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P-adic number

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are; this property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.p-adic numbers were first described by Kurt Hensel in 1897, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated by an attempt to bring the ideas and techniques of power series methods into number theory.

Their influence now extends far beyond this. For example, the field of p-adic analysis provides an alternative form of calculus. More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers; the field Qp is given a topology derived from a metric, itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense; this is what allows the development of calculus on Qp, it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility. The p in "p-adic" may be replaced with a prime or another placeholder variable; the "adic" of "p-adic" comes from the ending found in words such as triadic. This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals; the decadic numbers are not used in mathematics: since 10 is not prime or prime power, the decadics are not a field.

More formal constructions and properties are given below. In the standard decimal representation all real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows 1 3 = 0.333333 …. Informally, non-terminating decimals are understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are quite close to one another. 10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness". Whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10, thus 4739 and 5739, which differ by 103, are close in the 10-adic world, 72694473 and 82694473 are closer, differing by 107. More every positive rational number r can be uniquely expressed as r =: a/b·10d, where a and b are positive integers and gcd=1, gcd=1, gcd<10.

Let the 10-adic "absolute value" of 10d be | 10 d | 10:= 1 10 d. Additionally, we define | 0 | 10:= 0. Now, taking a/b = 1 and d = 0,1,2... we have |100|10 = 100, |101|10 = 10−1, |102|10 = 10−2...with the consequence that we have lim d → + ∞ | 10 d | 10 = 0. Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers x and y is given by |x − y|10. An interesting consequence of the 10-adic metric is that there is no longer a need for the negative sign; as an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1: 9 = − 1 + 10 so | 9 − | 10 = 1 10. 99 = − 1 + 10 2 so | 99 − | 10 = 1 100. 999 = − 1 + 10 3 so | 999 − | 10 = 1 1000. 9999 = − 1 + 10 4


Tattaguine is a town in the west of Senegal. It is the name of the rural community. Tattaguine derives its name from a bird called Tat in the Serer language. According to Serer mythology, prior to human habitation of this locality, it was these birds that lived there; the word Guine is the name of the bird's egg, i.e. the nesting of these birds. Several Serer religious festivals and customs are observed in Tattaguine; the Ndut rite of passage is just one of many rituals. It is in this rite that they receive their education about the paranormal world; the classical Ndut teachings prepare boys to be honourable men. The Ndut is the hut, it is the place they develop their artistic skills: singing and composing songs which are religious in nature. The initiation takes place at night around a camp fire, where they are taught about the mysteries of the Universe, Serer medicine, etc. Serer religion and culture forbids the circumcision of girls. Serer girls receive their initiation by tattooing of the gum. There is the Bok ceremony for young married women.

This is a fertility rite. The Misse is a religious dance, it takes place once year in April. It is a dance, it is sometimes referred to as the rain dance by some scholars. Somewhat similar to Misse is the Khoy ceremony; the Xooy is not a dance but a religious ceremony where the Serer priestly class gather once a year to divine the future and the rainy season. Tattaguine is one of the old Serer countries though much younger than the historic and spiritual village of Yaboyabo. In Serer medieval and dynastic history, in the pre-colonial Serer Kingdom of Sine was partnered with Ngouye because of their geographical affinity hence Ngouye-Tattaguine. Ngouye-Tattaguine gained particular prominence in the latter part of the 14th century, during the reigne of Maad a Sinig Diessanou Faye, it was one of the royal villages at that time. Diessanou Faye was the cousin and brother in law of Jaraff Boureh Gnilane Joof founder of the Royal House of Boureh Gnilane Joof, the first royal house founded by the Joof family.

During France's colonization of Senegal, old Tattaguine was divided due to population growth. This demarcation led to the creation of new localities: i.e. Tattaguine-Sérère and Tattaguine-Mbabara; the division of the old Tattaguine created Tattaguine-Sérère founded in 1901. Its first chief as of 1901 was Ngor Tine; the first chief of Tattaguine-Mbabara was Thierkoro Diakhate. Since the division, position for the chief has become hereditary. Tattaguine is the chief town of the Tattguine District in the Fatick Department, which lies within the Fatick Region of Senegal; the closest localities are Yaboyabo, Gaskor, Mboudaye Sek and Gaolamboura Diarab. Tattaguine's population consists of the Serer people, the original inhabitants of this area, they make up 99% of the total population. Other groups include the Bambara people. According to the site of PEPAM (Programme d'eau potable et d'assainissement du Millénaire, the rural community of Tattaguine consists of 22,561 people and 2,552 houses; that is equivalent to 2423 people for 274 homes.

As of May 2003, the estimated population was 20394. This Serer country used to be green and fertile, the peanut-monoculture has led to a decline in agricultural production and income; the main activity is seasonal subsistence farming. Crops growned include: millet, cowpea, peppers, sorrel and tomato. Philippe seck NGOM of Ngohé Ndoffongor is the current president of the rural community Doudou Diop, the former director of Ngohé Secco Primary School Babacar Ndiaye, the former director of El Hadji Moustapha Sarr Primary School of Tattaguine Communaute Rurale de Tattaguine "Données sociodémographiques" Fallingrain: Maps and airports for Tataguine PEPAM: La communauté rurale de Tattaguine sur le site du PEPAM PEPAM "Localité de Tattaguine Serere" PEPAM Project of « Terre des Hommes Association » Développement rural intégré. Tattaguine, Sénégal

Movement for the Liberation of the Congo

The Movement for the Liberation of the Congo is a political party in Democratic Republic of the Congo. It was a rebel group operating in the Democratic Republic of Congo that fought the government throughout the Second Congo War, it subsequently is one of the main opposition parties. During the war, the MLC was backed by the government of Rwanda and controlled much of the north of the country, in particular the province of Équateur, it was led by former businessman, Jean-Pierre Bemba, who became vice-president following the 2002 peace agreement. The town of Gbadolite was the headquarters of the MLC; the group was the supported by Uganda during the war, while the rival Rally for Congolese Democracy was dominated by Rwanda. The Movement for the Liberation of the Congo is the main suspect for perpetrating Effacer le tableau, an ethnic cleansing against Pygmy peoples; the MLC was found guilty of committing war crimes during fighting in the Central African Republic between 2002 and 2003 in General François Bozizé's attempted coup against the government of then-President Ange-Félix Patassé.

The MLC interceded at the behest of President Patassé's government and committed numerous acts of murder, rape and torture over the course of the conflict while attempting to suppress the coup attempt. The leader of the MLC, Jean Pierre-Bemba, was arrested in 2008 near Brussels and charged with three counts of crimes against humanity and five counts of war crimes in the neighboring Central African Republic between 200-2003 by the International Criminal Court, he was convicted of two counts of three counts of war crimes. He was sentenced to 18 years for war crimes but was acquitted by the ICC's appeal court on June 8, 2018; as part of the Inter-Congolese dialogue, Brigadier General Malik Kijege of the MLC was named head of military logistics, while Major General Dieudonné Amuli Bahigwa was named head of the navy. Two of the DRC's ten military districts were given to the MLC, Bemba was allowed to appoint and dismiss the foreign minister of the DRC. Bemba, as the MLC candidate, came second in the 2006 presidential election, the party gained 64 out of 500 seats in the parliament - the second highest number for any political party.

In the 19 January 2007 Senate elections, the party won 14 out of 108 seats. Fighting broke out in Kinshasa in March 2007 between the army and Bemba's guards, who were supposed to have been integrated into the army but had not been due to what were said to be concerns about Bemba's security; the army prevailed in the fighting, Bemba took refuge in the South African embassy. On April 8, the MLC released a statement in which it said that its headquarters had been occupied by government forces since the fighting and that it was being persecuted through arbitrary arrests and intimidation. On April 13, the party suspended its participation in the National Assembly due to what it described as a "climate of permanent insecurity"; this came shortly after the alleged looting of the home of a MLC member of parliament by government forces. On April 21, the party was allowed access to its previously-occupied buildings in the capital, which were found to have been plundered. On April 25, the party ended its boycott of the National Assembly after Kabila agreed to meet with representatives of the opposition.

Following the killing of Daniel Botethi, a member of the MLC, serving as Vice-President of the Provincial Assembly of Kinshasa, the MLC announced on July 6, 2008 that it was suspending its participation in the National Assembly, the Senate, the Provincial Assembly of Kinshasa. The MLC ended this boycott after a week. In the 2011 general election, the MLC lost its position as the second largest party in parliament and 42 of its seats in the lower house, ending the election as the fifth largest party in the National Assembly. President names top officers for unified national military, 20 August 2003 MLC, official website U. S. State Department, 04KINSHASA1753 FARDC on the move in Eastern DRC, 17 September 2004. Use of ex-MLC troops from Gbadolite in South Kivu

When a Woman Loves (album)

When a Woman Loves is the fourteenth solo album released by singer Patti LaBelle, her sixth on the MCA Records label. The album was released on October 24, 2000. By 2000, Patti LaBelle had achieved solo success with at least one platinum album and four gold albums. Four of the certified successes were with MCA Records, a company she had been an artist with since 1985. With the guidance of the label and with her husband, Armstead Edwards, LaBelle had achieved the solo success that had eluded her since leaving the flashy pop group, Labelle, in 1976, it had been three years since her last album, the platinum-certified Flame, which yielded the modestly popular hit, "When You Talk About Love". After winning a second Grammy Award for her live album, Live! One Night Only, LaBelle laid low; the marriage of LaBelle and Edwards seemed to be solid but in early 2000, the couple made news by announcing a trial separation after 31 years of marriage. The news shocked fans of the singer, who had told the media that the couple's relationship was built on their opposite differences.

Following news of the separation, LaBelle returned to the recording studio to work on her next studio album for MCA Records. Noting the modest success they had with LaBelle's original 1989 version of "If You Asked Me To", LaBelle and the song's writer, Diane Warren agreed to work together on LaBelle's new album; when A Woman Loves featured production from not only Warren and friend Denise Rich, who co-wrote several tracks, but longtime producers Jimmy Jam and Terry Lewis, producers of LaBelle's last two hits, "The Right Kinda Lover" and "When You Talk About Love". Upon its release, when critics heard the new tracks, they thought that LaBelle's impending divorce from her husband Edwards was the cause of the sound from the tracks and from the singer's voice, though LaBelle would deny such stories; because Edwards had served as LaBelle's manager for nearly 30 years, Edwards left his position as manager leaving that position to the couple's son Zuri. Upon its release, When a Woman Loves tanked on the pop and R&B charts after its release in October 2000.

The album's sole failed to chart and was given poor radio airplay. Though the title track did get airplay and LaBelle sometimes sung the song during the record's promotion, it was never released as a single and the album dropped out of the charts after only ten weeks, becoming the first LaBelle studio album since This Christmas, to not be certified. All tracks composed by Diane Warren "The Kitchen" "When a Woman Loves" "Make Tonight Beautiful" "If I Was a River" "Why Do We Hurt Each Other" "Too Many Tears, Too Many Times" "Call Me Gone" "Time Will" "Tell Me Where It Hurts" "I'll Still Love You More" "Love Will Lead You Back" "When a Woman Loves"

John Thomas Lockman

John Thomas Lockman was an American lawyer and soldier, brevetted Brigadier General for his efforts for the Union Army in the U. S. Civil War. Lockman was born on September 1834 in New York City, he was the son of Isaac Paul Mary Lockman. Among his siblings was the attorney Jacob Kennedy Lockman, Sarah H. Lockman, his paternal grandparents were Jacob Lockman and Catherine Lockman, his maternal grandparents were Thomas H. Kennedy, an intimate friend of Scottish poet Robert Burns, Margaret Kennedy, both of whom were born in Scotland. Lockman, who served in the old Volunteer Fire Department of New York City for seven years, was a law student when the U. S. Civil War broke out. On April 19, 1861, he enlisted as a private in the Union Army, first taking part in the Martinsburg campaign, under General Robert Patterson, at the Battle of Ball's Bluff, under General Charles Pomeroy Stone, he was promoted to Captain in the 83rd New York Volunteer Infantry and participated in the occupation of Winchester and the campaign of Virginia.

Lockman was again promoted to Lieutenant colonel of the 119th New York Volunteer Infantry and fought in the Army of the Potomac under multiple Union Army Generals, including General George B. McClellan, Ambrose Burnside, Joseph Hooker and George G. Meade. After Colonel Peisner was killed during the Battle of Chancellorsville, Lockman took charge of the regiment, he fought in the Battle of Gettysburg, where he was wounded. He was "ordered to the Southwest to reinforce General Thomas' command and fought in the Battles of Wauhatchie and Missionary Ridge and took part in the pursuit of General Bragg and in the relief of Knoxville."Lockman participated in the Battles of Cassville, Pine Hill, Kolb's Farm, Kennesaw Mountain, Peachtree Creek and the Siege of Atlanta. On March 13, 1865, he was brevetted a Brigadier General of the U. S. Volunteers for "meritorious conduct in the campaign ending with the occupation of Atlanta, Ga." After the War ended, Lockman resumed studying the law and graduated from Columbia Law School in April 1867.

Lockman became a member of the law firm of DeWitt, Lockman & Kip, based at 88 Nassau Street, with George Gosman DeWitt, his brother Jacob Kennedy Lockman, George Goelet Kip known as DeWitt, Lockman & DeWitt. The firm was known for its work defending the estates of New York's old Dutch families, he was a director of the Lawyers Mortgage Company and the Mortgage Bond Company. He was elected a member of the Saint Nicholas Society of the City of New York on March 4, 1889 and, in 1912, served as the Society's 42nd President, succeeding Charles Augustus Schermerhorn and remaining president until his death in September 1912. On October 14, 1862, while on furlough, Lockman was married to Harriet Hall, she was the daughter of Mary Hall, who were both born in England. Together and John were the parents of five children, three daughters and two sons, including: Mary Lockman, who married Pierre Joseph Smith, son of Benjamin Duval Smith. Jenat DeWitt Lockman, who married John Storm Appleby. Isabel Spalding Lockman, who married Dr. William Tod Helmuth Jr. editor of the New York Medical Times, in 1895.

John Quentin Lockman, a Yale University graduate who became a banker with Harvey Fisk & Sons. Frederick Irving Lockman, a Yale and Columbia Law School graduate who married Josephine Kernell, he served in the 12th Infantry of the New York Guard. He served as a vestryman of Trinity Church, a trustee of the New York Protestant Episcopal Public School, a member of the New-York Historical Society, the St. Andrew's Society, the Metropolitan Club, the Church Club, the Army and Navy Club, the Metropolitan Museum of Art and the American Museum of Natural History. Lockman died at his home, 140 West 73rd Street in New York City, on September 27, 1912, he was buried at Woodlawn Cemetery in the Bronx. John Thomas Lockman at Find a Grave

Desney Tan

Desney Tan is a Principal Researcher and Director of the Microsoft Research Medical Devices Group. He holds an affiliate faculty appointment in the Department of Computer Science and Engineering at the University of Washington. Tan was recipient of a National Science and Technology Board Fellowship in 2001, an Agency for Science and Research Fellowship in 2002, a Microsoft Research Fellowship in 2003, he was honored as one of MIT Technology Review's 2007 Young Innovators Under 35 in 2007 for his work on Brain Computer Interfaces. He was named one of SciFi Channel's Young Visionaries at TED 2009, as well as Forbes' Revolutionaries: Radical Thinkers and their World-Changing Ideas for his work on Whole Body Computing, he was the Technical Program Chair of the prestigious ACM SIGCHI Conference in 2008 and General Chair in 2011. His research interests focus on Human-Computer Interaction, Physiological Computing, Healthcare. Tan grew up in the Republic of Singapore, he moved to the United States for high school and received his Bachelor of Science in Computer Engineering from the University of Notre Dame in 1996.

He spent sometime in the Singapore Armed Forces. He attended Carnegie Mellon University, where he earned his PhD in Computer Science in 2004 under the supervision of Randy Pausch, popularly known for delivering "The Last Lecture: Really Achieving Your Childhood Dreams" after being diagnosed with terminal pancreatic cancer. Tan began his academic career working on robotic path planning, moved into augmented and virtual reality as well as large display and multiple device user experiences, his PhD dissertation demonstrated the cognitive and social effects of display size, irrespective of the field of view. He is most well known for his work utilizing bio-sensing to create novel forms of human-computer interfaces; this includes work on: brain-computer interfaces, muscle-computer interfaces, tongue-computer interfaces, bio-acoustic sensing, using the body as an antenna, as well as work on bionic contact lenses. Tan has published over 50 technical papers in these various domains, holds more than 30 associated patents.

Desney Tan's Professional Homepage at Microsoft Research Desney Tan's webpage at the University of Washington Desney Tan on Twitter