Development.
Advanced CLI & Analytical Tools.
Advanced ( MVC ) - ( HMVC ) Support.
Powerful Engines for DB, etc.
Yes, It's FREE and always will be!
Skytells PHP Framework is an open-source PHP framework licensed under MIT license.
Copyright © 2017 Skytells, Inc.


CLEVELAND -- An AMBER Alert issued for 1-month-old girl taken by her biological father has been canceled.
The baby was dropped off at University Hospitals by a family member. The suspect remains at large.
Police say the baby's father broke into the mother's Central Avenue home through a window, beat the mother and forcibly took his daughter, identified as Raniyah May. The incident happened around 12:45 p.m. Thursday.
The suspect, identified as Ronald Davis, 28, fled in the mother's car, which is a blue 2010 Chevrolet Malibu with an Ohio license plate reading HKY3562. Davis was last seen at 2457 E. 37th Street.
Anyone with information on Davis location should call police.


Style is not the only consideration when choosing what to wear on race day. Comfort is also an important consideration, especially for longer races, plus, if you can minimize what you put on and take off, you will have a faster races. We review what to wear for both men and women.
In this video David Glover, MS, CSCS reviews what men should wear on race day with a focus on comfort and minimizing clothing changes during the race.
In this video Krista Schultz, MEd, CSCS discusses options for what women wear on race day. Be sure to practice first in training what you plan to wear on race day.


text:
/**
 A collection of macros used throughout the FTUtils set of tools.

 This file need not be included directly since it's included by the other
 pieces of the library. The macros are useful throughout a project, and
 it's recommended that you just include the file in your prefix header.
*/

#pragma mark - Version

static const NSUInteger FTUtilsVersion = 010100;
static NSString *const FTUtilsVersionString = @"1.1.0";

#pragma mark - Math
///---------------------------------------------------------------------------
/// @name Math Helpers
///---------------------------------------------------------------------------

/** Convert from degrees to radians */
#define DEGREES_TO_RADIANS(d) (d * M_PI / 180)

/** Convert from radians to degrees */
#define RADIANS_TO_DEGREES(r) (r * 180 / M_PI)

#pragma mark - Colors
///---------------------------------------------------------------------------
/// @name Creating colors
///---------------------------------------------------------------------------

/**
 Create a UIColor from a hex value.

 For example, `UIColorFromRGB(0xFF0000)` creates a `UIColor` object representing
 the color red.
*/
#define UIColorFromRGB(rgbValue) \
[UIColor \
  colorWithRed:((float)((rgbValue & 0xFF0000) >> 16))/255.0 \
         green:((float)((rgbValue & 0x00FF00) >> 8))/255.0 \
          blue:((float)(rgbValue & 0x0000FF))/255.0 \
         alpha:1.0]

/**
 Create a UIColor with an alpha value from a hex value.

 For example, `UIColorFromRGBA(0xFF0000, .5)` creates a `UIColor` object
 representing a half-transparent red.
*/
#define UIColorFromRGBA(rgbValue, alphaValue) \
[UIColor \
  colorWithRed:((float)((rgbValue & 0xFF0000) >> 16))/255.0 \
         green:((float)((rgbValue & 0x00FF00) >> 8))/255.0 \
          blue:((float)(rgbValue & 0x0000FF))/255.0 \
         alpha:alphaValue]

#pragma mark - Delegates
///---------------------------------------------------------------------------
/// @name Calling Delegates
///---------------------------------------------------------------------------

/**
 Call a delegate method if the selector exists.
*/
#define FT_CALL_DELEGATE(_delegate, _selector) \
do { \
  id _theDelegate = _delegate; \
  if(_theDelegate != nil && [_theDelegate respondsToSelector:_selector]) { \
    [_theDelegate performSelector:_selector]; \
  } \
} while(0);

/**
 Call a delegate method that accepts one argument if the selector exists.
*/
#define FT_CALL_DELEGATE_WITH_ARG(_delegate, _selector, _argument) \
do { \
  id _theDelegate = _delegate; \
  if(_theDelegate != nil && [_theDelegate respondsToSelector:_selector]) { \
    [_theDelegate performSelector:_selector withObject:_argument]; \
  } \
} while(0);

/**
 Call a delegate method that accepts two arguments if the selector exists.
*/
#define FT_CALL_DELEGATE_WITH_ARGS(_delegate, _selector, _arg1, _arg2) \
do { \
  id _theDelegate = _delegate; \
  if(_theDelegate != nil && [_theDelegate respondsToSelector:_selector]) { \
    [_theDelegate performSelector:_selector withObject:_arg1 withObject:_arg2]; \
  } \
} while(0);

redpajama_set_name: RedPajamaGithub


text:
Burlington Free Press
A California-based semiconductor company has cut 78 jobs at an Essex chipmaking business immediately after closing a deal to buy the facility from GlobalFoundries.
Rebecca Kelley, spokeswoman for Gov. Phil Scott, confirmed Thursday that the state has received notice from the company with details on the number of jobs lost. Any time more than 50 jobs are cut, companies are required to inform the state Department of Labor.
Hanna Kang, a spokeswoman for Marvell in Santa Clara, California, wrote in an email that after completing its acquisition of Avera Semiconductor LLC on Wednesday for about $600 million, the company made the cuts.
"In joining the two companies, a number of functions between the Marvell and Avera teams were identified as redundant and have resulted in the elimination of a small percentage of positions worldwide," Kang said.
Avera has about 800 employees worldwide, with annual revenues in excess of $500 million and more than $3 billion in semiconductor designs underway. The company was spun off as a wholly owned subsidiary by GlobalFoundries a year ago to concentrate on integrated circuits made for specific uses, such as 5G, data centers and automotive applications.
More:GlobalFoundries 'streamlining' by shipping assets from Vermont to Germany
More:GlobalFoundries executive: Cheaper power a must for Essex Junction fab
Avera also offers access to cutting edge semiconductors measuring as small as 7 nanometers, a business that GlobalFoundries announced it was getting out of in August 2018, just before launching Avera in November of that year.
GlobalFoundries said it was moving away from chips approaching the size of atoms at 7 nanometers to focus on its 14/12 nanometer chips.
Going where the market is
Laurie Kelly, a spokeswoman for GlobalFoundries, said Thursday the company made the decision to stop chasing cutting-edge semiconductors because 75% of the market for chips is in the 12 nanometer range and above.
"The cost of development of 7 nanometers was crazy, to invest not only in research and development but also the equipment," Kelly said. "The return wasn't going to be there."
Kelly said part of the reason it made sense to sell Avera to Marvell was because of the decision not to produce 7 nanometer chips.
"A lot of the work and solutions coming from Avera are going to be on that 7 nanometer technology," she said.
Second biggest in the world
GlobalFoundries acquired the Essex Junction semiconductor plant from IBM in July 2015 in a deal that left many people scratching their heads. GlobalFoundries received a payment of $1.5 billion from IBM to take over the fab, which had been losing money. As part of the deal, GlobalFoundries agreed to be IBM's exclusive provider of semiconductor chips through 2025.
Kelly said GlobalFoundries' plant in Essex Junction still has 2,300 employees after about 800 employees made the transition to Avera, and ultimately Marvell. She said GlobalFoundries is the second largest chipmaker in the world, after TSMC, headquartered in Taiwan.
Kelly declined to share privately held GlobalFoundries' annual revenue, but said TSMC is about six times its size. TSMC generated revenue of $34.2 billion last year.
Contact Dan D'Ambrosio at 660-1841 or ddambrosio@freepressmedia.com. Follow him on Twitter @DanDambrosioVT.


text:
Darlington County School District
Brockington Elementary Magnet School for Science & Technology
Brunson-Dargan Elementary School
Cain Elementary School
Carolina Elementary School
Darlington County Institute of Technology
Darlington County Intervention School
Darlington High School
Darlington Middle School
Hartsville High School
Hartsville Middle School
Lamar Elementary School
Lamar High School
Mayo High School for Math, Science & Technology
North Hartsville Elementary School
Pate Elementary School
Rosenwald Elementary & Middle School
Southside Early Childhood Center
Spaulding Elementary School
Spaulding Middle School
St. John's Elementary School
Thornwell School for the Arts
Washington Street Elementary School
West Hartsville Elementary School
DCIS News
Darlington County Intervention School » About Us » Administration
Zenobia Edwards is the director of Darlington County Intervention School. She organized the program from inception in 1999. She is the founding Dean (2005) of Metropolitan College at Johnson C. Smith University in Charlotte, NC.. She is a career educator with over 20 years of experience in both secondary and post-secondary education. Dr. Edwards also worked as a school partnership manager with College Summit, a nonprofit that has helped 35,000 low-income high school seniors navigate the college application process. In 2003, she founded, I Am B.E.A.U.T.I.F.U.L., Inc., an award-winning educational enrichment nonprofit organization dedicated to building self-esteem and leadership capability in girls and women.
100 Magnolia St.Darlington, SC 29532
Darlington CountyIntervention School
Making The Right Choices


text:
Ep. #31 The Little Red Hen –🌾🐣—Mother's Love
This is a classic folktale. The story is about a hen. A hen is a chicken. In the story, the hen is very busy looking for food. Then she finds a seed. She wants to make the seed into bread. How does she do it? Let's listen to the story.
A little red hen lives on a farm. The farm also has a cat, a rat, and a pig. The cat sleeps in the sun every day. The rat plays in the field every day. The pig lies in the mud every day. They are all lazy. But the hen is busy. She looks for food for her baby chicks. Their favorite food is worms. So, every day she looks for worms.
One day the little red hen is looking for worms. But she finds something else. She bites it. This thing is not a worm. What is it? It is a seed. It is a wheat seed. The wheat seed can be planted in the ground. The wheat can be used to make bread. The little red hen says, "Who will plant this seed? I am too busy. I look for food all day for my baby chicks." The cat says, "I won't do it!" The rat says, "I won't!" The pig says, "I won't either!" So, the little red hen plants the seed.
The seed grows into a wheat plant. The wheat grows tall. The little red hen asks, "Who will cut down the wheat? I'm too busy!" The cat says, "I won't do it!" The rat says, "I won't!" The pig is sleeping. The little red hen says, "Ok, I will do it!" She cuts down the wheat.
Then the wheat is ready to go to the mill. At the mill the wheat will be made into flour. But the baby chicks are hungry. The little red hen needs to look for worms. She gives the worms to the baby chicks. She is tired. It's so much work taking care of baby chicks! The little red hen says, "Who will take the wheat to the mill? I am too busy!" The cat says, "I won't!" The pig is sleeping. The rat is playing in the field. The little red hen says, "Ok, I will do it then!" She takes the wheat to the mill. The wheat is made into soft flour.
The little red hen brings the flour home. She goes to bed. But she can't sleep. She thinks about the flour. Someone needs to make the flour into bread. She remembers how to make bread. She thinks about the bread all night. Then the sky turns purple and pink. The sun comes up. It is morning. The little red hen finds the cat and the rat and the pig. She says, "I will ask you for the last time. Will you help me? Will you make the bread?" The cat is still sleepy. The cat says, "I won't." The rat says, "I won't!" The pig says, "I won't either!" The little red hen says, "Ok, I will do it!" She goes to the kitchen.
The little red hen turns on the oven. The cat and the rat and the pig follow her into the kitchen. The cat sits by the warm oven. The rat runs around the kitchen looking for old pieces of food. The pig laughs at the hen. The pig says, "Little red hen, you are too little to make bread! The bread will be bigger than you!" The little red hen says, "I will do it! You'll see." The little red hen mixes the bread dough. She carries the bread dough to the oven. It's so heavy! She puts the bread dough into the oven.
Then she cleans the kitchen. She cleans the flour off of the table. She brushes the flour out of the cat's hair. She dusts the flour off of the pig's nose. Then the bread is ready. She takes the bread out of the oven. The bread smells delicious. The cat stands up. The pig moves closer. The rat runs to the oven. The little red hen says, "Who will eat this bread?" The cat says, "I will!" The rat says, "I will!" The pig says, "I will too!" The little red hen says, "No you won't!" She and her baby chicks eat the bread. The end.
The cat, the rat, and the pig smell the bread. They want to eat it. But it's too late. They should have helped the little red hen when they had the chance.
This story has been rewritten and read by Katrina Hao; it is a classic folk tale called "The Little Red Hen"
Ep. #30 The Ant and the Dove -🐜🕊- Kindness
Ep. #32 The Wolf at the Door --🐺🐐—Creativity


text:
\section{\label{SEC:Intro}Introduction}

This is the third paper of a series~\cite{Fried2009_QCD1,Fried2010_QCD2} describing a new, gauge-invariant, non-perturbative formulation of QCD, which provides analytic estimates of any amplitude which sums over all relevant Feynman graphs, including all possible gluon exchanges between appropriate quarks (Q) and antiquarks ($\bar{\mathrm{Q}}$).  All cubic and quartic gluon interactions are included.  Results are expressed in terms of Fradkin's most useful functional representations~\cite{Fradkin1966,HMF2} of the Green's function $\mathbf{G}_{c}(x,y|A) = \langle x | \mathbf{}G_{c}[A] | y \rangle$,
\begin{equation}\label{Eq:1}
\mathbf{G}_{c}[A] = [m + \gamma \cdot (\partial - ig A \cdot \lambda)]^{-1}
\end{equation}

\noindent and the closed-loop, or vacuum functional $\mathbf{L}[A]$,
\begin{equation}\label{Eq:2}
\mathbf{L}[A] =  {\mathrm{Tr}\, }{\ln{[1 - i g (\gamma \cdot A \cdot \lambda) \, \mathbf{S}_{c}]}}, \quad \mathbf{S}_{c} = \mathbf{G}_{c}[0],
\end{equation}

\noindent where $A_{\mu}^{a}(x)$ represents a given vector potential.  Since the Fradkin representations for these quantities are of Potential Theory origin, they have relatively simple approximations in different physical situations~\cite{HMF3}, especially at high energies where they effectively generate eikonal models.  It should be understood that we are here dealing with the simplest form of "textbook QCD": one type of quark coupled to multicolored gluons.  Flavors and electroweak interactions are to be added later on.

This new formulation provides an exact statement of what may be expected from conventional, or "ideal" QCD, where the quanta of the quark field operators are expected to have the same quantum-mechanical measurement properties as, \emph{e.g.}, electrons (perfect position measurement at the expense of momenta, and vice-versa).  For an important category of QCD processes which involve gluon exchange between quarks, it turns out that this theory is effectively empty, because attention has not been paid to the fact that asymptotic quarks are only found in bound states, and that their transverse coordinates can, in principle, never be exactly determined.  In all previous approximate calculations -- such as those corresponding to gauge-dependent perturbative approximations, or to a subset of Feynman graphs, or to machine calculations wherein gauge-invariance is enforced by breaking up space-time into plaquettes -- this difficulty has not been visible; but for the exact, non-perturbative theory, it becomes clear that quark transverse imprecision must be included as a basic feature of the theory.

The reason for this statement, as well as the relatively simple form of results for each and every QCD amplitude, is due to the novel and unexpected property we call "Effective Locality" (EL), which has been shown in Ref.~\cite{Fried2010_QCD2} to be rigorously true for the exact theory.  In exact QCD, the way in which EL forces the long-held paradigm of Abelian QFT -- where internal interactions are carried by a boson propagator from one space-time point to another -- to change to a new form of locality, will become clear below.

\section{The Fradkin Representations}\label{sec:FradkinRepresentations}

In the first paper on this subject~\cite{Fried2009_QCD1}, we replaced the exact Fradkin representations by their high-energy, eikonal limits; in this paper a more rigorous approach is maintained.  The aim here is to show how the conventional, or "ideal" QCD formulation becomes untenable when treated in this non-perturbative formulation; and then to adopt a more "realistic", though presently phenomenological, version of QCD in which "ideal" troubles disappear, and where the foundation is put in place for automatic quark binding into hadrons~\cite{Fried2011_QCD4}, as well as hadron-hadron interactions of the Yukawa-type~\cite{Fried2011_QCD5}.

For definiteness, we again consider the case of Q-Q or Q-$\bar{\mathrm{Q}}$ scattering, for which the corresponding 4-point function (unsymmetrized, for simplicity of presentation, and in configuration space, before mass-shell amputation) is proportional to
\begin{eqnarray}\label{Eq:3}
\mathcal{N} \int{\mathrm{d}[{\chi}] \, e^{\frac{i}{4} \int{\chi^{2}}}} &\cdot& e^{\mathfrak{D}_{A}} \cdot e^{\frac{i}{2} \int{\chi \cdot \mathbf{F}} + \frac{i}{2} \int{A \cdot \left( \mathbf{D}_{c} \right)^{-1} \cdot A} } \\ \nonumber \quad &\cdot& \left. \mathbf{G}_{c}^{\mathrm{I}}(x_{1}, y_{1} | A) \cdot  \mathbf{G}_{c}^{\mathrm{I\!I}}(x_{2}, y_{2} | A)  \cdot {e^{\mathbf{L}_{\mathrm{c}}[A]}} \right|_{A \rightarrow 0 },
\end{eqnarray}

\noindent where
\begin{equation}
\mathfrak{D}_{A} = - \frac{i}{2} \int {\mathrm{d}}^4x\int {\mathrm{d}}^4y{\frac{\delta}{\delta A^{a}_{\mu}(x)} \,  \mathbf{D}_{c,}{}^{ab}_{\mu \nu}(x-y) \, \frac{\delta}{\delta A^{b}_{\nu}(y)} },
\end{equation}

\noindent $\exp[\mathfrak{D}_{A}]$ is the "linkage operator" which links all pairs of $A_{\mu}^{a}$ to each other and provides the complete panoply of relevant Feynman graphs, $\mathbf{D}_{c,}{}^{ab}_{\mu \nu}$ is the gluon propagator in an arbitrary (relativistic) gauge, $\int{\mathrm{d}[{\chi}]}$ is Halpern's functional integral~\cite{Halpern1977a,Halpern1977b} which allows one to include all cubic and quartic gluon interactions, and $\mathcal{N}$ is a normalization constant such that
\begin{equation*}
\mathcal{N} \int{\mathrm{d}[{\chi}] \, e^{\frac{i}{4} \int{\chi^{2}}}} = 1.
\end{equation*}

\noindent The form of (\ref{Eq:3}), and the mechanism which generates a gauge-invariant result, have been derived, in detail, in Ref.~\cite{Fried2009_QCD1} and \cite{Fried2010_QCD2}, and need not be repeated here.

The Fradkin representations we use are exact variants of those introduced a half-century ago~\cite{Fradkin1966}, which have been re-derived and discussed in detail elsewhere~\cite{HMF2}.  In particular, we find it more convenient to employ a functional integral (FI) representation involving a space-time coordinate $u_{\mu}(s')$ rather than the original, functional differential representation of Fradkin, wherein his $v_{\mu}(s')$ denoted the proper-time-dependent 4-velocity of a real or virtual fermion; the relation between these two quantities is simply:
\begin{equation}\label{Eq:u}
u_{\mu}(s') = \int_{0}^{s'}{\mathrm{d}s'' \, v_{\mu}(s'')},
\end{equation}

\noindent where both $u_{\mu}(s')$ and $v_{\mu}(s')$ are understood to be continuous functions of their proper-time, although no such restriction is placed upon higher derivatives of $v_{\mu}(s')$.  The Fradkin representation of  $\mathbf{G}_{c}(x,y|A)$ is then
\begin{eqnarray}\label{Eq:4}
& & \mathbf{G}_{c}(x,y|A) \\ \nonumber &=&  i \int_{0}^{\infty}{\mathrm{d}s \ e^{-is m^{2}}} \, e^{- \frac{1}{2}  {\mathrm{Tr}\, }{\ln{\left( 2h \right)}} } \\ \nonumber & & \quad \times \mathcal{N}' \, \int{\mathrm{d}[u]} \, e^{ \frac{i}{2} \int{ u \cdot (2h)^{-1} \cdot u } }  \\ \nonumber & & \quad \times {\left[ m - \gamma_{\mu} \frac{\delta}{\delta u'_{\mu}(s)} \right]} \, \delta^{(4)}(x-y + u(s)) \\ \nonumber & & \quad \times \left( e^{ -ig \int_{0}^{s}{\mathrm{d}s' \, u'_{\mu}(s') \, A_{\mu}^{a}(w) \, \tau^{a}} + g \int_{0}^{s}{\mathrm{d}s' \sigma_{\mu \nu} \, \mathbf{F}_{\mu \nu}^{a}(w) \, \tau^{a}}} \right)_{+},
\end{eqnarray}

\noindent where
\begin{eqnarray*}
w &=& y-u(s'), \\
h(s_{1},s_{2}) &=& s_{2}\theta(s_{1} - s_{2})  + s_{1}\theta(s_{2} - s_{1}) , \\
h^{-1}(s_{1},s_{2}) &=& \frac{\partial}{\partial s_{1}} \frac{\partial}{\partial s_{2}} \delta(s_{1} - s_{2}), \\
\mathcal{N}^{\prime -1} &=& \int{\mathrm{d}[u] \, e^{ \frac{i}{2} \int{ u \cdot (2h)^{-1} \cdot u } }},
\end{eqnarray*}

\noindent and $m$ is the (bare) quark mass.

Explicit $A$-dependence can be extracted from the ordered exponential (OE) of (\ref{Eq:4}), and as in Ref.~\cite{Fried2009_QCD1}, this is illustrated by suppressing the spin-dependent part of the OE. This simplification is done simply for ease of presentation, since conclusions based upon EL are left unchanged~\cite{Fried2010_QCD2}.  We therefore consider the simple OE
\begin{eqnarray*}
& & \left( e^{ -ig \int_{0}^{s}{\mathrm{d}s' \, u'_{\mu}(s') \, A_{\mu}^{a}(y - u(s')) \, \tau^{a}} } \right)_{+} \\ \nonumber &=& \mathcal{N}'' \, \int{\mathrm{d}[\Omega] \, \int{\mathrm{d}[\alpha] \, e^{i\int_{0}^{s}{\alpha \cdot \Omega} } \, \left( e^{ -i \int_{0}^{s}{ \alpha \cdot \tau} } \right)_{+} }} \\ \nonumber & & \times
e^{ -ig \int_{0}^{s}{\mathrm{d}s' \, u'_{\mu}(s') \, A_{\mu}^{a}(y - u(s')) \, \Omega^{a}(s')} },
\end{eqnarray*}

\noindent where $\mathcal{N}''$ is that normalization constant needed for the delta-functional defined by the FI $\int{\mathrm{d}[\Omega]}$.

Because of its definition in terms of $v_{\mu}(s')$, and because of the $\delta^{(4)}(x-y + u(s))$ appearing in (\ref{Eq:4}), one has the "boundary conditions", $u_{\mu}(0)=0$ and $u_{\mu}(s)= - (x-y)_{\mu}$.  With or without that neglected spin-dependence of (\ref{Eq:4}), it is clear that the $A$-dependence of $\mathbf{G}_{c}(x,y|A)$ is not more complicated than Gaussian. This means -- suppressing the $\mathbf{L}[A]$ functional momentarily -- that the needed linkage operations of (\ref{Eq:3}) can be performed exactly, expressing the result in terms of the Halpern and Fradkin FI representations.  As shall become clear immediately, the Fradkin FI representation for $\mathbf{L}[A]$ is also not more complicated than Gaussian: If a simple expansion in powers of $\mathbf{L}$ is adopted, the $A$-dependence of every term in that expansion will not be more complicated than Gaussian either. In brief, the statement that the linkages can be calculated exactly means that the sum over all Feynman graphs is reduced to the summation of a set of FI's~\cite{Fried2010_QCD2}.

After such linkages have been performed, EL appears, which has the extraordinary and simplifying effect of converting the Halpern FI into a set of ordinary integrals.  If the most convenient cluster expansion is used for the linkage operation upon $\exp(\mathbf{L}[A])$, the previous expansion in powers of $\mathbf{L}[A]$ is brought into a much simpler form, and in the end, one need calculate a handful of overlapping proper-time integrals to achieve a reasonable estimate of the desired Physics.  In this paper such detail is not necessary, for to illustrate the distinction between "ideal" and "realistic" QCD, one can simply suppress all $\mathbf{L}[A]$ dependence, using, in effect, a quenched approximation.  In the subsequent papers~\cite{Fried2011_QCD4,Fried2011_QCD5}, explicit examples will be given for the construction of binding and scattering potentials, with and without quenching.

For completeness, we here state the Fradkin representation for $\mathbf{L}[A]$,
\begin{eqnarray}\label{Eq:5}
& & \mathbf{L}[A] \\ \nonumber &=&  - \frac{1}{2} \int_{0}^{\infty}{\frac{\mathrm{d}s}{s} \, e^{-is m^{2}}} \, e^{- \frac{1}{2}  {\mathrm{Tr}\, }{\ln{(2h)}} } \\ \nonumber && \quad \times \int{\mathrm{d}^{4}x} \, \mathcal{N}^{\prime} \, \int{\mathrm{d}[u]} \, \delta^{(4)}(u(s)) \, e^{ \frac{i}{2} \int{ u \cdot (2h)^{-1} \cdot u} } \\ \nonumber & & \quad \times \left[  {\mathrm{tr}\, }{\left( e^{ -ig \int_{0}^{s}{\mathrm{d}s' \, \left[ u'_{\mu}(s') \, A_{\mu}^{a}(w) + i \sigma_{\mu \nu} \, \mathbf{F}_{\mu \nu}^{a}(w) \right] \, \tau^{a}}} \right)_{+}} - 1 \right],
\end{eqnarray}

\noindent which in contrast to (\ref{Eq:4}) contains a Trace over all Dirac and color variables, as well as a four-dimensional $\int{d^{4}x}$ over this closed loop.  It should also be mentioned that $\mathbf{L}[A]$ is rigorously gauge-invariant under the full gauge group of QCD; an original and simple proof may be found in Ref.~\cite{HMF2}.

With the two simplifying assumptions noted above, we write the $A$-dependence of the exponential of the pair of Green's functions of (\ref{Eq:3}), with one $\mathbf{G}_{c}[A]$ representing Q and the other $\bar{\mathrm{Q}}$,
\begin{eqnarray}\label{Eq:6}
\tilde{\mathcal{Q}}_{\mu}^{a} =  &-& g \int_{0}^{s}{\mathrm{d}s_{1} \, u'_{\mu}(s_{1}) \, \Omega^{a}(s_{1}) \, \delta^{(4)}(w - y_{1} + u(s_{1})) } \\ \nonumber &-& g \int_{0}^{\bar{s}}{\mathrm{d}s_{2} \, \bar{u}'_{\mu}(s_{2}) \, \bar{\Omega}^{a}(s_{2}) \, \delta^{(4)}(w - y_{2} + \bar{u}(s_{2})) },
\end{eqnarray}
where the two lines of (\ref{Eq:6}) refer to distinct contributions from each $\mathbf{G}_{c}[A]$. In addition, there is the $A$-dependence from the Halpern FI,
\begin{eqnarray}\label{Eq:chiF}
\int{\mathrm{d}^{4}w \  \chi \cdot \mathbf{F}} = 2 \int{\mathrm{d}^{4}w \, (\partial_{\nu} \chi_{\mu\nu}^{a}) \, A_{\mu}^{a} } + g f^{abc} \int{\mathrm{d}^{4}w \, A_{\mu}^{b} \chi_{\mu\nu}^{a} A_{\nu}^{c} }.
\end{eqnarray}

\noindent The linkage operation to be performed is then
\begin{eqnarray}\label{Eq:7}
\left. e^{-\frac{i}{2} \int{\frac{\delta}{\delta A} \cdot \mathbf{D}_{c} \cdot \frac{\delta}{\delta A} } } \cdot e^{\frac{i}{2}\int{A \cdot \mathcal{K} \cdot A} + i \int{A \cdot (\tilde{\mathcal{Q}} + 2 \partial \chi)}} \right|_{A \rightarrow 0},
\end{eqnarray}

\noindent where $\langle w_{1} | \mathcal{K}_{\mu\nu}^{ab} | w_{2} \rangle = g f^{abc} \langle w_{1} | \chi_{\mu\nu}^{c} | w_{2} \rangle + \langle w_{1} | (\mathbf{D}_{c}^{-1})_{\mu\nu}^{ab} | w_{2} \rangle$.  The operation of (\ref{Eq:7}) is equivalent to a normalized Gaussian FI, and yields
\begin{eqnarray}\label{Eq:8}
e^{-\frac{1}{2}  {\mathrm{Tr}\, }{\ln{(1 - \mathcal{K} \cdot \mathbf{D}_{c})}} } \cdot e^{\frac{i}{2}\int{ (\tilde{\mathcal{Q}} + 2 \partial \chi) \cdot \mathbf{D}_{c} \cdot \left[ 1 - \mathcal{K} \cdot \mathbf{D}_{c} \right]^{-1} \cdot (\tilde{\mathcal{Q}} + 2 \partial \chi)}} ,
\end{eqnarray}

\noindent using an obvious notation.  And here is where a remarkable mechanism becomes apparent, since
\begin{eqnarray*}
\left[ 1 - \mathcal{K} \cdot \mathbf{D}_{c} \right]^{-1} &=& \left[ 1 - \left(\mathbf{D}_{c}^{-1} + g f \cdot \chi \right) \cdot \mathbf{D}_{c} \right]^{-1} \\ \nonumber &=& \left[ - g (f \cdot \chi) \cdot  \mathbf{D}_{c} \right]^{-1} \\ \nonumber &=& \mathbf{D}_{c}^{-1} \cdot  \left[ - g (f \cdot \chi) \right]^{-1},
\end{eqnarray*}

\noindent so that $\mathbf{D}_{c} \cdot \left[ 1 - \mathcal{K} \cdot \mathbf{D}_{c} \right]^{-1} = - \frac{1}{g} (f \cdot \chi)^{-1} $, and the gauge-dependent propagator has disappeared from the interaction, remaining only in the factor
\begin{eqnarray*}
e^{-\frac{1}{2}  {\mathrm{Tr}\, }{\ln{(1 - \mathcal{K} \cdot \mathbf{D}_{c})}} } = \left[ \det{(-g f \cdot \chi)} \right]^{-\frac{1}{2}} \cdot \left[ \det{\mathbf{D}_{c}} \right]^{-\frac{1}{2}},
\end{eqnarray*}

\noindent so that the $\left[ \det{\mathbf{D}_{c}} \right]^{-\frac{1}{2}}$ can be safely absorbed into an overall normalization.

We refer the reader to Ref.~\cite{Fried2009_QCD1} for details of exact and approximate evaluations of  $\langle w_{1} | (f \cdot \chi)^{-1} | w_{2} \rangle$.  The important point we here wish to stress comes from the property of locality \begin{eqnarray}\label{Eq:9}
\langle w_{1} | \left. (f \cdot \chi)^{-1} \right|_{\mu\nu}^{ab} | w_{2} \rangle = \left. (f \cdot \chi(w_{1}))^{-1} \right|_{\mu\nu}^{ab} \, \delta^{(4)}(w_{1} - w_{2})
\end{eqnarray}

\noindent We shall call the locality apparent in (\ref{Eq:9}) "Effective Locality" (EL), for it represents a contrast to the way in which ordinary propagators convey information, from one space-time point to another, from $w_{1}$ to $w_{2}$.  Here, (\ref{Eq:9}) is the object which replaces the conventional (perturbative) propagator $\mathbf{D}_{c}(w_{1} - w_{2})$. In Ref.~\cite{Fried2010_QCD2}, this locality is shown to hold for a large family of QCD processes; it simplifies all calculations tremendously, reducing Halpern's FI to a finite set of ordinary integrals, which can be studied numerically.

We illustrate the effects of EL in this paper by making two simplifications, in the interest of clarity:
\begin{enumerate}
  \item Neglect the $\partial \chi$ term compared to $\tilde{\mathcal{Q}}$, which approximation, in view of (\ref{Eq:6}) and (\ref{Eq:chiF}), one would make in a strong-coupling limit. This is just a convenient step for clarity and simplicity of presentation. Most importantly, conclusions are completely independent of this simplification.
  \item The product of the two $\tilde{\mathcal{Q}}$ terms of (\ref{Eq:8}) contains the self-interactions of each particle, as well as the cross-terms, corresponding to the interactions between the particles.  Again, for reasons of simplicity and clarity, we retain only the cross-terms, writing the relevant exponential factor of (\ref{Eq:8}) as
\begin{eqnarray}\label{Eq:10}
& & + \frac{i}{2} g \int{\mathrm{d}^{4}w_{1} \, \int_{0}^{s}{\mathrm{d}s_{1} \, \int_{0}^{\bar{s}}{\mathrm{d}s_{2} \, u'_{\mu}(s_{1}) \, \bar{u}'_{\nu}(s_{2}) \,}}} \\ \nonumber & & \quad \times \, \Omega^{a}(s_{1}) \, \bar{\Omega}^{b}(s_{2}) \,  \left. (f \cdot \chi(w_{1}))^{-1} \right|_{\mu\nu}^{ab} \\ \nonumber & & \quad \times \, \delta^{(4)}(w_{1} - y_{1} + u(s_{1})) \, \delta^{(4)}(w_{1} - y_{2} + \bar{u}(s_{2})),
\end{eqnarray}

\noindent where $u_{\mu}$, $\Omega^{a}$, and $s$ are variables associated with $\mathbf{G}_{c}^{\mathrm{I}}(x_{1}, y_{1}| A)$, whereas $\bar{u}_{\nu}$, $\bar{\Omega}^{b}$, and $\bar{s}$ are with $\mathbf{G}_{c}^{\mathrm{I\!I}}(x_{2}, y_{2}| A)$.  The last line of (\ref{Eq:10}) may be written as
\begin{eqnarray}\label{Eq:11}
\delta^{(4)}(w_{1} - y_{1} + u(s_{1})) \, \delta^{(4)}(y_{1} - y_{2} + \bar{u}(s_{2}) - u(s_{1}))
\end{eqnarray}
\noindent and one sees that, as a consequence of EL, this interaction is associated only with space-time point $w_{1}$.  This means that all the other $w \neq w_{1}$ are irrelevant to the interaction, and as a most important consequence, they are removed, along with their normalization factors, leaving dependence only upon the ordinary integral $\int{\mathrm{d}^{n} \chi(w_{1})}$. Further details are given in Appendix~\ref{AppA}.
\end{enumerate}

The point central to the argument of this article is now being reached, as one seeks to evaluate the support of the second delta-function in (\ref{Eq:11}), which it is convenient to display as a product of delta-functions, in time, longitudinal and transverse coordinates,
\begin{eqnarray}\label{Eq:12}
& & \delta(y_{10} - y_{20} + \bar{u}_{0}(s_{2}) - u_{0}(s_{1}))  \\ \nonumber & & \quad \times \, \delta(y_{1\mathrm{L}} - y_{2\mathrm{L}} + \bar{u}_{\mathrm{L}}(s_{2}) - u_{\mathrm{L}}(s_{1})) \\ \nonumber & & \quad \times \, \delta^{(2)}(\vec{y}_{1\perp} - \vec{y}_{2\perp} + \vec{\bar{u}}_{\perp}(s_{2}) - \vec{u}_{\perp}(s_{1})).
\end{eqnarray}

\noindent In the CM of Q and $\bar{\mathrm{Q}}$, one can choose the origin of each time coordinate as the time of their closest approach; the time difference $y_{10} - y_{20}$ is then always zero.  If the Q and $\bar{\mathrm{Q}}$ are scattering, then $y_{1\mathrm{L}} + y_{2\mathrm{L}} = 0$, since their longitudinal projections are in opposite directions; alternatively, if the Q and $\bar{\mathrm{Q}}$ are bound together, then $y_{1\mathrm{L}} = y_{2\mathrm{L}}$, and their difference vanishes.  Either choice makes no difference at all to the following analysis, and the simplest, second possibility can be adopted, that is, $y_{1\mathrm{L}} - y_{2\mathrm{L}} = 0$.

Consider the time-coordinate delta-function, $\delta(\bar{u}_{0}(s_{2}) - u_{0}(s_{1}))$, which can have a zero argument whenever $\bar{u}_{0}(s_{2})$ and $u_{0}(s_{1})$ coincide.  Assume this happens at a set of points $s_{l}$, so that
\begin{eqnarray}\label{Eq:13}
& & \delta(\bar{u}_{0}(s_{2}) - u_{0}(s_{1}))  \\ \nonumber &=& \sum_{\ell}{\delta(\bar{u}_{0}(s_{\ell}) - u_{0}(s_{1}) + (s_{2} - s_{\ell}) \cdot \bar{u}'_{0}(s_{\ell}) + \cdots)} \\ \nonumber &=& \sum_{\ell}{ \left. \frac{1}{|\bar{u}'_{0}(s_{\ell})|} \, \delta(s_{2} - s_{\ell}) \right|_{\bar{u}_{0}(s_{\ell}) = u_{0}(s_{1})}}.
\end{eqnarray}

\noindent In a similar way, the longitudinal delta-function may be evaluated as
\begin{eqnarray}\label{Eq:14}
\sum_{m}{ \left. \frac{1}{|u'_{\mathrm{L}}(s_{m})|} \, \delta(s_{1} - s_{m}) \right|_{u_{\mathrm{L}}(s_{m}) = \bar{u}_{\mathrm{L}}(s_{2}) \rightarrow  \bar{u}_{\mathrm{L}}(s_{\ell})}},
\end{eqnarray}

\noindent and the product of (\ref{Eq:13}) and (\ref{Eq:14}) as,
\begin{eqnarray}\label{Eq:15}
\sum_{\ell , m}{ \frac{1}{|u'_{\mathrm{L}}(s_{m})|} \, \frac{1}{|\bar{u}'_{0}(s_{\ell})|} \, \delta(s_{1} - s_{m}) \, \delta(s_{2} - s_{\ell}) }
\end{eqnarray}

\noindent under the restrictions $u_{0}(s_{m}) = \bar{u}_{0}(s_{\ell})$ and $u_{\mathrm{L}}(s_{m}) = \bar{u}_{\mathrm{L}}(s_{\ell})$.  Now, $u_{0}$ and $\bar{u}_{0}$, and $u_{\mathrm{L}}$ and $\bar{u}_{\mathrm{L}}$ are continuous but otherwise completely arbitrary functions: The probability that the intersections of $u_{0}(s_{1})$ with $\bar{u}_{0}(s_{2})$ and of $u_{\mathrm{L}}(s_{1})$ with $\bar{u}_{\mathrm{L}}(s_{2})$ occur at exactly the same points is therefore arbitrarily small.  The only place where all four of these continuous functions have the same value is at $s_{1} = s_{2} = 0$, where, by definition of these functions, $u_{\mu}(0) = \bar{u}_{\mu}(0) = 0$, and hence (\ref{Eq:15}) reduces to
\begin{eqnarray}\label{Eq:16}
\frac{1}{|u'_{\mathrm{L}}(0)|} \, \frac{1}{|\bar{u}'_{0}(0)|} \, \delta(s_{1}) \, \delta(s_{2})
\end{eqnarray}

\noindent where, it may be noted, corresponding $u'_{\mathrm{L}}(0)$ and $\bar{u}'_{0}(0)$ appearing in (\ref{Eq:10}) combine with those of (\ref{Eq:16}) to generate the signs, $\epsilon(u'_{\mathrm{L}}(0))$ and $\epsilon(\bar{u}'_{0}(0))$, which may be specified in comparison with those of an eikonal model.

For being crucial to the main argument of this article, it matters to support the heuristic derivation given above with a rigorous mathematical proof.  Since the Wiener functional space~\cite{Lapidus2000} is the most adopted realization of a functional space, the proof will be given in this framework. Here, this space is made out of proper-time parameterized paths: For example, starting from $y_{1\mathrm{L}}$ at $s_1=0$, one will consider paths passing through $u_{\mathrm{L}}(s_{1})$ at proper-time $s_{1} \in \, ]0,s]$; and likewise, paths starting from $y_{2}$ at $s_{2} = 0$ and passing through $\bar{u}_{\mathrm{L}}(s_{2})$ at proper-time $s_{2} \in \, ]0, \bar{s}]$.

Following the notation convention of Ref.~\cite{Lapidus2000}, $\mathcal{C}^{0,{s}}_0$ is the space ${\mathcal{C}}(]0, s],{\mathbb{R}})$ of continuous functions $u(s_{1})$ from the interval $]0, {s}]$ into ${\mathbb{R}}$, satisfying ${u}(0)=0$; and similar for "barred" quantities.  The $c$-functions $u(s_{1})$ and $\bar{u}(s_{2})$ are here understood to represent either of the 2 possibilities $u_{0}$ and $u_{\mathrm{L}}$.

Endowed with the Wiener measure $m$, $\mathcal{C}^{0,{s}}_0$ is the measurable space, and with the measure $m\!\otimes\! m$, so is the product space $\mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0$ endowed with the topology product, because $u$ and $\bar{u}$ are independent.

Then, we have the following,

\noindent \textit{Theorem:}

For all pair $(s_1,s_2) \in \, ]0,{s}] \, \times \, ]0,\bar{s}]$, one has
\begin{equation}\label{Eq:16A}
m \!\otimes\! m \!\left( \left\{ (u, \bar{u})\in \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \, | \, u(s_1)=\bar{u}(s_2)\right\} \right) = 0
\end{equation}

\noindent whereas
\begin{equation}\label{Eq:16B}
m \!\otimes\! m \!\left( \left\{ (u, \bar{u}) \in \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \, | \, u(0)=\bar{u}(0)=0 \right\} \right) = \left[ m\!\left(\left\{ u \in \mathcal{C}^{0,{s}}_0  \, | \, u(0)=0 \right\} \right) \right]^{2}=1
\end{equation}

\noindent The proof is as follows. Let $\mathrm{B}$ be the set $\left\{ (u, \bar{u}) \in \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \, | \, u(s_1)=\bar{u}(s_2) \right\}$.  One has $\mathrm{B} = \bigcap_{n=1}^{\infty} \mathrm{B}_{n}$, where
\begin{equation}\label{Eq:16C}
\mathrm{B}_{n} = \left\{ (u, \bar{u}) \in \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \, | \, {-\frac{1}{n}} \leq u(s_1)-\bar{u}(s_2) \leq +\frac{1}{n} \right\}
\end{equation}

\noindent Because of the obvious inclusion,
\begin{equation}\label{Eq:16D}
\mathrm{B}_{n+1} \subset \mathrm{B}_{n}, \quad \forall n,
\end{equation}

\noindent  one can write
\begin{equation}\label{Eq:16E}
m \!\otimes\! m(\mathrm{B}) = \lim_{n \rightarrow \infty} m \!\otimes\! m(\mathrm{B}_n).
\end{equation}

\noindent Next, one has~\cite{Lapidus2000}
\begin{eqnarray}\label{Eq:165}
m \!\otimes\! m (\mathrm{B}_n) &=& m \!\otimes\! m \!\left( \left\{ (u, \bar{u}) \in \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \, | \,  u(s_1) - \frac{1}{n} \leq \bar{u}(s_2) \leq u(s_1)+\frac{1}{n} \right\} \right) \\ \nonumber &=& \int_{-\infty}^{+\infty}{\frac{{\mathrm{d}}x}{\sqrt{2\pi s_1}} \, e^{-\frac{x^2}{2s_1}} \, \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} { \frac{{\rm{d}}y}{\sqrt{2\pi s_2}} \, e^{-\frac{y^2}{2s_2}} }} \\ \nonumber &=&  \int_{-\infty}^{+\infty}{\frac{{\rm{d}}x}{\sqrt{2\pi s_1}} \, e^{-\frac{x^2}{2s_1}} \, f_n(x)}.
\end{eqnarray}

\noindent Now, since
\begin{eqnarray}\label{Eq:166}
|\frac{1}{{\sqrt{2\pi s_1}}}\ e^{-\frac{x^2}{2s_1}}\ f_n(x)|\leq \frac{1}{{\sqrt{2\pi s_1}}}\ e^{-\frac{x^2}{2s_1}},
\end{eqnarray}

\noindent the \textit{dominated convergence theorem} can be used so as to take the limit $n\rightarrow \infty$ under the integral and get zero, which establishes (\ref{Eq:16A}), whereas (\ref{Eq:16B}) comes from the normalization procedure of the measure space $\left( m\!\otimes\! m, \mathcal{C}^{0,{s}}_0 \!\times \mathcal{C}^{0,{\bar{s}}}_0 \right)$ into a probability space; the Theorem is then demonstrated.
\par
The product of the first two delta distributions of (\ref{Eq:12}) is therefore proportional to $\delta(s_1)\delta(s_2)$, and further dimensional and symmetry arguments determine the overall multiplicative constant such as in (\ref{Eq:16}). One can observe that this determination of (\ref{Eq:12}) fits the eikonal approximated evaluation of Ref.~\cite{Fried2009_QCD1}.

It is the remaining, transverse delta-function of (\ref{Eq:12}) which is now most relevant, the term $\delta^{(2)}(\vec{y}_{1\perp} - \vec{y}_{2\perp}) = \delta^{(2)}(\vec{b})$, where $\vec{b}$ denotes the impact parameter, or transverse distance between the two particles.  This $\delta^{(2)}(\vec{b})$ is sitting in the exponential of (\ref{Eq:8}), and the question immediately arises as to what meaning it can be assigned.  Depending on its argument, a delta function is either zero or infinite: In the first case this means that there is no interaction, while the second case means that at $\vec{b}=0$, one has an infinite phase factor, suggestive of hard disc scattering~\cite{Schiff}.

The relevant question is therefore why such a delta-function $\delta^{(2)}(\vec{b})$ appears at all.  The answer is that the assumption has earlier been made, and in the conventional, Abelian way, that the $Qs$ and $\bar{Q}s$  may be treated as ordinary particles, whereas it is by now well-known that asymptotic $Qs$ and $\bar{Q}s$ exist only in bound states, and that their transverse coordinates cannot be specified with arbitrary precision. It is therefore unreasonable and unphysical to retain the conventional Abelian practice in which such measurement is assumed possible. This is the interpretation that will be posited here, taking the $\delta^{(2)}(\vec{b})$ outcome as a serious warning that some odd working hypothesis has popped out in exactly this way.

One may wonder why this happens in QCD.  Because QCD possesses EL, which conventional Abelian theories do not. The latter display sums over interconnected propagators, which provide a certain vagueness of position, whereas in the exact non-perturbative QCD, as described above and in Ref.~\cite{Fried2010_QCD2}, one finds the sharp determination of delta-functions corresponding to the EL property, and transverse imprecision must therefore be introduced separately.

In Ref.~\cite{Fried2009_QCD1}, it was suggested that this difficulty be treated in an \emph{ad hoc} phenomenological way, by replacing $\delta^{(2)}(\vec{b})$ by the smoothly varying, effective Gaussian
\begin{eqnarray*}
\varphi(\vec{b})=(2\pi)^{-2} \, \int{\mathrm{d}^{2}\vec{k} \, e^{i \vec{k} \cdot \vec{b} - \frac{k^{2}}{4\mu^{2}}}},
\end{eqnarray*}

\noindent where $\mu$ is a mass parameter on the order of the Q-$\bar{\mathrm{Q}}$ bound state (which we shall call a "model pion"), although we were able to draw the conclusion of that paper without specifying the precise form of $\varphi(\vec{b})$.  In this article, we face this question directly, by first developing a formalism in which transverse quark coordinates cannot be specified, and then showing how this formalism removes all such absurdities, such as that of the exponential factor of $\delta^{(2)}(\vec{b})$ above.  But it must be emphasized that our prescription for such a "realistic" QCD is phenomenological, for there remains to be shown how such an approach could be derived from a more fundamental, operator-field version of QCD, in which transverse imprecision would occur automatically, perhaps in relation to a possible non-commutative geometrical phase of non-perturbative QCD~\cite{Kouneiher2010}.

In a following article~\cite{Fried2011_QCD4}, it will be shown how a specific choice of $\varphi(\vec{b})$, slightly but importantly distinguishable from Gaussian, can serve to define physically reasonable quark binding, and in a subsequent paper~\cite{Fried2011_QCD5}, these techniques will be extended to the construction of nucleon-nucleon scattering and binding potentials.

\section{Towards a possible transverse adapted form of QCD}\label{sec:RealisticQCD}

Perhaps the simplest way of introducing transverse imprecision is to average that part of the QCD Lagrangian dealing with the quark-gluon interaction, so that the transverse position of the color-charge current operator $\bar{\psi} \, \gamma_{\mu} \tau^{a} \, \psi(x)$ should be averaged over a small range by means of an initially unspecified distribution.  One can also demand the same imprecision for the vector current $\bar{\psi} \, \gamma_{\mu} \, \psi(x)$ and scalar density $\bar{\psi} \psi(x)$, but these extra requirements seem to complicate the presentation, to no real advantage, and will not be considered here.

Instead of the conventional quark-gluon contribution to the Lagrangian density,
\begin{eqnarray}\label{Eq:17}
\mathcal{L}_{\mathrm{QG}} = - \bar{\psi} \, [m + \gamma_{\mu} \, (\partial_{\mu} - i g A_{\mu}^{a} \tau^{a})] \, \psi,
\end{eqnarray}

\noindent in which all field operators occur at the same space-time point, and for which gauge invariance under the standard QCD gauge transformations is obvious, we now adopt a local -- in time and longitudinal position -- but non-local in its transverse coordinates replacement,
\begin{eqnarray}\label{Eq:18}
\mathcal{L}_{\mathrm{QG}} &=& - \int{\mathrm{d}^{2} \vec{x}'_{\perp} \, \mathfrak{a}(\vec{x}_{\perp} - \vec{x}'_{\perp}) } \\ \nonumber & & \quad \times \bar{\psi}(x') \, [m + \gamma_{\mu} \, (\partial'_{\mu} - i g A_{\mu}^{a}(x) \tau^{a})] \, \psi(x'),
\end{eqnarray}

\noindent where the transverse imprecision function (TIF) $\mathfrak{a}(\vec{x}_{\perp} - {\vec{x}}'_{\perp})$ is a real, symmetric function of its arguments, of significant value only for distances on the order of the inverse of the pion mass, $x'_{\mu} = (x_{0}, x_{\mathrm{L}}; \vec{x}'_{\perp})$, $\partial'_{\mu} = (\frac{1}{i} \frac{\partial}{\partial x_{0}}, \frac{\partial}{\partial x_{\mathrm{L}}}; \frac{\partial}{\partial \vec{x}'_{\perp}})$, and $A_{\mu}^{a}(x)$ is left untouched.  In this formulation, rigorous local gauge-invariance is suppressed for the underlying quark fields, whose quanta have unmeasurable transverse positions, but the hadrons all constructed from these quanta will nevertheless be proper singlets under SU(3).

One notes that in the contribution of (\ref{Eq:18}) to its part of the Action operator, $\int{\mathrm{d}^{4}x \, \mathcal{L}_{\mathrm{QG}} }$, the $\vec{x}_{\perp}$ and $\vec{x}'_{\perp}$ coordinates can be interchanged, which yields an equivalent form in which every $A_{\mu}^{a}(x)$ of the original (\ref{Eq:17}) is replaced by $\int{\mathrm{d}^{2}\vec{x}'_{\perp} \, \mathfrak{a}(\vec{x}_{\perp} - \vec{x}'_{\perp}) \, A_{\mu}^{a}(x') }$. This interchange allows a very simple extraction of all such transverse imprecision, since both delta-functions of (\ref{Eq:10}) will now be replaced by
\begin{eqnarray}\label{Eq:19}
& & \int{\mathrm{d}^{2} {\vec{y}}_{1\perp}^{\, '} \, \mathfrak{a}(\vec{y}_{1\perp} - \vec{y}_{1\perp}^{\, '}) \, \int{\mathrm{d}^{2}\vec{y}^{\, '}_{2\perp} \, \mathfrak{a}(\vec{y}_{2\perp} - \vec{y}^{\, '}_{2\perp}) \, }}\\ \nonumber & & \quad \times  \delta^{(4)}(w_{1} - y'_{1} + u(s_{1})) \,  \delta^{(4)}(w_{1} - y'_{2} + \bar{u}(s_{2})).
\end{eqnarray}

\noindent This can be seen in the simplest way by noting that the presence of a TIF $\mathfrak{a}(\vec{y}_{\perp} - \vec{y}^{\, '}_{\perp})$ shall mean that the difference $|\vec{y}_{\perp} - \vec{y}^{\, '}_{\perp}|$ is effectively bounded by $1 / {\mu}$, where $\mu$ appears in the definition of $\varphi(b)$ below, and where, as shown in Ref.~\cite{Fried2011_QCD4}, $\mu$ is on the order of the pion mass, $m_{\pi}$.  Hence a negligible error is made by replacing the transverse coordinate $w = y'_{\perp}$ by $y_{\perp}$.  A more precise justification appears in Appendix~\ref{AppB}.  It should be noted that this approximation concerning the utility of EL can be justified in the same way as described in Appendix~\ref{AppB} for both the construction of the quark binding potential and the nucleon-nucleon scattering and binding potential (the exchange of a gluon bundle across a closed quark loop apparently involves a new set of $\int{\mathrm{d}^{n}\chi}$ integrals, and will be discussed elsewhere).

The first delta-function of (\ref{Eq:19}) defines the argument $w_{1}$ of $\chi(w_{1})$, and we again observe that the final output of the Halpern FI will be an ordinary integral $\int{\mathrm{d}^{n}\chi}$, independent of the choice of $w_{1}$.  The second delta-function of (\ref{Eq:19}) now involves the $\mathfrak{a}$-dependence, generating in place of the $\delta^{(2)}(\vec{y}_{1\perp} - \vec{y}_{2\perp})$ which follows from (\ref{Eq:16}), the combination
\begin{eqnarray}\label{mu}
& & \int{\mathrm{d}^{2}\vec{y}^{\, '}_{1\perp} \, \int{\mathrm{d}^{2}\vec{y}^{\, '}_{2\perp} \, \mathfrak{a}(\vec{y}_{1\perp} - \vec{y}^{\, '}_{1\perp}) \, \mathfrak{a}(\vec{y}_{2\perp} - \vec{y}^{\, '}_{2\perp}) }} \, \delta^{(2)}(\vec{y}^{\, '}_{1\perp} - \vec{y}^{\, '}_{2\perp}) \\ \nonumber &=& \int{\mathrm{d}^{2}\vec{y}^{\, '}_{\perp} \, \mathfrak{a}(\vec{y}_{1\perp} - \vec{y}^{\, '}_{\perp}) \, \mathfrak{a}(\vec{y}_{2\perp} - \vec{y}^{\, '}_{\perp})}.
\end{eqnarray}

\noindent Inserting 2-dimensional Fourier transforms of each
\begin{eqnarray*}
\mathfrak{a}(\vec{y}_{\perp} - \vec{y}^{\, '}_{\perp}) = \int{\frac{\mathrm{d}^{2}\vec{k}}{(2\pi)^{2}} \, e^{i\vec{k} \cdot (\vec{y}_{\perp} - \vec{y}^{\, '}_{\perp})} \, \tilde{\mathfrak{a}}(\vec{k}_{\perp})},
\end{eqnarray*}

\noindent the combination (\ref{mu}) becomes
\begin{eqnarray}\label{Eq:20}
\int{\frac{\mathrm{d}^{2}\vec{k}}{(2\pi)^{2}} \, \tilde{\mathfrak{a}}(\vec{k}) \, \tilde{\mathfrak{a}}(-\vec{k}) \, e^{i\vec{k} \cdot (\vec{y}_{1\perp} - \vec{y}_{2\perp})} }.
\end{eqnarray}

\noindent From its definition, $\mathfrak{a}$ is real, and hence (\ref{Eq:20}) becomes
\begin{eqnarray}\label{Eq:21}
\int{\frac{\mathrm{d}^{2}\vec{k}}{(2\pi)^{2}} \, \, e^{i\vec{k} \cdot \vec{b}} \, \left|\tilde{\mathfrak{a}}(\vec{k})\right|^{2} } \equiv \varphi(\vec{b}),
\end{eqnarray}

\noindent which provides the definition of $\varphi(\vec{b})$.  Note that while no restriction has been placed on the form of $\mathfrak{a}$ other than that it is real, from (\ref{Eq:21}), $\varphi$ turns out to be independent of the direction of $\vec{b}$, that is, $\varphi(\vec{b})=\varphi(b)$.

One might expect that an intuitive choice such as $\varphi(b) \sim e^{-\mu^{2} b^{2}}$ would suffice, but that is not true.  By trial and error it is found that a slight change to
\begin{eqnarray}\label{Levy}
\varphi(b) \sim e^{-(\mu b)^{2+\xi}}
\end{eqnarray}

\noindent with $\xi$ real, positive, and small, $\xi<1$, is most appropriate and leads to a binding potential of form $V_{B}(r)=\xi \mu (\mu r)^{1+\xi}$, for small $\xi$~\cite{Fried2011_QCD4}.  For the time being, the interpretation of the parameter $\xi$ is quite enigmatic. There are steps in the derived expression for $V_{B}(r)$ where $\xi$ may be neglected compared to unity; but $\xi$ cannot be set equal to zero.  And $\xi$ is not a coupling constant, for even when small, its effect on the Q-$\bar{\mathrm{Q}}$ binding is large.  It would seem that, in some sense, $\xi$ has something to do with the transverse correlations between bound quarks and/or antiquarks.  In this respect it is worth noticing that (\ref{Levy}) is a part of a Levy-flight probability distribution~\cite{levy}. At this point, a part of the enigma is precisely how such correlations would emerge from the assumed transverse imprecision of the quark field operators.

One other point deserves mention concerning the scale change used when $\int{\mathrm{d}^{4}w \, \chi^{2}(w)}$ in the exponent of the Halpern representation is broken up into small cells of volume $(\delta)^4$:
\begin{equation*}
\int d^4w \chi^2(w) \rightarrow (\delta)^4  \sum_ i  \chi^2_i\ , \ \ \ \  \chi_i \equiv \chi(w_i)
\end{equation*}

\noindent Upon rescaling, $\chi_i \rightarrow (\delta)^{-4} \, \chi'_i$, and re-expressing all interactions in terms of $\chi'$,  there appears in (\ref{Eq:10}) the factor $(\delta)^{2} \, \varphi(b)$.  In Ref.~\cite{Fried2009_QCD1}, where transverse imprecision was treated in an {\it{ad hoc}} way, the size of $\delta$ was taken to be $M^{-1}$, where $M$ corresponded to a very large energy associated with the eikonal limit.  Here, we ask the more physical question of just how small that $\delta$ may be chosen in the light of an actual measurement, and we let Quantum Physics provide the answer:  That contribution to the $\delta$ corresponding to a time separation should be chosen as $1/E$, that corresponding to a (CM) longitudinal coordinate should be $1/p_{\mathrm{L}} \simeq 1/E$, while that corresponding to each of the transverse coordinates should be $1/\mu$; and hence $\delta^{4} \sim 1/(E \, \mu)^{2}$.  An alternate way of expressing this is that, starting from arbitrarily small separations in each coordinate, we average each $\chi_{n}$ variable over a physically meaningful distance, and call that average the $\chi_{n}$ contained in the volume $(\delta)^{4}$.

Finally, we note that $\varphi(b)$ replaces the $\delta^{(2)}(b)$ of the conventional, `ideal' formulation of QCD, with obvious normalization: $\int{\mathrm{d}^{2}b \, \delta^{(2)}(b)} = 1$.  If we adopt the same normalization for $\varphi(b) = \varphi(0) \exp{[-(\mu \, b)^{2+\xi}]}$, then it is easy to see that, with $\delta^{2} = 1/(E \, \mu)$, $ \varphi(b)$ becomes
\begin{eqnarray}
\frac{\mu^{2}}{\pi} \, \frac{1+\xi/2}{\Gamma(\frac{1}{1+\xi/2})} \, e^{-(\mu b)^{2+\xi}} \simeq \frac{\mu^{2}}{\pi} \, e^{-(\mu b)^{2+\xi}}, \quad \xi \ll 1.
\end{eqnarray}

\section{Bundle Diagrams}\label{sec:BuddleDiagrams}

In the above example of quark and/or antiquark scattering, where the infinite number of exchanged gluons appears to originate and end at a single space-time point on a quark/antiquark line, it may be helpful to introduce the concept of an exchanged `gluon bundle', as in Figure~{\ref{Fig:1}}.  Because of the four-dimensional delta function $\delta^{(4)}(y'_{1} - y'_{2} - u(s_{1}) + \bar{u}(s_{2}))$, arising from the product of the pair of delta functions of (\ref{Eq:19}), and of the subsequent analysis which produces (\ref{Eq:20}), the transverse separation $\vec{b} = \vec{y}_{1\perp} - \vec{y}_{2\perp}$ satisfies the distribution of (\ref{Eq:21}).  But the argument $w$ in $(f \cdot \chi(w))^{-1}$ is given by $w = w_{1} = y'_{1} - u(s_{1}) \simeq y'_{1}$ in virtue of (\ref{Eq:16}), and hence the Halpern functional integral reduces to a set of ordinary integrals, as explained in Section~\ref{sec:FradkinRepresentations}, and represented by the Bundle Diagram of Figure~{\ref{Fig:1}}.
\begin{figure}
\includegraphics[height=35mm]{Fig1_bundlediagram_2quarks.eps}%
\caption{\label{Fig:1}A gluon bundle exchanged between quarks $\mathrm{I}$ and $\mathrm{I\!I}$}
\end{figure}

Here, the time and longitudinal coordinates of the end-points of the bundle are the same, although their transverse coordinates -- measured vertically in the figure -- are separated.  Bundle diagrams are not Feynman diagrams, but are perhaps a more efficient way of representing the sum over all of the Feynman graphs corresponding to such multiple gluon exchange.

A slightly more complicated expression describes gluon bundles exchanged between any two of three quarks, as in Figure~{\ref{Fig:2}}, where, because of EL, the $w$-coordinates of each of the $(f \cdot \chi)^{-1}$ entering into the appropriate Halpern functional integral are the same, even though the transverse coordinates of the three quarks can be quite different.
\begin{figure}
\includegraphics[keepaspectratio,height=35mm]{Fig2_bundlediagram_3quarks.eps}%
\caption{\label{Fig:2}Gluon bundles exchanged among three quarks.}
\end{figure}

In contrast, were a closed quark loop -- corresponding to a simple relaxation of the quenched approximation -- to appear between a pair of quarks, joined to each external quark lines by the exchange of a gluon bundle, as in Figure~{\ref{Fig:3}}, there will now be two distinct sets of ordinary Halpern integrals to be evaluated.

As will be seen elsewhere, the effective diagram of Figure~{\ref{Fig:3}} will provide us with the essential features of the Nucleon-Nucleon potential for separation lengths beyond 2 fm~\cite{Fried2011_QCD5}.
\begin{figure}
\includegraphics[keepaspectratio,height=35mm]{Fig3_bundlediagram_2quarkswithloop.eps}%
\caption{\label{Fig:3}Gluon bundles joining two quarks via a closed quark loop.}
\end{figure}

\section{Summary}\label{sec:Summary}

For a whole family of QCD processes, a remarkable property of Effective Locality was first observed on an approximated (quenched and eikonal) version of QCD~\cite{Fried2009_QCD1}. However, that property still holds without these approximations, so that Effective Locality appears as a genuine property of non-perturbative QCD~\cite{Fried2010_QCD2}.

In the present article, a thorough analysis of the constraints inherent to Effective Locality has displayed an odd $\delta^{(2)}(\vec{b})$-term in the argument of an exponential factor describing $Q/\bar{Q}$ scattering at impact parameter $\vec{b}$. This fact is here taken as an indication that an untenable hypothesis is lurking in the derivation leading to this $\delta^{(2)}(\vec{b})$-function, an assumption most certainly inherited from our long practice of perturbative QCD.

Accordingly, a replacement of the meaningless $\delta^{(2)}(\vec{b})$ in an exponential factor of an exact statement obtained in an 'ideal' QCD amplitude, by a phenomenologically defined $\varphi(\vec{b})$ is proposed. It can be performed for each and every QCD process and will be used for the derivation of quark binding in a following article~\cite{Fried2011_QCD4}.  As emphasized in the text, the proposed transition from 'ideal' to 'realistic' QCD is phenomenological, but it clearly reflects the necessary change in viewpoint, taking into account the fact that quark quanta are always bound.

It is only when a complete sum over all relevant Feynman graphs has been accomplished in a gauge-invariant way, that the new and exact property of Effective Locality appears.  That property, if taken seriously, then forces this change to a would-be more 'realistic' form of non-perturbative QCD where transverse imprecision would be built-in.

Clearly, work remains to be done in order to appreciate as deeply as possible the necessity of such a change as specified by (\ref{Levy}). This point will be dealt with, in detail, in Ref.~\cite{Fried2011_QCD4}.  At face value, though, it is certainly fascinating that starting from a notion of propagation, such as implemented through the familiar 2-point function $\mathbf{G}_{c}(x, y|A)$, with its obvious perturbative content, one could be lead, in the bound non-perturbative context, to another mode of behavior, more appropriately described in terms of Levy flight distributions!

redpajama_set_name: RedPajamaArXiv


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Autocorse | Is Adrafinil Legal?
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Is Adrafinil Legal?
May 30, 2019 In Default By Natalie Watkins
For those of us with big dreams, there are rarely enough hours in the day. Our lives are filled with all the tasks that we have to do and we often fall asleep tired and exhausted.
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While Adrafinil does have many benefits, those who suffer from epilepsy or any kidney or liver impairment should shy away from it. The same is recommended for pregnant women or those on anti-psychotic medication.
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It's also important to note Adrafinil is on a list of prohibited substances for those who engage in athletic competition. Thus, athletes shouldn't take it. Furthermore, because regulations change frequently, it's important to check with your country's regulating agency to determine its legal status in your specific jurisdiction.
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That said you could end up in such a position if you are not careful.
From financial challenges to consumers going to your competitors, things can unravel fast.
With this thought in mind, is your small business on borrowed time?
What Steps Can You Take to Reverse Course?
If you have concerns your business is on borrowed time, look at the areas where you could be susceptible.
Money woes – Given your finances are the backbone of your business, you can't afford to be swimming in debt. Doing so can lead you to have to put a closed sign on your business permanently. That said it may well be time to try and get a business line of credit. Note that you can apply in minutes and find out sooner than later if you will gain approval. With this being the case, you are closer to taking some of the financial pressures off your back. Part of your money woes may well be that you've acquired large debt over time. Often, this is due to too much credit card use. Do your best moving ahead to limit what expenditures you put on your credit card. Also, take the time to search for deals for your business. From the vendors you work with to the possibility of expanding your office needs, look for deals.
Customer service – How good of a job do you and your employees (if you have them) do of servicing customers? Keep in mind those customers more times than not have a fair amount of choices when it comes to who they shop with. As a result, you need to stand out to them for all the right reasons. As an example, do you do your best to go the extra mile for them time and time again? If you have not been doing so as of late, change this moving forward.
Lack of technology – Last, are you doing all you can to stay up to speed on technology? No matter what type of business you run, chances are technology plays a key role. As such, you can't go to sleep on your company's tech needs. Do all you can to see where technology can best help your business. For instance, it may be in speeding up how you make or sell products to your customers. Although there will almost always be a human component to your business, you can't turn a blind eye to tech needs.
When you come to the conclusion that your small business is on borrowed time, you need to act sooner than later.
Remember, you put a lot of time and effort into getting your small business started in the first place.
As a result, you do not want to see everything come crashing down around you.
If you feel like your company is on borrowed time, you need to give your small business a jump start.


text:
Albumin stimulates interleukin-8 expression in proximal tubular epithelial cells in vitro and in vivo
Sydney Tang, … , Tak Mao Chan, Kar Neng Lai
Citation Information: J Clin Invest. 2003;111(4):515-527. https://doi.org/10.1172/JCI16079.
Article Nephrology
Renal tubulointerstitial injury is characterized by inflammatory cell infiltrate; however, the stimuli for leukocyte recruitment are not fully understood. IL-8 is a potent chemokine produced by proximal tubular epithelial cells (PTECs). Whether nephrotic proteins stimulate tubular IL-8 expression remains unknown. Acute exposure of human PTECs to albumin induced IL-8 gene and protein expression time- and dose-dependently. Apical albumin predominantly stimulated basolateral IL-8 secretion. Electrophoretic mobility shift assay demonstrated nuclear translocation of NF-κB, and the p65/p50 subunits were activated. NF-κB activation and IL-8 secretion were attenuated by the NF-κB inhibitors pyrrolidine dithiocarbamate and cell-permeable peptide. Albumin upregulated intracellular reactive oxygen species (ROS) generation, while exogenous H2O2 stimulated NF-κB translocation and IL-8 secretion. Albumin-induced ROS generation, NF-κB activation, and IL-8 secretion were endocytosis- and PKC-dependent as these downstream events were abrogated by the PI3K inhibitors LY294002 and wortmannin, and the PKC inhibitors GF109203X and staurosporin, respectively. In vivo, IL-8 mRNA expression was localized by in situ hybridization to the proximal tubules in nephrotic kidney tissues. The intensity of IL-8 immunostaining was higher in nephrotic than non-nephrotic subjects. In conclusion, albumin is a strong stimulus for tubular IL-8 expression, which occurs via NF-κB–dependent pathways through PKC activation and ROS generation.
Sydney Tang, Joseph C.K. Leung, Katsushige Abe, Kwok Wah Chan, Loretta Y.Y. Chan, Tak Mao Chan, Kar Neng Lai
FACS analysis of nuclear expression of NF-κB/Rel proteins on PTECs. (a) Confluent PTECs were incubated in serum-free medium or medium containing escalating doses of albumin as shown. After 30 min, cells were lysed, and the cell nuclei were isolated and stained with rabbit polyclonal Ab against p65 (Rel A, filled circles), or p50 (NF-κB1, open circles), or nonimmune rabbit immunoglobulins for analysis of DNA-bound NF-κB/Rel proteins by flow cytometry. Results, expressed as MFI ± SD, are obtained from triplicate experiments. *P < 0.05, †P = 0.006, ‡P = 0.005, **P = 0.002, #P = 0.001, §P = 0.01 versus cells exposed to serum-free medium alone. (b) Representative flow-cytometry histograms of unstimulated and albumin-stimulated cells labeled with anti-p65 or anti-p50 subunits and FITC-conjugated secondary Ab's.


Crystal specializes in Blonding and Balayage. Creative and enthusiastic with her color technique she molds her technique to each individual client. Graduated from Paul Mitchell and is diligent in her education for herself her clients and her peers.


I had some friends that were doing CrossFit and raved about it so much. Eventually it just came time for me to give it a try. Fell in love instantly.
By far its seeing people's growth. It's not necessarily them hitting goals or numbers, but consistently watching how people develop into better shape and movers over months and years.
More gym stuff??? Also cook outs, writing, hiking, swimming, reading, learning, and of course hanging out with the wifey (and friends).


Volcano, California -- American Paranormal Investigations announced today that its first annual Ghost Rush conference will take place March 28-30, 2008 at the St George Hotel in Volcano, California.
Ghost Rush 2008 will be the first event of its kind held in Northern California. Conference Attendees will have the opportunity to attend lectures provided by some of the most respected names in the Paranormal Community, including: Jeff Belanger, Konstantinos, Loyd Auerbach, Todd Sheets, Mark Stinson and Al Tyas. Workshops will also be offered throughout the event and even more speakers are anticipated as the date nears.
"This will be a great event for novice and seasoned investigators alike. We'll have full access to some highly sought after and active locations and have the chance to learn from some of the best in the field" said Amy Bruni, conference organizer, Producer for Beyond Reality Radio and Investigator with API.
In addition to the informative lecture series, conference goers will spend two nights investigating the town of Volcano. Local business owners have teamed up with American Paranormal Investigations to allow access to their buildings after hours. At the St George Hotel alone, frequent reports of beds being unmade, knocking heard on walls, coins being thrown at bar patrons and full bodied apparitions are just a few of the happenings that drew conference organizers to Volcano in the first place.
" I'm looking forward to speaking at the Ghost Rush conference in Volcano, California," said Jeff Belanger, author, founder of Ghostvillage.com , and one of the event's featured speakers. "When you consider Volcano's mining history, remote location, and haunted reputation, we're sure to have a ghostly great time investigating and discussing the paranormal in a setting that hasn't been overly exploited like many of the better-known haunts on the West coast. It's an opportunity for the attendees and for me as well!"


Apple TV 2 (2010) review – a bit disappointing.
While some of the features in the new Apple TV will be appreciated, it is a BIG step back in terms of sharing your own content around your home.
After testing the first version of the Seagate 3TB with xbench on a 12-core Mac Pro, here are the results. It should be noted that this is the first version of the 3TB Seagate Barracuda XT and that the widely available versions due out soon(?) will likely have improved performance characteristics.
Screen shots of the 3TB Seagate Barracuda XT on the Apple Mac Pro without a RAID Card.
As you can see without a RAID Card the Mac Pro can make use of 3TB internal hard disks without problem.
We've been experimenting with the Seagate Barracuda XT 3000GB on various Macintosh configurations. In the past one could take out, for example, four 500 GB drives and replace them with four 1TB, 1.5TB, or 2TB internal drives.
The result for four 2TB drives configured in RAID 5 is a drive that is about 5.23TB with redundancy.
With the release of 3TB internal hard drives, such as the Seagate Barracuda XT above, many Windows PCs can't make use of more than approximately 2.2TB due to the use of MBR (see 1, 2). Apple on the other hand, has been using the GUID partition table (GPT) for some time (since 10.4.0 in early 2005) so the Mac OS X should be able to make use of larger than 2.2 TB hard drives.
1. On a 2008 Mac Pro (MacPro3,1 Quad-Core Intel Xeon, 3 GHz, 2 processors, 8 cores) with RAID Card (1.00, M-2.0.5.5), the drives only show up as 2.2 TB.
2. On a 2008 Mac Pro (MacPro3,1) without the RAID Card, the drives show up as 3.0 TB.
3. On a 2010 Mac Pro (MacPro5,1) with RAID Card (2.00, E-1.3.2.0, 0018) the drives show up as 2.2 TB.
4. On a 2010 Mac Pro (MacPro5,1) without the RAID Card, the drives show up as a full 3.0 TB.
In short, Mac OS X appears to have no difficultly using hard disks that are larger than 2.2TB, except when those drives are connected to an Apple RAID Card. Apple does warn "Do not assume that the block size is always going to be 512 bytes" and does not support SSD with the current RAID card. Apple states that "Solid-state drives are not compatible with the Mac Pro RAID Card in either RAID or Enhanced JBOD mode." My hypothesis as of now is that Apple did not heed their own warnings on block sizes on the Apple RAID card which is why hard drives attached to the RAID Card seem to be limited to 2.2TB.
Some performance notes on the 3TB Seagate Barracuda's will follow shortly.


The Speech Language Pathologist Home Health is responsible for modeling the 3 Company values of Compassion, Integrity, and Excellence, and for promoting the Compassus philosophy, using the 6 Pillars of success as the foundation. S/he is responsible for upholding the Code of Ethical Conduct and for promoting positive working relationships within the Program, between the Program and other Company departments, and between the Program and all external stakeholders. The Speech Language Pathologist Home Health is responsible for providing service to clients who have speech and language problems. The Home Therapy Program is provided under the direction of the attending physician by the Speech Therapist with participation by the family, nurse or other responsible person, as necessary.
Participates in the development of speech therapy plans of care for patients.
Makes visits to the patients for assessment and evaluation and communicates with the physician before start of care and as needed and completes appropriate OASIS assessments.
Provides education and instruction to team members, patients & families.
Documents care and submits paperwork in an accurate and timely manner to update and maintain the medical records.
May participate in agency quality improvement programs.
Ensures the growth and profitability of the company through the responsible use of company resources and educating the community to our services.
Accepts other assignments deemed appropriate.
Accepts the policies & procedures of the agency and the rules and regulations of State and Federal regulatory agencies and other certified agencies in providing care.
Willingly accepts direction from Clinical Service Director.
Meets or exceeds established productivity standards.
Provides speech/language pathology services in accordance with the interdisciplinary plan of care, recommending mechanisms which focus on alternative methods of communication, speech and swallowing exercises.
Participates in regularly scheduled interdisciplinary team meetings to coordinate the care of the patients and family, exchange information and problem solve & receive staff support and education.
Adheres to established Company policies.
Participates in agency quality improvement programs when requested.
Accepts assignments when deemed appropriate.
Evaluates outcomes of treatment plan and plans discharge as appropriate.
Supervises home health aide as appropriate.
Ensures that all care is provided with respect for patient rights.
Reports all grievances and complaints made by patients or families to the appropriate persons.
Adheres to established personnel policies.
Reports all allegations of patient abuse and/or misappropriation of patient property.
Follows standard precautions and infection control procedure.
Reports all accidents and incidents observed.
Identifies and responds appropriately to emergency situations.
Observes safety needs of the patients.
Educates staff in psychosocial aspects, as needed.
Attends in-services and meetings as required.
Collaborates with service provider furnishing contract services to the patient as needed.
Participates in developing and updating policies and procedures as requested.
Thinks and acts calmly and logically to meet unusual occurrences of the job.
Maintains proper documentation for billing.
Manages time efficiently and effectively.
Interfaces with patients, families, and staff to ensure customer satisfaction.
Communicates effectively with patients, families, and other health care providers.
Maintains clinical competency by assuming responsibility for acquiring knowledge and experience to meet learning needs.
Creates positive image between co-workers.
Provides role modeling to teamwork, accountability, and a positive attitude.
Instills confidence in staff members and builds self-esteem through acceptance and recognition of employees.
Demonstrates and encourages appropriate problem solving methods.
Strong oral and verbal and interpersonal communications are needed.
Currently licensed in state of practice. Minimum one year experience in a home health setting.
Ability to successfully complete competency evaluation program. This includes a written and a skills evaluation. Ability to work a flexible schedule, to include some evenings and weekends. High integrity, including maintenance of confidential information such as patient records. Must be able to travel.
Must be able to tend the geriatric and the terminally ill.
While performing the duties of this Job, the colleague is regularly required to sit; use hands to finger, handle, or feel; reach with hands and arms and talk or hear. The colleague is frequently required to walk. The colleague is occasionally required to stand; climb or balance; stoop, kneel, crouch, or crawl and taste or smell. The colleague must regularly lift and /or move up to 10 pounds, frequently lift and/or move up to 25 pounds and occasionally lift and/or move 50 or more pounds. Specific vision abilities required by this job include close vision, distance vision, color vision, peripheral vision, depth perception and ability to adjust focus.
WORK ENVIRONMENT The work environment characteristics described here are representative of those while performing the essential functions of this job. Reasonable accommodations may be made to enable individuals with disabilities to perform the essential functions.
There will be possible exposure to infectious diseases. Circumstances can be unpredictable and high stress. The noise level in the work environment is usually quiet. Frequent driving is required.